Ultra-cold Fermi Gases (Varenna lectures)

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Ultra-cold Fermi Gases, Course CLXIV

M. INGUSCIO W. KETTERLE C. SALOMON, Editors

IOS Press

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` ITALIANA DI FISICA SOCIETA

RENDICONTI DELLA

SCUOLA INTERNAZIONALE DI FISICA “ENRICO FERMI”

CLXIV Corso a cura di M. Inguscio, W. Ketterle and C. Salomon Direttori del Corso

VARENNA SUL LAGO DI COMO VILLA MONASTERO

20 – 30 Giugno 2006

Gas di Fermi ultrafreddi

2007

` ITALIANA DI FISICA SOCIETA BOLOGNA-ITALY

ITALIAN PHYSICAL SOCIETY

PROCEEDINGS OF THE

INTERNATIONAL SCHOOL OF PHYSICS “ENRICO FERMI”

Course CLXIV edited by M. Inguscio, W. Ketterle and C. Salomon Directors of the Course

VARENNA ON LAKE COMO VILLA MONASTERO

20 – 30 June 2006

Ultra-cold Fermi Gases

2007

AMSTERDAM, OXFORD, TOKIO, WASHINGTON DC

c 2007 by Societ` Copyright a Italiana di Fisica All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN 978-1-58603-846-5 (IOS) ISBN 978-88-7438-039-8 (SIF) Library of Congress Control Number: 2008922183

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INDICE

M. Inguscio, W. Ketterle and C. Salomon – Preface . . . . . . . . . . . . . .

pag. XXI

Gruppo fotograﬁco dei partecipanti al Corso . . . . . . . . . . . . . . . . . . . . . . . . . .

XXVI

D. S. Jin and C. A. Regal – Fermi gas experiments . . . . . . . . . . . . . . . . . .

1

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Why study ultracold Fermi gases? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Superﬂuidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3. Pairing of fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4. BCS-BEC crossover physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5. Status of ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Weakly interacting Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Creating a Fermi gas of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Thermometry using the momentum distribution . . . . . . . . . . . . . . . . . . 2 4. Thermometry using an impurity spin state . . . . . . . . . . . . . . . . . . . . . 3. Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Anisotropic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4. Interaction energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Feshbach molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Molecule creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Molecule binding energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Molecule conversion eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4. Long-lived molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Condensates in a Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Molecular condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Fermi condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Measurement of a phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Exploring the BCS-BEC crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Atom noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 6 8 9 9 10 14 15 16 16 19 21 22 24 25 27 29 30 32 32 35 38 39 40 41 44 47 VII

indice

VIII

S. Stringari – Dynamics and superﬂuidity of an ultracold Fermi gas . . . . . 1. 2. 3. 4. 5. 6. 7.

pag.

53

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideal Fermi gas in harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Role of interactions: The BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium properties of a trapped gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics and superﬂuidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotating Fermi gases and superﬂuidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 54 59 65 70 83 91

W. Ketterle and M. W. Zwierlein – Making, probing and understanding ultracold Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. State of the ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Strongly correlated fermions—a gift of nature? . . . . . . . . . . . . . . . . . . . 1 3. Some remarks on the history of fermionic superﬂuidity . . . . . . . . . . . . 1 3.1. BCS superﬂuidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3.2. The BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3.3. Experiments on fermionic gases . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3.4. High-temperature superﬂuidity . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4. Realizing model systems with ultracold atoms . . . . . . . . . . . . . . . . . . . . 1 5. Overview over the sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. The atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1. Hyperﬁne structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2. Collisional properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Cooling and trapping techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. Sympathetic cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2. Optical trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. RF spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.1. Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.2. Adiabatic rapid passage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.3. Clock shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.4. The special case of 6 Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.5. Preparation of a two-component system . . . . . . . . . . . . . . . . . . . 2 4. Using and characterizing Feshbach resonances . . . . . . . . . . . . . . . . . . . . 2 4.1. High magnetic ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.2. Methods for making molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.3. Observation of Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . 2 4.4. Determination of the coupling strength of Feshbach resonances . 2 4.5. The rapid ramp technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5. Techniques to observe cold atoms and molecules . . . . . . . . . . . . . . . . . . 2 5.1. Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5.2. Tomographic techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5.3. Distinguishing atoms from molecules . . . . . . . . . . . . . . . . . . . . 3. Quantitative analysis of density distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Trapped atomic gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. Ideal Bose and Fermi gases in a harmonic trap . . . . . . . . . . . . . 3 1.2. Trapped, interacting Fermi mixtures at zero temperature . . .

95 95 96 97 97 99 101 103 103 105 105 106 107 108 111 112 113 118 118 119 119 123 123 125 125 126 129 132 133 136 136 137 138 139 140 140 143

indice

IX

. 3 2.

Expansion of strongly interacting Fermi mixtures . . . . . . . . . . . . . . . . . 3 2.1. Free ballistic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Collisionally hydrodynamic expansion . . . . . . . . . . . . . . . . . . . . 3 2.3. Superﬂuid hydrodynamic expansion . . . . . . . . . . . . . . . . . . . . . . 3 3. Fitting functions for trapped and expanded Fermi gases . . . . . . . . . . . 3 3.1. Non-interacting Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2. Resonantly interacting Fermi gases . . . . . . . . . . . . . . . . . . . . . . . 3 3.3. Molecular clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Theory of the BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Elastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Pseudo-potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Cooper instability in a Fermi gas with attractive interactions . . . . . . . 4 3.1. Two-body bound states in 1D, 2D and 3D . . . . . . . . . . . . . . . . . 4 3.2. Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3. Pairing of fermions—The Cooper problem . . . . . . . . . . . . . . . . . 4 4. Crossover wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5. Gap and number equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6. Discussion of the three regimes—BCS, BEC and crossover . . . . . . . . . 4 6.1. BCS limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6.2. BEC limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6.3. Evolution from BCS to BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7. Single-particle and collective excitations . . . . . . . . . . . . . . . . . . . . . . . . . 4 7.1. Single-particle excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7.2. RF excitation into a third state . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7.3. Collective excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7.4. Landau criterion for superﬂuidity . . . . . . . . . . . . . . . . . . . . . . . . 4 8. Finite temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 8.1. Gap equation at ﬁnite temperature . . . . . . . . . . . . . . . . . . . . . . . 4 8.2. Temperature of pair formation . . . . . . . . . . . . . . . . . . . . . . . . . . 4 8.3. Critical temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 8.4. “Preformed” pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 9. Long-range order and condensate fraction . . . . . . . . . . . . . . . . . . . . . . . 4 10. Superﬂuid density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 11. Order parameter and Ginzburg-Landau equation . . . . . . . . . . . . . . . . . 4 12. Crossing over from BEC to BCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. History and experimental summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Scattering resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1. A model for Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . 5 4. Broad versus narrow Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . 5 4.1. Energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2. Criterion for a broad resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.3. Coupling energy scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.4. Narrow Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5. Open channel resonance and the case of 6 Li . . . . . . . . . . . . . . . . . . . . . 6. Condensation and superﬂuidity across the BEC-BCS crossover . . . . . . . . . . . 6 1. Bose-Einstein condensation and superﬂuidity . . . . . . . . . . . . . . . . . . . . . 6 2. Signatures for superﬂuidity in quantum gases . . . . . . . . . . . . . . . . . . . . 6 3. Pair condensation below the Feshbach resonance . . . . . . . . . . . . . . . . .

pag. 148 148 150 152 157 158 161 163 165 165 167 170 170 173 174 176 179 183 183 184 186 188 188 190 193 194 195 196 196 197 198 198 201 203 206 208 208 210 211 213 216 217 217 220 220 221 226 226 228 230

indice

X

. 6 4.

Pair condensation above the Feshbach resonance . . . . . . . . . . . . . . . . . . 6 4.1. Comparison with theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2. Formation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5. Direct observation of condensation in the density proﬁles . . . . . . . . . . . 6 5.1. Anomalous density proﬁles at unitarity . . . . . . . . . . . . . . . . . . . 6 5.2. Direct observation of the onset of condensation in Fermi mixtures with unequal spin populations . . . . . . . . . . . . . . . . . . . . . . 6 6. Observation of vortex lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.1. Some basic aspects of vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.2. Realization of vortices in superconductors and superﬂuids . . . . 6 6.3. Experimental concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.4. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.5. Observation of vortex lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.6. Vortex number and lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.7. A rotating bucket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.8. Superﬂuid expansion of a rotating gas . . . . . . . . . . . . . . . . . . . 7. BEC-BCS crossover: Energetics, excitations, and new systems . . . . . . . . . . . 7 1. Characterization of the equilibrium state . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1. Energy measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2. Momentum distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3. Molecular character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. Studies of excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1. Collective excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2. Speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3. Critical velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4. RF spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3. New systems with BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1. Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2. Population-imbalanced Fermi mixtures . . . . . . . . . . . . . . . . . . . 8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. 232 236 237 238 239

Y. Castin – Basic theory tools for degenerate Fermi gases . . . . . . . . . . . . . .

289

1. The . 1 1. . 1 2. . 1 3. . 1 4. . 1 5.

ideal Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coherence and correlation functions of the homogeneous gas . . . . . . . Fluctuations of the number of fermions in a given spatial zone . . . . . Application to the 1D gas of impenetrable bosons . . . . . . . . . . . . . . . . In a harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5.1. Semi-classical calculation of the entropy . . . . . . . . . . . . . . . . . . . 1 5.2. Semi-classical calculation of the density . . . . . . . . . . . . . . . . . . 2. Two-body aspects of the interaction potential . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Which model for the interaction potential? . . . . . . . . . . . . . . . . . . . . . . 2 2. Reminder of scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Eﬀective-range expansion and various physical regimes . . . . . . . . . . . . 2 4. A two-channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5. The Bethe-Peierls model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6. The lattice model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7. Application of Bethe-Peierls to a toy model: two macroscopic branches

241 244 244 245 246 247 250 253 255 257 258 258 258 259 260 261 261 262 262 263 268 268 268 274

289 289 293 295 297 298 298 300 304 304 306 308 310 314 318 321

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3. Zero-temperature BCS theory: Study of the ground branch . . . . . . . . . . . . . . 3 1. The BCS ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. A coherent state of pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2. A more convenient form from the Schmidt decomposition . . . . 3 1.3. As a squeezed vacuum: Wick’s theorem applies . . . . . . . . . . . . 3 1.4. Some basic properties of the BCS state . . . . . . . . . . . . . . . . . . . 3 2. Energy minimization within the BCS family . . . . . . . . . . . . . . . . . . . . . 3 3. Reminder on diagonalization of quadratic Hamiltonians . . . . . . . . . . . . 3 4. Summary of BCS results for the homogeneous system . . . . . . . . . . . . . 3 4.1. Gap equation in the thermodynamical limit . . . . . . . . . . . . . . . 3 4.2. In the limit of a vanishing lattice spacing . . . . . . . . . . . . . . . . . . 3 4.3. BCS prediction for an energy gap . . . . . . . . . . . . . . . . . . . . . . . . 3 4.4. BCS predictions in limiting cases . . . . . . . . . . . . . . . . . . . . . . . . 3 5. Derivation of superﬂuid hydrodynamic equations from BCS theory . . 3 5.1. Time-dependent BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5.2. Semi-classical approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5.3. Adiabatic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. 323 325 325 325 326 327 328 330 333 333 335 335 337 339 340 341 345

M. Holland and J. Wachter – Two-channel models of the BCS/BEC crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bose-Einstein condensation and superﬂuidity . . . . . . . . . . . . . . . . . . . . . . . . . Description of a superﬂuid in a dilute atomic gas . . . . . . . . . . . . . . . . . . . . . Breakdown of the mean-ﬁeld picture—resonance superﬂuids . . . . . . . . . . . . Single-channel vs. two-channel approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . Poles of the molecular propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The equivalent single-channel theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connection with the theory of Feshbach resonances . . . . . . . . . . . . . . . . . . . The BCS/BEC crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Momentum distribution in a dilute Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . Imaginary-time methods for single- and two-channel BCS models . . . . . . . . . 11 1. Single-channel BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2. Imaginary-time propagation for bosons . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3. Imaginary-time propagation for fermions . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1. Cauchy-Schwartz inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4. Imaginary-time algorithm for the single-channel model . . . . . . . . . . . . . 11 5. Imaginary-time propagation for the two-channel model . . . . . . . . . . . . 11 5.1. Two-channel equations of motion . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2. Imaginary-time algorithm for the two-channel model . . . . . . . 12. A mean-ﬁeld description for the crossover problem . . . . . . . . . . . . . . . . . . . . . 12 1. Boson scattering length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2. Beyond pair correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1. Four-particle correlations in the single-channel model . . . . . . . 12 2.2. Four-particle correlations in the two-channel model . . . . . . . . 13. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351 352 354 355 357 359 359 361 362 364 371 371 372 373 374 374 375 375 376 378 379 380 380 381 381

indice

XII

D. S. Petrov, C. Salomon and G. V. Shlyapnikov – Molecular regimes in ultracold Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Lecture 1. Diatomic molecules in a two-component Fermi gas . . . . . . . . . . . . 1 1. Feshbach resonances and diatomic molecules . . . . . . . . . . . . . . . . . . . . . 1 2. Weakly interacting gas of bosonic molecules. Molecule-molecule elastic interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3. Suppression of collisional relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4. Prospects for manipulations with weakly bound molecules . . . . . . . . . 2. Lecture 2. Molecular regimes in Fermi-Fermi mixtures . . . . . . . . . . . . . . . . . . 2 1. Inﬂuence of the mass ratio on the elastic intermolecular interaction . . 2 2. Collisional relaxation. Exact results and qualitative analysis . . . . . . . . 2 3. Molecules of heavy and light fermionic atoms . . . . . . . . . . . . . . . . . . . . 2 4. Crystalline molecular phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

R. Grimm – Ultracold Fermi gases in the BEC-BCS crossover: A review from the Innsbruck perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Brief history of experiments on strongly interacting Fermi gases . . . . . . . . . 3. Interactions in a 6 Li spin mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Energy levels of 6 Li atoms in a magnetic ﬁeld . . . . . . . . . . . . . . . . . . . . 3 2. Tunability at the marvelous 834 G Feshbach resonance . . . . . . . . . . . . . 3 3. Weakly bound dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The molecular route into Fermi degeneracy: creation of a molecular BoseEinstein condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. A brief review of diﬀerent approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. The all-optical Innsbruck approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Formation of weakly bound molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4. Evaporative cooling of an atom-molecule mixture . . . . . . . . . . . . . . . . . 4 5. The appearance of mBEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Crossover from mBEC to a fermionic superﬂuid . . . . . . . . . . . . . . . . . . . . . . . 5 1. BEC-BCS crossover physics: a brief introduction . . . . . . . . . . . . . . . . . 5 2. Basic deﬁnitions, typical experimental parameters . . . . . . . . . . . . . . . . 5 3. Universal Fermi gas in the unitarity limit . . . . . . . . . . . . . . . . . . . . . . . . 5 4. Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5. Phase diagram, relevant temperatures and energies . . . . . . . . . . . . . . . . 5 6. First Innsbruck crossover experiments: conservation of entropy, spatial proﬁles, and potential energy of the trapped gas . . . . . . . . . . . . . . 6. Collective excitations in the BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . . . . 6 1. Basics of collective modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Overview of recent experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Axial mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4. Radial breathing mode: breakdown of hydrodynamics . . . . . . . . . . . . . 6 5. Precision test of the equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6. Other modes of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. 385 386 387 387 390 395 398 400 400 402 405 408

413 413 414 415 416 417 418 420 420 421 422 425 428 428 429 431 432 432 433 435 437 438 440 441 443 444 445

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7. Pairing gap spectroscopy in the BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . 7 1. Basics of radio-frequency spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. RF spectroscopy on weakly bound molecules . . . . . . . . . . . . . . . . . . . . . 7 3. Observation of the pairing gap in the crossover . . . . . . . . . . . . . . . . . . 8. Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. 447 447 449 450 455

H. Moritz, T. St¨ oferle, K. G¨ unter, M. K¨ ohl and T. Esslinger – A lab in a trap: Fermionic quantum gases, Bose-Fermi mixtures and molecules in optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

463

1. 2. 3. 4. 5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concept of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Imaging Fermi surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interacting fermionic atoms in an optical lattice: the Hubbard model and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Weakly bound molecules in an optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . 7. Bose-Fermi mixtures in a three-dimensional optical lattice . . . . . . . . . . . . . . 8. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

468 471 473 474

A. Georges – Condensed-matter physics with light and atoms: Strongly correlated cold fermions in optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . .

477

1. 2. 3. 4.

5. 6. 7.

8.

Introduction: A novel condensed-matter physics . . . . . . . . . . . . . . . . . . . . . . Considerations on energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . When do we have a Hubbard model? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Mott phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Mean-ﬁeld theory of the bosonic Hubbard model . . . . . . . . . . . . . . . . . 4 2. Incompressibility of the Mott phase and “wedding-cake” structure of the density proﬁle in the trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Fermionic Mott insulators and the Mott transition in condensedmatter physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4. (Dynamical) mean-ﬁeld theory for fermionic systems . . . . . . . . . . . . . Ground state of the 2-component Mott insulator: Antiferromagnetism . . . Adiabatic cooling: Entropy as a thermometer . . . . . . . . . . . . . . . . . . . . . . . . The key role of frustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. Frustration can reveal “genuine” Mott physics . . . . . . . . . . . . . . . . . . . . 7 2. Frustration can lead to exotic quantum magnetism . . . . . . . . . . . . . . . Quasi-particle excitations in strongly correlated fermion systems, and how to measure them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1. Response functions and their relation to the spectrum of excitations . 8 2. Measuring one-particle excitations by stimulated Raman scattering . . 8 3. Excitations in interacting Fermi systems: A crash course . . . . . . . . . . . 8 4. Elusive quasi-particles and nodal-antinodal dichotomy: The puzzles of cuprate superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

463 464 465 466

477 478 480 486 486 488 489 491 493 495 498 499 501 502 502 504 505 507

XIV

indice

J. I. Cirac – Quantum information processing: Basic concepts and implementations with atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. 511

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Basic notions in quantum information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Quantum states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Observables and measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4. Examples and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.1. Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.2. Dense coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Quantum algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Quantum gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Requirements for a quantum computer . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4. Measurement-based quantum computing . . . . . . . . . . . . . . . . . . . . . . . 4. Quantum simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Physical implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Quantum optical systems for quantum computation . . . . . . . . . . . . . . . 5 2. Quantum computation with trapped ions . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Quantum computation with neutral atoms . . . . . . . . . . . . . . . . . . . . . . . 5 4. Quantum simulations with neutral atoms . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

511 512 512 514 515 516 516 517 518 519 520 522 522 525 527 528 529 530 532 533

A. Imambekov, V. Gritsev and E. Demler – Fundamental noise in matter interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

535

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Interference experiments with cold atoms . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Fundamental sources of noise in interference experiments with matter 2. Interference of ideal condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Interference of condensates with a well-deﬁned relative phase . . . . . . . 2 1.1. Basics of interference experiments. First quantized representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2. Second quantized representation . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Interference of independent clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Full counting statistics of shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Interference of two independent coherent condensates . . . . . . . . . . . . . . 3 2. Interference of independent clouds in number states . . . . . . . . . . . . . . . 3 3. Clouds with a well-deﬁned relative phase . . . . . . . . . . . . . . . . . . . . . . . 4. Interference of ﬂuctuating low-dimensional gases . . . . . . . . . . . . . . . . . . . . . . . 4 1. Interference amplitudes: from high moments to full distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Connection of the fringe visibility distribution functions to the partition functions of Sine-Gordon models . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Distribution functions for 1D gas with periodic boundary conditions . 4 3.1. Mapping to integrable structure of CFT and singular anharmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2. PT-symmetric quantum mechanics . . . . . . . . . . . . . . . . . . . . . .

535 535 541 543 544 544 546 549 551 555 557 558 559 564 567 570 571 574

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. 4 3.3. Analysis of distribution functions . . . . . . . . . . . . . . . . . . . . . . . Non-perturbative solution for the general case . . . . . . . . . . . . . . . . . . . . 4 4.1. Mapping to the statistics of random surfaces . . . . . . . . . . . . . . . 4 4.2. From interference of 1D Bose liquids of weakly interacting atoms to extreme value statistics . . . . . . . . . . . . . . . . . . . . . . . 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Some experimental issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Large K expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 1. Expansion to order (1/K)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 2. General properties of (1/K)m terms, and expansion to order (1/K)5 . A 3. Properties of the K → ∞ distribution . . . . . . . . . . . . . . . . . . . . . . . . . . A 4. D = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Jack polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C. Thermodynamic Bethe Ansatz of the quantum impurity model . . 4 4.

pag. 575 576 576 580 583 583 584 585 586 587 590 594 596 597 598

F. Chevy – Unitary polarized Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . . .

607

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Universal phase diagram of a homogeneous system . . . . . . . . . . . . . . . . . . . . 3. The N + 1 body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Trapped system and comparison with MIT experiment . . . . . . . . . . . . . . . . 5. Elongated systems and Rice’s experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix. Thermodynamical relations for the grand potential . . . . . . . . . . . . . . . A 1. Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

607 608 611 613 615 617 617 618

P. Pieri and G. C. Strinati – Exact treatment of trapped imbalanced fermions in the BEC limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

621

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Mean-ﬁeld treatment for the homogeneous case . . . . . . . . . . . . . . . . . . . . . . . 3. Mean-ﬁeld treatment for the trapped case . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Exact equations in the dilute case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Numerical results and comparison with experiments . . . . . . . . . . . . . . . . . . . 6. Extension to vortices (rotating frame) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Perspectives and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Mapping of the anisotropic onto the isotropic problem . . . . . . . . . Appendix B. Axial density proﬁles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

621 622 624 626 627 632 634 635 636

J. Tempere – Path integral description of the superﬂuid properties at the BEC/BCS crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

639

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Review: functional integral crossover theory . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Action functional for a dilute Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. The Hubbard-Stratonovic transformation . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Grassmann functional integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

639 640 640 641 642

indice

XVI

. 2 4. Saddle-point approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5. Gap and number equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6. Fluctuation corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7. Finite-temperature analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Application to optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Multilayer action functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Hydrodynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Application to multi-species Fermi mixtures . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

G. Modugno – Fermi-Bose mixture with tunable interactions . . . . . . . . . . . 1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feshbach resonances in the K-Rb mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . Feshbach spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-body losses at a Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . Tuning of the interaction in the quantum degenerate regime . . . . . . . . . . . . Formation of dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. 643 644 645 647 648 648 650 652 653 657 657 658 659 663 665 671 673

J. Hecker Denschlag and A. J. Daley – Exotic atom pairs: Repulsively bound states in an optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

677

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Repulsively bound pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Analytical solution of two-particle problem in an optical lattice . . . . . . . . . . 3 1. General discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Scattering states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Numerical approach for repulsively bound pairs . . . . . . . . . . . . . . . . . . . . . . . 4 1. Time-dependent DMRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1. Matrix product states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2. Time dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Numerical investigation of repulsively bound pairs . . . . . . . . . . . . . . . 5. Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. BEC production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Loading into lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Puriﬁcation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. Pair lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Quasimomentum distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Modulation spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4. Attractively bound pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Repulsively bound pairs of fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Other related physical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1. Pairing resonances in many-body systems . . . . . . . . . . . . . . . . . . . . . . . 8 2. Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3. Photonic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4. Gap solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

677 678 680 680 681 682 684 684 684 685 685 686 686 686 686 687 687 688 690 691 692 693 693 693 693 693 694

indice

XVII

R. Combescot – Introduction to FFLO phases and collective mode in the BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 2. 3. 4. 5.

pag. 697

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Clogston-Chandrasekhar limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Fulde-Ferrell-Larkin-Ovchinnikov phases . . . . . . . . . . . . . . . . . . . . . . . . . Vicinity of the tricritical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collective mode in the BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . . . . . . .

697 698 702 705 709

I. Bloch – Strongly correlated quantum phases of ultracold atoms in optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

715

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Optical dipole force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Optical lattice potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. 1D lattice potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2. 2D lattice potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3. 3D lattice potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Spin-dependent optical-lattice potentials . . . . . . . . . . . . . . . . . . . . . . . 3. Bose-Hubbard model of interacting bosons in optical lattices . . . . . . . . . . . . 3 1. Ground states of the Bose-Hubbard Hamiltonian . . . . . . . . . . . . . . . . . 3 2. Double-well case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Multiple-well case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4. Superﬂuid to Mott insulator transition . . . . . . . . . . . . . . . . . . . . . . . . . 4. Collapse and revival of a macroscopic quantum ﬁeld . . . . . . . . . . . . . . . . . . . 5. Quantum gate arrays via controlled collisions . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Spin-dependent transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Controlled collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Using controlled collisional quantum gates . . . . . . . . . . . . . . . . . . . . . . 6. Entanglement generation via spin-changing collisions . . . . . . . . . . . . . . . . . . 7. Quantum noise correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

715 716 716 719 719 719 720 722 723 724 725 726 727 731 735 735 737 740 740 742 746

K. Levin and Q. Chen – Finite-temperature eﬀects in ultracold Fermi gases

751

1. BCS-BEC crossover theory and the physical eﬀects of temperature . . . . . . 2. Theory outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Microscopic T -matrix scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Behavior of Tc and trap eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Experimental evidence for a pseudogap in cold gases . . . . . . . . . . . . . . . . . . 5. Establishing superﬂuidity in cold Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . 6. Fermi gases with imbalanced spin population . . . . . . . . . . . . . . . . . . . . . . . . 7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

751 754 756 761 763 766 769 775

N. V. Prokof’ev – Normal-superﬂuid transition temperature in the unitary Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

779

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

779 780

indice

XVIII

3. 4. 5. 6.

Extrapolation towards continuum system in the thermodynamic limit . . . . Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trapped Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. 783 786 792 794

G. Nardulli – Introduction to color superconductivity . . . . . . . . . . . . . . . .

797

1. 2. 3. 4. 5.

Nuclear matter and QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The true vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Homogeneous color superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Color superconductivity and compact stars . . . . . . . . . . . . . . . . . . . . . . . . . . Inhomogeneous color superconductivity: LOFF phase with two ﬂavors . . . . 5 1. Gap equation in the Ginzburg-Landau approximation . . . . . . . . . . . . . 5 2. Eﬀective gap equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. LOFF phase of QCD with three ﬂavors in the Ginzburg-Landau approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Stability of the LOFF phase of QCD with three ﬂavors . . . . . . . . . . . . . . . . 8. Neutrino emission by pulsars and the LOFF state . . . . . . . . . . . . . . . . . . . . .

W. Vassen, T. Jeltes, J. M. McNamara and A. S. Tychkov – Production of a degenerate Fermi gas of metastable helium-3 atoms . . . . . . .

797 798 800 803 806 808 809 811 813 814

817

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relevant atomic physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular and collision physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MOT results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Homonuclear and heteronuclear collisions in a MOT . . . . . . . . . . . . . . Magnetic trapping and one-dimensional Doppler cooling . . . . . . . . . . . . . . . Bose-Einstein condensation of helium-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermi degeneracy of helium-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

817 820 820 822 823 824 826 826 831 832

A. Kantian – Excited states on optical lattices: Atomic lattice excitons . .

835

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Description of atomic lattice excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. The qualitative picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. An eﬀective Hamiltonian description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. The single exciton: analytical solution for Jp = Jh . . . . . . . . . . . . . . . . 2 4. Interaction of two excitons: stability of the single-exciton picture . . . . 2 5. Lattice excitons at zero temperature: the exciton condensate . . . . . . 3. Probing atomic lattice excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Exciton formation on an optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

835 836 836 836 838 839 839 840 842 843

1. 2. 3. 4. 5. 6. 7. 8. 9.

indice

XIX

L. Tarruell, M. Teichmann, J. McKeever, T. Bourdel, J. Cubizolles, N. Navon, F. Chevy, C. Salomon, L. Khaykovich and J. Zhang – Expansion of a lithium gas in the BEC-BCS crossover . . . . . . . 1. 2. 3. 4. 5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Momentum distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Release energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

T. Fukuhara, S. Sugawa, Y. Takasu and Y. Takahashi – Quantum degenerate gases and the mixtures of ytterbium atoms . . . . . . . . . . . . . . . . . . 1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cooling and trapping of Yb atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermi degeneracy of 173 Yb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sympathetic cooling of 171 Yb with 174 Yb . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bose-Einstein condensation of 170 Yb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BEC-BEC mixture of 174 Yb and 176 Yb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. 845 845 846 847 850 851 853

857 857 859 860 863 863 864 865

F. Ferlaino, G. Modugno, G. Roati and M. Inguscio – Ultracold fermions in a 1D optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

867

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Geometrical localization in a 1D optical lattice . . . . . . . . . . . . . . . . . . . . . . . . 2 1. The “bent-tube” spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Localization and addressability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Semi-classical motion in the CHP potential . . . . . . . . . . . . . . . . . . . . . 3. Localized fermions and bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. K-Rb mixture in a combined harmonic and periodic potential . . . . . . . 3 2. Radio-frequency spectroscopy of localized states . . . . . . . . . . . . . . . . . 4. Dynamical response: localized vs. delocalized atoms . . . . . . . . . . . . . . . . . . . . 4 1. Transport of a Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conduction of a Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. From an insulating to a conducting Fermi gas: admixture of bosons . . 5 2. Two-component Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

867 868 868 871 872 873 874 875 876 876 879 880 881 882

Elenco dei partecipanti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Preface

All elementary constituents of everyday matter (electrons, protons and neutrons) are fermions, named after Enrico Fermi who introduced, in 1925 in Florence, the quantum statistics of half-integer spin particles. The Varenna school, which carries the name of Enrico Fermi, has witnessed all major advances in physics since 1953. It has been a special honour for us to organize an Enrico Fermi school on Ultracold Fermi Gases, yet another striking development that even the great scientist could not anticipate. The list of Varenna schools includes cornerstone courses in atomic physics. After the milestones of laser spectroscopy, the fast advances in the ﬁeld of cold atoms were timely covered by the 1991 School on Laser Manipulation of Atoms and the 1998 School on BoseEinstein Condensation in Atomic Gases. Following this tradition, the School on Ultracold Fermi Gases highlighted new developments and discussed exciting new directions. These three summer schools on cold atomic gases mark three distinct periods in the exploration of the ultralow temperature regime. The ﬁeld of cold atomic gases faced a revolution in 1995 when Bose-Einstein condensation was achieved. Since then, there has been an impressive progress, both experimental and theoretical. The quest for ultra-cold Fermi gases started shortly after the 1995 discovery, and quantum degeneracy in a gas of fermionic atoms was obtained in 1999. The Pauli exclusion principle plays a crucial role in many aspects of ultracold Fermi gases, including inhibited interactions with applications to precision measurements, and strong correlations. The path towards strong interactions and pairing of fermions opened up with the discovery in 2003 that molecules formed by fermions near a Feshbach resonance were surprisingly stable against inelastic decay, but featured strong elastic interactions. This remarkable combination was explained by the Pauli exclusion principle and the fact that only inelastic collisions require three fermions to come close to each other. The unexpected stability of strongly interacting fermions and fermion pairs triggered most of the research which was presented at this summer school. It is remarkable foresight (or XXI

XXII

Preface

good luck) that the ﬁrst steps to organize the summer school were already taken before this discovery. It speaks for the dynamics of the ﬁeld, how dramatically it can change course when new insight is obtained.

Fermi in Varenna.

This summer school took place after the quest for fermionic superﬂuidity with ultracold atoms has reached its goal, and high-temperature superﬂuidity was established in ultracold and ultradilute gases. These new superﬂuid atomic systems provide an ideal laboratory for investigating quantum many-body phenomena. Atomic physics brings to many-body physics the remarkable control and tunability of interactions, as well as of

Preface

XXIII

the spatial order provided by atom traps and optical lattices. This approach has stimulated an explosion of theoretical and experimental advances in the quantum physics of many-body systems. We are witnessing an important convergence of research eﬀorts dealing with open problems in many-body physics, covering ﬁelds as diverse as highenergy physics, condensed matter, astrophysics, quantum information, and of course quantum gases.

Fermi, the ﬁrst from the right, in Florence.

This school brought together many leaders in both the theory and experiments on ultracold Fermi gases as well as a very large number of enthusiastic students from all over the world and from diﬀerent ﬁelds of research. The lectures, which are written up in this volume, provided a detailed coverage of the experimental techniques for the creation and study of Fermi quantum gases, as well as the theoretical foundation for understanding the properties of these novel systems. Many exciting aspects were presented, including basic static and dynamical properties, molecule formation, superﬂuid behaviour and BECBCS crossover, fermions in optical lattices, and Fermi-Bose mixtures. The timing of the school was excellent since the ﬁeld is still small enough to be fully covered, but it is also undergoing a major expansion.

XXIV

Preface

This volume provides the ﬁrst systematic review of the many developments that have taken place since the early beginnings of the ﬁeld less than a decade ago. The exciting scientiﬁc program of the School was enhanced by the special atmosphere of Lake Como combining in a unique blend water and mountains with historical tradition and culture. We warmly thank our scientiﬁc secretary, Francesca Ferlaino for her enthusiastic support, and Barbara Alzani for the professional organisation and her dedication which were crucial to the success of the school.

M. Inguscio, W. Ketterle and C. Salomon

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Società Italiana di Fisica SCUOLA INTERNAZIONALE DI FISICA «E. FERMI» CLXIV CORSO - VARENNA SUL LAGO DI COMO VILLA MONASTERO 20- 30 Giugno 2006

81 82 83 84 85 95 96 86 87 88 89 90 91 92 93 94 79 80 68 66 76 63 64 54 73 74 45 75 67 55 56 57 58 59 60 61 62 47 48 49 50 51 52 77 53 22 65 46 43 44 20 23 21 27 26 25 37 36 35 34 33 32 31 38 30 29 28 40 39 24 13 10 12 11 9 8 4 6 7 3 2 5 78

70 41

72 42

71 1

1) M. Pigazzini 2) J. Catani 3) A. Schirotzek 4) T. Tiecke 5) B. Alzani 6) B. Peden 7) G. Bianchi Bazzi 8) J. Stewart 9) A. Nunnenkamp 10) A. Privitera 11) M. E. Pezzoli 12) G. Muradyan 13) J. K. Pachos 14) D. Sarchi 15) G. Szirmai 16) L. LeBlanc

17) A. Bezett 18) I. Kinski 19) E. Wille 20) L. Ferrari 21) A. Ludewig 22) A. Csordas 23) K. M. F. Magalhães 24) V. Guarrera 25) C. D’Errico 26) A. Kantian 27) M. R. Bakhtiari 28) F. Palestini 29) A. Zenesini 30) E. Fersino 31) M. Holland 32) D. Jin

33) Y. Castin 34) G. Strinati 35) S. Stringari 36) C. Salomon 37) W. Ketterle 38) M. Inguscio 39) F. Ferlaino 40) J. Tempere 41) M. Jona-Lasinio 42) S. Pilati 43) C.-C. Wang 44) T. L. Dao 45) D. Miller 46) L. Krzemien 47) T. Koponen 48) R. Nyman

49) G. Varoquaux 50) P. Windpassinger 51) S. Fagnocchi 52) M. Zaccanti 53) A. Kubasiak 54) K. Gubbels 55) F. Werner 56) A.-C. Voigt 57) M. Colome Tatche 58) L. Tarruell 59) A. Mering 60) T. Karpiuk 61) H. Kumar Pal 62) K. Gawryluk 63) R. Gati 64) U. Schneider

65) K. Kis-Szabó 66) M. Ciminale 67) M. Antezza 68) A. Schelle 69) T. Jeltes 70) K. Temme 71) J.-J. Su 72) M. Wright 73) S. Riedl 74) D. Murray 75) J. Fuchs 76) T. Fukuhara 77) A. Pouderous 78) N. Nooshi 79) M. Roghani 80) Y. Eksioglu

69 19 18

17 16

14

81) J. F. Clément 82) F. Impens 83) C. Schunck 84) C. Sanner 85) T. Rom 86) K. Günter 87) M. Taglieber 88) T. Volz 89) M. Teichmann 90) E. Dalla Torre 91) F. Chevy 92) T. Paananen 93) O. Zozulya 94) B. Oles 95) T. Henninger 96) C. Klempt

15

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Fermi gas experiments D. S. Jin and C. A. Regal JILA, University of Colorado and National Institute of Standards and Technology and Department of Physics, University of Colorado Boulder, CO, 80309-0440 USA

1. – Introduction These lectures endeavor to provide an overview of ultracold Fermi gas experiments, with an emphasis on the strongly interacting gas and the BCS-BEC crossover. This overview is not comprehensive, but rather presents selected experiments and topics. The discussion of experiments proceeds loosely in historical order and focuses on experiments done at JILA that explore an ultracold gas of 40 K atoms. Particular attention is paid to the development of novel experimental techniques that enable exploration of ultracold Fermi gases and the BCS-BEC crossover.

. 1 1. Why study ultracold Fermi gases? – Ultracold gases of atoms provide a new arena in which to explore interacting quantum systems. These gases have a number of unique features that facilitate such studies, as has been amply demonstrated by the many beautiful studies of dilute gas Bose-Einstein condensates (BEC). This superﬂuid phase was realized in 1995 when bosonic alkali atoms, such as 87 Rb, were cooled down to nanoKelvin temperatures via laser cooling and evaporative cooling [1,2]. The alkali BEC that was created was weakly interacting and the condensation was stunningly visible, as shown in ﬁg. 1. Starting with these images of the velocity distributions that demonstrated the ﬁrst BEC, many tools have been developed for probing and manipulating these ultracold quantum gases. c Societ` a Italiana di Fisica

1

2

D. S. Jin and C. A. Regal

(a)

(b) d

T
Tc : d ~ λdeBroglie T>Tc

Fig. 1. – Bose-Einstein condensation in a dilute gas of 87 Rb atoms. (a) The BEC forms when the thermal deBroglie wavelength of the particles is on the order of the interparticle spacing in the gas. (b) Velocity distributions of 87 Rb atoms at three values of the temperature compared to the critical temperature. (Figure adapted by M. R. Matthews from data in ref. [1].)

In addition to the unique experimental tools one has in ultracold atom gas experiments, these systems are also very accessible theoretically. Indeed many of the initial experiments with alkali BEC could be perfectly described by existing theories. This correspondence exists because the gas is weakly interacting, and the interaction between two atoms is extremely well understood. However, recent work in the ﬁeld of BEC has developed techniques to reach a regime that is more relevant for outstanding theoretical questions in condensed-matter physics; these questions arise most commonly in strongly correlated systems. For example, experiments have achieved BEC with much stronger interatomic interactions than typical alkali gases; furthermore, these interactions could even be controllably tuned [3,4]. A phase transition to the highly correlated Mott insulator state was observed through studies of quantum gases in optical lattice potentials [5]. These bosonic systems require theory that goes beyond mean-ﬁeld interactions; yet they have a controllability rarely found in solid-state materials. These characteristics lead to the exciting possibility of gaining an understanding of a many-body interacting quantum system that is built up from a complete understanding of the microscopic physics. At the same time as the creation of the ﬁrst strongly interacting Bose gases, the techniques used to create an alkali BEC were being applied to the other class of quantum particles, fermions. This greatly expanded the usefulness of quantum gases as model systems since fermions are the building blocks of visible matter; hence there are a large number of important many-body interacting quantum Fermi systems. These include, for example, electrons in metals and semiconductors, superconductors, superﬂuid liquid 3 He, nuclei, white dwarf and neutron stars, and the quark-gluon plasma in the early universe. Extending ultracold quantum gas experiments to Fermi gases was also driven by the exciting and challenging possibility of creating a Fermi condensate or, equivalently, a Fermi superﬂuid. This requires that the fermionic atoms form pairs such as the Cooper

Fermi gas experiments

3

Fig. 2. – Time-of-ﬂight images showing condensation of fermionic atom pairs. The images, taken after the projection of the fermionic system onto a molecule gas, are shown for ΔB = 0.12, 0.25, and 0.55 G (right to left) on the BCS side of the resonance. The original atom cloud starts at (T /TF )0 = 0.07. While these images are reminiscent of typical images of the BEC transition, note that the Fermi condensate here actually appears as a function of interaction strength, rather than temperature. 3D artistry is courtesy of Markus Greiner.

pairs of electrons in superconductivity. In conventional superconductors, s-wave pairing occurs between spin-up and spin-down electrons. The hope was that s-wave pairing would similarly occur with the creation of a two-component atomic gas with an equal Fermi energy for each component. Such a two-component gas would be realized using an equal mixture of alkali atoms in two diﬀerent hyperﬁne spin states. The simplistic view at the time was that a superﬂuid state involving Cooper pairs of atoms would appear if the temperature of this two-component gas were cold enough and the interaction between fermions were attractive and large enough. As we discuss in this article, the ﬁrst ultracold Fermi gas of atoms was created in 1999 [6], and in 2004 a phase transition to a Fermi condensate was achieved (see ﬁg. 2) [7]. As expected, this phase involved pairing of fermionic atoms with equal and opposite momentum [7], had an excitation gap [8], and exhibited the property of superﬂuidity [9]. . 1 2. Superﬂuidity. – The phenomenon of superconductivity/superﬂuidity has fascinated and occupied physicists since the beginning of the 20th century. In 1911, superconductivity was discovered when the resistance of mercury was observed to go to zero below a critical temperature [10]. Although liquid 4 He was actually used in this discovery, the superﬂuid phase of liquid 4 He was not revealed until the 1930s when the viscosity of the liquid below the λ point (2.17 K) was measured [11,12]. Much later, 3 He, the fermionic helium isotope, was also found to be superﬂuid at yet a much colder temperature than 4 He [13]. Relatively recently in 1986, high-temperature superconductors in Copper-oxide materials further enlarged the list of superconducting materials [14].

4

D. S. Jin and C. A. Regal

These “super” systems, which we will refer to in general as superﬂuids, are only classic examples. There are many other physical systems that display superﬂuid properties, ranging from astrophysical phenomena such as neutron stars through excitons in semiconductors to atomic nuclei [15]. Although the physical properties of these systems vary widely, all superﬂuids result from the macroscopic occupation of a single quantum state and exhibit counterintuitive behaviors such as frictionless ﬂow and quantized vorticity. The manifestation of these eﬀects depends upon, for example, whether the system in question is electrically charged (superconductors) or neutral (superﬂuids). Besides these intriguing properties, there are many practical reasons for the intense research in this ﬁeld; arguably the most useful supersystems are superconductors, and if a robust room temperature superconductor were created, it would be an amazing discovery. Condensation, i.e. the macroscopic occupation of a quantum state, is fundamentally associated with the quantum properties of bosons; however since all visible matter is made up of fermions, creating a superﬂuid most often requires pairing of fermions. This raises the intriguing question of how a bosonic degree of freedom emerges in a Fermi system. A closely related question is: What is the connection between the phenomena of superconductivity, which involves Cooper pairing of fermions, and Bose-Einstein condensation? To address this issue with ultracold atom gas experiments, one must start with a Fermi gas of atoms. For bosonic atoms, even though they are ultimately composed of fermionic constituents (protons, neutrons, and electrons), the underlying fermionic degrees of freedom are simply not accessible in ultracold atom gas experiments. In contrast, as we shall see, with a Fermi gas of atoms one can create conditions that enable pairing of atoms and introduce the boson degree of freedom required for superﬂuidity. . 1 3. Pairing of fermions. – Let us now consider how a superﬂuid might emerge from an ultracold atomic Fermi gas. The simplest (although historically not the most famous) way to imagine pairing fermions is to create a two-body bound state with two fermions. When paired, two half-integer-spin fermions will produce an integer spin particle, which is a composite boson. In the case of the experiments discussed in this article, the fermionic particles are atoms; this makes such a two-body bound state a diatomic molecule. Below a critical temperature, an ensemble of these diatomic molecules will form a BEC. The right side of ﬁg. 3 represents this type of pairing. The two shades of grey represent fermions in two diﬀerent spin states; two states are required if the fermions are to pair via s-wave (l = 0) interactions. One could also imagine creating Cooper pairs of atoms analogous to what happens in superconductors. In 1956, Cooper found that a pair of fermions in the presence of a ﬁlled Fermi sea (ﬁg. 4) will form a bound pair with an arbitrarily small attractive interaction [16]. This result is in surprising contrast to two fermions in a vacuum where there is no bound state until the interaction reaches a certain threshold. The key difference between the two situations arises from Pauli blocking, which in the Cooper pair case prevents the two fermions under consideration from occupying momentum states k < kF , where kF is the Fermi wave vector [17]. Thus in the Cooper-pairing mechanism the underlying fermionic nature of the system plays an essential role.

5

Fermi gas experiments

BEC

BCS kF

Cooper pairs

strongly interacting pairs

diatomic molecules

Fig. 3. – Cartoon illustration of the continuum of pairing in the BCS-BEC crossover.

Considering only one pair of electrons as free to pair on top of a static Fermi sea is not a suﬃcient solution to describe superconductivity. The 1957 Bardeen-CooperSchrieﬀer (BCS) theory of superconductivity addressed the many-body problem and has been amazingly successful [18, 19]. The result predicted (among other things) the formation of a minimum excitation energy, or energy gap, in the conductor below a critical temperature, Tc . A great many properties of conventional superconductors can be understood as consequences of this energy gap. A description of the full BCS theory is beyond the scope of this discussion but is presented in the original papers [18, 19] and discussed in numerous textbooks, for example refs. [17, 20]. Qualitatively, the BCS state consists of loose correlations between fermions at the Fermi surface in momentum space (ﬁg. 3 left side). Spatially the pairs are highly overlapping and cannot simply be considered to be composite bosons. It is interesting to consider what happens if the interaction energy of a Cooper-paired state is increased until it is close to EF . One could also consider diatomic molecules that are more and more weakly bound, to the point where the binding energy of the

bosons: integer spin

fermions: half-integer spin

EFermi spin spin Fig. 4. – Quantum statistics: Bosons vs. fermions with weak interactions at T = 0. Bosons form a BEC in which all of the bosons macroscopically occupy a single quantum state (the lowest energy state in the trapping potential). Because of the Pauli exclusion principle, fermions form a Fermi sea in which each energy state up to the Fermi energy is occupied.

6

D. S. Jin and C. A. Regal

molecules, Eb , becomes less than the Fermi energy, EF . The essence of the BCS-BEC crossover is that these two sentences describe the same physical state. As the interaction between fermions is increased, there will be a continual change, or crossover, between a BCS state and a BEC of diatomic molecules. The point where two fermions in vacuum would have zero binding energy is at the cusp of the crossover problem, and pairing in such a state is represented in the middle of ﬁg. 3. These pairs have some properties of diatomic molecules and some properties of Cooper pairs. Many-body eﬀects are required for the pairing, as with the BCS state, but there is some amount of spatial correlation, as with diatomic molecules. The pair size is on the order of the spacing between fermions, and the system is strongly interacting. . 1 4. BCS-BEC crossover physics. – The BCS-BEC crossover is a theoretical topic that dates back to the late 1960s. In a theory originally put forth by Eagles and later by Leggett, it was proposed that the BCS wave function was more generally applicable than just to the weakly interacting limit [21, 22]. As long as the chemical potential is found self-consistently as the interaction is increased, the BCS ground state can (at least qualitatively) describe everything from Cooper pairing to a BEC of composite bosons made up of two fermions [21-26]. More recent interest in crossover theories has come in response to the possibility that they could apply to high-Tc superconductors. These superconductors diﬀer from normal superconductors both in their high transition temperature and the apparent presence of a pseudogap; both are characteristics expected to be found in a Fermi gas in the crossover [27, 28]. The impact of this idea of a BCS-BEC crossover can be seen in ﬁg. 5, which sorts classic superﬂuid systems according to the strength of the interaction between the fermions. To put both BCS and BEC systems on one plot requires covering a very large range

Tc / TF

100

Alkali BEC Superfluid 4He BCS-BEC crossover regime High-Tc superconductors Superfluid 3He Superconductors

10-2 10-4 10-6

10

10

5

0

10 10 2Egap /kBTF

10

-5

Fig. 5. – Classic experimental realizations of superﬂuidity/superconductivity arranged according to the binding energy (twice the excitation gap, Egap ) of the constituent fermions. The vertical axis shows the corresponding transition temperature, Tc , to a superﬂuid/superconducting state compared to the Fermi temperature, TF . (Figure reproduced with permission from ref. [29].)

7

Fermi gas experiments

2.0

BEC limit BCS limit

μ /EF

Δ /EF

1.5 1.0 0.5 0

2

1

0

1/kF a

-1

-2

1 0 -1 -2 -3 -4 -5 -6

2

1

0

-1

-2

1/kF a

Fig. 6. – Gap parameter, Δ, and chemical potential, μ, of a homogeneous Fermi gas at T = 0 as determined through Nozieres Schmitt-Rink (NSR) theory. The dashed lines show the BCS and BEC limits of the theory. Note that the limiting theories only deviate signiﬁcantly from the full theory in approximately the range −1 < kF1 a < 1.

in both the energy gap (or binding energy of the pairs) and the transition temperature relative to the Fermi temperature. A key aspect of the classic BCS theory is that it applies to the perturbative limit of weak attractive interactions and hence is only an exact theory for the far left side of ﬁg. 5. The theory perfectly describes conventional superconductors for which the attraction between fermions is ∼ 10000 times less than the Fermi energy, EF . A key aspect of the usual theory of BEC is that it applies only to weakly interacting bosons that are assumed to be pointlike particles and hence is only an exact theory to the far right side of ﬁg. 5. In ﬁg. 6 we take a look at what happens in the region of the BCS-BEC crossover. The solid lines in ﬁg. 6 show the result of a calculation of the gap Δ and the chemical potential μ at T = 0 [30]. These are plotted as a function of the dimensionless parameter 1/kF a, √ where kF = 2mEF /¯ h. This parameter is typically used to characterize interaction strength in the crossover. We also plot the values of Δ and μ calculated in the BCS and BEC limits to ﬁnd that the crossover occurs in a relatively small region of the parameter 1/kF a, namely from approximately −1 < 1/kF a < 1. It is useful to explicitly understand the value and meaning of both μ and Δ in the two limits. μ is EF in the BCS limit and −π 1 2 16 1 kF |a| −Eb /2 = −( kF a ) EF in the BEC limit. Δ/EF is e in the BCS limit and 3π kF a in the BEC limit [30]. Although referred to as the gap parameter, it is only in the BCS limit that Δ has meaning as the excitation gap, i.e., the smallest possible energy that can create a hole (removea fermion) in the superﬂuid. In general, the excitation energy h ¯ 2 k2 is Egap = min Ek = min ( 2m − μ)2 + Δ2 [27]. This is Δ when μ is positive (as in the BCS limit) but becomes μ2 + Δ2 when μ is negative.

8

D. S. Jin and C. A. Regal

The experimental realization of a superﬂuid in the BCS-BEC crossover regime provides a physical link between superconductivity and superﬂuid 4 He. Moreover, ultracold low-density gases provide a very clean, strongly interacting Fermi system with the power to test these many-body theories. In principle the density and two-fermion interaction in the sample can be known precisely and the s-wave pairing fully characterized as a function of the interaction strength. The end result could be a fully understood physical system that connects the spectrum of pairing from BCS to BEC and unites the basic physics surrounding “super” systems. It should be noted that the complicated materials physics involved in high-Tc superconductors, for example, cannot be elucidated in these clean crossover experiments. Still the hope is that an understanding of the basic physics will provide a solid foundation for studies of real materials. . 1 5. Status of ﬁeld . – Experimental progress in ultracold Fermi gases has occurred at an amazingly fast rate with contributions from a large number of groups, in particular those of R. Grimm (Innsbruck), R. Hulet (Rice), D. Jin (JILA), W. Ketterle (MIT), C. Salomon (ENS), and J. Thomas (Duke). Experimenters discovered interesting properties of the normal state of a strongly interacting Fermi gas [31-35]. Then Fermi gases were reversibly converted to gases of diatomic molecules using Feshbach resonances [36-39]. The observation that these molecules were surprisingly long-lived created many opportunities for further study [37, 39, 38, 40]. Condensates of diatomic molecules in the BEC limit were achieved [41-45]; then these condensates were found to exist in the crossover regime [7, 46], signalling the existence of a phase transition in the BCS-BEC crossover regime. Collective excitations [47-49] and thermodynamic properties [50, 44, 51, 52] were also measured, and the nature of the pairs was probed in a variety of ways [8, 53, 45]. A vortex lattice was created in the crossover [9], and the eﬀects of unequal spin mixtures have been investigated [54, 55]. Developing techniques to access and probe the BCS-BEC crossover was a challenging adventure for the ﬁeld. Experiments in the crossover are inherently diﬃcult because the strong interactions make probing diﬃcult. Some of the techniques used in the end were borrowed from those developed for alkali BEC, while others were taken from condensedmatter physics. Still others were new inventions that relied on the unique ability to tune the interactions in the system at arbitrary rates using the Feshbach resonance. So far, the experiments that have been carried out with Fermi gases near Feshbach resonances have been qualitatively consistent with classic BCS-BEC crossover theory. The excitation gap is on the order of the Fermi energy; the system crosses a phase transition to a superﬂuid state. However, quantitatively there is much work to be done. In tandem with these experiments, sophisticated theories of the crossover have been developed that are too numerous to list here, but are actively being pursued in groups such as those of A. Bulgac, K. Burnett, J. Carlson, S. Giorgini, A. Griﬃn, H. Heiselberg, T. L. Ho, M. Holland, K. Levin, E. J. Mueller, M. Randeria, C. A. R. Sa de Melo, G. Shlyapnikov, S. Strinati, S. Stringari, B. Svistunov, E. Timmermans, and P. Torma. In time, it is expected that the BCS-BEC crossover system provided by dilute Fermi gases should be able to rigorously test these theories.

9

Fermi gas experiments

-11

σ (cm 2)

10

-12

10

-13

10

10

T (μ K)

100

Fig. 7. – Elastic cross-sections vs. temperature. The s-wave cross-section (◦), measured using a mixture of spin states, shows little temperature dependence. However, the p-wave cross-section (•), measured using spin-polarized atoms, exhibits the expected threshold behavior and is seen to vary by more than two orders of magnitude. The elastic cross-section was determined by measuring the cross-dimensional thermal relaxation rate in the trapped gas.

2. – Weakly interacting Fermi gas To create a Fermi gas of atoms, experimenters applied the same cooling techniques as those used to achieve BEC, simply replacing a bosonic atom, such as 87 Rb or 23 Na, with an alkali atom with an odd number of electrons, protons, and neutrons. (The two such stable alkali atoms are 40 K and 6 Li.) Still, evaporatively cooling fermions required ingenuity, and the ﬁrst gas of fermionic atoms to enter the quantum degenerate regime was created at JILA in 1999 using 40 K [6]. The observation in these experiments was not a phase transition, as in the Bose gas, but rather the presence of more energy than would be expected classically as the Fermi gas was cooled below the Fermi temperature. Many more Fermi gas experiments, using a variety of cooling techniques, followed [56-61, 50, 62-64]. . 2 1. Creating a Fermi gas of atoms. – The apparatus we use to cool 40 K employs the basic strategy used for some of the ﬁrst experiments with 87 Rb BEC [1, 65, 66]. We perform the “usual” combination of trapping and cooling in a magneto-optical trap (MOT) followed by evaporative cooling [67-69]. The laser cooling uses light from semiconductor diode lasers on the 40 K D2 line (4S1/2 to 4P3/2 transition at 766.7 nm), and a two-chamber apparatus allows for an ultrahigh vacuum region for evaporative cooling [70]. The major diﬀerence with respect to the 87 Rb experiments stems from the fact that elastic collisions between identical fermions are suppressed at ultracold temperatures. This suppression occurs because quantum statistics require antisymmetry of the total wave function for two colliding fermions. This requirement forbids s-wave collisions for identical fermions. While odd partial wave collisions, such as p-wave, are allowed, these collisions are suppressed below T ≈ 100 μK because of the angular momentum barrier (see ﬁg. 7) [71].

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D. S. Jin and C. A. Regal

40

f=7/2 7/2 = mf

K

1.3 GHz +7/2 -9/2 -7/2

-5/2

+9/2=mf

f=9/2

Fig. 8. – 40 K ground state level diagram, with exaggerated Zeeman splittings. The two levels represent the hyperﬁne structure, which originates from the coupling of the nuclear spin (I = 4) with the electron spin (S = 1/2). Note that the hyperﬁne structure of 40 K is inverted.

Since evaporative cooling requires collisions to rethermalize the gas, a mixture of two distinguishable particles is required to cool fermions. 40 K provides an elegant solution to this problem. Figure 8 shows the ground-state energy levels of 40 K. The large angular momentum of the lowest ground-state hyperﬁne level (f = 9/2) provides 10 distinct spin states. The two highest energy states, mf = +9/2 and mf = +7/2, can be held simultaneously with reasonable spatial overlap in a magnetic trap, which is the type of trap most proven for evaporative cooling when starting from a MOT. In this way, an apparatus designed for only one atomic species provided two distinguishable states for cooling. To remove the highest energy atoms for evaporative cooling, microwaves at ∼ 1.3 GHz were used to transfer atoms to untrapped spin states in the upper hyperﬁne state [6,70]. With this technique, quantum degeneracy was reached in 1999, and by 2001 two-component 40 K Fermi gases at temperatures of 0.25TF could be created [6, 72]. . 2 2. Thermodynamics. – As a gas of fermions is cooled from the classical regime to quantum degeneracy, the Pauli exclusion principle becomes manifest in the properties of the ultracold gas. The ﬁrst report of a Fermi gas of atoms in 1999 included measurements of the energy and the shape of the momentum distribution of the weakly interacting gas [6]. The quantum degeneracy of the Fermi gas is described by the temperature relative to the Fermi temperature T /TF . The Fermi temperature is proportional to the Fermi energy, which is deﬁned as the energy of the highest occupied level of the potential at T = 0, and is given by

(1)

TF =

EF ¯hω ¯ = (6N )1/3 . kB kB

11

Fermi gas experiments

E / EF

3 2 1 0

0

0.2

0.4 0.6 T / TF

0.8

1.0

Fig. 9. – Energy of an ideal, trapped Fermi gas.

δ E / E cl

Here kB is Boltzmann’s constant, N is the number of atoms in a particular spin-state, and ω ¯ /2π is the geometric mean trap frequency. In the classical regime (high T /TF ), the energy per particle is proportional to the temperature (E = 3kB T ), while in the Fermi gas limit, the energy asymptotically approaches 34 EF ( 38 EF kinetic energy and 3 8 EF potential energy) (see ﬁg. 9). Figure 10 shows the data from 1999 [6] for a single-component (spin polarized) Fermi gas at T /TF ≈ 0.4. We observed an energy per particle that was about 10% larger than the classical prediction. In 2001, we created two-component Fermi gases at temperatures

Fig. 10. – Data showing that the energy of the Fermi gas deviates from the classical expectation. We plot δE/Ecl = (E − 3kB T )/(3kB T ) vs. T /TF . This data was taken with spin polarized mf = 9/2 gases. The solid line shows the prediction for an ideal Fermi gas.

12

D. S. Jin and C. A. Regal

m f=9/2

(a)

E / 3k BT

m =7/2

(b)

2.0 1.6 1.2 1.0 0.0

0.5

1.0

1.5

T/ TF Fig. 11. – Pauli blocking in a degenerate Fermi gas. The average energy per particle E, extracted from absorption images such as the examples shown in the insets, is displayed for two spin mixtures, 46% mf = 9/2 (a) and 86% mf = 9/2 (b). In the quantum degenerate regime, the data deviate from the classical expectation (dashed line) as the atoms form a Fermi sea arrangement in the energy levels of the harmonic trapping potential. The data agree with the ideal Fermi gas prediction for a harmonic trap, as shown by the solid line. In (b) the shift of corresponding mf = 9/2 (•) and mf = 7/2(◦) points on the T /TF axis reﬂects a diﬀerence in the Fermi energies for the two components.

around 0.25TF [70]. Figure 11 shows the measured energy per particle in a two-component gas as a function of T /TF . In ﬁg. 11(a), which presents data for a roughly 50/50 mixture of atoms in the two trapped spin-states, one can see the excess energy of the Fermi gas compared to the classical prediction. Figure 11(b) presents the energy measurement for a Fermi gas with unequal spin populations and the ﬁrst demonstration of Pauli blocking in an atomic Fermi gas [70]. Here, the component gases, which are in thermal equilibrium with each other, have very diﬀerent average energies per particle. This ﬁnding is consistent with the fact that the Fermi temperatures of the two components diﬀer because of the population imbalance (eq. (1)). The fact that the equilibrium gas maintains this energy imbalance is due to Pauli blocking of elastic collisions. The Pauli exclusion principle forbids collisions for which the ﬁnal state would place fermions in already occupied levels, and in a degenerate Fermi sea, the low-lying energy states have a high probability of being occupied (see ﬁg. 4).

13

(a) (a)

Optical depth

Fermi gas experiments

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 0

data Fermi gas fit Gauss fit

T/TF = 0.1

(b) 10

20

30

40

50

radius (arb) Fig. 12. – Non-classical momentum distribution of Fermi gas. (a) Sample absorption image of the momentum distribution of a degenerate Fermi gas. White corresponds to many atoms and black to zero atoms. (b) Azimuthally averaged proﬁle of the absorption image.

The quantum degenerate Fermi gas was also shown to have a non-classical momentum distribution [70]. For a classical gas, the Maxwell-Boltzmann distribution gives rise to a Gaussian momentum distribution. However, for a Fermi gas, identical particles cannot occupy the same energy state. As the lowest energy states of the harmonic trap ﬁll up, atoms are forced to occupy higher energy states. This behavior leads to a momentum distribution that is wider (higher average energy per particle) and has a lower peak (fewer atoms at lower energy) as compared to a Gaussian. This eﬀect is measurable but not dramatic. In ﬁg. 12, we show the momentum distribution of a Fermi gas at T /TF = 0.1, which is near the lowest relative temperatures that can now be achieved in atomic Fermi gases. Figure 12(a) is a sample absorption image of an expanded Fermi gas. The points in ﬁg. 12(b) are the result of an azimuthal average of the image. The solid line shows the result of a surface ﬁt of the two-dimensional image to the FermiDirac distribution (eq. (2)), which reveals that the gas is at a temperature of 0.1TF . The classical distribution for the same temperature and number of particles would be dramatically diﬀerent. Furthermore, even if the temperature is a free-ﬁtting parameter, a Gaussian ﬁt (eq. (3)) (dashed line) clearly deviates from the measured distribution. In 2001 Truscott et al. reported the ﬁrst Fermi gas of 6 Li atoms and a measurement of the size of the trapped gas [56]. The 6 Li gas was cooled using a bosonic atom as the “second particle” to allow rethermalizing collisions. This approach, which is called sympathetic cooling, was ﬁrst demonstrated for fermionic atoms by Schreck et al. [57]. A feature of this type of cooling is that one can produce and explore Bose-Fermi mixtures. Indeed, in ref. [56] the size of the 6 Li Fermi gas was directly contrasted with the 7 Li Bose gas. The trapped Fermi gas was seen to have a larger size due to Fermi pressure. (Note that for a weakly interacting gas in a harmonic trap the potential and kinetic energies are equal. Therefore, the excess energy seen in expansion and the larger size in the trap go hand-in-hand.)

14

D. S. Jin and C. A. Regal

. 2 3. Thermometry using the momentum distribution. – The same thermodynamics that were used to demonstrate quantum degeneracy in a Fermi gas can be used to measure the temperature of a trapped Fermi gas. The development of reliable thermometry for the Fermi gas was an important issue and remains a challenge in the case of strong interactions. Our standard thermometry uses ﬁts to the observed momentum distribution of the gas [70]. In a typical experiment, we access the momentum distribution through absorption images of an expanded gas. The trap is suddenly turned oﬀ, and the gas is allowed to freely expand for some expansion time t. An absorption image is then taken by illuminating the atoms with a resonant laser beam pulse and imaging the shadow cast by the atoms onto a CCD camera. This method eﬀectively integrates through one dimension to give a two-dimensional image (for example, ﬁg. 12(a)). The appropriate function for this distribution, written in terms of the experimentally observed optical depth (OD), is the Fermi-Dirac distribution (2)

ODFD (y, z) = ODpk Li2

The function Lin (x) =

∞

−ζe

2

y − 2σ 2

y

2

z − 2σ 2

e

/Li2 (−ζ).

z

xk /k n appears often in the analysis of a harmonically trapped

k=1

Fermi gas. In the classical limit eq. (2) becomes a two-dimensional Gaussian function: 2

(3)

y − 2σ 2

ODgauss (y, z) = ODpk e

y

2

z − 2σ 2

e

z

.

These forms are applicable for both the spatial and momentum proﬁles and for arbitrary kB T kB T 2 2 2 expansion times through the relations σy2 = mω 2 [1 + (ωr t) ] and σz = mω 2 [1 + (ωz t) ]. r z Here m is the atom mass, and ωr /2π and ωz /2π are the radial and axial trap frequencies, respectively. Note that as the temperature is lowered far below TF , the changes in the distribution become small. Still, down to T /TF ≈ 0.1 the temperature can be determined from least-squared ﬁts to such distributions. To evaluate this thermometry, we examine the results of least-squared surface ﬁts for gases at a variety of expected temperatures. In the ﬁts, ODpk , σy , σz , and ζ are independent ﬁt parameters. The widths σy and σz tell us the temperature, and the fugacity ζ can be viewed as a shape parameter that is directly related to T /TF through Li3 (−ζ) = −(T /TF )−3 /6. As a check on the ﬁts, we compare the result for ζ to T /TF as calculated through (4)

σy2 mωr2 T = . TF ¯hω ¯ (6N )1/3 (1 + (ωr t)2 )

We use the measured trap frequencies for ωr and ωz and the number of atoms in each spin state N as determined from the total absorption in the image. Figure 13(a) shows a comparison of ζ with T /TF from eq. (4). The line shows the expected relationship for an ideal Fermi gas. In ﬁg. 13(b), we convert ζ to T /TF for a more direct comparison.

15

Fermi gas experiments

10

5

1.0 0.8 (b)

3

10

1

ζ

10

T/TF from ζ

(a)

10

-1

0.6 0.4 0.2

0 0.2 0.4 0.6 0.8 1.0

0 0 0.2 0.4 0.6 0.8 1.0

T/TF

T/TF

Fig. 13. – Analysis of ﬁts to eq. (2) for expansion images of an optically trapped gas with an equal mixture of mf = −9/2, −7/2 atoms [32]. For these data, the integration was in the x-direction, and T is extracted from σy .

In general, we ﬁnd that the two values agree, indicating that the ﬁts work well. Note that the noise in the points becomes large at temperatures > 0.5TF . This is expected because the changes in the shape of the distribution become small in this limit. A similar eﬀect occurs in the low-temperature limit where the distribution changes very little as the T = 0 Fermi gas limit is approached. However, the success of this thermometer in the 0.1 < T /TF < 0.5 range has made this method the workhorse of temperature measurements in our experiments. . 2 4. Thermometry using an impurity spin state. – We have also explored a second technique for measuring temperature using an impurity spin state. We have used this technique to check our Fermi-Dirac ﬁts; this is especially necessary for the lowest temperature gases at 0.1TF and below because of the decrease in the sensitivity of the FermiDirac ﬁts at these temperatures. We have not done extensive work using the impurity thermometer; however, as we will see here, this technique works quite well and has unexplored potential, in particular as a technique that could measure temperatures less than 0.1TF or temperatures in a strongly interacting gas. The idea of the impurity spin-state technique is to embed a small number of atoms in a third state within the usual two-component gas (ﬁg. 14). In the limit where the number of atoms in the impurity spin, Nim , is small compared to the particle number in the original states, the Fermi energy of the impurity state will be low enough that the impurity gas will be non-degenerate (see ﬁg. 11(b)). Provided all of the spin states mσ 2 in the system are in thermal equilibrium, the temperature of the system will be kB tim 2 , where σim is simply determined from a ﬁt of the impurity gas momentum distribution to a Gaussian distribution (eq. (3)).

16

D. S. Jin and C. A. Regal

non-degenerate "impurity" state

two-component degenerate Fermi gas

{

m f = 5/2

7/2 9/2

Fig. 14. – Measurement of T through an embedded impurity spin state. All three components are overlapped in the trap.

A diﬃculty with this method is that EF scales weakly with particle number. Suppose our original gas has a particle number of 105 at T /TF = 0.1. For T /TF of the impurity to be 1, Nim would need to be 100, and detecting the distribution of 100 atoms with a good signal-to-noise ratio is not trivial. However, the fully classical limit does not need to be reached to gain information about the temperature from the impurity state. It is only required that T /TF be large enough that the energy of the impurity gas changes signiﬁcantly with temperature. The range of T /TF for which this is the case can be seen in ﬁg. 9. To see if an impurity spin-state thermometer was feasible, we designed an experiment to test this thermometer against the surface-ﬁt technique described previously. We started with a (not necessarily equal) mixture of atoms in the mf = +7/2, +9/2 spin states. Part way through the evaporative cooling process, a small fraction of the mf = +7/2 atoms were transferred to the mf = +5/2 state, which serves as our impurity (ﬁg. 14). For this experiment, the gas was prepared at a low magnetic ﬁeld of a few gauss where the three-state mixture is fully stable and the scattering length between any pair of the three-spin states is 174a0 . The spin states were selectively imaged by applying a large magnetic ﬁeld gradient of ∼ 80 G/cm during the expansion to spatially separate the spin states (Stern-Gerlach imaging) [70]. Figure 15 shows the results of four measurements using impurity spin-state thermometry. The temperature result from the impurity measurement is compared to the result from the surface ﬁts applied to the mf = 9/2 cloud. We see that both methods agree to within the uncertainty for clouds for T /TF = 0.1–0.2. 3. – Feshbach resonance . 3 1. Predictions. – The next goal after the creation of a normal Fermi gas of atoms was to form a superﬂuid in a paired Fermi gas. This required considering the interactions between atoms in the ultracold gas. These interactions can be characterized by the swave scattering length a. The quantity a comes out of studying two-body low-energy

17

Fermi gas experiments

T/TF comparison: m f =9/2

T/ TF from ζ

0.3 0.2 0.1 0 0 0.1 0.2 0.3 T/TF from impurity technique Fig. 15. – Comparison of thermometers. The y-axis shows the result of the Fermi-Dirac ﬁts to the mf = 9/2 distribution, and the x-axis shows the result of the impurity spin-state technique.

scattering and is related to the s-wave collision cross-section through σ = 4πa2 . The top of ﬁg. 16 is a pictorial representation of a. As seen in the ﬁgure, the scattering length a is related to the phase shift of the scattering wave function because of the potential felt

V(R)

R

a<0 attractive V(R)

a>0 repulsive

R

Fig. 16. – Scattering wave functions in the presence of an attractive potential (top) and a more deeply attractive potential (bottom), in a regime where a bound state of the potential (dashed line) is near threshold. R here describes the relative position of two distinguishable fermions. The scattering length changes sign as the bound state moves through threshold.

18

D. S. Jin and C. A. Regal

(a)

(b) closed channel

Energy

open channel

-

Energy

C6

R6 internuclear separation R

magnetic field B

Fig. 17. – (a) Feshbach resonances are the result of coupling between a molecular state in one interatomic potential with the threshold of another. (b) The bare molecule state of the closed channel tunes diﬀerently with a magnetic ﬁeld than the open channel threshold. This can lead to a crossing of the two levels.

by the atoms at a small internuclear separation R. The sign of a determines whether the interaction is eﬀectively attractive (a < 0) or eﬀectively repulsive (a > 0). A BCS state would require a < 0, while for 40 K atoms a > 0. Moreover, for typical values of |a| for alkali atoms, the temperatures required to reach a true BCS state were far too low compared to achievable temperatures (at that point) to imagine creating Cooper pairs. Stoof et al. noted that the interaction between 6 Li atoms was strong (|a| ≈ 2000a0 ) and attractive, bringing the BCS transition temperature closer to realistic temperatures [73, 74]. It was then recognized that a type of scattering resonance, known as a Feshbach resonance [75-78], could allow arbitrary changes in the interaction strength. A magnetic-ﬁeld Feshbach resonance would, in principle, allow one to tune the s-wave scattering length a simply by varying the strength of an applied magnetic ﬁeld B. Figure 17 illustrates schematically the origin of a Feshbach resonance. The scattering length near a Feshbach resonance varies with the magnetic ﬁeld, B, according to [79] (5)

a(B) = abg

w 1− B − B0

.

Here abg is the background (non-resonant) scattering length for atoms scattering in the open channel, B0 is the magnetic ﬁeld position at which the molecular bound state of the coupled channel goes through threshold, and w is the width of the Feshbach resonance, deﬁned as the distance in magnetic ﬁeld between B0 and the magnetic ﬁeld at which a = 0. Figure 18 shows how a diverges according to eq. (5) for the 40 K resonance described in the next section. Theories of superﬂuidity in an atomic Fermi gas were developed that explicitly treated the case where the interactions were enhanced by a Feshbach resonance and relatively high transition temperatures were predicted [80, 29, 81]. Yet there were certainly reasons to be skeptical about the feasibility of experimentally realizing such a state. The Feshbach resonances that had been observed in Bose gases were associated with an extremely fast

19

Fermi gas experiments

3000 2000

a(a0)

1000 0 -1000 -2000 -3000 190

195

200

205

210

215

B (gauss) Fig. 18. – Behavior of the scattering length at a Feshbach resonance in −7/2, −9/2 spin states.

40

K between the mf =

inelastic decay of the trapped gas [3,82,83]. These decay processes, which most often stem from three-body collisions, can quickly turn a hard-earned quantum gas into a classical gas of atoms [84-87]. Carl Wieman’s group at JILA produced the only experiments studying BEC near a Feshbach resonance over long time scales [4]. In this group, 85 Rb BECs were studied at very low densities, where two-body elastic collisions dominate over three-body collisions. For two-component Fermi gases, three-body decay processes were expected to be suppressed compared to the Bose case [88, 89] because of the Pauli exclusion principle. However, there was a fair amount of contention about the degree of this suppression. The existence of Feshbach resonances had been predicted for both 6 Li and 40 K [90,91], and the ﬁrst step was to locate these resonances [92-95]. Another technical challenge was that the optimum spin states for accessing Feshbach resonances in the 40 K and 6 Li gases could not be conﬁned in a magnetic trap, which was the most proven trap in studies of ultracold gases up to that point. Instead the experiments would have to achieve an ultracold Fermi gas in an alternative trap such as an optical dipole trap. . 3 2. Collisions. – The ﬁrst Feshbach resonance observed for ultracold fermionic atoms was one that occurs between 40 K atoms in the mf = −9/2 and mf = −7/2 spin states [92]. The original theoretical prediction for the location of this resonance was B0 = 196+9 −21 G, based on available potassium potentials [91]. Our ﬁrst measurement of the position of this resonance used the technique of cross-dimensional rethermalization, which measures the elastic collision cross-section [96]. This was a technique that had provided much information about a Feshbach resonance in the bosonic 85 Rb gas [4, 97]. For this measurement, we started with a gas of fermions in the mf = −7/2, −9/2 spin states at T ≈ 2TF . The gas was taken out of thermal equilibrium by modulating the optical trap intensity at 2νy , which caused selective heating in the y-direction. (We

20

D. S. Jin and C. A. Regal -9

10

-10

σ (cm2 )

10

-11

10

-12

10

-13

10

160

180

200

220

240

260

B (gauss) Fig. 19. – Elastic collision cross-section measured near an s-wave Feshbach resonance between 40 K atoms in the mf = −7/2, −9/2 spin states at T = 4.4 μK [95]. Between the peak and dip in σ, the interaction is attractive; everywhere else it is repulsive.

could selectively modulate one radial direction because for this measurement our optical trap was not cylindrically symmetric (νx = 1.7νy , where νx and νy are the two radial trap frequencies).) The exponential time constant for energy transfer between the two radial directions, τ , was measured as a function of magnetic ﬁeld. τ is related to the s-wave collision cross-section through 1/τ = 2nσv/α. v = 4 kB T /πm is the mean relative speed between colliding fermions and n = N1tot n7 (r) n9 (r) d3 r is the densityweighted density. α is the calculated number of binary s-wave collisions required for rethermalization [71]. Figure 19 plots the result of this measurement as a function of the magnetic ﬁeld B. The magnetic ﬁeld was calibrated through radio-frequency (r.f.) transitions between mf levels in the 40 K system. An advantage of the cross-dimensional rethermalization technique is that it allows measurements of σ over a large range. Through cross-section measurements that extend over four orders of magnitude, both the position of the divergence of the scattering length, B0 , and the position of the zero crossing could be measured (ﬁg. 19). This allowed a measurement of the magnetic-ﬁeld width of the resonance w. The line in ﬁg. 19 is the result of a full coupled-channels calculation of σ carried out by C. Ticknor and J. Bohn, in which the parameters of the potassium potential were adjusted to achieve a best ﬁt to our data from two diﬀerent 40 K resonance [95, 98]. This calculation takes into account the distribution of collision energies in the gas by thermally averaging over a Gaussian distribution deﬁned by a temperature of 4.4 μK. The ﬁt result places the Feshbach resonance at B0 = 201.6 ± 0.6 G and the zero crossing at 209.9 ± 0.6 G.

21

Fermi gas experiments

1.2

σz / σy

1.0 0.8 0.6 0.4

220

224 B( gauss)

228

Fig. 20. – Anisotropic expansion of a strongly interacting Fermi gas near a Feshbach resonance between the mf = −5/2, −9/2 spin states [32].

. 3 3. Anisotropic expansion. – For 6 Li atoms in the two lowest energy spin states, two s-wave resonances were observed in a measurement of trap loss due to inelastic collisions [93]. The zero crossing of the resonance, where the elastic collision rate goes to zero, was seen in an experiment that measured the eﬃciency of evaporation vs. magnetic ﬁeld [94]. Using 6 Li, O’Hara et al. then reported the ﬁrst observation of anisotropic expansion of a low-temperature, strongly interacting Fermi gas [32]. This was an intriguing observation since anisotropic expansion was one of the ﬁrst signatures seen for a BEC [1]; it was suggested that this could be a signature of superﬂuidity in the Fermi gas. However, anisotropic expansion is only a clear signature of superﬂuidity in the case of a weakly interacting gas. The diﬃculty is that anistropic expansion can be caused either by collisional or superﬂuid hydrodynamics. A gas can be considered weakly interacting in the so-called collisionless regime where the trap oscillator period 1/ν is much shorter than the mean time between collisions in the gas, 1/Γ. In the opposite limit of strong interactions, where Γ ν, the gas is collisionally hydrodynamic, and anisotropic expansion results from elastic collisions during the expansion that transfer kinetic energy from the elongated axial (z) cloud dimension into the radial (r) direction. At the peak of a Feshbach resonance, the fermion-fermion interactions can easily become strong enough to make the gas collisionally hydrodynamic. After the results of ref. [31], we observed anisotropic expansion in a 40 K gas at T /TF = 0.34 (see ﬁg. 20), and Bourdel et al. reported anisotropic expansion for a 6 Li gas at T /TF ≈ 0.6 [35]. In the 40 K gas, we enhanced interactions between the mf = −9/2, −5/2 spin states using a Feshbach resonance at ∼ 224 G. Near the Feshbach resonance where the collision cross-section is large, the aspect ratio of the expanded cloud, σz /σy , decreases. As the gas becomes collisionless away from the resonance, the

22

D. S. Jin and C. A. Regal

aspect ratio smoothly evolves to the collisionless value. For this measurement, the magnetic ﬁeld was lowered after 5 ms of expansion, and an absorption image was taken after a total expansion time of 20 ms. The mf = −9/2, −5/2 gas was created at the ﬁeld B by starting with a mf = −9/2, −7/2 gas and applying a π pulse between the −5/2 and −7/2 states 0.3 ms before expansion. This technique avoids complications due to atom loss but creates a non-equilibrium gas. In general, the expected aspect ratio in the regime between collisionless and hydrodynamic behavior is diﬃcult to calculate. We can, however, check to see if some degree of hydrodynamic expansion of the normal gas is expected. We can calculate the elastic collision rate Γ = 2nσv in the gas using an elastic collision cross-section given by σ = 4πa259 and |a59 | = 2000 a0 (as was measured near the resonance peak [32]) to ﬁnd Γ = 46 kHz. Comparing this rate to the trapping frequencies, we ﬁnd Γ/νr = 37 and Γ/νz = 2400. Hence, it is not surprising that we observe anisotropic expansion. For a gas that was fully hydrodynamic, with Γ νr , νz , we would expect our measured aspect ratio to reach 0.4 [99, 100]. . 3 4. Interaction energy. – Measurements of the collision cross-section and observations of anisotropic expansion are useful for detecting the strength of the interactions but are not sensitive to whether the interactions are attractive or repulsive. The mean-ﬁeld energy, on the other hand, is a quantum-mechanical many-body eﬀect that is proportional to na. For Bose-Einstein condensates with repulsive interactions, the mean-ﬁeld energy (and therefore a) can be determined from the size of the trapped condensate [3, 101], while attractive interactions cause condensates with a large atom number to become mechanically unstable [102, 103]. For an atomic Fermi gas, the mean-ﬁeld interaction energy has a smaller impact on the thermodynamics. In 2003, we developed a novel r.f. spectroscopy technique that measures the mean-ﬁeld energy of a two-component Fermi gas directly [32] (see also [33]). In this measurement, we used the 40 K Feshbach resonance between the mf = −5/2, −9/2 spin states. First, optically trapped atoms were evaporatively cooled in a 72/28 mixture of the mf = −9/2 and mf = −7/2 spin states. After evaporation, the optical trap was recompressed to achieve a larger density, and the magnetic ﬁeld was ramped to the desired value near the mf = −5/2, −9/2 resonance. We then quickly turned on the resonant interaction by transferring atoms from the mf = −7/2 state to the mf = −5/2 state with a 73 μs r.f. π-pulse (ﬁg. 21(a)). The fraction of mf = −7/2 atoms remaining after the pulse was measured as a function of the r.f. frequency. The relative number of mf = −7/2 atoms was obtained from an absorption image of the gas taken after 1 ms of expansion from the optical trap. Atoms in the mf = −7/2 state were probed selectively by leaving the magnetic ﬁeld high and taking advantage of non-linear Zeeman shifts. Sample r.f. absorption spectra are shown in ﬁg. 21(b). At magnetic ﬁelds well away from the Feshbach resonance, we were able to transfer all of the mf = −7/2 atoms to the mf = −5/2 state, and the r.f. lineshape had a Fourier width deﬁned by the r.f. pulse duration. At the Feshbach resonance, we observed two changes to the r.f. spectra. First, the frequency for maximum transfer was shifted relative to

23

(a) energy

m f =-5/2

hνrf m f =-7/2 m f =-9/2

m f =-7/2 peak OD

Fermi gas experiments

1.0 0.8 0.6 0.4

(b)

0.2 0

-40

-20

0

20

40

Δν (kHz) Fig. 21. – Radio-frequency spectra. (a) Transition of interest; (b) r.f. lineshapes with (solid line) and without (dotted line) interactions [32].

the expected value from a magnetic-ﬁeld calibration. Second, the maximum transfer was reduced, and the measured lineshape was wider. Both of these eﬀects arise from the mean-ﬁeld energy caused by strong interactions between mf = −9/2, −5/2 atoms at the Feshbach resonance. The mean-ﬁeld energy produces a density-dependent frequency shift given by (6)

Δν =

2¯h n9 (a59 − a79 ), m

where n9 is the number density of atoms in the mf = −9/2 state, and a59 (a79 ) is the scattering length for collisions between atoms in the mf = −9/2 and mf = −5/2 (mf = −7/2) states [104]. Here we have ignored a non-resonant interaction term proportional to the population diﬀerence between the mf = −7/2 and mf = −5/2 states; this term equals 0 for a perfect π pulse. For our spatially inhomogeneous trapped gas, the density dependence broadens the lineshape and lowers the maximum transfer. This eﬀect occurs on both sides of the Feshbach resonance peak. In contrast, the frequency shift for maximum transfer reﬂects the scattering length and changes sign across the resonance. We measured the mean-ﬁeld shift, Δν, as a function of B near the Feshbach resonance peak. The r.f. frequency for maximum transfer was obtained from Lorentzian ﬁts to spectra like those shown in ﬁg. 21(b). The expected resonance frequency was then subtracted to yield Δν. The scattering length, a59 , was obtained using eq. (6) with n9 = 0.5npk and a79 = 174 a0 , where npk is the peak density of the mf = −9/2 gas [92]. The numerical factor 0.5 multiplying npk was determined by modelling the transfer with a pulse-width–limited Lorentzian integrated over a Gaussian density proﬁle. The measured scattering length as a function of B is shown in ﬁg. 22. The solid line shows a ﬁt to the expected form for a Feshbach resonance, eq. (5). Data within ±0.5 G of the peak were excluded from the ﬁt. We ﬁnd that the Feshbach resonance peak occurs at 224.21 ± 0.05 G, and the resonance has a width w of 9.7 ± 0.6 G.

24

D. S. Jin and C. A. Regal

scattering length (a0)

3000 2000 1000 0

-1000 -2000 -3000

215

220

225

230

B (gauss) Fig. 22. – Scattering length as measured through the mean-ﬁeld interaction [32]. These data were taken for a normal Fermi gas at T /TF = 0.4 and at two diﬀerent densities of the mf = −9/2 gas: npk = 1.8 × 1014 cm−3 (circles) and npk = 0.58 × 1014 cm−3 (squares).

When B is tuned very close to the Feshbach resonance peak, we expect the measured a59 to have a maximum value on the order of 1/kF because of the unitarity limit. This saturation can be seen in the data shown in ﬁg. 22. Two points that were taken within ±0.5 G of the Feshbach resonance peak, one on either side of the resonance, clearly lie below the ﬁt curve. We ﬁnd that the unitarity-limited point on the attractive interaction side of the resonance (higher B) has an eﬀective scattering length of ∼ 2/kF . (Here ¯hkF is the Fermi momentum for the mf = −9/2 gas.) 4. – Feshbach molecules After locating Feshbach resonances in our 40 K system, we set out to observe evidence of a molecular bound state near threshold on the low-ﬁeld side of the Feshbach resonances. Creating molecules in this bound state, referred to as “Feshbach molecules,” would be the ﬁrst step towards achieving the BEC limit of the BCS-BEC crossover. We were motivated to believe that it would be possible to create Feshbach molecules by experiments carried out in the Wieman group at JILA [105,106]. In this work the magnetic ﬁeld was quickly pulsed near the Feshbach resonance, and coherent oscillations between atoms and Feshbach molecules were observed in a 85 Rb BEC. We hoped to employ a slightly diﬀerent approach to creating molecules in which we would ramp the magnetic ﬁeld fully across the Feshbach resonance. In this section, we will present how, using this technique, we were able to eﬃciently and reversibly create Feshbach molecules from a Fermi gas of atoms. For completeness we also present our current understanding of the physics of the conversion of atoms to molecules using adiabatic magnetic-ﬁeld ramps; this understanding was gained through a study of the conversion dependences led by the Wieman group [107].

25

Fermi gas experiments

B(t) B0

atoms

Feshbach molecule Fig. 23. – Creating molecules using magnetic-ﬁeld ramps across a Feshbach resonance.

. 4 1. Molecule creation. – Figure 23 shows the behavior of the bound molecular state very near a Feshbach resonance, where the molecule binding energy is given by h ¯2 Eb = ma Given this picture, one would expect that atoms could be converted to 2. molecules simply by ramping the magnetic ﬁeld in time across the Feshbach resonance position B0 [108-110]. The only requirement for creating molecules in this way is that the magnetic-ﬁeld ramp must be slow enough to be adiabatic with respect to the two-body physics of the Feshbach resonance (two-body adiabatic). To a very good approximation, the Feshbach molecules would have twice the polarizability of the atoms [111] and therefore would be conﬁned in the optical dipole trap along with the atoms. In fact we would expect that the atoms and molecules have the same trapping frequency, but the molecule trap depth would be twice as large as the atom trap depth. We performed such an experiment using a magnetic-ﬁeld ramp across the mf = −5/2, −9/2 resonance. We started with a nearly equal mixture of the two spin states mf = −5/2, −9/2 at a magnetic ﬁeld of 227.8 G. The ﬁeld was ramped down at a rate of (40 μs/G)−1 across the resonance to various ﬁnal values. The number of atoms remaining following the ramp was determined from an absorption image of the cloud at ∼ 4 G after expansion from the optical trap. Since the light used for these images was resonant with the atomic transition, but not with any molecular transitions, we selectively detected only the atoms. In ﬁg. 24, we present the observed total atom number in the mf = −5/2, −9/2 states as a function of the ﬁnal magnetic-ﬁeld value of the ramp. We found that the atoms disappear abruptly at the Feshbach resonance peak (dashed line). We also found in similar experiments that we could recover the lost atoms with an immediate magnetic-ﬁeld ramp back above the Feshbach resonance. This result ruled out many atom loss processes and strongly suggested that all of the lost atoms were converted to Feshbach molecules. We were surprised at the eﬃciency of the conversion of our Fermi gas of atoms to a Bose gas of molecules; we could easily create hundreds of thousands of Feshbach molecules. We found that the conversion eﬃciency depends on the rate of the magnetic-ﬁeld ramp across the resonance (see ﬁg. 25), as one would expect for an adiabatic two-body conversion from atoms to molecules. If the ramp is too fast, no molecules will be created

26

D. S. Jin and C. A. Regal

B0

atom number (106)

1.5 1.0 0.5 0

215

220

225

230

B (gauss) B(G) Fig. 24. – Creation of molecules as seen through atom loss [36]. A ﬁt to an error function provides a guide to the eye. Atom loss occurs at precisely the expected position of the Feshbach resonance given a previous measurement of the scattering length divergence [32].

because the ramp will be diabatic with respect to the atom-molecule coupling. As the ramp is made slower, however, atoms will start to be coupled into molecules. Theoretical predictions ﬁnd that this eﬀect can be well modelled by the Landau-Zener formula for the transition probability at a two-level crossing

f = fm 1 − e−δLZ ,

(7)

molecule fraction

where f is the fraction of atoms converted to molecules, fm is the maximum fraction of

1.0 0.8 0.6 0.4 0.2 0 0

20

40

60

80

-1

(dB/dt) (μs/G) Fig. 25. – Time scale for two-body adiabaticity at a Feshbach resonance in

40

K [36].

27

Fermi gas experiments

kinetic energy (MHz)

transfer (arb)

1.0 0.8

(a)

0.6 0.4 0.2 0 0.4

(b)

0.3 0.2 0.1 0 49.8

50.1 50.4 50.7 rf frequency( MHz)

Fig. 26. – R.f. spectrum for an atom/Feshbach molecule mixture [36]. (a) Transfer to the mf = −7/2 states as a function of r.f. frequency. The left feature is the molecule dissociation spectrum, and the right feature represents the transfer of atoms between mf = −5/2 and mf = −7/2. (b) Corresponding kinetic energy of the mf = −7/2 state.

atoms that can be converted to molecules, and δLZ = b(dB/dt)−1 is the Landau-Zener parameter [110, 112]. We can ﬁt the data shown in ﬁg. 25 with b as the ﬁtting parameter to ﬁnd in this case b ≈ 20 μs/G [36, 107]. . 4 2. Molecule binding energy. – While suggestive of molecule creation, the measured atom loss was not conclusive proof for the existence of Feshbach molecules. We therefore developed a spectroscopic technique to probe the molecules. First, we created the molecules with a magnetic-ﬁeld ramp across the Feshbach resonance that stopped at a magnetic ﬁeld Bhold . At Bhold , a 13 μs r.f. pulse was applied to the cloud; the r.f. frequency was chosen so that the photon energy was near the energy splitting between the mf = −5/2 and mf = −7/2 atom states. The resulting population in the mf = −7/2 state, which was initially unoccupied, was then probed selectively either by separating the spin states spatially using a strong magnetic-ﬁeld gradient during free expansion (SternGerlach imaging) or by leaving the magnetic ﬁeld high (215 G) and taking advantage of non-linear Zeeman shifts. Figure 26(a) shows a sample r.f. spectrum at Bhold = 222.49 G; the resulting number of atoms in the mf = −7/2 state is plotted as a function of the frequency of the r.f. pulse.

28

D. S. Jin and C. A. Regal

We observe two distinct features in the spectrum. The sharp symmetric peak is very near the expected mf = −5/2 to mf = −7/2 transition frequency for free atoms. With the Stern-Gerlach imaging, we see that the total number of mf = −5/2 and mf = −7/2 atoms is constant, consistent with transfer between these two atom states. The width of this peak is deﬁned by the Fourier width of the applied r.f. pulse. Nearby is a broader asymmetric peak shifted lower in frequency. Here we ﬁnd that, after the r.f. pulse, the total number of observed atoms (mf = −5/2 and mf = −7/2) actually increases. Also, the resulting mf = −7/2 gas in this region has a signiﬁcantly increased kinetic energy, which grows linearly for larger frequency shifts from the atom peak (ﬁg. 26(b)). The asymmetric peak corresponds to the dissociation of molecules into free mf = −7/2 and mf = −9/2 atoms. Since the applied r.f. pulse stimulates a transition to a lower-energy Zeeman state, we expect hνr.f. = hν0 − Eb − ΔE, where Eb is the binding energy of the molecule, ν0 is the atom-atom transition frequency for non-interacting atoms, and we have ignored mean-ﬁeld interaction energy shifts. The remaining energy, ΔE, must be imparted to the dissociated atom pair as kinetic energy. Two separate linear ﬁts are applied to the kinetic energy data in ﬁg. 26(b) to determine the threshold position. The slope beyond threshold for the data is 0.49 ± 0.03; this indicates that the atom pair (mf = −7/2 + mf = −9/2) does indeed receive the additional energy, ΔE, beyond the binding energy when the molecule is dissociated. The observed lineshape of the asymmetric peak in ﬁg. 26(a) should depend upon the Franck-Condon factor, which gives the overlap of the molecular wave function with the atomic wave function. Ticknor and Bohn calculated this multichannel Franck-Condon overlap as a function of energy. The resulting transition rate, convolved with the frequency width of the applied r.f. and scaled vertically, is shown as the solid line in ﬁg. 26(a). The agreement between theory and experiment for the dissociation spectrum is quite good. This well-resolved spectrum provides much information about the molecular wave function. A useful discussion of the theoretical aspects of these dissociation spectra and their relation to the wave function of the initial and ﬁnal states can be found in ref. [113]. In ﬁg. 27, we plot the magnetic-ﬁeld dependence of the frequency shift Δν = νr.f. −ν0 , which to ﬁrst order should correspond to the molecular binding energy. While Δν could, in principle, be obtained directly from the transfer spectrum (ﬁg. 26(a)), we use the appearance of the threshold in the energy of the mf = −7/2 cloud, as it is more clear (ﬁg. 26(b)). We compare the position of this energy threshold to the expected atom-atom transition frequency, ν0 , based upon a calibration of the magnetic-ﬁeld strength. The data are consistent with a theoretical calculation of the binding energy (solid line) based upon a full coupled-channels calculation with no free parameters carried out by Ticknor and Bohn. The binding energy plot (ﬁg. 27) highlights the fact that these Feshbach molecules are not typical molecules. With binding energies on the order of h × 100 kHz (4 × 10−10 eV), they are extremely weakly bound as compared to the molecules chemists are accustomed to studying. The excellent agreement with theory in ﬁg. 27 left no doubt that we were creating Feshbach molecules from our fermionic atoms. In addition, the r.f. spectroscopy technique that we introduced has since been used for a variety of other measurements in paired

29

Fermi gas experiments

0

Δν (kHz)

-100 -200 -300 -400 -500 220

221

222

223

224

B(gauss) Fig. 27. – The frequency shift (Δν) from the expected mf = −5/2 → −7/2 transition plotted vs. magnetic ﬁeld for the mf = −7/2 atoms (squares) and the molecules (circles). The solid line corresponds to a calculation of the binding energy of the molecules as a function of detuning from the Feshbach resonance [36].

systems. It was proposed that r.f. spectroscopy could be used to measure the excitation gap in a superﬂuid Fermi gas [114, 115]; such a measurement is published in ref. [8]. R.f. spectroscopy was also used to detect conﬁnement-induced molecules in a one-dimensional Fermi gas [116]. Molecule dissociation via r.f. spectroscopy can also create correlated atoms with a large relative momentum, as in ref. [117]. In ﬁg. 28, we show an absorption image taken after dissociating molecules far above threshold to give them relative momentum. The dissociated atoms ﬂy out in a spherical shell, and the resulting absorption image is a ring structure. Finally, r.f. dissociation of molecules followed by spin-selective imaging provides a method of probing only the molecules (or only the atoms). . 4 3. Molecule conversion eﬃciency. – In our ﬁrst molecule creation data, we found that the conversion eﬃciency saturated at 50% (see ﬁg. 25) [36]. While we were obviously very pleased to observe such high conversion eﬃciency, this result also raised the question of what was the maximum conversion eﬃciency that one could achieve using adiabatic magnetic-ﬁeld sweeps across a Feshbach resonance. For completeness, we now discuss a recent study of the molecule conversion eﬃciency. Figure 29 shows the result of a measurement of the molecule conversion fraction for 40 a K gas as a function of T /TF . We ﬁnd that the conversion eﬃciency is monotonically related to the phase space density of the gas. The conversion fraction increases as T /TF decreases (and phase space density increases), with a maximum conversion of ∼ 90% at our lowest temperatures. In collaboration with Carl Wieman’s group at JILA, we developed a simple model that was able to describe our results and those for Feshbach

30

D. S. Jin and C. A. Regal

(a)

(b) +k -k

Fig. 28. – Dissociation of molecules with radio frequencies [117]. (a) The atoms resulting from the dissociation have equal and opposite momenta. (b) Absorption image of a dissociated molecular cloud.

molecules in a 85 Rb Bose gas [107]. The essential assumptions of this model are 1) that, to form a molecule, a pair of atoms must initially be suﬃciently close in phase space and 2) that all pairs satisfying this criterion will form molecules. The line in ﬁg. 29 is the result of the best ﬁt to this model. The model has a single ﬁtting parameter whose best-ﬁt value for the Fermi and Bose atomic gases agrees within the uncertainty. . 4 4. Long-lived molecules. – Having seen eﬃcient creation of Feshbach molecules from ultracold fermionic atoms, we wondered whether or not one could create a BEC of these

molecule fraction

1.0 0.8 0.6 0.4 0.2 0

0

0.2 0.4 0.6 0. 8 1.0 1.2

T/TF Fig. 29. – Dependence of molecule conversion on initial T /TF of a two-component Fermi gas [107].

31

N/N (ms-1)

Fermi gas experiments

0.1

0.01

0.001 1000

10000

a (a0 ) Fig. 30. – Feshbach molecule loss rate as a function of the atom-atom scattering length near a Feshbach resonance in 40 K [40]. N is the number of molecules. The line is a ﬁt of the closed circles (•) to a power law. The open circles (◦) are data for which the pair size predicted by two-body theory is larger than the interparticle spacing.

molecules. However, a critical issue was the lifetime of these molecules, especially since one might guess that these extremely weakly bound Feshbach molecules would have extremely short lifetimes. Therefore, it was an exciting development when in 2003 several groups reported very long-lived Feshbach molecules created in a 6 Li gas [37-39]. References [37,38] used magnetic-ﬁeld sweeps across a Feshbach resonance to create molecules. However, Jochim et al. [39] introduced a new technique that was possible because of the long lifetime of the molecules. Here molecules were created through the three-body recombination that occurs at a magnetic ﬁeld near the Feshbach resonance. For Feshbach molecules in a 40 K gas, we also found a surprisingly long lifetime [40], albeit signiﬁcantly shorter than what was seen for the 6 Li gas. Figure 30 shows the result of our measurement of the molecule decay rate at a variety of magnetic ﬁelds on the BEC side of the Feshbach resonance. To obtain these data, we created a molecule sample at the mf = −7/2, −9/2 Feshbach resonance in which typically 50% of the original atom gas was converted to molecules. We then measured the molecule number as a function of time while holding the molecule/atom mixture in a relatively shallow optical dipole trap [40]. The plot shows N˙ /N vs. the atom-atom scattering length, a. Here N is the number of molecules, and N˙ is the initial linear decay rate. We ﬁnd that far from resonance the molecules decay quickly, but the decay rate changes by orders of magnitude as the Feshbach resonance is approached. Close to the time that experimental results for both 6 Li and 40 K were reported, it was predicted that the Fermi statistics for the atoms would suppress collisional relaxation of Feshbach molecules to deeper bound states [88,118]. A scaling law for the dependence of the molecule decay rate on the atom-atom scattering length, a, was found in [118] and

32

D. S. Jin and C. A. Regal

later in [119]. The scaling law was found by solving the full few-body problem in the limit where the molecules are smaller than the interparticle spacing, yet a r0 , where r0 is the range of the van der Waals potential. For molecule-atom collisions, the decay rate should scale with a−3.33 and for molecule-molecule collisions with a−2.55 . Both the Fermi statistics of the atoms and the wave-function overlap between the Feshbach molecule and more tightly bound molecules were important in this result. Since our measurement was carried out with thermal molecules, the density of the molecule gas remained approximately constant over the a = 1000 to 3000 a0 range. (The peak atom density in one spin state in the weakly interacting regime was n0pk = 7.5 × 1012 cm−3 .) Thus, we could measure the power law by ﬁtting the data in ﬁg. 30 to the functional form Ca−p , where C and p are constants. We ﬁt only points for which the interatomic spacing at the peak of the cloud was larger than the expected size of a two-body molecule, a/2. We found p = 2.3 ± 0.4, consistent with the predicted power law for molecule-molecule collisions. A similar power law was observed in a gas of Li2 molecules at the 834 G Feshbach resonance [120]. In general, we found that the lifetime of the molecules is surprisingly long near the Feshbach resonance. For magnetic ﬁelds where a > 3000a0 , the molecule lifetime is greater than 100 ms. This is much longer than lifetimes observed in bosonic systems for similar densities and internal states [121, 122]. Moreover, 100 ms is actually a long time compared to many other time scales in our Fermi gas, e.g., the time scale for two-body adiabaticity, the mean time between elastic collisions, and the radial trap period. This comparison suggested that it would be possible to create a BEC of molecules starting with atomic 40 K gases. 5. – Condensates in a Fermi gas . 5 1. Molecular condensates. – The creation of a BEC from the bosonic Feshbach molecules was an obvious goal in Fermi gas experiments after it was observed that these molecules could be long lived [37-40]. Late in 2003, molecular BECs were reported simultaneously by our group at JILA using 40 K [41] and by the Grimm group in Innsbruck using 6 Li [42] (see also ref. [43]). The two experiments used very diﬀerent techniques to create and detect the molecule condensates. The Innsbruck group evaporatively cooled a 6 Li gas near a Feshbach resonance. During forced evaporation, molecules were created through three-body collisions and then were evaporated to form a BEC. The presence of the condensate was inferred from the changes in number of atoms conﬁned in a shallow trap potential [42]. At JILA, we started with a 40 K Fermi gas and used a relatively slow magnetic-ﬁeld sweep across a Feshbach resonance to create a molecular BEC without direct cooling of the molecules. The presence of the condensate was directly detected in a bimodal momentum distribution of the molecules. Previously, we had created molecules by applying a magnetic-ﬁeld ramp just slow enough to be two-body adiabatic; to create a molecular BEC, our approach was to apply a magnetic-ﬁeld ramp that was not only two-body adiabatic, but also slow with respect to the many-body physics time scale (many-body adiabatic). With such a magnetic-

33

optical density

Fermi gas experiments

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0

0 -200 -100

0

100 200

position (μm)

-200 -100

0

100

200

position (μm)

Fig. 31. – Momentum distribution of a molecule sample created by applying a magnetic-ﬁeld ramp to an atomic Fermi gas with an initial temperature of 0.19TF (0.1TF ) for the left (right) picture [41]. In the right sample, the molecules form a Bose-Einstein condensate. The lines illustrate the result of bimodal surface ﬁts.

ﬁeld ramp across the Feshbach resonance, the entropy of the original quantum Fermi gas should be conserved [38, 123]. For an initial atom gas with a low T /TF , the result should be a low entropy sample of bosonic molecules; for suﬃciently low entropy, the molecule gas must contain a BEC. To pursue this idea experimentally, we used the Feshbach resonance between the mf = −9/2 and mf = −7/2 spin states starting with a Fermi gas at temperatures below quantum degeneracy. We applied a time-dependent ramp of the magnetic ﬁeld starting above the Feshbach resonance and ending below the resonance. The magnetic ﬁeld was typically ramped in 7 ms from B = 202.78 G to either B = 201.54 G or B = 201.67 G, where a sample of 78 to 88% Feshbach molecules was observed. A critical element of this experiment was that the lifetime of the Feshbach molecules could be much longer than the typical collision time in the gas and longer than the radial trapping period (see previous section). The relatively long molecule lifetime near the Feshbach resonance allowed the atom/molecule mixture to achieve thermal equilibrium during the magnetic-ﬁeld ramp. Note, however, that since the optical trap was strongly anisotropic (νr /νz ≈ 80) we may have attained only local equilibrium in the axial direction. To study the resulting atom-molecule gas mixture after the magnetic-ﬁeld ramp, we measured the momentum distribution of the molecules using time-of-ﬂight absorption imaging. The molecules were selectively imaged using r.f. dissociation [36]. Below a (T /TF )0 of 0.17, we observed the sudden onset of a pronounced bimodal momentum distribution. Figure 31 shows such a bimodal distribution for an experiment starting with an initial temperature of 0.1TF ; for comparison we also show the resulting molecule momentum distribution for an experiment starting at 0.19TF . The bimodal momentum distribution is a striking indication that the cloud of weakly bound molecules has undergone a phase transition to a BEC [1, 2]. To see the bimodal momentum distribution, we found it necessary to reduce the interaction strength during the expansion of the gas. This was accomplished by rapidly

34

D. S. Jin and C. A. Regal

Fig. 32. – Molecular condensate fraction N0 /N vs. the scaled temperature T /Tc [41]. The temperature of the molecules was varied by changing the initial temperature of the fermionic atoms prior to the formation of the molecules. The temperature was measured through the momentum distribution of the molecular thermal gas.

changing the magnetic ﬁeld before we switched oﬀ the optical trap for expansion. The ﬁeld was typically lowered by 4 G in 10 μs. At this magnetic ﬁeld (farther away from the resonance), the atom-atom scattering length a was reduced to ∼ 500a0 . This magneticﬁeld jump resulted in a loss of typically 50% of the molecules, which we attribute to the reduced molecule lifetime away from the Feshbach resonance. In ﬁg. 32, the measured condensate fraction is plotted as a function of the ﬁtted temperature of the molecular thermal component in units of the critical temperature for an ideal Bose gas, Tc = 0.94(N νr2 νz )1/3 h/kB . In this calculated Tc , N is the total number of molecules measured without changing the magnetic ﬁeld for the expansion. The condensate fraction was extracted using a two-component ﬁt that is the sum of an inverted parabola describing the Thomas-Fermi momentum distribution of a bosonic condensate and a Gaussian momentum distribution describing the non-condensed component of the molecule cloud. From the data shown in ﬁg. 32, we determine an actual critical temperature for the strongly interacting molecules and for our trap geometry of 0.8 ± 0.1Tc . Such a decrease of the critical temperature relative to the ideal-gas prediction is expected for a strongly interacting gas [124]. We also examined the dependence on the ramp time for the magnetic-ﬁeld sweep across the Feshbach resonance. One expects that the creation of a BEC of molecules requires that the Feshbach resonance be traversed suﬃciently slowly to be many-body adiabatic. This many-body time scale should be determined by the time it takes atoms to collide and move in the trap. In ﬁg. 33, the measured condensate fraction is plotted vs. the ramp time starting with a Fermi gas at a temperature of ∼ 0.1TF . Our fastest ramps resulted in a much smaller condensate fraction, while the largest condensate fraction appeared for ramps slower than ∼ 3 ms/G.

35

Fermi gas experiments

N0 / N

0.15 0.10 0.05

0

0

2

4 6 8 ramp time (ms)

10

Fig. 33. – Time scale for many-body adiabaticity [41]. We plot the fraction of condensed molecules vs. the time in which the magnetic ﬁeld is ramped across the Feshbach resonance from 202.78 G to 201.54 G.

. 5 2. Fermi condensates. – For the molecular condensates in the previous sections, superﬂuidity occurs due to BEC of essentially local pairs whose binding energy is larger than the Fermi energy. (In hindsight, this is not strictly true —the weakly bound Feshbach molecules are too large to be considered pointlike particles, and therefore the usual theory of weakly interacting BEC does not perfectly describe these condensates.) However, there was a strong sense among ultracold Fermi gas researchers that one wanted to create a “Fermi condensate”, or equivalently a “Fermi superﬂuid”, that relied on a pairing of atoms that was more analogous to Cooper pairing in the BCS theory of superconductivity. In early 2004 we reported the creation and observation of the ﬁrst Fermi condensates [7]. To create the molecular BEC described in the previous sections, we started with a quantum Fermi gas, slowly traversed the BCS-BEC crossover regime, and ended up with a BEC of molecules. An obvious question was whether condensation also had occurred in the crossover regime that we had passed through. To answer this question, we needed to overcome a number of challenges. First, we had to show that we were not simply seeing condensation of pairs in the two-body bound state (two-body pairs), but rather condensation of pairs requiring many-body physics to form many-body pairs. A clear example of condensation of many-body pairs would be condensation on the BCS side of the Feshbach resonance. Here the two-body physics of the resonance no longer supports the weakly bound molecular state; hence, only many-body eﬀects can give rise to a condensation of fermion pairs. Second, we required a probe of the momentum distribution of many-body pairs in the crossover. On the BCS side of the Feshbach resonance, these pairs would not remain bound throughout expansion of the gas.

36

D. S. Jin and C. A. Regal

3x10 3x105

N molecules

5

2x10 2x105

dissociation of molecules at low density

5

1x105 1x10 0 -0.5

0.0

ΔB (gauss)

ΔB = 0.12 G

ΔB = 0.25 G

0.5

ΔB = 0.55 G

Fig. 34. – Time-of-ﬂight absorption images showing condensation of fermionic atoms pairs [7]. The images, taken after the projection of the fermionic system onto a molecule gas, are shown for ΔB = 0.12, 0.25, and 0.55 G (left to right) on the atom side of the resonance. The top plot shows a precise determination of the Feshbach resonance position from measurements of molecule dissociation in a low-density gas. A ﬁt of the data to an error function reveals B0 = 202.10 ± 0.07 G, where the uncertainty is given by the 10–90% width.

The key to the ﬁrst problem came from careful understanding of the two-body physics. We made a measurement to precisely determine the magnetic-ﬁeld position above which a two-body bound state no longer exists, B0 . If we observed condensation of fermionic atom pairs at B > B0 , we could be assured that these were pairs that were the result of manybody eﬀects. The top part of ﬁg. 34 shows the dissociation of Feshbach molecules in a lowdensity gas as a function of magnetic ﬁeld. Molecules were created by a slow magneticﬁeld ramp across the resonance and then were dissociated by raising the magnetic ﬁeld to a value Bprobe near the Feshbach resonance. To avoid many-body eﬀects, we dissociated the molecules after allowing the gas to expand from the trap to a much lower density. To determine if the molecules had been dissociated or not, we probed the gas at low magnetic ﬁeld; here atoms not bound in molecules can be selectively detected. The measured resonance position, B0 = 202.10±0.07 G, agreed well with previous less precise results [95, 40]. To solve the problem of measuring the momentum distribution of pairs in the crossover, we introduced a technique that takes advantage of the Feshbach resonance to pairwise project the fermionic atoms onto Feshbach molecules. We were able to probe the system by rapidly ramping the magnetic ﬁeld to the BEC side of the resonance,

Fermi gas experiments

37

where time-of-ﬂight imaging could be used to measure the momentum distribution of the weakly bound molecules. This ramp was completed on a time scale that allowed molecule formation but was still too brief for particles to collide or move signiﬁcantly in the trap. This is possible because of the clear separation of the two-body and many-body time scales. The time scale for many-body adiabaticity in ﬁg. 33 is two orders of magnitude longer than the time scale for two-body adiabaticity shown in ﬁg. 25. For these experiments, we created a weakly interacting Fermi gas as previously described. The magnetic ﬁeld was then slowly lowered at the many-body adiabatic rate of 10 ms/G to a value near the resonance. Whereas before we had considered only values of B below B0 on the BEC side, now we explored the behavior on both sides of the Feshbach resonance. We probed the gas by rapidly lowering the magnetic ﬁeld by ∼ 10 G at a rate of (40 μs/G)−1 as we simultaneously released the gas from the trap. This technique put the gas far on the BEC side of the resonance, where it was weakly interacting. After a total of typically 17 ms of expansion, the molecules were selectively detected using r.f. photodissociation immediately followed by spin-selective absorption imaging. To look for condensation, these absorption images were again surface ﬁt to a two-component function that is the sum of a Thomas-Fermi proﬁle for a condensate and a Gaussian function for non-condensed molecules. Figure 34 (bottom) shows momentum distributions of the fermion pairs, obtained with the projection technique described above, on the BCS side of the resonance. We clearly observed a bimodal momentum distribution characteristic of a condensate even when the two-body physics does not support a weakly bound molecule. In ﬁg. 35(a), we plot the measured condensate fraction N0 /N as a function of the magnetic-ﬁeld detuning from the resonance, ΔB = Bhold − B0 . The data in ﬁg. 35(a) were taken for a Fermi gas initially at T /TF = 0.08 and for two diﬀerent wait times at Bhold . Condensation was observed on both the BCS (ΔB > 0) and BEC (ΔB < 0) sides of the resonance. We further found that the condensation that occurs on the BCS side of the Feshbach resonance is distinguished by its longer lifetime (ﬁg. 35(a)). An essential aspect of these measurements is the fast magnetic-ﬁeld ramp that projects the fermionic atoms pairwise onto molecules. It is a potential concern that the condensation might occur during this ramp rather than at Bhold . To verify that condensation did not occur during the ramp, we studied the measured condensate fraction for diﬀerent magnetic-ﬁeld ramp rates. Figure 35 compares the condensate fraction measured using the (40 μs/G)−1 (circles) rate to that using a ramp that was ∼ 7 times faster (open squares). We found that the measured condensate fraction is identical for these two very diﬀerent rates, indicating that this measurement constitutes a projection with respect to the many-body physics. The validity of the magnetic-ﬁeld projection technique was also explored in studies of a 6 Li gas at MIT. Researchers there ﬁrst reproduced the observation of condensation using the pairwise projection technique with a 6 Li gas [46]. They also monitored the delayed response of the many-body system after modulating the interaction strength [125]. They found that the response time of the many-body system was slow compared to the rate of the rapid projection magnetic-ﬁeld ramp. There have also been a number of theoretical

38

D. S. Jin and C. A. Regal

N0 / N

0.15

(a)

0.10 0.05 0 0.15

N0 / N

(b) 0.10 0.05 0 -0.6 -0.4 -0.2

0

0.2 0.4 0.6 0.8

ΔB (gauss) Fig. 35. – Measured condensate fraction as a function of detuning from the Feshbach resonance ΔB = Bhold −B0 [7]. (a) Data for thold = 2 ms (•) and thold = 30 ms () with an initial cloud at T /TF = 0.08. (b) Data for two diﬀerent magnetic-ﬁeld ramp rates for the projection: 40 μs/G (circles) and ∼ 6 μs/G (squares). The dashed lines reﬂect the uncertainty in the Feshbach resonance position.

papers on the subject of the pairwise projection technique for measuring the condensate fraction in the crossover [126-128]. Work thus far has established that observation of condensation of molecules following a rapid projection ramp indicates a pre-existence of condensation of fermionic atom pairs before the projection ramp. To summarize, in this section we have discussed a method for probing the momentum distribution of fermionic atom pairs and seen how this technique was employed to observe condensation near a Feshbach resonance. By projecting the system onto a molecule gas, we observed condensation of fermionic pairs as a function of the magnetic-ﬁeld detuning from the resonance, as shown in ﬁg. 34. . 5 3. Measurement of a phase diagram. – In addition to varying ΔB and measuring the condensate fraction, we can also vary the initial temperature of the Fermi gas. Figure 36 is a phase diagram created from data from experiments varying both ΔB and (T /TF )0 . ΔB is converted to the dimensionless parameter 1/kF0 a, where a is calculated directly from ΔB through eq. (5), and kF0 is extracted from the weakly interacting Fermi gas.

39

Fermi gas experiments

0.20

N0/N -0.020 0.010 0.025 0.050 0.075 0.100 0.125 0.150 0.175

(T/TF)0

0.15 0.10 0.05 0 1.0

0.5

0

-0.5 -1.0

-1.5

1/(kF0 a) Fig. 36. – Transition to condensation as a function of both interaction and T /TF [7]. The contour plot is obtained using a Renka-Cline interpolation of approximately 200 distinct data points.

The shades of grey represent the measured condensate fraction using the projection technique. The boundary between the black and light grey regions shows where the phase transition occurs in the BCS-BEC crossover. On the BCS side of the resonance, we ﬁnd the condensate forms for higher initial T /TF as ΔB decreases (and the interaction strength increases); this result is expected based upon BCS-BEC crossover theories. The data lie precisely in the regime that is neither described by BCS nor by BEC physics, −1 < 1/kF a < 1 (see ﬁg. 6). The condensed pairs in these experiments are expected to be pairs with some qualities of diatomic molecules and some qualities of Cooper pairs. Thus, these experiments realize a phase transition in the BCS-BEC crossover regime and initiate experimental study of this physics. Finally we note that, as in our previous measurements performed in the BEC limit, the measured condensate fraction in ﬁg. 36 always remains well below one [41]. This is not observed in the case of 6 Li experiments [46], suggesting that technical issues particular to 40 K may play a role. 6. – Exploring the BCS-BEC crossover The observation of Fermi condensates opened up the possibility for experimental investigation of the physics of the BCS-BEC crossover. Many intriguing aspects of this new superﬂuid gas have been probed in experiments with both 40 K atoms and 6 Li atoms. In the following, we brieﬂy discuss three experiments that were performed at JILA using a Fermi gas of 40 K atoms. These measurements demonstrate some of the novel techniques that can be used to probe these strongly interacting Fermi condensates.

40

D. S. Jin and C. A. Regal

B

1/νpert

Bpert B0 t t pert Fig. 37. – Modulation sequence: After ramping to the ﬁnal magnetic-ﬁeld value, B0 , the magnetic ﬁeld is modulated at a frequency νpert . The envelope of the modulation is a haversine function.

. 6 1. Excitations. – The ability to control the interparticle interactions with a magneticﬁeld Feshbach resonance was clearly essential to both creating and detecting Fermi condensates. The Feshbach resonance also provides new ways to probe this system. For example, we studied excitations in the BCS-BEC crossover region using a small modulation of the magnetic ﬁeld, which, in turn, gives rise to a modulation of the interaction strength [53]. We found that the magnetic-ﬁeld modulation can couple molecules or atom pairs to the free-atom continuum. We measured the excitation spectrum of the gas at diﬀerent magnetic ﬁelds, B0 , by perturbatively modulating the magnetic ﬁeld around B0 with a frequency νpert (see ﬁg. 37). To quantify the response of the gas, we determined the increase in the gas temperature after the perturbation was applied. We found that a sensitive way to detect this increase in temperature was to measure the number of atoms that escaped from a shallow optical dipole trap because of evaporation [53]. By varying the frequency of the modulation, we could map out a pair dissociation spectrum and measure the dissociation threshold Δν. The result of this measurement for a variety of magnetic ﬁelds, B0 , is shown in ﬁg. 38. We observe several features. First, we ﬁnd a distinct peak in the excitation spectra for modulation frequencies νpert ≈ 500 Hz, which is close to twice the trap frequency. We attribute this peak to a collective excitation of the trapped gas driven by the periodic modulation of the interaction strength. Second, for frequencies larger than a threshold ν0 , the response increases. We interpret the threshold as a dissociation threshold: Pairs are only dissociated if hν0 is larger than the eﬀective pair binding energy. On the BEC side of the resonance, ν0 is non-zero and increases for decreasing B0 , consistent with twobody predictions for molecule binding energies. Third, for increasing frequency beyond ν0 , the response reaches a maximum and then slowly decreases. These results, which require further theoretical understanding, demonstrate a new probe of fermionic atom pairs and the superﬂuid state. Recently this magnetic-ﬁeld modulation method of dissociating atom pairs was extended to associate free bosonic 85 Rb atoms into Feshbach molecules in ref. [129].

41

a

Fermi gas experiments

Fig. 38. – Excitation spectra for diﬀerent detunings ΔB = B0 − 202.10 G with respect to ˜ is a measure for the Feshbach resonance. The normalized number of evaporated atoms N the response and is plotted vs. the modulation frequency f . Spectra on the BEC-side of the resonance are ﬁt to a simple model describing the dissociation process for molecules (solid line). The dashed line is a ﬁt to an empirical function, from which we can extract the threshold position ν0 and the position and height of the maximum.

. 6 2. Atom noise. – Fermi condensates are an example of a class of quantum systems that involve quantum entanglement and correlations, now being accessed by ultracold atom gas experiments. Other examples include the Mott insulator state for atoms in an optical lattice [130, 5] and proposed quantum Hall-like states for rapidly rotating condensates [131]. In these systems, the quantum state cannot always be seen in the density distributions probed in time-of-ﬂight (TOF) expansion with absorption imaging. However, Altman et al. [132] recently pointed out that atom cloud absorption images can

42

D. S. Jin and C. A. Regal

(a)

(b)

Fig. 39. – Atom shot noise in a time-of-ﬂight (TOF) absorption image. (a) One spin state of a weakly interacting two-component Fermi gas with 2.3 × 105 atoms per spin state is imaged after 19.2 ms of expansion. The grayscale shows the optical depth, OD. (b) The noise on the absorption image is dominated by atom shot noise. The noise was extracted using a ﬁlter with an eﬀective bin size of 15.5 microns.

hold information beyond the ﬁrst-order correlation provided by the density distribution. They proposed that density-density correlations can be directly measured by carefully analyzing the atom shot noise present in TOF absorption images of the atom gas. This analysis can reveal key properties of strongly correlated states of atoms such as fermionic superﬂuids or exotic states in optical lattices. With an ultracold 40 K Fermi gas, we demonstrated that analysis of the noise in absorption imaging could be used to directly probe atom-atom correlations in a quantum gas. We ﬁrst showed that we could take absorption images whose noise was dominated by atom shot noise rather than other noise sources (such as photon shot noise, laser interference fringes, and other technical noise). Atom shot noise arises because of the quantized nature of the atoms and causes a granularity in the observed density distribution (ﬁg. 39). We then created pair-correlated atoms by dissociating weakly bound diatomic molecules near a Feshbach resonance and detected the pairs through measurements of atom shot noise correlations in TOF absorption images. In the future, this method may provide a new way to probe Cooper pairs of atoms. Figure 40 shows the detection of local pairs by measuring correlations in the atom shot noise. To acquire this data, we created weakly bound Feshbach molecules, let them expand from the trap, and then dissociated them with a magnetic-ﬁeld sweep immediately before imaging. Two absorption images, corresponding to atoms in the two diﬀerent hyperﬁne spin states, were taken in rapid succession. We then calculated the correlation function, G, which is a pixel-by-pixel product of the noise in the two images averaged over the image. For uncorrelated noise, G = 0; for perfect correlations, G = 1. Figure 40

Fermi gas experiments

43

Fig. 40. – Pair-correlated atoms. We plot the measured noise correlation as a function of a relative angle of rotation between absorption images of atoms in the two spin states (inset) ˜ averaged over 11 images. The eﬀective bin size is 10.3 μm. Spatial pair correlations, G(0) > 1, can clearly be seen when the molecules are dissociated after expansion and then the atoms are immediately imaged.

shows G calculated as a function of an angle representing a relative rotation about the cloud center. We clearly see a non-zero value for G at an angle of 0 (or equivalently 2π). This corresponds to spatial correlations between atoms in the two spin states. In a second experiment, we are able to detect non-local pair correlations between atoms that have equal but opposite momentum and are therefore found at diametrically opposite points of the atom cloud in a TOF expansion (ﬁg. 41). These pair correlations are created by dissociation of molecules in the optical trap and expansion of the atom gas before imaging. Detecting these non-local pair correlations, which correspond to mo-

Fig. 41. – Atom pair correlations in momentum space. The averaged correlation signal for 102 image pairs shows a peak for atoms on opposite sides of the expanded image (corresponding to atoms with equal but opposite momentum). The eﬀective bin size is 15.5 μm.

44

D. S. Jin and C. A. Regal

mentum correlations, was signiﬁcantly more challenging experimentally, in part because any spread in center-of-mass motion of the pairs rapidly degrades the correlation signal because of blurring. With a similar method [132], it seems feasible to directly probe generalized Cooper pairs in the BCS-BEC crossover region [7, 46, 8]. These pairs would be detected as momentum correlations in the same way as presented here. For this measurement, it will be important to maximize the ratio between the relative and the center-of-mass momentum of the dissociated pairs and minimize the collision rate during the initial stage of TOF expansion. . 6 3. Thermodynamics. – In the previous section, we discussed a new technique that could be used to probe Cooper pairs of atoms. It would require analyzing the noise in absorption images whose density proﬁle corresponds to the atomic momentum distribution of the Fermi gas. This raises the question: What does the atomic momentum distribution look like for a strongly interacting Fermi gas? In this section, we describe the ﬁrst measurements of the atomic momentum distribution in the BCS-BEC crossover region. In these measurements, we observed large changes of the momentum distribution due to interparticle interactions. The momentum distribution of a Fermi gas in the crossover was measured using the standard technique of TOF expansion followed by absorption imaging [1]. The key to measuring the atom momentum distribution is that the gas must expand freely without any interatomic interactions; to achieve this, we used the magnetic-ﬁeld Feshbach resonance to quickly change the scattering length to zero for the expansion. This technique was particularly convenient using 40 K because the zero crossing of the scattering length occurs only 7.8 G above the resonance. Bourdel et al. pioneered this type of measurement using a gas of 6 Li atoms at T /TF ≈ 0.6, where TF is the Fermi temperature [35]. In this work we carried out measurements down to T /TF ≈ 0.1, where pairing becomes a signiﬁcant eﬀect and condensates have been observed [7, 46]. To understand what we expect for our trapped atom gas system, we can predict the atomic momentum distribution using a local density approximation and the results for the homogeneous case. In the trapped-gas case, in addition to the local broadening of the momentum distribution due to pairing, attractive interactions compress the density proﬁle and thereby enlarge the overall momentum distribution. Figure 42 shows a calculation of an integrated column density from the result of a mean-ﬁeld calculation at T = 0, as described in ref. [133]. Figure 43 shows the measured momentum distributions, from azimuthally averaged absorption images. The observed distributions are very similar to the prediction in ﬁg. 42. To acquire the data, we started with a weakly interacting mf = −7/2, −9/2 gas at T = 0.12TF . We then adiabatically increased the interaction strength by ramping the magnetic ﬁeld at a rate of (6.5 ms/G)−1 to near the mf = −7/2, −9/2 Feshbach resonance. After a delay of 1 ms, the optical trap was switched oﬀ and simultaneously a magnetic-ﬁeld ramp to a ≈ 0 (B = 209.6 G) at a rate of (2 μs/G)−1 was initiated. The rate of this magnetic-ﬁeld ramp was designed to be fast compared to the typical many-

45

Fermi gas experiments

1.0 0.8

0.8

n(k)

n(k/kF0 )k F0 2/N

1.0

0.6

0.4 0.2

0.4

0 0

0.2 0 0

0.6

0.5 1.0 1.5 2.0 2.5

k/kF

0.5

1.0

1.5

2.0

k/k F0 Fig. 42. – Theoretical column-integrated momentum distributions of a trapped Fermi gas n(k) R calculated from a mean-ﬁeld theory at T = 0 [133, 51]. The normalization is given by 2π n(k)kdk = N . The lines, in order of decreasing peak amplitude, correspond to a = 0, 1/kF0 a = −0.66, 1/kF0 a = 0, and 1/kF0 a = 0.59. Inset: corresponding distributions for a homogeneous system.

0.8

ODk 3 (arb)

OD normalized

1.0

0.6 0.4

4 2 0 -2 4 2 0 -2

0.2 0 0

0

0.5

1.0

1

1.5

2

3

2.0

k/kF0 Fig. 43. – Experimental azimuthally averaged momentum distributions of a trapped Fermi gas at (T /TF )0 = 0.12, normalized such that the area under the curves is the same as in ﬁg. 42 [51]. The curves, in order of decreasing peak OD, correspond to 1/kF0 a = −71, −0.66, 0, and 0.59, respectively. Error bars represent the standard deviation of the mean of averaged pixels. Inset: curves for 1/kF0 a = −71 (top) and 0 (bottom) weighted by k3 . The lines are the results of a ﬁt to eq. (2).

46

0 Ekin / Ekin

D. S. Jin and C. A. Regal

6 5 4 3 2 1 0

1.4 1.2 1.0

1.0 0.5

-2

-4

-6

0 -0.5 -1.0 -1.5 1/kF0 a

Fig. 44. – The measured energy Ekin of a Fermi gas at (T /TF )0 = 0.12 in the crossover normalized 0 = 0.25kB μK [51]. The dash-dot line is the expected energy ratio from a calculation only to Ekin valid in the weakly interacting regime (1/kF0 a < −1). In the molecule limit (1/kF0 a > 1), we calculate the expected energy for an isolated molecule (dashed line). Inset: a focus on the weakly interacting regime.

body time scales as determined by EhF = 90 μs. The cloud was allowed to freely expand for 12.2 ms, and then an absorption image of atoms in the mf = −9/2 state was taken. It is natural to consider extracting the kinetic energy from the momentum distribution. However, the measured kinetic energy is intrinsically dependent on the dynamics of the magnetic-ﬁeld ramp, with faster ramps corresponding to higher measured energies. To measure the atomic momentum distribution, the magnetic-ﬁeld ramp needs to be fast compared to many-body time scales. However, we can avoid a dependence on the details of the interatomic scattering potential by using a magnetic-ﬁeld ramp that is never fast enough to access features on the order of the interaction length of the van der Waals potential, r0 ≈ 60a0 for 40 K [134]. Thus, the results presented here represent a universal quantity, independent of the details of the interatomic potential. To exactly obtain the kinetic energy from the experimental data, we would need to 3 take the second moment of the distribution, which is proportional to k OD/ kOD. As illustrated in ﬁg. 43 (inset), this is diﬃcult because of the decreased signal-to-noise ratio for large k. Thus, our approach was to apply a 2D surface ﬁt to the image and extract an energy from the ﬁtted function. Empirically, our ﬁtting function describes the data reasonably well throughout the crossover, as illustrated in ﬁg. 43 (inset). Figure 44 shows the result of extracting Ekin as a function of 1/kF0 a; we see that Ekin more than doubles between the non-interacting regime and unitarity. Figure 44 compares the measured kinetic energy to theories corresponding to the BCS (dash-dot line) and BEC (dashed line) limits. A greater theoretical challenge is

47

Fermi gas experiments

to calculate the expected kinetic energy for all values of 1/kF0 a in the crossover. This is a diﬃcult problem because it requires an accurate many-body wave function at all points in the crossover and the ability to time-evolve this wave function. Recent work in ref. [135] has addressed this calculation using the Nozieres Schmitt-Rink (NSR) ground state [23]. In the strongly interacting regime, the result does not accurately reproduce the measured kinetic energy; this suggests that more sophisticated crossover theories are necessary. We have also recently studied the temperature dependence of the kinetic energy throughout the crossover [136]. 7. – Conclusion The experiments presented in this article, which focuses on a 40 K Fermi gas, represent only a fraction of the experimental work studying fascinating aspects of BCS-BEC crossover physics. ∗ ∗ ∗ We acknowledge support from the National Science Foundation, the National Aeronautical and Space Administration, and the National Institute of Standards and Technology. We have beneﬁted enormously from ongoing discussions with other members of the JILA BEC collaboration. REFERENCES [1] Anderson M. H., Ensher J. R., Matthews M. R., Wieman C. E. and Cornell E. A., Science, 269 (1995) 198. [2] Davis K. B. et al., Phys. Rev. Lett., 75 (1995) 3969. [3] Inouye S., Andrews M. R., Stenger J., Miesner H.-J., Stamper-Kurn D. M. and Ketterle W., Nature, 392 (1998) 151. [4] Roberts J. L., Claussen N. R., Burke J. P. jr., Greene C. H., Cornell E. A. and Wieman C. E., Phys. Rev. Lett., 81 (1998) 5109. ¨nsch T. W. and Bloch I., Nature, 415 [5] Greiner M., Mandel O., Esslinger T., H a (2002) 39. [6] DeMarco B. and Jin D. S., Science, 285 (1999) 1703. [7] Regal C. A., Greiner M. and Jin D. S., Phys. Rev. Lett., 92 (2004) 040403. [8] Chin C. et al., Science, 305 (2004) 1128. [9] Zwierlein M., Abo-Shaeer J., Schirotzek A., Schunck C. and Ketterle W., Nature, 435 (2005) 1047. [10] Onnes H. K., Akad. van Wetenschappen, 14 (1911) 818. [11] Allen J. F. and Misener A. D., Nature, 141 (1938) 75. [12] Kapitza P., Nature, 141 (1938) 74. [13] Osheroff D. D., Richardson R. C. and Lee D. M., Phys. Rev. Lett., 28 (1972) 885. [14] Bednorz J. G. and Mueller K., Z. Phys. B, 64 (1986) 189. [15] Snoke D. W. and Baym G., in Bose-Einstein Condensation, edited by Griffin A., Snoke D. W. and Stringari S. (Cambridge University Press, Cambridge) 1995, pp. 1–11. [16] Cooper L. N., Phys. Rev., 104 (1956) 1189.

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D. S. Jin and C. A. Regal

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Dynamics and superﬂuidity of an ultracold Fermi gas S. Stringari Dipartimento di Fisica, Universit` a di Trento and CNR-INFM BEC Center - 1-38050 Povo, Italy

1. – Introduction After the ﬁrst experimental realization of Bose-Einstein condensation in dilute atomic gases [1] the ﬁeld of ultracold gases has become a rapidly growing ﬁeld of research (for reviews see, for example, [2-7]). In the last years a considerable amount of experimental and theoretical work has focused on Fermi gases. With respect to Bose gases Fermi systems exhibit important diﬀerences which are the consequence of quantum statistics and of the role of interactions. A ﬁrst important diﬀerence is that, at low temperature, dilute Fermi gases occupying a single spin state practically do not interact since s-wave scattering is suppressed by the Pauli principle. This provides a unique opportunity for an almost perfect realization of the ideal Fermi gas, with useful applications to precision measurements and quantum information processes. A second important diﬀerence is that superﬂuidity in Fermi gases is the result of non-trivial many-body mechanisms where interactions play a crucial role giving rise to pairing eﬀects. The resulting manybody state exhibits a rich variety of features, depending on the sign and the value of the scattering length characterizing the interaction between atoms belonging to diﬀerent atomic species. A particularly interesting conﬁguration is the so-called unitary limit where the scattering length takes an inﬁnite value. At unitarity the system is found to be particularly stable and to exhibit clean manifestations of superﬂuidity. The possibility of tuning the value of the scattering length proﬁting of the existence of Feshbach resonances and the rich variety of trapping conﬁgurations, both of magnetic and optical nature, are making the study of ultracold Fermi gases a rich subject of research with many stimulating opportunities from both the experimental and theoretical side. The purpose of this paper is to review some of the dynamic and superﬂuid features c Societ` a Italiana di Fisica

53

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exhibited by ultracold Fermi gases with special emphasis on the eﬀects of the external conﬁnement which will be assumed in most cases of harmonic shape. After introducing the main features of the ideal Fermi gas in a harmonic trap (sect. 2) we will discuss the role of interactions and the general behavior exhibited by an interacting Fermi gas along the BEC-BCS crossover (sect. 3). We will then apply the many-body results of sect. 3 to trapped conﬁgurations (sect. 4). Sections 5 and 6 will be devoted to the study of superﬂuid eﬀects, both concerning the dynamic behavior (expansion and collective oscillations) and the rotational properties (sect. 6). 2. – Ideal Fermi gas in harmonic trap The ideal Fermi gas represents a natural starting point for discussing the physics of dilute Fermi gases. In some cases the role of interactions can be in fact neglected (this happens for spin polarized gases where the interaction is strongly suppressed by the antisymmetrization requirement). Vice versa, when the role of interactions becomes crucial, the comparison with the predictions of the ideal gas points out explicitly the new interesting features exhibited by Fermi gases. The ideal Fermi gas in the harmonic potential (1)

Vho =

1 1 1 mωx2 x2 + mωy2 y 2 + mωz2 z 2 2 2 2

is a subject with many applications in diﬀerent ﬁelds of physics, ranging from nuclear physics to quantum dots and is widely discussed in the literature. For this reason we will only focus on the most relevant features of the model, emphasizing the large-N behavior where the motion of the gas can be described in semiclassical terms. In the large-N limit many single particle states are in fact occupied and the role of the Heisenberg uncertainty principle can be safely ignored for most properties of the system. The simplest way to introduce the semiclassical description is to use a local density approximation for the Fermi distribution function of a given spin species: (2)

f (r, p) =

1 , exp[β (p2 /2m + Vho (r) − μ)] + 1

where μ is the chemical potential ﬁxed by the normalization condition ∞ g(E)dE 1 (3) Nσ = drdpf (r, p) = 3 (2π¯ h) exp[β(E − μ)] + 1 0 and Nσ is the number of atoms of the given spin species. Equation (2) explicitly shows that, the semiclassical approach accounts for the Pauli exclusion principle which represents the most relevant feature exhibited by Fermi gases. In eq. (3) we have introduced the single-particle density of states 1 (4) g(E) = drdpδ(E − sp (r, p)), (2π¯h)3

Dynamics and superfluidity of an ultracold Fermi gas

55

where sp (r, p) = p2 /2m + Vho (r) is the classical energy. The density of states depends on the dimensionality as well as on the actual form of the conﬁning potential. For the harmonic trapping potential (1) we ﬁnd (5)

g(E) =

1 E2 , 2(¯ hωho )3

where ωho = (ωx ωy ωz )1/3 is the geometrical average of the three trapping frequencies. The resulting energy dependence diﬀers from of the uniform 3D gas where the √ the one √ density of states takes the value g(E) = EV (m3/2 /( 2¯h3 π 2 )). The diﬀerence has its physical origin in the suppression of states in phase space due to the spatial conﬁnement produced by the oscillator potential. In terms of the density of states one can easily calculate the relevant thermodynamic functions. For example the energy of the gas is given by the expression (6)

E(T ) =

∞

d 0

g( ) . eβ(−μ) + 1

At T = 0 the chemical potential deﬁnes the Fermi energy (μ(T = 0) = EF ) and the normalization condition yields the result (7)

EF = kB TF = (6Nσ )1/3 ¯hωho

which ﬁxes an important energy (and temperature) scale in the problem, analog to expression EF = (¯ h2 /2m)(6π 2 nσ )2/3 holding for uniform matter where nσ is the density of a single spin component. It is worth noticing that the Fermi energy (7) has the same dependence on the number of atoms and on the oscillator frequency ωho as the critical temperature for BoseEinstein condensation, given by the well-known formula kB Tc = 0.94¯hωho N 1/3 . This is not a surprise since in a gas the eﬀects of quantum degeneracy become important when the thermal wavelength (2π¯ h2 /mkB T )1/2 is of the order of the average distance −1/3 n (0) between atoms where n(0) is the density in the center of the trap. Using a classical Gaussian distribution to provide an estimate of the density of the gas one ﬁnds 2 1/2 n−1/3 (0) ∼ N −1/3 (kB T /mωho ) , so that the scale of temperatures relevant for observing quantum eﬀects is given by h ¯ ωho N 1/3 /kB in both Fermi and Bose gases. In typical experiments the value of TF corresponds to microkelvin or fractions of microkelvins. It is however worth noticing that, diﬀerently from TBEC , the Fermi energy does not deﬁne the critical temperature of a phase transition, but just a crossover characterizing the onset of quantum degeneracy phenomena. The occurrence of a phase transition in a Fermi gas can be only the result of interaction eﬀects. An important quantity to calculate is also the release energy Erel given by the energy of the gas after switching oﬀ the conﬁning potential. A consequence of the equipartition theorem applied to the ideal gas trapped by the harmonic potential is that the release energy is always equal to Erel = E(T )/2, where E(T ) is the total energy. At T = 0 one

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S. Stringari

Fig. 1. – Release energy as a function of the temperature calculated for an ideal Fermi (solid line), classical (dotted line) and Bose (dashed line) gas with the same number of atoms and the same mass. Here Tc0 is the critical temperature for Bose-Einstein condensation for an ideal Bose gas. The release energy is given here in units of Tc0 . From [3].

has E(0) = (3/4)Nσ EF . In ﬁg. 1 we show how the release energy varies as a function of the temperature. The comparison with the prediction of the classical gas (dashed line) and of the ideal Bose gas reveals explicitly the eﬀects of quantum statistics on this measurable quantity. These eﬀects were clearly demonstrated in the experiment of [8, 9] which are reported in ﬁg. 2. The Fermi energy (7) can be used to deﬁne typical length and momentum scales characterizing the Fermi distribution in coordinate and momentum space, respectively: (8)

EF =

1 1 1 p2 mωx2 Rx2 = mωy2 Ry2 = mωz2 Rz2 = F , 2 2 2 2m

where Rx , Ry and Rz are the widths of the density distribution at zero temperature which can be directly calculated integrating the T = 0 distribution function f (r, p) = θ( (r, p) − EF ) in momentum space: (9)

n0 (r) =

Nσ 8 π 2 Rx Ry Rz

3/2 y2 z2 x2 , 1− 2 − 2 − 2 Rx Ry Rz

while the Fermi momentum pF ﬁxes the width of the corresponding momentum distri-

57

Dynamics and superfluidity of an ultracold Fermi gas

Fig. 2. – Evidence for quantum degeneracy eﬀects in trapped Fermi gases. The average energy per particle, extracted from absorption images, is shown for two spin mixtures. In the quantum degenerate regime the data agree well with the ideal Fermi gas prediction (solid line). From [9].

bution (10)

8 Nσ n0 (p) = 2 3 π pF

p2 1− 2 pF

3/2 ,

obtained by integrating the T = 0 distribution function in coordinate space. Equations (9) and (10) hold for positive values of their arguments and are often referred to as Thomas-Fermi distributions. Equation (10) is the analogue of the most familiar momentum distribution 3Nσ /(4πp3F )Θ(1 − p2 /p2F ) characterizing the uniform Fermi gas. The smoothing of n0 (p) with respect to the uniform case is the consequence of the harmonic trapping. Notice that the value of pF deﬁned above coincides with the Fermi momentum (11)

pF = h ¯ (6π 2 nσ )1/3

of a uniform gas evaluated in the center of trap. Using eqs. (7) and (8) one easiliy ﬁnds

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S. Stringari

the useful expressions (12)

1/6

Ri = aho (48Nσ )

ωho ωi

and (13)

pF =

¯ h 1.91Nσ1/6 aho

for the Thomas-Fermi radii and for the Fermi momentum, respectively, where aho = h/mωho is the average oscillator length. It is worth comparing eqs. (9)-(12) with the ¯ analogous results holding for a trapped Bose-Einstein condensed gas in the ThomasFermi limit [3]. For example, the density distribution of the BEC conﬁguration is given by the expression

(14)

N 15 n0 = 8π Rx Ry Rz

y2 z2 x2 1− 2 − 2 − 2 Rx Ry Rz

with the Thomas-Fermi radii given by Ri = aho (15N a/aho )1/5 ωho /ωi , where a is the s-wave scattering length characterizing the interaction between bosons. The density proﬁles (9) and (14) do not look dramatically diﬀerent. In both cases the radius of the atomic cloud increases with the number of atoms although the explicit dependence is slightly diﬀerent (N 1/5 for bosons and N 1/6 for fermions). Notice, however, that the form of the density proﬁles has a deeply diﬀerent physical origin in the two cases. For bosons it is ﬁxed by the two-body interaction, while in the Fermi case it is determined by quantum pressure eﬀects. In momentum space the Bose and Fermi distributions instead diﬀer in a profound way. First, as a consequence of the semiclassical picture, the momentum distribution of the Fermi gas is isotropic even if the trapping potential is deformed, diﬀerently from what happens in the BEC case. Second, the momentum width of a trapped Bose-Einstein condensed gas scales like 1/R and hence decreases by increasing N while, according to eqs. (8)-(13) the momentum width of a trapped Fermi gas scales like R and hence increases with the number of atoms. The diﬀerent behavior reﬂects the fact that the Heisenberg uncertainty inequality is close to an identity for a Bose-Einstein condensate, while in a Fermi gas one has ΔP ΔR ∼ EF /ωho ¯h. Despite its simplicity the ideal Fermi gas can exhibit non-trivial features. This is the case if one considers the combined presence of harmonic plus periodic potentials which gives rise to Bloch oscillations [10] and to insulating phenomena [11] of relatively easy experimental access. Furthermore the eﬀects of Fermi statistics of the ideal Fermi gas show up in a non-trivial way in an anti-bunching behaviour exhibited by the pair correlation function which has been the object of recent measurements [12, 13].

Dynamics and superfluidity of an ultracold Fermi gas

59

3. – Role of interactions: The BEC-BCS crossover The ideal gas model presented in sect. 2 provides a good description of a cold spin polarized Fermi gas. In this case interactions are in fact strongly inhibited by the Pauli exclusion principle. When atoms occupy diﬀerent spin states interactions instead deeply aﬀect the solution of the many-body problem. This is particularly true at very low temperature where they give rise to pairing eﬀects bringing the system into the superﬂuid phase. Let us consider the simplest case of a two-component system occupying two diﬀerent spin states hereafter called, for simplicity, spin-up (σ =↑) and spin-down (σ =↓). We will consider the grand-canonical many-body Hamiltonian

(15)

H=

σ

+

2 2

¯h ∇ † ˆ σ (r) ˆ + Vext (r) − μ Ψ dr Ψσ (r) − 2m

ˆ † (r)Ψ ˆ † (r )Ψ ˆ ↓ (r )Ψ ˆ ↑ (r), drdr V (r − r )Ψ ↑ ↓

written in second quantization where μ is the chemical potential and the ﬁeld operˆ σ (r), Ψ ˆ † (r )} = δσ,σ δ(r − r ). ators obey the fermionic anticommutation relations {Ψ σ The external potential Vext and the two-body potential V account, respectively, for the external conﬁnement and for the interaction between spin. The num atoms of diﬀerent 2 ˆ ber of atoms, ﬁxed by the normalization condition dr|Ψσ (r)| = Nσ , can in general be diﬀerent for the two spin species. In this section we will consider the unpolarized case N↑ = N↓ = N/2 in the absence of external trapping (Vext = 0). The inclusion of harmonic trapping will be discussed in the next section. We are interested in dilute gases where the range of the interatomic potential is much smaller than the interparticle distance. Furthermore, we assume that the temperature of the system is suﬃciently small so that only collisions in the s-wave channel are important. Under these conditions one expects that interaction eﬀects will be governed by a single parameter: the s-wave scattering length a. In this regard one should recall that the gaseous phase corresponds to a metastable solution of the many-body problem, the true ground state being in general a crystal conﬁguration where the microscopic details of correlations are important (to simplify the notation the metastable solution will be often also called the ground state). The above considerations explain why in eq. (15) we have ignored the interaction between atoms occupying the same spin state which, in most cases, is expected to give rise only to minor corrections, due to the quenching eﬀect produced by the Pauli principle. Of course this picture can change considerably in the presence of p-wave resonances. In order to investigate in a more tractable way the eﬀects of quantum correlations and better understand the role played by the scattering length, it is convenient to replace the microscopic potential V with an eﬀective potential Veﬀ . Diﬀerent types of eﬀective potentials can be considered. In many applications one introduces the regularized zero-

60

S. Stringari

range pseudo-potential [14]

(16)

Veﬀ (r) = gδ(r)

∂ r, ∂r

where the coupling constant g is related to the s-wave scattering length a characterizing the interaction between two atoms of diﬀerent spin by the relationship g = 4π¯h2 a/m. The regularization accounted for by the term (∂/∂r)r permits to cure the ultraviolet divergencies in the solution of the many-body Schr¨ odinger equation that arise from the vanishing range of the pseudo-potential. In general, the regularization is crucial to solve the many-body problem beyond lowest-order perturbation theory, as happens, for example, in the BCS theory of superﬂuidity. In this work we will often discuss the predictions of quantum Monte Carlo simulations. In these numerical approaches the use of the eﬀective potential (16) is hard to implement and one must resort to a diﬀerent eﬀective interatomic potential. A convenient choice is the attractive square-well potential deﬁned as follows: Veﬀ (r) = −V0 for r < R0 and Veﬀ (r) = 0 otherwise (other choices for Veﬀ have also been considered in the literature [15]). The interaction range R0 must be taken much smaller than the inverse Fermi wave vector, kF R0 1, in order to ensure that all the many-body properties of the system will not depend on its value. The depth V0 of the potential is instead varied so as to reproduce the actual value of the scattering length according to the relation a = R0 [1 − tan(K0 R0 )/(K0 R0 )], where K02 = mV0 /¯h2 . Notice that a becomes inﬁnite for every new two-body bound state entering the well. The above approaches permit to describe the many-body features uniquely in terms of the scattering length a. These schemes are adequate if the scatterng length is the only relevant interaction parameter and the two-body scattering amplitude, given by the expansion (17)

f (k) = −

1 a−1 + ik + R∗ k 2

can be safely evaluated by keeping only the ﬁrst two terms in the denominator and neglecting the term containing the eﬀective range R∗ . When R∗ becomes of the order of the inverse Fermi wave vector, as happens in the case of narrow Feshbach resonances, more complex eﬀective potentials should be introduced in the solution of the many-body problem [16]. In this section we will focus the discussion on the case of uniform systems (Vext = 0) where exact solutions of the many-body problem are available in some important limiting cases. A ﬁrst example is the dilute repulsive gas. In this case interactions can be treated through the eﬀective potential (16) with a positive scattering length a. Standard perturbation theory can be applied with the small parameter kF a 1 expressing the diluteness condition for the gas. The expansion of the energy per particle up to terms

Dynamics and superfluidity of an ultracold Fermi gas

61

quadratic in the dimensionless parameter kF a then yields, at T = 0, the result [17] (18)

3 E 4(11 − 2 log 2) 10 2 = EF 1 + kF a + (kF a) . . . , N 5 9π 21π 2

where EF is the Fermi energy. The above result is universal as it holds for any interatomic potential with a suﬃciently small eﬀective range R such that nR3 1. Higher-order terms in the expansion (18) will depend not only on the scattering length a, but also on the details of the potential. In the case of purely repulsive potentials, such as the hardsphere model, the expansion in eq. (18) corresponds to the energy of the “true” ground state of the system. For more realistic potentials with an attractive tail, the above result describes instead the metastable gas-like state of repulsive atoms. This distinction is particularly important in the presence of bound states in the two-body problem, as more stable many-body conﬁgurations could consist of a gas of dimers (see discussion below) with the same positive value of a. A second important case that can be solved exactly is the dilute Fermi gas interacting with negative scattering length (kF |a| 1). In this limit, hereafter called the BCS limit, the many-body problem can be solved both at T = 0 and at ﬁnite temperature and corresponds to the most famous BCS picture ﬁrst introduced to describe the phenomenon of superconductivity. There are many variants of the BCS theory available in the literature. We will report here the predictions of the complete BCS scheme which accounts for non trivial renormalizations of the physical quantities associated with quantum and thermal ﬂuctuations. The many body solution proceeds through a proper diagonalization of the Hamiltonian (15), (16) by applying the Bogoliubov transformation to the Fermi ﬁeld operators. This approach is non-perturbative and the ground state diﬀers in a profound way from the uncorrelated wave function of the ideal Fermi gas, being characterized by peculiar correlations associated with long-range order. The ground-state energy, expanded in terms of the small parameter kF a, takes the same form (18) of the repulsive gas, pairing eﬀects being responsible only for higher-order exponential corrections. Of course in this case the ﬁrst correction, linear in a, gives a negative contribution to the ground-state energy. BCS theory predicts a phase transition associated with the occurrence of long-range order. The corresponding critical temperature is given by the result [18] (19)

Tc = 0.28e2π/kF a TF

showing that the critical temperature becomes exponentially small as one decreases the value of kF |a|, making the observability of superﬂuid phenomena a diﬃcult task in dilute samples. Actually, in the experimentally relevant case of harmonically trapped conﬁgurations the predicted value for the critical temperature can become even smaller than the oscillator temperature h ¯ ωho /kB = TF /(6Nσ )1/3 . For example, choosing kF a = −0.2, −4 one ﬁnds Tc ∼ 10 TF , a value signiﬁcantly smaller than the oscillator temperature for realistic values of Nσ .

62

S. Stringari

Thanks to the Feshbach resonances exhibited by several atomic species it is now possible to change the scattering length in a highly controlled way by simply tuning the external magnetic ﬁeld. For example, starting from a negative and small value of a it is possible to increase the size of the scattering length, reach the resonance where a diverges and explore the other side of the resonance where the scattering length becomes positive and eventually small. One would naively expect to reach in this way the regime of the dilute repulsive gas discussed above. This is not the case because in the presence of the Feshbach resonance the positive value of the scattering length is associated with the emergence of a bound state and the formation of dimers. The size of the dimers is of the order of the scattering length a and their binding energy is ∼ ¯h2 /ma2 . These dimers have a bosonic nature, being composed of two fermions and if the gas is suﬃciently dilute they consequently give rise to Bose-Einstein condensation, as experimentally proven in [19]. The size of the dimers cannot be however too small otherwise collisions between dimers give rise to transitions to deeper molecular states [20]. The behaviour of the dilute gas of dimers (kF a 1), hereafter called the BEC limit, is properly described by the theory of Bose-Einsten condensed gases available in both uniform and harmonically trapped conﬁgurations [3, 6, 7]. In particular we can immediately evaluate the critical temperature Tc . In the uniform case this is given by the text-book relationship Tc = (2π¯ h2 /M )(nσ /g3/2 (1))2/3 where nσ is the density of dimers (equal to the density of each spin species), M = 2m is the mass of the dimers and g3/2 (1) = 2.612. In terms of the Fermi temperature of the uniform gas one can write (20)

Tc = 0.22TF

showing that the critical temperature for the onset of superﬂuidity, associated with the Bose-Einstein condensation of dimers, takes place at temperatures of the order of the Fermi temperature, i.e. at temperatures much higher than the exponentially small value (19) characterizing the BCS regime. For this reason one often speaks of high-Tc superﬂuidity. The inclusion of interactions between dimers, ﬁxed by the molecule-molecule scattering length aM according to the relationship aM = 0.6a [20], is also straightforward and is provided, at T = 0, by the Gross-Pitaevskii theory. The gas of dimers and the gas of repulsive fermions discussed above represent two diﬀerent branches of the many-body problem, both corresponding to positive values of the scattering length. The physical implementation of the repulsive gas conﬁguration can be achieved by switching on adiabatically the value of the scattering length starting from the value a = 0 (see [21]). Vice versa the gas of dimers is naturally implemented by crossing the Feshbach resonance starting from negative values of a (experimentally it is also realized by cooling down a gas at ﬁxed and positive value of the scattering length). The gas of dimers exploits the attractive nature of the force which is crucial in order to ensure the binding of the fermionic pairs. A more diﬃcult problem concerns the behavior of the many-body system when the scattering length becomes larger than the interparticle distance. This corresponds to the challenging situation of a dilute (in the sense that the range of the force is much smaller

63

Dynamics and superfluidity of an ultracold Fermi gas

than the interparticle distance) but strongly interacting system. Will the system be stable or will it collapse as happens for bosons interacting with large scattering lengths? At present an exact solution of the many-body problem is not available for kF |a| > 1 and one has to make use of approximate schemes or numerical simulations. Our present understanding, based on both theoretical estimates and experimental results, is that the system remains stable even in the so-called unitary limit, corresponding to kF |a| 1. Furthermore in this limit new interesting features are expected to take place. In fact the thermodynamic quantities should no longer depend on the actual value of the scattering length, the only relevant length scales of the problem being the inverse of the Fermi wave vector and the thermal wavelength [22]. An important consequence is that, at T = 0, the chemical potential can be parametrized in the simple way:

(21)

μ = (1 + β)EF = (1 + β)(3π 2 )2/3

¯ 2 2/3 h n , 2m

where β is a universal dimensionless parameter [23, 24] and n = n↑ + n↓ is the total density. The relationship (21) ﬁxes the density dependence of the equation of state with non-trivial consequence on the behaviour of the density proﬁles and of the collective frequencies of harmonically trapped superﬂuids (see the next sections). In general we have no reason to doubt that, at T = 0, the system will be superﬂuid for all values of kF a, i.e. along the so-called BEC-BCS crossover provided by the Feshbach resonance. This will be actually the basic point guiding our discussions in the following sections. The above results make also plausible to assume that for broad resonances, corresponding to kF R∗ 1, all the relevant properties of the system can be described in terms of the dimensionless combination kF a. This introduces a remarkable simpliﬁcation in the theoretical description of this non trivial many-body problem. While the exact solution of the many-body problem along the BCS-BEC crossover is not available, a useful approximation is provided by the BCS mean-ﬁeld theory. This approach was ﬁrst introduced to investigate the crossover by Eagles, Leggett and others [25] with the main motivation to explore the properties of superconductivity and superﬂuidity beyond the weak-coupling limit kF |a| 1. The main merit of this approach is that it provides a comprehensive, although approximate, description of the equation of state along the whole crossover, including the unitary limit 1/kF a → 0 and the BEC regime of small and positive a. The idea of BCS mean-ﬁeld theory is based on the reduction of the microscopic ˆ Ψ ˆ is taken into Hamiltonian to a mean-ﬁeld form where only the anomalous average Ψ †ˆ ˆ account, while the normal average Ψ Ψ is neglected in order to ensure convergency to the resulting equations along the whole crossover. The anomalous average characterizes the occurrence of long-range order in the oﬀ-diagonal 2-body density matrix and the non-vanishing value of the pairing function (22)

ˆ↑ r + s Ψ ˆ↓ r − s F (r, s) = Ψ . 2 2

64

S. Stringari

The diagonalization of the resulting Hamiltonian is obtained with the help of the Bogoliubov transformation which transforms particles into quasi-particles. This procedure, applied to the regularized potential (16), yields the non-trivial equation (23)

m = 4π¯ h2 a

dk (2π)3

m 1 2 2 − 2E ¯h k k

,

where Ek =

(24)

Δ2 + ηk2

is the energy of the elementary excitations ﬁxed by the order parameter Δ and ηk =

(25)

¯ 2 k2 h −μ 2m

is the energy of a free particle calculated with respect to the chemical potential. The order parameter is directly related to the pairing function (22) through the relation (26)

Δ=−

ˆ↑ r + s Ψ ˆ↓ r − s , ds Veﬀ (s) Ψ 2 2

where Veﬀ is the pseudopotential (16). For positive values of the chemical potential the order parameter Δ coincides with the energy gap of the particle excitation spectrum (24). 2 For negative values of μ the gap is instead given by Δ + μ2 . The gap has been measured in trapped gases [26] along the BEC-BCS crossover through radio-frequency transitions. Equation (23) provides an important relationship between Δ and the chemical potential μ entering the single-particle energy ηk . A second relation is given by the normalization condition which is preserved by the Bogoliubov transformation and takes the form

dk ηk (27) n= 1 − . (2π)3 Ek The two equations should be solved in a consistent way. It is remarkable that they admit a solution for any value of kF a. The resulting prediction for the equation of state μ(n) given by the BCS-mean ﬁeld theory is reported in ﬁg. 3 where it is compared with the ab initio calculation of [27], based on a Monte Carlo simulation, and the selfconsistent calculation of [28] based on a diagrammatic expansion. The ﬁgure reveals a general qualitative agreement between the two approaches. In particular in the BEC limit the mean-ﬁeld theory accounts for the existence of bound molecules of energy h ¯ 2 /ma2 . The BCS mean ﬁeld theory however misses some important features which are worth being mentioned:

Dynamics and superfluidity of an ultracold Fermi gas

65

Fig. 3. – Equation of state along the BEC-BCS crossover as a funtion of the dimensioneless parameter 1/kF a. The results of the ﬁxed-node diﬀused Monte Carlo calculations (FN-DMC) of [27] and of the diagrammatic expansion of [28] are compared with the prediction of the BCS mean-ﬁeld theory.

i) In the BEC limit the interaction between molecules is known to be governed by the value aM = 0.6a of the molecule-molecule scattering length [20]. BCS theory instead provides the wrong value aM = 2a. ii) In the BCS limit the mean-ﬁeld BCS theory does not account for the leading corrections to the ground state energy (see eq. (18)). This is due to the fact that the ˆ in the calculation of the energy. ˆ † Ψ theory ignores the Hartree terms Ψ iii) The value of the order parameter Δ in the BCS regime misses the Gorkov-Melik Barkhudarov correction which reduces the proportionality coeﬃcient between Δ and the Fermi energy by a signiﬁcant factor (∼ 0.45). iv) At unitarity the BCS theory predicts the value β = −0.42 to be compared with the value β = −0.58 given by the numerical Monte Carlo simulation. 4. – Equilibrium properties of a trapped gas Let us ﬁrst point out that in the deep BEC regime, where the interacting Fermi gas behaves like a gas of weakly interacting dimers, systematic information is available from our advanced knowledge of the physics of dilute Bose gases in harmonic traps [3]. Although the BEC regime is not easily achieved in present experiments with ultracold Fermi

66

S. Stringari

gases, the corresponding predictions nevertheless provide a useful reference for important physical quantities. For example the critical temperature for BEC in a harmonic trap is 1/3 given by the well-known expression kB TBEC = 0.94¯ hωho Nσ which, combined with the corresponding expression (7) for the Fermi energy, provides the useful relationship (28)

TBEC = 0.52TF .

Result (28) reveals that, once expressed in units of the Fermi temperature, the critical temperature in the BEC regime is higher than the corresponding value (20) holding in uniform matter. In this sense superﬂuidity is “favoured” by the presence of the harmonic trap. The critical temperature in trapped Fermi gases can be estimated also in other regimes, using the results of uniform matter within local density approximation. First estimates of Tc in the BCS regime were given in [29]. In the following we will show how the results of the previous section can be usefully employed to describe the equilibrium properties of a trapped gas along the BEC-BCS crossover at zero temperature. To this purpose we will make use of the Local Density Approximation (LDA). This approximation assumes that, locally, the system behaves like a uniform gas so that the energy of the trapped system can be written in the integral form (29) E = dr [ (n(r)) + Vho (r)n(r)] given by the sum of the internal, also called release, energy (30) Erel = dr (n(r)) and of the oscillator energy (31)

Eho =

drVho (r)n(r)

provided by the trapping potential (1). In eqs. (29)-(31) n(r) = n↑ (r) + n↓ (r) is the total density proﬁle determined by the variational relation δ(E − μ0 N )/δn(r) = 0 which yields the most relevant LDA equation (32)

μ0 = μ(n(r)) + Vho (r),

where (33)

μ(n) =

∂ (n) ∂n

is the chemical potential of uniform matter and μ0 is the chemical potential of the trapped gas, ﬁxed by the normalization condition drn(r) = N . Equation (32) provides an implicit equation for the density proﬁle n(r) holding at equilibrium.

67

Dynamics and superfluidity of an ultracold Fermi gas

The applicability of the LDA in a Fermi gas is justiﬁed if the relevant energies are much larger than the typical oscillator energies providing the quantization of the single particle levels (μ0 ¯ hω0 ). In the absence of interactions the equation of state is given by the ideal Fermi gas expression μ(n) = (3π 2 )2/3

(34)

¯ 2 2/3 h n 2m

yielding the result (35)

n(r) =

1 3π 2

2m ¯h2

3/2 (μ0 − Vho )(r)3/2

for the equilibrium density proﬁle. For harmonic trapping this corresponds to the 3/2-th power of an inverted parabola (see eq. (9)). The chemical potential μ0 takes the value μ0 = h ¯ ωho (6Nσ )1/3 ≡ EF

(36)

and, as expected, coincides with the Fermi energy (7) already introduced for the trapped ideal gas. Interactions modify the shape and the size of the density proﬁles. The eﬀects are accounted for by eq. (32) once the equation of state μ(n) is known. A simple result is obtained at unitarity where the equation of state (see eq. (21)) has the same density dependence (34) as for the ideal gas, apart from a dimensionless renormalization factor. By dividing (32) by (1+β) one then ﬁnds that the results at unitarity are simply obtained from the ones of the ideal Fermi gas by a simple rescaling of the trapping frequencies and of the chemical potential. In particular the density proﬁle at unitarity takes the form (37)

n0 (r) =

N 8 π 2 Rx Ry Rz

1−

y2 z2 x2 − 2 − 2 2 Rx Ry Rz

3/2

of the ideal gas, with the Thomas-Fermi radii given by the rescaled law (38)

Ri = (1 + β)1/4 Ri0 ,

where Ri0 are the Thomas-Fermi radii of the ideal gas (see eq. (12)). Another important case is the BEC limit where one treats the interaction between dimers using the mean-ﬁeld equation of state μ = gM n/4 with the coupling constant gM = 4π¯ h2 aM /2m ﬁxed by the molecule-molecule scattering length. According to the well-known results holding for weakly interacting bosons one ﬁnds (39)

Ri = aho

15 aM N 2 aho

1/5

ωho . ωi

68

S. Stringari

Fig. 4. – Experimental results for the column density proﬁles along the BEC-BCS crossover for a gas of 6 Li atoms. The continuous curve at 850 G, corresponding to unitarity, is the best ﬁt based on eq. (40). From [30].

In ﬁg. 4 we show the experimental results [30] for the in situ density proﬁles in a harmonically trapped Fermi gas at extremely low temperature. The results actually correspond to the axial column density, i.e. to the integral ncol = dxdyn(r) which is the quantity measured in the experiment of [30]. Very good agreement between eperiment and theory is found at unitarity where the column integral of (37) is easily calculated and yields the result (40)

16 N ncol (z) = 5π Rz

5/2 z2 1− 2 Rz

with Rz given by eq. (38). The best ﬁt to the experimental curve yields the value β = −0.68 revealing explicitly the attractive role of interactions at unitarity. In ﬁg. 5 we show the experimental results for the axial radius of the trapped cloud along the crossover obtained in the same experiment. The results on the BEC side are reasonably consistent with the prediction (39) (solid line) once one uses the value aM = 0.6a, provided by the exact solution of the dimer-dimer scattering problem [20]. For a systematic comparison between experimental and theoretical results of the density proﬁles see [31]. In addition to the in situ density proﬁles a valuable source of information comes from the measurement of the release energy, i.e. the energy of the system measured after switching oﬀ the conﬁning trap. The release energy is deﬁned by the sum (41)

Erel = Ekin + Eint

of the kinetic and the interaction terms and, in the local density approximation, is simply given by eq. (30), i.e. by the ﬁrst term of the total energy (29). The integral (30) should

Dynamics and superfluidity of an ultracold Fermi gas

69

Fig. 5. – Experimental results for the axial radius along the BEC-BCS crossover for a gas of 6 Li atoms. The data are plotted after normalization to the non-interacting Fermi gas. The full line corresponds to the prediction of eq. (39) with aM = 0.6a. From [30].

be in general calculated numerically. There are however important cases where analytical results are available. For example, at unitarity, one ﬁnds the useful result (42)

Erel =

3 (1 + β)1/2 N EF , 8

where EF = h ¯ ωho (3N )1/3 is the Fermi energy (36) of the ideal trapped gas. From the measurement of the release energy Bourdel et al. [19] were able to extract the value −0.64 for the parameter β in reasonably good agreement with the value extracted from the in situ measurement of the proﬁle as well as with the ab initio predictions of theory. Another quantity that has been recently measured in ultracold Fermi gases is the momentum distribution [32]. To this purpose one switches oﬀ the scattering length, proﬁting of the existence of a Feshbach resonance, just before the expansion of the gas. The images of the density proﬁle after expansion then provide direct access to the momentum distribution. The experimental and theoretical results are shown in ﬁg. 6 for diﬀerent values of kF0 a. The theoretical curves have been obtained by applying the local density approximation [33] n(k) = (2π)−3 drnk (r) to the particle distribution function (43)

1 ηk † nk ≡ ak ak = 1− 2 EK

calculated in uniform matter at the corresponding value of the density, within BCS meanﬁeld theory. Equation (43) reduces to the step function Θ(k − kF ) in the deep BCS limit, while it approaches the value (44)

nk =

1 4 (kF a)3 2 2 3 (k a + 1)2

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S. Stringari

Fig. 6. – Theoretical (left) vs. experimental (right) values of the momentum distribution in a gas of 40 K atoms along the BEC-BCS crossover. From [32].

in the opposite BEC limit where, as expected, it coincides with the momentum distribution of a particle in the dimer bound state. The ﬁgure reveals a rather satisfactory agreement between theory and experiments. 5. – Dynamics and superﬂuidity Superﬂuidity is one of the most important properties exhibited by ultracold Fermi gases, analog to the superconducting behaviour taking place in charged Fermi systems. It shows up in peculiar transport features. Among the most noticable manifestations one should recall the absence of viscosity, the hydrodynamic nature of macroscopic dynamics even at zero temperature, the existence of quantized vortices and the occurrence of pairing eﬀects. The last two features are typical of Fermi superﬂuids, while the ﬁrst ones characterize also the superﬂuid behavior of Bose systems. The possibility of exploring these phenomena in ultracold gases provides a unique opportunity to complement our present knowledge of superﬂuidity in neutral Fermi systems, previously limited to liquid 3 He. In this section we will discuss the hydrodynamic behaviour exhibited by superﬂuids and its implications on the dynamics of trapped Fermi gases. We will also discuss the dynamic behaviour at higher energies where pair breaking eﬀects become important. The implications of superﬂuidity on the rotational properties will be discussed in the following section. The macroscopic behaviour of a neutral superﬂuid is governed by the equations of irrotational hydrodynamics. At zero temperature they consist of coupled and closed equations for the density and the velocity ﬁeld. In fact, due to the absence of the normal component, the superﬂuid density coincides with the total density and the superﬂuid current with the total current. The equations take the form (45)

∂ n + ∇ · (nv) = 0 ∂t

71

Dynamics and superfluidity of an ultracold Fermi gas

for the density (equation of continuity) and (46)

∂ m v+∇ ∂t

1 mv2 + μ(n) + Vho 2

=0

for the velocity ﬁeld (Euler equation) where μ(n) is the chemical potential, ﬁxed by the equation of state of uniform matter. At equilibrium (v = 0) the Euler equation provides the Thomas-Fermi condition (32) for the ground state proﬁle. The irrotationality of the velocity ﬁeld, implied by eq. (46), is the consequence of the existence of the order parameter (26) whose phase Φ is related to the superﬂuid velocity ﬁeld by the relationship (47)

v=

¯h ∇Φ. 2m

The hydrodynamic equations (45)-(46) diﬀer from the corresponding equations holding in the collisionless regime of a non-superﬂuid system because of the irrotationality constraint (47). Despite the quantum origin underlying the superﬂuid behaviour, the hydrodynamic equations of motion have a classical form and do not depend explicitly on the Planck constant. This peculiarity raises the question whether the hydrodynamic behaviour of a cold Fermi gas can be used to test the achievement of the superﬂuid regime. As we will see, Fermi gases above the critical temperature can easily enter a collisional regime where the dynamic behaviour is governed by the same equations. In this respect it is important to stress that collisional hydrodynamics admits the possibility of rotational components in the velocity ﬁeld which are strictly absent in the superﬂuid. A distinction between classical and superﬂuid hydrodynamics is consequently possible only studying the rotational properties of the gas (see next section). The applicability of the hydrodynamic equations is restricted to the study of macroscopic phenomena, characterized by long-wavelength excitations. In particular the wavelengths should be larger than the so-called healing length. In the limit of BEC dimers the

healing length is proportional to ¯h2 /M gM n, where M = 2m and gM n is the molecular mean-ﬁeld energy. In the opposite BCS limit the healing length is instead proportional to h ¯ /Δ, where Δ is the pairing gap. At unitarity the healing length is of the order of the inter-particle distance. At the end of the section we will relate the healing length to the critical Landau’s velocity and discuss its behavior along the BEC-BCS crossover. Let us ﬁnally remark that the hydrodynamic equations of superﬂuids have the same form both for Bose and Fermi systems, the eﬀects of statistics entering only the form of the equation of state μ(n). Important applications of the hydrodynamic equations concern the expansion of the gas after release of the trap and the collective oscillations. In most experiments with ultracold atomic gases images are taken after expansion of the cloud. In the absence of interactions the expansion of a Fermi gas is asymptotically isotropic even if the gas is initially conﬁned by an anisotropic potential. This is

72

S. Stringari

the consequence of the isotropy of the momentum distribution n(p) which, for a noninteracting gas, ﬁxes the asymptotic beavior of the density n(r, t) of the expanded gas according to the law n(r, t) → (m/t)3 n(p) with r = tp/m. Deviations from isotropy are consequently an important indicator of the role of interactions. In the experiment of [34] the ﬁrst clear evidence of anisotropic expansion of an interacting ultracold Fermi gas was reported, opening an important debate in the scientiﬁc community aimed to understand the nature of these novel many-body conﬁgurations. Hydrodynamic theory has been extensively used in the past years to analyze the expansion of Bose-Einstein condensed gases. More recently it was proposed as a theory for the expansion of a Fermi superﬂuid [35]. The hydrodynamic solutions are obtained starting from the equilibrium conﬁguration, corresponding to a Thomas-Fermi proﬁle, and then solving eq. (46) by setting Vho = 0 for t > 0. For an important class of conﬁgurations the spatial dependence can be analytically inferred. In fact, if the chemical potential has the power law μ ∝ nγ dependence on the density, the Thomas-Fermi equilibrium proﬁles have the analytic form n0 ∝ (μ0 − Vho )1/γ and one can easily prove that the scaling ansatz (48)

n(x, y, z, t) = (bx by bz )−1 n0

x y z , , bx by bz

provides the exact solution for the expansion with the scaling parameters bi obeying the simple time-dependent equation (49)

¨bi −

ωi2 = 0. bi (bx by bz )γ

Equation (49) generalizes the scaling equations previously introduced in the case of an interacting Bose gas (γ = 1) [36, 37]. From the solutions of eq. (49) one can easily calculate the aspect ratio as a function of time. For an axially symmetric trap (ωx = ωy ≡ ω⊥ ; bx = by ≡ b⊥ ) this is deﬁned as the ratio between the radial and axial radii. In terms of the scaling parameters bi it can be written as (50)

R⊥ (t) b⊥ (t) ωz = . Z(t) bz (t) ω⊥

For an ideal gas the aspect ratio tends to unity, while the hydrodynamic equations yield an asymptotic value = 1. Furthermore hydrodynamics predicts a peculiar inversion of shape during the expansion caused by the hydrodynamic forces which are larger in the direction of larger density gradients. As a consequence an initial cigar shaped conﬁguration is brought into a disk proﬁle at large times and vice versa. One can easily estimate the typical time at which the inversion of shape takes place. For a highly elongated trap (ω⊥ ωz ) the axial radius is practically unchanged for short times since the relevant expansion time along the z-th axis is ﬁxed by 1/ωz 1/ω⊥ . Conversely the radial size increases fast, and, for ω⊥ t 1 one expects R⊥ (t) ∼ R⊥ (0)ω⊥ t. One then ﬁnds that the aspect ratio is equal to unity when ωz t ∼ 1.

Dynamics and superfluidity of an ultracold Fermi gas

73

Fig. 7. – Images of the expanding cloud of an ultracold Fermi gase of 6 Li atoms at diﬀerent times. The ﬁgure clearly shows the typical inversion from the cigar to the disc shape revealing the hydrodynamic nature of the expansion. From [34].

In ﬁg. 7 we show the experimental images [34] of the expanding cloud of an ultracold Fermi gas taken at diﬀerent times close to unitarity. The ﬁgure clearly shows the inversion of the shape from cigar to disc predicted by hydrodynamic theory. In ﬁg. 8 the predictions for the aspect ratio given by eqs. (49)-(50) at unitarity, where γ = 2/3, are shown together with the experimental results of [34]. The conﬁguration shown in these ﬁgures corresponds to an initial aspect ratio equal to R⊥ /Z = 0.035. The comparison strongly supports the hydrodynamic nature of the expansion of these ultracold Fermi gases. The experiment was repeated at higher temperatures and found to exhibit a similar hydrodynamic behaviour even at temperatures of the order of the Fermi temperature, where the system cannot be superﬂuid. One then concludes that in the normal phase the system exhibits a collisional regime. This is especially plausible close to unitarity where the scattering length is very large.

74

S. Stringari

Fig. 8. – Aspect ratio as a function of time during the expansion of an ultracold Fermi gas at unitarity (red points: experiment; red line: hydrodynamic theory). For comparison the ﬁgure also shows the results in the absence of interactions (blue points: experiment; blue line ballistic expansion). From [34].

The collective oscillations of a superﬂuid gas provide a further relevant source of information. These oscillations can be studied by considering the linearized form of the time dependent HD equations (45)-(46), corresponding to small oscillations n = n0 + δn exp[−iωt] of the density with respect to the equilibrium proﬁle n0 , where ω is the frequency of the oscillation. The linearized equations take the form (51)

−ω δn = ∇ · n0 ∇ 2

∂μ δn ∂n

,

the velocity ﬁeld being ﬁxed by the equation (52)

∂v = −∇ m ∂t

∂μ δn . ∂n

Let us ﬁrst consider the case of isotropic harmonic trapping (ωx = ωy = ωz ≡ ωho ). A general class of divergency free (also called surface) solutions is available in this case. They are characterized by the velocity ﬁeld v ∝ ∇(r Y m ), satisfying the condition ∇·v = 0 and corresponding to the behavior (∂μ/∂n)δn ∝ r Y m for the density variation. Using the identity (∂μ/∂n)∇n0 = −∇Vho , holding for the density proﬁle at equilibrium,

75

Dynamics and superfluidity of an ultracold Fermi gas

it is immediate to ﬁnd that these solutions obey the equation (53)

ω2

∂μ δn = ∇Vho · ∇ ∂n

∂μ δn ∂n

.

The resulting dispersion law (54)

ω() =

√ ωho

is independent of the form of the equation of state, as generally expected for the surface modes driven by an external force. This result provides a model-independent characterization of the hydrodynamic regime. The result in fact diﬀers from the prediction ω() = ωho of the ideal-gas model, revealing the importance of interactions accounted for by the hydrodynamic description. Only in the dipole case ( = 1), corresponding to the rigid oscillation of the center of mass, interactions do not aﬀect the frequency of these modes. In addition to the surface modes an important solution predicted by the hydrodynamic equations in isotropic harmonic traps is the = 0, m = 0 breathing radial mode whose solution can be found in analytic form if the equation of state is polytropic (μ ∝ nγ ). In this case the velocity ﬁeld has the radial form v ∝ r and the frequency is equal to (55)

ω(m = 0) =

3γ + 2ωho .

√ For γ = 1 one recovers the known BEC result 5ωho [38], while at unitarity one ﬁnds 2ωho . It is worth stressing that the unitary result ω = 2ωho is not limited to small amplitude oscillations and keeps its validity beyond the hydrodynamic approximation [39]. In the case of axi-symmetric trapping (ωx = ωy ≡ ω⊥ = ωz ) the third component h ¯m of angular momentum is still a good quantum number and one also ﬁnds simple solutions of eq. (53). The dipole modes, corresponding to the center of mass oscillation, have frequencies ω⊥ for m = ±1 and ωz for m = 0. The oscillations where the velocity ﬁeld is linear in the spatial coordinates exhibit a richer structure. The solutions with m = ±2 and m = ±1 are surface excitations of the form v ∝ ∇(x ± iy)2 and v ∝ ∇(x ± iy)z, with frequency given, respectively, by (56)

ω(m = ±2) =

√

2ω⊥

and (57)

ω(m = ±1) =

2 + ω2 , ω⊥ z

independent of the equation of state. The dispersion (56) of the radial quadruple mode has been recently tested experimentally along the crossover [40]. The m = 0 solutions instead depend on the equation of state. For a polytropic dependence of the chemical potential state (μ ∝ nγ ) the problem can be solved in an

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Fig. 9. – Frequency of the radial compression mode for an elongated Fermi gas in units of the radial frequency. The theory curves refer to mean ﬁeld BCS theory (lower curve) and quantum MC calculations (upper curve). Points are experimental data. From [46].

analytic way. The solutions are characterized by a velocity ﬁeld of the form v ∝ ∇[a(x2 + y 2 ) + bz 2 ], resulting from the coupling between the = 2 and = 0 modes caused by the deformation of the trap. The corresponding frequencies are given by [41] (58)

1 2 + (γ + 2)ωz2 ± ω 2 (m = 0) = 2(γ + 1)ω⊥ 2 4 + (γ + 2)2 ω 4 + 4(γ 2 − 3γ − 2)ω 2 ω 2 . 4(γ + 1)2 ω⊥ z ⊥ z

Equation (58) reduces to the one derived in [38] in the interacting Bose case (γ = 1), while at unitarity (γ = 2/3) it coincides with the HD prediction applied to the isoentropic oscillations of the ideal traps (ωz ω⊥ ) the two gas [37, 42-44]. For elongated solutions (58) reduce to ω = 2(γ + 1)ω⊥ and ω = (3γ + 2)/(γ + 1)ωz . Let us discuss in more detail the behavior of the compression modes at unitarity where, for elongated traps, one ﬁnds the prediction 10/3ω⊥ and 12/5ωz for the radial and axial oscillations, respectively. Experimentally both the two modes have been investigated in ultracold Fermi gases [45, 46]. In ﬁg. 9 we show the recent experimental results for the compressional radial mode taken from [46]. The agreement between theory and experiment at unitarity is remarkable conﬁrming our understanding of the dynamic behaviour in this highly correlated regime where the scattering length is much larger than the interparticle distance and the system exhibits a universal behavior. It is also worth noticing that the damping of the oscillations is smallest near unitarity.

Dynamics and superfluidity of an ultracold Fermi gas

77

When we move from unitarity the collective oscillations exhibits other interesting features. Theory predicts that in the BEC regime (γ = 1) the frequencies of both the axial and radial modes are higher than at unitarity (equal to 2ω⊥ and 5/2ωz respectively). Furthermore the ﬁrst corrections with respect to the BEC prediction can be calculated analytically, by accounting for the ﬁrst correction to the BEC equation of state μ = gn produced by quantum ﬂuctuations. This is the so-called Lee-Huang-Yang (LHY) correction ﬁrst derived in the framework of Bogoliubov theory of a uniform Bose gas [17]. The resulting shifts in the collective frequencies can be calculated analytically by solving the hydrodynamic equations (53) and treating the LHY term in a perturbative way. In the case of the most relevant m = 0 radial breathing mode in a cigar-like conﬁguration (ωz ω⊥ ) one ﬁnds the following result for the relative frequency shift [47, 48]: (59)

√ δω 105 π = a3M n(0) , ω 256

where ω = 2ω⊥ is the unperturbed value. The shift is positive reﬂecting the repulsive nature of the interaction between molecules. As a consequence, the dispersion law, when one moves from the BEC regime towards unitarity, exhibits a typical non-monotonic behavior. It ﬁrst increases, as a consequence of the LHY eﬀect, and eventually decreases to reach a lower value at unitarity [49]. In general the collective frequencies can be calculated numerically along the whole crossover by solving the hydrodynamic equations once the equation of state is known. Figure 9 shows the predictions obtained using the equation of state of the MC simulation [50] (see also [51]) and of BCS mean-ﬁeld theory [52]. The MC equation of state accounts for the LHY eﬀect while the mean-ﬁeld BCS theory misses it, providing a monotonic behavior for the compressional frequencies as one moves from the BEC regime to unitarity. The equation of state is in both cases consistent with the correct value 10/3ω⊥ at unitarity. The accurate measurements of the radial compression mode shown in ﬁg. 9 conﬁrm the prediction of the MC simulation, providing an important test of the equation of state and the ﬁrst observation of the LHY eﬀect. The behavior of the collective frequencies on the BCS side of the resonance exhibits diﬀerent features. Theoretically one expects that when the system reaches the BCS regime the frequencies should be the same as at unitarity, the equation of state being governed by the same 2/3 power law density dependence. However, things behave diﬀerently in experiments. The observed behavior is not a surprise and can be qualitatively understood by noticing that when one moves towards the BCS regime the critical temperature and the pairing gap become smaller and smaller and soon reach values of the order of the trapping oscillator frequencies. Under these conditions, even assuming that the system be at zero temperature, one looses superﬂuidity and the system is eventually expected to behave like a dilute collisionless gas whose collective frequencies, apart from minor mean-ﬁeld corrections [53], should approach the higher values 2ω⊥ and 2ωz , respectively, for the radial and axial modes. Experimentally this transition is observed for the radial mode where the relevant trapping frequency is higher and occurs at about kF |a| ∼ 1. It is also associated with a strong increase of the damping of the collective oscillations [45].

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As already pointed out, hydrodynamic theory describes correctly only the lowfrequency oscillations of macroscopic nature, corresponding to sound waves in uniform matter. When one considers higher excitation energies the dynamic response should also include the breaking of pairs into two fermionic excitations. The general picture of the excitations produced by a density probe can then be summarized as follows (for simplicity we consider a uniform gas): at low frequency the system exhibits a gapless phononic branch whose slope is ﬁxed by the sound velocity c and hence by the compressibility of the gas according to the equation (60)

mc2 = n

∂μ . ∂n

At high frequency one expects the emergence of a continuum of excitations starting from a given threshold frequency, above which one can break pairs. The value of the threshold frequency depends on the value of the total momentum carried by the perturbation. A ﬁrst estimate is provided by the BCS mean-ﬁeld theory which predicts the following result for the threshold: (61)

hωth = 2Δ for μ > 0 and q ≤ 2 2μ , ¯ hωth = 2 (q 2 /8m − μ)2 + Δ2 elsewhere. ¯

The interplay between phonon and pair breaking excitations gives rise to diﬀerent scenarios along the crossover. In the BCS regime the threshold occurs at low frequencies and the phonon branch very soon reaches the continuum of single-particle excitations. The behavior is quite diﬀerent in the opposite BEC regime where the gapless phonon branch extends up to high frequencies. At large momenta this branch actually looses its phononic character and approaches the dispersion q 2 /4m of a free molecule. In the deep BEC limit the gapless branch coincides with the Bogoliubov spectrum of a dilute gas of bosonic molecules. At unitarity the system is expected to exhibit an intermediate behavior, the discretized branch surviving up to momenta of the order of the Fermi momentum. A detailed calculation of the excitation spectrum [54], based on a proper time-dependent generalization of the mean ﬁeld BCS theory, is provided in ﬁg. 10. The results for the excitation spectrum provide a useful insight on the superﬂuid behavior of the gas in terms of Landau’s criterion according to which a system cannot give rise to energy dissipation if its velocity, with respect to a container at rest, is smaller than the Landau’s critical velocity deﬁned by the equation (62)

vcr = min q

¯hωq , q

where ¯hωq is the energy of an excitation carrying momentum q. According to this criterion the ideal Fermi gas is not superﬂuid because of the absence of a threshold for the single particle excitations, yielding vcr = 0. The interacting Fermi gas of ﬁg. 10 is

Dynamics and superfluidity of an ultracold Fermi gas

79

Fig. 10. – Excitation spectrum of the superﬂuid Fermi gas along the BEC-BCS crossover. Energy is given in units of the Fermi energy. Left: BCS regime. Center: unitarity. Right: BEC regime. From [55].

instead superﬂuid in all regimes. By inserting result (61) for the threshold frequency into eq. (62) one can calculate the critical value of v due to pair breaking. The result is

(63)

sp vcr

1/2 [ Δ2 + μ2 − μ] . = m

In the deep BCS limit |a|kF → 0 (corresponding to Δ μ) eq. (63) approaches the exponentially small value vcr = Δ/pF . On the BEC side the value (63) instead becomes larger and larger and the relevant excitations giving rise to Landau’s instability are no longer single-particle excitations but phonons, and the critical velocity coincides with the sound velocity. A simple estimate of the critical velocity along the whole crossover is then given by the expression (64)

sp vcr = min (c, vcr ).

Remarkably, one sees that vcr has a maximum near unitarity (see ﬁg. 11), further conﬁrming the robustness of superﬂuidity at unitarity. The large value of vcr should show up in a visible reduction of dissipation in experiments where one moves an external impurity in the medium at tunable velocities. Experiments of this type have been already performed in the case of Bose-Einstein condensed gases [56]. An important physical quantity, directly related to the critical velocity, is the healing length deﬁned as (65)

ξ=

¯h . mvcr

√ Apart from a trivial numerical factor it coincides with the usual deﬁnition h ¯ / M gM n of the healing length on the BEC side and with the size of Cooper pairs in the opposite BCS limit. The healing length provides the typical length scale above which the dynamic

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Fig. 11. – Landau’s critical velocity (in units of Fermi velocity) calculated along the crossover. The ﬁgure clearly shows that the critical velocity is largest close to unitarity. From [55].

description of the system is safely described by the hydrodynamic picture. It is smallest near unitarity. The knowledge of the excitation spectrum and of the corresponding matrix elements of the density operator allows one to calculate the dynamic structure factor [7] (66)

S(k, ω) = Q−1

e−βEmn | 0|δ ρˆk |n |2 δ(¯ hω − ¯hωmn ),

m,n

where ¯hk and h ¯ ω are the momentum and energy, respectively, transferred by the probe to the sample, δ ρˆ is the ﬂuctuation of the Fourier component ρˆk = j exp[−ik · rj ] of the density operator, ωmn = (Em − En )/¯h are the usual Bohr frequencies and Q is the partition function. The deﬁnition of the dynamic structure factor is immediately generalized to other excitation operators like, for example, the spin density operator. In dilute gases the dynamic structure factor can be measured with Bragg scattering experiments. The main features of the dynamic structure factor are best understood in uniform matter where the excitations are described in terms of their momentum. From the previous discussion on the excitation spectrum one expects that, for suﬃciently small momenta,

Dynamics and superfluidity of an ultracold Fermi gas

81

Fig. 12. – Dynamic structure factor of the superﬂuid Fermi gas along the BEC-BCS crossover at k = 4kF . From [54].

the dynamic structure factor be characterized by a δ-like peak and by a continuum of single-particle excitations at higher energy. At higher momentum transfer the behavior will depend crucially on the regime considered along the BEC-BCS crossover. In ﬁg. 12 we report the T = 0 predictions for the dynamic structure factor S(k, ω) at relatively high wave vectors (k = 4kF ) where the discretized branch is available only on the BEC side of the resonance. For such values of k the calculation of the dynamic response factor can be usefully applied, within a LDA procedure, to estimate the response of the system in a trapped conﬁguration. The results presented in these ﬁgures are based on a time-dependent generalization of the BCS mean ﬁeld theory. This theoretical approach accounts for both phononic and single-particle parts of the excitation spectrum as well as for the corresponding hybridization phenomena. On the BEC side of the resonance one clearly sees a discretized peak correponding to the free-molecule excitation energy h2 k 2 /4m. It is remarkable to see that even at unitarity, where molecules do not exist ¯ as independent excitations and the discretized peak has merged into the continuum of single-particle excitations, the dynamic structure factor exhibits a pronounced peak at ∼ ¯ h2 k 2 /4m. On the BCS side of the resonance the molecular signatures are instead

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Fig. 13. – Static structure factor of the superﬂuid Fermi gas along the BEC-BCS crossover. From [54].

completely lost and the response is very similar to the one of an ideal Fermi gas. In the same ﬁgure we also show the magnetic structure factor which is obtained replacing the density operator ρq = ρ↑ + ρ↓ with the spin density operator ρ↑ − ρ↓ in eq. (66). The magnetic operator does not excite the phonon mode and its strength is restricted to the continuum of single particle excitations. From the knowledge of the dynamic structure factor one can evaluate the static structure factor, given by its frequency integral (67)

¯ h S(k) = N

0

∞

1 S(k, ω)dω = N

eik·(ri −rj )

,

i,j

where, in deriving the last equality, we have used deﬁnition (66) and the completeness relationship n |nn| = 1. The static structure factor is related to the two-body correlation function g(r) by the relationship (68)

S(k) = 1 + n

dr[g(r) − 1]e−ik·r .

Its behavior, evaluated at diﬀerent points along the BEC-BCS crossover, is shown in ﬁg. 13 where we report both the calculation of the dynamic mean ﬁeld approach [54], which has been obtained by directly integrating the dynamic structure factor, and the results of the ab initio Monte Carlo calculations of [27], obtained by Fourier transforming

83

Dynamics and superfluidity of an ultracold Fermi gas

the pair correlation function g(r). The static structure factor decreases linearly at small k as a consequence of the phononic nature of the excitation spectrum, while it approaches the incoherent value S(k) = 1 for large wave vectors where only the terms i = j in the sum of eq. (67) survive. It worth noticing that at intermediate values of k the static structure factor exhibits a maximum which, in the BEC regime kF a 1, takes the value S(k) = 2 over an extended region of values of k. The origin of this plateau is directlly related to the molecular nature of elementary excitations. In fact, if k is larger than the Fermi momentum, but still small compared to the inverse of the size a of the molecules, the probe mainly excites free molecules with energy ¯h2 k 2 /4m. Using the model-independent f -sum rule result (69)

h ¯

2

dωS(k, ω)ω = N

¯ 2 k2 h 2m

and assuming, following Feynman, that a single molecular-like excitation with energy h2 k 2 /4m exhausts the integral, one ﬁnds the value S(k) = 2. Although this result holds ¯ only in the deep BEC limit, the ﬁgure shows that the enhancement of S(k) with respect to the incoherent atomic value S(k) = 1 is clearly visible also at unitarity. 6. – Rotating Fermi gases and superﬂuidity Superﬂuidity shows up in spectactular rotational features. In fact a superﬂuid cannot rotate like a rigid body, due to the irrotationality constraint (47) imposed by the existence of the order parameter. At low angular velocity an important macroscopic consequence of irrotationality is the quenching of the moment of inertia. At higher angular velocities the superﬂuid can instead carry angular momentum via the formation of singular vortex lines. The circulation of these lines is quantized. In the presence of many vortex lines a regular vortex lattice is formed and the angular momentum acquired by the system takes the classical rigid body value. Both the quenching of the moment of inertia and the formation of vortex lines have been the object of fundamental investigation in the physics of quantum liquids and have been more recently explored also in dilute Bose-Einstein condensed gases. In this section we summarize some of the main features exhibited by dilute Fermi gases where important experimental results are already available. We ﬁrst discuss the macroscopic consequences of the irrotationality constraint (moment of inertia and collective oscillations) and then some key features of quantized vortices. The moment of inertia Θ relative to the z-th axis is deﬁned as the response of the system to a rotational ﬁeld −ΩLz according to the relationship (70)

Lz = ΩΘ,

where Lz is the third component of angular momentum and the average is taken on the stationary conﬁguration in the presence of the perturbation. In the limit of small angular velocity one can employ the formalism of linear response theory and write the moment

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S. Stringari

of inertia in the form Θ = 2Q−1

(71)

m,n

e−βEM

|n|Lz |m|2 , En − E m

where |n and En are the eigenstates and eigenenergies of the unperturbed Hamiltonian and Q is the partition function. There is a simple case where the sum (71) can be calculated explicitly. This is the ideal gas trapped by a deformed harmonic potential where the moment of inertia takes the analytic form [57] (72)

Θ=

mN 2 (y − x2 )(ωx2 + ωy2 ) + 2 ωy2 y 2 − ωx2 x2 . 2 − ωy

ωx2

Result (72) holds for both the Bose and Fermi ideal gas. It assumes ωx = ωy , but admits a well-deﬁned limit when ωx → ωy . In the Fermi case, when the number of particles is large, one can use the semiclasssical estimate for the radii yielding x2 ∝ 1/ωx2 and y 2 ∝ 1/ωx2 . In this case eq. (72) reduces to the rigid value of the moment of inertia: Θrig = N mx2 + y 2

(73)

(for a non-interacting Bose-Einstein condensed gas, at T = 0, where the radii scale according to x2 ∝ 1/ωx and y 2 ∝ 1/ωx , one instead ﬁnds that Θ → 0 as ωx → ωy ). Interactions can change the value of the moment of inertia of a Fermi gas in a profound way. The simplest way to calculate Θ in the superﬂuid phase is to use the irrotational hydrodynamic equations developed in the previous section, by considering a trap rotating with angular velocity Ω and looking for the stationary solutions in the rotating frame. The resulting value for the angular momentum Lz then permits to evaluate the moment of inertia through deﬁnition (70). A similar procedure was implemented experimentally to generate the rotation of a dilute Bose gas [58]. The equations of motion in the frame rotating with the trap are easily obtained by adding the term −ΩLz to the Hamiltonian. The hydrodynamic equations, in the rotating frame, take the form ∂ n + ∇ · [n(v − Ω × r)] = 0 ∂t

(74) and (75)

∂ m v+∇ ∂t

1 2 mv + μ(n) + Vho − mv · (Ω × r) = 0, 2

where the rotating trap is now described by the time independent potential Vho . One sees that the rotation aﬀects both the equation of continuity and the Euler equation. Here v = (¯ h/2m)∇Φ, where Φ is the phase of the order parameter, is the superﬂuid velocity in the laboratory frame, expressed in terms of the coordinates of the rotating frame. In

Dynamics and superfluidity of an ultracold Fermi gas

85

the presence of harmonic trapping an important class of stationary solutions is obtained making the ansatz (76)

v = α∇(xy) .

The equilibrium density, derivable from eq. (75) by setting ∂v/∂t = 0, has the same Thomas-Fermi form as in the absence of rotation, with renormalized values of the oscillator frequencies: (77)

ω ˜ x2 = ωx2 + α2 − 2αΩ, ω ˜ y2 = ωy2 + α2 + 2αΩ

while the equation of continuity yields the relationship (78)

α = −δΩ,

where (79)

δ=

y 2 − x2 y 2 + x2

is the deformation of the atomic cloud in the (x, y)-plane. For small angular velocities one ﬁnds δ = where (80)

=

ωx2 − ωy2 ωx2 + ωy2

is the deformation of the trap. By evaluating the angular momentum Lz = m dr(r×v)n with the velocity ﬁeld (76) one ﬁnally ﬁnds that the moment of inertia is given by the irrotational form (81)

Θ = δ 2 Θrig

which vanishes for an axi-symmetric conﬁguration, pointing out the crucial role played by superﬂuidity in the rotation of the gas. The measurement of the moment of inertia is not directly accessible in dilute gases. However useful information on the rotational properties of the system can be obtained through the study of the collective oscillations. In this context a special role is played by the so-called scissors mode, an oscillation of the system caused by the sudden rotation of a deformed trap. If the angle of rotation φ0 of the trap is small compared to the deformation of the trap (80), the rotation of the potential (1) produces the perturbation δVpert = (ωx2 − ωy2 )φ0 xy which naturally excites the quadrupole mode. The behavior of the resulting oscillation depends in a crucial way on whether the system is normal or superﬂuid. In fact, while the restoring force is, in both cases, proportional to the

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square of the deformation parameter of the trapping potential, the mass parameter, being proportional to the moment of inertia, behaves quite diﬀerently in the two cases. In the superﬂuid the problem can be addressed by solving the hydrodynamic equations with the ansatz δn ∝ xy and v ∝ ∇xy. One easily ﬁnds that the oscillation around equilibrium is characterized by the frequency [59] (82)

ω=

ωx2 + ωy2 .

The result holds also for a tri-axially deformed trap where ωx = ωy = ωz . For a normal gas in the collisionless regime one ﬁnds two frequencies at ω± = |ωx ± ωy |. For a normal gas in the collisional regime one instead predicts an oscillation with the same frequency (82) in addition to a low-frequency mode of diﬀusive nature, caused by the viscosity of the ﬂuid. The diﬀerent behavior exhibited by the normal gas (both in the collisionless and collisional phases) reﬂects the rigid nature of the classical moment of inertia. The scissors mode, previously observed in a Bose-Einstein condensed gas [60], has been recently investigated also in ultracold Fermi gases [40]. At unitarity the experiment has conﬁrmed the correctness of the hydrodynamic frequency (82), while deeply in the BCS regime the beating between the frequencies ω± = |ωx ± ωy | has revealed the transition to the collisionless regime. If the gas is normal, but too deeply in the collisional hydrodynamic regime, the diﬀusive mode predicted by classical hydrodynamics would be located at too low frequencies to be observable in experiments and the scissors mode would look the same as in the superﬂuid phase. This is expected to be the case at unitarity just above the critical temperature, thereby making the distinction between the superﬂuid and the normal phase, based on the study of the scissors mode, a diﬃcult task. More promising perspectives to distinguish between superﬂuid and collisional hydrodynamics are provided by the study of the collective oscillations excited on top of a rotating conﬁguration. In fact, in the presence of vorticity ∇ × v = 0, the equations of collisonal hydrodynamics contain an addiditional term depending on the curl of the velocity ﬁeld, which is absent in the irrotational equations of superﬂuid hydrodynamics. In the laboratory frame the equation for the velocity ﬁeld given by collisional hydrodynamics is given by (83)

m

∂ v+∇ ∂t

1 mv2 + μ(n) + Vho 2

− mv × (∇ × v) = 0,

where we have omitted viscosity eﬀects. The collective oscilations correspond to linearized solutions with n = n0 + δn and v = v0 + δv. In the presence of a steady rotation of the trap at angular velocity Ω the oscillation frequencies resulting from the equations of rotational hydrodynamics diﬀer from the ones of superﬂuid hydrodynamics since in the former case the steady velocity ﬁeld v0 is given by the rigid body value Ω × r, while in the latter case is given by the irrotational value −Ωδ∇xy. After generating the steady rotation of the gas by adiabatically ramping the angular velocity of a deformed trap, one can suddenly stop the rotation and the system starts oscillating performing rotations

Dynamics and superfluidity of an ultracold Fermi gas

87

around the symmetry axis of √the trap. In the superﬂuid this procedure will excite the scissors mode with frequency 2ω⊥ . In the case of rotational hydrodynamcis the behavior will instead be diﬀerent, the scissors mode being coupled with the rigid rotation of the cloud. Under the condition Ω √ 2 ω⊥ the resulting oscillation is characterized by √ the beating law φ(t) = (Ω/ 2ω⊥ ) sin( 2ω⊥ t) cos(Ωt) [41]. The possibility of distinguishing between superﬂuid and rotational hydrodynamics is a unique opportunity provided by ultracold Fermi gases. In fact Bose-Einstein condensed gases, above Tc , are usually extremely dilute and collisionless. Vice versa in the Fermi case the normal gas can be easily dominated by collisions even at temperatures smaller than the Fermi energy as proven by the behavior of the aspect ratio during the expansion of the unitary gas (see the previous section). Let us now discuss the behavior of the superﬂuid Fermi gas at higher angular velocities where quantized vortices are formed. Recent experiments have conﬁrmed their existence along the BEC-BCS crossover (see ﬁg. 14). In these experiments vortices are produced by spinning the atomic cloud with a laser beam and are observed after expansion by ramping the value of the scattering length to positve values in order to increase their visibility. Quantized vortices were previously extensively investigated with Bose-Einstein condensed gases [61]. Quantized vortices emerge as stable conﬁgurations if the angular velocity exceeds a critical value ﬁxed by the energy cost needed for their production.

Fig. 14. – Experimental observation of quantized vortices in a superﬂuid Fermi gas along the BEC-BCS crossover [62].

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A quantized vortex along the z-axis is associated with the appearence of a phase in the order parameter (22) given by the form exp[iφ], where φ is the angle around the z-axis. This yields the complex form (84)

Δ(r) = Δ(r⊥ , z) exp[iφ]

for the order parameter Δ, where, for simplicity, we have assumed that the system exhibits axial symmetry and we have used cylindrical coordinates. The velocity ﬁeld v = (¯ h/2m)∇φ of the vortex conﬁguration has a tangential form with modulus (85)

v=

¯h 2mr⊥

which increases as one approaches the vortex line, in contrast to the rigid body dependence v = Ω×r characterizing the rotation of a normal ﬂuid. The circulation is quantized according to the rule π¯h (86) v · d = m which is smaller by a factor 2 with respect to the case of a Bose superﬂuid with the same atomic mass. The value of the circulation is independent of the radius of the contour. This is a consequence of the fact that the vorticity is concentrated on the z-axis according to the law (87)

∇×v =π

¯h (2) δ (r⊥ )ˆ z, m

ˆ is the unit vector in the z-th direction. It deeply diﬀers from the uniform vorticity where z ∇ × v = 2Ω of the rigid-body rotation. The angular momentum carried by the vortex is given by the expression ¯h (88) Lz = m drr × vn(r) = N 2 holding if the vortex line coincides with the symmetry axis of the density proﬁle. If the vortex is displaced towards the perifery of a trapped gas the angular momentum takes a smaller value. In this case the axial symmetry of the problem is lost and the order parameter cannot be written in the form (84). A ﬁrst estimate of the energy of the vortex line is obtained employing macroscopic arguments based on hydrodynamics and considering, for simplicity, a gas conﬁned in cylinder of radial size R. The energy Evacquired by the vortex is mainly determined by the hydrodynamic kinetic energy (m/2) drv 2 n which, employing the velocity ﬁeld (85), yields the following estimate for the vortex energy: (89)

Ev = N

¯h R ln , 4mR2 ξ

Dynamics and superfluidity of an ultracold Fermi gas

89

where we have introduced the radius ξ of the core of the vortex which ﬁxes the distance below which the hydrodynamic expression for the kinetic energy no longer applies. This size is identiﬁed with the healing length whose value varies signiﬁcantly along the BECBCS crossover, being particularly large in the BCS limit. Equation (89) can be used to evaluate the critical angular velocity Ωc for the existence of an energetically stable vortex line. This value is obtained by imposing that the change in the energy E −Ωc Lz acquired by the system in the frame rotating with angular velocity Ωc be equal to Ev . One ﬁnds (90)

Ωc =

¯ h R ln . 2 2mR ξ

Applying this estimate to a harmonically trapped conﬁguration with RTF ∼ R and neglecting the logarithmic term which provides only a correction of order of unity, we ﬁnd Ωc /ω⊥ ¯ hω⊥ /Eho , where ω⊥ is the radial frequency of the harmonic potential 2 2 and Eho ∼ mω⊥ RTF is the harmonic-oscillator energy of the trapped gas. The above estimate shows that in the Thomas-Fermi regime, where Eho ¯hω⊥ , the critical frequency is much smaller than the radial trapping frequency, thereby suggesting that vortices should be easily produced in slowing rotating traps. This conclusion, however, does not take into account the fact that the nucleation of vortices is strongly inhibited at low angular velocities by the occurrence of a barrier. For example in rotating Bose-Einstein condensates it has been experimentally shown that it is possible to increase the angular velocity of the trap up to values signiﬁcantly higher than Ωc without generating vortical states. Under these conditions the response of the superﬂuid is governed by the equations of irrotational hydrodynamics. A challenging problem concerns the visibility of the vortex lines. Due to the smallness of the healing length they cannot be observed in situ, but only after expansion. In particular the healing length is very small in the most interesting unitary regime. Another diﬃculty in revealing the vortex lines is the reduced contrast in the density with respect to the case of Bose-Einstein condensed gases. Actually, while the order parameter vanishes on the vortex line the density does not, unless one works in the deep BEC regime. Microscopic calculations of the vortex structure along the BEC-BCS crossover [63] are in most cases based on the generalization of the BCS mean-ﬁeld theory to include non uniform conﬁgurations. In ﬁg. 15 we report the predictions obtained in [63] for the density proﬁle and for the order parameter Δ as a function of the distance from the vortical line. The ﬁgure clearly shows that the contrast in the density proﬁle becomes weaker and weaker as one approaches the BCS regime where, diﬀerently from the opposite BEC regime, the density is not aﬀected by the presence of the vortex. Conversely the order parameter Δ always vanishes close to the vortex line. At higher angular velocities more vortices can be formed giving rise to a regular vortex lattice. In this limit the angular momentum acquired by the system takes the classical rigid-body value and the rotation will look similar to the one of a rigid body, characterized by the law ∇ × v = 2Ω. Using result (87) and averaging the vorticity over ˆ, where nv is the number of several vortex lines one ﬁnds the result ∇ × v = (h/2m)nv z

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Fig. 15. – Radial proﬁles of the density and of the order parameter of a vortex line along the BEC-BCS crossover. From Sensarma et al. [63].

vortices per unit area, so that the density of vortices is related to the angular velocity Ω by the useful relation (91)

nv =

2m Ω π¯h

which turns out to be a factor 2 larger than in the case of Bose superﬂuids with the same value of the atomic mass. Equation (91) shows that the distance between vortices √ (proportional to 1/ nv ) depends on the angular velocity but not on the density of the gas. In other words, the vortices form a regular lattice even if the average density is not uniform as happens in the presence of harmonic trapping. This feature, already pointed out in the case of Bose-Einstein condensed gases, has been conﬁrmed by the recent experiments on Fermi gases (see ﬁg. 14). The vortex lattice is responsible for an important bulge eﬀect associated with the increase of the radial size of the cloud. In fact in the presence of a rigid rotation the 2 eﬀective potential felt by the atoms is given by Vho − (m/2)Ω2 r⊥ , corresponding to a modiﬁed equilibrium density, whose Thomas-Fermi radii satisfy the new relationship (92)

2 − Ω2 ω⊥ Rz2 = 2 R⊥ ωz2

showing, by the way, that at equilibrium the angular velocity cannot overcome the radial trapping frequency. In experiments where the vortex lattice is not formed in the presence

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of a stationary rotating trap this formula can be used to evaluate the eﬀective value of the angular velocity by just measuring the aspect ratio. Important consequences of the vortex lines concern also the frequency of the collective oscillations. For example, using a sum rule approach [64] it is possible to show that the splitting between the m = ±2 quadrupole frequencies is given by the formula (93)

ω(m = +2) − ω(m = −2) = 2

z 2 , mr⊥

where z = Lz /N is the angular momentum per particle carried by the vortical conﬁguration. For a single vortex line z is equal to h ¯ /2, while for a vortex lattice z is 2 given by the rigid-body value Ωmr⊥ . In the latter case one recovers exactly the splitting 2Ω predicted by the equations of rotational hydrodynamics which, in the case of axi-symmetric conﬁgurations, yield the result [41] (94)

ω(m = ±2) =

2 − Ω2 ± Ω 2ω⊥

for the frequencies of the two m = ±2 quadrupole modes. The equations of rotational hydrodynamics actually provide the correct description of the collective oscillations of a gas containing a vortex lattice which, from a macroscopic point of view, behaves like a classical gas rotating in a rigid way. The experimental production and measurement of quantized vortices has provided a deﬁnitive proof of superﬂuidity in these ultracold Fermi gases. Actually the esixtence of vortices as stable conﬁgurations can be regarded as a proof of the absence of viscosity. In fact in the rotating frame the velocity ﬁeld of a vortical conﬁguration is not vanishing and any tiny presence of viscosity would bring the system into a rigid rotation, corresponding to a vanishing velocity in the rotating frame. 7. – Conclusions In this paper we have summarized some features of the superﬂuid behavior exhibited by ultracold Fermi gases with special emphasis on the behavior of the collective oscillations. Other subjects related to the dynamics of these novel systems that would deserve careful investigation concern Fermi gases in the presence of inbalance (N↑ = N↓ ) and Fermi gases of unequal masses (m↑ = m↓ ). ∗ ∗ ∗ It is a pleasure to acknowledge fruitful collaborations and discussions with the members of the CNR-INFM Center on Bose-Einstein Condensation in Trento. In particular this paper has beneﬁted from many discussions with S. Giorgini and L. Pitaevskii. Stimulating collaborations with R. Combescot are also acknowledged.

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Making, probing and understanding ultracold Fermi gases W. Ketterle and M. W. Zwierlein Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

1. – Introduction . 1 1. State of the ﬁeld . – This paper summarizes the experimental frontier of ultracold fermionic gases. It is based on three lectures which one of the authors gave at the Varenna Summer School describing the experimental techniques used to study ultracold fermionic gases, and some of the results obtained so far. In many ways, the area of ultracold fermionic gases has grown out of the study of Bose-Einstein condensates. After their ﬁrst experimental realizations in 1995 [1, 2], the ﬁeld of BEC has grown explosively. Most of the explored physics was governed by mean-ﬁeld interactions, conveniently described by the Gross-Pitaevskii equation. One novel feature of trapped inhomogeneous gases was the spatially varying density, that allowed for the direct observation of the condensate, but also led to new concepts of surface eﬀects and collective excitations which depended on the shape of the cloud. The experimental and theoretical explorations of these and other features have been a frontier area for a whole decade! A major goal had been to go beyond mean-ﬁeld physics, which is in essence single particle physics, and to ﬁnd manifestations of strong interactions and correlations. Three avenues have been identiﬁed: lower dimensions that enhance the role of ﬂuctuations and correlations, optical lattices that can suppress the kinetic energy in the form of tunnelling [3, 4], and Feshbach resonances [5-8] that enhance interactions by resonantly increasing the interparticle scattering length. In bosonic systems, the tuning of interactions near Feshbach resonances was of limited applicability due to rapid losses. Feshbach c Societ` a Italiana di Fisica

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resonances were used mainly to access molecular states of dimers and trimers. In contrast, for fermions, losses are heavily suppressed (see below), and most of this review focuses on strongly interacting fermions near Feshbach resonances. By addressing the physics of strongly correlated matter, the ﬁeld of ultracold atoms is entering a new stage where we expect major conceptional advances in, and challenges to many-body theory. We regard it as fortunate that BEC turned out to be a less complex target (both experimentally and theoretically), and over a decade, important techniques and methods have been developed and validated, including experimental techniques to conﬁne and cool nanokelvin atoms, the use of Feshbach resonances to modify their properties, and many theoretical concepts and methods to describe trapped ultracold gases and their interactions. What we are currently experiencing is the application of these powerful methods to strongly correlated systems, and due to the maturity of the ﬁeld, the developments have been breath-taking, in particular with bosons in optical lattices and fermions interacting via Feshbach resonances. It is possible that the most important conceptional advances triggered by the advent of Bose-Einstein condensation are yet to be discovered. It is amusing to note that in certain limits, strongly correlated fermion pairs are again described by a mean-ﬁeld theory. Their wave function is a product of identical pair wave functions (albeit correctly anti-symmetrized). Treating the pairing ﬁeld as mean ﬁeld provides the simplest description of the BEC-BCS crossover. Still, the fact that pairing has now become a many-body aﬀair stands for the advent of a new era in ultracold atom physics. . 1 2. Strongly correlated fermions — a gift of nature. – It shows the dynamics of the ﬁeld of ultracold atoms that the area of strongly interacting fermions has not been expected or predicted. This may remind us of the pre-BEC era, when many considered BEC to be an elusive goal, made inaccessible by inelastic interactions at the densities required [9]. When Feshbach resonances were explored in bosonic systems, strong interactions were always accompanied by strong losses, preventing the study of strongly interacting condensates [7, 10, 11]. The reason is that a Feshbach resonance couples the atomic Hilbert space to a resonant molecular state which is vibrationally highly excited. Collisions can couple this state to lower-lying states (vibrational relaxation). What occurred in Fermi gases, however, seemed too good to be true: all relaxation mechanisms were dramatically suppressed by the interplay of the Pauli exclusion principle and the large size of the Feshbach molecules. So what we have got is a Hilbert space which consists of atomic levels plus one single molecular level resonantly coupled to two colliding atoms. All other molecular states couple only weakly. As a result, pair condensation and fermionic superﬂuidity could be realized by simply ramping down the laser power in an optical trap containing 6 Li in two hyperﬁne states at a speciﬁc magnetic ﬁeld, thereby evaporatively cooling the system to the superﬂuid state. Even in our boldest moments we would not have dared to ask Nature for such an ideal system. Before the discovery of Feshbach resonances, suggestions to realize fermionic superﬂuidity focused on lithium because of the unusually large and negative triplet scattering

Making, probing and understanding ultracold Fermi gases

97

length [12-14]. However, a major concern was whether the gas would be stable against inelastic collisions. The stability of the strongly interacting Fermi gas was discovered in Paris in the spring of 2003, when long-lived Li2 molecules were observed despite their high vibrational excitation [15](1 ). This and subsequent observations [17, 18] were soon explained as a consequence of Pauli suppression [19]. Within the same year, this unexpected stability was exploited to achieve condensation of fermion pairs. This unique surprise has changed the ﬁeld completely. Currently, more than half of the research program of our group is dedicated to fermions interacting near Feshbach resonances. There is another aspect of Fermi gases, which turned out to be more favorable than expected. Early work on the BCS state in ultracold gases suggested a competition between superﬂuidity and collapse (for negative scattering length) or coexistence and phase separation (for positive scattering length) when the density or the absolute value of the scattering length a exceeded a certain value, given by kF |a| = π/2, where kF is the Fermi wave vector [13, 20, 21]. This would have implied that the highest transition temperatures to the superﬂuid state would be achieved close to the limit of mechanical stability, and that the BCS-BEC crossover would be interrupted by a window around the Feshbach resonance, where phase separation occurs. Fortunately, unitarity limits the maximum attractive energy to a fraction of the Fermi energy (βEF with β ≈ −0.58), completely eliminating the predicted mechanical instability. Finally, a third aspect received a lot of attention, namely how to detect the superﬂuid state. Since no major change in the spatial proﬁle of the cloud was expected [21], suggested detection schemes included a change in the decay rate of the gas [21], optical light scattering of Cooper pairs [22, 23], optical breakup of Cooper pairs [24], modiﬁcation of collective excitations [25, 26], or small changes in the spatial shape [27]. All these signatures are weak or complicated to detect. Fortunately, much clearer and more easily detectable signatures were discovered. One is the onset of pair condensation, observed through a bimodal density distribution in expanding clouds, observed either well below the Feshbach resonance or after rapid sweeps of the magnetic ﬁeld. Another striking signature was the sudden change in the cloud shape when fermion mixtures with population imbalance became superﬂuid, and ﬁnally, the smoking gun for superﬂuidity was obtained by observing superﬂuid ﬂow in the form of quantized vortices. Our ultimate goal is to control Nature and create and explore new forms of matter. But in the end, it is Nature who sets the rules, and in the case of ultracold fermions, she has been very kind to us. . 1 3. Some remarks on the history of fermionic superﬂuidity. . 1 3.1. BCS superﬂuidity. Many cold fermion clouds are cooled by sympathetic cooling with a bosonic atom. Popular combinations are 6 Li and 23 Na, and 40 K and 87 Rb. It is remarkable that the ﬁrst fermionic superﬂuids were also cooled by a Bose-Einstein (1 ) The observation of long lifetimes of molecules outside a narrow Feshbach resonance [16] is not yet understood and has not been used to realize a strongly interacting gas.

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condensate. Kamerlingh Onnes liqueﬁed 4 He in 1908, and lowered its temperature below the superﬂuid transition point (the λ-point) at Tλ = 2.2 K. In his Nobel lecture in 1913, he notes “that the density of the helium, which at ﬁrst quickly drops with the temperature, reaches a maximum at 2.2 K approximately, and if one goes down further even drops again. Such an extreme could possibly be connected with the quantum theory” [28]. But instead of studying, what we know now was the ﬁrst indication of superﬂuidity of bosons, he ﬁrst focused on the behavior of metals at low temperatures. In 1911, Onnes used 4 He to cool down mercury, ﬁnding that the resistivity of the metal suddenly dropped to nonmeasurable values at TC = 4.2 K, it became “superconducting”. Tin (at TC = 3.8 K) and lead (at TC = 6 K) showed the same remarkable phenomenon. This was the discovery of superﬂuidity in an electron gas. The fact that bosonic superﬂuidity and fermionic superﬂuidity were ﬁrst observed at very similar temperatures, is due to purely technical reasons (because of the available cryogenic methods) and rather obscures the very diﬀerent physics behind these two phenomena. Bosonic superﬂuidity occurs at the degeneracy temperature, i.e. the temperature T −1/3 at which the spacing between particles at density n becomes comparable to the n thermal de Broglie wavelength λ =

2π2 mkB T , where 2π2 2/3 ≈3K m n

m is the particle mass. The predicted

transition temperature of TBEC ∼ for liquid helium at a typical density 22 −3 of n = 10 cm coincides with the observed lambda point. In contrast, the degeneracy temperature (equal to the Fermi temperature TF ≡ EF /kB ) for conduction electrons is higher by the mass ratio m(4 He)/me , bringing it up to several ten-thousand degrees. It was only in 1957 when it became clear why in fermionic systems, superﬂuidity occurs only at temperatures much smaller than the degeneracy temperature. Of course, the main diﬀerence to Bose gases is that electrons, being fermions, cannot be in one and the same quantum state but instead must arrange themselves in diﬀerent states. An obvious scenario for superﬂuidity might be the formation of tightly bound pairs of electrons that can act as bosons and could form a condensate. But apart from the problem that the condensation temperature would still be on the order of EF /kB , there is no known interaction which could be suﬃcient to overcome the strong Coulomb repulsion and form tightly bound electron pairs (Schafroth pairs [29]). The idea itself of electrons forming pairs was indeed correct, but the conceptual diﬃculties were so profound that it took several decades from the discovery of superconductivity to the correct physical theory. In 1950, it became clear that there was indeed an eﬀective attractive interaction between electrons, mediated by the crystal lattice vibrations (phonons), that was responsible for superconductivity. The lattice vibrations left their mark in the characteristic √ variation TC ∝ 1/ M of the critical temperature TC with the isotope mass M of the crystal ions, the isotope eﬀect [30, 31] predicted by H. Fr¨ ohlich [32]. Vibrational energies

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in the lattice are a factor me /M smaller than the typical electronic energy(2 ) EF , on the order of kB × several 100 K (the Debye temperature TD of the metal). While the isotope eﬀect strongly argues for TC being proportional to TD , the Debye temperature is still one or two orders of magnitude higher than the observed critical temperature. A breakthrough came in 1956, when L. Cooper realized that fermions interacting via an arbitrarily weak attractive interaction on top of a ﬁlled Fermi sea can form a bound pair [33]. In other words, the Fermi sea is unstable towards pair formation. However, unlike the tightly bound pairs considered before, the “Cooper” pair is very large, much larger than the interparticle spacing. That is, a collection of these pairs necessarily needs to overlap very strongly in space. In this situation, it was far from obvious whether interactions between diﬀerent pairs could simply be neglected. But it was this simplifying idea that led to the ﬁnal goal: Bardeen, Cooper and Schrieﬀer (BCS) developed a full theory of superconductivity starting from a new, stable ground state in which pair formation was included in a self-consistent way [34]. Using the eﬀective phonon-mediated electron-electron interaction V , attractive for energies smaller than kB TD and assumed constant in this regime, the pair binding energy was found to be Δ = 2kB TD e−1/ρF |V | , with ρF = me kF /2π 2 2 the density of states at the Fermi energy and ρF |V | assumed small compared to 1. The bound-state energy or the pairing gap depended in the non-analytic fashion e−1/ρF |V | on the eﬀective electron-electron interaction V , explaining why earlier attempts using perturbation theory had to fail. Also, this exponential factor can now account for the small critical temperatures TC 5 K: Indeed, it is a result of BCS theory that kB TC is simply proportional to Δ0 , the pair binding energy at zero temperature: kB TC ≈ 0.57 Δ0 . Hence, the critical temperature TC ∼ TD e−1/ρF |V | is proportional to the Debye temperature TD , in accord with the isotope eﬀect, but the exponential factor suppresses TC by a factor that can easily be 100. . 1 3.2. The BEC-BCS crossover. Early work on BCS theory emphasized the diﬀerent nature of BEC and BCS type superﬂuidity. Already in 1950 Fritz London had suspected that fermionic superﬂuidity can be understood as a pair condensate in momentum space, in contrast to a BEC of tightly bound pairs in real space [35]. The former will occur for the slightest attraction between fermions, while the latter appears to require a true two-body bound state to be available to a fermion pair. Schrieﬀer points out that BCS superﬂuidity is not Bose-Einstein condensation of fermion pairs, as these pairs do not obey Bose-Einstein statistics [36]. However, it has become clear that BEC and BCS superﬂuidity are intimately connected. A BEC is a special limit of the BCS state. It was Popov [37], Keldysh and collaborators [38] and Eagles [39] who realized in diﬀerent contexts that the BCS formalism and its ansatz for the ground-state wave (2 ) The average distance between electrons r0 is on the order of atomic distances (several Bohr radii a0 ), the Fermi energy EF ∼ 2 /me r02 is thus on the scale of typical Coulomb q energies in an

atom. Vibrational energies of the lattice ions are then on the order ωD ≈ p p EF /M r02 ∼ me /M EF .

∂ 2 UCoulomb /M ∂r 2

∼

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W. Ketterle and M. W. Zwierlein

BEC of Molecules

Crossover Superfluid

BCS state

Fig. 1. – The BEC-BCS crossover. By tuning the interaction strength between the two fermionic spin states, one can smoothly cross over from a regime of tightly bound molecules to a regime of long-range Cooper pairs, whose characteristic size is much larger than the interparticle spacing. In between these two extremes, one encounters an intermediate regime where the pair size is comparable to the interparticle spacing.

function provides not only a good description for a condensate of Cooper pairs, but also for a Bose-Einstein condensate of a dilute gas of tightly bound pairs. For superconductors, Eagles [39] showed in 1969 that, in the limit of very high density, the BCS state evolves into a condensate of pairs that can become even smaller than the interparticle distance and should be described by Bose-Einstein statistics. In the language of Fermi gases, the scattering length was held ﬁxed, at positive and negative values, and the interparticle spacing was varied. He also noted that pairing without superconductivity can occur above the superﬂuid transition temperature. Using a generic two-body potential, Leggett showed in 1980 that the limits of tightly bound molecules and long-range Cooper pairs are connected in a smooth crossover [40]. Here it was the interparticle distance that was ﬁxed, while the scattering length was varied. The size of the fermion pairs changes smoothly from being much larger than the interparticle spacing in the BCS-limit to the small size of a molecular bound state in the BEC limit (see ﬁg. 1). Accordingly, the pair binding energy varies smoothly from its small BCS value (weak, fragile pairing) to the large binding energy of a molecule in the BEC limit (stable molecular pairing). The presence of a paired state is in sharp contrast to the case of two particles interacting in free (3D) space. Only at a critical interaction strength does a molecular state become available and a bound pair can form. Leggett’s result shows that in the many-body system the physics changes smoothly with interaction strength also at the point where the two-body bound state disappears. Nozi`eres and Schmitt-Rink extended Leggett’s model to ﬁnite temperatures and veriﬁed that the critical temperature for superﬂuidity varies smoothly from the BCS limit, where it is exponentially small, to the BEC-limit where one recovers the value for Bose-Einstein condensation of tightly bound molecules [41]. The interest in strongly interacting fermions and the BCS-BEC crossover increased with the discovery of novel superconducting materials. Up to 1986, BCS theory and its extensions and variations were largely successful in explaining the properties of supercon-

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ductors. The record critical temperature increased only slightly from 6 K in 1911 to 24 K in 1973 [42]. In 1986, however, Bednorz and M¨ uller [43] discovered superconductivity at 35 K in the compound La2−x Bax PCuO4 , triggering a focused search for even higher critical temperatures. Soon after, materials with transition temperatures above 100 K were found. Due to the strong interactions and quasi-2D structure, the exact mechanisms leading to high-TC superconductivity are still not fully understood. The physics of the BEC-BCS crossover in a gas of interacting fermions does not directly relate to the complicated phenomena observed in high-TC materials. However, the two problems share several features: In the crossover regime, the pair size is comparable to the interparticle distance. This relates to high-TC materials where the correlation length (“pair size”) is also not large compared to the average distance between electrons. Therefore, we are dealing here with a strongly correlated “soup” of particles, where interactions between diﬀerent pairs of fermions can no longer be neglected. In both systems the normal state above the phase transition temperature is far from being an ordinary Fermi gas. Correlations are still strong enough to form uncondensed pairs at ﬁnite momentum. In high-TC materials, this region in the phase diagram is referred to as the “Nernst regime”, part of a larger region called the “Pseudo-gap” [44]. One point in the BEC-BCS crossover is of special interest: When the interparticle potential is just about strong enough to bind two particles in free space, the bond length of this molecule tends to inﬁnity (unitarity regime). In the medium, this bond length will not play any role anymore in the description of the many-body state. The only length scale of importance is then the interparticle distance n−1/3 , the corresponding energy scale is the Fermi energy EF . In this case, physics is said to be universal [45]. The average energy content of the gas, the binding energy of a pair, and (kB times) the critical temperature must be related to the Fermi energy by universal numerical constants. The size of a fermion pair must be given by a universal constant times the interparticle distance. It is at the unitarity point that fermionic interactions are at their strongest. Further increase of attractive interactions will lead to the appearance of a bound state and turn fermion pairs into bosons. As a result, the highest transition temperatures for fermionic superﬂuidity are obtained around unitarity and are on the order of the degeneracy temperature. Finally, almost 100 years after Kamerlingh Onnes, it is not just an accidental coincidence anymore that bosonic and fermionic superﬂuidity occur at similar temperatures! . 1 3.3. Experiments on fermionic gases. After the accomplishment of quantum degeneracy in bosons, one important goal was the study of quantum degenerate fermions. Actually, already in 1993, one of us (WK) started to set up dye lasers to cool fermionic lithium as a complement to the existing experiment on bosons (sodium). However, in 1994 this experiment was shut down to concentrate all resources on the pursuit of BoseEinstein condensation, and it was only in early 2000 that a new eﬀort was launched at MIT to pursue research on fermions. Already around 1997, new fermion experiments were being built in Boulder (using 40 K, by Debbie Jin) and in Paris (using 6 Li, by Christophe

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Salomon, together with Marc-Oliver Mewes, a former MIT graduate student who had worked on the sodium BEC project). All techniques relevant to the study of fermionic gases had already been developed in the context of BEC, including magnetic trapping, evaporative cooling, sympathetic cooling [46,47], optical trapping [48] and Feshbach resonances [7,8]. The ﬁrst degenerate Fermi gas of atoms was created in 1999 by B. DeMarco and D. Jin at JILA using fermionic 40 K [49]. They exploited the rather unusual hyperﬁne structure in potassium that allows magnetic trapping of two hyperﬁne states without spin relaxation, thus providing an experimental “shortcut” to sympathetic cooling. All other schemes for sympathetic cooling required laser cooling of two species or optical trapping of two hyperﬁne states of the fermionic atom. Until the end of 2003, six more groups had succeeded in producing ultracold degenerate Fermi gases, one more using 40 K (M. Inguscio’s group in Florence, 2002 [50]) and ﬁve using fermionic 6 Li (R. Hulet’s group at Rice [51], C. Salomon’s group at the ENS in Paris [52], J. Thomas’ group at Duke [53], our group at MIT [54] in 2001 and R. Grimm’s group in Innsbruck in 2003 [55]). Between 1999 and 2001, the ideal Fermi gas and some collisional properties were studied. The year 2002 (and late 2001) was the year of Feshbach resonances when several groups managed to optically conﬁne a two-component mixture and tune an external magnetic ﬁeld to a Feshbach resonance [56-59]. Feshbach resonances were observed by enhanced elastic collisions [57], via an increase in loss rates [56], and by hydrodynamic expansion, the signature of a strongly interacting gas [60]. The following year, 2003, became the year of Feshbach molecules. By sweeping the magnetic ﬁeld across the Feshbach resonance, the energy of the Feshbach molecular state was tuned below that of two free atoms (“molecular” or “BEC” side of the Feshbach resonance) and molecules could be produced [61]. These sweep experiments were very soon implemented in Bose gases and resulted in the observation of Cs2 [62], Na2 [63] and Rb2 [64] molecules. Pure molecular gases made of bosonic atoms were created close to [62] or clearly in [63] the quantum-degenerate regime. Although quantum degenerate molecules were ﬁrst generated with bosonic atoms, they were not called Bose-Einstein condensates, because their lifetime was too short to reach full thermal equilibrium. Molecules consisting of fermionic atoms were much more long-lived [15,17, 16, 18] and were soon cooled into a Bose-Einstein condensate. In November 2003, three groups reported the realization of Bose-Einstein condensation of molecules [65, 66, 55]. All three experiments had some shortcomings, which were soon remedied in subsequent publications. In the 40 K experiment the eﬀective lifetime of 5 to 10 ms was suﬃcient to reach equilibrium in only two dimensions and to form a quasi- or nonequilibrium condensate [65]. In the original Innsbruck experiment [55], evidence for a long-lived condensate of lithium molecules was obtained indirectly, from the number of particles in a shallow trap and the magnetic ﬁeld dependence of the loss rate consistent with mean-ﬁeld effects. A direct observation followed soon after [67]. The condensate observed at MIT was distorted by an anharmonic trapping potential. To be precise, these experiments realized already crossover condensates (see sect. 6) consisting of large, extended molecules or fermion pairs. They all operated in the strongly

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interacting regime with kF a > 1, where the size of the pairs is not small compared to the interparticle spacing. When the interparticle spacing ∼ 1/kF becomes smaller than the scattering length ∼ a, the two-body molecular state is not relevant anymore and pairing is a many-body aﬀair. In fact, due to the increase of collisional losses on the “BEC” side, experiments have so far explored pair condensates only down to kF a ≈ 0.2 [68]. Soon after these ﬁrst experiments on fermion pair condensates, their observation was extended throughout the whole BEC-BCS crossover region by employing a rapid ramp to the “BEC”-side of the Feshbach resonance [69, 70]. During the following years, properties of this new crossover superﬂuid were studied in thermodynamic measurements [71, 72], experiments on collective excitations [73, 74], RF spectroscopy revealing the formation of pairs [75], and an analysis of the two-body part of the pair wave function was carried out [76]. Although all these studies were consistent with superﬂuid behavior, they did not address properties unique to superﬂuids, i.e. hydrodynamic excitations can reﬂect superﬂuid or classical hydrodynamics, and the RF spectrum shows no diﬀerence between the superﬂuid and normal state [77]. Finally, in April 2005, fermionic superﬂuidity and phase coherence was directly demonstrated at MIT through the observation of vortices [68]. More recent highlights (in 2006 and 2007) include the study of fermionic mixtures with population imbalance [78-82], the (indirect) observation of superﬂuidity of fermions in an optical lattice [83], the measurement of the speed of sound [84] and the measurement of critical velocities [85]. Other experiments focused on two-body physics including the formation of p-wave molecules [86] and the observation of fermion antibunching [87]. . 1 3.4. High-temperature superﬂuidity. The crossover condensates realized in the experiments on ultracold Fermi gases are a new type of fermionic superﬂuid. This superﬂuid diﬀers from 3 He, conventional and even high-TC superconductors in its high critical temperature TC when compared to the Fermi temperature TF . Indeed, while TC /TF is about 10−5 − 10−4 for conventional superconductors, 5 · 10−4 for 3 He and 10−2 for high-TC superconductors, the strong interactions induced by the Feshbach resonance allow atomic Fermi gases to enter the superﬂuid state already at TC /TF ≈ 0.2, as summarized in table I. It is this large value which allows us to call this phenomenon “high-temperature superﬂuidity”. Scaled to the density of electrons in a metal, this form of superﬂuidity would already occur far above room temperature (actually, even above the melting temperature). . 1 4. Realizing model systems with ultracold atoms. – Systems of ultracold atoms are ideal model systems for a host of phenomena. Their diluteness implies the absence of complicated or not well understood interactions. It also implies that they can be controlled, manipulated and probed with the precision of atomic physics. Fermions with strong, unitarity limited interactions are such a model system. One encounters strongly interacting fermions in a large variety of physical systems: inside a neutron star, in the quark-gluon plasma of the early Universe, in atomic nuclei, in strongly correlated electron systems. Some of the phenomena in such systems are captured by

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Table I. – Transition temperatures, Fermi temperatures and their ratio TC /TF for a variety of fermionic superﬂuids or superconductors. System Metallic lithium at ambient pressure [88] Metallic superconductors (typical) 3 He MgB2 High-TC superconductors Neutron stars Strongly interacting atomic Fermi gases

TC

TF

TC /TF

0.4 mK 1–10 K 2.6 mK 39 K 35–140 K 1010 K 200 nK

55 000 K 50 000 – 150 000 K 5K 6 000 K 2000 – 5000 K 1011 K 1 μK

10−8 10−4 − 10−5 5 · 10−4 10−2 1 − 5 · 10−2 10−1 0.2

assuming point-like fermions with very strong short-range interactions. The unitarity limit in the interaction strength is realized when the scattering length characterizing these interactions becomes longer than the interparticle spacing. For instance, in a neutron star, the neutron-neutron scattering length of about −18.8 fm is large compared to the few fm distance between neutrons at densities of 1038 cm−3 . Thus, there are analogies between results obtained in an ultracold gas at unitarity, at densities of 1012 cm−3 , and the physics inside a neutron star. Several communities are interested in the equation of state, in the value of the total energy and of the superﬂuid transition temperature of simple models of strongly interacting fermions [89]. Strongly interacting fermions can realize ﬂow deep in the hydrodynamic regime, i.e. with vanishing viscosity. As discussed in sect. 6, the viscosity can be so small that no change in the ﬂow behavior is observed when the superﬂuid phase transition is crossed. This kind of dissipationless hydrodynamic ﬂow allows to establish connections with other areas. For instance, the anisotropic expansion of an elongated Fermi gas shares features with the elliptical (also called radial) ﬂow of particles observed in heavy-ion collisions, which create strongly interacting quark matter [90]. The very low viscosity observed in strongly interacting Fermi gases [73, 91, 74] has attracted interest from the high-energy physics community. Using methods from string theory, it has been predicted that the ratio of the shear viscosity to the entropy density 1 cannot be smaller than 4π [92]. The two liquids that come closest to this lower bound are strongly interacting ultracold fermions and the quark gluon plasma [93]. Another idealization is the pairing of fermions with diﬀerent chemical potentials. This problem emerged from superconductivity in external ﬁelds, but also from superﬂuidity of quarks, where the heavy mass of the strange quark leads to “stressed pairing” due to a shift of the strange quark Fermi energy [94,95]. One of the authors (WK) still remembers vividly how an MIT particle physics colleague, Krishna Rajagopal, asked him about the possibility of realizing pairing between fermions with diﬀerent Fermi energies (see [96]), even before condensation and superﬂuidity in balanced mixtures had become possible. At this point, any realization seemed far away. With some satisfaction, we have included in these Varenna notes our recently observed phase diagram for population-imbalanced ultracold fermions [82].

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This overlap with other areas illustrates a special role of cold atom experiments: They can perform “quantum simulations” of simple models, the results of which may then inﬂuence research in other areas. Of course, those simulations cannot replace experiments with real quarks, nuclei and condensed matter systems. . 1 5. Overview over the sections. – With these notes we want to give a comprehensive introduction into experimental studies of ultracold fermions. The ﬁrst focus of this review is on the description of the experimental techniques to prepare and manipulate fermionic gases (sect. 2), and the methods to diagnose the system including image analysis (sect. 3). For those techniques which are identical to the ones used for bosons we refer to our review paper on bosons in the 1998 Varenna Proceedings [9]. The second focus is on the comprehensive description of the physics of the BEC-BCS crossover (sect. 4) and of Feshbach resonances (sect. 5), and a summary of the experimental studies performed so far (sects. 6 and 7). Concerning the presentation of the material we took a bimodal approach, sometimes presenting an in-depth discussion, when we felt that a similar description could not be found elsewhere, sometimes giving only a short summary with references to relevant literature. Of course, the selection of topics which are covered in more detail reﬂects also the contributions of the MIT group over the last six years. The theory section on the BCS-BEC crossover emphasizes physical concepts over formal rigor and is presented in a style that should be suitable for teaching an advanced graduate course in AMO physics. We resisted the temptation to include recent experimental work on optical lattices and a detailed discussion of population-imbalanced Fermi mixtures, because these areas are still in rapid development, and the value of reviewing these topics would be rather short lived. These notes include new material not presented elsewhere. Section 3 on analyzing density distributions in various regimes for trapped and expanding clouds summarizes many results that have not been presented together and can serve as a reference for how to ﬁt density proﬁles of fermions in all relevant limits. Section 4 on BCS pairing emphasizes the role of the density of states and the relation of Cooper pairs in three dimensions to a two-particle bound state in two dimensions. Many results of BCS theory are derived in a rigorous way without relying on complicated theoretical tools. In sect. 5, many non-trivial aspects of Feshbach resonances are obtained from a simple model. Section 6 presents density proﬁles, not published elsewhere, of a resonantly interacting Fermi gas after expansion, showing a direct signature of condensation. In sect. 6, we have included several unpublished ﬁgures related to the observation of vortices. 2. – Experimental techniques The “window” in density and temperature for achieving fermionic degeneracy is similar to the BEC window. At densities below 1011 cm−3 , thermalization is extremely slow, and evaporative cooling can no longer compete with (technical) sources of heating and loss. At densities above 1015 cm−3 , three-body losses usually become dominant. In this density window, degeneracy is achieved at temperatures between 100 nK and 50 μK.

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The cooling and trapping techniques to reach such low temperatures are the same as those that have been developed for Bose-Einstein condensates. We refer to our Varenna paper on BEC [9] for a description of these techniques. Table II summarizes the diﬀerent cooling stages used at MIT to reach fermionic superﬂuidity in dilute gases, starting with a hot atomic beam at 450 ◦ C and ending with a superﬂuid cloud of 10 million fermion pairs at 50 nK. Although no major new technique has been developed for fermionic atoms, the nature of fermionic gases emphasizes various aspects of the experimental methods: – Diﬀerent atomic species. The most popular atoms for BEC, Rb and Na, do not have any stable fermionic isotopes. The workhorses in the ﬁeld of ultracold fermions are 40 K and 6 Li. – Sympathetic cooling with a diﬀerent species (Na, Rb, 7 Li). This requires techniques to load and laser cool two diﬀerent kinds of atoms simultaneously, and raises the question of collisional stability. – All optical cooling. When cooling 6 Li, the need for a diﬀerent species can be avoided by all optical cooling using two diﬀerent hyperﬁne states. This required further development of optical traps with large trap depth. – Two-component fermionic systems. Pairing and superﬂuidity is observed in a twocomponent fermionic system equivalent to spin up and spin down. This raises issues of preparation using radiofrequency (RF) techniques, collisional stability, and detection of diﬀerent species. All these challenges were already encountered in spinor BECs, but their solutions have now been further developed. – Extensive use of Feshbach resonances. Feshbach resonances were ﬁrst observed and used in BECs. For Fermi gases, resonantly enhanced interactions were crucial to achieve superﬂuidity. This triggered developments in rapid switching and sweeping of magnetic ﬁelds across Feshbach resonances, and in generating homogeneous ﬁelds for ballistic expansion at high magnetic ﬁelds. – Lower temperatures. On the BCS side of the phase diagram, the critical temperature decreases exponentially with the interaction strength between the particles. This provides additional motivation to cool far below the degeneracy temperature. In this section, we discuss most of these points in detail. . 2 1. The atoms. – At very low temperatures, all elements turn into solids, with the exception of helium which remains a liquid even at zero temperature. For this reason, 3 He had been the only known neutral fermionic superﬂuid before the advent of laser cooling. Laser cooling and evaporative cooling prepare atomic clouds at very low densities, which are in a metastable gaseous phase for a time long enough to allow the formation of superﬂuids.

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Table II. – The various preparatory stages towards a superﬂuid Fermi gas in the MIT experiment. Through a combination of laser cooling, sympathetic cooling with sodium atoms, and evaporative cooling, the temperature is reduced by 10 orders of magnitude. The ﬁrst steps involve a spin-polarized gas. In the last step, strong attractive interactions are induced in a two-state Fermi mixture via a Feshbach resonance. This brings the critical temperature for superﬂuidity up to about 0.3 TF — the ultracold Fermi gas becomes superﬂuid. Stage

Temperature

Density

T /TF

Two-species oven

720 K

1014 cm−3

108

Laser cooling (Zeeman slower & MOT) Sympathetic cooling (Magnetic trap) Evaporative cooling (Optical trap)

1 mK

1010 cm−3

104

1 μK

1013 cm−3

0.3

50 nK

5 · 1012 cm−3

0.05

Neutral fermionic atoms have an odd number of neutrons. Since nuclei with an even number of neutrons are more stable, they are more abundant. With the exception of beryllium each element has at least one isotope, which as a neutral atom is a boson. However, there are still many choices for fermionic atoms throughout the periodic table. Because alkali atoms have a simple electronic structure and low-lying excited states, they are ideal systems for laser cooling. Among the alkali metals, there are two stable fermionic isotopes, 6 Li and 40 K, and they have become the main workhorses in the ﬁeld. Recently, degenerate Fermi gases have been produced in metastable 3 He∗ [97] and ytterbium [98], and experiments are underway in Innsbruck to reach degeneracy in strontium. . 2 1.1. Hyperﬁne structure. Pairing in fermions involves two hyperﬁne states, and the choice of states determines the collisional stability of the gas, e.g. whether there is a possible pathway for inelastic decay to lower-lying hyperﬁne states. Therefore, we brieﬂy introduce the hyperﬁne structure of 6 Li and 40 K. The electronic ground-state of atoms is split by the hyperﬁne interaction. The electrons create a magnetic ﬁeld that interacts with the nuclear spin I. As a result, the total electron angular momentum, sum of angular momentum and spin, J = L + S, is coupled to the nuclear spin to form the total angular momentum of the entire atom, F = J + I. Alkali atoms have a single valence electron, so S = 1/2, and in the electron’s orbital ground state, L = 0. Hence the ground state splits into two hyperﬁne manifolds with total angular momentum quantum numbers F = I + 1/2 and F = I − 1/2. In a magnetic ﬁeld B, these hyperﬁne states split again into a total of (2S + 1)(2I + 1) = 4I + 2 states. The Hamiltonian describing the various hyperﬁne states is (1)

Hhf = ahf I · S + gs μB B · S − gi μN B · I

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6Li

40K mS = +1/2

|6> |5> |4>

0 -200 F = 1/2

|3>

-400

|2>

mS = -1/2 0

100 200 Magnetic Field [G]

|1>

F = 7/2

1.0

F = 3/2

200

mS = +1/2

1.5

Energy [GHz]

Energy [MHz]

400

0.5 0.0 -0.5 F = 9/2

-1.0

mS = -1/2

-1.5

300

0

200 400 Magnetic Field [G]

600

Fig. 2. – Hyperﬁne states of 6 Li and 40 K. Energies are relative to the atomic ground state without hyperﬁne interaction. 6 Li has nuclear spin I = 1, for 40 K it is I = 4. The 6 Li hyperﬁne 6 40 splitting is ΔνhfLi = 228 MHz, for 40 K it is Δνhf K = −1.286 GHz. The minus sign indicates that the hyperﬁne structure is reversed for 40 K, with F = 9/2 being lower in energy than F = 7/2. Thick lines mark hyperﬁne states used during cooling to degeneracy.

Here, ahf is the hyperﬁne constant of the alkali atom, gs ≈ 2 and gi are the electron and nuclear g-factors, μB ≈ 1.4 MHz/G is the Bohr magneton and μN the nuclear magneton. The hyperﬁne states of 6 Li and 40 K are shown in ﬁg. 2. Good quantum numbers at low ﬁeld are the total spin F and its z-projection, mF . At high ﬁelds B ahf /μB , they are the electronic and nuclear spin projections mS and mI . . 2 1.2. Collisional Properties. The Pauli exclusion principle strongly suppresses collisions between two fermions in the same hyperﬁne state at low temperatures. Because of the antisymmetry of the total wave function for the two fermions, s-wave collisions are forbidden. Atoms in the same hyperﬁne state can collide only in odd partial waves with p-wave as the lowest angular-momentum channel. For p-wave collisions, with the relative angular momentum of , atomic mass m and a thermal velocity of vT , the impact parameter of a collision is /mvT , which is equal 2

2π . When the range of the interaction to the thermal de Broglie wavelength λT = mk BT potential r0 is smaller than λT , the atoms “ﬂy by” each other without interaction. For a van-der-Waals potential, the range is r0 ≈ (mC6 /2 )1/4 . Below the temperature kB Tp = 2 /mr02 , p-wave scattering freezes out, and the Fermi gas becomes collisionless, a truly ideal gas! For 6 Li, Tp ≈ 6 mK, much larger than the temperature in the magneto-optical trap (MOT). For 40 K, Tp = 300 μK. Since these values for Tp are much higher than the window for quantum degeneracy, a second species or second hyperﬁne state is needed for

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thermalization and evaporative cooling. We now discuss some general rules for inelastic two-body collisions. – Energy. Inelastic collisions require the internal energy of the ﬁnal states to be lower than that of the initial states. Therefore, a gas (of bosons or fermions) in the lowest hyperﬁne state is always stable with respect to two-body collisions. Since the lowest hyperﬁne state is a strong-magnetic-ﬁeld–seeking state, optical traps, or generally traps using ac magnetic or electric ﬁelds are required for conﬁnement. – Angular momentum. The z-component M of the total angular momentum of the two colliding atoms (1 and 2) is conserved. Here, M = Mint + Mrot , where the internal angular momentum Mint = mF,1 + mF,2 at low ﬁelds and Mint = mI,1 + mI,2 + mS,1 + mS,2 at high ﬁelds, and Mrot is the z-component of the angular momentum of the atom’s relative motion. – Spin relaxation. Spin relaxation occurs when an inelastic collision is possible by exchanging angular momentum between electrons and nuclei, without aﬀecting the motional angular momentum. Usually, the rate constant for this process is on the order of 10−11 cm3 s−1 which implies rapid decay on a ms scale for typical densities. As a general rule, mixtures of hyperﬁne states with allowed spin relaxation have to be avoided. An important exception is 87 Rb where spin relaxation is suppressed by about three orders of magnitude by quantum interference [46]. Spin relaxation is suppressed if there is no pair of states with lower internal energy with the same total Mint . Therefore, degenerate gases in a state with maximum Mint cannot undergo spin relaxation. – Dipolar relaxation. In dipolar relaxation, angular momentum is transferred from the electrons and/or nuclei to the atoms’ relative motion. Usually, the rate constant for this process is on the order of 10−15 cm3 s−1 and is suﬃciently slow (seconds) to allow the study of systems undergoing dipolar relaxation. For instance, all magnetically trapped Bose-Einstein condensates can decay by dipolar relaxation, when the spin ﬂips to a lower-lying state. – Feshbach resonances. Near Feshbach resonances, all inelastic processes are usually strongly enhanced. A Feshbach resonance enhances the wave function of the two colliding atoms at short distances, where inelastic processes occur (see sect. 5). In addition, the coupling to the Feshbach molecule may induce losses that are entirely due to the closed channel. It is possible that the two enhanced amplitudes for the same loss process interfere destructively. – (Anti-)Symmetry. At low temperature, we usually have to consider only atoms colliding in the s-wave incoming channel. Colliding fermions then have to be in two diﬀerent hyperﬁne states, to form an antisymmetric total wave function. Spin relaxation is not changing the relative motion. Therefore, for fermions, spin relaxation into a pair of identical states is not possible, as this would lead to a symmetric wave

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function. Two identical ﬁnal states are also forbidden for ultracold fermions undergoing dipolar relaxation, since dipolar relaxation obeys the selection rule ΔL = 0, 2 for the motional angular momentum and can therefore only connect even to even and odd to odd partial waves. We can now apply these rules to the hyperﬁne states of alkalis. For magnetic trapping, we search for a stable pair of magnetically trappable states (weak ﬁeld seekers, i.e. states with a positive slope in ﬁg. 2). For atoms with J = 1/2 and nuclear spin I = 1/2, 1 or 3/2 that have a normal hyperﬁne structure (i.e. the upper manifold has the larger F ), there is only one such state available in the lower hyperﬁne manifold. The partner state thus has to be in the upper manifold. However, a two-state mixture is not stable against spin relaxation when it involves a state in the upper hyperﬁne manifold, and there is a state leading to the same Mint in the lower manifold. Therefore, 6 Li (see ﬁg. 2) and also 23 Na and 87 Rb do not have a stable pair of magnetically trappable states. However, 40 K has an inverted hyperﬁne structure and also a nuclear spin of 4. It thus oﬀers several combinations of weak-ﬁeld seeking states that are stable against spin relaxation. Therefore, 40 K has the unique property that evaporative cooling of a two-state mixture is possible in a magnetic trap, which historically was the fastest route to achieve fermionic quantum degeneracy [49]. An optical trap can conﬁne both weak and strong ﬁeld seekers. Mixtures of the two lowest states are always stable against spin relaxation, and in the case of fermions, also against dipolar relaxation since the only allowed output channel has both atoms in the same state. Very recently, the MIT group has realized superﬂuidity in 6 Li using mixtures of the ﬁrst and third or the second and third state [99]. For the combination of the ﬁrst and third state, spin relaxation into the second state is Pauli suppressed. These two combinations can decay only by dipolar relaxation, and surprisingly, even near Feshbach resonances, the relaxation rate remained small. This might be caused by the small hyperﬁne energy, the small mass and the small van der Waals coeﬃcient C6 of 6 Li, which lead to a small release energy and a large centrifugal barrier in the d-wave exit channel. For Bose-Einstein condensates at typical densities of 1014 cm−3 or larger, the dominant decay is three-body recombination. Fortunately, this process is Pauli suppressed for any two-component mixture of fermions, since the probability to encounter three fermions in a small volume, of the size of the molecular state formed by recombination, is very small. In contrast, three-body relaxation is not suppressed if the molecular state has a size comparable to the Fermi wavelength. This has been used to produce molecular . clouds (see subsect. 2 4.2). After those general considerations, we turn back to the experimentally most relevant hyperﬁne states, which are marked with thick lines in ﬁg. 2. In the MIT experiment, sympathetic cooling of lithium with sodium atoms in the magnetic trap is performed in the upper, stretched state |6 ≡ |F = 3/2, mF = 3/2. In the ﬁnal stage of the experiment, the gas is transferred into an optical trap and prepared in the two lowest hyperﬁne states of 6 Li, labelled |1 and |2, to form a strongly interacting Fermi mixture around the Feshbach resonance at 834 G. The same two states have been used in all 6 Li experi-

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ments except for the very recent MIT experiments on mixtures between atoms in |1 and |3, as well as in |2 and |3 states. In experiments on 40 K at JILA, mutual sympathetic cooling of the |F = 9/2, mF = 9/2 and |F = 9/2, mF = 7/2 states is performed in the magnetic trap. The strongly interacting Fermi mixture is formed using the lowest two hyperﬁne states |F = 9/2, mF = −9/2 and |F = 9/2, mF = −7/2 close to a Feshbach resonance at 202 G. As we discussed above, evaporative cooling requires collisions with an atom in a different hyperﬁne state or with a diﬀerent species. For the latter approach, favorable properties for interspecies collisions are required. Here we brieﬂy summarize the approaches realized thus far. The stability of mixtures of two hyperﬁne states has been discussed above. Evaporation in such a system was done for 40 K in a magnetic trap [49] using RF-induced, simultaneous evaporation of both spin states. In the case of 6 Li, laser-cooled atoms were directly loaded into optical traps at Duke [53] and Innsbruck [17] in which a mixture of the lowest two hyperﬁne states was evaporatively cooled by lowering the laser intensity. Other experiments used two species. At the ENS [52] and at Rice [51], spin-polarized 6 Li is sympathetically cooled with the bosonic isotope of lithium, 7 Li, in a magnetic trap. At MIT, a diﬀerent element is used as a coolant, 23 Na. This approach is more complex, requiring a special double-species oven and two laser systems operating in two diﬀerent spectral regions (yellow and red). However, the 6 Li-23 Na interspecies collisional properties have turned out to be so favorable that this experiment has led to the largest degenerate Fermi mixtures to date with up to 50 million degenerate fermions [100]. Forced evaporation is selectively done on 23 Na alone, by using a hyperﬁne state changing transition around the 23 Na hyperﬁne splitting of 1.77 GHz. The number of 6 Li atoms is practically constant during sympathetic cooling with sodium. Other experiments on sympathetic cooling employ 87 Rb as a coolant for 40 K [101-103, 87] or for 6 Li [104, 105]. Another crucial aspect of collisions is the possibility to enhance elastic interactions via Feshbach resonances. Fortunately, for all atomic gases studied so far, Feshbach resonances of a reasonable width have been found at magnetic ﬁelds around or below one kilogauss, rather straightforward to produce in experiments. Since Feshbach resonances are of central importance for fermionic superﬂuidity, we discuss them in a separate section 5. . 2 2. Cooling and trapping techniques. – The techniques of laser cooling and magnetic trapping are identical to those used for bosonic atoms. We refer to the comprehensive discussion and references in our earlier Varenna notes [9] and comment only on recent advances. One development are experiments with two atomic species in order to perform sympathetic cooling in a magnetic trap. An important technical innovation are two-species ovens which create atomic beams of two diﬀerent species. The ﬂux of each species can be separately controlled using a two-chamber oven design [106]. When magneto-optical traps (MOTs) are operated simultaneously with two species, some attention has to be given to light-induced interspecies collisions leading to trap loss. Usually, the number of trapped atoms for each species after full loading is smaller than if the MOT is operated with only

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one species. These losses can be mitigated by using sequential loading processes, quickly loading the second species, or by deliberately applying an intensity imbalance between counter-propagating beams in order to displace the two trapped clouds [100]. Another development is the so-called all-optical cooling, where laser-cooled atoms are directly transferred into an optical trap for further evaporative cooling. This is done by ramping down the laser intensity in one or several of the beams forming the optical trap. All-optical cooling was introduced for bosonic atoms (rubidium [107], cesium [108], sodium [109], ytterbium [110]) and is especially popular for fermionic lithium, where evaporative cooling in a magnetic trap is possible only by sympathetic cooling with a second species. In the following two sections, we address in more detail issues of sympathetic cooling and new variants of optical traps, both of relevance for cooling and conﬁning fermions. . 2 2.1. Sympathetic cooling. Overlap between the two clouds. One limit to sympathetic cooling is the loss of overlap of the coolant with the cloud of fermions. Due to diﬀerent masses, the sag due to gravity is diﬀerent for the two species. This is most severe in experiments that employ 87 Rb to 2 cool 6 Li [104, 105]. For harmonic traps, the sag is given by Δx1,2 = g/ω1,2 for species 1 and 2, with g the Earth’s gravitational acceleration, and ω the trapping frequency along the vertical direction. The spring constant k = mω 2 ≈ μB B is essentially the same for all alkali atoms, when spin-polarized in their stretched state and conﬁned in magnetic traps with magnetic-ﬁeld curvature B . It is of the same order for alkalis conﬁned in optical traps, k = αI , where the polarizability α is similar for the alkalis and lasers far detuned from atomic resonances, and I is the curvature of the electric ﬁeld’s intensity. The thermal cloud size, given by kB T /k, is thus species-independent, while the sag Δx1,2 ≈ gm1,2 /k is proportional to the mass. The coolant separates from the cloud of fermions once g(m2 − m1 )/k ≈ kB T /k, or kB T ≈ g 2 (m2 − m1 )2 /k. For trapping frequencies of 100 Hz for 6 Li, and for 87 Rb as the coolant, this would make sympathetic cooling ineﬃcient at temperatures below 30 μK, more than an order of magnitude higher than the Fermi temperature for 10 million fermions. For 23 Na as the coolant, the degenerate regime is within reach for this conﬁnement. Using the bosonic isotope 7 Li as the coolant, gravitational sag evidently does not play a role. To avoid the problem of sag, one should provide strong conﬁnement along the axis of gravity. A tight overall conﬁnement is not desirable since it would enhance trap loss due to three-body collisions. Role of Fermi statistics. When fermions become degenerate, the collision rate slows down. The reason is that scattering into a low-lying momentum state requires this state to be empty, which has a probability 1 − f , with f the Fermi-Dirac occupation number. As the occupation of states below the Fermi energy gets close to unity at temperatures T TF , the collision rate is reduced. Initially, this eﬀect was assumed to severely limit cooling well below the Fermi temperature [49]. However, it was soon realized that although the onset of Fermi degeneracy changes the kinetics of evaporative cooling, it does not impede cooling well below the Fermi temperature [111,112]. The lowest temperature

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in evaporative cooling is always determined by heating and losses. For degenerate Fermi systems, particle losses (e.g. by background gas collisions) are more detrimental than for Bose gases, since they can create hole excitations deep in the Fermi sea [113-115]. Role of Bose statistics. If the coolant is a boson, the onset of Bose-Einstein condensation changes the kinetics of evaporation. It has been proposed that sympathetic cooling becomes highly ineﬃcient when the speciﬁc heat of the coolant becomes equal or smaller than that of the Fermi system [51,116]. However, although an almost pure Bose-Einstein condensate has almost zero speciﬁc heat, its capacity to remove energy by evaporating out of a trap with a given depth is even larger than that of a Boltzmann gas, since the initial energy of the Bose gas is lower. On the other hand, the rate of evaporation is lower for the Bose condensed gas, since the number of thermal particles is greatly reduced. In the presence of heating, a minimum rate of evaporation is required [116]. This might call for additional ﬂexibility to independently control the conﬁnement for bosons and fermions, which is possible via the use of a two-color trap [117]. In particular, on can then expand the bosonic coolant and suppress the onset of Bose-Einstein condensation. Other work discussed phenomena related to the interacting condensate. When the Fermi velocity becomes smaller than the critical velocity of a superﬂuid Bose-Einstein condensate, then the collisional transfer of energy between the fermions and bosons becomes ineﬃcient [118]. Another phenomenon for suﬃciently high boson density is mean-ﬁeld attraction or repulsion of the fermions, depending on the relative sign of the intraspecies scattering length [119]. Attractive interactions can even lead to a collapse of the condensate as too many fermions rush into the Bose cloud and cause three-body collisions, leading to losses and heating [120, 121]. Given all these considerations, it is remarkable that the simplest scheme of evaporating bosons in a magnetic trap in the presence of fermions has worked very well. In the MIT experiment, we are currently limited by the number of bosons used to cool the Fermi gas. Without payload (the fermions), we can create a sodium Bose-Einstein condensate of 10 million atoms. When the fermions outnumber the bosons, the cooling becomes less eﬃcient, and we observe a trade-oﬀ between ﬁnal number of fermions and their temperature. We can achieve a deeply degenerate Fermi gas of T /TF = 0.05 with up to 30 million fermions [100], or aim for even larger atom numbers at the cost of degeneracy. On a day-to-day basis, we achieve 50 million fermions at T /TF = 0.3. This degenerate, spin-polarized Fermi gas can subsequently be loaded into an optical trap for further evaporative cooling as a two-component Fermi mixture. The preparation of a two-component mixture by an RF pulse and decoherence (see . subsect. 2 3.5) lowers the maximum occupation number to 1/2 and increases the eﬀective T /TF to about 0.6. Therefore, there is no beneﬁt of cooling the spin polarized Fermi cloud to higher degeneracy. . 2 2.2. Optical trapping. Optical traps provide the conﬁnement for almost all experiments on ultracold fermions. The reason is that most of the current interest is on interacting two-component systems. Optical traps conﬁne both strong- and weak-ﬁeld–seeking states. Trapping atoms in the lowest-lying hyperﬁne states (which are always strong-

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. ﬁeld–seeking) suppresses or avoids inelastic collisions, as discussed in subsect. 2 1.2. Most importantly, using electric ﬁelds for trapping frees the magnetic ﬁeld to be tuned to Feshbach resonances and thereby to enhance elastic interactions. There is only one experiment on ultracold atoms that studied Feshbach resonances in a magnetic trap (in 85 Rb [11]), all others have been performed in optical traps. The important case of 6 Li has led to advances in optical traps with large volume and trap depth. All-optical cooling to BEC has been convenient in some experiments with rubidium and sodium Bose-Einstein condensates eliminating the need for magnetic traps [107, 109]. However, standard magnetic traps are not considerably increasing the complexity of the experiment. One could take the position that a magnetic trap is easier to operate and to maintain than a high power laser or an enhancement cavity. However, bypassing the magnetic trap for 6 Li also bypasses the need for another species (7 Li, Rb, Na) and therefore an additional laser system(3 ). In the following section, we discuss some optical trapping geometries used in ultracold Fermi experiments. For a more detailed discussion on optical trapping, we refer the reader to [9] and [122]. So far, all optical traps for fermions have used red detuned laser beams where the atoms are conﬁned in the intensity maximum of the laser beam(s). The trapping potential is given by the AC Stark shift (2)

U (r) = −

2 ωR (r) 4

1 1 + ω0 − ωL ω0 + ω L

2 (r) ωR , 4Δ

where ω0 is the atomic resonance frequency, ωL is the frequency of the laser light, and Δ = ωL − ω0 the laser’s detuning from resonance. The approximation on the right-hand side holds for |Δ| ω0 . ωR is the position-dependent Rabi frequency describing the strength of the atom-ﬁeld coupling. In terms of the intensity I(r) of the laser light and 2 atomic parameters, it is deﬁned by 2ωR (r)/Γ2 = I(r)/ISAT , where Γ is the natural decay rate of the atom’s excited state, and ISAT = ω03 Γ/12πc2 is the saturation intensity. For 6 Li, Γ = 2π · 6 MHz and ISAT = 3 mW/cm2 , for 40 K, Γ = 2π · 6 MHz and ISAT = 2 mW/cm2 . Single-beam optical trap. The simplest trap consists of a single, red-detuned, focused Gaussian laser beam, with intensity proﬁle (3)

2P 2 ρ2 I(ρ, z) = 2 ) exp − w2 (1 + z 2 /z 2 ) . πw2 (1 + z 2 /zR R

The beam parameters are the laser power P , the 1/e2 beam waist radius w, and the Rayleigh range zR . ρ and z are the distances from the beam focus along the radial and (3 ) In other cases magnetic trapping has not been an option due to inelastic collisions [108] or vanishing magnetic moment [110].

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axial directions, respectively. The Rayleigh range is related to the beam waist and the wavelength of the laser via zR = πw2 /λ. The bottom of the potential well formed by the laser beam can be approximated as a harmonic oscillator with trapping frequencies 2 . For 6 Li, a laser beam operating at ωρ /2π = 2P/π 3 mw4 and ωz /2π = P/π 3 mw2 zR a wavelength λ = 1064 nm with 100 mW of power, focused down to a waist of w = 25 μm, provides a√trap depth U 6 μK, a radial frequency ωρ /2π = 1.2 kHz and an aspect ratio ωρ /ωz = 2πw/λ 100. This is suﬃcient for loading atoms that were evaporatively or sympathetically pre-cooled in a magnetic trap. Loading atoms directly from a millimeter-sized MOT, at temperatures of several 100 μK, into a single beam optical trap requires larger trap depths, a larger waist and ideally a smaller aspect ratio to enhance the overlap with the rather spherical MOT region. One solution is the use of Quasi-Electrostatic Traps (QUEST) [123] formed by a focused CO2 laser at λ = 10.6 μm. Due to the large detuning from atomic resonance the trap operates in the quasi-electrostatic regime where ωL /ω0 → 0 and the dipole po2 I tential U = Γ 4ω0 ISAT no longer depends on the frequency of the laser light. The longer wavelength allows for a larger waist at still moderate aspect ratios. In the group at Duke University [53], 65 W of power was focused to a waist of 47 μm (Rayleigh range 660 μm), providing a trap depth for 6 Li atoms of 690 μK. The resulting radial and axial frequencies were 6.6 kHz and 340 Hz, respectively. This deep trap allowed to capture 1.5 × 106 atoms from the MOT at Doppler-limited temperatures of 150 μK. The tight conﬁnement ensured good starting conditions for evaporative cooling. Hybrid trap. A large beam waist is preferable for several purposes, e.g. for creating a large trap volume or for controlling any aberrations which would cause a deviation from cylindrical symmetry — this was crucial for the creation of vortices [68]. To avoid the large aspect ratio of the optical trap, a conﬁning magnetic curvature can be added along the axial direction by using two coils with a separation larger (smaller) than the Helmholtz conﬁguration (distance equals radius) for low ﬁeld (high ﬁeld) seekers. Maxwell’s equations then require an anti-conﬁning curvature along the radial direction, which, however, is negligible compared to the tight optical conﬁnement. As a result, this hybrid trap features optical radial conﬁnement and axial magnetic conﬁnement. In addition, high bias ﬁelds are needed to tune across the Feshbach resonance. Such a setup has been used in many experiments in Innsbruck, at Rice, and at MIT. Details of the MIT magnetic . ﬁeld conﬁguration are discussed in subsect. 2 4.1. In our experiments, the axial conﬁnement is almost purely magnetic (ωz /2π 23 Hz). The optical trap provides radial conﬁnement with ωr /2π in the range of 50 to 300 Hz, which varies the aspect ratio of the cloud between about 2 and 12. We will now discuss two other important aspects of optical traps. One is the compensation of gravity that is crucial for creating traps with cylindrical symmetry, the other one is the issue of the trap depth that controls evaporative cooling. In the MIT experiment [68], the hybrid trap has a typical aspect ratio of ωr /ωz = 6. The optical trapping beam and the magnetic ﬁeld coils are horizontally aligned.

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Fig. 3. – Alignment of the optical trap to achieve a radially symmetric potential. In the left image, the trap is still far from the “sweet spot”. In the right image, stray gradients are almost completely cancelled. The absorption images are of a lithium pair condensate after 10 ms of expansion. The ﬁeld of view for each image is 1.1 × 1.3 mm.

Compensation for gravity is ensured by “sitting on one side of the saddle”. Along the vertical (x-)axis, the combined potential of gravity+magnetic ﬁelds is √ − 41 mωz2 x2 − mgx, where we used iωz / 2 as the anti-conﬁning curvature. Thus, gravity shifts the saddle potential by an amount 2g/ωz2 ≈ 1 mm. The “sweet spot” in the radial plane to which the optical trap needs to be aligned is thus not the center of the magnetic ﬁeld coils, but about 1 mm above it. In this position no gradients act on the atoms. If the optical trap is round in the radial plane, the combined potential experienced by the atoms is round as well. Round traps are crucial for the observation of vortices, and also for the study of collective excitations with radial symmetry. The alignment procedure of the optical trap is shown in ﬁg. 3. At the end of evaporation of the lithium condensate, the trap depth is reduced in about 30 ms to a very shallow depth which is not suﬃcient to hold the atoms if they are not in the “sweet spot”. After 10 ms of expansion from the optical trap one clearly observes in which direction the atoms spill out, and one can counteract by moving the optical trap. A low intensity tightly focused beam and a larger intensity beam with a softer focus provide the same radial conﬁnement. However, the trap depth is very diﬀerent. This is important if the cloud needs to be cooled by evaporation, e.g. during the nucleation of a vortex lattice after stirring up the cloud. Cooling of the cloud will be eﬃcient if the trap depth U is not much higher than the Fermi energy EF . This condition sets a stringent constraint for the beam waist. We illustrate this by discussing the situation in the MIT vortex experiment, where we wanted to have a rather small aspect ratio a = ωr /ωz . The axial trapping frequency ωz was ﬁxed by the magnetic ﬁeld curvature. The relation between U and the waist w is (4)

U=

1 1 mωr2 w2 = mωz2 a2 w2 . 4 4

The Fermi energy per spin state for a total number of atoms N is given by (using the

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harmonic approximation for the radially Gaussian potential) (5)

EF = (ωr2 ωz )1/3 (3N )1/3 = ωz a2/3 (3N )1/3 .

Requiring U EF results in (6)

w2

−2/3 a (3N )1/6 . mωz

If we want to trap N = 1 × 107 atoms with an aspect ratio a = 5 and an axial trapping frequency ωz /2π = 20 Hz (typical values), we need the waist to be larger than 100 μm. Note that this requirement is quite stringent. Changing ωz is limited: Increasing the current in the curvature coils by a factor of two (which increases power dissipation in the curvature coils by four) only reduces the required waist by 20%. Allowing for an aspect ratio of 10 would give another reduction by only 35%. A longer aspect ratio would have had adverse eﬀects for the alignment of the stirring beam and the observation of vortex cores after expansion. For our choice of w = 120 μm, the Rayleigh range is z0 = πw2 /λ > 4 cm while a typical axial cloud size is 1 mm. The maximum power in the laser beam is 4 W, which limits the trap depth to about 10 μK. This is still deep enough to load about 3 × 107 degenerate fermions from the magnetic trap after the sympathetic cooling stage with sodium. (The Fermi temperature in the combined magnetic and optical trap during this loading is 5 μK, and the degenerate cloud at T /TF ≈ 0.3 is not much larger than a zero-temperature Fermi sea.) These numbers illustrate that optical traps for fermions need much more power than for a Bose-Einstein condensate because of the combined need for a deeper and larger trap. Crossed dipole trap. Another option for loading atoms from a MOT into an optical potential is the use of crossed laser beams. This geometry provides a roughly spherical trapping volume, and oﬀers a good trade-oﬀ between trap depth and volume. This conﬁguration allowed the ﬁrst demonstration of Bose-Einstein condensation of atoms by all-optical means [107]. Fermionic atoms were loaded into a crossed dipole trap by the Paris group [124] after pre-cooling in a magnetic trap. When magnetic ﬁelds are applied, e.g. for tuning near a Feshbach resonance, the tight optical conﬁnement in all three dimensions makes the trap more robust against potential magnetic ﬁeld gradients which could drag atoms out of the trap. Crossed dipole traps have also been used to prepare fermionic clouds for loading into optical lattices [83, 87]. Resonator-enhanced standing wave trap. The Innsbruck group enhanced the laser intensity and thus the trap depth by forming a standing-wave optical resonator [125]. The power of a 2 W Nd:YAG laser at λ = 1064 nm was resonantly enhanced by a factor of ∼ 120, resulting in a trap depth of ∼ 1 mK in the focus with 115 μm waist. This was deep enough to capture atoms directly from the MOT. The standing wave presented

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a 1D lattice potential to the atoms, that were thus tightly conﬁned in several pockets along the direction of the standing wave. The high density in each pocket provided good starting conditions for evaporative cooling. After some initial cooling, the atoms were transferred into a single-beam optical trap [55]. . 2 3. RF spectroscopy. – A single-component ultracold Fermi gas, with all atoms occupying the same spin states, is an almost perfect realization of an ideal non-interacting gas. s-wave collisions are forbidden due to the Pauli principle, and p-wave collisions are frozen out. In the absence of p-wave or higher partial wave scattering resonances, no phase transition occurs down to exponentially small temperatures (4 ). Physics becomes interesting only in the presence of interactions, and the obvious way to introduce interactions into a Fermi gas is by forming a two-component system, such as a mixture of the two lowest hyperﬁne states of 6 Li. s-wave scattering is then allowed between fermions of opposite spin. More accurately speaking, as the spin-part of the two-particle wave function can now be antisymmetric, symmetric s-wave scattering is now allowed. Such a two-state mixture can be created via optical pumping after the MOT phase, or via RF spectroscopy, starting from a pure single-component gas. Since RF spectroscopy is an invaluable tool to prepare, manipulate and probe ultracold gases, we review it here in more detail. First, we summarize basic aspects of RF spectroscopy, and then focus on clock shifts and mean ﬁeld energies. . 2 3.1. Basics. Let us note some important properties of RF spectroscopy: a) The RF ﬁeld has a very long wavelength (≈ 3m), so there is negligible momentum transfer. The coupling takes place only between internal states of each individual atom. b) The RF ﬁeld (typically from a ∼ cm-large antenna) is essentially constant over the size of the sample (∼ 100 μm). Thus, the entire cloud is simultaneously addressed by the same coupling. c) The RF pulse generally creates a superposition of the two coupled states. Such coherences can be long-lived in the absence of decay mechanisms. In many cases, one can approximate a system of two coupled states |1 and |2 with energies E1 and E2 as an isolated two level system driven by a ﬁeld V = V (t) |2 1| + eiφ |1 2| oscillating close to the resonant frequency ω0 = (E2 − E1 )/. Such a two-level system is conveniently described as a pseudo spin-1/2, for which |↑ ≡ |1 and |↓ ≡ |2. Keeping only the part of the interaction that resonantly drives the transition (“rotating wave approximation”), the Hamiltonian is written as (7)

H = H0 + V ;

H0 ≡ −

ω0 σz ; 2

V ≡−

ωR (σx cos ωt + σy sin ωt) , 2

where σi are the Pauli spin matrices and ωR is the Rabi frequency, giving the strength of the coupling. ωR depends on the drive ﬁeld (in our experiments a magnetic ﬁeld generated by an antenna) and the coupling matrix element between the two hyperﬁne (4 ) For attractive p-wave interactions with scattering length a, the critical temperature is TC ∼ (EF /kB ) exp[−π/2(kF |a|)3 ] [21].

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states. A typical value for ωR /2π is several kHz. The free Hamiltonian H0 has its natural interpretation as a constant magnetic ﬁeld in the z-direction of pseudospin-space. In the same way, the interaction V represents a (real or ﬁctitious) rotating magnetic ﬁeld in the transverse (x-y) plane. Transforming into the frame rotating at frequency ω, the direction of the transverse ﬁeld is constant, while the z-ﬁeld (and thus the energy splitting between the two states in the rotating frame) is reduced to −δ = (ω0 − ω). For a resonant drive with δ = 0, only the constant transverse ﬁeld is left, and — borrowing from the classical picture — the spin (or pseudospin) precesses around it at frequency ωR . A complete inversion of the spin-direction — and thus a complete transfer from state |1 into state |2 — is achieved for a RF pulse length t = π/ωR (so-called π-pulse). An equal superposition √12 (|1 + |2) is achieved for t = π/2ωR (π/2-pulse). . 2 3.2. Adiabatic rapid passage. For general detuning δ, the (ﬁctitious) magnetic ﬁeld ˆ ωR ˆ in the rotating frame is Brot = δ 2 z − 2 x. At large positive detuning δ ωR , it is predominantly pointing in the +z-direction, and the pseudospin precesses around it. If the initial state is either state |1 or state |2, then the pseudospin is pointing up or down, and the angle between it and the ﬁctitious magnetic ﬁeld is small. If the detuning is now slowly swept from δ ωR through resonance (δ = 0) and towards large and negative values, the pseudospin will adiabatically follow the direction of the changing magnetic ﬁeld and thus end up, for δ −ωR , aligned opposite to its original direction. One has thus adiabatically transferred the atom from state |1 to state |2 (or vice versa). The condition of adiabaticity requires that the pseudospin’s precession frequency is always fast compared to the change of the magnetic ﬁeld’s direction, given by the azimuthal angle θ = arctan ωδR . This condition is most stringent on resonance, where it reads ˙ ωR θ˙ = δ or δ˙ ω 2 . For a non-adiabatic transfer, the probability for a successful ωR

R

transfer is given by the formula due to Landau and Zener: (8)

ω2 P|1 →|2 = 1 − exp −2π R . δ˙

It is important to realize that both a short RF-pulse as well as a non-adiabatic LandauZener transfer will leave the atom in a superposition state cos α |1 + sin α eiφ |2. If these processes are applied to a Fermi gas initially polarized in a single spin state |1, it will still be a fully polarized Fermi gas after the RF-pulse, with the only diﬀerence that now all the atoms are in one and the same superposition state of |1 and |2. . It is only by decoherence discussed below (subsect. 2 3.5) that the superposition state transforms into an incoherent mixture of atoms in states |1 and |2. . 2 3.3. Clock shifts. Clock shifts are density dependent shifts of transition frequencies due to interactions between the atoms. The name derives from their presence in atomic clocks. Indeed, they are one of the dominant sources of systematic error in current cold atom fountain clocks [126]. The absence of clock shifts in two-state Fermi mixtures facilitates the use of RF transitions to create spin mixtures and allows to accurately

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calibrate magnetic ﬁelds. The emergence of clock shifts in three component Fermi systems provides an important tool to probe the many-body physics underlying such shifts. Absence of clock shifts in a fully polarized Fermi gas. It is tempting — but incorrect — to calculate clock shifts by considering the energy shift due to interactions of the two atomic states involved and then associating the resulting energy diﬀerence with a shifted transition frequency. Let us take for example a Fermi gas fully polarized in state |1, and let a12 be the scattering length for collisions between |1 and |2. An atom in state |2 2 would experience a mean-ﬁeld energy shift δE2 = 4π m a12 n1 due to interactions with the cloud of |1-atoms at density n1 . As an RF pulse transfers |1 atoms into state |2, one might incorrectly conclude that the RF resonance is shifted by an amount δE2 . However, the RF pulse does not incoherently transfer some atoms into state |2, where they would experience the shift δE2 . Such a process would increase entropy, while RF radiation is a unitary transformation which conserves entropy. Rather, the RF pulse transfers all atoms simultaneously into a new superposition state |α ≡ cos α |1 + sin α eiφ |2. The fermions are still fully spin-polarized, they all occupy the same (rotated) quantum state. Therefore, each pair of fermions has to be in an antisymmetric motional state which excludes s-wave collisions. As a consequence, there is no interaction energy in the ﬁnal state and the clock shift is zero. Clock shifts are absent in an ultracold spin-polarized Fermi gas. This was directly demonstrated in the MIT experiment [127]. A similar argument shows that in the case of thermal bosons (with intrastate scattering lengths a11 = a22 = 0), there is a clock shift, but it is twice the energy shift δE2 for an inﬁnitesimal RF transfer [128]. The factor of two results from correlations in a spin-polarized thermal Bose gas, which are preserved during the RF pulse: In a coherent collision, two indistinguishable thermal bosons either preserve their momenta or exchange them. If, in contrast, RF spectroscopy is performed on a pure Bose-Einstein condensate, the colliding bosons have the same momenta, hence exchange collisions are absent and the mean-ﬁeld shift is indeed δE2 [129, 130]. Absence of clock shifts in a two-state mixture of fermions. Switching back to fermions, one may ask whether there is a clock shift if the initial state is not spin-polarized, but a decohered mixture of populations in state |1 and |2. Interactions are now clearly present, the energy levels of atoms in states |1 and |2 are now truly shifted by δE1 = 4π2 m a12 n2 and δE2 , and one might (incorrectly) expect a clock shift δE2 − δE1 for transitions from state |1 to |2. However, similar to the case of the spin-polarized sample, one has to distinguish between the state which can be accessed by the transition, and some other incoherent state which can be reached only by an entropy increasing decoherence process (see ﬁg. 4). Even though the initial mixture had no coherence (i.e. the oﬀdiagonal elements of the density matrix were zero), the RF pulse reintroduces coherence into the system. The ﬁnal state after the RF pulse is not an incoherent mixture with diﬀerent particle numbers, but a state in which each |1 atom has been transferred into the superposition state |α, and each |2 atom into the orthogonal superposition state.

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a)

b)

2 C

B

2 C E

c)

B C

dq D

D

q

E

A

1

1

A

Fig. 4. – Bloch sphere representation of RF transitions. a) An RF pulse rotates a pure state A into B. The superposition state decoheres into a “ring” distribution which has “lost” its deﬁnite phase and is represented by the vertical vector C. b) A second RF pulse transforms the fully decohered state C into a partially coherent state D. The ﬁnal state E is reached only after further decoherence. c) Transfers A→B and C→D are coherent and reversible. B→C and D→E are irreversible. From [128].

One can show that for fermions, interaction energies are invariant under such a coherent transfer. The outcome is that in spite of possibly strong interactions between atoms in state |1 and |2, there is no interstate clock-shift in a two-state mixture of fermionic atoms [128]. Clock shifts in transitions to a third state. While RF transitions between two populated fermionic states do not reveal energy shifts, transitions from e.g. state |2 into a third, empty state reveal the presence of interactions. A priori, such transitions require knowledge of three scattering lengths, a12 , a13 and a23 . However, the preceding discussion shows that the coherent transfer from |2 to state |3 is not aﬀected by interactions between these two states. To ﬁrst approximation, the clock shift will be given by the diﬀerential mean-ﬁeld shift experienced by an atom in state |3 compared to that experienced by an atom in state |2. The clock shift should thus read Δω = 4π m (a13 − a12 ) n1 . This dependence was used to observe the change of the scattering length near a Feshbach resonance [131, 127]. However, this expression is valid only for small scattering lengths, much smaller than the range of the potential r0 and 1/kF . Its extension will be discussed in the next section. Sum rule expression for the average clock shift. It is possible to derive a general expression for the average clock shift of an RF transition, valid for any many-body state of bosons and fermions [132]. Of course, knowledge of the average shift may be of limited use in cases where the RF spectrum has a complex line shape, e.g. is asymmetric, has highenergy tails or shows several peaks, but it still provides an important consistency check. The starting point is an initial many-body state |12 with energy E12 that contains atoms in state |1 and |2. The RF pulse resonantly drives transitions between state |2 and

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|3. As before in the discussion of RF transitions in a two-state mixture, it is incorrect to calculate the clock shift via the energy diﬀerence between two fully decohered states with diﬀerent particle numbers in state |2 and |3. The expression for the diﬀerential mean-ﬁeld shift Δω = 4π m (a13 − a12 )n1 thus cannot be generally true. Rather, to obtain the average clock shift, one must calculate the average energy cost per atom for rotating all atoms in state |2 into a coherent superposition of |2 and |3, a state |ϑ = cos ϑ |2 + sin ϑ |3. The many-body state is then changed into |1ϑ, which has the same spatial many-body wave function as the original state. The total number of transferred atoms is N2 ϑ2 for small ϑ, with N2 the number of atoms originally in |2. The average energy cost per atom for this rotation is thus (9)

1ϑ| H |1ϑ − 12| H |12 , ϑ→0 N2 ϑ 2

¯ ω = ω23 + Δω = lim

where ω23 is the hyperﬁne+Zeeman energy diﬀerence between |2 and |3, and H is the total Hamiltonian of the interacting mixture in three hyperﬁne states. The latter can be split into the internal hyperﬁne+Zeeman Hamiltonian and the external Hamiltonian Hext describing the kinetic and interaction energy. The rotated state |1ϑ is generated by the many-body analogue of the transverse (ﬁctitious) magnetic ﬁeld above, Sx = 1 d3 r(Ψ3† Ψ2 + Ψ2† Ψ3 ). So we have |1ϑ = e−iϑSx |12 ≈ (1 − iϑSx ) |12, and eq. (9) gives 2 Δω =

(10)

1 12| [Sx , [Hext , Sx ]] |12 . 2N2

An identical expression for the clock shift can be calculated using Fermi’s Golden Rule [132]. This general sum rule for the spectral response can be applied to strongly interacting fermions [132, 133]. For weak interactions with the scattering lengths small compared to the characteristic size r0 of the interatomic potential, one indeed obtains the mean-ﬁeld expression of the previous section. For scattering lengths larger than r0 (but still smaller than 1/kF ) this expression is modiﬁed to (11)

Δω =

4π a12 (a13 − a12 )n1 . m a13

The general result, valid for all scattering lengths large than r0 , is (12)

Δω =

where ∂(E12 /N2 )/∂

1 kF a12

1

1 1 − kF a13 kF a12

∂(E12 /N2 ) , ∂ kF1a12

is the change in the energy of the original state |12 under

a change of the interaction strength 1/kF a12 . This change is varying smoothly as a function of 1/kF a12 and is well-behaved even for resonant interactions, 1/kF a12 = 0. This expression shows that for strong interactions, the clock shift is expected to approach

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zero. This explains, at least qualitatively, the observation of vanishing clock shifts in a strongly interacting, unpaired Fermi gas [127]. . 2 3.4. The special case of 6 Li. The usefulness of RF spectroscopy strongly depends on the spectral resolution one can achieve in the laboratory. The characterization of interaction eﬀects on the order of a tenth of the Fermi energy requires a resolution on the kilohertz level. At high magnetic ﬁelds around the Feshbach resonance in 6 Li, typical magnetic ﬁeld stabilities are about 10 mG, day-to-day ﬂuctuations can be ten times larger. It is one of the many fortuitous facts about the 6 Li atom that due to its small hyperﬁne interactions, magnetic ﬁelds of several hundred Gauss completely decouple the electron from the nuclear spin. Therefore there are several RF transitions which ﬂip only the nuclear spin and thus have only a very weak ﬁeld sensitivity. For example, the |1-|2 atomic resonance has a ﬁeld dependence smaller than 2.7 kHz/G above 600 G, which makes it easy to have sub-kHz resolution without any special ﬁeld stabilization. This unique property of 6 Li has allowed numerous RF experiments on unitarity limited interactions [127], on strong interaction eﬀects in resonantly interacting gases [75, 134], precision spectroscopy of atoms and molecules [135] and on imbalanced Fermi gases [77]. In contrast, 40 K has a ﬁeld dependence of 170 kHz/G for transitions between states |2 and |3 near the Feshbach resonance at 202 G. This was still suﬃcient for RF-dissociation of molecules [61] and the characterization of a Feshbach resonance [131]. . 2 3.5. Preparation of a two-component system. Here we discuss how we use RF pulses and magnetic ﬁelds to transform a single-component Fermi cloud at low magnetic ﬁelds into a strongly interacting two-component mixture near a high-ﬁeld Feshbach resonance. Experimental Procedure. In the MIT experiment, a spin-polarized Fermi gas is produced . by sympathetic cooling in a magnetic trap (see subsect. 2 4.1 and ﬁg. 6 for details). Loading into the optical trap is performed in several steps. First, the radial conﬁnement of the magnetic trap is removed by reducing the current in the cloverleaf coils to zero. This is a delicate process, as the center of the magnetic trap needs to remain aligned with the optical trap at all times during the current ramp-down. The atoms are still polarized in the stretched state |F, mF = |3/2, 3/2. They experience the radial conﬁnement of the optical trap plus the axial magnetic curvature. After the transfer into state |1 ≡ |F = 1/2, mF = 1/2 (an adiabatic Landau-Zener RF-transfer close to the zeroﬁeld hyperﬁne splitting of 228 MHz), the atoms are in a high-ﬁeld seeking state and thus experience an anti-trapping axial curvature. By quickly reversing the sign of the axial magnetic bias ﬁeld the atoms are trapped again (see ﬁg. 5). At this stage, the magnetic ﬁeld is increased to values near the Feshbach resonance between state |1 and |2, located at B = 834 G (see sect. 5). Starting with the fully polarized gas in state |1, a non-adiabatic Landau-Zener RF sweep (around the hyperﬁne splitting of ∼ 76 MHz on resonance) transfers atoms into a superposition of states |1 and |2. The admixture of state |2 can be freely controlled via the Landau-Zener sweep rate.

124

W. Ketterle and M. W. Zwierlein Spin Flip

Trapping Potential [a.u.]

Magnetic Field B [a.u.]

State:

|3/2,3/2>

Field Flip |1/2,1/2>

|1/2,1/2>

1

B0

B0

0

B1 -1

1

0 -1

0

1 -1

0

1 -1

0

1

Axial Position [a.u.]

Fig. 5. – Hyperﬁne transfer of the cloud in a magnetic ﬁeld curvature. The atoms are initially trapped in state |3/2, 3/2. After the spin transfer into state |1/2, 1/2, atoms are no longer trapped. A quick adiabatic reversal of the sign of the magnetic ﬁeld retraps the atoms.

Decoherence. RF spectroscopy will not produce a decohered two-state mixture, but a coherent superposition state, by applying a suitable RF pulse or via a non-adiabatic Landau-Zener sweep. A decoherence mechanism is required for the gas to develop into a mixture of two states, i.e. to incoherently populate two distinct quantum states described by a diagonal density matrix. Only such a mixture will interact via s-wave collisions and possibly show pairing and superﬂuidity at achievable temperatures. We found experimentally that an eﬃcient decoherence mechanism for the trapped gas is provided by the magnetic ﬁeld curvature of the optical/magnetic hybrid trap. Atoms that follow diﬀerent trajectories in the inhomogeneous ﬁeld will acquire diﬀerent phases. After some time, the relative phases of atoms are scrambled and one is left with an incoherent mixture. Being no longer in identical states, s-wave interactions between atoms are allowed. To demonstrate that decoherence has occurred, the emergence of clock shifts in transitions to a third, empty state has been recorded in [127]. The timescale for decoherence was found to be tens of milliseconds. We can estimate the decoherence time from the spread of magnetic ﬁelds across the sample. Since the axial potential is mainly magnetic, the atoms experience a spread of Zeeman energies equal to the Fermi energy. At high magnetic ﬁelds, the magnetic moment of the two lowest states diﬀers only by the nuclear magnetic moment, which is three orders of magnitude smaller than the electron’s magnetic moment. We thus

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Fig. 6. – Magnetic trap in the MIT experiment, used for sympathetic cooling of 6 Li with 23 Na. The trap consists of a “curvature coil” that produces an axially conﬁning potential. Its oﬀset magnetic ﬁeld is cancelled to about 0.5 G by the “anti-bias coil”. Radial conﬁnement is provided by the gradient coils which are wound in “cloverleaf” conﬁguration. They replace the four Ioﬀe bars in a standard Ioﬀe-Pritchard trap. After the fermions are loaded into a single-beam optical trap, the anti-bias coils access the wide Feshbach resonance between the two lowest hyperﬁne states of 6 Li at 834 G.

estimate the decoherence rate to be a thousand times smaller than the Fermi energy divided by . For a typical Fermi energy of × 100 kHz, we thus expect a decoherence time of 10 ms, in agreement with observations. . 2 4. Using and characterizing Feshbach resonances. – Feshbach resonances are crucial for realizing strongly interacting Fermi systems. In this section, we present the Feshbach resonance as an experimental tool to prepare and analyze such systems. This section assumes a basic understanding of the physics of a Feshbach resonance. A detailed discussion of the underlying concepts and a theoretical description of Feshbach resonances can be found in sect. 5. . 2 4.1. High magnetic ﬁelds. In 6 Li, the broad Feshbach resonance between the lowest two hyperﬁne states lies at 834 G [135]. To access the BEC-BCS crossover, magnetic ﬁelds of about 1000 G or more are necessary. To generate these ﬁelds with suﬃcient homogeneity while maintaining good optical access requires a careful design, usually with some compromises. If magnetic ﬁeld coils with N windings are placed in Helmholtz conﬁguration outside the vacuum chamber or glass cell of typical diameter d = 3 cm, a current of about I ∼ Bd/μ0 N ∼ 3000 A/N is required. For such current densities, water cooling is essential. For a given total coil cross-section A, the coil resistance is R = ρN 2 (2πd)/A, with ρ = 1.7 · 10−8 Ωm the resistivity of copper. The required electric power is P = RI 2 = EB /τ , where EB = B 2 d3 /2μ0 is the magnetic energy of a homogeneous ﬁeld B

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stored in a volume d3 , and τ ∼ μ0 A/ρ is the 1/e decay time of the ﬁeld energy if the coils were shorted (τ = coil inductance/resistance). Both τ and P are independent of the number of windings N . The division of the designated volume of copper into wires determines the voltage and current of the power supply at constant power. As the required magnetic ﬁeld and the dimension d are determined by design constraints, the only variable here is the total cross-section of the coils A which is often chosen to be a fraction of d2 . For A = (1 cm)2 , a power of P ∼ 300 W is dissipated in each coil and the time constant τ ∼ 2 ms. The time constant gives the fastest possible magnetic ﬁeld ramp-up time, unless the power supply has a higher maximum power than the power P for steady operation. Fortunately, the ﬁeld decay time can be reduced by using a “ring-down” resistance in parallel with the coil. A diode ensures that this ring-down path is opened once the power supply is switched oﬀ. Figure 6 shows the magnetic-ﬁeld conﬁguration used in the MIT experiment. It allows for independent control of the bias ﬁeld, the magnetic-ﬁeld curvature, and the radial gradient through the use of independent coils. The “cloverleaf” coils are needed for tight radial conﬁnement during the sympathetic cooling stage of 6 Li with 23 Na in the magnetic trap. In order to tune the interatomic interactions across the Feshbach resonance, the bias ﬁeld should be an independent parameter. This is accomplished by arranging a pair of coils as close as possible to the Helmholtz conﬁguration. Our “Feshbach” coils (which also serve as “anti-bias” coils [9] during magnetic trapping) generate a residual magnetic ﬁeld curvature that corresponds to an axial trapping frequency of 11.0 Hz for 6 Li at 834 G (resonance). If necessary we can compensate for this curvature by using the “curvature” coils. In practice, these two pairs of coils contribute both to curvature and bias ﬁeld, and controlling the two currents independently allows a wide range of possible values. In most of our experiments, the “curvature” coils provide the bulk of the axial conﬁnement. Thus, varying the bias ﬁeld with the “Feshbach” coils between B0 = 700 G to B0 = 1000 G changes the axial trap frequency by only 0.5 Hz around the value at the Feshbach resonance (ωz /2π = 22.8 Hz). . 2 4.2. Methods for making molecules. Molecules are one form of pairing, and therefore play a major role in studying pairing between fermions. Many of our experiments use a purely molecular cloud as an intermediate step. Several methods have been demonstrated to create molecules from ultracold atoms: – Photoassociation. In photoassociation two colliding atoms are optically excited to a bound state, which is electronically excited. By using a second step or a Raman transition, the electronic ground state can be accessed, usually with high vibrational excitation. This method is discussed in other recent reviews [136, 137]. So far, the phase space density of molecular clouds has been limited by heating from near resonant light and collisions involving the electronically excited intermediate state, or the vibrationally excited ﬁnal state. – Three-body recombination near a Feshbach resonance.

Two-Particle Energy

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127

kBT

EB

Magnetic Field Fig. 7. – Creating molecules via three-body collisions. A molecular state is coupled to the continuum. As the gas is cooled on the molecular side, the Feshbach molecular state is populated via three-body collisions. If the binding energy is not much larger than kB times the temperature, the energy carried away by the third body does not substantially heat the sample. For fermionic particles, further decay into lower lying vibrational states is strongly suppressed due to Pauli blocking.

– Coerent two-body transfer near a Feshbach resonance via (1) a magnetic ﬁeld sweep, (2) RF association, and (3) magnetic ﬁeld modulation. Note that many theoretical descriptions of photoassociation are directly applicable to Feshbach resonances, as they can be regarded as photoassociation resonances with zero frequency photons. Sweeps of the magnetic ﬁeld across the Feshbach resonance are equivalent to frequency sweeps across the photoassociation resonance. In a three-body recombination, two of the colliding atoms form a molecule, the third particle (atom or molecule) carrying away the leftover energy and momentum. This process preferentially populates the most weakly bound molecular states. Their binding energies lie between zero and ≈ −160 2 /mr02 (for an asymptotic van-der-Waals potential V (r) ∼ −C6 /r6 ), depending on rotational quantum numbers and boundary conditions at the inner turning point of the potential [138, 139]. With the van-der-Waals range r0 = (mC6 /2 )1/4 ≈ 60 a0 for 6 Li, these binding energies can be up to kB times 1 K. The released energy in such a collision heats up the cloud, leading to trap loss (hence the name “bad collisions”). However, in the case of the very weakly bound molecular state on the molecular side of a Feshbach resonance (scattering length a > 0), the binding energy can be on the order of the temperature, and molecules can eﬃciently form without severe heating and trap loss (ﬁg. 7). Subsequently, leftover atoms can be evaporated from the optical trap. This process is very eﬃcient, since weakly bound molecules have twice the atomic polarizability, hence the optical trap is twice as deep for molecules than it is for single atoms.

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BEC-Side

Magnetic Field

BCS-Side

Fig. 8. – Creating molecules via magnetic ﬁeld ramps. A magnetic ﬁeld sweep can transfer unbound atoms adiabatically into the molecular state, much like a two-level Landau-Zener transition.

The Feshbach molecules are in the highest vibrational state of the interatomic potential (see sect. 5). They are only stable if the decay into lower lying vibrational levels is slow. It turns out that for fermions this decay is suppressed by the Pauli principle (see . subsect. 5 1). Producing molecules coherently by a magnetic ﬁeld sweep is reversible and does not generate heat. It exploits the tunability of the Feshbach molecular state: Starting with unbound atoms in the continuum, one can sweep the magnetic ﬁeld across the resonance and form a bound molecule (ﬁg. 8). Some aspects of this sweep can be described as a twolevel Landau-Zener sweep through an avoided crossing. For a coupling matrix element ˙ one ﬁnds [140] V between two “bare” states, |a and |b, and an energy sweep rate E, (13)

P|a →|b = 1 − e−c

|V |2 ˙ E

for the probability P|a →|b to make a transition from |a to |b as the bare state energies are swept through resonance. Here c is a numerical constant on the order of 1. In the case of Feshbach resonances, the two “levels” are the molecular state and the state of two unbound atoms. V is the coupling matrix element between these states, V = N/Ω g0 , an expression that we will discuss in sect. 5. The number N of atom pairs that appears in |V |2 accounts for the fact that each spin up atom has N chances to form a molecule with a spin down atom per quantization volume Ω. Alternatively, one can consider two-body physics in a box of volume Ω/N , which emphasizes the local picture of two atoms forming a molecule. 2 If we take the simple Feshbach model of sect. 5, we can replace g02 = 4π m abg ΔμΔB, with abg the background scattering length, Δμ the diﬀerence in magnetic moments be-

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129

tween the molecular state and two free atoms, and ΔB describing the width of the Fesh˙ bach resonance. The bare state energies are tuned via the magnetic ﬁeld, so E˙ = Δμ B. We then have n (14) Patoms→molecules 1 − exp −A B˙ 4πa

ΔB

bg with A = c . The higher the density and the slower the magnetic ﬁeld ramp m across resonance, the more eﬃcient is the production of molecules [141, 142]. The schematic ﬁgure of the Feshbach resonance (ﬁg. 8) suggests that the simple twostate picture applies only to the lowest state of relative motion between the two atoms. Excited states of relative motion on the BCS side are adiabatically connected to the next lower-lying state of relative motion on the BEC side. Therefore, the Landau-Zener probability discussed above should have a prefactor which is the probability for two atoms to be in the same phase space cell, proportional to the phase space density. Indeed, it has been experimentally veriﬁed in [141] that the eﬃciency of forming molecules during a slow adiabatic sweep increases monotonously with the phase space density and that it can exceed 50 % for both bosonic and fermionic thermal clouds (up to 90% transfer was achieved for 40 K). The coherent conversion of two atoms into molecules can be accomplished not only by sweeping the Feshbach ﬁeld, but also by modulating the magnetic ﬁeld close to the Feshbach resonance, at a frequency corresponding to the molecular binding energy [143, 86]. Yet another method is to drive a free-bound RF transition [103], where initially one of the free atoms occupies a diﬀerent hyperﬁne state.

. 2 4.3. Observation of Feshbach resonances. A Feshbach resonance is an “intimate” encounter between two atoms, which collide and temporarily form a molecule before they separate again. Many collisional processes are enhanced and have been used to locate the magnetic ﬁeld position of these resonances. – Trap loss by enhanced inelastic collisions. The ﬁrst observations of Feshbach resonances were made by monitoring loss due to three-body recombination [7] and due to an enhanced photoassociation rate [8]. The broad Feshbach resonance in 6 Li was mapped out using trap loss [56]. However, since the molecules formed in three-body recombination are long-lived close to resonance, the center of the loss feature was found at ﬁelds well below the Feshbach resonance. In addition, an unpredicted narrow Feshbach resonance at 543 Gauss was found [56, 16, 144]. Trap loss spectroscopy is usually applied to ﬁnd new resonances and has been used, for example, to discover p-wave Feshbach resonances in 40 K [145] and 6 Li [146, 144] and interspecies Feshbach resonances [147, 101]. – Rapid thermalization. The increased scattering length leads to rapid thermalization of the gas. This method was used to study the resonance in 85 Rb [148], and in 40 K [57]. The absence of thermalization was used to locate the position of the zero-crossing of the scattering length in 6 Li at 528 G [58, 59].

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– Change of interaction energy. For Bose-Einstein condensates, this is observed by the change in mean-ﬁeld energy and therefore the size of the cloud, either in trap or in ballistic expansion [7, 11]. For fermions, the change of the interaction energy . has been monitored via clock shifts (see subsect. 7 2.4). The size of the fermionic cloud varies smoothly and monotonously across resonance, a direct consequence of the smooth change of the cloud’s energy in the BEC-BCS crossover (see sect. 4). – RF spectroscopy of Feshbach molecules. Using RF spectroscopy, one can determine the onset of molecular dissociation, and then, by extrapolation, ﬁnd the value of the magnetic ﬁeld at which the molecular binding energy vanishes [135, 61] . The most precise value for the broad 6 Li Feshbach resonance was derived from RF spectroscopy between weakly bound molecular states using a multi-channel scattering model [135]. – Threshold for molecule formation. When the magnetic ﬁeld is swept across the Feshbach resonance, molecules will appear with a sharp onset at the resonance. – Threshold for molecule dissociation. Feshbach molecules start to dissociate when the magnetic ﬁeld is raised to a value above the Feshbach resonance. Since the last two methods are directly related to the formation and detection of molecules, we discuss them in more detail.

a)

b)

1000

6

Atom Number [10 ]

Magnetic Field [G]

2.0 800 600 400 200

1.5 1.0 0.5 0.0

0 -4

0 4 Time [ms]

8

750

800

850

900

950

Magnetic Field [G]

Fig. 9. – Molecule formation by magnetic ﬁeld sweep across the Feshbach resonance. a) Experimental procedure. A Fermi mixture prepared on the BCS-side of the Feshbach resonance is swept across resonance (shown as the dashed line) to form molecules. The gas is released from the trap at the end point of the ramp at time t = 0 ms. Zero-ﬁeld imaging, indicated by the star, detects the leftover atoms. b) Atomic signal vs end point of the magnetic ﬁeld sweep. The line is a ﬁt to an error function, whose center is determined to be 838 ± 27 G, with an uncertainty given by the 10%-90% width (54 G).

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b) Atomic signal [arb. units]

1.0

950

Magnetic Field [G]

Atomic signal [arb. units]

a)

0.5

900 850 800 750

Time [ms]

700 0

0 820

860

10

900

Magnetic Field [G]

20

0.5

0.3

0.1

30

940

810

820

830

Magnetic Field [G]

Fig. 10. – Locating the Feshbach resonance by molecule dissociation. The experimental procedure is shown in the inset. A molecular cloud is prepared on the BEC-side of the Feshbach resonance, at 780 G, and released from the trap at t = 0 ms. After some expansion, the ﬁeld is ramped to a test value around resonance (shown as the dashed line), held constant and is ﬁnally brought to zero ﬁeld, where only unbound atoms are imaged. a) The atomic signal as a function of the test ﬁeld shows a sharp threshold behavior at 821 ± 1 G, where the uncertainty is the statistical error of a threshold ﬁt, shown in b).

To observe the onset of molecule formation, one prepares a Fermi mixture on the “BCS”-side of the Feshbach resonance, where no two-body molecular bound state exists in vacuum (see sect. 5). As the magnetic ﬁeld is swept across the resonance, molecules will form and, accordingly, the signal from unbound atoms will diminish (ﬁg. 9) [61, 62, 147]. From ﬁg. 9 we determine a value of B0 = 838 ± 27 G for the position of the resonance. The loss of atomic signal is reversible: Ramping back across the resonance will dissociate the molecules, and re-establish all or most of the atomic signal [61, 15-17, 62-64]. In fact, the dissociation method gives a more accurate determination of the location of the Feshbach resonance [69, 70]. To avoid eﬀects due to the high density in the trap (i.e. many-body physics), in [70] the molecular cloud is expanded to a 1000 times lower density, about 1010 cm−3 . Then the magnetic ﬁeld is ramped to a value Btest . If Btest lies above the Feshbach resonance, the molecules will dissociate into unbound atoms, which are subsequently detected at low ﬁeld. The very sharp onset of the atomic signal at Btest = 821 ± 1 G is striking (see ﬁg. 10) and suggests this magnetic ﬁeld value as the position of the Feshbach resonance. However, via molecular RF spectroscopy the location of the Feshbach resonance has been determined to lie at 834.1 ± 1.5 G [135]. The reason for this discrepancy is probably that molecules at threshold are extremely fragile and might break apart before the resonance is reached, thus shifting the observed threshold to lower values. See ref. [144] for a discussion and characterization of such shifts. RF spectroscopy addresses more tightly bound molecules and identiﬁes the resonance by extrapolation, thus avoiding stability issues very close to resonance.

132 b)

2.0

Molecule Fraction

6

Atomic Signal [10 ]

a)

W. Ketterle and M. W. Zwierlein

1.5

1.0

0.5

1.0 0.8 0.6 0.4 0.2 0.0

0.0 0 1 2 3 4 Expansion Time before Magnetic Field Ramp [ms]

0

1 2 3 4 5 12 -3 Density per Spin State [10 cm ]

Fig. 11. – Revival of the atomic signal during expansion and strength of Feshbach coupling. a) The magnetic ﬁeld is switched oﬀ after varying expansion times for a cloud released at 840 G. The ﬁeld ramp creates molecules more eﬃciently at the high densities of the trap than at low densities after long expansion. In b), all of the atomic signal loss is interpreted as molecular conversion and plotted as a function of density. The density was calibrated by imaging the cloud at high ﬁeld for varying expansion times. All ﬁts are for the simple Landau-Zener-model described in the text.

From ﬁg. 10 we can directly see that before dissociation, more than 99% of the gas exists in form of molecules. The reason is that this molecular cloud was formed via the three-body process, by simply cooling the gas at the ﬁxed ﬁeld of 780 G (the BEC-side of the resonance). The lifetime of the weakly bound molecules is so long, and the binding energy is so small, that losses and heating are negligible, and, after evaporation of leftover unbound atoms, essentially all particles are bound into molecules. . 2 4.4. Determination of the coupling strength of Feshbach resonances. The “strength” of a Feshbach resonance is determined by the square of the matrix element which couples the 2 . closed and open channels, proportional to g02 = 4π m abg ΔμΔB (see subsect. 2 4.2 and sect. 5). This expression depends on the background scattering length only because of ΔB the deﬁnition of ΔB in the formula for the scattering length a(B) = abg 1 − B−B0 . A Feshbach resonance with the same strength but on top of a larger background scattering length then has a narrower width ΔB. So one way to determine the strength of a Feshbach resonance is by measuring or knowing abg , Δμ and ΔB. The matrix element can be measured more directly from the dynamics of molecule dissociation and formation. When Feshbach molecules in 23 Na were dissociated with variable ﬁeld ramp, the kinetic energy of the fragments was shown to increase with the ramp speed [149]. This reﬂects the ﬁnite lifetime of the Feshbach molecules, which are “ramped up” in energy for about one lifetime, before they decay through their coupling to the open channel. This method was also applied to 87 Rb [150]. Here we describe experiments using the reverse process, i.e. the formation of molecules . by a variable ﬁeld ramp, as introduced in subsect. 2 4.2 above. Figure 11 demonstrates

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the extremely strong coupling strength of the 6 Li Feshbach resonance. In this experiment, a Fermi mixture is released from the trap at B = 840 G, slightly above the Feshbach resonance. When the magnetic ﬁeld is suddenly switched oﬀ at the same time (at an initial slew rate of B˙ = 30 G/μs), almost the entire atomic signal vanishes, i.e. the conversion into molecules is almost 95%. The 6 Li Feshbach resonance is so strong that the quantitative conversion from trapped fermions into molecules during a Feshbach sweep can only be avoided by using small magnetic ﬁeld coils with low self-inductance and correspondingly fast switch-oﬀ time [15]. However, when we allow the gas to expand and lower its density before the sweep, then the conversion to molecules is only partial, and we can determine the strength of the Feshbach coupling. In ﬁg. 11 b) we interpret all the disappeared atomic signal as conversion into the weakly-bound molecular state neglecting other loss-channels like unobserved molecular states. The conversion eﬃciency decreases with decreasing density and can be ﬁt using the simple the Landau-Zener formula eq. (14). We ﬁnd that the constant A in eq. (13) is A ≈ 24 G/(1012 cm−3 μs) with a relative error of 50% due to the uncertainty in the atom number. The theoretical prediction is eq. (14) with c = 2π [151]. With the parameters for 6 Li we ﬁnd (15)

A=

8π 2 abg ΔB G = 19 12 −3 . m 10 cm μs

For comparison, for the 40 K Feshbach resonance at B = 224.2 G used in some experiments G the prediction is A = 0.011 1012 cm −3 μs (abg = 174 a0 , ΔB = 9.7 G [131]). This is not G far from the value A ≈ 0.004 1012 cm [61] −3 μs one extracts from the measurement in (0.05 G/μs was the ramp speed that resulted in a 1/e transfer of molecules, at a peak density of npk = 1.4 × 1013 cm−3 ). The broad resonance in 6 Li can eﬃciently convert atoms into molecules at 2000 times faster sweep rates. The good agreement with the simple Landau-Zener model might be fortuitous. Reference [141] points out that the conversion eﬃciency must depend on the phase space density and presents data which, in the case of 85 Rb, disagree with simple theoretical predictions by a factor of eight. . 2 4.5. The rapid ramp technique. So far, we have discussed the time scale for two-body physics, namely the association of two atoms into a molecule. For isolated atom pairs, this process is independent of the total momentum of the pair, which is preserved due to Galilean invariance. In a many-body system, fermion pairs interact and collide, and their momentum changes. If the two-body time scale is faster than the many-body time scale, there is an interesting window of ramp rates for the sweep across the Feshbach resonance: One can be slow enough to quantitatively convert atom pairs into molecules, but also fast enough such that the momentum distribution of the ﬁnal molecules reﬂects the momentum distribution of the fermion pairs prior to the sweep (see ﬁg. 12). This method was introduced by the JILA group [69], and later adapted to 6 Li by our

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Fig. 12. – Rapid ramp to the molecular side to observe pair condensation. Immediately after switching oﬀ the trapping beam, the magnetic ﬁeld is ramped to zero ﬁeld. This converts longrange pairs into stable, tightly bound molecules, preserving the momentum distribution of the original pairs.

group [70]. It made it possible to analyze the momentum distribution of fermion pairs across the whole BEC-BCS crossover and detect the pair condensate (see sect. 6). The problem with the rapid ramp technique is that it is not clear what the many-body time scale is. In particular, one wants to rule out that the pair momentum distribution changes during the sweep or that a condensate is formed while ramping. We address this question by listing several time scales of the system (table III). For theoretical modelling of the ramp process, see refs. [152-155]. The ﬁnal demonstration that the rapid ramp does preserve the absence or presence of a pair condensate before the sweep has to come . from experiment (see subsect. 6 4.2). The fastest timescale, given by divided by the Feshbach coupling strength, governs

Table III. – Time scales involved in the rapid ramp technique. The given values are typical for the MIT experiment and assume a density of 1.5 · 1013 cm−3 . Time scale Two-body physics Magnetic ﬁeld ramp through anti-crossing Inverse Fermi energy Time required to leave strongly interacting region Evolution of the gap at kF |a| = 2 Gap at kF |a| = 2 Inverse collision rate at unitarity and T /TF = 0.1 Growth time of a pair condensate at kF |a| = 2 Radial trapping period

Formula √ /g √ 0 2πn g0 2πn/ΔμB˙ /EF δB/B˙ ˙ Δ/Δ /Δ ≈ 0.23 EF /(kB T )2 [156] ≈ EF /Δ2 [157] 2π/ωr

Value 20 ns 80 ns 3 μs 3 μs 10 μs 15 μs 70 μs 75 μs 2 ms

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the two-body physics (in the Landau-Zener picture, this is the inverse of the anti-crossing gap). The timescale at which the magnetic ﬁeld sweeps through the anti-crossing is given in the second line of table III. As pointed out above, in the MIT experiment, even switching oﬀ the current through the Feshbach coil is still slower than the in-trap two-body time scale, resulting in conversion eﬃciencies into molecules of larger than 90%. The next fastest time scale is set by the Fermi energy, which in the unitarity regime on resonance would set the timescale for collisions in the normal Fermi gas, were it not for Pauli blocking. Indeed, if we multiply the local density n with the rms velocity in the Fermi-Dirac distribution ∝ vF and with the unitarity limited cross section for elastic collisions ∼ 4π/kF2 , we obtain a “classical” collision rate of ∼ EF /. Also, the Fermi energy should set the time scale at which local ﬂuctuations of the gas density can “heal”, as the local chemical potential on resonance is given by μ ≈ 0.5EF . As the gas is brought into the weakly interacting regime on the BEC-side, where kF a < 1, this manybody relaxation rate μ/ should decrease to the smaller mean-ﬁeld rate of a molecular BEC. It is thus interesting to know whether the ramp is adiabatic with respect to this local “healing” or relaxation, averaged over the sweep. If we use /EF as an upper bound for the relaxation rate around resonance, and neglect relaxation outside this region, the relevant scale is the time it takes to leave the strongly interacting regime. For typical parameters in our experiment, kF a ≈ 1 around 750 G, δB ∼ 85 G away from resonance, ˙ and the time scale is ∼ δB/B. The time it takes to leave the unitarity limited region in our gas is on the order of the Fermi time scale, and should be smaller than the inverse collision rate. This would mean that the original momentum distribution of fermion pairs is “frozen in” during the ramp, and the momentum distribution of the molecules at the end of the sweep reﬂects that of the fermion pairs on the BCS-side. However, since a collisional model for a weakly interacting gas should not be taken too seriously to estimate the relaxation time, . experimental tests were required, which will be discussed in subsect. 6 4.2. The ramp is non-adiabatic with respect to the time scale of the gap, which is forced to evolve faster than it can adiabatically respond to the change in interaction strength, ˙ Δ/Δ Δ/. On the BCS-side of the resonance, the average binding energy of pairs is 34 Δ2 /EF . The last condition implies that pairs cannot adiabatically adjust their size during the fast ramp. On the BEC-side, the pair binding energy evolves into the molecular binding energy, EB = −2 /ma2 . If one ramps far enough on the molecular side, a becomes so small and EB becomes so large that the molecular state can follow the ramp adiabatically. This observation is used in [154] to split the discussion of the ﬁeld ramp into a “sudden” and an “adiabatic” part, connected at the scattering length a∗ for which E˙B /EB = EB /. The “sudden” part is then modelled as a projection of the initial to ˙ 1/3 , with A given by eq. (15), the ﬁnal pair wave function. One ﬁnds a∗ = (A/16π 2 B) ∗ 1/3 ˙ and kF a = (3An/16B) , which is just the third root of the Landau-Zener parameter entering eq. (14). The latter is 1 if the molecule conversion is eﬃcient, as it is in our case, directly implying that the “sudden” to “adiabatic” transition still occurs in the strongly interacting regime, kF a∗ 1. The ramp time needed to enter the adiabatic regime is thus smaller or about equal to the time required to leave the unitarity region.

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Finally, there is the relaxation time scale of the gas in response to a change in the particle distribution. In a normal Fermi gas of N particles at temperatures T TF , relaxation occurs via collisions of particles close to the Fermi surface, of number N T /TF . Pauli blocking reduces the available ﬁnal states for collisions by another factor of T /TF , giving a relaxation time τR ≈ EF /(kB T )2 . In general, if the Fermi surface is smeared over an energy width ΔE, the relaxation time is τ ≈ EF /ΔE 2 . In the case of a (BCStype) superﬂuid, ΔE = Δ, and the relaxation time thus scales as τR = EF /Δ2 [158]. . 2 5. Techniques to observe cold atoms and molecules. . 2 5.1. Basics. The basic techniques of imaging ultracold fermions are identical to those for imaging bosons, which were described in great detail in the 1999 Varenna lecture notes [9]. The two main techniques are absorption and dispersive imaging. In absorption imaging, a laser beam tuned to the atomic resonance is absorbed by the atoms, whose shadow image is recorded on a CCD-camera. It is often applied after expansion of the cloud from the atom trap, as the optical density of the trapped cloud is so high that the absorption is strongly saturated. Detuning the laser frequency to avoid strong absorption often results in image distortions due to dispersive eﬀects. Dispersive imaging relies on the phase shift that atoms impart on the laser light and is usually implemented with a suﬃciently large detuning δ so that the phase shift is on the order of unity. Both types of imaging heat the sample by the recoil of Rayleigh scattered photons. However, in dispersive imaging, the signal is due to forward scattering which is enhanced similarly to superradiance. As a result, for the same amount of heating, the number of signal photons is larger than in absorption imaging by the resonant optical density divided by four [9]. This factor can be big (on the order of one hundred) for large trapped clouds and has made it possible to take several dispersive images of the same sample without noticeable smearing (so-called non-destructive imaging). With regard to imaging, the main diﬀerence between experiments on ultracold fermions and bosons is that typically, in the boson case one deals with a single spin state (an exception are experiments on spinor-BEC [159-162]), while in Fermi gases at least two hyperﬁne states are involved. Especially for the study of imbalanced Fermi systems where the spatial proﬁle is diﬀerent for the two components, double-shot imaging techniques are essential. In such techniques, an image of one spin state is rapidly succeeded by an image of the second state. In case of residual oﬀ-resonant absorption of the ﬁrst imaging light pulse, the second image has to be taken after less then a few tens of μs, to avoid blurring as atoms move due to photon recoil. Current CCD cameras allow rapid successive exposure by shifting each pixel row of the ﬁrst image underneath a mask, where it is safely stored during the second exposure. Both absorption imaging [79] and dispersive imaging [80] have been used in this way. Another technique that has been employed for RF spectroscopy [127] was to use several independent laser beams, each resonant with a diﬀerent atomic transition, that were simultaneously recorded on diﬀerent parts of the CCD chip. The probe frequency in dispersive imaging can be chosen to record a weighted sum

Making, probing and understanding ultracold Fermi gases

137

(c)

Fig. 13. – In situ phase-contrast imaging of the density diﬀerence of two spin states of 6 Li at the 834 G Feshbach resonance. (a) The probe beam is tuned to the red of the resonance for state 1, and to the blue for state 2. The resulting optical signal in the phase-contrast image is proportional to the density diﬀerence of the atoms in the two states. (b) Phase-contrast images of trapped atomic clouds in state |1(left) and state |2(right) and of an equal mixture of the two states (middle). (c) The imbalance in the populations N1 , N2 of the two states, deﬁned as (N1 − N2 )/(N1 + N2 ), was chosen to be −50, −37, −30, −24, 0, 20, 30, 40 and 50%. The observation of a distinctive core shows the shell structure of the cloud caused by phase separation. The height of each image is about 1 mm. See Ref. [80] for further details.

of the column densities of the two components. In particular, by adjusting the laser detuning to lie in between the two resonance frequencies, phase-contrast imaging [9] then records directly the density diﬀerence [80] without the need of subtracting two large signals. Since spin polarization is proportional to the density diﬀerence, this technique was crucial in the study of imbalanced Fermi systems [80] (see ﬁg. 13). . 2 5.2. Tomographic techniques. Both absorption and dispersive imaging integrate along the line-of-sight and provide information about the column densities. However, by taking such projections along an inﬁnite number of angles, one can reconstruct the threedimensional density distribution tomographically using the so-called inverse Radon transformation. If the sample has cylindrical symmetry, then one line-of-sight integrated image n(x, z) is suﬃcient for the reconstruction of the atomic density n(r, z) using the inverse Abel

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transform [163] (16)

n(r, z) = −

1 π

r

∞

dn(x, z) 1 . dx √ x2 − r2 dx

This transformation takes the derivative of the column density image, and so-tospeak inverts the line-of-sight integration. In particular, discontinuous jumps in the three dimensional density appear as kinks (jumps in the derivative) of the column density. The derivative is sensitive to noise. Due to the derivative in eq. (16), this technique requires a very good signal-to-noise ratio. Reducing noise by averaging (blur ﬁlter) is not an option if high spatial resolution has to be maintained, i.e. for the reconstruction of sharp phase boundaries. The inverse Abel transformation has been used to reconstruct the propagation of phonons in a Bose-Einstein condensate [164], and to reconstruct s- and d-wave scattering halos in the collision of two Bose-Einstein condensates [165, 166]. Tomographic reconstruction was essential in distinguishing fermionic superﬂuids with equal densities of the two components from polarized superﬂuids (which have a density imbalance) [80]. The signal-to-noise was high enough to identify sharp phase boundaries between the superﬂuid and the normal phase in the reconstructed images, characteristic for a ﬁrst-order phase transition [82]. Tomographic reconstruction has been extended to RF spectroscopy. RF spectra have usually been recorded for the whole cloud. However, using suﬃciently short RF pulses followed immediately by spatial imaging of the cloud, it has been possible to record the spatial distribution of the RF induced changes in the population of the initial or ﬁnal state [127, 134]. When many such images are recorded for diﬀerent RF frequencies, and processed with an inverse Abel transformation, one obtains local RF spectra [134]. These spectra are no longer aﬀected by inhomogeneous broadening due to the spatially varying density, and one even obtains a series of spectra each at a diﬀerent local density. This method was developed to reveal a gap in the RF spectrum of a fermionic superﬂuid and to observe its homogeneous width and line shape [134]. . 2 5.3. Distinguishing atoms from molecules. On the BEC side of the Feshbach resonance, molecules are stable, and in order to verify the presence of molecules or to quantitatively characterize the system, it became necessary to distinguish atomic from molecular densities. The following properties of these Feshbach molecules are important for their detection. 1) The molecules are stable on the BEC side, not too far away from the Feshbach resonance, and are dissociated by sweeping the magnetic ﬁeld across it. 2) Close to the Feshbach resonance, the size of the molecules is approximately given by the atomic scattering length a and can become very large. Here, the molecular wave function has “open channel” character, i.e. the molecule is formed out of the same continuum of states in which two free atoms collide in. 3) As a consequence, the Feshbach molecule can be regarded as “two atoms on a stick”, and the frequency for resonant transitions is very close

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139

to the atomic frequencies. More precisely, the molecules are expected to absorb most strongly near the outer turning point R. The excited state potential is split by the resonant van der Waals interaction ζΓ(λ/2πR)3 where λ is the resonant wavelength and ζ is ±(3/4) or ±(3/2) for diﬀerent excited molecular states. Therefore, as long as a λ/2π, the Feshbach molecules resonate at the atomic transition frequencies. For a λ/2π, the molecules should be almost transparent at the atomic resonances. 4) Suﬃciently far away from the resonance, the Feshbach molecules assume more and more closed channel character, and due to the diﬀerent hyperﬁne interaction in the closed channel, have a magnetic moment diﬀerent from the free atoms. Various methods use these properties to obtain the molecular populations: – Subtract from the signal of (atoms+molecules) the signal of (atoms only). The atoms only signal is obtained by sweeping to low or zero magnetic ﬁeld, where the molecules no longer absorb at the atomic resonance. The ﬁeld ramp needs to be slow compared to two-body timescales (i.e. should not “rip” the molecule apart), but fast compared to losses. The atoms+molecules signal is obtained either at a magnetic ﬁeld close to resonance [66] or after a sweep across the resonance, which dissociates the molecules [61, 15, 62-64, 17, 66]. – Distinguish molecules by Stern-Gerlach separation. In ballistic expansion at magnetic ﬁelds suﬃciently far away from the Feshbach resonance, molecules can be spatially separated from the atoms and distinguished on absorption images. This technique was used to detect molecules formed of bosonic atoms [62-64]. For 6 Li, it was used to measure the magnetic moment and hence the contribution of the closed channel to the Feshbach molecule [17]. – Distinguish molecules by RF spectroscopy. The molecular RF spectrum is shifted from the atomic line by the molecular binding energy. Therefore, an RF pulse populating an unoccupied state [61] can be tuned to either spin ﬂip atoms or dissociate molecules. Imaging light in resonance with this initially unoccupied state can record the molecular population. In most of our studies at MIT in the BEC-regime of the Feshbach resonance, the temperature of the cloud was so low that it consisted purely of molecules, i.e. we then did not discern any atomic population using the ﬁrst of the methods listed above [66]. Therefore, we routinely image the whole cloud at ﬁelds slightly below the Feshbach resonance knowing that this (atoms+molecules) signal is purely molecular. 3. – Quantitative analysis of density distributions The purpose of imaging and image processing is to record density distributions of the atomic cloud, either trapped or during ballistic expansion. All our knowledge about the properties of cold atom systems comes from the analysis of such images. They are usually compared to the results of models of the atomic gas. Some models are exact (for

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the ideal gas), others are phenomenological or approximations. Many important models for bosonic atoms have been presented in our 1999 Varenna notes. Here we discuss important models for fermions, which allow us to infer properties of the system from recorded (column) density distributions. . 3 1. Trapped atomic gases. . 3 1.1. Ideal Bose and Fermi gases in a harmonic trap. The particles in an atom trap are isolated from the surroundings, thus the atom number N and total energy content Etot of the atomic cloud is ﬁxed. However, it is convenient to consider the system to be in contact with a reservoir, with which it can exchange particles and energy (grand canonical ensemble). For non-interacting particles with single-particle energies Ei , the average occupation of state i is ni =

(17)

1 e(Ei −μ)/kB T

∓1

with the upper sign for bosons, the lower sign for fermions. These are the Bose-Einstein and Fermi-Dirac distributions, respectively. For a ﬁxed number of particles N one chooses the chemical potential μ such that N = N = i ni . Let us now apply these distributions to particles conﬁned in a harmonic trap, with trapping potential (18)

V (r) =

1 m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 ) . 2

We assume that the thermal energy kB T ≡ 1/β is much larger than the quantummechanical level spacings ωx,y,z (Thomas-Fermi approximation). In this case, the occupation of a phase space cell {r, p} (which is the phase-space density times h3 ) is given by eq. (17) f (r, p) =

(19)

1 p2 ( 2m +V

e

(r)−μ)/kB T

∓1

.

The density distribution of the thermal gas is

d3 p f (r, p) (2π)3 1 = ± 3 Li3/2 ±eβ(μ−V (r)) , λdB

nth (r) = (20) where

2π2 mkB T

is the de Broglie wavelength. Lin (z) is the n-th–order Polylogarithm,

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deﬁned as (21)

1 Lin (z) ≡ n π

1 d r r2 e /z − 1

n =0

2n

=

1 Γ(n)

∞

dq 0

q n−1 , eq /z − 1

where the ﬁrst integral is over 2n dimensions, r is the radius vector in 2n dimensions, n is any positive half-integer or zero and Γ(n) is the Gamma-function (5 ). Note that expression (20) is correct for any potential V (r). The constraint on the number of thermal particles is (22) Nth = d3 r nth (r) For a harmonic potential (18), we obtain Nth = ±

(23)

kB T ¯ ω

3 Li3 (± eβμ )

with ω ¯ = (ωx ωy ωz )1/3 the geometric mean of the trapping frequencies. In the classical limit at high temperature, we recover the Maxwell-Boltzmann result of a Gaussian distribution, (24)

ncl (r) =

N π 3/2 σ

x σy σz

e−

P i

2 x2i /σx

i

2 with σx,y,z =

2kB T . 2 mωx,y,z

The regime of quantum degeneracy is reached when λdB ≈ n−1/3 , or when the tem2 perature T ≈ Tdeg . The degeneracy temperature Tdeg = 2mk n2/3 is around or below B one μK for typical experimental conditions. For bosons, it is at this point that the ground state becomes macroscopically occupied and the condensate forms. The density proﬁle of the ideal gas condensate is given by the (5 ) The Polylogarithm appears naturally in integrals over Bose-Einstein or Fermi-Dirac distributions. Some authors [167] use diﬀerent functions for bosons gn (z) = Lin (z) and for fermions P∞ z k fn (z) = −Lin (−z). The Polylogarithm can be expressed as a sum Lin (z) = k=1 kn which is often used as the deﬁnition of the Polylogarithm. This expression is valid for all complex numbers n and z where |z| ≤ 1. The deﬁnition given in the text is valid for all z ≤ l. 1 , Li1 (z) = − ln(1−z). f (r, p) can be written as ±Li0 (± exp[β(μ− Special cases: Li0 (z) = 1/z−1 p2 −V 2m

is

(r))]). When integrating density distributions to obtain column densities, a useful formula Z ∞ √ 2 dx Lin (z e−x ) = π Lin+1/2 (z). −∞

z1

z→∞

Limiting values: Lin (z) → z and −Lin (−z) →

1 Γ(n+1)

lnn (z).

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W. Ketterle and M. W. Zwierlein

square of the harmonic-oscillator ground-state wave function: nc (r) =

(25) where dxi =

mωxi

P N0 − i x2i /d2x i e π 3/2 dx dy dz

are the harmonic-oscillator lengths. The density proﬁle of the ther-

mal, non-condensed component can be obtained from eq. (20) if the chemical potential μ is known. As the number of condensed bosons N0 grows to be signiﬁcantly larger than 1, the chemical potential μ ≈ − kNB0T (from eq. (17) for E0 = 0) will be much closer to the ground state energy than the ﬁrst excited harmonic oscillator state. Thus we set μ = 0 in the expression for the non-condensed density nth and number Nth and obtain (26)

nth (r) =

(27)

Nth =

1 Li3/2 (e−V (r)/kB T ) , 3 λdB N (T /TC )3 for T < TC

with the critical temperature for Bose-Einstein condensation in a harmonic trap ω (N/ζ(3))1/3 = 0.94 ¯ ω N 1/3 , TC ≡ ¯

(28)

where ζ(3) = Li3 (1) ≈ 1.202. At T = TC , the condition for Bose condensation is fulﬁlled in the center of the trap, n = Li3/2 (1)/λ3dB = 2.612/λ3dB . For lower temperatures, the maximum density of the thermal cloud is “quantum saturated” at the critical value nth = 2.612/λ3dB ∝ T 3/2 . The condensate fraction in a harmonic trap is given by N0 /N = 1 − (T /TC )3 .

(29)

For T /TC = 0.5 the condensate fraction is already about 90%. For fermions, the occupation of available phase space cells smoothly approaches unity without any sudden transition: (30)

f (r, p) =

1 p2 ( 2m +V

e

(r)−μ)/kB T

T →0

+1

→

1, 0,

p2 2m p2 2m

+ V (r) < μ , + V (r) > μ .

Accordingly, also the density proﬁle changes smoothly from its gaussian form at high temperatures to its zero temperature shape:

(31)

d3 p d3 p T →0 f (r, p) → nF (r) = √ 3 (2π)3 |p|< 2m(μ−V (r)) (2π)

3/2 2m 1 3/2 (μ − V (r)) . = 2 6π 2

From eq. (30) we observe that at zero temperature, μ is the energy of the highest

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Making, probing and understanding ultracold Fermi gases

occupied state of the non-interacting Fermi gas, also called the Fermi energy EF . The √ (globally) largest momentum is pF ≡ kF ≡ 2mE F , the Fermi momentum. Locally, at position r in the trap, it is pF (r) ≡ kF (r) ≡ 2m F (r) ≡ (6π 2 nF (r))1/3 with the local Fermi energy F (r) which equals μ(r, T = 0) = EF − V (r). The value of EF is ﬁxed by the number of fermions N , occupying the N lowest energy states of the trap. For a harmonic trap we obtain N = (32)

1 d r nF (r) = 6 3

EF ¯ ω

3

⇒ EF = ¯ ω (6N )1/3

and for the zero-temperature proﬁle

(33)

N 8 nF (r) = 2 π RF x RF y RF z

with the Fermi radii RF x,y,z =

2EF . 2 mωx,y,z

3/2 x2 i max 1 − ,0 RF2 i i

The proﬁle of the degenerate Fermi gas has a

rather ﬂat top compared to the Gaussian proﬁle of a thermal cloud, as the occupancy of available phase space cells saturates at unity. At ﬁnite T TF , we can understand the shape of the cloud by comparing kB T with the local Fermi energy F (r). For the outer regions in the trap where kB T F (r), the gas shows a classical (Boltzmann) density distribution n(r) ∝ e−βV (r) . In the inner part of the cloud where kB T F (r), the density is of the zero-temperature form n(r) ∝ (EF − V (r))3/2 . The Polylogarithm smoothly interpolates between the two regimes. We notice here the diﬃculty of thermometry for very cold Fermi clouds: Temperature only aﬀects the far wings of the density distribution. While for thermal clouds above TF , the size of the cloud is a direct measure of temperature, for cold Fermi clouds one needs to extract the temperature from the shape of the distribution’s wings. Note that the validity of the above derivation required the Fermi energy EF to be much larger than the level spacing ωx,y,z . For example, in very elongated traps and for low atom numbers one can have a situation where this condition is violated in the tightly conﬁning radial dimensions. . 3 1.2. Trapped, interacting Fermi mixtures at zero temperature. We now consider the case of N fermionic atoms equally populating two hyperﬁne states (“spin up” and “spin down”). Atoms in diﬀerent spin states interact via s-wave collisions characterized by the scattering length a. A dimensionless parameter measuring the strength and sign of the interaction strength is 1/kF a, essentially the ratio of the interparticle spacing to the scattering length. For weak attractive interactions, 1/kF a → −∞, the ground state of the system is a BCS superﬂuid (see sect. 4). As the magnitude of the scattering length increases to a point where a → ∓∞ diverges (thus 1/kF a → 0), a two-body molecular bound state enters the interparticle potential. For weak repulsive interactions,

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W. Ketterle and M. W. Zwierlein

1.2

Strongly Interacting Fermions

Thermal Molecules

Weakly Interacting Fermions

T/TF 0.6

Thermal & Condensed Molecules

BCS Superfluid Resonance Superfluid

0.0 2

1

0 1/kFa

-1

-2

Fig. 14. – Phase diagram of interacting Fermi mixtures in a harmonic trap, as a function of temperature and interaction strength 1/kF a. Shown is the critical temperature TC for the formation of a superﬂuid as a function of 1/kF a (full line) as well as the characteristic temperature T ∗ at which fermion pairs start to form (dashed line), after [168]. The shading indicates that pair formation is a smooth process, not a phase transition.

1/kF a → +∞, the ground state of the system is then a Bose-Einstein condensate of weakly-interacting molecules of mass M = 2m, in which two fermions of opposite spin are tightly bound. Figure 14 summarizes the diﬀerent regimes within this BEC-BCS crossover. We see that the character of the Fermi mixture drastically changes as a function of temperature and interaction strength. For temperatures T T ∗ fermions are unpaired, and a free Fermi mixture exists on the BEC- and the BCS-side of the phase diagram. On resonance, the mixture might still be strongly interacting even at high temperatures, thus possibly requiring an eﬀective mass description of the interacting gas. The density distribution will have the same shape as a free Fermi gas at all interaction strengths. Below T ∗ , fermion pairs start to form. On the BEC-side, where fermions are tightly bound, the thermal distribution should now be that of a gas of bosons with mass M = 2m. As a consequence, the cloud will shrink. Below TC , we will ﬁnally observe a superﬂuid, condensed core, surrounded by a thermal cloud of molecules in the BEC-limit, or of unpaired fermions in the BCS-limit. In general, the calculation of density distributions in the strongly interacting regime is a diﬃcult aﬀair. Simple expressions for the densities can be derived for superﬂuid gases at zero temperature, for molecular gases on the “BEC”-side at large and positive 1/kF a, for weakly interacting Fermi gases on the “BCS”-side for large and negative 1/kF a, and in the classical limit at high temperatures.

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Making, probing and understanding ultracold Fermi gases

BEC limit. The molecular Bose-Einstein condensate is described by a many-body wave function ψ(r) which obeys the Gross-Pitaevskii equation [169] (34)

2 ∇2 ∂ 2 + VM (r) + g |ψ(r, t)| ψ(r, t) = i ψ(r, t) , − 2M ∂t 2

where VM (r) is the trapping potential experienced by the molecules, and g = 4πMaM de2 scribes the intermolecular interactions. We can identify |ψ| with the condensate density nc , which for weak interactions and at zero temperature equals the density of molecules nM . The validity of eq. (34) is limited to weakly interacting gases of molecules, for Fa 3 which the gas parameter nM a3M ≈ ( k6.5 ) 1. In typical experiments on BECs of bosonic atoms, the corresponding condition is very well fulﬁlled. For a sodium BEC with n ≈ 1014 cm−3 and a = 3.3 nm, we have na3 ≈ 4 · 10−6 . However, for molecular condensates near a Feshbach resonance, this condition can be easily violated (see sect. 6).

In equilibrium, the ground-state wave function is ψ(r, t) = e−iμM t/ ψ(r), where μM is the ground state energy and is identiﬁed with the molecular chemical potential, and ψ(r) is a solution of the stationary Gross-Pitaevskii equation (35)

2 2

∇ 2 + VM (r) + g |ψ(r)| ψ(r) = μM ψ(r) . − 2M

In the ideal gas limit, gnc ωx,y,z , we recover the harmonic oscillator result for the condensate’s density distribution nc (r). In the Thomas-Fermi limit, on the other hand, interactions dominate over the kinetic energy of the condensate wave function, gnc ωx,y,z . Already for weakly interacting alkali gases, this condition is very well fulﬁlled, with typical interaction energies of gnc ∼ kB × 150 nK and ωr ≈ kB × 5 nK. In this 2 approximation we obtain the condensate density nc (r) = |ψ(r)| : (36)

nc (r) = max

μM − VM (r) ,0 . g

Thus, a condensate in the Thomas-Fermi approximation “ﬁlls in” the bottom of the trapping potential up to an energy μM , which is determined by the total number of molecules, NM = N/2 = d3 r nc (r). Taking VM (r) = 2V (r) with the harmonic trapping potential for single atoms in eq. (18), one obtains a parabolic density proﬁle, (37)

x2 15 NM i nc (r) = max 1 − ,0 , 8π Rx Ry Rz Ri2 i

where the Thomas-Fermi radii Ri =

2μM M ωi2

give the half-lengths of the trapped conden-

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W. Ketterle and M. W. Zwierlein

sate where the density vanishes. The chemical potential is given by (38)

μM

1 ω = ¯ 2

15NM aM d¯h.o.

2/5 ,

¯ is the geometric mean of the harmonic oscillator where d¯h.o. = (dx dy dz )1/3 = /M ω lengths for molecules. Interactions thus have a major eﬀect on the shape of the Bose-Einstein condensate, changing the density proﬁle from the Gaussian harmonic-oscillator ground-state wave function to a broad parabola, as a result of the interparticle repulsion. The characteristic size of the condensate is no longer given by the harmonic-oscillator length but by the 1/5 15NM aM ω ¯ generally much larger Thomas-Fermi radius Rx,y,z = dx,y,z ωx,y,z . Also d¯h.o. ωy ωy Rx the aspect ratio changes, for example in the x-y plane from ddxy = ωx to Ry = ωx . Nevertheless, weakly interacting condensates are still considerably smaller in size than a thermal cloud at kB T > μM , and more dense. This leads to the clear separation between the dense condensate in the center of the cloud and the large surrounding thermal cloud, the “smoking gun” for Bose-Einstein condensation (both in the trapped and in . the expanding cloud, see subsect. 3 2 below). In the case of strong interactions, when the chemical potential μM becomes comparable to kB TC , this direct signature of condensation will be considerably weaker. In this regime we also have to account for the mutual . repulsion between the thermal cloud and the condensate (see subsect. 3 3.3 below). BCS limit. In the weakly interacting BCS limit (1/kF a → −∞), pairing of fermions and superﬂuidity have very small eﬀects on the density proﬁle of the gas. The sharp Fermi surface in k-space at kF is modiﬁed only in an exponentially narrow region of width ∼ kF exp(− 2kFπ|a| ). The density, i.e. the integral over occupied k-states, is thus essentially identical to that of a non-interacting Fermi gas. The result is eq. (33) with the number of spin-up (spin-down) atoms N↑,↓ = N/2 and Fermi energy EF = ¯ ω (6N↑,↓ )1/3 = ¯ ω (3N )1/3 . As one approaches the strongly interacting regime 1/kF a ≈ −1, it is conceivable that the formation of the superﬂuid leaves a distinct trace in the density proﬁle of the gas, as this is the situation in the BEC-limit, and the crossover between the two regimes is smooth. Indeed, several theoretical studies have predicted kinks in the density proﬁles signalling the onset of superﬂuidity [170, 168, 45, 171]. We were able to observe such a direct signature of condensation on resonance (1/kF a = 0) . and on the BCS-side (1/kF a < 0) in unequal Fermi mixtures (see subsect. 7 3.2). In equal mixtures, we detected a faint but distinct deviation from the Thomas-Fermi proﬁle . on resonance (see subsect. 6 5.1). Unitarity. The regime on resonance (1/kF a = 0) deserves special attention. The scattering length diverges and leaves the interparticle distance n−1/3 ∼ 1/kF as the only relevant length scale. Correspondingly, the only relevant energy scale is the Fermi en-

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Table IV. – Zero-temperature density proﬁles of a trapped, interacting Fermi mixture in the BEC-BCS crossover. The density is zero when the expressions are not positive. For deﬁnitions see the text. BEC-limit ∞

1 kF a

γ (in μ ∝ nγ ) n↑ (r)/n↑ (0) n↑ (0) Radii

Unitarity 0

1 P

x2 i i R2 i N↑ 15 8π Rx Ry Rz q 2μM Ri = M ωi2

1−

(1 −

2/3 P x2i

BCS-limit −∞ (1 −

)3/2

2/3 P x2i

3/2 i R2 ) Fi N↑ 8 π 2 RF x R F y RF z q 2EF RF i = mω 2 i

i R2 Ui N↑ 8 π 2 RU x RU y RU z RU i = ξ 1/4 RF i

ergy F = 2 kF2 /2m. The regime is thus said to be universal. The chemical potential μ can then be written as a universal constant times the Fermi energy: μ = ξ F . In the trapped case, we can use this relation locally (local density approximation) and relate the local chemical potential μ(r) = μ − V (r) to the local Fermi energy 2 F (r) ≡ 2 kF (r)2 /2m ≡ 2m (6π 2 n↑ (r))2/3 , where n↑ (r) is the density of atoms in one spin state. We then directly obtain a relation for the density proﬁle n↑U (r) of the unitary Fermi gas: 1 n↑U (r) = 6π 2

2m ξ2

3/2 (μ − V (r))

3/2

.

The constraint from the number of particles in spin up, N↑ = N/2, determines μ: N↑ = (39)

⇒μ=

1 d r n↑U (r) = 6 3

√

μ ξ¯ ω

3

ξEF .

The density proﬁle becomes

(40)

N↑ 8 n↑U (r) = 2 π RU x RU y RU z

3/2 x2 i max 1 − 2 ,0 RU i i

with the radii RU x,y,z = ξ 1/4 RF x,y,z . Table IV summarizes the various density proﬁles of interacting Fermi mixtures. Remarkably, the functional form n↑U (r) ∝ (μ − V (r))3/2 is identical to that of a non-interacting Fermi gas. The underlying reason is that the equation of state μ ∝ n2/3 has the same power law form as for non-interacting fermions. The universal constant ξ simply rescales the radii (by a factor ξ 1/4 ) and the central density (by a factor ξ −3/4 ). One thus has direct experimental access to the universal constant ξ by measuring the . size of the cloud at unitarity (see subsect. 7 1.1).

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. 3 2. Expansion of strongly interacting Fermi mixtures. – Intriguingly rich physics can be uncovered by the simple release of ultracold gases from their conﬁning trap. From the size of the expanded cloud and the known time-of-ﬂight one directly obtains the energy content of the gas: the temperature in the case of thermal clouds, the Fermi energy for non-interacting degenerate Fermi gases, the mean-ﬁeld energy for Bose-Einstein condensates. In the case of free ballistic expansion, where no collisions occur during expansion, the density distribution of the expanded cloud directly reveals the original momentum distribution of the particles in the trap. Thermal clouds will become spherical after ballistic expansion, reﬂecting their isotropic momentum distribution in the trap. The expansion of Bose-Einstein condensates is not ballistic but mean-ﬁeld driven, leading to superﬂuid hydrodynamic expansion. As mean-ﬁeld energy is preferentially released in the direction(s) of tight conﬁnement, this allows for the famous “smoking gun” signature of Bose-Einstein condensation: inversion of the condensate’s aspect ratio after expansion out of an anisotropic trap. In strongly interacting gases the normal, uncondensed cloud can be collisionally dense, and will expand according to classical hydrodynamics. As particles will preferentially leave the cloud along the narrower dimensions, where they undergo fewer collisions, this also leads to an inversion of the cloud’s initial aspect ratio. It is thus no longer a “smoking gun” for condensation, but merely for strong interactions. Expansion is also useful to measure correlations in momentum space [172]. Finally, in the case of harmonic trapping, expansion of a superﬂuid cloud can often be described as a “magnifying glass”, a mere scaling of the density distribution in the trap. This allows for example to observe quantized vortices [68], which are too small to be observable in the trap. In this section, we show how quantitative information can be derived from images of expanding clouds. . 3 2.1. Free ballistic expansion. Let us consider the expansion of a non-condensed thermal cloud. If the mean free path λc between collisions is longer than the size of the trapped cloud R, we can neglect collisions during expansion, which is hence ballistic. The collision rate is Γ = nσv, with density n, collisional cross-section σ and thermal (root mean square) velocity v, which gives λc = v/Γ = 1/nσ. As R = v/ω for a harmonic trap, the condition λc R is equivalent to having Γ ω, that is, the mean time interval between collisions should be larger than a period of oscillation in the trap. This condition can be fulﬁlled for the cloud of uncondensed molecules in the BEC limit where 1/kF a 1 and collisions are negligible (this has been the case also for atomic BECs with the exception of very large thermal clouds, see [173, 174]), and for the cloud of unpaired fermions in the BEC- and in the BCS-limit for kF |a| 1 (the exact criterion is still Γ ω). For molecules with mass M , we need to replace m → M in the following discussion. In the ballistic case, a particle initially at point r0 in the trap, will reach point r = r0 + pm0 t after expansion time t. We obtain the density at point r at time t by adding the contributions from particles at all points r0 that had the correct initial momentum

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p0 = m(r − r0 )/t. In terms of the semi-classical distribution f (r, p), eq. (19), this is

= (41)

p0 d3 p0 t f (r0 , p0 ) δ r − r0 − 3 (2π) m

d3 r0

n(r, t) =

=

d3 p0 p0 t, p0 f r− 3 (2π) m p0 p20 + βV r − t − βμ ∓ 1 exp β 2m m

d3 p0 (2π)3

−1

.

The integral can be carried out analytically in the case of a harmonic potential (eq. (18)): n(r, t) =

=±

d3 p0 (2π)3

exp β

i

1+

i

1 " λ3dB

1 1 + ωi2 t2

ωi2 t2

!−1 1

p20i ωi2 x2i +β m − βμ ∓ 1 2m 2 1 + ωi2 t2 i

Li3/2 ± exp βμ − β

1 i

ω 2 x2 m i 2i 2 2 1 + ωi t

.

Note that this has the same form as the density distribution in the trap, but with spatial dimension i = x, y, z rescaled by the factor bi (t) = 1 + ωi2 t2 . Ballistic expansion of a thermal (bosonic or fermionic) cloud from a harmonic trap is thus a scaling transformation:

(42)

n(r, t) =

1 n V(t)

x y z , , ,t = 0 , bx (t) by (t) bz (t)

where the unit volume scales as V(t) = bx by bz . After an expansion time long compared to the trapping periods (t 1/ωi ), we have

(43)

n(r, t 1/ωi ) = ±

1 λ3dB

1 1 r2 Li3/2 ± exp βμ − β m 2 . (¯ ω t)3 2 t

As expected, we obtain an isotropic density proﬁle, reﬂecting the original isotropic momentum distribution of the trapped gas. Importantly, the shape of the density proﬁle, i.e. its variation with r, becomes insensitive to the trapping potential. Equation (43) thus holds for a general trapping geometry, for expansion times long compared to the longest trapping period. Even if the trapping potential is not known in detail, one can still determine the cloud’s temperature and even decide whether the gas is degenerate. Note that the momentum distribution at point r after long expansion times t 1/ωi has

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W. Ketterle and M. W. Zwierlein

become anisotropic:

p0 t d3 r0 f (r0 , p0 ) δ r − r0 − m p0 = f r− t, p0 m −1

xi 2 2 p − m 1 r 0i t + m 2 −μ = exp β ωi2 t2 ∓1 . 2m 2 t i

f (r, p0 , t) =

(44)

t 1/ωi

→

¯ = m rt , and with charThe momentum distribution at point r is ellipsoidal, centered at p Δxi 1 acteristic widths Δpi ∝ m t ∝ ωi directly mirroring the ellipsoidal atomic distribution in the trap. Ballistic expansion into a saddle potential. In many experiments, atoms are released from an optical trap, but magnetic ﬁelds (Feshbach ﬁelds) are still left on. In general, these magnetic ﬁelds are inhomogeneous, either due to technical limitations, or deliberately, . e.g. in case of the optical-magnetic hybrid trap discussed in subsect. 2 2.2. We focus here on the important case of a magnetic ﬁeld created by pair of coils which generates a saddle point potential. So we assume that at t > 0, the gas is not released into free space, but into a new 2 2 2 potential. We deﬁne V (r, t > 0) = 12 m ωSx x2 + ωSz y 2 + ωSz z 2 , and can describe expansion into anticonﬁning potentials with imaginary frequencies. For example, for the magnetic saddle potentials relevant for the MIT experiments, the radial dimension is anticonﬁning and ωSx,y = i √12 ωSz . In the potential V (r, t > 0), particles with initial position r0 and momentum p0 will reach the point r with xi = cos(ωSi t)x0i + ω1Si sin(ωSi t) pm0i after expansion time t. The calculation of the density proﬁle is fully analogous to the case of free expansion, after the change of variables x0i → x ˜0i / cos(ωSi t) and the substitution t → sin(ωSi t)/ωSi . We again obtain a scaling transformation, eq. (42), but for this ω2

cos2 (ωSi t) + ω2i sin2 (ωSi t). For expansion into Si 2ω 2 the magnetic saddle potential, this gives b⊥ (t) = cosh2 ( √12 ωSz t) + ω2⊥ sinh2 ( √12 ωSz t) Sz ωz2 2 2 and bz (t) = cos (ωSz t) + ω2 sin (ωSz t). For the MIT trap, the initial axial trapping Sz potential is dominated by the magnetic-ﬁeld curvature, while the initial radial potential is almost entirely due to the optical trap. After switching oﬀ the optical trap, we have ωSz = ωz and ωSx = i √12 ωz . In this case, bz (t) = 1 and the cloud expands only into the radial direction. . 3 2.2. Collisionally hydrodynamic expansion. If the mean free path λc is short compared to the cloud size, the gas is in the hydrodynamic regime, and collisions during expansion can no longer be neglected. Collisions will tend to reestablish local thermal equilibrium, in particular an isotropic momentum distribution. For anisotropic traps, this directly leads to anisotropic expansion, in strong contrast to the ballistic case: Particles trying

case with scaling parameters bi (t) =

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151

to escape in one direction suﬀer collisions that redistribute their momenta equally in all directions. The escape is hindered more for the weakly conﬁned directions where the cloud is long initially and particles can undergo more collisions. For cylindrically symmetric clouds, this leads to an inversion of the aspect ratio of the cloud during expansion. Hydrodynamic expansion can take place for 1/kF |a| < 1, which includes (for a > 0) strongly interacting clouds of uncondensed molecules, and (for a < 0) a strongly interacting, normal Fermi mixture. There is no sharp boundary between molecular hydrodynamics and fermionic hydrodynamics, since 1/kF |a| < 1 is the strongly interacting regime where many-body physics dominates and the single-particle description (molecules in one limit, unbound fermions in the other) is no longer valid. In the hydrodynamic regime, the evolution of the gas is governed by the continuity equation for the density n(r, t) and, neglecting friction (viscosity), the Euler equation for the velocity ﬁeld v(r, t): (45) (46)

∂n + ∇ · (nv) = 0 , ∂t ∂v 1 dv =m + m(v · ∇)v = −∇V (r, t) − ∇P (r, t) , m dt ∂t n

where P is the pressure. Friction is negligible deep in the hydrodynamic regime, when the mean free path approaches zero. The Euler equation is simply Newton’s law for the collection of gas particles at r. In steady state, we recover the equilibrium solution (47)

∇P0 (r) = n0 (r)∇μ0 (r) = −n0 (r)∇V (r, 0) ,

where we have used the expression for the local chemical potential μ0 (r) = μ − V (r).

Scaling solution for harmonic potentials. In the case of free expansion, the potential V (r, t) is the initial harmonic trapping potential for t < 0, with radial and axial trapping frequencies ω⊥ (0) and ωz (0), and zero for t > 0. We can more generally consider here an arbitrary time variation ω⊥ (t) and ωz (t) of the trapping frequencies. For this case, the Euler equation allows a simple scaling solution for the coordinates and velocities [175]

(48)

xi (t) = bi (t) x0i , b˙ i vi (t) = xi (t) , bi

with initial conditions bi (0) = 1 and b˙ i (0) = 0. The unit volume scales as V(t) = bx by bz , the density varies as n(r, t) = n0 (r0 )/V, where the ﬂuid element at initial position r0 has propagated to r at time t.

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W. Ketterle and M. W. Zwierlein

Pressure. The thermodynamic properties of a simple ﬂuid or gas only depend on three variables, that are, in the grand canonical description, the temperature T , the chemical potential μ and the volume V . From the grand canonical partition function Z, one obtains in this case the pressure P = kB T lnVZ . For a non-interacting, ideal gas of bosons or fermions, the average energy is E = 32 kB T ln Z, leading to the relation P V = 2 3 E. This equation is no longer true for an interacting gas, for example the van der Waals gas. It is very remarkable, then, that this relation nevertheless holds also for the strongly interacting, unitary gas on resonance, for all temperatures [45, 176] (6 ). Under an adiabatic expansion, the energy E changes according to dE = −P dV . Hence 3 3 5/3 = const for adiabatic 2 d(P V ) = 2 (V dP + P dV ) = −P dV , which leads to the law P V −5/3 expansion. The pressure thus scales as V , and the force, using eq. (47), − (49)

1 ∂ ∂ V 1 ∂ P0 (r0 ) 1 P (r, t) = − = V (r0 , 0) n ∂xi n0 bi ∂x0i V 5/3 bi V 2/3 ∂xi0 1 = mωi2 (0)xi0 . bi V 2/3

The Euler equations then reduce to equations for the scaling parameters bi (t), which can be solved numerically: (50)

2 ¨bi = −ω 2 (t) bi + ωi (0) . i bi V 2/3

In the following section we will see that superﬂuid hydrodynamics leads to very similar scaling equations, with the exponent 2/3 for the volume scaling parameter V replaced by the parameter γ in the equation of state of the superﬂuid μ(n) = nγ . The discussion of free expansion, the long-time behavior, inversion of the aspect ratio etc. will be identical for superﬂuid hydrodynamics, so we defer the topic until the next section.

From ballistic to hydrodynamic expansion. The regime in between ballistic, collisionless expansion and pure hydrodynamic, collisional expansion can be treated approximately. For the eﬀects of interactions on a classical gas, see [177, 178], for the case of Fermi gases with attractive interactions, see [179]. . 3 2.3. Superﬂuid hydrodynamic expansion. In the simplest (scalar) a superﬂuid is case, iφ(r,t) described by a macroscopic, complex order parameter ψ(r, t) = n(r, t)e parameterized by the superﬂuid density n(r, t) and a phase φ(r, t). The dynamics of the order (6 ) On resonance, universality requires that the energy E = N F f (T /TF ) with a universal function f . Entropy can only be a function of T /TF , so adiabaticity requires this ratio to be constant. The pressure is then P = −∂E/∂V |S,N = −N f (T /TF )∂ F /∂V = 23 E/V .

Making, probing and understanding ultracold Fermi gases

153

parameter are well described by a time-dependent Schr¨ odinger equation of the type (51)

i

∂ ψ(r, t) = ∂t

−

2 2 ∇ + V (r, t) + μ(n(r, t)) ψ(r, t) , 2m

where μ(n) is the chemical potential given by the equation of state of the superﬂuid. In the case of weakly interacting BECs, this is the Gross-Pitaevskii equation for the . condensate wave function from subsect. 3 1.2. For fermionic superﬂuids, a formally similar equation is the Ginzburg-Landau equation, which is however valid only close to TC . Rewriting eq. (51) in terms of the superﬂuid density n and velocity v, neglecting √ the curvature ∇2 n of the magnitude of ψ and using the fact that the superﬂuid is irrotational ∇ × r = 0, we arrive again at the continuity equation and the Euler equation for classical inviscous ﬂow: (52) (53)

∂n + ∇ · (nv) = 0 , ∂t ∂v + m(v · ∇)v = −∇ (V + μ(n)) . m ∂t

The validity of these hydrodynamic equations is restricted to superﬂuids whose healing length is much smaller than the sample size and thus, for fermionic superﬂuids in a harmonic trap, for a superﬂuid gap larger than the harmonic-oscillator energies ωx,y,z [179]. For a power law equation of state μ(n) ∝ nγ , the equations allow a scaling solution for (possibly time-varying) harmonic potentials. The scaling parameters bi (t) are given by the diﬀerential equations [180, 175, 179, 181] (54)

2 ¨bi = −ω 2 (t) bi + ωi (0) . i bi V γ

Important limiting cases in the BEC-BCS crossover are: – BEC-limit (1/kF a 1): Here, the mean-ﬁeld repulsion between molecules leads 2 aM n to a chemical potential per fermion μ(n) = π m , so γ = 1. – BCS-limit (1/kF a −1): In the BCS-limit, the dominant contribution to the chemical potential comes from the kinetic energy of the constituent fermions, given by the Fermi energy. So here μ(n) = F ∝ n2/3 and γ = 2/3. – Unitarity limit (1/kF a = 0): In the unitarity limit, the only remaining energy scale is the Fermi energy. One necessarily has μ(n) ∝ F ∝ n2/3 and γ = 2/3, just as in the BCS-limit. Note that the scaling laws for the BCS- and the unitarity limit [182] are identical to . those found for a collisionally hydrodynamic gas in subsect. 3 2.2. For a derivation of superﬂuid hydrodynamics in the BCS-limit, we refer the reader to the contribution of Y. Castin to these proceedings.

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W. Ketterle and M. W. Zwierlein

1

J

0.9

0.8

2/3 0.6 2

1

0 1/kFa

-1

-2

Fig. 15. – The exponent γ as a function of the interaction parameter 1/kF a. γ approximately describes the superﬂuid equation of state μ(n) ∼ nγ in the BEC-BCS crossover. A similar ﬁgure can be found in [181].

. The Leggett ansatz (see subsect. 4 4) allows to interpolate between the BEC- and the BCS-regime and gives a chemical potential μ(n) that correctly captures the physics in ∂μ the two limits. With its help, we can deﬁne an eﬀective exponent γ = nμ ∂n and write γ μ(n) n , assuming that γ varies slowly with the interaction parameter 1/kF a. This exponent, shown in ﬁg. 15, attains the correct limiting values in the BEC- and the BCSlimit, as well as on resonance, so we may use it for the present purpose as an approximate description of the gas’ equation of state throughout the crossover.

In-trap density proﬁle. The in-trap density proﬁle of the superﬂuid at zero temperature can be deduced from the Euler equations in steady state. Neglecting kinetic energy 1 2 2 m v (Thomas-Fermi approximation), the equation simply reads V (r) + μ(n(r)) = const. = μ(n(0)). For the power law equation of state μ(n) ∝ nγ , we directly obtain (55)

1/γ

n(r) ∝ (μ(n(0)) − V (r))

for μ(n(0)) > V (r) and zero otherwise. For a BEC and harmonic trapping, we recover the inverted parabola, eq. (37), for a BCS superﬂuid in the limit of weak interactions the density distribution of an ideal Fermi gas, eq. (33). Note that in the crossover 1/kF |a| 1, the correct calculation of the density proﬁle is less straightforward, as the parameter 1/kF (r)a depends on position, and the equation of state varies across the cloud. The power law approach, using a ﬁxed γ = γ(1/kF (0)a), will only provide an approximate description. Fortunately, on resonance evidently 1/kF (r)a = 0 across the entire cloud, and the power law equation of state becomes exact at T = 0.

Making, probing and understanding ultracold Fermi gases

155

Free expansion out of a cylindrically symmetric trap. In this case ωi (t > 0) = 0, and ωx (0) = ωy (0) ≡ ω⊥ . We have (56)

¨b⊥ =

(57)

¨bz =

2 ω⊥

b2γ+1 bγz ⊥ ωz2 2γ γ+1 b⊥ bz

, .

The MIT trap is cigar-shaped, with an aspect ratio of short to long axes = ωz /ω⊥ 1. In such a case, expansion is fast in the radial, initially tightly conﬁned dimensions, whereas it is slow in the z-direction. For times short compared to τ = ω1z , many axial trapping periods, we can set bz ≈ 1 on the right side of eqs. (56) and (57), decoupling the transverse from the axial expansion. For γ = 1, the case of a Bose-Einstein condensate of tightly bound molecules, the simpliﬁed equations for t τ have an analytic solu 2 t2 and b (t) = 1+ 2 ω t arctan(ω t) − ln 2 t2 . tion [180,175]: b⊥ (t) = 1 + ω⊥ 1 + ω⊥ z ⊥ ⊥ For long times t, the expansion is linear in time: b⊥ (t) = ω⊥ t for t 1/ω⊥ and bz (t) = (π/2) 2 ω⊥ t for t τ . Note that the radial expansion accidentally follows the same scaling law as that of a ballistically expanding normal cloud. The general behavior of the expanding gas is the same for all relevant γ. Driven either by repulsive interactions (BEC-case) or by kinetic energy (BCS-case), the gas ﬁrst ¨ ⊥ (t 1/ω⊥ ) = R⊥ (0)ω 2 , and over a radial expands radially at constant acceleration R ⊥ ˙ 1/ω⊥ ) ≈ ω⊥ R⊥ (0). The axial trapping period reaches a ﬁnal expansion velocity R(t size grows as bz (t) − 1 ≈ 2 ω⊥ t, leading to an inversion of the cloud’s aspect ratio from initially to ∼ 1/ . This inversion is in contrast to the isotropic aspect ratio of a ballistically expanding gas, and is thus characteristic for hydrodynamic expansion, which can be of collisional or of superﬂuid origin. Figure 16 and table V summarize the time evolution of the cloud’s radii and aspect ratios for γ = 1 (BEC) and γ = 2/3 (BCS and unitarity), while ﬁg. 17 compares the long-time behavior of the velocities and aspect ratios across the BEC-BCS crossover. For expansion out of an elongated cigar-shaped trap and γ = 2/3, which holds in the BCS-limit, at unitarity, but also for a collisionally hydro-

dynamic gas, the asymptotic expansion velocity is v⊥ = 32 ω⊥ R⊥ (0) ≈ 1.22 ω⊥ R⊥ (0). This can be understood by noting that the cloud’s kinetic energy, initially distributed isotropically, is released only into the radial direction during hydrodynamic expansion, 2 2 so 12 mv⊥ = 32 μ = 34 mω⊥ R⊥ (0)2 . . Hydrodynamic expansion into a saddle potential . As discussed in subsect. 3 2.1, expansion may not occur into free space, but into an inhomogeneous magnetic ﬁeld which is often described by a saddle potential. The Euler equations (54) now read for t > 0 (58)

2 ¨bi = −ω 2 bi + ωi (0) . S,i bi V γ

156

W. Ketterle and M. W. Zwierlein γ=1 4

Aspect Ratio

γ = 2/3 3

2 ballistic

1

0 0

5

10 15 20 25 30

100

Time [1/ ω ⊥]

200

300

400

500

Time [1/ ω ⊥]

Fig. 16. – Aspect ratio (t) = Rx (t)/Rz (t) as a function of time for the MIT trap ( = 1/6) in ballistic, collisional or superﬂuid hydrodynamic expansion (γ = 2/3) and superﬂuid hydrodynamic expansion of a molecular BEC (γ = 1).

Asp. Ratio [2/SH] vx/(ZARA); vz/(H ZARAS

Here, ωS,i are the real or imaginary frequencies characterizing the saddle point potential. These equations typically need to be solved numerically. For a Bose-Einstein condensate of molecules expanding from long cigar-shaped traps ( 1), the radial equation again allows for an analytic solution identical to that for a ballistically expanding, non-

3.0 vz

2.5 2.0 1.5

vx

1.0 1.0 0.8 0.6 0.4 2

1

0 1/kFa

-1

-2

Fig. 17. – Asymptotic velocities and aspect ratio for hydrodynamic expansion out of a very elongated cigar-shaped trap ( = ωz /ωx 1), as a function of the interaction parameter 1/kF a. The dashed lines show the asymptotic values in the BCS-limit. A similar ﬁgure can be found in [181].

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Making, probing and understanding ultracold Fermi gases

Table V. – Comparison between ballistic and hydrodynamic expansion. Formulas for hydrodynamic expansion assume a long cigar-shaped trap ( = ωz /ωx 1), formulas for the aspect ratio (AR) and for the BCS, unitarity, collisional limit give the asymptotic behavior. The formula for BEC-expansion is valid at short times t ω⊥ /ωz2 , but also captures the correct long-time limit.

b⊥ (t) bz (t) AR

Ballistic p 2 2 t √1 + ω⊥ 1 + ωz2 t2

Hydrodynamic (BEC) p 2 2 1 + ω⊥ t 2 1 + (ω⊥ t arctan(ω ⊥ t) p 2 2 t ) − ln 1 + ω⊥

BCS, unitarity, collisional

2 1 π

1

∼ 1.22 ω⊥ t ∼ 2.05 π2 2 ω⊥ t ∼ 0.60

2 1 π

interacting gas. One obtains # (59)

b⊥ (t) =

cosh2

2ω 2 1 1 √ ωSz t + 2⊥ sinh2 √ ωSz t . ωSz 2 2

However, the axial cloud size behaves drastically diﬀerent from a non-interacting cloud. For ωSz = ωz , the axial cloud size of a non-interacting gas would never change (bz (t) = 1), whereas a hydrodynamic gas, released into the radial dimensions, will start to shrink axially under the inﬂuence of the conﬁning axial potential. The cloud’s energy (interaction energy for a BEC, kinetic energy for a BCS superﬂuid) escapes radially, hence there is not suﬃcient pressure to maintain the axial cloud size. Further discussions of superﬂuid hydrodynamics and scaling transformations can be found in the contributions of Y. Castin and S. Stringari to these proceedings. . 3 3. Fitting functions for trapped and expanded Fermi gases. – In the preceding sections we derived the 3D density distribution of a Fermi mixture in various regimes. However, all imaging techniques record column densities, density proﬁles integrated along the line of sight (the z-axis in the following). For condensed gases, where n(r) ≈ n(0) (1 − V (r)/μ)1/γ , one obtains the column density (60)

1+1 y2 γ 2 x2 n2D,c (x, y) = nc 1 − 2 − 2 . Rx Ry

For thermal Bose (molecular) and Fermi clouds, we have (61)

n2D (x, y) = n2D,0 Li2

1 2 2 2 2 ± exp βμ − β m(ωx x + ωy y ) / Li2 ± eβμ . 2

In the following, we will discuss the ﬁtting functions valid in the diﬀerent regimes of interaction, and the derived quantities.

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. 3 3.1. Non-interacting Fermi gases. Cloud size. In the classical regime at T /TF 1, the characteristic cloud size is given 2kB T by the Gaussian radius σi = . In the degenerate regime, however, the cloud mωi2 2EF size saturates at the Fermi radius RF i = . It is thus convenient to deﬁne a ﬁt mω 2 i

parameter that interpolates between the two limits: (62)

Ri2 =

2kB T f (eμβ ) → mωi2

σi , RF i ,

T /TF 1 , T /TF 1 ,

where f (x) =

1+x Li1 (−x) = ln(1 + x) . Li0 (−x) x

For all temperatures, Ri is thus directly related to the physical size of the cloud, and 2μ thus a better choice as a ﬁt parameter than σi , which goes to zero at T = 0, or mω 2, i

which goes to zero around T /TF = 0.57. Numerically, using Ri is easier to implement than using the root mean square radius of the cloud (63)

$ 2 % kB T Li4 (−eμβ ) . xi = mωi2 Li3 (−eμβ )

Fitting function. The ﬁt function used for the density proﬁles of Fermi clouds is then in 2D 2 y2 x f (eq ) Li2 ± exp q − R 2 + R2 x y (64) n2D (x, y) = n2D,0 Li2 (±eq ) and for 1D

(65)

n1D (x) = n1D,0

Li5/2 ± exp q −

x2 2 Rx

Li5/2 (±eq )

f (eq ) .

The parameter q = μβ, the logarithm of the fugacity, determines the shape of the cloud. For a small fugacity (large and negative q), the above functions reduce to the simple Gaussian distribution of thermal clouds. For high fugacity (large and positive q), they 2 2 tend to the zero-temperature distribution n2D,0 (1 − Rx2 )2 (in 2D) and n1D,0 (1 − Rx2 )5/2 Fx Fx (in 1D). Derived quantities. Degeneracy The degeneracy parameter T /TF can be calculated by combining eq. (23) with eq. (32): (66)

T −1/3 = [−6 Li3 (−eq )] . TF

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159

This parameter depends only on the shape of the cloud. A characteristic point where shape deviations due to quantum statistics start to play a role is the point where μ changes sign, and we see from eq. (66) that this occurs at T /TF ≈ 0.57. Many non-ideal aspects of imaging, such as ﬁnite resolution, out of focus imaging, saturation, heating of the cloud by the probe pulse etc., tend to wash out the non-Gaussian features of a highly degenerate Fermi cloud and hence lead to a larger value of T /TF . However, dispersive eﬀects due to non-resonant imaging light can potentially mimic sharp edges of the cloud, which the ﬁtting routine would then falsely interpret to result from a very low T /TF . It is clear that care has to be taken when determining the degeneracy parameter from the shape of the cloud alone. Temperature. The size of the cloud and the shape parameter q give the temperature as (67)

kB T =

R2 1 1 mωi2 i 2 , 2 bi (t) f (eq )

. where we have used the expansion factor bi (t) from subsect. 3 2. We recall that bi (t) = 2 2 1 + ωi t for the free expansion of a non-interacting Fermi gas. For low temperatures T TF , f (eμβ ) → μβ = μ/kB T and Ri = bi (t) RF i . In this case, temperature only aﬀects the wings of the density distribution, where the local T /TF (r) is still large. In fact, ⎧ 2 ⎨ (1 − x2 )5/2 for x RF x , RF x 2 (68) n1D (x) ∝ ⎩ e− σxx2 for x RF x , and we see that temperature only aﬀects the cloud’s wings beyond the zero-temperature Fermi radius. Thermometry of very low temperature Fermi clouds is thus diﬃcult, limited by the signal-to-noise ratio in the low-density wings of the distribution. This is diﬀerent from thermometry of thermal clouds at high temperature T TF , where the entire size of the cloud σi directly gives the temperature. Because of the sensitivity to the cloud’s wings, thermometry is more robust when the full 2D distribution is used for the ﬁt. Alternatively, one can rely on the known trapping geometry plus the local density approximation and perform an average over the elliptical equipotential lines in the x-y plane (line of sight integration necessarily mixes points at diﬀerent values of the potential energy.) As the number of points included in the average grows with the distance from the cloud’s center, the signal-to-noise will actually be best in the wings. Such an average is superior to a simple integration along the x-axis, for example, as this will more strongly mix regions that have diﬀerent local T /TF . The ideal gases (Fermi, Bose, Boltzmann) are the only systems for which we have an exact description. Therefore, they are attractive as a thermometer, when brought in contact with strongly interacting systems. This concept has been recently carried out by determining absolute temperatures for imbalanced Fermi gases at unitarity [82].

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In these systems, for suﬃciently high imbalance, the majority cloud extends beyond the minority cloud, and is (locally) an ideal gas. Therefore, in ref. [82] the spatial wings of these clouds could be ﬁtted with the functions for the ideal Fermi gas discussed in this section, and absolute temperatures for the superﬂuid phase transition could be determined. The ﬁtting of the majority wings had to be done with in-trap proﬁles, which required to address the eﬀect of anharmonicities of the optical trapping potential. Usually, for thermometry, ballistic expansion is preferable since velocity distributions are independent of the shape of the trapping potential. However, in the case of imbalanced Fermi gases, the atoms in the wings can collide with the strongly interacting core during expansion, modifying their velocity distribution. Another way to perform ideal gas thermometry is done by converting the sample to a non-interacting system by sweeping suﬃciently far away from the Feshbach resonance. If such magnetic ﬁeld sweeps are adiabatic, they conserve entropy (but not temperature). By ﬁtting the spatial proﬁles of the non-interacting cloud, the entropy S of the strongly interacting system can be determined. If it is possible to determine the energy of the strongly interacting system in a precise way (e.g. by using the virial theorem at unitarity [45, 176]) or to vary the energy by providing controlled heating [72], one can determine the derivative dS/dE which is equal to the inverse absolute temperature. So far this method could be implemented only for a balanced Fermi system at unitarity [183] and, due to the need of determining a derivative, could only provide temperatures averaged over a range of energies.

Number of atoms and Fermi energy. The number of atoms in the observed spin state can be obtained from the total absorption recorded in the cloud’s CCDR image. The transmission of resonant light at pixel (x, y) is given by T˜(x, y) = e−σ0 n3D (x,y,z) dz , where σ0 is the resonant atom-photon cross-section for light absorption. Thus, the number of atoms is (69)

N↑ =

A − ln(T˜(x, y)) , M σ0 pixels

where A is the area per pixel and M the optical magniﬁcation.

Typically, the ﬁtting functions are applied to the optical density σ n (x, y, z) = 0 z 3D ˜ − ln T (x, y) . The ﬁt parameter n2D,0 thus measures the peak optical density of the ˜ y have units of camera pixels. The number of atoms ˜ x and R cloud, while the radii R described by the ﬁtting function is thus given by q q A ˜xR ˜ y Li3 (−e ) Li0 (−e ) π n2D,0 R q M σ0 Li2 (−e ) Li1 − eq ) π ˜ ˜ A 3 n2D,0 RF x RF y , T TF , → π n2D,0 σ ˜x σ ˜y , T TF . σ0

Nﬁt = (70)

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From the number of atoms and the trapping frequencies, one can calculate the Fermi energy kB TF : kB TF = ¯ ω (6Nﬁt )1/3 .

(71)

An independent determination of the Fermi energy is provided by the measured (physical) size of the cloud Ri for highly degenerate clouds. For T → 0, Ri ≈ bi (t)RF i and R2 thus kB TF = 21 mωi2 bi (t)i 2 . As only the trapping frequencies and the magniﬁcation of the imaging system enter into this equation, this relation allows a calibration of the light absorption cross-section which may be reduced from the resonant cross-section by detuning, non-ideal polarization of the probe light, and saturation. For arbitrary temperature, the shape parameter q enters the relation for the Fermi energy: 1/3

(72)

kB TF = kB T

1 R2 (−6 Li3 (−eq )) TF = mωi2 i 2 T 2 bi (t) f (eq )

.

. 3 3.2. Resonantly interacting Fermi gases. The calculation of density proﬁles of interacting gases is delicate. Already above the superﬂuid transition temperature, attractive interactions lead to a shrinking of the cloud. Since interactions (parameterized by the local kF a) vary across the cloud, there is a priori no simple analytical function describing interacting Fermi gases. Experimentally, it turns out that the diﬀerence in the shape of a (balanced) interacting and a non-interacting Fermi mixture is minute around resonance and on the BCS-side. Especially for the resonant case (1/kF a = 0), this has led to the common practice of using the shape of the non-interacting Fermi gas as ﬁtting function, ˜ and quote an eﬀective temperature T˜ and eﬀective degeneracy TTF of resonantly interacting clouds [60,72]. In fact, universality on resonance implies that the gas’ chemical potential must be μ(r) = ξ(T /TF ) F (r), with a universal function ξ(T /TF ) which only depends on the reduced temperature T /TF [45]. The zero-temperature limit of ξ ≡ ξ0 has been . subject of extensive experimental and theoretical studies (see subsect. 7 1.1), and its value is ξ(0) ≈ 0.42. At T = 0, we have for a trapped gas μ(r) = μ0 −V (r) = ξ0 F (r) ∝ n2/3 (r). The density proﬁle will then have the exact same shape as a non-interacting Fermi gas, with a renormalized Fermi temperature. However, for ﬁnite temperature, ξ(T /TF ) differs from the temperature dependence of a non-interacting gas [184], and there is no a priori reason that the shape of the cloud at unitarity should be similar to that of a non-interacting Fermi gas. It turns out that the diﬀerence is very small. The shape similarity was an important issue in the quest for superﬂuidity in Fermi gases. In the case of weakly interacting BECs, condensation is apparent from the sudden appearance of a dense, central core in midst of a large thermal cloud. In contrast to that, Fermi gases do not show such a signature, at least at ﬁrst sight (see ﬁg. 18), and diﬀerent detection methods for superﬂuidity were explored. The only loophole that may allow seeing a signature of superﬂuidity in the spatial proﬁle of balanced Fermi gases would be a rapid variation of ξ(T /TF ) around the critical

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W. Ketterle and M. W. Zwierlein

-1.0

-0.5

0.0 0.5 Radius [mm]

1.0 -1.0

-0.5

0.0 0.5 Radius [mm]

1.0 -1.0

-0.5

0.0 0.5 Radius [mm]

1.0

Fig. 18. – The absence of a signature of condensation in the spatial proﬁle of strongly interacting Fermi gases. Shown are high-resolution images of spin-up atoms in a resonantly interacting, equal mixture of spin-up and spin-down for diﬀerent temperatures. The lower graphs show azimuthally averaged radial proﬁles (noise level well below 1% of the maximum optical density). All three clouds are very well ﬁt using a ﬁnite-temperature Thomas-Fermi distribution (with e fugacity eμ/kB T , central density n0 and mean square radius r2 as free parameters, see eq. (61)). However, the empirical temperatures of Te/TF = 0.22 (a), 0.13 (b) and 0.075 (c) determined from the proﬁles’ wings indicate that at least clouds b and c should be in the superﬂuid regime. Trap parameters νr = 162 Hz, νz = 22.8 Hz, 10 ms time of ﬂight, expansion factor 13.9, atom numbers N per spin state were 10.2 (a), 9.5 (b) and 7.5 (c) ×106 .

temperature TC . This would translate into a sudden variation of the density at the interface between the normal and superﬂuid region, e.g. where the gas is locally critical, T = TC (r). We have indeed found a faint signature of condensation in density proﬁles of the uni. tary gas on resonance after expansion. These results will be presented in subsect. 6 5.1. Note that the observation of such a feature in the density proﬁles draws into question the common practice of determining an “eﬀective temperature” from density proﬁles at unitarity using the ideal gas ﬁtting function. In contrast to balanced Fermi mixtures, a striking signature of condensation can be observed in the density proﬁles of mixtures with imbalanced populations of spin up and spin down fermions. This will be discussed . in subsect. 6 5.2.

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. 3 3.3. Molecular clouds. In partially condensed molecular gases that are weakly interacting, one can neglect the mutual repulsion between the condensate and the surrounding thermal cloud of molecules. The density distribution is typically well-ﬁt by a bimodal sum of an inverted parabola for the condensate

x2 y2 nc (x, y) = nc0 1 − 2 − 2 Rcx Rcy

(73)

and a Bose-function for the thermal cloud, as in eq. (61), with the parameter q = μβ often left as an adjustable parameter (instead of ﬁxing it via the condensate’s chemical potential μ = gnc0 ): (74)

nth (x, y) = nth0 Li2

y2 x2 exp q − 2 − 2 Rth,x Rth,y

) Li2 (exp [q]) .

For practical purposes, this is often simpliﬁed by a Gaussian, as if q 0. Then nth ≈ y2 x2 . nth0 exp − R2 − R2 th,x

th,y

2

Once the condensate mean-ﬁeld 2gnc (with g = 4πMaM ) experienced by thermal molecules is no longer small compared to kB T , the mutual repulsion can no longer be neglected. The thermal molecules will then experience a “Mexican-hat” potential, the sum of the conﬁning harmonic potential VM (r), and the repulsion from the condensate 2gnc (r) and from the surrounding thermal cloud, 2gnth (r). The thermal molecules themselves will in turn repel the condensate. The situation can be captured by two coupled equations for the condensate (in Thomas-Fermi approximation) and the thermal cloud: (75)

gnc (r) = Max (μ − VM (r) − 2gnth (r), 0) , 1 nth (r) = 3 Li3/2 (eβ(μ−VM (r)−2gnc (r)−2gnth (r)) ) λM 1 = 3 Li3/2 (e−β|μ−VM (r)−2gnth (r)| ) , λM

where λM is the thermal de Broglie wavelength for molecules. In the case of weakly interacting Bose gases, one can neglect the mean-ﬁeld term 2gnth (r) [185]. Note that these coupled equations are only an approximative way to describe the strongly correlated gas. The mean-ﬁeld approximation for the thermal molecules neglects phonons and other collective excitations. The above equations can be solved numerically. In the limit of strong interactions, the condensate almost fully expels the thermal molecules from the trap center, so that the thermal cloud forms a shell around the condensate. The practical implication of this discussion is that there is no simple analytic expression for the density distribution of partially condensed clouds in the strongly interacting regime. For ﬁtting purposes one may still choose for example the bimodal ﬁt of eq. (74),

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but one must be aware that quantities like the “condensate fraction” thus obtained depend on the model assumed in the ﬁt. For tests of many-body calculations, the full density distributions should be compared to those predicted by theory. Derived quantities. Temperature. For weakly interacting Bose gases, eq. (74) holds and the temperature is given by (76)

kB T =

2 Rth,i 1 mωi2 , 2 bi (t)2

where bi (t) = 1 + ωi2 t2 is the expansion factor of the thermal gas. To ensure modelindependent results, only the thermal gas should be included in the ﬁt, not the condensed core. For strongly interacting clouds, temperature can in principle still be obtained from the thermal wings of the trapped molecular distribution, which is Gaussian at distances r for which VM (r) μ. However, a possible systematic correction can occur in expansion due to interactions of molecules in the wings with the core, which may be either condensed or strongly interacting. Note that unless the whole cloud is deep in the hydrodynamic regime, there is no simple scaling law for the expansion of such strongly interacting molecular gases. Absolute thermometry of strongly interacting, balanced gases is still a challenging problem. Chemical potential In a conﬁning potential, and at zero temperature, the chemical potential is given by the size of the condensate, as V (r) = μ. It can be expressed by the ﬁt parameters according to eq. (73) as (77)

μ=

2 Rc,i 1 mωi2 2 bi (t)2

with bi (t) the expansion factor for superﬂuid hydrodynamic expansion into direction i. At ﬁnite temperatures and for strong interactions, the thermal cloud will mostly reside outside the condensate and can aﬀect the actual or ﬁtted condensate size.

Condensate fraction In the ﬁeld of dilute atomic gases, the condensate fraction is a key quantity to characterize the superﬂuid regime. In contrast to superﬂuid helium and superconductors, gaseous condensates can be directly observed in a dramatic way. However, unless interactions are negligible, the determination of the condensate fraction is model dependent. For weakly interacting gases (or those obtained after a rapid ramp into the weakly interacting regime), the density distribution can typically be well ﬁt with the bimodal distribution of eqs. (73) and (74). A robust way to deﬁne a “condensate fraction” is then to ascribe the total number of molecules in the narrower distribution to the condensate. For strong interactions however, the mean-ﬁeld repulsion of thermal and condensed

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molecules (see above) will lead to the expulsion of a large part of thermal molecules from the condensate. In addition, low-energy excitations such as phonons, as well as quantum depletion will modify the non-condensed fraction at the position of the condensate, and the ﬁtted condensate fraction depends on the form of the ﬁtting function for the bimodal ﬁt. In these cases, it is better to directly compare density distributions with theoretical predictions.

4. – Theory of the BEC-BCS crossover This section summarizes the concepts behind and the predictions of the “standard theory” of the BEC-BCS crossover. This provides a consistent and coherent reference for the interpretation of experimental results. . 4 1. Elastic collisions. – Due to their diluteness, most properties of systems of ultracold atoms are related to two-body collisions. If we neglect the weak magnetic dipole interaction between the spins, the interatomic interaction is described by a central potential V (r). At large distances from each other, atoms interact with the van der Waalspotential −C6 /r6 as they experience each other’s ﬂuctuating electric dipole(7 ). At short distances on the order of a few Bohr radii a0 , the two electron clouds strongly repel each other, leading to “hard-core” repulsion. If the spins of the two valence electrons (we are considering alkali atoms) are in a triplet conﬁguration, there is an additional repulsion due to Pauli’s exclusion principle. Hence, the triplet potential VT (r) is shallower than the singlet one VS (r). The exact inclusion of the interatomic potential in the description of the gas would be extremely complicated. However, the gases we are dealing with are ultracold and ultradilute, which implies that both the de Broglie wavelength λdB and the interparticle distance n−1/3 ∼ 5 000 − 10 000 a0 are much larger than the

the in range of teratomic potential r0 (on the order of the van der Waals length r0 ∼ μC6 /2 ∼ 50 a0 for 6 Li). As a result, scattering processes never explore the ﬁne details of the shortrange scattering potential. The entire collision process can thus be described by a single quantity, the scattering length. Since the description of Feshbach resonances and of the BCS-BEC crossover require the concept of the eﬀective range and renormalization of the scattering length, we quickly summarize some important results of scattering theory. The Schr¨ odinger equation for the reduced one-particle problem in the center-of-mass frame of the colliding atoms (with reduced mass m/2, distance vector r, and initial relative wave vector k) is (78)

(∇2 + k 2 )Ψk (r) = v(r)Ψk (r)

with k 2 =

mE 2

and v(r) =

mV (r) 2

(7 ) For distances on the order of or larger than the characteristic wavelength of radiation of the atom, λ r0 , retardation eﬀects change the potential to a −1/r7 law.

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Far away from the scattering potential, the wave function Ψk (r) is given by the sum of the incident plane wave eik·r and an outgoing scattered wave: (79)

Ψk (r) ∼ eik·r + f (k , k)

eikr r

f (k , k) is the scattering amplitude for scattering an incident plane wave with wave vector k into the direction k = k r/r (energy conservation implies k = k). Since we assume a central potential, the scattered wave must be axially symmetric with respect to the incident wave vector k, and we can perform the usual expansion into partial waves with angular momentum l [186]. For ultracold collisions, we are interested in describing the scattering process at low momenta k 1/r0 , where r0 is the range of the interatomic potential. In the absence of resonance phenomena for l = 0, s-wave scattering l = 0 is dominant over all other partial waves (if allowed by the Pauli principle): (80)

f ≈ fs =

1 2iδs 1 (e , − 1) = 2ik k cot δs − ik

where fs and δs are the s-wave scattering amplitude and phase shift, respectively, [186]. Time-reversal symmetry implies that k cot δs is an even function of k. For low momenta k 1/r0 , we may expand it to order k 2 : (81)

k2 1 , k cot δs ≈ − + reﬀ a 2

which deﬁnes the scattering length (82)

a = − lim

k1/r0

tan δs , k

and the eﬀective range reﬀ of the scattering potential. For example, for a spherical well 3 potential of depth V ≡ 2 K 2 /m and radius R, reﬀ = R − K12 a − 13 R a2 , which deviates from the potential range R only for |a| R or very shallow wells. For van der Waals potentials, reﬀ is of order r0 [187]. With the help of a and reﬀ , f is written as [186] (83)

f (k) =

1 − a1

2

+ reﬀ k2 − ik

.

In the limit k|a| 1 and |reﬀ | 1/k, f becomes independent of momentum and equals −a. For k|a| 1 and reﬀ 1/k, the scattering amplitude is f = ki and the crosssection for atom-atom collisions is σ = 4π k2 . This is the so-called unitarity limit. Such a divergence of a occurs whenever a new bound state is supported by the potential (see . subsect. 5 2).

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. 4 2. Pseudo-potentials. – If the de Broglie wavelength 2π k of the colliding particles is much larger than the ﬁne details of the interatomic potential, 1/k r0 , we can create a simpler description by modifying the potential in such a way that it is much easier to manipulate in the calculations, but still reproduces the correct s-wave scattering. An obvious candidate for such a “pseudo-potential” is a delta-potential δ(r).

However, there is a subtlety involved which we will address in the following. The goal is to ﬁnd an expression for the scattering amplitude f in terms of the potential 2 v(r) V (r) = m , so that we can try out diﬀerent pseudo-potentials, always ensuring that f → −a in the s-wave limit. For this, let us go back to the Schr¨ odinger equation eq. (78). If we knew the solution to the following equation: (∇2 + k 2 )Gk (r) = δ(r) ,

(84)

we could write an integral equation for the wave function Ψk (r) as follows: (85)

Ψk (r) = eik·r +

d3 r Gk (r − r )v(r )Ψk (r ) .

This can be simply checked by inserting this implicit solution for Ψk into eq. (78). G can be easily obtained from the Fourier transform of eq. (84), deﬁning Gk (p) = k (r) d3 re−ip·r Gk (r): (−p2 + k 2 )Gk (p) = 1 .

(86) The solution for Gk (r) is

(87)

Gk,+ (r) =

1 eikr eip·r d3 p = − , (2π)3 k 2 − p2 + iη 4π r

where we have chosen (by adding the inﬁnitesimal constant iη, with η > 0 in the denominator) the solution that corresponds to an outgoing spherical wave. Gk,+ (r) is the Green’s function of the scattering problem. Far away from the origin, |r − r | ∼ r − r · u, with the unit vector u = r/r, and (88)

Ψk (r) ∼ e

ik·r

eikr − 4πr

d3 r e−ik ·r v(r )Ψk (r ) ,

where k = ku. With eq. (79), this invites the deﬁnition of the scattering amplitude via (89)

f (k , k) = −

1 4π

d3 r e−ik ·r v(r)Ψk (r) .

Inserting the exact formula for Ψk (r), eq. (85), combined with eq. (87), leads to an

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integral equation for the scattering amplitude (90)

f (k , k) = −

v(k − k) + 4π

d3 q v(k − q)f (q, k) . (2π)3 k 2 − q 2 + iη

where v(k) is the Fourier transform of the potential v(r) (which we suppose to exist). This is the Lippmann-Schwinger equation, an exact integral equation for f in terms of the potential v, useful to perform a perturbation expansion. Note that it requires knowledge of f (q, k) for q 2 = k 2 (“oﬀ the energy shell”). However, the dominant contributions to the integral do come from wave vectors q such that q 2 = k 2 . For low-energy swave scattering, f (q, k) → f (k) then only depends on the magnitude of the wave vector k. With this approximation, we can take f (k) outside the integral. Taking the limit k 1/r0 , dividing by f (k) and by v0 ≡ v(0), we arrive at (91)

1 4π 4π ≈− + f (k) v0 v0

v(−q) d3 q . (2π)3 k 2 − q 2 + iη

If we only keep the ﬁrst order in v, we obtain the scattering length in Born approximation, v0 a = 4π . For a delta-potential V (r) = V0 δ(r), we obtain to ﬁrst order in V0 (92)

V0 =

4π2 a . m

However, already the second-order term in the expansion of eq. (91) would not converge, d3 q 1 as it involves the divergent integral (2π) 3 q 2 . The reason is that the Fourier transform of the δ-potential does not fall oﬀ at large momenta. Any physical potential does fall oﬀ at some large momentum, so this is not a “real” problem. For example, the van-derWaals potential varies on a characteristic length scale r0 and will thus have a natural momentum cut-oﬀ /r0 . A proper regularization of contact interactions employs the ∂ pseudo-potential [167] V (r)ψ(r) = V0 δ(r) ∂r (rψ(r)). It leads exactly to a scattering 2 4π a amplitude f (k) = −a/(1 + ika) if V0 = m . Here we will work with a Fourier transform that is equal to a constant V0 at all relevant momenta in the problem, but that falls oﬀ at very large momenta, to make the second-order term converge. The exact form is not important. If we are to calculate physical quantities, we will replace V0 in favor of the observable quantity a using the formal prescription (93)

m m 1 − 2 = 2 V0 4π a

d3 q 1 . (2π)3 q 2

We will always ﬁnd that the diverging term is exactly balanced by another diverging integral in the ﬁnal expressions, so this is a well-deﬁned procedure [188, 189]. Alternatively, one can introduce a “brute force” energy cut-oﬀ ER = 2 /mR2 (momentum cut-oﬀ /R), taken to be much larger than typical scattering energies. Equa-

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169

tion (91) then gives (94)

1 4π 2R 2 2 1 ≈− + k − ik . − f (k) v0 πR π

This is now exactly of the form eq. (83) with the scattering length (95)

a=

R π . 2 1 + 2πv2 R 0

For any physical, given scattering length a we can thus ﬁnd the correct strength v0 that reproduces the same a (provided that we choose R a for positive a). This approach implies an eﬀective range reﬀ = π4 R that should be chosen much smaller than all relevant distances. Note that as a function of v0 , only one pole of a and therefore only one bound state is obtained, at v0 = −2π 2 R. This prompts us to discuss the relation between eq. (93) and eq. (90): The LippmannSchwinger equation is an exact reformulation of Schr¨ odinger’s equation for the scattering problem. One can, for example, exactly solve for the scattering amplitude in the case of a spherical-well potential [190]. In particular, all bound states supported by the potential are recovered. However, to arrive at eq. (93), one ignores the oscillatory behavior of both v(q) and f (q, k) and replaces them by q-independent constants. As a result, eq. (93), with a cut-oﬀ for the diverging integral at a wave vector 1/R, only allows for one bound state to appear as the potential strength is increased (see eq. (95)). We will analyze this approximation for a spherical well of depth V and radius R. The true scattering length for a spherical well is given by [186] (96)

a tan(KR) =1− R KR

with K 2 = mV /2 , which one can write as (97)

*∞ 2 2 R (1 − K a n2 π 2 ) = 1 − *∞ n=1 2 R2 4K R n=1 (1 − (2n−1)2 π 2 )

← Zeros of a − R , ← Resonances of a .

3 In contrast, eq. (93) with V0 = − 4π 3 V R and the “brute force” cut-oﬀ at 1/R gives

(98)

a = R

K 2 R2 . 2 2 2 πK R − 3

The sudden cut-oﬀ strips the scattering length of all but one zero (at V = 0) and of all but one resonance. For a shallow well that does not support a bound state, the scattering length still behaves correctly as a = − 31 EVR R. However, the sudden cut-oﬀ v(q) ≈ const for q ≤ R1 and 0 beyond results in a shifted critical well depth to accommodate the ﬁrst

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W. Ketterle and M. W. Zwierlein 2

π bound state, V = 3π 2 ER , diﬀering from the exact result V = 4 ER . This could be cured by adjusting the cut-oﬀ. But for increasing well depth, no new bound state is found and a saturates at ∼ R, contrary to the exact result. At ﬁrst, such an approximation might be unsettling, as the van-der-Waals potentials of the atoms we deal with contain many bound states. However, the gas is in the ultracold regime, where the de Broglie-wavelength is much larger than the range r0 of the potential. The short-range physics, and whether the wave function has one or many nodes within r0 (i.e. whether the potential supports one or many bound states), is not important. All that matters is the phase shift δs modulo 2π that the atomic wave packets receive during a collision. We have seen that with a Fourier transform of the potential that is constant up to a momentum cut-oﬀ /R, we can reproduce any low-energy scattering behavior, which is described by the scattering length a. We can even realize a wide range of combinations of a and the eﬀective range reﬀ to capture scattering at ﬁnite values of k. An exception is the situation where 0 < a reﬀ or potentials that have a negative eﬀective range. This can be cured by more sophisticated models (see the model for Feshbach resonances in sect. 5). . 4 3. Cooper instability in a Fermi gas with attractive interactions. – In contrast to bosons, the non-interacting Fermi gas does not show any phase transition down to zero temperature. One might assume that this qualitative fact should not change as interactions are introduced, at least as long as they are weak. This is essentially true in the case of repulsive interactions (8 ). For attractive interactions, the situation is, however, dramatically diﬀerent. Even for very weak attraction, the fermions form pairs and become superﬂuid, due to pair condensation. The idea of pairing might be natural, as tightly bound pairs of fermions can be regarded as point-like bosons, which should form a Bose-Einstein condensate. However, for weak attractive interaction — as is the case for the residual, phonon-induced electronelectron interaction in metals — it is not evident that a paired state exists. Indeed, we will see in the following that in three dimensions there is no bound state for two isolated particles and arbitrarily weak interaction. However, by discussing exact solutions in 1D and 2D, where bound states exist for weak interactions, we gain insight into how a modiﬁed density of states will lead to bound states even in 3D — this is the famous Cooper instability. . 4 3.1. Two-body bound states in 1D, 2D and 3D. Localizing a quantum-mechanical particle of mass μ = m/2 to a certain range R leads to an increased momentum uncertainty of p ∼ /R at a kinetic energy cost of about ER = p2 /m = 2 /mR2 . Clearly, a shallow potential well of size R and depth V with V /ER ≡ 1 cannot conﬁne the particle within its borders. Butwe can search for a bound state at energy |EB | ER of much larger size rB = 1/κ ≡ 2 /m|EB | R.

(8 ) Repulsive interactions still allow for the possibility of induced p-wave superﬂuidity (Kohn and Luttinger [191], also see [192]) however at very low temperatures TC ≈ EF exp[−13(π/2kF |a|)2 ].

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Making, probing and understanding ultracold Fermi gases

a)

ψ(x)

1D

+κx

-κx

e 0

V1D

R rB ~ R/ε

b) -κr

e

V(x) -ER

ψ(r)

e x

0

2D -log κr

V2D

-ER

ψ(r) u(r)

-κr

e r

r R

0

-ER rB ~ R e1/ε

3D

c)

r V3D > ER rB ~ R ER/(V3D-Vc)

Fig. 19. – Bound state wave functions in 1D, 2D and 3D for a potential well of size R and depth V . In 1D and 2D, bound states exist for arbitrarily shallow wells. In terms of the small parameter = V /ER with ER = 2 /mR2 , the size of the bound state in 1D is R/ . In 2D, the bound state is exponentially large, of size Re−1/ . In 3D, due to the steep slope in u(r) = rψ(r), bound states can only exist for well depths V3D larger than a certain threshold Vc ≈ ER . The size of the bound state diverges as RER /(V3D − Vc ) for V3D > Vc .

– 1D: The bound state wave function far away from the well necessarily behaves like e±κx for negative (positive) x (see ﬁg. 19a). As we traverse the well, the wave function has to change its slope by 2κ over a range R. This costs kinetic energy ≈ 2 κ/mR that has to be provided by the potential energy −V . We deduce that κ ≈ mRV /2 = /R, where = V /ER is a small number for a weak potential. The size of the bound state rB ≈ R/ is indeed much larger than the size of the well, and the bound state energy EB ≈ −ER 2 /2 depends quadratically on the weak attraction −V . Importantly, we can always ﬁnd a bound state even for arbitrarily weak (purely) attractive potentials. – 2D: For a spherically symmetric well, the Schr¨ odinger equation for the radial wave function ψ(r) outside the well reads 1r ∂r (r∂r ψ) = κ2 ψ. The solution is the modiﬁed Bessel function which vanishes like e−κr as r 1/κ (see ﬁg. 19b). For R r 1/κ, we can neglect the small bound state energy EB ∝ −κ2 compared to the kinetic energy and have ∂r (rψ ) = 0 or ψ(r) ≈ log(κr)/ log(κR), where 1/κ is the natural scale of evolution for ψ(r) and we have normalized ψ to be of order 1 at R. Note that in 2D, it is not the change in the slope ψ of the wave function which costs kinetic energy, but the change in rψ . Inside the well, we can assume ψ(r) to be practically constant as V ER . Thus, rψ changes from ≈ 1/ log κR (outside) to ≈ 0 (inside) over a distance R. The corresponding kinetic energy cost 2 is mr ∂r (rψ )/ψ ≈ 2 /mR2 log(κR) = ER / log(κR), which has to be provided by the potential energy −V . We deduce κ ≈ R1 e−cER /V and EB ≈ −ER e−2cER /V with c on the order of 1. The particle is extremely weakly bound, with its boundstate energy depending exponentially on the shallow potential −V . Accordingly,

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W. Ketterle and M. W. Zwierlein

Table VI. – Bound-states in 1D, 2D and 3D for a potential well of size R and depth V . ψ(r > R) is the wave function outside the well, rB is the size of the bound state, and EB its energy (ER = 2 /mR2 ). 1D ER

V

j

2D

3D

ER

> Vc ≈ ER

− log r/rB , e−r/rB ,

R r rB r rB

e−r/rB r

ψ(r > R)

e−r/rB

rB

R EVR

R ecER /V

R R VE−V c

−V 2 /ER

−ER e−2cER /V

−(V − Vc )2 /ER

2

EB = − mr 2

B

K0 ( rrB ) =

the size of the bound state is exponentially large, rB ≈ R ecER /V . Nevertheless, we can always ﬁnd this weakly bound state, for arbitrarily small attraction. – 3D: For a spherically symmetric well, the Schr¨ odinger equation for the wave function ψ transforms into an eﬀective one-dimensional problem for the wave function u = rψ (see ﬁg. 19c). We might now be tempted to think that there must always be a bound state in 3D, as we already found this to be the case in 1D. However, the boundary condition on u(r) is now to vanish linearly at r = 0, in order for ψ(0) to be ﬁnite. Outside the potential well, we still have u ∝ e−κr for a bound state. Inside the well the wave function must fall oﬀ to zero at r = 0 and necessarily has to change its slope from −κ outside to ∼ 1/R inside the well over a distance R. This costs the large kinetic energy ∼ 2 u /2mu ≈ 2 /mR2 = ER . If the well depth V is smaller than a critical depth Vc on the order of ER , the particle cannot be bound. At V = Vc , the ﬁrst bound state enters at E = 0. As κ = 0, u is then constant outside the well. If the potential depth is further increased by a small amount ΔV Vc , u again falls oﬀ like e−κr for r > R. This requires an additional change in slope by κ over the distance R, provided by ΔV . So we ﬁnd analogously to the 1D case κ ∼ mRΔV /2 . Hence, the bound state energy EB ≈ −ΔV 2 /ER is quadratic in the “detuning” ΔV = (V − VC ), and the size of the bound state diverges as rB ≈ RER /(V − VC ). We will ﬁnd exactly the same behavior for a weakly bound state when discussing Feshbach resonances in sect. 5. The analysis holds for quite general shapes V (r) of the (purely attractive) potential well (in the equations, we only need to replace V by its average over the well — if it ∞ ∞ exists —, R1 −∞ V (x)dx in 1D, R12 0 rV (r)dr in 2D etc.). Table VI summarizes the diﬀerent cases. Applying these results to the equivalent problem of two interacting particles colliding in their center-of-mass frame, we see that in 1D and 2D, two isolated particles can bind for an arbitrarily weak purely attractive interaction. Hence in 1D and 2D, pairing of

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Making, probing and understanding ultracold Fermi gases

Table VII. – Link between the density of states and the existence of a bound state for arbitrarily weak interaction. The table shows the density of states, ρ( ), the equation relating the bound state energy E to V0 , and the result for √ E. It is assumed that ER |E|. To compare with table VI note that |V0 | ∼ V Rn . V0c = 2 π 2 ER R3 is the threshold interaction strength for the 3D case. The formula for the 3D bound state energy follows from the renormalization procedure . outlined in sect. 4 2, when expressing V0 in terms of the scattering length a using eq. (93).

2 ρ( ) mΩ 1 |V0 |

=

1 Ω

R <ER

1 π

ρn () d 2+|E|

2 2

E = − mκ

1D q

q

2D

2 2m

m 4 2 |E|

2 m − 4 2 V0

1 2π m 4π 2

1 2π 2

2ER +|E| |E|

log

2

4π − m|V 0|

−2ER e

√

1 m3/2 ( 2π 2 3

− π82 ER

3D q

2m 2

2ER −

(|V0 |−V0c )2 |V0 |2

π 2

p |E|)

= −2 /ma2

fermions can be understood already at the two-particle level. Indeed, one can show that the existence of a two-body bound state for isolated particles in 2D is a necessary and suﬃcient condition for the instability of the many-body Fermi sea (Cooper instability, see below) [193]. In 3D, however, there is a threshold interaction below which two isolated particles are unbound. We conclude that if pairing and condensation occur for arbitrarily weak interactions in 3D, then this must entirely be due to many-body eﬀects. . 4 3.2. Density of states. What physical quantity decides whether there are bound states or not? To answer this question, we formulate the problem of two interacting particles of mass m in momentum space. This allows a particularly transparent treatment for all three cases (1D, 2D, 3D) at once, and identiﬁes the density of states in the diﬀerent dimensions as the decisive factor for the existence of bound states. 2 2 Searching for a shallow bound state of energy E = − mκ (m/2 is the reduced mass), 2 we start by writing the Schr¨ odinger equation for the relative wave function m (∇2 − κ2 )ψ = V ψ in (n-dimensional) momentum space: (99)

ψκ (q) = −

1 m 2 2 q + κ2

dn q V (q − q )ψκ (q ) . (2π)n

For a short-range potential of range R 1/κ, V (q) is practically constant, V (q) ≈ V0 , for all relevant q, and falls oﬀ to zero on a large q-scale of ≈ 1/R. For example, for a potential well of depth V and size R, we have V0 ∼ −V Rn . Thus, ψκ (q) ≈ −

(100)

1 mV0 2 q 2 + κ 2

1 q R

dn q ψκ (q ) . (2π)n

We integrate once over q, applying the same cut-oﬀ 1/R, and then divide by the more dn q common factor q 1 (2π) n ψκ (q). We obtain the following equation for the bound state R

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W. Ketterle and M. W. Zwierlein

energy E: (101)

−

1 m = 2 V0

1 q R

dn q 1 1 = n 2 2 (2π) q + κ Ω

d <ER

ρn ( ) 2 + |E|

with the density of states in n dimensions ρn ( ), the energy cut-oﬀ ER = 2 /mR2 and the volume Ω of the system (note that V0 has units of energy times volume). The question on the existence of bound states for arbitrarily weak interaction has now been reformulated: As |V0 | → 0, the left-hand side of eq. (101) diverges. This equation has a solution for small |V0 | only if the right-hand side also diverges for vanishing boundstate energy |E| → 0, and this involves an integral over the density of states. Table VII presents the diﬀerent cases in 1D, 2D, 3D. In 1D, the integral diverges as 1/ |E|, so one can always ﬁnd a bound-state solution. The binding energy depends quadratically on the interaction, as we had found before. In 2D, where the density of states ρ2D is constant, the integral still diverges logarithmically as |E| → 0, so that again there is a solution |E| for any small |V0 |. The binding energy now depends exponentially on the interaction and ρ2D : (102)

−ρ

E2D = −2ER e

2Ω 2D |V0 |

.

However, in 3D the integral is ﬁnite for vanishing |E|, and there is a threshold for the interaction potential to bind the two particles. These results give us an idea why there might be a paired state for two fermions immersed in a medium, even for arbitrarily weak interactions: It could be that the density of available states to the two fermions is altered due to the presence of the other atoms. This is exactly what happens, as will be discussed in the next section. . 4 3.3. Pairing of fermions — The Cooper problem. Consider now two weakly interacting spin 1/2 fermions not in vacuum, but on top of a (non-interacting) ﬁlled Fermi sea, the Cooper problem [33]. Momentum states below the Fermi surface are not available to the two scattering particles due to Pauli blocking (ﬁg. 20a). For weak interactions, the particles’ momenta are essentially conﬁned to a narrow shell above the Fermi surface. The density of states at the Fermi surface is ρ3D (EF ), which is a constant just like in two dimensions. We should thus ﬁnd a bound state for the two-particle system for arbitrarily weak attractive interaction. In principle, the two fermions could form a pair at any ﬁnite momentum. However, considering the discussion in the previous section, the largest binding energy can be expected for the pairs with the largest density of scattering states. For zero-momentum pairs, the entire Fermi surface is available for scattering, as we can see from ﬁg. 20a. If the pairs have ﬁnite center-of-mass momentum q, the number of contributing states is strongly reduced, as they are conﬁned to a circle (see ﬁg. 20b). Consequently, pairs at rest experience the strongest binding. In the following we will calculate this energy. We can write the Schr¨ odinger equation for the two interacting particles as before, but

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Making, probing and understanding ultracold Fermi gases

Fig. 20. – Cooper problem: Two particles scattering on top of a Fermi sea. a) Weakly interacting particles with equal and opposite momenta can scatter into ﬁnal states in a narrow shell (blueshaded) on top of the Fermi sea (gray shaded), which blocks possible ﬁnal momentum states. b) For non-zero p total momentum 2q, particles can scatter only in a narrow band around a circle with radius kF2 − q 2 .

now we need to search for a small binding energy EB = E − 2EF < 0 on top of the large Fermi energy 2EF of the two particles. The equation for EB is 1 1 − = V0 Ω

(103)

d EF <<EF +ER

ρ3D ( ) . 2( − EF ) + |EB |

The eﬀect of Pauli blocking of momentum states below the Fermi surface is explicitly included by only integrating over energies > EF . In conventional superconductors, the natural cut-oﬀ energy ER is given by the Debye frequency ωD , ER = ωD , corresponding to the highest frequency at which ions in the crystal lattice can respond to a bypassing electron. Since we have ωD EF , we can approximate ρ3D ( ) ≈ ρ3D (EF ) and ﬁnd EB = −2ωD e−2Ω/ρ3D (EF )|V0 | .

(104)

In the case of an atomic Fermi gas, we should replace 1/V0 by the physically relevant scattering length a < 0 using the prescription in eq. 93. The equation for the bound state becomes (105)

−

m 1 = 4π2 a Ω

EF +ER

d EF

1 ρ3D ( ) − 2( − EF ) + |EB | Ω

EF +ER

d 0

ρ3D ( ) . 2

The right-hand expression is now ﬁnite as we let the cut-oﬀ ER → ∞, the result being (one assumes |EB | EF ) (106)

ρ3D (EF ) m = − 4π2 a 2Ω

− log

|EB | 8EF

−2 .

176

Inserting ρ3D (EF ) = (107)

W. Ketterle and M. W. Zwierlein ΩmkF 2π 2 2

with the Fermi wave vector kF = EB = −

2mEF /2 , one arrives at

8 EF e−π/kF |a| . e2

The binding energies eqs. (104) and (107) can be compared with the result for the bound state of two particles in 2D, eq. (102). The role of the constant density of states ρ2D is here played by the 3D density of states at the Fermi surface, ρ3D (EF ). The result is remarkable: Two weakly interacting fermions on top of a Fermi sea form a bound state due to Pauli blocking. However, in this artiﬁcial problem we neglected the interactions between particles in the Fermi sea. As we “switch on” the interactions for all particles from top to the bottom of the Fermi sea, the preceding discussion indicates that the gas will reorder itself into a completely new, paired state. The Fermi sea is thus unstable towards pairing (Cooper instability). The full many-body description of such a paired state, including the necessary anti-symmetrization of the full wave function, was achieved by Bardeen, Cooper and Schrieﬀer (BCS) in 1957 [34]. As we will see in the next section, the self-consistent inclusion of all fermion pairs leads to more available momentum space for pairing. The eﬀective density of states is then twice as large, giving a superﬂuid gap Δ that diﬀers from |EB | (eq. (107)) by a factor of 2 in the exponent: (108)

Δ=

8 EF e−π/2kF |a| . e2

It should be noted that the crucial diﬀerence to the situation of two particles in vacuum in 3D is the constant density of states at the Fermi energy (and not the 2D character of the Fermi surface). Therefore, if we were to consider the Cooper problem in higher dimensions n and have two fermions scatter on the (n − 1)-dimensional Fermi surface, the result would be similar to the 2D case (due to the constant density of states), and not to the case of (n − 1) dimensions. The conclusion of this section is that Cooper pairing is a many-body phenomenon, but the binding of two fermions can still be understood by two-body quantum mechanics, as it is similar to two isolated particles in two dimensions. To ﬁrst order, the many-body physics is not the modiﬁcation of interactions, but rather the modiﬁcation of the density of states due to Pauli blocking. . . 4 4. Crossover wave function. – From subsect. 4 3.1 we know that in 3D, two fermions in isolation can form a molecule for strong enough attractive interaction. The ground state of the system should be a Bose-Einstein condensate of these tightly bound pairs. However, if we increase the density of particles in the system, we will ultimately reach the point where the Pauli pressure of the fermionic constituents becomes important and modiﬁes the properties of the system. When the Fermi energy of the constituents exceeds the binding energy of the molecules, we expect that the equation of state will be fermionic, i.e. the chemical potential will be proportional to the density to the power of 2/3. Only when the size of the molecules is much smaller than the interparticle spacing,

Making, probing and understanding ultracold Fermi gases

177

i.e. when the binding energy largely exceeds the Fermi energy, is the fermionic nature of the constituents irrelevant — tightly bound fermions are spread out widely in momentum space and do not run into the Pauli limitation of unity occupation per momentum state. For too weak an attraction there is no bound state for two isolated fermions, but Cooper pairs can form in the medium as discussed above. The ground state of the system turns out to be a condensate of Cooper pairs as described by BCS theory. In contrast to the physics of molecular condensates, however, the binding energy of these pairs is much less than the Fermi energy and therefore Pauli pressure plays a major role. It was realized by Leggett [40], building upon work by Popov [37], Keldysh [38] and Eagles [39], that the crossover from the BCS- to the BEC-regime is smooth. This is somewhat surprising since the two-body physics shows a threshold behavior at a critical interaction strength, below which there is no bound state for two particles. In the presence of the Fermi sea, however, we simply cross over from a regime of tightly bound molecules to a regime where the pairs are of much larger size than the interparticle spacing. Closely following Leggett’s work [40], and its extension to ﬁnite temperatures by Nozi`eres and Schmitt-Rink [41], we will describe the BEC-BCS crossover in a simple “one-channel” model of a potential well. Rather than the interaction strength V0 as . in subsect. 4 3.1, we will take the scattering length a as the parameter that “tunes” the interaction. The relation between V0 and a is given by eq. (93) and its explicit form eq. (95). For positive a > 0, there is a two-body bound state available at EB = −2 /ma2 (see table VII), while small and negative a < 0 corresponds to weak attraction where Cooper pairs can form in the medium. In either case, for s-wave interactions, the orbital part of the pair wave function ϕ(r1 , r2 ) will be symmetric under exchange of the paired particles’ coordinates and, in a uniform system, will only depend on their distance |r1 − r2 |. We will explore the many-body wave function (109)

Ψ (r1 , . . . , rN ) = A {ϕ(|r1 − r2 |)χ12 . . . ϕ(|rN −1 − rN |)χN −1,N } ,

that describes a condensate of such fermion pairs, with the operator A denoting the correct antisymmetrization of all fermion coordinates, and the spin singlet χij = √1 (|↑ |↓ − |↓ |↑ ). In the experiment, “spin up” and “spin down” will correspond i j i j 2 to two atomic hyperﬁne states. In second quantization notation we write " d3 ri ϕ(r1 − r2 )Ψ†↑ (r1 )Ψ†↓ (r2 ) . . . ϕ(rN −1 − rN )Ψ†↑ (rN −1 )Ψ†↓ (rN ) |0 , (110) |ΨN = i

† e−ik·r √ where the ﬁelds Ψ†σ (r) = . With the Fourier transform ϕ(r1 − r2 ) = k ckσ Ω 1 −r2 ) eik·(r √ we can introduce the pair creation operator k ϕk Ω (111)

b† =

k

ϕk c†k↑ c†−k↓

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W. Ketterle and M. W. Zwierlein

and ﬁnd |ΨN = b†

(112)

N/2

|0 .

This expression for |ΨN is formally identical to the Gross-Pitaevskii ground state of a condensate of bosonic particles. However, the operators b† obey bosonic commutation relations only in the limit of tightly bound pairs. For the commutators, we obtain (113)

† † b , b − = kk ϕk ϕk c†k↑ c†−k↓ , c†k ↑ c†−k ↓ = 0 , − [b, b]− = kk ϕ∗k ϕ∗k [c−k↓ ck↑ , c−k ↓ ck ↑ ]− = 0 , † b, b − = kk ϕ∗k ϕk c−k↓ ck↑ , c†k ↑ c†−k ↓ = |ϕk |2 (1 − nk↑ − nk↓ ) . −

k

The third commutator is equal to one only in the limit where the pairs are tightly bound and occupy a wide region in momentum space. In this case, the occupation . nk of any momentum small (see section 4 6.3 below), and numbers 3 3state k are very † 2 2 b, b − ≈ k |ϕk | = d r1 d r2 |ϕ(r1 , r2 )| = 1. Working with the N -particle state |ΨN is inconvenient, as one would face a complicated combinatoric problem in manipulating the sum over all the c†k ’s (as one chooses a certain k for the ﬁrst fermion, the choices for the second depend on this k, etc.). It is preferable to use the grand canonical formalism, not ﬁxing the number of atoms but the chemical potential μ. A separate, crucial step is to deﬁne a many-body state which is a superposition of states with diﬀerent atom numbers. In the BEC limit, this is analogous to the use of coherent states (vs. Fock states) in quantum optics. Let Np = N/2 be the number of pairs. Then, (114)

N |Ψ =

√ 1 Np b† M/2 † M N b |0 = e |0 p Jeven M! M " † † * √ = k e Np ϕk ck↑ c−k↓ |0 = (1 + Np ϕk c†k↑ c†−k↓ ) |0 .

NpJ/4 (J/2)!

|ΨJ =

k

The second to last equation follows since the operators c†k↑ c†−k↓ commute for diﬀerent k, * 1 and the last equation follows from c†2 k uk = k = 0. If we choose the constant N = * 2 , then |Ψ becomes a properly normalized state 1 + N |ϕ | p k k (115)

|ΨBCS =

"

(uk + vk c†k↑ c†−k↓ ) |0

k

with vk = Np ϕk uk and |uk |2 + |vk |2 = 1. This is the famous BCS wave function, ﬁrst introduced as a variational Ansatz, later shown to be the exact solution of the simpliﬁed Hamiltonian equation (117) (below). It is a product of wave functions referring to the occupation of pairs of single-particle momentum states, (k, ↑, −k, ↓). As a special case,

Making, probing and understanding ultracold Fermi gases

179

it describes a non-interacting Fermi sea, with all momentum pairs occupied up to the Fermi momentum (uk = 0, vk = 1 for k < kF and uk = 1, vk = 0 for k > kF ). In general, for a suitable choice of the vk ’s and uk ’s, it describes a “molten” Fermi sea, modiﬁed by the coherent scattering of pairs with zero total momentum. Pairs of momentum states are seen to be in a superposition of being fully empty and fully occupied. The above derivation makes it clear that this wave function encompasses the entire regime of pairing, from point bosons (small molecules) to weakly and non-interacting fermions. . 4 5. Gap and number equation. – The variational parameters vk are derived in uk and ˆ ˆ the standard way by minimizing the free energy E − μN = H − μN . The many-body Hamiltonian for the system is (116)

ˆ = H

k c†kσ ckσ +

k,σ

V0 † ck+ q ↑ c†−k+ q ↓ ck + q2 ↓ c−k + q2 ↑ . 2 2 Ω k,k ,q

The dominant role in superﬂuidity is played by fermion pairs with zero total mo. mentum. Indeed, as we have seen in subsect. 4 3.3, Cooper pairs with zero momentum have the largest binding energy. Therefore, we simplify the mathematical description by neglecting interactions between pairs at ﬁnite momentum, i.e. we only keep the terms for q = 0. This is a very drastic simpliﬁcation, as hereby density ﬂuctuations are eliminated. It is less critical for charged superﬂuids, where density ﬂuctuations are suppressed by Coulomb interactions. However, for neutral superﬂuids, sound waves (the . Bogoliubov-Anderson mode, see subsect. 4 7.3) are eliminated by this approximation. The approximate Hamiltonian (“BCS Hamiltonian”) reads (117)

ˆ = H

k c†kσ ckσ +

k,σ

V0 † † ck↑ c−k↓ ck ↓ c−k ↑ . Ω k,k

The free energy becomes (118)

V0 ˆ − μN ˆ = F= H 2ξk vk2 + u k vk u k vk Ω k

k,k

with ξk = k − μ Minimizing E − μN leads to (119)

1 = 1− 2 1 u2k = 1+ 2 vk2

ξk , Ek

ξk , Ek

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W. Ketterle and M. W. Zwierlein

with Ek =

ξk2 + Δ2 ,

where Δ is given by the gap equation Δ ≡ Δ − VΩ0 k 2E or k −

(120)

1 = V0

V0 Ω

k

ck↑ c−k↓ = − VΩ0

k

u k vk =

d3 k

1 . (2π) 2Ek 3

Note the similarity to the bound state equation in free space, eq. (101), and in the simpliﬁed Cooper problem, eq. (103). An additional constraint is given by the number equation for the total particle density n = N/Ω (121)

n=2

d3 k (2π)

vk2 .

3

Gap and number equations have to be solved simultaneously to yield the two unknowns μ and Δ. We will once more replace V0 by the scattering length a using prescription eq. (93), so that the gap equation becomes (compare eq. (105)) (122)

m = − 4π2 a

d3 k (2π)

3

1 1 − 2Ek 2 k

,

where the integral is now well-deﬁned. The equations can be rewritten in dimensionless form with the Fermi energy EF = 2 kF2 /2m and wave vector kF = (3π 2 n)1/3 [195] (123)

μ Δ I1 , kF a EF Δ

3/2 μ 3 Δ , 1= I2 2 EF Δ 1

−

(124)

=

2 π

with (125)

∞

I1 (z) =

⎛ dx x2 ⎝

0

⎞ 1 2

(x2 − z) + 1

−

1⎠ , x2

and (126)

I2 (z) = 0

∞

⎛ dx x2 ⎝1 −

⎞ x2 − z 2

(x2 − z) + 1

⎠.

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Making, probing and understanding ultracold Fermi gases

1 1.5

-1 1.0 -2

Δ/EF

μ/EF

0

0.5

-3 -4 2

1

0 1/kFa

-1

0.0 -2

Fig. 21. – Chemical potential (dotted line) and gap (straight line, red) in the BEC-BCS crossover as a function of the interaction parameter 1/kF a. The BCS-limit of negative 1/kF a is to the right on the graph. The resonance where 1/kF a = 0 is indicated by the dashed line.

This gives

(127)

(128)

1/3 μ 2 2 μ = I1 − , kF a π 3I2 Δ Δ 2/3 Δ 2 μ = . EF 3I2 Δ 1

The ﬁrst equation can be inverted to obtain μ/Δ as a function of the interaction parameter 1/kF a, which can then be inserted into the second equation to yield the gap Δ. The result for μ and Δ as a function of 1/kF a is shown in ﬁg. 21. It is possible to obtain analytic expressions for the solutions in terms of complete elliptic integrals [195]. In this derivation, we have combined the simpliﬁed Hamiltonian, eq. (117) with the BCS variational Ansatz. Alternatively one can apply a decoupling (mean ﬁeld) approximation to the Hamiltonian [169]. Expecting that there will be some form of pair condensate, we assume that the pair creation and annihilation operator only weakly ﬂuctuates around its non-zero expectation value (129)

Ck = ck↑ c−k↓ = − c†k↑ c†−k↓

chosen to be real (since the relative phase of states which diﬀer in particle number by two can be arbitrarily chosen). That is, we write (130)

ck↑ c−k↓ = Ck + (ck↑ c−k↓ − Ck )

with the operator in parentheses giving rise to ﬂuctuations that are small on the scale of

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W. Ketterle and M. W. Zwierlein

Ck . The gap parameter Δ is now deﬁned as (131)

Δ=

V0 Ck . Ω k

We only include terms in the interaction part of the Hamiltonian which involve the Ck ’s at least once. That is, we neglect the correlation of ﬂuctuations of the pair creation and annihilation operators. One obtains (132)

ˆ = H

k (c†k↑ ck↑

+

c†k↓ ck↓ )

k

−Δ

c†k↑ c†−k↓

k

+ ck↓ c−k↑ +

Ck

.

k

This Hamiltonian is bilinear in the creation and annihilation operators and can easily be solved by a Bogoliubov transformation [196, 197, 169] from the particle operators ck↓ and ck↑ to new quasi-particle operators γk↑ and γk↓ : (133)

γk↑ = uk ck↑ − vk c†−k↓ , † = uk c†−k↓ + vk ck↑ . γ−k↓

The uk and vk are determined from the requirements that the new operators fulﬁll fermionic commutation relations and that the transformed Hamiltonian is diagonal with respect to the quasi-particle operators. This solution is identical to the one obtained before for the uk and vk , and the transformed Hamiltonian becomes (134)

2 † † ˆ − μN ˆ =− Δ + (ξk − Ek ) + Ek (γk↑ γk↑ + γk↓ γk↓ ) . H V0 /Ω k

k

The ﬁrst two terms give the free energy E − μN of the pair condensate, identical to eq. (118) when the correct uk and vk are inserted. The third term represents the energy of excited quasi-particles, and we identify Ek as excitation energy of a quasi-particle. The superﬂuid ground state is the quasi-particle vacuum: γk↑ |Ψ = 0 = γk↓ |Ψ. This approach via the pairing ﬁeld is analogous to the Bogoliubov treatment of an interacting Bose-Einstein condensate: There, the creation and annihilation operators √ for atoms with zero momentum are replaced by N0 , the square root of the number of condensed atoms (i.e. a coherent ﬁeld). In the interaction term of the Hamiltonian √ all terms are dropped that contain less than two factors of N0 . In other words, the Hamiltonian (eq. (117)) is solved by keeping only certain pair interactions, either by using a variational pairing wave function, or by introducing a mean pairing ﬁeld. It should be noted that these approximations not even be necessary as there is strong evidence that the BCS wave function is the exact solution to the reduced Hamiltonian eq. (117) [194, 198].

Making, probing and understanding ultracold Fermi gases

183

. 4 6. Discussion of the three regimes – BCS, BEC and crossover. . 4 6.1. BCS limit. In the BCS-limit of weak attractive interaction, kF a → 0− , we have(9 ) (135) (136)

μ ≈ EF , 8 Δ ≈ 2 e−π/2kF |a| . e

The ﬁrst equation tells us that adding a spin-up and spin-down particle to the system costs a Fermi energy per particle (with the implicit assumption that both a spin-up and a spin-down particle are added, raising the total energy by 2μ). In the weakly interacting BCS limit Pauli blocking still dominates over interactions, and hence the particles can only be added at the Fermi surface. The second equation is the classic result of BCS theory for the superﬂuid gap(10 ). Compared to the bound-state energy for a single Cooper pair on top of a non-interacting Fermi sea, eq. (107), the gap is larger (the negative exponent is smaller by a factor of two), as the entire collection of particles now takes part in the pairing(11 ). However, the gap is still exponentially small compared to the Fermi energy: Cooper pairing is fragile. The ground state energy of the BCS state can be calculated from eq. (118) and is (137)

EG, BCS =

3 1 N EF − ρ(EF ) Δ2 . 5 2

The ﬁrst term is the energy of the non-interacting normal state, where 35 EF is the average kinetic energy per fermion in the Fermi sea. The second term is the energy due to condensation, negative as it should be, indicating that the BCS state is energetically favorable compared to the normal state. Although the total kinetic energy of the Fermi gas has been increased (by populating momentum states above EF ), the total energy is lower due to the gain in potential energy. This is valid for any kind of pairing (i.e. proton and electron forming a hydrogen atom), since the localization of the pair wave function costs kinetic energy. The condensation energy of the BCS state, ES (N ) = − 12 ρ(EF ) Δ2 can be interpreted in two ways. One way refers to the wave function, eq. (109), which consists of N/2 identical fermion pairs. The average binding energy per pair is then − 34 Δ2 /EF . This S implies that the last pair added has a binding energy of ES (N ) − ES (N − 2) = 2 ∂E ∂N = (9 ) This follows by substituting ξ = x2 − z in the integrals and taking the limit z → ∞. One √ has I1 (z) ≈ z (log(8z) − 2) and I2 (z) = 23 z 3/2 . (10 ) The present mean-ﬁeld treatment does not include density ﬂuctuations, which modify the prefactor in the expression for the gap Δ [199, 169]. (11 ) In the self-consistent BCS solution, not only the momentum states above the Fermi surface contribute to pairing, but also those below it, in a symmetric shell around the Fermi momentum. In the Cooper problem the states below the Fermi surface were excluded, reducing the eﬀective density of states by a factor of two.

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W. Ketterle and M. W. Zwierlein

− 21 Δ2 /EF . The other interpretation refers to the BCS wave function, eq. (115). It is essentially a product of a “frozen” Fermi sea (as vk ≈ 1, uk ≈ 0 for low values of k) with a paired component consisting of ∼ ρ(EF ) Δ ∼ N Δ/EF pairs, located in an energy shell of width Δ around the Fermi energy. They each contribute a pairing energy on the order of Δ. The second interpretation justiﬁes the picture of a Cooper . pair condensate. In the solution of the Cooper problem (subsect. 4 3.3), the pair wave function has a peak occupation per momentum state of ∼ 1/ρ(EF )Δ. Therefore, one can stack up ∼ ρ(EF )Δ pairs with zero total momentum without getting into serious trouble with the Pauli exclusion principle and construct a Bose-Einstein condensate consisting of ∼ ρ(EF )Δ Cooper pairs (12 ). It depends on the experiment whether it reveals a pairing energy of 21 Δ2 /EF or of . Δ. In RF spectroscopy, all momentum states can be excited (see subsect. 2 3), and . 1 2 the spectrum shows a gap of 2 Δ /EF (see subsect. 4 7.2). Tunnelling experiments in superconductors probe the region close to the Fermi surface, and show a pairing gap of Δ. The two interpretations for the BCS energy carry along two possible choices of the . pairing wave function (see subsect. 4 6.3 and [194]). The ﬁrst one is ϕk = uk /vk Np , which can be shown to extend throughout the whole Fermi sea from zero to slightly above kF , whereas the second one, ψ(k) = uk vk , is concentrated around the Fermi surface (see ﬁg. 25 below). To give a sense of scale, Fermi energies in dilute atomic gases are on the order of a μK, corresponding to 1/kF ∼ 4 000 a0 . In the absence of scattering resonances, a typical scattering length would be about 50 − 100 a0 (on the order of the van der Waals-range). Even if a < 0, this would result in a vanishingly small gap Δ/kB ≈ 10−30 − 10−60 K. Therefore, the realization of superﬂuidity in Fermi gases requires scattering or Feshbach resonances to increase the scattering length, bringing the gas into the strongly interacting regime where kF |a| > 1 (see sect. 5). In this case, the above mean ﬁeld theory predicts Δ > 0.22 EF or Δ/kB > 200 nK for kF |a| > 1, and this is the regime where current experiments are operating. . 4 6.2. BEC limit. In the BEC limit of tightly bound pairs, for kF a → 0+ , one ﬁnds(13 ) (138) (139)

π2 an 2 , + 2 2ma m 16 EF √ Δ≈ . 3π kF a

μ=−

(12 ) Similarly to the fermion pairs described by the operator b† , the Cooper pairs from sub. sect. 4 3.3 are not bosons, as shown by the equivalent of eq. (113). However, if there were only a few Cooper pairs, much less than ρ(EF )Δ, the occupation of momentum states nk would still be very small compared to 1 and these pairs would be to a good approximation bosons. from the expansion of the integrals for z < 0 and |z| → ∞. One ﬁnds (13 ) This result p follows 1 π and I2 (z) = π8 √1 . I1 (z) = − π2 |z| − 32 |z|3/2 |z|

Making, probing and understanding ultracold Fermi gases

185

The ﬁrst term in the expression for the chemical potential is the binding energy per fermion in a tightly bound molecule (see table VII). This reﬂects again the implicit assumption (made by using the wave function in eq. (109)) that we always add two fermions of opposite spin at the same time to the system. The second term is a mean-ﬁeld contribution describing the repulsive interaction between molecules in the gas. Indeed, a condensate of molecules of mass mM = 2m, density nM = n/2 and a molecule-molecule scattering length aM will have a chemical 2 M nM potential μM = 4πmaM . Since μM is twice the chemical potential for each fermion, we obtain from the above expression the molecule-molecule scattering length aM = 2a. However, this result is not exact. Petrov, Shlyapnikov and Salomon [200] have performed an exact calculation for the interaction between four fermions and shown that aM = 0.6 a. The present mean-ﬁeld approach neglects correlations between diﬀerent pairs, or between one fermion and a pair. If those are included, the correct few-body physics is recovered [201-203]. The expression for the quantity Δ signiﬁes neither the binding energy of molecules nor does it correspond to a gap in the excitation spectrum. Indeed, in the BEC-regime, as soon as μ < 0, there is no longer a gap at non-zero k in the single-fermion excitation spectrum (see ﬁg. 23 below). Instead, we have for the quasi-particle energies Ek = Δ2 ( k − μ)2 + Δ2 ≈ |μ| + k + 2|μ| . So Δ itself does not play a role in the BEC-regime, 2 but only the combination Δ /|μ| is important. As we see from eq. (138), (140)

8 EF2 2ma2 4 2 3 4π2 Δ2 = kF a = n a, = 2 2|μ| 3π kF a 3π m m

. which is two times the molecular mean ﬁeld. In fact, we will show in subsect. 4 7 that it can be interpreted here as the mean ﬁeld energy experienced by a single fermion in a gas of molecules. It might surprise that the simpliﬁed Hamiltonian eq. (117), contains interactions between two molecules or between a molecule and a single fermion at all. In fact, a crucial part of the simpliﬁcation has been to explicitly neglect such three- and fourbody interactions. The solution to this puzzle lies in the Pauli principle, which acts as an eﬀective repulsive interaction: In a molecule, each constituent fermion is conﬁned to a region of size ∼ a around the molecule’s center of mass (see next section). The probability to ﬁnd another like fermion in that region is strongly reduced due to Pauli blocking. Thus, eﬀectively, the motion of molecules is constrained to a reduced volume Ω = Ω − cNM a3M , with the number of molecules NM and c on the order of 1. This is the same eﬀect one has for a gas of hard-sphere bosons of size aM , and generally for a Bose gas with scattering length aM . An analogous argument leads to the eﬀective interaction between a single fermion and a molecule. We see that the only way interactions between pairs, or between a pair and a single fermion, enter in the simpliﬁed description of the BEC-BCS crossover is via the anti-symmetry of the many-body wave function.

186

W. Ketterle and M. W. Zwierlein 1 -2 -0.5

0.8 0

nk

0.6 0.5 0.4 1

0.2

0.5

1

1.5

2

k/kF

Fig. 22. – Occupation nk of momentum states k in the BEC-BCS crossover. The numbers give the interaction parameter 1/kF a. After [41].

. 4 6.3. Evolution from BCS to BEC. Our variational approach smoothly interpolates between the two known regimes of a BCS-type superﬂuid and a BEC of molecules. It is a crossover, which occurs approximately between 1/kF a = −1 and +1 and is fully continuous. The occupation of momentum states nk = vk2 evolves smoothly from the step-function Θ(kF − k) of a degenerate Fermi gas, broadened over a width Δ EF due to pairing, to that of Np molecules, namely the number of molecules Np times the 3/2

1 1 √ probability |ϕk |2 to ﬁnd a molecule with momentum k (we have ϕk = (2πa) π 1+k2 a2 ) Ω (see ﬁg. 22). It is also interesting to follow the evolution of the “Cooper pair” wave function(14 ) both in k-space, where it is given by ΨBCS | c†k↑ c†−k↓ |ΨBCS = uk vk , and in real-space, where it is 1 ψ(r1 , r2 ) = ΨBCS | Ψ†↑ (r1 )Ψ†↓ (r2 ) |ΨBCS = (141) uk vk e−ik·(r1 −r2 ) Ω k 1 Δ −ik·(r1 −r2 ) e . = Ω 2Ek k

In the BCS limit, the pairing occurs near the Fermi surface k = kF , in a region Δ of width δk ∼ ∂k ∂ δ ≈ vF , where vF is the velocity of fermions at the Fermi surface. Therefore, the spatial wave function of Cooper pairs has a strong modulation at the inverse wave vector 1/kF , and an overall extent of the inverse width of the pairing region, ∼ 1/δk ∼ vΔF 1/kF . More quantitatively, eq. (141) gives (setting r = |r1 − r2 |) [34] (142)

ψ(r) =

kF Δ sin(kF r) K0 π 2 r vF

r πξBCS

r→∞

∼

sin (kF r) e−r/(πξBCS ) ,

(14 ) Note that this deﬁnition is not equal to the Fourier transform of the pair wave function p ϕ(r) introduced in eq. (109), which would be vk /uk Np . The deﬁnition given here is the twopoint correlation function. See [194] for a discussion of several possible deﬁnitions of pair wave functions.

187

Making, probing and understanding ultracold Fermi gases

0

1

2 3 k / kF

4

1.2 0.8 0.4 0.0

2

5

1/kFa = -1

P = 0.59 E F ' = 0.69 E F 4

0

1

2 3 k / kF

4

2

4 3 2 1 0

5 0

1

2 3 k / kF

4

P = 0.997 EF ' = 0.046 EF

20 15 10 5 0 0

5

1

2 3 k / kF

4

5

0

1

1

5

1/kFa = -2

P = 0.95 EF ' = 0.21 EF

\(k) [a.u.]

1/kFa = 0

P = - 0.8 EF ' = 1.33 EF 3

\ (k) [a.u.]

\(k) [a.u.]

r\ (r) / ' [arb. units]

2

0.5 0.4 0.3 0.2 0.1

\ (k) [a.u.]

1/kFa = 1 3

0

0 0 0

1

2

3

4

5 0

4

8 kFr

kFr

12

0

10

30 0

20 kFr

20

40

60

kFr

Fig. 23. – Evolution of the spatial pair wave function ψ(r) in the BEC-BCS crossover. The inset shows the Fourier transform ψ(k), showing clearly that in the BCS-limit, momentum states around the Fermi surface make the dominant contribution to the wave function. In the crossover, the entire Fermi sphere takes part in the pairing. In the BEC-limit, ψ(k) broadens as the pairs become more and more tightly bound. ψ(r) was obtained via numerical integration √ R∞ sin(r ξ+μ) √ (here, = 1 = m), an expression that follows from eq. (141). of −μ dξ 2 2 ξ +Δ

where K0 (kr) is the modiﬁed Bessel function that falls oﬀ as e−kr at inﬁnity. We have encountered a similar exponential envelope function for a two-body bound state (see table VI). The characteristic size of the Cooper pair, or the two-particle correlation

2

kF [0

10 6 4 2

1 6 4 2

4

2 1/kFa

0

-2

Fig. 24. tightly bound molecules to long-range Cooper pairs. Evolution of the pair size q – From ψ(r)|r 2 |ψ(r)

ξ0 = as a function of the interaction parameter 1/kF a. On resonance (dashed ψ(r)|ψ(r)

line), the pair size is on the order of the inverse wave vector, ξ0 (0) ∼ interparticle spacing.

1 kF

, about one-third of the

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W. Ketterle and M. W. Zwierlein

length ξ0 , can be deﬁned as ξ02 = (143)

ξ0 ≈ ξBCS ≡

In the BEC limit, uk vk ∝ (144)

ψ(r)|r 2 |ψ(r)

ψ(r)|ψ(r) ,

and this gives indeed ξ0 ∼ 1/δk,

vF 1/kF πΔ

1 1+(ka)2 ,

in the BCS-limit

and so

ψ(r1 , r2 ) ∼

e−|r1 −r2 |/a , |r1 − r2 |

which is simply the wave function of a molecule of size ∼ a (see table VI). The twoparticle correlation length(15 ) is thus ξ0 ∼ a. Figures 23 and 24 summarize the evolution of the pair wave function and pair size throughout the crossover. . 4 7. Single-particle and collective excitations. – Fermionic superﬂuids can be excited in two ways: Fermi-type excitations of single atoms or Bose-like excitations of fermion pairs. The ﬁrst is related to pair breaking, the second to density ﬂuctuations — sound waves. . 4 7.1. Single-particle excitations. The BCS-state |ΨBCS describes a collection of pairs, each momentum state pair (k ↑, −k ↓) having probability amplitude uk of being empty and vk of being populated. We now calculate the energy cost for adding a single fermion in state k ↑, which does not have a pairing partner, i.e. the state −k ↓ is empty. This requires a kinetic energy ξk (relative to the chemical potential). For the other particles, the states (k ↑, −k ↓) are no longer available, and according to eq. (118) the (negative) pairing energy is increased by −2ξk vk2 − 2 VΩ0 uk vk k uk vk , which equals 2 −ξk (1 − Eξkk ) + Δ Ek = Ek − ξk (see eq. (119)). The total cost for adding one fermion is thus simply ξk + (Ek − ξk ) = Ek (again relative to μ, i.e. this is the cost in free energy). In the same way, one calculates the cost for removing a fermion from the BCS-state (e.g. deep in the Fermi sea), and leaving behind an unpaired fermion in state −k ↓. The result is again Ek . This shows that adding or removing a particle creates a quasi-particle with energy Ek , as we had found already via the Bogoliubov transformation eq. (134). For example, the quasi-particle excitation (145)

† γk↑ |ΨBCS = c†k↑

"

ul + vl c†l↑ c†−l↓ |0

l =k

correctly describes the removal of the momentum pair at (k ↑, −k ↓), and the addition of a single fermion in k ↑. (15 ) This length scale should be distinguished from the coherence length ξphase that is associated with spatial ﬂuctuations of the order parameter. The two length scales coincide in the BCSlimit, but diﬀer in the BEC-limit, where ξphase is given by the healing length ∝ √1na . See [204] for a detailed discussion.

189

Making, probing and understanding ultracold Fermi gases 1/kFa = 1

1/kFa = 0.553

3 P = - 0.8 EF ' = 1.33 EF

Ek / EF

1/kFa = 0

P = 0 EF ' = 1.05 EF

1/kFa = -1

P = 0.59 EF ' = 0.69 EF

1/kFa = -2

P = 0.95 EF ' = 0.21 EF

P = 0.997 E F ' = 0.046 E F

2 1 0 0

1 k / kF

2 0

1 k / kF

2 0

1 k / kF

2 0

1 k / kF

2 0

1 k / kF

2

Fig. 25. – Evolution of the single-particle excitation spectrum in the BEC-BCS crossover. On p the BEC-side, for μ < 0, the minimum required energy to add a particle is μ2 + Δ2 and occurs at k = 0. This qualitatively changes at 1/kF a = 0.553 where μ = 0. For μ > 0, the minimum p energy is Δ and occurs at k = 2mμ/2 .

Figure 25 shows the single-particle excitation energy Ek for diﬀerent interaction strengths in the BEC-BCS crossover. For μ > 0, the minimum energy required to remove a particle from the condensate occurs for ξk = μ and is Δ, which gives Δ the name of the superﬂuid gap. One dramatic consequence of this gap is that it prevents single fermions to enter the superﬂuid, resulting in phase separation in imbalanced Fermi mixtures [80]. For μ < 0 the minimum energy to remove a particle becomes μ2 + Δ2 and occurs for k = 0. To excite the system without adding or removing particles can be done in two ways: One can remove a particle, requiring an energy Ek , and add it back at energy cost Ek , thus creating two unpaired particles with momenta k and k . The second possibility is to excite a fermion pair in (k ↑, −k ↓) into the state orthogonal to the ground state, which can be written " † † (146) γk↑ γ−k↓ |ΨBCS = (vk − uk c†k↑ c†−k↓ ) ul + vl c†l↑ c†−l↓ |0 . l =k

Instead of the pairing energy ξk − Ek for that state, the energy for such an excitation is ξk + Ek , that is, this excited pair state lies an energy 2Ek above the BCS ground state. The minimum energy required to excitethe system, without changing the particle number, is thus 2Δ in the BCS limit, and 2 μ2 + Δ2 in the BEC-limit. In the latter case, from eq. (138), one has (147)

2Δ2 . 2 μ2 + Δ2 ≈ |EB | − μM + |EB |

The ﬁrst two terms |EB | − μM = 2 /ma2 − μM give the energy required to remove a molecule (the positive mean-ﬁeld μM pushes this energy closer to threshold). The last term will then correspond to the energy required to add two unpaired fermions . into the system. From our discussion in subsect. 4 6.2, we expect that this should cost

190

W. Ketterle and M. W. Zwierlein 2

BF nM twice the mean ﬁeld energy μBF = 4πmaBF of a fermion interacting with a cloud of bosons, the molecules. Here, aBF is the Boson-Fermion scattering length and mBF = 2mB mF /(mB + mF ) = 4/3 m is twice the reduced mass of the boson-fermion system. With the help of eq. (140) we equate

(148)

3π2 aBF n2 4π2 an 4π2 aBF nM Δ2 = ≡ = |EB | m mBF m

and obtain the Boson-Fermion scattering length at the mean ﬁeld level, (149)

aBF =

8 a. 3

The exact value aBF = 1.18 a has been obtained already 50 years ago [205]. . 4 7.2. RF excitation into a third state. The hyperﬁne structure of ultracold atoms oﬀers more than just two states “spin up” and “spin down”. This allows for a new type of single-particle excitation, not available for electrons in superconductors, namely the transfer of, say, a spin up fermion into a third, empty state, |3, via a radiofrequency . . (RF) transition (see subsect. 2 3 and 7 2.4). We have all the tools ready to calculate the excitation spectrum for RF spectroscopy in the case where the third state does not interact with atoms in the spin-up or spin-down states. Due to its long wavelength, Doppler shifts are negligible and the RF excitation ﬂips the spin from |↑ to |3 and vice versa regardless of the momentum state of the atom, and without momentum transfer. The RF operator is thus (150)

Vˆ = V0

c†k3 ck↑ + c†k↑ ck3 ,

k

where V0 is the strength of the RF drive (the Rabi frequency ωR = 2V0 /) taken to be real. As the third state is initially empty, only the ﬁrst part contributes when acting on the initial state. To calculate the action of the spin ﬂip c†k3 ck↑ on the BCS state, † we express ck↑ = uk γk↑ + vk γ−k↓ in terms of the Bogoliubov quasi-particle operators (eq. (134)). As the BCS-state is the quasi-particle vacuum, γk↑ |ΨBCS = 0, and one has (151)

† |ΨBCS c†k3 ck↑ |ΨBCS = vk c†k3 γ−k↓

and thus (152)

Vˆ |ΨBCS = V0

† vk c†k3 γ−k↓ |ΨBCS .

k

When the RF excitation removes the particle from the BCS state, it creates a quasi. particle with a cost in total energy of Ek − μ (see subsect. 4 7.1). The energy cost for creating the particle in the third state is, apart from the bare hyperﬁne splitting ω↑3 ,

Making, probing and understanding ultracold Fermi gases

191

the kinetic energy k . In total, the RF photon has to provide the energy Ω(k) = ω↑3 + Ek + k − μ .

(153)

Fermi’s Golden Rule gives now the transition rate Γ(ω) at which particles leave the BCS state and arrive in state |3 (ω is the RF frequency): Γ(ω) ≡

(154)

/2 2π // // ˆ // / / f /V / ΨBCS / δ (ω − Ef ) , f

where the sum is over all eigenstates |f with energy Ef (relative to the energy of the BCS state). The relevant eigenstates are just the states calculated in eq. (151): |k ≡ c†k3 γ−k↑ |ΨBCS of energy Ω(k). The sum over ﬁnal states becomes a sum over momentum states, and, according to eq. (152), the matrix element is V0 vk . The condition for energy conservation, ω = Ω(k), can be inverted via eq. (153) to give k in k terms of ω. The delta function then becomes δ(ω − Ω(k)) = 1 d dΩ δ( k − (ω)). With ξk dΩ 2 dk = Ek + 1 = 2uk , we obtain the simple expression [152] (155)

/ / π 2 vk2 // Γ(ω) = V0 ρ( k ) 2 / = πNp V02 ρ( k ) |ϕk |2 / =(ω) . k uk k =(ω)

This result shows that RF spectroscopy of the BCS state directly measures the fermion pair wave function ϕk (see eq. (109) and eq. (114)). Note that it is ϕk = vk /uk Np , rather than the Cooper pair wave function ψk = uk vk , that appears here. While the two coincide in the BEC-limit of tightly bound molecules (apart from the normalization with Np ), they are quite diﬀerent in the BCS regime, where ϕk extends throughout the entire Fermi sea, while ψk is peaked in a narrow range around the Fermi surface (see ﬁg. 23). This goes back to the two possible interpretations of the BCS state discussed in . subsect. 4 6.1, either as a condensate of N/2 fermion pairs (eq. (109)) or as the product of a Fermi sea and a condensate of Cooper pairs (eq. (115)). In RF spectroscopy, the ﬁrst point of view is the natural choice, as the RF interaction couples to all momentum states in the entire Fermi sea. We now discuss the spectrum itself. From here on, frequencies ω are given relative to the hyperﬁne frequency ω↑3 . From eq. (153) we see that the minimum or threshold frequency required to excite a particle into state |3 is

(156)

ωth =

μ2 + Δ2 − μ →

⎧ ⎪ ⎨

Δ2 2EF

, in the BCS-limit, 0.31EF , on resonance, ⎪ ⎩ |E | = 2 , in the BEC-limit. 2 B ma

In either limit, the threshold for RF spectroscopy thus measures the binding energy of . fermion pairs (apart from a prefactor in the BCS-limit, see subsect. 4 6.1), and not the superﬂuid gap Δ, which would be the binding energy of Cooper pairs described by ψk .

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W. Ketterle and M. W. Zwierlein

1 To obtain explicitly, we calculate (ω) = 2ω (ω −ωth )(ω +ωth +2μ) / the spectrum 2 2/ 2 2 and vk /uk =(ω) = Δ /(ω) . With ρ(EF ) = 3N/4EF the spectrum ﬁnally becomes k

(157)

3π N V02 Δ2 Γ(ω) = √ 4 2 EF3/2

√

ω − ωth 2 ω 2

1+

2μ ωth + . ω ω

In the BEC-limit, this reduces to (see eq. (139)) 4 ΓBEC (ω) = NM V02 |EB |

(158)

ω − |EB | . 2 ω 2

This is exactly the dissociation spectrum of NM = N/2 non-interacting molecules (compare to the Feshbach association spectrum eq. (218) in sect. 5 for abg → 0). Figure 26 shows the RF spectra for various values of the interaction strength in the BEC-BCS crossover. Qualitatively, the shape of Γ(ω) does not change much, always staying close to the characteristic asymmetric shape of a molecular dissociation spectrum like eq. (158), with the pair binding energy as the only relevant energy scale. For example, the spectrum

1/ kF a Γ(ω) [a.u.]

1.5

-0.5 0

1.0

0.5 0.5

1 1.5

0.0 0

1

2

3

h ω /EF

4

5

6

Fig. 26. – RF spectra in the BEC-BCS crossover for transitions into a third, empty and noninteracting state. The threshold changes smoothly from the binding energy of molecules in the BEC-regime to the binding energy of fermion pairs ∼ Δ2 /EF in the BCS-regime.

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has a maximum at (159)

1 ωmax = −4μ + 16μ2 + 15Δ2 3 ⎧ 5 Δ2 ⎪ in the BCS-limit, ⎨ 8 EF = 54 ωth , → 0.40EF = 1.26 ωth , on resonance, ⎪ ⎩ 8 |E |, in the BEC-limit, B 3

which is always on the order of the fermion pair binding energy. The spectrum falls oﬀ like 1/ω 3/2 at large frequencies. This is due to the asymptotic momentum distribution vk2 ∼ 1/k 4 . The long tails lead to a divergence of the mean transition frequency, so . that the sum rule approach in 2 3.3 does not give a sensible result here. The divergence is removed if the third state interacts with atoms in state |↓. Note that the BCS formalism neglects interactions between spin up and spin down that are already present in the normal state, which may contribute additional shifts and broadening of the spectra. For the superﬂuid 6 Li system in the |1 and |2 states, the accessible ﬁnal state |3 has strong interactions with state |1. Therefore, the experimental spectra in the resonance region [206, 134, 77] are qualitatively diﬀerent from the idealized spectra presented here (see [99]). For recent theoretical studies on RF spectroscopy, incorporating ﬁnal state interactions, see [132, 133, 207]. . 4 7.3. Collective excitations. In addition to single-particle excitations, we have to consider collective excitations related to density ﬂuctuations or sound waves (16 ). Sound modes have a linear dispersion relation Ek = cs k. In the weakly interacting BEC /m limit, the speed of sound is given by the Bogoliubov solution c = μ M M = s 2 4π aM nM /mM . For stronger interactions, the Lee-Huang-Yangexpansion becomes n

a3

M M important, which increases the speed of sound by a factor 1 + 16 [167]. The π Bogoliubov sound mode ﬁnds its analogue in the BCS-regime, where it is called the √ Bogoliubov-Anderson mode, propagating at the speed of sound vF / 3, with vF = kF /m the Fermi velocity(17 ). The connection between the BEC and BCS results is smooth, as expected and found by [188, 208, 209]. Sound waves are described by hydrodynamic equations, which are identical for superﬂuid hydrodynamics and inviscid, classical collisional hydrodynamics. For trapped

(16 ) The reduced BCS Hamiltonian (eq. (117)) does not contain density ﬂuctuations. One needs to work with the Hamiltonian in eq. (116). q (17 ) This speed of sound can be calculated using the hydrodynamic equation c = 2 E 3V

2 E 5 F

2

∂P ∂ρ

2 (3π 2 )2/3 n5/3 . 5 2m

, ρ = mn

= n = Thus, the and the pressure of a normal Fermi gas P = sound mode is already present in the normal Fermi gas, the main eﬀect of superﬂuid pairing being to push low-lying single-particle excitations up in energy, which would otherwise provide damping. The low-temperature limit for the normal gas is peculiar. On the one hand, the damping vanishes at zero temperature, on the other hand, the sound mode cannot propagate, as collisions are absent and the gas can no longer maintain local equilibrium.

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clouds of ﬁnite size, collective modes are the solutions of the hydrodynamic equations for the geometry of the trapped cloud. In a harmonic trap, the size of the cloud depends on the square root of the chemical potential (not counting binding energies), just like the speed of sound. As a result, the lowest lying collective excitations are proportional to the trap frequency and, to leading order, independent of the density and size of the cloud [210]. Modes for which the velocity ﬁeld has zero divergence are called surface modes. Their frequencies are independent of the density of states and do not change across the BCSBEC crossover, as long as the system is hydrodynamic. However, the frequencies are diﬀerent from the collisionless regime (where all frequencies are integer multiples of the trap frequencies) and can be used to distinguish hydrodynamic from collisionless behavior. In contrast, compression modes depend on the compressibility of the gas and therefore on the equation of state. On both the BEC and BCS side and at unitarity, the chemical potential is proportional to a power of the density μ ∝ nγ . The frequency of breathing modes depends on γ, which has been used to verify that γ = 1 on the BEC side and γ = 2/3 at unitarity [73, 74, 211]. For an extensive discussion of collective modes we refer to the contributions of S. Stringari and R. Grimm to these proceedings. . 4 7.4. Landau criterion for superﬂuidity. The Landau criterion for superﬂuidity gives a critical velocity vc , beyond which it becomes energetically favorable to transfer momentum from the moving superﬂuid (or the moving object) to excitations [212]. As a result, superﬂuid ﬂow is damped. Creating an excitation carrying plane-wave momentum k costs an energy Ek + k · v in the rest frame (Doppler shift). The minimum cost occurs naturally for creating an excitation with k antiparallel to the velocity v of the superﬂuid. This is only energetically favored if Ek − kv < 0, leading to Landau’s criterion for the critical velocity: vc = min

(160)

k

Ek . k

The minimum has to be taken over all possible excitations, including single-particle excitations, collective excitations (and, for certain geometries, such as narrow channels and small moving objects, excitation of vortex pairs). On the BCS side, the single-particle excitation spectrum derived above gives a critical velocity of (161)

vc,BCS

Ek = = min k k

μ2 + Δ2 − μ /m

Δ μ →0+

→

Δ . kF

An object that is dragged through the superﬂuid faster than this velocity will break fermion pairs. For sound excitations, the Landau criterion gives the speed of sound as critical velocity. In a simple approximation (neglecting possible coupling between singleparticle and collective excitations), the critical velocity for the superﬂuid is given by the

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Making, probing and understanding ultracold Fermi gases

0.7

cS (Bog.-And.)

vc/vF, cs/vF

0.6 0.5 0.4

Sound waves

Pair breaking

cS (Bog.)

0.3

vC

0.2 0.1 0.0 4

2 1/kFa

0

-2

Fig. 27. – Critical velocity vc in the BEC-BCS crossover. The relevant excitations p in the μ = BEC-regime correspond to Bogoliubov (Bog.) sound waves with speed of sound cs = m √ v √F k a. This sound mode eventually becomes the Bogoliubov-Anderson (Bog.-And.) mode F 3π vF in the BCS-regime, with cs = √ . The evolution is smooth [188, 208, 209], but only the limiting 3 cases are shown here. In the BCS-regime the excitations with the lowest critical velocity are single-particle excitations that break a Cooper pair. Here, vc ≈ kΔF . After [209], also see [383].

smaller of the two velocities. On the BEC side, where the pairs are tightly bound, the speed of sound determines the critical velocity, whereas on the BCS side, the critical velocity comes from pair breaking. For the BEC-side, it has been shown in [209] that for small momenta k 1/a that do not resolve the composite nature of the molecules, the expression for the Bogoliubov 2 2

k + μM )2 − μ2M remains valid even well into the crossover dispersion Ek,BEC = ( 2m M region. This allows us to determine the speed of sound in an approximate way, which is shown in ﬁg. 27. Notable is the sharp peak in the critical velocity around resonance which reﬂects the rather narrow transition from a region where excitation of sound limits superﬂuid ﬂow, to a region where pair breaking dominates. At the same time, the onset of dissipation switches from low k’s (sound) to high k’s (pair breaking). It is near the Feshbach resonance that the superﬂuid is most stable [213, 209]. This makes the critical velocity one of the few quantities which show a pronounced peak across the BEC-BCS crossover, in contrast to the chemical potential, the transition temperature (except for a small hump), the speed of sound and the frequencies of shape oscillations, which all vary monotonically.

. 4 8. Finite temperatures. – At ﬁnite temperature, the superﬂuid state has thermal excitations in the form of the quasi-particles introduced in eq. (134). These quasi-particles modify the gap and number equations for the BCS state from which we derive an expression for the superﬂuid transition temperature.

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. 4 8.1. Gap equation at ﬁnite temperature. At ﬁnite temperature, the expectation value for the pairing ﬁeld Ck = ck↑ c−k↓ becomes † † . ck↑ c−k↓ = −uk vk 1 − γk↑ γk↑ − γk↓ γk↓

(162)

† As the quasi-particles are fermions, they obey the Fermi-Dirac distribution γk↑ γk↑ = 1 . The equation for the gap Δ = VΩ0 k ck↑ c−k↓ thus becomes (replacing V0 as 1+eβEk above by the scattering length a) −

(163)

m = 4π2 a

d3 k

1 tanh 2Ek

3

(2π)

βEk 2

−

1 2 k

.

. 4 8.2. Temperature of pair formation. We are interested in determining the temperature T ∗ = 1/β ∗ at which the gap vanishes. In the BCS-limit, this procedure gives the critical temperature for the normal-to-superﬂuid transition. Setting Δ = 0 in the gap equation, one needs to solve [41, 214, 188] m = − 4π2 a

(164)

d3 k (2π)

3

1 tanh 2ξk

β ∗ ξk 2

1 − 2 k

simultaneously with the constraint on the total number of atoms. Above the temperature T ∗ , we have a normal Fermi gas with a Fermi-Dirac distribution for the occupation of momentum states, so the number equation becomes (165)

n=2

d3 k

1 . ∗ (2π) 1 + eβ ξk 3

In the BCS-limit, we expect μ kB T ∗ and thus ﬁnd μ ≈ EF . Inserted into the gap equation, this gives the critical temperature for BCS superﬂuidity ∗ TBCS = TC,BCS =

(166)

eγ 8 −π/2kF |a| eγ e = Δ0 2 π e π

with Euler’s constant γ, and eγ ≈ 1.78. Here, we distinguish Δ0 , the value of the superﬂuid gap at zero temperature, from the temperature dependent gap Δ(T ). From eq. (163) one can show that (167)

Δ(T ) ≈

Δ 0 −

√ 2πΔ0 kB T e−Δ0 /kB T ,

8π 2 7ζ(3)

kB TC

1−

T TC ,

for T TC , for TC − T TC .

The full temperature dependence is shown in ﬁg. 28. In the BEC-limit, the chemical potential μ = EB /2 = −2 /2ma2 is again given by

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Making, probing and understanding ultracold Fermi gases 1.0

' (T)/'

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

T/TC

Fig. 28. – Temperature dependence of the gap in the BCS regime. Δ(T ) is normalized by its value Δ0 at zero temperature, and temperature is given in units of T ∗ ≈ TC .

half the molecular binding energy as before, and the temperature T ∗ is found to be ∗ TBEC ≈

(168)

|Eb | 1 , 3 W π 13 |Eb | 6

2EF

where W (x) is the Lambert W -function, solution to x = W eW with expansion W (x) ≈ ln(x) − ln(ln(x)) (useful for x 3). ∗ TBEC is not the critical temperature for the superﬂuid transition but simply the temperature around which pairs start to form. The factor involving W (x) has its origin in the entropy of the mixture of molecules and free fermions, which favors unbound fermions and lowers kB T ∗ below the binding energy Eb . There is no phase transition at T ∗ , but a smooth crossover. . 4 8.3. Critical temperature. Determining TC , the temperature at which long-range order is established, requires an additional term in the number equation, namely the inclusion of non-condensed pairs [41, 214, 188, 203, 215]. The result is that in the deep BECregime, the critical temperature is simply given by the non-interacting value for the BEC transition of a gas of molecules at density nM = n/2 and mass mM = 2m, (169)

TC,BEC

2π2 = mM

nM ζ( 23 )

2/3

π2 = m

n 2ζ( 32 )

2/3 = 0.22EF .

This result holds for weakly interacting gases. For stronger interactions, there is a small 1/3 positive correction TC /TC,BEC = 1 + 1.31nM aM , with aM = 0.60a [216-220]. On the BCS-side, the critical temperature should smoothly connect to the BCS result given above. This implies that there must thus be a local maximum of the critical temperature in the crossover [215]. The value of TC at unitarity has been calculated analytically [41, 188, 203, 215], via renormalization-group methods [221] and via Monte Carlo

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W. Ketterle and M. W. Zwierlein

TC / TF, T* / TF

0.25 0.2 0.15 0.1 0.05 0

-2

0

1/kFa

2

4

Fig. 29. – Superﬂuid transition temperature TC and pair creation temperature T ∗ (dashed line) in the BEC-BCS crossover. In the BEC regime, TC corresponds to the BEC transition temperature for a gas of molecules. In the BCS regime, the critical temperature depends exponentially on the interaction strength, drastically reducing TC . TC extracted from [215], T ∗ calculated from eq. (165).

simulations [184, 222]. The result is TC = 0.15 − 0.16TF [215, 222]. Note that these values hold for the homogeneous case, with kB TF = 2 (6π 2 n)2/3 /2m. In the trapped case, they apply locally, but require knowledge of the local TF and therefore the Fermi energy in the center of the trap. This requires knowledge of the central density nU as a function of temperature and the global Fermi energy EF = ¯ ω (3N )1/3 . Using, as a ﬁrst approximation, the zero-temperature relation nU = ξ −3/4 nF from eq. (40), with nF the density of a non-interacting Fermi gas of the same number of atoms, gives kB TC,Unitarity = 0.15 F (0) = 0.15 √1ξ EF ≈ 0.23EF . Figure 29 shows the behavior of TC as a function of the interaction strength. . 4 8.4. “Preformed” pairs. In the region between TC and T ∗ , we will already ﬁnd bound pairs in the gas, which are not yet condensed. In the BCS-limit, where T ∗ → TC , condensation occurs at the same time as pairing, which, as we see now, is no longer true for stronger interactions. Deep on the molecular side, it is of course not surprising to ﬁnd thermal molecules above TC . However, the qualitative picture of thermal (i.e. noncondensed) pairs still holds in the entire crossover region from −1 < 1/kF a < 1. These uncondensed pairs are sometimes called “preformed” (pairing occurs before condensation) and also occur in a part of the phase diagram of high-TC superconductors, the Nernst regime of the pseudogap [44]. . 4 9. Long-range order and condensate fraction. – In this and the following section, we discuss in detail the condensate and superﬂuid fractions. In dilute gas BECs, the diﬀerence between the two quantities is negligible, but their distinction is crucial in the BEC-BCS crossover and in the BCS limit.

Making, probing and understanding ultracold Fermi gases

199

Fritz London proposed in 1938 that superﬂuidity is a quantum-mechanical phenomenon. It should thus be possible to encode the properties of the superﬂuid in a macroscopic wave function that depends only on one or a few coordinates. In the case of Bose gases, it is the 1-particle density matrix that describes superﬂuidity [223, 212], ρ1 (r, r ) = Ψ†B (r)ΨB (r ) ,

(170)

where Ψ†B (r) is the creation operator for a boson at point r. The sum of all eigenvalues of this matrix is equal to the number of particles N . The criterion for Bose-Einstein condensation, as ﬁrst introduced by Onsager and Penrose [223], is the existence of exactly one macroscopic eigenvalue, i.e. with a value of order N . Such a macroscopically occupied state implies long-range order, signalled by an oﬀ-diagonal (r = r ) matrix element that does not vanish for large distances |r − r |. (171)

lim

|r−r |→∞

∗ ρ1 (r, r ) = ψB (r)ψB (r ) ,

where ψB (r) is the macroscopic wave function or order parameter describing the Bose superﬂuid (18 ). “Macroscopic” means that the number of condensed bosons N0 = 3 d r |ψB (r)|2 is extensive, i.e. large compared to 1, or more precisely that the condensate fraction N0 /N is ﬁnite in the thermodynamic limit. n0 (r) ≡ |ψB (r)|2 is the density of the condensed gas. Thus, an absorption image of a weakly interacting Bose-Einstein condensate directly reveals the magnitude of the wave function. This has led to the direct visualization of coherence between two Bose condensates [224], spatial coherence within a condensate [225] and of vortex lattices [226-229]. For fermionic gases, the 1-particle density matrix can never have a macroscopic matrix element, as the occupation number of a particular quantum state cannot exceed unity due to the Pauli principle. After our discussion of fermionic pairing, it does not come as a surprise that for fermionic superﬂuids, long-range order shows up in the 2-particle density matrix [212, 230] (172)

ρ2 (r1 , r2 , r1 , r2 ) = Ψ†↑ (r1 )Ψ†↓ (r2 )Ψ↓ (r2 )Ψ↑ (r1 ) ,

where we added spin labels corresponding to the case of s-wave pairing. Analogous to the Bose case, we can check for the presence of a pair condensate by increasing the distance between the primed and the unprimed coordinates, that is between the two centers of mass R = (r1 + r2 )/2 and R = (r1 + r2 )/2. If there is long-range order, one will ﬁnd a macroscopic “oﬀ-diagonal” matrix element (173)

lim

|R−R |→∞

ρ2 (r1 , r2 , r1 , r2 ) = ψ(r1 , r2 )ψ ∗ (r1 , r2 ) .

(18 ) For a discussion of the relation between condensation and superﬂuidity, see sect. 6.

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The function ψ(r1 , r2 ) = Ψ†↑ (r1 )Ψ†↓ (r2 ) is a macroscopic quantity in BCS theory. It is equal to the Cooper pair wave function discussed above and given by the Fourier † † ∗ transform of the pairing ﬁeld −Ck = ck↑ c−k↓ . The density of condensed fermion pairs n0 (R) is obtained from |ψ(r1 , r2 )|2 by separating center of mass R and relative coordinates r = r2 − r1 and integrating over r: (174) n0 (R) = d3 r |ψ(R − r/2, R + r/2)|2 . More accurately, n0 is the total density of pairs n/2, times the average occupancy of the k paired state. With −Ck∗ = uk vk tanh( βE 2 ), we can calculate the condensate density in a uniform system [231, 194]:

(175)

1 2 2 n0 = uk vk tanh2 Ω k

βEk 2

# √ μ + μ2 + Δ2 1 2 2 3π 2 Δ n = u k vk = . Ω 32 EF EF

T →0

k

The condensate fraction 2n0 /n is non-vanishing in the thermodynamic limit Ω → ∞, n → const, and therefore macroscopic. It is shown in ﬁg. 30 as a function of temperature in . the BCS-regime, and in ﬁg. 46 of subsect. 6 4.1, where it is compared to experimental results in the BEC-BCS crossover. In the BEC-limit, with the help of eq. (138), n0 becomes the density of molecules or half the total atomic density n, as expected, (176)

n0 = n/2

corresponding to a condensate fraction of 100%. As interactions increase, the Bogoliubov theory of the interacting Bose gas predicts that the zero-momentum state occupation den

a3

M M creases and higher momentum states are populated. This quantum depletion is 38 π for a molecular gas of density nM and scattering length aM . At kF a = 1, this would reduce the condensate fraction to 91% (using aM = 0.6 a). The mean-ﬁeld ansatz for the √ BEC-BCS crossover cannot recover this beyond-mean ﬁeld correction proportional to na3 . Indeed, the only way the repulsion between two molecules is built into the mean ﬁeld theory is via the Pauli exclusion principle for the constituent fermions. Rather, eq. (175) predicts a depletion proportional to na3 , which underestimates the true quantum depletion. Monte Carlo studies are consistent with the Bogoliubov correction [232]. On resonance, eq. (175) predicts a condensate fraction 2n0 /n = 70%, whereas the Monte Carlo value is 57(2)% [232]. In the BCS-regime, where μ ≈ EF and the gap is exponentially small, one ﬁnds from eq. (175):

(177)

n0 =

mkF 3π Δ n Δ= . 8π2 16 EF

The condensate fraction thus decreases exponentially with the interaction strength, like

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Making, probing and understanding ultracold Fermi gases 1.0

n0(T)/n, ns(T)/n

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

T/TC

Fig. 30. – Condensate density n0 (straight line) and superﬂuid density ns (dashed line) versus temperature in the BCS-regime (1/kF a = −1). The superﬂuid fraction is 100% at T = 0, while the condensate fraction saturates at 24%. Note that both densities vanish linearly with temperature (within mean-ﬁeld theory) as they approach TC .

the gap Δ. This strong depletion is entirely due to the Pauli principle, which can be seen from eq. (175). Fully occupied (uk = 0) and unoccupied (vk = 0) momentum states do not contribute to the condensate fraction. The bulk contribution comes from states in only a narrow energy range of width ∼ Δ around the Fermi surface, as they are in a superposition of being occupied (with amplitude vk ) or unoccupied (amplitude uk ). Their total number is ∼ N Δ/EF (see eq. (177)). The phase of this superposition state (the relative phase between the complex numbers uk and vk , the same for all k) deﬁnes the macroscopic phase of the superﬂuid state. Indeed, introducing a global phase * factor eiα into the BCS state, |ψBCS = k (uk + eiα vk c†k↑ c†−k↓ ) |0 is equivalent to a

rotation the “coherent state” in eq. (114) by an angle α, from exp Np b† |0 to of iα exp Np e b† |0. This is in direct analogy with BEC and the optical laser. . 4 10. Superﬂuid density. – It is important to distinguish the density of condensed fermion pairs n0 , which is smaller than the total density even at zero temperature, from the superﬂuid density ns . The superﬂuid density is the part of the system that does not respond to external rotation or shear motion. At zero temperature the entire system is superﬂuid and thus ns = n. As discussed above, one encounters this diﬀerence between n0 and ns already in BECs [233]. Figure 30 compares the two quantities for the BCSregime. The distinction between superﬂuid and normal density nn provides the basis of the two-ﬂuid hydrodynamic model of superﬂuids and superconductors [234, 235]. To obtain ns , one can place the system in a long tube that is slowly set in motion with velocity . v [236]. According to Landau’s criterion (4 7.4), as long as v < vc , no new excitations from the superﬂuid are created, so the superﬂuid stays at rest. However, due to friction with the walls of the tube, the collection of already existing excitations will be dragged along by the tube. The total momentum density P of the system is thus only due to

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W. Ketterle and M. W. Zwierlein

this normal gas of excitations, P = nn mv, which deﬁnes the normal density nn . The superﬂuid density is then ns = n − nn . . We have seen in subsect. 4 7 that there are two types of excitations in a fermionic superﬂuid: Excitations of fermionic (quasi-)particles, related to pair breaking, and bosonic excitations of pairs of fermions. Both types will contribute to the normal density [212, 237-239]. Single-particle excitations are frozen out for temperatures well below the characteristic temperature T ∗ for pair formation. In a molecular BEC, TC T ∗ , and fermions are strongly bound. The only relevant thermal excitations are thus due to non-condensed pairs. For kB T μM , the excitations are dominantly phonons. In the BCS-regime, TC = T ∗ , and the normal density contains both single-fermion excitations from broken pairs as well as bosonic excitations of pairs of fermions, the Bogoliubov. Anderson sound mode (see 4 7.3). Near TC , which is close to T ∗ in the BCS regime, single fermion excitations dominate. At low temperatures kB T Δ, they are frozen out and the contribution from sound waves dominates. At intermediate temperatures, the two types of excitations are coupled, leading to damping of the sound waves [239]. The normal density is obtained from the total momentum of the gas of excitations, that moves with velocity v with respect to the stationary superﬂuid part. In the reference frame moving with the normal gas, the excitation energies of the superﬂuid are Doppler shifted to − k · v [212]. The momentum is thus (178)

PB,F =

k fB,F ( − k · v) ,

k

where the subscripts B or F correspond to the bosonic and fermionic contribution, respectively, and fB,F ( ) is the Bose-Einstein and Fermi-Dirac distribution, respectively. For small velocities, this gives (179)

PB,F =

k

∂fB,F k(k · v) − ∂

∂fB,F 1 2 2 k − = v . 3 ∂ k

The last equation follows from spherical symmetry, obeyed by the energy levels and the gap Δ in an s-wave fermionic superﬂuid. It implies that P is in the direction of v and allows to replace (k · vv )2 by its angular average, 13 k 2 . The ﬁnal formula for the normal density is, with k = 2 k 2 /2m [212], nB,F n

(180)

21 ∂fB,F = k − . 3Ω ∂ k

Contribution from sound waves. Sound waves have = cs k and thus (181)

nB n

=− 3mcs

4 4 T 2π 2 kB π4 d3 k 2 ∂fB = n k = (2π)3 ∂k 45 m3 c5s 120

kB T EF

4

vF cs

5 .

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Making, probing and understanding ultracold Fermi gases

4 In the crossover and on the BCS-side, cs ≈ vF , and so nB n /n ≈ (kB T /EF ) , a small contribution that dominates only for kB T Δ (see below).

Contribution from fermionic quasi-particle excitations. Fermionic quasi-particle excitations have = Ek . Spin up and spin down excitations both contribute, giving a normal density (182)

nF n =

1 41 ∂fF 1 . k − β k = 2 3Ω ∂Ek 3Ω cosh βEk k

k

2

Via partial integration, it is not hard to see that for Δ = 0, nF n = n, that is, the entire system is normal and consists exclusively of thermally excited quasi-particles. This is because in mean-ﬁeld BCS theory, Δ = 0 implies that T > T ∗ , the temperature for pair formation. Below T ∗ , both quasi-particles and thermal pairs contribute to the normal gas. Below TC , the superﬂuid density ns becomes non-zero. In the BCS-regime, we ﬁnd [240, 212]

(183) (184)

2πΔ0 −Δ0 /kB T e , kB T

7ζ(3) T ns = n 2 2 Δ2 = 2 n 1 − , 4π TC TC nF n =n

for T TC , for T ≈ TC .

Close to TC , the superﬂuid density is proportional to the square of the gap. This provides a natural normalization of the superﬂuid order parameter in the next section. The exponential suppression of the quasi-particle contribution at low temperatures is characteristic for a gapped excitation spectrum. At temperatures kB T Δ, bosonic sound waves dominate the normal component. In the BEC-regime, the role of the excitation gap is played by |μ|, which is (half) the binding energy of molecules. Hence, already far above TC , fermionic excitations are frozen out and exponentially suppressed like e−|μ|/kB T . Bosonic excitations dominate at all temperatures T < TC . . 4 11. Order parameter and Ginzburg-Landau equation. – A Bose superﬂuid is described by ψB (r), the macroscopic wave function or order parameter. For fermionic superﬂuids, ψ(r1 , r2 ) is the wave function for fermions bound in Cooper pairs in the condensate. Then the function (185)

ψC (R) ≡ ψ(R, R)

describes the motion of the center of mass of these pairs and lends itself as the order parameter for a fermionic superﬂuid. In a uniform system, ψC (R) is a constant proportional

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to the gap Δ: (186)

ψC (R) =

1 † † 1 ck↑ c−k↓ = − Δ , Ω V0 k

where we have used the gap equation (19 ). This can be extended to a non-uniform system in which the density and Δ(R) does not vary rapidly (local density approximation). One should point out that it is the presence of a non-zero order parameter, deﬁned via the two-particle density matrix, that signals superﬂuidity, not the presence of a gap in the excitation spectrum. Gapless superﬂuidity might occur when the quasi-particle excitations are diﬀerent for spin-up and spin-down fermions, one branch touching zero (for example Ek↓ = 0) close to a second-order phase transition to the normal state. Such breaking of time-reversal symmetry leading to gapless superconductivity can occur for example in thin superconducting ﬁlms in a magnetic ﬁeld, or in the presence of magnetic impurities [241, 242]. Close to TC , the order parameter will be small, and after Ginzburg and Landau one can expand the free energy of the superﬂuid in terms of the small parameter ψC (r). From here, one derives the famous Ginzburg-Landau equation for the order parameter [240,212] (187)

−

2 ∇2 2 ψC + a ψC + b |ψC | ψC = 0 . 2m∗

The Ginzburg-Landau theory was developed for superconductors on purely phenomenological grounds in 1950, before the advent of BCS theory. m∗ was introduced as the mass of the “superelectrons” carrying the supercurrent. It is conventional to choose m∗ = 2m, the mass of a fermion pair. However, this choice modiﬁes the normalization of ψC from 2 eq. (186) to |ψC | = ns /2, one-half the superﬂuid density (20 ). This is consistent with eq. (184), which shows that ns ∝ Δ2 close to TC . Note that one could have equally well normalized |ψC |2 = n0 via the density of condensed fermion pairs, as this also vanishes like n0 ∝ T − TC ∝ Δ2 at TC . This would, however, change the mass m∗ into 2 m n0 /ns . The parameter b has to be positive, otherwise one could gain energy by making |ψC | arbitrarily large. In a uniform system, the squared magnitude of the order parameter, in 2 the superﬂuid state, is |ψC | = −a/b, which should start from zero at T = TC and then grow. Taylor expansion gives a(T ) = a (T − TC )/TC . The Ginzburg-Landau equation was later derived from BCS theory by Gorkov. With the choice m∗ = 2m, his derivation 6(πTC )2 and b = a /n. gives [240] a = 7ζ(3)E F (19 ) In the BEC-regime, one needs to include thermal molecules in the number equation if Δ is to vanish at T = TC [41, 188]. (20 ) The origin of this normalization is the free energy density F of a superﬂow with velocity vs = ∇φ/2m, where φ is the phase of the wave function ψC = |ψC | eiφ . By deﬁnition of the superﬂuid density, F = 21 mvs2 ns = 18 2 |∇φ|2 /m, but in terms of ψC we have F = 12 2 |ψC |2 |∇φ|2 /m∗ . From m∗ = 2m there follows |ψC |2 = ns /2.

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The Ginzburg-Landau equation has exactly the form of a non-linear Schr¨ odinger equation for the center-of-mass wave function of a fermion pair. In the BEC-regime at T = 0, a rigorous microscopic theory, which does not require a small order parameter is the Gross-Pitaevskii equation describing the condensate of molecules. It is formally identical to eq. (187) if we set −a = μM , the chemical potential of molecules, and b = 4π2 aM /mM , describing the interactions between molecules. In a uniform system and at T = 0, −a = b nM , as |ψC |2 = nM = n/2 in the BEC-regime. For a non-uniform system, eq. (187) deﬁnes a natural length scale over which the order parameter varies, the Ginzburg-Landau coherence length ⎧ 1/2 # ⎪ TC ⎨ 0.74 ξBCS , in BCS-regime , 2 TC −T = (188) ξGL (T ) = 1/2 4m |a| ⎪ TC ⎩ ξBEC , in the BEC-regime , TC −T 1 ξBEC = 8πaM nM is the healing length of the molecular condensate. ξGL becomes very large close to the critical temperature, and in particular it can be large compared to the BCS-coherence length ξBCS = vF /πΔ0 , deﬁned above via the zero-temperature gap Δ0 . Spatial variations of the wave function ψC then occur at a length scale much larger than the size of a Cooper pair, and in this regime, the wave function can be described by a local equation, although the pairs are extended [243]. While the G.-L. equation was originally derived close to TC , assuming a small order parameter, its validity can be extended to all temperatures under the only condition that Δ(r) varies slowly compared to ξGL (0) [244]. This condition is less and less stringent as we cross-over into the BEC-regime of tightly bound molecules. Note that very close to TC , ﬂuctuations of the order parameter are large and the G.-L. equations are no longer valid. The size of this critical region is given by |T − TC |/TC (TC /EF )4 in the BCS-regime, and |T − TC |/TC kF a in the BEC-regime [188]. The correlation length then diverges as (TC −T )−ν and the superﬂuid density vanishes as [212] (TC − T )(2−α)/3 with universal critical exponents α and ν, instead of the linear behavior ∝ TC − T implied in the Ginzburg-Landau theory (α ≈ 0 and ν ≈ 0.67 for a complex scalar order parameter in 3D [245, 246]).

Detecting the order parameter . One appealing feature of dilute gas experiments is the ability to directly visualize the order parameter. In the BEC-limit, the entire gas is condensed. As with atomic BECs, density proﬁles of the molecular gas then directly measure the condensate density n0 . In particular, the contrast of interference fringes and of vortex cores approaches 100%. However, in the BCS regime, the condensate fraction decreases. Furthermore, pairs dissociate in ballistic expansion. This can be avoided by ramping towards the BEC-regime during expansion. As described in sect. 6, it has been possible to observe condensates and vortices across the entire BEC-BCS crossover.

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. 4 12. Crossing over from BEC to BCS . – Throughout the BEC-BCS crossover, all quantities vary smoothly, many of them even monotonously with 1/kF a. Still, the question has often been raised in what region(s) of the crossover qualitative changes occur. When the initial observations of condensation of fermion pairs were announced [65, 55, 66, 67, 69, 70], the value 1/kF a = 0 was regarded as special, since this value separates the regimes where two atoms in isolation will or will not form a weakly bound pair. Observations at 1/kF a > 0 were classiﬁed as molecular condensates, and those at 1/kF a < 0 as fermionic condensates. However, it is clear that the absence or presence of an extremely weakly bound two-body state (with a 1/kF ) does not affect the many-body system, since many-body pairing is dominant in this regime. In the following section, we summarize all qualitative criteria we are aware of, which deﬁne speciﬁc values of 1/kF a where qualitative changes in physical properties occur. Of course, diﬀerent criteria lead to diﬀerent values. It appears that for the case of a broad Feshbach resonance, all important qualitative changes occur in the window 0.2 < 1/kF a < 0.9. We therefore suggest that one should refer to molecular BEC only in the regime 1/kF a > 1. Although BCS theory seems to be qualitatively correct already for 1/kF a < 0, we refer to the whole region 1/kF |a| < 1 as the crossover region, in accordance with most other authors. It seems most natural to use the word fermionic condensates for the regime with 1/kF a < −1 and apply it to superconductors, superﬂuid 3 He and the atomic Fermi gases. The big and unique accomplishment of the ﬁeld of ultracold atoms has been the creation of the ﬁrst crossover condensates, which connect two regimes that could be studied only separately before. In this crossover regime, bosonic and fermionic descriptions are merged or co-exist. – Excitation spectrum. At μ = 0, the character of single-particle excitations changes (see ﬁg. 25): For μ < 0, the minimum excitation energy lies at k = 0, while for μ > 0, the minimum occurs at non-zero momenta, around k = kF in the BCSregime. In the BCS mean-ﬁeld solution, this point lies at 1/kF a = 0.55, a more reﬁned theory gives 1/kF a = 0.41 [215]. – Critical velocity. In the BEC-limit, the critical velocity is due to excitations of sound waves, while in the BCS-regime, it is determined by pair breaking. Both types of excitations become more costly closer to resonance, and consequently there is a maximum in the critical velocity that occurs at 1/kF a ≈ 0.3 (see ﬁg. 27). – Normal density. Close to TC and in the BEC-limit, nn is dominated by bosonic excitations, thermal fermion pairs, while in the BCS-limit it is broken pairs that contribute mostly to the normal density. The point where the two contributions are equal lies at about 1/kF a ≈ 0.2, and the crossover between the two occurs rapidly, between 1/kF a ≈ 0 and 0.4 [238]. – Balanced superﬂuidity. The pairing gap Δ in the BCS-regime presents a natural barrier for excess atoms to enter the superﬂuid. A dramatic consequence of this is the observable phase separation in a trap between an equal superﬂuid mixture

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of spin up and spin down atoms and a normal imbalanced gas surrounding it [80]. In fact, on the BCS-side this is the consequence of a ﬁrst-order phase transition between the balanced superﬂuid and the normal gas [247]. Only if the chemical potential diﬀerence μ↑ − μ↓ between the spin-up and spin-down species becomes larger than 2Δ (or 2 μ2 + Δ2 if this occurs at μ < 0), can unpaired atoms (quasiparticles) enter the superﬂuid. This is likely to occur around μ = 0 [248], so again on the BEC-side of the resonance, around 1/kF a = 0.41. – Absence of unpaired minority fermions. As we already introduced imbalanced Fermi systems, we can go to the extreme case of a single spin-down atom emersed in a sea of spin-up atoms. A natural question to ask is: Will the single spin-down fermion still form a “monogamous” molecular pair with a spin-up fermion (BEC limit), or will it rather interact with an entire collection of majority atoms (“polaron” or polygamous pairing)? This intriguing question has recently been studied via a diagrammatic Monte Carlo calculation, and a critical interaction strength 1/kF a = 0.90(2) separating the two regimes of “pairing” has been found [249]. From that point on the situation can be described as an interacting Bose-Fermi mixture of bosonic molecules and single unpaired fermions. – Pair condensate in the presence of a Fermi sea. On the BCS side and on resonance, a large population imbalance destroys superﬂuidity (Clogston-Chandrasekhar-limit, . see subsect. 7 3.2). It is a feature of the BEC limit that a small number of molecules can condense even in the presence of a large Fermi sea of one of the two spin components. This “BEC” property is probably lost around the point where the chemical potential μ↑ + μ↓ becomes positive, as a very small molecular BEC will have μ ≈ 0. It is likely that a necessary and suﬃcient criterion for having a BEC is the existence of a “monogamous” molecular paired state, so again 1/kF a ≈ 0.9. – Critical temperature. On the BEC-side, the critical temperature is given by the value for a non-interacting gas of bosonic molecules, TC,BEC = 0.22 TF (see sub. sect. 4 8.3). For increasing interactions, the critical temperature ﬁrst increases, before it drops to TC,Unitarity ≈ 0.15 TF at unitarity and then to exponentially small values on the BCS-side of the resonance. There is thus a (low-contrast) maximum of the critical temperature in the crossover, which lies around 1/kF a ≈ 1.3 [215]. – Pair size. Another crossover occurs in the pair size, which can be smaller or larger than the interparticle spacing. With the deﬁnition for the pair size given above . (subsect. 4 6.3), we found ξ0 ≈ 1/kF on resonance, 1/3 of the interparticle spacing. At kF a ≈ −0.9, ξ0 ≈ n−1/3 . Of course, diﬀerent deﬁnitions of an average size can easily diﬀer by factors of 2 or 3, and it is not clear whether the pair size should be compared to n−1/3 or to 1/kF . However, it is clear that long-range Cooper pairs as found in superconductors — with many other particles ﬁtting in between — are only encountered for 1/kF a −1.

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. – Narrow Feshbach resonance. For a narrow Feshbach resonance (see subsect. 5 4.4), the crossover from closed channel dominated molecular BEC to open channel BCStype superﬂuidity occurs at 1/kF a −1. – Equation of state exponent (ﬁg. 15). The exponent γ in the approximate equation of state μ(n) ∼ nγ , as calculated in the BEC-BCS model, has a (low-contrast) minimum at 1/kF a ≈ −0.45. γ changes from the bosonic value (γ = 1) on the BECside to the fermionic value (γ = 2/3) on the BCS-side. However, since universality demands γ = 2/3 already on resonance, the crossover region is located mainly between 1/kF a = 1 and 1/kF a = 0. Near the Feshbach resonance, but still on the BEC side, the equation of state is then already fermionic (γ ≈ 2/3), which is a strong reason not to call this system a molecular BEC. 5. – Feshbach resonances Feshbach resonances are crucial for the study of strongly interacting fermions. Typical scattering lengths in alkali atoms are on the order of the van der Waals range r0 ≈ 50 − 100 a0 . Common interparticle spacings in ultradilute gases are n−1/3 ∼ 10 000 a0 , corresponding to kF = (2 500 a0 )−1 . For such small interaction strengths kF |a| ∼ 0.03, the critical temperature for achieving fermionic superﬂuidity is exponentially small, TC ≈ 10−23 TF . Clearly, one requires a way to enhance the interatomic interactions, for example via scattering resonances. Early on, 6 Li was considered as an exception and as a promising candidate to achieve fermionic superﬂuidity [13], as its triplet scattering length was found to be unusually large and negative, about −2 000 a0 [12]. The reason is that the (triplet or electronspin aligned) interatomic potential of 6 Li could, if it were just a bit deeper, support an additional bound state, so low-energy collisions are almost resonant. What ﬁrst seemed to be special for 6 Li, namely a large negative scattering length, can now be created in many two-atom systems by tuning the scattering length near a Feshbach resonance. These resonances occur as a bound state in the interatomic potential is tuned into resonance with the energy of two colliding atoms. This tuning is possible via an applied magnetic ﬁeld if the magnetic moment of the bound state diﬀers from that of the two unbound atoms. In this sect., we provide a quantitative description of Feshbach resonances. Our goal is to provide a thorough discussion of the conditions on which the closed channel molecular state can be eliminated, so that the physics is reduced to potential scattering (so-called single channel scattering). This is the case of the so-called broad Feshbach resonance. We start ﬁrst by summarizing the features of scattering resonances in a single channel, by using the attractive spherical-well as an exactly solvable example, and then present a model for Feshbach resonances. . 5 1. History and experimental summary. – Herman Feshbach introduced a formalism to treat nuclear scattering in a uniﬁed way [250, 251]. In elastic collisions, for example, a free nucleon colliding with a target nucleus can undergo resonant scattering. This occurs

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209

whenever the initial scattering energy is equal to that of a “closed channel” bound state between the nucleon and nucleus in the absence of the incoming “scattering channel”. A “closed channel” has a higher asymptotic energy than the “incoming” or initial scattering energy and inelastic decay into such a channel is energetically forbidden. The Feshbach formalism allowed to treat scattering entirely in the “open channel” by introducing an eﬀective potential that described coupling into and out of the closed channel. In atomic physics, a related type of resonance is encountered for example in highly excited atoms and ions, where a discrete autoionized state is coupled to a continuum of scattering states. Various aspects of such resonances were studied by Fano [252]. Feshbach resonances at zero energy are realized by tuning an external magnetic ﬁeld. This was predicted for hydrogen in 1976 [5] and for cold alkali atoms in 1993 [6]. In cold atom experiments, the initial emphasis was on modiﬁcation of elastic and inelastic atomic collisions [7, 8, 148], but it soon turned out that Feshbach resonances opened a new avenue towards ultracold molecules: Instead of cooling the molecules themselves, it became possible to create them cold by associating ultracold atoms. The ﬁrst observation of a Feshbach resonance in ultracold atoms [7, 10] showed strong losses in the atomic signal that were attributed [253, 140, 254] to the formation of ultracold, highly vibrationally excited molecules. However, it was predicted that these molecules, formed out of two bosonic atoms, would undergo fast vibrational relaxation into more tightly bound molecular states. Still, in experiments on 85 Rb, the presence of the molecules, as short-lived as they were (lifetime ∼ 100 μs), could be detected via coherent beats between the free atomic and the bound molecular state [255]. Studies of the decay of fermionic gases close to a Feshbach resonance [56, 124, 18] held a peculiar surprise: The maximum atom loss was not centered on resonance, but was shifted towards regions where the Feshbach molecular state was already quite deeply bound. The gas close to resonance was stable [56,60,124,18], in stark contrast to the bosonic case. This molecular state could be reversibly populated via a magnetic ﬁeld sweep across resonance [61], at a conversion eﬃciency exceeding 90% [15, 141]. Most importantly, it was found to be long-lived [15, 17, 16, 18], with lifetimes between about 100 ms (for 40 K) and several 10 s (for 6 Li). This is to be compared to the molecular lifetimes on the order of only 5 ms observed in bosonic gases [62-64]. The remarkable stability of fermion dimers near Feshbach resonances is directly linked to the Pauli principle [200]: The characteristic size of dimers is a, the scattering length for atom-atom collisions. A relaxation into more deeply bound molecular states of size r0 (roughly the van der Waals-range) requires at least three fermions to be within a distance r0 from each other. As two of them necessarily have the same spin, the relative wave function has to be antisymmetric, i.e. it has a node when the relative distance r = 0 and varies proportional to kr for small values of r, where k ∼ 1/a is the characteristic momentum spread of the dimer. This suppresses relaxation processes by a certain power of (kr0 ) ∼ (r0 /a). For dimer-dimer scattering, the power is 2.55 [200]. What is crucial for this suppression is the Pauli principle and the large ratio between initial and ﬁnal size. For bosons, the reverse is true, i.e. the relaxation rate diverges with a3.5 to a4 [256-260], although the overlap integral between initial and ﬁnal state decreases. The ratio between good to bad collisions can be very

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high for fermion dimers near Feshbach resonances, since in contrast to inelastic collisions, elastic scattering is not suppressed. For an extensive discussion of dimer stability, we refer the reader to the lecture notes of G. Shlyapnikov in these proceedings.

. 5 2. Scattering resonances. – We summarize ﬁrst some results for the attractive spherical-well potential which are derived in many text books. Our model for Feshbach resonances will have a region in detuning around the resonance, where the interaction and the scattering look very similar to the case of the spherical well. A three-dimensional spherical well potential of radius R and depth V has scattering states with energy E > 0 and also bound states with energy E < 0 when the depth is larger than a critical value Vc . We deﬁne E = 2 k 2 /m for E > 0 and wave vector k, |E| = 2 κ2 /m for E < 0, V ≡ 2 K 2 /m and ER = 2 /mR2 . The critical well depth is 2 Vc = π4 ER . New bound states appear when Kn R = (2n + 1)π/2 at Vn = (2n + 1)2 Vc . In the ultracold regime E ER , or equivalently kR 1, and for E V , scattering states have the same radial wave function inside the well as bound states with |E| V : u(r < R) = A sin(Kr). Outside the well, u(r) for scattering states is of the form u(r > R) = sin(kr + δs ). Matching value and slope of u(r > R) and u(r < R) at r = R ﬁxes the phase shift δs via the condition: k cot(kR + δs ) = K cot(KR). The scattering length is (189)

tan KR tan δs =R 1− . k KR k1/R

a = − lim

By expanding k cot δs = −1/a + 12 reﬀ k 2 + . . . , we obtain the eﬀective range reﬀ = R − 1/(K 2 a) − 13 R3 /a2 , which is small and on the order of R, unless the well is very shallow, K 2 Ra 1, or a is smaller than R. We will see below that Feshbach resonances lead to negative values of reﬀ that can be large. The scattering length in eq. (189) diverges whenever a new bound state enters the potential. This relationship applies to any potential scattering of ﬁnite range R: A diverging scattering length signiﬁes that the phase shift δs due to the potential well is approaching π/2. At R, we then have a normalized slope u (r)/u(r) ≈ k/ tan δs = −1/a for the scattering wave function. For positive a, this can just as well be continued by a bound-state wave function e−κr with matching slope, which gives κ = 1/a. So apart from the scattering solution, we ﬁnd a new bound state solution of Schr¨ odinger’s equation at negative energy (190)

EB = −2 /ma2 ,

for a > 0.

Away from resonances, the scattering length is close to the “background” scattering

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Energy

Energy Bound State Δμ B

Interatomic Distance

B0

Magnetic Field

Fig. 31. – Origin of Feshbach resonances. Atoms entering for example in the triplet potential are coupled to a singlet bound molecular state. By tuning the external magnetic ﬁeld, this bound state can be brought into resonance with the incoming state (at B0 in the graph on the right).

length R. Close to a resonance (at Vn = 2 Kn2 /m), the scattering length diverges as (191)

a≈

22 1 ≈ KR(K − Kn ) mR(V − Vn )

and for a > 0 the bound state energy depends on V like (192)

EB = −

2 1 (V − Vn )2 =− . 2 ma 4 ER

. This general behavior for weakly bound states was found already in subsect. 4 3.1: The binding energy depends quadratically on the “detuning”, and the scattering length is inversely proportional to the “detuning”. The beauty and power of Feshbach resonances is that this detuning is now controlled by an externally applied ﬁeld.

. 5 3. Feshbach resonances. – We now turn to realistic interaction potentials between alkali atoms. Here, the interaction actually depends on the internal structure of the two colliding atoms, namely on the relative spin orientation of their valence electrons, singlet or triplet. In ﬁg. 31 for example, the atoms enter in a triplet conﬁguration. If there was no coupling between the singlet VS and the triplet potential VT , the atoms would simply scatter oﬀ each other in VT (r), acquiring some certain, ﬁxed phase shift. However, the hyperﬁne interaction Vhf is not diagonal in the total electronic spin S = s1 + s2 of the

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|k

>

0

>

|m

|ϕ

0

>

Detuning δ

δ0

Fig. 32. – Simple model for a Feshbach resonance. The dashed lines show the uncoupled states: The closed channel molecular state |m and the scattering states |k of the continuum. The uncoupled resonance position lies at zero detuning, δ = 0. The solid lines show the coupled states: The state |ϕ connects the molecular state |m at δ 0 to the lowest state of the continuum above resonance. At positive detuning, the molecular state is “dissolved” in the continuum, merely causing an upshift of all continuum states as ϕ becomes the new lowest continuum state. In this illustration, the continuum is discretized in equidistant energy levels. In the continuum limit (ﬁg. 33), the dressed molecular energy reaches zero at a ﬁnite, shifted resonance position δ0 .

two atoms and thus provides a coupling between singlet and triplet potentials [261]: (193)

Vhf = ahf (s1 · i1 + s2 · i2 ) ahf ahf S (i1 + i2 ) + (s1 − s2 ) (i1 − i2 ) = 2 2 = Vhf+ + Vhf−

with the hyperﬁne constant ahf and the nuclear spins i1,2 of the two atoms. The coupling Vhf− connects singlet and triplet states since the operator s1 − s2 is antisymmetric in 1 and 2, and therefore couples symmetric (triplet) electronic spin states to antisymmetric (singlet) states. It is thus fully oﬀ-diagonal in the singlet/triplet basis, implying that coupling matrix elements are on the order of unity. Vhf− should thus have matrix elements on the order of ahf . The singlet potential is a “closed channel”, meaning that singlet continuum states are not available as ﬁnal scattering states by energy conservation. A Feshbach resonance occurs when the state that the atoms collide in (the “incoming” state) is resonant with a bound state in this singlet potential. The energy diﬀerence between the incoming and the Feshbach bound state can be tuned via an applied magnetic ﬁeld, due to their diﬀerent magnetic moments (see ﬁg. 31).

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. 5 3.1. A model for Feshbach resonances. Good insight into the Feshbach resonance mechanism can be gained by considering two coupled spherical-well potentials, one each for the open and closed channel [262]. Other models can be found in [142, 247]. Here, we will use an even simpler model of a Feshbach resonance, in which there is only one bound state of importance |m in the closed channel, the others being too far detuned in energy (see ﬁg. 32). The continuum of plane waves of relative momentum k between the two particles in the incoming channel will be denoted as |k. In the absence of coupling, these are eigenstates of the free Hamiltonian (194)

H0 |k = 2 k |k , H0 |m = δ |m ,

with k = with δ ≶ 0 ,

2 k 2 > 0, 2m

where δ, the bound state energy of the “bare” molecular state, is the parameter under experimental control. We consider interactions explicitly only between |m and the |k’s. If necessary, scattering that occurs exclusively in the incoming channel can be accounted for by including a phase shift into the scattering wave functions |k (see eq. (212)), i.e. using ψk (r) ∼ sin(kr + δbg )/r, where δbg and abg = − limk→0 tan δbg /k are the (so-called background) phase shift and scattering length, respectively, in the open channel. First, let us see how the molecular state is modiﬁed due to the coupling to the continuum |k. For this, solve (195)

H |ϕ = E |ϕ , with |ϕ = α |m +

ck |k ,

k

for E √ < 0, where H = H0 +V and the only non-zero matrix elements of V are m| V |k = gk / Ω and their complex conjugates (we will take gk to be real). Ω is the volume of the system and introduced in this deﬁnition for later convenience. We quickly ﬁnd (196)

gk (E − 2 k ) ck = √ α , Ω 1 1 gk2 α (E − δ) α = √ gk ck = , Ω E − 2 k Ω k k

and thus E−δ =

1 gk2 . Ω E − 2 k k

We only consider low-energy s-wave scattering, where the range of the potential r0 is much smaller than the de Broglie wave lengths, r0 1/k. The closed channel molecular state |m will have a size R on the order of r0 , that the de Broglie waves of colliding atoms cannot resolve. The couplings gk will not vary much for such low-energy collisions with

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k 1/R. One can thus take gk ≈ g0 constant, up to a natural cut-oﬀ ER = 2 /mR2 , and gk = 0 beyond. We then ﬁnd g2 |E| + δ = 0 Ω

(197)

ER

d 0

ρ( ) . 2 + |E|

The integral on the RHS is identical to that in the bound-state equation in one-channel scattering, eq. (101). There, the LHS was simply the inverse scattering strength −1/V0 . The two-channel problem introduces an energy-dependence in the strength of the potential, V0 → g02 /(E − δ). The integral gives |E| + δ =

(198)

g02 ρ(ER ) Ω

⎧ # ⎫ # ⎨ |E| 2ER ⎬ arctan 1− ⎩ 2ER |E| ⎭

δ0 − 2E0 |E| for |E| ER , R δ0 23 E for |E| ER , |E|

≈ with

δ0 ≡

g02 1 4 = E0 ER Ω 2 k π k

and E0 ≡

2 g02 m 3/2 . 2π 22

E0 is an energy scale associated with the coupling constant g0 . As illustrated in ﬁg. 32, for positive detuning δ > δ0 the original molecular state is “dissolved” in the continuum. Due to the coupling of the molecular state with the continuum, the resonance position is shifted by δ0 . For δ − δ0 < 0, we ﬁnd a true bound state at (199)

E=

−E0 + δ − δ0 + δ 2

−

δ2 4

+

(E0 − δ + δ0 )2 − (δ − δ0 )2 ,

2 3 δ 0 ER ,

for |E| ER , for |E| ER .

The “dressed” bound-state energy E is shown in ﬁg. 33. Far away from the resonance 3 region, for δ −(ER E0 )1/4 , one ﬁnds E ≈ δ, thus recovering the original bound 2 state. On the other hand, close to resonance, the energy E ≈ − 12 (δ − δ0 ) /E0 = 2 − π82 ER (δ − δ0 ) /δ02 depends quadratically on the detuning δ − δ0 , as expected. Scattering amplitude. To ﬁnd the scattering amplitude, we solve Schr¨ odinger’s equation for E > 0. A small imaginary part iη with η > 0 is added to the energy to ensure that the solution will correspond to an outgoing wave. The goal is to see how the coupling to the molecular state aﬀects scattering in the incoming channel. In an approach formally

Making, probing and understanding ultracold Fermi gases

215

equivalent to the solution for bound states, eq. (197), we ﬁnd for the amplitudes ck in the open channel: (200)

gk 1 g gk √ √q cq ≡ (E − 2 k ) ck = √ α = Veﬀ (k, q) cq . Ω Ω E − δ + iη Ω q q

By eliminating the closed-channel molecular amplitude α from the equations, the scattering problem is now entirely formulated in the open channel. The molecular state causes an eﬀective interaction Veﬀ that corresponds to two atoms colliding and forming a g molecule (matrix element √qΩ ), spending some small amount of time (of order /(E − δ)) 1 ) and exiting again as two unbound atoms in the molecular state (propagator E−δ+iη g k (matrix element √Ω ). The s-wave scattering amplitude can now be obtained using the . general expression we found in subsect. 4 1, eq. (91). We only need to insert the eﬀective potential Veﬀ (k, q) in place of V (k − q)/Ω. The problem is simpliﬁed by setting as before all gk = g0 for E < ER and gk = 0 for E > ER . The replacement is (201)

V0 → Veﬀ Ω =

g02 E−δ

as we had found for the bound-state problem, and eq. (91) becomes (202)

4π2 1 ≈− (E − δ) + 4π f0 (k) mg02

1 d3 q . 3 2 (2π) k − q 2 + iη

The integral on the RHS is identical to what one encounters in one-channel scattering: it generates the necessary −ik (see eqs. (83) and (94)), it determines the resonance position √ d3 q 1 2 2 by introducing a shift −4π (2π)3 q2 = − π R and it contributes to the eﬀective range reﬀ with a term ∝ R (which we neglect in the following, as ER is taken to be the largest energy scale in the problem). All the physics of the two-channel model is contained in the ﬁrst term, which includes the molecular state energy δ and a term proportional ∝ k2 d3 q 1 to E4π 2 that will give another contribution to the eﬀective range. Using −4π (2π)3 q2 = − mg2δ0 0 and replacing g0 in favor of E0 via eq. (199), we have (203)

# m 22 2 1 1 ≈− (δ0 − δ) − k − ik . 2 f0 (k) 2 E0 2 mE0

The scattering amplitude is now in the general form of eq. (83), and we can read oﬀ the scattering length and the eﬀective range of the model:

(204) (205)

22 E0 1 , m δ0 − δ # 22 =− . mE0

a= reﬀ

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W. Ketterle and M. W. Zwierlein

0.5

5

0.0

0

-5 δ0

Scattering Length [a.u.]

Energy [a.u.]

10

-10

-0.5 -1.0

-0.5 0.0 Detuning (δ − δ0 ) /δ0

0.5

Fig. 33. – Bare, uncoupled molecular state (dashed line), coupled, bound molecular state (solid line) and scattering length (dotted line) close to a Feshbach resonance. The shaded area represents the continuum of scattering states, starting at the collision threshold at E = 0. Interaction between the molecular state and the continuum shifts the position of the resonance by δ0 from the crossing of the uncoupled molecular state with threshold. Note the quadratic behavior of the bound-state energy with detuning (δ − δ0 ) close to resonance.

The scattering length, shown in ﬁg. 33, diverges at the shifted resonance position δ = δ0 . Not surprisingly, we recover E = −2 /ma2 for the bound state energy close to resonance for positive a > 0, as it should be (see eq. 199). In the experiment, the Feshbach resonance occurs for a certain magnetic ﬁeld B0 . With the magnetic moment diﬀerence Δμ between the incoming state and the closed (uncoupled) molecular state, we have δ − δ0 = Δμ(B − B0 ) (taking Δμ to be constant). Including the background scattering length abg for collisions that occur entirely in the open channel, the scattering length can be written in its usual form (206)

a = abg

ΔB 1− B − B0

with ΔB =

1 22 E0 . m Δμ abg

. 5 4. Broad versus narrow Feshbach resonances. – Initially it was assumed that all Feshbach resonances represent a novel mechanism for fermionic pairing and superﬂuidity. Treatments of “resonance superﬂuidity” [170] and “composite Fermi-Bose superﬂuid” [263] explicitly introduced coupled atomic and molecular ﬁelds as an extension of the standard BEC-BCS crossover theory presented in sect. 4. On the other hand, as discussed in the previous section, the closed channel molecular state can be formally eliminated from the description by introducing an eﬀective potential acting on the atoms

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217

in the open channel. We will see in this section that suﬃciently close to the Feshbach resonance, and for suﬃciently small Fermi energies, the physics is indistinguishable from a single channel model such as the attractive spherical well (with R 1/kF ) discussed . in subsect. 5 2 or a (suitably regularized) contact interaction. It turns out that this simple description applies to the experimental studies in 6 Li and 40 K. In these cases, that involve so-called broad Feshbach resonances, the resonance simply provides a knob to turn 1/kF a continuously from large positive values to negative values. The physics is independent of the nature of the molecular state. Therefore, these system are universal, i.e. they are ideal realizations of the “standard” BEC-BCS crossover physics described in sect. 4, and not a new form of “Feshbach” or “resonance” superﬂuidity. . 5 4.1. Energy scales. To address the question on the range of parameters where the molecular state does play a role, we consider the energy scales in the problem. They are E0 , the energy scale associated with the coupling strength, the detuning from resonance δ − δ0 , as a function of which we want to study the system, and EF , the Fermi energy. As we will show, the ratio EF /E0 of the Fermi energy to the coupling energy scale is the parameter that decides whether the physics around the resonance is universal (E0 EF ) or whether the closed molecular channel still plays a role (E0 EF ) [264-266,247]. With eq. (204) this can be equivalently expressed as kF reﬀ 1, that is, universality requires the eﬀective range of the potential to be much smaller than the interparticle distance. In principle, we have two more energy scales, the cutoﬀ energy scale ER and the shift δ0 . ER is much larger than the Fermi energy, as we deal with dilute gases where the interatomic distance is large compared to the range of the potential. Then, the shift √ δ0 ∼ ER E0 is much larger than EF if E0 EF , and does not lead to an additional criterion. . 5 4.2. Criterion for a broad resonance. The criterion E0 EF for a broad resonance is found in several diﬀerent ways, each of which is insightful. BEC-side. For a spherical-well potential, the bound-state energy is given by the universal relation EB = −2 /ma2 (as long as |EB | ER ). This signiﬁes that the character of the molecular state is entirely described by the scattering length, a property of the scattering states in the open channel. However, for the two-channel Feshbach model discussed above, this relation holds only for δ0 −δ E0 (see eq. (199)) or equivalently for 2 /ma2 E0 /2. To observe the universal version of the BEC-BCS crossover presented in the last sect., the bound state should obey the universal behavior already when kF a ≈ 1. This yields the condition EF E0 for the “BEC”-side of the resonance. A more quantitative way to see this is by calculating the contribution of the closed channel molecule to the “dressed” molecular state (207)

|ϕ = α |m +

ck |k .

k

This can be calculated from the magnetic moment of |ϕ, relative to two free atoms:

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W. Ketterle and M. W. Zwierlein

a)

b)

c)

Energy [EF]

1

0

-1 3

2

1

0 -1 1/kF a

-2

-3

3

2

1

0 -1 1/kF a

-2

-3

3

2

1

0 -1 1/kF a

-2

-3

Fig. 34. – Broad versus narrow Feshbach resonances. Shown are the two-body bound-state energy (straight line, in units of EF ), and scattering cross-section contours (in gray shades) for various Feshbach coupling strengthsp E0 . The dashed line is the bare molecular state (here, ER = 20EF , thus δ = 0 at 1/kF a = π2 ER /EF = 2.85). The dotted line marks the position of ﬁnite-energy maxima of the cross-section for 2 /ma2 > E0 /2 (see text). a) E0 = 0.005 EF , b) E0 = 0.1 EF , c) E0 = 15 EF . In c), bound-state and scattering cross-section contours closely approach those for a contact potential. A similar ﬁgure can be found in [247].

Bare molecules have a relative magnetic moment Δμ, so μ|ϕ = α2 Δμ. One ﬁnds with 2|EB | B eq. (206): μ|ϕ = ∂E = ∂B E0 Δμ. When the binding energy |EB | becomes comparable 2|EB | to the Fermi energy EF near resonance, the closed-channel contribution α2 = E0 should already be negligible in order for the physics to be dominated by the open channel. Again, this gives the criterion EF E0 . BCS-side. For δ > δ0 the molecular bound state has disappeared, but the closed channel molecule still leaves its mark in the scattering cross-section [247] (E = 2 k 2 /m): (208)

2 σ(E) = 4π|f (E)|2 = 4πreﬀ 2 = 4πreﬀ

E02

E02 (E − Eres )2 + Γ2 /4

with Eres = δ − δ0 − E0 and (209)

2

Γ =

4E02

2

(E − δ + δ0 ) + 2E0 E

2 (δ − δ0 ) −1 E0

Making, probing and understanding ultracold Fermi gases

219

For δ − δ0 > E0 (equivalently 2 /ma2 > E0 /2), Eres > 0 and a resonance appears in the scattering cross-section at ﬁnite energies. This resonance is just the (shifted) bare molecular state that has acquired a ﬁnite lifetime Γ−1 due to the coupling to the √ continuum. While the width Γ increases with detuning like ∼ δ − δ0 for Eres E0 , √ the relative width Γ/Eres ∼ 1/ δ − δ0 decreases, so that the relative position of the molecular state resonance can in principle be measured more and more accurately in scattering experiments. For large values of δ, the expression eq. (209) for Γ approaches the result of Fermi’s Golden Rule, Γ = 2π g02 ρ( )/Ω, where = (δ − δ0 )/2 is the energy of the fragments after molecular dissociation, and ρ( ) ∝ (δ − δ0 )1/2 is the density of ﬁnal states. This relation was veriﬁed by observing the decay of molecules after a rapid ramp across the resonance [149]. Clearly, in this region, the molecular state has a life of its own. The closed channel molecular state causes a ﬁnite-energy scattering resonance on the “BCS”-side starting at δ − δ0 = E0 that is non-universal. In single-channel scattering oﬀ a delta-potential or a spherical-well potential, no such ﬁnite-energy resonances exist. We require for universal behavior that such resonances do not occur within the BEC-BCS crossover, i.e. within the strongly interacting regime, where kF |a| 1, or equivalently δ − δ0 EF . This leads again to the condition E0 EF . Figure 34 summarizes these ﬁndings. The BEC-BCS crossover occurs for −1 1/kF a 1. If E0 EF , the “dressed” molecular state is almost completely dissolved in the open channel continuum throughout the crossover and the details of the original molecular state |m do not play a role (case of a “broad” Feshbach resonance). The binding energy is given by EB = −2 /ma2 on the BEC-side, and the scattering cross section a2 has the universal form σ(k) = 4π 1+k 2 a2 . On the other hand, if E0 is comparable to EF , then the molecular state aﬀects the many-body physics and it needs to be included in the description of the gas (case of a “narrow” Feshbach resonance) [267, 268, 247]. On resonance. A stringent criterion for universal behavior requires that all scattering properties for detunings δ − δ0 ≤ EF and for energies E < EF are identical to the case of a (regularized) delta-potential or a localized spherical-well potential, where the scattering amplitude f is given by 1/f = −1/a − ik, i.e. the contribution of the eﬀective range, 12 reﬀ k 2 to 1/f is negligible. For the total cross section, which is proportional to |f |2 (eq. 208), the eﬀective range correction is negligible for k 1/reﬀ , or equivalently, EF E0 . However, the real part of f , which determines the mean-ﬁeld energy for a dilute gas, depends more strongly on the eﬀective range: it is equal to −1/ak2 for large a, but approaches a constant reﬀ /2 if reﬀ = 0. This does not spoil universality, though, as the mean-ﬁeld energy associated with such a small eﬀective scattering length, ∝ 2 nreﬀ /m, has to be compared to the many-body interaction energy, βEF ∝ 2 kF2 /m ∝ 2 n/mkF (see sect. 4), which dominates as long as kF 1/reﬀ or E0 > EF . Magnetic moment. Finally, we want to come back to the schematic description of the Feshbach resonance in ﬁg. 32 that uses discrete states in a ﬁnite volume L3 . Each energy

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W. Ketterle and M. W. Zwierlein

curve has two avoided crossings. For small coupling, the slope in between these crossings is still given by the magnetic moment of the molecule, i.e. in this region, population can be purely in the closed channel. This picture is lost when many states couple, i.e. the resonant coupling gk /L3/2 is larger than the level spacing, which is about EL = 2 /mL2 . In order to still maintain some character of the closed channel molecule, one must have gk < 2 /mL1/2 or E0 < EL . However, any ﬁnite volume approximation has to choose L at least comparable to the interatomic spacing, or equivalently, the zero-point energy EL has to be less than the Fermi energy. Therefore, for the case of a broad resonance with E0 > EF , the simple picture of two-level avoided crossings no longer applies, the molecular state gets “smeared” out and “distributed” over many open-channel states. . 5 4.3. Coupling energy scale. We can relate the coupling energy scale E0 to experimentally observable parameters. Using eq. (206) and the deﬁnition for E0 , eq. (199), one has (210)

E0 =

1 (ΔμΔB)2 . 2 2 /ma2bg

The fraction of the dressed molecular wave function that is in the deeply bound state |m is (211)

2

α =

2EB EF 1 . =2 E0 E0 kF a

For the resonance used in the experiments by D. Jin on 40 K, E0 /kB ≈ 1 mK, which should be compared to a typical Fermi energy of EF /kB = 1 μK. This resonance is thus broad [269]. Still, at kF a = 1 the fraction of the wave function in the closed channel molecule is α2 ≈ 6%. This might possibly explain the shorter lifetime of the gas of molecules 40 K2 close to resonance [18] as compared to the case in 6 Li2 [270]. For the wide Feshbach resonance in 6 Li, one has E0 /kB ≈ 50 K, an unusually broad resonance. The strongly interacting regime where 1/kF |a| < 1 is thus completely in the universal regime. The simple relation EB = −2 /ma2 holds to better than 3% already at a magnetic ﬁeld of 600 G, 230 G away from resonance, while the strongly interacting regime is entered only above ≈ 750 G. Indeed, the closed channel contribution to the dressed molecular state has been measured in the group of R. Hulet [76] to be less than 1% at magnetic ﬁelds beyond 600 G and less than 10−3 throughout the entire strongly interacting regime beyond kF |a| ≈ 1. . 5 4.4. Narrow Feshbach resonance. The Feshbach resonance in 6 Li at 543 G, in turn, has E0 /kB ≈ 1 μK and is thus narrow. In the case of a narrow resonance, the many-body physics is qualitatively diﬀerent from the BEC-BCS crossover picture since molecular states will be populated even above the resonance. However, we have just shown how the molecular states have “disappeared” or have become scattering resonances. So how does many-body physics modify these results of two-body physics?

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For a narrow resonance and detunings δ − δ0 < 0, all fermion pairs are still tightly bound in the closed channel molecular state, where they form a condensate. For 0 < δ − δ0 < 2EF , the molecular condensate coexists with a BCS-type fermionic superﬂuid. Here, the molecular state (unstable in vacuum above threshold, represented by the resonances in the scattering cross-section in ﬁg. 34) is stabilized by Pauli blocking, as the outgoing momentum states are occupied by fermions in the BCS-state. Equilibrium between fermions and molecules requires that the chemical potential of the fermions is μ = (δ − δ0 )/2. This means that the molecular state “shaves oﬀ” all fermions above μ (they form molecules), and the Fermi sea is only ﬁlled up to this energy [271]. Only for δ − δ0 > 2EF is the molecular state no longer occupied and we are left with a BCS-type superﬂuid. However, since the resonance is narrow, the interactions for δ − δ0 > 2EF E0 will be

very small, kF |a| < EEF0 1, rendering the observation of such a state very diﬃcult. The transition from the narrow to the broad resonance requires a more complete twochannel description (see [247] and references therein), where even the two-body scattering physics is modiﬁed by the Fermi sea. One example is the transition from the narrow to the broad case right on resonance. In the two-body picture, no scattering resonances (and therefore identiﬁable molecular states) exist. This remains true for a small Fermi sea, with EF E0 , that cannot appreciably aﬀect the open channel states. However, as the Fermi energy becomes comparable to the coupling E0 , more and more k-states are occupied and Pauli blocked, and the closed-channel molecular state can no longer completely dissolve in the continuum states. For EF E0 , the closed-channel molecular state is present in its “undressed” form, and one expects a condensate of these “protected” closed channel molecules to coexist with a Fermi sea. For an extensive discussion of one and two-channel descriptions, we refer the reader to the contribution of M. Holland to these proceedings.

. 5 5. Open channel resonance and the case of 6 Li . – 6 Li stands out compared to all other fermionic atoms studied thus far by its enormously broad Feshbach resonance. It is this fact that has allowed direct evaporation of the gas at a ﬁxed magnetic ﬁeld directly into a molecular condensate, an experiment almost as straightforward in principle as Bose-Einstein condensation of bosonic atoms in a magnetic trap. Lithium is the fermion of choice at Duke, Rice, Innsbruck, ENS and MIT, and also in a growing number of new experimental groups. Figure 35 shows the s-wave scattering length for collisions between the two lowest hyperﬁne states of 6 Li, |F, m = |1/2, 1/2 and |1/2, −1/2. The prominent feature is the broad Feshbach resonance centered around B0 = 834.15 G. The resonance is approximately described by eq. (206) with abg = −1 405 a0 , ΔB = 300 G [135]. These values are very untypical when compared with scattering lengths and Feshbach resonance widths in other alkali atoms. Background scattering lengths are typically on the order of ±100 a0 or less, roughly the range of the van der Waals potential. Widths of other observed Feshbach resonances are two, rather three orders of magnitude smaller than ΔB. Clearly, the broad Feshbach resonance in 6 Li is a special case. The unusually large background scattering length of 6 Li that approaches −2 100 a0 at

222

W. Ketterle and M. W. Zwierlein

20

5

Energy [MHz]

0

0

-10

Scattering Length [a 0]

10

3

-5

-20x10 543

650 834 Magnetic Field [G]

Fig. 35. – Feshbach resonances in 6 Li between the two lowest hyperﬁne states |F, m = |1/2, 1/2 and |1/2, −1/2. A wide Feshbach resonance occurs at 834.15 G. The resonance position is shifted by an unusually large amount of ∼ 300 G from the crossing of the uncoupled molecular state at 543 G (thick dashed line). A second, narrow Feshbach resonance occurs right at 543 G, shifted by less than 200 mG. The solid line shows the energy of the bound molecular state, and the dotted line the scattering length.

high ﬁelds, signals a resonance phenomenon even away from the wide Feshbach resonance. Indeed, if the triplet potential of 6 Li were just about 2 /ma2bg ≈ h · 300 kHz deeper, it would support a new bound state. This “missing” potential depth should be compared to typical spacings between the highest-lying bound states of the van der Waals potential, several tens of GHz. The resulting very large background scattering length modiﬁes the free continuum states |k in a simple but important way: It increases the probability for the two colliding atoms to be close to each other. This leads to a much better wave function overlap between the free continuum states and the closed channel bound state — in the language of molecular spectroscopy, one has a much larger Franck-Condon factor. In the following, we want to show this quantitatively by directly calculating the coupling strength gk = m|V |k as a function of the background scattering length. The states |k are eigenstates of the Hamiltonian H0 , which includes the scattering potential in the open channel. Outside that potential, the wave function ψk (r) = r|k becomes (212)

1 sin(kr + δbg ) . ψk (r) = √ kr Ω

For a background scattering length much larger than the range of the potential abg r0 , we can neglect the short-range behavior of ψ at r r0 . The chosen normalization ensures the closure relation k ck k |k = ck to hold. The closed channel molecular state will

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Making, probing and understanding ultracold Fermi gases

r ψ(r) [a.u.]

a)

b)

1

0

1

0 abg ≈ r0

abg >> r0

-1

-1 0

2

4

r / r0

6

8

10

0

2

4

r / r0

6

8

10

Fig. 36. – Inﬂuence of the background scattering length on the Feshbach coupling. Shown is a spherical-well example, with a well of radius r0 . Solid line: Open-channel radial wave function rψ(r). Dashed line: Molecular state in the closed channel, with an assumed smaller well depth. a) “Typical” background scattering length, abg ≈ r0 , b) large scattering length, abg r0 . The probability to overlap with the bound state is resonantly enhanced.

be taken to be of the form (213)

ψm (r) = √

1 e−r/R . 2πR r

This also neglects the short-range behavior of ψm (r) for r r0 , permissible if the size of the molecule is much larger than the interatomic potential, R r0 . Figure 36 shows the situation for two diﬀerent background scattering lengths. For k 1/R we have rψk (r) ≈ √1Ω (r − abg ). For abg R, the probability for two colliding particles to be within a range R of each other is simply ∼ R3 /Ω, as it would be for noninteracting particles. However, for abg R, the probability increases to ∼ a2bg R/Ω. The coupling to the closed-channel molecular state should be enhanced by the same factor, 2 so we expect |m|V |k| ∝ a2bg . A simple calculation gives

(214) (215)

∗ (r)ψk (r) d3 r ψm 8πR sin δbg + kR cos δbg Vhf = , k Ω 1 + k 2 R2

m|V |k = Vhf

where Vhf is the amplitude of the hyperﬁne interaction Vhf− between the open and closed channel. In the s-wave limit of large de Broglie wavelengths, we can approximate tan δbg ≈ −k abg . We then get (216)

2

(1 − abg /R) , 2 1 + k 2 a2bg (1 + k 2 R2 )

2 2 gk2 = Ω |m|V |k| = 8π |Vhf | R3

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W. Ketterle and M. W. Zwierlein

0

-500 1

Evs

+

X Σg , v = 38

0

I=0 -1000

Energy [MHz]

Energy [MHz]

1

-1

I=2

ΔB 0

543.25 Magnetic Field [G]

834.15

Fig. 37. – Bound-state energies for 6 Li2 in a magnetic ﬁeld. The most weakly bound state of the singlet potential, X 1 Σ+ g , v = 38, splits into two hyperﬁne components with total nuclear spin I = 0 and I = 2. The state with I = 0 is almost not coupled to the triplet scattering continuum, causing the narrow resonance at 543.2 G. In turn, the state I = 2 is very strongly coupled and leads to the broad resonance at 834 G, a shift of ΔB ≈ 300 G. The strong coupling is caused by the large background scattering length abg in the triplet potential. The dashed line in the inset shows the associated energy of the “virtual”, almost bound state Evs ≈ h · 300 kHz, very close to threshold (E = 0).

gk2 is just proportional to the background scattering cross section σbg = 4πa2bg /(1+k 2 a2bg ), including its k-dependence and the unitarity limit: 2

gk2 = 2 |Vhf | R σbg

(217)

2

(1 − R/abg ) (1 + k 2 R2 )

2

.

. In subsect. 5 4.2 we have seen how gk2 determines the lifetime of a molecular state placed in the continuum at energy E. In our model, this lifetime becomes, by Fermi’s Golden Rule, (218) (219)

2π π1 2 2 ρ (E/2) gk(E/2) |m|V |k| δ(2 k − E) = Ω k √ √ E ER Ebg 4 abg 2 2 = |Vhf | 1 − 2 , R (Ebg + E) (ER + E)

Γ(E) =

where Ebg = 2 /ma2bg . The latter is exactly the same expression that one obtains for a bound-free radio-frequency transition that dissociates (or associates) a molecule into (from) two free atoms [206] (valid in the threshold regime abg , R r0 ). One merely has to replace the hyperﬁne coupling by the Rabi coupling 12 ΩR . The dependence on

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Making, probing and understanding ultracold Fermi gases

√ E is the usual Wigner threshold law. This reiterates the analogy between Feshbach resonances and photo or RF association. The k-dependent coupling gk2 rolls oﬀ over a characteristic range 1/R, as expected. So the natural cut-oﬀ energy is ER = 2 /mR2 , as before. For k 1/R, we obtain 2

g02 = 2 |Vhf | R σbg (1 − R/abg )

(220)

2

with the limiting cases 2

(221)

g02 = 8π |Vhf | R3 ,

(222)

g02

= 8π |Vhf |

2

R a2bg

for abg R , ,

for abg R .

The resonant limit abg R is valid if kabg 1. In the strongly interacting regime 2 where kabg 1, the Feshbach coupling is unitarity limited to g02 = 8π |Vhf | R/k 2 . We thus arrive at the conclusion that a large background scattering length resonantly enhances the coupling to the closed-channel molecular state. This is simply because of the increased probability for two particles colliding in the open channel to be near each other. Since g02 determines the eﬀective range of the Feshbach scattering amplitude, eq. (204), we can say that the background scattering length tunes the eﬀective range. In fact, for abg R, we ﬁnd (223)

reﬀ = −

2 ER R. ma2bg |Vhf |2

The larger the background scattering length, the smaller the eﬀective range. The criterion for a broad Feshbach resonance, E0 EF , now reads (224)

EF

4 2 |Vhf | m2 a4bg . ER 4

Knowing that 40 K does not have an unusually long background scattering length, this relation implies that for reasonably strong couplings Vhf (a sizeable fraction of the hyperﬁne splitting), but only “standard” background scattering lengths, molecular sizes R and for typical Fermi energies, Feshbach resonances are broad. We can now easily calculate the energy shift δ0 due to this enhanced Feshbach coupling. This directly gives the magnetic-ﬁeld shift B0 − B ∗ = δ0 /Δμ between the Feshbach resonance position at B0 , where the dressed bound-state energy vanishes, and the magnetic ﬁeld B ∗ where the uncoupled molecular state would cross threshold. In 6 Li, B0 = 834 G, whereas B ∗ = 543 G (in fact, an almost uncoupled, second closed-channel molecular state causes a narrow resonance at B ∗ , see ﬁg. 37). With the deﬁnition in

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W. Ketterle and M. W. Zwierlein

eq. (199), and again in the limit 1/k abg R, (225)

√ 2 4 2 4 1 m g02 2 mabg = |V δ0 = E0 ER = √ | . hf π π 2 2π 2 2 R

With the known magnetic-ﬁeld shift in 6 Li, using abg ≈ 2100 a0 and Δμ = 2μB = 2.8 MHz/G, we can now obtain an estimate of the hyperﬁne coupling strength, Vhf ≈ h · 10 MHz. This is indeed a “typical” coupling strength: The hyperﬁne constant for 6 Li is ahf ≈ h · 150 MHz, setting an upper bound on the matrix element Vhf which is less than ahf due to Clebsch-Gordan coeﬃcients. Of course, our model neglects short-range physics that may aﬀect g02 . Some authors [272, 273] arrive at the same conclusion of eq. (225) by introducing a “virtual state” at the energy Evs ∼ 2 /ma2bg above threshold and replacing the interaction of the molecular state with the scattering continuum by an eﬀective interaction between molecular and virtual state only. In this language, second-order perturbation 2 2 theory predicts an energy shift of δ0 ≈ |Vhf | /Evs = |Vhf | ma2bg /2 , exactly as we have obtained above. We point out that there is no ﬁnite energy scattering resonance associated with this “imaginary” state. Rather, it signiﬁes that if the potential were deeper by ∼ Evs , it would support a new bound state just below threshold. To summarize: the history of interactions in Fermi gases has gone full circle. At ﬁrst, 6 Li was thought to be a great candidate for fermionic superﬂuidity because of its large and negative background scattering length. Then it was realized that scattering lengths can be tuned at will close to a Feshbach resonance — so essentially any fermionic atom that could be laser cooled became a good candidate (as Feshbach resonances have so far been found for any atom, whenever experimentalists started to search for them). But in the end, it is still 6 Li that is the most robust choice, and this indeed because of its large scattering length — since this is what enhances the Feshbach coupling and makes the resonance abnormally — fantastically — large.

6. – Condensation and superﬂuidity across the BEC-BCS crossover In this section, we present experimental results on condensation and superﬂuid ﬂow across the BEC-BCS crossover. We will start with some general remarks on diﬀerent signatures for superﬂuidity, give some background on vortices and describe the experimental methods to observe condensation and vortex lattices in gases of fermionic atoms together with the results achieved. . 6 1. Bose-Einstein condensation and superﬂuidity. – Two phenomena occurring at low temperature have received special attention: Bose-Einstein condensation and superﬂuidity. An interesting question is how the two are related. K. Huang has pointed out [274] that Bose-Einstein condensation is not necessary for superﬂuidity, but also not suﬃcient. This is illustrated by the examples in table VIII.

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Table VIII. – Condensation versus superﬂuidity. Condensation and superﬂuidity are two different, but related phenomena. The ideal gas is Bose condensed, but not superﬂuid. In lower dimensions, ﬂuctuations can destroy the condensate, but still allow for superﬂuidity. System 3D Ideal gas 2D, T = 0 2D, T = 0 1D, T = 0

BEC √ √ Ø √ Ø

Superﬂuid √ Ø √ √ √

The ideal Bose gas can undergo Bose-Einstein condensation, but it does not show superﬂuid behavior since its critical velocity is zero. Superﬂuidity requires interactions. The opposite case (superﬂuidity without BEC) occurs in lower dimensions. In 1D at T = 0 [275, 276] and in 2D at ﬁnite temperature, superﬂuidity occurs [277], but the condensate is destroyed by phase ﬂuctuations [278-280]. In 2D at zero temperature, there is both a condensate and superﬂuidity [281]. In 3D, condensation and superﬂuidity occur together. An interesting case that has been widely discussed are bosons in a random potential. For weak disorder and weak interactions, there is an unusual regime where the superﬂuid fraction is smaller than the condensate fraction [282, 283]. It appears that some part of the condensate is pinned by the disorder and does not contribute to the superﬂuid ﬂow. However, the extrapolation to strong disorder and the conclusion that the system can be Bose condensed without being superﬂuid [282,274] is not correct [284,285]. The condensate and superﬂuid fraction disappear together when the disorder is suﬃciently strong [284, 285]. A very comprehensive discussion on the relation between superﬂuidity and BEC is presented in the Appendix of ref. [286]. When condensation is generalized to quasicondensation in lower dimensions the two phenomena become equivalent. It is shown that superﬂuidity plus ﬁnite compressibility are suﬃcient conditions for either condensation or quasi-condensation. The reverse is also true, i.e. condensation or quasi-condensation are necessary for superﬂuidity. Here, superﬂuidity is deﬁned by the rigidity of the system against changes in the phase of the boundary condition and condensation by the presence of a macroscopic eigenvalue of the density matrix, which, for translationally invariant systems, implies oﬀ-diagonal long range order. Quasi-condensates are local condensates without long range order. This discussion on bosons applies directly to a gas of bosonic molecules created at a Feshbach resonance. For a Fermi gas, the examples for lower dimensions apply as well [278] (for a discussion of superconductivity in 2D ﬁlms and arrays, see [242]). The example of the non-interacting Bose gas, however, does not carry over: A non-interacting Fermi gas does not form a pair condensate. The eﬀect of disorder on a BCS superﬂuid is complex. The pair condensate survives in the presence of local impurities (weak disorder), with the order parameter and TC unchanged [287], while condensate fraction and superﬂuid density are reduced [288].

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These examples lead to the conclusion that experimentalists need to study both condensation and superﬂuidity! . 6 2. Signatures for superﬂuidity in quantum gases. – What constitutes an observation of superﬂuidity? Even theoretically, superﬂuidity is deﬁned in several diﬀerent ways. The most frequent deﬁnition employs the concept of rigidity against phase-twisting [289,290]. In some deﬁnitions, even a non-interacting BEC qualiﬁes as a superﬂuid [291]. From the experimentalists’ point of view, superﬂuidity consists of a host of phenomena, including phase coherence, transport without dissipation, an excitation spectrum which results in a non-zero value of Landau’s critical velocity (usually a phonon spectrum), the Meissner eﬀect, the existence of quantized vortices, and a reduction of the moment of inertia. After the discovery of Bose-Einstein condensation in 1995, it still took until 1999 before researchers agreed that superﬂuidity was established, through the observation of vortices [292, 226] and a critical velocity in a stirred condensate [293, 294]. The general consensus was that the experimental veriﬁcation of superﬂuidity required the observation of some aspect of superﬂuid ﬂow that would not be possible in a classical system. Therefore, neither the hydrodynamic expansion of a condensate was regarded as evidence (since collisionally dense classical clouds would behave in the same way), nor the observation of phonon-like excitations, nor the interference of condensates, which established phase coherence only for a stationary cloud. The observation of a critical velocity [293, 294] provided evidence for superﬂuid ﬂow, although the contrast between the behavior in the superﬂuid and normal regime did not even come close to the drop in resistivity or viscosity that was observed when superconductors or superﬂuids were discovered. Long-lived ﬂow in the form of vortices has been regarded as a smoking gun for superﬂuids. However, vortices can be long lived even in classical liquids [295]. What sets the superﬂuid apart is the quantization of vortices and the fact that the ground state with angular momentum is necessarily a state with vortices. The emergence of vortex arrays and vortex lattices [226-229] after driving surface excitations [296] dramatically demonstrated both properties. Although there is no rigorous derivation showing that ordered lattices of uniformly charged vortices prove superﬂuidity, we are not aware of any system or observation that could provide a counter example. The reduction of the moment of inertia is another distinguishing feature of superﬂuid ﬂow. It can be observed through the so-called scissors mode, a collective excitation created by a sudden rotation [297, 298, 26], or by observing the expansion of a rotating superﬂuid [299-301]. Both methods have been regarded as a way to directly observe superﬂuidity. However, studies with normal Fermi gases have impressively demonstrated that both features, originally regarded as a unique signature of superﬂuids, occur already for normal gases deep in the hydrodynamic regime where dissipation is extremely small [302, 303]. It appears that superﬂuid and low-viscosity collisional hydrodynamics can only be distinguished if there is suﬃcient time for the small but ﬁnite viscosity in the normal phase to create vorticity, a velocity ﬁeld with ∇×v = 0 whereas the superﬂuid will always continue to be irrotational (unless quantized vortices are nucleated). For instance, if the

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ﬂow ﬁeld has equilibrated with a slowly rotating container, then collective excitations will reveal the diﬀerence between a superﬂuid and a normal ﬂuid [304]. The last example shows how the physics of strong interactions can “obscure” the seemingly dramatic transition to a superﬂuid state. The normal state of a Fermi gas around the Feshbach resonance is already almost “super” due to its very low viscosity. Many experiments uncovered the unique properties of this strongly interacting gas, and eventually its transition into the superﬂuid state. The observation of anisotropic expansion [60] was initially believed to provide evidence for superﬂuidity. However, such an expansion was also observed in a normal strongly interacting Fermi gas [60,124,131] and was predicted to occur even at T = 0, since Pauli blocking is no longer eﬀective during expansion [305]. Can the damping of collective modes distinguish between superﬂuid and normal ﬂow? In a simple picture, damping in collisional hydrodynamics increases with lower temperature, because Pauli blocking lowers the collision rate and increases the mean free path. In contrast, damping of superﬂuid hydrodynamics decreases with lower temperature, because the normal density, which provides friction, decreases. The observation of such a decrease of the damping rate of collective excitations was regarded as evidence for superﬂuidity [73]. Later, however, it was found that a similar positive slope of damping vs. temperature occurs for a normal strongly interacting Fermi gas [91]. The observation of a “paring gap” in RF spectroscopy was regarded as strong evidence for superﬂuidity [75, 306], mainly based on the theoretical interpretation of the experimental data. However, experiments with population imbalanced Fermi gases showed that RF spectra of normal and superﬂuid clouds are identical, and that RF spectroscopy cannot distinguish between the two phases, at least not at the current level of resolution [77]. The reason is the presence of strong pair correlations in the normal phase and possibly also strong interactions in the ﬁnal state used for spectroscopy, which were not included in the models used to interpret the data. When collective excitations were studied as a function of the scattering length, intriguing sudden peaks in the damping rate were observed [74, 307]. The conjecture is that this may reﬂect a resonance of the collective mode with the pairing energy Δ, and damping would occur due to pair breaking. This phenomenon remains to be systematically studied. The observation of pair correlations across the BEC-BCS crossover was consistent with predictions of a theory for the superﬂuid state [76], but it seems that similar pair correlations also exist in the normal state [308]. Finally, a kink was observed when the speciﬁc heat [72] or the entropy [183] were determined as a function of the temperature or the energy of the cloud. Due to the signal-to-noise ratio, these kinks could be distilled only by separately ﬁtting the low- and high-temperature regions. The discussion above has summarized many aspects of superﬂuid systems, some of which are shared with strongly interacting normal gases. In the following sections we will focus on the two phenomena that do not occur in a normal gas: condensation and the formation of quantized vortices in rotating superﬂuids.

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Fig. 38. – Lifetime of molecules in partially condensed clouds. The cloud with initially about 1 × 106 molecules was held for 9.9 s (initial density about 5 × 1012 cm−3 , slightly varying with the interaction strength). Shown is the remaining fraction as a function of scattering length. The lifetime 1/Γ is calculated under the simplifying assumption of a pure exponential decay e−Γt . The line is a ﬁt with a power law for Γ = ca−p , giving p = −0.9. The clouds were partially condensed (up to 80% condensate fraction at the largest scattering length), and all measurements were done in the strongly interacting regime where a > 1/kF , so the expression for the relaxation rate diﬀers from the prediction for weakly interacting, thermal molecules (Γ = c a−2.55 ) [200].

. 6 3. Pair condensation below the Feshbach resonance. – The successful creation of ultracold molecules out of ultracold atoms via Feshbach resonances in gases of fermions [61, 15, 17, 16] and bosons [255, 62-64] brought the goal of Bose-Einstein condensation of molecules into close reach. Indeed, molecular samples in cesium close to [62] and in sodium clearly within [63] the regime of quantum degeneracy were generated. However, their lifetime was too short to observe an equilibrium Bose-Einstein condensate. Molecules formed of fermions turned out to have a much longer lifetime due to . greater stability against inelastic decay (see sect. 2 and subsect. 5 1). Within a few months, this favorable property allowed the successful Bose-Einstein condensation of molecules, or more precisely of strongly interacting fermion pairs [55, 65, 66]. In the case of 6 Li, the long lifetime of molecules [15] (see ﬁg. 38) enables us to evaporate the Fermi mixture at a ﬁxed magnetic ﬁeld, just like cooling a cloud of bosonic atoms towards BEC. As the mixture is cooled by ramping down the trapping laser power, molecules form as the temperature becomes comparable to the binding energy. Accordingly, the atomic signal observed in zero-ﬁeld imaging vanishes: We can see this in ﬁg. 10 for ﬁelds below resonance, where essentially no atomic signal is measured. Below a certain temperature, one observes the striking onset of a bimodal density distribution, the hallmark of Bose-Einstein condensation (see ﬁgs. 39 and 40). The emergence of the bi-

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Molecule Number per Unit Length [ 1/mm ]

4 mW

10 mW

11 mW

14 mW

19 mW

6

1x10

38 mW

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Fig. 39. – Bimodal density distribution emerging in a cloud of molecules. Shown are radially integrated proﬁles of absorption images such as those in ﬁg. 40, as a function of ﬁnal laser power. The dashed lines are ﬁts to the thermal clouds.

modality was actually accentuated by an anharmonic trapping potential where a shallow minimum of the potential was oﬀset with respect to the deeper potential which held the thermal cloud. Figure 40 shows the gallery of molecular Bose-Einstein condensates observed at JILA [65], MIT [66], Innsbruck [67], at the ENS [71] and at Rice [76]. In contrast to weakly interacting Bose gases, the condensate peak is not much narrower than the thermal cloud, indicating a large mean-ﬁeld energy of the BEC, comparable to kB times the condensation temperature. As we move closer to the Feshbach resonance, the size of the condensate grows to be almost that of a degenerate Fermi −1/3 gas (see ﬁg. 41). The average distance nM between molecules becomes comparable to the molecular size in free space, given approximately by the scattering length. Thus we have entered the strongly interacting regime of the BEC-BCS crossover where two-body pairing is modiﬁed by Pauli pressure.

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Fig. 40. – Observation of Bose-Einstein condensation of molecules. The gallery shows bimodal density distributions observed after expansion and molecule dissociation at JILA [65], after expansion, dissociation and zero-ﬁeld imaging at MIT [66] and at the ENS [71], and in situ proﬁles from Innsbruck [67] and Rice [76].

. 6 4. Pair condensation above the Feshbach resonance. – When the Feshbach resonance is approached, the bimodality of the cloud becomes almost undetectable. There is no strong spatial signature of the phase transition, even at a much better signal-to-noise . ratio than in the initial observations (see ﬁg. 18 in subsect. 3 3.2). We have observed . weak signatures in these spatial proﬁles (see subsect. 6 5.1 below). A second diﬃculty with fermion pair condensates on the BCS-side is the instability of the pairs during expansion. When the gas becomes more dilute the pair binding energy can decrease below (kB times) the local temperature, causing pairs to break during time of ﬂight. See . subsect. 6 6.8 below for a study of this eﬀect using vortices. To extend the study of pair condensation from below to above the Feshbach resonance, a new detection method was needed. Such a method was introduced by the JILA group [69] and later adapted to 6 Li by our group [70]. The rapid ramp tech. nique is discussed in detail in subsect. 2 4.5 and also in the contribution of D. Jin to these proceedings. The concept of this technique is to prevent the fragile fermion pairs from dissociating by sweeping the magnetic ﬁeld towards the BEC-side of the resonance, thereby transforming them into stable molecules (see ﬁg. 12). This is done in the moment the trap is switched oﬀ for expansion. If each fermion pair is transferred into a tightly

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Radial density [a.u.]

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Fig. 41. – Comparison between a molecular BEC (a) and a degenerate Fermi sea (b). The condensate containing Nm = 6×106 molecules is in the strongly interacting regime at a magnetic ﬁeld of 780 G (1/kF a = 0.6). Its expanded size is almost as large (factor ∼ 0.7) as an expanded non-interacting Fermi gas containing Nm atoms, indicated by the white circle. Image and proﬁle b) show an essentially spin-polarized Fermi sea (minority component of < 2% not shown) containing N = 8 × 106 atoms at the same ﬁeld. The images were taken after 12 ms expansion with the probe light aligned with the long axis of the cigar-shaped clouds.

bound molecule, the momentum information of the original pair is preserved. Time-ofﬂight analysis of the resulting molecules should thus allow one to infer the momentum distribution of pairs in the gas above resonance. The momentum distribution might be broadened by the residual mean-ﬁeld interaction of molecules after the ramp. However, these interactions are greatly reduced by sweeping suﬃciently far away from the Feshbach resonance into the weakly interacting regime where kF a 1. This technique enabled us to demonstrate fermion pair condensation in the entire BEC-BCS crossover. Sample images and proﬁles of the resulting molecular clouds are shown in ﬁg. 42. The drastically reduced interaction results in a clear separation of the condensate from the “thermal” or uncondensed part of the cloud (21 ). (21 ) At zero ﬁeld, the scattering length between molecules should be on the order of the singlet scattering length of lithium atoms, which is about 40 a0 . The exact value is not known. In fact, the residual mean-ﬁeld interaction at zero ﬁeld is so low that the condensate practically does not expand if the rapid ramp is performed immediately after switching oﬀ the trap. For this reason, it is sometimes beneﬁcial to let the cloud expand by some amount before the rapid ramp is performed. This converts some of the interaction energy in the cloud into kinetic energy, which allows one to “choose” the ﬁnal expanded size of the molecular condensate.

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W. Ketterle and M. W. Zwierlein BEC-side

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Fig. 42. – Fermion pair condensates. Axial density of the atomic cloud after switching oﬀ the optical trap, a rapid ramp to zero ﬁeld (in < 100 μs), further expansion (for 10 ms), and dissociation of the resulting molecules by ramping back across resonance. The initial ﬁeld B0 , the number of fermion pairs N , the condensate fraction and the interaction parameter 1/kF a where a) 745 G, 700 000, 47%, 1.2; b) 835 G, 1.4 × 106 , 81%, 0.0 (resonance); c) 912 G, 1 × 106 , 49%, −0.5.

The condensate fraction was determined by ﬁtting a bimodal distribution to the proﬁles like those in ﬁg. 42, a parabola for the central dense part and a Gaussian for the thermal background (see sect. 3). Remarkably large condensate fractions were found throughout the entire BEC-BCS crossover, with a peak of 80% at B ≈ 820 G, close to the resonance, but still on its BEC-side (see ﬁg. 43).

Condensate Fraction

0.8

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Fig. 43. – Condensate fraction in the BEC-BCS crossover as a function of the magnetic ﬁeld before the rapid ramp. The symbols correspond to diﬀerent hold times, 2 ms (crosses), 100 ms (squares) and 10 s (circles). From [70].

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Fig. 44. – Condensate fraction in the BEC-BCS crossover obtained by the JILA group using 40 K, as a function of degeneracy T /TF and magnetic ﬁeld (interaction strength) around the Feshbach resonance. From [69].

The high condensate fraction is a hint that the pairs in the strongly interacting regime on the BCS-side of the resonance are still smaller than the interparticle spacing, not larger, as one would expect for conventional Cooper pairs. An intuitive assumption is that during the magnetic ﬁeld sweep an atom preferably forms a molecule with its nearest neighbor (22 ). In the case of localized pairs, molecules are then formed from the original “Cooper partners”. In the case of delocalized Cooper pairs, molecules might rather form out of uncorrelated atoms, resulting in a thermal cloud after the ramp. In accord with this argument, BEC-BCS crossover theory predicts that the pair size . ξ will be smaller than the interparticle spacing n−1/3 up to kF a ≈ −1 (see section 4 6.3). So far no experiment on Fermi gases has shown condensation or superﬂuidity in a regime where kF |a| (a < 0) is signiﬁcantly less than 1 and hence where pairing is truly long-range. Observing superﬂuidity for kF a < −1 would require exponentially lower temperatures of T /TF < 0.28 e−π/2kF |a| 0.06 and, furthermore, the sweep technique may no longer allow the observation of pair condensation. A simple theoretical model (see below) agrees with the high condensate fraction, but the latter is in stark contrast to the maximum fraction of about 14% found in experiments with 40 K [69] (see ﬁg. 44). The reason for this discrepancy might be related to the shorter lifetime of the Fermi mixture in 40 K close to resonance, on the order of 100 ms [18]. In (22 ) This should happen as long as the relative momentum is not larger than the inverse distance, i.e. the neighbor populates the same phase space cell. This is the case in the regime of quantum degeneracy, and it is experimentally conﬁrmed by the almost 100% conversion from atoms into . molecules (see 2 4.2).

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Fig. 45. – Condensate fraction as a function of magnetic ﬁeld and temperature in the MIT experiments on 6 Li. Condensates are obtained in the entire BEC-BCS crossover. The highest condensate fraction and highest onset temperature are obtained on the BEC-side close to resonance. As a model-independent measure of temperature, the condensate fraction at 822 G (see arrow) is used as the vertical axis. The Feshbach resonance lies close to this point, at 834 G. From [70].

addition, technical issues particular to 40 K may play a role including strong losses during the probing procedure. In our experiments, the condensates were found to be very long-lived. For a hold time of 10 s, the condensate fraction on resonance was observed to be still close to its initial value. In fact, these lifetimes can compare very favorably to those found for atomic Bose-Einstein condensates. On the BEC-side, the condensate decayed more rapidly due to the increasing rate of vibrational relaxation of the molecules away from resonance. The decay of the condensate fraction on the BCS-side can be caused by heating and atom loss due to inelastic collisions or a larger sensitivity to ﬂuctuations of the trapping ﬁelds. . 6 4.1. Comparison with theory. Figures 44 and 45 show “phase diagrams” for the condensate fraction as a function of temperature and interaction strength. Several theoretical studies [152, 309, 310] have conﬁrmed the general behavior of the “critical temperature” of the observed condensation phenomenon in 40 K and in 6 Li. In sect. 4, we discussed the BEC-BCS crossover theory which predicts a condensate fraction (see eq. (175)) of

(226)

1 2 2 N0 m3/2 3/2 = Δ n0 = u k vk = Ω Ω 8π3 k

# μ + Δ

1+

μ2 . Δ2

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Condensate Fraction

1.0 0.8 0.6 0.4 0.2

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0

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1 / kFa

Fig. 46. – Condensate fraction as a function of the interaction strength in the BEC-BCS crossover. The circles show the 100 ms data of ﬁg. 43. The interaction strength is calculated using the known scattering length as a function of magnetic ﬁeld and the experimental value 1/kF = 2 000 a0 . The curve shows the variational BCS prediction for the condensate fraction. On the BEC-side, heating due to vibrational relaxation leads to fast decay on the condensate. Figure adapted from [231] using eq. (175).

Figure 46 compares the variational BCS prediction to our results. The very close agreement must be considered fortuitous since the simple crossover theory is only qualitatively correct near resonance. Furthermore, it is not clear how accurately the observed molecular condensate fraction after the ramp reﬂects the pair condensate fraction before the ramp. The strongest conﬁrmation that the bimodal density distributions observed after the ramp are an indicator of a phase transition comes from the direct detection of con. densation in population imbalanced clouds (see subsect. 6 5.2). In the next section we summarize experimental evidence that the condensate fraction cannot change strongly during the sweep time. On the other hand, some evidence has been reported [311], that the conversion eﬃciency into molecules is higher for condensates. This eﬀect increases the condensate fraction during the sweep, but the eﬀect is small [311]. As discussed . in subsect. 2 4.5, the system’s dynamics during the sweep poses a diﬃcult challenge to theory, due to the presence of several timescales for coherent and incoherent evolution. . 6 4.2. Formation Dynamics. The underlying assumption for the rapid sweep technique is that the momentum distribution of pairs is not changing during the sweep time. Evidence for this was obtained both in the JILA and MIT experiments, where it was shown that the condensate fraction did not change when the sweep rate was varied or when the density was changed. However, since in both experiments the sweep time through the strongly interacting regime was comparable to the inverse Fermi energy, dynamics during the sweep could not be fully ruled out. We addressed this issue by directly measuring the relaxation time of the strongly interacting Fermi system. This was done by modulating

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Fig. 47. – Relaxation time of fermionic pair condensates. Shown is the delayed response of the condensate fraction to a 250 Hz magnetic ﬁeld modulation on the BCS side of the Feshbach resonance. From [311].

the magnetic ﬁeld (and therefore the scattering length) and observing the growth or decay of the condensate. In the experiment, we used a periodic modulation of the magnetic ﬁeld and measured the phase shift of the induced condensate modulation (see ﬁg. 47). The observed relaxation time of 500 μs was much longer than the time to sweep through the strongly interacting regime (10 μs). Therefore, it should not be possible for the condensate fraction to change noticeably during the sweep. Still, a possible loophole is some exotic mechanism for such a change that is not captured by the simple relaxation models assumed in the experimental tests. In conclusion, the rapid ramp to the BEC-side has proven to be a very valuable tool for the detection of condensation in the BEC-BCS crossover. Moreover, the ramp provides us with a way to preserve the topology of the pair wave function on the BCSside. This allows the observation of vortex lattices in the entire BEC-BCS crossover, as . will be discussed in subsect. 6 6. In the following, we will show that condensates can be detected without any ramp, by direct absorption imaging. . 6 5. Direct observation of condensation in the density proﬁles. – The hallmark of Bose-Einstein condensation in atomic Bose gases was the sudden appearance of a dense central core in the midst of a large thermal cloud [1, 2]. This direct signature in the density distribution derives from a clear separation of energy scales in weakly interacting gases. The condensate’s repulsive mean-ﬁeld μ ∝ na is much smaller than the critical temperature (times kB ) at which condensation occurs, TC ∝ n2/3 : The gas parameter na3 is much less than 1 (about 4 × 10−6 for 23 Na condensates). In a harmonic trap, the diﬀerent energy scales directly translate into the diﬀerent sizes of a thermal cloud, √ √ Rth ∝ T , and of a condensate RC ∝ μ. This is the situation we encounter with weakly interacting molecular clouds in Fermi mixtures on the “BEC”-side of the Feshbach resonance. However, as the interactions between molecules are increased by moving closer

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Fig. 48. – Density proﬁles of an equal Fermi mixture on resonance. The temperature in a) was T /TF ≈ 0.15, whereas in b) it was T /TF ≈ 0.09. Temperatures were determined from the thermal molecular cloud after the rapid ramp, and might not be quantitatively accurate. Both gas clouds contained a condensate after the rapid ramp to the BEC-side. The condensate fraction was: a) 7%, b) 60%.

to the Feshbach resonance, the size of the molecular condensate grows and the bimodal feature close to TC becomes almost invisible. In strongly interacting Fermi gases, the separation of energy scales is no longer given. On resonance, the size of the condensate is governed by μ ≈ 0.5EF , while kB TC ≈ 0.15EF EF , so that the normal cloud’s size is dominated not by temperature, but by the Fermi energy. The question arises whether the condensate still leaves a trace in the cloud as the gas undergoes the phase transition. . 6 5.1. Anomalous density proﬁles at unitarity. (23 ) We have indeed found a faint signature of condensation in density proﬁles of the unitary gas on resonance after expansion. To a very good approximation, the trapping potential was cylindrically symmetric (see . subsect. 2 2.2). This allowed us to obtain low-noise proﬁles via azimuthal averaging. Sample proﬁles are shown in ﬁg. 48. To observe a deviation from the shape of a non-interacting Fermi cloud, an unconstrained ﬁnite-temperature ﬁt is performed on the proﬁles. The relevant information is now contained in the residuals of such a ﬁtting procedure. The ﬁt residuals deviate at most by 2% from the non-interacting Fermi shape. This explains why this eﬀect has not been observed in earlier experiments. Despite of the rather small deviation, the non-interacting ﬁt is aﬀected by the “kinks”. This draws into question whether the “eﬀective temperature” typically obtained from such ﬁts to the whole proﬁle, is a well-deﬁned quantity. For a well controlled determination of an eﬀective temperature, only the proﬁle’s wings should be ﬁt, where the gas is normal. A convenient way to graph ﬁt residuals as a function of temperature is by means of a “density” plot of gray shades, with white and black corresponding to positive or negative deviations of the measured proﬁle from the ﬁt. This is shown in ﬁg. 49. Also (23 ) The results of this section have not been published elsewhere.

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Fig. 49. – Density proﬁles, their curvature and their ﬁt-residuals on resonance. a) Density proﬁles on resonance as a function of trap depth. There is no sign of a phase transition at this resolution. b) After the rapid ramp to the BEC-side and expansion, a condensate is clearly visible below a certain trap depth. c) The curvature of the density proﬁles on resonance in a) carries a signature of the condensate. No ﬁeld ramp is required. d) The ﬁt residuals for a ﬁnite-temperature Thomas-Fermi ﬁt. Also here, the condensate’s imprint in the density proﬁle is clearly visible. To obtain the curvature, the noisy central region of ±0.1RF in each proﬁle was replaced by a ﬁt.

included in this ﬁgure is the information on the density proﬁles and their curvature. While the proﬁles themselves do not appear to change with temperature (trap depth) on the scale of the plot, we observe an intriguing structure appearing in the residuals at an evaporation depth of about U = 2 μK. The curvature of the density proﬁles shows a similar qualitative behavior. To indicate that the observed feature indeed stems from the superﬂuid, we also include . a density plot of the proﬁles obtained with the rapid ramp method from subsect. 6 4 above. This allows to clearly separate the condensate and thermal cloud in expansion. The condensate fraction is shown in ﬁg. 50, and shows that the condensate appears around a trap depth of 4.2 μK. We observe that a small condensate does not leave a strong signature in the gas cloud, unlike the case of weakly interacting Bose gases. Only when the condensate has grown to an appreciable size (about 20% in our data) does it signiﬁcantly deform the density proﬁles. At the lowest temperatures and high condensate fractions, the quality of Fermi ﬁts improves again, indicating that now a large fraction of the gas is in the superﬂuid state. The size of the cloud is then RTF = 0.83 RF , which gives ξ(0) ≈ 0.47, in accord with . other experiments and theory (see subsect. 3 1.2 above).

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Fig. 50. – Condensate fraction n0 and temperature as a function of evaporation depth on resonance. The condensate starts to form around a trap depth of U ∼ 2.2 corresponding to 4.2 μK. The temperature was determined from the thermal wings of expanding molecular clouds after the rapid ramp. The Fermi temperature decreased slowly from 1.5 μK for U = 2.6 to 1.4 μK at U = 0.4, and dropped quickly due to atom spilling below U = 0.2. All measurements were done after recompression into a deeper trap with U = 2.0.

To conclude, the density proﬁles in resonantly interacting Fermi gases are modiﬁed in the presence of a superﬂuid core. Such features have been predicted by several authors [170, 168, 45, 171], but had previously been too small to be observable. In the next section, we will demonstrate how an imbalance in the spin-up versus spin-down population in the gas greatly enhances the visibility of the condensate and leads to a striking signature of condensation. . 6 5.2. Direct observation of the onset of condensation in Fermi mixtures with unequal spin populations. We have seen that a balanced mixture of spin-up and spin-down fermions at unitarity does not show a strong signature of condensation. The reason is that on resonance, due to the symmetry in atom numbers, only one energy scale is available, the Fermi energy. In stark contrast, breaking the symmetry in atom numbers and working with Fermi mixtures with unequal spin populations produces a direct and striking signature of the superﬂuid phase transition in the spatial density proﬁles both in expansion [312] and in trap [80]. A similar situation has been encountered in Bose-Einstein condensation, where breaking the symmetry of a spherical trap resulted in dramatic anisotropic expansion of the condensate, now a hallmark of the BEC phase transition. Part of the reason for the direct signature is the new hierarchy of energy scales. The normal majority and minority cloud sizes are governed by the respective Fermi energies (rather, chemical potentials) μ↑ and μ↓ , while the cloud size of fermion pairs is governed by the average chemical potential, (μ↑ + μ↓ )/2. The deeper reason for the sudden change of the spatial proﬁle at the phase transition is that fermionic superﬂuids (around resonance and in the BCS-regime) do not tolerate unpaired fermions, at least at zero temperature. The superﬂuid gap presents an energy barrier for these “singles” to enter the superﬂuid. This leads to a superﬂuid central region of equal spin populations

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Fig. 51. – Direct observation of condensation in imbalanced clouds on resonance. The upper row shows majority clouds, the lower row minority clouds, for an imbalance of δ = 60%. The dashed line is a ﬁt to the wings of the minority cloud to a Thomas-Fermi proﬁle, clearly missing the central feature. Temperature was varied by lowering the trapping power. To within 20%, temperatures can be obtained from the ballistically expanding wings of the majority cloud. We have T /TF = 0.14 (a), 0.09 (b) and 0.06 (c). Here, kB TF is the Fermi temperature of an equal mixture containing the same total atom number. The ﬁgure shows data from [312].

surrounded by the polarized normal phase. The two regimes are separated by a ﬁrstorder phase transition, at which the density imbalance jumps. The presence of such a sudden change in the density distribution allowed the ﬁrst direct observation of the phase transition, without the need for sweeps to the BEC side of the Feshbach resonance [312]. We present here, side by side, the density proﬁles of an imbalanced Fermi mixture at unitarity (ﬁg. 51) and on the BEC-side of the Feshbach resonance (ﬁg. 52). In the BECregime, the sharp phase boundary between a balanced superﬂuid and the normal region no longer occurs. It is replaced by an expulsion of the normal cloud of unpaired atoms from the molecular condensate, which can be understood from a mean-ﬁeld picture. The imbalanced gas has turned into a Bose-Fermi mixture of molecules and unpaired

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Fig. 52. – Condensation in a strongly interacting, imbalanced Fermi mixture on the BEC-side, at B = 780 G or 1/kF a ≈ 0.5. In this regime one may start to describe the imbalanced gas as a strongly interacting atom-molecule mixture. Unlike on resonance, essentially all minority atoms are part of condensed fermion pairs. Temperatures were T /TF = 0.2 (a), T /TF = 0.12 (b) and T /TF ≤ 0.05 (c). The ﬁgure shows data from [312].

. fermions. We know from subsect. 4 7 that molecules repel unpaired fermions with a “Bose-Fermi” scattering length aBF = 1.18 a [205]. As a result, unpaired fermions expe2 rience a “Mexican-hat” potential V (r) + 4π aBF mBF nM (r) in the presence of molecules at density nM . . The physics of imbalanced fermionic superﬂuids is discussed in subsect. 7 3.2. The detailed analysis of the spatial density proﬁles for population imbalanced Fermi clouds is still an area of current research, and will not be covered in these lecture notes. The sudden change in the density proﬁle of imbalanced mixtures — as a function of temperature at ﬁxed imbalance, or as a function of imbalance at ﬁxed temperature — occurred simultaneously with the appearance of a condensate peak after a rapid ramp to the BEC side [312, 80]. This provides strong conﬁrmation that condensates observed

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via the rapid ramp technique truly mark a phase transition. In the following section, we will present the demonstration that these condensates are indeed superﬂuid — the observation of vortex lattices. . 6 6. Observation of vortex lattices. – The most dramatic demonstration of superﬂuidity in Bose-Einstein condensates is the observation of vortex lattices in rotating systems . (see subsect. 6 6.1). It was a natural goal to repeat such experiments for ultracold Fermi gases and to demonstrate superﬂuidity due to fermionic pairing. In this section, based on the Ph.D. thesis of one of the authors, we include details on the experimental techniques and results that were not included in the original publication [68]. Before we discuss the experimental realization, we will summarize some basic properties of vortices. In particular, we will show how a macroscopic wave function can accommodate vortices, and emphasize that it is not the existence of vortices, but rather the quantization of circulation, that is unique to superﬂuids and superconductors. . 6 6.1. Some basic aspects of vortices. Superﬂuids are described by a macroscopic wave function ψ(r) which is zero in the normal state and non-zero in the superﬂuid state, so . it qualiﬁes as the order parameter of the superﬂuid phase transition (see subsect. 4 11). As a wave function, it is a complex quantity with a magnitude and phase φ (227)

ψ(r) = |ψ(r)| eiφ(r) .

The velocity of the superﬂuid is the gradient of its phase, (228)

v=

∇φ , m∗

where m∗ is the mass of the bosonic entities forming the superﬂuid. In the case of fermionic superﬂuids, we have m∗ = 2m, where m is the fermion mass. Integrating eq. (228) around a closed loop inside the superﬂuid, we immediately arrive at the OnsagerFeynman quantization condition [313-315], (229)

v · dl = n

h m∗

with integer n. If the superﬂuid wave function has no nodal lines and the loop fully lies in a simply connected region of space, we must have n = 0. However, eq. (229) can be fulﬁlled with n = 0 if the wave function contains a vortex, that is, a ﬂow ﬁeld that depends on the vortex core distance r like v ∼ 1/r. At the location of the vortex, the wave function vanishes, it has a nodal line. This is the way a superﬂuid can carry angular momentum. In case of cylindrical symmetry (with the vortex core at the center), the angular momentum per boson or fermion pair is quantized in units of . Note that vortices are a property of the superﬂuid in the ground state at given angular momentum. This is in marked contrast to classical vortices, which exist only in metastable or nonequilibrium situations. Vortex patterns will ultimately decay into rigid body rotation

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whenever the viscosity is non-zero. Vortices of equal charge repel each other. This immediately follows from kineticenergy considerations. Two vortices on top of each other double the velocities and quadruple the energy. Two vortices far separated have only twice the energy of a single vortex. As a result, vortices with charge |n| > 1 will quickly decay into singly charged vortices [316]. If many vortices are created, they minimize the total kinetic energy of the cloud by arranging themselves into a regular hexagonal lattice, called Abrikosov lattice [317]. How can quantized vortices nucleate? Vortices cannot suddenly appear within the condensate, as the angular momentum contained within a closed loop inside the condensate cannot abruptly jump. Rather, the nodal lines have to enter the condensate from a surface, where the condensate’s wave function is zero. This surface can also be the surface of a stirrer, if it fully expels the condensate. One pathway to generate vortices is to excite surface modes. They are generated by moving a boundary condition (stirrer or container walls) faster than the local critical velocity vc for such excitations [318]. Which surface excitations are eﬃciently created depends on the shape of the stirrer [319, 320], or, in the case of a rotating container, the roughness of the container walls. Accordingly, the necessary critical angular velocity Ωc to nucleate vortices will depend on the stirrer’s shape. Note that Ωc can be much higher than the thermodynamic critical angular velocity Ωth . The latter is the angular velocity at which, in the rotating frame, the ground state of the condensate contains a single vortex. But simply rotating the condensate at Ωth will not lead to this ground state, because a vortex has to form on the surface where its energy is higher than in the center presenting an energy barrier. Driving a surface excitation provides the necessary coupling mechanism to “pump” angular momentum into the condensate, which can subsequently relax into a state containing vortices. . 6 6.2. Realization of vortices in superconductors and superﬂuids. The Lorentz force on charged particles due to a magnetic ﬁeld is equivalent to the Coriolis force on neutral particles due to rotation. Therefore, a magnetic ﬁeld does to a superconductor what rotation does to a neutral superﬂuid. Weak magnetic ﬁelds are completely expelled by a superconductor (the Meissner eﬀect), analogous to a slow rotation with angular velocity less than Ωth for which the neutral superﬂuid does not acquire angular momentum. For higher magnetic ﬁelds, quantized magnetic ﬂux lines, vortices, penetrate the superconductor. Quantized circulation in superﬂuid 4 He was observed by Vinen in 1958 [321] by measuring the frequency of a thin wire’s circular motion placed at the center of the rotating superﬂuid. Quantized magnetic ﬂux was measured by Deaver and Fairbanks [322] and Doll and N¨ abauer in 1961 [323] by moving a thin superconducting cylinder of tin toward and away from a conducting coil and measuring the electromotive force induced in the coil as a function of applied ﬁeld. Entire Abrikosov lattices of magnetic ﬂux lines were observed by using ferromagnetic particles that were trapped at the lines’ end-points (Tr¨ auble and Essmann [324], Sarma [325], independently in 1967). The di-

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Fig. 53. – Fate of a quadrupole oscillation in a rotating atomic Bose-Einstein condensate. The images show a sodium condensate in the magnetic trap after stirring slightly above the quadrupole resonance (at 52 Hz, trapping frequencies ν⊥ = 73 Hz and νz = 18 Hz) and equilibrating for a certain time t (time given in ms). First, the condensate rotates in the form of a perfect quadrupolar collective excitation. After about 100 ms, density depletions looking like vortex cores start to appear at the edges of the condensate. Between 500 ms and 1 s, some of these penetrate into the condensate as vortex lines, which arrange themselves into an ordered lattice after about 1-2 s.

rect observation of vortex lattices in superﬂuid 4 He was achieved in 1979 by Yarmchuk, Gordon and Packard [326] by imaging ions trapped in the core of the vortex lines. In gaseous Bose-Einstein condensates, single vortices were created by a phase imprinting technique [292], and vortex lattices were created by exposing the condensate to a rotating potential [226-229]. Using the method of the vibrating wire, the presence of quantized circulation was conﬁrmed for the fermionic superﬂuid 3 He in 1990 by Davis, Close, Zieve and Packard [327]. The MIT work described here represents the ﬁrst direct imaging of vortices in a fermionic superﬂuid. It is worth adding that glitches in the frequency of pulsars, fast rotating neutron stars, have been attributed to the spontaneous decay of vortex lines leaving the neutron pair superﬂuid [328, 329]. . 6 6.3. Experimental concept. For weakly interacting Bose gases in magnetic traps, the techniques for setting the cloud in rotation are well established [296, 226, 227, 330, 228]. In [330,228], the initially axially symmetric magnetic potential is deformed into an ellipse in the radial plane, which is then set in rotation. In [226, 296], an asymmetric optical dipole potential is superimposed with a cylindrically symmetric magnetic trap, again resulting in an elliptically deformed potential. In these cases, the role of the “rotating container walls” needed to nucleate vortices is played by the smooth elliptical deformation. This potential can excite only a speciﬁc surface excitation of the condensate, a rotating quadrupole mode. This collective excitation carries angular momentum m = ±2 (the axial component of angular momentum) and can only be excited around a resonant

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√ angular frequency (24 ) ΩQ = ω⊥ / 2, where ω⊥ is the radial trapping frequency (25 ). The excited quadrupole mode will eventually decay (via a dynamical instability) into vortices [332] (see ﬁg. 53). In the MIT experiments [296,227,333], two (or more) small, focused laser beams were symmetrically rotated around the cloud. Vortices could be created eﬃciently over a large range of stirring frequencies [227, 320]. The small beams presented a sharp obstacle to the superﬂuid, most likely creating vortices locally at their surface [320], corresponding to high angular momentum excitations with low critical angular velocities. This is the strategy followed in our experiment on rotating Fermi gases. A major technical challenge was to create a trapping potential which had a high degree of cylindrical symmetry. In Bose gases, this was provided by a magnetic trap, a TOP trap in [226, 229] that can be accurately adjusted for a very symmetric potential, or a Ioﬀe-Pritchard trap in [227] that has a high degree of cylindrical symmetry builtin. For fermions, one had to engineer an optical trap with a very round laser beam for optical trapping and carefully align it parallel to the symmetry axis of the magnetic saddle point potential, formed by the magnetic Feshbach ﬁelds and gravity. In this “sweet spot”, gravity is balanced by magnetic-ﬁeld gradients, and the only remaining . force acting on the atoms is from the laser beam (see subsect. 2 2.2). Experimentally, the trapping potential was designed using a sodium BEC as a test object. This had the advantage that the experimental parameters for the creation of vortex lattices were well known. After this had been accomplished, the next challenge was to identify the window in parameter space that would allow to observe vortices in a Fermi gas. It was not evident whether such a window existed at all, where heating during the stirring would not destroy the superﬂuid, and where the decay rate of vortices would be slow enough to allow their crystallization and observation. . 6 6.4. Experimental setup. The experimental setup is presented in ﬁg. 54. It was tested and optimized using sodium Bose-Einstein condensates. Figure 55 shows how we determined the parameters for stirring and equilibration using a sodium BEC in a magnetic trap. The next step was to repeat these experiments in an optical trap, initially without high magnetic ﬁeld, optimizing the shape of the optical trap. Then magnetic ﬁelds were added. This required a careful alignment of the optical trap to the magnetic saddle point. We estimate the residual ellipticity of the transverse potential to be less than 2% (26 ). Magnetic ﬁelds were left on during the expansion. For lithium this was crucial: molecules at the initial densities are only stable against collisions at magnetic ﬁelds close to the Feshbach resonance. During the expansion at high magnetic ﬁeld, the √ (24 ) While a collective excitation carrying angular momentum m has an energy ω⊥ m, it is √ m-fold symmetric and is thus excited at a frequency Ω = ω⊥ / m, see [331]. p (25 ) In the presence of an elliptic deformation, one needs to replace ω⊥ by (ωx2 + ωy2 )/2, where ωx,y are the trapping frequencies in the direction of the long and short axis of the ellipse. (26 ) Of course, this cannot compare with the almost perfect roundness of a (magnetic) TOP trap, with residual ellipticity of less than 0.1% [330].

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b) 10 W IR 1064 nm

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Fig. 54. – Experimental setup for the observation of vortices in a Fermi gas. a) Sketch of the geometry. The atomic cloud (in red) is trapped in a weakly focused optical dipole trap (pink). The coils (blue) provide the high magnetic oﬀset ﬁeld to access the Feshbach resonance as well as the axial conﬁnement (additional curvature coils not shown). Two blue-detuned laser beams (green) rotate symmetrically around the cloud. An absorption image of the expanded cloud shows the vortices. b) Optical setup for the vortex experiment. The laser beam forming the dipole trap is spatially ﬁltered using a high-power optical ﬁber. Care is taken not to deteriorate the quality of the Gaussian beam’s roundness when passing through several lenses after the ﬁber exit. The stirring beam (green) passes through two crossed AOMs that deﬂect it in the transverse (XY) plane. These beams are overlapped with the imaging light by dichroic mirrors. The light for the magneto-optical trap (MOT) is overlapped on a polarizing beam splitter cube (PBS).

cloud could tilt, revealing small misalignments between the optical trap and the saddle point. Even more importantly, additional steps were necessary to ensure vortex visibility after expansion into the saddle point potential, as discussed below. Finally, large vortex lattices containing about 120 vortices were created in sodium Bose-Einstein condensates both in magnetic and optical traps with similar lifetimes (see ﬁg. 56), and we were ready to proceed with lithium. This required two changes to the trap geometry: 1) Lithium is lighter, and keeping the saddle point of the combined magnetic and gravitational potential in place requires reducing the ﬁeld curvature. 2) The higher chemical potential of lithium required a larger beam waist to obtain the correct trap . depth for evaporative cooling (see eq. (6) in subsect. 2 2.2). Expansion of vortex lattices. In the trap, the vortex size is on the order of the healing length (for an atomic or a molecular BEC) or of the inverse Fermi wave vector (for

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Fig. 55. – Vortex nucleation for violent stirring in an atomic BEC. The upper row shows expansion images of sodium condensates after 500 ms of stirring at the quadrupole frequency, for diﬀerent laser powers of the stirring beam. The lower row shows the resulting BEC after 300 ms of equilibration time. This suggests that the condensate has to be severely excited to generate many vortices. From left to right, the laser power was increased for each subsequent image by a factor of two. The cloud was held in a magnetic trap.

a strongly interacting Fermi gas), about 200 nm. This small size is prohibitive for in situ detection using optical techniques. Fortunately, angular-momentum conservation allows vortices to survive the expansion of the condensate, which we can thus use as a “magnifying glass”. However, only in simple geometries is the expansion a faithful magniﬁcation. Complications arise due to the expansion into a saddle potential. Hydrodynamic expansion into a saddle point potential has been discussed in sect. 3. How does the vortex core size change during expansion? There are two regimes one can simply understand, the initial hydrodynamic expansion and the ballistic expansion at long times of ﬂight. In the ﬁrst part of the expansion, the mean-ﬁeld μ ∝ na changes so slowly that the condensate wave function can still react to the change in density: Adjustments on the healing length scale ξ — about the size of a vortex in equilibrium — can occur at a rate /mξ 2 = μ/. As long as the rate of change of μ — essentially the rate of change of the radial Thomas-Fermi radius Rr — is smaller than μ/, the vortex core can still adjust in size to the local mean-ﬁeld. It thus grows as ξ ∝ 1/ n(t)a ∝ Rr (t) Raa(t) , where Ra (t) is the axial Thomas-Fermi radius after expansion time t. If Ra does not vary appreciably, ξ/Rr (t) will remain constant during the expansion, the vortex core grows just as the size of the condensate, and the magniﬁcation is faithful. Once the rate of change of μ(t) becomes comparable to μ(t)/, the condensate can no longer adiabatically adapt to the lowering density. The characteristic expansion rate being ωr , this occurs when μ(t) ≈ ωr . For much longer expansion times, we are in the limit of ballistic expansion. Here, each particle escapes outward with the given velocity (in free space) or, in the case of the saddle potential, with a radial acceleration

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Fig. 56. – Vortex lattice in a Bose-Einstein condensate of sodium atoms in the magnetic trap (left) and the optical trap (right image). The optical trap (highest number obtained ∼ 120 vortices) can favorably compare with the magnetic trap (highest number in our experiment ∼ 150 vortices).

proportional to its distance from the origin. This simply rescales the radial dimension, and thus stretches the vortex core and the cloud size by the same factor. Again the magniﬁcation is faithful. However, in our experiment we are not in the quasi-2D regime where Ra Rr . The saddle potential “squishes” the cloud in the axial dimension, as the decreasing meanﬁeld no longer stabilizes the condensate’s axial size. According to the above estimate, the vortex cores will shrink in comparison to the cloud size by a factor ∝ Ra (t). We can see the eﬀect on a sodium condensate in our optical trap in ﬁg. 57, where the axial curvature was left on for longer and longer times during expansion. To work around this problem, the magnetic-ﬁeld curvature was quickly reduced after releasing the cloud from the trap, by ramping down the current in the curvature coils (in about 1 ms). As this increases the overall oﬀset ﬁeld (the curvature bias ﬁeld is aligned opposite to the Feshbach bias), the current in the Feshbach coils is decreased accordingly, so as to leave the oﬀset ﬁeld B0 — and the interaction parameter of the Fermi mixture — constant. The radial expansion is speed up even further in comparison to the axial evolution by actively “squishing” the cloud about 3 ms before release. This is done by simply ramping up the power in the optical trapping beam by a factor of 4. Not only does this increase the radial trapping frequency, but it also excites a “breathing” mode in the condensate. The result is that the condensate expands almost twice as fast radially as without these steps. . 6 6.5. Observation of vortex lattices. The search for vortices in Fermi gases started on the BEC side of the Feshbach resonance. We hoped that far on the molecular side, in the regime where kF a is small, the situation would be fully analogous to the case of atomic condensates. However, too far away from resonance, the molecules (which are in the highest vibrational state of the interatomic potential!) can undergo rapid vibrational

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Fig. 57. – Decrease of vortex visibility for a sodium condensate expanding from the optical trap into a magnetic saddle potential. Top and bottom row show axially and radially integrated optical densities, respectively. The saddle potential is conﬁning in the axial and anti-conﬁning in the radial direction. As the condensate expands radially, it collapses in the axial dimension, a direct consequence of hydrodynamic ﬂow. The vortex cores shrink and collapse onto themselves, thereby ﬁlling in completely and forming ring-like structures (see text for details). For the images, the magnetic ﬁeld curvature (νz = 26 Hz) was switched oﬀ after, from left to right, 0, 2, 3, 4, 5, 7 and 10 ms. The total time of ﬂight was constant at 35 ms.

relaxation via three-body collisions, leading to heating and trap loss. The lifetime of the gas needs to be longer than the vortex nucleation and equilibration time (typically 1 s). If the entire preparation of vortex lattices is to happen at the same magnetic ﬁeld, this limits the smallest values of kF a one can study (kF a 2 in our experiment). On the other hand, closer to resonance, the vortex cores become smaller. The core size in the BEC regime is given by the healing length ξ ∝ k1F √k1 a and decreases for F increasing kF a, eventually saturating in the unitarity regime at a value on the order of 1/kF . Furthermore, closer to resonance, quantum depletion becomes important: The condensate density n0 is no longer equal to the total density, and the vortex core loses contrast. Fortunately, it turned out that a window existed, and at a ﬁeld of 766 G (1/kF a = 1.3), we were successful: After stirring the cloud for 800 ms and letting the cloud equilibrate in 400 ms, we observed a vortex lattice in the density proﬁle (ﬁg. 58). This established superﬂuidity for fermion pairs. Starting from here, diﬀerent methods were developed to improve the vortex contrast. Not surprisingly it turned out that reducing the interaction strength had the largest impact. In the moment the vortex lattice is released from the trap, the magnetic ﬁeld is lowered to ﬁelds around 700 G (1/kF a ≈ 3 initially, further increasing during expansion). If the condensate still has time to react to this change in scattering length, the vortex √ size ξ ∝ Rr (t)/ a will increase relative to the condensate’s radius (the expression for ξ is valid in the BEC-limit, and assumes radial expansion, see previous section). On the other hand, we found that the ramp should not move too far into the weakly interacting regime: The condensate would simply not expand anymore as practically all the repulsive mean ﬁeld has been taken out of the cloud. We also explored delaying the ramp until some expansion has taken place. However, if the delay was too long, the condensate had reduced its density to the point where it was not able to adjust quickly enough to the

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Fig. 58. – Vortex lattice in a 6 Li2 molecular condensate. In a), stirring, equilibration and imaging of the vortex lattice all took place at a ﬁxed ﬁeld of 766 G (1/kF a = 1.3). Image a) shows the very ﬁrst clear signature we observed. The vortex core depletion is barely 10%. b) A Fourier ﬁlter applied to a) clearly shows the Abrikosov vortex lattice. c) The improved scheme of “squishing and release” (see text), as well as a sudden reduction of the interaction strength led to an improved vortex contrast. From [68].

new interaction strength and increase its vortex size. For this, the “reaction rate” of the condensate wave function, μ/ at the ﬁnal ﬁeld should be faster than the rate of change of μ, that is, the rate of change of Rr (t). This technique of observing vortices worked on both sides of the Feshbach resonance. Stirring, equilibration and initial expansion could be performed at magnetic ﬁelds between 740 G (1/kF a = 2) and 860 G (1/kF a = −0.35), before switching to the BEC-side during expansion for imaging. The observation of ordered vortex lattices above the Feshbach resonance at 834 G, on the BCS-side of the resonance, establishes superﬂuidity and phase coherence of fermionic gases at interaction strengths where there is no two-body bound state available for pairing. The sweep down to 735 G solved another potential diﬃculty in detecting vortices on the BCS side. The condensate fraction is reduced by quantum depletion on the BEC-side, . and on the BCS side it is only ∼ Δ/EF (see 4 9). In a simpliﬁed picture, it is only this “coherent” part of the atomic density which vanishes at the vortex core, reducing the contrast (for more elaborate treatments, see [334, 213, 335]). By sweeping to the weakly interacting BEC regime, the low contrast vortices on the BCS-side are transformed into BEC-type vortices with high contrast. Since a ramp is involved in the detection of vortex lattices, the relevant time scales need to be analyzed, as in the case of the observation of condensation via rapid ramps. The conclusion is that vortex lattices cannot form during the 10 ms of expansion at the imaging ﬁeld, on the BEC-side of the resonance. We observed that the vortex lattice needs many hundreds of milliseconds to form in the stirred cloud. This is the same time scale found for the lattice formation in atomic BECs [332, 333]. This time scale was found to be independent of temperature [333] and seems to represent an intrinsic

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Fig. 59. – Observation of vortices in a strongly interacting Fermi gas, below, at and above the Feshbach resonance. This establishes superﬂuidity and phase coherence in fermionic gases. After a vortex lattice was created at 812 G, the ﬁeld was ramped in 100 ms to 792G (BEC-side), 833G (resonance) and 853G (BCS-side), where the cloud was held for 50 ms. After 2ms of ballistic expansion, the magnetic ﬁeld was ramped to 735G for imaging (see text for details). The ﬁeld of view of each image is 880 μm × 880μm. More recent version of ﬁg. 3 in [68].

time scale of superﬂuid hydrodynamics, dependent only on the trapping frequencies. It is also in agreement with a theoretical study of vortex formation in strongly interacting Fermi gases [336]. When a thermal cloud is slowly cooled through the transition temperature [330], the condensate ﬁrst forms without a vortex. As the condensate grows, vortices are nucleated at the surface and then enter the condensate [318]. When a thermal cloud is suddenly cooled, a condensate with phase ﬂuctuations will form [337, 338] which can arrange themselves into a vortex tangle. In either case, one would expect a crystallization time of at least several hundred milliseconds before a regular vortex lattice would emerge. Also, it takes several axial trapping periods for the vortex tangle to stretch out. Even if these time scales were not known, it is not possible to establish a regular vortex lattice with long-range order in a gas that expands at the speed of sound of the trapped gas. Opposing edges of the expanding cloud simply cannot “communicate” fast enough with each other. The regularity of the lattice proves that all vortices have the same vorticity. From their number, the size of the cloud and the quantum of circulation h/2m for each vortex, we can estimate the rotational frequency of the lattice. For an optimized stirring procedure, we ﬁnd that it is close to the stirring frequency. . 6 6.6. Vortex number and lifetime. The number of vortices that could directly be created on the BCS-side was rather low in the ﬁrst experiments, as the stirring seems to have

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1.5

1 / k Fa 0 -0.4

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834 880 930 Magnetic Field [G]

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Vortex Lifetime [s]

Vortex Decay Rate [1/s]

100

b) -0.7

Number of Vortices

0.6

a)

60

40

20

0 700

750

800 834 900 Magnetic Field [G]

950

Fig. 60. – 1/e-Lifetime (a) and number (b) of vortices in the BEC-BCS crossover. Vortices are long-lived across the entire BEC-BCS crossover. A narrow dip in the lifetime on resonance is presumably due to pair breaking (see text). Open symbols are data from April 2005 [68]. Optimization of the system using a deeper trap resulted in improved characteristics on the BCS-side, shown by the full symbols (data from June 2005). In b), the triangles give the number obtained by stirring and equilibrating both at the given ﬁeld. Stirring at 812 G and subsequently ramping to the ﬁnal ﬁeld for equilibration resulted in the data shown as circles. Figure adapted from [68], incorporating more recent results.

had an adverse eﬀect on the stability of the pairs. This corresponds to the expectation that the gas is more robust on the BEC-side, where the lowest excitations are sound waves, while on the BCS-side it is pair breaking. To optimize the vortex number on the BCS-side, ﬁrst a large vortex lattice was produced close to resonance, at 812 G, before ramping the magnetic ﬁeld beyond the Feshbach resonance. In this way, large numbers of vortices could be obtained in the entire BEC-BCS crossover (see ﬁg. 59). By monitoring the number of vortices after a variable delay time, the vortex lifetime was determined (ﬁg. 60a). The vortex lifetime around the Feshbach resonance is on the same order of what was found for atomic BECs. This displays the high degree of metastability of vortices in superﬂuids. One picture for vortex decay assumes that thermal excitations (or the normal component) provide friction between some residual trap anisotropies and the rotating superﬂuid [339, 340]. The diﬀerence in lifetimes observed in two diﬀerent data sets (ﬁg. 60a) can be explained by changes in the trap geometry, diﬀerent atom number and temperature. A deeper trap can hold large Fermi clouds on the BCS-side better, leading to a longer lifetime in this regime. There is a peculiar dip in the lifetime on resonance, which may be caused by the coupling of external motion to internal degrees of freedom. One possibility is a resonance between the pair binding energy and the rotation frequency. This requires pairs with a very small binding energy, which should exist only in the far outside wings of the cloud. For example, the (two-body) molecular binding energy at 830 G, 4 G away from

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a)

b)

Number of Vortices

25 20 15 10 5 0 0

5

10 15 Time Spent in Bucket [s]

20

after 10 s in Rotating Bucket

Fig. 61. – Rotating bucket for superﬂuid Fermi gases. a) Vortex number vs time spent in the rotating trap. After am equilibration time, the number of vortices stays constant. The ﬁnal vortex number depends on the power of the green stirring beam, indicating slippage due to residual friction with the “container walls”. The lower and upper curve correspond to lower and higher green beam power. b) Absorption image of an optimized vortex lattice, containing about 75 vortices, after 10 s hold time in the rotating bucket. The magnetic ﬁeld for all data was 812 G, corresponding to 1/kF a ≈ 0.2.

resonance, is only kB ×3 nK or h×60 Hz. This is on the order of × the rotation frequency Ω of the vortex lattice. If the molecules rotate around the trap, trap anisotropies may excite them resonantly and cause dissociation. On the BCS-side, any possible resonance may be suppressed by density-dependent broadening. Of course, the fraction of pairs at such low binding energies is very small, and they can contribute to damping in a major way only if surface eﬀects are important. . 6 6.7. A rotating bucket. (27 ) All the experiments described so far ﬁrst set the cloud in rotation using the stirring beam and then let the gas equilibrate in the stationary trap into a vortex lattice. In the stationary trap, the vortex lattice is of course only metastable (lasting as long as the angular momentum Lz is conserved), whereas in a trap rotating at a constant angular frequency Ω, the vortex lattice would be the true ground state. Mathematically speaking, both situations are described by the Hamiltonian H − ΩLz , where the second term is, in the latter case, the usual transformation to a frame rotating at frequency Ω, whereas in the ﬁrst case, the Lagrangian multiplier Ω enforces the conservation of Lz . In some previous experiments with BECs the rotating anisotropy needed to be switched oﬀ before an ordered vortex lattice could form [227], possibly because the rotating laser beam was not moving smoothly enough to allow equilibration into a vortex (27 ) The results of this section have not been published elsewhere.

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Fig. 62. – Superﬂuid expansion of a strongly interacting rotating Fermi gas. Shown are absorption images for diﬀerent expansion times on the BCS-side of the Feshbach resonance at 910 G (0, 1, 2, 3, 3.5, 4, and 4.5 ms) and at 960 G (0, 0.5, 1, 1.5, 2, 2.5, and 3 ms), before ramping to the BEC-side for further expansion and imaging. The total time of ﬂight was constant. The vortices served as markers for the superﬂuid parts of the cloud. Superﬂuidity survived during expansion for several ms, and was gradually lost from the low-density edges of the cloud towards its center. The ﬁeld of view of each image is 1.2 mm × 1.2 mm. From [341].

lattice. In our setup, it has become possible to observe vortex lattices in the presence of a rotating bucket and even increase the lifetime of the vortex lattice by maintaining the rotating drive. The experiment was performed at 812 G in a trap with radial trapping frequency νR = 90 Hz. The two stirring laser beams (power in each beam ≈ 100 μW, waist w = 16 μm) created only a weak potential of about 20 nK each on the cloud (mean ﬁeld μ ≈ 400 nK). They were rotated around the cloud at a frequency of 70 Hz. For imaging, the atoms were released from the combined trap, the conﬁning optical potential plus the repulsive stirring beam. We found that it was possible to stabilize a vortex lattice containing 19 vortices for 20 s (see ﬁg. 61), limited only by the computer memory controlling the experiment. The ﬁnal vortex number depended on the laser power of the stirrer. Increasing the power in the stirring laser by 60% increased the equilibrium vortex number to 29. This suggests that the equilibrium vortex lattice, at least at the weaker laser power, had not reached the angular velocity of stirrer. It appears that the drive was necessary to compensate for friction with some residual trap anisotropy. At later stages of the experiment, we were able to stabilize 75 vortices for 10 s in a deeper trap with νr = 120 Hz. These experiments are analogous to the pioneering “rotating bucket” experiments on 4 He [326], where it was possible to maintain a rotating superﬂuid containing four vortices for eleven hours, only limited by the eventual exhaustion of the refrigerator helium supply.

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Peak kFa

-1.0

-2.0 Number of Vortices 40 -3.0 880

900

920

30

20 940

10

0 960

Magnetic Field [G]

Fig. 63. – Central interaction strength kF a during superﬂuid expansion. Starting at a central kF a in the optical trap, vortices survive up to an interaction strength kF a ≈ −0.8, almost independent of the magnetic ﬁeld (scattering length a). Filled circles correspond to partially superﬂuid, open circles to normal clouds. The observed number of vortices is color coded. From [341].

. 6 6.8. Superﬂuid expansion of a rotating gas. What will happen when a fermionic superﬂuid expands? First, it should follow the hydrodynamic equations for a superﬂuid ﬂow [342]. However, ultimately, pairing is a many-body eﬀect, and when the cloud becomes very dilute, at ﬁnite temperature, the pairs will eventually break and superﬂuidity will be quenched. This is diﬀerent from the situation for a BEC and at unitarity. Since phase space density is preserved during expansion, T /TF or T /TC is, here, constant and the gas remains superﬂuid (28 ). By using rotating Fermi gases, vortices serve now as a convenient marker for the superﬂuid phase. When the superﬂuid is quenched, the vortices will disappear [341]. We allow the Fermi gas to expand on the BCS-side for a certain time tBCS , then ramp down to the BEC-side for further expansion and imaging. Vortices can be observed only when the gas is still a superﬂuid at the moment of the magnetic ﬁeld sweep. The total expansion time is kept constant. It is found that superﬂuid ﬂow initially persists during the expansion. Then, vortices start to disappear ﬁrst at the edges of the cloud, then, for longer BCS-expansion tBCS , further inwards until the last vortex disappears at the cloud’s center (see ﬁg. 62). The time tBCS for which the last vortex disappears, increases the closer we are to resonance, that is, the larger the interaction strength and the stronger the fermion pairs are bound. Vortices and therefore superﬂuid ﬂow in free expansion were observed up to expansion times, for example, of 2.5 ms at 960 G and of 5 ms at 910 G. By varying the magnetic ﬁeld and thus the scattering length, we ﬁnd that the last vortex always disappears at about the same value of the interaction parameter kF a ≈ (28 ) For interacting condensates, there is a small variation of critical phase space density as function of particle density, and therefore the gas can cross the phase transition during adiabatic expansion [343].

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−0.8 (see ﬁg. 63). The simplest explanation for this observation is that we cross the phase transition line during expansion. While T /TF is an adiabatic constant for the expansion, T /TC is not, as TC /TF depends exponentially on the density. As the density decreases, the critical temperature in the outer regions of the cloud eventually drops below T , superﬂuidity is lost starting from the edges inwards. We can estimate the critical interaction strength for this breakdown to occur. At our lowest temperatures, T /TF = 0.05. The formula for TC due to Gorkov and Melikγ 7/3 Barkhudarov [199] gives TC = eπ 2e TF e−π/2kF |a| = 0.28 TF e−π/2kF |a| . This formula should be valid in the BCS-regime where kF |a| 1. The equation (230)

1=

T T TF TF = ≈ 0.05 = 0.18 eπ/2kF |a| TC TF TC TC

gives a critical interaction strength kF |a| = 0.9, close to the observed value. For a discussion of other mechanisms which can explain the observed disappearance of vortices, we refer to the original publication [341]. On resonance we have 1/kF a = 0 during the whole expansion, and the gas should remain superﬂuid. Indeed, in the experiment we do not see evidence for a sudden quenching of vortices, but rather a gradual loss in contrast. Vortices could be discerned at total densities as low as 1.2 × 1011 cm−3 , thus providing evidence for fermionic pairing and superﬂuidity at average interatomic distances of 2 μm. 7. – BEC-BCS crossover: Energetics, excitations, and new systems In the previous section, we focused on two key observations in the BEC-BCS crossover, pair condensation and vortices, that provide direct access to the phenomena of coherence, a macroscopic wave function, and superﬂuid ﬂow. A host of other studies have been performed in this regime, which we summarize here. We divide these experiments into three diﬀerent categories: characterization of the equilibrium state by energy, entropy and momentum distribution, dynamic measurements addressing collective excitations, sound and the critical velocity for superﬂuidity, and thirdly new systems, where the original two-component fermion system has been modiﬁed, either by an optical lattice, or by imbalanced populations. Both of these modiﬁcations add another “dimension” to the system, which is the lattice depth and the imbalance. . 7 1. Characterization of the equilibrium state. . 7 1.1. Energy measurements. The total energy of the cloud determines how large the cloud is in the harmonic-oscillator potential, or how fast it expands after switching oﬀ the trap (29 ). On resonance, the virial theorem provides a simple relationship between cloud size and total energy. Using the universality hypothesis that the only relevant energy scale is the Fermi energy, it follows that the potential energy is half the total (29 ) The binding energy of molecules must be subtracted, as this internal energy cannot be converted into external, mechanical energy.

Making, probing and understanding ultracold Fermi gases

259

energy, as for a non-interacting gas [176]. In a homogenous system (and locally for a trapped system), the energy content of an interacting Fermi gas is parameterized as E = (1 + β)EF , where βEF is the contribution of interactions. For unitarity limited interactions, β is an important universal parameter characterizing the ground state of strongly interacting fermions. The total energy of an interacting Fermi gas at or close to resonance was derived from measurements of the cloud size either in trap or after expansion [60, 344, 67, 72, 79, 345]. In an interesting variant, the Paris group applied a rapid switching of the magnetic ﬁeld to zero, which was faster than the trap period [124]. In this case, the interaction energy could be removed before it had been converted to kinetic energy. By comparing expansion with immediate or delayed magnetic ﬁeld switching, the interaction energy could be directly measured. This work showed a surprising behavior of the interaction energy in a wide region of magnetic ﬁelds below the Feshbach resonance. This behavior was later explained by the formation of molecules and was probably the ﬁrst hint that these molecules would be stable. Current experiments (including a measurement of the speed of sound [84]) give β ≈ 0.58, diﬀerent experiments agree to within 10%, and most importantly agree with theoretical predictions (via analytical methods [346-349, 203, 215, 350], Monte Carlo calculations [351-353] and renormalization group methods [221]). A table summarizing all experimental and theoretical determinations of β can be found in the contribution of C. Salomon to these proceedings. The fact that the same value of β was found for 6 Li and 40 K is an impressive conﬁrmation of universality [350] and is a powerful demonstration for the use of ultracold atoms as a model system for many-body physics. It is possible to obtain the entropy of the cloud from size or energy measurements. For this, the magnetic ﬁeld is adiabatically swept to transform the system into a weakly interacting Fermi gas. The observed size or energy in this regime gives the entropy, since their relation is accurately known for an ideal Fermi gas. This allows the determination of entropy vs. energy for the strongly interacting gas [183] (see ﬁg. 64). The results of this study agreed well with the predictions from Monte Carlo simulations that vary smoothly across the phase transition. By using a split power law ﬁt, a value for a critical energy Ec had been obtained. However, since the ﬁt did not address the behavior in the critical region, it is not clear how accurately the split power law ﬁt can determine the transition point. When energy measurements were combined with empirical thermometry and theoretical corrections, the dependence of the heat capacity of the Fermi gas on temperature could be deduced [72]. Further discussion of energy measurements can be found in the contributions of C. Salomon and S. Stringari to these lecture notes. . 7 1.2. Momentum distribution. The momentum distribution of the atoms in the cloud can be determined by releasing them from the trap and simultaneously switching the scattering length to zero. Such studies have been performed in both 40 K [354] and 6 Li (see C. Salomon’s contribution to these proceedings).

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Fig. 64. – Measured entropy per particle of a strongly interacting Fermi gas at 840 G versus its total energy per particle. Various theoretical predictions are compared. Reprinted from [183].

Far on the BCS side, one ﬁnds the momentum distribution of an ideal Fermi gas in a harmonic trap. On the BEC side, the momentum distribution approaches the squared magnitude of the molecular wave function’s Fourier transform. The crossover region smoothly interpolates between these two limits. The modiﬁcation of the momentum distribution due to the superﬂuid phase transition is too small to be discernable in these measurements [355]. Momentum distributions are discussed in the contributions of D. Jin and C. Salomon to these proceedings. . 7 1.3. Molecular character. Near a Feshbach resonance, the closed-channel molecular state responsible for the resonance is mixed with the continuum of scattering states in the open channel. In the case of 6 Li, those two channels have singlet and triplet character, respectively. Close to the Feshbach resonance, the loosely bound molecular state becomes completely dominated or “dressed” by the open channel. This was conﬁrmed by applying a molecular probe to a cold Fermi gas, thereby exciting atom pairs to an electronically excited, singlet molecular state at a rate that was proportional to their closed-channel character Z [76]. The wave function describing the dressed molecule or fermion pair can be written [76] (231)

|ψp =

√ √ Z |ψv=38 (S = 0) + 1 − Z |φa (S = 1) ,

(S = 1) where |ψv=38 (S = 0) denotes the closed channel, singlet molecular state, and |φa√ the open channel, triplet contribution, with relative probability amplitude Z and √ 1 − Z, respectively. In the singlet channel, only the v = 38 vibrational state is relevant due to its near-resonant energy.

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10

-1

10

-2

10

-3

Z

10

-4

10

-5

10

-6

10

-7

10

600

650

700

750

800

850

900

950

Magnetic Field (G) Fig. 65. – Closed-channel character Z of lithium atom pairs versus magnetic ﬁeld. The dotted line shows the closed-channel character of the bound molecular state below the Feshbach resonance. Reprinted from [76].

By monitoring trap loss during the excitation, Z could be determined (see ﬁg. 65) and it was veriﬁed that the Feshbach resonance in 6 Li is indeed broad, that is, the closedchannel contribution to the pair wave function is negligible throughout the crossover region. In the two-channel BEC-BCS crossover description, Z is proportional to Δ2 [76], and one might hope that by measuring Z one actually measures the magnitude of the macroscopic order parameter. However, a spectroscopic determination of the singlet character is a local probe, sensitive only to g 2 (0), the two-particle correlation function at zero distance. As such, it can measure local pair correlations, but not global phase coherence or condensation of pairs. In fact, the BEC-BCS crossover allows for pair . correlations above TC , that can be seen in RF spectra (see subsect. 7 2.4). A closedchannel character of these pairs has indeed been identiﬁed even in the normal phase, on both sides of the Feshbach resonance [308]. . 7 2. Studies of excitations. – To explore a new form of matter, one should probe its response to perturbations. In this section we summarize experimental studies on excitations of Fermi gases in the BEC-BCS crossover region. These studies include sound waves (resonant standing waves and sound propagation), observation of the critical velocity for superﬂuid ﬂow, and single-particle excitations probed via RF spectroscopy. . 7 2.1. Collective excitations. By a sudden or periodic modulation of the trapping potential, eigenmodes of the trapped cloud can be excited. The eigenfrequencies can be sensitive to the equation of state, μ(n), and thus characterize the interactions in the system. Collective excitations were among the ﬁrst properties studied in the case of atomic

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BECs. They provided stringent tests of the Gross-Pitaevskii-equation governing those condensates [356-358, 173, 359], and posed challenges to ﬁnite-temperature theories. In the case of fermions, collective excitations were studied already for weakly interacting gases in the “pre-Feshbach era” [360]. After the realization of fermion pair condensation, extensive studies were carried out at Duke and Innsbruck. These are discussed in detail in the contribution of R. Grimm to these proceedings. Section 3 discusses the equations of motion for strongly interacting gases, including collisional and superﬂuid hydrodynamics. To obtain the response to a small perturbation of the conﬁning (harmonic) potential, one can directly use eq. (50) in the case of classical hydrodynamics, or eq. (54) for superﬂuid hydrodynamics, and linearize the equations for small oscillatory changes of the cloud radius. For instance, a “breathing” mode of a cigarshaped cloud can be excited by suddenly squeezing the cigar in both radial directions. By only squeezing one radial dimension, one excites a “standing quadrupole” mode. Depending on the symmetry of the excited mode, diﬀerent eigenfrequencies are found that depend more or less strongly (or do not depend at all) on the equation of state. The latter is parameterized by the exponent γ in μ(n) ∝ nγ . Both the Duke [73] and Innsbruck [74] group have conﬁrmed the value of γ = 2/3 at the Feshbach resonance, which is the same as for a weakly interacting Fermi gas. A precision study of collective oscillations on the BEC side of crossover has veriﬁed the famous Lee-Yang-Huang correction to the equation of state of a strongly interacting Bose gas [211]. . We refer the reader to subsect. 6 2 for a discussion of further experimental studies of collective excitations. They characterized the strongly interacting and superﬂuid regimes as a function of scattering length and temperature and revealed an intriguing (and not yet fully understood) picture of hydrodynamic behavior and smooth or sudden transitions to collisionless dynamics. . 7 2.2. Speed of sound. Density perturbations propagate at the speed of sound. In . subsect. 4 7.3 we discussed that the Bogoliubov sound mode on the BEC side smoothly evolves into the Bogoliubov-Anderson mode on the BCS side. A laser beam focused into the center of the cloud can create a localized density perturbation, which then propagates along a cigar-shaped atom cloud [361]. Using this technique, the Duke group has recently measured the speed of sound in a Fermi gas across the BEC-BCS crossover and found very good agreement with Monte Carlo predictions [84] (see ﬁg. 66). . 7 2.3. Critical velocity. Superﬂuid ﬂow breaks down above a critical velocity. This critical velocity is a threshold velocity for creating excitations. For density ﬂuctuations, it is the speed of sound, discussed in the previous paragraph, and this provides the critical velocity on the BEC side. It monotonously increases towards resonance. On the BCS . side, as discussed in subsect. 4 7.4, pair breaking becomes the dominant mechanism. The pairing energy is largest near resonance, resulting in a maximum of the critical velocity around resonance.

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Making, probing and understanding ultracold Fermi gases

c0/vF 0.5 0.4 0.3 0.2 -1

0

1

2

3

4

5

6

1/kFa Fig. 66. – Sound velocity normalized by the Fermi velocity vF versus the interaction parameter, . 1/kF a. Black dotted curve: mean-ﬁeld theory based on the Leggett ground state (see 4 7.4). Gray solid curve: quantum Monte Carlo calculation [362]. Black dashed curve: Thomas-Fermi theory for a molecular BEC. Reprinted from [84].

This has been recently observed at MIT [85]. By recording the onset of dissipation in a Fermi cloud exposed to a weak one-dimensional lattice moving at a variable velocity, critical velocities were obtained. When the magnetic ﬁeld was varied, they showed a peak near resonance. In these experiments the lattice was created by two focused laser beams crossing at an angle of about 90 degrees exposing only the central part of the cloud to the moving lattice (ﬁg. 67). When the whole cloud was excited by the moving lattice, much lower critical velocities were found, most likely due the breakdown of superﬂuidity in the low-density spatial wings of the cloud. Using larger depths of the moving lattice, smaller values of the critical velocity were found. This shows that the lattice is not only a way to probe the Fermi gas, it is also a way to create new systems with interesting . properties (see subsect. 7 3). . 7 2.4. RF spectroscopy. Single-particle excitations can reveal the nature of pairing. On the BEC side, the excitation of a single atom requires breaking a molecular bond, thus providing information about the binding energy. On the BCS side, single-particle excitations reveal the superﬂuid energy gap and give access to the microscopic physics underlying these Fermi mixtures. In condensed-matter samples, the presence of an excitation gap is clearly seen in tunnelling experiments between a superconductor and a normal metal, divided by a thin insulating barrier. The tunnel eﬀect allows individual electrons to pass through the barrier. For this to occur, electrons must ﬁrst be excited from the pair condensate, which costs an energy Δ. For applied voltages smaller than Δ/e, no tunnelling occurs. Abstracting from the tunnelling example, what is required to measure an excitation spectrum is the coupling (= tunnelling) between the initial many-body state of interest and a well-characterized reference state (= metal). In atomic Fermi gases, this situation

264

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Condensate Number N0 (x 105)

5

4

3

v 2

1

0 0

2

4

6

8

Lattice Velocity v (mm/s)

Fig. 67. – Onset of dissipation for superﬂuid fermions in a moving optical lattice. (Inset) Schematic of the experiment in which two intersecting laser beams produced a moving optical lattice at the center of an optically trapped cloud (trapping beams not shown). Number of fermion pairs which remained in the condensate after being subjected to an optical lattice of depth 0.2 EF for 500 ms, at a magnetic ﬁeld of 822 G (1/kF a = 0.15). An abrupt onset of dissipation occurred above a critical velocity. Reprinted from [85].

can be established to some degree using RF spectroscopy. Starting with a Fermi mixture of atoms in, say, the hyperﬁne states |1 and |2, an RF pulse couples atoms from state |2 into an empty state |3. If state |3 is non-interacting with states |1 and |2, it serves as a reference state. This situation is illustrated in ﬁg. 68. It is important to note that, because the ﬁnal state is empty, the RF pulse can excite the entire Fermi sea, and not just atoms at the Fermi surface as in tunnelling experiments. In subsect. 2 we presented RF spectroscopy as an experimental tool, summarized the basics of RF spectroscopy in a two-level system (where no line shifts due to interactions can be observed), and discussed simple (weak interactions, mean-ﬁeld shift) and exact . (sum rule for average shift) limits for the case of three levels. In subsect. 4 7.2, we calculated the spectrum for an RF excitation of spin-up (|2) atoms out of the BCS state and into the empty state |3. In this section, we want to summarize how this “RF tool” has been applied to strongly interacting Fermi systems. The full interpretation of these results is still an open question. RF spectroscopy of normal Fermi gases. For an isolated atom, the resonant frequency ω23 for this transition is known to an extreme accuracy. In the presence of a surrounding cloud of interacting atoms, however, the transition can be shifted and broadened. The shifts can originate from the atom experiencing the “mean ﬁeld” of the surrounding gas. Pairing between fermions can lead to additional frequency shifts, as the RF ﬁeld has

265

Making, probing and understanding ultracold Fermi gases a) 1000

b)

F

3 2

|4>

1 2

|3>

|5>

νRF |2>

|2> |1>

~76 MHz

Energy / h [MHz]

|6>

~82 MHz

|3>

|1>

-1000 0

500 Magnetic Field [G]

Fig. 68. – RF spectroscopy using a three-level system in 6 Li. a) Hyperﬁne structure of lithium. b) Close-up on the three lowest hyperﬁne states involved in RF spectroscopy. Typically, the Fermi mixture is prepared in state |1 and |2, and the RF pulse drives the transition to the initially empty state |3. Figure adapted from [127].

to provide the necessary energy to ﬁrst “break” the pair before the excited atom can be transferred into the ﬁnal state (this picture implies that ﬁnal state interactions are negligible). Broadening can be inhomogeneous, for example due to averaging over a range of densities in a trapped sample, or intrinsic (homogeneous), reﬂecting the local correlations (and thus pairing) between atoms. When a |2 atom is transferred into state |3 in the presence of a cloud of |1 atoms, its “mean-ﬁeld” interaction energy changes: The ﬁnal state interacts diﬀerently with |1 than the initial state. This leads to a clock shift in the RF spectrum (see sect. 2). The ﬁrst experiments on RF spectroscopy in Fermi gases observed such “mean-ﬁeld” interaction shifts close to a Feshbach resonance [131, 127] (see ﬁg. 69) and demonstrated the tunability of interactions around such resonances. Furthermore, it was found that near the Feshbach resonance, the mean-ﬁeld shifts did not diverge [127], contrary to the simple picture that shifts should be proportional to the diﬀerence in scattering lengths. In fact, in the case of 6 Li, mean-ﬁeld shifts were found to be practically absent close to the Feshbach resonances. In this experiment, both the initial and the ﬁnal state were strongly interacting with state |1 and it was supposed that the two, unitarity limited energy shifts cancel out in the transition. This interpretation was recently conﬁrmed in [132], where it was found that average clock shifts depend on the inverse of scattering lengths and thus become small near Feshbach resonances. RF spectroscopy was used to detect and to study Feshbach molecules. Potassium molecules formed via a sweep across the Feshbach resonance were detected by RF spectroscopy [61]. The molecular line was shifted with respect to the atomic resonance by the molecular binding energy. Bound-bound and bound-free transitions in 6 Li were

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Fig. 69. – Mean-ﬁeld clock shift in an interacting Fermi mixture. RF spectroscopy is performed on the transition from |2 → |3 in the presence (open circles) and absence (ﬁlled circles) of atoms in state |1. The broadening reﬂects the inhomogeneous density distribution in the trap, as can be veriﬁed via absorption images of atoms in the ﬁnal state shortly after the RF pulse. Atoms in the low-density wings have smaller shifts than at the center at high density. Figure adapted from [127].

used to precisely determine the position of the Feshbach resonance and other scattering properties [135, 206]. RF spectroscopy of superﬂuid Fermi gases. After the arrival of fermion pair condensates, the Innsbruck group traced the evolution of the molecular spectrum all the way across resonance [75] (see also the article by R. Grimm in these proceedings). Although twobody physics no longer supports a bound state beyond the resonance, the spectra were still shifted and broad, providing evidence for a pairing gap in the strongly interacting Fermi mixture. As expected for fermionic pairing, the observed feature scaled with the Fermi energy, whereas an “atomic” peak, observed at higher temperature, was narrow and showed only a small shift from the resonance position for isolated atoms. Theoretical modelling suggested that the gas was in the superﬂuid regime at the lowest temperatures, where the “atomic” peak had fully disappeared [363]. However, the interpretation of the spectra relied on a theory that neglected interactions in the ﬁnal state, between states . |3 and |1 (such spectra were calculated in subsect. 4 7.2). Recent theoretical work [132, 133, 207] and also experimental studies by the MIT group [99] have demonstrated the importance of such ﬁnal-state interactions. Furthermore, using fermion mixtures with population imbalance it was shown that RF spectra of the |2 → |3 transition do not change as the gas undergoes the superﬂuid to normal transition [77] (see ﬁg. 70). The gas can be normal without any “atomic peak”

267

B

T/TF = 1.9

0

250

500 0.03

T/TF = 1.0 Decreasing Temperature

A

Atom Number in |3〉 [a. u.]

Making, probing and understanding ultracold Fermi gases

D

T/TF = 0.9

0.025

0.020

0.04 T/TF = 0.5

C

Atom Number Fraction in |2〉

0.02

0.03 0.02 -40

0

40

80

120

Radio-Frequency Offset [kHz]

Fig. 70. – Radio-frequency spectroscopy of the minority component in a strongly interacting mixture of fermionic atoms at an imbalance of 0.9, clearly above the Chandrasekhar-Clogston . limit of superﬂuidity (see subsect. 7 3.2). As the temperature is lowered, the spectrum shows the transition from an “atomic peak” (A, with some asymmetry and broadening) to an almost pure “pairing” peak (D). Figure adapted from ref. [77].

in the RF spectrum, in contrast to earlier interpretations of such “pure” pairing spectra. The conclusion is that RF spectra probe correlations and pairing only locally and, at the current level of sensitivity, cannot distinguish between a normal and a superﬂuid phase. One important technical advance is the introduction of spatially resolved RF “tomography” [134] that allows the reconstruction of local spectra free of inhomogeneous broad. ening (see also subsect. 2 5.2). For the resonantly interacting superﬂuid, this method was used to demonstrate a true frequency gap and a sharp onset of the spectrum at a frequency shift corresponding to about 40% of the Fermi energy. Final-state interactions are currently a major limitation in the interpretation of RF spectra. In lithium, for the resonant superﬂuid in states |1 and |2 the ﬁnal-state

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interaction between states |1 and |3 has a large scattering length of a13 = −3300a0 . Recent results show that Fermi mixture initially in states |1 and |3 provide clearer spectra, presumably because the ﬁnal state |2 is less strongly interacting with either state |1 or |3 (a23 = 1100a0 and a12 = 1400a0 at the |1-|3 resonance at B = 690 G) [99]. . 7 3. New systems with BEC-BCS crossover . – The ﬁeld of physics stays vibrant by creating new systems to ﬁnd new phenomena. Two major extensions of the BEC-BCS crossover in an equal mixture of two fermionic states are the addition of optical lattices and population imbalanced Fermi mixtures. . 7 3.1. Optical lattices. Early studies of 40 K in an optical lattice were carried out at Z¨ urich [102]. The band structure and Fermi surfaces for non-interacting fermions and interacting fermion mixtures were observed and a normal conductor and band insulator were realized. For a discussion of these experiments, see the contribution of T. Esslinger to these proceedings. Loading a superﬂuid fermion mixture into a weak optical lattice should not destroy superﬂuidity. The only eﬀect of the lattice is to replace the bare mass by an eﬀective mass. Evidence for superﬂuid behavior was recently observed at MIT [83]. When the fermionic cloud was released from the lattice, and a rapid magnetic ﬁeld sweep converted atom pairs into molecules, the characteristic lattice interference pattern was observed, the signature of long-range coherence providing indirect evidence for superﬂuidity (ﬁg. 71). Delayed rapid switching of the magnetic ﬁeld out of the strongly interacting region was necessary to prevent collisions during ballistic expansion, which would have destroyed the interference pattern. For deeper lattice depths, the interference pattern disappeared. This is analogous to the superﬂuid-to-Mott-insulator transition in bosons, but now in the regime of strong interactions, which will need a multi-band picture for its full description [364]. . 7 3.2. Population-imbalanced Fermi mixtures. The subject of superﬂuidity with population-imbalanced Fermi gases is almost as old as BCS theory itself. Over the last three years, with its realization in ultracold gases, it became a new frontier with major theoretical and experimental activities, and would deserve a review article on its own. Here we can only summarize some basic aspects of this rich system. For an extensive discussion of imbalanced gases see the contribution of F. Ch´evy to these proceedings. Often, breaking a symmetry provides additional insight into a system, even into its symmetric conﬁguration. Breaking the equality of the two spin populations has given us new and very direct ways to observe the superﬂuid phase transition, without need . for magnetic ﬁeld sweeps [312, 80] (see ﬁgs. 51 and 52 in subsect. 6 5.2, and ﬁg. 13 in . subsect. 2 5). In addition, it opened a way to measure the absolute temperature for a strongly interacting system, using the non-interacting wings of the majority component . as a thermometer (see sect. 3 3.1). Scientiﬁcally, imbalance is another way (besides temperature, and scattering length) to probe how stable the superﬂuid is. In addition, in an imbalanced gas, there is no longer a smooth crossover between the BEC- and the BCS-

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269

Fig. 71. – Observation of high-contrast interference of fermion pairs released from an optical lattice below and above the Feshbach resonance. a) The orientation of the reciprocal lattice, also with respect to the imaging light. b-d) Interference peaks are observed for magnetic ﬁelds of 822 G (b), 867 G (c) and 917 G (d). The lattice depth for all images is 5 recoil energies. The ﬁeld of view is 1 mm. Density proﬁles through the vertical interference peaks are shown for each image. Figure reprinted from [83].

regimes. Instead, a ﬁrst-order transition takes place: If the attractive interactions become too weak, the superﬂuid state becomes normal. This is in contrast to the population balanced case, where superﬂuidity occurs for arbitrarily small interactions. The stronger the imbalance, the smaller is the window of superﬂuidity. Or phrased diﬀerently, on the BCS side (and also on the BEC side close to the resonance) there is a critical imbalance, beyond which superﬂuidity breaks down. This quantum phase transition (30 ) is called the Chandrasekhar-Clogston limit, which we will derive in the next paragraph. Imbalance introduces a much richer phase diagram. The quantum phase transition at zero temperature continues as a ﬁrst-order phase transition at ﬁnite temperature up to tricritical point, where it becomes second order [365]. At high imbalance, the system is normal even at zero temperature, and one can study a highly correlated normal phase (30 ) The name “quantum” is deserved as it occurs at zero temperature. Sometimes, the term “quantum phase transition” is reserved for second-order phase transitions.

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without the complications of thermal excitations [77]. Imbalance also oﬀers new opportunities to investigate pairing. Using RF spectroscopy, our group is currently studying diﬀerences in the RF spectrum of the majority and minority component. One question, which we address, is whether the majority atoms show a bimodal spectrum reﬂecting a paired and unpaired component. This will occur in a molecular picture far on the BEC side, but one may expect that on the BCS side the distinction between unpaired and paired atoms is blurred or vanishes.

Chandrasekhar-Clogston limit. The Chandrasekhar-Clogston limit follows from a simple model. A two component Fermi gas can either be in a normal state with two diﬀerent Fermi energies for the two components, or in a superﬂuid state that requires balanced populations. The superﬂuid is stabilized by the condensation energy density δEs , which we will later set equal to the BCS result 12 ρF Δ2 where Δ is the superﬂuid gap, and ρF ≡ ρ(EF )/Ω is the density of states (per spin state and per volume) at the Fermi energy. A balanced superﬂuid region can only be created by expelling majority atoms which requires extra kinetic energy. It is the interplay between superﬂuid stabilization energy and kinetic energy which determines the phase boundary. We consider N↑ and N↓ fermions in a box of volume Ω. Since the imbalance can only be accommodated in the normal phase, we assign a volume fraction η to it, and the rest of the volume is superﬂuid. The superﬂuid and (average) normal densities, ns and nn , may be diﬀerent, and are constrained by the constant total number of atoms N = N↑ + N↓ = 2ns (1 − η) Ω + 2nn η Ω.

(232)

The energy density of the superﬂuid gas is 3 Es = 2 EF [ns ] ns − δEs , 5

(233)

where EF [n] ∝ n2/3 is the Fermi energy of a Fermi gas at density n. For the normal phase we have (234) (235)

Δn Δn Δn Δn EF nn + nn + + EF nn − nn − η η η η

2 1 Δn 3 , ≈ 2 EF [nn ] nn + 5 ρF η

3 En = 5

where we have assumed that the density diﬀerence Δn = (N↑ − N↓ )/2Ω is much smaller than the total average density per spin state n = (N↑ + N↓ )/2Ω. The total energy is minimized as a function of η and ns − nn . For η = 1 the whole system becomes normal. The calculation is simpliﬁed if we introduce a Lagrangian multiplier μ to account for the constraint on the total number of atoms, and then minimize

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Fig. 72. – Observation of vortices in a strongly interacting Fermi gas with imbalanced spin populations. For the 812 G data, the population imbalance (N↑ − N↓ )/(N↑ + N↓ ) was (from left to right) 100, 90, 80, 62, 28, 18, 10 and 0%. For the 853 G data, the mismatch was 100, 74, 58, 48, 32, 16, 7 and 0%. From [78].

the total free energy Etot − μN with respect to η, ns and nn :

(236)

3 2 EF [ns ]ns − δEs Ω (1 − η) 5

2 1 Δn 3 Ωη + 2 EF [nn ]nn + 5 ρF η

Etot − μN =

− μ 2Ω ((1 − η)ns + η nn ) . Using 35 EF [ns ]ns − 35 EF [nn ]nn = EF [nav ](ns − nm ), where nav is between nn and ns , and setting the η derivative to zero, we obtain (237)

η2 =

(Δn)2 . ρF {δEs + 2 (μ − EF [nav ]) (ns − nn )}

Setting the other two derivatives (with respect to ns and nn ) of eq. (236) to zero provides expressions for the density diﬀerence and the chemical potential. They show that both μ − EF [nav ] and ns − nn scale linearly with δEs , i.e. that the second term in the denominator of eq. (237) is second order in δEs and negligible for weak interactions

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Population Imbalance δ

1.0 0.8 0.6

Superfluid

Normal

0.4 0.2 0.0 2

1

0 1/kF a

-1

-2

Fig. 73. – Critical population imbalance δ = (N↑ − N↓ )/(N↑ + N↓ ) between the two spin states for which the superﬂuid-to-normal transition is observed. The proﬁles indicate the distribution of the gas in the harmonic trap. Data from [78].

(BCS limit). Substituting 12 ρF Δ2 for δEs , one obtains for the normal volume fraction (238)

√ 2Δn . η= ρF Δ

√ This becomes one for Δn = ρF Δ/ 2, which is the Chandrasekhar-Clogston limit for the superﬂuid-to-normal quantum phase transition. In case of superconductors, the number imbalance can be created by a magnetic ﬁeld B (assuming that the Meissner eﬀect is suppressed, e.g. in heavy fermion or layered superconductors [94]) Δn = ρ√F μB B which leads to the Chandrasekhar-Clogston limit in its original form μB Bc = Δ/ 2, where Bc is the critical magnetic ﬁeld [366, 367]. Using eq. (166), one obtains Bc = 18500 G TKC , much larger than the thermodynamic critical √ ﬁeld of a conventional superconductor, which is Bc = μ0 ρF Δ ≈ 50 G TKC . This shows that conventional superconductors will be quenched by orbital eﬀects of the magnetic ﬁeld (Meissner eﬀect and ﬂux quanta), and not by spin eﬀects (Chandrasekhar-Clogston limit). Experimental observations. The proof for the occurrence of superﬂuidity in imbalanced gases was obtained, as in the balanced case, by the observation of superﬂuid ﬂow in the form of vortices (ﬁg. 72). Since vortices can be diﬃcult to create and observe near the phase boundaries, the superﬂuid phase diagram has been mapped out by using pair condensation as an indicator for superﬂuidity. The phase diagram (ﬁg. 73) shows that on the BEC side, the Chandrasekhar-Clogston limit approaches 100%. So even if there are only a few minority atoms in a majority Fermi sea, they can form bosonic molecules

273

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0.2

Temperature (T/TF )

Normal

0.1

Tricritical Point

Superfluid

Unstable 0.0 0.0

0.1

0.2

0.3

0.4

Spin Polarization

Fig. 74. – Phase diagram for a homogeneous spin-polarized Fermi gas with resonant interactions, as a function of spin polarization and temperature. Below the tricritical point phase boundaries have been observed, characteristic for a ﬁrst-order phase transition. The polarization jumps from the value in the superﬂuid (shown by the gray solid circles) to the higher value in the normal phase (black solid circles). Above the tricritical point, no abrupt change in spin polarization was observed, and the phase transition in the center of the cloud was determined from the onset of pair condensation (black square). The open symbols show theoretical predictions for the critical temperature of a homogeneous equal mixture (see ref. [82] for a full discussion).

and Bose-Einstein condense. The observed deviation from 100% is probably due to ﬁnite temperature. On resonance, the critical population imbalance converged towards ≈ 70% when the temperature was varied [78]. First hints for phase separation between the normal and superﬂuid phase were seen in refs. [78, 79]. Using tomographic techniques, a sharp separation between a superﬂuid core and a partially polarized normal phase was found [80]. Finally, the phase diagram of a spin-polarized Fermi gas at unitarity was obtained, by mapping out the superﬂuid phase versus temperature and density imbalance [82]. Using tomographic techniques, spatial discontinuities in the spin polarization were revealed, the signature of a ﬁrstorder phase transition that disappears at a tricritical point (ﬁg. 74). These results are in excellent agreement with recent theoretical predictions [368, 369]. The ChandrasekharClogston limit was not observed in the work at Rice, from which it was concluded that the critical imbalance on resonance was close to 100%. The discrepancy to the MIT results is probably related to the lower atom number and higher aspect ratio used in these experiments [79, 81].

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The phase diagram in ﬁg. 74 highlights how far experimental studies on degenerate Fermi gases have progressed since their ﬁrst observation in 1999. It is a rich diagram including ﬁrst- and second-order phase transitions, phase separation and a tricritical point. It is expressed using only local quantities (density, spin polarization, local Fermi temperature) and therefore applies directly to the homogeneous case. Experimentally, it required 3D spatial resolution using tomographic techniques. Finally, it is worth pointing out that the position of the tricritical point could not be predicted and has even been controversial until it had been experimentally determined. 8. – Conclusion In this review paper, we have given a comprehensive description of the concepts, methods and results involved in the exploration of the BEC-BCS crossover with ultracold fermionic atoms. Of course, many of the techniques and concepts apply to other important areas where experiments with ultracold fermions are conducted. One area is atom optics and precision measurements. The important feature of fermions is the suppression of interactions in a one-component gas due to Pauli blocking. The Florence group studied transport behavior of fermions in an optical lattice and observed collisionally induced transport [370], and conducting and insulating behavior of ideal Fermi gases [371]. The realization of atom interferometry and observation of Bloch oscillation in a Fermi gas [372] demonstrated the potential of fermions for precision measurements unaﬀected by atomic collisions [373]. Another area is the study of mixtures of fermions with other species. Bose-Fermi mixtures have been used to study how the addition of fermionic atoms aﬀects the bosonic superﬂuid-to-Mott-insulator transition [374, 375]. Also intriguing is the study of bosonmediated interactions between fermions [376, 377]. Interspecies Feshbach resonances between fermionic and bosonic atoms have already been identiﬁed [147,101], and heteronuclear molecules observed [103,378]. Mixtures between two fermionic species (e.g. 6 Li and 40 K) may allow the study of pairing and superﬂuidity where the pairs are made from atoms with diﬀerent masses [105, 379]. p-wave interactions provide a diﬀerent way of pairing atoms, and ultimately may connect the study of superﬂuidity in 3 He (which is based on p-wave pairing) with ultracold atoms. p-wave Feshbach resonances have already been observed by various groups [145, 144, 146, 380], and p-wave molecules have been produced [86]. And ﬁnally, a whole new area is the study of fermions with repulsive interactions in optical lattices. At low temperature and half ﬁlling, the system should be antiferromagnetic [381, 382], and at lower ﬁlling, it may show d-wave superﬂuidity [381] and help to elucidate the nature of pairing in high-temperature superconductors. For a discussion of these eﬀects, we refer to the contribution of A. Georges to these proceedings. With many new systems on the drawing board or already in preparation, and with a ﬂurry of theoretical papers predicting new aspects of superﬂuidity and other novel phenomena, it seems certain that we have exciting years ahead of us.

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∗ ∗ ∗ Work on ultracold fermions at MIT has been a tremendous team eﬀort, and we are grateful to the past and present collaborators who have shared both the excitement and ¨ rlitz, S. the hard work: J. R. Abo-Shaeer, J. K. Chin, K. Dieckmann, A. Go Gupta, Z. Hadzibabic, A. J. Kerman, Y. Liu, D. E. Miller, S. M. F. Raupach, C. Sanner, A. Schirotzek, C. H. Schunck, W. Setiawan, Y.-I. Shin, C. A. Stan, and K. Xu. We also acknowledge the fruitful interactions with our colleagues in this rich and exciting ﬁeld, and we want to thank the organizers and participants of the Varenna summer school for the stimulating atmosphere. We are grateful to R. Gommers, A. Keshet, C. Sanner, A. Schirotzek, Y.-I. Shin, C. H. Schunck and W. Zwerger for comments on the manuscript. We want to thank the National Science Foundation, the Oﬃce of Naval Research, NASA, the Army Research Oﬃce, DARPA, and the David and Lucile Packard Foundation for their encouragement and ﬁnancial support of this work during the past eight years.

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Basic theory tools for degenerate Fermi gases Y. Castin ´ Laboratoire Kastler Brossel, Ecole normale sup´erieure 24 rue Lhomond, 75231 Paris Cedex 5, France

1. – The ideal Fermi gas We consider in this section a non-interacting Fermi gas in a single spin component, at thermal equilibrium in the grand-canonical ensemble. We review basic properties of such a gas. We concentrate on the degenerate regime ρλ3 1, with ρ the density and λ the thermal de Broglie wavelength deﬁned as (1)

λ2 =

2π¯h2 . mkB T

. 1 1. Basic facts. – In ﬁrst quantization, the Hamiltonian of the non-interacting system is the sum of one-body terms, (2)

H=

ˆ N

ˆ 0 (i), h

i=1

ˆ 0 is the one-body Hamilˆ is the operator giving the number of particles, and h where N tonian given here by (3)

2 ˆ 0 = p + U (r ), h 2m

where p is the momentum operator, m is the mass of a particle and U (r ) is a position dependent external potential. We shall consider two classes of external potential. Either c Societ` a Italiana di Fisica

289

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Y. Castin

a zero external potential, supplemented by periodic boundary conditions in a box of size L, deﬁned by r ∈ [0, L[d , or a harmonic potential mimicking the traps used in experiments: (4)

U (r ) =

d 1 mωα2 rα2 , 2 α=1

where ωα is the oscillation frequency of a particle in the trap along one of the d dimensions of space. The system is made of indistinguishable fermions, all in the same spin state. To implement the antisymmetry of the many-body wave function, we move to second quantization. Introducing the orthonormal basis of the eigenmodes φλ of the one-body Hamiltonian, with eigenenergy λ , we expand the ﬁeld operator as ˆ r) = φλ (r )aλ , (5) ψ( λ

where aλ annihilates one particle in the mode φλ and obeys anticommutation relations with the creation and annihilation operators: (6)

{aλ , aλ } = 0,

(7)

{aλ , a†λ } = δλλ .

The Hamiltonian then reads in second quantized form: (8) H= λ a†λ aλ , λ

and the operator giving the number of particles: † ˆ = (9) N aλ aλ . λ

The system is assumed to be at thermal equilibrium in the grand-canonical ensemble, with a many-body density operator given by (10)

ˆ

σ ˆ = Ξ−1 e−β(H−μN ) ,

where the factor Ξ ensures that σ ˆ has unit trace, β = 1/kB T and μ is the chemical potential. The choice of the grand-canonical ensemble allows to decouple one mode from the other and makes the density operator Gaussian in the ﬁeld variables, so that Wick’s theorem may be used to calculate expectation values, which are thus (possibly complicated) functions of the only non-zero quadratic moments (11)

a†λ aλ = nλ = n( λ ) ≡

1 . exp[β( λ − μ)] + 1

291

Basic theory tools for degenerate Fermi gases

n(H) and 1-n(H)

1 0.8 0.6 0.4 0.2 0 0

1 HP

0.5

1.5

2

Fig. 1. – Fermi-Dirac distribution in the degenerate regime. Solid line: occupation numbers n( ) for the particles. Dashed line: occupation numbers 1 − n( ) for the holes. The temperature is kB T = μ/10.

This is the famous Fermi-Dirac law for the occupation numbers nλ . An elementary derivation of Wick’s theorem can be found in an appendix of [1]. We note that, because of the fermionic nature of the system, the grand-canonical ensemble cannot be subject to unphysically large ﬂuctuations of the number of particles, contrarily to the ideal Bose gas case. Using Wick’s theorem we ﬁnd that the variance of the particle number is (12)

ˆ = Var N

nλ (1 − nλ ),

λ

ˆ . In the zero-temperature limit, for μ which is indeed always below the Poisson value N not coinciding with an eigenmode energy λ , one ﬁnds a vanishing variance of the particle number, so that the grand-canonical ensemble becomes equivalent to the canonical one. In this lecture, we shall be concerned with the degenerate limit, where the temperature is much smaller than the chemical potential: (13)

kB T μ.

The Fermi-Dirac distribution has then the shape presented in ﬁg. 1. We shall repeatedly use the fact that the occupation number of holes 1 − nλ for λ < μ and the occupation number of particles nλ for λ > μ are narrow functions of λ − μ, of energy width ∼ kB T .

292

Y. Castin

More precisely, a standard low-T expansion technique proceeds as follows [2]. One writes the Fermi-Dirac formula as (14)

n( ) = nT =0 ( ) +

sign( − μ) , exp[β| − μ|] + 1

singling out as a second term a narrow function of − μ. In the case of a continuous density of states ρ( ), an integral over appears in thermodynamic quantities. One then extends the integration over to ] − ∞, +∞[ for the second term of the right-hand side, neglecting a contribution O[exp[−μ/kB T ]], after having expanded in powers of − μ the functions of multiplicating this second term, such as ρ( ). We illustrate this technique for the calculation of the density ρ in a spatially homogeneous system in the thermodynamic limit: (15)

dd k nk = (2π)d

ρhom (μ, T ) =

+∞

d ρ( )n( ), 0

where we introduced the free space density of states (16)

ρ( ) =

√ dd k d 2 2 −d (2π) δ( − h ¯ k /2m) = V ( 2m /¯h) d (2π)d 2

with Vd (u) is the volume enclosed by the sphere of radius u in dimension d. We go to the limit T → 0 for a ﬁxed chemical potential, and we take μ > 0 in order to have a non-zero density at zero temperature. From the rewriting eq. (14) of the Fermi-Dirac formula we obtain (17)

ρhom (μ, T ) = ρhom (μ, 0) +

μ

d(δ ) 0

ρ(μ + δ ) − ρ(μ − δ ) + O e−βμ , exp[βδ ] + 1

where we cut the integration over to 2μ, introducing an exponentially small error. The zero-temperature contribution is readily evaluated by the integral in k space: (18)

ρhom (μ, 0) =

h ¯ 2 k2 /2m<μ

dd k Vd (1) = (2π)d (2π)d

2mμ ¯h2

d/2 .

The thermal contribution is obtained as a low-T expansion by expanding ρ(μ ± δ ) in powers of δ : only odd powers of δ contribute; in the resulting integrals over δ , we extend the upper bound from μ to +∞, paying the price of an exponentially small error. Finally a series expansion with even powers of T is obtained:

(19)

π2 d(d − 2) ρhom (μ, T ) = ρhom (μ, 0) 1 + 24

kB T μ

2

4

+ O (kB T /μ)

.

293

Basic theory tools for degenerate Fermi gases

The bidimensional case d = 2 deserves a particular discussion. One ﬁnds that all the powers of T in the expansion have vanishing coeﬃcients, since the density of states is constant, ρ( ) = m/(2π¯ h2 ). The integral over in eq. (15) can be performed exactly, (20)

ρd=2 hom (μ, T ) =

mkB T βμ . 2 ln 1 + e 2π¯h

For a ﬁxed and negative chemical potential, one sees that the density tends exponentially rapidly to zero when T → 0. For μ > 0 a more convenient rewriting in the low-temperature limit is (21)

ρd=2 hom (μ, T )

kB T mμ −βμ 1+ ln 1 + e = . μ 2π¯h2

This explicitly shows that the deviation of the spatial density from the zero-temperature value ρhom (μ, 0) is exponentially small for T → 0 if d = 2. . 1 2. Coherence and correlation functions of the homogeneous gas. – We consider here a gas in a box with periodic boundary conditions. The eigenmodes are thus plane waves φk (r ) = exp[ik · r ]/Ld/2 , with a wave vector equal to 2π/L times a vector with integer components. We shall immediately go to the thermodynamic limit, setting L to inﬁnity for a ﬁxed μ and T . The ﬁrst-order coherence function of the ﬁeld is deﬁned as the following thermal average: (22)

ˆ 0). g1 (r ) = ψˆ† (r )ψ(

In the thermodynamic limit, it is given by the Fourier transform of the momentum distribution of the gas, which is an isotropic function: (23)

g1 (r ) = φ(r) ≡

dd k nk eik·r . (2π)d

It is easy to see that no long-range order exists for this coherence function, even at zero temperature. At T = 0, nk = 1 for k < k0 and nk = 0 for k > k0 , where the wave vector k0 is deﬁned by

(24)

¯ 2 k02 h =μ 2m

and is simply the Fermi wave vector, in the present case of zero temperature. In the

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Y. Castin

dimensions for d = 1 to d = 3 this leads to (25) (26) (27)

sin k0 r , πr k0 φ2D (r) = J1 (k0 r), 2πr 1 ∂r φ1D (r), φ3D (r) = − 2πr φ1D (r) =

where J1 (x) is a Bessel function, behaving as − cos(x + π/4)(2/πx)1/2 for x → +∞. One then gets an algebraic decay of φ at large distances, in all dimensions, because of the discontinuity of the momentum distribution. Note that eq. (27) holds at all temperatures. At low but non-zero temperature, the Fermi surface is no longer inﬁnitely sharp but has a smooth variation with an energy width ∼ kB T . This corresponds to a momentum width δk0 such that (28)

k0 +δk0 − k0

¯ 2 k0 δk0 h ≡ kB T. m

We then expect a decreasing envelope for the oscillating function φ(r) with a decay faster than algebraic with a length scale 1/δk0 . This can be checked by the following approximate calculation in 1D: we restrict to k > 0 using parity, we single out from nk the zero-temperature contribution θ(k0 − k), where θ is the Heaviside function, and we linearize the kinetic-energy dispersion relation around k0 , writing k0 +δk μ + h ¯ k0 δk/m. Extending the integration over δk to −∞ we obtain (29)

φ1D (r) − φT1D=0 (r) −

2δk0 sin(k0 r) π

+∞

du 0

sin(uδk0 r) , exp[u] + 1

where u originates from the change of variable u = |δk|/δk0 . Integrating by parts over u, taking the integral of the sine function, gives a fully integrated term exactly compensating the T = 0 contribution, so that one is left with (30)

φ1D (r)

sin(k0 r) 2πr

+∞

du 0

cos(uδk0 r) . cosh2 (u/2)

After extension to an integral over the whole real line (thanks to the parity of the integrand), one may close the integration contour at inﬁnity and sum the residues over all the poles with Im u > 0 to get (31)

φ1D (r)

δk0 sin(k0 r) . sinh(πδk0 r)

We have recovered a result derived in [3]. This approximate formula can also be used to calculate the 3D case, by virtue of eq. (27).

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The correlation function of the gas, also called the pair distribution function, is deﬁned as the following thermal average: (32)

ˆ 0 )ψ( ˆ r ). g2 (r ) = ψˆ† (r )ψˆ† (0 )ψ(

By virtue of Wick’s theorem, g2 is related to the function φ as (33)

g2 (r ) = φ2 (0) − φ2 (r),

where φ(0) is simply the mean particle density ρ. A key feature is that g2 vanishes in the limit r → 0 (quadratically in r): this is a direct consequence of the Pauli exclusion principle, and expresses the fact that one cannot ﬁnd (with a ﬁnite probability density) two fermions with the same spin state at the same location. This spatial antibunching is intuitively expected to lead to reduced density ﬂuctuations for an ideal degenerate Fermi gas, as compared to an ideal Bose gas. This is quantiﬁed in the next subsection. . 1 3. Fluctuations of the number of fermions in a given spatial zone. – We deﬁne the operator giving the number of particles within the sphere of radius R: ˆ r ), ˆ (34) NR = dd r ψˆ† (r )ψ( r
the gas being spatially homogeneous and taken in the thermodynamic limit. We now show ˆR has sub-Poissonian ﬂuctuations for a degenerate ideal Fermi gas. that the number N The mean value of the number of particles within the sphere is simply ˆR = ρVd (R), N

(35)

where ρ = φ(0) is the mean density and Vd (R) the volume enclosed by the sphere of radius R in dimension d. From Wick’s theorem we ﬁnd a variance ˆR = N ˆR − (36) Var N dd r dd r φ2 (|r − r |). r
r
This immediately reveals the sub-Poissonian nature of the ﬂuctuations. We quantify this subpoissonian nature in the limit of a large radius R. Replacing φ by its expression as a Fourier transform of nk , see eq. (23), we obtain (37)

ˆR = N ˆR − Var N

dd k (2π)d

dd k nk nk F (k − k ), (2π)d

where F is the modulus squared of the Fourier transform of the characteristic function of the sphere: (38)

/ / F (q ) = //

r
d

d re

/2 / / .

i q · r/

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Y. Castin

One can show that F is an integrable function peaked in q = 0 and with a width ∼ 1/R. From the Parseval-Plancherel identity the integral over the whole momentum space of F is (2π)d Vd (R). When Rδk0 1, where δk0 is the momentum width of the ﬁnite temperature Fermi surface, see eq. (28), we can replace F by a Dirac distribution: F (q ) (2π)d Vd (R)δ d (q ).

(39) This leads to

(40)

dd k nk (1 − nk ). (2π)d

ˆR Vd (R) Var N

To easily calculate the resulting integral to leading order in T , we note that ∂μ nk = βnk (1 − nk ),

(41)

so that the T = 0 value of the density only is required, eq. (18). We ﬁnally obtain ˆR ˆR N Var N

(42)

d kB T . 2μ

The above result does not apply when Rδk0 1. In the range of radius R (δk0 )−1 we may simply perform a T = 0 calculation. The technique to take advantage of the translational and rotational invariance of φ2 (|r − r |) is to introduce the purely geometrical quantity

(43)

dd r

K(X) = r
r
dd r δ(|r − r | − X),

so that (44)

ˆR = ρVd (R) − Var N

2R

dX K(X)φ2 (X). 0

An exact calculation of K(X) is possible: for X ≤ 2R, one obtains (45) (46) (47)

K1D (X) = 4(R − X/2), X − X (4R2 − X 2 )1/2 , K2D (X) = πX 4R2 arccos 2R π2 2 X (X + 4R)(X − 2R)2 . K3D (X) = 3

When combined with eqs. (25), (27), this leads to exact expressions for the variance in 1D and 3D, in terms of the complex integral Ci and Si functions. The large k0 R expressions are then readily obtained. The asymptotic expansion of the 2D has to be worked out

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Basic theory tools for degenerate Fermi gases

by hand, singling out the contribution of the low-X quadratic expansion of K(X). We obtain (48) (49) (50)

ˆR = γ + 1 + ln 4k0 R + O(1/(k0 R)2 ), d = 1 : Var N π2 R k 0 ˆR = [ln(4k0 R) + C] + O(ln(k0 R)/k0 R), d = 2 : Var N π2 ˆR = 1 ln(A3 k0 R) (k0 R)2 − 1 ln(B3 k0 R) + O(1/k0 R) d = 3 : Var N 2π 2 24π 2

with

(52)

3 3 5 π 3 u 2 F3 , ; 2, , 3; −u2 − ln u 0.656657, u→+∞ 12 2 2 2 A3 = exp[γ + 2 ln 2 − 1/2] = 4.321100 . . . ,

(53)

B3 = exp[γ + 2 ln 2 − 5/12] = 4.696621 . . .

(51)

C = lim

and 2 F3 is a hypergeometric function, γ = 0.5772156649 . . . is Euler’s constant. . 1 4. Application to the 1D gas of impenetrable bosons. – Consider a 1D homogeneous gas of bosons interacting via a delta potential gδ(x1 − x2 ) in the limit g → +∞. In this so-called impenetrable boson limit, the interaction potential can be replaced by the contact condition that the many-body wave function vanishes when two bosons are in the same point. On the fundamental domain of ordered positions of the N bosons, x1 < . . . < xN , one then realizes that the eigenwavefunction for bosons coincides with an eigenwavefunction of N non-interacting fermions, with the same eigenenergy. Out of the fundamental domain, the bosonic and fermionic wave functions may diﬀer by a sign. As a consequence, the spatial distribution of the impenetrable bosons, being sensitive to the modulus squared of the wave function, will be the same as for the ideal Fermi gas. The discussion on the variance of the number of particles in an interval [−R, R] in . subsect. 1 3 immediately applies. On the contrary, the ﬁrst-order coherence function for the impenetrable bosons g1B (x) is sensitive to the phase of the wave function and will diﬀer from the g1 fermionic function. More details, obtained with the Jordan-Wigner transformation, are given in [1] and references therein. At zero temperature, we simply recall the mathematical fact, showing the absence of condensate, that at large distances [4] (54)

g1B (x) ∼

Aρ , |kF x|1/2

where kF = πρ, ρ being the density of bosons, and A = 0.92418 . . .. We also use the fact (that one can check numerically [1]) that the power law behavior of g1B (x) is the same as the one of the simpler function (55)

ˆ

G(x) = eiπN[0,x] ,

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Y. Castin

ˆ[0,x] is the operator giving the number of fermions in the interval from 0 to x where N (here x > 0) and the expectation value is taken in the ground state of the ideal Fermi gas. In this way, one relates the long-range behavior of g1B to the counting statistics of an ideal Fermi gas: G(x) is the diﬀerence between the probabilities of having an even number and an odd number of fermions in the interval [0, x]. One may assume that the probability distribution of the number n of fermions in the interval has a Gaussian envelope [5]. Using the Poisson formula +∞

(56)

f (n) =

n=−∞

+∞

f˜(2πn),

n=−∞

where f˜(k) = dx f (x) exp[−ikx] is the Fourier transform of the arbitrary function f , and restricting to the leading terms n = 0 and n = 1, we obtain (57)

G(x) 2 cos(πρx)e−π

2

ˆ[0,x] )/2 (Var N

.

From the asymptotic expression eq. (48) we obtain (58)

G(x) ∝

cos(πρx) (ρx)1/2

which explains the 1/x1/2 decay of g1B (x). . 1 5. In a harmonic trap. – In a harmonic trap, simple approximate results can be obtained in the limit where the chemical potential is much larger than the trapping frequencies ωα times ¯h, in a temperature range to be speciﬁed, in the so-called semi-classical approximation. We illustrate this semi-classical approximation for two quantities, a thermodynamic one, the entropy, and a local observable, the density. . 1 5.1. Semi-classical calculation of the entropy. The entropy S is a physically appealing way to evaluate to which extent a system is cold, as it is a constant in thermodynamically adiabatic transformations, contrarily to the temperature. To calculate thermodynamic quantities in the grand-canonical ensemble, it is convenient to start from the grand potential ˆ (59) Ω(μ, T ) = −kB T ln Tr e−β(H−μN ) . Then one uses the fact that (60)

dΩ = −S dT − N dμ ,

ˆ is the mean number of particles. Since the various ﬁeld eigenmodes where N = N decouple in the grand-canonical ensemble, one ﬁnds (61) Ω(μ, T ) = kB T d ρ( ) ln[1 − n( )],

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Basic theory tools for degenerate Fermi gases

where n( ) is the Fermi-Dirac formula and ρ( ) is the density of states for a particle in the trap: (62)

ρ( ) =

δ −

d

(nα + 1/2)¯ hωα ,

α=1

n

where the sum is over all vectors with non-negative integer coordinates. In the semiclassical limit one replaces the sum by an integral (63) ρ( ) ρsc ( ) = dd n δ( − nα ¯hωα ), [0,+∞[d

α

where we dropped the ground mode energy 0 under the assumption 0 . By the change of variables nα = uα /¯ hωα one obtains (64)

d−1 ρsc ( ) = * hωα α¯

d uδ 1 − d

[0,+∞[d

uα

.

α

The integral can be calculated by integrating over ud and by performing the change of variables τ1 = u1 , τ2 = u1 + u2 , . . . , τd−1 = u1 + . . . + ud−1 . We ﬁnally obtain (65)

ρsc ( ) =

d−1 , (d − 1)!(¯ hω ¯ )d

* where ω ¯ = ( α ωα )1/d is the geometric mean of the trap frequencies. We replace ρ by ρsc in eq. (61), and we integrate by parts, taking the derivative of the ln[1 − n( )], to obtain the semi-classical approximation (66)

−d

Ω Ωsc = −(¯ hω ¯)

+∞

d 0

d n( ). d!

In the degenerate regime, we use the technique sketched around eq. (14), based on the fact that n( )−nT =0 ( ) is a narrow function of the energy, to obtain the low-T expansion: (67)

2 kB T μd+1 π2 d(d + 1) + ... 1+ Ωsc (μ, T ) = − (d + 1)!(¯ hω ¯ )d 6 μ

Calculating S = −∂T Ω and N = −∂μ Ω, we obtain the semi-classical approximation for the number of particles and for the entropy per particle: (68) (69)

μd , d!(¯ hω ¯ )d dπ 2 kB T /TF S/N 3 N

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Y. Castin

with the Fermi energy (70)

kB TF ≡ ¯hω ¯ (N d!)1/d .

What are the validity conditions of this approximate formula for the entropy? The temperature should be T TF but, contrarily to the thermodynamic limit case, the temperature should remain high enough, to allow the approximation of the density of states (which is a sum of Dirac peaks) by a smooth function. The typical distance between the Dirac peaks at the Fermi energy should be smaller than the thermal energy width ∼ kB T of the Fermi surface. In the case of an isotropic harmonic trap, ωα = ω for all d dimensions, the Dirac peaks are regularly spaced by h ¯ ω so that we impose (71)

kB T > ¯hω.

In the case of incommensurable trap frequencies, the peaks are irregularly spaced and we estimate a typical spacing from the inverse of the smoothed density of states at the Fermi surface: we require (72)

kB T >

1 , ρsc (μ)

with μ kB TF . This leads to the condition (73)

kB TF (d!)1/d =h ¯ω ¯ . dN dN 1−1/d

kB T

One sees that this is a much weaker condition than in the isotropic case, for d > 1, since we are here in the large N limit. This fact is illustrated by a numerical example in 3D, calculating the entropy for an isotropic trap and an anisotropic trap, see ﬁg. 2. . 1 5.2. Semi-classical calculation of the density. The semi-classical approximation for the density proﬁle in the external potential U (r ) is strictly equivalent to the so-called Local Density Approximation (LDA), as can be revealed by the writings

(74) (75)

dd p n[U (r ) + p2 /2m], (2π¯h)d ρLDA (r ) = ρhom [μ − U (r ), T ], ρsc (r ) =

where ρhom is deﬁned in eq. (15) and n( ) is the Fermi-Dirac formula eq. (11). The philosophies of the two approximations of course diﬀer. In the semi-classical case, one relies on a phase space density of the particles, which is an approximation to the Wigner representation W (r, p ) of the one-body density operator of the gas [6]: (76)

ρˆ =

1 ˆ 0 − μ)] + 1 exp[β(h

,

301

Basic theory tools for degenerate Fermi gases 0.25

S/NkB

0.2

0.15

0.1

0.05

0

0

0.2

0.4

0.6

0.8

1

kB T/h Z Fig. 2. – Entropy S of a spin-polarized ideal Fermi gas in a 3D harmonic trap. Black circles: ¯ (0.5981188 . . . , 1.2252355 . . . , 1.36440128 . . .). Empty exact result for trap frequencies ωα = ω ¯ (1, 1, 1). Solid line: approximate formula circles: exact result for an isotropic trap ωα = ω hω ¯ , where 0 is the trap ground mode eq. (69). The chemical potential is ﬁxed to μ = 0 + 47.6¯ energy. In order to test the accuracy of eq. (69), in a situation where the number of particles N rather than the chemical potential μ is known, as it is the case in experiments with atomic gases, we use the exact, numerical value of N to evaluate eq. (69) rather than its approximation eq. (68). Here N is about 2 × 104 .

ˆ 0 is deﬁned in eq. (3). One assumes where the one-body Hamiltonian h 4 5 u dd u u |ˆ ρ | r + r − eip·u/¯h (2π¯h)d 2 2 1 1 . (2π¯h)d exp[β[U (r ) + p2 − μ]] + 1 2m

(77)

W (r, p ) ≡

It remains to integrate this approximation of W (r, p ) over p to obtain ρsc (r ). In the local density approximation, the gas is considered as a collection of quasimacroscopic pieces that have the same properties as the homogeneous gas of temperature T and chemical potential μLDA = μ − U (r ). Note that this last equation intuitively expresses the fact that the chemical potential is uniform in a gas at thermal equilibrium. These approximations assume that the external potential varies weakly over the correlation length of the homogeneous gas, ∼ 1/k0 where k0 (r ) is the local Fermi wave vector, such that h ¯ 2 k02 /2m = μLDA . In a harmonic trap, the scale of variation of the potential may be taken as the spatial extension of the gas, the so-called Thomas-Fermi

302

Y. Castin

length Rα along direction α such that (78)

μ=

1 2 mωα2 Rα . 2

The condition k0 (r )Rα 1, and correspondingly the semi-classical approximation and the LDA, will fail close to the borders of the cloud where k0 diverges. At T = 0, as is apparent from eq. (18), these approximations predict the density proﬁles:

(79)

d/2 Vd (1) 2m(μ − U (r )) ρ(r ) = . (2π)d ¯2 h

Integrating this density proﬁle over the domain |rα | ≤ Rα gives an expression of the total number of particles as a function of the chemical potential: this expression coincides with the right-hand side of eq. (68), whatever the dimension of space d, which illustrates the consistency of the various approaches. It is instructive to see if the LDA for the density proﬁle can be used at T = 0 strictly, or if a lower bound on T is required, as was the case for the entropy. An answer to this question is obtained by calculating exactly the density of the gas and its second order derivatives in the trap center and comparing to the LDA. The value of ρ(0 ) is simple to obtain from a numerical summation over the harmonic oscillator quantum numbers nα , since the value in x = 0 of the 1D harmonic-oscillator eigenstate wave function of quantum number n ≥ 0 is known exactly: for n odd, φn (0) = 0, and for n even, (80)

φn (0) = (−1)n/2

(n!)1/2

−1/2

2n/2 (n/2)!π 1/4

aho

,

h/mω)1/2 . This may be obtained from where the harmonic-oscillator length is aho = (¯ properties of the Hermite polynomials Hn , using φn (x) = Nn Hn (x/aho ) exp[−x2 /2a2ho ] with a positive normalisation factor such that Nn−2 = aho π 1/2 2n n!, or from the recursion relation φn (0) = −(1 − 1/n)1/2 φn−2 (0), obtained from the a and a† formalism. The second-order √ derivatives of the density in r = 0 are also easily obtained: they involve φn (0) = 2nφn−1 (0)/aho and the second-order derivative of φn , which is related to φn by Schr¨ odinger’s equation, φn (0) = −(2n + 1)φn (0)/a2ho . We apply this check in 1D. At zero temperature, explicit expressions can be obtained. For a chemical potential n + 1/2 < μ/¯hω < n + 3/2, where n is an even non-negative integer, we obtain (81) (82)

ρ(0) = (n + 1)φ2n (0) = ρ (0) = −2a−2 ho ρ(0).

1

(n + 1)!

π 1/2 aho

2n [(n/2)!]2

,

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Basic theory tools for degenerate Fermi gases

0

2.5

(b) 3 ] ho

(a) d U/dx (x=0) [1/a

1

n=4, N=5 & 6 n=2, N=3 & 4

0.5 0 0

n=0, N=1 & 2

5

10

P/hZ

-0.05 -0.1

2

1.5

2

U(x=0) [1/aho]

2

15

20

-0.15 -0.2 0

25

5

10

P/hZ

15

20

25

Fig. 3. – Central density of a 1D harmonically trapped ideal Fermi gas, in the grand-canonical ensemble, as a function of the chemical potential μ. (a) Density in the trap center x = 0: hω (smooth solid exact result for T = 0 (solid line with steps), exact numerical sum for kB T = ¯ line) vs. the T = 0 LDA (dashed line). (b) Second-order derivative of the density in the trap hω: exact numerical sum (solid line) vs. the T = 0 LDA center, d2 ρ/dx2 (x = 0), for kB T = ¯ (dashed line).

For a chemical potential n + 3/2 < μ/¯hω < n + 5/2, where n is still even, the x = 0 density assumes the same value (since φn+1 (0) = 0), whereas the second-order derivative changes sign: (83) (84)

ρ(0) =

1 (n + 1)! , π 1/2 aho 2n [(n/2)!]2

ρ (0) = 2a−2 ho ρ(0).

The T = 0 exact value of the density as a function of the chemical potential is compared to the LDA in ﬁg. 3a: the LDA does not reproduce the steps but nicely interpolates between them. By using Stirling’s formula for the exact result, one ﬁnds that the relative error in the LDA is O(1/N ), where N is the particle number. If one cleverly approximates the value of a step in ρ(0) by taking the median value of the chemical potential, μ = (n + 3/2)¯ hω, one even ﬁnds that the relative error in the LDA vanishes as O(1/N 2 ). The LDA, on the contrary, gives a totally incorrect prediction for ρ (0) (not shown in the ﬁgure): one has (85)

ρLDA (0) = −

1 π 2 a4ho ρLDA (0)

,

which tends to zero in the large N limit, whereas the exact result oscillates with a diverging amplitude. All this can be understood from the fact that the exact density proﬁle at T = 0 is an oscillating function of x, oscillating at smaller and smaller scales in the large N limit. This fact can be revealed numerically [7], but more elegantly results

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Y. Castin

from the exact summation formula [8]: (86)

m

φ2k (x) = (m + 1)φ2m (x) −

m(m + 1)φm+1 (x)φm−1 (x).

k=0

The LDA of course misses these oscillations. What happens at ﬁnite temperature ? We expect that, at kB T ∼ ¯hω, these oscillations are suﬃciently reduced to make the LDA more accurate, while the temperature is low enough to allow the use of the T = 0 limit of the LDA. This is conﬁrmed by the kB T = h ¯ω data in ﬁg. 3: the plateaus in ρ(0) as a function of μ are hardly visible (see a), so are the oscillations of ρ(x) as a function of x [9], and the oscillations of ρ (0) are so strongly reduced that the LDA approximation becomes acceptable (see b). 2. – Two-body aspects of the interaction potential In real gases, there are interactions among the particles. In fermionic atomic gases it is even possible to reach the maximally interacting regime allowed by quantum mechanics in a gas, the so-called unitary regime, without altering the stability and lifetime of the gas. This section reviews possible models that can be used to represent the interaction potential and recalls basic facts of two-body scattering theory. We restrict to a three-dimensional two–spin-state, single-species Fermi gas; the p-wave interactions among atoms of a common spin state are neglected, usually an excellent approximation away from p-wave Feshbach resonances. The s-wave interactions are on the contrary supposed to be strong, the scattering length being much larger than the potential range, in the vicinity of a Feshbach resonance. . 2 1. Which model for the interaction potential? – The detailed description of the interaction of two atoms is involved. At large enough interatomic distances, in particular much larger than the size of the electronic orbitals of an atom, one can hope to represent this interaction by a position-dependent potential V (r ), which includes the van der Waals interaction term (87)

V (r) −

C6 , r6

a simple formula that actually neglects retardation eﬀects and long-range dipole-dipole magnetic interactions. Forgetting about these complications, we can use Schr¨ odinger’s equation and the C6 coeﬃcient to construct a length b, called the van der Waals length, that we shall consider as the “true” range of the interaction potential: (88)

C6 ¯2 h = 6 . mb2 b

For alkali atoms, b is in the nanometer range.

Basic theory tools for degenerate Fermi gases

305

At short interatomic distances, however, this simple picture of a scalar interaction potential has to be abandoned, and one has to include the various Born-Oppenheimer potentials curves coming from the QED Hamiltonian for two atomic nuclei at ﬁxed distance, including all the electronic and nuclear spin states, and the motional couplings among the Born-Oppenheimer curves due to the ﬁnite mass of an atom. Fortunately we are dealing with gaseous systems: the mean interparticle distance is much larger than the potential range b: (89)

ρ1/3 b 1,

where ρ is the mean density. We shall also consider atomic dimers, but these dimers shall be very weakly bound, with a size on the order of the scattering length a, which is much larger than b in the vicinity of a Feshbach resonance. As a consequence, the central postulate for the theory of quantum gases is that the short-range details of the atomic interactions are unimportant, only the low-momentum behavior of the scattering amplitude between two atoms is relevant. For fermions of equal masses, this postulate has proved robust in its confrontation to experiments, even in the unitary regime. A practical consequence of this postulate is that any short-range model for the interaction leading to almost the same scattering amplitude fk as the true interaction, in the typical relative momentum range k of atoms in a quantum gas, is an acceptable model. We then put the constraint on any acceptable model for the interaction: (90)

fkmodel fk

for the relevant values of the relative momenta k populated in the gas. We insist here that we impose similar scattering amplitudes over some momentum range, not just equal scattering lengths. Typical values of k can be the following ones: (91)

ktyp ∈ {a−1 , kF , λ−1 },

where a is the s-wave scattering length between opposite spin fermions, kF = (6π 2 ρ↑ )1/3 is the Fermi momentum expressed in terms of the density of a single-spin component ρ↑ = ρ↓ , and λ is the thermal de Broglie wavelength eq. (1). The choice of the correct value of ktyp is left to the user and depends on the physical situation. The ﬁrst choice ktyp ∼ a−1 is well suited to the case of a condensate of dimers (a > 0) since it is the relative momentum of two atoms forming the dimer. The second choice ktyp ∼ kF is well suited to a degenerate Fermi gas of atoms (not dimers). The third choice ktyp ∼ λ−1 is relevant for a non-degenerate Fermi gas, a case not discussed in this lecture. A condition for the gas not to be sensitive to the microscopic details of the potential is then that (92)

ktyp b 1.

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Y. Castin

As we shall see, the scattering amplitude in this momentum range should be described by a very limited number of parameters, which allows to use very simple models in the many-body problem. The situation is more subtle for bosons, or for mixtures of fermions with widely diﬀerent masses, where the Eﬁmov phenomenon takes place (see the lecture by Gora Shlyapnikov in this volume): at the unitary limit, an eﬀective 3-body attractive interaction occurs and accelerates the atoms; the wave vector k can then, for some type of Feshbach resonances (the broad Feshbach resonances to be deﬁned below), reach the range ∼ b−1 , in which case the details of the true interaction may become important. We shall not consider this case here. . 2 2. Reminder of scattering theory. – Suppose for simplicity that two particles of mass m interact via the short potential V (r1 − r2 ) that tends to zero at inﬁnity faster 3 than 1/r12 . Moving to the center-of-mass frame, with a center of mass with vanishing momentum, one gets the Schr¨ odinger equation at energy E: (93)

2 ¯h Eφ(r ) = − Δr + V (r ) φ(r ). m

At negative energies E < 0, one looks for the discrete values of E such that φ is square integrable: these eigenstates correspond to two-body bound states, that we call dimers. At positive energies E > 0, one sets (94)

E=

¯ 2 k02 h m

with k0 > 0, one looks for scattering states that obey the boundary conditions suited to describe a scattering experiment:

(96)

φ(r ) = φ0 (r ) + φs (r ), φ0 (r ) = r |k0 ≡ eik0 ·r ,

(97)

φs (r ) ∼ fk0 (n )

(95)

eik0 r r

for r → +∞.

The wave function φ0 represents the free wave coming from inﬁnity, taken here to be a plane wave of wave vector k0 of modulus k0 . The function φs (r ) represents the scattered wave which, at inﬁnity, depends on the distance as an outgoing spherical wave and possibly depends on the direction of observation n ≡ r/r through the scattering amplitude fk0 . It is important to keep in mind that the Schr¨ odinger equation with the above boundary conditions is formally solved by (98)

|φ = [1 + G(E + i0+ )V ]|φ0 ,

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Basic theory tools for degenerate Fermi gases

where G(z) = (z − H)−1 is the resolvent of the Hamiltonian, z a complex number. Using the identity G = G0 + G0 V G, where G0 (z) = (z − H0 )−1 is the resolvent of the free, kinetic energy Hamiltonian, one gets (99)

|φ = [1 + G0 (E + i0+ )T (E + i0+ )]|φ0 ,

where we have introduced the T operator (or T -matrix) (100)

T (z) = V + V G(z)V.

Equation (99) allows to obtain the equivalence in momentum space of the asymptotic conditions (95)–(97) in real space: (101)

k |φ = (2π)3 δ 3 (k − k0 ) +

1 k|T (E + i0+ )|k0 , E + i0+ − ¯h2 k 2 /m

where the matrix element of T is an a priori unknown smooth, regular function of k. The link between the k-space and the r-space points of view is completed by the following relation: (102)

fk0 (n ) = −

m k0n|T (E + i0+ )|k0 , 4π¯h2

which follows from the expression of the kinetic-energy Green’s function for E > 0:

(103)

m eik0 |r−r | . r |G0 (E + i0 )|r = − 4π¯h2 |r − r | +

Clearly, the central object is the scattering amplitude. It obeys the optical theorem 4π (104) σscatt (k0 ) = d2 n |fk0 (n )|2 = Im fk0 (k0 /k0 ), k0 where σscatt is the total scattering cross-section for distinguishable particles. This theorem is a direct consequence of the conservation of probability: the matter current j in the stationary state φ has a vanishing divergence, that is a zero ﬂux through a sphere of radius r → +∞: using the asymptotic form of φ, one gets the announced theorem. Here we shall consider model potentials that scatter only in the s-wave: φs is isotropic and so is the scattering amplitude, which reduces to an energy-dependent complex number: (105)

fk0 (n ) = fk0 .

The optical theorem simpliﬁes to (106)

|fk0 |2 =

Im fk0 . k0

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Y. Castin

Using Im z −1 = − Im z/|z|2 , for a complex number z, one realizes that this simply imposes: fk0 = −

(107)

1 , u(k0 ) + ik0

where u(k0 ) is at this stage an arbitrary real function. The scattering amplitude takes its maximal modulus when u is negligible as compared to k0 , this constitutes the unitary limit fk0 −

(108)

1 . ik0

A last point concerns the two-body bound states: their eigenenergies correspond to poles of the resolvent G, which are also poles of the T matrix. A simple way to ﬁnd the dimer eigenenergies is therefore to look for poles in the analytic continuation of the scattering amplitude to negative energies, setting k0 = iq0 , q0 > √ 0, so that E = −¯ h2 q02 /m < 0. Note that the determination +iq0 is chosen for k0 = mE/¯h so that the matrix elements of G0 , see eq. (103), assume the correct value (that tends to zero at inﬁnity). Furthermore one can easily have access to the asymptotic behavior of the dimer wave functions φj , including the correct normalization factor. One uses the closure relation (109)

d3 k0 |φk0 φk0 | + |φj φj | = I 3 (2π) j

where I is the identity operator. One takes the matrix elements of this identity between r | and |r , for large values of r and r so that the asymptotic expression eq. (97) may be used. After angular average (for a s-wave scattering), one gets factors e±ik0 (r+r ) and eik0 (r−r ) . Using the optical theorem one shows that the coeﬃcient of eik0 (r−r ) vanishes. Assuming that u(k) is an even analytical function of k one can extend the integral over k0 to ] − ∞, +∞[, setting fk∗0 = f−k0 , and use a contour integral technique by closing the contour with an inﬁnite half-circle in the Im z > 0 part of the complex plane. From eq. (107) the residue of fk in the pole k = iqj is obtained in terms of u (iqj ). We ﬁnally obtain the large r behavior of any s-wave correctly normalized dimer wave function: (110)

φj (r) ∼

qj 1 − iu (iqj )

1/2

e−qj r √ . 2πr

. 2 3. Eﬀective-range expansion and various physical regimes. – For the quantum gases, it is very convenient to introduce the usual low-k expansion of the scattering amplitude: (111)

fk0 = −

1 , a−1 + ik0 − k02 re /2 + . . .

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Basic theory tools for degenerate Fermi gases

where the parameter a is called the scattering length, the parameter re is called the eﬀective range of the interaction, not to be confused with the true range b. As discussed below, this expansion is interesting if the omitted terms . . . in the denominator of eq. (111) are indeed negligible when eq. (92) is satisﬁed. We take the opportunity to point out that the general allowed values for the eﬀective range can go from −∞ to +∞. This can be demonstrated on a speciﬁc example, the square well potential, with V (r) = −V0 for r < b, V (r) = 0 otherwise, the true range b being ﬁxed and V0 being a variable positive quantity. Setting κ0 = (mV0 /¯h2 )1/2 , we obtain the exact formulas: tan κ0 b , κ0 b3 1 re = b − 2 − 2 . 3a κ0 a a = b−

(112) (113)

This easily shows that for all non-zero values of κ0 such that the scattering length vanishes, the eﬀective range re tends to −∞. When κ0 tends to zero, the scattering length tends to zero (more precisely, 0− ), quadratically in κ20 , as is evident also from the Born approximation; in the expression for re , the last two terms diverge to inﬁnity, with opposite sign; a systematic series expansion in powers of κ0 b leads to (114)

re = −

176 2b2 + b + O(κ20 b3 ) 5a 175

for κ0 → 0+ .

One then sees that re tends to +∞ for κ0 → 0+ since a → 0− in this limit. Note that the leading term in eq. (114) can also be obtained in the Born approximation. As already mentioned, a very favorable case is when the . . . in the denominator of eq. (111) is negligible as compared to the sum of the ﬁrst three terms, in the lowmomentum regime k0 b 1: one can then characterize the true interaction by only two parameters, the scattering length and the eﬀective range. In what follows, it is implicitly assumed that this favorable case is obtained, otherwise more realistic modeling of the interaction should be performed. One can then identify interesting limiting cases. The zero-range limit: This is a limit where the only relevant parameter of the true interaction is the scattering length a, i.e. we shall assume that the typical momentum ktyp satisﬁes (115)

2 ktyp |re | |a−1 + iktyp |,

where re is the eﬀective range of the true interaction. In practice, in present experiments and for typical atomic densities, this is true for the so-called broad Feshbach resonances, where re ∼ b: condition (115) is then a direct consequence of eq. (92). For narrow Feshbach resonances, |re | can be considerably larger than b and eq. (115) may be violated for typical densities.

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Y. Castin

What type of model can be chosen in this zero-range limit? A natural idea is to use a strictly zero-range model, having both a vanishing true range and eﬀective range, and characterized by the scattering length as the unique parameter. This “ideal world” idea . corresponds to the Bethe-Peierls model of subsect. 2 5, where the interaction is replaced by contact conditions. In practice, the contact conditions are not always convenient to implement, in an approximate or numerical calculation. It is therefore also useful to introduce a ﬁniterange model, usually with a true range of the order of its eﬀective range remodel . One then usually puts the model range to zero at the end of the calculation. Note that this is not always strictly speaking necessary: it may be suﬃcient that the model is also in the zero-range limit, (116)

2 ktyp |remodel | |a−1 + iktyp |.

In sect. 3 on BCS theory we shall restrict to the zero-range limit. We shall then approximate the true interaction by an on-site interaction in a Hubbard lattice model, . see subsect. 2 6: the eﬀective range of the model is then of the order of the lattice spacing, and one may easily take the limit of a vanishing lattice spacing in the BCS theory. The unitary limit: What is the condition to reach the unitary limit in a gas, in a regime where the low-k expansion holds? One should have (117)

2 |a−1 |, ktyp |re | ktyp .

The condition on a is in principle possible to fulﬁll, as a diverges close to a Feshbach resonance. The condition on re shows that the zero-range limit eq. (115) is a necessary condition to reach the unitary limit. This is why the broad Feshbach resonances are favored in present experiments (over the narrow ones) to reach the strongly interacting regime. To make it simple, one may say that the unitary limit is simply the zero-range limit with inﬁnite scattering length. . 2 4. A two-channel model . – We have the oversimpliﬁed view of a magnetically induced Feshbach resonance, see ﬁg. 4: two atoms interact via two potential curves, Vopen (r) and Vclosed (r). At short distances these two curves correspond to the electronic spin singlet and triplet, respectively, for atoms of electronic spin 1/2, so that they experience diﬀerent shifts in the presence of an external magnetic ﬁeld. At large distances, Vopen (r) tends to zero, whereas Vclosed (r) tends to a strictly positive value V∞ . The atoms are supposed to come from inﬁnity in the internal state corresponding to the curve Vopen (r), the so-called open channel. The atoms are ultracold and have a low kinetic energy, (118)

E V∞ .

In real life, the two curves are actually weakly coupled. Due to this coupling, the atoms can have access to the internal state corresponding to the curve Vclosed (r), but only

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3

V(r12) /kB [10 K]

0.2

Emol

Vf

0

/ -0.2 1

1.5

2

r12 [nm]

2.5

3

Fig. 4. – Oversimpliﬁed view of a magnetically induced Feshbach resonance. The interaction between two atoms is described by two Born-Oppenheimer curves. Solid line: open-channel potential curve. Dashed line: closed-channel potential curve. When one neglects the coupling Λ between the two curves, the closed channel has a molecular state of energy Emol with respect to the dissociation limit of the open channel. Note that the energy spacing between the solid curve and the dashed curve was greatly exaggerated, for clarity.

at short distances; at long distances, the external atomic wave function in this so-called closed channel will be an evanescent wave that decays exponentially with r since E < V∞ . Now assume that, in the absence of coupling between the channels, the closed channel supports a bound state of energy Emol , denoted in this text as the molecular state. Assume also that, by applying a judicious magnetic ﬁeld, one sets the energy of this molecular state close to zero, that is to the dissociation limit of the open channel. In this case one may expect that the scattering amplitude of two atoms may be strongly aﬀected, by a resonance eﬀect, when a weak coupling exists between the two channels. A quite quantitative discussion is obtained thanks to a very simple two-channel model. This model is the most realistic of the models presented here, and we shall consider it as a “reality” against which the other models should be confronted. Assuming that the particles are free in the open channel, and can populate only the molecular state in the closed channel, the only non-trivial ingredient is a coupling between the molecule with center of mass momentum zero, |mol, and a pair of atoms of opposite momenta and spin states ||k, with (119)

||k ≡ a† a† |0, k↑ −k↓

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Y. Castin

where the creation and annihilation operators obey the free-space anticommutation relation 6 7 akσ , a† = (2π)3 δ(k − k )δσσ .

(120)

k σ

This coupling in the momentum representation is deﬁned as V |mol =

(121)

d3 k Λχ(k)||k, (2π)3

V ||k = Λχ(k)|mol.

(122)

We have introduced a real coupling constant Λ, and a real and isotropic cut-oﬀ function χ(k), equal to 1 for k = 0 and to 0 for k = ∞, to avoid ultraviolet divergencies. It is convenient to take χ(k) = e−k

(123)

2

σ 2 /2

,

where the length σ is a priori of the order of the true potential range b. The two-channel model is thus deﬁned by the three parameters Emol , Λ and σ. It scatters in the s-wave channel only, since χ is isotropic. An eigenstate of energy E of the two-channel Hamiltonian is of the form |φ = β|mol +

(124)

d3 k α(k )||k. (2π)3

Including the molecular state energy and the kinetic energy of a pair of atoms, the stationary Schr¨ odinger equation leads to (125) (126)

¯ 2 k2 h α(k ) + Λχ(k)β = Eα(k ) m d3 k Emol β + Λχ(k)α(k ) = Eβ. (2π)3

We restrict to a scattering problem, E > 0. Instructed by the previous discussion around eq. (101), we correctly solve for α in terms of β in eq. (125): (127)

α(k ) = (2π)3 δ(k − k0 ) +

Λχ(k) β, E + i0+ − ¯h2 k 2 /m

where k0 is the wave vector of the incident wave, and E = h ¯ 2 k02 /m. Insertion of this solution in eq. (126) leads to a closed equation for β that is readily solved. From eq. (101) and eq. (102) we obtain the scattering amplitude (128)

fk0 =

Λ2 χ2 (k0 ) m . d3 k 2 Λ2 χ2 (k) 4π¯ h Emol − E + (2π) 3 E+i0+ −¯ h2 k2 /m

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Basic theory tools for degenerate Fermi gases

Let us work a little bit on this scattering amplitude. First, using the identity 1 1 = P − iπδ(X), X + i0+ X

(129)

where P denotes the principal value, one can check that fk0 is indeed of the form (107), so that it obeys the optical theorem. Second, the scattering length is readily obtained by taking the k0 → 0 limit: (130)

−a−1 =

d3 k mχ2 (k) 4π¯ h2 Emol − . m Λ2 (2π)3 ¯h2 k 2

This shows that the location of the resonance (where a−1 = 0) is not Emol = 0 but a 0 positive value Emol , this shift being due to the coupling between the two channels. For Emol larger than this value, a < 0. For Emol below this value, a > 0. Another general result, not speciﬁc to a Gaussian choice for χ(k) (but assuming that χ(k) varies quadratically around k = 0), concerns the zero range limit σ → 0 for ﬁxed values of k0 , Λ and the scattering length a (note that the molecular energy is then not ﬁxed): (131)

fk0 → −

1 a−1

+ ik0 +

4π¯ h2 E m Λ2

.

+∞ This can be seen formally as resulting from the identity 0 dX P1/(X 2 − 1) = 0. This shows that the two-channel model has a well-deﬁned limit, in the zero true-range limit, characterized only by the scattering length and by a non-zero and negative eﬀective range, (132)

re0 = −

8π¯h4 . m2 Λ 2

This limit may be described by generalized Bethe-Peierls contact conditions [10]. This illustrates dramatically the diﬀerence between the true range and the eﬀective range of an interaction. This also shows that, in this limit, the two-channel model can support a weakly bound state, that is a dimer with a size much larger than the true range σ. Setting k0 = iq0 , one ﬁnds that the right-hand side of eq. (131) has a pole with q0 real and positive if and only if a > 0: (133)

q0 =

1−

1 − 2re0 /a . re0

Finally, an exact calculation of the scattering amplitude for the Gaussian choice eq. (123) is possible: the trick put forward by Mattia Jona-Lasinio is to calculate the

314

Y. Castin

analytic continuation to imaginary k0 = iq0 , so that one no longer needs to introduce the principal part distribution. One gets (134)

2 2 1 (fiq0 )−1 = −a−1 − re0 q02 e−q0 σ + q0 erfc(q0 σ), 2

where erfc is the complementary error function, that tends to unity in zero, erfc(x) = 1 − 2π −1/2 x + O(x3 ). The scattering length is given by (135)

−a−1 =

4π¯h2 Emol 1 − √ . 2 mΛ σ π

The calculation of the eﬀective range for a ﬁnite σ is then straightforward: (136)

re =

re0

4σ +√ π

√

πσ 1− 2a

.

We shall then classify the Feshbach resonances as follows: for a = ∞, the ones with re re0 < 0, much larger than the true range in absolute value, constitute narrow Feshbach resonances. The ones with re σ constitute broad Feshbach resonances. This classiﬁcation is not only theoretical but also reﬂects the magnetic width ΔB of the resonance. If one assumes that Emol is an aﬃne function of the magnetic ﬁeld B, with a slope −μ, one ﬁnds from eq. (130) that the scattering length a of the two-channel model varies with B as (137)

a=

abg ΔB , B0 − B

where B0 is the magnetic ﬁeld value of the resonance center, and we have singled out the value abg of the background scattering length (that is the value of the scattering length of the true interaction far from the resonance), motivated by the fact that a = abg [1 + ΔB/(B0 − B)] in a more complete theory including the direct interaction in the open channel [11]. This deﬁnes the width ΔB, such that (138)

μΔB =

2¯h2 . mabg re0

¯ 2 /mb2 amounts to So, when abg ∼ σ ∼ b, comparing μΔB to the natural energy scale h 0 comparing |re | to b. . 2 5. The Bethe-Peierls model. – In this model, the true interaction is replaced by contact conditions on the wave function. For two particles in free space, in the center of mass frame, the contact condition is that there exists a constant A such that (139)

φ(r ) = A r−1 − a−1 + O(r)

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Basic theory tools for degenerate Fermi gases

in the limit r → 0. For r > 0 the wave function is an eigenstate of the free Hamiltonian: Eφ(r ) = −

(140)

¯2 h Δr φ(r ). m

In mathematical terms, this deﬁnes the domain of the Hamiltonian. The Hilbert space remains however the same, with the usual scalar product, because a 1/r divergence in 3D is square integrable. An equation valid for all values of r can be written using the theory of distributions: (141)

Eφ(r ) = −

¯2 h 4π¯h2 a Δr φ(r ) + φreg δ(r ), m m

where the so-called regular part of φ is (142)

φreg = −

A = ∂r [rφ(r )]r=0 = −a−1 lim rφ(r ). r→0 a

This establishes the equivalence with the regularized pseudo-potential model [12-15]. The last form of φreg is convenient to use in the limit a = ∞, since a−1 simpliﬁes with the factor a in front of the Dirac δ(r ). The solution of eq. (140) with the boundary conditions (97) and the contact condition is simply (143)

φ(r ) = eik0 ·r + fk0

eik0 r , r

with the scattering amplitude (144)

fk0 = −

1 . a−1 + ik

This shows that the model has both a zero true range and zero eﬀective range. It may be applied to mimic the eﬀect of the true interaction potential when condition (115) is satisﬁed. Following the discussion above eq. (110), one ﬁnds that the model supports a dimer if and only if a > 0. The pole of fiq0 is then q0 = 1/a, resulting in a dimer energy (145)

E0 = −

¯2 h . ma2

Since the model has a zero true range, it turns out that the dimer wave function coincides everywhere with its asymptotic form eq. (110), (146)

1 e−r/a . φ0 (r) = √ 2πa r

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Y. Castin

The advantage of the Bethe-Peierls model for the N -body problem is that it introduces the minimal number of parameters (the scattering length). In the unitary limit, it is a zero-parameter model, which is ideal to describe universal states. This advantage actually really exists if one restricts to analytical solutions of the model (numerical solutions tend to introduce extra parameters like a discretization step or a momentum cut-oﬀ). Analytical solutions in free space are available up to N = 3 in the unitary limit [16, 17]. For particles trapped in harmonic potentials, analytical solutions are available for two particles at any a for isotropic traps [18] and cylindrically symmetric traps [19]. For three particles in an isotropic harmonic trap, the problem was solved analytically only in the unitary limit a = ∞ [20, 21]. In the many-body problem, for the universal unitary gas in an isotropic harmonic trap, a scaling solution can be found for the many-body wave function in the time-dependent case [22] and an exact mapping to free-space zero-energy eigenstates can be constructed [23-25]. It is worth mentioning that, for N = 3 bosons, the Bethe-Peirls model does not deﬁne a self-adjoint operator and has to be supplemented by extra contact conditions; this is related to the Eﬁmov phenomenon [16, 17]; the problem of course persists for N > 3, making the model non satisfactory. Fortunately this lecture is devoted to equal-mass spin-(1/2) fermions, for which no Eﬁmov eﬀect appears; the corresponding unitary gas is then called universal, to emphasize this aspect that no extra parameter has to be added to the Bethe-Peirls model. Note that this absence of Eﬁmov eﬀect is not simply due to the fact that the Pauli principle prevents one from having three fermions of spin 1/2 in the same point of space: if the spin-up and spin-down fermions have widely diﬀerent masses, an Eﬁmov eﬀect may appear [26]. We take the opportunity to mention another trap to avoid, even for fermions. With the formulation of the model in terms of binary interactions with the pseudo-potential VPP (r ) = gδ(r )∂r (r ·), where g = 4π¯h2 a/m, it is easy to “forget” that contact conditions are imposed on the many-body wave function. One may then be tempted to perform a variational calculation with a N -body trial wave function which does not satisfy the contact conditions (139), that is which is not in the domain of the Hamiltonian [27]. Whereas in the low a-limit the corresponding prediction for the energy may make sense as a perturbative prediction, it may become totally wrong in the large a limit. To illustrate this statement, let us consider two identical particles in a harmonic isotropic trap interacting via the pseudo-potential. After separation of the center-ofmass coordinates, one is left with the following Hamiltonian for the relative motion: (147)

H=−

¯2 h 1 Δr + μω 2 r2 + gδ(r )∂r (r ·), 2μ 2

where μ = m/2 is the reduced mass. Let us take as a trial wave function |φt the Gaussian of the ground state of the harmonic oscillator, (148)

φt (r) =

e−r

2

/(4a2ho ) 3/2

(2π)3/4 aho

,

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Basic theory tools for degenerate Fermi gases

where aho = ¯ h/mω. This trial wave function does not obey the contact conditions, so it is not in the domain of the Hamiltonian. It is therefore mathematically incorrect to calculate the mean energy of φt by representing the action of H on φt by the diﬀerential operator eq. (147). One rather has to calculate the overlap ψn |φt of |φt with each exact eigenstate ψn of H of eigenenergy En , and calculate the sum n En |ψn |φt |2 . From [18] and for a ﬁnite a one ﬁnds that the overlap squared scales as 1/n3/2 whereas the energy En scales as n, for large n. The correct mean energy of φt is thus inﬁnite. If one, however, falls in the trap, one gets the wrong mean energy Ewrong = φt |H|φt =

(149)

3 ¯hω + 2

1/2 2 a ¯hω . π aho

In the low-a limit this can be compared to the expansion of the exact minimal positive eigenenergy [18] (150)

Eexact

3 hω + h ¯ω = ¯ 2

2 1/2 2 a a 2 + (1 − ln 2) + ... . π aho π aho

One sees that the wrong variational calculation gives the correct linear correction in a, that is the correct perturbative result. One may then be tempted to use the wrong variational calculation to predict the sign of the coeﬃcient of a2 ; one would naively assume Eexact ≤ Ewrong and one would wrongly predict a negative sign for the coeﬃcient of a2 (1 ). In the opposite limit of an inﬁnite scattering length, one directly realizes the absurdity of the assumption Eexact ≤ Ewrong : for a → −∞, one ﬁnds Ewrong → −∞, 2 2 whereas Eexact → ¯ hω/2, with the exact wave function ∝ e−r /4aho /r. Note that the same pathology occurs for the many-body problem. Consider, for example, a two–spin-component spatially homogeneous Fermi gas, with interactions modeled by the pseudo-potential. If one uses the Hartree-Fock variational ansatz, which is not in the domain of the Hamiltonian, one correctly obtains, in the weakly attractive regime kF a → 0− , the mean-ﬁeld shift of the ground-state energy, but one wrongly predicts a negative sign for the coeﬃcient of a2 in the low kF a expansion of the ground-state energy, by wrongly arguing that the exact ground-state energy has to be below the “variational” Hartree-Fock energy. As can be checked on the systematic expansion [28] the coeﬃcient of a2 is actually positive, because 11 > 2 ln 2. One should not leave this section with the impression that any variational calculation is impossible within the Bethe-Peierls model. To be safe, one simply has to take a variational ansatz which is in the domain of the Hamiltonian. An example of a successful variational calculation is the derivation of the virial theorem for a unitary gas in a nonnecessarily isotropic harmonic trap; this derivation, due to Fr´ ed´eric Chevy, is reported in [25], and uses the fact, for the universal unitary gas 1/a = 0, that the function of the (1 ) This is in a way doubly naive for a → 0+ , as far as the pseudo-potential is concerned, since for a > 0 the exact problem admits an eigenstate of energy < 3¯ hω/2, that → −∞ for a → 0+ .

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rescaled coordinates Ψ(r1 /λ, . . . , rN /λ), where λ > 0, is in the domain of the Hamiltonian if the wave function Ψ(r1 , . . . , rN ) is in the domain of the Hamiltonian. . 2 6. The lattice model. – The last model we consider is at the intersection of two very popular classes of models in condensed-matter physics, the separable potentials and the lattice models. It is slightly simpler than the two-channel model, but it does not apply to narrow Feshbach resonances in a situation where the eﬀective-range term cannot be neglected in the scattering amplitude. It is more tractable than the Bethe-Peierls model for variational treatments (like the BCS method, see sect. 3) or for exact numerical treatments since the domain of the Hamiltonian is the one of usual quantum mechanics. In particular, since it is a sort of Hubbard model, the machinery of quantum Monte Carlo methods applicable to the Hubbard model [29-32] could be applied to it. In this model, the spatial coordinates r of the particles are discretized on a cubic grid of step l. As a consequence, the components of the wave vector of a particle have a meaning modulo 2π/l only, since the function r → exp[ik · r ] deﬁned on the grid is not changed if a component of k is shifted by an integer multiple of 2π/l. We shall therefore restrict the wave vectors to the ﬁrst Brillouin zone: 3 k ∈ D ≡ − π , π . (151) l l This also shows that the lattice structure in real space provides a cut-oﬀ in momentum space. The interaction between opposite-spin particles takes place when two particles are at the same lattice site, as in the Hubbard model. In ﬁrst quantized form, it is represented by a discrete delta potential: (152)

V =

g0 δr ,r . l3 1 2

The coupling constant g0 is a function of the grid spacing l. It is adjusted to reproduce the scattering length of the true interaction. The scattering amplitude of two atoms on the lattice with vanishing total momentum is given in [33], and a detailed discussion is presented in [1]. We give here only the result, generalized to an arbitrary even dispersion relation k → Ek for the kinetic energy on the lattice: (153)

fk0 = −

m 4π¯h2 g0−1 − D

1 1 d3 k (2π)3 E+i0+ −2Ek

.

Adjusting g0 to recover the correct scattering length then gives (154)

1 1 = − g0 g

D

d3 k 1 , (2π)3 2Ek

with g = 4π¯ h2 a/m. This formula is reminiscent of the technique of renormalization of the coupling constant.

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Basic theory tools for degenerate Fermi gases

A natural case to consider is the one of the usual parabolic dispersion relation, Ek = ¯ 2 k 2 /2m. A more explicit form of eq. (154) is then h (155)

g0 =

4π¯h2 a/m 1 − Ka/l

with a numerical constant given by (156)

12 K= π

π/4

dθ ln(1 + 1/ cos2 θ) = 2.442 749 607 806 335 . . . , 0

and that may be expressed analytically in terms of the dilog special function. The eﬀective range of the model is easily calculated with the complexiﬁcation technique k0 = iq0 ; it is positive and proportional to l: (157)

remodel =

l 2π 3

3 k∈[−1/2,1/2[ /

d3 k = 0.336 868 47 . . . l. k4

In the l → 0 limit, for a ﬁxed a (not ﬁxed g0 ) the lattice model then reproduces the scattering amplitude of the Bethe-Peierls model, (158)

lim fk0 = −

l→0

1 , a−1 + ik0

and admits a dimer for a > 0. Note that g0 indeed assumes negative values that tend to 0− linearly with l in this limit: the interaction is attractive, whatever the sign of the scattering length (2 ). Studying the weakly interacting regime: In the opposite case of l |a|, the bare coupling constant is (159)

g0

4π¯h2 a , m

so that g0 has the sign of the scattering length. For a > 0, the model interaction is repulsive, for a < 0 it is attractive. In both cases, it does not support any two-body bound state. Its scattering length can be calculated in the Born approximation, since the term in a in the denominator of eq. (155) is small as compared to unity. This makes it an appropriate model to study the weakly interacting regime, as was done for Bose gases in [33]: the Hartree-Fock theory, for example, which relies on the Born approximation, may be applied, and its accuracy may be supplemented by keeping higher-order terms in a perturbative expansion of the interaction. In the context of fermions, it may be (2 ) More generally, the lattice model admits a dimer if g0 < g0c , where g0c is the a → ∞ limit of eq. (155).

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Y. Castin

used in the weakly attractive or weakly repulsive regime; in the latter case, it may be an alternative to the celebrated hard-sphere model (with hard spheres of diameter a > 0). It is instructive to apply the philosophy around eq. (90) to understand why the lattice model with l |a| is restricted to the weakly interacting regime, for a quantum degenerate gas of atoms with ktyp = kF . – In an eﬀort to have fkmodel fk , one ﬁrst adjusts the scattering length of the model to the correct value a by tuning g0 . – If one further adjusts the lattice spacing l to have remodel = re : this assumes a broad Feshbach resonance with re ∼ b, since the narrow Feshbach resonance has re < 0; one then ﬁnds l ∼ b; since ktyp b 1 (otherwise microscopic details of the interaction will come in, and the . . . in eq. (111) is not negligible), having |a| l leads to kF |a| 1, which is the weakly interacting regime. – Alternatively, one may consider the more favorable case of a true interaction being in the zero-range limit eq. (115). The model should also be in the zero-range limit eq. (116), which can be written here as / / / 1 / / kF l / + i// , kF a

(160)

and this is suﬃcient (remodel ∝ l may diﬀer a lot from re ). Since we wish to have |a| l, the above equation leads to kF |a| |(kF a)−1 + i|, a condition that may be satisﬁed only in the weakly interacting limit kF |a| 1. – The same result may be obtained in the following much faster way: since we are dealing here with fermions, there will be at most one fermion per spin component per lattice site, so that kF l < (6π 2 )1/3 ; the condition |a| l then immediately leads to kF |a| < 1. This reasoning is however speciﬁc to the lattice model, whereas the previous reasoning is easily generalized to the hard-sphere model (where remodel ∝ a). The “true” Hubbard model : To be complete, we consider a second case for the dispersion relation, the one of the true Hubbard model: this makes the link with condensed-matter physics, and this also shows that a universal quantum gas may be described by an attractive Hubbard model in the limit of a vanishing ﬁlling factor. The Hubbard model is usually presented in terms of the tunneling amplitude between neighboring sites, here t = −¯ h2 /2ml2 , and of the on-site interaction U = g0 /l3 . The dispersion relation is then (161)

Ek =

¯2 h [1 − cos(kα l)] , ml2 α

where the summation is over the three dimensions of space. Close to k = 0, the Hubbard dispersion relation by construction reproduces the free space one h ¯ 2 k 2 /2m. The explicit

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321

version of eq. (154) is obtained from eq. (155) by replacing the numerical constant K by K Hub = 3.175911 . . .. At the unitary limit this leads to U/t = 7.913552 . . ., which corresponds to an attractive Hubbard model since t < 0, lending itself to a quantum Monte Carlo analysis for equal-spin populations with no sign problem [29-31]. We have also calculated the eﬀective range of the Hubbard model, which remarkably is negative: (162)

reHub −0.3056l.

. 2 7. Application of Bethe-Peierls to a toy model: two macroscopic branches. – In this subsection, we move to the problem of N interacting fermions, N/2 with spin component ↑ and N/2 with spin component ↓. In order to have a global view of what the BCS theory will be useful for, it is instructive to start with a purely qualitative model [34]. Consider a matter wave of isotropic wave function φ(r) in a hard-wall spherical cavity of radius R, in the presence of a point-like scatterer of ﬁxed position in the center of the cavity. The eﬀect of the point-like scatterer is characterized by the scattering length a and is treated by imposing the Bethe-Peierls contact condition eq. (139). What is the link between this model and the many-body problem of N interacting fermions? The wave function φ(r) describes the relative motion of a, let us say, spin ↑ atom, with respect to the nearest spin ↓ atom modelized by the point-like scatterer. The cavity represents i) the interaction eﬀect of the other N/2 − 1 spin ↓ atoms and ii) the Pauli blocking eﬀect of the other N/2 − 1 spin ↑ atoms. Interaction eﬀect i): the radius of the cavity should then be of the order of the mean interparticle separation in the gas, R∝

(163)

1 . kF

Pauli blocking eﬀect (ii): in the case g = 0, the zero-point energy of φ should be of the order of the Fermi energy so that one has also the choice (163). This explains also the choice of hard walls for the cavity; in the case of bosons, imposing that the derivative ∂r φ vanishes in r = R [35], or taking a cubic cavity with periodic boundary conditions would be appropriate. Finally, the total energy of the gas is related to by E ∝ N .

(164)

Speciﬁc but arbitrary proportionality coeﬃcients are used in eqs. (163), (164) to produce ﬁg. 5 to come, as detailed in [34]. Let us proceed with the calculation of the eigenenergies of the matter wave in the cavity. An eigenmode of the cavity with energy solves Schr¨odinger’s equation for 0
−

¯2 h Δφ(r) = φ(r) m

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Y. Castin

2

4

(a)

(b) 3

P [PF]

E/N [EF]

1 0

2

-1 1

-2 -10

-5

0

5

10

0 -10

-1/kFa

-5

0

5

10

-1/kFa

Fig. 5. – In the toy model with a scatterer in a hard-wall spherical cavity: (a) energy per particle and (b) pressure of the gas in the ﬁrst two branches, as functions of −1/kF a. kF , EF and PF are, respectively, the Fermi wave vector, the Fermi energy and the pressure of the T = 0 ideal Fermi gas with the same density as the interacting gas.

with the Bethe-Peierls contact condition (139). If > 0, we set = h ¯ 2 k 2 /m and k solves (166)

tan kR = ka.

If < 0, we set = −¯ h2 κ2 /m and κ solves (167)

tanh κR = κa.

In the cavity, there is an inﬁnite number of discrete modes. To each of this mode we associate a distinct macroscopic state of the gas. In ﬁg. 5 we have plotted the energy per particle and the pressure P = −∂V E, where V = N/ρ is the total volume of the gas, for the ﬁrst two branches, as functions of −1/kF a. We have taken −1/kF a as the abscissa because it allows an almost direct mapping with the B ﬁeld axis in a real experiment around the location B0 of the Feshbach resonance, see eq. (137). The ﬁrst excited branch is metastable. It starts with a weakly repulsive Fermi gas on the extreme left of the ﬁgure and has a larger energy than the ideal Fermi gas, indicating eﬀective repulsion. When 0 < a kF−1 , three-body collisions (not included in the toy model) lead in a real gas to the formation of dimers φ0 : in the language of the toy model, the system starts populating the ground branch. This suggests a ﬁrst experimental way to produce a gas of dimers. The ground branch continuously connects the weakly attractive Fermi gas (on the right) to a gas of dimers (on the left). This provides a second experimental way to produce a condensate of dimers, by adiabatic following of the ground branch. The sharp decrease of the total energy on the left part of the ﬁgure reﬂects the 1/a2 dependence of the dimer binding energy 0 . The pressure is always less than the Fermi pressure of the ideal Fermi gas, indicating eﬀective attraction with respect to the ideal Fermi gas. Note

Basic theory tools for degenerate Fermi gases

323

that P drops very rapidly on the left side, due to the absence of interaction between the molecules in the toy model. In real life, this interaction occurs with a scattering length that was calculated by solution of the 4-body problem [36]. As discussed in other lectures of this volume, the molecular condensates have been obtained experimentally [37-40]. The regime of inﬁnite scattering length |a| = +∞ is universal in the toy model: one ﬁnds that the energy of the corresponding unitary gas has to be proportional to the ground-state energy of the ideal Fermi gas: (168)

E unitary = ηE0ideal ,

where E0ideal = 3N EF /5, η is a numerical constant depending on the branch, and EF is the Fermi energy of the ideal gas. In the present state of the art of the ﬁeld, it is accepted that this universality property also holds for a real ground-state Fermi gas. Approximate ﬁxed node Monte Carlo calculations [41, 42] give for the ground branch η 0.4, in disagreement with early experiments [43] but in good agreement with recent experimental measures [44, 37, 45] and with recent exact Quantum Monte Carlo results [32]. The toy model allows to easily solve the following paradox: – A common saying is that a gas with a positive scattering length a > 0 experiences eﬀective repulsion, and that it experiences eﬀective attraction for a < 0. – Another saying is that for |a| = +∞ the energy of the gas is universal and does not depend on the sign of a. – However, if a → +∞ (scenario 1), one expects that the universal state has eﬀective repulsion, whereas for a → −∞ (scenario 2), one expects that the universal state experiences attraction. How can it be that there is no dependence on the sign of a in the unitary limit? The answer provided by the toy model is simple: scenario 1 and scenario 2 are predicted to lead in the unitary limit to two diﬀerent universal states, belonging respectively to the ﬁrst excited branch and to the ground branch, one experiencing eﬀective repulsion, the other eﬀective attraction. 3. – Zero-temperature BCS theory: Study of the ground branch In this section, we consider the many-body problem of a two-component Fermi gas with an interaction characterized only by the s-wave scattering length a. To be able to use the BCS theory for all values of kF a, while getting sensible results, we restrict to the zero-temperature case: the usual static BCS theory is indeed unable to get a fair approximation to the critical temperature on the bosonic side of the Feshbach resonance (a > 0, kF a 1), for reasons that will become clear at the end of this section. An improved ﬁnite-temperature theory can be found in [46, 47]. For simplicity we also assume that the two spin states ↑ and ↓ have the same number of particles, so that they have a common chemical potential μ. The case of imbalanced

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Y. Castin

chemical potentials and particle numbers is presently the object of experimental and theoretical studies, and may lead to a variety of observed and/or predicted phenomena, as the not yet observed non-demixed BCS phases with spatially modulated order parameter (the so-called LOFF phases) [48, 49] or the already observed spatial demixing of the two spin components [50, 45, 51, 52]. Going beyond the toy model of the previous section, the zero-temperature BCS theory predicts that the particles arrange in pairs and gives several properties of the gas of pairs: – The existence of long-range order in the pair coherence function: ˆ † (r )Ψ( ˆ 0 ), g1pair (r ) = Ψ ˆ r ) = ψˆ↓ (r )ψˆ↑ (r ) annihilates a pair of particles with opposite spin in where Ψ( r. In the thermodynamic limit, the zero-temperature BCS theory predicts that g1pair has a non-zero limit for large r: the pairs form a condensate. Except in the regime kF a → 0+ , the pairs are not bosons, so that this condensate is not simply a Bose-Einstein condensate. – The BCS theory predicts an energy required to break a pair. – The time-dependent BCS theory also predicts collective modes for the motion of the pairs, like sound waves, associated to the famous Anderson-Bogoliubov phonon branch [53, 54]. In a trapped system, the time dependent BCS theory predicts the equivalent of superﬂuid hydrodynamic modes. . In what follows, we shall use the lattice model of subsection 2 6. In second quantized form we therefore take the grand-canonical Hamiltonian (169)

H=

h ¯ 2 k2 − μ a† akσ + l3 U (r )ψˆσ† (r )ψˆσ (r ) kσ 2m r,σ k∈D σ=↑,↓ † + g0 l 3 ψˆ↑ (r )ψˆ↓† (r )ψˆ↓ (r )ψˆ↑ (r ). r

The external potential U (r ) may be accompanied by a rotational term if one wishes to study vortices. Here the akσ ’s obey the usual discrete anticommutation relations eqs. (6), (7). The ﬁeld operators ψˆσ (r ) obey anticommutation relations mimicking the ones in continuous space in the limit l → 0: (170)

{ψˆσ (r ), ψˆσ† (r )} = l−3 δrr δσσ .

. As discussed in subsect. 2 6, this lattice model is very close to the usual Hubbard model of condensed-matter physics. What is unusual here is that we take a quadratic dispersion relation, and more important the model makes sense (to describe a real atomic gas with continuous positions) in the low ﬁlling factor limit, an unusual limit in solid-state

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Basic theory tools for degenerate Fermi gases

physics. We note that the original Hubbard model (with the usual cosine dispersion relation eq. (161) and with a ﬁlling factor of the order of unity) may be realized in a real experiment with atoms in an optical lattice [55-58]. . 3 1. The BCS ansatz . 3 1.1. A coherent state of pairs. Let us recall the Glauber coherent state of a bosonic ﬁeld, |α = e−|α|

(171)

2

/2 αa†

e

|0,

where α is a complex number and a† creates a boson in some normalized one-body wave function φ(r ), a† =

(172)

l3 φ(r )ψˆ† (r ).

r

By analogy, the BCS theory [59], which is a variational theory, takes as a trial state a coherent state of pairs: †

|ΨBCS = N eγC |0,

(173)

where N is a normalization factor, γ is a complex number and C † creates two particles in the normalized pair wave function φ(r1 , r2 ): (174)

C† =

l6 φ(r1 , r2 ) ψˆ↑† (r1 )ψˆ↓† (r2 ).

r1 , r2

Note that in general C and C † do not obey bosonic commutation relations. For simplicity, we omit terms creating two particles in the same spin state. This shall be suﬃcient for our purposes as we restrict here to the case of balanced spin state populations. . 3 1.2. A more convenient form from the Schmidt decomposition. To brieﬂy review the main properties of this ansatz, it is convenient to introduce the Schmidt decomposition of γ|φ, which is, in the physics of entanglement, routinely applied to the state vector of two arbitrary quantum particles: (175) γ|φ = Γα |Aα ⊗ |Bα , α

where the coeﬃcients Γα are real numbers, the set of |Aα is an orthonormal basis here for a spinless particle, and the set of |Bα is also an orthonormal basis for a spinless particle. Note that in general these two basis are distinct. As |φ is normalized to unity, one has α Γ2α = |γ|2 . Then the pair creation operator takes the simple expression (176)

γ C† =

α

Γα cˆ†α↑ cˆ†α↓

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Y. Castin

where cˆ†α↑ is the creation operator of a particle in the state |Aα | ↑ and cˆ†α↓ is the creation operator of a particle in the state |Bα | ↓. Note that these creation operators and the associated annihilation operators obey the usual fermionic anticommutation relations. The key advantage of the Schmidt decomposition is to allow a rewriting of the BCS state, more familiar to the reader: (177)

|ΨBCS = N

"

1+

Γα cˆ†α↑ cˆ†α↓

|0.

α

To obtain this expression, we have used the fact that any binary product cˆ†α↑ cˆ†α↓ commutes

c†α↑ cˆ†α↓ ] terminates to with any cˆ†β↑ cˆ†β↓ , and the fact that the series expansion of exp[γˆ ﬁrst order in γ since (ˆ c† )2 = 0 for fermions. The form eq. (177) shows that one can consider each pair of modes {α ↑, α ↓} as independent, since each factor in the product commutes with any other factor. The normalization factor is then readily calculated, * N = α 1/ 1 + Γ2α . It can be absorbed in the following rewriting, that may be the one directly familiar to the reader:

(178)

|ΨBCS =

"

Uα −

Vα cˆ†α↑ cˆ†α↓

|0,

α

where Uα and Vα are the real numbers (179)

Uα =

1 1 + Γ2α

−Γα Vα = . 1 + Γ2α

and

Note that they satisfy Uα2 + Vα2 = 1, and the minus sign in Vα was introduced to ensure consistency with coming notations. . 3 1.3. As a squeezed vacuum: Wick’s theorem applies. A third interesting rewriting of the BCS state can be obtained from the identity (180)

eθ(b

† †

c −cb)

|0 = cos θ|0 + sin θ b† c† |0,

where θ is a real number, b and c are two fermionic annihilation operators with standard anticommutation relations (3 ). Then |ΨBCS = S|0 where the unitary operator is (181)

S=

"

†

†

eθα (cˆα↑ cˆα↓ −h.c.)

α

(3 ) This identity can be proved by a direct expansion of the exponential in powers of θ, or by obtaining a diﬀerential equation satisﬁed by the left-hand side considered as a function of θ.

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Basic theory tools for degenerate Fermi gases

and the angles θα are such that (182)

Vα = − sin θα .

Uα = cos θα

For bosons, S would be interpreted as a squeezing operator [60]. The BCS state is therefore the equivalent for fermions of the squeezed vacuum for bosons. Calculating the expectation value in the BCS state of a product of operators cˆ and cˆ† is therefore equivalent to calculating the expectation value in the vacuum state of the product of the transformed operators S † cˆS and S † cˆ† S. These transformed operators have a linear expression in terms of the original cˆ and cˆ† , since one has (183)

S † cˆα↑ S = Uα cˆα↑ − Vα cˆ†α↓ ,

(184)

S † cˆ†α↓ S = Vα cˆα↑ + Uα cˆ†α↓ .

The linearity of these transformations is evident since S can be formally considered as the time evolution operator for a quadratic Hamiltonian, which generates indeed linear equations of motion of the creation and annihilation operators. As a consequence, Wick’s theorem can be applied to calculate expectation values in the BCS state, since it applies for the vacuum. . 3 1.4. Some basic properties of the BCS state. Before moving to the energy minimization within the BCS ansatz, we calculate some interesting quantities. Since Wick’s theorem applies, the expectation value of any quantity is a function of the only non-zero two-point averages: (185) (186)

Γ2α , 1 + Γ2α Γα c†α↑ cˆ†α↓ = , ˆ cα↓ cˆα↑ = ˆ 1 + Γ2α

ˆ c†α↑ cˆα↑ = ˆ c†α↓ cˆα↓ =

where the expectation values are taken in the BCS state. For the total number of particles in the BCS state, we obtain the mean value and the variance (187)

ˆ = N

(188)

ˆ = Var N

2Γ2 α , 2 1 + Γ α α α

4Γ2α . (1 + Γ2α )2

One then immediately obtains the inequality (189)

ˆ ≤ 2N ˆ , Var N

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Y. Castin

an inequality that becomes an equality in the limit where all Γα 1. This shows that, in the large N limit, the fact that the BCS state has not a well-deﬁned number of particles √ is not a problem for most practical purposes, since the relative ﬂuctuations are O(1/ N ). For the expectation value of the commutator of C and C † in the BCS state we obtain †

[C, C ] = 1 −

(190)

α

2Γ4α /(1 + Γ2α ) 2 . α Γα

We see that, in the limit where all Γ2α 1, this expectation value is close to unity. In this limit, one may consider that C † creates a bosonic pair: the BCS state becomes a Glauber coherent state for this bosonic pair, one obtains Poissonian ﬂuctuations in the number ˆ /2 of pairs, which explains the upper limit of eq. (189). This also shows that the BCS N ansatz will have no diﬃculty to predict the formation of a Bose-Einstein condensate of dimers in the a → 0+ limit [61]. A last interesting property is that the vacuum of an arbitrary quadratic Hamiltonian (quadratic in the quantum ﬁeld) is actually a BCS state. We defer the proof of this . statement to subsect. 3 3. A similar property is that the vacuum of a set of arbitrary operators bα obeying anticommutation relations is a BCS state: in other words, the BCS state is a quasi-particle vacuum. We refer to [62] for a detailed description of this aspect. . 3 2. Energy minimization within the BCS family. – We now proceed with the minimization of the energy of the lattice model Hamiltonian within the family of not necessarily normalized BCS states. We deﬁne (191)

E[Φ] =

ΨBCS |H|ΨBCS , ΨBCS |ΨBCS

where the BCS state |ΨBCS is of the form eq. (173) parametrized by the unnormalized pair wave function Φ = γφ and by the factor N now considered as an independent variable. If one performs a variation of Φ around a minimizer of E[Φ], Φ = Φ0 + δΦ, this induces a variation δ|ΨBCS of the BCS state around |Ψ0BCS , (192)

δ|ΨBCS =

ˆ 0BCS . l6 δΦ(r1 , r2 )ψˆ↑† (r1 )ψˆ↓† (r2 )|Ψ0BCS ≡ δ X|Ψ

r1 , r2

The corresponding ﬁrst-order variation of the energy function E[Φ] has to vanish for all possible values of δΦ, (193)

δE = (δΨBCS |)(H − E[Φ0 ])|Ψ0BCS + c.c. = 0

∀ δΦ,

where we assumed that the minimizer |Ψ0BCS is normalized to unity. The second step is to introduce the quadratic Hamiltonian deduced from the full Hamiltonian H by performing incomplete Wick’s contractions. This recipe has to be applied to the interaction term only, since the other terms of the Hamiltonian are quadratic.

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Basic theory tools for degenerate Fermi gases

The quartic on-site interaction is replaced by the quadratic expression: g0 ψˆ↑† ψˆ↓† ψˆ↓ ψˆ↑ → ψˆ↑† ψˆ↓† g0 ψˆ↓ ψˆ↑ + h.c.

(194)

+ ψˆ↑† ψˆ↑ g0 ψˆ↓† ψˆ↓ + ↑↔↓, where the expectation values are taken in the state |Ψ0BCS . The ﬁrst line of the righthand side consists in a pairing term, that creates/annihilates a pair of particles; it involves the following pairing ﬁeld, also called “gap” for historical reasons that will become clear: Δ(r ) ≡ g0 ψˆ↓ (r )ψˆ↑ (r ).

(195)

The second line of the right-hand side is simply the Hartree mean ﬁeld term: each spin-up particle sees a mean-ﬁeld potential which is g0 times the density of spin-down particle. Note that, in the lattice model, this mean-ﬁeld term does not diverge at the unitary limit, since g0 remains bounded in this limit, see eq. (155). The quadratic Hamiltonian associated by Wick’s contractions to the full Hamiltonian is the so-called BCS Hamiltonian, and it has the structure † (196) H≡ l3 ψˆσ† (r )hr,r ψˆσ (r ) + l3 ψˆ↑ (r )ψˆ↓† (r ) Δ(r ) + h.c. + Eadj . r, r ,σ

r

Here the matrix h is the sum of the one-body part of the Hamiltonian H and of the Hartree mean-ﬁeld terms, (197) hr,r = [kin]r,r + U (r ) − μ + g0 ψˆ↑† (r )ψˆ↑ (r ) δr,r , where [kin] is the matrix representing the kinetic-energy operator. We have assumed for simplicity that the one-body Hamiltonian is spin independent and that the two spin density proﬁles are identical, so that the matrix h does not depend on the spin state. Finally, the additive constant Eadj is adjusted to the value (198)

Eadj = −g0

l3 ψˆ↑† ψˆ↑ 2 + |ψˆ↓ ψˆ↑ |2

r

to ensure that the mean energy of H and of H coincide in the minimizer |Ψ0BCS , (199)

H = H.

ˆ is a quadratic function of Using Wick’s theorem and the fact that the operator δ X the quantum ﬁeld, one can then show the marvelous property (200)

(δΨBCS |)H|Ψ0BCS = (δΨBCS |)H|Ψ0BCS ,

whatever the variation δ|ΨBCS of the BCS state within the BCS family.

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Y. Castin

There is the following key consequence. Considering the quadratic Hamiltonian H as a given Hamiltonian, it turns out that the BCS state |Ψ0BCS , obeying eq. (193), also obeys the condition to be a stationary state of the energy functional (201)

E[Φ] =

ΨBCS |H|ΨBCS , ΨBCS |ΨBCS

noting that one has E[Φ0 ] = E[Φ0 ]. Condition (193) is thus equivalent to the condition (202)

δE = (δΨBCS |)(H − E[Φ0 ])|Ψ0BCS + c.c. = 0,

∀ δΦ.

That is, to minimize E[Φ] within the BCS family, we shall take the BCS state that minimizes the energy of H. We are left with the study of a quadratic Hamiltonian. As reminded to the reader in . subsect. 3 3, the quadratic Hamiltonian H may be diagonalized by a canonical transformation, the so-called Bogoliubov transformation. This allows to show that the ground . state |ground of H is in fact a BCS state (see subsect. 3 3), and to calculate the ex† pectation values ψˆσ ψˆσ ground and ψˆ↓ ψˆ↑ ground in |ground. Since |Ψ0BCS and |ground actually coincide, one is left with the self-consistency conditions: (203) (204)

ψˆσ† ψˆσ ground = ψˆσ† ψˆσ , ψˆ↓ ψˆ↑ ground = ψˆ↓ ψˆ↑ .

Since the expectation values in the ground state of H are non-linear functionals of the coeﬃcients ψˆσ† ψˆσ and ψˆ↓ ψˆ↑ of H, this constitutes a non-linear self-consistent problem, the so-called BCS equations for the density and the gap function. Another crucial consequence is to realize that one may write the self-consistency conditions taking an excited eigenstate of the quadratic BCS Hamiltonian H rather than the ground state (4 ): in this case, the variational BCS calculation is used to predict excited states of the gas! This shows that BCS theory immediately predicts elementary excitations of the gas. As we shall see in the thermodynamic limit, these excitations correspond to pair breaking and are characterized by an energy spectrum with a gap. . 3 3. Reminder on diagonalization of quadratic Hamiltonians. – We consider here a quadratic Hamiltonian of the form eq. (196), where, at this stage, h is an arbitrary (spin independent) hermitian matrix and Δ(r ) an arbitrary complex ﬁeld. We wish to put this Hamiltonian in canonical form by performing a Bogoliubov transformation. (4 ) One may wonder if condition eq. (202) really implies that |Ψ0BCS is an eigenstate of H. This is not evident a priori since the variation δ|ΨBCS does not explore the whole Hilbert space but only the manifold tangent to the BCS family in |Ψ0BCS . One can show however that the answer is yes: as sketched below eq. (251), (H − E[Φ0 ])|Ψ0BCS also belongs to this tangent manifold, so that one may choose δ|ΨBCS proportional to it.

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Basic theory tools for degenerate Fermi gases

The key property of a quadratic Hamiltonian is that it leads in Heisenberg picture to linear equations of motion. Collecting the ﬁeld operators and their Hermitian conjugate at all points of the lattice in big vectors, we get (205)

i¯ h∂t

ψˆσ † ˆ ψ−σ

σ

=L

ψˆσ † ˆ ψ−σ

,

where −σ stands for the spin component opposite to σ and where we have introduced two matrices

h diag(Δ) h diag(−Δ) ↑ ↓ (206) L = and L = . diag(Δ∗ ) diag(−Δ∗ ) −h∗ −h∗ These two matrices are Hermitian: they can be diagonalized in an orthonormal basis. Furthermore they obey the following symmetry property: if the vector (u, v) is an eigenstate of L↑ with the energy , then – the vector (−v ∗ , u∗ ) is an eigenvector of L↑ , with the eigenvalue − , – the vector (u, −v) is an eigenstate of L↓ , with the eigenvalue , – the vector (v ∗ , u∗ ) is an eigenvector of L↓ , with the eigenvalue − . Assuming for simplicity that all the eigenenergies are non-zero, we can collect the † eigenvectors in pairs of opposite eigenenergies. Expanding (ψˆσ , ψˆ−σ ) in the eigenbasis σ ↑ of L , and expressing the negative energy modes of L and all the eigenmodes of L↓ in terms of the positive energy modes of L↑ , noted as (uα , vα ), we obtain (207)

ψˆ↑ (r ) =

(208)

ψˆ↓ (r ) =

bα↑ uα (r ) − b†α↓ vα∗ (r ),

α

bα↓ uα (r ) + b†α↑ vα∗ (r ),

α

where the positive energy eigenvectors of L↑ are normalized in a way mimicking the continuous space limit, with Dirac’s notation r |uα = uα (r ): (209)

uα |uβ + vα |vβ = l3

u∗α (r )uβ (r ) + vα∗ (r )vβ (r ) = δαβ .

r

The coeﬃcients in the expansion of the ﬁeld operators are themselves operators that are easy to express using the orthonormal nature of the eigenbasis of Lσ : (210)

bα↑ = l3

(211)

3

u∗α (r )ψˆ↑ (r ) + vα∗ (r )ψˆ↓† (r ),

r

bα↓ = l

r

u∗α (r )ψˆ↓ (r ) − vα∗ (r )ψˆ↑† (r ).

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Y. Castin

Using eq. (209) again, one can check that the b’s and their Hermitian conjugates obey the standard fermionic anticommutation relations. The last step is to express the Hamiltonian in normal form, using the bα ’s. One ﬁrst realizes that the quadratic Hamiltonian can be expressed as a quadratic form deﬁned by the matrices Lσ , 1 3 ˆ† ˆ σ ψˆσ , (212) H = Tr h + Eadj + l ψσ , ψ−σ L 2 ψˆ† −σ

σ

where the extra constant term, given by the trace of the matrix h, originates from the anticommutator of ψˆσ† with ψˆσ . Inserting the modal decomposition eqs. (207), (208) and using the fact that (uα , vα ) is an eigenvector of L↑ , etc., we obtain the canonical form (213)

H = E0 +

α b†ασ bασ ,

α,σ

where we recall that all the α > 0 by construction. The immediate expression for E0 is E0 = Tr h + Eadj − α α ; it involves two quantities that are not well behaved in the zero lattice spacing limit. Expressing the matrix L↑ as a sum of ± α times dyadics involving the eigenvectors as ket and bra, and projecting over the upper left block leads to α [uα (r )u∗α (r ) − vα∗ (r )vα (r )] , (214) hr,r = l3 α

where we used the orthonormal nature of the eigenbasis of L↑ , as deﬁned in eq. (209). Taking the trace of this expression in the spatial basis of the lattice leads to the more convenient form for E0 : α vα |vα , (215) E0 = Eadj − 2 α

where Eadj is given by eq. (198). The canonical form eq. (213), with all the α > 0 by construction, immediately shows that the ground state |Ψ0 of H is the vacuum of all the bα ’s, with an eigenenergy E0 . Please remember that we are dealing here with the grand-canonical Hamiltonian; this is why E0 < 0 for the ideal gas g0 = 0. To show that this ground state is a BCS state, we take for |Ψ0 an ansatz of the BCS-type eq. (173) to solve the equations (216)

bασ |Ψ0 = 0

for all mode index α and spin component σ =↑, ↓. The commutator of a bασ with the pair creation operator C † is a linear combination of the ψˆ† ’s so it commutes with C † . As a consequence, (217)

†

†

[bασ , eγC ] = γeγC [bασ , C † ].

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Since exp[γC † ] is an invertible operator, eq. (216) reduces to (218)

bασ + γ[bασ , C † ] |0 = 0.

From expressions (210), (211) we calculate the commutator with C † and obtain the linear system (219)

vα∗ (r ) + γ

l3 φ(r , r )u∗α (r ) = 0,

r

(220)

vα∗ (r ) + γ

l3 φ(r, r )u∗α (r ) = 0.

r

The solution may be written in matrix form as follows. Since the lattice has discrete positions, we consider φ(r, r ) as the element of a square matrix [φ] on row r and column r . Similarly we form the square matrix [U ] (respectively [V ]) such that the element on row r and column α is uα (r ) (respectively vα (r )). Then one has the explicit writing (221)

γ [φ] = γ t [φ] = −l−3 ([U ]† )−1 [V ]† ,

where t M is the transposed matrix of a matrix M . Since the set of all the eigenvectors of L↑ forms an orthonormal basis, one has l3 (U † U + V † V ) = Id and t U V = t V U , where Id is the identity matrix. This ensures that the solution [φ] is indeed a symmetric matrix (222)

φ(r1 , r2 ) = φ(r2 , r1 ).

This symmetry property could be expected from the fact that the quadratic Hamiltonian H is invariant by an arbitrary rotation R of the spin degree of freedom, and that its ground state is not degenerate since we have assumed α strictly positive. The condition that C † = RC † R† for all spin rotations indeed imposes that φ is a symmetric function of r1 and r2 . Using Wick’s theorem one can easily calculate expectation values of observables in |Ψ0 using the modal decompositions eqs. (207), (208). Two important examples are the mean density and the anomalous average: (223) |vα (r )|2 , ρ↑ (r ) = ρ↓ (r ) = α

(224)

ψˆ↓ (r )ψˆ↑ (r ) = −

uα (r )vα∗ (r ).

α

. 3 4. Summary of BCS results for the homogeneous system . 3 4.1. Gap equation in the thermodynamical limit. In the case of a spatially homogeneous system (cubic box with periodic boundary conditions) explicit predictions are easily extracted from the BCS theory. For a spatially uniform gap parameter Δ, assumed to be real non-negative, and for a uniform density proﬁle ρ↑ = ρ↓ the spectral decomposition

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Y. Castin

of L↑ is readily obtained: from the translational invariance one seeks eigenvectors of the form

(uk (r ), vk (r )) = L−3/2 eik·r (Uk , Vk ).

(225)

Restricting to the positive part of the spectrum, one ﬁnds the eigenenergies (226)

k =

1/2 / /

2 / / ¯h2 k 2 ¯ 2 k2 h 2 −μ ˜ +Δ −μ ˜ + iΔ// , = // 2m 2m

where μ ˜ = μ − g0 ρ↑ is the chemical potential minus the Hartree mean-ﬁeld term. The amplitudes Uk , Vk , chosen to be real, are normalized as Uk2 + V 2 = 1 and are given by (Uk + iVk )2 =

(227)

h ¯ 2 k2 2m

−μ ˜ + iΔ . k

. To make the link with the quantities Γα of subsubsect. 3 1.2, we take for |Aα = |Ak the plane wave of the wave vector k and for |Bα = |Bk the plane wave of the wave vector −k. This gives Γk = −

(228)

Vk Δ = − h¯ 2 k2 , Uk − μ ˜ + k 2m

and a pair wave function (5 ) (229)

γφ(r1 , r2 ) =

1 Γk eik·(r1 −r2 ) . L3 k∈D

Remember that the regime where all the Γ2k 1 corresponds to the case where the BCS state is a condensate of bosonic pairs. Taking the real and imaginary parts of eq. (227), and going directly to the thermodynamic limit, one obtains from eqs. (223), (224) h ¯ 2 k2 − μ ˜ d3 k ρ= (230) 1 − 2m , 3 k D (2π) d3 k Δ Δ = −g0 (231) , 3 D (2π) 2 k where ρ is the total density. This makes the self-consistent character of the conditions (203), (204) explicit. The last equation (231) is called the gap equation. (5 ) This expression coincides with eq. (221), since one may check that ([U ]−1 ) k, r =

e−ik· r l3 /(L3/2 Uk ).

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Basic theory tools for degenerate Fermi gases

. 3 4.2. In the limit of a vanishing lattice spacing. An important point is to discuss the dependence of these predictions on the lattice spacing l, since the relevant regime of a continuous gas is described by the lattice model only in the l → 0 limit. The integral giving ρ converges at large k, since the integrand is O(1/k 4 ), so that one may directly set l → 0 in eq. (230). On the contrary the integral in eq. (231) has an ultraviolet divergence when integrated over the whole momentum space. However g0 tends to zero in the l → 0 limit, compensating exactly this divergence. Dividing eq. (231) by g0 and expressing 1/g0 from eq. (154) leads to a more convenient expression for the gap equation, in which one may take directly l → 0: Δ =Δ g

(232)

D

m d3 k 1 − . (2π)3 h 2 k ¯ 2 k2

Furthermore, we note that the BCS theory becomes simpler when l → 0: since the coupling constant g0 vanishes in this limit, the Hartree mean-ﬁeld term also vanishes, so that μ ˜ → μ and the matrix h in the quadratic Hamiltonian H reduces to the matrix of the kinetic energy minus μ. The a priori unknown coeﬃcients of H thus reduce to the anomalous average ψˆ↓ ψˆ↑ , and eq. (204) remains as the only self-consistent equation, the other one eq. (203) giving explicitly the density as a function of μ and of the anomalous average. In general, one has in practice to use the BCS theory in this l → 0 limit, to be in the . continuous gas regime. As we shall see in subsubsect. 3 4.4, a notable exception is the weakly attractive regime kF a → 0− . . 3 4.3. BCS prediction for an energy gap. As sketched in the last paragraph of sub. sect. 3 2, the excited states of the BCS Hamiltonian H correspond to a variational BCS approximation for excited states of the full Hamiltonian of the gas. Here we come back to this statement, and show how to derive one of the most celebrated predictions of BCS theory, the minimum energy required to break a pair. . First we revisit brieﬂy subsect. 3 2: in the process of energy minimization within the BCS family of states, we ended up looking for a BCS state that is the ground state of a quadratic Hamiltonian of the form eq. (196). Taking here for simplicity the continuous limit l → 0, for a spatially homogeneous solution, we can omit the Hartree mean-ﬁeld term and perform a variational calculation that immediately restricts to BCS states being the ground state of the following Hamiltonian: (233)

Hλ =

k∈D σ

(Ek − μ) a† akσ + l3 kσ

† λψˆ↑ (r )ψˆ↓† (r ) + h.c. , r

¯ 2 k 2 /2m and the real non-negative number λ is now the only variational where Ek = h parameter. The expectation value of the full Hamiltonian H in the ground state |Ψgλ of

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Y. Castin

. Hλ is readily evaluated from subsect. 3 3 and from eqs. (226), (227),

2 Ek − μ (Ek − μ) 1 − (234) Hgλ = + g0 L3 ψˆ↓ ψˆ↑ gλ , |Ek − μ + iλ| k∈D

with the anomalous average ψˆ↓ ψˆ↑ gλ = −L−3 k∈D λ/(2|Ek − μ + iλ|). In these expressions, the apex g stands for “ground state”. It remains to require that the ﬁrst-order derivative of Hgλ with respect to λ vanishes, which reduces, after explicit calculation of the derivative, to the following equation satisﬁed by the optimal value λg of λ: g0 ψˆ↓ ψˆ↑ gλg = λg .

(235)

One has recovered, in a slightly diﬀerent way, the self-consistency condition (204), and one ﬁnds that λg satisﬁes the gap equation (with μ ˜ = μ). This optimal λg was called earlier Δ. We can now perform a similar variational calculation, taking as a variational ansatz an excited state of Hλ . To preserve the symmetry between the two spin components ↑ and ↓, we consider the excited state |Ψeλ = bq† ↑ b†−q↓ |Ψgλ ,

(236)

where q is a ﬁxed wave vector. Two Bogoliubov modes of the Hamiltonian Hλ are now in the excited state, with occupation number one, rather than being in the vacuum state of the corresponding b’s. The mean energy of H in this excited state is simply, in the thermodynamic limit, (237)

Heλ = Hgλ +

2(Eq − μ)2 + 2λg0 ψˆ↓ ψˆ↑ gλ + O(L−3 ). |Eq − μ + iλ|

It diﬀers from Hgλ by a term O(1), whereas the full energy is O(L3 ), so that the optimal value λe of λ for the minimization of Heλ diﬀers from λg = Δ by a small term δλ. We can then simply expand eq. (237) in powers of δλ. The key point now is that the ground-state energy varies only to second order in δλ since it is minimal in Δ; since the second-order derivative of Hgλ is O(L3 ), δλ is O(1/L3 ) and the contribution to the total energy of the corresponding second-order variation of Hgλ is O(1/L3 ) and negligible. In the remaining terms of eq. (237) we can simply set λ = Δ. Using eq. (235), we are left with (238)

Heλe = HgΔ + 2 q + O(1/L3 ),

where q is the BCS spectrum eq. (226). To summarize, one considers excited states of H as variational states, with a small number of Bogoliubov excitations. One should then in principle solve again the selfconsistency conditions (203), (204), but in the thermodynamic limit the excitation of a

Basic theory tools for degenerate Fermi gases

337

few Bogoliubov modes has a small eﬀect on the density and gap parameter, an eﬀect that we have shown to be negligible on the energy. One can then directly consider that the elementary excitations of the quadratic Hamiltonian H associated to the ground energy BCS state actually give the energy of the elementary excitations of BCS theory (6 ). When Δ > 0, we see that the minimal value of the BCS spectrum k with respect to k is non-zero: it has a gap Egap = mink k . When μ ˜ > 0, which contains the usual regime of the BCS theory, the regime of condensation of Cooper pairs (see below), the gap is Egap = Δ, hence the name of Δ. When μ ˜ < 0, which contains the regime of a

1/2 2 ˜ + Δ2 . Finally, we get Bose-Einstein condensate of dimers, the gap is Egap = μ from BCS theory a prediction for the miminal energy 2Egap required to break a pair, that is to get from a condensate of N/2 pairs, a condensate of N/2 − 1 pairs and two unpaired atoms [63]. . 3 4.4. BCS predictions in limiting cases. We shall not discuss the full solution of the BCS equations but we brieﬂy review simple limiting cases. We introduce the Fermi momentum kF of the ideal Fermi gas, such that kF3 /(6π 2 ) = ρ↑ = ρ↓ . Limit kF a → 0− : the gap parameter Δ tends exponentially to zero in this limit, as we shall see. For such a small value of Δ, one may set Δ = 0 in eq. (230), to obtain that μ ˜ is the Fermi energy of the ideal gas h ¯ 2 kF2 /2m, so that (239)

μ

¯2 h (6π 2 ρ↑ )2/3 + g0 ρ↑ . 2m

If one takes the mathematical limit l → 0 the Hartree mean-ﬁeld term disappears since g0 → 0 in this limit, as already mentioned. However in the present weakly attractive limit one can choose |a| l kF−1 so that the continuous space limit and the zero-range condition (116) for the lattice model are obtained while g0 g, see eq. (155). The on-site interaction potential is then treatable in the Born approximation, which makes the BCS approach more accurate. One then indeed recovers the ﬁrst term gρ↑ of a systematic expansion of μ in powers of kF a [28]. Now turning to the gap equation, one ﬁnds a gap parameter [47] (240)

˜ e−π/(2kF |a|) . Δ 8e−2 μ

(6 ) The reasoning was done for a ﬁxed chemical potential and for the excitation energy of the grand-canonical Hamiltonian H. What is the corresponding excitation energy of the canonical Hamiltonian for a ﬁxed number N of particles? We have a canonical energy in the ground state g e = Hg (μ) + μN , and in the excited state Ecan = He (μe ) + μe N . Here the chemical Ecan 3 potentials μ and μe diﬀer by O(1/L ) for a few Bogoliubov excitations, so that one may expand e g d −Ecan = He (μ)− Hg (μ)+δμ[N + dμ

Hg (μ)]+ to ﬁrst order in δμ = μe −μ. This leads to Ecan 3 O(1/L ). Now the quantity in between brackets vanishes, a standard thermodynamic result. One can also use (d/dμ)[ Hgλ=Δ(μ) ] = (∂μ Hgλ )λ=Δ(μ) and calculate the partial derivative of eq. (234) with respect to μ for a ﬁxed λ. Setting λ = Δ in the resulting expression leads to the standard thermodynamic result.

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Y. Castin

Since μ ˜=h ¯ 2 kF2 /2m > 0 in this weakly attractive limit, |Γk | can assume extremely large values, for h ¯ 2 k 2 /2m < μ ˜: the pairs that are condensed are not bosons. Note the nonanalytic dependence of the gap on the small parameter kF |a|, which indicates that the BCS state in the thermodynamic limit cannot be obtained by a perturbative treatment of the interaction potential. This non-analytic dependence can be readily seen from √ eq. (232), whose integrand diverges in k = 2m˜ μ/¯h, in the limit Δ → 0. Replacing the k integration variable by the ideal gas mode energy E = h ¯ 2 k 2 /2m, and approximating the ideal gas density of modes by a constant in the energy interval of half-width δ μ ˜ around μ ˜, one gets a contribution (241)

μ ˜ +δ

μ ˜ −δ

dE Δ2

Δ→0

+ (E −

μ ˜)2

∼ 2 ln

2δ . Δ

The same technique applied over, e.g., the interval E ∈ [0, 4˜ μ] leads to eq. (240), when one also takes into account (to lowest order in Δ) the diﬀerence between the exact integral in the gap equation and the approximate one. Limit kF a → 0+ : the coeﬃcients Γk tend uniformly to zero in this limit, because μ ˜ < 0 and Δ |˜ μ|, so that the pair creation operator C † obeys approximately bosonic commutation relations. This was expected physically, since the ground state of the gas is a condensate of almost bosonic dimers. Since two atoms in a dimer are at a relative distance ∼ a, one should ensure l a in the lattice model, so that one takes the mathematical limit l → 0, which implies μ ˜ → μ. Let us check that the BCS theory correctly predicts a condensate of such dimers. We ﬁrst simplify the gap equation (232) by using k −μ + h ¯ 2 k 2 /2m. After division by Δ, we get an equation for the chemical potential

(242)

1 − g

d3 k (2π)3

m − 2 2 − 2μ ¯h k

1 h ¯ 2 k2 m

which leads to (243)

μ−

¯2 h . 2ma2

This is minus half the binding energy of a dimer, exactly what was expected (keep in mind that μN = μmol Nmol , where Nmol is the total number of dimers and is equal to N/2, so that the molecular chemical potential μmol is twice the atomic one). The next step is to expand the integrand of eq. (230) to leading order in Δ to calculate the gap parameter [47]: (244)

2 ¯h2 ¯h2 Δ √ (kF a)3/2

. ma2 ma2 3π

Basic theory tools for degenerate Fermi gases

339

Note that in this molecular BEC regime, Δ is not proportional to the energy required to break a pair. One sees from eq. (226) that the gap in k is |μ|, since μ is negative and much larger in absolute value than Δ. The energy to break a pair is then 2|μ|, which is indeed the binding energy of a dimer. Finally, by performing the Fourier transform in eq. (229), approximating Vk /Uk to leading order in Δ, one obtains the pair wave function (245)

φ(r1 − r2 )

1 φ0 (|r1 − r2 |), L3/2

where φ0 is the wave function of the bound state of two atoms given by eq. (146), and where we used γ 2 = k (Vk /Uk )2 . Limit kF |a| = +∞: the numerical solution of the gap equation (232) and of the density equation (230) in the limit l → 0 gives the BCS estimate of the numerical coeﬃcient η of eq. (168) for the ground branch. This estimate is an upper bound [64]: (246)

η ≤ ηBCS = 0.5906 . . .

A much better estimate was obtained by approximate (ﬁxed node) Monte Carlo calculations [41, 42], η 0.4, a value conﬁrmed by recent exact quantum Monte Carlo calculations [32]. Early measurements of η were in contradiction with these theoretical values [43], but more precise measurement performed in Innsbruck [44], in Paris [37] and at Rice University [45] are consistent with them. . 3 5. Derivation of superﬂuid hydrodynamic equations from BCS theory. – A key point of the BCS theory is to predict a spectrum of elementary excitations having a gap, see eq. (226). One could then be tempted to infer strong predictions on the thermodynamic properties, e.g., that the entropy obeys an activation law O(e−Egap /kB T ) at low temperature, where Egap is the energy gap. However, in the BEC limit, where the gas is simply a Bose-Einstein condensate of bosonic dimers at low temperature, this prediction of an activation law is obviously wrong, since one knows that the relevant excitation spectrum is the Bogoliubov spectrum, which has no gap. In reality, the spatially homogeneous gas, whatever the considered regime (BEC or BCS), is expected to have a branch of collective excitations with no gap, behaving as sound waves at low momenta. These collective excitations correspond to coherent oscillations of the density of the pairs, which are not gapped, to be distinguished from the pair breaking excitations which are gapped. The key point brieﬂy addressed in this section is that the time-dependent BCS theory contains such a branch of collective, nonpair-breaking oscillations [65]. This is similar to what happens for the weakly interacting Bose gas: the Bogoliubov spectrum is obtained from a linearization of the time dependent Gross-Pitaevskii equation. Here, rather than performing an exact linearization of the time-dependent BCS equations, which leads to the so-called RPA approach [62,66-70], we go through a sequence of simple approximations allowing one to derive superﬂuid hydrodynamic equations from the time-dependent BCS theory. This has several advantages: it is more physical, it easily

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Y. Castin

applies to harmonic trapping (7 ), and it is applicable to the non-linear time-dependent regime, to study analytically low-energy collective excitations of a trapped superﬂuid Fermi gas [71, 72], but also the expansion of the gas after the trap was switched oﬀ [73] and the response of the gas to a rotation of the harmonic trap [74, 75] even in the nonlinear regime [75]. Coming back to thermodynamical aspects, one may fear that the straightforward ﬁnite-temperature extension of the BCS theory, with a variational density operator which is Gaussian in the ﬁeld operators [62], is not able to calculate the critical temperature Tc with a good accuracy, because it does not correctly include the collective excitations. It turns out that this simple ﬁnite temperature BCS theory correctly gives Tc in the BCS limit kF a → 0− within a numerical factor [76], but is indeed totally wrong in the BEC limit kF a → 0+ where it does not reproduce at all Einstein’s prediction for the critical temperature of the ideal Bose gas. Reﬁnements of the BCS theory have been developed to recover Einstein’s prediction and to obtain a calculation of Tc approximately valid over the whole range of values of kF a [46, 47]. . 3 5.1. Time-dependent BCS theory. This is a direct generalization of the static case of . subsect. 3 2. The exact N -body Schr¨ odinger equation can be obtained from extremalisation over the time-dependent state vector |Ψ(t) of the following action: (247)

tf

S=

dt ti

i¯ h d Ψ(t)| |Ψ(t) − c.c. − Ψ(t)|H(t)|Ψ(t) 2 dt

for ﬁxed initial |Ψ(ti ) and ﬁnal |Ψ(tf ) values of the state vector. For an arbitrary variation |δΨ(t) of the ket |Ψ(t), subject to the condition |δΨ(ti ) = |δΨ(tf ) = 0,

(248)

we calculate the ﬁrst-order variation of the action, (249)

tf

δS =

dt ti

d δΨ(t)| i¯h − H(t) |Ψ(t) + c.c. . dt

We have integrated by parts over t to get rid of (d/dt)|δΨ and we have used eq. (248) to show that the fully integrated term vanishes. The condition that δS vanishes for all ket variations indeed leads to Schr¨ odinger’s equation. For time-dependent BCS theory, one forces the ket to be of the BCS form eq. (173), |Ψ = |ΨBCS . The variation of the ket is performed within the BCS family, by a variation of Φ = γφ and of N , so that |δΨ = δ(|ΨBCS ). We can then introduce the (7 ) The RPA approach can be applied in an isotropic harmonic trap, with some non-negligible numerical eﬀort, see [67].

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Basic theory tools for degenerate Fermi gases

. now time-dependent quadratic Hamiltonian H(t) constructed in subsect. 3 2 such that, for all variations within BCS family, (250)

(δΨBCS (t)|)H(t)|ΨBCS (t) = (δΨBCS (t)|)H(t)|ΨBCS (t).

One then sees that δS in eq. (249) identically vanishes if the ket evolves with the quadratic Hamiltonian, (251)

i¯ h

d |ΨBCS (t) = H(t)|ΨBCS (t). dt

It remains to check that the ket evolving this way remains of the BCS form. To this end, one shows, with the reasoning having led to eq. (217), that ψˆσ |ΨBCS (t) is equal to a linear combination of the ψˆσ† (r ) acting on |ΨBCS (t). One then sees that H|ΨBCS is of the form “constant plus polynomial of degree exactly two in ψˆ† ” acting on |ΨBCS (t), which can be reproduced by an appropriate time dependence of Φ. In practice, the equation of motion for Φ is not useful. One rather moves to the Heisenberg picture with respect to the time-dependent Hamiltonian H(t), as we did in . subsect. 3 3. Since this Hamiltonian is quadratic, the equations of motion for the ﬁelds are linear, of the form eq. (205), where L↑ is now time dependent. Since these equations are linear, we can solve them by evolving in eqs. (207), (208) the mode functions (uα , vα ) while keeping constant the quantum coeﬃcients ˆbασ where σ =↑ or ↓: (252)

i¯ h∂t

uα vα

uα = L (t) . vα ↑

These equations on the mode functions are eﬀectively non-linear since coeﬃcients of L↑ , involving the mean density ψˆ↑† ψˆ↑ and the anomalous average ψˆ↓ ψˆ↑ , depend on the mode functions; since the ˆbασ are constants of motion, this dependence is still given by eqs. (223), (224), now taking the time-dependent uα ’s and vα ’s. . 3 5.2. Semi-classical approximation. The physical situation that we have in mind here is a gas in the BCS regime (a < 0, μ > 0) in a time-dependent harmonic trap. The trap may be rotating, with an angular velocity Ω(t), in which case one moves to the rotating frame to eliminate the time dependence of the trapping potential due to rotation; this in the one-body Hamiltonian, where L is the introduces an additional term −Ω(t) ·L angular-momentum operator of a particle. In this case one may expect that quantum vortices form [77] for a high enough rotation frequency; the semi-classical approximation that we present is however restricted to a vortex-free gas. We wish to treat the equation of motion of (uα , vα ) in the semi-classical approximation. The general validity condition of a semi-classical approximation is that the applied potentials vary slowly over the coherence lengths of the gas, a coherence length being the typical width of a correlation function of the gas.

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A ﬁrst correlation function here is ψ↑† (r )ψ↑ (r ). Using the homogeneous gas expression of this function, one sees that it is the Fourier transform of the function Vk2 of width the Fermi momentum, Δk = kF , where ¯h2 kF2 /2m = μ. The associated coherence length is 1/Δk = kF−1 . A second correlation function is ψˆ↓ (r )ψˆ↑ (r ). For the homogeneous gas this is the Fourier transform of the function Uk Vk , which according to eq. (227) has a momentum width Δk = m|Δ|/¯ h2 kF . The associated coherence length 1/Δk corresponds in BCS theory to the length of a Cooper pair, (253)

lBCS =

¯ 2 kF h . m|Δ|

Since Δ < μ in the BCS regime, we keep lBCS as the relevant coherence length. A ﬁrst typical length scale of variation of the matrix elements in eq. (252) originates from the position dependence of |Δ|: in the absence of rotation and for an isotropic trap, this is expected to be the Thomas-Fermi radius RTF of the gas, deﬁned as μ = 2 mω 2 RTF /2, where ω is the atomic oscillation frequency. This assumes that the scale of variation of the modulus of the gap is the same as the one of the density, a point conﬁrmed in the adiabatic approximation to come. The condition that the mean-ﬁeld Hartree term and the harmonic potential have a weak relative variation over lBCS for typical values of the position also leads to the condition (254)

lBCS RTF .

For an isotropic harmonic trap this is equivalent to the condition (255)

|Δ| ¯hω.

In the general time-dependent case, however, this is not the whole story, since the phase of Δ can also become position dependent. In the rotating case for example, with a rotation frequency of the order of ω, the phase of Δ may vary as mωxy/¯h; this introduces a local wave vector mωRTF /¯h kF , making a semi-classical approximation hopeless. We eliminate this problem with a gauge transform of the u’s and v’s [75]: (256)

u ˜α (r, t) ≡ uα (r, t)e−iS(r,t)/¯h ,

(257)

v˜α (r, t) ≡ vα (r, t)e+iS(r,t)/¯h ,

where S is deﬁned as (258)

Δ(r, t) = |Δ(r, t)| e2iS(r,t)/¯h .

The time-dependent BCS equations (252) are modiﬁed, Δ being replaced by |Δ| and the single-particle Hamiltonian h being replaced by the gauge transformed Hamiltonian (259)

˜ = e−iS/¯h he+iS/¯h + ∂t S. h

343

Basic theory tools for degenerate Fermi gases

An eﬃcient frame to perform a semi-classical approximation in a systematic way for the evolution of a wave in a potential is to use the Wigner representation [6] of the density operator of the wave. Here the wave is represented by the two-component spinor (˜ uα , v˜α ) so that we introduce the corresponding density operator (260)

σ ˆ=

σ ˆ11 σ ˆ21

σ ˆ12 σ ˆ22

≡

|˜ uα | uα ˜ α

|˜ vα ˜ uα |

|˜ uα ˜ vα | . |˜ vα ˜ vα |

Note that the matrix elements of σ ˆ in position space are related to the previously mentioned correlation functions of the gas, up to the gauge transform, which makes the discussion consistent: (261)

|r |ˆ σ22 |r | = |ψˆ↑† (r )ψˆ↑ (r )|,

(262)

|r |ˆ σ12 |r | = |ψˆ↓ (r )ψˆ↑ (r )|.

We introduce the Wigner representation of σ ˆ [6], assuming for simplicity a continuous position space: (263)

W (r, p, t) = Wigner{ˆ σ} ≡

d3 x r − x/2|ˆ σ |r + x/2eip·x/¯h . (2π¯h)3

Since σ ˆ is the density operator of a two-component spinor, the Wigner distribution W is a two-by-two matrix. Within this representation, the key quantities of BCS theory, the total density, the modulus of the gap function and the total matter current have the following expressions in the general case of a rotating frame: (264)

ρ(r, t) = 2

(265)

|Δ|(r, t) = −g0

h ¯D

d3 p W22 (r, p, t),

(266)

h ¯D

d3 p W12 (r, p, t),

2 j (r, t) = ρ v (r, t) − Ω(t) × r − d3 p p W22 (r, p, t), m h¯ D

is the angular velocity of the rotating frame, and the so-called velocity ﬁeld is where Ω deﬁned as (267)

v (r, t) ≡

S(r, t) grad . m

We have introduced here the total matter current j(r, t), that obeys by construction (268)

∂t ρ + divj = 0.

344

Y. Castin

In the rotating frame, in a many-body state invariant by exchange of the spin states ↑ and ↓, it is very generally given by (269)

j(r, t) =

¯ ˆ† h ψˆ↑ (r, t) − c.c. − ρ(r, t) Ω(t) × r. ψ↑ (r, t)grad im

Within BCS theory, these last two relations still hold [62] and lead to eq. (266). The semi-classical expansion then consists, e.g., in (270)

i¯h ˆ Wigner{V (r )ˆ σ } = V (r ) + ∂r V · ∂p + . . . W (r, p, t), 2

where V is a position-dependent potential. The successive terms in this expansion we call zeroth order, ﬁrst order, etc., in the semi-classical approximation. Assuming that the momentum derivative of W is W/Δp, where Δp is the momentum width of W , we recover the fact that the ﬁrst-order term is small as compared to the zeroth-order one if the variation of V over the coherence length ¯h/Δp, which is of the order of the spatial derivative of V times the coherence length, is small as compared to V . Here we shall need only the equations of motion of the Wigner distribution W up to zeroth order in the semi-classical approximation: (271)

i¯ h∂t W (r, p, t) L↑0 (r, p, t), W (r, p, t) ,

where the two-by-two matrix L↑0 is equal to (272)

L↑0 (r, p, t)

=

p2 2m

− μeﬀ (r, t) |Δ|(r, t) . p2 |Δ|(r, t) − 2m + μeﬀ (r, t)

We have introduced the position- and time-dependent function, (273)

1 μeﬀ (r, t) ≡ μ − U (r, t) − mv 2 (r, t) + mv (r, t) · (Ω(t) × r ) − ∂t S(r, t), 2

that may be called eﬀective chemical potential for reasons that will become clear later. At time t = 0, the gas is at zero temperature. By introducing the spectral decomposition of L↑ (t = 0) one can then check that (274)

σ ˆ (t = 0) = θ[L↑ (t = 0)],

where θ(x) is the Heaviside function. Since L↑0 (t = 0) is the classical limit of the operator L↑ (t = 0), the leading-order semi-classical approximation for the corresponding Wigner function is, in a standard way, given by (275)

W (r, p, t = 0)

1 θ[L↑0 (r, p, t = 0)]. (2π¯h)3

345

Basic theory tools for degenerate Fermi gases

This implies that the 2 × 2 matrix W is proportional to a pure spin-1/2 state |ψψ| with |ψ(r, p, t = 0) =

(276)

U0 (r, p ) , V0 (r, p )

where (U0 , V0 ) is the eigenvector of L↑0 (r, p, t = 0) of positive energy and normalized to unity. At time t, according to the zeroth-order evolution eq. (271), each two-by-two matrix W remains a pure state, with components U and V solving (277)

i¯ h∂ t

U (r, p, t) V (r, p, t)

=

L↑0 (r, p, t)

U (r, p, t) . V (r, p, t)

. 3 5.3. Adiabatic approximation. We then introduce the so-called adiabatic approximation: the vector (U, V ), being initially an eigenstate of L↑0 (r, p, t = 0), will be an instantaneous eigenvector of L↑0 (r, p, t) at all later times t. This approximation holds under the validity condition of the adiabaticity theorem [78], discussed for our speciﬁc case in [75], generically requiring that the energy diﬀerence between the two eigenvalues of L↑0 (r, p, t) be large enough. As this energy diﬀerence can be as small as the gap parameter, this imposes a minimal value to the gap, not necessarily coinciding with the one of eq. (255), as shown in [75]. In this adiabatic approximation, one can take (278)

W (r, p, t)

1 1 θ[L↑0 (r, p, t)] = | + (r, p, t)+(r, p, t)|, (2π¯ h)3 (2π¯h)3

where | + (r, p, t), having real components (Uinst , Vinst ), is the instantaneous eigenvector with positive eigenvalue of the matrix L↑0 deﬁned in eq. (272). (Uinst , Vinst ) are simply the amplitudes on the plane wave exp[i p · r/¯h] of the BCS mode functions of a spatially uniform BCS gas of chemical potential μeﬀ and of gap parameter |Δ(r, t)|. Using eqs. (264) and (265) with the approximate Wigner distribution eq. (278), one further ﬁnds that this ﬁctitious spatially uniform BCS gas is at equilibrium at zero temperature so that the zero-temperature equations of state may be used, relating the chemical potential to the density, μ0 [ρ], and the gap to the density, Δ0 [ρ], where the functions μ0 and Δ0 may be calculated from the spatially homogeneous BCS theory. In particular, the equation of state relating the chemical potential to the density leads to (279)

μeﬀ (r, t) = μ0 [ρ(r, t)]

which leads, together with eq. (273), to one of the time-dependent hydrodynamic equations, the Euler-type one. Also, Uinst and Vinst are even functions of p, so that the integral on the right-hand side of eq. (266) vanishes and eq. (268) reduces to the hydrodynamic continuity equation in a rotating frame, (280)

8 9 ∂t ρ(r, t) + div ρ(r, t) v (r, t) − Ω(t) × r = 0.

346

Y. Castin

Under the adiabatic approximation, the superﬂuid hydrodynamic equations are thus derived from BCS theory, remarkably without having to postulate the expression of the superﬂuid current in terms of the gradient of the phase of the order parameter. We refer to [75] for a discussion of the validity condition of the adiabatic approximation. The presence of superﬂuid hydrodynamic equations allows to conclude in the existence of collective modes in the BCS theory, in a transparent way, by a simple linearization around steady state. Although we have considered here a trapped system, we note that the formalism may also be developed in the spatially homogeneous case. To create a sound wave of wave vector q, one may apply an external potential varying with spatial harmonics e±iq·r , e.g., with a Bragg pulse [79, 80]. The semi-classical approximation holds if this external potential varies slowly at the scale of the Cooper pair size lBCS , that is qlBCS 1. Linearization of hydrodynamic equations in the linear response regime predicts a linear dispersion relation ωq = cs q, with a sound velocity cs such that (281)

mc2s = ρ

dμ0 [ρ], dρ

of the order of h ¯ kF /m in the regime a < 0. The semi-classicality condition qlBCS 1 then results in h ¯ ωq Δ. This is satisfactory, as for h ¯ ωq > 2Δ one observes in the full RPA theory a coupling of the sound wave to the elementary, pair breaking excitations, which may distort the dispersion relation and even damp the sound wave, see, e.g., [68-70] and references therein, a phenomenon not included in superﬂuid hydrodynamics. ∗ ∗ ∗ We acknowledge useful suggestions from F. Werner, M. Jona-Lasinio, C. Weiss and A. Sinatra. REFERENCES [1] Castin Y., Simple theoretical tools for low dimension Bose gases, Lecture Notes of the 2003 Les Houches Spring School, Quantum Gases in Low Dimensions, edited by Olshanii M., Perrin H. and Pricoupenko L., J. Phys. IV (France), 116 (2004) 89. [2] Diu B., Guthman C., Lederer D. and Roulet B., Physique Statistique (Hermann, Paris, France) 1989. [3] Efetov K. B. and Larkin A. I., Sov. Phys. JETP, 42 (1976) 390. [4] Vaidya H. G. and Tracy C. A., Phys. Rev. Lett., 42 (1979) 3; J. Math. Phys., 20 (1979) 2291. [5] See the paragraph below eq. (31) in the paper by Levitov L. S., Lee H.-W. and Lesovik G. B., J. Math. Phys., 37 (1996) 4845. [6] Wigner E. P., Phys. Rev., 40 (1932) 749. [7] Vignolo P., Minguzzi A. and Tosi M. P., Phys. Rev. Lett., 85 (2000) 2850. [8] Baranov M., private communication, October 2000. [9] Akdeniz Z., Vignolo P., Minguzzi A. and Tosi M. P., Phys. Rev. A, 66 (2002) 055601. [10] Petrov D. S., Phys. Rev. Lett., 93 (2004) 143201. [11] Moerdijk A. J., Verhaar B. J. and Axelsson A., Phys. Rev. A, 51 (1995) 4852.

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Two-channel models of the BCS/BEC crossover M. Holland and J. Wachter JILA, Department of Physics at the University of Colorado at Boulder and the National Institute for Science and Technology - Boulder, CO 80309-0440, USA

1. – Introduction In this paper, we formulate a many-body theory of a dilute gas focusing speciﬁcally on the description of a scattering resonance. This situation is relevant to current experiments on quantum gases in atomic physics involving Feshbach resonances. Utilizing a Feshbach resonance, experimentalists are able to probe the crossover physics between the BardeenCooper-Schrieﬀer (BCS) superﬂuidity of a two-spin Fermi gas, and the Bose-Einstein condensation (BEC) of composite bosons. The physical situation giving rise to a Feshbach resonance, a two-channel system, is the principal focus of this paper. In particular, we highlight the criteria necessary for a two-channel model to be reducible to a single-channel situation. When such reduction is not possible, additional microscopic parameters which characterize the resonance itself are found to play an essential role in the many-body problem. The structure of the paper is as follows. We begin with an overview of the main ideas of superﬂuid quantum gases, and examine the general problem of describing resonance interactions. We proceed to look at a speciﬁc example of the crossover physics in the momentum distribution and the released energy of an interacting Fermi gas. Finally, we discuss extensions to the simple mean-ﬁeld approach, including the introduction of an imaginary-time procedure for fermions to ﬁnd the zero-temperature ground state, and how this may be utilized to include higher-order correlations in the theory, for example, the correlation between fermions and composite bosons. c Societ` a Italiana di Fisica

351

352

M. Holland and J. Wachter

2. – Bose-Einstein condensation and superﬂuidity Bose-Einstein condensation, as the textbook problem was originally posed, occurs in a non-interacting gas at equilibrium as a consequence of the bosonic statistics of the particles. Perhaps the most striking aspect is the phenomenon that below a critical temperature a ﬁnite fraction of all the particles are found in a single quantum state, the ground state of the system, representing the Bose-Einstein condensate. This persists even when one considers the limit in which the single quantum ground state is a state of zero integral measure. Although Bose-Einstein condensation is remarkable in itself, there is an even more subtle and profound aspect which is the connection with superﬂuidity. The phenomenon of superﬂuidity requires that there be interactions between the particles. Superﬂudity occurs in a variety of physical systems bridging many ﬁelds of physics and a wide range of energetic and spatial scales. Superﬂuidity can be deﬁned in a number of ways, but perhaps the most powerful is the connection of superﬂuidity with the presence of an order parameter or macroscopic wave function which describes all the superﬂuid particles. As in any scalar ﬁeld of complex numbers, a macroscopic wave function ψ(x) contains precisely two degrees of freedom at each point in space, which may be interpreted in terms of the local superﬂuid density n(x) and the local superﬂuid phase φ(x), that is, ψ(x) = n(x) exp[iφ(x)]. Since the global phase of a wave function is not measurable and plays no physical role, it is only the variation in space of the superﬂuid phase that is of physical signiﬁcance. To make this clear, we point out that for the simple situation of a de Broglie matter wave of a particle of mass m with wave vector k, the wave function is simply a plane wave proportional to exp[ik·x]. In this case it is the gradient of the quantum-mechanical phase multiplied by h ¯ /m which leads to the particle’s velocity, that is, h ¯ k/m. We deﬁne the superﬂuid velocity in analogous manner from the gradient in the phase of the macroscopic wave function (1)

v=

¯ h ∇φ . m

While this may appear straightforward, such a construction has truly non-trivial consequences. Any vector ﬁeld formed from the gradient of a scalar ﬁeld is irrotational, that is, ∇ × v = 0. This means that a superﬂuid in which the density is non-zero everywhere cannot contain circulation. The natural question which immediately arises therefore is how can a superﬂuid possibly rotate? The dilemma is naturally resolved by accounting for the fact that it is not required that the density be non-zero everywhere, and indeed a superﬂuid can rotate if there are lines of zero superﬂuid density which penetrate the ﬂuid. These are known as vortex lines and have been imaged in a number of physical systems, including by direct methods in dilute atomic gases. Interestingly, when there are many vortex lines, it is generally the case that the motion of the vortex lines themselves mimics that of rigid-body rotation of the system. In any case, when one performs a circulation integral inside any superﬂuid,

353

Two-channel models of the BCS/BEC crossover

(a)

(b)

S vortex S

core

S Fig. 1. – (a) The circulation integral in a superﬂuid depends on the number of vortex lines enclosed by the loop, in this case 7. (b) A characteristic phase pattern around a vortex core.

as illustrated in ﬁg. 1, the circulation is quantized v · dl = n

(2) loop

h , m

where n is an integer and is equal to the number of vortex lines enclosed by the loop. Figure 2 shows an example of the density and phase proﬁle of a vortex in a dilute atomic gas trapped in a harmonic potential. For the state |1 the situation shown is a Bose-Einstein condensate with no vortex cores and a ﬂat phase proﬁle. In |2, a single quantized vortex is shown, and the characteristic 2π phase winding around the vortex line of zero density is evident. The density drops to zero in the center of the vortex core since the superﬂuid velocity diverges in the region. The characteristic size of this density

|2

2S

density

phase

|1

y

y

x

x

Fig. 2. – |1: The phase (top) and density (bottom) proﬁle of a Bose-Einstein condensate in a harmonic trap in the ground state. |2: As for |1 but showing a condensate containing a single quantized vortex [1].

354

M. Holland and J. Wachter

dip —the vortex core— coincides with the region in which the superﬂuid velocity exceeds the speed of sound. Of course, it is never guaranteed that in any given quantum system a single macroscopic wave function exists to deﬁne the superﬂuid. The condensate can, at least in principle, be fragmented. Indeed, there are many situations in which a more sophisticated description is necessary. A few examples include strongly correlated gases where the interaction eﬀects are large or resonantly enhanced, gases where the ground state has a macroscopic degeneracy (a situation which occurs for harmonically conﬁned gases at high rotation), and gases in the vicinity of critical points where the ground-state symmetry changes. 3. – Description of a superﬂuid in a dilute atomic gas In a dilute atomic gas, typically composed of ground state alkali-metal atoms, the description of interactions in the superﬂuid phase is typically straightforward and depends on very few microscopic quantities. The primary physical reason for this is that the de Broglie wavelength associated with the atoms in the temperature scale at which Bose-Einstein condensation occurs is much larger than the characteristic range R of the interaction potential. This characteristic range is determined by matching the spatial scale at which the kinetic energy coincides with the potential energy for collisions of alkali-metal atoms. If we assume the thermal collision energy to be suﬃciently low, and the atoms to be in the ground internal state, then only the long-range part of the van der Waals potential plays an important role. The van der Waals potential is given in general form by C6 /R6 , where C6 is a constant coeﬃcient which encapsulates the induced polarizability. In this case the matching is given by the spatial scale R for which (3)

¯2 h C6 = 6. mR2 R

The resulting scale R for alkali-metal atoms is generally much less than the typical interparticle spacing so that the simple picture of contact interactions in a dilute quantum gas is relevant. Furthermore, when the temperature corresponds to a de Broglie wave length much larger than R, one may anticipate that the detailed structure of the interaction potential at small internuclear separation will not be revealed in the scattering properties. A further simpliﬁcation occurs when one considers the eﬀects of quantized angular momentum. Collisions between atoms which have a non-zero value for the orbital angular momentum quantum number l see a centrifugal barrier proportional to h ¯ 2 l(l + 1)/mr2 which becomes large and repulsive at short internuclear separation r. At suﬃciently low temperature, the presence of a centrifugal barrier prevents the atoms from reaching small enough separation for the true interatomic potential to have appreciable eﬀect. In that case, collisions only occur in the l = 0 channel where the centrifugal barrier is absent. This channel is known as the s-wave channel.

355

Two-channel models of the BCS/BEC crossover

Closed Channel

Energy

Q

Open Channel

Internuclear Separation Fig. 3. – A Feshbach resonance. A bound state of a closed potential is in close proximity (with detuning ν¯) to the scattering threshold (dashed line).

Thus, in ultracold quantum gases, the interactions are generally parametrized by the s-wave scattering phase shift which may be expressed in length units as the swave scattering length a, which then determines the zero-energy scattering T -matrix, T = 4π¯ h2 a/m. A consequence is that the theory of superﬂuidity in dilute Bose-Einstein condensates, for the most part, has involved solution of the non-linear Schr¨ odinger equation known as the Gross-Pitaevskii equation (4)

i¯ h

dψ = dt

−

¯2 2 h ∇ + V (x) + T |ψ|2 ψ. 2m

The full condensate evolution depends here on three energy contributions in the bracketed expression on the right-hand side of the equation: the kinetic energy, the potential energy V (x) of an externally applied potential, and the internal mean-ﬁeld energy proportional to both the T -matrix and condensate density |ψ|2 . The fact that interactions are parametrized by this particularly concise form of a constant T -matrix is not the only simpliﬁcation. The many-body state is also taken to be completely factorisable into orbitals which depend only on a single coordinate; an approximation equivalent to completely dropping explicit two-particle and higher correlations. In the next section, we begin to consider the eﬀects of a failure of this approximation. 4. – Breakdown of the mean-ﬁeld picture—resonance superﬂuids There are important and relevant situations in which this approach fails. Scattering resonances can modify the qualitative character since it is possible to tune the scattering length through inﬁnity by appropriate modiﬁcation of the details of the potential. When the scattering length is inﬁnity, clearly the Gross-Pitaevskii equation cannot be applied as written. The resonance can be of many types: a Feshbach resonance, a shape resonance, a potential resonance, or even a resonance induced through a photoassociative laser coupling. Regardless of the detailed mechanism, the principles we now outline are almost universally applicable. The case of a Feshbach resonance is illustrated in ﬁg. 3. A closed

356

M. Holland and J. Wachter

0 Ŧ1 Ŧ2

Ŧ4

Ŧ2

Ŧ1

'B (Gauss) 0 1

2

10 Ŧ5 a (103 a0)

TŦMatrix (103 a0)

Ŧ3

Ŧ6 Ŧ7 Ŧ8

0 Ŧ5

Ŧ10 Ŧ0.2

Ŧ9 Ŧ10 0

5

0.1

0.2

Ŧ0.1

0 Q0 (mK)

0.3 0.4 0.5 Scattering Energy (PK)

0.1

0.6

0.2

0.7

Fig. 4. – Real (solid line) and imaginary (dashed line) components of the T -matrix for collisions of the lowest two spin states of 40 K at a detuning of 20 F ( F is a typical Fermi energy for a dilute gas), shown in length dimensions, that is, T /(4π¯ h2 /m) [2]. The scattering length is the intercept at zero scattering energy which for this case is approximately −10000a0 , where a0 is the Bohr radius. The inset shows the scattering length as a function of detuning, with 20 F detuning indicated by the dot-dashed line.

channel potential, typically corresponding to a distinct spin conﬁguration, can support bound states with energies in close proximity to the scattering threshold. The diﬀerence in the magnetic moments of the open and closed channels allow the detuning ν¯ to be varied by application of an external magnetic ﬁeld. When the bound state crosses threshold, the scattering length passes from positive inﬁnity to negative inﬁnity. In vicinity of this point, the T -matrix is not constant and becomes strongly dependent on the scattering energy. The T -matrix may even acquire a substantial imaginary component as shown in ﬁg. 4. The assumption that the ﬁelds factorize into single-particle orbitals is no longer valid and quantum correlations must be included as an essential part of the description. For a single Feshbach resonance, as considered here, the behavior of the scattering length as a function of magnetic ﬁeld is universal and is shown in the inset of ﬁg. 4. Note that the separation of scales arguments that were discussed previously apply here as well. In its minimal form only two parameters are required to characterize this resonance: the matrix element between the open and closed channels g, and the detuning from resonance ν¯. The behavior of the scattering length in this approximation is given by (5)

T =

4π¯h2 a g2 =− . m ν¯

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Two-channel models of the BCS/BEC crossover

5. – Single-channel vs. two-channel approaches We now turn to the formulation of a many-body description of a dilute gas including scattering resonances. In accomplishing this task, it is necessary to ensure that the microscopic physics just explained is correctly incorporated. We begin by presenting two alternative starting points and then proceed to establish their connection. We focus our attention solely on a system of fermions, rather than considering directly bosons as described by the Gross-Pitaevskii theory. This is a good starting point, since a dilute gas of bosons emerges indirectly when the interaction properties are tuned, such that the fermions pair-up to form composite bosonic molecules. Since fermions require an antisymmetric wave function under exchange, s-wave interactions require at least two spin components, which we label as ↑ and ↓. We may then write a Hamiltonian for the system, keeping pairwise or binary interactions of general form, (6)

ˆ single = H

kσ

k cˆ†kσ cˆkσ +

qkk

Uk,k cˆ†q/2+k↑ cˆ†q/2−k↓ cˆq/2−k ↓ cˆq/2+k ↑ , (†)

where k = h ¯ 2 k 2 /2m is the free-fermion dispersion relation. The operators cˆkσ annihilate (create) fermions with momentum k and spin σ, and the momentum indices enforce momentum conservation. The interaction potential U need not be the physical interaction potential for the atoms being considered. This is because there is no unique potential that will reproduce the physical T -matrix for the given atoms over the relevant energy scale. In other words, one is free to choose U to be of a particularly simple and convenient form, a procedure known as renormalization. One way of doing this is to impose the constraint that the potential be independent of momentum and to cut all momentum sums in the theory oﬀ at a maximum momentum, K. This leads to the following relationship between U and T : (7)

T =

U , 1 − αU

where α = mK/(2π 2 ¯ h2 ). There exists an alternative approach to the many-body formulation. This is based on constructing the many-body Hamiltonian by considering the microscopic formation and dissociation of molecules in the Feshbach resonance state [3, 4] (8)

ˆ res = H

k,σ=↑,↓

k a ˆ†kσ a ˆkσ +

q q

2

+ ν ˆb†q ˆbq + gk ˆb†q a ˆq/2−k↓ a ˆq/2+k↑ + H.c. , qk

where gk is the matrix element relating two free fermions in the open channel to the closed-channel bound state near threshold, and ν is the bare detuning of the bound state, which we relate to the physical detuning by renormalization below. The operators

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M. Holland and J. Wachter

(†) (†) a ˆkσ annihilate (create) open channel fermions with momentum k and spin σ, while ˆbk annihilate (create) closed channel bosons. So we may pose the question: what is the connection between the single- and twochannel formulations? The relationship can be well understood by considering the twofermion relative wave function. An eigenstate of eq. (8) can be constructed by considering the following linear combination of basis states:

(9)

Ψk = χk + ck φ,

where χk = a−k↓ ak↑ and φ = b0 . The constant ck will be determined later. The evolution of Ψk is given by taking vacuum expectation values of the Heisenberg equations of motion generated by the resonance Hamiltonian in eq. (8): (10)

d dΨk = i¯ h (χk + ck φ) = 2 k χk + gk φ + ck i¯ h dt dt

νφ +

gk χk

.

k

This can be rewritten to eliminate the explicit dependence on the open channel fermions by substituting χk = Ψk − ck φ to give (11)

i¯ h

dΨk = 2 k Ψk + ck gk Ψk + (. . .)φ, dt k

where (. . .)φ contains the residual explicit dependence on the bosons in the Feshbach resonance state. In order for Ψk to represent the dressed eigenstate solution, giving the description of an eﬀective single-channel theory, it is required that the last (. . .)φ term be zero. We will now show this to be satisﬁed by the choice (12)

ck = P

gk , 2 k − E

with P denoting the Cauchy Principal Value and E deﬁned by the solution of an integral equation (13)

E =ν−P

k

gk2 . 2 k − E

The nature of the solution depends on the presence or absence of a bound state indicated by the sign of the renormalized detuning ν¯. This is deﬁned as (14)

ν¯ = ν −

g2 k , 2 k k

and is physically related to the magnetic-ﬁeld shift from the Feshbach resonance [5]. The case of ν¯ < 0 corresponds to the side of the resonance in which the scattering length is

359

Two-channel models of the BCS/BEC crossover

positive. There a bosonic dimer bound state exists and the solution of eq. (13) coincides at small detuning with the bound-state energy E = −¯h2 /ma2 [6-8]. For ν¯ > 0 there is no bound state and the solution is E = ν¯. 6. – Poles of the molecular propagator Finding the correct physical solutions for the energy in eq. (13) requires some care. If we simply were to drop the principal value from eq. (13), we would ﬁnd the following equations for a complex-frequency pole of the molecular propagator ∞ g2 k2 ω=ν− 2 dk 2 2 (15a) , δ =0+ 2π 0 ¯h k /m − ω + iδ ω g2 m ∞ dk 2 2 (15b) = ν¯ − 2 2 2π ¯ h 0 ¯h k /m − ω + iδ g 2 m3/2 ω (15c) . dz 2 = ν¯ − 2 3 4π ¯ z − ω + iδ h The subtle point here is that the solution one ﬁnds depends on the integration path in the complex plane. If we perform the integral, the poles will arise as roots of the following quadratic equation: z2 ± i

(16)

g 2 m3/2 z − ν¯ , 4π ¯h3

√

ω = z.

Figure 5 shows the various possible solutions. Note the prediction of two bound-state solutions on the positive side of the resonance (¯ ν > 0), and the shift of the real part of the pole to higher detunings than ν¯ at large detuning. With the exception of the true bound state at negative detuning (¯ ν < 0), these curves are misleading and do not correspond to physical solutions of the two-channel scattering problem. 7. – The equivalent single-channel theory Utilizing the deﬁnition for E, we may write the (. . .) prefactor of φ in eq. (11) as gk gk gk (17) (2 k − E) + ν − 2 k − P , 2 k − E 2 k − E k

and substituting the equation which deﬁnes E, this is (18)

gk (2 k − E + ν − 2 k − ν + E) = 0, 2 k − E

as required. The evolution of Ψk is then given by (19)

i¯ h

gk gk dΨk = 2 k Ψk + P Ψk . dt 2 k − E k

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M. Holland and J. Wachter

4 3 2 1

E

0 Ŧ1 Ŧ2 Ŧ3 Ŧ4 Ŧ2

Ŧ1

0

_ Q

1

2

Fig. 5. – Poles of the molecular propagator as a function of detuning; solid line: real part, dashed line: imaginary part. The insets show the various integration contours which lead to the solutions shown. The true bound state is the upper solid curve for ν¯ < 0.

This is nothing more than a time-dependent Schr¨ odinger equation for an eﬀective singlechannel problem. In other words, an eﬀective single-channel theory has now been shown to be encapsulated by the resonance Hamiltonian theory. The interaction potential, deﬁned in eq. (6), can be directly read oﬀ from eq. (19) Uk,k = P

(20)

gk gk . 2 k − E

What remains is to show that this potential generates the correct scattering length at all detunings ν¯. To this end, we must obtain the T -matrix by solving the LippmannSchwinger equation [9] (21)

Tk,k = Uk,k +

q

Uk,q Tq,k , 2 k − 2 q + iδ

δ →0+ .

We wish to solve this in the limit of zero scattering energy k → 0 and constant gk → g. Equation (21) can then be rewritten as a series by recursive substitutions (22)

T =−

g2 g2 g2 + P + ... . E E 2 k (2 k − E) k

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Two-channel models of the BCS/BEC crossover

Now from the deﬁnition of E in eq. (13) (23a)

E = ν−P

k

= ν−P

(23b)

g 2 (2 k − E + E) k

= ν¯ − P

(23c)

g2 2 k − E

k

2 k (2 k − E) g2 E , 2 k (2 k − E)

which leads to P

(24)

k

ν¯ − E g2 = . 2 k (2 k − E) E

Substituting this expression into eq. (22) and continuing similarly for the rest of the terms, we arrive at the geometric series (25a)

g2 T =− E

(25b)

=−

(25c)

E − ν¯ + 1+ E

E − ν¯ E

2

+ ...

g2 E(1 − ((1 − ν¯)/E)) g2 =− . ν¯

Equation (25) provides the correct behavior of the tuning of the scattering length around resonance, with the usual deﬁnition T = 4π¯h2 a/m, and conﬁrms that the potential Uk,k leads to the correct eﬀective fermion interaction properties. We have thus presented a detailed mathematical proof of the equivalence of the two initial Hamiltonians for the single- and two-channel models describing the scattering of two fermions in vacuum. One must extend this result to consider the equivalence in systems that contain more than two fermions. The structure of the mathematical proof can be continued along the presented lines. The result for the important case of four fermions is that the equivalence between the single and two-channel theories has been shown to hold, but requires that the width of the Feshbach resonance be suﬃciently broad [10]. It should be emphasized that no single-channel picture can be formulated when the equivalence does not hold, and one must include explicitly the Feshbach formulation in the many-body theory. 8. – Connection with the theory of Feshbach resonances We may equivalently express the two-channel model in terms of the original language of the open P and closed Q channels as originally done by Feshbach [11]. In terms of

362

M. Holland and J. Wachter

these separate Hilbert subspaces, the time-independent Schr¨ odinger equation takes the following coupled form: ˆ P P |ΨP + H ˆ P Q |ΦQ , E|ΨP = H ˆ ˆ E|ΦQ = HQQ |ΦQ + HQP |ΨP .

(26a) (26b)

We may formally solve eq. (26a), |ΨP =

(27)

1 ˆP P E−H

ˆ P Q |ΦQ , H

and substitute the result into eq. (26b) and obtain (28)

ˆ QP ˆ QQ − H E−H

1 ˆ P Q |ΦQ = 0. H ˆP P E−H

The eﬀective interaction due to the coupled spaces is therefore (29)

ˆ QQ + H ˆ QP ˆ eﬀ = H H QQ

1 ˆ H . ˆP P P Q E−H

This expression is, in fact, similar to eq. (13); except that in this case it is in operator form, whereas eq. (13) is represented in a basis. The chosen basis involves a continuum |k for the P subspace, a single quantum resonance state |φ for the Q subspace, and an explicit form for the matrix elements of momentum-dependent coupling. The mapping ˆ QQ |φ = ν, k|H ˆ P P |k = 2 k , and φ|H ˆ QP |k = k|H ˆ P Q |φ = gk . is thus φ|H 9. – The BCS/BEC crossover One of the ﬁrst attempts to understand the crossover between the phenomena of BCS and BEC was put forth by Eagles in a 1969 paper on pairing in superconducting semiconductors [12]. He proposed moving between these two limits by doping samples, in this case by decreasing the carrier density in systems of SrTiO3 doped with Zr. In a 1980 paper by Leggett [13], motivated by the early ideas of quasi-chemical equilibrium theory, he modeled the crossover at zero temperature by way of a variational wave function: (30)

ψBCS =

"

(uk + vk a†k a†−k )|0.

k

This wave function is simply the BCS wave function and assumes that at T = 0 all the fermions form Cooper pairs. What Leggett was able to show was that he could smoothly interpolate between conventional BCS theory and the occurrence of BEC. In 1985, Nozi`eres and Schmitt-Rink (NSR) extended this theory to ﬁnite temperatures, in order to calculate the critical temperature TC [14]. NSR derived the conventional BCS gap and number equations, but introduced into the number equation the

Two-channel models of the BCS/BEC crossover

363

Fig. 6. – Schematic comparison of BCS theory and the BCS/BEC crossover theory of resonance superﬂuidity. Resonance superﬂuidity describes closed-channel, tightly bound pairs of fermions in addition to the loosely bound BCS pairs. Below the transition temperature TC , the closedchannel pairs condense and also mediate pairing of open-channel fermions away from the Fermi sphere.

self-energy associated with the particle-particle ladder diagram (or scattering T -matrix) to lowest order. This very inﬂuential paper was built upon by many other groups and was transformed into a functional form by Randeria et al. [15]. A compelling motivation for understanding the crossover problem comes from the fact that many high-TC superconductors seem to fall within the intermediate region between BCS and BEC. In the copper oxides, for instance, the coherence length of the Cooper pairs has been measured to be only a few times the lattice spacing. In contrast, in conventional superconductors, the coherence lengths are usually much greater than the lattice spacings. An understanding of the crossover may be one of the keys to understanding and manipulating high-TC materials. Dilute quantum gases have already played a very important role is probing experimentally the BCS/BEC crossover. This crossover is in fact a special case of a more general framework of resonance superﬂuids. When the resonance is suﬃciently broad, as discussed above, a single-channel approach is suﬃcient to describe many of the important eﬀects. In this case, which is generally the experimentally relevant one, the system maps on to the BCS/BEC crossover problem originally introduced in the context of condensed matter systems. Figure 6 illustrates some of the important distinctions of resonance superﬂuids. In particular, the pairing in a weakly coupled BCS superconductor occurs primarily at the Fermi surface in momentum space and the superﬂuid appears out of the degenerate Fermi sea at a critical temperature TC much less than the Fermi temperature. As ﬁg. 6 illustrates, the physical situation for resonance superﬂuids can be quite diﬀerent, with pairing throughout the Fermi surface, and molecular condensation of the composite

364

M. Holland and J. Wachter

bosons. Furthermore the critical temperature for superﬂuidity in this case can be comparable to the Fermi temperature. This is very important, since current experiments in dilute quantum gases can, at the lowest, reach temperatures on the order of a tenth of the Fermi temperature, which is far above the critical temperatures predicted by BCS theory in the region in which it can be applied. 10. – Momentum distribution in a dilute Fermi gas [16] The study of the momentum distribution of an atomic gas in the quantum degenerate regime carries a wealth of information on the role played by interactions and on the existence of a superﬂuid order parameter. As an example, in a homogeneous Bose gas at T = 0, the momentum distribution nk exhibits a singular behavior at small wave vectors nk mc/2¯ hk, which is determined by the excitation of phonons propagating with the speed of sound c and is a signature of Bose-Einstein condensation [17]. In a corresponding system of fermions with attractive interactions, the broadening of the Fermi surface is instead a consequence of the formation of pairs and of the presence of a superﬂuid gap [18]. This latter eﬀect becomes dramatic in the BCS/BEC crossover region where the pairing gap is of the order of the Fermi energy of the system [13, 14, 19]. The second moment of the momentum distribution deﬁnes the kinetic energy of the system: Ekin = k nk ¯ h2 k 2 /(2m), where m is the mass of the atoms. This quantity, which also plays a central role in the many-body description of ultracold gases, is very sensitive to the large-k behavior of nk . For interacting systems the dominant contribution to Ekin comes from short-range correlations, where the details of the interatomic potential are relevant. In the case of a zero-range potential it is well known that the momentum distribution decreases like 1/k 4 for large momenta and the kinetic energy diverges in dimensionality greater than one. This unphysical divergence can be understood recalling that the zerorange approximation is only correct to describe the region of momenta k 1/r0 , where r0 denotes the physical range of interactions [20]. This behavior of the kinetic energy is a general feature of quantum-degenerate gases, where interactions are well described by the s-wave scattering length a, holding both for fermions and bosons and for repulsive and attractive interactions. At T = 0 for a Bose gas√ with repulsive interactions one ﬁnds nk 1/(16k 4 ξ 4 ) at large momenta, where ξ = 1/ 8πna is the healing length. For a Fermi gas one ﬁnds instead nk m2 Δ2 /(¯ h4 k 4 ), where Δ is the BCS gap, for attractive interactions and nk (4akF /3π)2 (kF /k)4 for repulsive interactions [21]. As discussion in sect. 3, the physics of ultracold gases is characterized by a clear separation of energy scales: the energy scale associated with the two-body physics as ﬁxed for example by h ¯ 2 /mr02 ∼ 10 mK, being r0 ∼ 100a0 the typical interaction length of the van der Waals potential, and the energy scale associated with the many-body physics as determined by the typical Fermi energy F ∼ 1 μK. This separation of energy scales provides a very large range of timescales for which the dynamical process can be safely considered fast (diabatic) as the many-body dynamics is concerned and slow (adiabatic) with respect to the two-body dynamics. This feature is exploited in recent experiments aiming to measure the momentum distribution, that are based on the ballistic expansion

Two-channel models of the BCS/BEC crossover

365

of the cloud after the scattering length has been quickly set to zero by a fast magneticﬁeld ramp [22,23]. These experiments give access to the released momentum distribution, which is a non-equilibrium quantity deﬁned as the momentum distribution of the system after the scattering length has been rapidly ramped to a = 0. Provided the timescale of the ramp satisﬁes the conditions given above, the released momentum distribution does not depend on the detailed structure of the interatomic potential, being in this sense universal, but it does depend on the timescale of the ramping process. In this section we investigate the behavior of a Fermi gas at T = 0 in the BCS-BEC crossover. We calculate the released momentum distribution and its second moment for a homogeneous system as a function of the interaction strength 1/(kF a), where kF is the Fermi wave vector. For harmonically trapped systems we give an explicit prediction of the column-integrated released momentum distribution and of the released energy for values of the interaction strength ranging from the BCS to the BEC regime and we compare our results with recently obtained experimental data [23]. We consider an unpolarized two-component Fermi gas with equal populations of the ↑ and ↓ components: N↑ = N↓ = N/2, where N is the total number of particles. We determine the dynamical evolution of such a system starting from the equations of motion for the non-equilibrium density matrices of the ↑ and ↓ components interacting through the Hamiltonian 2 2

¯h ∇x † H= (31) dx ψσ (x) − ψσ (x) + 2m σ + dx dx ψ↑† (x)ψ↓† (x )V (x, x )ψ↓ (x )ψ↑ (x), where σ =↑, ↓ labels the spins and V (x, x ) is the interaction potential to be speciﬁed later. The correlations are treated within a mean-ﬁeld approach [19], where they are expressed in terms of the normal GNσ (x, x , t) = ψσ† (x , t)ψσ (x, t) and the anomalous GA (x, x , t) = ψ↓ (x , t)ψ↑ (x, t) density matrices. By neglecting the Hartree terms, one obtains the following coupled equations of motion [18] (from now on GN↑ (x, x , t) = GN↓ (x, x , t) ≡ GN (x, x , t)): (32)

i¯ h

dGN (x, x , t) = dt

h2 ∇2x ¯ ¯h2 ∇2x − + GN (x, x , t) + 2m 2m + dx [V (x, x ) − V (x , x )]GA (x, x , t)G∗A (x , x , t)

and (33)

dGA (x, x , t) = i¯ h dt

¯ 2 ∇2x h ¯h2 ∇2x − − GA (x, x , t) + V (x, x )GA (x, x , t) − 2m 2m − dx V (x, x )GN (x , x , t)GA (x, x , t) − − dx V (x, x )GN (x, x , t)GA (x , x , t) .

366

M. Holland and J. Wachter

The short-range nature of the interaction potential V (x, x ) can be properly described through the regularized pseudopotential V (r) = (4πa¯h2 /m)δ(r)(∂/∂r)r [24, 25], with a the s-wave scattering length and r ≡ |x − x |. During the magnetic-ﬁeld ramp the value of the scattering length changes in time according to the relation a(t) = abg 1 −

(34)

Γ B(t) − B0

,

valid close to the Feshbach resonance. In the above expression, abg denotes the background scattering length, B0 and Γ the position and width of the resonance, respectively, and B(t) is the instantaneous value of the magnetic ﬁeld. Under the dynamical conditions that we are considering, where the non-equilibrium processes take place over a timescale adiabatic with respect to the two-particle problem and diabatic with respect to the many-particle system, the time evolution does not depend on the details of the short-range potential and the eﬀect of interactions results in a boundary condition at short length scales (35)

(rGA (r, t)) rGA (r, t)

=− r=0

1 , a(t)

where the prime indicates the derivative with respect to r, which must be fulﬁlled at any time t. For small values of r, many-body eﬀects in eq. (33) can be neglected and the boundary condition (35) corresponds to the one of the two-body problem with the pseudopotential V (r), where GA (r, t) is the wave function of the relative motion. We notice that the contact boundary condition limrij →0 ∂(rij Ψ)/∂rij /(rij Ψ) = −1/a, where rij is the distance between particles i and j, holds in general for the exact many-body wave function Ψ if the eﬀective range of the potential can be neglected [26]. In the case of a homogeneous system and by using the pseudopotential approximation for the interatomic potential V (x, x ) and the boundary condition (35), eqs. (32)-(33) ˜ N (r, t) ≡ can be greatly simpliﬁed. One obtains the following coupled equations for G ˜ A (r, t) ≡ rGA (r, t): rGN (r, t) and G (36a)

i¯ h

˜ N (r, t) 8π¯h2 dG ˜ A (r, t)[G ˜ ∗A (t)]r=0 = i G dt m

and (36b)

i¯ h

˜ A (r, t) h2 ∂ 2 ˜ ¯ dG 8π¯h2 ˜ ˜ A (t)]r=0 , G =− GN (r, t)[G (r, t) + A dt m∂r2 m

˜ A ) /G ˜ A ]r=0 = −1/a(t). Notice that interaction eﬀects with the boundary condition [(G only enter eqs. (36) through the boundary condition (35). We determine the initial ˜ N (r, t = 0) and G ˜ A (r, t = 0) of eqs. (36) from the mean-ﬁeld gap and conditions G

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Two-channel models of the BCS/BEC crossover

number equations corresponding to the equilibrium state of the gas with the initial value of the scattering length a(0) (37a)

∞

dkk2 2π 2

n= 0

1−

k − μ

( k − μ)2 + Δ2

and (37b)

m = 4π¯ h2 a(0)

0

∞

dkk2 4π 2

1 1 − , k ( k − μ)2 + Δ2

where k = h ¯ 2 k 2 /2m, μ is the chemical potential, Δ the superﬂuid gap and n = n↑ + ˜ N and G ˜ A are then calculated from the n↓ the total particle density. The functions G 2 2 Bogoliubov quasiparticle amplitudes uk = 1 − vk = [1 + ( k − μ)/ ( k − μ)2 + Δ2 ]/2, as ∞ ∞ 2 2 ˜ ˜ GN (r, t = 0) = 0 dkk sin(kr)vk /(2π ), and GA (r, t = 0) = 0 dkk sin(kr)uk vk /(2π 2 ). ˜ N (r, t = 0), G ˜ A (r, t = 0) We solve the dynamic equations (36) with the initial conditions G given above and a(t) given by (34) from the initial time t = 0 to the ﬁnal time t = tf , where a(tf ) = 0. The released momentum distribution is then calculated from the Fourier transform of GN at the time t = tf (38)

nk (t = tf ) =

dreik·r GN (r, t = tf ) .

The results for the homogeneous gas are shown in ﬁgs. 7 and 8. In ﬁg. 7 we compare the equilibrium momentum distribution nk (t = 0) in the unitary limit, 1/(kF a(0)) = 0, with the corresponding released momentum distribution (38) calculated for a magneticﬁeld ramp rate of 2 μs/G. For values of k < kF the shape of the distribution does not change appreciably. The large-k tail is instead greatly suppressed (as is shown in the inset) and the second moment of the released nk is a convergent integral. Notice that the fast decaying tail of the released nk aﬀects the normalization constant. In ﬁg. 8 we show the results of the released energy as a function of the initial interaction strength 1/(kF a(0)) for two diﬀerent values of the magnetic-ﬁeld ramp rate. We notice that on the BCS side of the crossover, kF a(0) < 0, the dependence on the ramp rate is weak, while on the BEC side, kF a(0) > 0, a faster ramp produces a signiﬁcantly larger energy. In the BEC regime the system is in fact more sensitive to changes of the high-energy tail of the momentum distribution. Deep in the BCS regime, −1/(kF a(0)) 1, the released 0 energy reduces to the kinetic energy of the non-interacting gas Ekin = 3 F /5. In the opposite BEC regime, −1/(kF a(0)) −1, many-body eﬀects become less relevant and the released energy coincides with the one obtained from the dissociation of the molecular state [23] (see ﬁg. 10). In order to make quantitative comparison with the experiment, we now consider harmonically trapped systems conﬁned by the external potential Vext (r) = m(ωx2 x2 + ωy2 y 2 +ωz2 z 2 )/2. Within the local density approximation (LDA) we introduce the rescaled

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M. Holland and J. Wachter

1

0.8 k4nk

0.1

0.05

nk

0.6

0 2

0.4

4

6 k/kF

8

10

0.2

0 0

0.5

1

1.5

2

2.5

3

k/k

F

Fig. 7. – Released momentum distribution (solid lines) of a homogeneous gas at unitarity, 1/(kF a(0)) = 0, for a ramp rate of 2 μs/G [16]. The large-k behavior of nk weighted by k4 is shown in the inset, where the dotted line corresponds to the equilibrium asymptotic value (Δ/2 F )2 . The initial equilibrium distribution is also shown (dashed lines).

1

4 2Ps/G 4Ps/G

3 2

rel

E /E

0 kin

0.8

0.6 nk

1 0 Ŧ1

0.4

Ŧ0.5

0 Ŧ1/(k a)

0.5

1

F

0.2

0 0

0.5

1

1.5

2

2.5

3

k/kF Fig. 8. – Released energy of a homogeneous gas as a function of the interaction strength for two values of the ramp rate [16]. The energy is normalized to the kinetic energy of the non-interacting 0 = 3 F /5. gas Ekin

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Two-channel models of the BCS/BEC crossover

1 0.1

0.9

0

0 F

n(k/k )k /(k N)

0.8

0.6

02 F

n(k/k )k /N

0 F

3

0.7

Ŧ0.1 0.1

0 F

0.5 0.4

0

0.3

Ŧ0.1 0

0.2

1

2

3

0.1 0 0

0.5

1

0 F

k/k

1.5

2

2.5

Fig. 9. – Column-integrated released momentum distribution of a harmonically trapped gas [16]. From top to bottom, the lines correspond to 1/(kF0 a(0)) = −71, 1/(kF0 a(0)) = −0.66, 1/(kF0 a(0)) = 0 and 1/(kF0 a(0)) = 0.59. The magnetic-ﬁeld ramp rate is 2 μs/G. The symbols correspond to the experimental results of ref. [23]. Inset: Results for 1/(kF0 a(0)) = 0 (top) and 0.59 (bottom) weighted by k3 .

spatial variables x ˜ = x ωx /ω, y˜ = y ωy /ω and z˜ = x ωz /ω, so that the conﬁning ˜ 2 /2, where ω = potential becomes isotropic in the new coordinates Vext (r) = mω 2 R 1/3 (ωx ωy ωz ) is the geometric average of the harmonic oscillator frequencies. For each ˜ = (x+ ˜ ˜ x ˜ )/2, eqs. (37) are solved for the local spatial slice R potential μlocal (R) chemical 3 ˜ subject to the normalization d R ˜ n(R) ˜ = N and the local and the local density n(R), ˜ + Vext (R). ˜ Each slice is then evolved according to equilibrium condition μ = μlocal (R) ˜ N (˜ ˜ t = 0) and G ˜ A (˜ ˜ t = 0), where r˜ = x ˜−x ˜ is eqs. (36) with initial conditions G r, R, r, R, ˜ t) the relative coordinate. The released momentum distribution is obtained from GN (˜ r, R, ˜ at the ﬁnal time t = tf through the integral over the rescaled coordinates R and r˜, ˜ d3 r˜eik·r˜GN (r, R, t = tf ). (39) n(k, t = tf ) = d3 R In ﬁg. 9 we compare the column-integrated released momentum distribution ∞ kx2 + ky2 , tf = dkz n(k, tf ), (40) n −∞

calculated from eq. (39), with the experimental results obtained in ref. [23]. The values

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M. Holland and J. Wachter

6

5

0

Erel/Ekin

4

3

2

1

0 Ŧ1

Ŧ0.5

0

0.5

1

1.5

Ŧ1/(k(0) a) F

Fig. 10. – Released energy of a harmonically trapped gas as a function of the interaction strength 1/(kF0 a(0)) for a ramp rate of 2 μs/G (upper line) [16]. The lower line is the corresponding result solving the two-body problem associated with the molecular state. The symbols are the experimental results from ref. [23]. The energy is normalized to the kinetic energy of the 0 = 3 0F /8. non-interacting gas Ekin

of the interaction strength are 1/(kF0 a(0)) = −71, −0.66, 0 and 0.59 as in the experiments and the magnetic-ﬁeld ramp rate is 2 μs/G. The agreement is quite good for the momentum distribution on the BEC side of the resonance and in the unitary limit (see the inset of ﬁg. 9 to compare the large-k tails of the distributions). For 1/(kF0 a(0)) = −0.66 the experimental n(k) is more broadened because the Hartree mean-ﬁeld term, which enhances the shrinking of the cloud due to attraction, is neglected in the calculation. In ﬁg. 10 we show the results for the released energy of the inhomogeneous gas as a func 0 0 1/6 tion of the interaction strength 1/(kF a(0)), where kF = (24N ) mω/¯h is the Fermi wave vector in the center of the trap corresponding to a non-interacting gas. Experimental results from ref. [23] and theoretical results obtained by solving the time-dependent Schr¨ odinger equation for the molecular state (see [23]) are also shown in ﬁg. 10. The mean-ﬁeld calculation reduces to the molecular two-body result only in the deep BEC regime, −1/kF0 a(0) −1, and agrees better with the experimental results. Given that there are no adjustable parameters, theory and experiment are in reasonable agreement over the whole crossover region. In the unitary limit the present mean-ﬁeld approach is known to overestimate the equilibrium energy per particle compared to more advanced quantum Monte Carlo calculations [27,28]. This might be the reason for the larger energy

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Two-channel models of the BCS/BEC crossover

obtained around resonance compared to the observed one. On the deep BEC side, the underestimate may instead be due to the eﬀect of ﬁnite temperature on the experimental data in that regime. Finally, on the BCS side of the resonance, the present approach neglects the mean-ﬁeld Hartree term and we expect a faster convergence to the kinetic 0 energy of the non-interacting gas Ekin = 3 0F /8, where 0F = (¯ hkF0 )2 /2m, as −1/kF0 a(0) becomes large. In the section, we have developed a theoretical approach which allows one to calculate the released momentum distribution, released energy and, in principle, other nonequilibrium properties of a superﬂuid Fermi gas if the scattering length is set to zero using a fast magnetic-ﬁeld ramp. The approach is fully time dependent, accounts at the mean-ﬁeld level of many-body pairing eﬀects and can be applied for any value of the interaction strength along the BCS-BEC crossover. For harmonically trapped systems we compared our theoretical predictions with the recently obtained experimental results of ref. [23]. Qualitatively, we reproduced the data well, but signiﬁcant quantitative discrepancies are found which indicate the inadequacy of mean-ﬁeld theory for describing the gas in the crossover region. This would illustrate a need to include in a time-dependent formalism correlations beyond the simple pair correlations in order to explain the observed experimental expansion energetics on resonance. 11. – Imaginary-time methods for single- and two-channel BCS models The method of steepest descents has been widely applied for ﬁnding condensate wave functions in Boson systems. In this section, we want to generalize this method and calculate the single- and two-channel BCS solution for interacting fermions. Our imaginarytime approach can be generalized to include beyond-BCS interactions. The most important advantage of our imaginary-time method for fermions is that it gives direct access to zero-temperature ground states for fermion systems without diagonalizing the BCS self-energy matrix. One could, for example, study topological excitations of the BCS superﬂuid by imposing symmetry constraints. . 11 1. Single-channel BCS theory. – BCS theory is a single-channel theory for fermions, whose interactions are characterized by the scattering length a as the single microscopic parameter [29]. The BCS Hamiltonian can be diagonalized analytically by solving the following number and gap equations (41a)

∞

n= 0

k2 dk 2 2π

1−

k − μF

( k − μF )2 + Δ2

and (41b)

m = 4π¯ h2 a

0

∞

k2 dk 2 4π

1 1 − k ( k − μF )2 + Δ2

,

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M. Holland and J. Wachter

where k = h ¯ 2 k 2 /(2m) is the kinetic energy, μF the chemical potential, Δ the superﬂuid gap and n = n↑ +n↓ the total particle density. These equations are self-consistently solved for the chemical potential and the gap, and one obtains the Bogoliubov quasi-particle modes uk and vk , which are given by (42)

u2k

=1−

vk2

1 = 2

1−

k − μF ( k − μF )2 + Δ2

.

The normal and anomalous averages fk and mk at zero temperature are then given by % $ † ˆk↑ a ˆk↑ = vk2 fk↑ = a

(43a) and

% $ mk = a ˆ−k↓ a ˆk↑ = uk vk .

(43b)

The normal average fk↑ is the density of spin-up atoms at momentum k, and the anomalous average a pair correlation function between atoms of opposite momentum and spin. We want to ﬁnd the solutions (43) for the averages by using imaginary-time propagation. . 11 2. Imaginary-time propagation for bosons. – How does imaginary-time propagation work for bosons? The basic idea is to replace the time variable t in the Gross-Pitaevskii (GP) equation for the condensate wave function ψ, (44)

dψ i¯ h = dt

¯2 2 h 2 − ∇ + V + T |ψ| ψ = HGP ψ , 2m

with the imaginary time variable −it. The time evolution under the GP equation (44) can be written in terms of its eigenstates φn , which are deﬁned by HGP φn = En φn , with the eigenenergies En : (45)

ψ(t) =

n

iEn t cn exp − φn , ¯h

where the coeﬃcients cn are deﬁned by the expansion of the initial condition ψ(t = 0) = n cn φn . Propagating the GP equation in imaginary time changes the above time evolution to (46)

ψ(t) =

n

En t cn exp − φn . ¯h

The unitary time evolution in eq. (45) has turned into an exponential decay.

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Two-channel models of the BCS/BEC crossover

The algorithm for the imaginary-time method is now to use the imaginary-time evolution over a time interval and renormalizing the resulting wave function after each step using a normalization condition or number equation, in this case, (47)

N=

d3 x|ψ(x)|2 .

This procedure will converge on the lowest-energy ground-state solution φ0 (48)

ψ(t)

−it

−−→

φ0 .

Due to numerical errors, this even works if the initial wave function ψ(t = 0) does not contain a contribution of the ground state, that is if c0 = 0. The convergence can, however, be accelerated in practice by choosing ψ(t = 0) appropriately. Imaginary-time propagation can include symmetry, topological, or orthogonality constraints, and one can thus calculate topological condensate states or higher-excited states of the Hamiltonian. See ﬁg. 2 for an example of a vortex state calculated using imaginarytime propagation. We will now generalize this powerful approach to fermions. . 11 3. Imaginary-time propagation for fermions. – In the case of bosons above, we learned how to propagate a wave function equation in imaginary time and thus ﬁnd ground-state solutions. Time-dependent BCS theory, which is the simplest single-channel theory for interacting fermions, has two equations for the normal fk and anomalous density mk , (49a)

i¯ h

dfk↑ = i2U (p∗ mk ) dt

and (49b)

i¯ h

dmk = 2( k − μF )mk + U p (1 − fk↑ − fk↓ ) , dt

with the pairing ﬁeld p = mk and the renormalized potential U , which is derived from the T -matrix. The second equation for the pairing correlation can be evolved like the GP . equation in subsect. 11 2. The ﬁrst equation for the density, however, is a density matrix equation, which does not evolve like a wave function. Density matrices evolve under two time-evolution operators with positive and negative energies, such that the diagonal elements do not evolve at all. The conventional imaginary-time algorithm would thus not change the initial particle distribution function. We here propose a new solution to ﬁnding the evolution of the density matrix equation by using a Cauchy-Schwartz inequality at zero temperature.

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M. Holland and J. Wachter

. 11 3.1. Cauchy-Schwartz inequality. In this section, we will use the Cauchy-Schwartz inequality to ﬁnd a relation between the density fk and the pairing correlation mk that we can use instead of the density matrix equation to determine the evolution of fk . The Cauchy-Schwartz inequality holds for any inner-product space and can be written in the usual bra-ket notation as α|αβ|β ≥ |β|α|2 .

(50)

Choosing values for |α and |β, we can prove the following relation: $

(51)

%$ % /$ %/2 ˆ−k↓ a ˆ−k↓ a a ˆ†k↑ a ˆk↑ a ˆ†−k↓ ≥ / a ˆk↑ / .

At zero temperature, this relation becomes an identity fk↑ (1 − fk↓ ) = |m2k | ,

(52)

as one can see from the quasi-particle vacuum relations eqs. (43) and the properties of the Bogoliubov modes in eq. (42). One can prove the identity in eq. (52) for any set of evolution equations for which one can ﬁnd a quasi-particle transformation. In order to be able to use eq. (52) as planned, we have to assume spin symmetry (53)

fk↑ = fk↓ = fk .

On closer examination, we note that the solution for fk of eq. (52) has two branches 1 1 − |mk |2 . fk = + sgn(μF − k ) 2 4

(54)

The sign function here picks the positive branch for energies below the chemical potential and the negative branch for higher energies, as is the case for the BCS solutions given in eqs. (43). . 11 4. Imaginary-time algorithm for the single-channel model. – With these ingredients, we can now formulate the new algorithm for ﬁnding zero-temperature ground states in interacting fermion systems. 1) Pick an initial pairing correlation mk and chemical potential μF . 2) Calculate the pairing ﬁeld p = mk and the density fk according to eq. (54). 3) Evolve the anomalous density mk for a time step dt in imaginary time using (55)

h ¯

dmk = −2( k − μF )mk − U p (1 − fk↑ − fk↓ ) . dt

4) Repeat 2) and 3) until convergence. 5) Adjust chemical potential μF in 1) until total density n = k,σ fkσ is correct.

375

Two-channel models of the BCS/BEC crossover 1

Normal density f

k

Anomalous density |m |

0.9

k

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

Momentum k/kF

2

2.5

3

Fig. 11. – Normal (full line) and anomalous (dashed line) density for a single-channel model at kF a = −1. The paired fermions live predominantly at the Fermi energy.

We have veriﬁed that this algorithm yields the BCS solution both for local and a Gaussian, non-local potential. One can, in fact, show analytically that the BCS equations are a solution to the imaginary-time equations. See ﬁg. 11 for the results for the local potential. Can we generalize this algorithm to the two-channel case? . 11 5. Imaginary-time propagation for the two-channel model. – The two-channel model with contact interactions is equivalent to a single-channel model with non-local interactions. However, models with contact interactions are much easier computationally. We will further show in the following sections how we can extend the two-channel contact model to suﬃciently high-order correlations to properly reproduce the composite boson-boson scattering length. This extension is not feasible for the single-channel model. . 11 5.1. Two-channel equations of motion. We begin by deriving the equations of motion of the relevant mean ﬁelds for the two-channel model. The crossover Hamiltonian for a homogeneous system is, again, given by (56)

ˆ = H

k,σ=↑,↓

k a ˆ†kσ a ˆkσ +

q q

2

+ ν ˆb†q ˆbq + gk ˆb†q a ˆq/2−k↓ a ˆq/2+k↑ + H.c. , qk

where we now have composite-boson ﬁelds bq coupling to the fermions. The minimal set of mean ﬁelds that we now have to derive equations of motion for is the anomalous density mk and now also the condensate wave function φm = ˆb0 . The normal density is again given by the Cauchy-Schwartz relation in eq. (54). We $$ %% ignore the lowest-order thermal molecular mean ﬁelds ˆb†q ˆbq = ˆb†q ˆbq − |φm |2 and

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M. Holland and J. Wachter

$$ %% ˆb−q ˆbq = ˆb−q ˆbq − φ2 , which neglects the quantum depletion of the molecular m condensate. Note that there is no thermal depletion, since we are only interested in zero-temperature ground states. To derive the equations of motion for the relevant mean ﬁelds, we ﬁrst write the Heisenberg equations of motion for the three individual operators (57a)

i¯ h

(57b)

i¯ h

dˆ ak↑ = k a ˆk↑ + g−q/2+k a ˆ†q−k↓ˆbq , dt q dˆ ak↓ = k a ˆk↓ − gq/2−k a ˆ†q−k↑ˆbq dt q

and i¯ h

(57c)

dˆbq q = + ν ˆbq + gk a ˆq/2−k↓ a ˆq/2+k↑ . dt 2 k

We then take the average of eq. (57c) to obtain the equation of motion for the condensate wave function φm (58)

i¯ h

dφm = νφm + g mk = νφm + gp. dt k

We similarly combine eqs. (57a) and (57b) to obtain the equation of motion for the anomalous density mk (59)

i¯ h

$$ † %% dmk ˆbq a = 2 k mk + gφm (1 − fk↑ − fk↓ ) − 2g ˆq+k↑ a ˆk↑ . dt q

$$ This %% equation for the anomalous density mk couples to the three-operator cumulant ˆbˆ a† a ˆ . The cumulant notation again indicates that the lower-order factorized averages have been subtracted out. We drop this cumulant for now, but sect. 12 will discuss its importance for reproducing the correct molecule-molecule scattering length on the BEC side of the crossover. With these two equations of motion (58) and (59), we can now update the algorithm for the steepest-descent method. . 11 5.2. Imaginary-time algorithm for the two-channel model. The algorithm now has two coupled wave functions that need to be evolved. 1) Pick an initial pairing correlation mk , condensate wave function φm , and chemical potential μF . 2) Calculate the pairing ﬁeld p = mk and the density fk according to eq. (54).

377

Two-channel models of the BCS/BEC crossover

3) Evolve the anomalous density mk and condensate wave function φm for a time step dt in imaginary time using h ¯

(60a)

dφm = −(ν − 2μF )φm − gp dt

and (60b)

h ¯

dmk = −2( k − μF )mk − gφm (1 − fk↑ − fk↓ ) . dt

4) Repeat 2) and 3) until convergence. 5) Adjust chemical potential μF in 1) until total density n = correct.

k,σ

fkσ + 2|φm |2 is

In ﬁg. 12, we show results for this algorithm for a contact two-channel model. They look very similar to the ones we found in the single-channel case in ﬁg. 11. Is that what we would expect? The superﬂuid gaps of both theories turn out to be the same (61)

Δ = U p = gφm .

1

Normal density f

k

Anomalous density |m |

0.9

k

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

Momentum k/kF

2

2.5

3

Fig. 12. – Normal (full line) and anomalous (dashed line) density for a two-channel model at kF a = −1. This is calculated for the broad resonance in 6 Li at 834 G, with a bare molecular fraction of 4 · 10−6 . One can see the good agreement with the single-channel result in ﬁg. 11 in the broad resonance limit.

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M. Holland and J. Wachter

Fig. 13. – Schematic illustrating the crossover between fermions, whose interactions can be described by BCS theory with scattering length a, and composite bosonic molecules, with interactions given by 0.6a, as a function of detuning ν¯, that is, magnetic ﬁeld.

However, the number equations are slightly diﬀerent. In the single-channel case, only the dressed fermions are summed over (62)

n1C =

fkσ ,

kσ

whereas, in the two-channel case, both the bare fermions and bare molecules contribute to the total fermion density (63)

n2C =

fkσ + 2|φm |2 .

kσ

Each molecule contributes two fermions to the total density. This diﬀerence results in a correction to the chemical potential μF ; a small correction in the broad-resonance case. 12. – A mean-ﬁeld description for the crossover problem In this section, we want to determine the minimal ingredients for a mean-ﬁeld theory that wants to correctly reproduce the molecule-molecule scattering between composite bosons on the BEC side of the resonance. Consider ﬁrst a schematic picture of the crossover in ﬁg. 13. The picture shows how the overlapping, loosely bound Cooper pairs on the right side of the resonance contract as the detuning is lowered and changes sign. On the BEC side, at negative detuning, the pairs turn into tightly bound molecules, which are interacting with a molecule-molecule scattering length of approximately 0.6a [30], where a is the atom-atom scattering length.

379

Two-channel models of the BCS/BEC crossover

Fig. 14. – Schematic of the interaction between two dimers of paired fermions.

. 12 1. Boson scattering length. – To ﬁnd an expression for the dimer-dimer scattering length add , which is the eﬀective interaction of the composite bosonic molecules, Petrov et al. [30] start with a four-body Schr¨ odinger equation in the set of coordinates deﬁned in ﬁg. 14, (64)

−

1 mE ∇2r1 + ∇2r2 + ∇2R + 2 Ψ = 2 ¯h

r1 + r2 m ±R Ψ, U − 2 U (r1 ) + U (r2 ) + 2 h ¯ ±

where U (r) is the two-body potential in real space. This equation is simpliﬁed by assuming a pseudopotential boundary condition (65)

r1 →0

Ψ(r1 , r2 , R) −−−→ f (r2 , R)

1 1 − r1 a

,

which is valid, because the eﬀective range of the interatomic potential U is small compared to the scattering length a. The factor multiplying f (r2 , R) on the right-hand side of eq. (65) is an expansion of the bound-state wave function exp[−r1 /a]/r1 near threshold. This boundary condition eq. (65) implies that we do not need the full four-body wave function Ψ, which is six-dimensional in a homogeneous system, to describe the dimer-dimer scattering correctly. Instead, it suﬃces to solve for the reduced wave function f (r2 , R), which has only three independent dimensions in a

Fig. 15. – This atom-molecule correlation function is the minimum ingredient needed to recover the boson-boson scattering length for the composite molecules as 0.6a. The schematic on the right shows the dimensionality of f (r2 , R) in momentum space.

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M. Holland and J. Wachter

Table I. – A more complete picture of the crossover. φm = ˆb0 ⇑ ¸¸ ˙˙ ˆb−q a ˆq/2−k↓ a ˆq/2+k↑ ⇓ a−k↓ a ˆk↑ mk = ˆ

BEC: Interactions mediated by fermions Crossover BCS: Interactions mediated by bosons

homogeneous system. This simpliﬁcation allowed the authors of [30] to solve the scattering equation (64) and ﬁnd the dimer-dimer scattering length as add ≈ 0.6a, a result that has been supported experimentally [31-33]. What is the physical meaning of the wave function f (r2 , R)? The schematic in ﬁg. 15 depicts f (r2 , R) as an atom-molecule correlation function between a tightly bound dimer and two loosely bound fermions. As we have seen in sect. 11, this correlation function is not part of BCS theory or the lowest-level mean-ﬁeld picture of the crossover we discussed . in subsubsect. 11 5.1. We did, however, see in that section how to extend the equations of motion: Equation (59) couples to a three-operator correlation function that is of the . same vector structure as f (r2 , R). To extend the set of equations in subsubsect.$$11 5.1, %% † we would have to derive an equation of motion for the new correlation function ˆbˆ a a ˆ . This correlation in turn couples to other three-operator correlation functions. We can drop all couplings to still higher-order correlations and solve the coupled three-operator equations. This yields a theory that includes the Hartree self-energy shift in the crossover and yields the correct dimer-dimer scattering length discussed above. We may thus anticipate that one should be able to combine the quantity f (r2 , R) with . the mean-ﬁeld description of the crossover that we began to present in subsubsect. 11 5.1 to get a more complete picture of the crossover as presented in table I. . 12 2. Beyond pair correlations. – In the last section, we learned that we need to include four-particle correlation functions in order ﬁnd the correct value for the molecule-molecule interactions on the BEC side. Here, we want to revisit the single- and two-channel models discussed in the context of BCS theory in sect. 11 and see how they can be extended to include these beyond-pair correlations. . 12 2.1. Four-particle correlations in the single-channel model. The Hamiltonian of the single-channel model is (66)

ˆ = H

kσ

k a ˆ†kσ a ˆkσ +

qkk

Uk−k a ˆ†q/2+k↑ a ˆ†q/2−k↓ a ˆq/2−k ↓ a ˆq/2+k ↑ .

The minimum necessary mean ﬁeld to include the required four-particle correlations in this model is (67)

$$

%% a ˆ−q/2−k↓ a ˆ−q/2+k↑ a ˆq/2−k ↓ a ˆq/2+k ↑ .

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Two-channel models of the BCS/BEC crossover

With this Hamiltonian, it is impossible to contract a fermion pair into a boson directly, so we have to treat all four particles explicitly. The four-particle correlation above is a function of three vectors, and thus has six degrees of freedom in a homogeneous system, which is numerically intractable. . 12 2.2. Four-particle correlations in the two-channel model. Let us now see whether the two-channel model has an advantage in describing the necessary four-particle correlations. The Hamiltonian for this model is q ˆb† a ˆ = (68) H + ν ˆb†q ˆbq + g k a ˆ†kσ a ˆkσ + ˆq/2+k↑ + H.c. , q ˆq/2−k↓ a 2 q k,σ=↑,↓

qk

which shows that this model contains composite molecules explicitly. The minimum correlation function to include four-particle interactions is now (69)

$$ %% ˆb−q a ˆq/2−k↓ a ˆq/2+k↑ ,

which is a function of only two momentum vectors. The dimensionality of this correlation function is thus only three in a homogeneous system, which is directly accessible in numerical calculations. The two-channel model thus gives naturally a minimal description that is consistent with the vacuum scattering properties of four-particle scattering discussed in sub. sect. 12 1. 13. – Summary Atomic physics has provided a wealth of information on a variety of aspects of superﬂuidity, both in bosonic and fermionic systems. We have presented in this article the foundation concepts of superﬂuids, and discussed the vortices which support rotation in superﬂuid systems. We have shown how the separation of scales, both in energy and in physical space, leads to a simpliﬁed parametrization of the interaction eﬀects in dilute quantum gases. Of particular interest has been Feshbach resonances, which allow the collision eﬀects to be resonantly enhanced. We have shown that for two fermions in vacuum, one is able to prove the equivalence between the two-channel approach which arises naturally in the description of Feshbach resonances, and the single-channel approach which is a typical starting point for condensed-matter theories. We should emphasize here that in the case of a many-body system, the single-channel and two-channel theories do not generally coincide. The description of Feshbach resonances in dilute atomic gases has required the development of a many-body theory able to describe strong correlations and speciﬁcally the point of inﬁnite scattering length. Careful consideration must therefore be made of the breakdown of simple mean-ﬁeld approaches which contain the scattering length explicitly. We have shown how one may include the two-channel Feshbach formulation in the many-body Hamiltonian. This problem is relevant to the theoretical description of

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M. Holland and J. Wachter

many current experimental eﬀorts, exploring the formation and dissociation of molecules around Feshbach resonances, and the crossover from fermionic to bosonic superﬂuidity. ∗ ∗ ∗ The material presented in this article has been derived from work done in collaboration with a number of people. Speciﬁcally we would like to pay special acknowledgement to the contributions of M. Chiofalo, S. Giorgini, C. Menotti, L. Viverit, C. Regal, D. Jin, S. Stringari, L. Pitaevskii, S. Kokkelmans, J. Milstein, and J. Cooper. This work was supported by the Department of Energy, Oﬃce of Basic Energy Sciences via the Chemical Sciences, Geosciences, and Biosciences Division (M.H.) and by the National Science Foundation (J.W.). REFERENCES [1] Williams J. and Holland M., Nature, 401 (1999) 568. [2] Bohn J. L., Phys. Rev. A, 61 (2000) 053409. [3] Holland M., Kokkelmans S. J. J. M. F., Chiofalo M. L. and Walser R., Phys. Rev. Lett., 87 (2001) 120406. [4] Timmermans E., Furuya K., Milonni P. W. and Kerman A. K., Phys. Lett. A, 285 (2001) 228. [5] Kokkelmans S. J. J. M. F., Milstein J. N., Chiofalo M. L., Walser R. and Holland M. J., Phys. Rev. A, 65 (2002) 053617. [6] Duine R. A. and Stoof H. T. C., J. Opt. B., 5 (2003) S212. [7] Bruun G. M. and Pethick C. J., Phys. Rev. Lett., 92 (2004) 140404. [8] Kokkelmans S. J. J. M. F. and Holland M. J., Phys. Rev. Lett., 89 (2002) 180401. [9] Pethick C. J. and Smith H., Bose-Einstein Condensation in Dilute Gases (Cambridge University Press) 2002. [10] Holland M., Menotti C. and Viverit L., preprint cond-mat/0404234 (2004). [11] Feshbach H., Ann. Phys. (N.Y.), 5 (1958) 357; 19 (1962) 287; Feshbach H., Theoretical Nuclear Physics (Wiley, New York) 1992. [12] Eagles D. M., Phys. Rev., 186 (1969) 456. [13] Leggett A. J., in Modern Trends in the Theory of Condensed Matter, edited by Pekalski A. and Przystawa R. (Springer-Verlag, Berlin) 1980, pp. 13–27. `res P. and Schmitt-Rink S., J. Low Temp. Phys., 59 (1985) 195. [14] Nozie [15] Randeria M., in Bose-Einstein Condensation, edited by Griffin A., Snoke D. and Stringari S. (Cambridge University Press, Cambridge) 1995, pp. 355–92. [16] Chiofalo M. L., Giorgini S. and Holland M., Phys. Rev. Lett., 97 (2006) 070404. [17] See, e.g., Pitaevskii L. P. and Stringari S., Bose-Einstein Condensation (Oxford University Press, Oxford) 2003. [18] See, e.g., de Gennes P. G., Superconductivity of Metals and Alloys (Addison-Wesley, California) 1966. ´ de Melo C. A. R., Phys. Rev. B, 55 (1997) [19] Engelbrecht J. R., Randeria M. and Sa 15153. [20] Nikitin E. E. and Pitaevskii L. P., preprint cond-mat/0508684. [21] Belyakov V. A., Sov. Phys. JETP, 13 (1961) 850. ˜es K. M. F., Kokkelmans [22] Bourdel T., Cubizolles J., Khaykovich L., Magalh a S. J. J. M. F., Shlyapnikov G. V. and Salomon C., Phys. Rev. Lett., 91 (2003) 020402.

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[23] Regal C. A., Greiner M., Giorgini S., Holland M. and Jin D. S., Phys. Rev. Lett., 95 (2005) 250404. [24] Huang K. and Yang C. N., Phys. Rev., 105 (1957) 767; Lee T. D., Huang K. and Yang C. N., ibid., 106 (1957) 1135. [25] Bruun G., Castin Y., Dum R. and Burnett K., Eur. Phys. J. D, 7 (1999) 433. [26] Bethe H. and Peierls R., Proc. R. Soc. London, Ser. A, 148 (1935) 146. [27] Carlson J., Chang S.-Y., Pandharipande V. R. and Schmidt K. E., Phys. Rev. Lett., 91 (2003) 050401. [28] Astrakharchik G. E., Boronat J., Casulleras J. and Giorgini S., Phys. Rev. Lett., 93 (2004) 200404. [29] Bardeen J., Cooper L. N. and Schrieffer J. R., Phys. Rev., 108 (1957) 1175. [30] Petrov D. S., Salomon C. and Shlyapnikov G. V., Phys. Rev. Lett., 93 (2004) 090404. [31] Bartenstein M., Altmeyer A., Riedl S., Jochim S., Chin C., Hecker Denschlag J. and Grimm R., Phys. Rev. Lett., 92 (2004) 120401. [32] Zwierlein M. W., Stan C. A., Schunck C. H., Raupach S. M. F., Kerman A. J. and Ketterle W., Phys. Rev. Lett., 92 (2004) 120403. [33] Bourdel T., Khaykovich L., Cubizolles J., Zhang J., Chevy F., Teichmann M., Tarruell L., Kokkelmans S. J. J. M. F. and Salomon C., Phys. Rev. Lett., 93 (2004) 050401.

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Molecular regimes in ultracold Fermi gases D. S. Petrov Laboratoire de Physique Th´ eorique et Mod`eles Statistiques, Universit´ e Paris-Sud 91405 Orsay Cedex, France Russian Research Center Kurchatov Institute - Kurchatov Square, 123182 Moscow, Russia

C. Salomon Laboratoire Kastler Brossel, Ecole Normale Sup´ erieure - 24 rue Lhomond, 75231 Paris, France

G. V. Shlyapnikov Laboratoire de Physique Th´ eorique et Mod`eles Statistiques, Universit´ e Paris-Sud 91405 Orsay Cedex, France Van der Waals-Zeeman Institute, University of Amsterdam - Valckenierstraat 65/67 1018 XE Amsterdam, The Netherlands

The use of Feshbach resonances for tuning the interparticle interaction in ultracold Fermi gases has led to remarkable developments, in particular to the creation and BoseEinstein condensation of weakly bound diatomic molecules of fermionic atoms. These are the largest diatomic molecules obtained so far, with a size of the order of thousands of angstroms. They represent novel composite bosons, which exhibit features of Fermi statistics at short intermolecular distances. Being highly excited, these molecules are remarkably stable with respect to collisional relaxation, which is a consequence of the Pauli exclusion principle for identical fermionic atoms. The purpose of these lectures is to describe molecular regimes in two-component Fermi gases and Fermi-Fermi mixtures, focusing attention on quantum statistical eﬀects.

c Societ` a Italiana di Fisica

385

386

D. S. Petrov, C. Salomon and G. V. Shlyapnikov

Introduction The ﬁeld of quantum gases is now strongly expanding in the direction of ultracold clouds of fermionic atoms, with the goal of revealing novel macroscopic quantum states and achieving various regimes of superﬂuidity. This provides tight links with condensed matter and nuclear physics, where superﬂuid pairing between fermions lies in the background of many-body quantum eﬀects and has been extensively studied during several decades. The initial idea in the ﬁeld of cold atoms was to achieve the Bardeen-CooperSchrieﬀer (BCS) superﬂuid phase transition in a two-component Fermi gas, which requires attractive interaction between the atoms of diﬀerent components. Then, in the simplest version of this transition, at suﬃciently low temperatures fermions belonging to diﬀerent components and having opposite momenta on the Fermi surface form correlated (Cooper) pairs in the momentum space. This leads to the appearance of a gap in the excitation spectrum and to the phenomenon of superﬂuidity (see, for example, [1]). In a dilute ultracold two-component Fermi gas, most eﬃcient is the formation of Cooper pairs due to the attractive intercomponent interaction in the s-wave channel (negative s-wave scattering length a). However, for ordinary values of a, the superﬂuid transition temperature is extremely low. For this reason, the eﬀorts of experimental groups have been focused on modifying the intercomponent interaction by using Feshbach resonances (see below). In this case, one can switch the sign and tune the absolute value of a, which at resonance changes from +∞ to −∞. This has led to exciting developments, such as the direct observation of superﬂuid behavior in the strongly interacting regime (n|a|3 1, where n is the gas density) through vortex formation [2], and revealing the inﬂuence of imbalance between the two components of the Fermi gas on superﬂuidity [3-7]. The physics of the strongly interacting regime is covered in other lecturing courses, and the present lectures will be focused on another interesting topic, the remarkable physics of weakly bound diatomic molecules of fermionic atoms. These molecules are formed on the positive side of the resonance (a > 0) [8-11] and they are the largest diatomic molecules obtained so far. Their size is of the order of a and it reaches thousands of angstroems in current experiments. Accordingly, their binding energy is exceedingly small (10 μK or less). Being composite bosons, these molecules obey Bose statistics, and they have been Bose-condensed in JILA experiments with 40 K2 [12,13] and in 6 Li2 experiments at Innsbruck [14,15], MIT [16,17], ENS [18], and Rice [19]. Nevertheless, some of the interaction properties of these molecules reﬂect Fermi statistics of the individual atoms forming the molecule. In particular, these molecules are found remarkably stable with respect to collisional decay. Being in the highest rovibrational state, they do not undergo collisional relaxation to deeply bound states on a time scale exceeding seconds at densities of about 1013 cm−3 , which is more than four orders of magnitude longer than the lifetime of similar molecules consisting of bosonic atoms. Currently, a new generation of experiments is being set up for studying degenerate mixtures of diﬀerent fermionic atoms, with the idea of revealing the inﬂuence of the mass diﬀerence on superﬂuid properties and ﬁnding novel types of superﬂuid pairing. On the

Molecular regimes in ultracold Fermi gases

387

positive side of the resonance one expects the formation of heteronuclear weakly bound molecules which attract a great deal of interest, in particular for creating dipolar gases. In the ﬁrst lecture we discuss diatomic molecules formed in a two-component Fermi gas by atoms in diﬀerent internal (hyperﬁne) states. They key idea is to show how one obtains an exact universal result for the elastic interaction between these weakly bound molecules and how the Fermi statistics for the atoms provides a strong suppression of their collisional relaxation into deep bound states. It is emphasized that the repulsive character of the elastic intermolecular interaction and remarkable collisional stability of the molecules are the main issues for their Bose-Einstein condensation and for prospects related to interesting manipulations with these molecular condensates. The second lecture is dedicated to heteronuclear molecules formed in mixtures of diﬀerent fermionic atoms. It is analyzed how the mass ratio for constituent atoms inﬂuences the elastic interaction between the molecules and their collisional stability. Special attention is focused on molecules of heavy and light fermions and it is shown that this system can undergo a phase transition to a crystalline state. The crystalline ordering is due to a relatively long-range interaction between the molecules originating from exchange of light fermions. Remarkably, the atomic system itself is dilute and interatomic forces are strictly short-range. This occurs at densities where na3 1, and in this respect the atomic system remains dilute. 1. – Lecture 1. Diatomic molecules in a two-component Fermi gas . 1 1. Feshbach resonances and diatomic molecules. – At ultralow temperatures, where the de Broglie wavelength of atoms greatly exceeds the characteristic radius of interatomic interaction forces, the s-wave scattering between atoms is generally the most important. Therefore, in two-component Fermi gases one may consider only the interaction between atoms of diﬀerent components, which can be tuned by using Feshbach resonances. The description of a many-body system near a Feshbach resonance requires a detailed knowledge of the 2-body problem. In the vicinity of the resonance, the energy of a colliding pair of atoms in the open channel is close to the energy of a molecular state in another hyperﬁne domain (closed channel). The coupling between these channels leads to a resonant dependence of the scattering amplitude on the detuning δ of the closed channel state from the threshold of the open channel, which can be controlled by an external magnetic (or laser) ﬁeld. Thus, the scattering length becomes ﬁeld dependent (see ﬁg. 1). The Feshbach eﬀect is a two-channel problem which can be described in terms of the Breit-Wigner scattering [20,21], and various aspects of such problems have been discussed by Feshbach [22] and Fano [23]. In cold atom physics the idea of Feshbach resonances was introduced in ref. [24], and optically induced resonances have been discussed in refs. [25, 26]. At resonance the scattering length changes from +∞ to −∞, and in the vicinity of the resonance one has the inequality n|a|3 1, where n is the gas density. The gas then is in the strongly interacting regime. It is still dilute in the sense that the

388

D. S. Petrov, C. Salomon and G. V. Shlyapnikov

a a<0

a>0

BCS

Wekly bound Molecules BEC

II

B I

III

Fig. 1. – The dependence of the scattering length on the magnetic ﬁeld. The symbols I, II, and III stand for the regime of a weakly interacting degenerate atomic Fermi gas, strongly interacting regime, and the regime of weakly bound molecules. At suﬃciently low temperatures region I corresponds to the BCS superﬂuid pairing, and region III to Bose-Einstein condensation of molecules.

mean interparticle separation greatly exceeds the characteristic radius of interparticle interaction Re . However, the amplitude of binary interactions (scattering length) is larger than the mean separation between particles, and in the quantum degenerate regime the ordinary mean ﬁeld approach is no longer valid. For a large detuning from resonance the gas is still in the weakly interacting regime, i.e. the inequality n|a|3 1 is satisﬁed. On the negative side of the resonance (a < 0), at suﬃciently low temperatures of the two-species Fermi gas one expects the BCS pairing between distinguishable fermions, well described in literature [1]. On the positive side (a > 0) two fermions belonging to diﬀerent components form diatomic molecules. For a Re these molecules are weakly bound, with a size of the order of a. The crossover from the BCS to BEC behavior attracts now a great deal of interest, in particular with respect to the nature of superﬂuid pairing, transition temperature, and elementary excitations. This type of crossover has been earlier discussed in literature in the context of superconductivity [27-30] and in relation to superﬂuidity in two-dimensional ﬁlms of 3 He [31, 32]. The idea of resonance coupling through a Feshbach resonance for achieving a superﬂuid phase transition in ultracold two-component Fermi gases has been proposed in refs. [33, 34]. The two-body physics of the Feshbach resonance is the most transparent if one can omit the (small) background scattering length. Then for low collision energies ε the scattering amplitude is given by [21] (1)

F (ε) = −

√ γ/ 2μ √ , ε + δ + iγ ε

√ where the quantity γ/ 2μ ≡ W characterizes the coupling between the two hyperﬁne domains and μ is the reduced mass of the two atoms. The scattering length is a = −F (0). In eq. (1) the detuning δ is positive if the bound molecular state is below the continuum

389

Molecular regimes in ultracold Fermi gases

of colliding atoms. Then for δ > 0 the scattering length is positive, and for δ < 0 it is negative. Introducing a characteristic length R∗ = 2 /2μW

(2)

and expressing the scattering amplitude through the relative momentum of particles √ k = 2με/, eq. (1) takes the form (3)

F (k) = −

a−1

1 . + R∗ k 2 + ik

The validity of eq. (3) does not require the inequality kR∗ 1. At the same time, this equation formally coincides with the amplitude of scattering of slow particles by a potential with the same scattering length a and an eﬀective range R = −2R∗ , obtained under the condition k|R| 1. The length R∗ is an intrinsic parameter of the Feshbach resonance problem. It characterizes the width of the resonance. From eqs. (1) and (2) we see that small W and, hence, large R∗ correspond to a narrow resonance, whereas large W and small R∗ lead to a wide resonance. The term “wide” is generally used when the length R∗ drops out of the problem, which according to eq. (3) requires the condition kR∗ 1. In a quantum degenerate atomic Fermi gas the characteristic momentum of particles is the Fermi momentum kF = (3π 2 n)1/3 . Thus, in the strongly interacting regime and on the negative side of the resonance (a < 0), for a given R∗ the condition of the wide resonance depends on the gas density n and takes the form kF R∗ 1 [35-39]. For a > 0 one has weakly bound molecular states (it is certainly assumed that the characteristic radius of interaction Re a), and for such molecular system the criterion of the wide resonance is diﬀerent [40, 41]. The binding energy of the weakly bound molecule state is determined by the pole of the scattering amplitude (3). One then ﬁnds [40, 41] that this state exists only for a > 0 and under the condition (4)

R∗ a,

the binding energy is given by (5)

ε0 = 2 /2μa2 .

The wave function of such weakly bound molecular state has only a small admixture of the closed channel, and the size of the molecule is ∼ a. Characteristic momenta of the atoms in the molecule are of the order of a−1 and in this respect the inequality (4) represents the criterion of a wide resonance for the molecular system. Under these conditions atom-molecule and molecule-molecule interactions are determined by a single parameter —the atom-atom scattering length a. In this sense, the problem becomes universal. It is equivalent to the interaction problem for the two-body

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D. S. Petrov, C. Salomon and G. V. Shlyapnikov

potential which is characterized by a large positive scattering length a and has a potential well with a weakly bound molecular state. The picture remains the same when the background scattering length can not be neglected, although the condition of a wide resonance can be somewhat modiﬁed [42]. Most ongoing experiments with Fermi gases of atoms in two diﬀerent internal (hyperﬁne) states use wide Feshbach resonances [43]. For example, weakly bound molecules 6 Li2 and 40 K2 have been produced in experiments [9-13,16,17,14,15,18,19] by using Feshbach resonances with a length R∗ 20 ˚ A, and for the achieved values of the scattering length a (from 500 to 2000 ˚ A) the ratio R∗ /a was smaller than 0.1. In these lectures we will consider the case of a wide Feshbach resonance. . 1 2. Weakly interacting gas of bosonic molecules. Molecule-molecule elastic interaction. – As we showed in the previous section, the size of weakly bound bosonic molecules formed at a positive atom-atom scattering length a in a two-species Fermi gas (region III in ﬁg. 1) is of the order of a. Therefore, at densities satisfying the inequality na3 1, one expects a weakly interacting gas of these molecules. Moreover, under this condition at temperatures suﬃciently lower than the molecular binding energy ε0 and for equal concentrations of the two atomic components, practically all atoms are converted into molecules [44]. This is deﬁnitely the case at temperatures below the temperature of quantum degeneracy Td = 2π2 n2/3 /M (the lowest one in the case of fermionic atoms with diﬀerent masses, with M being the mass of the heaviest atom). One can clearly see this by comparing Td with ε0 given by eq. (5). Thus, one has a weakly interacting molecular Bose gas and the ﬁrst question is related to the elastic interaction between the molecules. For a weakly interacting gas the interaction energy in the system is equal to the sum of pair interactions and the energy per particle is ng (2ng for a non-condensed Bose gas), with g being the coupling constant. In our case this coupling constant is given by g = 4π2 add /(M + m), where add is the scattering length for the moleculemolecule (dimer-dimer) elastic s-wave scattering, and M , m are the masses of heavy and light atoms, respectively. The value of add is important for evaporative cooling of the molecular gas to the regime of Bose-Einstein condensation and for the stability of the condensate. The Bose-Einstein condensate is stable for repulsive intermolecular interaction (add > 0), and for add < 0 it undergoes a collapse. We thus see that for analyzing macroscopic properties of the molecular bose gas one should ﬁrst solve the problem of elastic interaction (scattering) between two molecules. In the rest of this lecture we present the exact solution of this problem for homonuclear molecules formed by fermionic atoms of diﬀerent components (diﬀerent internal states) in a two-component Fermi gas. The case of M = m will be dealt with in Lecture 2. The solution for M = m was obtained in refs. [45,41] assuming that the atom-atom scattering length a greatly exceeds the characteristic radius of interatomic potential: (6)

a Re .

391

Molecular regimes in ultracold Fermi gases

Then, as in the case of the 3-body problem with fermions [46-49], the amplitude of elastic interaction is determined only by a and can be found in the zero-range approximation for the interatomic potential. This approach was introduced in the two-body physics by Bethe and Peierls [50]. The leading idea is to solve the equation for the free relative motion of two particles placing a boundary condition on the wave function ψ at a vanishing interparticle distance r: (rψ) 1 =− , rψ a

(7)

r → 0,

which can also be rewritten as ψ ∝ (1/r − 1/a),

(8)

r → 0.

One then gets the correct expression for the wave function at distances r Re . For the case where a Re , eq. (8) correctly describes the wave function of weakly bound and continuum states even at distances much smaller than a. We now use the Bethe-Peierls approach for the problem of elastic molecule-molecule (dimer-dimer) elastic scattering which is a 4-body problem described by the Schr¨ odinger equation (9)

−

√ 2 2 (∇r1 − ∇2r2 − ∇2R ) + U (r1 ) + U (r2 ) + U [(r1 + r2 ± 2R)/2] − E Ψ = 0, m ±

where m is the atom mass. Labeling fermionic atoms in diﬀerent internal states by the symbols ↑ and ↓, the distance between two given ↑ and ↓ fermions is r1 , and r2 is the distance√between the other two. √ The distance between the centers of mass of these pairs is R/ 2, and (r1 + r2 ± 2R)/2 are the separations between ↑ and ↓ fermions in the other two possible ↑↓ pairs (see ﬁg. 2). The total energy is E = −2ε0 + ε, with ε = 2 k 2 /m being the collision energy, and ε0 = −2 /ma2 the binding energy of a dimer. The wave function Ψ is symmetric with respect to the permutation of bosonic ↑↓ pairs and antisymmetric with respect to permutations of identical fermions: (10) √ √ r1 − r2 r1 + r2 ± 2R r1 + r2 ∓ 2R Ψ(r1 , r2 , R) = Ψ(r2 , r1 , −R) = −Ψ ,± √ . 2 2 2 For the weak binding of atoms in the molecule assuming that the 2-body scattering length satisﬁes the inequality (6), at all interatomic distances (even much smaller than a) except for very short separations of the order of or smaller than Re , the motion of atoms in the 4-body system obeys the free-particle Schr¨ odinger equation (11)

∇2r1

+

∇2r2

+

∇2R

mE + 2 Ψ = 0.

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D. S. Petrov, C. Salomon and G. V. Shlyapnikov

2 r1 1

1

R

2 r2 2

Fig. 2. – Set of coordinates for the four-body problem.

The correct description of this motion requires the 4-body wave function Ψ to satisfy the Bethe-Peierls boundary condition for a vanishing distance in any pair of ↑ and ↓ fermions, √ i.e. for r1 → 0, r2 → 0, and r1 + r2 ± 2R → 0. Due to the symmetry condition (10) it is necessary to require a proper behavior of Ψ only at one of these boundaries. For r1 → 0 the boundary condition reads: (12)

Ψ(r1 , r2 , R) → f (r2 , R)(1/4πr1 − 1/4πa).

The function f (r2 , R) contains the information about the second pair of particles when the ﬁrst two are on top of each other. In the ultracold limit, where the condition (13)

ka 1

is satisﬁed, the scattering is dominated by the contribution of the s-wave channel. The inequality (13) is equivalent to ε ε0 and, hence, the s-wave scattering can be analyzed from the solution of eq. (11) with E = −2ε0 < 0. For large R the corresponding wave function takes the form (14)

Ψ ≈ φ0 (r1 )φ0 (r2 )(1 −

√

2add /R);

R a,

where the wave function of a weakly bound molecule is given by (15)

1 φ0 (r) = √ exp[−r/a]. 2πa r

393

Molecular regimes in ultracold Fermi gases

Combining eqs. (12) and (14) we obtain the asymptotic expression for f at large distances R: f (r2 , R) ≈ (2/r2 a) exp[−r2 /a](1 −

(16)

√

2add /R);

R a.

In the case of s-wave scattering the function f depends only on three variables: the absolute values of r2 and R, and the angle between them. We now derive and solve the equation for f . The value of the molecule-molecule scattering length add is then deduced from the behavior of f at large R governed by eq. (16). We ﬁrst establish a general form of the wave function Ψ satisfying eq. (11), with the boundary condition (12) and symmetry relations (10). In our case the total energy E = −22 /ma2 < 0, and the Green’s function of eq. (11) reads √ √ G(X) = (2π)−9/2 (Xa/ 2)−7/2 K7/2 ( 2 X/a),

(17)

where X = |S − S |, and S = {r1 , r2 , R} is a 9-component vector. Accordingly, one has |S − S | = (r1 − r 1 )2 + (r2 − r 2 )2 + (R − R )2 . The 4-body wave function Ψ is regular everywhere except for vanishing distances between ↑ and ↓ fermions. Therefore, it can be expressed through G(|S − S |) with coordinates S corresponding √ to a vanishing distance between ↑ and ↓ fermions, i.e. for r 1 → 0, r 2 → 0, and (r 1 +r 2 ± 2R )/2 → 0. Thus, for the wave function Ψ satisfying the symmetry relations (10) we have

d3 r d3 R G(|S − S1 |) + G(|S − S2 |) − −G(|S − S+ |) − G(|S − S− |) h(r , R ),

Ψ(S) = Ψ0 +

(18)

√ √ √ where S1 = {0, r , R }, S2 = {r , 0, −R }, and S± = {r /2±R / 2, r /2∓R / 2∓r 2}. The function Ψ0 is a properly symmetrized ﬁnite solution of eq. (11), regular at any distances between atoms. For E < 0, non-trivial solutions of this type do not exist and we have to put Ψ0 = 0. The function h(r2 , R) has to be determined by comparing Ψ (18) at r1 → 0, with the boundary condition (12). Considering the limit r1 → 0 we extract the leading terms on the right-hand side of eq. (18). These are the terms that behave as 1/r1 or remain ﬁnite in this limit. The last three terms in the square brackets in eq. (18) provide a ﬁnite contribution (19)

d3 r d3 R h(r , R ) G(|S¯2 − S2 |) − G(|S¯2 − S+ |) − G(|S¯2 − S− |) ,

where S¯2 = {0, r2 , R}. For ﬁnding the contribution of the ﬁrst term in the square brackets we subtract from this term and add to it an auxiliary quantity (20)

h(r2 , R)

G(|S − S1 |)d3 r d3 R =

√ h(r2 , R) exp[− 2r1 /a]. 4πr1

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D. S. Petrov, C. Salomon and G. V. Shlyapnikov

The result of the subtraction yields a ﬁnite contribution which for r1 → 0 can be written as d3 r d3 R [h(r , R ) − h(r2 , R)]G(|S − S1 |) = (21) = P d3 r d3 R [h(r , R ) − h(r2 , R)]G(|S¯2 − S1 |); r1 → 0, with the symbol P standing for the principal value of the integral over dr (or dR ). A detailed derivation of eq. (21) and the proof that the integral in the second line of this equation is convergent are given in ref. [41] and are omitted in this lecture. In the limit r1 → 0, the right-hand side of eq. (20) is equal to h(r2 , R)(1/4πr1 −

(22)

√

2/4πa).

We thus ﬁnd that for r1 → 0 the wave function Ψ of eq. (18) takes the form Ψ(r1 , r2 , R) =

(23)

h(r2 , R) + R; 4πr1

r1 → 0,

where R is the sum of regular r1 -independent terms given by eqs. (19) and (21), and by the second term on the right-hand side of eq. (22). Equation (23) should coincide with eq. (12), and comparing the singular terms of these equations we ﬁnd h(r2 , R) = f (r2 , R). As the quantity R should coincide with the regular term of eq. (12), equal to −f (r2 R)/4πa, we obtain the following equation for the function f :

(24)

8 d3 r d3 R G(|S¯ − S1 |)[f (r , R ) − f (r, R)] + [G(|S¯ − S2 |) − 9 √ − G(|S¯ − S± |)]f (r , R ) = ( 2 − 1)f (r, R)/4πa. ±

Here S¯ = {0, r, R}, and we omitted the symbol of principal value for the integral in the ﬁrst line of eq. (24). As we already mentioned above, for the s-wave scattering the function f (r, R) depends only on the absolute values of r and R and on the angle between them. Thus, eq. (24) is an integral equation for the function of three variables. This equation is solved numerically for all distances R and r, and all angles between the vectors R and r. Fitting the asymptotic expression (16) at R a with the function f (r, R) obtained numerically from eq. (24), we ﬁnd with 2% accuracy that the dimer-dimer scattering length is [45,41] (25)

add = 0.6a > 0.

The calculations show the absence of 4-body weakly bound states, and the behavior of f at small R suggests a soft-core repulsion between dimers, with a range ∼ a.

395

Molecular regimes in ultracold Fermi gases

U h2 H 0 = ma 2 R

Fig. 3. – Interaction potential U as a function of the distance R between two distinguishable fermionic atoms. The dashed line shows the energy level of the weakly bound molecule, and the solid line the energy level of a deeply bound state.

The result of eq. (25) is exact, and eq. (25) indicates the stability of molecular BEC with respect to collapse. Compared to earlier studies which assumed add = 2a [30, 51], eq. (25) gives almost twice as small a sound velocity of the molecular condensate and a rate of elastic collisions smaller by an order of magnitude. The result of eq. (25) has been conﬁrmed by Monte Carlo calculations [52] and by calculations within the diagrammatic approach [53, 54]. . 1 3. Suppression of collisional relaxation. – Weakly bound dimers that we are considering are diatomic molecules in the highest ro-vibrational state (see ﬁg. 3). Hence, they can undergo relaxation into deep bound states in their collisions with each other: for example, one of the colliding molecules goes to a deep bound state and the other one gets dissociated(1 ). The released energy is the binding energy of the deep state, which is of the order of 2 /mRe2 . It is transformed into the kinetic energy of particles in the outgoing collisional channel and they escape from the trapped sample. Therefore, the process of collisional relaxation of weakly bound molecules determines the lifetime of a gas of these molecules and possibilities to Bose-condense this gas. We now show that collisional relaxation is suppressed due to Fermi statistics for atoms in combination with a large size of weakly bound molecules [45, 41]. The binding energy of the molecules is ε0 = 2 /ma2 and their size is ∼ a Re . The size of deep bound states is of the order of Re . Hence, the relaxation requires the presence of at least three fermionic atoms at distances ∼ Re from each other. As two of them are necessarily identical, due to the Pauli exclusion principle the relaxation probability acquires a small (1 ) Including p-wave interactions, one can think of the formation of deeply bound states by two identical (↑ or ↓) fermions. So, the collision of two weakly bound molecules can lead to the creation of a deep bound state by two ↑ (or ↓) fermionic atoms, and two ↓ (or ↑) atoms become unbound.

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R e <
a Fig. 4. – Conﬁguration space contributing to the relaxation probability.

factor proportional to a power of (qRe ), where q ∼ 1/a is a characteristic momentum of the atoms in the weakly bound molecular state. Relying on the inequality a Re we outline a method that allows us to establish the dependence of the relaxation rate on the scattering length a, without going into a detailed analysis of the short-range behavior of the systems. It is assumed that the amplitude of the inelastic process of relaxation is much smaller than the amplitude of elastic scattering. Then the dependence of the relaxation rate on a is related only to the a-dependence of the initial-state 4-body wave function Ψ. We again consider the ultracold limit described by the condition (13), where the relaxation process is dominated by the contribution of the s-wave molecule-molecule scattering. The key point is that the relaxation process requires only three atoms to approach each other to short distances of the order of Re . The fourth particle can be at a large distance from these three and, in this respect, does not participate in the relaxation process. This distance is of the order of the size of a molecule, which is ∼ a Re . We thus see that the conﬁguration space contributing to the relaxation probability can be viewed as a system of three atoms at short distances ∼ Re from each other and a fourth atom separated from this system by a large distance ∼ a (see ﬁg. 4). In this case the 4-body wave function decomposes into a product: (26)

Ψ = η(z)Ψ(3) (ρ, Ω),

where Ψ(3) is the wave function of the 3-fermion system, ρ and Ω are the hyperradius and the set of hyperangles for these fermions, z is the distance between their center of mass and the fourth atom. The wave function η(z) describes the motion of this atom and is normalized to unity. Note that eq. (26) remains valid for any hyperradius ρ |z| ∼ a. The transition to a deep bound 2-body state occurs in the system of three atoms and does not change the wave function of the fourth atom, η(z). Therefore, averaging the transition probability over the motion of the fourth particle, the rate constant of relaxation in dimer-dimer collisions can be written as (27)

αrel = α(3)

|η(z)|2 d3 z = α(3) ,

where α(3) is the rate constant of relaxation for the 3-atom system.

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Molecular regimes in ultracold Fermi gases

At interatomic distances ∼ Re where the relaxation occurs, as well as at all distances where the hyperradius ρ a, the wave function Ψ(3) is determined by the Schr¨ odinger equation with zero energy and, hence, depends on the scattering length a only through a normalization coeﬃcient: (28)

Ψ(3) = A(a)ψ;

ρ a,

where the function ψ is independent of a. The probability of relaxation and, hence, the relaxation rate constant are proportional to |Ψ|2 at distances ∼ Re . We thus have (29)

αrel = α(3) ∝ |A(a)|2 .

The goal then is to ﬁnd the coeﬃcient A(a), which determines the dependence of the relaxation rate on a. For this purpose it is suﬃcient to consider distances where a ρ Re and eq. (28) is still valid. Then, using the zero-range approximation one ﬁnds the coordinate dependence of the three-body wave function Ψ(3) . The derivation is made in ref. [41] and here we only present the result: (30)

Ψ(3) = A(a)Φν (Ω)(ρ/a)ν−1 ,

ρ a,

where Φν (Ω) is a normalized function of hyperangles, and the coeﬃcient ν depends on the symmetry of Ψ(3) . The a-dependence of the prefactor A(a) can be determined from the following scaling arguments for the 4-body problem. The scattering length a is the only length scale in our problem and we can measure all distances in units of a. Then, turning to rescaled coordinates ρ = aρ and z = az we see that Ψ(3) in eq. (37) becomes a function of ρ/a, multiplied by A(a)aν−1 . The wave function η(z) is normalized to unity and hence it is a function of z/a, multiplied by a−3/2 . Accordingly, the 4-body wave function Ψ of eq. (26) will be a function of rescaled coordinates, multiplied by the coeﬃcient A(a)aν−5/2 . By applying the same rescaling to eq. (14) with the account of eq. (15) we see that the same coeﬃcient should be proportional to a−3 . Therefore, A(a) ∝ a−ν−1/2 and αrel ∝ a−s , where s = 2ν + 1. The strongest relaxation channel corresponds to the lowest value of ν. It is achieved in the case of p-wave symmetry in the three-body system described by wave function Ψ(3) and is equal to ν = 0.773, which leads to s = 2.55. Assuming that the short-range physics is characterized by the length scale Re and the energy scale 2 /mRe2 we can restore the dimensions and write (31)

αrel = C(Re /m)(Re /a)s ;

s = 2.55,

where the coeﬃcient C depends on a particular system and cannot be obtained using the zero-range approximation. Note that the p-wave symmetry in the three-body system corresponds to the p-wave scattering of a fermionic atom of one of the molecules (referred to as the 3rd fermion)

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on the other molecule. Then the 4th particle also undergoes p-wave scattering on this molecule in such a way that the total orbital angular momentum of the molecule-molecule collision is equal to zero. Since the 3rd and 4th fermions are bound to each other in the molecular state with a size ∼ a, the relative momentum of their collisions with the other molecule is ∼ 1/a and such p-wave collisions are not at all suppressed. The relaxation channel corresponding to the s-wave scattering of the 3rd fermion on the molecule leads to ν = 1.1662 and hence to the relaxation rate proportional to a−3.33 as in the case of ultracold atom-molecule collisions [45, 41]. Thus, for large a this mechanism can be omitted. The channels, where the 3rd fermion (and the 4th one) scatters on the molecule with orbital angular momentum l > 1, lead to even stronger decrease of the relaxation rate with increasing a and hence can be neglected. Equation (31) shows a remarkable collisional stability of weakly bound molecules consisting of fermionic atoms in two diﬀerent internal states and a counterintuitive decrease of the relaxation rate with increasing the two-body scattering length a. For currently achieved values of the 2-body scattering length a ∼ 1000 ˚ A, the suppression factor (Re /a)s for the relaxation process is about 4 orders of magnitude. This eﬀect is due to Fermi statistics for the atoms. It is not present for weakly bound molecules of bosonic atoms, even if they have the same large size. Indeed, as the size of weakly bound molecules is ∼ a, identical fermionic atoms participating in the relaxation process have very small relative momenta k ∼ 1/a. Hence, the probability that they approach each other to short distances ∼ Re where the relaxation transitions occur, should be suppressed as (kRe )2 ∼ (Re /a)2 compared to the case of molecules of bosonic atoms. The exponent s in eq. (31) is diﬀerent from 2 due to the Frank-Condon factor for the relaxation transition and three-body dynamics. The remarkable collisional stability of weakly bound molecules K2 and Li2 consisting of two fermionic atoms has been observed in experiments at JILA [11-13], Innsbruck [10, 14, 15], MIT [16, 17], ENS [9, 18], and Rice [19]. At molecular densities n ∼ 1013 cm−3 the lifetime of the gas ranges from tens of milliseconds to tens of seconds, depending on the value of the scattering length a. A strong decrease of the relaxation rate with increasing a, following from eq. (31), is consistent with experimental data. The potassium experiment at JILA [11] and lithium experiment at ENS [18] give the relaxation rate constant αrel ∝ a−s , with s ≈ 2.3 for K2 , and s ≈ 1.9 for Li2 , in agreement with theory (s ≈ 2.55) within experimental uncertainty. The experimental and theoretical results for lithium are shown in ﬁg. 5. The absolute value of the rate constant for a 6 Li2 condensate is αrel ≈ 1 × 10−13 cm3 /s for the scattering length a ≈ 110 nm. For K2 it is by an order of magnitude higher at the same value of a [11], which can be a consequence of a larger value of the characteristic radius of interaction Re . . 1 4. Prospects for manipulations with weakly bound molecules. – The suppression of the relaxation decay rate of weakly bound molecules of fermionic atoms has a crucial consequence for the physics of these molecules. At realistic temperatures the relaxation rate constant αrel is much smaller than the rate constant of elastic collisions 8πa2dd vT , where vT is the thermal velocity. For example, for the Li2 weakly bound molecules at a

399

Molecular regimes in ultracold Fermi gases

=rel [10-13 cm3.s-1]

10

1

0.1 40

100

a [nm]

200

Fig. 5. – Two-body decay rate αrel of a 6 Li2 molecular condensate as a function of interatomic scattering length near the 6 Li Feshbach resonance at 834 G. Solid line: least-square ﬁt, αrel ∝ a−1.9±0.8 . Dashed line, theory: αrel ∝ a−2.55 . The theoretical relaxation rate has been normalized to the experimental value at a = 78 nm.

temperature T ∼ 3 μK and a ∼ 800 ˚ A, the corresponding ratio is of the order of 10−4 or −5 10 . This opens wide possibilities for reaching BEC of the molecules and cooling the Bose-condensed gas to temperatures of the order of its chemical potential. Long-lived BEC of weakly bound molecules has been recently observed for 40 K2 at JILA [12,13] and for 6 Li2 at Innsbruck [14, 15], MIT [16, 17], ENS [18], and Rice [19]. Measurements of the molecule-molecule scattering length conﬁrm the result add = 0.6a with accuracy up to 30% [15, 18]. The long lifetime of weakly bound bosonic molecules of fermionic atoms allows interesting manipulations with these molecules. One of them is related to achieving the superﬂuid BCS regime which has not been obtained so far because of diﬃculties with evaporative of fermionic atoms due to Pauli blocking of their elastic collisions. In the ﬁrst stage, one arranges a deep evaporative cooling of the molecular Bose-condensed gas to temperatures of the order of the chemical potential and then converts the molecular BEC into fermionic atoms by adiabatically changing the scattering length to negative values. This provides an additional cooling, and the obtained atomic Fermi gas will have extremely low temperatures T ∼ 10−2 TF , where TF is the Fermi temperature. The gas can then enter the superﬂuid BCS regime [55]. Moreover, at such temperatures one has a very strong Pauli blocking of elastic collisions and expects the collisionless regime for the thermal cloud. This is promising for identifying the BCS-paired state through the observation of collective oscillations or free expansion [56, 57, 18, 15, 58]. It will also be interesting to transfer the weakly bound molecules of fermionic atoms to their ground (or less excited) ro-vibrational state by using two-photon spectroscopy, as has been done in refs. [59-61] for molecules of bosonic atoms. Long lifetime of weakly bound molecules of fermionic atoms at densities ∼ 1013 cm−3 should provide more eﬃcient production of ground-state molecules compared to the case of molecules of bosonic atoms in the gas phase, where one has severe limitations on achievable densities and

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lifetimes. One can then extensively study the physics of molecular Bose-Einstein condensation. Moreover, heteronuclear molecules which are supposed to be formed in mixtures of diﬀerent fermionic atoms, can be polarized by an electric ﬁeld. One then gets a gas of dipoles interacting via anisotropic long-range forces, which drastically changes the physics of Bose-Einstein condensation (see, e.g., [62] and references therein). 2. – Lecture 2. Molecular regimes in Fermi-Fermi mixtures In this lecture we mostly focus on novel physics of heteronuclear molecules which are expected to be formed in a mixture of two diﬀerent fermionic atoms (Fermi-Fermi mixture) at a large positive 2-body scattering length a for the interaction between them. In several aspects, the physics is similar to that discussed in Lecture 1 for homonuclear molecules of fermionic atoms in diﬀerent internal states. However, for a large mass ratio of the atoms, the situation drastically changes. This is related to the existence of 3-body bound states (Eﬁmov states), which in general makes it impossible to describe the molecule-molecule scattering relying only on the known value of the 2-body scattering length a. On the other hand, a long-range interaction between the molecules consisting of heavy and light fermions is characterized by a strong exchange repulsion, which can lead to the appearance of a molecular crystalline phase in the system. . 2 1. Inﬂuence of the mass ratio on the elastic intermolecular interaction. – We start with calculating the amplitude of elastic interaction (scattering) between weakly bound heteronuclear molecules consisting of a heavy (mass M ) and light (mass m) fermionic atoms, assuming that the atom-atom scattering length satisﬁes the inequality a Re and again considering the ultracold limit determined by the condition (13). In this case the scattering is dominated by the contribution of the s-wave channel, and we present here the exact results obtained in ref. [63] using the zero-range approximation. The inequality ka 1 is equivalent to the collision energy much smaller than the molecular binding energy ε0 . Hence, the s-wave molecule-molecule elastic scattering can be analyzed putting the total energy equal to −2ε0 = −2 /μa2 . In the zero-range approximation one should solve the four-body free-particle Schr¨ odinger equation which again can be written in the form (11): −∇2r1 − ∇2r2 − ∇2R + 2/a2 Ψ = 0, where r1 is the distance between two given heavy and light fermions, and r2 the distance between the other two (see ﬁg. 6). However, it is now more convenient to deﬁne the distance between the centers of mass of these pairs as βR, and the separations between heavy and light fermions in the other two possible heavy-light pairs as r± = α± r1 + √ α∓ r2 ± βR, with β = 2α+ α− , α+ = μ/M , α− = μ/m, and μ = mM/(m + M ) being the reduced mass. The symmetry condition (10) then takes the form (32)

Ψ(r1 , r2 , R) = Ψ(r2 , r1 , −R) = −Ψ(r± , r∓ , ±β(r1 − r2 ) ∓ (α+ − α− )R),

401

Molecular regimes in ultracold Fermi gases

m

r1

m

r2

ER M

M

Fig. 6. – Set of coordinates for the four-body problem with two heteronuclear molecules.

and the Bethe-Peierls boundary condition should be placed for a vanishing distance in any pair of heavy and light fermions, i.e. for r1 → 0, r2 → 0, and r± → 0. For r1 → 0 it is again given by eq. (12). Due to the redeﬁnition of the coordinates the asymptotic expression for the wave function Ψ at large distances R now reads Ψ ≈ φ0 (r1 )φ0 (r2 )(1 − add /βR);

(33)

R a,

where the notation add is again used for the molecule-molecule scattering length, and the wave function of a weakly bound molecule is given by eq. (15). Then the asymptotic expression for the function f (r2 , R) at large R is given by f (r2 , R) ≈ (2/r2 a) exp[−r2 /a](1 − add /βR);

(34)

R a.

For the s-wave scattering the function f depends only on three variables: the absolute values of r2 and R, and the angle between them. Using the procedure described in . subsect. 1 2 we obtain for f the same integral equation (24). The eﬀect of diﬀerent masses is contained in the expressions for the vectors S± , which now read S± = {α∓ r ± βR , α± r ∓ βR , ∓βr ∓ (α+ − α− )R }. For ﬁnding the molecule-molecule scattering length as a function of the mass ratio M/m one has to perform an extended set of calculations, and it is more convenient to do this the space, transforming eq. (24) into an equation for the function momentum 3 3 f (k, p) = d rd Rf (r, R) exp[ik · r/a + iβp · R/a]: (35)

±

=

2+

β 2 p2

f (k ± α∓ (p − p), p )d3 p = + (k ± α∓ (p − p))2 + (k ± α± (p + p))2

2π 2 (1 + k 2 + β 2 p2 )f (k, p) f (k , −p)d3 k . − 2 2 2 2 2+k +k +β p 2 + k 2 + β 2 p2 + 1

402

D. S. Petrov, C. Salomon and G. V. Shlyapnikov 1.2 1.1

add /a

1.0 0.9 0.8 0.7 0.6 1

2

4

8

16

M/m

Fig. 7. – The ratio add /a vs. M/m.

By making the substitution f (k, p) = (δ(p) + g(k, p)/p2 )/(1 + k 2 ) we reduce eq. (35) to an inhomogeneous equation for the function g(k, p). For p → 0 this function tends to a ﬁnite value independent of k. The molecule-molecule scattering length is given by add = −2π 2 a limp→0 g(k, p), and it was calculated numerically from eq. (35) [63]. In ﬁg. 7 we display the ratio add /a vs. the mass ratio M/m. For the case of homonuclear molecules (m = M ) we recover the molecule-molecule scattering length add = 0.6a given in Lecture 1. The universal dependence of add /a on the mass ratio, presented in ﬁg. 7, can be established in the zero-range approximation only if M/m is smaller than 13.6. Calculations then show the absence of four-body weakly bound states, and for M/m ∼ 1 the behavior of f suggests a soft-core repulsion between molecules, with a range ∼ a. For the mass ratio larger than the limiting value 13.6, the description of the molecule-molecule scattering requires a three-body parameter coming from the short-range behavior of the three-body subsystem consisting of one light and two heavy fermions [46, 47]. A qualitative explanation of this behavior will be given in next section of this lecture. . 2 2. Collisional relaxation. Exact results and qualitative analysis. – The most exciting physics with weakly bound (heteronuclear) bosonic molecules consisting of diﬀerent fermionic atoms is related to their collisional stability. As well as homonuclear molecules discussed in Lecture 1, they are in the highest rovibrational state and hence undergo relaxation into deeply bound states in molecule-molecule collisions, which leads to decay of the sample. The collisional relaxation determines the lifetime of the Bose gas of weakly bound molecules and there is a subtle question of whether and how the mass ratio M/m can inﬂuence the suppression of this process [63], originating from the Fermi statistics for the atoms and playing a crucial role in the case of homonuclear molecules discussed . in subsect. 1 3. In a similar way, behaving themselves as point-like bosons at large intermolecular distances, heteronuclear molecules “start remembering” that they consist of fermions when the intermolecular separation becomes smaller than the molecule size

Molecular regimes in ultracold Fermi gases

403

(∼ a). The relaxation requires the presence of at least three fermions at distances ∼ Re from each other and two of them are necessarily identical, so that due to the Pauli exclusion principle the relaxation probability acquires a small factor proportional to a power of (qRe ), where q ∼ 1/a is a characteristic momentum of the atoms in the weakly bound molecular state. What changes in this picture when the fermionic atoms forming a molecule have diﬀerent masses? We ﬁrst consider molecule-molecule relaxation collisions for the case where the mass ratio is smaller than the limiting value 13.6 and short-range physics is not supposed to inﬂuence the dependence of the relaxation rate on the 2-body scattering length a. As . well as in the case of homonuclear molecules in subsect. 1 3, we assume the inequality a Re and consider the ultracold limit described in eq. (13). The conﬁguration space contributing to the relaxation probability can be again viewed as a system of only three atoms at short distances ∼ Re from each other and a fourth atom separated from this system by a large distance ∼ a. Hence, the four-body wave function decomposes into a product according to eq. (26): Ψ = η(z)Ψ(3) (ρ, Ω), with Ψ(3) being the wave function of the three-fermion system, ρ a and Ω the hyperradius and the set of hyperangles for these fermions, z the distance between their center of mass and the fourth atom, and the function η(z) describing the motion of this atom. In the case of fermionic atoms with diﬀerent masses one has two possible choices of a three-body subsystem out of four fermions. The most important is the relaxation in the system of one atom with the mass m and two heavier atoms with masses M . . We then use the same arguments as in subsect. 1 3 of Lecture 1 and obtain eq. (30) for the function Ψ(3) at distances where Re ρ a: Ψ(3) = A(a)Φν (Ω)(ρ/a)ν−1 , with the coeﬃcient A(a) determining the a-dependence of the relaxation rate according to . eq. (29): αrel = α(3) ∝ |A(a)|2 . A similar scaling procedure as in subsect. 1 3 again leads −s to αrel ∝ a , where s = 2ν + 1, and restoring the dimensions we can write the relaxation rate in the form (31): αrel = C(Re /m)(Re /a)s , with a coeﬃcient C depending on the mass ratio and on short-range physics. However, the exponent s now depends not only on the symmetry of the three-body wave function Ψ(3) , but also on the mass ratio M/m. The smallest value of ν, i.e. the one corresponding to the leading relaxation channel at large a, is achieved for the pwave symmetry in the system of one light and two heavy fermions [47]. In the interval −1 ≤ ν < 2 it is given by the root of the function [47] (36)

λ(ν) =

ν(ν + 2) πν ν sin γ cos(νγ + γ) − sin(νγ) cot + , ν+1 2 (ν + 1) sin2 γ cos γ sin(πν/2)

where γ = arcsin[M/(M + m)]. A detailed derivation of eq. (36) is given in ref. [47]. . In the case of equal masses we recover s = 2ν + 1 ≈ 2.55 obtained in subsect. 1 3, and it slowly decreases with increasing the mass ratio (see ﬁg. 8). For M/m ∼ 1 nothing dramatic happens: the suppression of the relaxation rate when increasing the 2-body scattering length a becomes slightly weaker than for homonuclear molecules. However, for the mass ratio approaching the limiting value 13.6 the exponent s ﬁrst reaches zero

404

D. S. Petrov, C. Salomon and G. V. Shlyapnikov

2

s

1

0

-1 2

4

6

8

10

12

14

M/m

Fig. 8. – The dependence of the exponent s = 2ν + 1 in eq. (31) on the mass ratio M/m.

and then becomes negative, showing even an increase of the relaxation rate with a. We will give a qualitative explanation of this phenomenon relying on the Born-Oppenheimer approximation for the system of two heavy and one light atom. In the Born-Oppenheimer picture one assumes that the heavy atoms move much slower than the light atom and one ﬁrst ﬁnds the wave function and energy of a bound state of the light atom with two heavy atoms ﬁxed at their positions x1 and x2 . This energy adiabatically ajusts itself to a given separation between the heavy atoms, x = x1 − x2 , and represents an eﬀective potential Ueﬀ (x) in which they undergo their relative motion. For x < a there is only one bound state of a light atom interacting with two ﬁxed heavy atoms and the wave function of this state is symmetric with respect to their permutation. It can be written as (37)

ψx (y) ∝

e−κ(x)|y−x1 |/x e−κ(x)|y−x2 |/x + |y − x1 | |y − x2 |

,

where y is the coordinate of the light atom, and the function κ(x) follows from the BethePeierls boundary condition (8) placed on ψx (y) at vanishing distances |y − x1,2 |. The limit of κ(x) for x a this leads to κ = 0.56, and the energy of the bound state given by ε(x) = −2 κ2 (x)/2mx2 represents the eﬀective potential Ueﬀ (x) = −0.162 /mx2 . We thus see that when the heavy atoms are separated from each other by a distance x a, the light atom mediates an eﬀective 1/x2 attraction between them. Actually, this follows from the Eﬁmov picture of eﬀective interaction in a three-body system [46] and the Born-Oppenheimer approximation only gives a physically transparent illustration of this picture [63, 64]. For a large mass ratio the mediated attractive potential Ueﬀ (x) = −0.162 /mx2 strongly modiﬁes the physics of the relaxation process. It competes with the Pauli principle which in terms of eﬀective interaction manifests itself in the centrifugal 1/x2 repulsion between the heavy atoms. The presence of this repulsion is clearly seen from the fact that the light-atom wave function ψx (y) does not change sign

Molecular regimes in ultracold Fermi gases

405

under permutation of heavy fermions. As the total wave function of the three-body system ψx (y)χ(x) is antisymmetric with respect to this permutation, the wave function of the relative motion of heavy atoms χ(x) should change its sign. Therefore, χ(x) contains only partial waves with odd angular momenta, and for the lowest angular momentum (p-wave) the centrifugal barrier is Uc (x) = 22 /M x2 . For comparable masses it is significantly stronger than Ueﬀ (x). Thus, we have the physical picture discussed in the case of homonuclear molecules in Lecture 1: the Pauli principle (centrifugal barrier) reduces the probability for the atoms to be at short distances and, as a consequence, the relaxation rate decreases with increasing the atom-atom scattering length a. The role of the eﬀective attraction increases with M/m. As a result, the decrease of the relaxation rate with increasing a becomes weaker. The exponent s in eq. (31) continuously decreases with increasing M/m and becomes zero for M/m = 12.33 (see ﬁg. 8). In the Born-Oppenheimer picture this means that at this point one has a balance between the mediated attraction and the centrifugal repulsion. A further increase in M/m makes s negative and it reaches the value s = −1 for the critical mass ratio M/m = 13.6. Thus, in the range 12.33 < M/m < 13.6 the relaxation rate increases with a. For an overcritical mass ratio M/m > 13.6 we have a well-known phenomenon of the fall of a particle to the center in an attractive 1/x2 potential [21]. As has been . mentioned in subsect. 2 1, in this case the shape of the wave function at distances of the order of Re can signiﬁcantly inﬂuence the large-scale behavior and a short-range three-body parameter is required to describe the system. The wave function of heavy atoms χ(x) acquires many nodes at short distances x, which indicates the appearance of 3-body bound (Eﬁmov) states. . . 2 3. Molecules of heavy and light fermionic atoms. – The discussion of subsect. 2 2 shows that weakly bound molecules of heavy and light fermions become collisionally unstable for the mass ratio M/m close to the limiting value 13.6. The eﬀect of the Pauli principle becomes weaker than the attraction between heavy atoms at distances x a, mediated by light fermions. However, this picture explains only the dependence of the relaxation rate on the 2-body scattering length a, whereas for heteronuclear molecules the relaxation rate can also depend on the mass ratio irrespective of the value of a and short-range physics. For revealing this dependence, we will look at the interaction between the molecules of heavy and light fermions at large intermolecular separations. We consider the interaction between two such molecules in the Born-Oppenheimer approximation, assuming that heavy fermionic atoms are moving slowly in the ﬁeld produced by the exchange of fast light atoms. We ﬁrst calculate the wave functions and binding energies of two light fermions in the ﬁeld of two heavy atoms ﬁxed at their positions x1 and x2 . Assuming that the bound states of the light atoms adiabatically adjust themselves to a given value of the separation between the heavy ones, x = |x1 −x2 |, the sum of the corresponding binding energies gives an eﬀective interaction potential Ueﬀ (x) for the heavy fermions. For x > a, there are two bound states of a light atom interacting with a pair of ﬁxed heavy atoms. One of these states is symmetric and the other one antisymmetric with

406

D. S. Petrov, C. Salomon and G. V. Shlyapnikov

respect to permutation of heavy atoms. In analogy with eq. (37), the corresponding light-atom wave functions are given by (38)

ψx,± (y) ∝

e−κ± (x)|y−x1 |/x e−κ± (x)|y−x2 |/x ± |y − x1 | |y − x2 |

,

where y is again the coordinate of the light atom, and the symbols + and − label the symmetric and antisymmetric states, respectively. The functions κ± (x) > 0 are determined by the requirement that for y → x1,2 the wave function (38) satisﬁes the Bethe-Peierls boundary condition (8) in which r is substituted by |y − x1,2 |. This gives the algebraic equations (39)

κ± (x) ∓ e−κ± (x) = x/a,

and the energies of the bound states are given by ε± (x) = −2 κ2± (x)/2mx2 . Since the light fermions are identical, their two-body wave function can be constructed as an antisymmetrized product of ψx,+ and ψx,− : (40)

ψx (y1 , y2 ) ∝ ψx,+ (y1 )ψx,− (y2 ) − ψx,+ (y2 )ψx,− (y1 ),

and their total energy gives the eﬀective potential Ueﬀ (x) = ε+ (x) + ε− (x). Note that the wave function (40) is antisymmetric with respect to the transformation x → −x. This means that solving the Shr¨ odinger equation for the heavy fermions, (41)

2 − ∇2x + Ueﬀ (x) χ(x) = Eχ(x), M

we have to look for a solution that is symmetric with respect to their permutation (in our case it corresponds to the s-wave scattering). Then the total four-body wave function Ψ(x1 , x2 , y1 , y2 ) ∝ χ(x)ψx (y1 , y2 ) has a proper symmetry. Solving eqs. (39) at distances x > a gives a purely repulsive potential monotonically decreasing with x (see ﬁg. 9). At distances x a it takes the form of the Yukawa potential: (42)

Ueﬀ (x) ≈ −2ε0 + (22 /m) exp[−2x/a]/ax;

x a.

We see that the ratio of the eﬀective repulsion to the kinetic energy term in eq. (41) increases with M/m. This is consistent with exact calculations based on eqs. (24) and (35) for the mass ratio smaller than the limiting value 13.6. It is also worth noting that for a very large M/m the repulsive eﬀective potential at x > a is very strong compared to the relative kinetic energy in eq. (41), and due to exponential decay of Ueﬀ (x) at large distances the molecule-molecule scattering length should increase logarithmically with M/m.

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U

a

x

Fig. 9. – Interaction potential for two molecules of heavy and light fermions as a function of the separation x between the heavy atoms.

The Born-Oppenheimer approach fails at distances close to a as the energy ε− (x) vanishes at x = a, and the light atom in the state “−” moves slower than the heavy atoms. This leads to a contradiction with the adiabatic approximation, and for this reason in ﬁg. 9 we present the eﬀective potential Ueﬀ (x) at x > a. For x < a the light atom in the state “−” becomes essentially delocalized and at distances x a the problem reduces to a three-body problem with two heavy fermions and one light fermion in the state “+”. As has been shown in the previous section, this problem is characterized by a limiting mass ratio ≈ 13.6 above which the behavior of the system drastically changes. The eﬀective potential at distances x < a becomes attractive and can support bound (trimer) states of one light and two heavy fermions. Such a trimer state can be formed in ultracold intermolecular collisions if its binding energy exceeds the sum of molecular binding energies. The strong exchange repulsion between heteronuclear molecules of heavy and light fermions at distances x larger than a, described by eq. (42), is crucial for the relaxation process. The potential barrier Ueﬀ (x) reduces the amplitude of the wave function at intermolecular distances ∼ a and, hence, leads to a suppression of relaxation into deep bound states and trimer formation. Estimating the corresponding tunneling probability P in the WKB approach, we obtain P ∝ exp[−B M/m] where B ≈ 0.6. We thus see that one also expects the suppression of inelastic processes of collisional relaxation and trimer formation for weakly bound molecules of light and heavy fermionic atoms for a very large mass ratio M/m. However, the suppression factor is independent of the atom-atom scattering length a and is governed by the mass ratio M/m. The mechanism of this suppression originates from Fermi statistics for the light atoms, which leads to a strong repulsion between molecules at large intermolecular distances. It is worth noting that one also expects collisional stability of a mixture of light and heavy fermionic atoms on the BCS side of a Feshbach resonance (a < 0 and kF |a|

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1). Three-body recombination to deeply bound states will be suppressed as (kF |a|)2 compared to the case of bosonic atoms with the same densities, masses, and scattering length. The stability of a mixture of light and heavy fermionic atoms in the strongly interacting regime requires a separate investigation. . 2 4. Crystalline molecular phase. – Strong long-distance repulsive interaction between weakly bound molecules of light and heavy fermionic atoms has important consequence not only for the relaxation process, but also for macroscopic properties of the molecular system. So far it was believed that dilute heteronuclear Fermi-Fermi mixtures will be in the gas phase, similarly to two-component Fermi gases of atoms in diﬀerent internal states. In this section we show that they can form a molecular crystalline phase, even in the limit where (43)

na3 1.

Let us consider a mixture of heavy and light fermionic atoms with equal concentrations and a positive scattering length for the interaction between them, satisfying the inequality a Re . At zero temperature all atoms will be converted into molecules and under the condition (43) the mean intermolecular separation will be much larger than the size of a molecule (∼ a). Then, using the Born-Oppenheimer approximation and integrating out the motion of light atoms we are left with a system of identical (composite) bosons which can be described by the Hamiltonian (44)

2 1 ˆ =− H Δri + Ueﬀ (rij ), 2M i 2 i =j

where indices i and j label the bosons, their coordinates are denoted by ri and rj , and rij = |ri − rj | is the separation between the i-th and j-th bosons. The repulsive Yukawa potential for the binary interaction between the bosons is given by eq. (42). We now turn to rescaled coordinates ˜ri = ri /a and write the Hamiltonian (44) in the form

2 m 1˜ ˜ ˜ri , M ; H ˆ = H ˜ ˜ri , M = − U (˜ rij ), (45) H Δ˜ri + 2 2ma m m M 2 i,j i where the rescaled interaction potential is (46)

4 ˜ (˜ U rij ) = exp[−2˜ rij ]. r˜ij

˜ shows that for a given density the structure The form of the rescaled Hamiltonian H of the ground state is determined by a single parameter, the mass ratio M/m. The ratio of the potential to kinetic energy increases with M/m and for the mass ratio exceeding a critical value (M/m)c we expect the formation of a crystalline phase. The crystalline

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cr ys ta l

ga s

a

?

B

Fig. 10. – The phase diagram for the system of molecules of heavy and light fermions for the mass ratio (M/m) > (M/m)c and ﬁxed density. The solid curve shows the 2-body scattering length as a function of the magnetic ﬁeld. The arrow indicates the critical point corresponding to the gas-crystal transition, and the question mark shows the point at a large a where the crystalline phase is destroyed.

phase surely emerges for M/m well above 100 [65]. For a given density and the mass ratio much larger than (M/m)c one will ﬁnd the crystalline phase by increasing the 2-body scattering length to a suﬃciently large value. Importantly, the distance between the neighboring molecules will be always signiﬁcantly larger than a, which justiﬁes the use of the repulsive intermolecular potential in the form (42). This circumstance also prevents the heavy atoms from approaching each other to short distances where they can form deep bound states or bound trimer states with a light atom. Let us now make a gedanken experiment showing the presence of two critical points (see ﬁg. 10). Assume that the density and volume are ﬁxed, the mass ratio (M/m) > (M/m)c , and the scattering length a is continiously increasing. Then for a small a one has a molecular Bose-condensed gas. The ﬁrst transition occurs when a is suﬃciently large and the crystalline phase is formed. This phase then survives a certain increase in a but then gets destroyed, which at least follows from the fact that the picture of the eﬀective repulsion between the molecules breaks down at a larger than the intermolecular separation. We do not yet have an answer whether this will be a transition to a gas phase or the system will decay due to the formation of deeply bound dimer or trimer states. Therefore, this point is indicated by the question mark in ﬁg. 10. A similar crystalline phase is expected for molecules consisting of light fermions and heavy bosons. In this case the eﬀective repulsive potential Ueﬀ (x) remains the same as it is determined by the statistics of light atoms. The last question that we touch here is related to realization of such a crystalline

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phase. For the mass ratio above 100 one can not think of molecules in the gas phase: the lightest alkaline fermionic atom is 6 Li and there is no atomic partner which is heavier by two orders of magnitude. The same holds even if we consider metastable 3 He, and one can hardly hope on doing this type of experiments with deuterium. The way to observe the physics discussed in this section will be to arrange an optical lattice for heavy fermionic atoms with a small ﬁlling factor. The light atoms will be present in the system because they form diatomic molecules with heavy ones. The exchange repulsion between the molecules will be still described by eq. (42), with the bare light-atom mass m. On the other hand, the eﬀective mass of heavy atoms (molecules) in an optical lattice can be increased by raising the barrier for intersite tunneling. The corresponding mass ratio can be made very large, and the crystalline phase discussed above will appear as a superlattice. The distance between neighboring sites of this superlattice will be a few times larger than a and can approach microns, signiﬁcantly exceeding the lattice constant of the optical lattice employed for increasing the eﬀective mass of heavy atoms. ∗ ∗ ∗ These lectures have been given by Gora Shlyapnikov at the Enrico Fermi Summer School “Ultracold Fermi Gases”, organized in Varenna in 2006 by Wolfgang Ketterle, Massimo Inguscio, and Christophe Salomon. The authors are grateful to the Italian Physical Society for providing this unique location and organization for very stimulating discussions. The work on these lectures was ﬁnancially supported by the IFRAF Institute, by ANR (grants NT05-2-42103 and 05-Nano-008-02), by Nederlandse Stichtung voor Fundamenteel Onderzoek der Materie (FOM), and by the Russian Foundation for Fundamental Research. LKB is a research unit no. 8552 of CNRS, ENS, and of the University of Pierre et Marie Curie. LPTMS is a mixed research unit no. 8626 of CNRS and University Paris-Sud. REFERENCES [1] Lifshitz E. M. and Pitaevskii L. P., Statistical Physics (Pergamon Press, Oxford) 1980, Part 2. [2] Zwierlein M. W., Abo-Shaeer J. R., Schirotzek A., Schunck C. H. and Ketterle W., Nature, 435 (2005) 1047. [3] Zwierlein M. W., Schirotzek A., Schunck C. H. and Ketterle W., Science, 311 (2006) 492. [4] Partridge G. B., Li W. H., Kamar R. I., Liao Y. A. and Hulet R. G., Science, 311 (2006) 503. [5] Zwierlein M. W., Schirotzek A., Schunck C. H. and Ketterle W., Nature, 442 (2006) 54. [6] Shin Y., Zwierlein M. W., Schunck C. H., Schirotzek A. and Ketterle W., Phys. Rev. Lett., 97 (2006) 030401. [7] Partridge G. B., Li W. H., Liao Y. A., Hulet R. G., Haque M. and Stoof H. T. C., Phys. Rev. Lett., 97 (2006) 190407. [8] Regal C. A., Ticknor C., Bohn J. L. and Jin D. S., Nature, 424 (2003) 47. [9] Cubizolles J., Bourdel T., Kokkelmans S. J. J. M. F., Shlyapnikov G. V. and Salomon C., Phys. Rev. Lett., 91 (2003) 240401.

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[10] Jochim S., Bartenstein M., Altmeyer A., Hendl G., Chin C., Hecker Denschlag J. and Grimm R., Phys. Rev. Lett., 91 (2003) 240402. [11] Regal C. A., Greiner M. and Jin D. S., Phys. Rev. Lett., 92 (2004) 083201. [12] Greiner M., Regal C. and Jin D. S., Nature, 426 (2003) 537. [13] Regal C. A., Greiner M. and Jin D. S., Phys. Rev. Lett., 92 (2004) 040403. [14] Jochim S., Bartenstein M., Altmeyer A., Hendl G., Riedl S., Chin C., Denschlag J. H. and Grimm R., Science, 302 (2003) 2101; Bartenstein M., Altmeyer A., Riedl S., Jochim S., Chin C., Hecker Denschlag J. and Grimm R., Phys. Rev. Lett., 92 (2004) 120401. [15] Bartenstein M., Altmeyer A., Riedl S., Jochim S., Chin C., Hecker Denschlag J. and Grimm R., Phys. Rev. Lett., 92 (2004) 203201. [16] Zwierlein M. W., Stan C. A., Schunck C. H., Raupach S. M. F., Gupta S., Hadzibabic Z. and Ketterle W., Phys. Rev. Lett., 91 (2003) 250401. [17] Zwierlein M. W., Stan C. A., Schunck C. H., Raupach S. M. F., Kerman A. J. and Ketterle W., Phys. Rev. Lett., 92 (2004) 120403. [18] Bourdel T., Khaykovich L., Cubizolles J., Zhang J., Chevy F., Teichmann M., Tarruell L., Kokkelmans S. J. J. M. F. and Salomon C., Phys. Rev. Lett., 93 (2004) 050401. [19] Partridge G. B., Streker K. E., Kamar R. I., Jack M. W. and Hulet R. G., Phys. Rev. Lett., 95 (2005) 020404. [20] Breit G. and Wigner E., Phys. Rev., 49 (1936) 519. [21] Landau L. D. and Lifshitz E. M., Quantum Mechanics (Butterworth-Heinemann, Oxford) 1999. [22] Feshbach H., Ann. Phys. (N.Y.), 19 (1962) 287; Theoretical Nuclear Physics (Wiley, New York) 1992. [23] Fano U., Phys. Rev., 124 (1961) 1866. [24] Moerdijk A. J., Verhaar B. J. and Axelsson A., Phys. Rev. A, 51 (1995) 4852; see also Tiesinga E., Verhaar B. J. and Stoof H. T. C., Phys. Rev. A, 47 (1993) 4114. [25] Fedichev P. O., Kagan Yu., Shlyapnikov G. V. and Walraven J. T. M., Phys. Rev. Lett., 77 (1996) 2913. [26] Bohn J. L. and Julienne P. S., Phys. Rev. A, 60 (1999) 414. [27] Eagles D. M., Phys. Rev., 186 (1969) 456. [28] Leggett A. J., in Modern Trends in the Theory of Condensed Matter, edited by Pekalski A. and Przystawa J. (Springer, Berlin) 1980. [29] Nozieres P. and Schmitt-Rink S., J. Low Temp. Phys., 59 (1985) 195. [30] See for review Randeria M., in Bose-Einstein Condensation, edited by Griffin A., Snoke D. W. and Stringari S. (Cambridge University Press, Cambridge) 1995. [31] Miyake K., Prog. Theor. Phys., 69 (1983) 1794. [32] See for review Kagan M. Yu., Sov. Phys. Usp., 37 (1994) 69. [33] Holland M., Kokkelmans S. J. J. M. F., Chiofalo M. L. and Walser R., Phys. Rev. Lett., 87 (2001) 120406. [34] Timmermans E., Furuya K., Milonni P. W. and Kerman A. K., Phys. Lett. A, 285 (2001) 228. [35] Bruun G. M. and Pethik C., Phys. Rev. Lett., 92 (2004) 140404. [36] Bruun G. M., Phys. Rev. A, 70 (2004) 053602. [37] De Palo S., Chiofalo M. L., Holland M. J. and Kokkelmans S. J. J. M. F., Phys. Lett. A, 327 (2004) 490. [38] Cornell E. A., KITP Conference on Quantum Gases, Santa Barbara, May 10-14, 2004, http://online.kitp.ucsb.edu/online/gases c0/discussion2/ [39] Diener R. and Ho T.-L., cond-mat/0405174.

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[40] [41] [42] [43]

Petrov D. S., Phys. Rev. Lett., 93 (2004) 143201. Petrov D. S., Salomon C. and Shlyapnikov G. V., Phys. Rev. A, 71 (2005) 012708. Drummond P. D. and Kheruntsyan K., Phys. Rev. A, 70 (2004) 033609. Experimental studies of a narrow resonance with 6 Li2 molecules have been performed at Rice: Strecker K. E., Partridge G. B. and Hulet R. G., Phys. Rev. Lett., 91 (2003) 080406. Kokkelmans S. J. J. M. F., Shlyapnikov G. V. and Salomon C., Phys. Rev. A, 69 (2004) 031602. Petrov D. S., Salomon C. and Shlyapnikov G. V., Phys. Rev. Lett., 93 (2004) 090404. Efimov V. N., Sov. J. Nucl. Phys., 12 (1971) 589; Nucl. Phys. A, 210 (1973) 157. Petrov D. S., Phys. Rev. A, 67 (2003) 010703. Skorniakov G. V. and Ter-Martirosian K. A., Sov. Phys. JETP, 4 (1957) 648. Danilov G. S., Sov. Phys. JETP, 13 (1961) 349. Bethe H. and Peierls R., Proc. R. Soc. London, Ser. A, 148 (1935) 146. An approximate diagrammatic approach leading to a = 0.75a has been developed in Pieri P. and Strinati G. C., Phys. Rev. B, 61 (2000) 15370. Astrakharchik G. E., Boronat J., Casulleras J. and Giorgini S., Phys. Rev. Lett., 93 (2004) 200404. Brodsky I. V., Kagan M. Y., Klaptsov A. V., Combescot R. and Leyronas X., Phys. Rev. A, 73 (2006) 032724. Levinsen J. and Gurarie V., Phys. Rev. A, 73 (2006) 053607. Carr L. D., Shlyapnikov G. V. and Castin Y., Phys. Rev. Lett., 92 (2004) 150404. Menotti C., Pedri P. and Stringari S., Phys. Rev. Lett., 89 (2002) 250402. O’Hara K. M., Hemmer S. L., Gehm M. E., Granade S. R. and Thomas J. E., Science, 298 (2002) 2179. Kinast J., Hemmer S. L., Gehm M. E., Turlapov A. and Thomas J. E., Phys. Rev. Lett., 92 (2004) 150402. Kerman A. J., Sage J. M., Sainis S., Bergeman T. and DeMille D., Phys. Rev. Lett., 92 (2004) 153001. Sage J. M., Sainis S., Bergeman T. and DeMille D., Phys. Rev. Lett., 94 (2005) 203001. Winkler K., Lang F., Thalhammer G., Straten P. V. D., Grimm R. and Denschlag J. H., Phys. Rev. Lett., 98 (2007) 043201. In this experiment the presence of an optical lattice suppressed inelastic collisions between molecules of bosonic 87 Rb atoms, which provided a highly eﬃcient transfer of these molecules to a less excited ro-vibrational state and a long molecular lifetime of about 1 second. Santos L. and Pfau T., Phys. Rev. Lett., 96 (2006) 190404. Petrov D. S., Salomon C. and Shlyapnikov G. V., J. Phys. B, 38 (2005) S645. The Born-Oppenheimer approach for the three-body system of one light and two heavy atoms was discussed in Fonseca A. C., Redish E. F. and Shanley P. E., Nucl. Phys. A, 320 (1979) 273. The FCC crystalline phase for bosons interacting with each other via the Yukawa potential has been found in Ceperley D., Chester G. V. and Kalos M. H., Phys. Rev. B, 17 (1978) 1070. Using the results of this early work one ﬁnds that in our case the crystalline phase already exists for M/m close to 10. However, recent calculations do not reproduce this result and indicate that the crystallization may require signiﬁcantly larger mass ratios: Petrov D. S., Astrakharchik G. E., Salomon C. and Shlyapnikov G. V., in preparation.

[44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61]

[62] [63] [64]

[65]

Ultracold Fermi gases in the BEC-BCS crossover: A review from the Innsbruck perspective R. Grimm Institute of Experimental Physics and Center for Quantum Physics University of Innsbruck - Technikerstraße 25, A-6020 Innsbruck, Austria Institute for Quantum Optics and Quantum Information (IQOQI) Austrian Academy of Sciences, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria

1. – Introduction By the time of the “Enrico Fermi” Summer School in June 2006, quantum degeneracy in ultracold Fermi gases has been reported by 13 groups worldwide [1-13]. The ﬁeld is rapidly expanding similar to the situation of Bose-Einstein condensation at the time of the “Enrico Fermi” Summer School in 1998 [14]. The main two species for the creation of ultracold Fermi gases are the alkali atoms potassium (40 K) [1, 6, 8, 10, 11] and lithium (6 Li) [2-5, 7, 9]. At the time of the School, degeneracy was reported for two new species, 3 He∗ [12] and 173 Yb [13], adding metastable and rare earth species to the list. Fermionic particles represent the basic building blocks of matter, which connects the physics of interacting fermions to very fundamental questions. Fermions can pair up to form composite bosons. Therefore, the physics of bosons can be regarded as a special case of fermion physics, where pairs are tightly bound and the fermionic character of the constituents is no longer relevant. This simple argument already shows that the physics of fermions is in general much richer than the physics of bosons. Systems of interacting fermions are found in many areas of physics, like in condensedmatter physics (e.g., superconductors), in atomic nuclei (protons and neutrons), in primordial matter (quark-gluon plasma), and in astrophysics (white dwarfs and neutron stars). Strongly interacting fermions pose great challenges for many-body quantum thec Societ` a Italiana di Fisica

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ories. With the advent of ultracold Fermi gases with tunable interactions and controllable conﬁnement, unique model systems have now become experimentally available to study the rich physics of fermions. In this contribution, we will review a series of experiments on ultracold, strongly interacting Fermi gases of 6 Li which we conducted at the University of Innsbruck. We will put our experiments into context with related work and discuss them according to the present state-of-the art knowledge in the ﬁeld. After giving a brief overview of experiments on strongly interacting Fermi gases (sect. 2), we will discuss the basic interaction properties of 6 Li near a Feshbach resonance (sect. 3). Then we will discuss the main experimental results on the formation and Bose-Einstein condensation of weakly bound molecules (sect. 4), the crossover from a molecular Bose-Einstein condensate to a fermionic superﬂuid (sect. 5), and detailed studies on the crossover by collective modes (sect. 6) and pairing-gap spectroscopy (sect. 7).

2. – Brief history of experiments on strongly interacting Fermi gases To set the stage for a more detailed presentation of our results, let us start with a brief general overview of the main experimental developments in the ﬁeld of ultracold, strongly interacting Fermi gases; see also the contributions by D. Jin and W. Ketterle in this volume. The strongly interacting regime is realized when the scattering length, characterizing the two-body interacting strength, is tuned to large values by means of Feshbach resonances [15, 16]. In the case of Bose gases with large scattering lengths rapid three-body decay [17-19] prevents the experiments to reach the strongly interacting regime(1 ). Experiments with ultracold Fermi gases thus opened up a door to the new, exciting regime of many-body physics with ultracold gases. The creation of a strongly interacting Fermi gas was ﬁrst reported in 2002 by the group at Duke University [21]. They studied the expansion of a 6 Li gas with resonant interactions after release from the trap and observed hydrodynamic behavior. In similar experiments, the group at the ENS Paris provided measurements of the interaction energy of ultracold 6 Li in the strongly interacting region [22]. In 2003 ultracold diatomic molecules entered the stage. Their formation is of particular importance in atomic Fermi gases, as their bosonic nature is connected with a fundamental change of the quantum statistics of the gas. The JILA group demonstrated molecule formation in an ultracold Fermi gas of 40 K [23], followed by three groups working with 6 Li: Rice University [24], the ENS Paris [25], and Innsbruck University [26]. The latter experiments on 6 Li also demonstrated an amazing fact. Molecules made of fermionic atoms can be remarkably stable against inelastic decay, allowing for the formation of stable molecular quantum gases. (1 ) This statement refers to macroscopically trapped gases of a large number of atoms. Highly correlated systems of bosons can be created in optical lattices [20].

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In late 2003 three groups reported on the achievement of molecular Bose-Einstein condensation, our group (6 Li) [7], the JILA group (40 K) [27], and the MIT group (6 Li) [28], followed early in 2004 by the ENS group (6 Li) [29]. Early in 2004, the JILA group [30] and the MIT group [31] demonstrated pair condensation in strongly interacting Fermi gases with resonant interactions, i.e. beyond the BEC regime. These experiments demonstrated a new macroscopic quantum state of ultracold matter beyond well-established BEC physics, which has stimulated an enormous interest in the ﬁeld. The experiments then started to explore the crossover from a BEC-type system to a fermionic superﬂuid with Bardeen-Cooper-Schrieﬀer (BCS)–type pairing. Elementary properties of the Fermi gas in the BEC-BCS crossover were studied by several groups. In Innsbruck, we showed that the crossover proceeds smoothly and can be experimentally realized in an adiabatic and reversible way [32]. At the ENS the crossover was investigated in the free expansion of the gas after release from the trap. Measurements of collective excitation modes at Duke University and in Innsbruck showed exciting observations and provided pieces of evidence for superﬂuidity in the strongly interacting gas. The Duke group measured very low damping rates, which could not be explained without invoking superﬂuidity [33]. Our work on collective oscillations [34] showed a striking breakdown of the hydrodynamic behavior of the gas when the interaction strength was changed, suggesting a superﬂuid-normal transition. Spectroscopy on fermionic pairing based on a radio-frequency method showed the “pairing gap” of the strongly interacting gas along the BEC-BCS crossover [35]. In these experiments performed in Innsbruck, temperature-dependent spectra suggested that the resonantly interacting Fermi gas was cooled down deep into the superﬂuid regime. A molecular probe technique of pairing developed at Rice University provided clear evidence for pairing extending through the whole crossover into the weakly interacting BCS regime [36]. Measurements of the heat capacity of the strongly interacting gas performed at Duke University showed a transition at a temperature where superﬂuidity was expected [37]. After several pieces of experimental evidence provided by diﬀerent groups, the ﬁnal proof of superﬂuidity in strongly interacting Fermi gases was given by the MIT group in 2005 [38]. They observed vortices and vortex arrays in a strongly interacting Fermi gas in various interaction regimes. New phenomena were recently explored in studies on imbalanced spin-mixtures at Rice University [39] and at MIT [40]. These experiements have approached a new frontier, as such systems may oﬀer access to novel superﬂuid phases. Experiments with imbalanced spin-mixtures also revealed the superﬂuid phase transition in spatial proﬁles of the ultracold cloud [41]. 3. – Interactions in a 6 Li spin mixture Controllable interactions play a crucial role in all experiments on strongly interacting Fermi gases. To exploit an s-wave interaction at ultralow temperatures, non-identical particles are needed; thus the experiments are performed on mixtures of two diﬀerent spin states. Feshbach resonances [15,16,42] allow tuning the interactions through variations of

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Fig. 1. – Energy levels for the electronic ground state of 6 Li atoms in a magnetic ﬁeld. The experiments on strongly interacting Fermi gases are performed in the high magnetic ﬁeld range, where the nuclear spin essentially decouples from the electron spin. The two-component atomic mixture is created in the lowest two states, labelled with 1 and 2 (inset), close to the broad Feshbach resonance centered at 834 G.

an external magnetic ﬁeld. In this section, we review the two-body interaction properties of 6 Li. In particular, we discuss the behavior close to a wide Feshbach resonance with very favorable properties for interaction tuning in strongly interacting Fermi gases. . 3 1. Energy levels of 6 Li atoms in a magnetic ﬁeld. – The magnetic-ﬁeld dependence of the energy structure of 6 Li atoms in the electronic S1/2 ground state is shown in ﬁg. 1. The general behavior is similar to any alkali atom [43] and is described by the well-known Breit-Rabi formula. At zero magnetic ﬁeld, the coupling of the 6 Li nuclear spin (I = 1) to the angular momentum of the electron (J = 1/2) leads to the hyperﬁne splitting of 228.2 MHz between the states with quantum numbers F = I + J and F = I − J. Already at quite moderate magnetic ﬁelds the Zeeman eﬀect turns over into the highﬁeld regime, where the Zeeman energy becomes larger than the energy of the hyperﬁne interaction. Here the nuclear spin essentially decouples from the electron spin. In atomic physics this eﬀect is well known as the “Paschen-Back eﬀect of the hyperﬁne structure” or “Back-Goudsmit eﬀect” [43]. In the high-ﬁeld region the states form two triplets, depending on the orientation of the electron spin (ms = ±1/2), where the states are characterized by the orientation of the nuclear spin with quantum number mI . For simplicity, we label the states with numbers according to increasing energy (see inset in ﬁg. 1). The lowest two states 1 and 2 are of particular interest for creating stable spin mixtures. These two states ms = −1/2, mI = +1 (ms = −1/2, mI = 0) are adiabatically connected with the states F = 1/2, mF = 1/2 (F = 1/2, mF = −1/2) at low magnetic ﬁelds.

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Fig. 2. – Tunability of the s-wave interactions in a spin mixture of 6 Li atoms in the two lowest spin states 1 and 2. The s-wave scattering length a shows a pronounced, broad resonance as a function of the magnetic ﬁeld [44, 45]. The vertical dotted line indicates the exact resonance ﬁeld (834 G) where a goes to inﬁnity and the interaction is only limited through unitarity.

. 3 2. Tunability at the marvelous 834 G Feshbach resonance. – Interactions between 6 Li atoms in states 1 and 2 show a pronounced resonance in s-wave scattering [44, 45] with favorable properties for the experiments on strongly interacting Fermi gases. Figure 2 displays the scattering length a as a function of the magnetic ﬁeld B. The center of the resonance, i.e. the point where a diverges, is located at 834 G. This resonance center is of great importance to realize the particularly interesting situation of a universal Fermi . gas in the unitarity limit; see discussion in subsect. 5 3. The investigation of the broad Feshbach resonance in 6 Li has a history of almost ten years. In 1997, photoassociation spectroscopy performed at Rice University revealed a triplet scattering length that is negative and very large [46]. A theoretical collaboration between Rice and the Univ. of Utrecht [44] then led to the prediction of the resonance near 800 G. In 2002, ﬁrst experimental evidence for the resonance was found at MIT [47], at Duke University [48], and in Innsbruck [49]. At about 530 G, experiments at Duke and in Innsbruck showed the zero crossing of the scattering length that is associated with the broad resonance. The MIT group observed an inelastic decay feature in a broad magneticﬁeld region around 680 G. The decay feature was also observed at the ENS Paris, but at higher ﬁelds around 720 G [22]. The ENS group also reported indications of the resonance position being close to 800 G. In Innsbruck the decay feature was found [50] in a broad region around 640 G(2 ). Molecule dissociation experiments at MIT [31, 51] provided a (2 ) The interpretation of these inelastic decay features involves diﬀerent processes, which depend . on the particular experimental conditions, see also subsect. 4 3. In a three-body recombination

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lower bound of 822 G for the resonance point. To date the most accurate knowledge on a(B) in 6 Li spin mixtures results from an experiment-theory collaboration between Innsbruck and NIST on radio-frequency spectroscopy on weakly bound molecules [45]. This work puts the resonance point to 834.1 G within an uncertainty of ±1.5 G. The dependence a(B) near the Feshbach resonance can be conveniently described by a ﬁt formula [45], which approximates the scattering length in a range between 600 and 1200 G to better than 99%, (1)

a(B) = abg 1 +

ΔB B − B0

(1 + α(B − B0 ))

with abg = −1405 a0 , B0 = 834.15 G, ΔB = 300 G, and α = 0.040 kG−1 ; here a0 = 0.529177 nm is Bohr’s radius. Concerning further Feshbach resonances in 6 Li, we note that besides the broad 834 G resonance in the (1, 2) spin mixture, similar broad s-wave resonances are found in (1, 3) and in (2, 3) mixtures with resonance centers at 690 G and at 811 G, respectively [45]. The (1, 2) spin mixture also features a narrow Feshbach resonance near 543 G with a width of roughly 100 mG [24, 51]. Moreover, Feshbach resonances in p-wave scattering of 6 Li have been observed in (1, 1), (1, 2), and (2, 2) collisions at the ENS [52] and at MIT [51]. . 3 3. Weakly bound dimers. – A regime of particular interest is realized when the scattering length a is very large and positive. The scale for “very large” is set by the van der Waals interaction between two 6 Li atoms, characterized by a length RvdW = (mC6 /¯ h2 )1/4 /2 = 31.26 a0 (for 6 Li, C6 = 1393 a.u. and the atomic mass is m = 6.015 u). For a RvdW , a weakly bound molecular state exists with a binding energy given by the universal formula(3 )

(2)

Eb =

¯2 h . ma2

In this regime, the molecular wave function extends over a much larger range than the interaction potential and, for large interatomic distances r RvdW , falls oﬀ exponentially as exp[−r/a]. The regime, in which a bound quantum object is much larger than a classical system, is also referred to as the “quantum halo regime” [54]. For quantum halo event, immediate loss occurs when the release of molecular binding energy ejects the particles out of the trap. Another mechanism of loss is vibrational quenching of trapped, weakly bound molecules. The fact that, in contrast to bosonic quantum gases, maximum inelastic decay loss does not occur at the resonance point, but somewhere in the region of positive scattering length is crucial for the stability of strongly interacting Fermi gases with resonant interactions. (3 ) A useful correction to the universal expression for the non-zero range of the van der Waals h2 /(m(a − a ¯)2 ) where a ¯ = 0.956 RvdW [53]. potential is Eb = ¯

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Fig. 3. – Binding energy Eb of weakly bound 6 Li molecules, which exist on the lower side of the 834 G Feshbach resonance. Here we use temperature units (kB × 1 μK ≈ h × 20.8 kHz) for a convenient comparison with our experimental conditions.

states, the details of the short-range interaction are no longer relevant and the physics acquires universal character [55]. Here two-body interactions are completely characterized by a as a single parameter. From these considerations, we understand that the lower side of the 834 G Feshbach resonance in 6 Li is associated with the regime of weakly bound (quantum halo) molecules. The binding energy of the weakly bound 6 Li molecular state is plotted in ﬁg. 3 as a function of the magnetic ﬁeld. Weakly bound molecules made of fermionic atoms exhibt striking scattering properties [56]. As a big surprise, which enormously boosted the ﬁeld of ultracold fermions in 2003, these dimers turned out to be highly stable against inelastic decay in atom-dimer and dimer-dimer collisions. The reason for this stunning behavior is a Pauli suppression eﬀect. The collisional quenching of a weakly bound dimer to a lower bound state requires a close encounter of three particles. As this necessarily involves a pair of identical fermions the process is Pauli blocked. The resulting collisional stability is in sharp contrast to weakly bound dimers made of bosonic atoms [57-60], which are very sensitive to inelastic decay. The amazing properties of weakly bound dimers made of fermions were ﬁrst described in ref. [56]. Here we just summarize the main ﬁndings, referring the reader to the lecture of G. Shlyapnikov in these proceedings for more details. For elastic atom-dimer and dimer-dimer collisions, Petrov et al. [56] calculated the scattering lengths (3)

aad = 1.2 a,

(4)

add = 0.6 a,

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respectively. Inelastic processes, described by the loss-rate coeﬃcients αad and αdd , follow the general scaling behavior [56] (5) (6)

αad = cad αdd

¯hRvdW m

¯hRvdW = cdd m

RvdW a RvdW a

3.33 ,

2.55 .

Here the dimensionless coeﬃcients cad and cdd depend on non-universal short-range physics. We point out that, for typical experimental conditions in molecular BEC experiments (see sect. 4), the factor (RvdW /a)2.55 results in a gigantic suppression of ﬁve orders of magnitude in inelastic dimer-dimer collisions! The general scaling behavior of inelastic loss is universal and should be the same for 6 Li and 40 K, consistent with measurements on both species [25,26,61] The pre-factors Cad and Cdd , however, are non-universal as they depend on short-range three-body physics. A comparison of the experiments on both species shows that inelastic decay of weakly bound molecules is typically two orders of magnitude faster for 40 K than for 6 Li. This diﬀerence can be attributed to the larger van der Waals length of 40 K in combination with its less favorable short-range interactions. This important diﬀerence in inelastic decay is the main reason why experiments on 6 Li and 40 K follow diﬀerent strategies for the creation of degenerate Fermi gases. In 6 Li, the regime of weakly bound dimers on the molecular side of the Feshbach resonance opens up a unique route into deep degeneracy, as we will discuss in the following section. 4. – The molecular route into Fermi degeneracy: creation of a molecular Bose-Einstein condensate In experiments on 6 Li gases, a molecular Bose-Einstein condensate (mBEC) can serve as an excellent starting point for the creation of strongly interacting Fermi gases in the BEC-BCS crossover regime. In this section, after discussing the various approaches followed by diﬀerent groups, we describe the strategy that we follow in Innsbruck to create the mBEC. . 4 1. A brief review of diﬀerent approaches. – The experiments on strongly interacting gases of 6 Li in diﬀerent laboratories (in alphabetical order: Duke University, ENS Paris, Innsbruck University, MIT, Rice University) are based on somewhat diﬀerent approaches. The ﬁrst and the ﬁnal stages of all experiments are essentially the same. In the ﬁrst stage, standard laser cooling techniques [62] are applied to decelerate the atoms in an atomic beam and to accumulate them in a magneto-optic trap (MOT); for a description of our particular set-up see refs. [63, 50]. In the ﬁnal stage, far-detuned optical dipole traps [64] are used to store and manipulate the strongly interacting spin mixture. The creation of such a mixture requires trapping in the high-ﬁeld seeking spin states 1 and 2 (see ﬁg. 1), which cannot be achieved magnetically. The main diﬀerences in the experimental

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approaches pursued in the ﬁve laboratories concern the intermediate stages of trapping and cooling. The general problem is to achieve an eﬃcient loading of many 6 Li atoms into the small volume of a far-detuned optical dipole trap. At Rice Univ., ENS, and MIT, magnetic traps are used as an intermediate stage [2, 3, 5]. This approach oﬀers the advantage of a large volume and eﬃcient transfer from a MOT with minimum loading losses. To achieve eﬃcient cooling in the magnetic trap, the experiments then use bosonic atoms as a cooling agent. At Rice Univ. and at ENS, the 6 Li atoms are trapped together with the bosonic isotope 7 Li [2, 3]. The isotope mixture can be eﬃciently cooled to degeneracy by radio-frequency induced evaporation. Finally the sample is loaded into an optical dipole trap, and the atoms are transferred from their magnetically trappable, low-ﬁeld seeking spin state into the high-ﬁeld seeking states 1 and 2. The internal transfer is achieved through microwave and radio-frequency transitions. In this process it is important to create an incoherent spin mixture, which requires deliberate decoherence in the sample. At MIT the approach is basically similar [5], but a huge BEC of Na atoms is used as the cooling agent. This results in an exceptionally large number of atoms in the degenerate Fermi gas [65]. In all three groups (Rice, ENS, MIT), ﬁnal evaporative cooling is performed on the strongly interacting spin mixture by reducing the power of the optical trap. The experiments at Duke University [21] and in Innsbruck [7] proceed in an all-optical way without any intermediate magnetic traps. To facilitate direct loading from the MOT, the optical dipole traps used in these experiments have to start with initially very high laser power. For the ﬁnal stage of the evaporation much weaker traps are needed. Therefore, the all-optical approach in general requires a large dynamical range in the optical trapping power. The Duke group uses a powerful 100-W CO2 laser source [4] both for evaporative cooling and for the ﬁnal experiments. In Innsbruck we employ two diﬀerent optical trapping stages to optimize the diﬀerent phases of the experiment. . 4 2. The all-optical Innsbruck approach. – An eﬃcient transfer of magneto-optically trapped lithium atoms into an optical dipole trap is generally much more diﬃcult than for the heavy alkali atoms. The much higher temperatures of lithium in a MOT of typically a few hundred microkelvin [63] require deep traps with a potential depth of the order of 1 mK. We overcome this bottleneck of dipole trap loading by means of a deep large-volume dipole trap serving as a “funnel”. The trap is realized inside a build-up cavity constructed around the glass cell [66]. The linear resonator enhances the power of a 2-W infrared laser (Nd:YAG at a wavelength of 1064 nm) by a factor of ∼ 150 and, with a Gaussian beam waist of 160 μm, allows us to create a 1 mK deep optical standing-wave trapping potential. Almost 107 atoms in the lower hyperﬁne level with F = 1/2 can be loaded from the 6 Li MOT into the resonator-enhanced dipole trap at a temperature of typically a few 100 μK. Note that, when loaded from the MOT, the spin mixture of states 1 and 2 in the optical dipole trap is incoherent from the very beginning. Then we apply a single beam from a 10-W near-infrared laser (wavelength 1030 nm), which is focussed to a waist of typically a few ten μm (∼ 25 μm in our earlier experiments [7,32,34], ∼ 50 μm in more recent work [67]), overlapping it with the atom cloud in

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Fig. 4. – Illustration of the “dimple trick”. The atoms are ﬁrst transferred from the MOT into a large-volume optical reservoir trap (a), here implemented inside of an optical resonator. A narrow “dimple” potential (b) is then added and thermalization leads to a huge increase of the local density and phase-space density according to the Boltzmann factor as the temperature is set by the reservoir. After removal of the reservoir trap (c) one obtains a very dense sample optically trapped sample. Forced evaporative cooling can then be implemented (d) by ramping down the trap power. The dimple trick, originating from work in refs. [68] and [69], has proven a very powerful tool for the all-optical creation of degenerate quantum gases [70, 71].

the standing-wave trapping potential. The total optical potential can then be regarded as a combination of a large-volume “reservoir” trap in combination with a narrow “dimple” potential. The dimple is eﬃciently ﬁlled through elastic collisions resulting in a large increase in local density, phase-space density, and elastic collisions rate; this “dimple trick” is illustrated in ﬁg. 4. After removal of the reservoir, i.e. turning oﬀ the standing-wave trap, we obtain a very dense cloud of ∼ 1.5 × 106 atoms at a temperature T ≈ 80 μK, a peak density of ∼ 1014 cm−3 , a peak phase-space density of 5 × 10−3 , and a very high elastic collision rate of 5 × 104 s−1 . In this way, excellent starting conditions are realized for evaporative cooling. A highly eﬃcient evaporation process is then forced by ramping down the laser power by typically three orders of magnitude within a few seconds. The formation of weakly bound molecules turns out to play a very favorable role in this process and eventually leads to the formation of a molecular BEC. The details of this amazing process will be elucidated in the following. . 4 3. Formation of weakly bound molecules. – The formation of weakly bound molecules in a chemical atom-molecule equilibrium [72,73] plays an essential role in the evaporative cooling process; see illustration in ﬁg. 5. In the 6 Li gas, molecules are formed through three-body recombination. As the molecular binding energy Eb is released into kinetic

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Fig. 5. – Illustration of the atom-molecule thermal equilibrium in a trapped 6 Li gas at the molecular side of the Feshbach resonance. Atoms in the two spin states and molecules represent three sub-ensembles in thermal contact (thermal energy kB T ). The molecules are energetically favored because of the binding energy Eb , which is reﬂected in the Boltzmann factor in eq. (7). The equilibrium can also be understood in terms of a balance of the chemical processes of exoergic recombination and endoergic dissociation [72].

energy, this process is exoergic and thus leads to heating of the sample(4 ). The inverse chemical process is dissociation of molecules through atom-dimer and dimer-dimer collisions. These two-body processes are endoergic and can only happen when the kinetic energy of the collision partners is suﬃcient to break up the molecular bond. From a balance of recombination (exoergic three-body process) and dissociation (endoergic twobody processes) one can intuitively understand that molecule formation is favored at low temperatures and high number densities, i.e. at high phase-space densities. For a non-degenerate gas, the atom-molecule equilibrium follows a simple relation [72] (7)

φmol = φ2at exp

Eb , kB T

where φmol and φat denote the molecular and atomic phase-space densities, respectively. (4 ) The relation of released binding energy Eb to the trap depth is crucial whether the recombination products remain trapped and further participate in the thermalization processes. For low trap depth the recombination leads to immediate loss. This explains why the loss features observed by diﬀerent groups [47, 22, 50] shift towards lower ﬁelds at higher trap depths.

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R. Grimm

Fig. 6. – Experimental results [26] demonstrating how an ultracold 6 Li gas approaches a chemical atom-molecule equilibrium on the molecular side of the Feshbach resonance. The experiment starts with a non-degenerate, purely atomic gas at a temperature of 2.5 μK and a peak atomic phase-space density of 0.04. The magnetic ﬁeld is set to 690 G, where a = 1420 a0 and Eb /kB = 15 μK. Nat and Nmol denote the number of unbound atoms and the number of molecules, respectively. The total number of unbound and bound atoms 2Nmol + Nat slowly decreases . because of residual inelastic loss, see subsect. 3 3.

The Boltzmann factor enhances the fraction of molecules in a trapped sample and can (partially) compensate for a low atomic phase-space density. Including the eﬀect of Fermi degeneracy, the thermal atom-molecule equilibrium was theoretically investigated in ref. [73]. We have experimentally studied the thermal atom-molecule equilibrium in ref. [26]. Figure 6 illustrates how an initially pure atomic gas tends to an atom-molecule equilibrium. The experiment was performed at a magnetic ﬁeld of 690 G and a temperature T = 2.5 μK with a molecular binding energy of Eb /kB = 15 μK. The observation that more than 50% of the atoms tend to form molecules at a phase-space density of a factor of thirty from degeneracy, highlights the role of the Boltzmann factor (see eq. (7)) in the equilibrium. Note that in ﬁg. 6, the total number of particles decreases slowly because of residual inelastic decay of the molecules. The magnetic ﬁeld of 690 G is too far away from resonance to obtain a full suppression of inelastic collisions. Further experiments in ref. [26] also demonstrated how an atom-molecule thermal equilibrium is approached from an initially pure molecular sample. In this case atoms are produced through dissociation of molecules at small molecular binding energies closer to the Feshbach resonance. An experiment at ENS [25] demonstrated the adiabatic conversion of a degenerate 6 Li Fermi gas produced at a < 0 into a molecular gas by slowly sweeping across the Feshbach resonance. This resulted in a large molecular fraction of up to 85% and experimental conditions close to mBEC. Before the work in 6 Li, molecule formation in an ultracold Fermi

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425

Fig. 7. – Stages of evaporative cooling on the molecular side of the Feshbach resonance. (a) For a hot gas, very few molecules are present and the evaporation can be understood in terms of elastic collisions in the atomic spin mixture. (b) As the gas gets colder the chemical atommolecule equilibrium begins to favor the molecules. (c) Further evaporation removes atoms but not molecules because of the two times diﬀerent trap depths. (d) After disappearance of the atoms, evaporation can be fully understood in terms of the molecular gas. This eventually leads to molecular Bose-Einstein condensation.

gas through a Feshbach sweep was demonstrated with 40 K at JILA [23]. The long-lived nature of the 40 K molecules close to the Feshbach resonance was demonstrated in later work [61]. Note that long-lived molecules of 6 Li were also produced from a degenerate gas at Rice Univ. [24]. This experiment, however, was performed by sweeping across the narrow Feshbach resonance at 543 G. The observed stability cannot be explained in . terms of the Pauli suppression arguments in subsect. 3 3 and, to the best knowledge of the author, still awaits a full interpretation. . 4 4. Evaporative cooling of an atom-molecule mixture. – Based on the thermal atommolecule equilibrium arguments discussed before, we can now understand why the evaporation process works so well on the molecular side of the Feshbach resonance. Experimentally, we found that highly eﬃcient evaporative cooling can be performed at a ﬁxed magnetic ﬁeld around 764 G [7]. At this optimum ﬁeld, the large scattering length a = + 4500 a0 warrants a large stability of the molecules against inelastic decay . (see subsect. 3 3). The corresponding binding energy Eb /kB = 1.5 μK is small enough to minimize recombination heating during the cooling process. However, it is larger than the typical Fermi energies in the ﬁnal evaporation stage of a few hundred nK, which favors the molecule formation in the last stage of the cooling process. The diﬀerent stages of evaporative cooling are illustrated in ﬁg. 7. In the ﬁrst stage (a) molecule formation is negligible. As the cooling process proceeds (b, c), an increasing part of the trapped sample consists of molecules. Here, it is important to note that the optical trap is twice as deep for the molecules. This is due to the weakly bound dimers having twice the polarizability. Therefore, evaporation in an atom-molecule mixture near

426

R. Grimm

thermal equilibrium essentially removes atoms and not the molecules. This predominant evaporation of unpaired atoms also has the interesting eﬀect that the sample reaches a balanced 50/50 spin mixture, even if one starts the evaporation with some imbalance in the spin composition(5 ). In the ﬁnal stage of the evaporation process, only molecules are left and the process can be essentially understood in terms of elastic molecule-molecule interactions. This leads to the formation of a molecular Bose-Einstein condensate (mBEC), . as we will discuss in more detail in subsect. 4 5 We point out two more facts to fully understand the eﬃciency of the evaporative cooling process in our set-up. The magnetic ﬁeld that we use for Feshbach tuning of the scattering properties [50] exhibits a curvature(6 ), which provides us with a magnetic trapping potential for the high-ﬁeld seeking atoms along the laser beam axis (corresponding trapping frequency of 24.5 Hz × B/kG). When the optical trap is very week at the end of the evaporation process, the trap is a hybrid (optically for the transverse motion and magnetically for the axial motion). The cooling then results in an axial compression of the cloud which helps to maintain high enough number densities. The second interesting fact, which makes the all-optical route to degeneracy diﬀerent for fermions and bosons [74, 70, 71, 75], is that evaporative cooling of fermions can be performed at very large scattering lengths. For bosons this is impossible because of very fast threebody decay [17, 70]. For a very large scattering length, a substantial part of the cooling process proceeds in the unitarity limit, where the scattering cross-section is limited by the relative momentum of the particles. Decreasing temperature leads to an increase in the elastic scattering rate, which counteracts the eﬀect of the decreasing number density when the sample is decompressed. Axial magnetic trapping and cooling in the unitarity limit help us to maintain the high elastic collision rate needed for a fast cooling process to degeneracy. It is very interesting to compare evaporative cooling on the molecular side of the Feshbach resonance (a < 0) to the cooling on the other side of the resonance (a > 0). For similar values of |a|, one obtains a comparable cross-section σ = 4πa2 for elastic collisions between atoms in the two spin states. However, a striking diﬀerence shows up at low optical trap depth in the ﬁnal stage of the evaporative cooling process. Figure 8 shows how the number of trapped atoms (including the ones bound to molecules) decreases with the trap power. On the molecular side of the Feshbach resonance, a shallow trap can contain about ten times more atoms than on the other side of the resonance. Obviously, this cannot be understood in terms of the scattering cross-section of atoms and highlights a dramatic dependence on the sign of the scattering length. At the negative-a side of the resonance (open symbols in ﬁg. 8) a sharp decrease of the number of trapped particles is observed when the Fermi energy reaches the trap (5 ) A large initial imbalance, however, is detrimental as the cooling process already breaks down in the ﬁrst stage where only atoms are present. (6 ) For technical reasons the coils were not realized in the Helmholtz conﬁguration, where the curvature disappears. At the end this turned out to be a lucky choice for the creation of the mBEC.

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Fig. 8. – Evaporative cooling on both sides of the Feshbach resonance exhibits a strikingly diﬀerent behavior [7]. The ﬁlled and open circles refer to magnetic ﬁelds of 764 G (a = +4500 a0 ) and 1176 G (a = −3000 a0 ). We plot the total number of trapped particles 2Nmol + Nat as a function of the laser power. The power p is given relative to the initial laser power of 10.5 W of an exponential evaporation ramp with a 1/e time constant of 230 ms; the corresponding initial trap depth for the atoms is ∼ 850 μK. The solid line shows the maximum number of trapped atoms according to the number of motional quantum states of the trap. The dashed lines indicate the corresponding uncertainty range due to the limited knowledge of the trap parameters. The inset shows the optimum production of molecules in the magnetic-ﬁeld range where a weakly bound molecular state exists. Here the total number of particles is measured for various magnetic ﬁelds at a ﬁxed ﬁnal ramp power p = 2.8 × 10−4 , corresponding to a trap depth of ∼ 440 nK for the molecules.

depth. Lowering the trap power below this critical level leads to a spilling of atoms out of the trap. The trapping potential does simply not oﬀer enough quantum states for the atoms. The observed spilling is consistent with the number of quantum states calculated for a non-interacting Fermi gas (solid line). A similar spilling eﬀect is observed at the molecular side of the resonance (a > 0, ﬁlled symbols), but at much lower trap power. Before this spilling sets in, the trap contains nearly ten times more atoms as it would be possible for a non-interacting Fermi gas. This striking eﬀect is explained by the formation and Bose-Einstein condensation of molecules. The spilling eﬀect observed for the molecules with decreasing trap depth shows the chemical potential of the molecular condensate. The strategy to evaporatively cool on the molecular side of the Feshbach resonance and to produce an mBEC as the starting point for further experiments is also followed at MIT, ENS, and Rice University. The Duke group performs forced evaporation very close to the resonance, which we believe to be a better strategy when the dynamical

428

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range for the trap power reduction is technically limited. Comparing the performance of evaporative cooling at diﬀerent magnetic ﬁelds, we observed that the cooling process is somewhat more eﬃcient and more robust on the molecular side of the resonance than very close to resonance. . 4 5. The appearance of mBEC . – At the time of our early mBEC experiments in fall 2003 [7] we had no imaging system to detect the spatial distribution of the gas at high magnetic ﬁelds, where we performed the evaporation experiments described in the preceding section. Nevertheless, by measuring the dependence of the total number of trapped particles on diﬀerent parameters, we compiled various pieces of evidence for the formation of mBEC: 1. We observed that a very shallow trap can contain much more atoms than it oﬀers quantum states for a weakly interacting atomic Fermi gas. 2. We observed very long lifetimes of up to 40 s for the trapped sample after a fast and highly eﬃcient evaporation process. This shows that the sample has enough time to thermalize into an equilibrium state. 3. We measured the frequency of a collective oscillation mode (see also sect. 6), which clearly revealed hydrodynamic behavior. 4. By controlled spilling of the quantum gas out of the trap applying a variable magnetic gradient, we could demonstrate that the chemical potential of the trapped sample depends on the magnetic ﬁeld in the way expected for a mBEC from the prediction of the dimer-dimer scattering lenghts, see eq. (4). These observations, together with our previous knowledge on molecule formation in the gas [26] and the general properties of the weakly bound dimers [56], led us to a consistent interpretation in terms of mBEC. At the same time mBEC was observed in a 40 K gas at JILA in Boulder [27]. It is an amazing coincidence that our manuscript was submitted for publication on exactly the same day (Nov. 3, 2003) as the Boulder work on mBEC in 40 K. Very shortly afterwards the MIT group observed the formation of mBEC in 6 Li by detecting bimodal spatial distributions of the gas expanding after release from the trap [28]. A few weeks later, we observed the appearance of bimodal distributions in in situ absorption images of the trapped cloud [32]. At about the same time also the ENS group reported on mBEC. Figure 9 shows a gallery of diﬀerent observations of bimodal distributions in formation of 6 Li mBECs at MIT [28], in Innsbruck [32], at ENS [29], and at Rice University [36]. 5. – Crossover from mBEC to a fermionic superﬂuid With the advent of ultracold Fermi gases with tunable interactions a unique way has opened up to explore a long-standing problem in many-body quantum physics, which has attracted considerable attention since the seminal work by Eagles [76], Leggett [77]

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Fig. 9. – Gallery of 6 Li molecular BEC experiments. Bimodal spatial distributions were observed for expanding gases at MIT [28] and at ENS [29], and in in situ proﬁles of the trapped cloud in Innsbruck [32] and at Rice University [36].

. and Nozi`eres and Schmitt-Rink [78]. Here we give a brief introduction (5 1) into the physics of the BEC-BCS crossover(7 ) and we introduce some basic deﬁnitions and typical . experimental parameters (5 2). We then consider a universal Fermi gas with resonant . . interactions (5 3) and the equation of state in the crossover (5 4). Next we discuss the crossover at non-zero temperatures, including the isentropic conversion between diﬀerent . interaction regimes (5 5). We ﬁnally review our ﬁrst crossover experiments where we have observed how spatial proﬁles and the size of the strongly interacting, trapped cloud . changed with variable interaction strength (5 6). . 5 1. BEC-BCS crossover physics: a brief introduction. – The crossover of a superﬂuid system from the BEC regime into the Bardeen-Cooper-Schrieﬀer (BCS) regime can be (7 ) In the condensed-matter literature, the crossover is commonly referred to as the “BCS-BEC crossover”, because BCS theory served as the starting point. In our work on ultracold gases, we use “BEC-BCS crossover”, because we start out with the molecular BEC. The physics is one and the same.

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Fig. 10. – Illustration of the BEC-BCS crossover in a zero-temperature 6 Li Fermi gas with tunable interactions. For positive scattering length (a > 0, BEC side of the Feshbach resonance) the ground state of the system is a Bose-Einstein condensate of molecules. On resonance (a → ±∞, unitarity limit) a strongly interacting Fermi gas with universal properties is realized. For negative scattering length (a < 0, BCS side of the resonance) the system approaches the BCS regime.

intuitively understood by ﬁrst considering the two limits, which can be described in the framework of well-established theory (see illustration in ﬁg. 10). For moderate positive scattering lengths, the fermions form bosonic molecules, and the ground state at T = 0 is a BEC. For moderate negative scattering lengths, the ground state at T = 0 is the wellknown BCS state [79, 80]. With variable interaction strength across a resonance, both regimes are smoothly connected through the strongly interacting regime. Here both BEC and BCS approaches break down and the description of the strongly interacting system is a diﬃcult task. This situation poses great challenges for many-body quantum theories [81]. The nature of pairing is the key to understanding how the system changes through the crossover. On the BEC side, the pairs are molecules which can be understood in the framework of two-body physics. The molecular binding energy Eb is large compared with all other energies, and the molecules are small compared with the typical interparticle spacing. In this case, the interaction can be simply described in terms of molecule. molecule collisions, for which the scattering length is known (see subsect. 3 3). On the BCS side, two atoms with opposite momentum form Cooper pairs on the surface of the Fermi sphere. The pairing energy, i.e. the “pairing gap”, is small compared with the Fermi energy EF and the Cooper pairs are large objects with a size greatly exceeding the typical interparticle spacing. In the strongly interacting regime, the pairs are no longer pure molecules or Cooper pairs. Their binding energy is comparable to the Fermi energy and their size is about the interparticle spacing. One may consider them either as generalized molecules, stabilized by many-body eﬀects, or alternatively as generalized Cooper pairs. The ground state at T = 0 is a superﬂuid throughout the whole crossover. In the BEC limit, the fermionic degrees of freedom are irrelevant and superﬂuidity can be fully

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understood in terms of the bosonic nature of the system. In the opposite limit, superﬂuidity is described in the framework of BCS theory [79, 80]. In the strongly interacting regime, a novel type of superﬂuidity (“resonance superﬂuidity” [82, 83]) occurs where both bosonic and fermionic degrees of freedom are important. . 5 2. Basic deﬁnitions, typical experimental parameters. – Let us start with some basic deﬁnitions, which we will need to describe the physics in the rest of this contribution. The Fermi energy of a trapped, non-interacting two-component gas is given by EF = h ¯ω ¯ (3N )1/3 ;

(8)

here N is the total number of atoms in both spin states, and ω ¯ = (ωx ωy ωz )1/3 is the geometrically averaged oscillation frequency in the harmonic trapping potential. This expression can be derived within the Thomas-Fermi approximation for a suﬃciently large number of trapped atoms [84]. The chemical potential μ of the non-interacting gas is equal to EF . We use the Fermi energy of a non-interacting gas, EF , to deﬁne an energy scale for the whole BEC-BCS crossover, i.e. for any regime of interactions. The corresponding Fermi temperature is TF = EF /kB .

(9)

We now introduce a Fermi wave number kF , following the relation (10)

¯ 2 kF2 h = EF . 2m

The inverse Fermi wave number kF−1 deﬁnes a typical length scale for the crossover problem. For the non-interacting case, kF is related to the peak number density n0 in the center of the trap [84] by (11)

n0 =

kF3 . 3π 2

To characterize the interaction regime, we introduce the dimensionless interaction parameter 1/kF a, which is commonly used to discuss crossover physics. We can now easily distinguish between three diﬀerent regimes. The BEC regime is realized for 1/kF a 1, whereas the BCS regime is obtained for 1/kF a −1. The strongly interacting regime lies between these two limits where 1/kF |a| being small or not greatly exceeding unity. Let us consider typical experimental parameters for our 6 Li spin mixture: an atom number N of a few 105 , and a mean trap frequency ω ¯ /2π near 200 Hz. This corresponds to a typical Fermi temperature TF ≈ 1 μK and to kF−1 ≈ 200 nm ≈ 4000 a0 . A comparison of kF−1 with the scattering lengths close to the 834 G Feshbach resonance (see ﬁg. 2) shows that there is a broad crossover region where the 6 Li system is strongly interacting. The

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peak number then considerably exceeds the typical value n0 ≈ 4 × 1012 cm−3 calculated for the non-interacting case. . 5 3. Universal Fermi gas in the unitarity limit. – The resonance where a(B) diverges and 1/kF a = 0, is at the heart of BEC-BCS crossover physics(8 ). Here the s-wave interaction between colliding fermions is as strong as quantum mechanics allows within the fundamental limit of unitarity. In this situation, EF and 1/kF represent the only energy and length scales in the problem and the system acquires universal properties [8587]. The broad Feshbach resonance in the ultracold 6 Li gas oﬀers excellent possibilities to study the properties of the universal Fermi gas [88] and the situation has attracted a great deal of experimental interest, as described in various parts of this volume. At T = 0, universality implies a simple scaling behavior with respect to the situation of a non-interacting Fermi gas. Following the arguments in refs. [85, 86, 21] the atomic mass m can be simply replaced by an eﬀective mass (12)

meﬀ =

m , 1+β

where β −0.57 [89, 90] is a dimensionless, universal many-body parameter. For a har√ monic trapping potential, eq. (12) results in an eﬀective trap frequency ω eﬀ = 1 + β ω, and the chemical potential for a zero-temperature gas in the unitarity limit is then given by (13)

μ=

1 + β EF .

The density proﬁle of the universal Fermi gas with resonant interactions is just a simple rescaled version of the density proﬁle of the non-interacting gas, smaller by a factor of (1 + β)1/4 0.81. The universal many-body parameter was recently calculated based on quantum Monte-Carlo methods, yielding β = −0.56(1) [89] and −0.58(1) [90]. A diagrammatic theoretical approach [91] gave a value −0.545 very close to these numerical results. Several experiments in 6 Li [32, 29, 37, 39] and in 40 K [92] have provided measurements of β in good agreement with the theoretical predictions. We will discuss our experimental . results on β in some more detail in subsect. 5 6. At T = 0, the Fermi gas with unitarity-limited interactions obeys a universal thermodynamics with T /TF being the relevant dimensionless temperature parameter [87]. Thermodynamic properties of the system have been experimentally studied in ref. [37]. . 5 4. Equation of state. – The equation of state is of central interest to characterize the interaction properties of the Fermi gas in the BEC-BCS crossover. For a system at (8 ) In nuclear physics this situation is known as the “Bertsch problem”. G. F. Bertsch raised the question on the ground-state properties of neutron matter under conditions where the scattering length between the two neutron spin states is large compared to the interparticle spacing.

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T = 0, the equation of state is described by the chemical potential μ as a function of the number density n. For μ(n) at T = 0, we now consider three special cases. For a non-interacting Fermi gas, μ = EF , and one thus obtains μ = (3π 2 )2/3

(14)

¯ 2 2/3 h n . 2m

For a Fermi gas with resonant interactions, universality implies that one obtains the same expression with a prefactor 1 + β (eq. (12)). In the mBEC regime, the chemical potential for the dimers is μd = 4π¯h2 add m−1 d nd . With the simple relations between mass (md = 2m), number density (nd = n/2), and scattering length (add = 0.6a, see eq. (4)) for dimers and atoms, and after substraction of the molecular binding energy Eb = h ¯ 2 /(ma2 ), we obtain (15)

μ=

1 ¯2 h (μd − Eb ) = 0.6π a n − a−2 . 2 2m

For the general BEC-BCS crossover problem one can introduce a “polytropic” equation of state [93] in the form (16)

μ ∝ nγ .

Here the “polytropic index” γ depends on the interaction parameter 1/kF a. By comparing this equation of state with the above expressions one immediately sees that γ = 1 for the mBEC case (1/kF a 1), γ = 2/3 both for the unitarity limit (1/kF a = 0) and for the non-interacting case (1/kF a −1). These three values are ﬁxed boundary conditions for any crossover theory describing γ as a function of 1/kF a. In the experiments, the Fermi gases are usually conﬁned in nearly harmonic trapping potentials, which leads to an inhomogeneous density distribution. If the trap is not too small one can introduce the local-density approximation and consider a local chemical potential μ(r) = μ − U (r), which includes the trapping potential U (r) at the position r. This assumption holds if the energy quantization of the trap is irrelevant with respect to the chemical potential and the pair-size is small compared to the ﬁnite size of the trapped sample. This approximation is well fulﬁlled for all crossover experiments performed in Innsbruck. . 5 5. Phase diagram, relevant temperatures and energies. – At ﬁnite temperatures the BEC-BCS crossover problem becomes very challenging and it is of fundamental interest to understand the phase diagram of the gas. Two temperatures play an important role, the temperature Tc for the superﬂuid phase transition and a pairing temperature T ∗ , characterizing the onset of pairing. Let us ﬁrst discuss these two temperatures in the three limits of the crossover (BEC, unitarity, and BCS), see ﬁrst two rows in table I. The critical temperature Tc in the mBEC limit follows directly from the usual expres1/3 sion for the BEC transition temperature in a harmonic trap kB Tc 0.94¯hω ¯ Nm [94], Nm = N/2, and kB TF = h ¯ω ¯ (3N )1/3 . The given value for the critical temperature in the

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Table I. – Overview of important temperatures and energies in the three crossover limits. The expressions are valid for a harmonically trapped Fermi gas. mBEC (1/kF a 1) crit. temp.

Tc

pair. temp.

T∗

gap energy

2Δ

chem. pot.

μ

“ 12

T TF

”3

0.518 TF h = exp TTF −2

Unitarity (1/kF a = 0)

2 (kF a)2

i

2 (kF a) EF ” “ 0.294(kF a)2/5 − (kF a)−2 EF

∼ 0.3 TF

BCS (1/kF a −1) i h 0.277 TF exp 2kπF a

∼ 0.4 TF

Tc

1.8EF

3.528 kB Tc

0.66EF

EF

unitarity limit was derived in various crossover theories [95,37]. For the BCS regime, the critical temperature is a well-known result from ref. [96]; see also refs. [97, 98]. For the pairing temperature T ∗ , typical numbers are given in the second row of table I. In the framework of BCS theory, there is no diﬀerence between T ∗ and Tc , which means that as soon as Cooper pairs are formed the system is also superﬂuid. On the BEC side, however, molecules are formed at much higher temperatures as the phase transition to . molecular BEC occurs (see discussion in 4 3). Setting φmol = φat in eq. (7), one can derive the implicit equation for T ∗ /TF given in the table. In the unitarity limit, T ∗ is not much higher than Tc ; ref. [95] suggests T ∗ /Tc ≈ 1.3. The phase diagram in ﬁg. 11 illustrates the behavior of Tc and T ∗ , as discussed before for the three limits. We point out that, in strongly interacting Fermi gases, there is a

Fig. 11. – Schematic phase diagram for the BEC-BCS crossover in a harmonic trapping potential [95]. The critical temperature Tc marks the phase transition from the normal to the superﬂuid phase. Pair formation sets in gradually at a typical temperature T ∗ > Tc .

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certain region where pairing occurs without superﬂuidity. In the language of high-Tc superconductivity [99, 81], “preformed pairs” are present in the “pseudo-gap regime”. For overview purposes, table I also gives the pairing energy 2Δ and the chemical potential μ. In the mBEC regime, the pairing energy just corresponds to the molecular binding energy Eb = 2(kF a)−2 EF . The chemical potential μ = (1/2)(μd − Eb ) (see eq. (15)) can be derived from μd /¯hω ¯ = (1/2)(15Nd ad /aho )2/5 with aho = (¯ h/md ω ¯ )1/2 , valid for an mBEC in the Thomas-Fermi limit [94]. In the BCS limit, there is the ﬁxed relation of the “gap” Δ to the critical temperature Tc given in table I. For the unitarity limit, the value given for the pairing energy stems from quantum Monte-Carlo calcula. tions [89]. The behavior of Δ in the crossover is extensively discussed in subsect. 7 3. The table also presents the chemical potential μ according to eqs. (14) and (15), rewritten in terms of the parameters 1/kF a and EF . . 5 6. First Innsbruck crossover experiments: conservation of entropy, spatial proﬁles, and potential energy of the trapped gas. – The possibility to continuously vary the interaction parameter 1/kF a through Feshbach tuning oﬀers the fascinating possibility to convert the Fermi gas between diﬀerent regimes and thus to explore the BEC-BCS crossover. We performed our ﬁrst experiments on crossover physics [32] in December 2003 shortly after the ﬁrst creation of the mBEC [7]. Here we summarize the main results of these early experiments, which are of general importance for BEC-BCS crossover experiments with 6 Li. We performed slow conversion-reconversion cycles, in which the strongly interacting gas was adiabatically converted from the BEC side of the crossover to the BCS side and vice versa. We found that this conversion took place in a lossless way and that the spatial proﬁles of the trapped cloud did not show any signiﬁcant heating. We could thus demonstrate that, under appropriate experimental conditions, the conversion process can proceed in an essentially adiabatic and reversible way, which means that the entropy of the gas is conserved. The conservation of entropy has important consequences for the experiments: Because of the diﬀerent relations between entropy and temperature in various interaction regimes, an isentropic conversion in general changes the temperature. As a substantial beneﬁt, a drastic temperature reduction occurs when the degenerate gas is converted from mBEC into the BCS regime. This is very favorable for the achievement of a superﬂuid state on the BCS side of the resonance [98] or in the unitarity-limited resonance regime [100]. In our experiments, we typically start out with a condensate fraction of more than 90% in the weakly interacting mBEC regime. Based on the isentropic conversion process and the thermodynamics discussed in ref. [100] we estimate that we obtain typical temperatures between 0.05 TF and 0.1 TF for the Fermi gas in the unitarity limit(9 ). (9 ) We note that the large stability of 6 Li in the mBEC regime oﬀers an advantage over 40 K in that one can evaporatively cool in the mBEC regime and exploit the temperature-reduction eﬀect in conversion onto the BCS side of the resonance.

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Fig. 12. – Axial density proﬁles of a trapped 6 Li Fermi gas in the crossover region [32]. The middle proﬁle, taken very close to resonance (850 G), is compared to the Thomas-Fermi proﬁle of a universal Fermi gas (solid line). The small deviation on the top is due to a residual interference pattern in the images.

Using slow magnetic ﬁeld ramps we isentropically converted the trapped gas into diﬀerent interaction regimes covering the whole resonance region and beyond. By in situ imaging we recorded the axial density proﬁles of the trapped cloud. The results (see proﬁles in ﬁg. 12) demonstrated the smooth behavior in the crossover. The cloud just became larger without showing any particular features, and we found simple ThomasFermi proﬁles to ﬁt our observations very well. The one-dimensional spatial proﬁles did not show any signatures of a superﬂuid phase transition(10 ), in agreement with theoretical expectations [95, 101]. To quantitatively characterize the behavior, we measured the root-mean-square axial size zrms of the cloud as a function of the magnetic ﬁeld B. The normalized quantity ζ = zrms /z0 gives the relative size as compared to a non-interacting Fermi gas, where z0 = (EF /4mωz2 )1/2 . The potential energy of the harmonically trapped gas relative to a non-interacting Fermi gas is then simply given by ζ 2 . Within the local density approximation, this is also valid for the three-dimensional situation. Our experimental results can thus be interpreted as the ﬁrst measurements of the potential energy of a trapped Fermi gas near T = 0 in the BEC-BCS crossover(11 ). (10 ) This is diﬀerent in an imbalanced spin-mixture, where the superﬂuid phase transition was observed by changes in the spatial proﬁles [41]. (11 ) We note that a later thorough analysis of the conditions of the experiments in ref. [32] conﬁrmed the atom number N = 4 × 105 to within an uncertainty of ±30%. However, we found that the horizontal trap frequency was only 80% of the value that we used based on the . assumption of a cylindrically symmetric trap (see subsect. 6 4). Moreover, the exact position . of the Feshbach resonance was located at 834 G (see subsect. 3 2) instead of 850 G as assumed in the ﬁrst analysis of the experiment. The up-to-date values are used for ﬁg. 13, causing slight deviations from the original presentation of our data.

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Fig. 13. – Results of the ﬁrst Innsbruck BEC-BCS crossover experiments [32] on the axial size normalized to the theoretical size of a non-interacting Fermi gas (ζ = zrms /z0 ). The solid line is a theoretical prediction for the size of a mBEC in the Thomas-Fermi limit, and the star indicates the theoretical value for the unitarity limit.

The measured values for the relative size ζ are plotted in ﬁg. 13 and compared to the predictions for a weakly interacting molecular BEC with add = 0.6a in the in the ThomasFermi limit (solid line) and a universal Fermi gas in the unitarity limit (star). The experimental data on the mBEC side are consistent with the theoretical prediction. On resonance, the measured size was found somewhat below the prediction (1 + β)1/4 0.81 . (see subsect. 5 3); see star in ﬁg. 13. This slight discrepancy, however, may be explained by possible calibration errors in the measured number of atoms and in the magniﬁcation of the imaging system in combination with the anharmonicity of the radial trapping potential. Beyond resonance are results stayed well below the non-interacting value ζ = 1, showing that we did not reach weakly interacting conditions. This is a general consequence of the large background scattering length of 6 Li, which (in contrast to 40 K) makes it very diﬃcult to realize a weakly interacting Fermi gas on the BCS side of the Feshbach resonance. In general, the dependence of the size and thus the potential energy of the trapped gas in the BEC-BCS crossover that we observed in our ﬁrst experiments [32] was found to ﬁt well to corresponding theoretical predictions [91]. Later experiments by other groups provided more accurate measurements on the size of the gas for the particulary interesting unitarity limit [37, 39, 92]. 6. – Collective excitations in the BEC-BCS crossover Elementary excitation modes provide fundamental insight into the properties of quantum-degenerate gases. In particular, they provide unique experimental access to study

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Fig. 14. – Illustration of elementary collective modes of a cigar-shaped quantum gas, conﬁned in an elongated trap. The axial mode corresponds to a slow oscillation with both compression and surface character. The two low-lying radial modes correspond to fast oscillations with strong compression character (“radial breathing mode”) and with pure surface character (“radial quadrupole mode”).

the hydrodynamic behavior that is associated with superﬂuidity. Collective modes have been studied very early in atomic BEC research, both in experiments [102, 103] and in theory [104]. Measurements on collective oscillations have proven powerful tools for the investigation of various phenomena in atomic BECs [105-109]. Building on this rich experience, collective modes attracted immediate attention to study strongly interacting Fermi gases [110,33,34] as soon as these systems became experimentally available. Here, . we give a basic introduction into collective modes in the BEC-BCS crossover (6 1), and we present an overview of the major experimental results obtained in our laboratory . in Innsbruck and at Duke University (6 2), before we discuss our results in some more . . detail (6 3–6 6). . 6 1. Basics of collective modes. – We will focus our discussion on the geometry of elongated traps with cylindrical symmetry, because this is the relevant geometry for strongly interacting Fermi gases in single-beam optical traps. Besides the simple sloshing modes that correspond to center-of-mass oscillations in the trap, the cigar-shaped quantum gas exhibits three elementary, low-lying collective modes, which are illustrated in ﬁg. 14. The axial mode corresponds to an oscillation of the length of the “cigar” with a frequency of the order of the axial trap frequency ωz . This oscillation is accompanied by a 180◦ phase-shifted oscillation of the cigar’s radius, which reﬂects a quadrupolar character of the mode. Thus, the mode has the mixed character of a compression and a surface mode. The frequencies of the two low-lying radial modes are of the order of the radial trap frequency ωr . The “radial breathing mode” is a compression mode, for which

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the radius of the sample oscillates. The “radial quadrupole mode” is a pure surface mode where a transverse deformation oscillates without any change of the volume. To understand collective modes in a Fermi gas, it is crucial to distinguish between two fundamentally diﬀerent regimes. Which regime is realized in an experiment depends on the interaction strength 1/kF a and the temperature T of the gas. – The collisionsless, non-superﬂuid regime – In a weakly interacting degenerate Fermi gas, elastic collisions are Pauli blocked [111-113]. This is due to the fact that the ﬁnal states for elastic-scattering processes are already occupied. This Pauli blocking eﬀect has dramatic consequences for the dynamics of a two-component Fermi gas when it is cooled down to degeneracy. In the non-degenerate case, the inﬂuence of collisions between the two diﬀerent spin states can be very strong, as it . is highlighted by our eﬃcient evaporative cooling process (see subsect. 4 3). In the degenerate case, however, collisions are strongly suppressed. A substantial increase in relaxation times [114] shows up as an important consequence. – The hydrodynamic regime – When a superﬂuid is formed at suﬃciently low temperatures, hydrodynamic behavior occurs as an intrinsic property of the system, and the gas follows the equations of superﬂuid hydrodynamics (see contribution by S. Stringari in this volume). However, in a strongly interacting Fermi gas bosonic pairs can be formed and their elastic interactions are no longer Pauli blocked; this may lead to classical hydrodynamics in a degenerate gas. In this case, the sample follows basically the same hydrodynamic equations as in the superﬂuid case. Therefore, it is not possible to draw an immediate conclusion on superﬂuidity just from the observation of hydrodynamic behavior. The existence of these two diﬀerent regimes has important consequences for collective oscillations. In the (non-superﬂuid) collisionless case, the fermionic atoms perform independent oscillations in the trapping potential and the eﬀect of elastic collisions and collisional relaxation is small [114, 115]. The ensemble then shows decoupled oscillations along the diﬀerent degrees of freedoms with frequencies that are twice the respective trap frequencies(12 ). In the hydrodynamic regime, a solution of the equations of motion (see lecture of S. Stringari) yields the following expressions for the collective mode frequencies in the elongated trap limit, ωz /ωr → 0(13 ): (17) (18) (19)

(3¯ γ + 2)/(¯ γ + 1) ωz , ωc = 2¯ γ + 2 ωr , √ ωq = 2 ω r .

ωax =

(12 ) We neglect small interaction shifts, which are discussed in [115]. (13 ) For all experiments reported here, the traps fulﬁlled ωz /ωr < 0.1, which makes the elongated trap limit a valid approximation.

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Table II. – Overview of collective mode frequencies in diﬀerent regimes. Hydrodynamic

Axial mode Radial compression mode Radial quadrupole mode

ωax /ωz ωc /ωr ωq /ωr

mBEC (1/kF a 1) p 5/2 = 1.581.. 2 √ 2 = 1.414..

unitarity (1/kF a = 0) p p12/5 = 1.549.. 10/3 = 1.826.. √ 2 = 1.414..

Collisionless, non-superﬂuid

2 2 2

Here γ¯ is an eﬀective polytropic index for the equation of state (eq. (16)), which takes into account the variation of the density across the inhomogeneous sample in the harmonic trap [116, 117]. For the mBEC case γ¯ = 1, and for the unitarity limit γ¯ = 2/3. The theory of collective modes in the BEC-BCS crossover has attracted considerable interest and is extensively discussed in refs. [110, 93, 116-123]. Table II presents an overview of the frequencies of the three low-lying modes in different regimes. When the interaction is varied from mBEC to the unitarity limit, the axial mode mode changes its frequency by just ∼ 2%. However, for the radial breathing mode the relative change is ﬁve times larger (∼ 10%). This diﬀerence reﬂects the much stronger compression character of the radial breathing mode, which is why this mode is . a prime tool to experimentally investigate the equation of state (see subsect. 6 5). The fact that the radial quadrupole mode is a pure surface mode makes it insensitive to the equation of state. This mode can thus serve as a powerful tool for investigating the large diﬀerences between hydrodynamic and collisionless behavior [124]. . 6 2. Overview of recent experiments. – Here we give a brief overview of the major results of collective-mode experiments performed at Duke University and in Innsbruck. Already in our early work on mBEC [7] we measured the axial-mode frequency to show that the trapped sample behaved hydrodynamically. The ﬁrst experimental results on collective modes in the BEC-BCS crossover were reported by our team and the Duke group at the Workshop on Ultracold Fermi Gases in Levico (4-6 March 2004). These results were published in refs. [33, 34]. The Duke group investigated the radial breathing mode for resonant interactions and measured a frequency that was consistent with the theoretical prediction (see sub. sect. 6 1) for a hydrodynamic Fermi gas with unitarity-limited interactions. They also investigated the temperature-dependent damping behavior and observed strongly increasing damping times when the sample was cooled well below the Fermi temperature TF ; the main result is shown in ﬁg. 15. By comparing the results with available theories on Fermi gases in the collisionless, non-superﬂuid regime and with theories on collisional hydrodynamics they found the observed behavior to be inconsistent with these two regimes [33]. Superﬂuidity provided a plausible explanation for these observations, and the Duke group thus interpreted the results as evidence for superﬂuidity. Later experiments [35, 37, 38]

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Fig. 15. – Evidence for superﬂuidity in a strongly interacting Fermi gas obtained at Duke University from measurements of the damping of the radial breathing mode [33]. The damping time 1/Γc (in units of the oscillation period 2π/ωc ) is plotted vs. temperature T . The inset shows a breathing oscillation (ωc /2π = 2830 Hz) at the lowest temperatures reached in the experiment. This ﬁgure was adapted from ref. [33].

indeed provided a consistent picture of superﬂuidity for the conditions under which these collective mode experiments were performed. In our early experiments [34], we measured the frequencies of the axial mode and the radial compression mode in the BEC-BCS crossover. Here we observed the frequency variations that result from the changing equation of state. On the BCS side of the Feshbach resonance, we observed a transition from hydrodynamic to non-superﬂuid, collisonless . behavior. The transition occurred rather smoothly in the axial mode (see subsect. 6 3) . but abruptly in the radial breathing mode (see subsect. 6 4). We also observed ultralow damping in the axial mode, which nicely ﬁts into the picture of superﬂuidity. The abrupt breakdown of hydrodynamics in the radial breathing mode was also observed at Duke University [125]. Further experiments on collective modes at Duke University [126] provided more information on the temperature dependence of damping for unitarity-limited interactions. This experiment also hinted on diﬀerent damping regimes. At Innsbruck University, we carried out a series of precision measurements on the frequencies of collective modes in the crossover [67]. This provided a test of the equation of state and resolved seeming discrepancies between state-of-the-art theoretical predictions [117] and the early experiments [34, 125]. . 6 3. Axial mode. – Our measurements of frequency and damping of the axial mode [34] are shown in ﬁg. 16. To tune the two-body interaction we varied the magnetic ﬁeld in a range between 700 and 1150 G, corresponding to a variation of the interaction parameter 1/kF a between 2.5 and −1.2. For magnetic ﬁelds up to ∼ 900 G (1/kF a ≈ −0.45), the oscillation shows the hydrodynamic frequencies and very low damping. For higher ﬁelds,

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Fig. 16. – Axial mode in the BEC-BCS crossover. The ﬁgure shows our measurements [34] of the frequency ωax and the damping rate Γax in units of the axial trap frequency ωz (ωz /2π = 22.6 Hz at B = 834 G, ωr /2π ≈ 700 Hz). The horizontal, dashed lines indicate the theoretically expected frequencies in the BEC limit and in the collisionless limit (cf. table II). The ﬁgure on the righthand side shows a blow-up of the resonance region; here the star refers to the frequency expected for the unitarity limit.

damping strongly increases and the frequency gets closer to the collisionless value, but never reaches it completely. These observations are consistent with a gradual transition from hydrodynamic to collisionless behavior [114]. Even far on the BCS side of the resonance, the true collisionless regime is not reached, as the Pauli blocking eﬀect is not strong enough to suppress elastic collisions on a time scale below the very long axial oscillation period of about 50 ms. On the right-hand side of ﬁg. 16, we show a blow-up of the resonance region. One clearly sees that the axial mode frequency changes from the BEC value ω /ω = 5/2 = ax z 1.581 to the value of a universal Fermi gas in the unitarity limit of 12/5 = 1.549. We were able to detect this small 2% eﬀect because of the very low damping of the mode, allowing long observation times. Moreover, the magnetic axial conﬁnement was perfectly harmonic, and the corresponding trap frequency ωz could be measured with a relative uncertainty of below 10−3 [127]. It is also very interesting to consider the damping of the axial mode. The minimum damping rate was observed at ∼ 815 G, which is slightly below the exact resonance (834 G). Here we measured the very low value of Γz /ωz ≈ 0.0015, which corresponds to a 1/e damping time as large as ∼ 5 s. According to our present knowledge of the system, this ultralow damping is a result of superﬂuidity of the strongly interacting Fermi gas.

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Fig. 17. – Measurements of the frequency ωc and the damping rate Γc of the radial compression mode in the BEC-BCS crossover [34]. Here the oscillation frequencies were determined relative to the vertical trap frequency ωy ≈ 750 Hz. As we found in later experiments, the trap was somewhat elliptic with a horizontal trap frequency ωx ≈ 600 Hz. This is about 20% below the vertical one and has a substantial eﬀect on the hydrodynamic frequencies [128]. On the BEC side of the Feshbach resonance, the dashed line indicates the frequency theoretically expected in the BEC limit (ωc /ωy = 1.85 for ωx /ωy = 0.8). The star marks the frequency in the unitarity limit (ωc /ωy = 1.805 for ωx /ωy = 0.8). On the BCS side, the dashed line indicates the frequency 2ωy for the collisionless case.

The damping observed for lower magnetic ﬁelds can be understood as a consequence of heating due to inelastic processes in the gas(14 ). In general, damping rates are very sensitive to the residual temperature of the sample. . 6 4. Radial breathing mode: breakdown of hydrodynamics. – Our early measurements [34] of frequency and damping of the radial breathing mode are shown in ﬁg. 17. The most striking feature is a sharp transition from the hydrodynamic to the collisionless regime. This occurs at a magnetic ﬁeld of ∼ 900 G (1/kF a ≈ −0.45). Apparently, the hydrodynamic regime extends from the mBEC region across the unitarity limit onto the BCS side of the resonance. This behavior is consistent with the direct observation of superﬂuidity in the crossover region through vortices [38]. The breakdown of superﬂuid (14 ) Precise frequency measurements of the slow axial mode require long observation times. On the BEC side of the resonance, heating due to inelastic decay then becomes a hardly avoidable problem.

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hydrodynamics is accompanied by very fast damping, which indicates a fast dissipation mechanism in the sample. We will come back to this in our discussion of the pairing gap . (last paragraph in subsect. 7 3). The breakdown of hydrodynamics on the BCS side of the resonance was also observed by the Duke group [125]. To quantitatively understand the frequencies in the hydrodynamic regime as measured in our early collective mode experiments [34], one has to take into account an unintended ellipticity of the transverse trapping potential [128]. We found out later after technical upgrades to our apparatus that the ratio of horizontal and vertical trap frequencies was ωx /ωy ≈ 0.8. Due to the fact that the gas was not completely cylindrically symmetric, the collective-mode frequencies deviate from the simple expressions presented . in subsect. 6 1. For small ellipticities, eq. (18) still provides a reasonable approximation √ when an eﬀective transverse oscillation frequency of ωx ωy is used for ωr [33]. For an accurate interpretation of the measurements, however, a more careful consideration of ellipticity eﬀects is necessary [129, 67, 128]. In ﬁg. 17, we indicate the expected normalized compression mode frequencies ωc /ωy for the limits of mBEC (dashed line below resonance), unitarity (star), and for a noninteracting collisionless gas (dashed line above resonance). For normalization we have used the vertical trap frequency ωy , which was directly measured in the experiments. Moreover, we assumed ωx /ωy = 0.8 to calculate the eigenfrequencies of the collective modes [128]. We see that, within the experimental uncertainties, the measurements agree reasonably well with those limits(15 ). For a quantitative comparison of our early compression mode measurement with theory and also with the experiments of the Duke group, the ellipticity turned out to be the main problem. However, as an unintended beneﬁt of this experimental imperfection, the larger diﬀerence between the frequencies in the hydrodynamic and the collisionsless regime strongly enhanced the visibility of the transition between these two regimes. . 6 5. Precision test of the equation of state. – Collective modes with compression character can serve as sensitive probes to test the equation of state of a superﬂuid gas in the BEC-BCS crossover. The fact that a compression mode frequency is generally lower for a Fermi gas in the unitarity limit than in the mBEC case simply reﬂects the larger compressibility of a Fermi gas as compared to a BEC. The data provided by the experiments in Innsbruck [34] and at Duke University [33,125] in 2004 opened up an intriguing possibility for quantitative tests of BEC-BCS crossover physics. For such precision tests, frequency measurements of collective modes are superior to the simple size measurements discussed in sect. 5. It is an important lesson that one learns from metrology that it is often very advantageous to convert the quantity to be measured into a frequency. In this spirit, the radial breathing mode can be seen as an excellent instrument to convert compressibility into a frequency for accurate measurements. (15 ) The slight deviation in the unitarity limit is likely due to the anharmonicity of the trapping potential, which is not taken into account in the calculation of the frequencies.

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A comparison of the axial mode data from Innsbruck and the measurements on the radial breathing mode from Duke University with mean-ﬁeld BCS theory showed reasonable agreement [116, 121, 117]. This, however, was somewhat surprising as mean-ﬁeld BCS theory has the obvious shortcoming that it does not account for beyond-mean-ﬁeld eﬀects [130, 131]. The latter were expected to up-shift the compression mode frequencies in the strongly interacting mBEC regime [110], but they seemed to be absent in the experiments. Advanced theoretical calculations based on a quantum Monte Carlo approach [90] conﬁrmed the expectation of beyond-mean-ﬁeld eﬀects in the equation of state and corresponding up-shifts in the collective mode frequencies as compared to mean-ﬁeld BCS theory [121, 117]. The apparent discrepancy between theory and experiments(16 ) motivated us to perform a new generation of collective mode experiments [67] with much higher precision and with much better control of systematic eﬀects. To achieve a 10−3 accuracy level, small ellipticity and anharmonicity corrections had to be taken into account. In ﬁg. 18 we present our measurements on the frequency of the radial breathing mode in the BECBCS crossover. Because of the very low uncertainties it can be clearly seen that our data agrees with the quantum Monte Carlo equation of state, thus ruling out mean-ﬁeld BCS theory. Our experimental results also demonstrate the presence of the long-sought beyond-mean-ﬁeld eﬀects in the strongly interacting BEC regime, which shift the normalized frequency somewhat above the value of two, which one would obtain for a weakly interacting BEC. To obtain experimental results valid for the zero-temperature limit (ﬁg. 18) it was crucial to optimize the timing sequence to prepare the gas in the BEC-BCS crossover with a minimum of heating after the production of the mBEC as a starting point. A comparison of the ultralow damping rates observed in our new measurements with the previous data from 2004 shows that the new experiments were indeed performed at much lower temperatures. We are convinced that temperature-induced shifts provide a plausible explanation for the earlier measurements being closer to the predictions of mean-ﬁeld BCS theory than to the more advanced quantum Monte Carlo results. For the strongly interacting mBEC regime, we indeed observed heating (presumably due to inelastic processes) to cause signiﬁcant down-shifts of the breathing mode frequency [67]. To put these results into a broader perspective, our precision measurements on collective modes in the BEC-BCS crossover show that ultracold Fermi gases provide a unique testing ground for advanced many-body theories for strongly interacting systems. . 6 6. Other modes of interest. – At the time of the Varenna Summer School we had started a set of measurements on the radial quadrupole mode in the BEC-BCS crossover [124]. This mode had not been investigated before. We implemented a twodimensional acousto-optical scanning system for the trapping laser beam; this allows us to (16 ) The discrepancy between the ﬁrst experiments at Duke [33] and in Innsbruck [34] disap. peared when we understood the problem of ellipticity in our setup (see subsect. 6 4).

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Fig. 18. – Precision measurements of the radial breathing mode frequency versus interaction parameter 1/kF a in comparison with theoretical calculations [67]. The frequency is normalized to the radial trap frequency. The experimental data include small corrections for trap ellipticity and anharmonicity and can thus be directly compared to theory in the limit of an elongated trap . with cylindrical symmetry (see subsect. 6 1). The open and ﬁlled circles refer to measurements 1/2 of 290 Hz and 590 Hz, respectively. Here ωx /ωy was typically at trap frequencies (ωx ωy ) between 0.91 and 0.94. The ﬁlled triangle shows a zero-temperature extrapolation of a set of measurements on the temperature dependence of the frequency. The theory curves refer to mean-ﬁeld BCS theory (lower curve) and quantum Monte Carlo calculations (upper curve) and correspond to the data presented in ref. [117]. The horizontal dashed lines indicate the values for the BEC limit and the unitarity limit (see table II).

produce time-averaged optical potentials [132,133], in particular potentials with variable ellipticities. With this new system, it is straightforward to create an appropriate deformation of the trapping potential to excite the quadrupole mode and other interesting modes. Here we just show the oscillation of the radial quadrupole mode for a universal Fermi gas right on resonance at B = 834 G for the lowest temperatures√that we can achieve (ﬁg. 19). The mode indeed exhibits the expected frequency (ωq = 2 ωr ), which nicely demonstrates the hydrodynamic behavior. Moreover, we ﬁnd that the damping is considerably faster than for the radial compression mode at the same temperature. Scissors modes [134, 108, 135, 136] represent another interesting class of collective excitations which we can investigate with our new system. A scissors mode is excited by a sudden rotation of an elliptic trapping potential. Scissors modes may serve as a new tool to study the temperature dependence of hydrodynamics in the BEC-BCS crossover. Scissors modes are closely related to rotations [137] and may thus provide additional insight into the collisional or superﬂuid nature of hydrodynamics.

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Fig. 19. – Radial quadrupole oscillation of the universal Fermi gas with unitarity-limited interactions at B = 834 G. We plot the diﬀerence in horizontal and vertical widths after a free expansion time of 2 ms as a function of the variable hold time in the trap. The measured √ oscillation frequency ωq /2π = 499 Hz exactly corresponds to 2 ωr .

7. – Pairing gap spectroscopy in the BEC-BCS crossover In the preceding sections we have discussed our experiments on important macroscopic properties of the strongly interacting Fermi gas, like potential energy, hydrodynamics, and the equation of state. We will now present our experimental results on the observation of the “pairing gap” [35], which is a microscopic property essential in the context of superﬂuidity. The gap shows the pairing energy and thus characterizes the nature of . pairing in the crossover; see discussion in subsect. 5 1. Historically, the observation of a pairing gap marked an important experimental breakthrough in research on superconductivity in the 1950s [138, 139, 80]. The gap measurements provided a key to investigating the paired nature of the particles responsible for the frictionless current in metals at very low temperatures. The ground-breaking BCS theory [79, 80], developed at about the same time, showed that two electrons in the degenerate Fermi sea can be coupled by an eﬀectively attractive interaction and form a delocalized, composite particle with bosonic character. BCS theory predicted that the gap in the low-temperature limit is proportional to the critical temperature Tc (see table I), which was in agreement with the experimental observations from gap spectroscopy. Here we will ﬁrst discuss radio-frequency spectroscopy as our method to investigate . pairing in diﬀerent regimes (see subsect. 7 1). We will then show how molecular pairing can be investigated and precise data on the binding energy can be obtained (see sub. sect. 7 2). Finally, we will discuss our results on pairing in the many-body regimes of . the crossover (see subsect. 7 3), including the temperature dependence of the gap. . 7 1. Basics of radio-frequency spectroscopy. – Radio-frequency (RF) spectroscopy has proven a powerful tool for investigating interactions in ultracold Fermi gases. In 2003, the method was introduced by the JILA group for 40 K [140, 23] and by the MIT group for 6 Li [141].

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Fig. 20. – Illustration of the basic principle of RF spectroscopy for 6 Li at high-magnetic ﬁelds (see also ﬁg. 1). The three states 1, 2, and 3 essentially diﬀer by the orientation of the nuclear spin (mI = 1, 0, −1, respectively). By driving RF transitions the spins can be ﬂipped and atoms are transferred from state 2 to the empty state 3. In the region of the broad Feshbach resonance, the splitting between the RF coupled states 2 and 3 is about 82 MHz. The RF does not couple states 1 and 2 because of their smaller splitting of about 76 MHz. In the mean-ﬁeld regime, interactions result in eﬀective level shifts (dashed lines).

The basic idea of RF spectroscopy can be easily understood by looking at the simple . Zeeman diagram of 6 Li in the high-ﬁeld region; see also subsect. 3 1 for a more detailed discussion of the energy levels. The 6 Li spin mixture populates states 1 and 2 (magnetic quantum numbers mI = 1, 0), whereas state 3 (mI = −1) is empty. RF-induced transitions can transfer atoms from state 2 to the empty state 3; see ﬁg. 20. The experimental signature in a state-selective detection scheme (e.g., absorption imaging) is the appearance of particles in state 3 or the disappearance of particles in state 2(17 ). In the non-interacting case, the transition frequency is determined by the magnetic ﬁeld through the well-known Breit-Rabi formula. We found that interactions are in general very small for “high” temperatures of a few TF . We perform such measurements for the calibration of the magnetic ﬁeld used for interaction tuning. In the experiment, the transition frequency can be determined within an uncertainty of ∼ 100 Hz, which corresponds to magnetic-ﬁeld uncertainties as low as a ∼ 20 mG. In the mean-ﬁeld regime of a weakly interacting Fermi gas, interactions lead to a shift of the RF transition frequency [140] given by Δνmf = 2¯ hm−1 n1 (a − a), where n1 = n/2 is the number density of atoms in state 1 and a is the scattering length for interactions between atoms in 1 and 3. In experimental work on 40 K [140], this mean-ﬁeld shift was used to measure the change of the scattering length a near a Feshbach resonance (17 ) In a dense gas of 6 Li, atoms in state 3 show a very rapid decay, which we attribute to three-body collisions with 1 and 2. With atoms in three diﬀerent spin states, a three-body recombination event is not Pauli suppressed and therefore very fast. This is the reason why all our measurements show the loss from state 2 instead of atoms appearing in 3.

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under conditions where a just gives a constant, non-resonant oﬀset value. In 6 Li the interpretation of the diﬀerential mean-ﬁeld shift is somewhat more complicated because both scattering length a and a show resonant behavior [141, 45]. In the strongly interacting regime, the MIT group made the striking observation of the absence of interaction shifts [141]. This experimental ﬁnding, which is also of great relevance for the interpretation of our results on RF spectroscopy in the crossover (see sub. sect. 7 3), is related to the fact that for 6 Li both a and a are very large. In this case all resonant interactions are unitarity-limited, so that diﬀerential interaction shifts are absent. Regarding the sensitivity of RF spectroscopy to small interaction eﬀects, which typically occur on the kHz scale or even below, 6 Li features an important practical advantage over 40 K. In the relevant magnetic-ﬁeld region around 834 G the 6 Li RF-transition frequency changes by −5.6 kHz/G, in contrast to 170 kHz/G near the 202 G Feshbach resonance used in the crossover experiments in 40 K. This large diﬀerence results from decoupling of the nuclear spin from the electron spin in 6 Li at high magnetic ﬁelds. The fact that a strongly interacting Fermi gas of 6 Li is about 30 times less susceptible to magnetic ﬁeld imperfections, like ﬂuctuations, drifts, and inhomogeneities, facilitates precise measurements of small interaction eﬀects. . 7 2. RF spectroscopy on weakly bound molecules. – The application of RF spectroscopy to measure binding energies of ultracold molecules was introduced by the JILA group in ref. [23]. We have applied RF spectroscopy to precisely determine the molecular energy structure of 6 Li, which also yields precise knowledge of the two-body scattering properties [45]. Meanwhile, RF spectroscopy has found various applications to ultracold Feshbach molecules [142-145]. The basic idea of RF spectroscopy applied to weakly bound molecules is illustrated on the left-hand side of ﬁg. 21. Transferring an atom from state 2 to state 3 breaks up the dimer. The RF photon with energy hνRF has to provide at least the molecular binding energy Eb in addition to the bare transition energy hν23 . Therefore, the dissociation sets in sharply at a threshold ν23 + Eb /h. Above this threshold, the RF couples molecules to atom pairs in the continuum with a kinetic energy Ekin = E−Eb , where E = h(νRF −ν23 ). The dissociation lineshape can be understood in terms of the wave function overlap of ¯ /(ma2 ) the molecular state with the continuum. For weakly bound dimers, where Eb = h (eq. (2)), this lineshape is described by [146] (20)

f (E) ∝ E −2 (E − Eb )1/2 (E − Eb + E )−1 ,

¯ /(ma2 ) is an energy associated with the (positive or negative) scattering where E = h length a between states 1 and 3. The energy E becomes important when a is comparable to a, i.e. when both scattering channels show resonant behavior. This is the case for 6 Li [141,45], but not for 40 K [23]. In ﬁg. 21 (right-hand side) we show two RF-dissociation spectra taken at diﬀerent magnetic ﬁelds [45]. The spectra show both the change of the binding energy Eb and the variation of the lineshape (parameter E ) with the magnetic ﬁeld. The experimental data is well ﬁt by the theoretical lineshapes of eq. (20).

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Fig. 21. – Radio-frequency spectroscopy in the molecular regime: basic principle (left-hand side) and experimental results for 6 Li [45] (right-hand side). To dissociate a molecule, the RF photon with energy hν has to provide at least the molecular binding energy Eb in addition to the bare transition energy hν0 . In the experimental spectra, the onset of dissociation thus occurs shifted from the bare atomic transition frequencies, which for the two diﬀerent magnetic ﬁelds are indicated by the dashed vertical lines. The solid curves show ﬁts by the theoretical dissociation lineshapes according to eq. (20).

To precisely determine the scattering properties of 6 Li [45], we used measurements of the binding energy Eb in the (1, 2) channel obtained through RF-induced dissociation spectroscopy described above. In addition, we also identiﬁed bound-bound molecular transitions at magnetic ﬁelds where the channel (1, 3) also supports a weakly bound molecular level (a > 0). These transitions do not involve continuum states and are thus much narrower than the broad dissociation spectra. This fact facilitated very precise measurements of magnetic-ﬁeld–dependent molecular transition frequencies. The combined spectroscopic data from bound-free and bound-bound transitions provided the necessary input to adjust the calculations based on a multi-channel quantum scattering model by our collaborators at NIST. This led to a precise characterization of the two-body scattering properties of 6 Li in all combinations of the loweset three spin states. This included . the broad Feshbach resonance in the (1, 2) channel at 834 G (see discussion in 3 2) and further broad resonances in the channels (1, 3) and (2, 3) at 690 G and 811 G, respectively. . 7 3. Observation of the pairing gap in the crossover . – After having discussed the application of RF spectroscopy to ultracold molecules in the preceding section, we now turn our attention to pairing in the many-body regime of the BEC-BCS crossover. The basic idea remains the same: Breaking pairs costs energy, which leads to corresponding shifts in the RF spectra. We now discuss our results of ref. [35], where we have observed the “pairing gap” in a strongly interacting Fermi gas. Spectral signatures of pairing have been theoretically considered in refs. [147-152]. A clear signature of the pairing process is the emergence of a double-peak structure in the spectral response as a result of the

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Fig. 22. – RF spectra for various magnetic ﬁelds and diﬀerent degrees of evaporative cooling. The RF oﬀset (kB × 1 μK h × 20.8 kHz) is given relative to the atomic transition 2 → 3. The molecular limit is realized for B = 720 G (ﬁrst column). The resonance regime is studied for B = 822 G and 837 G (second and third column). The data at 875 G (fourth column) explore the crossover on the BCS side. Upper row, signals of unpaired atoms at T ≈ 6TF (TF = 15 μK); middle row, signals for a mixture of unpaired and paired atoms at T = 0.5TF (TF = 3.4 μK); lower row, signals for paired atoms at T < 0.2TF (TF = 1.2 μK). Note that the true temperature T of the atomic Fermi gas is below the temperature T which we measure in the BEC limit (see text). The solid lines are introduced to guide the eye.

coexistence of unpaired and paired atoms. The pair-related peak is located at a higher frequency than the unpaired-atoms signal. The important experimental parameters are temperature, Fermi energy, and interaction strength. The temperature T can be controlled by variation of the ﬁnal laser power of the evaporation ramp. Lacking a reliable method to determine the temperature T of a deeply degenerate, strongly interacting Fermi gas in a direct way, we measured the temperature T after an isentropic conversion into the BEC limit. Note that, for a deeply degenerate Fermi gas, the true temperature T is substantially below our temperature parameter T [98, 100]. The Fermi energy EF can be controlled after the cooling process by an adiabatic recompression of the gas. The interaction strength is varied, as in our experiments described before, by slowly changing the magnetic ﬁeld to the desired ﬁnal value. We recorded the RF spectra shown in ﬁg. 22 for diﬀerent temperatures and in various coupling regimes. We studied the molecular regime at B = 720 G (a = +2170 a0 ). For the resonance region, we examined two diﬀerent magnetic ﬁelds 822 G (+33 000 a0 ) and . 837 G (−15 0000 a0 ), because the exact resonance location (834.1±1.5 G, see subsect. 3 2) was not exactly known at the time of our pairing gap experiments. We also studied

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the regime beyond the resonance with large negative scattering length at B = 875 G (a ≈ −12 000 a0 ). Spectra taken in a “hot” thermal sample at T ≈ 6 TF (TF = 15 μK) show the narrow atomic 2 → 3 transition line (upper row in ﬁg. 22) and serve as a frequency reference. We present our spectra as a function of the RF oﬀset with respect to the bare atomic transition frequency. To understand the spectra both the homogeneous lineshape of the pair signal [148] and the inhomogeneous line broadening due to the density distribution in the harmonic trap need to be taken into account [150]. As an eﬀect of inhomogeneity, fermionic pairing due to many-body eﬀects takes place predominantly in the central high-density region of the trap, and unpaired atoms mostly populate the outer region of the trap where the density is low [95, 150, 101]. The spectral component corresponding to the pairs shows a large inhomogeneous broadening in addition to the homogeneous width of the pairbreaking signal. For the unpaired atoms the homogeneous line is narrow and the eﬀects of inhomogeneity and mean-ﬁeld shifts are negligible. These arguments explain why the RF spectra in general show a relatively sharp peak for the unpaired atoms together with a broader peak attributed to paired atoms. We observed a clear double-peak structure at T /TF = 0.5 (middle row in ﬁg. 22, TF = 3.4 μK). In the molecular regime (720 G), the sharp atomic peak was well separated . from the broad dissociation signal; see discussion in 7 2. As the scattering length was tuned to resonance, the peaks began to overlap. In the resonance region (822 and 837 G), we still observed a relatively narrow atomic peak at the original position together with a pair signal. For magnetic ﬁelds beyond the resonance, we could resolve the double-peak structure for ﬁelds up to ∼ 900 G. For T /TF < 0.2, we observed a disappearance of the narrow atomic peak in the RF spectra (lower row in ﬁg. 22, TF = 1.2 μK). This showed that essentially all atoms were paired. In the BEC regime (720 G) the dissociation lineshape is identical to the one observed in the trap at higher temperature and Fermi energy. Here the localized pairs are molecules with a size much smaller than the mean interparticle spacing, and the dissociation signal is independent of the density. In the resonance region (822 and 837 G) the pairing signal showed a clear dependence on the density, which became even more pronounced beyond the resonance (875 G). To quantitatively investigate the crossover from the two-body molecular regime to the fermionic many-body regime we measured the pairing energy in a range between 720 G and 905 G. The experiments were performed after deep evaporative cooling (T /TF < 0.2) for two diﬀerent Fermi temperatures TF = 1.2 μK and 3.6 μK (ﬁg. 23). As an eﬀective pairing gap we deﬁne Δν as the frequency diﬀerence between the pair-signal maximum and the bare atomic resonance. In the BEC limit, the eﬀective pairing gap Δν simply reﬂects the molecular binding energy Eb , as shown by the solid line in ﬁg. 23(18 ). With (18 ) The maximum of the dissociation signal, which deﬁnes hΔν in the molecular regime, varies between EB and (4/3) EB , depending on E /Eb in eq. (20). The solid line takes this small variation into account [153].

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Fig. 23. – Measurements of the eﬀective pairing gap Δν as a function of the magnetic ﬁeld B for deep evaporative cooling and two diﬀerent Fermi temperatures TF = 1.2 μK (ﬁlled symbols) and 3.6 μK (open symbols). The solid line shows Δν for the low-density limit, where it is essentially given by the molecular binding energy [153]. The two dotted lines at higher magnetic ﬁelds . correspond to the condition 2Δν = ωc for the coupling of the compression mode (see subsect. 6 4) to the gap at our two diﬀerent trap settings. The inset displays the ratio of the eﬀective pairing gaps measured at the two diﬀerent Fermi energies.

increasing magnetic ﬁeld, in the BEC-BCS crossover, Δν showed an increasing deviation from this low-density molecular limit and smoothly evolved into a density-dependent many-body regime where hΔν < EF . A comparison of the pairing energies at the two diﬀerent Fermi energies (inset in ﬁg. 23) provides further insight into the nature of the pairs. In the BEC limit, Δν is solely determined by Eb and thus does not depend on EF . In the universal regime on resonance, EF is the only energy scale and we indeed observed the eﬀective pairing gap Δν to increase linearly with the Fermi energy (see ref. [127] for more details). We found a corresponding relation hΔν ≈ 0.2 EF (19 ). Beyond the resonance, where the system is expected to change from a resonant to a BCS-type behavior, Δν is found to depend more strongly on the Fermi energy and the observed gap ratio further increases. We interpret this in terms of the increasing BCS character of pairing, for which an exponential dependence hΔν/EF ∝ exp(−π/2kF |a|) (see table I) is expected. In another series of measurements (ﬁg. 24), we applied a controlled heating method to study the temperature dependence of the gap in a way which allowed us to keep all (19 ) Note that there is a quantitative deviation between this experimental result for the unitarity limit (see also [127]) and theoretical spectra [150-152], which suggest hΔν ≈ 0.35 EF . This discrepancy is still an open question. We speculate that interactions between atoms in state 1 and 3 may be responsible for this, which have not been fully accounted for in theory.

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Fig. 24. – RF spectra measured at B = 837 G, i.e. very close to the unitarity limit, for diﬀerent temperatures (TF = 2.5 μK). The temperature parameter T was determined by measurements in the mBEC regime after an isentropic conversion of the gas. Based on the entropy calculations of ref. [100] we also provide estimates for the true temperature T . The solid lines are ﬁts to guide the eye using a Lorentzian curve for the atom peak and a Gaussian curve for the pair signal. The vertical dotted line marks the atomic transition and the arrows indicate the eﬀective pairing gap Δν.

other parameters constant. After production of a pure molecular BEC (T < 0.2TF ) in the usual way, we adiabatically changed the conditions to B = 837 G and TF = 1.2 μK. We then increased the trap laser power by a factor of nine (TF increased to 2.5 μK) using exponential ramps of diﬀerent durations. For fast ramps this recompression is non-adiabatic and increases the entropy. By variation of the ramp time, we explore a range from our lowest temperatures up to T /TF = 0.8. The emergence of the gap with decreasing temperature is clearly visible in the RF spectra (ﬁg. 24). The marked increase of Δν for decreasing temperature is in good agreement with theoretical expectations for the pairing gap energy [81]. Our pairing gap experiments were theoretically analyzed in refs. [150-152]. The calculated RF spectra are in agreement with our experimental results and demonstrate how a double-peak structure emerges as the gas is cooled below T /TF ≈ 0.5 and how the atomic peak disappears with further decreasing temperature. In particular, the theoretical work clariﬁes the role of the pseudo-gap regime [81, 99] in our experiments, where pairs are

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formed before superﬂuidity is reached. We believe that the upper spectrum of ﬁg. 24 (T = 0.8 TF , corresponding to T = 0.3 TF [100]) shows the pseudo-gap regime. The lower spectrum, however, which was taken at a much lower temperature (T < 0.2 TF , T < 0.1 TF ), is deep in the superﬂuid regime. Here, the nearly complete disappearance of the atom peak shows that fermionic pairing took place even in the outer region of the trapped gas where the density and the local Fermi energy are low. According to theory [150, 151] this happens well below the critical temperature for the formation of a resonance superﬂuid in the center of the trap. This conclusion [35] ﬁts well to the other early observations that suggested superﬂuidity in experiments performed under similar conditions [33, 34], and also to the observation of superﬂuidity by vortex formation in ref. [38]. We ﬁnally point to an interesting connection to our measurements on radial collec. tive excitations (see subsect. 6 4), where an abrupt breakdown of hydrodynamics was observed at a magnetic ﬁeld of about 910 G [34]. The hydrodynamic equations which describe collective excitations implicitly assume a large gap, and their application becomes questionable when the gap is comparable to the radial oscillation frequency [154]. We suggest a pair breaking condition ωc = 2hΔν(20 ), which roughly corresponds to ωr = hΔν (ωc ≈ 2ωr ). Our pair breaking condition is illustrated by the dashed lines in ﬁg. 23 for the two diﬀerent Fermi energies of the experiment. In both cases the eﬀective gap Δν reaches the pair breaking condition somewhere slightly above 900 G. This is in striking agreement with our observations on collective excitations at various Fermi temperatures [34, 127]. This supports the explanation that pair breaking through coupling of oscillations to the gap leads to strong heating and large damping and thus to a breakdown of superﬂuidity on the BCS side of the resonance. 8. – Conclusion and outlook Ultracold Fermi gases represent one of the most exciting ﬁelds in present-day physics. Here experimental methods of atomic, molecular, and optical physics oﬀer unprecedented possibilities to explore fundamental questions related to many diﬀerent ﬁelds of physics. In the last few years, we have seen dramatic and also surprising developments, which have already substantially improved our understanding of the interaction properties of fermions. Amazing progress has been achieved in the exploration of the crossover of strongly interacting system from BEC-type to BCS-type behavior. Resonance superﬂuidity now is well established. Recent experimental achievements have made detailed precision tests of advanced many-body quantum theories possible. The majority of experiments have so far been focused on bulk systems of two-component spin mixtures in macroscopic traps; however, ultracold gases oﬀer many more possibilities to realized intriguing new situations. The recent experiments on imbalanced (20 ) The factor of 2 in this condition results from the fact that here pair breaking creates two in-gap excitations, instead of one in-gap excitation in the case of RF spectroscopy, where one particle is removed by transfer into an empty state.

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systems [39, 40] give us a ﬁrst impression how rich the physics of fermionic systems will be when more degrees of freedom will be present. As another important example, optical lattices [155, 8, 156] allows us to model the periodic environment of crystalline materials, providing experimental access to many interesting questions in condensed-matter physics [157]. Also, fermionic mixtures of diﬀerent atomic species open up many new possibilities. In Bose-Fermi mixtures [158-160, 145], fermionic pairing and superﬂuidity can be mediated through a bosonic background [161, 162]. In the case of Fermi-Fermi mixtures [163], pairing between particles of diﬀerent masses [164] and novel regimes of superﬂuidity represent intriguing prospects for future research. With the recent experiments on the physics of ultracold fermions, we have just opened the door to an exciting new research ﬁeld. On the large and widely unexplored terrain, many new challenges (and surprises) are surely waiting for us! ∗ ∗ ∗ Our work on ultracold Fermi gases, which developed in such an exciting way, is the result of a tremendous team eﬀort over the past eight years. Its origin dates back to my former life in Heidelberg (Germany), where our activities on ultracold fermions began ¨ller, in the late 1990s. I thank the team of these early days (A. Mosk, M. Weidemu ¨sser) for the pioneering work to start our adventures with H. Moritz and T. Elsa 6 Li. The experiment moved to Innsbruck in 2001, and many people have contributed to its success there. For their great work and achievements, I thank the Ph.D. students S. Jochim (who moved with the experiment from Heidelberg to Innsbruck) and M. Bartenstein, A. Altmeyer and S. Riedl, along with the diploma students G. Hendl and C. Kohstall. Also, I acknowledge the important contributions by the post-docs R. Geursen and M. Wright (thanks, Matt, also for the many useful comments on the manuscript). I am greatly indebted to C. Chin, who shared a very exciting time with us and stimulated the experiment with many great ideas, and my long-standing colleague J. Hecker Denschlag for their invaluable contributions. The experiment strongly beneﬁted from the great synergy in a larger group (www.ultracold.at) and from the outstanding scientiﬁc environment in Innsbruck. Finally, I thank the Austrian Science Fund FWF for funding the experiment through various programs, and the European Union for support within the Research Training Network “Cold Molecules”. REFERENCES [1] DeMarco B. and Jin D. S., Science, 285 (1999) 1703. [2] Truscott A. G., Strecker K. E., McAlexander W. I., Partridge G. B. and Hulet R. G., Science, 291 (2001) 2570. [3] Schreck F., Khaykovich L., Corwin K. L., Ferrari G., Bourdel T., Cubizolles J. and Salomon C., Phys. Rev. Lett., 87 (2001) 080403. [4] Granade S. R., Gehm M. E., O’Hara K. M. and Thomas J. E., Phys. Rev. Lett., 88 (2002) 120405. ¨ rlitz [5] Hadzibabic Z., Stan C. A., Dieckmann K., Gupta S., Zwierlein M. W., Go A. and Ketterle W., Phys. Rev. Lett., 88 (2002) 160401.

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A lab in a trap: Fermionic quantum gases, Bose-Fermi mixtures and molecules in optical lattices ¨ ferle, K. Gu ¨nter, M. Ko ¨ hl and T. Esslinger H. Moritz, T. Sto Institute of Quantum Electronics, ETH Z¨ urich - CH-8093 Z¨ urich, Switzerland

1. – Introduction The ﬁeld of ultracold atoms has seen two major inspirations: the use of laser light to cool and trap atoms [1], and the accomplishment of Bose-Einstein condensation [2]. The latter has opened a window on the beautiful physics of weakly interacting quantum gases. Many fundamental phenomena could be studied in a text-book like manner, e.g., superﬂuidity, vortices, collective excitations and atom lasers. During the last years the ﬁeld of ultracold quantum gases has witnessed a further development which has gone beyond the most optimistic expectations. The ﬁeld has seen a wealth of novel atomic quantum many-body systems, mostly in the strongly interacting regime [3]. Quantum degenerate Fermi gases, Bose-Fermi mixtures, and low-dimensional systems were created and the superﬂuid to Mott-insulator transition has been observed in a Bose gas. Furthermore, a remarkable series of experiments on the BEC-BCS crossover has been carried out, following the realisation of molecular Bose-Einstein condensates. The pace with which new research directions have been created has almost outstripped the capabilities to investigate these new systems at the required depths. A tool which has played a crucial role in the recent rapid progress is the optical lattice. The idea of an optical lattice experiment is to trap a quantum degenerate gas in the periodic potential of one, two or three standing laser waves, see ﬁg. 1. Even though the concept is simple, the versatility of the system is enormous and eﬀective. Variation of a single parameter can be enough to access and study very diﬀerent types of physics. c Societ` a Italiana di Fisica

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¨ ferle, K. Gu ¨nter, M. Ko ¨ hl and T. Esslinger H. Moritz, T. Sto

Fig. 1. – Three-dimensional optical lattice. An ultracold gas of atoms is trapped by three pairs of counter-propagating laser beams. Each pair produces a standing laser wave in which the atoms experience a periodic potential. All three pairs generate a three-dimensional simple cubic lattice structure, where the atoms are trapped in the intensity maxima of the light.

2. – Optical lattices The concept of optical lattices goes back to a proposal of Letokhov in 1968 [4]. In order to carry out precision spectroscopy of atoms using laser light, he suggested reducing the eﬀect of line broadening due to the Doppler eﬀect by trapping the atoms in the dipole potential created by a laser standing wave. The mechanism behind this idea is that laser light induces an electric dipole in an atom which then experiences a mechanical force in the ﬁeld gradient of the standing wave. If the laser frequency is lower than the atomic transition frequency, the atoms are attracted to the intensity maxima of the standing wave, which acts as a periodic potential for the atoms. These ideas could not immediately be realised in experiments since the depth of the standing wave potential is on the order of only 10 to 100 μK, for realistic parameters. However, with the method of laser cooling it became possible to prepare atomic samples cold enough to be trapped in the periodic potential of a standing light wave. Indeed, eﬀective sub-Doppler laser cooling relies on optical pumping of atoms into the nodes of the standing wave potential [5]. In ﬁrst optical lattice experiments the quantization of the atomic motion in these potentials was investigated [6-9]. The link between optical lattices and the physics in solids was realized very early. It was pointed out that the description of a bandstructure is adequate for atoms in optical lattices [10]. By observing laser light scattered under the Bragg angle from the atoms in the optical lattice their periodic arrangement has been demonstrated [11, 12]. These experiments worked even though only 0.1% of the lattice sites were occupied. Furthermore, Bloch oscillations [13, 14] and the Wannier Stark ladder [15] could be observed with laser cooled atoms in optical lattices, detuned far oﬀ the atomic resonance as to suppress spontaneous emission.

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A new chapter was opened up by putting Bose-Einstein condensates into optical lattices [16]. In a ﬁrst experiment a Bose-Einstein condensed gas of rubidium atoms was loaded into a vertically aligned standing laser wave [17]. A periodic Landau-Zener– type coupling between the two lowest bands could be monitored through the emission of a pulsed train of coherent matter waves. A further study of the system showed that the spatial coherence of the condensate disappears for deep lattice potentials but can be regained after subsequent lowering of the periodic potential [18]. Research eﬀorts with one-dimensional optical lattices have recently been extended to quantum degenerate Fermi gases [19] and to Bose-Fermi mixtures [20]. The strongly interacting regime has become accessible in a novel generation of experiments designed for optical lattices consisting of two and three mutually perpendicular standing waves. In particular, it was proposed [21] that the physics of interacting bosonic atoms in a three-dimensional optical lattice is well described by the Bose-Hubbard model, which shows a quantum phase transition between a spatially coherent superﬂuid phase and a Mott-insulating state. Indeed, the experiment [22] showed that the spatial coherence fades away when the ratio between collisional and kinetic energy reaches the value of the critical point predicted by mean-ﬁeld theory. In addition, the reversibility of the process and the discreteness of the excitation spectrum of the Mott-insulator could be shown. Many fascinating aspects of this strongly correlated Bose system are currently investigated [23]. Most recently a very close link between solid-state physics and quantum gases has been established by loading a quantum degenerate gas of atomic fermions into a threedimensional optical lattice potential [24]. With the fermions prepared in two diﬀerent spin-states the gas realizes a fermionic Hubbard model, which plays a key role in the description of many intriguing phenomena in modern condensed-matter physics, most notably in modelling high-temperature superconductivity [25]. The following discussion will focus on concepts of recent experiments carried out in our group at the ETH Z¨ urich in which we have explored the physics of quantum degenerate Fermi gases and Bose-Fermi mixtures in optical lattices. 3. – Concept of the experiment Excellent optical access is crucial for an experiment aiming at the study of quantum degenerate gases in optical lattices. Compared to other experiments with ultracold atoms, the challenge is to create standing-wave potentials along all three spatial dimensions as well as to image along them while still being able to produce quantum degenerate gases in the ﬁrst place. In most apparatuses the magneto-optical trapping takes place in the same region of space as the subsequent creation of the quantum degenerate gas and its investigation, thus blocking optical access needed for the optical lattices. In our experiment, we circumvent this diﬃculty by magnetically transporting the atoms [26,27] collected in a magneto-optical trap into an adjacent ultra-high vacuum chamber where the gas is ﬁnally cooled to quantum degeneracy and loaded into the optical lattice.

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4

3

3

2

3

3

1

2

4

2

3

3 4

3

2

3

4

Fig. 2. – The Brillouin zones in a two-dimensional cubic lattice. The reciprocal lattice vectors are marked by black dots and the zones are labelled according to the index of the band that they contain. Several Fermi surfaces within the ﬁrst zone for diﬀerent ﬁllings are shown as white dotted lines.

More speciﬁcally, our ﬁrst step on the way to quantum degeneracy is the sympathetic cooling of a gas of fermionic 40 K atoms by bosonic 87 Rb atoms, the latter being subjected to forced evaporation. This allows us to reach quantum degeneracy simultaneously for both species. In the next step either one or both of the species are loaded into a crossed beam optical dipole trap. There the atoms are prepared in the desired spin states using tailored radio-frequency pulses and cooled by optical evaporation. With a pure sample of potassium atoms, we reach ﬁnal temperatures well below T /TF = 0.25 (TF : Fermi temperature) with up to 5 · 105 particles before loading them into a three-dimensional optical lattice. The optical lattice is created by up to three mutually perpendicular standing laser waves which produce a cubic lattice structure. The separation of adjacent lattice sites is half the laser wavelength, in our case either 413 nm or 532 nm depending on the laser used for the optical lattice. The transverse Gaussian proﬁle of the lattice beams gives rise to a force pointing towards the beam centre, where the atoms are harmonically conﬁned. 4. – Imaging Fermi surfaces A periodic potential as it is experienced by an atom in the optical lattice modiﬁes the parabolic dispersion relation and the spectrum splits up in diﬀerent bands, with the corresponding eigenstates —the Bloch functions— all having well deﬁned quasimomenta. The entire volume in quasi-momentum space which belongs to one particular band is called a Brillouin zone and an example is depicted in ﬁg. 2. Assuming zero temperature, we expect that all states up to the Fermi energy EF are occupied in a non-interacting Fermi gas. In quasi-momentum space, this means that all states within the volume enclosed by the Fermi surface, the equipotential surface with energy EF , are ﬁlled. For the homogeneous two-dimensional case, several Fermi surfaces within the ﬁrst band for diﬀerent ﬁllings are indicated in ﬁg. 2 with white lines. When the Fermi energy is much lower than the width of the band, the lowest band is only partially ﬁlled and the Fermi surface is round. As the ﬁlling is increased the Fermi

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Fig. 3. – Observing the Fermi surface. In the ﬁrst row the experimentally observed quasimomentum distributions are displayed with the eﬀective ﬁlling in the lattice increasing from left to right. a) At low densities the Fermi surface is of spherical shape and the corresponding projection is seen. This distribution corresponds to a conducting state. With increasing ﬁlling in the trap the Fermi surface develops a more complex structure (b + c) showing extensions towards the boundary of the ﬁrst Brillouin zone. For very high ﬁllings a band insulating state emerges and the entire ﬁrst Brillouin zone is ﬁlled (d + e). In order to access diﬀerent eﬀective ﬁllings, optical lattices of varying depths and particle numbers were prepared. The quasimomentum distribution was obtained by lowering the lattice adiabatically with respect to single particle physics followed by free expansion and absorption imaging. The second row shows the numerically calculated quasi-momentum distribution for the given experimental parameters and a temperature of T /TF = 0.2, which corresponds to the temperature of the atoms before loading into the optical lattice. Parts of the ﬁgure are reproduced from [24]. Copyright (2005) by the American Physical Society.

surface becomes a rhombus at half ﬁlling before developing extensions towards the zone edge and ﬁnally ﬁlling the entire ﬁrst Brillouin zone. In the last case the ﬁlling is unity and a band insulator forms. A direct observation of the Fermi surfaces is possible using a method originally developed to characterize laser cooled atoms in optical lattices [28]. The intensity of the optical lattice is ramped down in such a way that the quasi-momentum is mapped to the real momentum of the expanding cloud. This distribution is then measured by absorption imaging of the cloud after ballistic expansion. In ﬁg. 3, several quasi-momentum distributions, which were measured with this method, are displayed [24]. The ﬁlling increases from left to right. For the lowest ﬁllings the distribution is round, corresponding to a conducting state in a metal. The distribution then develops the characteristic extensions towards the zone boundaries before becoming a square in the band insulating regime for full ﬁlling. In the experiment an underlying harmonic trapping potential is present, which is caused by the Gaussian proﬁle of the laser beams. Hence, for a given Fermi energy the ﬁlling of the lattice is maximal in the centre of the trap and decreases towards the edges of the trap. Moreover, the system is three-dimensional and the absorption imaging constitutes an integration along the line of sight. To compare the measured

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Fig. 4. – A Hubbard model with ultracold atoms. Fermionic atoms trapped in the lowest band of an optical lattice can tunnel between lattice sites with a tunnelling rate J. Due to Pauli’s principle, tunnelling is only possible if the ﬁnal lattice site is empty or occupied with an atom with a diﬀerent spin. Two atoms with opposite spin localized at the same lattice site have an interaction energy U , which can be likewise positive (repulsive interaction) or negative (attractive interaction). The interplay between J and U and the ﬁlling determines the physics of the system.

distributions to theory, we have performed a numerical simulation of the system taking into account the underlying harmonic conﬁnement as well as the ﬁnite temperature. The calculated distributions integrated along line of sight are shown in the second row of ﬁg. 3, demonstrating very good agreement between experiment and theory. 5. – Interacting fermionic atoms in an optical lattice: the Hubbard model and beyond The low temperatures in the experiment allow only for collisions with zero angular momentum, i.e. s-wave collisions. Consequently, a spin-polarized Fermi gas is eﬀectively non-interacting, since Pauli’s principle does not allow s-wave collisions, which are of even parity. The situation is diﬀerent if the Fermi gas is prepared in an equal mixture of two diﬀerent spin states, between which s-wave collisions are permitted. The s-wave collisions are characterised by a scattering length a, which is positive for repulsive and negative for attractive interactions. In the experiment the potassium gas is prepared in two diﬀerent magnetic sublevels of the hyperﬁne ground state, which represent the two spin states. The physics of interacting atoms in the optical lattice can be accessed by an important simpliﬁcation. It is possible to prepare all atoms in the lowest band and regard the atoms as hopping from one lattice site to the next, as illustrated in ﬁg. 4. This motion is characterized by the tunnelling matrix element J between adjacent sites. If two atoms happen to be on the same site the atom-atom collisions give rise to a short-range interaction U , which is proportional to the scattering length a. This was pointed out by Jaksch and coworkers who suggested that neutral atoms in an optical lattice are well

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described by a Hubbard Hamiltonian [21]. The proposed ideas led to the experimental observation of the transition from a superﬂuid to a Mott-insulating phase for bosons, using a Bose-Einstein condensate loaded into a three-dimensional optical lattice [22, 29]. For fermionic atoms the Hubbard Hamiltonian in an optical lattice reads (1)

H = −J

{i,j},σ

cˆ†i,σ cˆj,σ +

i,σ

εi n ˆ i,σ + U

n ˆ i,↑ n ˆ i,↓ .

i

The ﬁrst term contains the kinetic energy and is proportional to the tunnelling matrix element J. The operators cˆ†i,σ and cˆi,σ are the fermionic creation and annihilation operators for a particle in the spin state (up or down) at lattice site i. The occupation number of the site i is given by n ˆ i,σ = cˆ†i,σ cˆi,σ . The second term takes the additional harmonic conﬁnement of the optical lattice into account. Here εi denotes the site speciﬁc oﬀset. The last term describes the interaction energy in the system and is determined by the on-site interaction U . It is the control of parameters which makes the atomic realisation of Hubbard models unique. The intensity of the laser standing waves controls the barrier between the lattice sites, i.e. the tunnel coupling J. This allows tuning of the kinetic energy and of the time scale for transport. It also gives direct access to the dimensionality of the system. For example, a one-dimensional situation is created by suppressing tunnelling in two directions, using two standing waves with very high intensities. Further, the on-site interaction U can be tuned to negative or positive values (see below). In the extreme case where the U approaches the band separation physics beyond the single-band Hubbard model can be accessed. This can lead to an interaction induced occupation of higher bands [24]. Moreover, by forming bosonic diatomic molecules from two fermionic atoms of diﬀerent spin [30] even the statistics of the particles on the lattice can be changed. Experiments demonstrating these two cases will be discussed next. In the experiment we make use of Feshbach resonances to tune the collisional interaction between two atoms, i.e. the U in the Hubbard model. Near a Feshbach resonance an applied magnetic ﬁeld induces a coupling between diﬀerent collisional channels which leads to a resonant behaviour of the scattering length a as a function of the magnetic ﬁeld. As shown in ﬁg. 5, starting from a background value abg the scattering length increases with increasing magnetic ﬁeld and diverges to plus inﬁnity at the resonance position, then it switches to minus inﬁnity before smoothly approaching the background value again. To investigate the atom-atom interactions in the optical lattice we prepare the Fermi gas in the two spin states |F = 9/2, mF = −9/2 and |F = 9/2, mF = −5/2 and produce a band insulator for each component, i.e. in the center of the trap there is one particle in each spin state per lattice site (F : total hyperﬁne spin; mF : corresponding magnetic quantum number). Starting from a weakly repulsively interacting situation we ramp the magnetic ﬁeld over the Feshbach resonance at 224 G. After crossing the resonance we have measured the population in each band and observed an increased population in the higher bands, see ﬁg. 6. Due to the increase of the interaction to the order of the band

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scattering length [a0]

1000

500

0

-500

mF = -9/2 + mF = -7/2 -1000

180

200

mF = -9/2 + mF = -5/2 220

240

magnetic field [G]

Fig. 5. – Two Feshbach resonances [31] are used to tune the interaction strength. By sweeping across the scattering resonance between the |F = 9/2, mF = −7/2 and the |F = 9/2, mF = −5/2 state from below, atoms can be transferred in to higher bands. To create molecules, a sweep across the scattering resonance between the |F = 9/2, mF = −7/2 and the |F = 9/2, mF = −7/2 state from above is employed.

Fig. 6. – Interaction-induced coupling between Bloch bands. If the interaction energy U between particles on the same lattice site becomes comparable to the band gap, the single-band Hubbard model breaks down. The strong interaction leads to a coupling between diﬀerent bands. The two false color images show the measured quasi-momentum distribution before (left) and after (right) strong interaction between the atoms. The interaction has been induced by sweeping the magnetic ﬁeld across a Feshbach resonance. Quasi-momentum states outside the ﬁrst Brillouin zone become occupied which demonstrates the interaction-induced coupling between the bands. The transfer of population occurs in one direction only, since a lower value for the intensity of the lattice beam has been chosen for this direction. Parts of the ﬁgure are reproduced from [24]. Copyright (2005) by the American Physical Society.

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gap we entered a regime beyond the standard Hubbard model. For a full description of the experiment higher bands would have to be taken into account. These models are notoriously diﬃcult and a simpler approach to get a rough understanding of the experimental observations is to consider the low-tunneling limit, where we can describe each lattice site as a harmonic oscillator with two interacting particles. This model can be solved analytically [32] and shows that crossing the Feshbach resonance transfers part of the population to higher harmonic oscillator states. When the tunneling between the harmonic oscillators is not negligible, the higher oscillators states correspond to the observed population in higher bands. 6. – Weakly bound molecules in an optical lattice Approaching the Feshbach resonance from high magnetic ﬁelds and then crossing the resonance converts fermionic atoms into weakly bound bosonic molecules. We have studied this process in the three-dimensional optical lattice with the fermionic gas initially prepared in a band insulator for the two spin states |F = 9/2, mF = −9/2 and |F = 9/2, mF = −7/2. Starting at a value of 210 G, we ramp the magnetic ﬁeld to different ﬁnal values in the vicinity of the Feshbach resonance at 202.1 G. In the unconﬁned case, bound states exist only for positive scattering lengths. For magnetic ﬁelds above the Feshbach resonance the scattering length is negative. However, the interaction between the atoms in this case is attractive and their energy is lowered if they are conﬁned in space, as for example in a doubly occupied lattice site. Therefore bound states can form in the lattice already above the Feshbach resonance, i.e. for negative scattering lengths. These so-called conﬁnement induced molecules are stabilized by the lattice and dissociate when the lattice potential is removed. When crossing the Feshbach resonance to the side of positive scattering length weakly bound molecules form, which survive the lattice turn oﬀ. In order to gain spectroscopic information on the energy shift of doubly occupied sites as a function of magnetic ﬁeld we use a radio-frequency pulse to drive the transition from the |F = 9/2, mF = −7/2 to the empty |F = 9/2, mF = −5/2 state. The spectroscopic signal shown in ﬁg. 7 exhibits two main features for the atom number in the |F = 9/2, mF = −5/2 state, recorded after switching oﬀ the lattice potential. The increase in the atom number in the |F = 9/2, mF = −5/2 state at zero detuning corresponds to the free atom transition and indicates singly occupied sites. A second increase in this number is observed for higher frequencies of the radio-frequency pulse, corresponding to doubly occupied sites. From the energy diﬀerence between the two peaks we can deduce the binding energy of the doubly occupied sites [30] and ﬁnd very good quantitative agreement with the theoretical prediction [32] for two interacting particles in a harmonic oscillator, as shown in ﬁg. 8. The spectra of the |F = 9/2, mF = −7/2 state displayed in ﬁg. 7 show a decrease in the atom number for the free atom transition. A second decrease at a shifted position due to the dissociation of conﬁnement-induced molecules is observed for magnetic-ﬁeld values which lie above the Feshbach resonance. This dip is absent for magnetic-ﬁeld

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Fig. 7. – a) Illustration of the r.f. spectroscopy between two bound states within a single well of the optical lattice. The atoms in the initial states |mF = −7/2 and |mF = −9/2 are converted into a bound dimer by sweeping across a Feshbach resonance. Subsequently we drive an r.f. transition |mF = −7/2 → |mF = −5/2 to dissociate the molecule. b) r.f. spectrum taken at B = 202.9 G, i.e. for a < 0. c) r.f. spectrum taken at B = 202.0 G, i.e. for a > 0. The lines are Lorentzian ﬁts to the data. The ﬁgure is reproduced from [30]. Copyright (2006) by the American Physical Society.

values corresponding to a positive scattering length where weakly bound molecules are formed and the constituting atoms do not contribute to the detected atom number. The radio-frequency dissociation of the molecules therefore causes a net increase in atom number, observed in the states |F = 9/2, mF = −9/2 and |F = 9/2, mF = −5/2.

Fig. 8. – The measured binding energy of molecules in a three-dimensional optical lattice in units of the harmonic-oscillator frequency ω within a single well. The data are taken for several potential depths of the optical lattice of 6 Er (triangles), 10 Er (stars), 15 Er (circles), and 22 Er (squares). Here Er = h2 /2mλ2 is the recoil energy experienced by the 40 K atoms of mass m in the lattice of periodicity λ/2. The solid line is the calculation with no free parameters of the energy of two particles interacting via a contact interaction in a harmonic oscillator according to [32]. The ﬁgure is reproduced from [30]. Copyright (2006) by the American Physical Society.

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b

1

c

optical density

a

0 NF / NB = 0

NF / NB = 0.08

NF / NB = 0.8

Fig. 9. – Interference pattern of bosonic atoms released from a three-dimensional optical lattice for varying admixture of NF fermionic atoms at a value UBB /zJB = 5. The bosonic atom numbers are NB = 1.2×105 (a and b) and NB = 8×104 (c) and the image size is 660 μm×660 μm. The ﬁgure is reproduced from [38]. Copyright (2006) by the American Physical Society.

7. – Bose-Fermi mixtures in a three-dimensional optical lattice The interaction between the bosonic and the fermionic atoms interconnects two systems of fundamentally diﬀerent quantum statistics and a wealth of physics becomes accessible which is beyond that of the purely bosonic [21, 22] or purely fermionic [24, 25] case. We have realized a mixture of spin-polarized bosonic 87 Rb and fermionic 40 K quantum gases trapped in the periodic potential of a three-dimensional optical lattice. The optical lattice allows us to change the character of the system by tuning the depth of the periodic potential. This leads to a change of the eﬀective mass and varies the role played by atom-atom interactions. In particular, a diﬀerent eﬀective mass for the two constituents can be realized making the fermions much more mobile than the bosons. In principle, tuning of the wavelength of the optical lattice would allow for tuning of the relative mobilities of the diﬀerent species. A variety of theoretical work has been devoted to Bose-Fermi mixtures in optical lattices, and new quantum phases have been predicted at zero temperature [33-37]. The physics of the Bose-Fermi mixture in an optical lattice can be described by the BoseFermi Hubbard model (e.g., [34]). The parameters of the model are the tunnelling matrix elements JB for bosons and JF for fermions and the on-site interaction strength UBB between two bosons and UBF between bosons and fermions. Using the most recent experimental value of the K-Rb s-wave scattering length, the ratio between the onsite Bose-Fermi interaction and the onsite boson-boson interaction is approximately UBF /UBB = −2. In our experiment [38], we studied the behavior when changing the mixing ratio between bosons and fermions (ﬁg. 9). The momentum distribution of the pure bosonic sample shows a high contrast interference pattern reﬂecting the long-range phase coherence of the system (see also [39]). Adding fermionic particles to the system results in the loss of phase coherence of the Bose gas. The visibility of the interference pattern

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diminishes and the coherence length is reduced. The observed depletion of the BoseEinstein condensate may have various reasons. At low depth of the optical lattice, the interaction of the Bose-Einstein condensate with the Fermi gas leads to the depletion of the condensate [40] and to the formation of polarons where a fermion couples to a phonon excitation of the condensate [36]. At a larger depth of the optical lattice, composite fermions consisting of one fermion and nB bosons form when the binding energy of the composite fermion exceeds the gain in kinetic energy that the particles would encounter by delocalizing. An eﬀective Hamiltonian for these (spinless) composite fermions with renormalized tunneling and nearest-neighbor interaction has been derived, and their quantum phases have been investigated theoretically [33,34]. In this situation, the BoseEinstein condensate can be completely depleted by the interactions between bosons and fermions. Last but not least, the thermodynamics of the mixture may lead to an interesting eﬀect because bosonic and the fermionic gases have signiﬁcantly diﬀerent entropies. While the lattice is ramped up their temperatures remain equal to each other due to thermal contact but the eﬀective masses of the species evolve diﬀerently. Accordingly, both TF and Tc evolve diﬀerently and in fact, using a model for a noninteracting gas, we ﬁnd that Tc decreases more rapidly than TF . This suggests that the Bose-Einstein condensate is heated adiabatically due to transfer of entropy from the fermionic atom cloud which is in qualitative agreement with our observation. 8. – Outlook The physics of fermionic atoms in optical lattices covers a wide range of concepts in modern condensed-matter physics and beyond. Experiments which now appear to be within reach are the creation of a Mott-insulating or an anti-ferromagnetic phase, where the repulsive interaction between atoms in diﬀerent spin states should cause a pattern with alternating spin up and spin down. Moreover, it has become possible to study the superﬂuid properties in the BEC-BCS crossover regime inside an optical lattice [41]. In general, fermionic atoms in optical lattices are more closely connected to the behaviour of electrons in solid-state materials and provide a richer physics than their bosonic counterparts, but they are also more diﬃcult to understand. A particular tantalizing prospect is that fermionic atoms in optical lattices may provide solutions to unanswered question in condensed matter physics, such as high-temperature superconductivity [42]. The challenge here is twofold. One central requirement is to reach extremely low temperatures inside the optical lattice. The second challenge is how to extract the information on the quantum many-body state from the experiment. To test new approaches and techniques with optical lattices one-dimensional systems will play a crucial role since they allow a comparison between exactly solvable models and the experimental ﬁndings. The experiments can then easily be extended to two or three dimensions. By preparing mixtures of bosonic and fermionic atoms in optical lattices a novel quantum system with a very rich phase diagram has become accessible. It is conceivable that in the future the hetero-nuclear molecules which have already been formed in these mixtures [43] will serve as a starting point for the creation of polar molecules. Moreover,

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optical lattices of diﬀerent geometric structure, superlattices or lattices with disorder will most likely be implemented in experiments. Besides simulating quantum systems, optical lattices are a promising system for the development of a quantum computer. The optical lattice can be regarded as a quantum register with each atom on a lattice site acting as a quantum bit. Whilst the initial preparation of such a quantum register with thousands of qubits seems manageable, it is the controlled interaction between diﬀerent atoms and the readout of single bits which represents the challenge. ∗ ∗ ∗ We would like to thank OLAQUI and SNF for funding.

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[19] Modugno G., Ferlaino F., Heidemann R., Roati G. and Inguscio M., Phys. Rev. A, 68 (2003) 011601(R). [20] Ott H., de Mirandes E., Ferlaino F., Roati G., Modugno G. and Inguscio M., Phys. Rev. Lett., 92 (2004) 160601. [21] Jaksch D., Bruder C., Cirac J. I., Gardiner C. W. and Zoller P., Phys. Rev. Lett., 81 (1998) 3108. ¨nsch T. and Bloch I., Nature, 415 (2002) [22] Greiner M., Mandel O., Esslinger T., H a 39. [23] Bloch I., Nature Physics, 1 (2006) 23. ¨ hl M., Moritz H., Sto ¨ ferle T., Gu ¨nter K. and Esslinger T., Phys. Rev. Lett., [24] Ko 94 (2005) 080403. [25] Hofstetter W., Cirac J. I., Zoller P., Demler E. and Lukin M. D., Phys. Rev. Lett., 89 (2002) 220407. ¨nsch T. W. and Esslinger T., Phys. Rev. A, 63 (2001) [26] Greiner M., Bloch I., Ha 031401(R). [27] Lewandowski H. J., Harber D. M., Whitaker D. L. and Cornell E. A., J. Low Temp. Phys., 132 (2003) 309. [28] Kastberg A., Phillips W. D., Rolston S. L., Spreeuw R. J. C. and Jessen P. S., Phys. Rev. Lett., 74 (1995) 1542. ¨ ferle T., Moritz H., Schori C., Ko ¨ hl M. and Esslinger T., Phys. Rev. Lett., 92 [29] Sto (2004) 130403. ¨ ferle T., Moritz H., Gu ¨nter K., Ko ¨ hl M. and Esslinger T., Phys. Rev. Lett., [30] Sto 96 (2006) 030401. [31] Loftus T., Regal C. A., Ticknor C., Bohn J. L. and Jin D. S., Phys. Rev. Lett., 88 (2002) 173201. [32] Busch T., Englert B.-G., Rzazewski K. and Wilkens M., Found. Phys., 28 (1998) 549. [33] Kuklov A. B. and Svistunov B. V., Phys. Rev. Lett., 90 (2003) 100401. [34] Lewenstein M., Santos L., Baranov M. A. and Fehrmann H., Phys. Rev. Lett., 92 (2004) 050401. [35] Cramer M., Eisert J. and Illuminati F., Phys. Rev. Lett., 93 (2004) 090405. [36] Mathey L., Wang D. W., Hofstetter W., Lukin M. D. and Demler E., Phys. Rev. Lett., 93 (2004) 120404. ¨chler H. P. and Blatter G., Phys. Rev. Lett., 91 (2003) 130404. [37] Bu ¨ hl M. and Esslinger T., Phys. Rev. Lett., ¨nter K., Sto ¨ ferle T., Moritz H., Ko [38] Gu 96 (2006) 180402. [39] Ospelkaus C., Ospelkaus S., Wille O., Succo M., Ernst P., Sengstock K. and Bongs K., Phys. Rev. Lett., 96 (2006) 180403. ¨chler H. P., Phys. Rev. B, 72 (2005) 024534. [40] Powell S., Sachdev S. and Bu [41] Chin J. K., Miller D. E., Liu Y., Stan C., Setiawan W., Sanner C., Xu K. and Ketterle W., Nature, 443 (2006) 961. ¨ ck U., Troyer M. and Zoller P., Phys. Rev. Lett., 96 (2006) [42] Trebst S., Schollwo 250402. [43] Ospelkaus C., Ospelkaus S., Humbert L., Ernst P., Sengstock K. and Bongs K., Phys. Rev. Lett., 97 (2006) 120402.

Condensed-matter physics with light and atoms: Strongly correlated cold fermions in optical lattices A. Georges Centre de Physique Th´ eorique, Ecole Polytechnique - 91128 Palaiseau Cedex, France

Various topics at the interface between condensed-matter physics and the physics of ultra-cold fermionic atoms in optical lattices are discussed. This article starts with basic considerations on energy scales, and on the regimes in which a description by an eﬀective Hubbard model is valid. Qualitative ideas about the Mott transition are then presented, both for bosons and fermions, as well as mean-ﬁeld theories of this phenomenon. Antiferromagnetism of the fermionic Hubbard model at half-ﬁlling is brieﬂy reviewed. The possibility that interaction eﬀects facilitate adiabatic cooling is discussed, and the importance of using entropy as a thermometer is emphasized. Geometrical frustration of the lattice, by suppressing spin long-range order, helps revealing genuine Mott physics and exploring unconventional quantum magnetism. The importance of measurement techniques to probe quasi-particle excitations in cold fermionic systems is emphasized, and a recent proposal based on stimulated Raman scattering brieﬂy reviewed. The unconventional nature of these excitations in cuprate superconductors is emphasized. 1. – Introduction: A novel condensed-matter physics The remarkable recent advances in handling ultra-cold atomic gases have given birth to a new ﬁeld: condensed-matter physics with light and atoms. Artiﬁcial solids with unprecedented degree of controllability can be realized by trapping bosonic or fermionic atoms in the periodic potential created by interfering laser beams (for a recent review, see ref. [1], and other contributions to this volume). Key issues in the physics of strongly correlated quantum systems can be addressed from a new perspective in this context. The observation of the Mott transition of bosons c Societ` a Italiana di Fisica

477

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in optical lattices [2, 3] and of the superﬂuidity of fermionic gases [4-7] have been important milestones in this respect, as well as the recent imaging of Fermi surfaces [8]. To quote just a few of the many promising roads for research with ultra-cold fermionic atoms in optical lattices, I would emphasize: – The possibility of studying and hopefully understanding better some outstanding open problems of condensed-matter physics, particularly in strongly correlated regimes, such as high-temperature superconductivity and its interplay with Mott localization. – The possibility of studying these systems in regimes which are not usually reachable in condensed-matter physics (e.g., under time-dependent perturbations bringing the system out of equilibrium), and to do this within a highly controllable and clean setting. – The possibility of “engineering” the many-body wave function of large quantum systems by manipulating atoms individually or globally. The present lecture notes certainly do not aim at covering all these topics! Rather, they represent an idiosyncratic choice reﬂecting the current interests of the author. Hopefully, they will contribute in a positive manner to the rapidly developing dialogue between condensed matter physics and the physics of ultra-cold atoms. Finally, a warning and an apology: these are lecture notes and not a review article. Even though I do quote some of the original work I refer to, I have certainly omitted important relevant references, for which I apologize in advance. 2. – Considerations on energy scales In the context of optical lattices, it is convenient to express energies in units of the recoil energy: ER =

2 2 kL 2m

in which kL = 2π/λL is the wave vector of the laser and m the mass of the atoms. This is typically of the order of a few micro-Kelvins (for a YAG laser with λL = 1.06 μm and 6 Li atoms, ER 1.4 μK). When venturing in the cold atoms community, condensed matter physicists who usually express energy scales in Kelvins (or electron-Volts. . . !) will need to remember that, in units of frequency: 1 μK 20.8 kHz. The natural scale for the kinetic energy (and Fermi energy) of atoms in the optical lattice is not the recoil energy however, but rather the bandwidth W of the Bloch band under consideration, which strongly depends on the laser intensity V0 . For a weak intensity V0 ER , the bandwidth W of the lowest Bloch band in the optical lattice is

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Condensed-matter physics with light and atoms: etc.

Fig. 1. – Width of the lowest Bloch band and gap between the ﬁrst two bands for a 3-dimensional potential, as a function of laser intensity (in units of ER ) (adapted from ref. [9]). Note that in 3 dimensions, the two lowest bands overlap for a weak lattice potential, and become separated only for V0 2.3ER .

of order ER itself (the free-space parabolic dispersion 2 k 2 /2m reaches the boundary of the ﬁrst Brillouin zone at k = π/d = kL with d = λL /2 the lattice spacing, so that W ER for small V0 /ER ). In contrast, for strong laser intensities, the bandwidth can be much smaller than the recoil energy (ﬁg. 1). This is because in this limit the motion of atoms in the lattice corresponds to tunneling between two neighboring potential wells (lattice sites), and the hopping amplitude (1 ) t has the typical exponential dependence of a tunnel process. Speciﬁcally, for a simple separable potential in D (=1, 2, 3) dimensions,

(1)

V (r) = V0

D

sin2 kL ri ,

i=1

one has [12] (2)

t/ER = 4π −1/2 (V0 /ER )3/4 e−2(V0 /ER )

1/2

,

V0 ER .

The dispersion of the lowest band is well approximated by a simple tight-binding expres(1 ) I could not force myself to use the notation J for the hopping amplitude in the lattice, as often done in the quantum optics community. Indeed, J is so commonly used in condensed matter physics to denote the magnetic superexchange interaction that this can be confusing. I therefore stick to the condensed matter notation t, not to be confused of course with time t, but it is usually clear from the context.

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sion in this limit: (3)

εk = −2t

D

cos ki

i=1

corresponding to a bandwidth W = 4Dt ER . The dependence of the bandwidth, and of the gap between the ﬁrst two bands, on V0 /ER are displayed on ﬁg. 1. Since W is much smaller than ER for deep lattices, one may worry that cooling the gas into the degenerate regime might become very diﬃcult. For non-interacting atoms, this indeed requires T εF , with εF the Fermi energy (energy of the highest occupied state), with εF ≤ W for densities such that only the lowest band is partially occupied. Adiabatic cooling may however come to the rescue when the lattice is gradually turned on [9]. This can be understood from a very simple argument, observing that the entropy of a non-interacting Fermi gas in the degenerate regime is limited by the Pauli principle to have a linear dependence on temperature: S ∝ T D(εF ), where D(ε) is the density of states. Hence, T D(εF ) is expected to be conserved along constant entropy trajectories. D(εF ) is inversely proportional to the bandwidth W (with a proportionality factor depending on the density, or band ﬁlling): the density of states is enhanced considerably as the band shrinks since the one-particle states all ﬁt in a smaller and smaller energy window. Thus, T /W is expected to be essentially constant as the lattice is adiabatically turned on: the degree of degeneracy is preserved and adiabatic cooling is expected to take place. For more details on this issue, see ref. [9] in which it is also shown that when the second band is populated, heating can take place when the lattice is turned on (because of the increase of the inter-band gap, cf. ﬁg. 1). For other ideas about cooling and heating eﬀects upon turning on the lattice, see also ref. [10]. Interactions can signiﬁcantly modify these eﬀects, and lead to additional mechanisms of adiabatic cooling, as discussed later in this article (sect. 6). Finally, it is important to note that, in a strongly correlated system, characteristic energy scales are in general strongly modiﬁed by interaction eﬀects in comparison to their bare, non-interacting values. The eﬀective mass of quasiparticle excitations, for example, can become very large due to strong interaction eﬀects, and correspondingly the scale associated with kinetic energy may become very small. This will also be the scale below which coherent quasiparticle excitations exist, and hence the eﬀective scale for Fermi degeneracy. Interaction eﬀects may also help in adiabatically cooling the system however, as discussed later in these notes. 3. – When do we have a Hubbard model? I do not intend to review here in details the basic principles behind the trapping and manipulation of cold atoms in optical lattices. Other lectures at this school are covering

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Condensed-matter physics with light and atoms: etc.

this, and there are also excellent reviews on the subject, see, e.g., refs. [1, 11, 12]. I will instead only summarize the basic principles behind the derivation of the eﬀective Hamiltonian. The focus of this section will be to emphasize that there are some limits on the range of parameters in which the eﬀective Hamiltonian takes the simple single-band Hubbard form [13, 14]. I consider two-component fermions (e.g., two hyperﬁne states of identical atomic species). The Hamiltonian consists in a one-body term and an interaction term: (4)

H = H0 + Hint .

Let me ﬁrst discuss the one-body part, which involves the lattice potential VL (r) as well as the potential of the trap (or of the Gaussian waist of the laser) VT (r): (5)

H0 =

σ

2 2 ∇ + VL (r) + VT (r) ψσ (r) ≡ H0L + H0T . dr ψσ† (r) − 2m

The trapping potential having a shallow curvature as compared to the lattice spacing, the standard procedure consists in ﬁnding ﬁrst the Bloch states of the periodic potential (e.g. treating afterwards the trap in the local density approximation). The Bloch functions φkν (r) (with ν an index labelling the band) satisfy: H0L |φkν = εkν |φkν

(6)

with φkν (r) = eik·r ukν (r) and ukν a function having the periodicity of the lattice. From the Bloch functions, one can construct Wannier functions wRν (r) = wν (r − R), which are localized around a speciﬁc lattice site R: (7) wRν (r) = wν (r − R) = e−ik·R φkν (r) = eik·(r−R) ukν (r). k

k

In ﬁg. 2, I display a contour plot of the Wannier function corresponding to the lowest band of the 2-dimensional potential (1). The characteristic spatial extension of the Wannier function associated with the lowest band is l1 ∼ d (the lattice spacing itself) for a weak potential V0 ER , while l1 /d ∼ (ER /V0 )1/4 1 for a deep lattice V0 ER . The latter estimate is simply the extent of the ground-state wave function of the harmonic oscillator in the quadratic well approximating the bottom of the potential. The fermion ﬁeld operator can be decomposed on the localised Wannier functions basis set, or alternatively on the Bloch functions as follows: (8) ψσ† (r) = wν∗ (r − R) c†Rνσ = φ∗kν (r) c†kνσ . Rν

kν

This leads to the following expression for the lattice part of the one-particle Hamiltonian: (ν) † εkν c†kνσ ckνσ = − tRR cRνσ cR νσ + (9) H0L = εν0 c†Rνσ cRνσ kνσ

RR νσ

Rνσ

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Fig. 2. – Contour plot of the Wannier function corresponding to the lowest band in the twodimensional separable potential (1) with V0 /ER = 10. The function has the symmetry of the square lattice, and has secondary maxima on nearest-neighbor sites. The intensity of these secondary maxima control the hopping amplitude. From ref. [14].

with the hopping parameters and on-site energies given by (10)

(ν) tRR

=−

ik·(R−R )

e

εkν = −

k

(11)

εν0 =

dr wν∗ (r

2 2 ∇ + VL (r) wν (r − R ), − R) − 2m

εkν .

k

Because the Bloch functions diagonalize the one-body Hamiltonian, there are no interband hoppings terms in the Wannier representation considered here. Furthermore, for a separable potential such as (1), close examination of (10) show that the oppings are only along the principal axis of the lattice: the hopping amplitudes along diagonals vanish for a separable potential (see also sect. 7). Let us now turn to the interaction Hamiltonian. The inter-particle distance and lattice spacing are generally much larger than the hard-core radius of the inter-atomic potential. Hence, the details of the potential at short distance do not matter. Long distance properties of the potential are characterized by the scattering length as . As is well known, and described elsewhere in these lectures, as can be tuned over a wide range of positive or negative values by varying the magnetic ﬁeld close to a Feshbach resonance. Provided the extent of the Wannier function is larger than the scattering length (l1 as ), the following pseudopotential can be used: (12)

σ,−σ Vint (r − r ) = g δ(r − r ) g ≡

4π2 as . m

483

Condensed-matter physics with light and atoms: etc.

The interaction Hamiltonian then reads dr ψ↑† (r)ψ↑ (r)ψ↓† (r)ψ↓ (r), (13) Hint = g which can be written in the basis set of Wannier functions (assumed for simplicity to be real) as follows: (14)

Hint =

R 1 R 2 R 3 R 4 ν1 ν2 ν3 ν4

ν1 ν2 ν3 ν4 UR c† c c† c 1 R 2 R 3 R 4 R 1 ν1 ↑ R 2 ν2 ↑ R 3 ν3 ↓ R 4 ν 4 ↓

with (15)

ν1 ν2 ν3 ν4 UR 1 R2 R3 R 4

=g

dr wν1 (r − R1 )wν2 (r − R2 )wν3 (r − R3 )wν4 (r − R4 ).

The largest interaction term corresponds to two atoms on the same lattice site. Furthermore, for a deep enough lattice, with less than two atoms per site on average, the second band is well separated from the lowest one. Nelecting all other bands, and all interaction terms except the largest on-site one, one obtains the single-band Hubbard model with a local interaction term: (16)

HH = −

tRR c†Rσ cR σ + U

RR σ

n ˆ R↑ n ˆ R↓

R

with (17)

U =g

dr w1 (r)4 .

For a deep lattice, using the above estimate of the extension l1 of the Wannier function of the lowest band, this leads to [12] (compare to the hopping amplitude (2) which decays exponentially) (18)

U ER

8 as kL π

V0 ER

3/4 .

The hopping amplitude and the on-site interaction strength U , calculated for the lowest band of a three-dimensional separable potential, are plotted as a function of V0 /ER in ﬁg. 3. Let us ﬁnally discuss the conditions under which this derivation of a simple singleband Hubbard model is indeed valid. We have made 3 assumptions: i) neglect the second band, ii) neglect other interactions besides the Hubbard U and iii) replace the actual interatomic potential by the pseudopotential approximation. Assumption i) is justiﬁed provided the second band is not populated (less than two fermions per site, and V0 not too small so that the two bands do not overlap, i.e. V0 2.3ER cf. ﬁg. 1), but

484

A. Georges

Fig. 3. – Hopping amplitude t and on-site interaction energy U , as a function of V0 /ER , for the three-dimensional separable potential (1) corresponding to a cubic lattice. t is expressed in units of ER and U in units of 100 ER as /d, with as the scattering length and d the lattice spacing. From ref. [14].

also provided the energy cost for adding a second atom on a given lattice site which already has one is indeed set by the interaction energy. If U as given by (17) becomes larger than the separation Δ = k (εk2 − εk1 ) between the ﬁrst two bands, then it is more favorable to add the second atom in the second band (which then cannot be neglected, even if not populated). Hence one must have U < Δ. For the pseudopotential to be valid (assumption iii)), the typical distance between two atoms in a lattice well (which is given by the extension of the Wannier function l1 ) must be larger than the scattering length: as l1 . Amusingly, for deep lattices, this actually coincides with the requirement U Δ and boils down to (at large V0 /ER ): (19)

as d

V0 ER

−1/4 .

In order to see this, one simply has to use the above estimates of l1 and U and that of the separation Δ (ER V0 )1/2 in this limit. Equation (19) actually shows that for a deep lattice, the scattering length should not be increased too much if one wants to keep a Hubbard model with an interaction set by the scattering length itself and given by (18). For larger values of as , it may be that a one-band Hubbard description still applies (see however below for the possible appearance of new interaction terms), but with an eﬀective U given by the inter-band separation Δ rather than set by as . This requires a more precise investigation of the speciﬁc case at hand (2 ). Finally, the possible existence of other interaction terms besides the on-site U (ii)), and when they can be neglected, requires a more careful examination. These interactions (2 ) This is reminiscent of the so-called Mott insulator to charge-transfer insulator crossover in condensed-matter physics.

485

Condensed-matter physics with light and atoms: etc.

0.01

Heisenberg

U/ t=

20

10

Spin-density wave

t=

U/

0

t=1

U/

=5

U/t

=1 U/t

as / a

.05

0

0.001 0

10

20

30

V0 / ER Fig. 4. – Range of validity of the simple one-band Hubbard model description, for a separable three-dimensional potential (1), as a function of lattice depth (normalized to recoil energy) V0 /ER , and scattering length (normalized to lattice spacing) as /d. In the shaded region, the one-band Hubbard description is questionable. The dashed line corresponds to the condition U/Δ = 0.1, with Δ the gap to the second band: above this line, other bands may have to be taken into account and the pseudopotential approximation fails, so that U is no longer given by (17). The dash-dotted line corresponds to Vh /t = 0.1: above this line, Vh becomes sizeable. These conditions may be somewhat too restrictive, but are meant to emphasize the points raised in the text. Also indicated on the ﬁgure are: contour plots of the values of the Hubbard coupling U/t, and the regions corresponding to the spin-density wave and Heisenberg regimes of the antiferromagnetic ground state at half-ﬁlling (sect. 5). The crossover between these regimes is indicated by the dotted line (U/t = 10), where TN /t is maximum. Figure from ref. [13].

must be smaller than U but also than the hopping t which we have kept in the Hamiltonian. In refs. [13, 14], we considered this in more details and concluded that the most “dangerous” coupling turns out to be a kind of “density-assisted” hopping between two nearest-neighbor sites, of the form (20) Vh n ˆ R,−σ c†Rσ cR ,σ + h.c., RR σ

where d denotes a lattice translation between nearest-neighbor sites, and the last formula holds for a separable potential with Vh = g dr w1 (r)3 w1 (r + d) =

3 4 4 =g dxwx (x) wx (x + d) dywy (y) dzwz (z) . The validity of the single-band Hubbard model also requires that Vh t, U . All these requirements insuring that a simple Hubbard model description is valid are summarized on ﬁg. 4.

486

A. Georges

4. – The Mott phenomenon Strong-correlation eﬀects appear when atoms “hesitate” between localized and itinerant behaviour. In such a circumstance, one of the key diﬃculties is to describe consistently an entity which is behaving simultaneously in a wave-like (delocalized) and particle-like (localized) manner. Viewed from this perspective, strongly correlated quantum systems raise questions which are at the heart of the quantum-mechanical world. The most dramatic example is the possibility of a phase transition between two states: one in which atoms behave in an itinerant manner, and one in which they are localized by the strong on-site repulsion in the potential wells of a deep lattice. In the Mott insulating case, the energy gain which could be obtained by tunneling between lattice sites (∼ Dt W ) becomes unfavorable in comparison to the cost of creating doubly occupied lattice sites (∼ U ). This cost will have to be paid for sure if there is, for example, one atom per lattice site on average. This is the famous Mott transition. The proximity of a Mott insulating phase is in fact responsible for many of the intriguing properties of strongly correlated electron materials in condensed-matter physics, as illustrated below in more details. This is why the theoretical proposal [3] and experimental observation [2] of the Mott transition in a gas of ultra-cold bosonic atoms in an optical lattice have truly been pioneering works establishing a bridge between modern issues in condensed matter physics and ultra-cold atomic systems. . 4 1. Mean-ﬁeld theory of the bosonic Hubbard model. – Even though this school is devoted to fermions, I ﬁnd it useful to brieﬂy describe the essentials of the mean-ﬁeld theory of the Mott transition in the bosonic Hubbard model. Indeed, this allows to focus on the key phenomenon (namely, the blocking of tunneling by the on-site repulsive interaction) without having to deal with the extra complexities of fermionic statistics and spin degrees of freedom which complicate the issue in the case of fermions (see below). Consider the Hubbard model for single-component bosonic atoms: (21)

H=−

ij

tij b†i bj +

U n ˆ i (ˆ ni − 1) − μ n ˆi. 2 i i

As usually is the case in statistical mechanics, a mean-ﬁeld theory can be constructed by replacing this Hamiltonian on the lattice by an eﬀective single-site problem subject to a self-consistency condition. Here, this is naturally achieved by factorizing the hopping term [15,16]: b†i bj → const + b†i bj + b†i bj + ﬂuct. Another essentially equivalent formulation is based on the Gutzwiller wave function [17, 18]. The eﬀective 1-site Hamiltonian for site i reads (22)

heﬀ = −λi b† − λi b + (i)

U n ˆ (ˆ n − 1) − μˆ n. 2

In this expression, λi is a “Weiss ﬁeld” which is determined self-consistently by the boson

Condensed-matter physics with light and atoms: etc.

487

amplitude on the other sites of the lattice through the condition (23) λi = tij bj . j

For nearest-neighbour hopping on a uniform lattice of connectivity z, with all sites being equivalent, this reads λ = z t b.

(24)

These equations are easily solved numerically, by diagonalizing the eﬀective single-site Hamiltonian (22), calculating b and iterating the procedure such that (24) is satisﬁed. The boson amplitude b is an order-parameter which is non-zero in the superﬂuid phase. For densities corresponding to an integer number n of bosons per site on average, one ﬁnds that b is non-zero only when the coupling constant U/t is smaller than a critical ratio (U/t)c (which depends on the ﬁlling n). For U/t > (U/t)c , b (and λ) vanishes, signalling the onset of a non-superﬂuid phase in which the bosons are localised on the lattice sites. For non-integer values of the density, the system remains a superﬂuid for arbitrary couplings. It is instructive to analyze these mean-ﬁeld equations close to the critical value of the coupling: because λ is then small, it can be treated in perturbation theory in the eﬀective Hamiltonian (22). Let us start with λ = 0. We then have a collection of disconnected lattice sites (i.e. no eﬀective hopping, often called the “atomic limit” in condensed-matter physics). The ground state of an isolated site is the number state |n when the chemical potential is in the range μ ∈ [(n − 1)U, nU ]. When λ is small, the perturbed ground state becomes √ √ n n+1 |n − 1 + |n + 1 , (25) |ψ0 = |n − λ U (n − 1) − μ μ − Un so that (26)

ψ0 |b|ψ0 = −λ

n n+1 + . U (n − 1) − μ μ − U n

Inserting this in the self-consistency condition yields n n+1 + + ··· , (27) λ = −z t λ U (n − 1) − μ μ − U n where “· · · ” denotes higher-order terms in λ. Hence, the critical value of the coupling corresponds to the vanishing of the coeﬃcient of the term linear in λ (corresponding to the mass term of the expansion of the Landau free-energy), i.e. hence the critical boundary for a ﬁxed average (integer) density n is given by (28)

(n − μ/U )(μ/U − n + 1) zt = . U 1 + μ/U

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A. Georges

Fig. 5. – Left: phase diagram of the Bose Hubbard model as a function of chemical potential μ/U and coupling t/U . An incompressible Mott insulator is found within each lobe of integer density. Right: density proﬁles in a harmonic trap. The “wedding cake” structure (see text) is due to the incompressibility of the Mott insulator (numerical calculations courtesy of H. Niemeyer and H. Monien, ﬁgure courtesy F. Gerbier).

This expression gives the location of the critical boundary as a function of the chemical potential. In the (t/U, μ/U )-plane, the phase diagram (ﬁg. 5) consists of lobes inside which the density is integer and the system is a Mott insulator. Outside these lobes, the system is a superﬂuid. The tip of a given lobe corresponds to the the maximum value of the hopping at which an insulating state can be found. For n atoms per site, this is given by / zt // 1 (n − x)[x − n + 1] (29) = . = Maxx∈[n−1,n] U /c,n 1+x 2n + 1 + 2 n(n + 1) So that the critical interaction strength is (U/zt)c 5.8 for n = 1, and increases as n increases ((U/zt)c ∼ 4n for large n). . 4 2. Incompressibility of the Mott phase and “wedding-cake” structure of the density proﬁle in the trap. – The Mott insulator has a gap to density excitations and is therefore an incompressible state: adding an extra particle costs a ﬁnite amount of energy. This is clear from the mean-ﬁeld calculation above: if we want to vary the average density from inﬁnitesimally below an integer value n to inﬁnitesimally above, we have to change the chemical potential across the Mott gap: (30)

Δg (n) = μ+ (n) − μ− (n),

where μ± are the solutions of the quadratic equation corresponding to (28), i.e. (31)

(μ/U )2 − [2n − 1 − (zt/U )](μ/U ) + n(n − 1) + (zt/U ) = 0

Condensed-matter physics with light and atoms: etc.

489

yielding (32)

Δg (n) = U

zt U

2

1/2 zt − 2(2n + 1) + 1 . U

√ The Mott gap is ∼ U at large U and vanishes at the critical coupling (∝ U − Uc within mean-ﬁeld theory). The existence of a gap means that the chemical potential can be changed within the gap without changing the density. As a result, when the system is placed in a trap, it displays density plateaus corresponding to the Mott state, leading to a “wedding cake” structure of the density proﬁle (ﬁg. 5). This is easily understood in the local density approximation, in which the local chemical potential is given by: μ(r) = μ ¯ − mω02 r2 /2, 2 1/2 yielding a maximum extension of the plateau: ∼ (2Δg /mω0 ) . Several authors have studied these density plateaus beyond the LDA by numerical simulation (see, e.g., [19]), and they have been recently observed experimentally [20]. . 4 3. Fermionic Mott insulators and the Mott transition in condensed-matter physics. – The discussion of Mott physics in the fermionic case is somewhat complicated by the presence of the spin degrees of freedom (corresponding, e.g., to 2 hyperﬁne states in the context of cold atoms). Of course, we could consider single-component fermions, but two of those cannot be put on the same lattice site because of the Pauli principle, hence spinless fermions with one atom per site on average simply form a band insulator. Mott and charge density wave physics would show up in this context when we have, e.g., one fermion out of two sites, but this requires inter-site (e.g., dipolar) interactions. The basic physics underlying the Mott phenomenon in the case of two-component fermions with one particle per site on average is the same as in the bosonic case however: the strong on-site repulsion overcomes the kinetic energy and makes it unfavorable for the particles to form an itinerant (metallic) state. From the point of view of band theory, we would have a metal, with one atom per unit cell and a half-ﬁlled band. Instead, at large enough values of U/t, a Mott insulating state with a charge gap develops. This is purely charge physics, not spin physics. One must however face the fact that the naive Mott insulating state has a huge spin entropy: it is a paramagnet in which the spin of the atom localized on a given site can point in either direction. This huge degeneracy must be lifted as one cools down the system into its ground state (Nernst). How this happens will depend on the details of the model and of the residual interactions between the spin degrees of freedom. In the simplest case of a two-component model on an unfrustrated (e.g. bipartite) lattice the spins order into an antiferromagnetic ground state. This is easily understood in strong coupling U t by Anderson’s superexchange mechanism: in a single-band model, a nearest-neighbor magnetic exchange is generated, which reads on each lattice bond (33)

JAF =

4t2ij . U

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A. Georges

This expression is easily understood from second-order degenerate perturbation theory in the hopping, starting from the limit of decoupled sites (t = 0). Then, two given sites have a 4-fold degenerate ground state. For small t, this degeneracy is lifted: the singlet state is favoured because a high-energy virtual state is allowed in the perturbation expansion (corresponding to a doubly occupied state), while no virtual excited state is connected to the triplet state because of the Pauli principles (an atom with a given spin cannot hop to a site on which another atom with the same spin already exists). If we focus only on low-energies, much smaller than the gap to density excitations (∼ U at large U ), we can consider the reduced Hilbert space of states with exactly one particle per site. Within this low-energy Hilbert space, the Hubbard model with one particle per site on average reduces to the quantum Heisenberg model: (34)

HJ = JAF

Si · Sj .

ij

Hence, there is a clear separation of scales at strong coupling: for temperatures/energies T U , density ﬂuctuations are suppressed and the physics of a paramagnetic Mott insulator (with a large spin entropy) sets in. At a much lower scale T JAF , the residual spin interactions set in and the true ground state of the system is eventually reached (corresponding, in the simplest case, to an ordered antiferromagnetic state). At this point, it is instructive to pause for a moment and ask what real materials do in the condensed-matter physics world. Materials with strong electronic correlations are those in which the relevant electronic orbitals (those corresponding to energies close to the Fermi energy) are quite strongly localized around the nuclei, so that a band theory description in terms of Bloch waves is not fully adequate (and may even fail completely). This happens in practice for materials containing partially ﬁlled d- and f -shells, such as transition metals, transition-metal oxides, rare earths, actinides and their compounds, as well as many organic conductors (which have small bandwidths). In all these materials, Mott physics and the proximity to a Mott insulating phase plays a key role. In certain cases, these materials are poised rather close to the localisation/delocalisation transition so that a small perturbation can tip oﬀ the balance. This is the case, for example, of a material such as V2 O3 (vanadium sesquioxide), whose phase diagram is displayed in ﬁg. 6. The control parameter in this material is the applied pressure (or chemical substitution by other atoms on vanadium sites), which change the unit-cell volume and hence the bandwidth (as well, in fact, as other characteristics of the electronic structure, such as the crystal-ﬁeld splitting). It is seen from ﬁg. 6 that all three phases discussed above are realized in this material. At low pressure and high temperature one has a paramagnetic Mott insulator with ﬂuctuating spins. As the pressure is increased, this insulator evolves abruptly into a metallic state, through a ﬁrst-order transition line (which ends at a critical endpoint at Tc 450 K). At low temperature T < TN 170 K, the paramagnetic Mott insulator orders into an antiferromagnetic Mott insulator. Note that the characteristic temperatures at which these transitions take place are considerably smaller than the bare electronic energy scales (∼ 1 eV 12000 K).

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491

Fig. 6. – Phase diagram of V2 O3 as a function of pressure of Cr-substitution, and temperature. The cartoons illustrate the nature of each phase (paramagnetic Mott insulator, parmagnetic metal, antiferromagnetic Mott insulator).

On ﬁg. 6, I have given for each phase a (much oversimpliﬁed) cartoon of what the phase looks like in real space. The paramagnetic Mott insulator is a superposition of essentially random spin conﬁgurations, with almost only one electron per site and very few holes and double occupancy. The antiferromagnetic insulator has N´eel-like long-range order (but of course the wavefunction is a complicated object, not the simple N´eel classical wave function). The metal is the most complicated when looking at it in real space: it is a superposition of conﬁgurations with singly occupied sites, holes, and double occupancies. Of course, such a material is far less controllable than ultra-cold atomic systems: as we apply pressure many things change in the material, not only, e.g., the electronic bandwidth. Also, not only the electrons are involved: increasing the lattice spacing as pressure is reduced decreases the electronic cohesion of the crystal and the ions of the lattice may want to take advantage of that to gain elastic energy: there is indeed a discontinuous change of lattice spacing through the ﬁrst-order Mott transition line. Atomic substitutions introduce furthermore some disorder into the material. Hence, ultra-cold atomic systems oﬀer an opportunity to disentangle the various phenomena and study these eﬀects in a much more controllable setting. . . 4 4. (Dynamical) mean-ﬁeld theory for fermionic systems. – In subsect. 4 1, we saw how a very simple mean-ﬁeld theory of the Mott phenomenon can be constructed for bosons, by using b as an order parameter of the superﬂuid phase and making an eﬀective ﬁeld (Weiss) approximation for the inter-site hopping term. Unfortunately, this cannot be immediately extended to fermions. Indeed, we cannot give an expectation value to the

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A. Georges

single fermion operator, and c is not an order parameter of the metallic phase anyhow. A generalization of the mean-ﬁeld concept to many-body fermion systems does exist however, and is known as the “dynamical mean-ﬁeld theory” (DMFT) approach. There are many review articles on the subject (e.g., [21-23]), so I will only describe it very brieﬂy here. The basic idea is still to replace the lattice system by a single-site problem in a self-consistent eﬀective bath. The exchange of atoms between this single site and the eﬀective bath is described by an amplitude, or hybridization function (3 ), Δ(iωn ), which is a function of energy (or time). It is a quantum-mechanical generalization of the static Weiss ﬁeld in classical statistical mechanics, and physically describes the tendancy of an atom to leave the site and wander in the rest of the lattice. In a metallic phase, we expect Δ(ω) to be large at low-energy, while in the Mott insulator, we expect it to vanish at low-energy so that motion to other sites is blocked. The (site+eﬀective bath) problem is described by an eﬀective action, which for the paramagnetic phase of the Hubbard model reads (35)

Seﬀ = −

n

c†σ (iωn )[iωn + μ − Δ(iωn )]cσ (iωn ) + U

σ

β

dτ n↑ n↓ . 0

From this local eﬀective action, a one-particle Green’s function and self-energy can be obtained as G(τ − τ ) = −T cσ (τ )c†σ (τ )eﬀ ,

(36)

Σ(iωn ) = iωn + μ − Δ(iωn ) − G(iωn )−1 .

(37)

The self-consistency condition, which closes the set of dynamical mean-ﬁeld theory equations, states that the Green’s function and self-energy of the (single-site+bath) problem coincides with the corresponding local (on-site) quantities in the original lattice model. This yields (38)

G(iωn ) =

k

1 1 = . iωn + μ − Σ(iωn ) − εk Δ(iωn ) + G(iωn )−1 − εk k

Equations (35), (38) form a set of two equations which determine self-consistently both the local Green’s function G and the dynamical Weiss ﬁeld Δ. Numerical methods are necessary to solve these equations, since one has to calculate the Green’s function of a many-body (albeit local) problem. Fortunately, there are several computational algorithms which can be used for this purpose. On ﬁg. 7, I display the schematic shape of the generic phase diagram obtained with dynamical mean-ﬁeld theory, for the one band Hubbard model with one particle per site. At high temperature, there is a crossover from a Fermi liquid (metallic) state at (3 ) Here, I use the Matsubara quantization formalism at ﬁnite temperature, with ωn = (2n + 1)π/β and β = 1/kT .

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493

Fig. 7. – Schematic phase diagram of the half-ﬁlled fermionic Hubbard model, as obtained from DMFT. It is depicted here for the case of a frustrated lattice (e.g., with next-nearest neighbour hopping), which reduces the transition temperature into phases with long-range spin ordering. Then, a ﬁrst-order transition from a metal to a paramagnetic Mott insulator becomes apparent. For the unfrustrated case, see next section. Adapted from [24].

weak coupling to a paramagnetic Mott insulator at strong coupling. Below some critical temperature Tc , this crossover turns into a ﬁrst-order transition line. Note that Tc is a very low energy scale: Tc W/80, almost two orders of magnitude smaller than the bandwidth. Whether this critical temperature associated with the Mott transition can be actually reached depends on the concrete model under consideration. In the simplest case, i.e. for a single band with nearest-neighbor hopping on an unfrustrated lattice, long-range antiferromagnetic spin ordering takes place already at a temperature far above Tc , as studied in more details in the next section. Hence, only a ﬁnite-temperature crossover, not a true phase transition, into a paramagnetic Mott insulator will be seen in this case. However, if antiferromagnetism becomes frustrated, the N´eel temperature can be strongly suppressed, revealing genuine Mott physics, as shown in the schematic phase diagram of ﬁg. 7. 5. – Ground state of the 2-component Mott insulator: Antiferromagnetism Here, I consider in more details the simplest possible case of a one-band Hubbard model, with nearest-neighbor hopping on a bipartite (e.g., cubic) lattice and one atom per site on average. The phase diagram, as determined by various methods (Quantum Monte Carlo, as well as the DMFT approximation) is displayed on ﬁg. 8. There are only two phases: a high-temperature paramagnetic phase, and a low-temperature antiferromagnetic phase which is an insulator with a charge gap. Naturally, within the high-temperature phase, a gradual crossover from itinerant to Mott localized is observed

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T

*

0.8

1.5

s=

0.7 5

s=

F

s=0

.7 s=

T/t

0.7

1

PM 0.5

s=0

.4

AF 0

0

10

20

U/t Fig. 8. – Phase diagram of the half-ﬁlled Hubbard model on the cubic lattice: antiferromagnetic (AF) and paramagnetic (PM) phases. Transition temperature within the DMFT approximation (plain curve, open circles) and from the QMC calculation of ref. [25] (dot-dashed curve, squares). Dashed lines: isentropic curves (s = 0.4, 0.7, 0.75, 0.8), computed within DMFT. Dotted line: quasi-particle coherence scale TF∗ (U ). See ref. [13] for more details.

as the coupling U/t is increased, or as the temperature is decreased below the Mott gap (∼ U at large U/t). Note that the mean-ﬁeld estimate of the Mott critical temperature Tc W/80 is roughly a factor of two lower than that of the maximum value of the N´ eel temperature for this model (∼ W/40), so we do not expect the ﬁrst-order Mott transition line and critical endpoint to be apparent in this unfrustrated situation. Both the weak coupling and strong coupling sides of the phase diagram are rather easy to understand. At weak coupling, we can treat U/t by a Hartree-Fock decoupling, and construct a static mean-ﬁeld theory of the antiferromagnetic transition. The broken symmetry into (A, B) sublattices reduces the Brillouin zone to half of its original value, and two bands are formed which read (39)

Ek± = ±

ε2k + Δ2g /4.

In this expression, Δ is the Mott gap, which within this Hartree approximation is directly related to the staggered magnetization of the ground state ms = nA↑ − nA↓ = nB↓ − nB↑ by (40)

Δg = U ms .

This leads to a self-consistent equation for the gap (or staggered magnetization): (41)

U 1 = 1. 2 2 εk + Δ2g /4 k∈RBZ

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495

At weak coupling, where this Hartree approximation is a reasonable starting point, the antiferromagnetic instability occurs for arbitrary small U/t and the gap, staggered magnetization and N´eel temperature are all exponentially small. In this regime, the antiferromagnetism is a “spin density-wave” with wave vector Q = (π, · · · , π) and a very weak modulation of the order parameter. It should be noted that this spin-density wave mean-ﬁeld theory provides a band theory (Slater) description of the insulating ground state: because translational invariance is broken in the antiferromagnetic ground state, the Brillouin zone is halved, and the ground state amounts to fully occupy the lowest Hartree-Fock band. This is because there is no separation of energy scales at weak coupling: the spin and charge degrees of freedom get frozen at the same energy scale. The existence of a band-like description in the weak-coupling limit is often a source of confusion, leading some people to overlook that Mott physics is primarily a charge phenomenon, as it becomes clear at intermediate and strong coupling. In the opposite regime of strong coupling U t, we have already seen that the Hubbard model reduces to the Heisenberg model at low energy. In this regime, the N´eel temperature is proportional to JAF , with quantum ﬂuctuations signiﬁcantly reducing TN /JAF from its mean-ﬁeld value: numerical simulations [25] yield TN /JAF 0.957 on the cubic lattice. Hence, TN /t becomes small (as ∼ t/U ) in strong coupling. In between these two regimes, TN reaches a maximum value (ﬁg. 8). On ﬁg. 4, we have indicated the two regimes corresponding to spin-density wave and Heisenberg antiferromagnetism, in the (V0 /ER , as /d)-plane. In fact, the crossover between these two regimes is directly equivalent to the BCS-BEC crossover for an attractive interaction. For one particle per site, and a bipartite lattice, the Hubbard model with U > 0 maps onto the same model with U < 0 under the particle-hole transformation (on only one spin species):

(42)

† ci↑ → : ci↑ , ci↓ → (−1)i : ci↓

with (−1)i = +1 on the A-sublattice and = −1 on the B-sublattice. The spin density wave (weak coupling) regime corresponds to the BCS one and the Heisenberg (strongcoupling) regime to the BEC one. 6. – Adiabatic cooling: Entropy as a thermometer As discussed above, the N´eel ordering temperature is a rather low scale as compared to the bandwidth. Considering the value of TN at maximum and taking into account the appropriate range of V0 /ER and the constraints on the Hubbard model description, one would estimate that temperatures on the scale of ∼ 10−2 ER must be reached. This is at ﬁrst sight a bit deceptive, and one might conclude that the prospects for cooling down to low enough temperatures to reach the antiferromagnetic Mott insulator are not so promising.

496

A. Georges

In ref. [13] however, we have argued that one should in fact think in terms of entropy rather than temperature, and that interaction eﬀects in the optical lattice lead to adiabatic cooling mechanisms which should help. Consider the entropy per particle of the homogeneous half-ﬁlled Hubbard model: this is a function s(T, U ) of the temperature and coupling (4 ). The entropy itself is a good thermometer since it is an increasing function of temperature (∂s/∂T > 0). Assuming that an adiabatic process is possible, the key point to reach the AF phase is to be able to prepare the system in a state which has a smaller entropy than the entropy at the N´eel transition, i.e. along the critical boundary: (43)

sN (U ) ≡ s (TN (U ), U ) .

It is instructive to think of the behaviour of this quantity as a function of U . At weak coupling (spin-density wave regime), sN (U ) is expected to be exponentially small. In contrast, in the opposite Heisenberg regime at large U/t, sN will reach a ﬁnite value sH , which is the entropy of the quantum Heisenberg model at its critical point. sH is a pure number which depends only on the speciﬁc lattice of interest. Mean-ﬁeld theory of the Heisenberg model yields sH = ln 2, but quantum ﬂuctuations will reduce this number. In [13], this reduction was estimated to be of order 50% on the cubic lattice, i.e. sH ln 2/2, but a precise numerical calculation would certainly be welcome. How does sN evolve from weak to strong coupling? A rather general argument suggests that it should go through a maximum smax > sH . In order to see this, we take a derivative of sN (U ) with respect to coupling, observing that (44)

∂p 2 ∂s =− . ∂U ∂T

In this expression, p 2 is the probability that a given site is doubly occupied: p 2 ≡ ni↑ ni↓ . This relation stems from the relation between entropy and free-energy: s = −∂f /∂T and ∂f /∂U = p 2 Hence, one obtains (45)

/ c(TN ) dTN ∂p 2 // dsN = − dU TN dU ∂T /T =TN

in which c(T, U ) is the speciﬁc heat per particle: c = T ∂s/∂T . If only the ﬁrst term was present on the r.h.s of this equation, it would imply that sN is maximum exactly at the value of the coupling where TN is maximum (note that c(TN ) is ﬁnite (α < 0) for the 3D Heisenberg model). Properties of the double occupancy discussed below show that the second term on the r.h.s has a similar variation than the ﬁrst one. These considerations suggest that sN (U ) does reach a maximum value smax at intermediate (4 ) The entropy depends only on the ratios T /t and U/t: here we express for simplicity the temperature and coupling strength in units of the hopping amplitude t.

497

Condensed-matter physics with light and atoms: etc. 1

PM smax

sH

s

0.5

AF

0

0

10

20

U/t Fig. 9. – Schematic phase diagram of the one-band Hubbard model at half ﬁlling, as a function of entropy and coupling constant. The marked dots are from a DMFT calculation (in which case sH = ln 2), but the shape of the critical boundary is expected to be general (with sH < ln 2 reduced by quantum ﬂuctuations).

coupling, in the same range of U where TN reaches a maximum. Hence, sN (U ) has the general form sketched on ﬁg. 9. This ﬁgure can be viewed as a phase diagram of the half-ﬁlled Hubbard model, in which entropy itself is used as a thermometer, a very natural representation when addressing adiabatic cooling. Experimentally, one may ﬁrst cool down the gas (in the absence of the optical lattice) down to a temperature where the entropy per particle is lower than smax (this corresponds to T /TF < smax /π 2 for a trapped ideal gas). Then, by branching on the optical lattice adiabatically, one could increase U/t until one particle per site is reached over most of the trap: this should allow to reach the antiferromagnetic phase. Assuming that the timescale for adiabaticity is simply set by the hopping, we observe that typically /t ∼ 1 ms. The shape of the isentropic curves in the plane (U/t, T /t), represented on ﬁg. 8, can also be discussed on the basis of eq. (45). Taking a derivative of the equation deﬁning the isentropic curves: s(Ti (U ), U ) = const, one obtains (46)

c(Ti )

∂p 2 ∂Ti = Ti |T =Ti . ∂U ∂T

The temperature dependence of the probability of double occupancy p 2 (T ) has been studied in details using DMFT (i.e. in the mean-ﬁeld limit of large dimensions). When U/t is not too large, the double occupancy ﬁrst decreases as temperature is increased from T = 0 (indicating a higher degree of localisation), and then turns around and grows again. This apparently counter-intuitive behavior is a direct analogue of the Pomeranchuk eﬀect in liquid Helium 3: since the (spin-) entropy is larger in a localised state than when the fermions form a Fermi-liquid (in which s ∝ T ), it is favorable to increase the degree of localisation upon heating. The minimum of p 2 (T ) essentially coincides with

498

A. Georges

the quasiparticle coherence scale TF (U ): the scale below which coherent (i.e. long-lived) quasi-particles exist and Fermi liquid theory applies (see sect. 8). Mott localisation implies that TF (U ) is a rapidly decreasing function of U/t (see ﬁg. 8). The “Pomeranchuk cooling” phenomenon therefore applies only as long as TF > TN , and hence when U/t is not too large. For large U/t, Mott localisation dominates for all temperatures T < U and suppresses this eﬀect. Since ∂p 2 /∂T < 0 for T < TF (U ), while ∂p 2 /∂T > 0 for T > TF (U ), eq. (46) implies that the isentropic curves of the half-ﬁlled Hubbard model (for not too high values of the entropy) must have a negative slope at weak to intermediate coupling, before turning around at stronger coupling, as shown on ﬁg. 8. It is clear from the results of ﬁg. 8 that, starting from a low enough initial value of the entropy per site, adiabatic cooling can be achieved by either increasing U/t starting from a small value, or decreasing U/t starting from a large value (the latter requires however to cool down the gas while the lattice is already present). We emphasize that this cooling mechanism is an interaction-driven phenomenon: indeed, as U/t is increased, it allows to lower the reduced temperature T /t, normalized to the natural scale for the Fermi energy in the presence of the lattice. Hence, cooling is not simply due here to the tunneling amplitude t becoming smaller as the lattice is turned on, which is the eﬀect for non-interacting fermions discussed in ref. [9] and sect. 2 above. At weak coupling and low temperature, the cooling mechanism can be related to the eﬀective mass of quasi-particles (∝ 1/TF ) becoming heavier as U/t is increased, due to Mott localisation. Indeed, in this regime, the entropy is proportional to T /TF (U ). Hence, conserving the entropy while increasing U/t adiabatically from (U/t)i to (U/t)f will reduce the ﬁnal temperature in comparison to the initial one Ti according to: Tf /Ti = TF (Uf )/TF (Ui ). This discussion is based on the mean-ﬁeld behaviour of the probability of double occupancy p 2 (T, U ). Recently [26], a direct study in three dimensions conﬁrmed the possibility of “Pomeranchuk cooling”, albeit with a somewhat reduced eﬃciency as compared to mean-ﬁeld estimates. In two dimensions however, this eﬀect is not expected to apply, due to the rapid growth of antiferromagnetic correlations which quench the spin entropy. A ﬁnal note is that the eﬀect of the trapping potential has not been taken into account in this discussion, and further investigation of this eﬀect in a trap would certainly be worthwile. 7. – The key role of frustration In the previous section, we have seen that, for an optical lattice without geometrical frustration (e.g., a bipartite lattice with nearest-neighbour hopping amplitudes), the ground state of the half-ﬁlled Hubbard model is a Mott insulator with long-range antiferromagnetic spin ordering. Mott physics has to do with the blocking of density (charge) ﬂuctuations however, and spin ordering is just a consequence. It would be nice to be able to emphasize Mott physics by getting rid of the spin ordering, or at least reduce the temperature scale for spin ordering. One way to achieve this is by geometrical frustration of the lattice, i.e. having next–nearest-neighbor hoppings (t ) as well. Indeed, such a hopping will induce a next–nearest-neighbor antiferromagnetic superexchange, which

Condensed-matter physics with light and atoms: etc.

499

Fig. 10. – Laser setup (top) proposed in ref. [28] to realize a trimerized Kagome optical lattice (bottom). Figure adapted from [28].

obviously leads to a frustrating eﬀect for the antiferromagnetic arrangement of spins on each triangular plaquette of the lattice. It is immediately seen that inducing next–nearest-neighbour hopping along a diagonal link of the lattice requires a non-separable optical potential however. Indeed, in a separable potential, the Wannier functions are products over each coordinate axis: *D W (r − R) = i=1 wi (ri − Ri ). The matrix elements of the kinetic energy i 2 ∇2i /2m between two Wannier functions centered at next–nearest-neighbor sites along a diagonal link thus vanish because of the orthogonality of the wi ’s between nearest neighbors. Engineering the optical potential such as to obtain a desired set of tight-binding parameters is an interesting issue which I shall not discuss in details in these notes however. A classic reference on this subject is the detailed paper by Petsas et al. [27]. Recently, Santos et al. [28] demonstrated the possibility of generating a “trimerized” Kagome lattice, a highly frustrated two-dimensional lattice, with a tunable ratio of the intra-triangle to inter-triangle exchange (ﬁg. 10). . 7 1. Frustration can reveal “genuine” Mott physics. – As mentioned above, frustration can help revealing Mott physics by pushing spin ordering to lower temperatures. One of the possible consequences is the appearance of a genuine (ﬁrst-order) phase transition at ﬁnite temperature between a metallic (itinerant) phase at smaller U/t and a paramagnetic Mott insulating phase at large U/t, as depicted in ﬁg. 7. Such a transition is indeed found

500

A. Georges

within dynamical mean-ﬁeld theory (DMFT), i.e. in the limit of large lattice connectivity, for frustrated lattices. A ﬁrst-order transition is observed in real materials as well (e.g. in V2 O3 , cf. ﬁg. 6) but in this case the lattice degrees of freedom also participate (although the transition is indeed electronically driven). There are theoretical indications that, in the presence of frustration, a ﬁrst order Mott transition at ﬁnite temperature exists for a rigid lattice beyond mean-ﬁeld theory (see, e.g., [29]), but no solid proof either. In solid-state physics, it is not possible to prevent electronic instabilities to couple to lattice degrees of freedom, hence the experimental demonstration of this is impossible. This is a question that ultra-cold atomic systems might help answering. The ﬁrst-order transition line ends at a second-order critical endpoint: there is indeed no symmetry distinction between a metal and an insulator at ﬁnite temperature and it is logical that one can then ﬁnd a continuous path from one to the other around the critical point. The situation is similar to the liquid-gas transition, and in fact it is expected that this phase transition is in the same universality class: that of the Ising model (this has been experimentally demonstrated for V2 O3 [30]). A qualitative analogy with the liquid-gas transition can actually been drawn here: the Mott insulating phase has very few doubly occupied, or empty, sites (cf. the cartoons in ﬁg. 6) and hence corresponds to a low-density or gas phase (for double occupancies), while the metallic phase has many of them and corresponds to the higher-density liquid phase. One can also ask whether it is possible to stabilize a paramagnetic Mott phase as

Fig. 11. – Ground-state phase diagram of the two-dimensional Hubbard model with nearestneighbor and next nearest-neighbor hopping, as obtained in ref. [31] from the “path-integral renormalization group method”. A non-magnetic Mott insulator (NMI) is stabilized for large enough frustration t /t and intermediate coupling U/t. A similar model with n.n.n hopping along only one of the diagonals (anisotropic triangular lattice) was studied in ref. [32] using a cluster extension of DMFT, and an additional d-wave superconducting phase was found in this study.

501

Condensed-matter physics with light and atoms: etc.

the ground state, i.e. suppress spin ordering down to T = 0. Several recent studies of frustrated two-dimensional models found this to happen at intermediate coupling U/t and for large enough frustration t /t, with non-magnetic insulating and possibly d-wave superconducting ground states arising (ﬁg. 11). . 7 2. Frustration can lead to exotic quantum magnetism. – The above question of suppressing magnetic ordering down to T = 0 due to frustration can also be asked in a more radical manner by considering the strong-coupling limit U/t → ∞. There, charge (density) ﬂuctuations are entirely suppressed and the Hubbard model reduces to a quantum Heisenberg model. The question is then whether quantum ﬂuctuations of the spin degrees of freedom only, can lead to a ground state without long-range order. Studying this issue for frustrated Heisenberg models or related models has been a very active ﬁeld of theoretical condensed matter physics for the past 20 years or so, and I simply direct the reader to existing reviews on the subject, e.g., ref. [33, 34]. Possible disordered phases are valence bond crystals, in which translational symmetry is broken and the ground state can be qualitatively thought of as a speciﬁc paving of the lattice by singlets living on bonds. Another, more exotic, possibility is that the ground state can be thought of as a resonant superposition of singlets (a sort of giant benzene molecule): this is the “resonating valence bond” idea proposed in the pioneering work of Anderson and Fazekas. There are a few examples of this, one candidate being the Heisenberg model on the kagome lattice (ﬁg. 10). Naturally, obtaining such unconventional states in ultra-cold atomic systems, and more importantly being able to measure the spin-spin correlations and excitation spectrum experimentally would be fascinating. One last remark in this respect, which establishes an interesting connection between exotic quantum magnetism and Bose condensation. A spin-(1/2) quantum Heisenberg model with a ground state which is not ordered and does not break translational symmetry (e.g., a resonating valence bond ground state) is analogous, in a precise formal sense, to a speciﬁc interacting model of hard-core bosons which would remain a normal liquid (not a crystal, not a superﬂuid) down to T = 0. Hence, somewhat ironically, an unconventional ground state means, in the context of quantum magnetism, preventing Bose condensation. To see this, we observe that a quantum spin-1/2 can be represented with a hard-core boson operator bi as Si+ = b†i

Si− = bi

Siz = b†i bi −

1 2

with the constraint that at most one boson can live on a given site b†i bi = 0, 1 (inﬁnite hard-core repulsion). The anisotropic Heisenberg (XXZ) model then reads H = J⊥

† † [bi bj + b†j bi ] + Jz (bi bi − 1/2)(b†j bj − 1/2). ij

ij

Hence, it is an inﬁnite-U bosonic Hubbard model with an additional interaction between nearest-neighbor sites (note that dipolar interactions can generate those for real bosons).

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The superﬂuid phase for the bosons correspond to a phase with XY-order in the spin language, a crystalline (density-wave) phase with broken translational symmetry to a phase with antiferromagnetic ordering of the S z components, and a normal Bose ﬂuid to a phase without any of these kinds of orders. 8. – Quasi-particle excitations in strongly correlated fermion systems, and how to measure them . 8 1. Response functions and their relation to the spectrum of excitations. – Perhaps even more important than the nature of the ground state of a many-body system is to understand the nature of the excited states, and particularly of the low-energy excited states (i.e. close to the ground state). Those are the states which control the response of the system to a weak perturbation, which is what we do when we perform a measurement without disturbing the system too far out of equilibrium (5 ). When the perturbation is weak, linear response theory can be used, and in the end what is measured is the ˆ correlation function of some observable (i.e. of some operator O): ˆ t) O ˆ † (r , t ). CO (r, r ; t, t ) = O(r,

(47)

In this expression, the operators evolve in the Heisenberg representation, and the brackets denote either an average in the ground-state (many-body) wave function (for a measurement at T = 0) or, at ﬁnite temperature, a thermally weighted average with the equilibrium Boltzmann weight. How the behaviour of this correlation function is controlled by the spectrum of excited states is easily understood by inserting a complete set of states in the above expression (in order to make the time evolution explicit) and obtaining the following spectral representation (given here at T = 0 for simplicity): (48)

CO (r, r ; t, t ) =

ˆ ˆ † e− (En −E0 )(t−t ) Φ0 |O(r)|Φ n Φn |O(r ) |Φ0 . i

n

In this expression, Φ0 is the ground-state (many-body) wave function, and the summation is over all admissible many-body excited states (i.e. having non-zero matrix elements). A key issue in the study of ultra-cold atomic systems is to devise measurement techniques in order to probe the nature of these many-body states. In many cases, one can resort to spectroscopic techniques, quite similar in spirit to what is done in condensedˆ we want to access is matter physics. This is the case, for example, when the observable O a local observable such as the local density or the local spin-density. Light (possibly polarized) directly couples to those, and light scattering is obviously the tool of choice in the context of cold atomic systems. Bragg scattering [35] can be indeed used to measure the density-density dynamical correlation function ρ(r, t)ρ(r , t ) and polarized light also (5 ) Ultra-cold atomic systems, as already stated in the introduction, also oﬀer the possibility of performing measurements far from equilibrium quite easily, which is another fascinating story.

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allows one to probe [36] the spin-spin response S(r, t)S (r , t ). In condensed matter physics, analogous measurements can be done by light or neutron scattering. One point is worth emphasizing here, for condensed-matter physicists. In condensedmatter physics, we are used to thinking of visible or infra-red light (not X-ray!) as a zeromomentum probe, because the wavelength is much bigger than inter-atomic distances. This is not the case for atoms in optical lattices! For those, the lattice spacing is set by the wavelength of the laser, hence lasers in the same range of wavelength can be used to sample the momentum-dependence of various observables, with momentum transfers possibly spanning the full extent of the Brillouin zone. Other innovative measurements techniques of various two-particle correlation functions have recently been proposed and experimentally demonstrated in the context of ultra-cold atomic systems, some of which are reviewed elsewhere in this set of lectures, e.g. noise correlation measurements [37-39], or periodic modulations of the lattice [40,41]. The simplest examples we have just discussed involve two-particle correlation functions (density-density, spin-spin), and hence probe at low energy the spectrum of particlehole excitations, i.e. excited states Φn which are coupled to the ground state via an operator conserving particle number. In contrast, one may want to probe one-particle correlation functions, which probe excited states of the many-body system with one atom added to it, or one atom removed, i.e. coupled to the ground state via a single particle process. Such a correlation function (also called the single-particle Green’s function G1 ) reads Tt ψ(r, t)ψ † (r , t ) ≡ i G1 (r, r ; t, t ),

(49)

in which Tt denotes time ordering. The corresponding spectral decomposition involves the one-particle spectral function (written here, for simplicity, for a homogeneous system —so that crystal momentum is a good quantum number— and at T = 0): −1 2 A(k, ω) = (50) |ΦN |c k |ΦN n 0 | δ(ω + μ + En − E0 ) (ω < 0), n

=

2 +1 † |ΦN |c k |ΦN n 0 | δ(ω + μ + E0 − En ) (ω > 0).

n

The spectral function is normalized to unity for each momentum, due to the anticommutation of fermionic operators: +∞ (51) A(k, ω) dω = 1. −∞

As explained in the next section, the momentum and frequency dependence of this quantity contains key information about the important low-energy excitations of fermionic systems (hole-like, i.e. corresponding to the removal of one atom, for ω < 0, and particle-like for ω > 0). Let us note that, for Bose systems with a ﬁnite condensate density n0 , the two-particle correlators are closely related to the one-particle correlators via terms such as n0 ψ † (r, t) ψ(r , t ). By contrast, in Fermi systems the distinction between one- and two-particle correlators is essential.

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Fig. 12. – Raman process: transfer from an internal state α to another internal state β through an excited state γ. The momentum-resolved spectral function is schematized, consisting of a quasiparticle peak and an incoherent background. From ref. [45].

A particular case is the equal-time correlator, ψ † (r, t)ψ(r , t), i.e. the one-body density matrix, whose Fourier transform is the momentum distribution in the ground state: (52)

N (k) = Φ0 |c†k ck |Φ0 =

0

−∞

A(k, ω)dω.

For ultra-cold atoms, this can be measured in time-of-ﬂight experiments. Conversely, rf-spectroscopy experiments [42] give some access to the frequency dependence of the one-particle spectral function, but not to its momentum dependence. . 8 2. Measuring one-particle excitations by stimulated Raman scattering. – In condensed matter physics, angle-resolved photoemission spectroscopy (ARPES) provides a direct probe of the one-particle spectral function (for a pedagogical introduction, see [43]). This technique has played a key role in revaling the highly unconventional nature of single-particle excitations in cuprate superconductors [44]. It consists in measuring the energy and momentum of electrons emitted out of the solid exposed to an incident photon beam. In the simplest approximation, the emitted intensity is directly proportional to the single-electron spectral function (multiplied by the Fermi function and by some matrix elements). In ref. [45], it was recently proposed to use stimulated Raman spectroscopy as a probe of one-particle excitations, and of the frequency and momentum dependence of the spectral function, in a two-component mixture of ultracold fermionic atoms in two internal states α and α . Stimulated Raman spectroscopy has been considered previously in the context of cold atomic gases, both as an outcoupling technique to produce an atom laser [46] and as a measurement technique for bosons [47-50] and fermions [51, 52]. In the Raman process of ﬁg. 12, atoms are transferred from α into another internal state

505

Condensed-matter physics with light and atoms: etc.

β = α, α , through an intermediate excited state γ, using two laser beams of wave vectors k1,2 and frequencies ω1,2 . If ω1 is suﬃciently far from single-photon resonance to the excited γ state, we can neglect spontaneous emission. The total transfer rate to state β can be calculated [47-49] using the Fermi golden rule and eliminating the excited state: R(q, Ω) = |C| n1 (n2 + 1) 2

∞ −∞

dt

drdr ei[ Ω t−q·(r−r )] gβ (r, r ; t)ψα† (r, t)ψα (r , 0).

Here q = k1 − k2 and Ω = ω1 − ω2 + μ with μ the chemical potential of the interacting gas, and n1,2 the photon numbers present in the laser beams. Assuming that no atoms are initially present in β and that the scattered atoms in β do not interact with the atoms in the initial α, α states, the free propagator for β-state atoms in vacuum is to be taken: gβ (r, r ; t) ≡ 0β | ψβ (r, t)ψβ† (r , 0)|0β . For a uniform system, the transfer rate can be related to the spectral function A(k, ω) of atoms in the internal state α by [48] (53) R(q, Ω) ∝ dk nF (εkβ − Ω) A(k − q, εkβ − Ω) in which the Green’s function has been expressed in terms of the spectral function and the Fermi factor nF . In the presence of a trap, the conﬁning potential can be treated in the local density approximation by integrating the above expression over the radial coordinate, with a position-dependent chemical potential. From (53), the similarities and diﬀerences with ARPES are clear: in both cases, an atom is eﬀectively removed from the interacting gas, and the signal probes the spectral function. In the case of ARPES, it is directly proportional to it, while here an additional momentum integration is involved if the atoms in state β remain in the trap. One the other hand, in the present context, one can in principle vary the momentum transfer q and regain momentum resolution in this manner. Alternatively, one can cut oﬀ the trap and perform a time of ﬂight experiment [45], in which case the measured rate is directly proportional to nF (εkβ − Ω) A(k − q, εkβ − Ω), in closer analogy to ARPES. Varying the frequency shift Ω then allows to sample diﬀerent regions of the Brillouin zone [45]. . 8 3. Excitations in interacting Fermi systems: A crash course. – Most interacting fermion systems have low-energy excitations which are well-described by “Fermi liquid theory” which is a low-energy eﬀective theory of these excitations. In this description, the low-energy excitations are built out of quasi-particles, long-lived (coherent) entities carrying the same quantum numbers than the original particles. There are three key quantities characterizing the quasi-particle excitations: – Their dispersion relation, i.e. the energy ξk (measured from the ground-state energy) necessary to create such an excitation with (quasi-) momentum k. The interacting system possesses a Fermi surface (FS) deﬁned by the location in momentum space on which the excitation energy vanishes: ξkF = 0. Close to a given point on the FS, the quasiparticle energy vanishes as: ξk ∼ vF (kF ) · (k − kF ) + · · · , with vF the local Fermi velocity at that given point of the Fermi surface.

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– The spectral weight Zk ≤ 1 carried by these quasi-particle excitations, in comparison to the total spectral weight (= 1, see above) of all one-particle excited states of arbitrary energy and ﬁxed momentum. – Their lifetime Γ−1 k . It is ﬁnite away from the Fermi surface, as well as at ﬁnite temperature. The quasi-particle lifetime diverges however at T = 0 as k gets close to the Fermi surface. Within Fermi liquid theory, this happens in a speciﬁc manner (for phase-space reasons), as Γk ∼ ξk2 . This insures the overall coherence of the description in terms of quasiparticles, since their inverse lifetime vanishes faster than their energy. Typical signatures of strong correlations are the following eﬀects (not necessarily occurring simultaneously in a given system): i) strongly renormalized Fermi velocities, as compared to the non-interacting (band) value, corresponding, e.g., to a large interactioninduced enhancement of the eﬀective mass of the quasi-particles, ii) a strongly suppressed quasi-particle spectral weight Zk 1, possibly non-uniform along the Fermi surface, iii) short quasi-particle lifetimes. These strong deviations from the non-interacting system can sometimes be considerable: the “heavy fermion” materials for example (rareearth compounds) have quasiparticle eﬀective masses which are several hundred times bigger than the mass from band theory, and in spite of this are mostly well described by Fermi liquid theory. The quasi-particle description applies only at low energy, below some characteristic energy (and temperature) scale TF , the quasiparticle coherence scale. Close to the Fermi surface, the one-particle spectral function displays a clear separation of energy scales, with a sharp coherent peak carrying spectral weight Zk corresponding to quasi-particles (a peak well resolved in energy means long-lived excitations), and an “incoherent” background carrying spectral weight 1 − Zk . A convenient form to have in mind (ﬁg. 12) is: (54)

A(k, ω) Zk

Γk + Ainc (k, ω). π[(ω − ξk )2 + Γ2k ]

Hence, measuring the spectral function, and most notably the evolution of the quasiparticle peak as the momentum is swept through the Fermi surface, allows one to probe the key properties of the quasi-particle excitations: their dispersion (position of the peak), lifetime (width of the peak) and spectral weight (normalized to the incoherent background, when possible), as well of course as the location of the Fermi surface of the interacting system in the Brillouin zone. In [45], it was shown that the shape of the Fermi surface, as well as some of the quasi-particle properties can be determined, in the cold atoms context, from the Raman spectroscopy described above. For the pioneering experimental determination of Fermi surfaces in weakly or non-interacting fermionic gases in optical lattices, see [8]. What about the “incoherent” part of the spectrum (which in a strongly correlated system may carry most of the spectral weight. . . )? Close to the Mott transition, we

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expect at least one kind of well-deﬁned high-energy excitations to show up in this incoherent spectrum. These are the excitations which consist in removing a particle from a site which is already occupied, or adding a particle on such a site. The energy difference separating these two excitations is precisely the Hubbard interaction U . These excitations, which are easier to think about in a local picture in real-space (in contrast to the wave-like, quasiparticle excitations), form two broad dispersing peaks in the spectral functions: the so-called Hubbard “bands”. In the mean-ﬁeld (DMFT) description of interacting fermions and of the Mott transition, the quasi-particle weight Z is uniform along the Fermi surface. Close to the Mott transition, Z vanishes and the eﬀective mass (m /m = 1/Z in this theory) of quasiparticles diverges. The quasiparticle coherence scale is TF Z TF , with TF the Fermi energy (∼ bandwidth) of the non-interacting system: this coherence scale also becomes very small close to the transition, and Hubbard bands carry most of the spectral weight in this regime. . 8 4. Elusive quasi-particles and nodal-antinodal dichotomy: The puzzles of cuprate superconductors. – The cuprate superconductors, which are quasi two-dimensional doped Mott insulators, raise some fundamental questions about the description of excitations in strongly interacting fermion systems. In the “normal” (i.e. non-superconducting) state of these materials, strong departure from Fermi liquid theory is observed. Most notably, at doping levels smaller than the optimal doping (where the superconducting Tc is maximum), i.e. in the so-called “underdoped” regime: – Reasonably well-deﬁned quasiparticles are only observed close to the diagonals of the Brillouin zone, i.e. close to the “nodal points” of the Fermi surface where the superconducting gap vanishes. Even there, the lifetimes are shorter and appear to have a diﬀerent energy and temperature dependence than that of Fermi liquid theory. – In the opposite regions of the Fermi surface (“antinodal” regions), the spectral function shows no sign of a quasiparticle pleak in the normal state. Instead, a very broad lineshape is found in ARPES, whose leading edge is not centered at ω = 0, but rather at a ﬁnite energy scale. The spectral function appears to have its maximum away from the Fermi surface, i.e. the density of low-energy excitations is strongly depleted at low energy: this is the “pseudo-gap” phenomenon. The pseudo-gap shows up in many other kinds of measurements in the under-doped regime. Hence, there is a strong dichotomy between the nodal and antinodal regions in the normal state. The origin of this dichotomy is one of the key issues in the ﬁeld. One possibility is that the pseudo-gap is due to a hidden form of long-range order which competes with superconductivity and is responsible for suppressing excitations except in nodal regions. Another possibility is that, because of the proximity to the Mott transition in such low-dimensional systems, the quasiparticle coherence scale (and most likely

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Fig. 13. – Illustration of the dichotomy between “nodal” and “antinodal” regions of the Fermi surface, as observed in cuprate superconductors. Colour coding corresponds to increasing intensity of the quasiparticle peak. Such eﬀects could be revealed in cold atomic systems by stimulated Raman spectroscopy measurements, as proposed in ref. [45].

also the quasiparticle weight) varies strongly along the Fermi surface, hence suppressing quasiparticles in regions where the coherence scale is smaller than temperature. This nodal-antinodal dichotomy is illustrated in ﬁg. 13. This ﬁgure has actually been obtained from a simulated intensity plot of the Raman rate (53), using a phenomenological form of the spectral function appropriate for cuprates. It is meant to illustrate how future experiments on ultra-cold fermionic atoms in two-dimensional optical lattices might be able to address some of the fundamental issues in the physics of strongly correlated quantum systems. ∗ ∗ ∗ I am grateful to C. Salomon, M. Inguscio and W. Ketterle for the opportunity to lecture at the wonderful Varenna school on “Ultracold Fermi Gases”, to J. Dalibard and C. Salomon at the Laboratoire Kastler-Brossel of Ecole Normale Sup´erieure for stimulating my interest in this ﬁeld and for collaborations, and to M. Capone, I. Carusotto, T.-L. Dao, S. Hassan, O. Parcollet and F. Werner for collaborations related to the topics of these lectures. I also acknowledge useful discussions with I. Bloch, F. Chevy, E. Demler, T. Esslinger and Th. Giamarchi. My work is supported by the Centre National de la Recherche Scientiﬁque, by Ecole Polytechnique and by the Agence Nationale de la Recherche under contract “GASCOR”.

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Quantum information processing: Basic concepts and implementations with atoms J. I. Cirac Max-Planck Institute for Quantum Optics, Hans-Kopfermannstr. 1 D-85748 Garching, Germany

We review some of the basic concepts in quantum information processing, including both the gate-based and the measurement-based quantum computation set-ups. We also show how one can perform quantum simulations using a quantum computer. Finally, we review how one can implement some of these ideas using neutral atoms and trapped ions. 1. – Introduction Quantum Information is a rapidly developing area of current research where the goal is to use some of the intriguing laws of Quantum Mechanics to process and transmit information in a more eﬃcient way. In particular, concepts like superpositions, entanglement or collapse of the wave function are exploited in order to carry out tasks which would not be possible otherwise [1]. For instance, using these features of Quantum Mechanics, it is possible to send secret information making sure that nobody is listening to the communication, something which has evident applications in cryptography. It is also possible, at least in principle, to build a quantum computer which would use superpositions and entangled states to perform operations with a number of basic steps which is much smaller than the ones required with standard algorithms, something which also has a large variety of applications. Thus, currently there is a strong eﬀort worldwide to ﬁnd out which are the tasks that a quantum system may perform better than classical systems, as well as to investigate ways of implementing small prototypes of quantum computers with diﬀerent physical systems [2]. c Societ` a Italiana di Fisica

511

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Atomic systems are among the most promising candidates where to observe the main features which form the basis of a quantum computer. They can be very well isolated from the external world and can also be manipulated using laser, electric and magnetic ﬁelds. This extremely high degree of control which has been achieved with atomic system is one of the main requirements to build a quantum computer. Among the atomic systems, trapped ions and neutral atoms stand out (see, for example, ref. [3]). In particular, recent experiments with atoms in optical lattices, dipole and magnetic traps have already displayed single-particle control and even entanglement, and there are at the moment several experimental groups trying to perform quantum gates (the building blocks of quantum computation) with those systems. In this paper we will introduce the basic concepts of quantum computation and show how it should be possible to create small prototypes of a quantum computer using neutral atoms. We will mostly concentrate on trapped ions and atoms loaded in optical lattices, since those are the most advanced systems at the moment. However, most of the techniques explained here can be easily extended to other atomic set-ups. We will assume that the reader is familiar with the basic principles of Quantum Mechanics, although we will brieﬂy mention them at the beginning in order to deﬁne our notation. Because of lack of space (and time), we will leave out some important aspects of quantum computation, like the eﬀects of decoherence or methods to circumvent the problems it causes. 2. – Basic notions in quantum information In this section we introduce some of the basic concepts of quantum information theory [4]. We will basically review the postulates of Quantum Mechanics using the language of quantum information. . 2 1. Quantum states. – A qubit (quantum bit) is a two-level system, which can be in state |0, |1, or in any linear superposition of them c0 |0 + c1 |1, with |c0 |2 + |c1 |2 = 1. There are many physical situations where the two-level description suﬃces. For example: – Two-level atom: An atom has inﬁnite internal levels (|n, l, m, ms for the hydrogen atom, for example). However, when the dynamics is such that only two of them are occupied, one can consider only those two. For example, in hydrogen we can take |0 ≡ |1, 0, 0, 1/2 and |1 ≡ |2, 1, 1, 1/2. – Two polarizations of a photon: Consider the states of the electromagnetic ﬁeld of a ﬁxed wave vector k that contain one photon. There are two, corresponding to two polarizations. If the processes involving this ﬁeld do not change the frequency of the photons, one can describe them in terms of two levels, corresponding to two orthogonal polarizations. – spin-(1/2) particle: An electron, proton, neutron, etc., has spin 1/2, and therefore can be described as a two-level system.

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In quantum information we will often have a collection of qubits. For example, one can have several atoms, several photons, several electrons, etc. The Hilbert space corresponding to N qubits is H2N = H2 ⊗ H2 ⊗ . . . ⊗ H2 . An orthonormal basis (the so-called computational basis) in that Hilbert space is (1a)

|0 = |0, 0, . . . , 0 = |01 ⊗ |02 ⊗ . . . |0N ,

(1b)

|1 = |0, 0, . . . , 1 = |01 ⊗ |02 ⊗ . . . |1N ,

(1c)

|2 − 1 = |1, 1, . . . , 1 = |11 ⊗ |12 ⊗ . . . |1N .

... N

As it is shown here, we will sometimes denote the states of the computational basis in terms of the states of the qubits interpreted as a number in binary notation. Thus, a general pure state of the N qubits can be written as |Ψ =

(2)

N 2 −1

cx |x,

with

x=0

N 2 −1

|cx |2 = 1.

x=0

When we have more than on qubit, we can distinguish two kinds of pure states: – Product states: Those are states of the form |Ψ = |Ψ1 1 ⊗ |Ψ2 2 ⊗ . . . ⊗ |ΨN N . That is, they are states that can be factored. For example, all the states in the computational basis are product states. – Entangled states: Those that cannot be written as product states. For example, if N = 2 the Bell states (3)

1 |Ψ± = √ (|0, 1 ± |1, 0), 2

1 |Φ± = √ (|0, 0 ± |1, 1) 2

are all entangled. Another example of entangled states of N two-level systems are the so-called GHZ states (4)

1 |Ψ = √ (|0, 0, . . . , 0 − |1, 1, . . . , 1). 2

All the states (pure or mixed) of a set of qubits can be described in terms of a density operator ρ, fulﬁlling ρ = ρ† , Tr(ρ) = 1 and ρ ≥ 0. A state is pure if and only if the corresponding density operator can be written as ρ = |ΨΨ| = ρ2 . On the other hand, one can also classify mixed states according to whether they are entangled or not. However, the situation is a bit subtler and will not be discussed here [5]. If one is interested in the properties of one of the systems alone, say the ﬁrst qubit, all the information about that system is contained in its reduced density operator, which is obtained after partial tracing the other systems: ρ1 = Tr23...N (|ΨΨ|). This operator represents a pure state if and only if the ﬁrst qubit is not entangled to the rest.

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. 2 2. Observables and measurement. – In Quantum Mechanics, observables are represented in terms of Hermitian operators. The possible outcomes of a measurement are given by the corresponding eigenvalues, whereas the state of the system after the measurement is modiﬁed. For example, consider an observable whose corresponding operator O has non-degenerate eigenvalues oi and the corresponding eigenstates are |oi . If our system is in a state |Ψ, the probability that we obtain oi when measuring the observable is Pi = |oi |Ψ|2 . The state after this outcome will be |oi . We will often use the term “measure in the basis {|oi }”, meaning that we measure an observable of this sort. More generally, if the state of the system is represented by a density operator ρ and one of the eigenvalues oi is degenerate (i.e. there are several eigenstates corresponding to it, |oi,α ), the probability of obtaining oi is Pi = Tr(ρQi ), where Qi = α |oi,α oi,α | is the projector onto the subspace spanned by the degenerate eigenstates. The state after obtaining the outcome oi is ρi = Qi ρQi /Pi . We will say that we perform an incomplete measurement when the spectrum of the corresponding observable is degenerate. On the other hand, suppose we prepare a set of N qubits in a given state ρ, and measure an observable O. Then we repeat the preparation and the measurement many times. The averaged value of the outcomes of the measurements is given by the expectation value O = Tr(Oρ). All observables of a single qubit can be expressed in terms of the following operators: (5)

P0 = |00|,

σ+ = |10|,

P1 = |11|,

σ− = |01|.

The ﬁrst two are projectors (i.e., Pi2 = Pi ), whereas the second two are called excitation (σ+ ) and de-excitation (σ− ) operators, respectively. Instead of using these operators, sometimes it is more convenient to use the identity operator I = P0 + P1 , together with the Pauli operators (6)

σx = σ+ + σ− ,

σy = −i(σ+ − σ− ),

σz = P1 − P0 .

These operators satisfy angular momentum commutation relations (e.g. [σx , σy ] = 2iσz ) as well as σi2 = 1. Therefore their eigenvalues are ±1. The eigenstates can be easily calculated: σα |0α = −|0α , and σα |1α = |1α , where α = x, y, z, and (7a) (7b) (7c)

1 1 |0x = √ (|0 − |1), |1x = √ (|0 + |1), 2 2 1 1 |0y = √ (|0 + i|1), |1y = √ (|0 − i|1), 2 2 |0z = |0, |1z = |1.

The set A = {Ai , i = 0, . . . , 3} ≡ {1, σx , σy , σz } forms an orthonormal basis of the Hilbert space of linear operators acting in H2 , L(H2 ). That is, any linear operator can be written as a linear combination of the elements of the set, and these elements are mutually orthogonal (with the scalar product (A, B) = 12 Tr(A† B)). In the same way, we

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can construct a basis for all the operators acting on the Hilbert space of N qubits. We just have to construct all the operators of the form An = An1 ⊗ An2 ⊗ . . . ⊗ AnN . All observables can be expressed in terms of those operators. When we have several qubits, we can distinguish three kinds of measurements: – Local measurements: If a measurement is performed on system A only, this is called a local measurement on A. For example, for N = 2, if one measures the observable O = OA ⊗ 1B this would correspond to a local measurement in A. All the results of such measurements can be determined by the reduced density operators. The reason is that Tr(ρO) = TrA [TrB (ρO)] = TrA [TrB (ρ)OA ] = TrA (ρA OA ). – Correlation measurements: One can perform simultaneous local measurements on several qubits and compare the corresponding results. For example, for N = 2 one can measure OA in system A and OB in system B, and multiply the corresponding results. This is equivalent to measuring the observable O = OA ⊗ OB . Note that only for uncorrelated states O = OA OB . – Joint measurements: These are measurements that are not performed locally, i.e. they do not correspond to observables O = O1 ⊗ O2 ⊗ . . . ⊗ ON . For example, for N = 2 we can take O = σxA σyB + σyA σxB . Another important example consists in the so-called measurements in the Bell basis. They correspond to the measurement of an observable whose eigenstates are the Bell states, so that as a result of the measurement one of the Bell states is found (or equivalently, to measuring the 4 projectors onto the Bell basis states). . 2 3. Evolution. – If we have a system of isolated qubits, its dynamics are characterized by a Hamiltonian operator H. The evolution of a state can then be expressed in terms of a unitary operator U (t) (with U U † = U † U = 1) as |Ψ(t) = U (t)|Ψ(0) or ρ(t) = U (t)ρ(0)U (t)† . Unitary operators conserve scalar products (and therefore the norm); that is, if |Ψ1 is orthogonal to |Ψ2 , then U (t)|Ψ1 is orthogonal to U (t)|Ψ2 . Naturally, the evolution operator as well as the Hamiltonian can be expressed in terms of Pauli operators (the operators An deﬁned above). Depending on the form of the Hamiltonian, we can distinguish two kinds of evolution operators: – Local operators: Those are the ones corresponding to Hamiltonians of the form N H = i=1 Hi , where Hi only act on system i. In that case, since [Hi , Hj ] = 0, one can write U (t) = U1 (t) ⊗ U2 (t) ⊗ . . . ⊗ UN (t), where Ui is the evolution operators for the system i. A local evolution cannot entangle two systems, if they are initially in a product state. Similarly, it cannot disentangle two systems that are initially entangled. – Non-local operators: Those operators coming from a Hamiltonian that cannot be written as a sum of operators acting on diﬀerent systems. They are able to entangle states which are initially product states. For example, for N = 2 we can have H =

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¯ ασxA ⊗ σyB which gives the evolution operator U (t) = cos(αt) − iσxA ⊗ σyB sin(αt). h For αt = π/4, the evolution of the state |0, 0 is U (t)|0, 0 = |Φ− , i.e. it corresponds to the preparation of a Bell state. As opposed to what occurs in classical physics, according to Quantum Mechanics states cannot be cloned (copied). This can be proved as follows. Imagine we want to copy an unknown state of one two-level system |ψ = c0 |0 + c1 |1, to another two-level system, prepared in the known state |ψ0 . Thus, we have to ﬁnd a unitary transformation such that U |ψ|ψ0 = |ψ|ψ for any |ψ. Take two states |ψ1 and |ψ2 . Using this operator for both of them, and the fact that any unitary operator conserves the scalar product, we deduce that ψ1 |ψ2 = ψ1 |ψ2 2 which is always false unless ψ1 |ψ2 = 0, 1. Thus, only states that are orthogonal can be copied. . 2 4. Examples and applications. – In the following we will apply the concepts reviewed above to some tasks that naturally appear in the context of Quantum Information. . 2 4.1. Teleportation. By teleportation we deﬁne transfering an intact quantum state from one place to another, by a sender who knows neither the state to be teleported nor the location of the intended receiver [6]. The term teleportation comes from Science Fiction meaning to make a person of object disappear while an exact replica appears somewhere else. Several teleportation experiments using photons, ions, and atoms and photons have already taken place [7]. Consider two partners, Alice and Bob, located at diﬀerent places. Alice has a qubit in an unknown state |φ, and she wants to teleport it to Bob, whose location is not known. Prior to the teleportation process, Alice and Bob share two qubits in a Bell state. The idea is that Alice performs a joint measurement of the two-level system to be teleported and her particle. Due to the non-local correlations, the eﬀect of the measurement is that the unknown state appears instantaneously in Bob’s hands, except for a unitary operation which depends on the outcome of the measurement. If Alice communicates to Bob the result of her measurement, then Bob can perform that operation and therefore recover the unknown state. Let us call particle 1 that which has the unknown state |φ1 , particle 2 the member of the EPR that Alice possesses and particle 3 that of Bob. We write the state of particle 1 as |φ1 = a|01 +b|11 where a and b are (unknown) complex coeﬃcients. The state of particles 2 and 3 is the Bell state |Ψ− . The complete state of particles 1, 2 and 3 is therefore (8)

b a |Ψ123 = √ (|01 |02 |13 − |01 |12 |03 ) + √ (|11 |02 |13 − |11 |12 |03 ). 2 2

Using the Bell basis for particles 1 and 2, we can write (9)

|Ψ123 =

1 − [|Ψ 12 (−a|03 − b|13 ) + |Ψ+ 12 (−a|03 + b|13 ) + 2 +|Φ− 12 (a|13 + b|03 ) + |Φ+ 12 (a|13 − b|03 )].

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In order to teleport the state, Alice and Bob follow this procedure: 1) Alice measurement: Alice makes a joint measurement of her particles (1 and 2) in the Bell basis. 2) Alice broadcasting: Then she broadcasts (classically) the outcome of her measurement. 3) Bob restoration: Bob then applies a unitary operation to his particle to obtain |ψ3 . According to the state of the particles (9), the possible outcomes are: – With probability 1/4, Alice ﬁnds |Ψ− 12 . The state of the third particle is automatically projected onto a|03 + b|1. Thus, in this case Bob does not have to perform any operation. – With probability 1/4, Alice ﬁnds |Ψ+ 12 . The state of the third particle is automatically projected onto −a|03 + b|1. Teleportation occurs if Bob applies σz to his particle. – With probability 1/4, Alice ﬁnds |Φ− 12 . The state of the third particle is automatically projected onto a|13 + b|0. Teleportation occurs if Bob applies σx to his particle. – With probability 1/4, Alice ﬁnds |Φ+ 12 . The state of the third particle is automatically projected onto a|13 − b|0. Teleportation occurs if Bob applies σy to his particle. Note that Alice ends up with no information about her original state so that no violation of the no-cloning theorem occurs. In this sense, the state of particle 1 has been transferred to particle 3. On the other hand, there is no instantaneous propagation of information. Bob has to wait until he receives the (classical) message from Alice with her outcome. Before he receives the message, his lack of knowledge prevents him from having the state. Note that no measurement can tell him whether Alice has performed her measurement or not. Since teleportation is a linear operation applied to a state, it will also work for statistical mixtures, or in the case in which particle 1 is entangled with other particles. Finally, one can also generalize teleportation to N -level systems. . 2 4.2. Dense coding. Given a Bell state, one can prepare any other Bell state using local operations. In particular, one can prepare all four elements of the Bell basis. Thus if Alice sends the member of an EPR pair to Bob, he can apply one of four unitary operations and obtain 4 orthogonal states of the particles. Then, sending the particle back to Alice, she can know the operation performed by Bob. In this sense, Bob can send two bits of information by only acting on one particle. This process is called dense coding [8], and has been experimentally observed [9]. Dense coding is based on the following procedure. Alice has two particles in a Bell state |Ψ− 12 . Then sends one to Bob, who applies one of the following local operations (i.e. a unitary operator on particle 2): 1, σx,y,z . One can readily see that (10)

σxB |Ψ− ∝ |Φ− ,

σyB |Ψ− ∝ |Φ+ ,

σzB |Ψ− ∝ |Ψ+ .

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Thus, by choosing diﬀerent Pauli opertors (or the identity), Bob can prepare any of the state of Bell’s basis. Since these states are orthogonal, he can encode 2 bits in his action. Then he sends back the particle, and Alice measures in the Bell basis. 3. – Quantum computation A computation can be considered as a physical process that transforms an input into an output. A classical computation is that in which the physical process is based on classical laws (without coherent quantum phenomena). A quantum computation is that based on quantum laws (and in particular on the superposition principle) [10]. In quantum computation, inputs and outputs are represented by states of the system. For example, enumerating the state of a given basis as |1, |2, . . ., the number N would be represented by the N -th state of this basis. A quantum computation consists of evolving the system with a designed Hamiltonian interaction, such that the states are transformed as we want. Note that the operation that transforms input into outputs has to be unitary. For example, the operation that gives 1 if a number is odd and 2 if it is even could not be implemented: |2n + 1 → |1 and |2n → |2 (where n is an integer). This operation cannot be unitary since it does not conserve the scalar product (i.e. 1|3 = 0 but the corresponded mapped states are not orthogonal). One can, however, use an auxiliary system so that the output is written in that system while keeping the unitarity of the operation: |2n + 1 ⊗ |0 → |2n + 1 ⊗ |1 and |2n ⊗ |0 → |2n ⊗ |2 (where n is an integer). In general, if our algorithm consists of evaluating a given function f , we can design an interaction Hamiltonian such that the evolution operator transforms the input states according to the following equations: |1|0 → |1|f (1), |2|0 → |2|f (2), (11)

... |n|0 → |n|f (n).

Note that using this transformation we can, at least, do the same computations with quantum computers as with classical computers. However, with quantum computers we can do even more. We can prepare the input state that in a superposition 1 |ψ = √ |n|0, n n

(12)

k=1

1 √ |n|f (n) n n

obtaining

k=1

after a single run. In principle, all the values of f are present in this superposition. Note, however, that we do not have access to this information since if we perform a measurement we will only obtain a result (with certain probability). Nevertheless, we see that with a quantum computer we can do at least the same as with a quantum computer, . . . and even more. This property of using quantum superpositions and running only once the computer was termed by Feynman quantum parallelism.

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As in a classical computer, in a quantum computer we use several qubits, say N , in order to store a number. This is a very eﬃcient way of storing, since if we had only a single system instead, in order to store the same number we would need to have 2N levels, something which will probably be very hard to control. Thus, a quantum computation consists of preparing the initial state of the qubits, generally |0, 0, . . . , 0, then applying a unitary operation, and then measuring an observable, typically σz , on each qubit. . 3 1. Quantum algorithms. – Given a problem, we can look for algorithms to solve it. There are algorithms which are eﬃcient, and others which are not. In particular, algorithms can be classiﬁed in terms of the relationship between the number of steps required ns and the size of the input (number of bits; the size of a number N is log2 (N )): – Fast: The number of steps scale as a polynomial of the size of the input, that is ns ≤ poly[log2 (N )] for all N . – Slow : The number of steps cannot be bounded by any polynomial of the size of the input, that is, for any polynomial, ns > poly[log2 (N )] for all N . Now, depending on whether we use a classical or a quantum computer, a given problem can be easy or diﬃcult to solve. In fact, for all the problems for which there exists a fast algorithm with a classical computer, there also exist a fast algorithm with a quantum computer. The reason is that, as we mentioned before, with a quantum computer we can do at least the same as with a classical computer. For example, multiplication by the number 123 requires ns < k log2 (N ) (for a given k) and it is therefore fast in any computer. However, there are problems for which there only exist slow algorithms in a classical computer but in a quantum computer there exists a fast one. The most famous example is the problem of factorization, which consists of ﬁnding the prime factors of an input number N . Note, √ for example, that √ if we use a simple algorithm that checks if it is a divisor of 1, 2, . . . , N it requires N = 2log2 (N )/2 steps. However, there exists a quantum algorithm which can solve the problem in O[(log2 (N ))3 ] steps [11]. Another quantum algorithm can ﬁnd √ an item in an unsorted database composed of N items by only making of the order of N look-ups [12] (in contrast to any classical algorithm, which would need of the order of N ). Both algorithms are complicated and their explanation goes beyond the scope of the present paper. We will just give here a simple illustrative example of the power of quantum computation. In the following we will illustrate this statement with an artiﬁcial example, the socalled Deutsch’s algorithm. Imagine we have a quantum system formed by two qubits. Then a physical process is applied to the qubits, which transforms them according to some unitary operation. The unitary operation can be completely characterized by its action on the elements of the computational basis as follows: Uf |x, y = |x, y ⊕ f (x), with x, y = 0, 1. That is, in that basis, the ﬁrst qubit remains untouched whereas the state of the second qubit is obtained by adding modulo two its input value plus the action of a function f on the ﬁrst qubit. There exist four possible functions: f1 : x → x, f2 : x → 1 ⊕ x, f3 : x → 0, and f4 : x → 1. The operation corresponding to the ﬁrst

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Fig. 1. – Some quantum gates. Left: NOT gate (α = π/2, φ = π/2, γ = 0), Hadamard gate (α = π/4, φ = π/2, γ = 0), phase-ϕ gate (α = φ = 0, γ = ϕ). Right: quantum-XOR gate.

two functions will be called balanced (since the output can be 0 or 1), whereas the one of the functions f3,4 will be called constant. We can prepare any initial state we want and then a unitary operation will be applied once and only once. The goal is to ﬁnd out whether it was balanced or constant. Now, we can try to solve this problem either using superpositions or without them. In the latter case, one can easily convince oneself that this task is impossible. For any state |x, y we input, the corresponding output is compatible with the action of one of the balanced unitaries and one of √ the constant ones. However, setting as an input the state |+, − with |± = (|0 ± |1)/ 2 it can be readily checked that by measuring σx on the ﬁrst qubit one can check right away if the unitary was balanced or not. In summary, in a world where there exist superpositions one can do more than without them. In particular, we can solve this artiﬁcial problem which otherwise could not be solved. Of course, this problem is useless in practice. However, computer scientists have found out some other problems which are useful and that by using superpositions and entanglement one can solve in a much more eﬃcient way than without them. . 3 2. Quantum gates. – In order to solve a particular problem with a quantum computer, we have to be able to apply an arbitrary unitary operation on a set of N qubits. This is, in general, very complicated (as it would be to evaluate any arbitrary classical function on N bits). This is why, as in the classical case, one decomposes any unitary operation in terms of simple gates. A quantum gate is a quantum process that transforms the state of the qubits. For example, a general single-qubit gate has the form (13a)

|0 → cos(α)eiγ/2 |0 − ieiφ sin(α)|1,

(13b)

|1 → −ie−iφ sin(α)e−iγ/2 |0 + cos(α)|1.

In particular, for α = φ = π/2 and γ = 0, we have the quantum version of the NOT gate (see ﬁg. 1).

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Fig. 2. – Some examples of concatenations of fundamental gates: control-phase, where a gate is applied to the second qubit if the ﬁrst one is in 1; control-unitary, where a unitary operator Q = U DU † is applied, with D a diagonal operator with eigenvalues e±iϕ/2 ; two gates that permute the elements of the computational basis.

We can also have two-qubit gates, like the quantum-XOR gate (see ﬁg. 1), (14a)

|0, 0 → |0, 0,

(14b)

|0, 1 → |0, 0,

(14c)

|1, 0 → |1, 1,

(14d )

|1, 1 → |1, 0.

Note that this corresponds to evaluating the function XOR in the way given in (11). It can be shown that any unitary operation acting on a set of qubits can be written as a sequence of XOR and single-qubit gates (for some examples, see ﬁgs. 2 and 3) [10]. Note that the XOR gate is a non-local operation, and therefore it requires interaction between the qubits. This can be shown, for example, noticing that the input product state |0(|0 + |1) is transformed into |00 + |11, which is an (entangled) Bell state. The complexity of a quantum algorithms is measured in terms of the number of elementary gates that have to be carried out. For an arbitrary unitary operation, this number will be larger than 2N (as can be guessed by counting the number of parameters required to describe a unitary operator). However, for certain problems this number may grow only polynomially with M , the number of digits of the number one wants to factorize. This is exactly what happens in Shor’s factoring algorithm. On the other hand, with a classical system, the number of basic logic gates (NOT, XOR) scales exponentially with the M . There exist already several problems for which we know that a quantum computer is more eﬃcient, and there are several computer scientists nowadays looking for new ones.

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Fig. 3. – Any unitary operation can be decomposed in terms of fundamental gates.

. 3 3. Requirements for a quantum computer. – From what we have said, in order to build a quantum computer one needs: – Qubits: A set of two-level systems. – Quantum gates: One has to be able to perform single-qubit gates and the XOR gate. – Erase: One has to be able to erase the state of the qubits. That is, to prepare the state |0, 0, . . . , 0. – Read out: One has to be able to perform local measurements on the qubits. – Scalability: The error per time step (e.g. gate) cannot exceed some value (typically 10−4 ), so that fault tolerant error correction [13-15] is possible. Also, the cost in time, energy, etc. cannot increase exponentially with the number of qubits, since otherwise the advantages of a quantum computer would be lost. . 3 4. Measurement-based quantum computing. – According to what we have said before, quantum computing shares many analogies with its classical counterpart. One starts out with an initial state, performs a sequence of logic gates according to the algorithm one is running, and then one sees what is the result of the computation. The main diﬀerence is that the logic gates are quantum in the sense that they can give rise to superpositions and entangled states. However, the philosophies behind both methods are very similar. There exist another way of performing quantum computations which has no classical analogy and that uses very explicitly the non-locality contained in entangled states (that is also responsible for phenomena like teleportation). This method was introduced by Rausendorﬀ and Briegel [16] motivated by certain states (the so-called cluster states) that can be created with atoms in optical lattices [17]. The main idea is to prepare a cluster state of N qubits on a lattice and then measure each individual qubit in a basis that depends on the algorithm one is running, as well as on the outcomes of the previous measurements. That is, the state that is prepared is universal in the sense that it is

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always the same, independent of the problem one is trying to solve. The problem (i.e. the algorithm) is contained in the way one has to perform the measurements on the qubits. After the state has been prepared, there is no quantum gate required, and only the fact that after a measurement the state of the rest of the qubits collapses through the entanglement is suﬃcient to perform the computation. Classically, this would be impossible since once a state of the bits is prepared (for example, 0001000) if we measure the bits we will obtain a ﬁxed value. The cluster state cannot be easily speciﬁed by writing it in the computational basis. It is better to describe it in terms of how it can be prepared. Let us assume that we have a rectangular lattice of L × M qubits. We start with all of them in the state |0 + |1. Then we evolve them according to the Ising Hamiltonian (15)

H=

σzi σzj ,

i,j

for a time π/4, where the sum is taken over nearest neighbors. The resulting state is the cluster state. Note that the Ising Hamiltonian is a sum of terms that commute, and thus it is possible to create the cluster state in very diﬀerent ways. For example, we can ﬁrst switch on the interactions horizontally, and then vertically. Or we can perform the required interactions sequentially. In order to show the idea behind the measurement-based quantum computation, we will follow ref. [18]. In fact, we will see that it is very much related to teleportation. As we saw, if we have one qubit in some general state and we perform a joint measurement in the Bell basis with another qubit that is entangled to a third, we will obtain that, up to some unitary operation, the state of the ﬁrst qubit is transferred to the third particle. Noting that the four Bell states can be written as |Φα = σα1 |Φ+ 12 , where α = 0, x, y, z and σ0 = 1, we have that a Bell measurement on the state |φ1 |Φ+ 23 with outcome α will produce the state (16)

+ 12 Φα |φ1 |Φ 23

= 12 Φ+ |(σα |φ1 )|Φ+ 23 = σα |φ3

in the third particle (we have left out normalization factors and used the fact that σα = | σα† ). If instead of measuring in the Bell basis we would measure in the basis U1 Φα 12 , where U is a unitary operator, we would obtain the state (17)

σα U |φ3 .

That is, up to the σ’s, we would have applied a single-qubit gate, U , to the ﬁrst particle, and transferred the state to the third particle. So, we can perform single-qubit gates using teleportation by simply measuring in the appropriate basis (see ﬁg. 4a). In a very similar way it is possible to perform a two-qubit gate (see ﬁg. 4b). Let us assume that we have two particles, 1 and 2 in an unknown state |φ. Near particle 1 (2) we have two other particles, 1a and 1b (2a and 2b). Particles 1b and 2b are in a state |Φ+ , whereas particles

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Fig. 4. – (a) By choosing the measurement basis, we can apply a quantum gate to the teleported state. (b) We can also apply a two-qubit gate to particles 1 and 2 by using the auxiliary entangled states, so that the ﬁnal state appears in particles 1c and 2c.

1a and 2a are, respectively, entangled with some particles, 1c and 2c, respectively (also in the same Bell state). Now, performing a joint measurement in particle 1, 1a and 1b, and another one in particles 2, 2a and 2b, we will collapse the state of particles 1c and 2c. By choosing appropriately the joint measurements [18], it is possible to have that the state of those particles is σα1 σβ2 Uxor |ψ1c,2c , where Uxor is the quantum XOR gate. That is, also up to the σ’s, we can perform two qubit gates to our qubits by using entangled particles. Now, imagine that we have a rectangular lattice, with four qubits per sites (except at the borders), as shown in ﬁg. 5. Each qubit is entangled to its neighbor, except the ﬁrst rows of qubits, which are in the state |0. Now, we can perform quantum computations as follows. Imagine that we want to perform a single-qubit gate on the ﬁrst qubit on the left. Then we perform an appropriate joint measurement on the corresponding node so that the state of the qubit is teleported, after applying the operation, to the next node on its right. On the rest of the qubits of the ﬁrst column we can also apply other single-qubit gates, or simply teleport their states if we do not have to apply any. Now, the active qubits for the computation are all in the second columns of nodes (in the left-most qubits in each node). If we want to apply a two-qubit gate to a pair of neighboring qubits we can do it as explained above, making the corresponding joint measurements in their nodes. Of course, since in the previous teleportations we will have obtained some random (but known) outcome, we have to take this into account when we design the measurements in each node. For example, if in the ﬁrst qubit we obtained the outcome α and in the second β, and now the two qubit gate that we want to apply to them is Uxor , then we will have to apply the measurements that apply the operators Uxor σα1 σβ2 , such that the σ’s undo the random operations that have been applied before (recall that (σα )2 = 1). In this way, the quantum computation proceeds from left to right and at the end we will have the output state |Ψout in the last column. Now, we can perform a normal measurement on each qubit in order to get the results of the quantum computation. The above idea illustrates how a measurement-based quantum computer really works. In order to give this explanation, we have considered that we have four qubits per node.

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Fig. 5. – Measurement-based quantum computation is possible by having four qubits per node, which are entangled to their nearest neighbors according to the lattice structure. By performing local measurements in each node, it is possible to perform any sequence of logic gates and thus any quantum computation. In a cluster state, there is a single qubit per node, but still the same procedure is possible.

In a cluster state there is a single qubit per node. However, it behaves as if there where indeed four qubits which are projected onto a two-dimensional subspace [18]. In fact, the cluster state can also be understood in this way, as the state of the ﬁgure which is then projected in each node to a qubit subspace. Now, it turns out that the measurements that are required for the teleportations (and the gates) commute with the projectors, so that they can be made by simply measuring the qubit that is on each node. 4. – Quantum simulators Twenty-ﬁve years ago, R. Feynman [19] already realized that quantum systems are hard to simulate using classical computers. In order to store the state of N qubits, one has to give the value of 2N coeﬃcients if one expresses it in the computational basis. Thus, already for N = 100 one would need a memory larger than the largest we can ever build. Feynman also realized that with a quantum system one may be able to do this task much more eﬃciently. Let us see how one can use a quantum computer to simulate the evolution of a quantum many-body system [20]. Let us assume that the system involves N qubits, initially in the state |0, 0, . . . , 0 interacting according to a Hamiltonian (18)

H=

i,j

Hi,j ,

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where Hi,j describes the interaction between qubits i and j. We want to know what is the value of some correlation function oi (t)oj (t) after some time t. This problem is intractable for N > 100 with a classical computer since already to describe the state at each moment we would need to store 2N coeﬃcients, which is impossible. However, if we had a quantum one we could proceed as follows. We write

(19)

e−iHt

⎛ ⎞ ¯ 2 2 2 M " −iHt/M M h t N ⎠ , = e = ⎝ e−iHi,j /M 1 + O M2 i,j

¯ 2 is the typical (or maximal where we have used the Campbell-Hausdorﬀ formula and h eigen) value taken by the product of two Hamltonians Hi,j . Now, if M h2 t2 N 2 , we can thus approximate ⎛ (20)

e−iHt = ⎝

"

⎞M e−iHi,j /M ⎠

,

i,j

where the error in the approximation can be made arbitrarily small by choosing M . Since e−iHi,j /M is nothing but a two-qubit gate, this means that we can obtain the evolution by applying of the order of N 2 M gates, which is polynomial in N . In summary, if we had a quantum computer we could eﬃciently simulate the evoultion of any state. After the evolution, we can perform measurements of oi and oj , multiply them, and then average with respect to diﬀerent runs of the quantum computer. In order to obtain the required value with a small error , we just have to repeat the procedure of the order of 1/ times. So far we have been talking about the simulation of the time evolution of a quantum system. Sometimes we are interested in studying the thermal equilibrium properties instead. For example, we may be interested in the value of some correlation function oi oj in the ground state of H. This may be achieved with a quantum computer by using adiabatic algorithms [21]. The main idea is very simple. We write H(s) = (1−s)H0 +sH, where H0 is a Hamiltonian for which we know its ground state |ψ0 , and that is somehow close to H. Now, we proceed as follows: we prepare the state of the qubits in |ψ0 ; then, we evolve according to H(s), where we change s(t) very “slowly” in such a way that the state follows adiabatically the changes in the Hamiltonian. This is done with the method sketched above for the time evolution. If the process is adiabatic, then when we reach s = 1, according to the adiabatic theorem we will end up in the ground state of H. In order for the process to be successful, we need that there is no crossing (i.e. that the ground state of H(s) is never degenerate), and that the energy gap Δ(s) between the ground and the ﬁrst excited state is suﬃciently large. In fact, the condition to remain adiabatic is that the time for the whole process is much larger than the minimal gap. Thus, if the gap only decreases as the inverse of a polynomial in N , then the quantum computer will be able to ﬁnd the ground state eﬃciently. Note that for adiabatic quantum computation one has to choose H0 and that, typically, we do not know in advance how

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the gap will behave. Thus, one possibility is to ﬁrst evolve for some time (until certain value of s = s0 ) and then back and check that we end up in |ψ0 . If this is not the case, we do it more slowly until we manage to end up in that state. Then we repeat the same procedure with s1 > s0 , and then with s2 , etc., so that at the end we are able to go forth and back and end up in the ground state of ψ0 . Note that if there is some crossing, or that if we are not adiabatic, it is very unlikely that we will end up in the original state. In this way we make sure that the process has been adiabatic. In practice, it may be simpler to build certain kinds of quantum simulators, even if one is not able to build a quantum computer. For example, if one is able to engineer certain kinds of interactions in a many-body system, which are eﬀectively described by a Hamiltonian H, then one should be able to simulate systems that are described by that Hamiltonian. This is what happens with atoms in optical lattices, where it is nowadays possible to engineer several Hubbard models and spin Hamiltonians. By changing the parameters of the system adiabatically, one should also be able to study the lowtemperature properties of other systems, or the time evolution. On the other hand, so far we have considered general two-body Hamiltonians. In practice, one is interested in systems with certain symmetries, for example translational invariance (homogeneous). In that case, the quantum simulation can be done in a simpler way if one has a translationally invariant system. One may wonder if we really need quantum simulators. It may be possible to simulate the interesting many-body quantum systems using a classical commputer, but with more sophisticated algorithms. For example, one may use Monte Carlo methods [22], or density matrix renormalization group methods (DMRG) [23], which work very well in certain situations. For example, it is very likely that all ground states of homogeneous systems with low range interactions in 1D can be well approximated using DMRG. However, it is very unlikely that we will be able to simulate more general systems with a classical computer. First of all, if it were possible to simulate general time evolutions, then we would be able to simulate a quantum computer itself, something which seems impossible. Moreover, even if the system were homogeneous with local interactions and in 1D, classical simulation would have the same implication, since it has been shown that then one could also simulate any general quantum computation [24]. 5. – Physical implementations For the moment, we know very few systems which fulﬁll the requirements to implement a quantum computer with them. Perhaps, the most important problem is related to the necessity of ﬁnding a quantum system which is suﬃciently isolated, and for which the required controlled interactions can be produced. For the moment, there exist three kinds of physical systems that fulﬁll, at least, most of the requirements: 1) Quantum-optical systems [25, 26, 3]: Qubits are atoms, and the manipulation takes place with the help of a laser. These systems are very clean in the sense that with them it is possible to observe quantum phenomena very clearly. In fact, with them several groups have managed to prepare certain states which lead to phenomena that present

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certain analogies with the Schr¨ odinger cat paradox, Zeno eﬀect, etc. Moreover, those systems are currently used to create atomic clocks, and with them one can perform the most precise measurements that exist nowadays. For the moment, experimentalists have been able to perform certain quantum gates, and to entangle up to 8 atoms [27,28]. The most important diﬃculty with those systems is to scale up the models so that one can perform computations with many atoms. 2) Solid-state system [29-31]: There have been several important proposals to construct quantum computers using Cooper pairs or quantum dots as qubits. The highest diﬃculty in these proposals is to ﬁnd the proper isolation of the system, since in a solid it seems hard to avoid interactions with other atoms, impurities, phonons, etc. For the moment, both single and two-qubit gates have been experimentally realized. However, these systems posses the advantage that they may be easy to scale up. 3) Nuclear-magnetic-resonance systems [32,33]: In this case the qubits are represented by atoms within the same molecule, and the manipulation takes place using the NMR technique. Initially, these systems seemed to be very promising for quantum computation, since it was thought that the cooling of the molecules was not required, which otherwise would make the experimental realization very diﬃcult. However, it seems that without cooling, these systems lose all the advantages for quantum computation. 4) Photons [34]: The qubits are stored in the polarization of photons, and the quantum gates are carried out by using linear optical elements and photodetectors. The basic steps of single and two-qubit gates have been experimentally realized, but it is still a challenge to produce single photons and to develop very eﬃcient detectors. At the moment it is very diﬃcult to predict which will be the technology with which a quantum computer will be built. As it happened with classical computers, where the technologies that were used in the ﬁrst prototypes have been completely overcome with new technologies, the ﬁnal quantum technologies are still to be discovered. However, as experimentalists try to build small prototypes of quantum computers, we are learning about the main obstacles we can ﬁnd and ways to overcome them. . 5 1. Quantum optical systems for quantum computation. – In the following we will describe how to perform quantum computations with quantum optical systems [3, 35]. As we have mentioned before, the qubits are atoms, and the states |0 and |1 are two internal levels. In order to avoid spontaneous emission, those states must correspond to two stable electronic conﬁgurations. For example, in atoms with only one electron in the last shell, one can take two ground hyperﬁne levels with diﬀerent magnetic numbers. In order to isolate them from the environment, one uses high vacuum chambers, so that there are practically no other atoms or molecules that can collide with them. The initialization of the state is achieved using optical pumping, which consists of exciting the atom with a laser if it is in a diﬀerent state than the |0 (this is achieved by tuning appropriately the laser frequency and polarization). In this way, the atom will change the state via stimulated absorption and spontaneous emission until it decays in the state |0, in which the laser does not excite it anymore. The same method, with small modiﬁcations, can also be used to read out the state of each qubit. The idea is to use a laser in such a

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way that if the atom is in the state |0, then it does not absorb light; if it is in the state |1, then absorption-emission cycles occur such that the atom is excited by the laser and then it comes back to the state |1 via spontaneous emission. In this way, if at the end of the computation we switch the lasers on and observe light coming from some of the atoms, we will have measured the state |1 in that particular atom. If we do not observe light coming out, we will deduce that it is in the state |0. The single-qubit gates can be also carried out using a laser in such a way that it gives rise to stimulated absorption and emission (but not spontaneous emission, which would lead to decoherence). This is achieved by using two lasers whose frequencies are very far from resonance with respect to all atomic transitions. The absorption of one laser photon, followed by a stimulated emission of a photon in the other laser achieves the transition |0 ↔ |1. By choosing appropriately the laser intensities and phases, one can carry out any arbitrary singlequbit operation. The quantum XOR gate is usually the hardest part, since it requires a controlled interaction between the atoms. One way of achieving it is by manipulating the atoms in such a way that they exchange a photon (that is, one atom emits a photon and the other absorbs it). In order to do that, one needs high-quality cavities, so that the photons emitted by the atoms always go to a single resonant cavity mode, and not in any other direction. Other methods to perform the quantum XOR gate will be explained in the next subsection. . 5 2. Quantum computation with trapped ions. – Ions conﬁned in electric traps provide us with one of the most appropriate systems for quantum computation. The ions can be easily trapped in a region of space in the following way. One heats up an oven ﬁlled with atoms (typically Be, Ca or Ba) in such a way that they leave the oven towards a region which contains some electric ﬁelds (trap). Since the atoms are neutral, they are not aﬀected by those ﬁelds. However, if one targets the atoms with an electron beam, they may be ionized. As soon as this occurs, they start feeling the electric ﬁelds, which conﬁne them in that region. Those ﬁelds are generated by some electrodes, whose parameters can be changed in such a way that the potential felt by the ions is harmonic, but in which the restoring forces along two directions (say x and y) are much stronger than in the other direction. In this way, and due to the Coulomb repulsion, the ions tend to align along the z axis. Once the ions are trapped, one can cool them (i.e. stop them) using laser light. The idea is to drive the atoms with a laser of frequency ω, which is quasi-resonant with some other ω0 corresponding to a certain atomic transition. This happens in such a way that the atom absorbs photons from the laser and emits them spontaneously. Choosing appropriately the laser parameters (ω < ω0 ), in each absorption-emission cycle, the ions lose the energy ¯h(ω0 − ω), which is extracted from the ions motion. In this way one can achieve that the ions practically stop in space; well, in reality they end up in the ground state of the potential created by the trap and the Coulomb interaction. In this way we have a set of ions, separated by a distance of the order of 20 μm, which are basically stopped. As mentioned before, two internal states of each ion represent the qubit states |0 and |1. The single-qubit gates can be performed as indicated in the previous subsection. There exist several methods to realize the quantum XOR gate.

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Here we will mention one which is based in a conceptually simple eﬀect [36]. Atoms, when they interact with light, apart from absorbing and emitting photons, feel pushed by the laser. In particular, if an atom absorbs a photon of energy h ¯ ω and is transferred from some state |g to some other |e, its momentum increases by ¯hk, where k is the laser wave vector (k = ω/c). On the contrary, if the atom goes from |e to |g, the momentum of the atom decreases by ¯hk. If we have an atom practically stopped in the internal state |g and we send a laser pulse propagating from left to right, then it will be transferred into the state |e and will start moving to the right. Similarly, if it was in the state |e, then it will be transferred to |g and will move to the left. Of course, if it is √ initially in the state (|g + |e)/ 2, then the motional state will be a superposition of a state of the atom moving to the right and another moving to the left. In fact, this is the way in which some atomic interferometers operate, where the atomic wave function is split into two wave packets which are later on recombined to obtain an interference pattern. Let us analyze how we can use this eﬀect to produce a quantum XOR gate between to ions. In order to simplify the argument, we will assume that we just have two ions (1 and 2) in the trap. After laser cooling, the ﬁrst ion is located to the left of the other one. The quantum logic gate takes place in three steps: – Pushing ion 1: Using a laser pulse, the ﬁrst ion is pushed to the right or left depending on its internal state (for example, |0 = |g and |1 = |e). Note that due to the Coulomb repulsion, if the ﬁrst ion is pushed to the right (left) then the second one (pushed by the ﬁrst ion) will be also moving to the right (left). That is, after this step the second ion will move to the right or to the left depending on the internal state of the ﬁrst one. – Transition in the second ion: Focalizing another laser beam to the right of the equilibrium position of the second ion, one can change its internal state |0 ↔ |1, but only provided the ion is there. Otherwise, if the ion is moving to the left, it will not be aﬀected by this laser. – Pushing ion 1 back: Due to the external electric potential, the ions will oscillate back to their original positions. At that moment, one can use the same laser pulse as in the ﬁrst step so that it is reversed. The ions will stop, and the ﬁrst one will come back to its original internal state. . 5 3. Quantum computation with neutral atoms. – With neutral atoms, there are also several ways in which one can perform the two-qubit gates. Here we will also present one which is conceptually simple [26], even if in practice it may be hard to implement. Let us assume that we have two neutral atoms trapped in two diﬀerent regions of space. If the atoms are brought together, then they will start feeling each other, and thus their internal state will change. Then if their are brought back to their original positions the net eﬀect will be a two-qubit gate. If the interactions are such that, due to selection rules (or energy conservation), the states |0 and |1 are not changed (i.e. we have an elastic collision in this basis), then the eﬀect of the interaction will be just to include phases in

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Fig. 6. – The gate performed with neutral atoms (control-R), together with single-qubit gates, is equivalent to the XOR gate.

the wave functions that will depend on the internal states of the atoms. For example, if all the phases are multiples of 2π except when the atoms are in the states |1 and |0, where it is π, then we will have implemented the gate (21a)

|0, 0 → |0, 0,

(21b)

|0, 1 → |0, 0,

(21c)

|1, 0 → −|1, 0,

(21d )

|1, 1 → |1, 1.

Up to single-qubit operations, this gate is equivalent to the XOR gate mentioned above (see ﬁg. 6). It may be hard in practice to choose all the phases such that the above gate is implemented. Apart from that, when the atoms are brought together, they will feel the potential that is holding the other atom (unless they are diﬀerent) so that it will not be possible to bring them to the same spatial position. A way of overcoming those problems is as follows (see ﬁg. 7). Imagine that the trapping potential depends on the internal state of the particles. That is, if an atom is in state |0, then it feels a potential, V0 , which is diﬀerent than if it is in |1, V1 . In optical traps, this can be done by using lasers with diﬀerent polarizations, since the internal atomic states will feel diﬀerently the diﬀerent polarizations [26, 37]. For example, the potentials V0 and V1 may form both a double-well potential whose minima are occupied by the atoms. Initially, V0 = V1 . If we want to perform the gate, we can move the potential V1 to the right (keeping V0 constant) until the potential well of V1 coincides with the second of V0 . In that case, only if the ﬁrst atom is in the state |1 and the second in |0 a collision will take place. By choosing the interaction time, one can make the corresponding collisional phase shift equal to π, so that one implements the above two-qubit gate. Note that one has to move the atoms adiabatically, in such a way that their motional state does not get entangled with the motion since otherwise this will spoil the gate. The best way of achieving that is to have the atoms in the ground state of the corresponding potentials and to move the potentials adiabatically such that the motional state is always the instantaneous ground state of the conﬁning potential. Note also that moving the atoms also adds some extra phases into the wave functions (which are given by the time integration of the kinetic

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Fig. 7. – A collision only occurs if the ﬁrst atom is in |1 and the second in |0.

energy of an atom at the center of the potential during the process). These phases are nothing else but single-qubit gates, which can be undone at the end of the process. . 5 4. Quantum simulations with neutral atoms. – Neutral atoms are specially attractive to perform quantum simulations of lattice problems. In fact, atoms can be loaded in optical lattices so that they ﬁll the lowest Bloch band of the periodic potential. Since atoms can tunnel between neighboring sites, and they also interact when they meet at some site, their dynamics in the lattice is well described by the Hubbard Hamiltonian, which, depending on the bosonic or fermionic character, displays very diﬀerent behaviors. The tunneling amplitude as well as the on-site interaction energy can be easily changed by modifying the intensity of the trapping lasers. Also, one can use diﬀerent internal atomic levels, or diﬀerent species (e.g., bosons and fermions at the same time, or molecules), magnetic and electric ﬁelds, laser ﬁelds which induce virtual transtions, Feschbach resonaces, superlattices, etc. Thus, with atoms in optical lattices one can simulate a great variety of Hubbard-type Hamiltonians. On the other hand, if one decreases the tunneling amplitude one may end up with a Mott state, where there is one atom per lattice site. In this case, virtual tunneling to nearest neighbors will induce eﬀective interactions giving rise to pseudo-spin Hamiltonians. In this way, diﬀerent Heisenberg models on diﬀerent geometries (square, triangular, Kagome, etc.) can also be simulated. In fact, atoms in optical lattices seem to be the most versatile way of performing quantum simulations at the moment. With them, it may be possible to simulate Hamiltonians for which there exists no classical algorithm to go beyond, say, 30 particles. Among the most interesting problems one could attack with this system is the fermionic Hubbard model in 2 dimensions, since it is closely related to several open questions in the ﬁeld of high-Tc superconductivity. Finally, also with atoms in optical lattices it is possible to create the cluster state [16]. One just has to use the method mentioned in the previous subsection in order to build quantum gates. In fact, it is simple to see that the quantum gate mentioned above is (up to local operations) nothing but the one generated by an Ising interaction which is

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required to create such a state. In particular, let us assume that the atoms are trapped in a 2-dimensional square lattice with diﬀerent lattice potentials depending on their internal states, and one moves the potentials by changing the laser conﬁguration [26] as follows: ﬁrst one step to the right and back, and then one step to the front and back. In this way, one obtains the cluster state with which measurement-based quantum computation should be possible. 6. – Conclusions Quantum Mechanics, apart from providing us with a new description of the physical reality, allows us to use some new eﬀects in the ﬁeld of communication and computation. These eﬀects are related to the superposition principle, and in particular to the existence of entangled states. Furthermore, the fact that a computer can accept input and create output states in superpositions can be used in order to solve problems that otherwise would be intractable. The implementation of these ideas with particular physical systems is, however, very complicated. The ﬁrst experiments on quantum computation are being carried out. It is not clear yet, when the advantages of quantum computation will be practical. However, it seems that quantum simulations can indeed give rise to the study of very interesting phenomena soon. In this respect, atoms in optical lattices seem to be an ideal playground for this kind of experiments. ∗ ∗ ∗ This work was supported by DFG, and the European Union projects SCALA and CONQUEST. REFERENCES [1] Nielsen M. and Chuang I., Quantum Computation and Quantum Information (Cambridge University Press) 2000. [2] See, for example, Quantum Information Processing and Communication in Europe (European Commission) November 2005. [3] Cirac J. I. and Zoller P., Phys. Today, 57 (2004) 38. [4] A very good introduction to the main concepts of quantum information can be found in Peres A. in Quantum Theory: Concepts and Methods (Kluwer Academic) 1993. [5] See, for example, the ﬁrst issue of Quantum Inf. Comput., September 2001. [6] Bennett C. H. et al., Phys. Rev. Lett., 70 (1993) 1895. [7] Bouwmeester D. et al., Nature, 390 (1997) 575; Boschi D. et al., Phys. Rev. Lett., 80 (1998) 1121; Furusaw A. et al., Science, 282 (1998) 706; Barret M. et al., Nature, 429 (2004) 737; Riebe M. et al., Nature, 429 (2004) 734; Sherson J. et al., Nature, 443 (2006) 557. [8] Bennett C. H. and Wiesner S. J., Phys. Rev. Lett., 69 (1992) 2881. [9] Mattle K. et al., Phys. Rev. Lett., 76 (1996) 4656. [10] Ekert A. and Josza R., Rev. Mod. Phys., 68 (1995) 733, and references therein. [11] Shor P. W., in Proceedings of the 3rd Annual Symposium on the Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, CA) 1994, p. 124.

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[12] [13] [14] [15]

Grover Lov K., Phys. Rev. Lett., 79 (1997) 325. Shor P. W., Phys. Rev. A, 52 (1995) 2493. Steane A. M., Phys. Rev. Lett., 77 (1996) 793. Shor P. W., in 37th Symposium on Foundations of Computing (IEEE Computer Society Press, Los Alamitos, CA) 1996, p. 56; Preskill J., Proc. R. Soc. London, 454 (1998) 385. Raussendorf R. and Briegel H. J., Phys. Rev. Lett., 86 (1998) 5188. Briegel H. J. et al., J. Mod. Opt., 47 (1999) 415. Verstraete F. and Cirac J. I., Phys. Rev. A, 70 (2004) 060302. Feynman R. P., Int. J. Theor. Phys., 21 (1982) 467. Lloyd S., Science, 273 (1996) 1073. Fahri E. et al., Sience, 292 (2001) 472. Ceperley D. M. and Alder B. J., Phys. Rev. Lett., 45 (1980) 566. White S. R., Phys. Rev. Lett., 69 (1992) 2863. Vollbrecht K. and Cirac J. I., Phys. Rev. A, 73 (2006) 012324. Cirac J. I. and Zoller P., Phys. Rev. Lett., 74 (1995) 4091. Jaksch D. et al., Phys. Rev. Lett., 82 (1999) 1975. Leibfried D. et al., Nature, 438 (2005) 639. ¨fner H. et al., Nature, 438 (2005) 643. Ha Kane B. E., Nature, 393 (133) 1998. Loss D. and DiVincenzo D. P., Phys. Rev. A, 57 (1998) 120. ¨ n G., Nature, 398 (1999) 305. Makhlin Y. and Scho Gershenfeld N. A. and Chuang I. L., Science, 275 (1997) 350. Cory D. G., Fahmy A. F. and Havel T. F., Proc. Natl. Acad. Sci. U.S.A., 94 (1997) 1634. Knill E. et al., Nature, 409 (2001) 46. Cirac J. I. and Zoller P., Science, 301 (2003) 176. Poyatos J. F., Cirac J. I. and Zoller P., Phys. Rev. Lett., 81 (1998) 1322. Brennen G. et al., Phys. Rev. Lett., 82 (1999) 1060.

[16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

Fundamental noise in matter interferometers A. Imambekov, V. Gritsev and E. Demler Department of Physics, Harvard University - Cambridge MA 02138, USA

These lecture notes discuss two eﬀects which contribute to the reduction of the interference fringe contrast in matter interferometers. The ﬁrst eﬀect is the shot noise arising from a ﬁnite number of atoms used in experiments. Focusing on a single-shot measurement, we provide explicit calculations of the full distribution functions of the fringe contrast for the interference of either the coherent or the number states of atoms. Another mechanism of the suppression of the amplitude of interference fringes discussed in these lecture notes is the quantum and thermal ﬂuctuations of the order parameter in low-dimensional condensates. We summarize recent theoretical and experimental studies demonstrating that suppression of the interference fringe contrast and its shot to shot variations can be used to study correlation functions within individual condensates. We also discuss full distribution functions of the fringe amplitudes for one and twodimensional condensates and review their connection to high-order correlation functions. We point out intriguing mathematical connections between the distribution functions of interference fringe amplitudes and several other problems in ﬁeld theory, systems of correlated electrons, and statistical physics. 1. – Introduction . 1 1. Interference experiments with cold atoms. – From the earliest days of quantum mechanics its probabilistic nature was the cause of many surprises and controversies [1]. Perhaps the most unusual manifestation of the quantum uncertainty is a quantum noise: measurements performed on identical quantum-mechanical systems can produce results which are diﬀerent from one experimental run to another. At the level of a single particle quantum mechanics, the quantum noise is no longer a research topic but is discussed in undergraduate physics textbooks [2]. However, the situation is diﬀerent when we c Societ` a Italiana di Fisica

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talk about quantum mechanics of many-body systems. One can ask seemingly simple questions to which there is no obvious answer: does it still make sense to talk about quantum noise when discussing measurements on many-body quantum states? How does the quantum noise manifest itself? Can one use this noise to extract nontrivial information about the system? The idea of the quantum noise analysis of many-particle systems is common to many areas of condensed-matter physics [3-6] and quantum optics [7-12]. In the ﬁeld of ultracold atoms it has been successfully employed in a variety of recent experiments [13-19] with many more theoretical proposals awaiting their turn [20-26]. These lecture notes address a very speciﬁc experimental probe of the cold-atomic-ensembles interference experiments. Our discussion focuses on a variety of interesting and important phenomena which originate from the fundamental quantum and/or thermal noise of cold atoms condensates and can be studied in interference experiments. Although focused on the speciﬁc type of experiments, the general methodology discussed in these lecture notes can be extended to a variety of other measurements on systems of cold atoms. Interference experiments constitute an important part of the modern toolbox for studying ultracold atoms. Original experiments used large three-dimensional BoseEinstein condensates (BEC) to demonstrate macroscopic coherence [27]. More recently interference experiments have been done with one and two-dimensional condensates [28-33] and demonstrated the important role of ﬂuctuations in low-dimensional systems. Matter interferometers using cold atoms [29,30,34-50] have been considered for applications in accelerometry, gravitometry, search for quantum gravity, and many other areas (for a review see ref. [51]). Interference experiments have been used to measure the condensate formation [52, 53] as well as the critical properties of the BEC transition [54]. What is common to most interference experiments is the focus on the phase of interference patterns. Suppression of the fringe contrast is considered to be a spurious eﬀect caused by noise and ﬂuctuations. On the contrary, these lecture notes focus on understanding physical phenomena that underlie the imperfect visibility of interference fringes. As we discuss below, suppression of the fringe visibility comes from fundamental physical phenomena, such as the noise intrinsic to performing a classical measurement on a quantum-mechanical wave function (shot noise) or classical and quantum ﬂuctuations of the order parameter. In these lecture notes we discuss how one can use analysis of the contrast of interference fringes to learn about ﬂuctuations of the order parameter [32, 33, 55-58]. We will demonstrate that important information is contained not only in the average contrast but also in its shot-to-shot variations. For example, when we discuss ﬂuctuating condensates, we will show that high moments of the interference fringe amplitudes contain information about high-order correlation functions and thus provide valuable information about the system. The basic scheme of interference experiments is shown in ﬁg. 1. Originally two condensates are located at a distance d away from each other. At some point they are

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Fig. 1. – Schematic view of the interference experiments with ultracold atoms. Two condensates, originally separated by the distance d, are released from the traps and expand until the clouds overlap. The imaging beam measures the density of the atoms after the expansion. Quantum interference leads to the periodically modulated density projected on the screen. Projected density image is taken from the actual experimental data of Hadzibabic et al. [32].

allowed to expand ballistically(1 ) until sizes of the clouds become much larger than the original separation between the clouds d. After the expansion the density is measured by shining a laser beam through the cloud. Interference leads to the appearance of the density modulation at a wave vector Q = md/t (see ﬁg. 1 and discussion in sect. 2). When the two condensates are coherent, the position of interference fringes is determined by the relative phase between the two clouds. Surprisingly the interference pattern will be observed even for two independent condensates which do not have a well-deﬁned relative phase (see, e.g., ﬁg. 2). To beginning readers it may seem confusing that we can observe interference in the absence of coherence between the two clouds. Or even more confusing, we discuss interference patterns when both clouds are number states and phases of individual condensates are not well deﬁned. Several theoretical frameworks have been introduced to understand the origin of interference patterns in the absence of phase coherence [63-67]. In this paper we explain the origin of interference fringes using the language of correlation functions and point out connections to Hanbury Brown and Twiss (HBT) experiments [7] in optics. This section provides a simple heuristic picture and a more formal discussion is left to sect. 2. (1 ) Readers may be concerned that the initial part of expansion may not be purely ballistic and atomic collisions may take place in expanding clouds. In the case of tight conﬁnement such collisions are rare and have minimal eﬀect. One can use such approximation in a variety of experimental situations, because the density of the condensate falls of rapidly during the ﬁrst stages of the expansion. Eﬀects of the interactions on the interference signal have been recently discussed in refs. [59-61]. It is worth noting that the eﬀect of collisions can be minimized by tuning the scattering length to zero at the moment of the release using magnetic Feshbach resonances [62]. See also discussion after eq. (43).

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Fig. 2. – Interference pattern from the ﬁrst set of experiments with sodium atoms. Taken from ref. [27].

What one measures in experiments is the density proﬁle after the expansion ρ(r). Interference pattern appears as the density modulation ρ(r) = ρQ e−iQr +c.c.+const. The absolute value of ρQ determines the amplitude of interference fringes and its phase deﬁnes the position of the fringe maxima and minima. Schematically we can write ρQ ∼ eiφ1 −iφ2 , where φ1,2 are phases of the two condensates before the expansion (see discussion in sect. 2). In the absence of coherence eiφ1 −iφ2 = 0, which implies that ρQ = 0. Vanishing of the average, however, does not mean that interference fringes are absent in each individual shot. In the present case it only shows that the phase of interference fringes is random in each shot. We remind the readers that taking an expectation value in quantum mechanics implies averaging over many measurements. On the other hand, we can focus on the amplitude of interference fringes and accept the fact that we cannot predict their phase. Then we need to consider the quantity which does not vanish after averaging over the unpredictable phase diﬀerence. One such quantity is given by the density-density correlation function ρ(r)ρ(r ) = |ρQ |2 (eiQ(r−r ) +c.c.)+other terms. The right-hand side of the last equation does not vanish when we average over the random phases φ1,2 , and we ﬁnd the ﬁnite expectation value (1)

ρ(r)ρ(r ) = 2|ρQ |2 cos(Q(r − r )) + const.

What this correlation function tells us is that in a single shot we can not predict whether at a given point r we will have a minimum or a maximum of the density modulation. However what we can say is that if there is a maximum at point r, it will be followed by another maximum a distance 2π/Q away. While this simple argument explains the origin of interference patterns from independent condensates, it leaves many questions unanswered. For example, it is not obvious how accurately one can represent two independent condensates using states with a well-deﬁned but unknown phase diﬀerence. Also it is not clear how to generalize this analysis to elongated condensates, when we need to go beyond the single-mode approximation and include phase ﬂuctuations within

Fundamental noise in matter interferometers

539

individual systems. These lecture notes will present a uniform approach for addressing these and many other questions. When the focus of interference experiments is on measuring the phase, one usually averages interference patterns obtained in several shots. The result is easy to interpret: an average of many experimental runs is precisely what we deﬁne as a quantum-mechanical average. However in experiments with independent condensates, summing interference patterns is not appropriate. The phase of interference patterns is random from shot to shot and adding individual images washes out interference fringes completely (for a nice experimental demonstration see ref. [68]). Hence, in this case one needs to focus on interference patterns obtained in individual shots. In the absence of averaging, a single-shot measurement contains noise. Thus to characterize such experiments, we need to provide both the average value and the shot-to-shot ﬂuctuations of the fringe contrast. The most comprehensive description of the ﬂuctuating variable comes from providing its full distribution function. Theoretical calculations of the distribution functions of the fringe contrast will be the central part of these lecture notes. It is useful to point out the analogy between the approach discussed in this paper and the famous Hanbury Brown and Twiss experiments in optics. The original motivation for HBT experiments came from astronomy: the goal was to measure the angle between two incoherent stellar sources such as two diﬀerent points on the surface of the star. Since the two sources are incoherent, this cannot be done using a single detector: ﬁrst-order interference is absent and the measured signal is simply the sum of the two intensities [12]. The insight of HBT was to use two detectors and measure the correlation function of the two intensities as a function of the relative distance between the two detectors. One ﬁnds that this correlation function is given by (2)

I(r)I(r ) ∼ cos ((k1 − k2 )(r − r )) + const,

where k1,2 are wave vectors of photons arriving from the two points on the surface of the star, r, r are positions of the detectors, and I(r), I(r ) are the intensities measured in the two detectors (see ﬁg. 3). Hence, the main idea of HBT experiments is that information is contained not only in the average signal I(r), but also in the noise. Such noise can be characterized by looking at higher-order correlations. In astronomy, HBT experiments were used to measure several important properties of distant stars, including their angular sizes and the surface temperature [69]. There is an obvious analogy between eqs. (1) and (2), but there is also an important diﬀerence. HBT stellar interferometers operate in real time, and averaging over time is built into the measuring procedure. In these experiments, ﬂuctuations of I(r)I(r ) − I(r)I(r ), which would correspond to higher-order correlations in I(r), are not easy to measure. On the other hand, interference experiments with cold atoms are of a singleshot type: each measurement is destructive and gives a certain density proﬁle ρ(r). A single image contains information not only about the two point correlation function ρ(r)ρ(r ), but about higher-order correlations as well. The most important information for our purposes is contained in the interference pattern at wave vector Q. Essentially

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Fig. 3. – Schematic view of Hanbury Brown and Twiss noise correlation experiment as an example of intensity interferometry. Detectors at positions r, r measure the intensity of light coming from two distant incoherent sources. The “correlator” (denoted by the box) measures the coincidence events and thus the intensity-intensity correlation function.

each interference pattern constitutes a classical measurement of the quantum-mechanical operator ρQ . We remind the readers that we expect the phase of ρQ to be random, so the quantity of interest will be |ρQ |2 . By performing measurements several times we will ﬁnd not only the average value of this operator, but also its higher moments. Ultimately we should be able to reconstruct the entire distribution function for |ρQ |2 . So the simpliﬁed and idealized procedure that we analyze is the following: one performs interference experiments many times. Each experiment is analyzed by doing a Fourier transform of the density to extract ρQ . The histogram of the measured values of |ρQ |2 will be the main subject of these lecture notes. We will demonstrate the wealth of information that can be extracted from analysis of such histograms. As a passing note, we mention that the setup considered in ﬁg. 1 is not the only possible conﬁguration for interference experiments. In another common setup one makes several copies of the same cloud using Bragg pulses, and observes an interference between them [70-76]. It is useful to put our work in the general perspective of noise analysis in physical systems. Understanding photon ﬂuctuations is at the heart of modern quantum optics and provides a basis for creation, detection, and manipulation of non-classical states of light [11]. The ﬁeld of quantum optics has a long and fruitful tradition of using the higher-order correlation functions as well as the shot noise to characterize the quantum states of light. The notion of higher-order degree of coherence was ﬁrst introduced by R. Glauber in 1963 [8], also by Klauder and Sudarshan [9] and by Mandel and Wolf [10]. The knowledge of photoelectron counting distribution function reveals such non-classical features of light as antibunching [77], sub-Poissonian statistics and probe of violation of Bell inequalities. In particular, third-order correlations provide a test for distinguishing between quantum and hidden-variable theories in a way analogous to that provided by the Greenberger-Horne-Zeilinger test of local hidden-variable theories [78]. Interference

Fundamental noise in matter interferometers

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of independent laser beams was ﬁrst observed in ref. [79] and stimulated a number of theoretical studies (for reviews see refs. [10, 80, 81]). In condensed-matter physics, noise analysis was also suggested as a powerful approach for analyzing electron systems [3]. It was demonstrated theoretically that in certain mesoscopic systems current ﬂuctuations should contain more information than the average current itself. In particular, the third and higher moments contain important quantum information on interaction eﬀects, entanglement and relaxation processes (see, e.g., refs. [82, 83]). Speciﬁc proposals exist for detecting statistics of quasi-particles [84], understanding transmission properties of small conductors [4] and observing entanglement between electrons [5]. Perhaps the most spectacular experimental success of the noise analysis in electronic systems has been the demonstration of the fractional charge of quasiparticles in the fractional quantum Hall regime [6]. However generally the noise analysis in condensed-matter systems did not become the detection tool of the same prominence as in quantum optics. The main reason for this is the excruciating diﬃculty of the noise measurements in solid state experiments. One often needs to measure a signal which is only a part in a million of the unwanted technical noise. In the ﬁeld of ultracold atoms experiments analyzing quantum noise are only starting. However, we have already seen spectacular successes in several recent experiments. Analysis of noise correlations in the time-of-ﬂight experiments [20] was used to demonstrate fermionic pairing [13] as well as HBT-type correlations for atoms in optical lattices [14-16] (see also I. Bloch’s lectures in this volume). Single-atom detectors have been used to demonstrate HBT noise correlations for cold atoms [17-19]. Strongly interacting systems of cold atoms are expected to realize analogues of important models of condensed-matter systems [85-87]. Being able to study noise in such systems should provide an important new perspective on strongly correlated states of matter and have a profound eﬀect on many areas of physics. We hope that these lecture notes will stimulate more experimental work in analyzing noise in interference experiments. The ﬁrst success in this direction was the recent observation of the Berezinskii-Kosterlitz-Thouless (BKT) transition [88, 89] by Hadzibabic et al. in ref. [32]. . 1 2. Fundamental sources of noise in interference experiments with matter . – Two fundamental sources of ﬂuctuations in the amplitude of interference fringes are the shot noise and the order parameter ﬂuctuations within individual condensates. Shot noise comes from the ﬁnite number of atoms used in the experiments. Let us discuss limiting cases ﬁrst. Consider an interference experiment with one atom. Before the expansion the atom is in a perfect superposition of being between the two wells. After the expansion we get a perfectly periodic wave function ψ(r) = 2C cos( Qr+φ 2 ) (for a more detailed discussion see sect. 2). The expectation value of the density operator is ρ(r) = |ψ(r)|2 = 2|C|2 (cos(Qr + φ) + 1). However, this average value will not be measured in a single shot. A single measurement ﬁnds the atom at a single point. The expectation value of the density determines probabilities with which we can ﬁnd atom at any given point, but in a single measurement we collapse a quantum-mechanical wave function and observe the atom at one point only. Can one reconstruct the entire amplitude of the

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Fig. 4. – Simpliﬁed setup of interference experiments with 1D Bose liquids (see, e.g., ref. [29]). Two parallel condensates are extended in the x-direction. After atoms are released from the trap, clouds are imaged by the laser beam propagating along the z-axis. Meandering structure of the interference pattern arises from phase ﬂuctuations along the condensates. The net interference amplitude ρQ (L) is deﬁned from the density integrated along the section of length L.

interference pattern ρQ = |C|2 e−iφ from a single measurement? Obviously the answer is no. In the opposite case of a very large number of atoms in the same single particle state one should be able to reconstruct a complete interference pattern already from a single measurement, since the measurement of positions of many atoms performs a statistical averaging implicit in quantum mechanics. In the general case of experiments with a ﬁnite number of atoms, the question arises how well one can determine the amplitude of interference fringes from doing a single-shot measurement. Formulated more accurately the problem is to determine probabilities of ﬁnding a certain amplitude of interference fringes, |ρQ |2 , in a single measurement. Fluctuations of the order parameter are particularly important for low-dimensional systems. If the condensates are conﬁned in one [90, 91] or two [31-33, 90, 92] dimensions, then the true long-range order may not exist. Rigorous theorems forbid true long-range order in two-dimensional systems at ﬁnite temperature and in one-dimension even at zero temperature [93-95]. What this means is that low-dimensional condensates cannot be characterized by a single phase and we need to take into account spatial ﬂuctuations of the order parameter. Eﬀects of such ﬂuctuations on interference experiments are illustrated schematically in ﬁg. 4. Two one-dimensional clouds expand in the transverse direction. Each point along the condensates has a local interference pattern, but in the presence of phase ﬂuctuations (either thermal or quantum), these patterns are not in phase with each other. It is natural to deﬁne the net interference amplitude from the density integrated over the axis of the system (the so-called columnar density). In many experiments such integration is done by the measurement procedure itself. For example, systems such as shown in ﬁg. 4 originally had imaging done along the axis of the interferometer [29]. Then the laser beam integrates the atomic densities within the

Fundamental noise in matter interferometers

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imaging length. Integrating over local interference patterns which are not in phase with each other leads to a reduced contrast of the net interference fringes. In earlier literature smearing of interference patterns by ﬂuctuations was considered an unwanted eﬀect [27]. The point of view presented in these notes is quite the opposite. Suppression of the fringe contrast is an interesting eﬀect which tells us about important phenomena in Bose condensates, such as thermal and quantum ﬂuctuations of the order parameter. By analyzing such suppression we can extract non-trivial information about the system. In particular, it has been shown in ref. [55] that the scaling of the average interference signal with the observation area contains information about the two-point correlation functions within each cloud. Recent experiments [32] by Hadzibabic et al. used this approach to observe the BKT transition in two-dimensional condensates (see discussion in sect. 4). We also note that such experiments can be used to extract information which is diﬃcult to obtain by other means. For example, in sect. 4 we discuss that analysis of the high moments of the contrast tells us about high-order correlations within individual clouds. These lecture notes are organized as follows. In sect. 2 we discuss how interference fringes appear for ideal non-interacting 3D BECs at zero temperature. In sect. 3 we analyze the shot noise for ideal condensates. In these lecture notes ideal condensates are understood as clouds of non-interacting atoms which before the expansion occupy a single mode within each of the traps. The problem of interference of independent condensates of ideal bosons has been extensively analyzed in the literature before [63-67, 96-101]. In particular in an important recent paper [102], Polkovnikov showed that the variance of the fringe amplitude decreases as an inverse power of N , with a non-universal coeﬃcient which contains information about the state of each cloud (e.g. coherent states vs. Fock states). In this paper we develop a general formalism for calculating the full distribution functions of the fringe amplitudes in interference experiments with ideal condensates with a ﬁnite number of atoms. We apply this formalism to obtain distribution functions for several experimentally relevant cases such as states with a well-deﬁned phase diﬀerence between the two clouds and Fock states of atoms. Eﬀects of the order parameter ﬂuctuations are discussed in sect. 4. We obtain distribution functions for both one and two dimensional condensates in the limit when the number of particles is large and the shot noise can be neglected. We also discuss intriguing mathematical connections between these distribution functions and a number of important problems in physics, such as the quantum impurity problem in a low-dimensional interacting electron system [103] or the distribution of roughness in systems with 1/f noise [104]. In these lecture notes we do not address the issue of technical noise which is obviously important for understanding real experiments. In the concluding sect. 5, however, we comment on the experimental requirements for observing some of the phenomena discussed in these lecture notes. 2. – Interference of ideal condensates In this section we discuss why interference fringes appear for ideal non-interacting BECs at zero temperature. We follow ref. [67], and introduce notations for subsequent

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Fig. 5. – Schematic view of the interference experiment with 3D condensates.

sections. First we consider the case of two clouds with a well-deﬁned relative phase, where appearance of interference fringes can be understood at a single-particle level. Then we show that almost ideal interference fringes appear even when two expanding clouds are uncorrelated, provided that the number of particles in each cloud is large. . 2 1. Interference of condensates with a well-deﬁned relative phase . 2 1.1. Basics of interference experiments. First quantized representation. To illustrate how interference fringes arise, let us start by considering a simple case of two BEC clouds with a well-deﬁned relative phase. Here we neglect interactions between atoms, so initially all atoms are assumed to be in the same single-particle state (single-mode approximation). After the conﬁning potential is removed, the single-particle state evolves with time, but many-body wave function remains in the product state. The interference appears as a result of single-particle wave function evolution, which can be studied in detail (corresponding setup is shown in ﬁg. 5). Normalized single-particle wave functions for two clouds will be denoted as ψ1 (r, t) and ψ2 (r, t), and the initial relative phase is ϕ. If the total number of particles equals N , then the complete wave function of the system in the ﬁrst quantized notations at any moment of time is given by

(3)

Ψ(r1 , . . . , rN , t) =

N " 1 √ (ψ1 (rn , t)eiϕ/2 + ψ2 (rn , t)e−iϕ/2 ). 2 n=1

This wave function satisﬁes the proper symmetry requirements for permutations of ri and rj , and evolution of ψ1 (r, t) and ψ2 (r, t) is controlled by the single-particle Schr¨ odinger equation. Initial overlap of the states ψ1 (r) ≡ ψ1 (r, 0) and ψ2 (r) ≡ ψ2 (r, 0) is assumed to be negligible: (4)

drψ1∗ (r)ψ2 (r) ≈ 0.

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Fundamental noise in matter interferometers

The expectation value of the total density corresponding to the wave function (3) is (5)

ρ(r, t) =

N |ψ1 (r, t)|2 + |ψ2 (r, t)|2 + 2 Re eiϕ ψ1 (r, t)ψ2∗ (r, t) . 2

The expectation value of the total density displays an interference pattern due to the last term of eq. (5). As a simple example, let us assume that ψ1 (r, t) and ψ2 (r, t) are initially in the Gaussian states centered at points ±d/2, and their widths are R0 d. Then the evolution of the single-particle wave functions can be simply calculated, and the result is 2 (1+it/mR2 ) 0 2 2Rt

,

2 (1+it/mR2 ) 0 2 2Rt

,

(6)

(r−d/2) − 1 ψ1 (r, t) = e 2 (πRt )3/4

(7)

ψ2 (r, t) =

(r+d/2) − 1 e (πRt2 )3/4

where the widths of the wave packets, Rt , at time t are given by Rt2

(8)

=

R02

+

t mR0

2 .

We will be interested in the regime, when the sizes of the clouds Rt are much larger than the original distance between the clouds, that is Rt d R0 .

(9)

In this regime, the clouds overlap strongly, and the real parts of the exponents in eqs. (6) and (7) are responsible for the broad overall density proﬁle. Imaginary parts in the same exponents give rise to interference eﬀects in the last term of eq. (5). Thus the interference part of the density is equal to

2 +d2 /4

(10)

r − N e (πRt2 )3/2

2 Rt

cos

rd t+ϕ . m R02 Rt2

For suﬃciently large t, one can substitute Rt ≈ t/mR0 , and obtain oscillations of the density at wave vector Q = md/t, with positions of the minima and the maxima controlled by the relative phase ϕ. The Fourier transform of the density at wave vector Q is 4 (11)

ρQ =

5 iQr

drρ(r, t)e

≈

N −iϕ e . 2

Physically Q can be understood as the momentum diﬀerence of the two particles which have been released from the two traps and arrive simultaneously at the detection

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point r. This can be seen from the following quasiclassical argument. A particle released from the condensate one and detected at time t at point r has momentum Q1 = m(r − d/2)/(t).

(12)

During the expansion this particle picks up a phase Q1 (r − d/2). Analogously a particle originating from the condensate two has momentum (13)

Q2 = m(r + d/2)/(t)

and picks up a phase Q2 (r + d/2). The interference pattern arises from the oscillating structure of the phase diﬀerence with the wave vector of oscillations Q = Q1 − Q2 = md/t.

(14)

This simple argument shows that as long as the original sizes of the clouds are much smaller than the distance between them, after suﬃcient expansion one should observe oscillations of the density at the wave vector Q determined by the distance between the clouds. The interference patterns which we introduced up to this point appear as a result of the time evolution of single-particle states. The many-body nature of the state comes into eq. (5) only as a prefactor N . In principle, one could have done the same experiment with only one particle. The same result as eq. (5) can be obtained in this case by doing experiments many times and averaging over individual experiments: in each particular realization the particle is observed at some random point r with probability ρ(r, t). In experiments with a large number of atoms, N , used in each shot, each absorption image is a result of N measurements of single-particle wave functions. This performs the statistical averaging implicit in quantum mechanics, and leads to the density proﬁle close to eq. (5) in each shot. . 2 1.2. Second quantized representation. To set up the stage for later we will now present the discussion leading to eq. (11) using the second-quantized formalism. Wave function (3) in the second quantization at t = 0 can be written as (15)

|ϕ, N =

1 (a† eiϕ/2 + a†2 e−iϕ/2 )N |0. (2N N !)1/2 1

Here a†1 and a†2 are creation operators for clouds one and two: (16) a†i = drψi (r)ψˆ† (r). ˆ ψ(r) is the second quantized operator for the boson ﬁeld, which satisﬁes the usual commutation relations ˆ ), ψˆ† (r) = δ(r − r), ˆ ˆ ), ψ(r) (17) = ψˆ† (r ), ψˆ† (r) = 0. ψ(r ψ(r

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Fundamental noise in matter interferometers

Operators a1 , a2 and their conjugates satisfy the canonical boson commutation relations: (18)

ai , a†j = δij ,

[ai , aj ] = a†i , a†j = 0.

Diﬀerent initial states can be simply written using the Fock basis of operators a†1 and a†2 . For example, the initial state for two independent condensates with N1 and N2 particles in clouds 1 and 2 can be conveniently written as 1 (a† )N1 (a†2 )N2 |0. |N1 , N2 = √ N1 !N2 ! 1

(19)

The initial state written in the Fock basis of operators a†1 and a†2 contains all information about properties of the interference amplitudes. During the free expansion the occupation numbers of states one and two do not change, and only the single-particle wave functions ψ1 (r, t) and ψ2 (r, t) evolve. After the expansion, the many-body wave function at time t can be obtained from the initial state written in the Fock basis of operators a†1 and a†2 using the substitutions (20)

a†1 →

(21)

a†2

→

drψ1 (r, t)ψˆ† (r), drψ2 (r, t)ψˆ† (r).

For example, substituting eqs. (20)-(21) into eq. (15), the wave function (3) considered earlier is written as (22)

|ϕ, N, t =

1 N (2 N !)1/2

ˆ † dr(ψ1 (r, t)eiϕ/2 + ψ2 (r, t)e−iϕ/2 )ψ(r)

N |0.

In the long time limit (9) considered earlier, single-particle wave functions ψ1 (r, t), ψ2 (r, t) can be written as (23)

ψ1 (r, t) = u1 (r, t)eiQ1 r ,

(24)

ψ2 (r, t) = u2 (r, t)eiQ2 r ,

where Q1 , Q2 are deﬁned by eqs. (12)-(13) and u1 (r, t), u2 (r, t) are slowly varying real functions, which determine the overall density proﬁles. Since clouds overlap strongly after the expansion, these functions are normalized according to (25) (26)

u1 (r, t)2 dr = 1, u1 (r, t)u2 (r, t)dr ≈ 1.

u2 (r, t)2 dr = 1,

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The operator, which corresponds to the amplitude of density oscillation at wave vector Q is written in the second quantized notations as (27)

ρˆQ =

drˆ ρ(r)eiQr =

iQr ˆ drψˆ† (r)ψ(r)e .

To ﬁnd out the statistical average of the amplitude of density oscillations for state (22), we need to evaluate the following matrix element: (28)

ρQ = ϕ, N, t|ˆ ρQ |ϕ, N, t = ϕ, N, t|

iQr ˆ drψˆ† (r)ψ(r)e |ϕ, N, t.

ˆ To evaluate such matrix elements, ﬁrst we need to know how annihilation operator ψ(r) acts on a state |ϕ, N, t. Since |ϕ, N, t is obtained from |ϕ, N by substitutions (20)-(21), it is easy to see that ˆ N, t = ϕ, N | a†1 u1 (r, t)e−iQ1 r + a†2 u2 (r, t)e−iQ2 r × ϕ, N, t|ψˆ† (r)ψ(r)|ϕ,

× a1 u1 (r, t)eiQ1 r + a2 u2 (r, t)eiQ2 r |ϕ, N , where |ϕ, N is deﬁned in eq. (15) and is written only in terms of a†1 and a†2 . Integration over dr in (28) can be done using normalization (26), and assuming that ui (r, t) vary at scales much larger than 1/Q. Since Q = Q1 − Q2 , evaluation of Fourier transform picks only the product of the ﬁrst term in the ﬁrst parentheses and of the second term in the second parentheses of the equation above. We obtain (29)

ρQ =

ϕ, N |a†1 a2

u1 (r, t)u2 (r, t)dr|ϕ, N =

= ϕ, N |a†1 a2 |ϕ, N = ϕ, N − 1|

N −iϕ/2 e 2

2 |ϕ, N − 1 =

N −iϕ e , 2

which is the same as obtained from a single particle discussion in eq. (11). The example above illustrates how the matrix elements of many-particle operators at time t can be evaluated using initial states written in the Fock basis of operators a†1 and a†2 . In general, when one needs to evaluate an expectation value of some normal ordered ˆ combination of operators ψ(r), and ψˆ† (r) over the ﬁnal state at time t, one needs to make substitutions (30) (31)

ˆ ψ(r) → a1 u1 (r, t)eiQ1 r + a2 u2 (r, t)eiQ2 r , ψˆ† (r) → a† u1 (r, t)e−iQ1 r + a† u2 (r, t)e−iQ2 r , 1

2

and evaluate matrix elements over the t = 0 state, written in the Fock basis of operators a†1 and a†2 . It is important that the expression needs to be normal ordered using

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Fundamental noise in matter interferometers

commutation relations (17) before making substitutions (30)-(31), since after substituˆ tions (30)-(31) ﬁelds ψ(r), ψˆ† (r) do not satisfy the exact commutation relations (17). Another way of seeing this is to realize that in Heisenberg representation (see, e.g., chap. 6 of ref. [105]) substitutions (30)-(31) perform the time evolution of a product of boson annihilation operators ˆ ˆ ˆ ˆ 1 , t) . . . ψ(r ˆ n , t) = eiHt ˆ n )e−iHt ψ(r , ψ(r1 ) . . . ψ(r

(32)

only when this product acts on states of the form 1 |Ψ0 = √ (a†1 )N1 (a†2 )N2 |0. N1 !N2 !

(33)

Thus for calculating the expectation values of the form ˆ

ˆ

Ψ0 |eiHt ρˆ(r1 ) . . . ρˆ(rn )e−iHt |Ψ0 ,

(34)

one ﬁrst needs to normal order ρˆ(r1 ) . . . ρˆ(rn ) using commutation relations (17) as (35)

ρˆ(r1 ) . . . ρˆ(rn ) =

ˆ 1 ), fm (r1 , . . . , rn )ψ † (r1 ) . . . ψˆ† (rm )ψ(rm ) . . . ψ(r

m≤n

and only then use substitutions (30)-(31) to evaluate matrix elements: (36)

ˆ

ˆ

Ψ0 |eiHt ρˆ(r1 ) . . . ρˆ(rn )e−iHt |Ψ0 = ˆ fm (r1 , . . . , rn )Ψ0 |eiHt ψˆ† (r1 ) . . . ψˆ† (rm ) × = m≤n ˆ ˆ ˆ ˆ −iHt ˆ ×e−iHt eiHt ψ(r |Ψ0 = m ) . . . ψ(r1 )e † = fm (r1 , . . . , rn )Ψ0 | a1 u1 (r1 , t)e−iQ1 r1 + a†2 u2 (r1 , t)e−iQ2 r1 m≤n

. . . a†1 u1 (rm , t)e−iQ1 rm + a†2 u2 (rm , t)e−iQ2 rm ×

× a1 u1 (rm , t)eiQ1 rm + a2 u2 (rm , t)eiQ2 rm

. . . a1 u1 (r1 , t)eiQ1 r1 + a2 u2 (r1 , t)e−iQ2 r1 |Ψ0 . . 2 2. Interference of independent clouds. – The surprising phenomenon which was observed in ref. [27] is the appearance of interference fringes in the case when condensates are completely independent. To illustrate how interference fringes appear for independent clouds, let us now discuss the case when the numbers of particles in each of the clouds, N1 and N2 , are ﬁxed, hence the phase diﬀerence between the two clouds is not well deﬁned. Initial state in the Fock basis in this case is given by (19): (37)

1 |N1 , N2 = √ (a† )N1 (a†2 )N2 |0. N1 !N2 ! 1

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Using the formalism of second quantization explained earlier, one can evaluate ρQ by analogy to eq. (29): ρQ = N1 , N2 |a†1 a2 |N1 , N2 = 0.

(38)

However, ρQ = 0 does not imply that there are no interference eﬀects for independent condensates. Indeed, ρQ gives only the statistical average over many experiments, according to the usual interpretation of expectation values of operators in quantum mechanics. Being a quantum operator, ρˆQ has non-vanishing quantum ﬂuctuations. In each particular realization of experiment, complex number ρQ can have a nonzero value. To show this, let us evaluate |ρQ |2 , which is the density-density correlation function at wave vector Q: (39)

4 5 ˆ ψˆ† (r )ψ(r ˆ )eiQ(r−r ) = ρQ ρˆ−Q = drdr ψˆ† (r)ψ(r) |ρQ |2 = ˆ 5 4 ˆ ψ(r ˆ )eiQ(r−r ) + drψ † (r)ψ(r) ˆ . = drdr ψˆ† (r)ψˆ† (r )ψ(r)

These matrix elements can be evaluated using the second quantization prescription of the previous section, and the result is (40)

|ρQ |2 = N1 , N2 |a†1 a†2 a1 a2 + a†1 a1 + a†2 a2 |N1 , N2 = N1 N2 + N1 + N2 .

In the limit of large N1 = N2 = N/2, the leading contribution to |ρQ |2 is the same as for the state with the ﬁxed phase. Information about the full distribution of the quantum operator ρQ is contained in higher moments of the distribution. If one considers higher moments of the type |ρQ |2n = ρnQ ρn−Q , the leading contribution in the limit of large N1 and N2 will again have the form (41)

1 1 |ρQ | = (N1 N2 ) 1 + O +O . N1 N2 2n

n

Corrections which appear because of the normal ordering result in subleading terms which are denoted by O(1/N1 ) + O(1/N2 ). The leading term implies that in the limit of large N1 and N2 , the distribution function of |ρQ |2 is highly peaked near the value N1 N2 , with the relative width which is proportional to the inverse square root of number of particles. Any operator of the form ρnQ ρm −Q will have zero expectation value for m = n similar to ρQ , which means that the phase of the complex number ρQ is uniformly distributed from 0 to 2π. The expectation value of any operator which depends on the phase of ρQ becomes zero due to the averaging over the phase. The physical picture which emerges from the calculations is the following [67, 64, 65]: for two independent ideal clouds in the limit of large N the absolute value of interference fringe amplitude is the same as for the state with a ﬁxed relative phase, but the position

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Fundamental noise in matter interferometers

of the intensity minima ﬂuctuates from shot to shot. The state with a ﬁxed number of particles is a superposition of states with ﬁxed relative phases. For example, (42)

|N/2, N/2 =

πN 2

1/4 0

2π

dϕ |ϕ, N . 2π

In the limit of large N the phase states are almost orthogonal, and the measurement picks some value of the relative phase. Since the relative phase is not well deﬁned for independent clouds, in each particular experiment the positions of the minima will ﬂuctuate from shot to shot. To distinguish independent clouds from states which have correlated relative phases, one needs to do a series of experiments and measure not only the absolute magnitude of interference fringes, but also the positions of the minima. Experiments which distinguish states with a ﬁxed relative phase from some other manybody states are already being done, and can be used i.e. to measure the temperature [106] or to study the dynamical evolution [29, 30, 45, 48, 49] of the relative phase. 3. – Full counting statistics of shot noise As has been explained in the previous section, for experiments with independent clouds the average interference amplitude depends only on the number of particles per cloud. In this section we consider not only the average interference amplitude, but also its shot-to-shot ﬂuctuations due to a ﬁnite number of atoms in the clouds. We will demonstrate that while the average value of |ρQ |2 depends only on the number of particles per cloud, the full distribution function of the variable (43)

R = |ρQ |2

contains information about the states of individual clouds. Our analysis is motivated by the earlier work of Polkovnikov [102], who showed that the variance of the fringe amplitude decreases as the inverse power of the number of particles per cloud, N , with a non-universal coeﬃcient which contains information about the state of the clouds. Experimental observation of the eﬀects discussed in this section requires systems with a small number of atoms. This may be realized with micro-BECs on chips [29, 30, 45-50]. The shot noise for ﬁnite N has a fundamental nature, which stems from the probabilistic nature of quantum mechanics. Distributions of R obtained below correspond to the following “idealized” experimental procedure: release the conﬁning potential and take an absorption image of the columnar density on an ideal CCD camera with 100% eﬃciency (photon shot noise is ignored). To obtain the amplitude of interference fringes, ρQ , extract a Fourier component of the density at wave vector Q from each image separately. The results of many experiments give the histogram W (R) of the values of R = |ρQ |2 . ˆ deﬁned in such way is a many-body operator, We note that the quantum observable R and calculation of its full distribution is a non-trivial task, even when all atoms are in the same state, such as for the case of a well-deﬁned relative phase between atoms in the

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Fig. 6. – Rescaled distribution functions of R = |ρQ |2 for independent clouds in the coherent states (solid) or in the states with a well-deﬁned numbers of atoms (dashed). Here N1 = N2 = 100.

two wells. In this section we develop a general method to ﬁnd distribution functions of R analytically. We note that in our idealized setting we ﬁnd interference patterns at a well-deﬁned wave vector Q. We expect that the ﬁnite size of the systems in transverse direction after expansion and collisions during the initial stage of expansion broaden the peak in the Fourier space to a ﬁnite, but small range of wave vectors around Q [107]. Hence a one should consider R = |ρQ |2 as an integral over the peak in the Fourier image of the density. Also in this paper we will discuss the amplitude of the interference fringes whereas experimental papers typically discuss visibility of the interference patterns. The two quantities diﬀer only by the trivial rescaling. Results for independent clouds in coherent (solid) and number (dashed) states for N1 = N2 = 100 are presented in ﬁg. 6. One can see, that even for a relatively large number of atoms, N = 100, ﬂuctuations due to shot noise are appreciable. In ﬁg. 7 we compare the full counting statistics of R = |ρQ |2 for independent clouds in coherent (dashed) and number (dotted) states with N1 = N2 = 20, and for clouds with a ﬁxed relative phase (solid) with total number of atoms N = N1 + N2 = 40. Distribution functions for the cases of i) well-deﬁned relative phase between the clouds and ii) ﬁxed number of atoms in each cloud are very close. They become indistinguishable when R is rescaled by its average value R, although each of them diﬀers considerably from the Gaussian distribution. The distribution function of R is wider for coherent states compared to number states, as was suggested in ref. [102] based on the study of the variance of the two distributions. Hence the conclusion is that coherent and number states can be easily distinguished based on the statistics of ﬂuctuations of R relative to its average value. In principle, when the two clouds are prepared with the same relative phase over many experiments, it is possible to distinguish independent condensates in number states from states with a ﬁxed relative phase using a set of several interference experiments: minima

Fundamental noise in matter interferometers

553

Fig. 7. – Distribution functions of R = |ρQ |2 for independent clouds in the coherent (dashed) and in the number (dotted) states with N1 = N2 = 20. Solid line is a distribution function of R for clouds with a ﬁxed relative phase with total number N = N1 + N2 = 40. Distribution functions for states with a ﬁxed relative phase and with ﬁxed numbers are very close, and become indistinguishable when R is normalized by its average value R.

positions are uniformly distributed for independent clouds, while for states with a ﬁxed relative phase positions of the interference minima are always at the same points in space. However one can imagine the situation when the clouds are prepared in state with a ﬁxed relative phase but the relative phase itself is random from realization to realization. Our results show that it is practically impossible to distinguish such states from number states by looking at the distribution of the amplitude of interference fringes. Our calculations below provide additional support to the physical interpretation [64-67] presented in the previous section. Fluctuations of the absolute value of the interference amplitude are the same for the cases when clouds have random relative phase and when clouds are prepared in number states so the random relative phase is “measurement induced”. The method developed in this section can be generalized to a variety of experimental situations, i.e. several independent condensates. Diﬀerent squeezed states within individual condensates can be considered, and measurement of full counting statistics of shot noise can be used as an experimental probe to distinguish between diﬀerent correlated states. As noted earlier, we will be interested in the full distribution function of the positive deﬁnite quantum observable (44)

ˆ = |ˆ R ρQ |2 = ρˆQ ρˆ−Q ,

deﬁned by eq. (39). To calculate its full distribution function, W (R), one needs to ˆ n . After that, one has to solve the “problem know expressions for higher moments R of moments”, i.e. to recover the distribution function on the (0, ∞) interval using all moments. In general, this procedure is numerically hard and unstable, unless higher

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moments have a certain analytical form. If the expression for higher moments is known analytically, then one can sometimes avoid the “problem of moments” by calculating the so-called characteristic function, χ(λ), which is the Laplace transform of W (R): (45)

χ(λ) =

∞

e−λR W (R)dR =

0

∞ ∞

0

i=0

∞ ˆn (−λ)n R (−λR)n W (R)dR = . n! n! i=0

If χ(λ) can be calculated analytically, then W (R) can be recovered by the inverse Laplace transform. In our case it is more practical to calculate not the characteristic function, but the analog of the Hankel transformation [108, 109] of W (R), given by (46)

Z(iλ) =

n=∞ n=0

(iλ)2n ˆ n R . (n!)2

Using the expansion of the zeroth order Bessel function, one can write (47)

∞

Z(iλ) =

√ W (R)J0 (2λ R)dR.

0

The inversion of the transformation ∞can be found using the orthogonality condition for the zeroth-order Bessel functions 0 J0 (λx)J0 (λy)|x|λdλ = δ(|x| − |y|), which gives (48)

∞

W (R) = 2

√ Z(iλ)J0 (2λ R)λdλ.

0

By the end of this section we will provide analytical expressions for Z(iλ) for certain cases (see eqs. (60), (67), (70)), from which W (R) can be obtained by simple numerical integration according to eq. (48). To proceed, we note that ρˆQ and ρˆ−Q commute with each other: (49)

[ˆ ρQ , ρˆ−Q ] = 0.

Operators ρˆQ are understood as in eq. (27), without the projection on single particle states ψ1 (r, t), ψ2 (r, t) as in eqs. (30)-(31). The latter substitutions can be only done . after normal ordering (see discussion in subsect. 2 1.2). Hence we ﬁnd 4 (50)

2π

Z(iλ) = 0

5 dϕ iλ(ρˆQ eiϕ +ρˆ−Q e−iϕ ) e . 2π

Indeed, after expanding the exponent and integration, only even degrees of iλ survive, and non vanishing terms are exactly what is needed for eq. (46). The normal ordering of eq. (50) can be done using the following identity: R

(51)

e

ˆ ˆ† (r)ψ(r)dr f (r)ψ

R

=: e

ˆ† (r)ψ(r)dr ˆ (ef (r) −1)ψ

:.

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Fundamental noise in matter interferometers

Here we have assumed that operators ψ † (r), ψ(r), have the canonical commutation relations given by eq. (17). Equation (51) is a generalization of the simpler identity [110]: †

eλa

(52)

a

=: e(e

λ

−1)a† a

:

for operators which obey the commutation relations [a, a† ] = 1.

(53)

The normal ordering signs : : mean that all creation operators should be put to the left of annihilation operators in Taylor expansion of expressions being ordered. To illustrate the meaning of eq. (52), let us consider the expansions of left and right side up to λ2 . The left hand side is

λ2 † † λ2 λ2 † 3 a† a + a† a† aa + O(λ3 ). (54) 1 + λa a + a aa a + O(λ ) = 1 + λ + 2 2 2 The right hand side is (eλ − 1)2 1 + (eλ − 1) : a† a : + : a† aa† a : +O(λ3 ) = 2

λ2 + O(λ3 ) † † λ2 + O(λ3 ) : a† a : + : a aa a : +O(λ3 ) = =1+ λ+ 2 2

λ2 λ2 =1+ λ+ a† a + a† a† aa + O(λ3 ). 2 2 Equation (52) holds not only up to λ2 , but to all orders in λ and plays an important role in quantum optics. Using the deﬁnition of ρQ given by eq. (27), one can apply eq. (51) with f (r) = 2iλ cos(Qr + ϕ) and rewrite Z(iλ) in eq. (50) as (55)

Z(iλ) =

4 : 0

2π

5 dϕ R (e2iλ cos(Qr+ϕ) −1)ψˆ† (r)ψ(r)dr ˆ e : . 2π

. 3 1. Interference of two independent coherent condensates. – Let us ﬁrst explain how to evaluate Z(iλ) for √ independent√clouds in coherent states [8, 111] of operators a1 and a2 with eigenvalues N 1 eiψ1 and N 2 eiψ2 . Since coherent states form a complete basis, any initial state can be expanded in this basis, and thus the problem of calculating of W (R) is essentially solved for arbitrary initial states. Coherent states are convenient, since they are the eigen states of the annihilation operator, and the annihilation operator acts on them as a c-number. Hence after making substitutions (30)-(31) √ into the √ normal ordered expression, one can substitute operators ai , a†i by numbers N i eiψi , N i e−iψi . Since the normal-ordered expression is obtained by the normal ordering of the Taylor expansion of eq. (55), one needs to collect the Taylor series back. For coherent states, the

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A. Imambekov, V. Gritsev and E. Demler

whole procedure is equivalent to removing the normal ordering signs √ making √in eq. (55), substitutions (30)-(31) and treating operators ai , a†i as numbers N i eiψi , N i e−iψi . Thus we obtain (56)

√ √ Z(iλ; N 1 eiψ1 , N 2 eiψ2 ) = 2π

= 0

dϕ R (e2iλ cos(Qr+ϕ) −1)(N1 u1 (r,t)2 +N2 u2 (r,t)2 +2√N1 N2 cos(ψ1 −ψ2 +Qr)u1 (r,t)u2 (r,t))dr e . 2π

. Similar to subsect. 2 1.2 we assume the that normalized functions u1 (r, t) and u2 (r, t) strongly overlap and vary at scales much larger than 1/Q, which is equivalent to einQr uα (r, t)uβ (r, t)dr = δn0 .

(57)

Then integration over dr in the exponent of eq. (56) can be done using the following equations: (58)

e2iλ cos(Qr+ϕ) − 1 (N1 u1 (r, t)2 + N2 u1 (r, t)2 )dr = =

m=∞ m=1 m=∞

(iλ)2m (2m)!

(N1 u1 (r, t)2 + N2 u1 (r, t)2 )(ei(Qr+ϕ) + e−i(Qr+ϕ) )2m dr =

(iλ)2m (2m)! = (J0 (2λ) − 1)(N1 + N2 ); (2m)! m!m! m=1 e2iλ cos(Qr+ϕ) − 1 cos(ψ1 − ψ2 + Qr)u1 (r, t)u2 (r, t)dr = (59) 2 N1 N2 =

(N1 + N2 )

m=∞ (iλ)2m+1 = N1 N2 (ei(Qr+ϕ) + e−i(Qr+ϕ) )2m+1 × (2m + 1)! m=0 i(ψ1 −ψ2 +Qr) × e + e−i(ψ1 −ψ2 +Qr) u1 (r, t)u2 (r, t)dr = m=∞ (iλ)2m+1 (2m + 1)! ei(ψ1 −ψ2 −ϕ) + e−i(ψ1 −ψ2 −ϕ) = N1 N2 (2m + 1)! m!(m + 1)! m=0 = 2i N1 N2 J1 (2λ) cos(ψ1 − ψ2 − ϕ).

=

Substituting eq. (58) and eq. (59) into eq. (56), and doing the integral over ϕ, we ﬁnally obtain the central result of this section: (60)

√ √ Z(iλ; N 1 eiψ1 , N 2 eiψ2 ) = 2π dϕ (J0 (2λ)−1)(N1 +N2 )+2i√N1 N2 J1 (2λ) cos(ψ1 −ψ2 −ϕ) e = = 2π 0 = e(J0 (2λ)−1)(N1 +N2 ) J0 2 N1 N2 J1 (2λ) .

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Fundamental noise in matter interferometers

. 3 2. Interference of independent clouds in number states. – Let us now explain how to calculate Z f (iλ, N1 , N2 ) for the Fock states with the number of particles equal to N1 and N2 . First, we need to expand the Fock states |N1 , N2 using the coherent states basis. Since the basis of coherent states is overcomplete [111], there are many ways to do a decomposition. For our purposes it is convenient to use (61)

2 (a† )N1 (a†2 )N2 √ |N1 , N2 = √1 |0 = N1 !N2 !α−N1 −N2 eα × N1 ! N2 ! 2π 2π dϕ1R dϕ2R −iN1 ϕ1R −iN2 ϕ2R iϕ1R × e |αe , αeiϕ2R , 2 (2π) 0 0

where α is an arbitrary real positive number. Coherent states are given by 2

|αeiϕ1R , αeiϕ2R = e−α

(62)

+αeiϕ1R a†1 +αeiϕ2R a†2

|0,

and the overlap between them equals (63)

2

αeiϕ1L , αeiϕ2L |αeiϕ1R , αeiϕ2R = eα

(eiϕ1R −iϕ1L +eiϕ2R −iϕ2L −2)

.

One can also expand the bra state N1 , N2 | similarly to (61) by introducing integration variables ϕ1L , ϕ2L . For any given values of ϕ, ϕ1L , ϕ2L , ϕ1R , and ϕ2R , matrix elements between coherent states can be evaluated as in previous section, and by simple modiﬁcation of eq. (60) we obtain dϕdϕ1L dϕ2L dϕ1R dϕ2R (64) Z f (iλ, N1 , N2 ) = × (2π)5 2

×N1 !N2 !α−2N1 −2N2 e2α eiN1 (ϕ1L −ϕ1R )+iN2 (ϕ2L −ϕ2R ) × ×αeiϕ1L , αeiϕ2L |αeiϕ1R , αeiϕ2R × 2

×e(J0 (2λ)−1)α

(eiϕ1R −iϕ1L +eiϕ2R −iϕ2L ) iα2 J1 (2λ)(ei(ϕ−ϕ1L +ϕ2R ) +e−i(ϕ−ϕ1R +ϕ2L ) )

e

.

One can now substitute eq. (63) into the equation above, introduce variables (65)

ϕ1 = ϕ1L − ϕ1R

and ϕ2 = ϕ2L − ϕ2R ,

and integrate over ϕ, ϕ1R , ϕ2R . Multiple cancellations occur, and eventually we obtain (66)

Z f (iλ, N1 , N2 ) = 2 −iϕ1 dϕ1 dϕ2 +e−iϕ2 ) N1 !N2 !α−2N1 −2N2 eiN1 ϕ1 +iN2 ϕ2 eJ0 (2λ)α (e × = (2π)2 ×J0 2α2 J1 (2λ)e−i(ϕ1 +ϕ2 )/2 .

Both integrations in the equation above can be done in a closed form for arbitrary positive integer N1 and N2 using hypergeometric functions. Here we will present the results only

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A. Imambekov, V. Gritsev and E. Demler

for N1 = N2 = N . One needs to expand the last exponent and J0 (2α2 J1 (2λ)e−i(ϕ1 +ϕ2 )/2 ) using Taylor series. After integration over dϕ1 and dϕ2 dependence on α disappears, and we obtain the ﬁnal result for Fock states: N 2(N −k) J0 (2λ)2k (iJ1 (2λ)) (67) (N !)2 = Z f (iλ, N, N ) = k!2 (N − k)!2 k=0

J0 (2λ)2 2N = 2 F1 −n, −n; 1; − (iJ1 (2λ)) , J1 (2λ)2 where 2 F1 (a, b; c; x) in a hypergeometric function deﬁned by (68)

2 F1

ab x a(a + 1)b(b + 1) x2 + + ... c 1! c(c + 1) 2!

(a, b; c; x) = 1 +

. 3 3. Clouds with a well-deﬁned relative phase. – Let us now consider the case of clouds with a ﬁxed relative phase, when the initial state |ϕ0 , N at t = 0 is given by eq. (15). This state can be expanded using the coherent states basis as |ϕ0 , N = =

1 (2N N !)1/2

(a†1 eiϕ0 /2 + a†2 e−iϕ0 /2 )N |0 =

√ √ 2 N !( 2α)−N eα

2π

0

dϕR −iN ϕR iϕR +iϕ0 /2 e |αe , αeiϕR −iϕ0 /2 . 2π

A similar expansion can be written for bra- vector ϕ0 , N | using the phase variable ϕL . Coherent states and their overlaps are given by eqs. (62)-(63), and one obtains an expression for the generating function Z(iλ, N ) similar to eq. (64): √ 2 dϕdϕL dϕR N !( 2α)−2N e2α eiN (ϕL −ϕR ) × Z(iλ, N ) = 3 (2π) ϕ0 2

, αeiϕL −i

ϕ0 2

ϕ0

ϕ0

|αeiϕR +i 2 , αeiϕR −i 2 × 2 i(ϕ−ϕ0 +ϕR −ϕL ) 2 iϕR −iϕL +e−i(ϕ−ϕ0 +ϕL −ϕR ) ) . eiα J1 (2λ)(e ×e2(J0 (2λ)−1)α e

×αeiϕL +i

The integrand depends only on the diﬀerence Δϕ = ϕL − ϕR , and the integral over ϕ can be done analytically. Dependence of Z(iλ, N ) on ϕ0 drops out, as expected:

2 −iΔϕ dΔϕ (69) Z(iλ, N ) = N !α−2N 2−N eiN Δϕ e2J0 (2λ)α e J0 2α2 J1 (2λ)e−iΔϕ . 2π Expanding the last exponent and J0 (2α2 J1 (2λ)e−iΔϕ ) in the expression above and integrating over Δϕ, we obtain the ﬁnal expression for even N : (70)

N/2 (iJ1 (2λ))2k (2J0 (2λ))N −2k J0 (2λ) N −N −N Z(iλ, N ) = N !2−N = 2 (iJ (2λ)) C , 1 N (k!)2 (N − 2k)! iJ1 (2λ) k=0

where Cnα (x) is a Gegenbauer polynomial [109].

Fundamental noise in matter interferometers

559

Fig. 8. – Experimental setup for interference of 2D gases. Note that the interference patterns are straight at low temperatures indicating suppressed phase ﬂuctuations. Meandering patterns at high temperatures come from strong phase ﬂuctuations. Taken from ref. [32].

4. – Interference of ﬂuctuating low-dimensional gases As we discussed in previous sections, for macroscopic three-dimensional Bose-Einstein Condensates the long-range phase coherence manifests itself in the nearly perfect interference fringes between two independent condensates [27]. For low dimensional Bose gases, the situation is diﬀerent, since phase ﬂuctuations are very eﬀective in destroying the long-range order. In one dimension, long-range coherence is prohibited even at zero temperature [95], while in two dimensions any nonzero temperature destroys long-range order [94]. In addition, the Berezinskii-Kosterlitz-Thouless (BKT) phase transition occurs [88, 89], which separates the low-temperature phase with power law correlations from the high-temperature phase with short-range correlations. Phase ﬂuctuations reduce the average visibility of the interference fringes, and result in the shot-to-shot ﬂuctuations of the visibility. In this section, we discuss how measurements of interference fringes can reveal information about spatial correlations within individual condensates. The typical experimental setups are shown in ﬁgs. 4 and 8. They correspond to the so-called open boundary

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A. Imambekov, V. Gritsev and E. Demler

conditions (OBC). Essentially the OBC mean that the imaged area is cut out of a larger system. As a theoretical model one can also consider a one-dimensional condensate with periodic boundary conditions (PBC), which corresponds to interference experiments with two coaxial rings lying in two parallel xy-planes. While this model is somewhat artiﬁcial from the experimental point of view (see however ref. [112]), it allows a very elegant theoretical analysis, hence we will discuss it in these lecture notes as well. The conﬁning potential is highly anisotropic, and after it is switched oﬀ, the clouds predominantly expand in the transverse direction, while no signiﬁcant expansion occurs in the axial (for 1D gases) or in-plane (for 2D gases) directions. For low-dimensional gases the phase of the condensate does not have a long-range order due to quantum or thermal ﬂuctuations. Locally the phase determines the positions of the minima of the absorption intensity, and ﬂuctuations of the phase lead to ﬂuctuations of the interference fringe positions along the condensates, as shown in ﬁg. 4. Fluctuations of the fringe positions contain information about the original phase ﬂuctuations present in the system, which are preserved during expansion. To extract information about fringe position ﬂuctuations for the 1D case, we will integrate the intensity along the axes of the clouds. Fluctuations of the relative phase result in ﬂuctuations of the minima positions for diﬀerent x. For each y, the image can be integrated along the x direction to give the integrated fringe amplitude ρQ (L) (see ﬁg. 4). Note that the integrated fringe amplitude depends on the integration length L. One experimental image can be used to extract information for diﬀerent values of L. Many images are still required to obtain distribution functions for each L. For 2D gases, the setup is analogous and is shown in ﬁg. 8. Here part of the integration is performed by the imaging beam itself. The size of the integration area along the direction of the imaging beam can be controlled by applying magnetic ﬁeld gradients, so that only a speciﬁed section of the cloud is resonant with the probe light. The operator which corresponds to the fringe amplitude ρQ (L), illustrated in ﬁg. 4, is the same as ρQ deﬁned by eq. (27), where the integration along x-dimension is limited to the section of length L. Let us ﬁrst consider the expectation value |ρQ (L)|2 (expectation values of operators which depend on the phase of ρQ (L) vanish, similar to 3D case, since two clouds are assumed to be independent). One has to use modiﬁed formulas (30)-(31), where operators a†i , ai are now allowed to have x-dependence. In the limit when the number of particles in the section of size L is large, the average value of |ρQ (L)|2 is given by [55] (71)

|ρQ (L)|2 = 0

L

0

L

dx1 dx1 a†1 (x1 )a†2 (x1 )a1 (x1 )a2 (x1 ).

Note that in eq. (71) we used the normal ordered form of the operators which means that we neglect the shot noise considered in sect. 3. This is justiﬁed for long condensates as we discuss below. From now on, we will concentrate on the case when independent clouds are identical and have the same density of particles with equal interaction strengths.

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Fundamental noise in matter interferometers

Then (72)

|ρQ (L)|2 = 0

L

0

L

dx1 dx1 a† (x1 )a(x1 )2 .

To gain intuition into the physical meaning of the average amplitude of interference fringes, we address two limiting cases. First, consider the situation when a† (x)a(0) decays exponentially with distance and the correlation length is given by ξ L. Then √ eq. (72) implies that |ρQ (L)| ∝ Lξ, which has a simple physical interpretation. Since the phase is only coherent over a length ξ, the system is eﬀectively equivalent to a series of L/ξ pairs of independent condensates. Each pair contributes interference fringes with a constant amplitude proportional to ξ and a random phase. The total amplitude ρQ (L) is the result of adding L/ξ independent vectors of constant length ξ and random direction. Adding random uncorrelated vectors gives a zero average except for a typical square root √ ﬂuctuation. Thus scaling of the absolute value of the net interference amplitude is Lξ. This observation is similar in spirit to that made in ref. [68] of interference between 30 independent condensates in a√chain. Fringes can be seen, though their average amplitude is suppressed by a factor of 30 compared to the interference between two condensates. Now consider the opposite limit of perfect condensates, for which a† (x)a(0) is constant. In this case eq. (72) implies that |ρQ (L)| ∝ L. Pictorially this is the result of adding vectors which are all aligned, resulting in a fringe amplitude absolute value of which scales as the total size of the system. Methods developed in this section for analyzing |ρQ (L)|2 can be applied to condensates with either uniform and non-uniform densities. For simplicity, we concentrate on the case when L is much smaller than the size of the clouds, so the change in the atomic density along the clouds can be ignored. In this case correlation functions for 1D gases are described by the Luttinger-liquid theory [113, 114]. For OBC at zero temperature two-point correlation functions are given by (73)

a† (x)a(y) ∼ ρ (ξh /|x − y|)

1/2K

.

Here ρ is the particle density, ξh is the healing length, which also serves as the short range cutoﬀ, and K is the so-called Luttinger parameter, which characterizes the strength of interactions. For bosons with a repulsive short-range potential, K ranges between 1 and ∞, with K = 1 corresponding to strong interactions, or “impenetrable” bosons, while K → ∞ for weakly interacting bosons. Substituting eq. (73) into eq. (72) and assuming that L ξh , we obtain [55] (74)

˜ 2 L2 |ρQ (L)|2 = Cρ

ξh L

1/K ,

where C˜ is a constant of order unity. We see that the amplitude of the interference fringes |ρQ (L)| scales as a non-trivial power of the imaging length. In the non interacting limit

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(K → ∞) the scaling is linear |ρQ (L)| ∼ L, as expected for a fully coherent system. √ Interestingly, |ρQ (L)| ∼ L appears in the hard core limit (K = 1), as in systems with short-range correlations which were discussed above. One may be concerned that eq. (74) gives only the long-distance asymptotic behavior of the correlation functions, and does not describe the short-distance behavior. From eq. (72) one ﬁnds that the contribution of the short-distance part of the correlation functions to |ρQ (L)|2 scales as L. In the physically relevant case of K > 1 and in the limit of large L this contribution is smaller than eq. (74). We note that in principle one can use the exact Bethe ansatz solution of the Lieb-Liniger model [115] to obtain correlation functions valid at all distances [116]. Another contribution which has been neglected is the shot noise. The shot noise contribution to |ρQ (L)|2 comes from the normal ordering of operators, and equals L

0 L L

L

dx1 dx1 a†1 (x1 )a2 (x1 )a†2 (x1 )a1 (x1 ) −

(75) 0

− 0

0

L

= 0

0

L

dx1 dx1 a†1 (x1 )a†2 (x1 )a1 (x1 )a2 (x1 ) = dx1 dx1 δ(x1 − x1 )a†1 (x1 )a1 (x1 ) = n1D L.

In the limit of large L and K > 1 this is again a subleading contribution and can be neglected. For 2D, one can use similar approach to describe the contrast distribution at ﬁnite temperature below the BKT transition. We note that we assume that the temperature is small enough such that 2D gas is in a quasicondensate regime [117,118], when only phase ﬂuctuations are present. In this case, correlation functions are given by [88, 89, 119] (76)

†

a (r)a(0) ∼ ρ

ξh r

η(T ) ,

where η(T ) = mT /(2π2 ρs (T )) depends on the temperature and the superﬂuid density ρs (T ). The BKT transition takes place at the universal value ηc (Tc ) = 1/4. To keep connection to the 1D case, we introduce (77)

K = 1/(2η(T )),

and restrict our attention to K > Kc = 2. For temperatures above the BKT transition eq. (76) does not hold, and correlations decay exponentially. This means that the integrated interference amplitude will only increase as the square root of the integration area [55]. Figure 8 illustrates experiments performed with 2D gases to identify the BKT transition [32]. Two independent 2D condensates are conﬁned in transverse directions using an optical lattice potential. After the optical potential is switched oﬀ and clouds expand,

Fundamental noise in matter interferometers

563

Fig. 9. – Emergence of quasi-long-range order in a 2D gas. a, Examples of average integrated ˜ 2 are shown for low (blue circles) and high (red squares) temperatures; interference contrasts C Lx is the integration length along x-direction. The lines are ﬁts to the data using the power-law functions 1/(Lx )2α . b, Exponent α as a function of the central contrast c0 . Central contrast c0 serves as a “thermometer”, such that smaller values of c0 correspond to higher temperatures. Dashed lines indicate theoretically expected values of α above and below the BKT transition in a uniform system. Taken from ref. [32].

the density is imaged on a CCD camera. When temperature is small, interference fringes are straight lines. As the temperature is increased, the fringes start to meander due to spatial ﬂuctuations of the phase. Integrating the image along the section of variable length L in x-direction gives L-dependent fringe amplitude |ρQ (L)|. Scaling of this amplitude with L contains information about η(T ), which is expected to have a universal value ηc (Tc ) = 1/4 at the BKT transition. Figure 9 illustrates the procedure used to extract scaling exponents in experiments. |ρQ (L)|2 (denoted as C˜ 2 in ﬁg. 9A) is plotted as a function of L, and its scaling with

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A. Imambekov, V. Gritsev and E. Demler

L in a certain range (see ref. [32] for more details) is used to extract the exponent η(T ) (denoted as α in ﬁg. 9B). The average central contrast c0 serves as a “thermometer”, such that smaller values of c0 correspond to higher temperatures. Above the BKT temperature the value of η(T ) extracted from interference experiments is expected [55] to be equal to 0.5, while at the transition point it is equal to ηc (Tc ) = 1/4. Figure 9B shows a sudden change of the exponent in a relatively narrow range of temperatures. This change is reminiscent of the universal jump in the superﬂuid density for 2D helium ﬁlms [120]. Of course, since experiments are done with ﬁnite systems and imaging done along the y-direction performs averaging over the inhomogeneous density proﬁle, one should not expect the universal jump, but rather a crossover. The presence of the trap also aﬀects parameters of the BKT transition [121, 122]. The BKT transition is an example of a topological transition which is driven by the unbinding of vortices, and the remarkable feature of experiments in ref. [32] is the ability to independently resolve the vortices. When only one of the condensates has a vortex, the interference pattern will have a disclination-like structure [31]. It was shown experimentally in ref. [32] that proliferation of vortices occurs at the same point in the parameter space as the jump in the scaling exponent. In following sections we will show that in uniform systems not only the scaling exponent, but also the full distribution function of the fringe contrast has a universal form at the BKT transition. . 4 1. Interference amplitudes: from high moments to full distribution functions. – Measuring atom density to obtain an interference pattern is a classical measurement on a quantum-mechanical wave function. The process of the measurement itself introduces an intrinsic quantum-mechanical noise. Said diﬀerently, from shot to shot we will not get precisely the same value of the amplitude of interference fringes. Expressions for |ρQ (L)|2 which we derived in eqs. (71)-(74) correspond to averaging over many shots. For example, data points from ﬁg. 9A correspond to averaging over approximately one hundred measurements [33]. However each individual shot will give the value of |ρQ (L)|2 which may be diﬀerent from its average value. An interesting question to consider is how this amplitude ﬂuctuates from one experimental run to another. To address this question we need to consider higher moments of the operator |ρQ (L)|2 . Generalizing the argument which lead to eq. (72), we obtain (78)

|ρQ (L)|2n = L L = ... dx1 . . . dxn dx1 . . . dxn |a† (x1 ) . . . a† (xn )a(x1 ) . . . a(xn )|2 . 0

0

In eq. (78) we used a normal ordered correlation function similar to eq. (72). One can calculate [102] corrections due to normal ordering for higher moments of |ρQ (L)|2 , and show that in the limit of large L and K > 1 they can be neglected. From eq. (72) we observe that |ρQ (L)|2 contains information about two-point correlation functions of individual clouds. Equation (78) shows that higher moments of |ρQ (L)|2 contain information about higher-order correlation functions. The full distri-

565

Fundamental noise in matter interferometers

bution of the ﬂuctuating variable |ρQ (L)|2 contains information about all high-order correlation functions. In the Luttinger liquid theory ﬂuctuations of the phase are described by the Gaussian action. For Gaussian actions higher-order correlation functions are simply related to two-point correlation functions (see, e.g., ref. [119]): *

(79)

†

†

a (x1 ) . . . a

(xn )a(x1 ) . . . a(xn )

=*

† ij a (xi )a(xj ) * † † i<j a (xi )a(xj ) i<j a (xi )a(xj )

.

Using this formula together with eq. (73), the higher moments of fringe amplitudes can be written as 1/K (80) |ρQ (L)|2n = A2n Z , where A = Cρ2 ξh L2−1/K , 2n 0 0 C is a constant of the order of unity and for OBC in 1D (81)

1

Z2n =

1

... 0

=

0

1

... 0

1

/* /1 * / − uj | i<j |vi − vj | // K / i<j |ui* du1 . . . dvn / / = / / ij |ui − vj | 1

P

du1 . . . dvn e K (

i<j

G(ui ,uj )+

P i<j

G(vi ,vj )−

P ij

G(ui ,vj ))

.

0

Here for OBC G(x, y) is given by (82)

G(x, y) = log |x − y|.

Integrals similar to eq. (81) appeared in the literature before [123, 124], but they are not easy to compute. The Gaussian model possesses a powerful conformal symmetry [114, 119], which dictates the form of the correlation functions for periodic boundary conditions or nonzero temperatures. For PBC with circumference of the condensates equal to the imaging length, the change in the correlation functions leads to [114] (83)

Gper (x, y) = log

1 sin π|x − y|. π

For nonzero temperature, Z2n depends on K and the thermal length ξT = vs /(kB T ), where vs is the sound velocity:

ξT π|x − y|L ξT sinh = log (84) G x, y, . L πL ξT . The analysis in this section (except for subsect. 4 3) does not depend on the particular form of G(x, y). The only general restriction is (85)

G(x, y) = G(y, x).

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A. Imambekov, V. Gritsev and E. Demler

Equation (81) is also valid for 2D case below the BKT transition temperature, with ui and vi being 2D variables on a rectangle with G(x, y ) = log |x − y |, and with a properly redeﬁned A0 . In what follows, we will be interested in distribution functions W (α) of the variable α = |ρQ (L)|2 /A20 and of its normalized version α ˜ = |ρQ (L)|2 /ρQ (L)2 . By the deﬁnition of higher moments (86)

Z2n =

∞

n

W (α)α dα, 0

and

Z2n = Z2n

∞

˜ (˜ W α)˜ αn dα. ˜

0

The general problem we consider now is how to construct distribution functions W (α) given by eq. (86) with Z2n given by eq. (81). G(x, y) can be an arbitrary symmetric function of x and y, and not necessarily the function of their diﬀerence. This allows us to study systems with non-uniform density in external traps, where one needs to . use modiﬁed correlation functions [118]. In subsect. 4 2 we will show that W (α) is connected to the partition functions of various Sine-Gordon models and Coulomb gases, and methods developed in this section provide a new non-perturbative tool for calculating partition functions of such models. One can think of Z2n in eq. (81) as partition functions of a classical two-component gas of ﬁctitious charged “particles” in a microcanonical ensemble, with K being the “temperature”. We ﬁrst brieﬂy comment on the limiting cases K 1, and K → 1. For K 1, expansion of Z2n gives Z2n ≈ 1 in the zeroth order. This means that ˜ (˜ W α) ≈ δ(˜ α − 1), i.e. a very narrow function peaked near its average value. Higher˜ (˜ order terms in the expansion give W α) a small width of the order of 1/K, and are studied in detail in Appendix A. For 1D gases at zero temperature with K → 1 or ˜ (˜ nonzero temperatures with ξT K/L 1 the distribution function W α) is Poissonian −α ˜ ˜ irrespectiveof OBC or PBC: W (˜ α) = e . To demonstrate this, we need to verify that ∞ n −α Z2n /Z2n = 0 α ˜ e ˜ dα ˜ = n!. This can be shown using the classical gas analogy: as K → 1, Z2 = 1/|x − y|1/K dxdy starts to diverge for x → y, and the main contribution to Z2n comes from “molecular” states of the two-component gas, i.e. from the parts of the conﬁguration space in which each “particle” has a “particle” of the opposite charge in its neighborhood; n! is just the number of ways to form such pairs. In this language K 1 corresponds to the “plasma” phase of the classical charged gas, and the evolution of the distribution function can be understood as a formation of “molecules” out of the “plasma” phase as the “temperature” (i.e. the Luttinger parameter K) is lowered. For ﬁnite ξT , the main contribution to Z2n comes from distances ∼ ξT K/L, and if this parameter is much smaller than 1, then again “molecular” contributions dominate. There is a simple physical interpretation to W (α) for both K 1 and K → 1. When bosons do not interact and K = ∞, there should be no phase ﬂuctuations within individual condensates. Hence in each experiment we should ﬁnd a perfect interference pattern although positions of the density minima are unpredictable (see discussion in sect. 2). Alternatively in the regime of strong interactions when K → 1 we can think of the net interference as a result of adding many random uncorrelated two-dimensional

Fundamental noise in matter interferometers

567

vectors (see discussion after eq. (72)). Earlier we used the fact that for a random walk the net displacement is proportional to the square root of the number of steps. But we also know that for 2D random walks the distribution function of the square of the net ˜ (˜ displacement is Poissonian, which is what we ﬁnd for W α). ˜ As K varies from 1 to ∞, the distribution function W (˜ α) should evolve from being a very broad Poissonian function to a narrow delta-function. The evolution of the distribution in the intermediate regimes will be studied in detail below. In Appendix A we develop a systematic expansion of Z2n in powers of 1/K for large K which works for diﬀerent dimensions and boundary conditions, and investigate the distribution functions . in this limit. The rest of this section is organized as follows. In subsect. 4 2 we discuss the connection of our problem to the Sine-Gordon models, which describe various physical problems, ranging from ﬂux line in superconductors [125] to string theory [126]. . In subsect. 4 3 we obtain full distributions for the 1D case with PBC using the exact . solution of the boundary Sine-Gordon model on a circle [56]. In subsect. 4 4 we present a novel non perturbative solution which is applicable for any value of K and for various dimensions and boundary conditions. We discuss the connection of the distribution functions of fringe visibilities to the statistics of random surfaces [58], and prove that for 1D case with periodic boundary conditions in the limit of large K the distribution of fringe visibilities is given by the Gumbel distribution, one of the extreme value statistical distributions [127]. . 4 2. Connection of the fringe visibility distribution functions to the partition functions of Sine-Gordon models. – To illustrate the connection of the fringe visibility distribution functions to the partition functions of Sine-Gordon models we start from a 1D system with periodic boundary conditions. Shortly we will demonstrate that this example provides a unique possibility to make an excursion into very diﬀerent subjects of physics and mathematics. As shown earlier, for a one-dimensional ring condensate with PBC and circumference equal to the imaging length, the correlation functions lead to / / / /1 (87) Gper (x, y) = log // sin(π(x − y))// . π This function describes a two-component gas of particles interacting via a 1D Coulomb potential, which is a logarithmic in the interparticle separation. For PBC the distance is a chord function sin(π|x − y|)/π. It is common to deﬁne the grand partition function of such Coulomb gas as (88)

Z0 (K, g) =

∞ g 2n per Z , (n!)2 2n n=0

per where Z2n is given by eq. (81) with G(x, y) substituted by Gper (x, y). Several diﬀerent physical problems can be related to the partition function given by eq. (88). Here we mention only a few examples: a) the anisotropic Kondo model [128]; b) the quantum impurity model in one-dimensional interacting electron systems introduced

568

A. Imambekov, V. Gritsev and E. Demler

in ref. [103], which has been extensively studied in the context of edge states in Quantum Hall systems [129]; c) the background-independent string theory and the model of strings attached to a D-brane originally introduced in ref. [126] (see ref. [130] for a recent review); d) Calogero-Sutherland model [131], which has numerous application as an eﬀective model; e) ﬂux line pinning in superconductors [125]; f) quantum tunneling in the presence of dissipation within Caldeira-Leggett model [132]; g) interference of two one-dimensional condensates [55, 56, 58]. We now brieﬂy describe how the expression given by eq. (88) appears as a partition function in physical systems. We consider the imaginary time action (89) Sper [g] =

1 2

∞

−∞

1

1

dτ [(∂τ φ)2 + (∂x φ)2 ] + 2g

dx 0

dτ cos[βφ(x = 0, τ ) + 2πpτ ] 0

known as the boundary Sine-Gordon model [133]. The quantum ﬁeld φ(x, τ ) in eq. (89) is deﬁned on an inﬁnite line in the x-direction and is assumed to be periodic along the τ direction: φ(x, τ ) ≡ φ(x, τ + 1). The interaction term is present only at x = 0. A typical physical system which is described by action (89) is an interacting 1D electron liquid scattered by an impurity [103]. In this case the cos-term describes backscattering of electrons within the Luttinger liquid formalism. In eq. (89) we also added a p-dependent phase winding term. For a quantum impurity problem this is somewhat reminiscent of having a ﬁnite voltage [134], while in interference experiments this term corresponds to a . relative momentum along the condensates [56] (except for subsect. 4 3 we only consider the case with p = 0). Partition function of eq. (89) is deﬁned as [123] (90)

Zp (K, g) Zp (K, g) = , Zp (K, g = 0)

Zp (K, g) =

Dφe−Sper [g] .

One can expand Zp (K, g) in Taylor series of the coupling g. Nonvanishing contributions come from combinations which have equal number of exp[+iβφ] and exp[−iβφ] terms. This is essentially a charge neutrality condition for the Coulomb gas. If we identify (91)

β2 1 = , 2π K

1

x = g (2π) 2K ,

we obtain (92)

Zp (K, g) =

∞ x2n (p) Z . (n!)2 2n n=0 (p)

Here the microcanonical partition functions Z2n are represented by the following integrals: (93) /* ui −uj * vk −vl // K1 2π 2π " n 2 sin dui dvi i2p Pi (ui −vi ) // i<j 2 sin / k
569

Fundamental noise in matter interferometers (p)

per Z2n for p = 0 is related to Z2n deﬁned above by per Z2n = Z2n (2π)−n/K . (0)

(94) Then (95)

∞ ∞ g 2n per g 2n Z = αn = 2n 2 2 (n!) (n!) n=0 n=0 ∞ ∞ ∞ √ g 2n n = W (α)α dα = W (α) I0 (2g α) dα. 2 (n!) 0 0 n=0

Z0 (K, g) =

Z0 (K, g) is essentially a Hankel transformation of W (α), and inverting eq. (95) we can express the probability W (α) through the partition function Z0 (K, g). Noting ∞ that I0 (ix) = J0 (x) and using the completeness relation for Bessel functions, J0 (λx)J0 (λy)|x|λdλ = δ(|x| − |y|), we obtain 0 (96)

∞

W (α) = 2

√ Z0 (K, ig)J0 (2g α)gdg.

0

It is important that the last equation has the partition function at imaginary value of the coupling constant. This should be understood as an analytic continuation of Z0 (K, g). For PBC in 1D, partition functions Zp (K, g) can be evaluated using the exact solution of the boundary Sine-Gordon model with periodic boundary conditions, and this approach . will be presented in subsect. 4 3. For open boundary conditions one can also write the grand canonical partition function (97)

Z(g) =

∞ g 2n Z2n , (n!)2 n=0

where Z2n are given by eqs. (81), (82). We can express eq. (97) as a partition function of a certain Sine-Gordon model: Dφe−S(g) , (98) Z(g) = Dφe−S(0) where (99)

S(g) = πK

∞

−∞

dx

∞

−∞

2

2

dτ [(∂τ φ) + (∂x φ) ] + 2g

1

dτ cos[2πφ(x = 0, τ )]. 0

Note diﬀerent limits of the τ integration in the ﬁrst and the second terms. One can see that eqs. (97) and (98) deﬁne the same Z(g) by observing that − log(|x − y|)/(2π) is a free propagator of the Gaussian action on a plane.

570

A. Imambekov, V. Gritsev and E. Demler

So far we established the relation between the distribution functions of the amplitude of interference fringes of 1D condensates and the boundary Sine-Gordon models (89), (99). Generalization of this argument to the case of 2D condensates is straightforward. Higher moments of the interference amplitude are given by integrals of type (81), but each u and v is now a two-dimensional coordinate. Distribution function of fringe amplitudes is then related to the partition function of the bulk Sine-Gordon model (100) S2D (g) = πK

∞

−∞

dx

∞

−∞

1

dτ [(∂τ φ)2 + (∂x φ)2 ] + 2g

1

dτ dx cos[2πφ(x, τ )]. 0

0

Expanding the partition function corresponding to action (100) in powers of g, one ﬁnds the expression identical to eq. (95), but with W (α) corresponding to interference of 2D condensates. When we describe interference of systems with OBC, we use eqs. (99) and (100), in which the ﬁeld φ(x, τ ) is deﬁned on a whole plane, and is not periodic in τ . In both these cases the interaction is present only in some part of the system, so translational invariance is lost in both x and τ dimensions. Thus to calculate W (α) using eq. (96) one needs to calculate partition functions of inhomogeneous Sine-Gordon models. Exact solution of Sine-Gordon model is available only for periodic boundary conditions, so to treat open boundary conditions one needs to develop alternative methods. Conversely, if one has a solution for W (α), this provides a tool for calculating partition functions of . inhomogeneous Sine-Gordon models and Coulomb gases using eq. (95). In subsect. 4 4 we present a novel mapping of W (α) for arbitrary G(x, y) to the statistical properties of random surfaces, which provides a new tool for calculating partition functions of a wide class of Sine-Gordon models and Coulomb gases. In general, partition functions with inhomogeneous g(x, τ ) can be evaluated using this mapping as well. In this case the function g(x, τ ) will appear in the integrand of eq. (100). We point out that the mapping which we use is not related to the existence of the exact solution of Sine-Gordon models, but relies only on the structure of the correlation functions in the absence of interactions. We also note that a suitable extension of our method can be used to compute correlation functions of Coulomb-gas models in equilibrium and non-equilibrium situations. . 4 3. Distribution functions for 1D gas with periodic boundary conditions. – There are three natural ways to compute partition function Z0 (K, g) for complex g, which can be used to construct distribution functions for 1D gas with PBC using eq. (96). The ﬁrst one is related to the theory of symmetric polynomials and is described in appendix B. The second one relies on the integrability of the quantum impurity model deﬁned by the action (89), and makes use of the thermodynamic Bethe ansatz (TBA). This approach is discussed in appendix C. These two approaches have been presented in refs. [123], [135]. They are diﬃcult to implement directly and are included for completeness. The third . approach described below in subsect. 4 3.1 leads to the most transparent answer. Although it is intrinsically related to the ﬁrst two, it has a broader applicability and will be studied in detail.

571

Fundamental noise in matter interferometers

At imaginary g the theory (89) is apparently non-Hermitian, but as we will see later the model belongs to a special class of non-Hermitian ﬁeld theories, which have real spectrum. We describe an interesting connection with PT -symmetric quantum mechanics and conformal ﬁeld theories with negative central charges. Here we also make close contact with the impurity problem in a Luttinger liquid [103]. This connection is based on the following observation: when we consider Taylor expansion of the quantum impurity partition function in powers of g, we need to take multi-point correlation functions at the same x but at diﬀerent times. To calculate higher moments of the interference amplitude signal we need to use correlation functions for equal time, but at diﬀerent points in space. The two are the same because of the relativistic invariance of the Luttinger liquid [114, 119]. . 4 3.1. Mapping to integrable structure of CFT and singular anharmonic oscillator. To introduce our method, we need to make several formal remarks regarding one dimensional conformal ﬁeld theories. In a recent series of papers Bazhanov, Lukyanov, and Zamolodchikov explored an integrable structure of conformal ﬁeld theories focusing on connections to solvable problems on lattices [136]. The key ingredients of the solvability of lattice models are the so-called transfer matrix operators T(λ). These operators contain information about all integrals of motion as well as excitation spectra of the system. Transfer matrices are deﬁned as a function of the so-called spectral parameter λ (in the continuum limit λ corresponds to rapidity) and commute for diﬀerent values of λ. The latter property is a direct manifestation of the existence of an inﬁnite number of commuting integrals of motion. In his studies of the 8-vertex model, Baxter [137] introduced the operator Q(λ) which helps to ﬁnd eigenvalues of T. Operators T and Q± satisfy a set of commutation relations, in particular [136] T(λ)Q± (λ) = Q± (qλ) + Q± (q −1 λ),

(101)

where q = exp(iπ/2K). So T matrices can be obtained explicitly when one knows the Q operators. By construction, these operators act in the representation space of Virasoro algebra which can be constructed from the Fock space of free bosonic operators a±n deﬁned such that (102)

an |p = 0,

for n > 0,

P |p = p|p,

[an , am ] =

n 2π δn+m,0 . 2K

Here |p denotes the vacuum vector and p is the zero mode of the corresponding bosonic ﬁeld φ(u): (103)

φ(u) = iQ + iP u +

a−n einu . n

n =0

Operators A± (λ) = Q± (λ)λ∓4p/K act in the representation space of Virasoro algebra, which can be constructed from the Fock space of bosonic operators a±n satisfying

572

A. Imambekov, V. Gritsev and E. Demler

an |p = 0, (n > 0). The Fock vacuum state |p is an eigenstate of the momentum operator, P |p = p|p. For p = N/2 (N = 0, 1, 2 . . .) the vacuum eigenvalues of the (vac) operator A± (λ), A± (λ)|p = A± |p, are given by (below we consider only the quantities with the + subscript which correspond to the positive p) A(vac) (λ) = Zp (K, −ig),

(104) where (105)

λ=

π x sin . π 2K

At this point it is not clear what we gained by connecting the analytically continued partition function of the impurity problem Zp (K, ig) to the expectation value of the operator A(vac) (λ). As we discuss below, a considerable number of important results have been derived for A(vac) (λ). We will be able to make use of these results to obtain the distribution functions of the fringe amplitudes. The function A(vac) has known large-λ asymptotics [136]

K log(A(vac) )(λ) ∼ M (K) −λ2 2K−1 ,

(106)

where the constant M (K) is given by √ (107)

M (K) =

2K

1 2K−1 π Γ( 4K−2 )[Γ( 2K−1 2K )] π K cos( 4K−2 )Γ( 2K−1 )

.

The function A(vac) (λ) is entire function for K > 1 and is completely determined by its zeros λk , k = 0, 1, . . . . Therefore A(vac) (λ) can be represented by the convergent product (108)

(vac)

A

(λ) =

∞ " k=0

λ2 1− 2 λk

,

A(vac) (0) = 1.

On the basis of analysis of a certain class of exactly solvable model, corresponding to the integrable perturbation of the conformal ﬁeld theory, it was conjectured in [138] that the so-called Y -system and related T system (where Y = er and r are the Bethe-ansatz energies parametrized by r, the nodes of the Dynkin diagrams (see TBA-section) satisfy the same functional equations and possess the same analytical structure and asymptotics as the spectral determinant of the one-dimensional anharmonic oscillator. Further, the same functional equations, analytical properties (108) and asymptotics (106) are satisﬁed for the vacuum eigenvalues of Q-operator for special values of p and the latter are given by the spectral determinant of the following Schr¨ odinger equation (109)

(−∂x2 + x2α )Ψ(x) = EΨ(x).

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Fundamental noise in matter interferometers

The spectral determinant is deﬁned as (110)

D(E) =

∞ " n=1

1−

E En

.

Soon after, in ref. [139], this conjecture has been extended to all values of p: A(vac) (λ, p) = D(ρλ2 ),

(111)

where now D(E) is the spectral determinant of the Schr¨ odinger equation −∂x2 Ψ(x)

(112)

4K−2

+ x

l(l + 1) + x2

Ψ(x) = EΨ(x),

with l = 4pK − 1/2. Here ρ = (4K)2−1/K [Γ(1− 1/(2K))]2 . In relation to the interference problem, p = (md/t) tan θ, where m is the atom’s mass, d is the separation between the two condensates and t is the time when the measurement was done after the free expansion started. For some values of the parameters the eq. (112) can be solved exactly: a) K = 1. In this case, corresponding to a singular harmonic oscillator En = 4n+2l−1, n = 1, 2, . . . and (113)

D(E, l) =

Γ( 34 + 2l )e−CE Γ( 34 +

l 2

−

E 4)

,

where C is nonuniversal renormalization constant. b) The case K → ∞ is recovered by the rigid-well potential for which the eigen energies are given by the zeroes of the Bessel function, and therefore (114)

√ √ D(E, l) = Γ(l + 3/2)( E/2)−l−1/2 Jl+1/2 ( E).

c) For K = 3/4 and l = 0 the result is expressed in terms of Airy function. For generic values of K and l the Schr¨ odinger equation can be solved numerically with a very good precision with subsequent computation of the spectral determinant. Alternatively, for n ≥ 5 − 10 the spectrum of the equation (112) is very well approximated by the standard WKB expression (see, e.g., [140]) (115)

En = (K)(n − γl (K))

2K−1 K

,

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A. Imambekov, V. Gritsev and E. Demler

where n = 1, 2, . . . . Here γl (K) is the Maslov index. For 1/2 < K < ∞, γl (K) = 14 − 2l , for K = ∞, γl (K) = −l/2 and for 0 < K < 12 , γl (K) = 4K−2l−1 . The function (K) in 8K eq. (115) reads

(116)

(K) =

√ 2 π Γ( 32 + Γ(1 +

1 4K−2 ) 1 4K−2 )

2K−1 K .

In principle, the function γl (K) can be a smooth function interpolating between limiting values given above and can be considered as a noninteger Maslov index [141]. This interpolation allows to (approximately) evaluate the partition function and the distribution function in many cases. In the limiting cases K → 1 and K → ∞ the WKB approximation gives the exact spectrum. We point out however, that using the approximate WKB spectrum carelessly can result in non-physical results, e.g. negative values of the distribution function W (α), and thus have to be used with care. WE emphasize that the solution of the ODE (112) for several potentials gives us already analytically continued function Zp (K, ig). It seems that this approach to ﬁnding the solution is easier and more elegant than the solution of TBA equations with subsequent analytical continuation. Moreover a variety of approximate methods are available for solving Schr¨ odinger-like 1D equations. Note that eq. (112) has an interesting duality symmetry which generalizes the Coulomb-harmonic oscillator duality and which allows to relate the K > 1/2 and 0 < K < 1/2 sectors. Making the transformation x → y 1/A , Ψ(x) → y λ φ(y) with √ λ = −(1/2) + 1/(2A), and A = 2K and then rescaling y → αy with α = (−4K 2 /E)2K we obtain an equation of the same form with parameters K , l , E given by (117)

1 , 4K = K

l = 2p − 1/2,

1 E = 4K 2

4K 2 E

2K .

The point K = 1/2 is a self-dual point of this transformation. Presumably this symmetry is the origin of the Seiberg-Witten type duality in the impurity problem observed in ref. [142]. . 4 3.2. PT-symmetric quantum mechanics. The surprising link between the ordinary diﬀerential equation (112) and the exactly solvable problems related to conformal ﬁeld theories with negative central charge and/or two-color Coulomb gas on a circle is not limited to the examples considered above. There is a deep connection between these problems and a non-Hermitian PT -symmetric quantum mechanics. The latter has been formulated and studied by C. Bender in various contexts [143]. This story goes back to the (unpublished) work of Bessis and Zinn-Justin where, inspired by studies of Lee-Yang model, they conjectured that the 1D Schr¨ odinger equation with potential V (x) = (ix)3 has a real spectrum. Later in ref. [144] this statement was generalized for arbitrary powerlaw potentials, V (x) = (ix)n for real n. It was also conjectured that the Schr¨ odinger equation is invariant with respect to combined action of the PT symmetry: P (parity inversion): x → −x, p → −p; T : x → x, p → −p, i → −i.

575

Fundamental noise in matter interferometers

As we have seen, the partition function of the boundary Sine-Gordon model (as well as of the other related models) is real for the imaginary value of the coupling constant. This is of course not a generic property of ﬁeld theory models. Therefore the boundary sine-Gordon model with imaginary coupling also belongs to the class of PT -symmetric systems. It is then not surprising why the ODE (112) appears in such a theory. This intriguing relation between ordinary diﬀerential equations and various integrable models has been recently extended to a large class of models (for recent review and references see [145]) by making a close link to the PT -symmetric systems. These relationships have been used [146] for the so-called circular brane model which in certain limit describes the so-called Ambegaokar-Eckern-Sch¨ on model [147]. It is therefore reasonable to expect other new interesting applications of these ﬁndings. . 4 3.3. Analysis of distribution functions. Using exact analytic expressions for Zp (K, ig) at K = 1 after the integral transform we obtain the Poissonian distribution for Wp (α). The case K → ∞ is recovered easily as well. One sees that when p = 0, the distribution function is a delta-function in the leading order (corrections to this result will be considered in Appendix A). As p increases, Wp (α) rapidly broadens. In the limit of large K, the function Wp (α) depends only on the product Kp and takes a simple form

(118)

Wp (α) ≈

4Kp(1 − α)4Kp−1 , 0,

α < 1, α > 1.

This function is peaked at α = 1 for Kp < 1/4, it becomes a step function exactly at Kp = 1/4, and for Kp > 1/4. Wp (α) is a monotonically decreasing function of α. When the product Kp becomes large Kp 1 the function Wp becomes Poissonian: Wp (α) ≈ 4Kp exp[−4KP α]. In general, the tendency of broadening of the distribution function remains true for all values of K. The distribution given in (118) is a particular representative of class of extreme-value distributions called Weibull distributions. At p = 0 in the limit of K → ∞ it will be . analytically proven in subsect. 4 4.2 that the normalized amplitude α ˜ is characterized by the universal Gumbel distribution, for which (119)

K(α−1)−γ ˜ ˜ ˜ (˜ W α) = KeK(α−1)−γ−e ,

where γ ≈ 0.577 is the Euler gamma-constant. The Gumbel distribution, which also belongs to a class of extreme value statistics frequently appears in various problems (see, e.g., books [127] for historic introduction), including number theory [148], 1/f noise [104], Kardar-Parisi-Zhang growth [149], free Bose gas [150], etc. Probably the whole distribution function in our interference context can in general be characterized as a certain extreme-value statistics.

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A. Imambekov, V. Gritsev and E. Demler

. 4 4. Non-perturbative solution for the general case. – In this section we present a novel approach for calculating W (α), which is based on a mapping to the statistics of random surfaces [58]. We use this method to evaluate W (α) numerically for a variety of situations. We point out that our method is not related to the existence of the exact solution of Sine-Gordon models, but relies only on the structure of the multipoint correlation functions in the absence of interactions. As has been discussed in . subsect. 4 2, W (α) is connected to the partition functions of Sine-Gordon models and the partition functions of Coulomb gases. Our mapping provides a new non-perturbative tool for calculating partition functions of such models. We expect that there should be numerous applications of our new method to other physical problems which can be related to Sine-Gordon and Coulomb gas models. As a concrete application of our approach we prove analytically that for periodic boundary conditions in the limit of large K the distribution of fringe visibilities is given by the Gumbel distribution, one of the extreme value statistical distributions [127]. . 4 4.1. Mapping to the statistics of random surfaces. We start by observing that G(x, y) is real and symmetric. Hence it can be diagonalized on (0, 1) by solving the eigenvalue equations (120)

1

G(x, y)Ψf (y)dy = G(f )Ψf (x). 0

Here f is an integer index, which goes from 1 to ∞. Ψf (x) can be chosen to be real and normalized according to (121)

1

Ψf (x)Ψk (x)dx = δ(f, k). 0

Then, G(x, y) is given by

(122)

G(x, y) =

f =∞

G(f )Ψf (x)Ψf (y).

f =1

Decomposition given by eqs. (120)-(122) is similar to diagonalization of a symmetric matrix by ﬁnding its eigenvectors and eigenvalues. Now we can write Z2n from eq. (81)

577

Fundamental noise in matter interferometers

as

1

(123) Z2n =

du1 . . . dun dv1 . . . dvn

0 P

1

1

... 0

P

P

× e K ( i<j G(ui ,uj )+ i<j G(vi ,vj )− 1 1 = ... du1 . . . dvn 0 "

0 P

×e =

f

G(f ) K

1

2

2

(Pi Ψf (ui )) +(Pi Ψf (vi ))

−

G(ui ,vj ))

h P G(f ) i=n

P

1

du1 . . . dvn e

(

2K

f

=

P 2 P 2 i Ψf (ui ) − i Ψf (vi )

2

...

0

ij

i=1

P P −( i Ψf (ui ))( i Ψf (vi ))

2

Ψf (ui )−Ψf (vi )) −

Pi=n i=1

#

= i

(Ψf (ui )2 +Ψf (vi )2 ) .

0

Square of the ﬁrst i-sum in the last line above can be decoupled by introducing Hubbard-Stratonovich integrations over auxiliary variables tf : (124)

×

1

... 0

f" =∞

1

Z2n =

∞

du1 . . . dvn 0

dtf e− −∞

P

t2 f 2

e

q i tf

G(f ) G(f ) K (Ψf (ui )−Ψf (vi ))− 2K

√ 2π

f =1

(Ψf (ui )2 +Ψf (vi )2 ) .

Now we can simply integrate over du1 , . . . , dvn , since all integrals over u-variables are the same (integrals over v-variables are also identical): ⎛ (125)

⎜ Z2n = ⎝

∞

f" =∞

⎞

t2 f 2

e dtf ⎟ n n √ ⎠ g({tf }) g({−tf }) , 2π

−∞

f =1

−

where (126)

g({tf }) =

q

P

1

dx e

f

tf

G(f ) G(f ) 2 K Ψf (x)− 2K Ψf (x)

.

0

If all eigenvalues G(f ) are negative, then g(−{tf }) = g({tf })∗ ,

g({tf })g({−tf }) = |g({tf })|2 .

From comparison of eqs. (86) and (125) we obtain the central result of this section

(127)

W (α) =

f" =∞ f =1

∞

t2 f

e− 2 dtf −∞ √ δ α − |g({tf })|2 . 2π

578

A. Imambekov, V. Gritsev and E. Demler

˜ (α) Fig. 10. – Scaled distribution functions of the normalized interference amplitude W ˜ at T = 0 for 1D gases with open boundary conditions, shown for Luttinger parameters K = 2 (dashed), K = 3 (dotted), and K = 5 (solid). The inset shows a comparison between open (solid) and periodic (dashed) boundary conditions for K = 5. The ﬁgure is taken from ref. [58].

Equation (127) can be used to simulate distributions W (α) using a Monte Carlo approach. First, one needs to solve the integral equations (120)-(122) numerically to obtain eigenfunctions and eigenvectors. Then one needs to choose random numbers {tf } from the Gaussian ensemble, and plot the histogram of the results for |g({tf })|2 . In what follows we perform simulations of W (α) with up to N = 106 − 107 realizations of {tf } and smooth the data. We use a ﬁnite value of fmax and check for convergence with fmax , typically ∼ 30. α is always kept within 1% from its expected value. For most of the presented results, all eigenvalues G(f ) are negative, and eq. (127) can be directly applied. Special care should be taken for 1D case with nonzero temperature, where one of the eigenvalues can be positive. This situation can be handled by subtracting suﬃciently large positive constant C from G(x, y, ξLT ), which makes all eigenvalues negative. According to (81), this leads to rescaling of α by a factor e−C/K , which can be easily taken into account. ˜ (˜ In ﬁg. 10 we show scaled distribution of contrast W α) at T = 0 for 1D gases with OBC for various K. The inset shows a comparison between OBC and PBC for K = 5. In ﬁg. 11 we show the scaled distribution of contrast for 1D gas with OBC at nonzero temperature and K = 5. As has been discussed earlier, for ξT K/L 1 distribution is Poissonian and wide, while for K 1 and ξT K/L 1 it is very narrow. Evolution of the full distribution function of the visibilities while L is varied can be used to measure the thermal length, ξT = vs /(kB T ), precisely and to extract the temperature. As seen in ﬁg. 11, at T = 0 the distribution function has characteristic features, i.e. it is generally non-symmetric and can have a minimum. These features can be used to distinguish the noise due to ﬂuctuations of the phase from technical noise. Finally, in ﬁg. 12 we show scaled distribution of contrast for 2D gas with unity aspect ratio of imaging area and OBC below BKT temperature. Above BKT temperature distribution functions become Poissonian for L ξ, where ξ is the correlation length. In 2D one cannot describe

Fundamental noise in matter interferometers

579

˜ (α) Fig. 11. – Scaled distribution functions of the normalized interference amplitude W ˜ for a 1D Bose gas with open boundary conditions at nonzero temperature and K = 5. Diﬀerent curves correspond to ratios KξT /L = ∞ (solid), KξT /L = 1 (dotted), and KξT /L = 0.25 (dashed). ξT is the thermal correlation length, K is the Luttinger parameter and L is the imaging length. The ﬁgure is taken from ref. [58].

the crossover at L ∼ ξ similar to 1D, since the action which describes the ﬂuctuations of the phase is not quadratic in this region, and eq. (81) does not hold. We note that distribution functions of magnetic ﬂuctuations in the classical 2D XY model well below the BKT transition with PBC have been considered in the literature before in the diﬀerent physical context [151]. Result (127) can be interpreted as the mapping of our problem to the statistics of random surfaces (strings in 1D), subject to classical noise. In this mapping, Ψf (x) are the

˜ (α) Fig. 12. – Scaled distribution functions of the normalized interference amplitude W ˜ for a two-dimensional Bose gas with the aspect ratio of the imaging area equal to unity and open boundary conditions. Temperature is below the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature. Diﬀerent curves correspond to T = TBKT (BKT transition point, solid), T = (2/3)TBKT (dashed line) and T = (2/5)TBKT (dotted line). Above the BKT transition temperature the distribution function is Poissonian. The ﬁgure is taken from ref. [58].

580

A. Imambekov, V. Gritsev and E. Demler

eigenmodes of the surface vibrations, tf are the ﬂuctuating mode amplitudes, and |G(f )| is the noise power. This mapping holds as long as all eigenvalues G(f ) are negative, as discussed earlier. Inﬁnite dimensional integral over {tf } variables can be understood as an averaging over ﬂuctuations of the surface. For particular realization of noise variables {tf }, complex-valued surface coordinate at point x is given by (128)

h(x; {tf }) =

tf

f

G(f ) G(f ) Ψf (x) − Ψf (x)2 . K 2K

g({tf }) is a number deﬁned for each realization of a random surface {tf }, given by eq. (126). Fringe amplitude α for each realization of {tf } variables is given by / / α = |g({tf })| = // 2

(129)

1

/2 / / .

h(x;{tf }) /

dxe

0

. 4 4.2. From interference of 1D Bose liquids of weakly interacting atoms to extreme value statistics. To illustrate the power of the interpretation of the interference fringe amplitudes in terms of random surfaces, we now prove analytically that for periodic boundary ˜ (˜ conditions in 1D and in the limit of large K, the normalized distribution W α) is given by the Gumbel distribution (119), one of the extreme value statistical distributions. We note that to prove such result using 1/K expansion described in appendix A, one needs to go to the inﬁnite order of the perturbation theory, so essentially our result is non-perturbative in K. The Gumbel function −(x+γ)

PG (x) = e−(x+γ)−e

(130)

plays the same role in the extreme value statistics [127] as the usual Gaussian distribution plays in the statistics of the average value. According to the central limit theorem, the average value of N numbers taken from the same ensemble in the limit of large N is distributed according to Gaussian function. One can prove similar theorems for the distribution of the extreme, i.e. largest value of N 1 numbers taken from the same ensemble. Gumbel function (130) is one of the universal functions describing ensembles which decay faster than any algebraic function at inﬁnity. The Gumbel distribution often appears in applied mathematics, including problems in ﬁnance or climate studies, where it is used to describe rare events such as stock market crashes or earthquakes. For the periodic case corresponding to eq. (83), the eigenvalue problem (120)-(122) can be solved analytically, and the noise has 1/f power spectrum: 1 sin π|x − y| = π f =∞ √ √ √ 1 √ − log 2π − ( 2 cos 2πf x)( 2 cos 2πf y) + ( 2 sin 2πf x)( 2 sin 2πf y) . 2f

(131) Gper (x, y) = log

f =1

581

Fundamental noise in matter interferometers

Eigenmodes are given by simple harmonic functions, and all of them, except one, are doubly degenerate: we will denote corresponding noise variables as t1,f , t2,f and eigenvalues as G(f ) = −1/(2f ). Simpliﬁcation in the limit K 1 stems from the fact that exponent in eq. (129) can be expanded in Taylor series, since h(x; {tf }) is small for large K according to its deﬁnition (128). In this case it can be shown that the distribution of α is linearly related to roughness, or mean square ﬂuctuation of the surface, as deﬁned in ref. [104]. It has been shown in ref. [104] that 1/f noise results in the Gumbel distribution of the roughness, which has been interpreted in terms of extreme value statistics in ref. [152]. √ Terms of the order of 1/ K vanish in the Taylor expansion of eq. (129), since the average values of cos(2πf x) and sin(2πf x) on interval (0, 1) are equal to 0. To order 1/K, we obtain ∞

(132)

α≈1+

1 1 log 2π − (t2 + t22,f − 2). K 2f K 1,f f =1

The constant term − log 2π in Gper (x, y) gives a constant rescaling of the distribution function of α, and does not show up in the normalized fringe amplitude α: ˜ α ˜ ≈1−

(133)

∞ f =1

1 (t2 + t22,f − 2). 2f K 1,f

Thus to the leading order in 1/K expansion, the distribution Y (x) of the rescaled variable

(134)

x = −K(˜ α − 1) =

∞ 1 2 (t + t22,f − 2) 2f 1,f

f =1

equals

(135)

Y (x) =

f" =∞ ∞

∞

2

2

t1,f +t2,f dt1,f dt2,f − Pff =∞ 2 =1 e δ× 2π −∞ −∞ f =1 ⎛ ⎞ f =∞ 1 2 × ⎝x − (t + t22,f − 2)⎠ . 2f 1,f

f =1

To prove eq. (119), we need to show Y (x) = PG (x), where Gumbel function PG (x) is given by eq. (130). Let us introduce new positive variables

(136)

uf =

(t21,f + t22,f ) > 0, 2f

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A. Imambekov, V. Gritsev and E. Demler

then (137)

Y (x) =

f" =∞ ∞

⎛ f duf e−

Pf =∞ f =1

f uf

δ ⎝x −

0

f =1

⎞

f =∞

(uf − 1/f )⎠ .

f =1

To prove that Y (x) = PG (x) we will calculate their Fourier transforms: (138)

Y (is) =

=

∞

−∞ f" =∞ f =1 ∞

(139)

PG (is) =

−∞

isx

dxY (x)e

=

f" =∞

f duf e−f uf eis(uf −1/f ) =

f =1

f e−is/f = e−iγs Γ[1 − is], f − is ∞ −(x+γ) isx dxe−(x+γ)−e eisx = e−iγs Γ[1 − is]. dxPG (x)e = −∞

The proof above does not illustrate the meaning of Y (x) as of a distribution of extreme value. Here we will follow the method of ref. [152] and explicitly construct the variable, extreme value of which generates Gumbel distribution. Let us impose a ﬁnite cutoﬀ fmax = N , and at the end of calculation we will send N to inﬁnity. If one identiﬁes (140)

z1 = u N

(141)

z2 = uN −1 + uN ,

(142)

...

(143)

zN = u 1 + u 2 + . . . + u N ,

then {z1 , . . . , zN } is an ordered set (since ui > 0) of outcomes taken from Poissonian distribution, since e−

(144)

Pf =N f =1

f uf

= e−

Pf =N f =1

zf

,

and Jacobian of transformation from variables {uf } to {zf } variables is unity. Then (145)

YN (x) = ∞ −z1 e dz1 = N! 0

∞

z1

⎞⎞ N 1 ⎠⎠ e−z2 dz2 . . . e−zN dzN δ ⎝x − ⎝zN − f zN −1

∞

⎛

⎛

f =1

is nothing but the shifted distribution of the largest of N numbers taken from the Poissonian distribution, and in the limit of large N this distribution converges to Gumbel function. One can understand the appearance of the Gumbel distribution by noting that for K 1 the distribution function of the interference amplitude is dominated by rare

Fundamental noise in matter interferometers

583

ﬂuctuations of the random periodic 1D string, which are spatially well localized. The Gumbel distribution was introduced precisely to describe similar rare events such as stock market crashes or earthquakes. For open boundary conditions the universal distribution for large K is slightly diﬀerent from Gumbel function, similar to 1/f noise in other systems [104]. But the main properties, like the presence of asymmetry or the asymptotic form of the tails are preserved. 5. – Conclusions . 5 1. Summary. – When we discuss interference experiments with ultracold atoms, the conventional idea of the particle-wave duality takes a new meaning. On the one hand, these experiments probe phase coherence which is typically associated with coherent non-interacting waves. On the other hand, one can use powerful tools of atomic physics to control interactions between atoms in a wide range and to reach the regime of strong correlations. One can also prepare atomic systems in states which would be diﬃcult if not impossible to obtain in optics, e.g. low-dimensional condensates with strong thermal or quantum ﬂuctuations. This remarkable combination places interference experiments with ultracold atoms in a unique position: they can address a problem of how the interactions, correlations, and ﬂuctuations aﬀect the coherent properties of matter. This question appears in many areas of physics, including high-energy and condensed-matter physics, nonlinear quantum optics, and quantum information. While the naive answer that interactions suppress interference turns out to be correct in most cases, the goal of these lecture notes was to demonstrate that the quantitative analysis of this suppression can provide a lot of nontrivial information about the original systems. We discussed two eﬀects which contribute to the reduction of the interference fringe contrast in matter interferometers. The ﬁrst eﬀect is the shot noise arising from a ﬁnite number of atoms used in a single measurement. This analysis is particularly important for interference experiments with independent condensates in which the position of interference fringes is random and averaging over many shots can not be performed. In this case one needs to rely on single-shot measurements to observe interference patterns. While interference of independent condensates has been discussed before [63-67, 96-101], to our knowledge, we provide the ﬁrst derivation of the full distribution function of the amplitude of interference fringes. Another mechanism of the suppression of the amplitude of interference fringes discussed in these lecture notes is the quantum and thermal ﬂuctuations of the order parameter in low-dimensional condensates. The motivation for this discussion comes from the observation that interference experiments between independent ﬂuctuating condensates can be used to study correlation functions in such systems [55]. For example, one can use the scaling of the integrated amplitude of interference patterns to analyze two point correlation functions. This method has been successfully applied by Hadzibabic et al. [32] to observe the Berezinskii-Kosterlitz-Thouless transition in two-dimensional condensates. One conceptual approach to understanding interference experiments with independent condensates is to consider them as analogues of the Hanbury Brown and Twiss experiments in optics [7]. In the latter experiments

584

A. Imambekov, V. Gritsev and E. Demler

interference between incoherent light sources appears not in the average signal but in the higher-order correlation function. One important diﬀerence however is that matter interference experiments are of a single-shot type and information is contained not only in the average fringe contrast but also in the variation of the signal between individual shots. In particular higher moments of the amplitude of interference fringes contain information about higher-order correlation functions [55]. A complete theoretical description of the fringe contrast variations is contained in the full distribution functions of the fringe amplitudes, which we calculate for one- and two-dimensional condensates [56, 58] in the limit when the number of atoms is large and the shot noise can be neglected [102]. An important aspect of these lecture notes was identifying intriguing mathematical connections which exist between the problem of calculating distribution functions of interference fringe amplitudes and several other problems in ﬁeld theory and statistical physics, such as the quantum impurity problem [103], tunneling in the presence of the dissipation [132], non-hermitian PT-symmetric quantum mechanics [144, 143] and various conformal ﬁeld theories. We developed a novel mapping of a wide class of such problems to the statistics of random surfaces, which provided a complete non-perturbative solution. In certain cases we have analytically proven [58] the relationship between the distribution function of fringe amplitudes and the universal extreme value statistical distribution [127]. . 5 2. Some experimental issues. – We now comment on a few issues relevant for experimental analysis of noise in interference experiments. The amount of information contained in the experimentally measured distribution function is directly related to the number of cumulants which can be accurately extracted. This includes the second cumulant k2 , which corresponds to the width of the distribution; the third cumulant k3 , which 3/2 is related to skewness, g1 = k3 /k2 , and describes the asymmetry of the distribution function, and so on. In general, the statistical error in determining the n-th–order cumu lant after N measurements scales as An /N , where An is a constant which grows with n and depends on the higher moments of the distribution. For example, to experimentally distinguish the normal and Gumbel distributions it is necessary that the statistical error in skewness is at least a factor of two smaller than the mean skewness, which for the Gumbel distribution is g1 ≈ −1.14. Thus the minimal number of measurements required is Nmin ≈ 24/g12 ≈ 20, where we used A3 ≈ 6, appropriate for the normal distribution [153]. In practice the required number of measurements may be higher because of the inﬂuence of other possible sources of noise. However, it is certainly experimentally feasible. Another experimentally relevant issue is the eﬀect of the inhomogeneous density due to the parabolic conﬁning potential. While the approach discussed in these lecture notes can be extended to include the inhomogeneous density proﬁle, interpretation of the experimental results is more straightforward when density variations can be neglected. We note that when the condensate density varies gradually in space, the power-law decay of the correlation functions is not strongly aﬀected [154], except that the exponent may be diﬀerent in diﬀerent parts of the trap (correlation function exponents typically depend on the density). We expect that qualitatively our results will not change provided that the power law decay of the correlation functions is much stronger than the change of

Fundamental noise in matter interferometers

585

the condensate density in the measured part of the cloud. To be more precise, the best comparison with theory can be achieved when the observation region L is much smaller than the size of the condensate, determined by the eﬀective Thomas-Fermi length [155], RT F = (3N 2 /(m2 ω 2 a1D ))1/3 (here N is the number of atoms of mass m and a1D is the one-dimensional scattering length). As long as L remains much larger the healing length, our analysis is valid. In the regime of weakly interacting atoms, one can show that the ratio between the eﬀective Luttinger parameter K at the center of the harmonic trap and at the boundary of the observation region is given by 1 − L2 /8RT2 F , thus giving only a small correction to the distribution function computed in the central region. One can also reach similar conclusions in the strongly interacting regime. It is also worth pointing out that we expect the limiting case of the Poissonian distribution to be particularly robust to the inhomogeneous density of atoms. Indeed the Poissonian distribution is related to √ the fast 1/ x decay of the one-particle correlation functions in the strongly interacting limit. This scaling is a universal feature of the Tonks-Girardeau limit of bosons and is not aﬀected by the weak harmonic trap [156]. . 5 3. Outlook . – Before concluding these lecture notes we would like to discuss questions which still need to be understood in the context of interference experiments with ultracold atoms. We also suggest an outlook for future theoretical work. Combining shot noise with the order parameter ﬂuctuations. A careful reader has undoubtedly noticed that we discussed either the shot noise or the order parameter ﬂuctuations. At this point we are still lacking theoretical tools which would allow to include both eﬀects simultaneously. One of the diﬃculties is that such analysis requires the knowledge of the correlation functions for all distances rather than the long-distance asymptotic form. Indeed, in sect. 4 we showed that the short-distance part of the correlation functions gives contribution of the same order as the shot noise. In the particular case of the interference of 2D condensates, the knowledge of the short distance behavior of the correlation functions is needed to include the eﬀect of the vortex excitations below the BKT transition. Stacks of independent condensates. In these lecture notes we focused on interference patterns from a single pair of condensates. However in experiments one often has a stack of several condensates (see, e.g., ref. [31]). In this case interference arises from all possible pairs, and the system provides intrinsic averaging and suppression of the noise. For a ﬁnite number of condensates selfaveraging is not complete and one expects ﬁnite ﬂuctuations of the fringe contrast. It would be useful to generalize analysis of the shot noise and order parameter ﬂuctuations to such systems. Dynamics of interacting atoms. One of the advantages of the cold atoms systems is the possibility to study non equilibrium coherent dynamics of interacting systems. In particular dynamical splitting of a

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single condensate into a pair of condensates has been performed in experiments on microchips [30, 45, 48, 49] and stimulated theoretical work on the subject [157-162]. Similar experiments can also be done using superlattice potentials in optical lattices which are now available in experiments [163, 164]. While analysis of fringe amplitude distribution functions presented in these lecture notes dealt exclusively with systems in the thermodynamic equilibrium, it would be interesting to generalize it to systems undergoing nonequilibrium dynamical evolution. Interference experiments with fermions. The discussion presented in these notes was limited to the case of interference of bosons. Such experiments can also be done with fermions [165], which are available experimentally in diﬀerent dimensions [166-168]. For fermions, modulation of the density can be related to fermion antibunching [20,169,170]. Analysis of the noise of the fringe contrast visibility for fermions would be an interesting problem too. Generalization to other systems. We note that mapping of the Coulomb gas into the statistics of random surfaces intro. duced in sect. 4 4 should have applications beyond calculating the distribution functions of the interference fringe amplitudes. This is a new non-perturbative tool to calculate partition functions of a variety of other systems that can be represented as Coulomb gas models. Examples include quantum impurity-related problems [103], Sine-Gordon models where interaction strength can depend on position, and many others. Our mapping is not related to the existence of the exact solution of Sine-Gordon models, but relies only on the factorable structure of the many-point correlation functions in the absence of interactions, which is a general property of a Gaussian action. ∗ ∗ ∗ Many results presented in this review originally appeared in the research papers published together with E. Altman and A. Polkovnikov, to whom we owe a special gratitude. We also thank I. Bloch, R. Cherng, M. Greiner, M. Lukin, G. Morigi, T. Porto, J. Schmiedmayer, J. Thywissen, V. Vuletic and P. Zoller for numerous enlightening discussions. This work was partially supported by the NSF Grant No. DMR-0132874, MIT-Harvard CUA and AFOSR. VG was also supported by the Swiss National Science Foundation, grant PBFR2-110423. Appendix A. Large K expansion In this Appendix, we will describe a systematic “diagrammatic technique” to calculate per Z2n or Z2n as an expansion in small parameter 1/K. It corresponds to the “high . temperature” limit of the classical gas analogy discussed in sect. 4 1. This expansion can be applied both in 1D or 2D, and can be used to study the limiting distribution at large

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Fundamental noise in matter interferometers

K, which for PBC in 1D has been conjectured [56] and proven [58] to be the Gumbel distribution [127]. . A 1. Expansion to order (1/K)2 . – We will start from the 1D case by expanding the exponent in eq. (81):

1

1

1

P

P

P

du1 . . . dun dv1 . . . dvn e K ( i<j G(ui ,uj )+ i<j G(vi ,vj )− ij G(ui ,vj )) = 0 0 ⎛ ⎞m 1 1 ∞ 1 ⎝ . . . du1 . . . dun dv1 . . . dvn G(ui , uj )+ G(vi , vj )− G(ui , vj )⎠ . m m!K 0 0 m=0 i<j i<j ij

(A.1) Z2n =

...

In the ﬁrst order of expansion in powers of 1/K, G(x, y) dependence comes only through one integral

1

1

I0 =

(A.2)

dxdyG(x, y) 0

0

after the integration in (A.1). The prefactor depends only on n, and can be calculated analytically: the total number of u-u terms is n(n − 1)/2, the total number of v-v terms is also n(n − 1)/2, and the total number of u-v terms is n2 . The latter come with −1 sign, so the total expression for Z2n up to O(1/K 2 ) is (A.3)

Z2n

1 = 1 − nI0 + O K

1 K2

.

In second order, dependence of Z2n on G(x, y) comes through three diﬀerent integrals: (A.4)

1

0

0

1

1

0

1

I2,2 =

1

1

0

0

0

1

2 dxdyG(x, y) = I02 ,

1

dxdydzG(x, y)G(y, z), 0

0

(A.6)

1

dxdydzdtG(x, y)G(z, t) =

(A.5)

I1,2 =

1

I3,2 =

0 1

dxdyG(x, y)G(x, y). 0

0

Pictorially, these expressions can be represented by the corresponding diagrams, shown in ﬁg. 13. There, horizontal solid line corresponds to G(xi , xj ). If two ends are connected by a dashed line, then in the integral these two ends should correspond to the same variable. To write the expression for Z2n in 1/K 2 order, one has to calculate the total number of terms corresponding to I1,2 , I2,2 , I3,2 . Clearly, it will be some universal polynomial depending on n, which is determined by combinatorics. Here we describe how to do this combinatorics in detail.

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Fig. 13. – Topologically inequivalent diagrams, corresponding to 1/K 2 terms in expansion of Z2n .

When parenthesis are multiplied in (A.1) there are several types of expressions which appear, and come with diﬀerent signs to the integral: ⎛ ⎞2 G(ui , uj ) + (A.7) ⎝ G(vi , vj ) − G(ui , vj )⎠ =

+

i<j

i<j

ij

G(ui1 , vj1 )G(ui2 , vj2 ) +

i1,j1,i2,j2

(G(ui1 , uj1 )G(ui2 , uj2 ) + G(vi1 , vj1 )G(vi2 , vj2 ) + 2G(ui1 , uj1 )G(vi2 , vj2 )) −

i1<j1,i2<j2

−2

(G(ui1 , uj1 )G(ui2 , vj2 ) + G(vi1 , vj1 )G(ui2 , vj2 )).

i1<j1,i2,j2

These are (signs are indicated) (A.8) (A.9) (A.10) (A.11)

+G(ui1 , uj1 )G(ui2 , uj2 ), and equivalent after integration G(vi1 , vj1 )G(vi2 , vj2 ); +G(ui1 , vj1 )G(ui2 , vj2 ); −G(ui1 , uj1 )G(ui2 , vj2 ), and equivalent after integration − G(vi1 , vj1 )G(ui2 , vj2 ); +G(ui1 , uj1 )G(vi2 , vj2 ).

For terms which have G(uik , ujk ) or G(vik , vjk ), k = {1, 2} there is a restriction that ik < jk, but there is no such restriction for G(uik , vjk ). Let us calculate the total number of I1,2 , I2,2 and I3,2 terms for u−u−u−u expressions of type (A.8). Total number of I1,2 terms compatible with restrictions is (A.12)

1 n(n − 1)(n − 2)(n − 3). 2!2

1/2!2 appears since pairs {i1, j1} and {i2, j2} are ordered, while n(n − 1)(n − 2)(n − 3) is the total number of ways to choose a sequence of four diﬀerent not ordered numbers out of the set of size n. Total number of I2,2 terms compatible with restrictions is (A.13)

1 n(n − 1)(n − 2) ∗ 4. 2!2

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Fundamental noise in matter interferometers

Fig. 14. – Topologically equivalent diagrams, corresponding to a) I2,2 and b) I3,2 .

Here the situation is similar to I1,2 , but there is a “topological” prefactor of 4, which corresponds to four topologically equivalent diagrams for I2,2 , as shown in ﬁg. 14.a. Finally, total number of I3,2 terms is (A.14)

1 n(n − 1) ∗ 2. 2!2

Here, 2 is again a “topological” prefactor, corresponding to two topologically equivalent diagrams shown in ﬁg. 14.b. Overall, u − u − u − u term of (A.8) gives (A.15)

1 1 n(n − 1)(n − 2)(n − 3)I1,2 + n(n − 1)(n − 2)I2,2 + n(n − 1)I3,2 . 4 2

As a simple check of combinatorics one can calculate the total numbers of u − u − u − u terms of type (A.8). It is 1 1 1 (A.16) 2 n(n−1)(n−2)(n−3)+ 2 n(n−1)(n−2)∗4+ 2 n(n−1)∗2 = 2! 2! 2!

n(n − 1) 2!

2 ,

as it should be from the calculation which neglects the dashed lines, and does not impose any restrictions on the terms at diﬀerent horizontal rows. Analogously, one can calculate polynomials for expressions (A.9)-(A.11), and results are as follows, respectively: (A.17) (A.18) (A.19)

n2 (n − 1)2 I1,2 + 2n2 (n − 1)I2,2 + n2 I3,2 , 1 − n2 (n − 1)(n − 2)I1,2 − n2 (n − 1)I2,2 , 2 1 2 n (n − 1)2 I1,2 . 22

Summing all the terms with corresponding prefactors from (A.7), we obtain (A.20)

1 1 1 Z2n = 1 − nI0 + (P1,2 I1,2 + P2,2 I2,2 + P3,2 I3,2 ) + O = K 2!K 2 K3

1 1 1 (3n(n−1)I1,2 −4n(n−1)I2,2 +(2n−1)nI3,2 )+O , = 1− nI0 + K 2!K 2 K3

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where P1,2 = 3n(n−1), P2,2 = −4n(n−1), and P3,2 = (2n−1)n are universal polynomials of n. Notice, that due to sign cancellations of terms in (A.8)-(A.11), the overall degree of polynomials in 1/K 2 order is m = 2, compared to naively expected 2m = 4. Using (A.20), one can calculate Z2n /Z2n up to O(1/K 3 ): Z2n n(n − 1) =1+ (I1,2 − 2I2,2 + I3,2 ) + O n Z2 K2

(A.21)

1 K3

per per per Calculated values of {I0 , I1,2 , I2,2 , I3,2 } and {I0per , I1,2 , I2,2 , I3,2 } are

(A.22)

I0 = −

3 = −1.5, 2

I0per = − log 2π ≈ −1.83788,

9 = 2.25, 4 51 − π 2 ≈ 2.28502, = 18 7 = = 3.5, 2

(A.23)

I1,2 =

per I1,2 = (log 2π)2 ≈ 3.37779;

(A.24)

I2,2

per I2,2 = (log 2π)2 ≈ 3.37779;

(A.25)

I3,2

per I3,2 =

π 2 + 12(log 2π)2 ≈ 4.20026. 12

Thus, for K → ∞, the limiting ratio of the widths of the distributions for OBC and PBC are (A.26)

Z4 /(Z2 )2 − 1 per Z4 /(Z2per )2 −

1

→

I1,2 − 2I2,2 + I3,2 per per per ≈ 1.43465. (I1,2 − 2I2,2 + I3,2 )

. A 2. General properties of (1/K)m terms, and expansion to order (1/K)5 . – From the expansion of the previous subsection, one can formulate the general properties of the “diagram technique” to calculate terms up to (1/K)m : a) First, one has to draw all possible topologically inequivalent diagrams, which consist of m horizontal solid lines, with some of the ends connected by dashed lines. Each end can have at most two dashed lines coming out of it. b) Ends which are connected by a dashed line correspond to the same variable. Diagrams for which two opposite ends of the horizontal line correspond to the same variable are excluded. c) Expression which corresponds to a diagram is constructed the following way: if variables at the end of a given solid horizontal line are x and y, then G(x, y) should be put as one of the terms in the product under the integrand. Thus the integrand consists of the product of function G of some variables m times. Diagrams for which the integrands are the same up to relabeling of the variables are considered to be identical. d) all free variables should be integrated from 0 to 1. Diagrams can be connected or disconnected. For example, in ﬁg. 13 diagram corresponding to I1,2 is disconnected, and diagrams corresponding to I2,2 and I3,2 are

Fundamental noise in matter interferometers

591

connected. The integral which corresponds to a disconnected diagram is a product of expressions, corresponding to its parts: for example, I1,2 = I0 ∗ I0 . If the number of topologically inequivalent diagrams of order m is g(m), then the term of the order 1/K m in Z2n has the following form: ⎛ ⎞m 1 ⎝ G(ui , uj )+ G(vi , vj )− G(ui , vj )⎠ = du1 . . . dvn m!K m i<j 0 0 i<j ij ⎛ ⎞ g(m) 1 ⎝ Pr,m (n)Ir,m ⎠ , = m!K m r=1

(A.27)

1

...

1

where Pr,m (n) are universal polynomials of n, which can be calculated combinatorially, . as described in sect. A 1. Polynomials Pr,m (n) should satisfy the following requirements: a) For positive integer n, the values of Pr,m (n) are integer, and

Pr,m (0) = 0.

(A.28)

b) If one sets G(x, y) = 1, then Ir,m = 1, and lhs of (A.27) can be trivially calculated. This implies

g(m)

(A.29)

Pr,m (n) = (−n)m .

r=1

c) Degree of each polynomial Pr,m (n) is not larger than m. This has been shown above for m = 1 and m = 2, and has been checked up to m = 5, although we did not succeed in proving it directly. This conjecture is supported by the fact, that it guarantees that for any G(x, y) for K → ∞ distributions ﬁt on the top of each other after proper rescaling (see next section). One can do the combinatorial calculations similar to previous section for m = 3, and it takes about a day “by hand”. We will only show the results here. For m = 3 there are g(3) = 8 topologically inequivalent diagrams, which are shown in ﬁg. 15, out of which 5 diagrams (I1,3 to I5,3 ) are connected. Expressions corresponding to each diagram and

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A. Imambekov, V. Gritsev and E. Demler

Fig. 15. – Topologically inequivalent diagrams for m = 3. Diagrams I1,3 -I5,3 are connected and I6,3 -I8,3 are disconnected.

universal polynomials are, respectively, (A.30)

1

1

1

1

I1,3 =

dxdydzdtG(x, y)G(x, z)G(z, t), 0

0

0

0

(A.31)

P1,3 (n) = −12n(n − 1)(2n − 3), 1 1 1 1 I2,3 = dxdydzdtG(x, y)G(x, z)G(x, t),

(A.32)

P2,3 (n) = 12n(n − 1), 1 1 1 I3,3 = dxdydzG(x, y)2 G(x, z),

(A.33)

P3,3 (n) = −12n(n − 1), 1 1 1 dxdydzG(x, y)G(x, z)G(y, z), I4,3 =

(A.34)

P4,3 (n) = 4n(n − 1)(2n − 1), 1 1 dxdyG(x, y)3 , I5,3 =

0

0

0

0

0

0

0

0

0

0

0

P5,3 (n) = −n,

0

(A.35) (A.36)

I6,3 = I2,2 ∗ I0 , I7,3 = I3,2 ∗ I0 ,

P6,3 (n) = 36n(n − 1)(n − 2), P7,3 (n) = −3n(n − 1)(2n − 3),

(A.37)

I8,3 = I1,2 ∗ I0 = I03 ,

P8,3 (n) = −15n(n − 1)(n − 2).

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Fundamental noise in matter interferometers

Fig. 16. – Irreducible diagrams for m = 4.

Numerically evaluated integrals for OBC and PBC are (A.38)

I1,3 = −3.49399,

per I1,3 = −6.20797;

(A.39)

I2,3 = −3.5268,

per I2,3 = −6.20797;

(A.40)

I3,3 = −5.32704,

per I3,3 = −7.71956;

(A.41)

I4,3 = −4.21255,

per I4,3 = −6.50848;

(A.42)

I5,3 = −11.25,

per I5,3 = −12.5458;

(A.43)

I6,3 = −3.42753,

per I6,3 = −6.20797;

(A.44)

I7,3 = −5.25,

per I7,3 = −7.71956;

(A.45)

I8,3 = −3.375,

per I8,3 = −6.20797.

For m > 3 it becomes too cumbersome to manually calculate universal polynomials Pr,m . We wrote a program in Mathematica, which expands the m-th term of (A.1) directly, and calculates the values {Pr,m (0), . . . , Pr,m (n)} using powerful pattern recognition tools. After that, Pr,m (n) is recovered using Newton’s formula. Results for m = 3 can be recovered that way. One can also check in each order that the degree of Pr,m (n) is not larger than m. For each m the program needs as an input all topologically inequivalent diagrams, and currently the results have been extended to m = 5. For m = 4, the overall number of diagrams is 23, out of which 12 are irreducible, and shown in ﬁg. 16. For m = 5, the overall number of diagrams is 66, out of which 33 are irreducible. Numerical prefactors of polynomials Pr,m (n) grow with m, while their overall sum has a prefactor 1, as follows from (A.29). For example, one of the polynomials for m = 5 has a prefactor of 384. This puts stringent requirements on the errors in calculation of Ir,m (n), so going beyond m = 5 will require additional numerical eﬀort. For m = 5 results are reliable, since we can check the numerical accuracy of the calculation of integrals by comparison with analytically proven [58] Gumbel distribution. While calculation of original Z2n requires 2n-dimensional numerical integration, calculation of the Ir,m requires at most m-dimensional numerical integration. This can be seen from the fact that even for irreducible diagrams, only the parts which contain “loops” have to be integrated numerically. Horizontal lines which have a free end can be “integrated out” analytically, and the dimension of the numerical integral has at most

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dimension of the loop. For example, ﬁrst and third horizontal bars in I1,3 of ﬁg. 15 can be integrated out using the following identity: (A.46)

1

dy log |x − y| = −1 + log(1 − x) − x log(1 − x) + x log x,

0

so one has to do only 2-dimensional integral numerically. Analogously, diagram I2,3 requires only one-dimensional numerical integration. In each order m there is only one diagram which requires m-dimensional integration, i.e. I4,3 in ﬁg. 15 for m = 3. All the rest require at most (m − 1)-dimensional integration. . A 3. Properties of the K → ∞ distribution. – For K → ∞ distribution function becomes very narrow, and it is an interesting question to investigate the limiting behavior of the distribution function. Let us consider the distribution of normalized contrast α, ˜ deﬁned by (86). Due to normalization the following relations hold: (A.47) 0

∞

˜ (˜ W α)dα ˜ ≡ 1,

∞

˜ (˜ α ˜W α)dα ˜ ≡ 1.

0

For large K distribution function is peaked near α ˜ = 1, and its width is proportional to 1/K. To calculate the properties of the universal function we assume that it decays relatively fast (exponential decay is enough) away from α ˜ = 1 for large K, so we can extend integrations to inﬁnity. If we deﬁne the ﬂuctuation of the normalized contrast ˜ =W ˜ then ˜ 0 (β) ˜ (1 + β), β˜ = α ˜ − 1 with distribution function W (A.48)

(A.49)

(A.50)

(A.51)

∞ ∞ Z2 ˜ β˜ ≈ ˜ (˜ ˜ 0 (β)d −1 + ≡0= (˜ α − 1)W α)dα ˜= β˜W Z2 0 −1 ∞ ˜ β˜ ≡ M ˜1 , ˜ 0 (β)d β˜W −∞ ∞ Z4 ˜ (˜ −1 + 2 = (˜ α2 − 1)W α)dα ˜≈ Z2 0 ∞ ˜ β˜ ≡ 2M ˜ 0 (β)d ˜1 + M ˜2 , (2β˜ + β˜2 )W −∞ ∞ ∞ Z6 3 ˜ β˜ ≡ ˜ ˜ 0 (β)d −1 + 3 = (˜ α − 1)W (˜ α)dα ˜≈ (3β˜ + 3β˜2 + β˜3 )W Z2 0 −∞ ˜2 + M ˜3 , ˜ 1 + 3M 3M ...,

˜ Inverting (A.48)-(A.50), one can ﬁnd ˜ 1, M ˜ 2, M ˜ 3 are the moments of W ˜ 0 (β). where M ˜ ˜ M2 , M3 , . . . as series expansion in powers of 1/K. If for large K the distribution functions collapse on the top of each other after proper rescaling, as has been conjectured in [56], ˜ m in powers of 1/K should start from 1/K m . Below we will then the expansion of M show, that this is necessarily true for all n, if the degrees of universal polynomials Pr,m (n) are not larger than m and Pr,m (0) = 0.

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Fundamental noise in matter interferometers

To show this, let us consider an analog of eqs. (A.48)-(A.50) for the ﬂuctuation of the unnormalized contrast β = α − 1 with distribution W0 (β) = W (1 + β) (normalized contrast considered here is related to α by simple rescaling). Then (A.52)

Z2n =

∞

n

α W (α)dα = 0

≈

i=n ∞

−∞ i=0

∞

−1

(1 + β)n W0 (β)dβ ≈

Cni β i W0 (β)dβ =

i=n

Cni Mi ,

i=0

where Mi is the i-th moment of W0 (β). We need to show that expansion of Mi in powers of 1/K starts only from the terms of the order of 1/K i . This can be seen from expansion (A.27) together with (A.52). If degrees of Pr,m (n) are not larger than m, then (A.27) means that 1/K m terms in Z2n grow at most as nm for large n. On the n! other hand, Cni = i!(n−i)! grows as ni for large n, and if expansion of Mi in powers of i 1/K starts before 1/K , this will contradict the previous sentence. ˜ m , one has to go to order 1/K m in the To ﬁnd the ﬁrst nontrivial contribution to M expansion. Using results for m = 5 calculated above, one can calculate for K → ∞ ˜ 2, . . . , K 5M ˜ 5 for periodic and non-periodic cases, and compare limiting behavior of K 2 M it with the result of the Gumbel distribution

(A.53)

˜

K β−γ ˜ ˜ = KeK β−γ−e ˜ G (β) W ,

where γ ≈ 0.577 is the Euler gamma-constant. One obtains ˜ 2 → 2.35991, K 3 M ˜ 3 → −5.105577, (A.54) Non-periodic: K 2 M ˜2 ˜ 5 → −242.492, M3 → 1.98336; K 5M ˜3 M

˜ 4 → 37.5258, K 4M

2

˜ per → 1.64493, K 3 M ˜ per → −2.40411, (A.55) Periodic: K 2 M 2 3 ˜ per → 14.6114, K 5 M ˜ per → −64.4321; K 4M 4 5 ˜ G → 1.64493, K 3 M ˜ G → −2.40411, K 4 M ˜ G → 14.6114, (A.56) Gumbel: K 2 M 2 3 4 ˜ G )2 ( M 3 ˜ 5G → −64.4321, K 5M → 1.29857. ˜ G )3 (M 2

Clearly, for periodic case the agreement with the Gumbel distribution is excellent, but for the non-periodic case, one can unambiguously conclude that the limiting function is NOT a Gumbel function. For the non-periodic case the function is more widely distributed compared to the Gumbel function, as evident from the higher moments. Here

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per for completeness we provide numerical results for Z2n , Z2n up to m = 5:

1.8379n −0.4112n + 2.5114n2 + + K K2 0.1002n − 0.1548n2 + 2.1456n3 −0.03382n + 1.0804n2 − 0.7399n3 + 1.7369n4 + + + K3 K4 0.01535n + 1.05941n2 − 0.161727n3 + 0.172302n4 + 0.93098n5 + , K5 1.5n −0.554956n + 2.30496n2 0.049657n + 0.343838n2 + 1.481505n3 Z2n = 1 + + + + K K2 K3 2 3 4 −0.04741n + 2.0329n − 1.8735n + 1.8256n + + K4 0.0106468n + 3.14868n2 − 4.1829n3 + 2.8980n4 + 0.0943039n5 + . K5 per =1+ Z2n

(A.57)

. A 4. D = 2. – Here we will brieﬂy consider results of the expansion in powers of 1/K for 2-dimensional case. Correlation function below the BKT transition is given by eq. (76), and for square imaging area with unity aspect ratio, all discussions of one-dimensional case carry over, with substitutions (A.58) 0

1

dui →

1

1

d2 ui , 0

0

0

1

dvi →

1

1

d2vi , 0

G(x, y ) = log |x − y |,

0

and (77). In 2D case, there is one extra degree of freedom which can be controlled in experiments, which is the aspect ratio of the observation region. If the aspect ratio of the observation region is very large, then the distribution function is essentially the same as in the one-dimensional case. Below we concentrate on the case with unity aspect ratio. Similarly to one-dimensional case, the integral

1

(A.59) 0

0

1

1 dx1 dy1 log (x1 − x2 )2 + (y1 − y2 )2 2

can be evaluated analytically, which somewhat simpliﬁes the numerical evaluation. However, the dimensions of integrals grow fast with m, and here we present results only up to m = 3: (A.60)

0.805087n −0.0740n + 0.3342n2 + K K2

2 1 0.015n + 0.045n + 0.241n3 + +O . K3 K4

Z2n = 1 +

The error in numerical coeﬃcient due to integration is of the order of the last reported digit. Compared to 1D case, convergence is faster, which is consistent with our general ideology that the role of ﬂuctuations is larger in systems with lower dimensions.

597

Fundamental noise in matter interferometers

Appendix B. Jack polynomials (p)

The microcanonical partition functions Z2n can be computed using so-called Jack polynomials. The Jack polynomials belong to a class of symmetric polynomials [131,171] and are a one-parametric generalization of Schur symmetric functions and generically can be deﬁned as (B.1) Jλ (x; g) = vλ,μ (β)mμ (x), μ≤λ

where the monomial symmetric function mλ (x) mλ (x) =

(B.2)

λ

λ

x1 σ(1) · · · xnσ(n)

σ

includes sum over all permutations σ. Here vλ,μ are numerical coeﬃcients and vλ,λ = 1. The ordering in sum (B.1) means the ordering on the set of partitions λ = (λ1 , λ2 , . . . , λn ), λ1 ≥ λ2 ≥ . . . ≥ λn ≥ 0 and μ = (μ1 , μ2 , . . . , μn ), μ1 ≥ μ2 ≥ . . . ≥ μn ≥ 0 of natural k k numbers λ and μ. The ordering μ ≤ λ here means that j=1 μj ≤ j λj for any k ≤ n − 1. The partitions can be represented using Young diagrams as follows: we put λ1 boxes in the ﬁrst row, λ2 boxes in the second, and so on. The deﬁnition of Jack polynomials thus includes a sum over all Young diagrams for which the number of rows is smaller or equal to the number of variables n. The Jack polynomials have the property of orthogonality, which allows to bring the microcanonical partition function into the following form (see [123] for more details on derivation) (B.3)

(p)

Z2n = c2n

n " Γ[λi + λ i=1

1 − i + 1)]Γ[p + λi + 2K (n − i + 1)] . 1 1 Γ[λi + 1 + 2K (n − i)]Γ[p + λi + 2K (n − i)] 1 2K (n

Here cn = Γ[n + 1]/Γn [1/2K] and sum goes over Young diagrams labeled by integers λ1 ≥ λ2 ≥ . . . λn ≥ 0. In particular, the Jack polynomials expressions give the following results for the lowest microcanonical partition functions: (p)

(B.4) and (B.5)

Z2

=

1 sin π( 2K

π )Γ(1 − 1/K) sin( 2K 1 + p)Γ(1 − 2K + p)Γ(1 −

1 2K

− p)

1 1 1 2−1/K Γ( 12 − 2K )Γ2 ( 2K )Γ2 ( K ) 1 = 4 1 √ 1 1 2 Γ ( 2K ) πΓ(1 − 2K )Γ (1 + 2K )

1 1 1 1 ,1 + 1, , , 1+ ,1 ×p Fq K K 2K 2K

2 ∞ 1 Γ( K + n) 1 1 + n, 1 + + n , {2 + n, 2 + n}, 1 , 1, 1 + − p Fq Γ(2 + n) 2K 2K n=0

(0) Z4

598

A. Imambekov, V. Gritsev and E. Demler

where p Fq is a generalized hypergeometric function [172]. Apparently, for larger n, (p) expressions for Z2n become more and more cumbersome and are diﬃcult to use for real computations. Jack polynomials appear particularly in connection to Calogero-Sutherland model, i.e. the model which describes n particles (the same n as in formulas above) on a line interacting via inverse square interaction. In the ﬁrst quantized form the Hamiltonian of this model can be deﬁned as [131] 1 ∂2 + π2 2 j=1 ∂x2j n

HCS = −

(B.6)

g(g − 1) . sin π(xi − xj )2 1≤i<j≤n 2

Wave functions for the excited states of this model are Jack polynomials. Appendix C. Thermodynamic Bethe Ansatz of the quantum impurity model The model described by the partition function (90) can be solved exactly by thermodynamic Bethe ansatz (TBA) [173, 174]. This solution is non perturbative in g. One should be careful about the correspondence between the perturbative expansion and the results from the non perturbative TBA calculations. The correct correspondence between Z(K, g)pert and the non-perturbative result of the TBA is given by (C.1)

Z0 (K, g)pert = √

1 exp[F˜T BA (K, g)], 2K

where F˜T BA (K, g) is the free energy. It can be expressed in terms of the energies (θ) of elementary excitations which depend on rapidity θ: (C.2)

F˜T BA (K, g) =

∞

−∞

2K − 1 dθ log(1 + exp( + (θ))). 2π cosh[(2K − 1)(θ − α)]

Here, (C.3)

α = log

x 1 Γ( 2K )

√ 2K 2K−1 1 2 πΓ( 4K−2 ) K Γ( 2K−1 )

and the variable x is the fugacity in the perturbative expansion (92). To clarify the issue further we brieﬂy describe the formalism of TBA. The TBA equations are parametrized by the Dynkin diagrams corresponding to the classical algebras An , Dn , En [175]. Our case under consideration is related to the diagrams Dn , which describes the sine-Gordon model. To each node of the diagram we associate the particle with the mass given by the Coxeter number of the corresponding diagram (which is equal to 2K − 2) shown in ﬁg. 17. These are denoted by μ+ = μ− (corresponding to the kink

599

Fundamental noise in matter interferometers

Fig. 17. – Dynkin diagram for the algebra Dn . In the context of sine-Gordon problem nodes 1 . . . 2K − 2 correspond to breathers, whereas nodes ± correspond to kink/antikink.

and antikink) and μj , j = 1 . . . 2K − 2, (note that we consider K to be of the form n/2, n = 3, 4, 5, . . .) corresponding to breathers, bound states of kink and antikink. Explicitly, (C.4)

μj = 2μ+ sin

jπ 4K − 2

for breathers j = 1 . . . 2K − 2.

Due to a great simpliﬁcation of the structure of scattering matrices in integrable theory [175] the free energy of the boundary sine-Gordon problem can be expressed entirely in terms of the energies + (see eq. (C.2) above). This can further be reduced as follows. Consider the TBA equations for the energies: (C.5)

r (θ) =

∞

−∞

2K − 1 dθ Nrs log(1 + exp[ s (θ )]), 2π cosh[(2K − 1)(θ − θ )] s

where Nrs is the incidence matrix of the corresponding Dynkin diagram. To clarify these notations consider few examples. For K = 3/2 we have kink, antikink and one breather. Therefore the diagram consists of the three nodes, say +, 1, −. The vertex + (kink) is connected to the node 1 and the vertex 1 is connected to the vertex −. Vertices + and − are disconnected. The incidence matrix is therefore ⎞ ⎛ 1 0 ⎝1 1⎠ (C.6) 0

1

(the number of rows is equal to the number of vertices, the number of columns is equal to the number of bonds between these vertices). Using this matrix we obtain the following set of coupled TBA equations:

(C.7) (C.8)

∞

dθ 2 log(1 + exp[ 1 (θ )]), −∞ 2π cosh[2(θ − θ )] ∞ 2 dθ 1 (θ) = 2 log(1 + exp[ + (θ )]), −∞ 2π cosh[2(θ − θ )]

± (θ) =

where we use the fact that + = − . Now, from the second equation we can see that the 1 (θ)/2 gives precisely the function F˜ from the eq. (C.2). Therefore, for the case of

600

A. Imambekov, V. Gritsev and E. Demler

K = 3/2, (C.9)

Z0

3 ,g 2

pert

1 1 = √ exp 1 (α) , 2 3

where 1 must be determined from the set of coupled equations (C.7). To continue, consider the case of K = 4/2. Here we have 2 breathers, kink, and antikink. The incidence matrix is ⎛ 1 ⎜1 ⎜ ⎝0 0

(C.10)

0 1 1 0

⎞ 0 1⎟ ⎟. 0⎠ 1

The set of TBA equations is therefore

(C.11) (C.12) (C.13)

∞

dθ 3 log(1 + exp( 2 (θ ))), 2π cosh[3(θ − θ )] −∞ ∞ dθ 3 1 (θ) = log(1 + exp( 2 (θ ))), 2π cosh[3(θ − θ )] −∞ ∞ 3 dθ 2 (θ) = log(1 + exp( 1 (θ ))) + 2π cosh[3(θ − θ )] −∞ ∞ 3 dθ +2 log(1 + exp( + (θ ))), 2π cosh[3(θ − θ )] −∞

± (θ) =

from which we conclude that the 2 /3 gives the function F˜ from the eq. (C.2). Therefore, for the case of K = 4/2, (C.14)

Z0

4 ,g 2

pert

1 1 = √ exp 2 (α) , 3 4

where 2 must be determined from the set of coupled equations (C.11). For K = 5/2 we can ﬁnd that (C.15)

Z0

5 ,g 2

pert

1 1 = √ exp ( 3 (α) − 1 (α)) , 2 5

where 1,3 must be determined from the set of corresponding coupled equations (easy to write down explicitly). For K = 6/2 we have (C.16)

Z0

6 ,g 2

pert

1 1 = √ exp ( 4 − 2 + F˜ ( 1 )) , 2 6

601

Fundamental noise in matter interferometers

and therefore it includes the function F˜ ( 1 ). This is not convenient from the point of view of numerics since it contains an additional integration. Continuing the above computations one can observe that for the values of K = 2k/2, k = 2, 3, 4, . . . we will always have an additional function F˜ ( 1 ) in the expression for the perturbative function Z0 (g, K)pert . On the other hand for the values of K = (2k + 1)/2, k = 1, 2, 3, . . . one can always express the result for partition function in terms of energies i . It seems that this is more convenient for numerics. The result for this case is the following: (C.17)

Z0

2n + 1 g, 2

pert

n−1 1 1 =√ (−1)s (2n−1)−2s , exp 2 s=0 2n + 1

where the set of energies must be determined from the solution of coupled integral equations. It is useful to consider the asymptotic behavior. For a large θ → ±∞ values of the integration parameter, the energies of breathers may be taken in the following form (see eq. (C.4)): (C.18)

s (α) =

μs = 2 sin μ+

sπ 4K − 2

exp[α],

as α → ∞.

The general strategy is clear now: one can solve numerically the set of coupled integral equations for some values of K = (2n + 1)/2 and compute the partition function using eq. (C.17). At this point the diﬃculty arises: how to unambiguously deﬁne the analytic continuation of Zp (K, g) into imaginary-g domain. To avoid this problem it is better to . use the alternative approach described in subsect. 4 3.1.

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Unitary polarized Fermi gases F. Chevy ´ Ecole Normale Sup´erieure - 24 rue Lhomond, Paris, France

Although recent theoretical and experimental progress has considerably clariﬁed pairing mechanisms in spin-(1/2) fermionic superﬂuid with equally populated internal states, many open questions remain when the two spin populations are mismatched. We show here that, taking advantage of the universal behavior characterizing the regime of inﬁnite scattering length, the macroscopic properties of these systems can be simply and quantitatively understood in the regime of strong interactions. 1. – Introduction Pairing lies at the core of the standard Bardeen-Cooper-Schrieﬀer mechanism for metal superconductivity, and the very natural question to know whether it could survive population imbalances between the two spin states naturally arose very soon after its development [1, 2]. It was pointed out that pairing was indeed robust to some amount of mismatch between the chemical potentials of the two species, but the fate of the system after the critical imbalance is reached has long been a mystery. The absence of clear answer to this problem was due in particular to the absence of an experimental system on which the various scenarios envisioned could be tested: existence of a spatially modulated order parameter (Fulde, Ferrel, Larkin and Ovshinikov, or FFLO, phases) [3-5], or the extension to trapped systems [6-9], deformed Fermi surfaces [10], interior gap superﬂuidity [11], phase separation between a normal and a superﬂuid state through a ﬁrst-order phase transition [12-15], BCS quasi-particle interactions [16] or onset of pwave pairing [17]. When the strength of the interactions is varied, a complicated phase diagram mixing several of these scenarios is expected [18-20]. This issue was revived by the possibility of obtaining fermionic superﬂuids with ultracold atoms [21-26], where spin imbalance could be controlled and maintained for a c Societ` a Italiana di Fisica

607

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F. Chevy

long time. This led to a series of experiments performed at MIT [27,28] and Rice [29,30] which clearly demonstrated a phase separation between regions characterized by diﬀerent polarizations (i.e. spin population imbalances, by analogy with magnetism). The number of phases obtained by the two groups is however diﬀerent. In Rice experiment, the cloud is constituted of a core where both spin populations are equal, surrounded by a shell of majority atoms only while at MIT a third phase mixing both species with unequal densities is intercalated between the previous ones, a discrepancy which is not yet fully explained [31-38]. In what follows we wish to explore the various consequences of these experiments. By contrast to most recent works on the subject, we would like to avoid the use of BCS mean ﬁeld, which is known to give good qualitative insight to the problem under study, but fails when precise quantitative estimates are needed. Our scheme is based on a combination of exact variational analysis and Monte Carlo simulations. We will demonstrate that, in agreement with MIT experiments, three phases are expected in homogeneous systems. To compare with experimental results, we will make use of Local Density Approximation (LDA) which leads to quantitative agreement with MIT’s data. Finally, following [31], we will show how Rice’s apparently contradictory results can be interpreted as a breakdown of local density approximation in elongated traps. 2. – Universal phase diagram of a homogeneous system Let us ﬁrst consider an ensemble of spin-(1/2) fermions of mass m trapped in a box = is given by of volume V . In the s-wave approximation, the Hamiltonian H (1)

= = H

k,σ

k = a†k,σ = ak,σ +

gB † = ak+q,↑ a†k −q,↓ = ak ,↓ = ak,↑ , V k,k ,q

where k = 2 k 2 /2m, = ak,σ annihilates a particle of spin σ and momentum k and gb is the coupling constant characterizing s-wave interactions between atoms. This choice of interaction potential is singular and yields unphysical results and to get rid of the divergencies resulting by the zero range of the potential, we introduce an ultraviolet cutoﬀ qc in momentum space (or equivalently, we work on a lattice of step 1/qc ). When qc goes to inﬁnity, the Lippmann-Schwinger formula obtained by the resolution of the two-body problem yields the following relationship between the bare coupling constant and the scattering length a: (2)

1 1 1 m − = , 2 gb 4π a V k k

where the sum over k is restricted to k < qc . To anticipate the analysis of inhomogeneous systems, we work in the grand-canonical ensemble, where the atom numbers ﬂuctuate and only their expectation values are kept constant. Introducing the chemical potentials μ↑,↓ as Lagrange multipliers associated

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with the constraints on atom numbers, we need to ﬁnd the ground state of the grand = given by potential Ω ==H = − μ↑ N =↑ − μ↓ N =↓ . Ω

(3)

= by a maximization In what follows, we replace the minimization condition on Ω = Ω problem on the pressure P , using the thermodynamical relation Ω = −P V . Moreover, we assume μ↑ > μ↓ and we restrict ourselves to the unitary limit where a = ∞. This choice of scattering length leads to a deep simpliﬁcation of the formalism, due to the universality characterizing this regime. Indeed, from dimensional analysis [39], we can show that for an arbitrary scattering length, the pressure P of a given phase is given by some relation P (m, , a, μ↑ , μ↓ ) = P0 (μ↑ , , m)f (μ↓ /μ↑ , 1/kF↑ a), where P0 is the pressure of an ideal Fermi gas with chemical potential μ↑ and kF↑ is the Fermi wave vector associated with μ↑ . At unitarity, 1/kF a = 0 and f is therefore function of η = μ↓ /μ↑ yielding the universal relation P = g(η), P0

(4)

where g(μ↓ /μ↑ ) = f (μ↓ /μ↑ , 0). Although the general minimization of the grand potential is an extremely challenging and still open problem, we ﬁrst note that two exact eigenstates of the system can be found. 1) Fully polarized ideal gas. If we consider a fully polarized system containing no = disappears, and we are left with a pure minority atoms, the interaction term in H ideal gas of majority atoms. The pressure of this normal phase is simply the Fermi pressure, and we have in particular P/P0 = 1. 2) Fully paired superﬂuid. Let |SFμ be the ground state of the balanced potential = − μ(N =↑ + N =↓ ). Since Ω = commutes with the atom number operators, |SFμ = = H Ω =↑,↓ , with N =↑ |SFμ = N =↓ |SFμ . Going back can be chosen as an eigenstate of both N = to the unbalanced problem, we write Ω as (5)

=↑ + N =↑ − N =↓ ) + μ↑ − μ↓ (N =↓ ). ==H = + μ↑ + μ↓ (N Ω 2 2

= = We see readily that for μ = (μ1 + μ2 )/2 we have Ω|SF μ = Ω |SFμ , which proves that |SFμ is also an eigenstate of the imbalanced grand potential. The pressure in this superﬂuid phase can be calculated using known results for the unitary balanced superﬂuid for which the universal relationship between chemical potential

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c

Fig. 1. – Sketch of the grand potential Ω as a function of η = μ↓ /μ↑ . Ω is normalized to the grand potential Ω0 of the pure ideal gas of chemical potential μ↑ . Full line: paired superﬂuid; dotted line: fully polarized normal phase; dashed line: intermediate mixed phase. ηα and ηβ designate the critical values for the two transitions between the superﬂuid/mixed phase and the mixed phase/fully polarized Fermi gas.

and density reads (6)

μ↑ = μ↓ = ξ

2 (6π 2 n↑ )2/3 , 2m

where ξ ∼ 0.42 is a universal number that was evaluated both experimentally [24, 29, 40-42] and theoretically [43-45, 14]. Integrating Gibbs-Duhem identity (see Appendix), one then obtains for the imbalanced system (7)

PSF =

1 15π 2

m ξ2

3/2 (μ↑ + μ↓ )5/2 ,

hence PSF /P0 = (1 + η)5/2 /(2ξ)3/2 . The variation of the pressure vs. η is displayed in ﬁg. 1. We see that for small imbalances, i.e. η smaller than ηc = (2ξ)3/5 − 1 ∼ −0.10, the fully paired superﬂuid is more stable than the fully paired normal phase, conﬁrming the stability of pairing against a small mismatch of the Fermi surfaces. The experimental results presented in ref. [28] suggest that the two classes of states we have until now restricted ourselves are not suﬃcient to fully capture the physics of imbalanced systems. In particular, a mixed normal phase, containing atoms of both species in unequal proportions, must be taken into account. A sketch of g(η) for this intermediate phase is shown in ﬁg. 1. On this more general phase diagram, the parameters ηα and ηβ are of special importance, since

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they characterize the phase transitions between the three diﬀerent phases. A glance at ﬁg. 1 shows that they must satisfy the inequality ηα < ηc < ηβ , and the next section is devoted to an improvement of these bounds. 3. – The N + 1 body problem Theoretically, the existence of the intermediate phase can be demonstrated by the study of the N + 1 body problem, in other words the study of the ground state of the majority Fermi sea in the presence of a single minority atom. This particular system corresponds to an intermediate phase with η → ηβ+ and we will prove that it yields the inequality ηβ < ηc . To address the N +1 body problem, we use a variational scheme, that we will compare to recent predictions based on Monte Carlo simulations [46]. Let us consider the following trial state |ψ: |ψ = φ0 |FS +

φk,q |k, q,

k,q

where |FS is a spin-up Fermi sea plus a spin-down impurity with 0 momentum, and |k, q is the perturbed Fermi sea with a spin-up atom with momentum q (with q lower than kF ) excited to momentum k (with k > kF ). To satisfy momentum conservation, the impurity acquires a momentum q − k. = = The energy of this state with respect to the non-interacting ground state is H = 0 + V= , with H = 0 |ψ = ψ|H

|φk,q |2 ( k + q−k − q ),

k,q

and ⎛ ⎞ g B ⎝ |φ0 |2 + φk ,q φ∗k,q + φk,q φ∗k,q + (φ∗0 φk,q + φ0 φ∗k,q )⎠ , ψ|V= |ψ = V q k,k ,q

k,q,q

q,k

where k = 2 k 2 /2, and the sums on q and k are implicitly limited to q < kF and k > kF . As we will check later, φk,q ∼ 1/k 2 for large momenta (see below, eq. (10)), in order to satisfy the short-range behavior 1/r of the pair wave function in real space. This means that most of the sums on k diverge for k → ∞. This singular behavior is regularized by the renormalization of the coupling constant gB using the Lippman-Schwinger formula. It implies that gB vanishes for large cut-oﬀ, thus yielding a ﬁnite energy. However, it must be noted that the third sum in ψ|V= |ψ is convergent and when multiplied by gB

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F. Chevy

will give a zero contribution to the ﬁnal energy and can therefore be omitted in the rest of the calculation. = with respect to φ0 and φk,q is straightforward and yields The minimization of H the following set of equations: gB gB φ0 + φk,q = Eφ0 , V q V q,k gB gB φ0 = Eφk,q , ( k + q−k − q )φk,q + φk ,q + V V

(8) (9)

k

where E is the Lagrange multiplier associated to the normalization of |ψ, and can also be identiﬁed with the trial energy. Let us introduce χ(q) = φ0 + k φk,q . We see from eq. (9) that φk,q = −

(10)

χ(q) gB . V k + q−k − q − E

As expected, we note here the 1/ k ∼ 1/k 2 dependence for large k. Inserting this expression in the deﬁnition of χ, we obtain χ(q) = φ0 −

χ(q) gB , V k + q−k − q − E k

that is χ(q) =

1 gB

+

1 V

φ0 /gB

1 k>kF k +q−k −q −E

Finally, eq. (8) can be recast as Eφ0 /gB = expression for χ(q), E=

1 V

1 gB

q
+

1 V

q
.

χ(q)/V , that is, using the explicit

1 1 k>kF k +q−k −q −E

.

We get rid of the bare coupling constant gB by using the Lippman-Schwinger equation, which ﬁnally yields the following implicit equation for E: (11)

E=

1 V

q
m 4π2 a

−

1 V

k
1 2k

+

1 V

1

k>kF

1

k +q−k −q −E

−

1 2k

.

Before addressing the unitary limit case, let us show that this formula allows us to recover the known exact results in the limit of small scattering lengths where the

613

Unitary polarized Fermi gases

denominator is dominated by the 1/a term. The correction to the energy is therefore E∼

4π2 a N 1 4π2 a = , V m m V q
where N is the total number of majority atoms. We thus see that the trial state recovers the mean-ﬁeld prediction for low interactions. For a → 0+ (BEC regime), a little more involved calculation allows one to recover the classical molecular binding energy E ∼ −2 /ma2 . Finally in the case of the unitary regime relevant to experiments, eq. (11) is solved numerically and yields E ∼ −0.32 kF2 /m, that is ηβ < −0.60, a value remarkably close to that obtained in Monte Carlo simulations [46]. 4. – Trapped system and comparison with MIT experiment The model presented in the previous section adresses only the situation of a homogeneous system and to compare with experiments, we need to extend the formalism developed in the previous section to the case of trapped systems. To this purpose we make use of the Local Density Approximation (LDA), in which we assume that the chemical potential of species σ varies as μσ (r) = μ0σ − V (r),

(12)

where V is the trapping potential. From this relation, we see that varying r is equivalent to varying the chemical potentials of the two species, and in particular their ratio η(r). The two-phase transitions described in the previous section will happen for radii r = Rα,β such that μ↓ (Rα,β )/μ↑ (Rα,β ) = ηα,β . Moreover, since the outer rim is constituted by a normal ideal gas, the boundary R↑ of the majority component is given by the condition μ↑ (R↑ ) = 0. In an isotropic harmonic trap, we can combine these three relations to eliminate the (0) parameters μσ , thus obtaining the general relation relating the three radii Rα,β,↑ : (13)

Rα R↑

2 =

(Rβ /R↑ )2 − q , 1−q

where q = (ηα − ηβ )/(1 − ηβ ). One striking consequence of this equation is the prediction of a threshold at which Rα vanishes, corresponding to the disappearance of the fully paired superﬂuid. This transition happens when the ratio (Rβ /R↑ )2 reaches the critical value q. From the upper and lower bounds obtained for ηα and ηβ , we see that q > 0.30. This prediction of LDA is remarkably well veriﬁed in MIT’s experiments [28] for which the three phases discussed above were indeed observed, and eq. (13) could be tested experimentally (ﬁg. 2). On this graph, we see that for large imbalance, the linear scaling predicted by eq. (13) is indeed satisﬁed, with q ∼ 0.32, in agreement with the

614

F. Chevy

0,5

n

↑

0,4

0,3

Rα Rβ

R↑

r

0,2

0,1

0,0 0,00

0,25

0,50

0,75

(R /R↑ Fig. 2. – Comparison of eq. (13) with experimental data from MIT. A ﬁt to the data yields q ∼ 0.32. Inset: sketch of the density proﬁle. The full (respectively dashed) line corresponds to the density of the majority (respectively minority) component. Rα marks the end of the superﬂuid region, Rβ that of the mixture and R↑ is the frontier of the majority cloud.

lower bound obtained earlier. The deviation from theory observed for (Rβ /R↑ )2 0.5 is not yet fully understood. However, it must be noted that the discrepancy takes place in a regime of low imbalance, where the phase transitions take place in the tail of the density distribution. In these regions of low density, we may observe a breakdown of the LDA, or of the hydrodynamical expansion that was used to infer the experimental radii. The value q ∼ 0.32 obtained from the comparison with experimental data can help us improve the bounds for ηα,β . Indeed, this relation ﬁxes the relative values of ηα and ηβ . When combined with the bounds found in the previous section, we obtain indeed

(14)

−0.62 < ηβ < −0.60,

(15)

−0.10 < ηα < −0.088.

From the previous analysis, we see that the combination of theoretical arguments and analysis of experimental data allows for a precise determination of the thresholds of the diﬀerent phase transitions. Knowing the values of ηα,β as well as the exact equation of state in the fully polarized and fully paired phases, we can even obtain some upper and lower bounds for the equation of state of the mixed phase, using the concavity of the grand potential [36].

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Unitary polarized Fermi gases

1,2 1,0

R/R TF

0,8 0,6 0,4 0,2 0,0 0,0

0,2

0,4

0,6

0,8

1,0

P Fig. 3. – Rice’s radius measurement and comparison with a two-phase model. The radius Ri is scaled in units of the Thomas-Fermi radius of an ideal gas with a the same atom number Ni .

5. – Elongated systems and Rice’s experiment Surprisingly, similar experiments performed at Rice University showed no evidence of an intermediate phase, but rather the coexistence of the fully paired and fully polarized phases only. Measurements of the axial radii of the two phases from ref. [29] are presented in ﬁg. 3 and can be compared with the model presented above when omitting the intermediate mixed phase [37]. In these conditions, the inner superﬂuid region is now deﬁned by the condition μ↓ (r)/μ↑ (r) < ηc and is bounded by the radius R↓ deﬁned by R↓2

(16)

2 = m¯ ω2

μ0↓ − ηc μ0↑ 1 − ηc

.

Atoms of the minority species are located in the paired superﬂuid phase only. We thus have

(17)

2 N↓ = n↓ (r) d r = 3πξ 3/2 r

μ0↑ + μ0↓ ¯ ω

3 ¯ g(R↓ /R),

¯ 2 = (μ0 + μ0 )/m¯ where R ω 2 and ↑ ↓ (18)

√

x 1 − x2 −3 + 14 x2 − 8 x4 + 3 arcsin(x) . g(x) = 48

Excess atoms of the majority species are located between r = R↓ and r = R↑ such R that m¯ ω 2 R↑2 /2 = μ0↑ . The number of excess atoms is therefore N↑ − N↓ = R↓↑ n1 (r) d3 r,

616

F. Chevy

hence (19)

2 N↑ − N ↓ = 3π

2μ0↑ ¯ ω

3 (g(1) − g(R↓ /R↑ )).

Dividing (19) by (17) yields the implicit equation for η0 = μ0↓ /μ0↑ as a function of N↑ /N↓ (20)

8 g(1) − g(R↓ /R↑ ) N↑ = 1 + ξ 3/2 . 3 ¯ N↓ (1 + η0 ) g(R↓ /R)

Equation (20) is solved numerically and the value obtained for η0 is then used to calculate the radii R↑ and R↓ . The predicted evolution of the Ri vs. the population imbalance P = (N↑ − N↓ )/(N↑ + N↓ ) is shown in ﬁg. 2. To follow ref. [29], we have normalized each Ri to the Thomas-Fermi radius RTF associated to an ideal gas containing Ni atoms. The agreement with the experimental data is quite good as soon as P 0.1, a remarkable result, since the model presented here contains no adjustable parameters, as soon as the value of ξ is known. Despite this remarkable agreement, this simple two phase+local density approximation model fails to captures all experimental features. In particular, a qualitative discrepancy occurs in the comparison between the theoretical and integrated density proﬁles. Indeed, as shown in [47], LDA at unitarity implies a constant density diﬀerence in the paired superﬂuid region, in contradiction with experimental data. One solution to this problem was presented in [32, 31]. In these papers, it is noted that in the presence of phase transitions, the description of the sharp frontier separating to adjacent phases involves the introduction of density gradient terms in the energy. When the interface in thin enough, they can be encapsulated in a new surface tension energy term reading [31] ΩST =

S

γ(μ↑↓ (r))d2 S,

where S is the interface between the two phases, and γ is the surface tension constant, which should dimensionally vary as γ=λ

mμ2↑ . 2

Here, λ is a numerical factor that will be determined by comparison with experiments and we have used the fact that at the coincidence between the phases, the ratio μ↓ /μ↑ is ﬁxed and equal to ηc , meaning that the two chemical potentials are no longer independent. We can minimize the total grand potential Ω = Ωbulk + ΩST , where Ωbulk = − (PN + PSF )d3 r is the bulk contribution to the energy. Following [31] we simplify the analysis by assuming that the interfaces are ellipsoidal, and for λ ∼ 10−4 one obtains the results presented in ﬁg. 4, which coincides with experimental data. The absence of capillary

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Unitary polarized Fermi gases

Fig. 4. – Integrated density diﬀerence in Rice experiments, and comparison with the surface tension model (data from [31]). Dashed line, LDA prediction: the density diﬀerence is ﬂat in the superﬂuid region, in contradiction with experimental date. Full line: two-phase model incorporating surface tension eﬀect. The same surface tension parameter λ =∼ 10−4 is used for all three graphs.

eﬀects at MIT can be explained by a smaller trap aspect ratio and a larger atom number of atoms compared with Rice’s experimental situation, as shown by a simple scaling argument [31]. 6. – Conclusions The formalism presented here allows for a simple and quantitative description of macroscopic properties of polarized Fermi gases in the regime of strong interaction. This analysis is nevertheless far from being complete, since it does not give any information on the superﬂuid nature of the various phases. For instance, the mixed region of the phase diagram may contain superﬂuid and normal subdomains, the transition between this two regimes being characterized by a universal number ηγ ∈ [ηβ , ηα ]. The quantitative understanding of these superﬂuid properties will require beyond mean-ﬁeld theories, such as the Monte Carlo calculations of [46]. ∗ ∗ ∗ The author gratefully acknowledges support by the IFRAF institute and the ACI Nanosciences 2004 NR 2019. The author thanks the ENS ultracold atoms group, S. Stringari, C. Lobo, A. Recati, A. Bulgac, E. A. Mueller, X. Leyronas, C. Mora and R. Combescot for stimulating discussions. Laboratoire Kastler Brossel is a research unit No. 8552 of CNRS, ENS, and Universit´ e Paris 6. Appendix Thermodynamical relations for the grand potential Let us consider a homogeneous many-body system characterized by a Hamiltonian = 0 and containing particles of p diﬀerent species labelled by i = 1 . . . p. In the grandH canonical ensemble, one looks for the ground state of this system by letting the atom =i=1...p ﬁxed. This therefore numbers ﬂuctuate, but keeping the expectation values N =i , where = − p μi N = requires to ﬁnd the ground state of the grand potential Ω(μi ) = H i=1 the μi are Lagrange multiplier that we interpret as chemical potentials.

618

F. Chevy

Let |ψ(μi , V ) be the ground state of the grand potential, we set Ω(μi , V ) = = ψ(μi , V )|Ω|ψ(μ i , V ). Using Hellman-Feynman relation, we can write that (A.1)

= ∂Ω ∂Ω =i , = ψ(μi , V )| |ψ(μi , V ) = −N ∂μi ∂μi

from which we deduce that (A.2)

dΩ =

−Ni dμi +

i

∂Ω dV. ∂V

By deﬁnition, and by analogy with classical thermodynamics, we identify ∂V Ω with −P , the pressure in the system. Let us now us the extensivity of the potential: when the volume is multiplied by some scaling factor λ, Ω is multiplied by the same factor. In other words, we have Ω(λV, μi ) = λΩ(V, μi ). Taking λ = 1/V , we get Ω(V, μi ) = V Ω(1, μi ). Diﬀerentiating this with respect to V , we note that Ω(1, μi ) = −P , hence Ω = −P V.

(A.3)

From this equation, we see that the minimum grand potential state has also the highest pressure. P can moreover be calculated by diﬀerentiating Ω and using eqs. (A.3) and (A.2). We then obtain the Gibbs-Duhem relation (A.4) dP = ni dμi , i

where ni = Ni /V is the density of species i. From eq. (A.4), we see that the pressure (hence the grand potential) can be obtained simply from the knowledge of the equation of state ni (μj ). . = i ), we have A 1. Concavity. – Since, by deﬁnition, |ψ(μi ) is the ground state of Ω(μ for any δμi (A.5)

= i )|ψ(μi + δμi ) ≥ ψ(μi )|Ω(μ = i )|ψ(μi ). ψ(μi + δμi )|Ω(μ

=j , we see that for any δμi = i + δμi ) + δμj N = i ) = Ω(μ Moreover, if one notes that Ω(μ j (A.6)

Ω(μi + δμi ) +

δμj Nj (μi + δμi ) ≥ Ω(μi ).

j

Finally, recalling that Ni = ∂μi Ω and after expansion of eq. (A.6) to second order in δμi , we obtain (A.7)

∂2Ω δμi δμj ≤ 0, ∂μi ∂μj

hence proving the concavity of the grand potential (or conversely the convexity of the pressure).

Unitary polarized Fermi gases

619

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[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

[25] [26] [27] [28] [29] [30] [31]

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[32] Imambekov A., Bolech C. J., Lukin M. and Demler E., Phys. Rev. A, 74 (2006) 053626. [33] Pieri P. and Strinati G. C., Phys. Rev. Lett., 96 (2006) 150404. [34] Yi W. and Duan L.-M., Phys. Rev. A, 73 (2006) 031604. [35] Haque M. and Stoof H. T. C., Phys. Rev. A, 74 (2006) 011602. [36] Bulgac A. and McNeil Forbes M., preprint cond-mat/0606043. [37] Chevy F., Phys. Rev. Lett., 96 (2006) 130401. [38] Chevy F., Phys. Rev. A, 74 (2006) 063628. [39] Vaschy A., Ann. T´ el, 12 (1892) 25. [40] O’Hara K. M., Hemmer S. L., Gehm M. E., Granade S. R. and Thomas J. E., Science, 298 (2002) 2179. [41] Bartenstein M., Altmeyer A., Riedl S., Jochim S., Chin C., Denschlag J. H. and Grimm R., Phys. Rev. Lett., 92 (2004) 120401. [42] Kinast J., Turlapov A., Thomas J. E., Chen Q., Stajic J. and Levin K., Science, 307 (2005) 1296. [43] Carlson J., Chang S.-Y., Pandharipande V. R. and Schmidt K. E., Phys. Rev. Lett., 91 (2003) 050401. [44] Perali A., Pieri P. and Strinati G. C., Phys. Rev. Lett., 93 (2004) 100404. [45] Astrakharchik G. E., Boronat J., Casulleras J. and Giorgini S., Phys. Rev. Lett., 93 (2004) 200404. [46] Lobo C., Recati A., Giorgini S. and Stringari S., Phys. Rev. Lett., 97 (2006) 200403. [47] De Silva T. N. and Mueller E. J., Phys. Rev. A, 73 (2006) 051602(R).

Exact treatment of trapped imbalanced fermions in the BEC limit P. Pieri and G. C. Strinati Dipartimento di Fisica, Universit` a di Camerino - I-62032 Camerino, Italy

1. – Introduction In the present contribution we shall analyze the eﬀects of imbalancing the populations of two-component trapped fermions in the BEC (strong-coupling) limit of the attractive interaction between fermions of diﬀerent components. In particular, we shall derive a set of coupled equations which describe composite bosons and excess fermions in this limit, starting from the gap equation with two diﬀerent fermionic chemical potentials. Care will be used to include in these equations the processes leading to the correct dimer-dimer and dimer-fermion scattering lengths, which require us to consider beyond-mean-ﬁeld eﬀects. Numerical results will be presented for the density proﬁles of composite bosons and excess fermions, which are relevant to the recent experiments with trapped Fermi atoms. Results for the formation of vortex patterns in the presence of density imbalance will also be presented. The interest in imbalanced populations of fermions with diﬀerent spins originated long time ago from condensed-matter physics, where a magnetic ﬁeld can alter the populations of spin ↑ and ↓ electrons [1, 2]. Quite generally, density imbalance δn = n↑ − n↓ between “spin up” and “spin down” fermions introduces a new degree of freedom in the system, and may thus possibly lead to the removal of degeneracies and the occurrence of novel phases [3, 4]. In this context, trapped cold Fermi atoms oﬀer a unique possibility for observing the consequences of density imbalance. These systems have been lately the object of mounting interest, both experimentally and theoretically, as they allow one to explore c Societ` a Italiana di Fisica

621

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P. Pieri and G. C. Strinati

the BCS-BEC crossover by controlling the interaction between fermions of diﬀerent components via the use of Fano-Feshbach resonances. Recently, two experimental studies with imbalanced populations have raised novel interest in these systems [5, 6]. Density proﬁles of the two fermionic species as well as vortices have been detected. A quantum phase transition to the normal state on the BCS side of the crossover as well as phase separation in the crossover region have been identiﬁed. From a theoretical point of view, the BCS-BEC crossover gets modiﬁed as the manybody problem becomes richer in the presence of density imbalance. The interest in this problem has then involved not only cold-atom and condensed-matter physics, but also nuclear and subnuclear physics. On the BEC side of the crossover, it further gives one the opportunity of embedding into the diagrammatic structure the processes [7,8] leading to the correct values of the scattering lengths for two composite bosons (aB ) and for a composite boson with an excess fermion (aBF ). The eﬀects of density imbalance on fermionic superﬂuids were originally studied in the weak-coupling (BCS) limit of the crossover both for the homogeneous [9] and trapped case [10]. Only recently these calculations have been extended to cover the BCS-BEC crossover [11], and to consider the eﬀects of the trap [12-14] which are essential to account for the experimental results with density imbalance. We thus begin by giving in the next section a brief account of the mean-ﬁeld approach for the homogenous case when diﬀerent spin populations are considered, with emphasis to the strong-coupling (BEC) limit to which we shall eventually restrict in the later sections when considering the trapped case. 2. – Mean-ﬁeld treatment for the homogeneous case The microscopic BCS theory of superconductivity is most conveniently formulated in terms of fermionic single-particle Green’s functions [15]. Due to the presence of spontaneous broken symmetry, anomalous averages need be considered together with normal averages, leading to a 2 × 2 matrix for the Green’s functions. This formulation can be readily extended to include diﬀerent populations for the two fermionic species (labeled by spin-↑ and spin-↓) which mutually interact via a contact potential. This is done by considering two diﬀerent chemical potentials μ↑ and μ↓ for the two species, so that the equation of motion for the fermionic single-particle Green’s functions Gij (k, ωs ) within mean ﬁeld reads

(1)

⎡ k2 ⎢iωs − 2m + μ↑ ⎣ −Δ∗

⎤ −Δ iωs +

k2 − μ↓ 2m

⎥ G11 ⎦ G21

1 G12 = 0 G22

0 . 1

Here, ωs = (2s+1)π/(kB T ) (s integer) is a fermionic Matsubara frequency at temperature T , k a wave vector, m the fermion mass, and Δ the gap function. The novelty introduced in eq. (1) by population imbalance is the presence of two diﬀerent chemical potentials in the diagonal matrix elements on its left-hand side. This seemingly minor diﬀerence will, however, yield signiﬁcant diﬀerent results upon inverting eq. (1) in favor of Gij .

Exact treatment of trapped imbalanced fermions in the BEC limit

623

From this inversion one obtains, in particular (2)

n = n↑ + n↓ =

eiωs η G11 (k) − e−iωs η G22 (k) k

ξ(k) [1 − f (E+ (k)) − f (E− (k))] 1− E(k)

dk (2π)3

=

for the total density (where η is a positive inﬁnitesimal), (3)

δn = n↑ − n↓ = =

eiωs η G11 (k) + e−iωs η G22 (k) k

dk {f (E+ (k)) − f (E− (k))} (2π)3

for the density diﬀerence, and (4)

m = − 4πaF

dk (2π)3

1 − f (E+ (k)) − f (E− (k)) m − 2 2E(k) k

for the gap equation where the strength of the contact fermionic attraction has been replaced with the fermionic scattering length aF by suitable regularization [16]. In these expressions, f (E) = (eβE + 1)−1 is the Fermi distribution function with β = 1/(kB T ) and we have introduced the notation (5)

E± (k) = E(k) ± δξ,

E(k) =

ξ(k)2 + |Δ|2

and (6)

ξ(k) =

μ↑ + μ↓ k2 − , 2m 2

δξ =

μ↓ − μ↑ 2

(only the case of equal fermion masses will be considered throughout). Note from eq. (3) that, in the low-temperature limit, the only way to sustain a nonvanishing value of δn is to have either E+ (k) or E− (k) negative. For deﬁniteness, we shall assume n↑ ≥ n↓ . It is known from the theory of the BCS-BEC crossover that the dimensionless parameter (kF aF )−1 controls the evolution from the weak-coupling BCS regime (where aF < 0 and (kF aF )−1 −1) to the strong-coupling BEC regime (where aF > 0 and (kF aF )−1 +1), with the “crossover” region being limited in practice to the interval −1 (kF aF )−1 +1. Here, kF is the Fermi wave vector related in the standard way to the total density. In the following, we shall mostly be interested in the strong-coupling (BEC) regime at low temperature. In this limit, one expects the presence of density imbalance to produce a density n↓ of composite bosons formed by pairing a fermion of spin ↑ with a fermion

624

P. Pieri and G. C. Strinati

of spin ↓, plus a density δn = n↑ − n↓ of excess fermions of spin ↑. Correspondingly, one ﬁnds from eqs. (2)-(3) μ↓ = − 0 − μ↑ + μB ,

(7) where 0 =

1 m a2F

is the binding energy of the associated two-body problem and

2πaBF μ↑ ∼ n0 , = F (δn) + mBF

2πaBF 4πaB n0 + δn . μB ∼ = mB mBF

(8) (9)

In these expressions, F (δn) is the Fermi energy corresponding to δn, μB plays the role of the chemical potential of the composite bosons, and n0 is the condensate density which equals (n − δn)/2 under the present circumstances. In addition, mB = 2m is the mass of a composite boson, mBF = 32 m is the reduced mass of the two-body system made up of a composite boson and an excess fermion, while aB = 2aF is the scattering length for the scattering of two composite bosons and aBF = (8/3)aF for the scattering of a composite boson and an excess fermion. These values for aB and aBF are speciﬁc to the mean-ﬁeld treatment and correspond to the Born approximation for the scattering processes. After these preliminary considerations of general relevance in the presence of imbalanced fermion populations, we pass now to consider the trapped case which is speciﬁcally relevant to the experiments with cold atoms. 3. – Mean-ﬁeld treatment for the trapped case At a formal level the mean-ﬁeld treatment for the inhomogeneous case proceeds along similar lines as for the homogenous case discussed in the previous section, the diﬀerence being that the equation of motion for the fermionic single-particle Green’s functions Gij (r, r ; ωs ) is now given by (10)

iωs − H↑ (r) −Δ(r)∗ 2

−Δ(r) iωs + H↓ (r)

G11 G21

1 G12 = δ(r − r ) 0 G22

0 . 1

∇ + V (r) − μσ is the single-particle Hamiltonian in the presence Here, Hσ (r) = − 2m of the trapping potential V (r). Equation (10) is the Green’s function version of the Bogoliubov-de Gennes equations [17], which are often used to describe inhomogeneous superconductors. As in the previous section, we are mostly interested in the strong-coupling BEC regime at low temperature. In the case of equal fermion populations (whereby μ↑ = μ↓ ), it has been shown [18] that the Gross-Pitaevskii equation for composite bosons can be derived in this limit from the Bogoliubov-de Gennes equations (10). This mapping enables one to exploit the results obtained directly from the more manageable bosonic Gross-Pitaevskii

Exact treatment of trapped imbalanced fermions in the BEC limit

625

equation and, when needed, to use them as benchmarks for the fermionic calculation in the BEC limit. This analysis has recently been extended to the imbalanced case when μ↑ = μ↓ . In this case, in the place of the Gross-Pitaevskii equation one ends up with two coupled equations that describe the simultaneous presence of composite bosons and excess fermions [12]. As the analysis proceeds along similar lines in both (balanced and imbalanced) cases, we indicate it here schematically for convenience. One starts by rewriting the Bogoliubov-de Gennes equations (10) in the integral form (11)

ˆ r ; ωs ) = Gˆ0 (r, r ; ωs ) + G(r,

ˆ ) G(r ˆ , r ; ωs ), dr Gˆ0 (r, r ; ωs ) B(r

where (12)

G˜0 (r, r ; ωs |μ↑ ) 0 ˆ G0 (r, r ; ωs ) = 0 −G˜0 (r , r; −ωs |μ↓ )

is the matrix of the noninteracting Green’s functions which satisfy the equation (13)

[iωs − Hσ (r)] G˜0 (r, r ; ωs |μσ ) = δ(r − r ),

being subject to the same trapping potential V (r) entering the single-particle Hamiltonian Hσ (r). In addition, the matrix (14)

ˆ B(r) =

0 Δ∗ (r)

Δ(r) 0

contains the eﬀects of the interaction via the gap function. At this point, expansion of G11 (r, r ; ωs ) and G22 (r, r ; ωs ) up to order Δ2 yields the expressions for the local densities n↑ (r) and n↓ (r), in the order, while expansion of G12 (r, r ; ωs ) up to order Δ3 yields an equation for the local gap Δ(r). The gap equation can be cast in the form [12] (15)

4 π aB 3πaBF ∇2 δn(r) Φ(r) + Φ(r) + 2V (r) + |Φ(r)|2 Φ(r) = μB Φ(r), − 2mB m mB

where again mB = 2m, aB = 2aF , aBF = 83 aF , μB = μ↑ + μ↓ + 0 , while Φ(r) = m2 aF 8π Δ(r) plays the role of the bosonic condensate wave function. By a similar token, the density of excess fermions reads (16)

δn(r) = n↑ (r) − n↓ (r) ∼ =

dk f (2π)3

k2 3πaBF 2 + V (r) + |Φ(r)| − μ↑ . 2m m

Note that the two coupled equations (15) and (16) embody the mutual eﬀects of the bosonic distribution |Φ(r)|2 and the fermionic distribution δn(r). Finally, the expression

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P. Pieri and G. C. Strinati

for the density is (17)

n(r) = n↑ (r) + n↓ (r) = δn(r) + 2 |Φ(r)|2 ,

which completes the set of three coupled equations for Φ(r), δn(r), and n(r). The remaining problem is that the scattering between composite bosons (as embodied by aB ) and between a composite boson and an excess fermion (as embodied by aBF ) has been treated so far at the lowest order within the Born approximation, corresponding to the values aB = 2aF and aBF = 38 aF obtained by our derivation. This points to the need of going beyond the mean-ﬁeld treatment to include the full set of scattering processes for aB and aBF . In diagrammatic language, this implies identifying additional fermionic diagrams containing diagrammatic sequences for aB and aBF as sub-units. 4. – Exact equations in the dilute case The validity of the equations obtained in the previous section can be extended by improving on the values of the scattering lengths aB and aBF . From the exact solutions (in real-space representation) of the three- [19] and four-body [20] problems, it is known that the correct values are aBF = 1.18aF and aB = 0.6aF , in the order. These values have also been determined diagrammatically (in wave vector representation) in the limit of vanishing density, for aBF in ref. [8] and for aB in ref. [7]. To improve on the derivation of the equations obtained in the previous section, one has thus to embed the diagrammatic sub-units, which identify aB and aBF in the limit of vanishing density, into the many-body structure at ﬁnite density. This can be achieved as follows. As a natural extension of what is done for determining the condensate density of pointlike bosons [21], in the BEC limit of interest here the gap equation can be interpreted as the condition of vanishing “tadpole” insertions for composite bosons. At the meanﬁeld level considered in the previous section, the diagrams representing this condition are depicted in ﬁg. 1(a), with the understanding that a composite-boson propagator with zero four-momentum can be inserted from the left while the gap Δ corresponds to a condensate line. These diagrams account for the dimer-fermion and dimer-dimer scattering processes within the Born approximation. The additional tadpole diagrams, which yield the correct values of aBF and aB , are depicted in ﬁgs. 1(b) and (c), in the order. Here, the dimer-fermion and dimer-dimer scattering processes exclude the Bornapproximation contributions already included at the mean-ﬁeld level in ﬁg. 1(a). With this procedure, one ends up formally with the same coupled equations (15)(17) which, however, now contain the correct values of aB and aBF . In this context, a comment is worth on how to factor out from the diagrams of ﬁg. 1(b) the product aBF times δn that enters eq. (15). The point is that the integration over the wave vector P is bounded within the Fermi sphere of radius 2mμ↑ , while the remaining integrations over q, q , · · · , extend outside this Fermi sphere. One may, accordingly, neglect the P -dependence everywhere in the diagrams of ﬁg. 1(b) except in the fermion propagator labeled by P and corresponding to spin-↑ fermions. The density of excess fermions results

627

Exact treatment of trapped imbalanced fermions in the BEC limit

' q

q

(a)

'

Ŧq

+ Ŧq

' Ŧq

q

' q

q T’(0) 3

(b)

' =

'

Ŧq

Ŧq

Ŧq P

P

q ŦqŦq’

+

Ŧq

Ŧq’ q’

+ ...

' P

' q

q

(c)

T’(0) 4

' =

Ŧq

Ŧq

'

' + ...

Ŧq’ q’

'

' Fig. 1. – Tadpole diagrams for composite bosons describing the gap equation in the BEC limit. (a) Mean-ﬁeld contributions; contributions beyond the Born approximation which include the scattering processes yielding the correct values of the (b) dimer-fermion and (c) dimer-dimer scattering lengths. Full lines represent the fermion propagators of given spin and broken lines represent the bare fermionic interaction. (Reproduced from ref. [12].)

in this way from the P -integration, with the remaining parts of the diagrams yielding the exact dimer-fermion scattering matrix T3 (0). This excludes, by deﬁnition, the Born contribution resulting from mean ﬁeld. A similar analysis can be carried out for the density equation (16), where only the value of the dimer-fermion scattering length aBF requires corrections beyond mean ﬁeld [12]. 5. – Numerical results and comparison with experiments We pass now to report on the numerical solution of the coupled equations (15)-(17) with the exact values of aB and aBF . By their very derivation, these equations are

628

P. Pieri and G. C. Strinati

3.5

density

1.5 2.5 0.5 1.5 0

0.4

0.8

1.2

0

0.4

0.8

1.2

0

0.4

0.8

1.2

0.5 3.5

density

1.5 2.5 0.5 1.5 0.5 3.5

density

1.5 2.5 0.5 1.5 0.5 0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

r Fig. 2. – Density proﬁles δn(r) (full lines) and n0 (r) (broken lines) vs. r for (kF aF )−1 = 3 and α = 0.2 (upper panel), α = 0.5 (middle panel), α = 0.8 (lower panel). The insets show the results when (kF aF )−1 = 1 for the same values of α. Here, r is in units of RTF and densities are 3 . The Thomas-Fermi radius RTF and the Fermi wave vector kF are in units of (N↑ + N↓ )/RTF deﬁned for equal populations. (Reproduced from ref. [12].)

expected to be valid in the strong-coupling (BEC) regime where aF > 0, which however extends as down as to (kF aF )−1 ≈ +1 for all practical purposes. We determine the density proﬁles δn(r) and n0 (r) = |Φ(r)|2 in the Thomas-Fermi (LDA) approximation (which corresponds to neglecting the kinetic energy term in eq. (15)) for a spherical trap, as functions of the asymmetry parameter α = (N↑ − N↓ )/(N↑ + N↓ ) (where 0 ≤ α ≤ 1). Numerical results [12] are shown in i) Figure 2 for the density proﬁles δn(r) and n0 (r) vs. the distance r = |r| from the center of the trap, for three characteristic values of α and for the coupling (kF aF )−1 = 3 on the BEC side of the crossover. The insets show the results for the smaller coupling

Exact treatment of trapped imbalanced fermions in the BEC limit

629

column density

1.5 1.5

1 0.5

1 0.5

0

0.4

0.8

1.2

0

0.4

0.8

1.2

0

0.4

0.8

1.2

0.8

1

column density

1.5 1.5

1 0.5

1 0.5

column density

1.5 1.5

1 0.5

1 0.5

0

0.2

0.4

0.6

1.2

1.4

U Fig. 3. – Column density proﬁles δn(ρ) (full lines) and n0 (ρ) (broken lines) vs. ρ for (kF aF )−1 = 3 and α = 0.2 (upper panel), α = 0.5 (middle panel), α = 0.8 (lower panel). The insets show the results when (kF aF )−1 = 1 for the same values of α. Here, ρ is in units of RTF and densities 2 . are in units of (N↑ + N↓ )/RTF

(kF aF )−1 = 1. A spatial separation results between the condensed composite bosons and the excess fermions, which appears sharper for the smaller coupling, corresponding to enhanced eﬀects of the dimer-fermion repulsion. Upon approaching unitarity [(kF aF )−1 = 0], there thus appears a tendency toward phase separation with a superconducting core of fully paired fermions surrounded by a cloud of excess fermions. For the couplings here considered, this phase separation occurs for all values of α. Note that, for each value of α, the maximum of δn(r) occurs where n0 (r) vanishes, and that there occurs a progressive size shrinking of n0 (r) for increasing α, with a simultaneous penetration of δn(r) toward the center of the trap.

630

P. Pieri and G. C. Strinati

1.2 E column density

1 0.8 MF

0.6

E

0.4

MF

0.2 0 0

0.2

0.4

0.6 r/RF

0.8

1

1.2

Fig. 4. – Column density proﬁles δn(ρ) (full lines) and n0 (ρ) (broken lines) vs. ρ when (kF aF )−1 = 1 and α = 0.5, as obtained with the exact (E) and mean-ﬁeld (MF) values of the scattering lengths. Units are the same as in ﬁg. 3.

ii)Figure 3 for the corresponding column density proﬁles n(ρ) = dz n(ρ, z) vs. ρ = x2 + y 2 . Phase separation appears now less visible even for (kF aF )−1 = 1, since n(ρ) leaks toward ρ = 0 where it acquires a ﬁnite value. iii) Figure 4 for the comparison of the column density proﬁles when (kF aF )−1 = 1 and α = 0.4, as obtained with the exact and mean-ﬁeld values of the scattering lengths. The use of the exact values of the scattering lengths results in density proﬁles which are more compressed toward ρ = 0. This feature can be conveniently characterized by the values of the critical radius Rc where the condensate vanishes. iv) Figure 5 for the critical radius Rc of the condensate density, which is plotted vs. α for diﬀerent couplings and using the exact (a) or mean-ﬁeld (b) values of the scattering lengths. This quantity identiﬁes also the position of the maximum of the density of excess fermions. Note the inverted sequence of the curves corresponding to the diﬀerent values of (kF aF )−1 , as calculated with the exact or with the mean-ﬁeld values of the scattering lengths. This feature could be subject to experimental veriﬁcation. v) Figure 6 for the ratio ν between the column density of the excess fermions at the center of the trap and its maximum value at the critical radius Rc vs. aBF /aF for α = 0.5 and diﬀerent couplings. The use of column density to obtain the ratio ν stems from our ﬁnding that column density proﬁles have a more marked dependence on aBF . Indeed, the marked dependence on aBF should make it possible to extract the expected value 1.18 of aBF /aF from the experimental data, using the plots of ﬁg. 6 as calibration curves. This procedure is indicated schematically in the ﬁgure. Note that, although the results presented in this section have been obtained for an isotropic (spherical) trap, they can be also utilized for an anisotropic (ellipsoidal) trap, as indicated in Appendix A.

631

Exact treatment of trapped imbalanced fermions in the BEC limit 1

1

(a)

0.6 0.4

(kF aF)-1= 1

0.2 (k a )-1= 2 F F (kF aF)-1= 3 0 0 0.2

0.4 D

0.6

0.8

(b)

0.8

Rc(D)/Rc(0)

Rc(D)/Rc(0)

0.8

0.6 0.4

(kF aF)-1= 1 0.2 (k a )-1= 2 F F (kF aF)-1= 3 0 0 0.2

0.4 D

0.6

0.8

Fig. 5. – Critical radius Rc of the condensate density vs. the asymmetry parameter α (normalized to the value Rc (0) at α = 0) for three coupling values, obtained by using the exact (a) or meanﬁeld (b) values of the scattering lengths. (Adapted from ref. [12].)

Comparison with the experimental results by the MIT [5] and Rice [6] groups for the density proﬁles of the two imbalanced fermion species can be established at this point, at least as far as the BEC side of the crossover region is concerned. In this regime and at low temperatures, the experiments indicate that the dominant eﬀect of density imbalance is to phase separate the system in two components, with an inner superﬂuid region where fermions of diﬀerent species balance each other and an outer normal region where the excess fermions reside. This eﬀect was correctly reported for the ﬁrst time in ref. [12] and it is clearly evidenced in the above ﬁgures.

1

(kF aF)-1= 4 -1

(kF aF) = 3 -1 (kF aF) = 2

Q

0.9 0.8 0.7 0.6 0.8

1.2 aBF/aF

1.6

Fig. 6. – Ratio ν between the column density of excess fermions at the center of the trap and its maximum value at Rc vs. the ratio aBF /aF , for α = 0.5 and three coupling values. (Adapted from ref. [12].)

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6. – Extension to vortices (rotating frame) Unambiguous detection of the superﬂuid phase for trapped Fermi atoms with balanced populations has been achieved by the observation of vortex lattices when the system is put into rotation [22]. By a similar token, in the presence of density imbalance between the two fermionic species, detection of the superﬂuid region has relied on the observation of vortices in that region [5]. To deal with vortex patterns originating from fermions in the presence of density imbalance, we extend the previous treatment holding on the BEC side of the crossover in the following way. We begin by noting that the two coupled equations (15) and (16) for the condensate wave function Φ(r) and the density of excess fermions δn(r) could alternatively be derived from a suitable energy functional E[Φ, δn]. When the system is further put into rotation, this energy functional reads 1 πaB E[Φ, δn] = Z dx dy |∇Φ(r)|2 + mωr2 r2 |Φ(r)|2 + |Φ(r)|4 − (18) 4m m 65/3 π 4/3 δn(r)5/3 + 20m 3πaBF 1 |Φ(r)|2 δn(r) − mΩ2 r2 δn(r) + m 2

− Ω Φ(r)∗ Lz Φ(r) +

1 mωr2 r2 δn(r) + 2 .

Here, Ω is the rotation frequency, Lz the angular momentum operator of the composite bosons, r = (x, y, 0), while for calculation convenience the original ellipsoidal trap has been replaced with a cylinder of height Z with a harmonic radial potential. Note that in eq. (18) the excess fermions in the rotating frame have been treated within a semi-classical description. Let, in fact, 1 dp (19) nR (r) = (2π)3 eβ[R (p,r)−μ] + 1 be the semi-classical fermionic distribution in the rotating (R) frame, where R (p, r) = p2 2m +U (r)−Ω·l the associated semi-classical dispersion with angular momentum l. Since the last expression can be manipulated as follows: (20)

p2 + U (r) − p · (Ω × r) 2m 2 1 p + U (r) − m(Ω × r)2 , = 2m 2

R (p, r) =

where p = p − mΩ × r in the last line, eq. (19) can eventually be cast in the form 2 1 p dp 2 + U (r) − m(Ω × r) , f (21) nR (r) = (2π)3 2m 2 which agrees with the expression for δn(r) one obtains from eq. (18).

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Exact treatment of trapped imbalanced fermions in the BEC limit

0.5 1.6 1.2 0.8 0.4 0

0

-0.5 -0.5

0 X/RF

(b)

Y/RF

(a)

Y/RF

0.5

0

-0.5 -0.5

0.5

3.6 2.7 1.8 0.9 0

0 X/RF

0.5

Fig. 7. – Pattern of an isolated vortex in the balanced situation (α = 0) at the lower critical frequency for coupling values (a) (kF aF )−1 = 1 and (b) (kF aF )−1 = 4.

Sticking to this equation, one has the choice of either solving the associated coupled diﬀerential equations for Φ(r) and δn(r) or minimizing directly E[Φ, δn] by varying Φ(r) and δn(r). In practice, a “mixed” approach may sometimes be preferred [23]. Note that in the present treatment the Thomas-Fermi (LDA) approximation is altogether avoided. (We have also veriﬁed that this reﬁnement leads only to minor changes for the density proﬁles presented in sect. 5 in the absence of rotation.) We present numerical results for the parameters of the MIT trap [5] for which N = 2 × 106 , ω⊥ = 57 Hz, and ωz = 23 Hz. We begin by showing in ﬁg. 7 the pattern of an isolated vortex in the balanced (α = 0) situation at the lower critical frequency Ωc1 when the ﬁrst vortex enters the system, for the two cases (kF aF )−1 = 1 and Ωc1 = 3.95 Hz (a), and (kF aF )−1 = 4 and Ωc1 = 6.11 Hz (b). When the rotation frequency increases further, this pattern evolves in a triangular-shape vortex lattice shown in ﬁg. 8 when

0.5 1.6 1.2 0.8 0.4 0

0

-0.5 -0.5

0 X/RF

0.5

(b)

Y/RF

(a)

Y/RF

0.5

3.6 2.7 1.8 0.9 0

0

-0.5 -0.5

0 X/RF

0.5

Fig. 8. – Triangular-shape vortex lattice in the balanced situation (α = 0) when Ω = 19 Hz and for coupling values (a) (kF aF )−1 = 1 and (b) (kF aF )−1 = 4.

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0.75 1.6 1.2 0.8 0.4 0

0

-0.75 -0.75

(a)

0 X/RF

Y/RF

Y/RF

0.75

(b)

0 X/RF

0.75

0.75 0.6

0.9

0.4

0.6

0.2

0

0

0 X/RF

0.75

Y/RF

Y/RF

0.75

-0.75 -0.75

0

-0.75 -0.75

0.75

3.6 2.7 1.8 0.9 0

0.3

0

0

-0.75 -0.75

0 X/RF

0.75

Fig. 9. – Triangular-shape vortex lattice in the imbalanced situation with α = 0.4, Ω = 19 Hz, and coupling values (a) (kF aF )−1 = 1 and (b) (kF aF )−1 = 4. The upper (lower) plots refer to the density of condensed composite bosons (excess fermions).

Ω = 19 Hz for the same coupling values as in ﬁg. 7. Such triangular pattern persists in the imbalanced situation, as shown in ﬁg. 9 for α = 0.4 and the same frequency Ω = 19 Hz, when (a) (kF aF )−1 = 1 and (b) (kF aF )−1 = 4. In ﬁg. 9 the upper plots picture the density of condensed composite bosons while the lower plots refer to excess fermions. Note that the excess fermions tend to pile up in the vortex cores, taking thus advantage of the fact that the density of composite bosons is depressed there. Further study along these lines is in progress [23]. 7. – Perspectives and open problems In this contribution, we have presented a theoretical account of the eﬀects produced by imbalancing the populations of trapped fermions. Special emphasis has been placed to the strong-coupling (BEC) limit, where theoretical studies can be extended beyond BCS mean ﬁeld by exploiting the diluteness condition of the system. Several eﬀects were, however, left out from our analysis.

Exact treatment of trapped imbalanced fermions in the BEC limit

635

First of all, it appears relevant to combine density imbalance with mass imbalance of the two fermionic species, which may act to promote still novel phenomena in the system and it is being subject to experimental investigations at present. The present approach was conﬁned to the BEC region where (kF aF )−1 +1. As the experiments [5, 6] actually span the whole crossover region −1 (kF aF )−1 +1, theoretical calculations should cover this region, too. In this context, a mean-ﬁeld approach could readily produce a qualitative overview of the eﬀects taking place across the crossover region, including both density and mass imbalance. As in the case of the BEC region, however, mean-ﬁeld calculations may lead to quantitative incorrect results. For instance, at unitarity and for T 0, mean-ﬁeld calculations [14] result in the value αcMF 1 for the critical asymmetry parameter αc past which the superﬂuid region disappears from the system, while the experiment [5] yields αcexp 0.70. This is a clear indication that inclusion of pairing ﬂuctuations beyond mean ﬁeld is relevant in the crossover region to account for the experimental data. At higher temperatures, density imbalance may give access to “precursor” pairing (pseudo-gap) eﬀects for the excess fermions, which could thus be evidenced in the normal phase even below the critical temperature Tc . This may result in a strong suppression of thermal ﬂuctuations, leading to a possible detection of an underlying Quantum Critical Point. Finally, we mention that a compelling test for theories would result from calculating the dependence of Tc on α, as the experiments were able to determine the temperature of the system directly from the “tail” of the density proﬁles of the excess fermions [24]. ∗ ∗ ∗ We thank S. Simonucci, who performed the numerical calculations on vortices and prepared ﬁgs. 7-9. This work was partially supported by the Italian MIUR (contract Coﬁn-2005 “Ultracold Fermi Gases and Optical Lattices”). Appendix A. Mapping of the anisotropic onto the isotropic problem The calculations presented in sect. 5 refer to an isotropic harmonic potential (spherical trap). In the experiments with cold Fermi atoms, however, anisotropic harmonic traps are most often used. The question thus naturally arises to what an extent the theoretical calculations done for a spherical trap can account for the experimental situation. The answer to this question relies on the validity of the Thomas-Fermi (LDA) approximation, which applies when the number N of trapped atoms is large enough and was explicitly used in the calculations of sect. 5. (As a matter of fact, when N is large enough, avoiding this approximation leads only to minor changes of the density proﬁles, as mentioned in sect. 6.) This is because, within LDA, physical quantities acquire an r-dependence only through the local chemical potential (A.1)

μ(x, y, z) = μ −

1 m ωx2 (x2 + λ2y y 2 + λ2z z 2 ). 2

636

P. Pieri and G. C. Strinati

Here, the anisotropy of the harmonic potential is speciﬁed by the parameters λy and λz . In particular, the density proﬁles of interest can be expressed as n(x, y, z) = n[μ(x, y, z)]. To establish the desired mapping with the isotropic case, we introduce an equivalent isotropic problem with a total number of atoms Niso = λy λz N , the harmonic frequency ωiso = ωx , and the same Fermi energy EFiso = EF = (3ωx ωy ωz N )1/3 of the original anisotropic problem. To determine the corresponding chemical potential μiso for this problem, we argue that the ratio μiso /EF must be a universal function of the dimension√ less quantities (kF aF )−1 and T /EF (where, by deﬁnition, kF = 2mEF ). This implies that the ratio μiso /EF can depend on N only via EF . As a consequence, μiso of the equivalent isotropic problem must coincide with the chemical potential μ of the original anisotropic problem. For the density proﬁle n(x, y, z) of the original anisotropic problem, this equivalence eventually implies that it can be expressed in terms of the density proﬁle niso (x, Y, Z) of the equivalent isotropic problem as follows: 1 2 2 2 2 n(x, y, z) = niso μiso − mωiso (x + Y + Z ) , 2

(A.2)

where Y = λy y and Z = λz z. Appendix B. Axial density proﬁles The density proﬁles reported in sect. 5 were obtained within the Thomas-Fermi (LDA) approximation. This approximation tends to sharpen the density proﬁles at their edges, but is otherwise appropriate when the number of trapped atoms is large enough. From those results we have also concluded that the dominant eﬀect of density imbalance is to phase separate the system into two components, an inner superﬂuid region and an outer normal region with excess fermions only. In this context, an argument was proposed by De Silva and Mueller [25] to evidence the occurrence of phase separation in the density proﬁles when the trapping potential is harmonic. Their argument relies on the validity of the LDA approximation and applies to the axial density nA (z), obtained from the density through a double integration: (B.1)

nA (z) =

dx dy n(x, y, z) = π

∞

2

2

2

dρ n(ρ + z ) = π 0

∞

z2

dζ n(ζ),

where ζ = ρ2 + z 2 and n(ζ) ≥ 0. From the last form of eq. (B.1) one readily concludes that nA (z) is a decreasing function of z. In addition, when n(x, y, z) presents a “hole” in the core region (say, for r < r0 ), nA (z) is constant in that region. This situation is pictured in ﬁg. 10, where schematic plots of the radial density n(ζ) with a hole in the core region (upper panel) and of the corresponding axial density nA (z) with a ﬂat behavior in the core (lower panel) are shown. This situation is a ﬁngerprint for the occurrence of phase separation and applies to the density proﬁles shown in the insets of ﬁg. 2.

Exact treatment of trapped imbalanced fermions in the BEC limit

637

n(])

r

r

]

nA( z)

r

r z

Fig. 10. – Schematic plot of the density n(ζ) with a hole in the core region and of the corresponding axial density nA (z) with a ﬂat behavior in the core.

REFERENCES [1] [2] [3] [4] [5]

[6]

[7] [8] [9]

Clogston A. M., Phys. Rev. Lett., 9 (1962) 266. Sarma G., J. Phys. Chem. Solids, 24 (1963) 1029. Fulde P. and Ferrell R. A., Phys. Rev., 135 (1964) A550. Larkin A. I. and Ovchinnikov Y. N., Sov. Phys. JETP, 20 (1965) 762. Zwierlein M. W., Schirotzek A., Schunck C. H. and Ketterle W., Science, 311 (2006) 492; Shin Y., Zwierlein M. W., Schunck C. H., Schirotzek A. and Ketterle W., Phys. Rev. Lett., 97 (2006) 030401. Partridge G. B., Li W., Kamar R. I., Liao Y.-An and Hulet R. G., Science, 311 (2006) 503; Partridge G. B., Li W., Liao Y. A., Hulet R. G., Haque M. and Stoof H. T. C., Phys. Rev. Lett., 97 (2006) 190407. Brodsky I. V., Klaptsov A. V., Kagan M. Yu., Combescot R. and Leyronas X., JETP Lett., 82 (2005) 273. Bedaque P. F. and Van Kolck U., Phys. Lett. B, 428 (1998) 221. Combescot R., Europhys. Lett., 55 (2001) 150; Liu W. V. and Wilczek F., Phys. Rev. Lett., 90 (2003) 047002; Caldas H., Phys. Rev. A, 69 (2004) 063602; Sedrakian A., ¨ther H., Phys. Rev. A, 72 (2005) 013613. Mur-Petit J., Polls A. and Mu

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[10] Mizushima T., Machida K. and Ichioka M., Phys. Rev. Lett., 94 (2005) 060404; ` D., Phys. Rev. A, Castorina P., Grasso M., Oertel M., Urban M. and Zappala 72 (2005) 025601. `res P., Schuck P., Schulze H.-J. and Sedrakian A., Phys. Rev. [11] Lombardo U., Nozie C, 64 (2001) 064314; Carlson J. and Reddy S., Phys. Rev. Lett., 95 (2005) 060401; Pao C.-H., Wu Shin-Tza and Yip S.-K., Phys. Rev. B, 73 (2006) 132506; Sheehy D. E. and Radzihovsky L., Phys. Rev. Lett., 96 (2006) 060401. [12] Pieri P. and Strinati G. C., Phys. Rev. Lett., 96 (2006) 150404. ¨ rma ¨ P., Phys. Rev. Lett., 96 (2006) 110403; Yi W. [13] Kinnunen J., Jensen L. M. and To and Duan L. M., Phys. Rev. A, 73 (2006) 031604(R); Chevy F., Phys. Rev. Lett., 96 (2006) 130401; De Silva T. N. and Mueller E. J., Phys. Rev. A, 73 (2006) 051602(R); Haque M. and Stoof H. T. C., Phys. Rev. A, 74 (2006) 011602(R); Martikainen J.-P., Phys. Rev. A, 74 (2006) 013602; Chien C.-C., Chen Q., He Y. and Levin K., Phys. Rev. A, 74 (2006) 021602(R). [14] Yi W. and Duan L. M., Phys. Rev. A, 74 (2006) 013610; Pao C.-H. and Yip S.-K., J. Phys. Condens. Matter, 18 (2006) 5567. [15] Fetter A. L. and Walecka J. D., Quantum Theory of Many-Particle Systems (McGrawHill, New York) 1971. [16] Pieri P. and Strinati G. C., Phys. Rev. B, 61 (2000) 15370. [17] De Gennes P. G., Superconductivity of Metals and Alloys (Benjamin, New York) 1966. [18] Pieri P. and Strinati G. C., Phys. Rev. Lett., 91 (2003) 030401. [19] Skorniakov G. V. and Ter-Martirosian K. A., Sov. Phys. JETP, 4 (1957) 648. [20] Petrov D. S., Salomon C. and Shlyapnikov G. V., Phys. Rev. Lett., 93 (2004) 090404. [21] Popov V. N., Functional Integrals and Collective Excitations (Cambridge University Press, Cambridge) 1987. [22] Zwierlein M. W., Abo-Shaeer J. R., Schirotzek A., Schunck C. H. and Ketterle W., Nature, 435 (2005) 1047. [23] Simonucci S., Pieri P. and Strinati G. C., unpublished. [24] Zwierlein M. W., Schunck C. H., Schirotzek A. and Ketterle W., Nature, 442 (2006) 54. [25] De Silva T. N. and Mueller E. J., Phys. Rev. A, 73 (2006) 051602.

Path integral description of the superﬂuid properties at the BEC/BCS crossover J. Tempere TFVS, Universiteit Antwerpen - Groenenborglerlaan 171, B2020 Antwerpen, Belgium

1. – Introduction Since their introduction by Feynman in 1949 [1], path integrals have been applied successfully in diverse ﬁelds of physics, from cosmology to condensed-matter physics, and from elementary-particle physics to econophysics. One of the reasons for this ubiquity is that path integrals oﬀer an alternative to the descriptions based on diﬀerential equations such as the (nonlinear) Schr¨ odinger equation. Whereas diﬀerential equations build solutions by starting from local results and adding (inﬁnitesimal) changes, the path integral description takes a global view of the whole system. Moreover, in treating quantummechanical systems, the path integral description oﬀers the advantage that position and momentum need not be expressed as (not commuting) operators, and that covariance is established straightforwardly. The interested reader can ﬁnd some excellent textbooks on the path integral technique in the bibliography [2-4]. In relation to the transition between a Bardeen-Cooper-Schrieﬀer (BCS) state and a Bose-Einstein condensate (BEC) of strongly bound pairs, the path integral technique has been applied ﬁrst by Sa de Melo, Randeria and Engelbrecht [5, 6], in the context of high-Tc cuprate superconductors. Among the competing theories for the still elusive microscopic mechanism of high-temperature superconductivity, one candidate theory envisages high-Tc superconductivity as resulting from Bose-Einstein condensation of bipolarons [7]. Bipolarons are bound states of two electrons due to the electron-phonon interaction. In this theory, conventional BCS superconductors and the high-temperature superconductors with their BEC of bipolarons are situated on diﬀerent sides of the Feshc Societ` a Italiana di Fisica

639

640

J. Tempere

bach resonance, but share the electron-phonon interaction as the underlying mechanism for superconductivity. The recent breakthroughs in creating superﬂuid Fermi gases in the BEC/BCS crossover regime with controllable interactions [8] will lead to new tests and reﬁnements of this model for high-Tc superconductivity. In the next section, we give a review of the path integral treatment of refs. [5, 6] for the BEC/BCS transition for a uniform order parameter. In sect. 3, we discuss extensions of the method to optical lattices, and illustrate how equations of motion for the density and phase ﬁelds are obtained from the action functional. Finally, in sect. 4, we address the question of pairing in a Fermi mixture of more than two spin states. 2. – Review: functional integral crossover theory . 2 1. Action functional for a dilute Fermi gas. – Consider the Hamiltonian functional of a two-spin-state atomic Fermi gas with interactions only between diﬀerent spin states (1)

6 7 H ψ¯x,τ,σ , ψx,τ,σ =

dx

σ

+

dx

2 2 ¯ ∇ ψx,τ,σ + ψx,τ,σ − 2m x

dy ψ¯x,τ,↑ ψ¯y,τ,↓ V (x − y)ψy,τ,↓ ψx,τ,↑ .

The factor (1/2) that usually precedes the interaction part is not present here: since we ﬁxed the choice of spins, the double counting is avoided. Note that in the functional integral description ﬁeld operators are not present. If the Hamiltonian functional (1) is to describe bosons, the variables ψx,τ,σ and ψ¯x,τ,σ would be complex functions of position x, imaginary time τ and spin σ =↑, ↓. However, fermionic ﬁelds have to anticommute. Imposing anticommutation rules on complex functions yields the Grassmann algebra, 2 and ψx,τ,σ and ψ¯x,τ,σ are called Grassmann variables. As a consequence, ψx,τ,σ = 0, and all analytic functions have a ﬁnite Taylor expansion. The action functional corresponding to the Hamiltonian (1) is (2)

7 6 S ψ¯x,τ,σ , ψx,τ,σ =

∂ dτ dx ψ¯x,τ,σ ψx,τ,σ + ∂τ σ ! 7 6 ¯ . +H ψx,τ,σ , ψx,τ,σ

Furthermore, we introduce a Lagrangian multiplier μ to ﬁx the number of particles, N = dx σ ψ¯x,τ,σ ψx,τ,σ so that (3)

6 7 S ψ¯x,τ,σ , ψx,τ,σ =

∂ 2 2 ¯ ψx,τ,σ − ∇ − μ ψx,τ,σ + dτ dx ∂τ 2m x 0 σ β + dτ dx dy ψ¯x,τ,↑ ψ¯y,τ,↓ V (x − y)ψy,τ,↓ ψx,τ,↑ .

β

0

Path integral description of the superfluid properties at the BEC/BCS crossover

641

The (low-energy part of the) interaction between ultracold atoms is well described by an instantaneous contact interaction. The strength of this interaction needs to be renormalized so that the solution of the two-body problem can be retrieved. Thus, we use V (r) = gδ(r) with (from the two-body problem) 1 m 1 = . − 2 g 4π as (k)2 /m

(4)

k

The scattering length as can be tuned through the use of a Feshbach resonance. This resonance occurs when the energy of two atoms scattering in an open channel coincides with the energy of a (molecular) bound state in a closed scattering channel. For broad Feshbach resonances, such as those currently used to achieve superﬂuidity in 40 K and 6 Li, a single channel model with an eﬀective scattering length that depends on the magnetic ﬁeld oﬀers a good description [9]. In what follows we investigate superﬂuid properties as a function of the tunable scattering length. For negative scattering length (and attraction between the atoms) one expects the BCS scenario to hold, whereas for positive scattering length bound pairs can condense. The current experiments with ultracold fermions probe the BEC/BCS crossover regime, where typically −1 < (kF as )−1 < 1 (with kF the Fermi wave number). The partition function of the ultracold Fermi gas is given by the sum over all possible Grassmann ﬁeld conﬁgurations, weighed by the action functional of the ﬁelds Z=

(5)

7 6 , Dψ¯x,τ,σ Dψx,τ,σ exp −S ψ¯x,τ,σ , ψx,τ,σ

with the action functional given by

(6)

∂ 1 2 ¯ ψx,τ,σ − ∇ − μ ψx,τ,σ + dτ dx ∂τ 2m x 0 σ β dτ dx g ψ¯x,τ,↑ ψ¯x,τ,↓ ψx,τ,↓ ψx,τ,↑ . +

7 6 S ψ¯x,τ,σ , ψx,τ,σ =

β

0

We use units such that = 1. . 2 2. The Hubbard-Stratonovic transformation. – Analytic solutions for functional integrals over Grassmann variables are only available for action functionals that are quadratic in the Grassmann variables, so we need to get rid of the four-product of ﬁelds in (6). The four-ﬁeld product can be decoupled by making use of the following identity: (7)

Dφ¯x Dφx exp

x

¯bx bx ¯ ¯ ¯ ax φx φx + bx φx + bx φx ∝ exp − ax x

642

J. Tempere

valid if φ¯x , φx are bosonic ﬁelds. In the present case β dτ dx ψ¯x,τ,↑ ψ¯x,τ,↓ ψx,τ,↓ ψx,τ,↑ = (8) exp −g 0

¯ Δx,τ Δx,τ + g 0 ¯ x,τ ψx,τ,↓ ψx,τ,↑ + Δx,τ ψ¯x,τ,↑ ψ¯x,τ,↓ . +Δ

=

¯ x,τ DΔx,τ exp DΔ

β

dτ

dx

¯ x,τ and Δx,τ are bosonic. The partition sum can then The Hubbard-Stratonovic ﬁelds Δ be written as a double integral 7 6 ¯ x,τ , Δx,τ ¯ x,τ DΔx,τ exp −S ψ¯x,τ,σ , ψx,τ,σ , Δ , (9) Z = Dψ¯x,τ,σ Dψx,τ,σ DΔ with (10)

S=

∂ 1 2 − ∇x − μ ψx,τ,σ β ψ¯x,τ,σ ∂τ 2m 0 σ ! ¯ x,τ Δx,τ Δ ¯ ¯ ¯ −Δx,τ ψx,τ,↓ ψx,τ,↑ − Δx,τ ψx,τ,↑ ψx,τ,↓ − . g

β

dτ

dx

. 2 3. Grassmann functional integration. – This action is quadratic in the fermionic variables. As a result, the functional integration over the Grassmann variables can be performed analytically, using the identity ¯ ¯ ψx Axy ψy = det[Axy ] = exp [Tr [ln(A)]] , (11) Dψx Dψx exp x,y

where Tr denotes the trace. We ﬁnd ¯ x,τ DΔx,τ exp (12) Z = DΔ 0

β

dτ

¯ x,τ Δx,τ

Δ dx ln −G−1 + , g

where

(13)

−G−1

⎛∂ 1 2 − ∇ −μ ⎜ ∂τ 2m x =⎝ ¯ x,τ Δ

⎞ Δx,τ

⎟ ⎠. ∂ 1 2 + ∇x + μ ∂τ 2m

¯ x,τ cannot be solved The remaining path integral over the bosonic ﬁelds Δx,τ and Δ exactly —if this were possible, an analytic solution for the interacting Fermi gas would

Path integral description of the superfluid properties at the BEC/BCS crossover

643

be available. So what is the advantage of introducing the Hubbard-Stratonovic ﬁelds? From eq. (10) it is clear that these ﬁelds represent a (bosonic) pair formed from two fermions with opposite spin. . 2 4. Saddle-point approximation. – The saddle-point approximation consists in replacing the functional integral over the bosonic ﬁelds by the single most contributing term to the integral. If one assumes that all pairs are Bose condensed in the k = 0 ground state, the major contribution to the functional integral will come from the terms with ¯ x,τ = Δ∗ . This corresponds to an order parameter that is constant in space Δx,τ = Δ, Δ (i.e. corresponding to a momentum-zero ground state). (14)

β

Zsp = exp

dτ

dx ln

0

−G−1 sp

|Δ| + g

2

,

with

(15)

−G−1 sp

⎛∂ 1 2 − ∇ −μ ⎜ 2m x = ⎝ ∂τ Δ∗

⎞ Δ

⎟ ⎠. ∂ 1 2 + ∇ +μ ∂τ 2m x

The inverse Green’s function at the saddle-point can be decomposed in Pauli matrices (16)

−G−1 sp

∂ 1 2 − σ1 Δ + σ3 − ∇ −μ . = σ0 ∂τ 2m x

One can get rid of the time and position partial derivatives appearing in this result by Fourier transforming to reciprocal space k and fermionic Matsubara frequencies ωn . The trace over ln(−G−1 sp ) can be taken in either x, τ or in k, ωn representation. The result can be expressed as Zsp = exp[−Ssp ],

(17)

with an eﬀective saddle-point action (after Matsubara summation) 2

(18)

Ssp = −

|Δ| −2 g

dk ln[2 cosh(βEk /2)], (2π)3

with # (19)

Ek =

2 k2 2 − μ + |Δ| . 2m

644

J. Tempere

Fig. 1. – The chemical potential as a function of the interaction parameter 1/(kF as ).

. 2 5. Gap and number equations. – In the eﬀective action (18), the value of |Δ| still needs to be determined, as well as the chemical potential μ. The former is found from extremizing the action functional with respect to |Δ|, to identify the saddle-point value. This results in

tanh(βEk /2) 1 1 1 ∞ 2 − 2 , (20) − = dk k kF as π 0 Ek /2 k where the scattering length appears after introducing eq. (4) for the renormalized contact potential strength. In eq. (20), we express wave numbers in units kF and energies in units of the Fermi energy. The chemical potential is ﬁxed, as usual, by ﬁxing the number of atoms, or the density n = kF3 /(3π 2 ). The thermodynamic potential Ω, obtained from Zsp = e−βΩ , satisﬁes n = −∂Ω/∂μ from which (21)

3 1= 2

0

∞

k dk k 1 − tanh(βEk /2) , Ek 2

with k = k 2 /(2m) − μ. The result for the chemical potential, in the limit of temperature T → 0, is shown in ﬁg. 1. In the BCS limit, μ → EF , whereas in the BEC limit, μ → −2 /(2ma2s ), the binding energy of the molecule. In ﬁg. 2 the gap is plotted as a function of the interaction parameter and temperature. As the temperature is increased, the region of interaction strengths 1/(kF as ) where the gap becomes zero grows. First pairing is suppressed in the BCS state, at higher temperature also in the BEC state. For any interaction strength, the temperature above

Path integral description of the superfluid properties at the BEC/BCS crossover

Fig. 2. – The temperatures. critical value. which the gap

645

gap is shown as a function of the interaction parameter 1/(kF as ), at diﬀerent For temperatures T > 0, the gap becomes zero for 1/(kF as ) smaller than a Alternatively, for any value of 1/(kF as ) one can identify a temperature above becomes zero. This temperature is plotted in the inset.

which the gap vanishes can be extracted —this is plotted in the inset of ﬁg. 2. In the BCS regime, superﬂuidity is destroyed by breaking up Cooper pairs and making the gap vanish. Thus, the temperature at which |Δ| = 0 corresponds to the critical temperature. Indeed, in this regime, one ﬁnds that the gap and Tc are proportional to exp[−1/(kF as )] as in the conventional BCS theory. In the BEC regime, however, superﬂuidity is not destroyed by breaking up the bosons or molecules. Rather, it is destroyed by quantum ﬂuctuations and the resulting depletion of the ground-state condensate. The temperature at which |Δ| vanishes corresponds to the temperature Td where the bosonic molecules dissociate, and no longer to the critical temperature. To ﬁnd the critical temperature Tc in the BEC regime, ﬂuctuations need to be taken into account. In the context of the bipolaronic theory of high-Tc superconductors, the dissociation temperature Td would correspond to the pseudogap temperature. From the dependence of the pseudogap temperature on the doping, it is possible to conjecture that the underdoped regime is deeper in the BEC regime whereas the overdoped regime moves closer to the Feshbach resonance. . 2 6. Fluctuation corrections. – Many schemes exist to take into account ﬂuctuations and ﬁnite-temperature eﬀects. Treating the interactions in a perturbative scheme, selected sets of Feynman diagrams can be resummed and diﬀerent choices lead to diﬀerent crossover theories [10, 11]. Alternatively, quantum Monte Carlo simulations [12] can be used to obtain results on the crossover physics and put the crossover theories to the test. At the unitarity limit, 1/(kF as ), the existing theories do not succeed to ﬁnd the free en-

646

J. Tempere

ergy of the system with more than 10% accuracy with respect to the Monte Carlo results. In the saddle point approximation we replace an entire path integral by just one value of the integrand, of which we hope it will be the most strongly contributing part. We can do slightly better and take into account a small region around that value, and take the integral over that small region. When the ﬂuctuations are small enough to be contained in this region, this prescription will take ﬂuctuation eﬀects suitably into account. To achieve this, we set Δx,τ = Δ + δx,τ , ¯ x,τ = Δ∗ + δ¯x,τ , Δ

(22) (23) obtaining

Dδ¯x,τ Dδx,τ exp [−Ssp+ﬂ ] ,

Zsp+ﬂ =

(24) with (25)

Ssp+ﬂ

= − Tr ln −G−1 − sp + F

β

dτ

dx

0

(Δ∗ + δ¯x,τ )(Δ + δx,τ ) , g

where G−1 sp is given by (15) and F=

(26)

0 −δ¯x,τ

−δx,τ 0

.

The functional integral over δ¯x,τ , δx,τ can be performed only for a quadratic action. Thus, we expand the action Sﬂ up to second order in the ﬂuctuation ﬁelds, using (27)

∞

1 −1 (Gsp F )m + ln −G−1 + F = ln −G sp sp m m=1

and breaking oﬀ the sum at m = 2. The correction to the saddle-point partition function Zsp can then be written as (28)

Zsp+ﬂ ∝ Zsp ×

Dδx,τ Dδ¯x,τ exp −Sﬂ [δx,τ , δ¯x,τ ] ,

with Zsp = exp[−Ssp /] the saddle point result (eq. (18)) and (29)

Sﬂ = − Tr

β 1 δ¯x,τ δx,τ Gsp FGsp F − . dτ dx 2 g 0

Path integral description of the superfluid properties at the BEC/BCS crossover

647

. 2 7. Finite-temperature analysis. – This action function is quadratic in the ﬂuctuation operators and thus the functional integral over those ﬁelds can be taken analytically. The result is / / /M11 M12 / / , / (30) Zﬂ ∝ exp[−βΩﬂ ] = exp − dq log / M21 M22 / n

where the sum over n is a sum over bosonic Matsubara frequencies ωn = 2πn/β, and Ωﬂ is the thermodynamic potential of the gas of excitations. The ﬂuctuation matrix ∗ ∗ elements satisfy M22 (q, n) = M11 (−q, −n) = M11 (q, n) and M21 (q, n) = M12 (q, n). They are given by (31)

M21 (q, n) = −Δ2

1 dk tanh(βEk /2) 3 (2π) Ek (iωn + Ek + Ek+q )(iωn + Ek − Ek+q )

and (32)

M11 (q, n) =

Ek + k dk (iωn + Ek − k+q ) 1 − g (2π)3 2Ek (iωn + Ek + Ek+q )(iωn + Ek − Ek+q ) Ek − k (iωn − Ek − k+q ) − tanh(βEk /2), 2Ek (iωn − Ek + Ek+q )(iωn − Ek − Ek+q )

where, as before, Ek is the Bogoliubov spectrum and k = k 2 /(2m) − μ. These matrix elements are equivalent to the results derived by Strinati and co-workers [10] using a diagrammatic technique. In the framework of the diagrammatic technique, the matrix elements describe the “polarization bubble” that results from splitting a propagating pair into its fermion components and then recombining those components into a pair. The wave number integrals that have to be evaluated in this result are nontrivial; solutions are known at T = Tc and at T = 0. The critical temperature can be found using Δ(T → Tc ) → 0; in this limit M21 → 0 and (33)

M11 =

1 − g

dk tanh[ k /(2kB Tc )] . (2π)3 iωn + k + k+q

Note that the action for the ﬂuctuations depends (through M11 and M21 ) on the choice of the saddle-point Δ. This means that the spectrum of excitations (obtained from the diagonalized ﬂuctuation action) also depends on the choice of the saddle-point. Excitations for a vortex condensate may be diﬀerent from excitations on top of a ground-state condensate. At nonzero temperature those excitations will be populated through Bose statistics. But the physics is more complex than just Bose populating excitations: the excitation spectrum itself (the dispersion and lifetime of those excitations) is temperature dependent: new single-particle and collective modes appear and shift as a function of temperature.

648

J. Tempere

At zero temperature the only single-particle excitations are Ek , the energy spectrum for breaking a Cooper pair, but at ﬁnite temperature, we also have the excitations of the thermal gas. These consist in taking the atoms of a broken Cooper pair, and giving those atoms an extra kick: Ek+q − Ek . Besides those single-particle excitations, we will have collective excitations whenever M11 M22 − M21 M12 = 0 —these are the excitations that correspond to the plasmons (roots of the dielectric function) when the Fermi gas is in the normal state. Several authors have studied the BEC/BCS transition at ﬁnite temperatures. Nozi`eres and Schmitt-Rink solve the quadratic ﬂuctuations in the peculiar case T = Tc to obtain a corrected expression for the transition temperature [13]. Strinati and co-workers [10] work diagrammatically to improve on the results of Nozi`eres and Schmitt-Rink; treating the ﬂuctuations in a perturbative scheme leaves room to choose which diagrams to take into account and which not. Levin and co-workers [11] construct a ﬁnite-temperature theory similar to that of Strinati and co-workers but include diﬀerent diagrams in the summation. Up to this moment and as far as the author knows, there is not yet a complete ﬁnite-temperature theory suited for all coupling strengths and all temperatures, as each choice of diagrams has its limitation. 3. – Application to optical lattices Optical lattices oﬀer the possibility to investigate artiﬁcial crystals, with tunable lattice parameters, and containing bosons or fermions. They allow to create an experimental realization of the (Bose- or Fermi-)Hubbard model. So, can the path formalism presented in the previous sections be adapted for use with optical lattices? . 3 1. Multilayer action functional. – The periodic potential of the optical lattice is formed by standing laser waves with wavelength λ. Here we focus on the case of a one-dimensional periodic potential along the z-direction. Vopt (z) = sER sin2 (2πz/λ),

(34)

where ER = h2 /(2mλ2 ) is the recoil energy and s is the laser intensity expressed in units of the recoil energy. For a suﬃciently strong intensity (e.g., s > 10), the fermionic gas loaded into the optical potential can be treated as a stack of coupled, “pancake”-shaped clouds. Typically, there are a few 100 layers containing on the order of 1000 atoms each. Each of the “pancakes” can be described by the action functional for a (quasi) 2D Fermi gas (35)

Sjlayer

=

dτ 0

β

dx

σ

ψ¯x,j,σ

∂ 1 2 − ∇x − μj ψx,j,σ + ∂τ 2m

! ¯ ¯ +U ψx,j,↑ ψx,j,↓ ψx,j,↓ ψx,j,↑ .

Path integral description of the superfluid properties at the BEC/BCS crossover

649

Here ψ¯x,j,σ is the Grassmann variable for a fermion with spin σ, in layer j, and at x = {x, τ }. The local chemical potential is μj = μ − Vj , where μ is the overall chemical potential and Vj is an external (conﬁnement) potential such as a magnetic trap, evaluated in layer j. Care should be taken in relating the interaction strength U of the 2D contact pseudopotential to the scattering length as . Renormalization leads to (36)

1 m −1 = ln [E/Eb ] + i , U 4 π

where Eb is the energy of the 2D bound state that always exists in two dimensions. In an optical potential the binding energy of the quasi-2D bound state is given by [14] (37)

√ 1 λ Eb = 0.583 sER exp √ . 2π s−1/4 as

The interlayer coupling results from tunneling of atoms from one layer to another (38)

Sjtunnel = t1

† † ψx,j,σ ψx,j+1,σ + ψx,j+1,σ ψx,j,σ . σ

The tunneling amplitude is given by [15] (39)

t1 = sER

√ 2 π2 − 1 e− sπ /4 . 4

The total action functional is then given by (40)

S=

Sjlayer + Sjtunnel .

j

Having obtained the action functional for this system, we can go through the steps listed in the previous section: i) perform a Hubbard-Stratonovic transformation to introduce the pairing ﬁeld Δx,j , ii) integrate over the Grassmann variables and iii) make a saddle-point approximation, Δx,j → |Δj |eiθj . In the previous section, we set the phase of the saddle-point to zero: it does not appear in the free energy (only changes in phase do). In the present case, we want to be able to investigate the ﬂow of atoms through the optical lattice. Although the absolute phase does not matter, the relative phase between two layers will matter. The resulting saddle point action can still be separated in terms relating to one layer only, and terms coupling neighboring layers (see ref. [16]). (41)

Ssp =

layer tunnel Ssp,j , + Ssp,j

j

650

J. Tempere

with ⎛ ⎞ μj t21 ⎝ ⎠ cos(θj+1 − θj ), 1+ dτ dx =− 4π 0 |Δj |2 + μ2j β

|Δj |2 + =− dτ dx ln −G−1 , sp,j g 0

(42)

layer Stunnel,j

(43)

layer Ssp,j

β

where the saddle-point Greens function for layer j is given by (44)

(−G−1 sp,j )

∂ 1 − i (∇θj ) · ∇ σ0 = −|Δj |σ1 + ∂τ 2m 2 1 i 2 1 2 i ∂θj 1 (∇θj ) ∇ + − μj + − ∇ θj σ3 . − − 2m 2 ∂τ 2m 4 2m 2

. 3 2. Hydrodynamic equations. – One of the strengths of the path integral formalism is that it allows to derive equations of motion for collective ﬁelds such as the particle density or the pairing ﬁeld. These equations of motion often take the form of hydrodynamic equations for the pair density and the pair velocity ﬁeld. The equations are derived by extremizing the saddle-point action with respect to the order parameter and the chemical potential. For a smoothly varying density (ρj+1 ≈ ρj ≈ ρj−1 ) we ﬁnd the following equations for density and phase [16]: (45)

t21 ρj 1 ∂t ρj = [sin (θj − θj−1 ) − sin (θj+1 − θj )] 2 2πρj /m + Eb

and (46)

Eb 1 t2 − ∂t θ j = − 1 [cos (θj+1 − θj ) + cos (θj − θj−1 )] + 2 2 (2πρj /m + Eb )2 +

Eb πρj − − μj . m 2

These equations can be used to study the ﬂow of a superﬂuid Fermi gas through an optical lattice. For Bose gases, the ﬂow through a deep optical lattice heralds the onset of superﬂuidity [17]: the thermal gas is trapped while the condensate ﬂows freely. Phase coherence over the diﬀerent lattice sites in the superﬂuid state has also been demonstrated for a three-dimensional optical lattice, and this phenomenon was used to detect the Mottsuperﬂuid transition [18]. Can optical lattices be used to distinguish the normal and superﬂuid state also for Fermi gases? A Fermi gas in the normal state [19] will perform damped Bloch oscillations when placed in a deep optical lattice, but this behavior can be distinguished from a superﬂuid ﬂow through the lattice in that the center of mass of the Fermi cloud can be trapped away from the deepest lattice site. In the superﬂuid state, the ﬂow through

Path integral description of the superfluid properties at the BEC/BCS crossover

651

Fig. 3. – Critical velocity of a superﬂuid 40 K gas in an optical lattice generated by a λ = 795 nm laser beam with intensity s = 0.5. From ref. [20].

the lattice is only limited by the critical velocity. The critical velocity can be found straightforwardly from the equations of motion by ﬁxing the interlayer phase diﬀerence Δθ = θj+1 −θj and keeping the density constant over the layers. The equations of motion reduce to the familiar Josephson equations for a coupled Josephson junction array [20] (47)

J=

t21 ρj 2 sin(Δθ) 2πρj /m + Eb

and (48)

h ∂Δθ = ΔV, λ ∂t

where ΔV is the energy diﬀerence between two consecutive wells of the optical lattice. The critical current occurs at Δθ = π/2 and this leads to a critical velocity (49)

vc =

t21 ρj λ . 2πρj /m + Eb

The critical velocity is shown in ﬁg. 3 as a function of the interaction parameter 1/(kF as ) and the Fermi wave vector kF in the two-dimensional layer. This shows that (as long as the ﬂow velocity does not exceed a few micron/ms) the superﬂuid fermions will be able to tunnel coherently from lattice site to lattice site and ﬂow through the lattice. Near resonance, an interplay between the molecular binding energy and the tunneling energy leads to a more complicated situation as molecules or pairs can delocalize over several layers [21].

652

J. Tempere

4. – Application to multi-species Fermi mixtures Fermionic superﬂuids, up till now, arise from two-component fermionic systems. It is nevertheless possible to simultaneously trap more than two hyperﬁne states, and create a multicomponent fermionic system. Recently, Jin and co-worked used a three-component fermion system (a mixture 40 K in the mF = 5/2, 7/2 and 9/2 states) for impurity spinstate thermometry [22]. Adding a third (impurity) component will not only allow for thermometry, but it will also inﬂuence the Cooper pairing between the two original components. How will the pairing properties be aﬀected as the interaction strengths between the diﬀerent components are changed (albeit not independently)? Is it possible to create a “resonating bond” pairing, a superposition between pair formation between diﬀerent combinations of partners? These questions can be straightforwardly addressed by the path integral formalism, following the steps highlighted in the Review section. First, we identify the action functional describing an M -component system S=

(50)

β

dτ

dx

0

+

M M

M ∂ 1 2 − ∇x − μσ ψx,τ,σ + ψ¯x,τ,σ ∂τ 2m σ=1 ! ¯ ¯ gσσ ψx,τ,σ ψx,τ,σ ψx,τ,σ ψx,τ,σ .

σ=1 σ =σ=1

Here, gσσ represents the strength of the contact pseudopotential for the interaction between fermion species σ and σ . The number of atoms in each hyperﬁne spin state σ is ﬁxed by the chemical potential μσ . Secondly, the Hubbard-Stratonovic ﬁelds Δx,τ,σσ are introduced S=

β

dτ

dx

0

M ∂ 1 2 ¯ − ∇ − μσ ψx,τ,σ ψx,τ,σ ∂τ 2m x σ=1

! M M ¯ x,τ,σσ Δx,τ,σσ Δ ¯ x,τ,σσ ψx,τ,σ ψx,τ,σ + Δx,τ ·σσ ψ¯x,τ,σ ψ¯x,τ,σ + Δ − . gσσ σ=1 σ =σ=1

Next, the integration over the Grassmann variables is taken, and a saddle-point approximation is made for the pair ﬁeld, resulting in the eﬀective action (51)

Ssp =

β

dτ 0

⎧ ⎨ −G−1 0 dx − Tr ln D∗ ⎩

D G−1 0,τ →−τ

−

M M σ=1 σ =σ=1

⎫

2⎬

|Δσσ | . gσσ ⎭

is an M × M diagonal matrix containing the unperturbed single-particle Here −G−1 0 contributions. For M = 3 (i.e. three simultaneously trapped hyperﬁne states σ = 1, 2, 3)

Path integral description of the superfluid properties at the BEC/BCS crossover

653

we have ⎛

(52)

−G−1 0

1 = ⎝0 0

0 1 0

⎞ ⎛

μ1 0 ∂ 1 2 ⎠ ⎝ × − ∇ − 0 0 ∂τ 2m x 0 1

0 μ2 0

⎞ 0 0 ⎠. μ3

The Hubbard-Stratonovic ﬁelds are arranged in an antisymmetric M × M matrix D. For M = 3 we have ⎛ ⎞ 0 Δ12 Δ13 (53) D = ⎝−Δ12 0 Δ23 ⎠ . −Δ13 −Δ23 0 For the case M = 3, we have three components to the order parameter, {Δ12 , Δ13 , Δ23 }. To determine these, the saddle-point action Ssp is extremized with respect to Δσσ , yielding M (M − 1)/2 gap equations. In the case M = 3, and for equal populations of all three spin components μ1 = μ2 = μ3 , one obtains (54)

Δσσ = gσσ

k

#

k2 2m

Δσσ .

2 − μ + |Δ12 |2 + |Δ13 |2 + |Δ23 |2

These equations are solvable by setting all Δσσ = 0: this is the normal state. Cooper pairing between two partners σ and σ will occur when there is a solution with Δσσ = 0. Can the set of gap equations be solved simultaneously when more than one component of the order parameter is diﬀerent from zero? Suppose both Δ12 = 0 and Δ13 = 0. Then, both the equation with g12 and the equation with g13 need to be satisﬁed simultaneously, which is impossible unless g12 = g13 . So, unless the interaction strengths are exactly the same, the mixture of three fermions will be either in the normal state, or in a state where two types of atoms pair up, leaving the third in the normal state. The excitation spectrum consists of a (doubly degenerate) branch of Bogoliubov excitations and a branch of freeparticle excitations. During this “Enrico Fermi” Summer School, the author learned that the results presented here for the three-species case have been extended to the case of population imbalance between the three diﬀerent spin-states by Paananen et al. [23] —in that case order parameters with several nonzero components are possible. 5. – Conclusion In this contribution, the path integral description of the BEC/BCS crossover was reviewed and applied to ultracold Fermi gases. The strength of the present experiments with dilute Fermi gases lies in the level of control over the conﬁnement geometry and over the interaction strength. The recent experiments [17-19] with atomic gases in optical lattices illustrate the importance of the conﬁnement geometry. The tunability of the interaction strength allows to investigate the crossover regime, and may also allow to

654

J. Tempere

construct exotic order parameters for multicomponent gases. The path integral formalism provides a convenient framework to extend the crossover theory to optical lattices and to multicomponent gases. The short review presented here is by no means an exhaustive list of the applications of path integrals to cold atomic gases. Notable other applications to Fermi gases in the crossover regime include the description of vorticity [24], the study of p-wave superﬂuidity [25], and the derivation of hydrodynamic equations [26]. ∗ ∗ ∗ The author acknowledges support of the Special Research Fund of the University of Antwerp, BOF NOI UA 2004, and of the FWO-Vlaanderen in the form of a mandaat “Postdoctoraal Onderzoeker van het FWO-Vlaanderen”. This research has been supported ﬁnancially by the FWO-V projects Nos. G.0356.06, G0115.06, G.0435.03, the W.O.G. project WO.025.99N, the GOA BOF UA 2000 UA and the IUAP.

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¨nsch T. W. and Bloch I., Nature, 415 [18] Greiner M., Mandel O., Esslinger T., H a (2002) 39. [19] Modugno G., Ferlaino F., Heidemann R., Roati G. and Inguscio M., Phys. Rev. A, 68 (2003) 011601. [20] Tempere J., Wouters M. and Devreese J. T., Phys. Rev. A, 72 (2005) 063601. [21] Wouters M. and Orso G., Phys. Rev. A, 73 (2006) 012707. [22] Regal C. A., Experimental Realization of BCS-BEC Crossover Physics with a Fermi Gas of Atoms, Dissertation, December 2005 (electronic version available at: http://jilawww.colorado.edu/~jin/papers.html). ¨ rma ¨ P., Phys. Rev. A, 73 (2006) 053606. [23] Paananen T., Martikainen J.-P. and To [24] Botelho S. S. and Sa de Melo C. A. R., Phys. Rev. Lett., 96 (2006) 040404; Tempere J., Wouters M. and Devreese J. T., Phys. Rev. A, 71 (2006) 033631. [25] Iskin M. and Sa de Melo C. A. R., Phys. Rev. Lett., 96 (2006) 040402. [26] De Palo S., Castellani C. Di Castro C. and Chakraverty B. K., Phys. Rev. B, 60 (1999) 564.

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Fermi-Bose mixture with tunable interactions G. Modugno LENS and Dipartimento di Fisica, Universit` a di Firenze Via Nello Carrara 1, 50019 Sesto Fiorentino (FI), Italy

1. – Introduction The possibility of controlling the atom-atom interaction in ultracold quantum gases has recently opened various research directions, ranging from molecular physics in the nanoKelvin regime [1] to strongly interacting superﬂuid Fermi gases [2] and to strongly correlated atomic systems [3]. The interaction is usually controlled through Feshbach resonances [4], which consist in a resonant coupling of atomic and molecular levels that can be tuned through an external magnetic ﬁeld. This kind of tool has been extensively applied so far only to homonuclear Bose and Fermi systems. Tuning of the interactions in heteronuclear mixtures via Feshbach resonances can give access to an even broader range of phenomena. Interaction eﬀects such as collapse or phase separation [5] would be accurately studied. A tunable interaction between the fermionic and bosonic components would also give access to a wealth of condensed-matter phenomena in strongly correlated systems that have been recently proposed [6]. Moreover, Feshbach resonances can be used to associate ultracold heteronuclear molecules. These would possess a permanent electric-dipole moment, allowing to manipulate the samples via electric ﬁelds, and to study dipole-dipole interactions in the context of ultracold quantum gases. The presence of a long-range anisotropic interaction between the particles in Bose and Fermi quantum gases is expected to profoundly modify the properties of such systems, and to give access to new phenomenology [7, 8]. The study of Feshbach resonances in heteronuclear systems has been delayed by a few years with respect to the homonuclear case, mainly because of the larger complexity of the experiments and because of the lack of accurate theoretical models for Feshbach c Societ` a Italiana di Fisica

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resonances in most of the alkali pairs. Currently, Feshbach resonances have been experimentally detected in three alkali fermion-boson mixtures: 6 Li-23 Na [9], 6 Li-7 Li [10], 40 K-87 Rb [11,12]. However, a ﬁne tuning of the interaction has so far been achieved only in the 40 K-87 Rb system [13, 14]. In this work we ﬁrst describe the various steps that have led to the observation of Feshbach resonances in the K-Rb system and their accurate characterization. We then describe recent experiments in which Feshbach resonances are exploited to study interaction eﬀects and to perform ﬁrst experiments on association of weakly bound KRb dimers. 2. – Feshbach resonances in the K-Rb mixture Among ultracold Fermi-Bose atomic mixtures, the 40 K-87 Rb system has so far been the most investigated one. Since its ﬁrst production [15], it has been employed for a series of experiments on interaction phenomena and on Fermi and Bose gases in optical lattices. Nowadays it is by far the most employed ultracold atomic mixture. One of the peculiarities of this system that made it particularly interesting from the very beginning is the naturally attractive interaction between fermions and bosons. This is interesting to study mean-ﬁeld eﬀects such as collapse and bright solitons or boson-induced superﬂuidity. The possibility of tuning of the interspecies interaction via Feshbach resonances will further enlarge the spectrum of phenomena that can be investigated with this mixture. Let us remind here a few general aspects of interactions in ultracold atomic gases and Feshbach resonances. The two-body interaction between atoms is described by a contact potential and the collisional events can be expanded in partial waves. At the low temperature of actual experiments, only the lowest-order, energy-independent partial wave, i.e. the s-wave, need to be taken into account. Just one parameter, the s-wave scattering length a, is needed to deﬁne quantities such as the collisional cross-section and the coupling constant for interacting quantum gases in the mean-ﬁeld approach. In a mixture of spin-polarized fermions and bosons, two diﬀerent interactions have to be considered: the fermion-boson interaction, described by aFB , and the boson-boson one, described by aBB . Fermion-fermion interaction must not be taken into account, since Pauli principle strictly forbids s-wave interaction between identical fermions. Throughout this work we consider the case in which only the interspecies scattering length aFB can be tuned, while aBB is ﬁxed to its natural value (this is about 100a0 for 87 Rb). In general, the possibility of tuning at the same time both aFB and aBB in a fermion-boson system is marginal. The scattering length aFB depends on the phase shift that the wave function of a pair of free atoms experiences when the pair separation approaches those typical of the molecular potential, i.e. several tens of a0 . As a matter of fact, the natural value of aFB is determined just by the binding energy of the last vibrational state of the molecular potential. For example, in the case of the 40 K-87 Rb pair in its ground state aFB is negative and relatively large (aFB = −185a0 ) because the closest molecular level to the atomic threshold is actually just above the threshold. By applying an external magnetic ﬁeld to the system, it is possible to tune in a diﬀerent way the energy of the atomic threshold and that of the molecular states, and eventually obtain a situation in which

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a molecular level crosses the atomic threshold. At this crossing the scattering length of the system shows a resonant enhancement, the so-called Feshbach resonance, and passes from −∞ to +∞ as the molecular state passes from above to below the atomic threshold. At one of such resonances, the magnetic-ﬁeld dependence of aFB can be parametrized as (1)

aFB (B) = abg (1 − Δ/(B − B0 )),

where abg is the background value of the scattering length, B0 is the magnetic-ﬁeld location of the resonance center, and Δ is a width parameter that accounts for the coupling strength of the speciﬁc atomic and molecular levels. The 40 K-87 Rb has a large number of internal states (the total angular momentum is F = 9/2 for K and F = 1 for Rb in their ground hyperﬁne state) and therefore in principle a large number of Feshbach resonances is available. In a series of experiments on cold collision measurements [16-18] we have determined to a good accuracy the scattering length aFB of the system, which has been used to ﬁne tune the best approximation of the K-Rb molecular potential, and in particular to determine the binding energy of the states close to dissociation. Actually, two diﬀerent molecular potentials appear for alkali atoms, i.e. the singlet X 1 Σ+ and triplet a3 Σ+ potentials with scattering lengths as and at , respectively. This study has then allowed to predict the magnetic-ﬁeld position of Feshbach resonances [18]. In particular, for most of the states we expected several resonances in the region between 400 and 700 G. This prediction was conﬁrmed by a ﬁrst study of heteronuclear resonances in the absolute ground states performed at JILA [11]. In a successive detailed investigation of Feshbach resonances in various internal states [12] we got the complete picture of such resonances that is necessary for their exploitation in experiment. In the next section we describe the experimental and theoretical methods used. 3. – Feshbach spectroscopy In our experiment the K-Rb Fermi-Bose mixture is routinely produced via laser and evaporative cooling in a magnetic trap [19, 15]. In order to study magnetic Feshbach resonances, one has to decouple the trapping mechanism from the magnetic moment of the atoms. We achieved this by transferring the atoms from the magnetic trap to an optical dipole trap created by focused laser beams crossing in the horizontal plane. In a ﬁrst series of experiments [12] we have used a titanium:sapphire laser at a wavelength of 830 nm, which is suﬃciently detuned from the atomic resonance to provide a lifetime of the atomic sample exceeding 1 s. The trap depth was about 10 μK, which is suﬃciently large to hold “hot” samples at temperatures above quantum degeneracy. Typically 105 K fermions and 5 × 105 Rb bosons were cooled in a magnetic trap down to temperatures of a few hundreds nK and then adiabatically transferred to the optical trap. The typical density and temperature of the bosonic sample in the optical trap throughout this experiment were 5 × 1012 cm−3 and 1 μK, respectively. The fermionic sample was in thermal equilibrium with the bosonic one.

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Rb The mixture was initially prepared in the state |F K = 9/2, mK = F = 9/2 ⊗ |F Rb 2, mF = 2, which is the only stable combination of low-ﬁeld seeking states. This state does not possess Feshbach resonances, since it has the maximum projection of the angular momentum. In the choice of a K ⊗ Rb state to explore for Feshbach resonances, one has to take into account its stability against spin-exchange collisions. These are inelastic collisions which can take place every time a second state with the same projection of the Rb total angular momentum mK F + mF with a lower energy is available. The excess kinetic energy is tipically enough to have an immediate loss of the collision partners. Due to the particular level structure of this K-Rb mixture, all the states where both species are in their ground hyperﬁne state and either Rb or K are in their absolute ground state are stable against such collisions and have a lifetime that is limited mainly by three-body recombination. According to our theoretical study [18], most of these states were also expected to feature Feshbach resonances. To prepare the atoms in these states we have used a series of radio-frequency (RF) and microwave (μw) adiabatic rapid passages. A ﬁrst combination of sweeps was used to transfer the system to its ground state |9/2, −9/2 ⊗ |1, 1 in the presence of a bias ﬁeld of about 10 G. The ﬁeld was then raised to 100 G to perform the additional transfer of K or Rb to excited states such as the |9/2, −7/2, |9/2, 7/2 or |1, 0. The homogeneous ﬁeld was then changed in few ms to any value in the range 0–1000 G and actively stabilized there with an accuracy of about 200 mG. The ﬁeld was calibrated by means of RF spectroscopy of the |2, 2-|2, 1 transition of Rb. As we will describe in detail in the next section, a Feshbach resonance is usually accompanied by an enhancement of the inelastic decay rate. The easiest way to look for Feshbach resonances is therefore to record the fraction of atoms lost from the trap when changing the magnetic ﬁeld. In our mixture the best sensitivity to resonances was achieved by recording the atom number of the minority component, i.e. potassium, after about 1 s of permanence at a ﬁxed magnetic ﬁeld. The typical experimental signature of interspecies resonance is shown in ﬁg. 1 for the absolute ground state of the mixture: due to the lower abundance of K in our sample, we usually observe a complete loss of K atoms at resonance. To avoid any possible confusion with homonuclear resonances, we also check for the absence of losses after removing either K or Rb from the mixture before applying the magnetic ﬁeld. Actually, in our investigation we have observed several Rb resonances that have been previously studied in dedicated experiments [20]. In a ﬁrst survey we have detected 14 Feshbach resonances in three diﬀerent states, and used them to reﬁne our theoretical quantum collisional model. To do this, the singlet X 1 Σ+ and triplet a3 Σ+ interaction potentials were parameterized in terms of the as and at scattering lengths, respectively. The two-body elastic cross-section was then computed for diﬀerent as,t and its maxima, corresponding to Feshbach resonances, were compared to four sample experimental features. Once a good agreement was found, a least-square ﬁt to all 14 experimental features was then performed, letting also the van der Waals coeﬃcient C6 free to vary. The best-ﬁt parameters with one standard deviation we obtained are as = (−111 ± 5)a0 , at = (−215 ± 10)a0 and C6 = (4292 ± 19) a.u., which agrees to better than one standard deviation with the high-precision ab initio calculation

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Fermi-Bose mixture with tunable interactions

Fig. 1. – Relative inelastic losses of potassium atoms in a 40 K-87 Rb mixture in its absolute ground state at interspecies Feshbach resonances. The two features near 456 G and 515 G are p-wave resonances, the others are s-wave resonances.

obtained in [21]. Error bars also include a typical ±10% uncertainty in C8 . The position and width of the resonances are listed in table I. The average theory-experiment deviation for the resonance positions is about 0.3 G only.

Table I. – Magnetic-ﬁeld positions and widths of the observed 40 K-87 Rb resonances compared to the corresponding theoretical predictions of our best-ﬁt model. Δth is deﬁned in the text, and is the orbital angular momentum of the molecular state associated to each resonance. |mf a ⊗ |mf b

Bexp (G)

Bth (G)

−Δth (G) −3

| − 9/2 ⊗ |1

456.0 – 495.6 515.7 546.7 547.4 658.9 663.7

456.5 462.2 495.7 515.4 546.8 547.6 659.2 663.9

2 · 10 0.067 0.16 0.25 2.9 0.08 1.0 0.018

1 0 0 1 0 2 0 2

| − 7/2 ⊗ |1

469.2 – 584.0 591.0 598.3 697.3 705.0

469.2 521.6 584.1 591.0 598.2 697.3 704.5

0.27 0.051 0.67 2 · 10−3 2.5 0.16 0.78

0 0 0 2 0 0 0

|7/2 ⊗ |1

299.1 852.4

298.6 852.1

0.59 0.065

0 0

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A determination of the K-Rb scattering lengths based on extensive Feshbach spectroscopy is by far the most precise one can obtain. A large number of detected resonances indeed unequivocally maps the last few molecular states close to dissociation, and therefore determines the scattering lengths. Our determination of as,t helped in solving some existing inconsistencies between previous experiments on cold collisions [17,22], collapse [23, 24] and Feshbach resonances [11] in this mixture. The optimized 40 K-87 Rb model can also be used to determine singlet and triplet scattering lengths for any K-Rb isotopic pair [12]. Within the Born-Oppenheimer approximation this can be simply achieved by using the appropriate reduced mass in the Hamiltonian. Such mass-scaling procedure depends in a sensitive way on the actual number of bound states supported by the potentials. They are nominally Nbs = 98 and Nbt = 32 for the singlet and triplet ab initio potentials we use, with an expected uncertainty of ±2 [25]. Other quantities of general interest for experiments with K-Rb mixtures are the eﬀective elastic scattering length a and the location of Feshbach resonances for the absolute ground state. We have determined also these quantities with relatively high accuracy using the collisional model above; they can be found in ref. [12]. These data complete our previous investigation [16], and are of great interest for forthcoming experiments on Bose-Bose mixtures with K and Rb atoms [26, 27]. Let us now discuss the properties of the Feshbach resonances we detect in the experiment, listed in table I. The nature of the molecular states associated to the resonances can be better understood through multichannel bound-state calculations. Since as and at are comparable, the spacing between singlet and triplet vibrational levels is small compared to the hyperﬁne interaction. Strong singlet/triplet mixing then occurs at least for the two vibrational states closest to dissociation, resulting in molecular levels labeled as (F K F Rb F ) in zero magnetic ﬁeld, where is the rotational quantum number and F = F K + F Rb . The features below ≈ 600 G arise from these strongly mixed levels. At such magnetic ﬁelds, however, the Zeeman magnetic energy is comparable to the smaller hyperﬁne splitting in the system, that of 40 K. Therefore F K is not a good quantum number to label the resonances, whereas F Rb is approximately good and equal to 2. Resonances at higher magnetic ﬁeld correlate with more deeply bound states and tend to assume singlet or triplet character. In all cases is an almost exact quantum number and is also shown in table I. The measurement of inelastic decay features described above allows to access only the location of Feshbach resonance, but not the exact magnetic-ﬁeld dependence of the fermion-boson scattering length. This can be calculated by our collisional model: isolated resonances are well described by the simple parametrization introduced above: aFB (B) = abg (1 − Δth /(B − B0 )). The background scattering length for a particular state is determined by the model with an accuracy comparable to that obtained for at,s . For example, the determined value for the ground state is abg = (−185±7)a0 . The width Δth can also be calculated for all resonances, and is shown in table I, together with the angular momentum of the corresponding molecular states. The = 0 molecules tend to give rise to broad resonances due to strong spin-exchange coupling to incoming s-wave atoms. We also observe a few narrow resonances due to coupling of a = 2 molecule to

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incoming s-wave atoms through weaker anisotropic spin-spin interactions [28, 29]. The two resonances associated with = 1 molecules couple by spin-exchange to incoming p-wave atoms. These resonances have an energy-dependent width [30] which we compute at collision energy E/kB ≈ 1 μK, with kB the Boltzmann constant. Table I also shows two narrow not yet observed resonances. We ﬁnd that several stable states of the mixture present at least one broad resonance, analogous to that in the ground state near 546 G. Any of these resonances can be very well suited for control of the interaction and molecule formation. The resonance width can also be measured in the experiment by locating the so-called zero-crossing, i.e. the magnetic-ﬁeld value at which aFB = 0. For example, we have done this for one of the broadest resonances of our system by studying the eﬃciency of sympathetic cooling of fermions close to the resonance. A mixture composed of about 106 bosons and 105 fermions is loaded in a crossed dipole trap at a temperature around 1 μK. We performed this experiment using a Yb:YAG laser emitting at 1030 nm. We chose relatively large beam waist radii of about 100 μm, in order to have a relevant vertical sag of the rubidium cloud into the non-parabolic part of the optical trap. This resulted in a more eﬃcient evaporation of Rb than of K when lowering the overall trap depth via acusto-optic modulators. Figure 2 shows the temperature of the fermionic component after a 2.4 s evaporation in the vicinity of the broad Feshbach resonance in the ground state located at 546.7 G. Thermalization to lower temperatures of spin-polarized fermionic atoms can take place only via eﬃcient fermion-boson elastic collisions. Since the collisional cross-section at these low temperature is well described by σ = 4πa2FB , it vanishes when aFB crosses zero. The zero-crossing location extracted from a Gaussian ﬁt to the temperature of the fermionic component is 543.4(1) G. This gives a width Δ = 3.3(2) G which is in good agreement with the theoretical expectation of Δ = 3 G. We have checked that such agreement between model and experiment persists on a few other broad resonances of the mixture. 4. – Three-body losses at a Feshbach resonance The resonance in the ground state of the mixture around 546 G is in principle the most appropriate to achieve a ﬁne tuning of the fermion-boson interaction. This is indeed one of the broadest resonances, and moreover it can be expected to have the slowest inelastic decay achievable for this mixture. Two-body inelastic processes are indeed forbidden for the ground state, and only three-body recombination is responsible for decay in proximity of Feshbach resonances. In a ﬁrst series of experiments we have explored the main features of three-body recombination in our system in the thermal regime, in the absence of possible interaction eﬀects. At a boson-fermion Feshbach resonance, three-body processes involving two bosons and one fermion are the dominant decay channel, while those involving two fermions and one boson are suppressed by the Pauli principle [31]. During a three-body event, one boson and one fermion are associated into a deeply bound dimer whose binding energy is shared as kinetic energy by the dimer and the second boson. Conservation

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Fig. 2. – a) Theoretical expectation for the fermion-boson scattering length aFB at the broadest Feshbach resonance in the ground state of the K-Rb mixture. b) Eﬃciency of sympathetic cooling of fermions around the zero-crossing location. The feature at 547.4 G is a narrow spin resonance.

Fig. 3. – Three-body recombination in a thermal fermion-boson mixture after 100 ms of hold time at a Feshbach resonance. a) Remaining atom number for fermions (triangles) and bosons (circles). b) Final temperature for bosons measured in the experiment (triangles) and calculated (circles).

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Fermi-Bose mixture with tunable interactions

of energy and momentum requires this energy to be shared by the KRb dimer and the Rb atom in a ratio of approximately 2:3. Typically this energy is at least of the order of 100 MHz×¯ h, hence much larger than the trap depth, and both atom and dimer are immediately lost. In the regime of very large scattering lengths the loss rate is expected to depend on bosons and fermions density distributions nB and nF and on the interspecies scattering length as (2)

Γ3 = K3

nB (x)2 nF (x)dx,

where K3 ∝ a4FB [31]. In general, these losses result also in a heating of the remaining sample, since the probability of ﬁnding two bosons and one fermion in the same location is larger at the center of the distributions, where the coldest atoms reside [32]. The average energy per lost particle can be computed as (3)

El =

U (x)nB (x)2 nF (x)dx , nB (x)2 nF (x)dx

where U is the potential energy of the trapped atoms. If the boson and fermion distributions are equal, one ﬁnds the same heating rate found for homonuclear bosonic systems: the energy per lost particle is 0.5kB T . This implies 3kB T excess energy for each loss event, to be redistributed in the remaining sample. In the experiment we have investigated the heating related to three-body losses. For example, ﬁg. 3 shows the evolution of atom numbers and temperature for a nondegenerate K-Rb sample that was held for 100 ms at various magnetic ﬁelds around 546 G. We clearly see a heating of the sample as the number of atoms drops close to the resonance. Here we compare the temperature measured in the experiment to the one calculated with an intuitive model. Such model evaluates the time evolution of the mixture as a series of single three-body recombination events, each one followed by rethermalization at a higher temperature due to the 3kB T excess energy. Experiment and theory are in qualitative agreement, although the latter slightly overestimates the heating. 5. – Tuning of the interaction in the quantum degenerate regime When the sample is cooled into the quantum degenerate regime in proximity of the Feshbach resonance, the properties of the system are largely modiﬁed by the resonant interaction between the two components. The Fermi-Bose interaction energy (4)

4π¯h2 UFB (x, a) = aFB μ

nB (x, aFB ) nF (x, aFB )dx

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can get comparable to or even much larger than the trap potential, and therefore determines the distributions of the two components in the trap [5]. In the case of aFB < 0 the large attractive interaction can beat the natural repulsion within the Bose and Fermi gases and lead to a collapse of the system. For large aFB > 0 the two components can instead undergo phase separation. Collapse of a Fermi-Bose mixture has already been observed in this system as a sudden loss of a relevant fraction of the atoms when the number of atoms in the condensate was increased along the evaporation path [24]. The background K-Rb scattering length is indeed negative and suﬃciently large to reach the unstable region for large atom numbers in tightly conﬁning traps. No evidence of phase separation has instead been reported in mixtures with positive scattering length. The possibility of a rapid, ﬁne tuning of the interaction at a Feshbach resonance allows to access and characterize both regimes of phase separation and collapse. In the experiment we have investigated these regimes through a study of three-body losses in the quantum degenerate regime. We indeed found that the loss behavior can give strong indications on the overlap of the two components in the trap. To perform this experiment, we produced a fully degenerate mixture by evaporative cooling in the crossed dipole trap at 1030 nm. The trap depth was lowered exponentially from 5 μK to 0.5 μK in 2.4 s, with a time constant of 1 s. The trap was then recompressed to the full depth in 150 ms. The trap frequencies for Rb were (120, 92, 126) Hz and a factor about (87/40) larger for K. The Fermi gas typically contains 5 · 104 atoms at a temperature T < 0.3TF , where TF = 500 nK. The Bose-Einstein condensate contains instead about 105 atoms and no thermal component is discernible, i.e. the temperature is below 30 nK. The chemical potential of the Bose gas is μ/kB = 250 nK. Due to the diﬀerent energy scale, the linear sizes of the Fermi gas are almost twice those of the Bose gas. The latter is totally contained in the volume of the Fermi gas, and is shifted down by about 7 μm by the larger gravitational force. In a ﬁrst experiment we have compared the time evolution of the mixture for two values of the magnetic ﬁelds corresponding to opposite values of the fermion-boson scattering length: aFB ≈ ±800a0 . They were obtained by preparing the mixture far from the resonance and then rapidly shifting the ﬁeld to the ﬁnal value. For the positive aFB the mixture was prepared at 539 G and then brought to 546.0 G, where the expectation is aFB = +740+80 −70 . It was instead prepared at 551 G for negative a and then brought to 547.6 G, where the expectation is aFB = −820+40 −40 . In such way crossing of the resonance center during the rapid sweep was avoided. In ﬁg. 4 we show the evolution of the condensed fraction of the Bose gas, which is the most sensitive component of the mixture to excitations and perturbations. On the negative aFB side we see a very rapid depletion of the condensate on a timescale shorter than the trap period, while for positive aFB the condensate remains stable for a much longer time interval. At a longer time in both cases the Bose gas is heated up into a pure thermal cloud. This, however, happens already at about 20 ms for negative aFB , and only at about 100 ms for positive aFB . During the whole time span, the total atom number in the bosonic sample decreases by about 50% in the case of negative aFB and about 30% for positive aFB . For the Fermi gas (not shown in the picture) we similarly observe both atom loss and heating, which are larger for negative aFB .

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Fig. 4. – Time evolution of the boson number (triangles) and of the condensed fraction (circles) +80 for large positive and negative scattering lengths: a) aFB = −820+40 −40 ; b) aFB = +740−70 . In the ﬁrst case the inelastic decay and heating is sped up by interaction-induced collapse; in the latter case it is slowed down by phase separation.

To understand the dramatically diﬀerent behaviors observed for positive and negative aFB ’s, we rely on a zero-temperature mean-ﬁeld model of our system [33]. We use it to study the density distribution of the mixture in the two regimes, and to predict the evolution of three-body losses. This model calculates the local interaction energy and uses it as an additional eﬀective potential for both species to evaluate the distribution of the mixture in the trap. This procedure is done recursively until the true ground state of the system in the presence of interaction is found. For example, in ﬁg. 5 we plot a cut along the z-direction of the distribution of the two components when the number of fermions and bosons is the same and equal to 5×104 , for three diﬀerent values of aFB : 0, −400a0 and +250a0 . In the case of large negative aFB the density overlap of the two components is clearly increased with respect to the noninteracting case, while for large positive aFB the two components tend to phase separate and it is strongly reduced. Note that the model would not converge for aFB = −800a0 , since the system is collapsed, i.e. it does not possess a stable ground state.

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Fig. 5. – Calculated relative energy gain (E B − El )/E B for each particle lost in a three-body collision event for a K-Rb Fermi-Bose mixture with variable interspecies interaction. For scattering lengths below 500a0 the system energy increases as a consequence of losses, while above that value it decreases. This is a consequence of an increased overlap in the ﬁrst case, and of phase separation in the second one, as shown by the calculated distributions along the vertical direction of bosons and fermions at T = 0.

A reduced or increased overlap will obviously aﬀect the three-body loss rate Γ3 , which increases in the regime of collapse and is reduced in the phase-separation regime. This will aﬀect the ratio between the mean energy of particles in the overlap region and the mean energy of the whole system, which determine the heating ratio. We have used the model to calculate numerically the evolution of the overlap integral, the mean energy per lost particle and the mean energy per particle in the system, at T = 0. For example, in ﬁg. 5 we show the excess energy per lost particle, normalized to the mean energy in the Bose condensate (5)

E B − El , EB

where

EB =

U (x)nB (x)dx . nB (x)dx

For aFB = 0 the model predicts a relative energy gain of about 0.5, close tho the classical value discussed above of 2/3. The increase of heating on the aFB < 0 side and the corresponding reduction on the aFB > 0 side are apparent. Actually, this model indicates that three-body losses should eventually cool down the system as aFB gets larger than 500a0 , where the relative energy gain becomes negative.

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Let us now interpret the behavior shown in ﬁg. 4 on the basis of the model’s predictions. For aFB ≈ −800a0 the system presumably starts a compression phase just after the interaction energy is switched to a large and negative value. After a quarter of the trap period, i.e. 2.5 ms, we have a maximum of three-body loss rate and a large heating of the sample. The condensate is therefore rapidly heated into a thermal cloud, and the loss rate decreases because of the decreased density of the samples. In the opposite case, aFB ≈ +800a0 , the system is in the phase separation regime. For the ﬁrst 100 ms the condensate is not heated up despite a 15% loss of atoms, indicating that high-energy atoms are preferentially removed by the loss events. Note that in order to have a more realistic, quantitative description of the system’s behavior, one would need a more complex model which can also track the dynamics of the system at ﬁnite T . For example, according to the simple model described above, one would expect to see a cooling of the system as the loss process goes on. We think this is not observed because of the presence of the weakly bound molecular state on this side of the resonance that is responsible for the existence of the resonance itself. This can be expected to result in the production of just moderately energetic atom-molecule pairs, i.e. with a binding energy comparable to the trap depth. These would have time to scatter with the remaining atoms before leaving the trap, given the large collisional rate expected in this regime (of the order of 1000 s−1 ). The dramatically diﬀerent behavior of losses on the two sides of the resonance appears also in the measurements reported in ﬁg. 6. Here we report the evolution of the Bose-Einstein condensate in the mixture following a sweep over the Feshbach resonance. The ﬁeld was increased in 50 ms from Bi = 543.4 G to a ﬁnal ﬁeld B that was varied from 543.4 G to 548 G, and held there for 10 ms. Inelastic losses start to deplete the system as the sweep approaches the resonance center, but the condensate survives as long as aFB > 0. It is instead very rapidly destroyed by collapse when the sweep crosses the resonance center into the aFB < 0 region. The evolution of the width of the gas after ballistic expansion with B can be used to ﬁnd accurately the resonance center, as shown in ﬁg. 6. A phenomenological ﬁt with a Boltzmann growth function gives B0 = 546.65(20) G, which is in good agreement with the measured position of the loss maximum. The possibility of a ﬁne tuning of the interaction allows also to explore the threshold for the collapse instability in the region aFB < 0. According to our mean-ﬁeld model, the system is expected to remain stable as long as aFB > −400a0 , and to collapse for larger negative scattering lengths. Figure 7 shows the evolution of the width of the BoseEinstein condensate after a sweep from above the resonance into the instability region. The ﬁeld was decreased in 50 ms from Bi = 551 G to a lower ﬁnal ﬁeld B, and held there for 20 ms. The width of the condensate starts to increase around Bf = 549.4 G until around Bf = 548.6 G the condensate is totally heated into a thermal sample as a consequence of collapse. In the highlighted region we observe a smooth transition between stable and unstable conditions: the condensate is heated up by three-body recombination and shows shape excitations due to the rapid change of the interaction energy. The corresponding scattering length range is aFB = −600÷−350a0 in qualitative accordance with the theory prediction.

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Fig. 6. – Evolution of the Bose-Einstein condensate in the mixture following a slow sweep across the Feshbach resonance from aFB > 0 to aFB < 0. Despite a strong reduction in the atom number due to three-body losses, the system remains stable until the ﬁeld crosses the resonance center, where it is totally heated up into a thermal sample. A phenomenological ﬁt of the Gaussian width of the condensate with a Boltzmann growth function gives an accurate estimation of the resonance center.

Fig. 7. – Evolution of the width of the Bose-Einstein condensate in the mixture following a sweep in the region of aFB < 0 close to a Feshbach resonance. The highlighted area indicates the transition magnetic-ﬁeld region from a stable to an unstable (collapsed) system.

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Fermi-Bose mixture with tunable interactions

Note that in this simple experiment it is diﬃcult to make a more precise determination of the threshold scattering length. It is indeed diﬃcult to separate an increase of the width due to a complete collapse of the system from an increase due to ordinary threebody losses in the regime of large attractive interactions or simply a modiﬁed expansion of the Bose gas in the presence of the Fermi gas. Further investigation is needed in order to characterize the whole phenomenology and possibly test mean-ﬁeld models of this system. 6. – Formation of dimers An interspecies Feshbach resonance can also be exploited to associate pairs of atoms into KRb dimers, using the same technique that has proven successful in the case of homonuclear systems. The idea is simple: since the resonance takes place in coincidence with a crossing of an atomic and a molecular state, one can adiabatically convert pairs of atoms into molecules with a magnetic-ﬁeld sweep. The magnetic-ﬁeld dependence of the atomic and molecular state involved in the resonance we studied are plotted in ﬁg. 8. The sweep needs to originate in the region B > B0 , where Ea < Em , and end on the other side. A maximum ramp speed can be evaluated with a simple Landau-Zener model developed for homonuclear gases, which describes the number of molecules as (6)

Nmol = Nmax 1 − e−δLZ ,

˙ where δLZ = αnΔ/B,

and α = 4.5(4) × 104 m2 s−1 is an experimentally determined coupling constant [34]. The maximum conversion is reached when δLZ gets larger than one, which in our case corresponds to ramp speeds smaller than 50 G/ms.

Fig. 8. – Magnetic-ﬁeld dependence of the atomic (dashed line) and molecular (continuous line) states involved in the ground-state K-Rb Feshbach resonance. The atomic state is |9/2, −9/2 ⊗ |1, 1, while the molecular state is labeled as F Rb = 2, = 0, and has mixed singlet-triplet nature.

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G. Modugno

Fig. 9. – Evolution of the atom numbers in the mixture during a downward magnetic-ﬁeld sweep at the ground-state Feshbach resonance. The sudden decrease in atom number for both components when the ﬁeld crossed the resonance center at 546.6 indicates that atom pairs are associated into KRb dimers.

Figure 9 shows a series of absorption images of the mixture taken at various intermediate magnetic ﬁelds during a sweep over the Feshbach resonances. The sweep was originating 4 G above the resonance, and ended after about 5 ms at a variable magnetic ﬁeld across the resonance. The clouds were released from the optical trap right at the end of the sweep, and the images were taken at zero magnetic ﬁeld, after an appropriate ballistic expansion. Note how the number of atoms in both components drops as the ﬁeld is brought below 547 G. As in previous experiments, we interpret this reduction in atom numbers as the result of molecule formation. The transition energy of the molecules is indeed no longer resonant with the light used to image the atoms, and molecules are therefore not detected. It is important to note that the atoms are not lost because of three-body recombination while sweeping over the resonance center, where aFB → ±∞. Indeed, in that case one would detect also a strong heating of the system, which is not apparent in ﬁg. 9. Qualitative information on the process of molecule formation in this system can be obtained through simple measurements. The maximum conversion eﬃciency we were able to observe is about 30–40%, that is not obtained at the lowest temperatures achievable in the experiment, but at temperatures around the condensation temperature of bosons Tc . This can be qualitatively understood in terms of the simple model and of the experiments on homonuclear systems presented in [34]. One expects to reach the maximum conversion eﬃciency when the phase space overlap of the two components is maximum. This is reached for T = 0 in the homonuclear Fermi or Bose cases, but not for a Fermi-Bose system, where the spatial overlap of the two samples starts to decrease rapidly as soon as T gets smaller than Tc . A crucial information for future experiments on such molecules is obviously the stability of the molecular sample. This can be investigated by reconverting the molecules into atoms via a backward sweep across the resonance. We have been able to see molecules converted back into atoms only under special conditions: very short permanence on the molecular side of the resonance (shorter than 500 μs), and very low number of unpaired

Fermi-Bose mixture with tunable interactions

673

bosonic atoms. This apparently indicates that the main decay channel of the molecular sample is inelastic collisions with free bosonic atoms. Further experiments in which one is able to remove the free atoms of both species will be helpful to give a quantitative assessment of this preliminary indication. A possible solution to the short lifetime of the molecular sample is also the use of a tight 3D optical lattice. This environment allows to prepare isolated K-Rb atomic pairs into individual lattice sites, allowing for a loss-free production of KRb molecules when sweeping over the resonance. As recently shown in experiments [35], this allows to have a lifetime exceeding 100 ms, although this is obtained at the expenses of a lower conversion eﬃciency. 7. – Outlook In this contribution we have discussed experiment on a K-Rb Fermi-Bose mixture with tunable interaction at interspecies Feshbach resonances. The capability of controlling the boson-fermion scattering length to a high extent and the possibility of associating pairs of atoms into molecules open few diﬀerent research directions. The most important one is probably the study of ultracold dipolar molecules. Weakly bound KRb molecules produced at Feshbach resonances do not possess a relevant electric dipole moment, because their range is so large that only a negligible distortion of the electronic clouds is present. On the other hand, ground state (X1 Σ+ (ν = 0)) KRb molecules are predicted to have a dipole moment of the order of 1D [36]. These groundstate molecules can in principle be produced by transferring “Feshbach molecules” with optical Raman schemes [37] that are already being developed [38]. This is large enough to be used to investigate the fundamental properties of ultracold and quantum degenerate dipolar gases, and to test proposed schemes for quantum computing applications, in which electric ﬁelds are used to address and manipulate the particles [39]. The possibility of controlling at will the Fermi-Bose interaction will also allow to investigate novel quantum phases that have been proposed for strongly correlated atomic systems in optical lattices [6]. It can also allow to study in a more eﬀective way the phenomenology of disordered systems [40]. Fine tuning of aFB will also allow to test the predictions of present theories of Fermi-Bose systems for collapse [41] and phase separation, collective excitations [42], eﬀects beyond mean ﬁeld [43] and boson-induced superﬂuidity [44]. ∗ ∗ ∗ I thank M. Inguscio, a continuous source of ideas and encouragement, and G. Roati, the driving force of the Fermi-Bose experiment at LENS. I also thank the other people that contributed to the work described in this lecture: C. D’Errico, F. Ferlaino, M. Modugno, A. Simoni, and M. Zaccanti. This work was supported by MIUR, by EU under contract MEIF-CT-2004-009939, by Ente CRF, Firenze and by CNISM, Progetti di Innesco 2005.

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REFERENCES [1] Jochim S., Bartenstein M., Altmeyer A., Hendl G., Riedl S., Chin C., Hecker Denschlag J. and Grimm R., Science, 302 (2003) 2101; Greiner M., Regal C. A. and Jin D. S., Nature, 426 (2003) 537; Zwierlein M., Stan C. A., Schunck C. H., Raupach S. M. F., Gupta S., Hadzibabic Z. and Ketterle W., Phys. Rev. Lett., 91 (2003) 250401. [2] Regal C. A., Greiner M. and Jin D. S., Phys. Rev. Lett., 92 (2004) 040403; Chin C., Bartenstein M., Altmeyer A., Riedl S., Jochim S., Hecker Denschlag J. and Grimm R., Science, 305 (2004) 1128; Bourdel T. et al., Phys. Rev. Lett., 93 (2004) 050401; Partridge G. B. et al., Phys. Rev. Lett., 95 (2005) 020404; Zwierlein M. W., Abo-Shaeer J. R., Schirotzek A., Schunck C. H. and Ketterle W., Nature, 435 (2005) 1047. ¨rle T. et al., Phys. Rev. Lett., 96 [3] Winkler K. et al., Nature, 441 (2006) 853; Stofe (2006) 030401; Chin J. K. et al., Nature (London), 443 (2006) 961. [4] Inoyue S. et al., Nature (London), 392 (1998) 151. [5] Molmer K., Phys. Rev. Lett., 80 (1998) 1804. ¨chler [6] Albus A., Illuminati F. and Eisert J., Phys. Rev. A, 68 (2003) 023606; Bu H. P., Blatter G. and Zwerger W., Phys. Rev. Lett., 90 (2003) 130401; Lewenstein M., Santos L., Baranov M. A. and Fehrmann H., Phys. Rev. Lett., 92 (2004) 050401. [7] Baranov M. A., Marenko M. S., Rychkov V. S. and Shlyapnikov G. V., Phys. Rev. A, 66 (2002) 013606; Damski B. et al., Phys. Rev. Lett., 90 (2003) 110401. [8] Micheli A., Brennen G. K. and Zoller P., Nature Phys., 2 (2006) 341. [9] Stan C. A. et al., Phys. Rev. Lett., 93 (2004) 143001. [10] Zhang J. et al., in Proceedings of the XIX International Conference on Atomic Physics, edited by Marcassa L. G., Bagnato V. S. and Helmerson K. (AIP, New York) 2005. [11] Inouye S. et al., Phys. Rev. Lett., 93 (2004) 183201. [12] Ferlaino F., D’Errico C., Roati G., Zaccanti M., Inguscio M., Modugno G. and Simoni A., Phys. Rev. A, 73 (2006) 040702. [13] Zaccanti M., D’Errico C., Ferlaino F., Roati G., Inguscio M. and Modugno G., Phys. Rev. A, 73 (2006) 040702. [14] Ospelkaus S., Ospelkaus C., Humbert L., Sengstock K. and Bongs K., Phys. Rev. Lett., 97 (2006) 120403. [15] Roati G., Riboli F., Modugno G. and Inguscio M., Phys. Rev. Lett., 89 (2002) 150403. [16] Ferrari G., Jastrebski W., Modugno G., Roati G., Simoni A. and Inguscio M., Phys. Rev. Lett., 89 (2002) 053202. [17] Ferlaino F., Brecha R. J., Hannaford P., Riboli F., Roati G., Modugno G. and Inguscio M., J. Opt. B: Quantum Semiclass. Opt., 5 (2003) s3. [18] Simoni A., Ferlaino F., Roati G., Modugno G. and Inguscio M., Phys. Rev. Lett., 90 (2003) 163202. [19] Modugno G., Ferrari G., Roati G., Brecha R. and Inguscio M., Science, 294 (2001) 1320. [20] Marte A. et al., Phys. Rev. Lett., 89 (2002) 283202. [21] Derevianko A., Babb J. F. and Dalgarno A., Phys. Rev. A, 63 (2001) 052704. [22] Goldwin J., Inouye S., Olsen M. L., Newman B., DePaola B. D. and Jin D. S., Phys. Rev. A, 70 (2004) 021601. [23] Modugno G., Roati G., Riboli F., Ferlaino F., Brecha R. J. and Inguscio M., Science, 297 (2002) 2200. [24] Ospelkaus C., Ospelkaus S., Sengstock K. and Bongs K., Phys. Rev. Lett., 96 (2006) 020401.

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´ R. and Stwalley W. C., Phys. Rev. A, 71 (2005) 062706. [25] Zemke W. T., Cote [26] Modugno G., Modugno M., Riboli F., Roati G. and Inguscio M., Phys. Rev. Lett., 89 (2002) 190404. [27] Catani J., Maioli P., De Sarlo L., Minardi F. and Inguscio M., Phys. Rev. A, 73 (2006) 033415. [28] Leo P., Williams C. J. and Julienne P. S., Phys. Rev. Lett., 85 (2000) 2721. [29] Werner J. et al., Phys. Rev. Lett., 94 (2005) 183201. [30] Ticknor C., Regal C. A., Jin D. S. and Bohn J. L., Phys. Rev. A, 69 (2004) 042712. [31] D’Incao J. P. and Esry B. D., Phys. Rev. A, 73 (2006) 030702(R). ¨gerl H.-C. and Grimm R., Phys. Rev. Lett., 91 [32] Weber T., Herbig J., Mark M., Na (2003) 123201. [33] Modugno M., Ferlaino F., Roati G., Modugno G. and Inguscio M., Phys. Rev. A, 68 (2003) 043626. [34] Hodby E. et al., Phys. Rev. Lett., 94 (2005) 120402. [35] Ospelkaus C. et al., Phys. Rev. Lett., 97 (2006) 120402. [36] Kotochigova S., Julienne P. S. and Tiesinga E., Phys. Rev. A, 68 (2003) 022501. [37] Stwalley W. C., Eur. Phys. J. D, 31 (2004) 221. [38] Wang D., Qi J., Stone M. F., Nikolayeva O., Wang H., Hattaway B., Gensemer S. D., Gould P. L., Eyler E. E. and Stwalley W. C., Phys. Rev. Lett., 93 (2005) 243005; Sage J. M., Sainis S., Bergeman T. and DeMille D., Phys. Rev. Lett., 94 (2005) 203001. [39] DeMille D., Phys. Rev. Lett., 88 (2002) 067901. ¨nther K., Stho ¨ ferle [40] Gavish U. and Castin Y., Phys. Rev. Lett., 95 (2005) 020401; Gu ¨ hl M. and Esslinger T., Phys. Rev. Lett., 96 (2006) 180402; T., Moritz H., Ko Ospelkaus S. et al., Phys. Rev. Lett., 96 (2006) 180403. [41] Adhikari S. K., Phys. Rev. A, 70 (2004) 043617. [42] Liu X.-J., Modugno M. and Hu H., Phys. Rev. A, 68 (2003) 053605. [43] Albus A. P., Illuminati F. and Wilkens M., Phys. Rev. A, 67 (2003) 063606. [44] Heiselberg H., Pethick C. J., Smith H. and Viverit L., Phys. Rev. Lett., 85 (2000) 2418; Bijlsma M. J., Heringa A. and Stoof H. T. C., Phys. Rev. A, 61 (2000) 052601; Viverit L., Phys. Rev. A, 66 (2002) 023605; Matera F., Phys. Rev. A, 68 (2003) 043624.

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Exotic atom pairs: Repulsively bound states in an optical lattice J. Hecker Denschlag Institut f¨ ur Experimentalphysik, Universit¨ at Innsbruck - 6020 Innsbruck, Austria

A. J. Daley Institut f¨ ur Theoretische Physik, Universit¨ at Innsbruck - 6020 Innsbruck, Austria

1. – Introduction Ultracold atoms in 3D optical lattices provide an intriguing environment to study strongly correlated condensed-matter systems and quantum information. Unique features of these atomic many-body systems include the complete control of system parameters, and —in particular contrast to solid-state physics— weak couplings to dissipative environments. This so-called quantum lattice gas [1, 2] is described by a Bose or Fermi Hubbard Hamiltonian. The high control over the atoms opens the possibility to engineer a wide class of interesting many-body quantum states. Seminal experiments have already demonstrated the superﬂuid-Mott insulator transition [3], the realisation of 1D quantum liquids with atomic gases [4, 5] (see also [6, 7]), and a Bose spin glass [8]. Here we review another recent experiment [9] where we have observed a novel kind of bound state of two atoms which is based on repulsive interactions between the particles. These repulsively bound pairs exhibit long lifetimes, even under conditions when they collide with one another. Stable repulsively bound objects should be viewed as a general phenomenon and their existence will be ubiquitous in cold atoms lattice physics. Although the experiment described here is based on bosonic 87 Rb atoms, other composites with fermions [10] or Bose-Fermi mixtures [11] should exist in an analogous manner. Furthermore, repulsively bound objects could also be formed with more than two particles. c Societ` a Italiana di Fisica

677

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J. Hecker Denschlag and A. J. Daley

10

V 0 = 0 Er

V 0 = 4 Er

V0 = 10 Er

Energy (Er)

8

6

4

2

4J 1

0

-1

0

1 -1 0 1 -1 Quasi-momentum (S/a)

0

1

Fig. 1. – Energy spectrum for a single particle in a 1D lattice for three diﬀerent potential depths V0 . The band width of the lowest Bloch band is given by 4J, where J is the hopping energy. h2 /2ma2 denotes the recoil energy. a is the lattice period and Er = π 2 ¯

In the following we will ﬁrst explain the theoretical background of repulsively bound atom pairs. Afterwards we will present the experiments which demonstrate several key properties of the pairs. Finally we give a short discussion of how these repulsively bound pairs relate to bound states in some other physical systems. 2. – Repulsively bound pairs Optical lattices are generated by pairs of counterpropagating laser beams, where the resulting standing wave intensity pattern forms a periodic array of microtraps for the cold atoms, with period given by half the wavelength of the light, λ/2. This periodicity of the potential gives rise to a band structure for the atom dynamics with Bloch bands separated by band gaps, which can be controlled via the laser parameters and conﬁguration, as shown in ﬁg. 1. The dynamics of an atomic Bose-Einstein condensate loaded into the lowest band of a suﬃciently deep optical lattice [1, 2] is well described by a single-band Bose Hubbard model [12] with Hamiltonian (1)

ˆ = −J H

ij

† † ˆb†ˆbj + U ˆb ˆbj ˆb ˆbj − 1 + iˆb†j ˆbj . i j j 2 j i

Here ˆbi (ˆb†i ) are destruction (creation) operators for the bosonic atoms at site i. J/¯h and U denote, respectively, the tunnelling rate of atoms between neighbouring sites, and the collisional energy shift from interactions between atoms on the same site. The resulting width of the Bloch band is 4J, and this single-band model is valid because the kinetic

Exotic atom pairs: Repulsively bound states in an optical lattice

679

Fig. 2. – Left: a state with two atoms located on the same site of an optical lattice has an energy oﬀset ≈ U with respect to states where the atoms are separated. Breaking up of the pair is suppressed due to the lattice band structure and energy conservation, so that the pair remains bound as a composite object, which can tunnel through the lattice. In the ﬁgure, n = 0 denotes the lowest Bloch band and n = 1 the ﬁrst excited band. Right: breaking up of atom pairs in a shallow 3D optical lattice. (The potential depth is (10 ± 0.5) Er in one direction and (35 ± 1.5) Er in the perpendicular directions.) Shown is the remaining fraction of pairs for a scattering length of 100a0 (open diamonds) and a scattering length of about (0 ± 10)a0 (ﬁlled circles) as a function of the hold time. The atom pairs quickly break up within a few ms if they do not interact, but stay together for a long time in case of repulsive interaction. The lines are ﬁt curves of an exponential (dashed line) and the sum of two exponentials (solid line).

energy and interaction energy in this system are much smaller than the separation of the Bloch bands ω. The Bose-Hubbard Hamiltonian (1) predicts the existence of stable repulsively bound atom pairs. These are most intuitively understood in the limit of strong repulsive interaction U J (where U > 0 but this energy is still smaller than the separation to the ﬁrst excited Bloch band, U √

ω). If a state is prepared with two atoms occupying a single †2 ˆ site, |2i ≡ (bi | vac)/ 2, then it will have a potential energy oﬀset ≈ U with respect to states where the atoms are separated (see ﬁg. 2(left)). This state will be unable to decay by converting the potential energy into kinetic energy, as the Bloch band provides a maximum kinetic energy for two atoms both at the edge of the Brillouin zone given by 8J U . Instead, the atoms will remain together, and tunnel through the lattice as a bound composite object —a repulsively bound pair. We can observe this nature of repulsive binding in the experiment (see ﬁg. 2(right)). After production of the atom pairs, as is discussed in detail in sect. 5, we allow the atoms to tunnel through the lattice along one dimension. If the on-site interaction of the atoms is tuned to zero with the help of a Feshbach resonance, the pairs break up within a few ms, corresponding to the tunneling timescale. However, if the eﬀective interaction between the atoms is repulsive, we observe a remarkably long lifetime of t = 700 ms (determined by an exponential ﬁt). This lifetime is mainly limited by inelastic scattering of lattice photons.

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3. – Analytical solution of two-particle problem in an optical lattice . 3 1. General discussion. – Our understanding of these stable pairs can be made more formal by an exact solution of the two-particle Lippmann-Schwinger scattering equation on the lattice corresponding to the Bose-Hubbard Hamiltonian eq. (1). Denoting the primitive lattice vectors in each of the d dimensions by ei , we can d d write the position of the two atoms by x = i=1 xi ei and y = i=1 yi ei , where xi , yi are integers, and we can write the two atom wave function in the form Ψ(x, y). The related Schr¨ odinger equation from the Bose-Hubbard model (eq. (1)) with homogeneous background, i = 0 then takes the form ˜x + Δ ˜ y + U δx,y Ψ(x, y) = E Ψ(x, y), −J Δ

(2) where the operator (3)

˜ x Ψ(x, y) = Δ

d

[Ψ(x + ei , y) + Ψ(x − ei , y) − 2Ψ(x, y)]

i=1

denotes a discrete lattice Laplacian on a cubic lattice. Note that in order to express this in terms of the discrete lattice Laplacian, we have added 4dJΨ(x, y) to each side of the Schr¨ odinger equation. This eﬀectively changes the zero of energy, so that E → E + 4Jd. We then introduce relative coordinates r = x − y existing on the same lattice structure as the coordinate x, and center of mass coordinates R = (x + y)/2, existing on a lattice with the same symmetry as the original lattice but smaller lattice constant a/2. We then separate the wave function using the ansatz (4)

Ψ(x, y) = exp[iKR]ψK (r),

with K the centre-of-mass quasimomentum. This allows us to reduce the Schr¨odinger equation to a single-particle problem in the relative coordinate (5)

˜ K + EK + U δr,0 ψK (r) = EψK (r), −2J Δ r

d where EK = 4J i=1 [1 − cos(Kei /2)] is the kinetic energy of the center-of-mass motion, and where the discrete lattice Laplacian for a square lattice is now given by (6)

˜K Δ r Ψ(r) =

d

cos (Kei /2) [Ψ(r + ei ) + Ψ(r − ei ) − 2Ψ(r)] .

i=1

The solutions of this Schr¨ odinger equation can be found using the Greens function of the non-interacting problem with U = 0, which is deﬁned by (7)

[E − H0 ] GK (E, r) = δr,0 ,

Exotic atom pairs: Repulsively bound states in an optical lattice

681

with δr,0 a three-dimensional Kronecker delta, and H0 = −2JΔK r the Hamiltonian of the non-interacting system. This equation can be easily solved via Fourier transformation, ˜ G(E, r) = [1/(2π)d ] dd k G(E, k) exp[ikr], and we obtain the solution ˜ K (E, k) = G

(8)

1 , E − K (k) + iη

where K (k) accounts for the dispersion relation of the non-interacting system (9)

K (k) = 4J

d i=1

cos

Ki a [1 − cos(ki a)] . 2

The solutions of eq. (5) can be divided into two classes: scattering states, and bound (localised) states. We will ﬁrst analyze the scattering states. . 3 2. Scattering states. – Similarly to scattering problems involving particles in free space, the scattering states of particles on the lattice with energy E obey the LippmannSchwinger equation 0 (10) ψE (r) = ψE (r) + GK (E, r − r )V (r )ψE (r ), r 0 = exp[ikr] an eigenstate of H0 with energy E = K (k). In the present situation with ψE with a short-range potential V (r) = U δr,0 , the Lippmann-Schwinger equation can be solved via a resummation of the Born expansion and we obtain

ψE (r) = exp[ikr] − 8πJfE (K)GK (E, r),

(11)

with scattering amplitude U/(2J) 1 , 4π 1 − GK (E, 0)U where the total energy is E = k,K + EK , k,K = 4J i=1 cos(Kei /2)[1 − cos(kei )], and fE (K) = −

(12)

(13)

GK (E, 0) =

4π 2J

1 dk . (2π)d E/(2J) − 2 di=1 cos Ki a (1 − cos ki a) 2

The scattering states ψE (r) correspond to two free atoms moving on the lattice and undergoing scattering processes. The corresponding energies appear as a continuum in ﬁg. 3. In order to make a connection to the scattering length in free space, we can consider the limit of small momenta of the incoming plane wave, i.e., k → 0, K → 0 and E → 0. Then the solution (11) reduces in the limit r → ∞ to (14)

0 ψE (r) ∼ ψE (r) + f (k, k )

exp[ikr] , r

682

J. Hecker Denschlag and A. J. Daley

1.2

a)

b)

Energy (Er)

0.8

U/J = 2 V0 ~ 2 Er

0.4

U/J = 5 V0 ~ 4 Er

0 1.2

c)

d)

U/J = 9 V0 ~ 6 Er

0.8

U/J = 29 V0 ~ 10 Er

0.4 0

0

-0.5

0

1 0

0.5

-0.5

0

0.5

1

Quasi-momentum K (a/S) Fig. 3. – Two-particle energy spectrum in a 1D lattice for four diﬀerent potential depths V0 as a function of centre-of-mass quasimomentum K. The Bloch band for repulsively bound pairs is located above the continuum of unbound scattering states. The grey level for the shading of the continuum is proportional to the density of states.

with the scattering amplitude (15)

f (k, k ) = −as = −

U/(2J) 1 , 4π 1 − αU/(2J)

equivalent to the s-wave scattering length as , while the constant α = limE→0 G(E, 0) is α ≈ −0.25 [13]. . 3 3. Bound states. – Note that the scattering amplitude in eq. (11) contains a pole, associated with a bound state. We now focus on these bound states in the regime, U > 0, which will correspond to a repulsively bound pair. First we note that we can write eq. (7) in the form (16)

[E − H0 ] GK (E, r) =

1 δr,0 GK (E, r). GK (E, 0)

BS As a consequence, the function ψK (r) = GK (E, r) is a solution of the Schr¨ odinger equation if the self-consistency relation is satisﬁed

(17)

U=

1 , GK (E, 0)

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Fig. 4. – Wave functions ψK (r) showing the amplitude for various site separations for repulsively bound pairs (as = 100a0 ) in 1D with K = 0. r denotes the separation between the two atoms. (a) U/J = 30 (V0 = 10Er ) and (b) U/J = 3 (V0 = 3Er ).

BS which determines the energy EBS of the bound state ψK . The resulting bound state wave BS function, ψK (r) falls oﬀ exponentially for large r, and describes a bound two-particle state travelling with center-of-mass momenta K through the lattice. The momentum distribution of this bound state is then given by GK (EBS , k)

(18)

BS (k) = ψK

1 . EBS − K (k)

Note, that this wave function is not normalized. In three dimensions, eq. (17) only has a solution for interaction strengths above a critical value, U > Uc = −2J/G(0, 0) ≈ 8J, and thus we require U > Uc for the formation of the bound two-particle state. The wave function ψK (r) is then square-integrable, as shown in ﬁg. 4. For a deep lattice, i.e. U/J 1, bound pairs essentially consist of two atoms occupying the same site, whereas for small U/J, the pair is delocalized over several lattice sites. A main feature of the repulsive pair wave function is its oscillating character: the wave function amplitude alternates sign from one site to the next, as shown in ﬁg. 4. In quasimomentum space this corresponds to a wave function that is peaked at the edges . of the ﬁrst Brillouin zone, which is discussed in detail in subsect. 6 2. When motion is conﬁned to one dimension, the bound two-particle state exists for arbitrarily small repulsive interaction U > 0, in contrast to the three-dimensional situation. Here the energy of the bound pairs, computed from eq. (17) is ELS (K) =

2 2 4(cos Ka 2J 2 ) + (U/2J) + 2 , which can be seen plotted in ﬁg. 3 as Bloch band of a stable composite object above the continuum of two-particle scattering states. The ﬁgure shows how the binding energy (separation of these states from the continuum) increases as U/J is increased, and how the curvature of the band becomes less pronounced. In the limit of strong interaction, U J, the bound-state energy reduces to

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E(K) ∼ 4J + U + (4J 2 /U )(1 + cos Ka), which is consistent with our expectation of a positive binding energy U , and the center-of-mass energy of a composite object with an eﬀective tunnelling matrix element J 2 /U . 4. – Numerical approach for repulsively bound pairs We want to be able to treat not just a single repulsively bound pair, but a lattice gas of many interacting repulsively bound pairs. This is important both in order to properly describe the eﬀects of interactions on experimental measurements, and to investigate many-body eﬀects on the behaviour of the pairs. Whilst perturbation theory can be used to produce useful analytical models in some limits [14], it is possible in one dimension to treat the system in more general regimes using recently developed numerical techniques. . 4 1. Time-dependent DMRG. – In one spatial dimension, the system of interacting repulsively bound pairs can be treated by directly simulating the Bose-Hubbard model time-dependently, using time-dependent Density Matrix Renormalisation Group (DMRG) methods [15-18]. These methods allow near-exact integration of the Schr¨ odinger equation for the 1D many-body system by an adaptive decimation of the Hilbert space, provided that the resulting states can be written eﬃciently in the form of a truncated matrix product state (this will be explained in more detail below). This method has been successfully applied to several lattice and spin models of interest in condensed-matter physics, including systems realisable using cold atoms in optical lattices [19-25]. The algorithm, both in the form originally proposed by Vidal and similar methods proposed by Verstrate and Cirac have also been generalised to the treatment of master equations for dissipative systems and systems at ﬁnite temperature [26, 27]. . 4 1.1. Matrix product states. DMRG methods are based on a decomposition of the many-body wave function into a matrix product representation [18]. This requires that the original state can be expressed on a Hilbert space that is the product of local Hilbert spaces. For the Bose-Hubbard model, each local Hilbert space corresponds to a particular lattice site, and the basis states in the local Hilbert space correspond to diﬀerent occupation numbers, running from 0 to S − 1. We then write the coeﬃcients of the wave function expanded in terms of local Hilbert spaces of dimension S,

(19)

S

|Ψ =

ci1 i2 ...iM |i1 ⊗ |i2 ⊗ . . . ⊗ |iM ,

i1 i2 ...iM =1

as a matrix product state given by

(20)

ci1 i2 ...iM =

χ α1 ...αM −1

i1 [1] [2] i2 [2] [2] i2 [M ] iM Γ[1] α1 λα1 Γα1 α2 λα2 Γα3 α4 . . . ΓαM −1 .

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[l]

The Γ and λ tensors are chosen so that the tensor λα speciﬁes the coeﬃcients of the Schmidt decomposition [15] for the bipartite splitting of the system at site l, (21)

|ψ =

χl

/ / [l] / [1...l] / [l+1...M ] , λα /φα /φα

α=1

where χl is the Schmidt rank, and the sum over the remaining tensors speciﬁes the [1...l] [l+1...M ] Schmidt eigenstates, |φα and |φα . The key to the usefulness of this representation is that we can truncate the representation at relatively small χ and still provide an inner product of almost unity with the exact state of the system |ψ. This is possible because for many states corresponding to a low energy in 1D systems, the Schmidt [l] coeﬃcients λα decay rapidly as a function of their index α (when ordered in decreasing magnitude) [15]. Thus we can truncate the representation at relatively small χ and still provide an inner product of almost unity with the exact state of the system |Ψ. In implementations of this method we perform convergence tests for the state representation, that is, we vary the values of χ and S to check that the point at which the representation is being truncated does not aﬀect the ﬁnal results. We also make use of an optimised version of the code in which the Schmidt eigenstates are forced to correspond to ﬁxed numbers of particles [16,18]. This allows us to make use of the total number conservation in the Hamiltonian to substantially increase the speed of the code, and also improve the scaling with χ and S. With this number-conserving code we are able to compute results with much higher values of χ in a much shorter time than the original algorithm. . 4 1.2. Time dependence. Time dependence of these states can be computed for Hamiltonians acting only on neighbouring lattice sites because when an operator acts on the local Hilbert state of two neighbouring sites, the representation can be eﬃciently updated to the Matrix product state that best approximates the new state of the system. To do this, the Γ tensors corresponding to those two sites must be updated, which can be achieved using a number of operations that scales as χ3 S 3 for suﬃciently large χ [15]. In this way, we represent the state on a systematically truncated Hilbert space, which changes adaptively as we perform operations on the state. The time evolution operator ˆ is then split into a product of operators, each of which acts only on a pair of exp[−iHt] neighbouring sites by means of a Suzuki-Trotter decomposition [28, 29]. This is done in small time steps δt. Initial states can also be found using an imaginary time evolution, ˆ i.e., the repeated application of the operator exp[−Hδt], together with renormalisation of the state. . 4 2. Numerical investigation of repulsively bound pairs. – In investigation of repulsively bound pairs using these methods we are able to use parameters U and J directly corresponding to values of the lattice depth V0 in the experiments, and are also able to account for the background trapping potential. We typically study 10–30 pairs in 60 lattice sites, and begin with an initial product state, corresponding to a random distribution of doubly occupied and unoccupied lattice sites. We then reduce the value of U and

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increase the value of J, using the same time dependence for the depth of the lattice V0 (t) as in the experiment. The single-particle momentum distributions can then be calculated eﬃciently from the matrix product state representation, and we also average the results over diﬀerent initial conﬁgurations, to match the averaging over diﬀerent 1D tubes in the experiment. We can also perform lattice modulation spectroscopy, computing the time evolution of the many-body state when the parameters U and J vary as a function of time, based on the time dependence of the lattice depth V (t) used in the experiments (these calculations are similar to those in refs. [22, 25]). 5. – Experimental realization In this section we describe the experimental steps to produce a pure ensemble of repulsively bound atom pairs in an optical lattice. We use 87 Rb as the atomic species for our experiments. . 5 1. BEC production. – We begin by creating a 87 Rb Bose-Einstein condensate (BEC) of 6 × 105 87 Rb atoms in spin state |F = 1, mF = −1 in a vacuum apparatus featuring a magnetic transport line [30, 31]. This transport initially transfers laser-cooled atoms from the chamber of the magneto-optical trap (MOT) into a UHV glass cell (pressure < 10−11 mbar) which oﬀers good optical access from all sides. Here the BEC is produced in a QUIC trap [32,30] with trapping frequencies ωx,y,z = 2π × (15, 15, 150) Hz at a magnetic bias ﬁeld of 2 G. Afterwards the QUIC trap is converted into a Ioﬀe-type magnetic trap with trap frequencies ωx,y,z = 2π × (7, 19, 20) Hz by adjusting the currents through the quadrupole and Ioﬀe coils and by applying additional magnetic ﬁeld gradients. This moves the BEC over a distance of 8 mm into the center of the QUIC quadrupole coils which are later used to generate the homogeneous magnetic ﬁeld for Feshbach ramping. . 5 2. Loading into lattice. – Within 100 ms the BEC is adiabatically loaded into the vibrational ground state of an optical lattice which is 35Er deep (Er = 2π 2 ¯h2 /mλ2 , where m is the mass of the atoms). Our 3D lattice is cubic and consists of three pairs of retro-reﬂected intensity-stabilized laser beams, which propagate orthogonally to each other. They are derived from a frequency-stable single-mode Ti:Sapphire laser (< 500 kHz linewidth) with a wavelength of λ = 830.44 nm. For this wavelength, the laser is detuned by about 100 GHz from the closest transition to an excited molecular level, minimizing light-induced losses as a precondition for long lifetimes of pairs and molecules. The laser beams are polarized perpendicularly to each other and their frequencies diﬀer by several tens of MHz to avoid disturbing interference eﬀects. The waists of all three beams are about 160 μm and the maximum obtainable power is about 110 mW per beam. At this stage about 20% of the condensate atoms are grouped in pairs of two into the lattice sites. 60% of the condensate atoms are found in singly occupied sites, and another 20% percent of atoms are located in triply and more highly occupied lattice sites [33]. . 5 3. Puriﬁcation scheme. – In order to remove all atoms from those lattice sites that are not occupied by exactly two atoms, we use a puriﬁcation scheme which involves

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an intermediate step in which Feshbach molecules are produced. For this, we turn oﬀ the magnetic trap and ﬂip the spins of the 87 Rb atoms from their initial state |F = 1, mF = −1 to |F = 1, mF = +1 by suddenly reversing the bias magnetic ﬁeld of a few G. This spin state features a 210 mG wide Feshbach resonance at 1007.40 G [34]. By adiabatically ramping over this resonance, we convert pairs of atoms in multiply occupied lattice sites into Rb2 Feshbach molecules with almost unit eﬃciency [33]. After the Feshbach ramp, fast inelastic collisions will occur in sites that were initially occupied with more than two atoms. These exothermic collisions between either a created molecule and an atom or between two molecules will remove these particles from the lattice. At this stage the lattice consists only of sites which are either empty, ﬁlled with a single atom, or ﬁlled with a single Feshbach molecule. A subsequent 3 ms long combined radiofrequency (r.f.) and optical puriﬁcation pulse removes all chemically unbound atoms, thus creating a pure molecular sample of about 2 × 104 molecules [33]. The microwave drives the transition at a frequency of 9113 MHz between levels which correlate with |F = 1, mF = +1 and |F = 2, mF = +2. The light pulse drives the closed transition |F = 2, mF = +2 → |F = 3, mF = +3. The optical transition frequency is 1402 MHz blue detuned compared to the transition at zero magnetic ﬁeld. The light literally blows the atoms out of the lattice, while the direct eﬀect of the microwave and light ﬁeld pulse on the molecules is negligible because the radiation is oﬀ resonance. Finally, sweeping back across the Feshbach resonance we adiabatically dissociate the dimers and obtain a lattice where 2 × 104 sites are ﬁlled with exactly two atoms. According to sect. 3, at the deep lattice depth of 35Er (U/J ≈ 3700), the corresponding two-atom wavepacket matches perfectly with the wave function of the repulsively bound atom pair. By adiabatically lowering the lattice depth (typically within a few ms) in a horizontal direction we can then produce 1D repulsively bound atom pairs states for arbitrary values of U/J. 6. – Experiments In the following we discuss the properties of the repulsively bound pairs that were experimentally investigated by measuring their lifetime, quasi-momentum distribution and binding energies. By varying the eﬀective interaction between the atoms with the help of the Feshbach resonance, we can also create lattice-induced bound atom pairs which are based on attractive interactions. . 6 1. Pair lifetime. – We have already seen from ﬁg. 2 (right) that for a repulsive interaction with as = 100a0 the lifetime of the pairs is remarkably long (700 ms, exponential ﬁt). This lifetime is mainly limited by inelastic scattering of lattice photons [33], and greatly exceeds the calculated time for an atom to tunnel from one site to the next, 2π¯ h/(4J) ∼ 4 ms. The lifetime measurements are based on lowering the lattice depth to a chosen height, and then measuring the number of remaining pairs after a variable hold time. In order to make this measurement, the lattice is adiabatically raised again to its full initial depth of V0 = 35Er . Using the Feshbach resonance, atoms in doubly occupied sites are converted to Feshbach molecules with near unit eﬃciency [33], and another com-

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Particle density [a.u.]

J. Hecker Denschlag and A. J. Daley

experiment - - theory

Momentum (S/a) Fig. 5. – Quasimomentum distributions of atoms in the lattice for (a) V0 = 5Er and (b) V0 = 20Er . Above: images show absorption images of the atomic distribution after release from the 3D lattice and a subsequent 15 ms time of ﬂight. The horizontal and vertical dashed lines enclose the ﬁrst Brillouin zone. Below: corresponding quasimomentum distributions in the horizontal direction, after integration over the vertical direction. For comparison, numerical simulations (see text) are also shown (dashed lines). The density values have been scaled to facilitate comparison between experimental and theoretical results.

bined r.f.-light puriﬁcation pulse then removes all remaining atoms (which stem from now dissociated pairs) as in the original preparation step. Afterwards the molecules are again converted back into atoms, and can then be detected via conventional absorption imaging. . 6 2. Quasimomentum distribution. – We have experimentally investigated the quasimomentum distribution of the pairs in various regimes by mapping it onto a spatial distribution, which we measured using standard absorption imaging. For this, we ﬁrst adiabatically lower the lattice depth in a horizontal direction at a rate of 1.3Er /ms to a pre-chosen height while the lattice depth in the other two directions are kept high (35Er ). This will prepare repulsively bound pairs at the chosen lattice depth. We then turn oﬀ the lattice rapidly enough so that the quasimomentum distribution cannot change, but slowly with respect to the bandgap, so that single-particle quasimomenta are mapped to real momenta [31, 35]. We have typically employed linear ramps with rates of 0.2Er /μs. The resulting momentum distribution is converted to a spatial distribution after ∼ 15 ms time of ﬂight. Figure 5 shows two measured quasimomentum distributions for lattice depths V0 /Er = 6, 20. The top row shows the bare images of the atomic density taken in the laboratory. Below are the corresponding quasimomentum distributions in horizontal

Exotic atom pairs: Repulsively bound states in an optical lattice

V0 (Er )

689

V0 (Er )

20

Momentum (S/a)

Momentum (S/a)

Fig. 6. – Momentum distributions similar to the ones shown in ﬁg. 5 are plotted here as a function of lattice depth V0 . Left: experiment. Right: numerical calculation.

direction. It is clearly visible that for low lattice depths the quasimomentum distributions are peaked at the edges of the ﬁrst Brillouin zone. For deep lattices, however, the ﬁrst Brillouin zone is homogenously ﬁlled and the quasimomentum distribution has a ﬂat top shape. This latter distribution is reminiscent of the one observed for a dephased ensemble of ultracold atoms in the lowest band of a lattice [31]. The agreement between experimental data and theoretical calculations is quite good. Note, however, that the experimental distributions appear to extend beyond the ﬁrst Brillouin zone. This is an experimental artifact related to repulsion between atoms during expansion (before imaging) and also relatively long imaging times (many photons are scattered from each atom, which performs a random walk). This leads to smearing out of the sharp structure at the edge of the Brillouin zone. Figure 6 shows in a more continuous fashion the dependence on lattice depth V0 of the quasimomentum distribution for repulsively bound pairs for both experiment and numerical simulation. As discussed before, the peak structure is more pronounced for lower values of V0 , and diminishes for larger V0 . This characteristic is a clear signature of the pair wave function for repulsively bound pairs. It is important to note that in all cases here we measure the distribution of single-atom quasimomenta in a large sample. That we still obtain the peaked distribution characteristic of repulsively bound pairs is non-trivial. In fact, if we just take a single repulsively bound pair with centre-of-mass quasimomentum K = 0, its single-atom momentum distribution will not be peaked anymore at the edges of the ﬁrst Brillouin zone. The peak will be somewhat translated towards the center of the ﬁrst Brillouin zone. Fortunately, with increasing |K|, the peak in the single-particle quasimomentum distribution also becomes less pronounced. As a result, when we average over a roughly uniform distribution of centre-of-mass quasimomenta K for a dilute gas of repulsively bound pairs, we still observe the pronounced peaks at the edges of the Brillouin zone. This is conﬁrmed by the numerical simulations and their good agreement with experiments.

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Fig. 7. – Modulation spectroscopy of repulsively bound pairs. Left: by modulating the optical lattice amplitude with the proper frequency, the pair can dump its binding energy into the lattice motion and subsequently break up. Right: typical resonance dip showing the remaining number of atom pairs as a function of the modulation frequency, for V0 ≈ 6Er . The black line is a Gaussian ﬁt, a choice which was justiﬁed by numerical calculations.

. 6 3. Modulation spectroscopy. – By modulating the depth of the lattice at a chosen frequency, we can determine the binding energy of the pairs (see ﬁg. 7, left). For appropriate modulation frequencies, the pairs can dump their binding energy into the lattice

Fig. 8. – Measured resonance frequencies of the modulation spectroscopy as a function of the lattice depth. The resonance frequency was determined from resonance curves similar to the one in ﬁg. 7 (right). The experimental points (ﬁlled circles) show good agreement with numerical simulations (crosses) and also coincide with the on-site collisional energy shift U (line). Experimental error bars correspond to the 95% conﬁdence interval for the Gaussian ﬁt parameters of the resonance dips.

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Scattering length (a0)

500 300 100 0 -100 -300 -500

0

0.2

0.4

0.6

0.8

1

Quasi-momentum (π/a)

B- 1007 G (G) Fig. 9. – From repulsively to attractively bound atom pairs. Left: with the help of a Feshbach resonance around 1007.4 G, we can control the eﬀective interaction of the paired atoms by adjusting the scattering length a. a0 is the Bohr radius. The inserted images show quasimomentum distributions similar to the ones of ﬁg. 5. For eﬀectively attractive interaction the quasimomentum distribution is peaked around 0 momentum. Right: the momentum distribution for atom pairs as a function of magnetic ﬁeld (scattering length.). At zero scattering length the distribution has a ﬂat top shape. The shown data correspond to experiments where the lattice depth V0 had been adiabatically lowered in 1D below 3Er .

motion and dissociate. Figure 7 (right) shows a typical resonance curve of the number of remaining pairs as a function of the modulation frequency. The resonance frequency of about 1.5 kHz (for a lattice depth of V0 = 6Er ) agrees well with the calculated binding energy of a pair. The width of the resonance curve can be understood, as the pair will decay into a continuum of scattering states which has an energy width of up to 8J (depending on the initial centre-of-mass quasimomentum K (see ﬁg. 3)). In addition to this width, broadening due to Fourier-limited modulation pulses and inhomogeneity eﬀects in the lattice will occur. Modulation spectroscopy measurements were carried out for a variety of lattice depths (see ﬁg. 8). The resonance positions are in good agreement with numerical simulations and essentially coincide with interaction energy, U . . 6 4. Attractively bound pairs. – Making use of the Feshbach resonance at 1007.40 G, we can tune the eﬀective interaction of the atoms within the pair (see ﬁg. 9, left). It is then possible to also create bound atom pairs which are based on attractive interaction. After initial production of repulsively bound atom pairs in the deep lattice (V0 = 35Er ), we applied an appropriate nearly homogenous magnetic oﬀset ﬁeld. This tuned the scattering length of the atomic pair from its default value of as = 100a0 to negative scattering length of up to as = −110a0 . Afterwards the optical lattice was lowered as before. In contrast to repulsively bound pairs where the momentum distribution is peaked at the edges of the ﬁrst Brillouin zone, the momentum distribution for attractively

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Fig. 10. – Possible realization of repulsively bound pairs with fermionic atoms. Left lattice site: in the case of two identical fermions (same spin) the atoms have to be in diﬀerent bands due to the Pauli exclusion principle. Right lattice site: fermions of diﬀerent spins or Bose-Fermi mixtures, however, can occupy the same band.

bound pairs is peaked in the center of the ﬁrst Brillouin zone. This goes along with having a bound state with minimal internal energy. Figure 9 (right) shows how the quasimomentum distribution of the pairs changes continuously as the scattering length is changed. Interestingly, for non-interacting atoms (as = 0) the distribution again becomes a ﬂat top shape. With respect to stability, we ﬁnd that lifetimes of bound atom pairs are similar for scattering lengths of equal size but opposite sign. 7. – Repulsively bound pairs of fermions Although so far we have only discussed repulsively bound pairs which are composed of bosonic atoms, analogous pairs can also be formed from fermions or boson-fermion mixtures. These systems will exhibit interesting physics based on their composite nature and the quantum statistics of their components. Of course, in a single-species fermion experiment it is not possible to put two identical fermions into the same site and band due to the Pauli exclusion principle. The two atoms would have to be at least in diﬀerent bands (see ﬁg. 10, left), and even then the interaction between them typically would be very small in the ultracold regime. These problems do not arise using a two-component spin mix of fermions (see ﬁg. 10, right), as two atoms of diﬀerent spin can share the same site and band and can also interact strongly. The fact that a higher-site occupancy than two is again strictly forbidden could be advantageous in the initial production of pairs. Furthermore, the pairing of two fermions can result in a pair with bosonic character. In a 3D environment pairing of fermions recently led to interesting experiments studying the BEC-BCS transition (see, e.g., [36-41]). It would be interesting to study similar properties to this transition with repulsive pairing, investigating the system as the interaction strength is changed. Also it would be very interesting to study repulsive pairs which are composite objects of a fermion and a boson. One question would be how these bound states would interact

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with each other, and how, for example, the bosonic atoms within the pair would mediate next-neighbor interactions [11]. 8. – Other related physical systems Although no stable repulsively bound pairs have previously been observed, their physics is partially related to other physical systems. Here we brieﬂy discuss a few examples of such systems with bound states which are in a way reminiscent of repulsively bound states. . 8 1. Pairing resonances in many-body systems. – For example, resonance behaviour based on similar pairing of fermions of diﬀerent spin in the Hubbard model was ﬁrst discussed by Yang [42], and plays an important role in SO(5) theories of superconductivity [43]. There are several examples of many-body bound states that can occur for repulsive as well as attractive interactions, such as the resonances discussed in the context of the Hubbard model by Demler et al. [44]. Such resonance behaviour is common in manybody physics, although states of this type are normally very short-lived. Optical lattice experiments will now provide an opportunity to prepare and investigate stable versions of such states, which until now have only appeared virtually as part of complex processes. . 8 2. Excitons. – The stability and many-body physics of repulsively bound pairs is perhaps most closely associated with that of excitons, which are attractively bound pairs of a particle in the conduction band and a hole in the valence band of a periodic system [45]. These bind to form a composite boson, a gas of which can, in principle, Bose-condense. Excitons are excited states of the many-body system, but are bound by an attractive interaction between the particle and hole that form the pair. They are also discussed in the speciﬁc context of fermionic systems. However, a single exciton on a lattice could have a description very similar to that of a single repulsively bound pair, and could be realised and probed in optical lattices experiments [46]. . 8 3. Photonic crystals. – Repulsively bound atom pairs in an optical lattice are also reminiscent of photons being trapped by impurities in photonic crystals [47], which consist of transparent material with periodically changing index of refraction. An impurity in that crystal in the form of a local region with a diﬀerent index of refraction can then give rise to a localized ﬁeld eigenmode. In an analogous sense, each atom in a repulsively bound pair could be seen as an impurity that “traps” the other atom. . 8 4. Gap solitons. – An analogy can also be drawn between repulsively bound atom pairs and gap solitons, especially as found in atomic gases [48-51]. Solitons are normally a non-linear wave phenomenon, and in this sense have a very diﬀerent behaviour to repulsively bound pairs, which exhibit properties characteristic of many-body quantum systems. However, there has been increasing recent interest in discussing the limit of solitons in atomic systems where very few atoms are present, giving rise to objects that are often referred to as quantum solitons [52-54]. These are N -body bound states in 1D,

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and thus a 2-atom bright quantum soliton is a bound state of two atoms moving in 1D. In this sense, the solution for a single repulsively bound pair in 1D is related to a single quantum soliton on a lattice. 9. – Conclusion We have reviewed theoretically and experimentally the physics of repulsively bound pairs of atoms in an optical lattice. The good agreement between experiment and theory exempliﬁes the strong correspondence between the optical lattice physics of ultracold atoms and the Hubbard model on a new level, a connection which has particular importance for applications of these cold atom systems to a more general simulation of condensed-matter models and to quantum computing. The existence of such metastable bound objects will be ubiquitous in cold-atoms lattice physics, giving rise to new potential composite objects also in fermions or in systems with mixed Bose-Fermi statistics. These states could also be formed with more than two particles, or as bound states of existing composite particles. Repulsively bound pairs have no direct counterpart in condensedmatter physics due to the strong inelastic decay channels observed in solid-state lattices, and could be a building block of yet unstudied quantum many-body states or phases. ∗ ∗ ∗ We would like to thank our fellow team members who contributed to the research work ¨chler, R. Grimm, A. Kantian, F. Lang, on repulsively bound atom pairs: H. Bu G. Thalhammer, K. Winkler and P. Zoller. We would like to thank E. Demler for interesting discussions. We acknowledge support from the Austrian Science Fund (FWF) within the Spezialforschungsbereich 15, from the European Union within the OLAQUI and SCALA networks, from the TMR network “Cold Molecules”, and the Tiroler Zukunftsstiftung. REFERENCES [1] Jaksch D. and Zoller P., Ann. Phys. (N.Y.), 315 (2005) 52 and references therein. [2] Bloch I., Nature Phys., 1 (2005) 23. ¨nsch T. W. and Bloch I., Nature, 415 [3] Greiner M., Mandel O., Esslinger T., H a (2002) 39. ¨ lling S., Cirac I., Shlyapnikov [4] Paredes B., Widera A., Murg V., Mandel O., F o ¨nsch T. W. and Bloch I., Nature, 429 (2004) 277. G. V., Ha ¨ ferle T., Moritz H., Schori C., Ko ¨ hl M. and Esslinger T., Phys. Rev. Lett., 92 [5] Sto (2004) 130403. [6] Kinoshita T., Wenger T. and Weiss D. S., Science, 305 (2004) 1125. [7] Laburthe Tolra B., O’Hara K. M., Huckans J. H., Phillips W. D., Rolston S. L. and Porto J. V., Phys. Rev. Lett., 92 (2004) 190401. [8] Fallani L., Lye J. E., Guarrera V., Fort C. and Inguscio M., cond-mat/0603655 (2006). [9] Winkler K., Thalhammer G., Lang F., Grimm R., Hecker Denschlag J., Daley ¨chler H. P. and Zoller P., Nature, 441 (2006) 853. A. J., Kantian A., Bu

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Introduction to FFLO phases and collective mode in the BEC-BCS crossover R. Combescot Laboratoire de Physique Statistique, Ecole Normale Sup´ erieure 24 rue Lhomond, 75231 Paris Cedex 05, France Institut Universitaire de France - 103 Boulevard Saint-Michel, 75005 Paris, France

1. – Introduction My main focus in this lecture will be the discussion of what might happen in a BCS superﬂuid when there is an imbalance between the two populations of fermionic particles forming Cooper pairs. Naturally we have in mind speciﬁcally in this course the ultracold Fermi gases which are its subject and in particular the BEC-BCS crossover. In the last part I will consider brieﬂy the evolution in the crossover of the collective mode arising in such a superﬂuid (with balanced atomic populations), which is not a completely unrelated matter. Actually I will even much more restrain my scope since I will only tell about what is known on this topic from past research in the domain of superconductivity, which spans almost the entire ﬁfty years period of time since BCS theory came out, this ﬁeld being still very active with a large number of problems still open. This research is driven in particular by the search for more interesting or exotic superconductors, but also by the hope to master superconductors with extremely high critical ﬁelds, such as high-Tc superconductors, which are of obvious practical interest. But it turns out to be also of interest for elementary-particle physicists and astrophysicists, since similar phenomena might arise in the core of neutron stars, the involved fermions being quarks in this case. This has recently been reviewed [1]. c Societ` a Italiana di Fisica

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This means in particular that I will not speak about the very recent huge activity in this ﬁeld for ultracold gases and leave it to other speakers in this course, such as G. Strinati and K. Levin. This is quite appropriate since I have not contributed to this activity. I have rather been interested in this problem earlier [2], when it seemed to be a major problem in the way of obtaining a BCS condensation. Indeed, in contrast to the case of electrons in superconductors where there are fast spin relaxation processes leading to balanced spin populations, in cold gases there are no such processes between various hyperﬁne states which play the role of spins in superconductors. Hence, a mixture of diﬀerent hyperﬁne states is diﬃcult to balance precisely, and if the critical temperature for BCS superﬂuidity is low, this might kill altogether the transition. Fortunately experimentalists have found a very nice way around this problem by making use of Feshbach resonance and obtaining the associated molecules. By getting rid of the remaining atoms, this ensures an exact balance between the hyperﬁne states populations. Nevertheless, this choice will not be so frustrating because the situation we will consider is not simple at all, and gives rise to very rich and interesting physics, so much that the only thing I will be able to do is to give just a brief overview. However it must be kept in mind that we will deal only with weakly interacting superﬂuids, since this is the range of validity of standard BCS theory and the physics ruling standard superconductors. This corresponds to looking only, in the BEC-BCS crossover, at the BCS limit where the scattering length takes small negative values a → 0− . Naturally we are more interested in what may happen around unitarity, and the physics may indeed be quite diﬀerent for such a strongly interacting superﬂuid. However, there is no simple way to extrapolate what we will see toward unitarity. On the other hand, this same physics cannot be simply dismissed as some peculiarity of the weak coupling BCS limit, since we will see that the occurring phenomena have a fairly general origin. In particular, it happens in this limit that various competing superﬂuid states exist only in a small range of the phase diagram. But this does not imply that the same is necessarily true around unitarity if the same kind of physics occurs. In the same spirit, it is worth recalling that states separated by a small energy diﬀerence may display very diﬀerent physical properties. For example it is well known that the relative change of energy between a normal metal and its superconducting phase is typically of order 10−6 . Nevertheless the physical properties of these two phases are vastly diﬀerent. 2. – The Clogston-Chandrasekhar limit The natural idea which comes to mind in order to produce an imbalance between electronic populations in a superconductor is to apply a magnetic ﬁeld. Indeed it will couple to the electronic magnetic moment and, by producing a diﬀerence between the chemical potentials of the two spin populations, it will induce a diﬀerence between the two spin populations. However, it is well known that superconductivity disappears at a critical magnetic ﬁeld, but this is usually not because of the unfavorable eﬀect of electronic population imbalance. This is due to a much stronger coupling than the one we are interested in here, namely the coupling of the magnetic ﬁeld to the orbital,

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rather than the spin, electronic degrees of freedom. Basically the magnetic ﬁeld induces supercurrents (this is the Meissner eﬀect), and at some stage the kinetic energy associated with these currents becomes too large for the superconducting state to be energetically favorable, and the metal reverts accordingly to the normal state. However, it is also known that there are two ways for the superconductor to yield to the unfavorable eﬀect of the magnetic ﬁeld. Either it switches directly to the normal state, which is the case of type-I superconductors, and this leads to a ﬁrst-order transition. Or, in the case of type-II superconductors, the order parameter (OP) ﬁnds a more subtle way to adjust to the existence of the magnetic ﬁeld, by letting some regions become normal (there the OP is equal to zero) while some others remain superconducting, with a non-zero OP. This is the famous mixed state where vortices are present and allow a partial penetration of the magnetic ﬂux in the superconductor. Naturally for high enough magnetic ﬁeld the superconducting state disappears, but this occurs more progressively by a second-order transition. This phenomenon is quite remarkable since it corresponds to a spontaneous breaking of translational invariance. What happens essentially is that the OP is ﬂexible enough (and the OP space is large enough) to ﬁnd some appropriate state to adjust to the presence of the magnetic ﬁeld, rather than merely disappearing. Naturally, this phenomenon is completely analogous to the appearance of vortices in an ultracold superﬂuid gas when it is set in rotation, as has been observed now experimentally for fermionic as well as for bosonic gases. The coupling of the magnetic ﬁeld to the orbital degrees of freedom is due to the charge of the electron, and the resulting term in the Hamiltonian is much larger in standard superconductors than the coupling E = −M · B to the magnetic moment, which is usually completely negligible. However, there are geometries where one can avoid the existence of the orbital currents, responsible for the standard critical ﬁeld. Indeed, if one has a quasi-two-dimensional geometry, where the superconductor is essentially a stack of conducting planes, with very small tunnelling probability between planes, a magnetic ﬁeld parallel to these planes would produce supercurrents ﬂowing perpendicular to the planes. But the very small coupling between the planes will prohibit the existence of such supercurrents. This is as if the coupling to the orbital degrees of freedom had disappeared, and in this case the critical ﬁeld will be controlled by the above coupling of the magnetic ﬁeld to the electronic magnetic moment. This kind of situation is practically extremely interesting because the resulting critical ﬁelds will be much higher than in the standard situation, and people are naturally extremely interested for applications by superconductors able to stand very high magnetic ﬁelds. It is worth noting that the above geometry is realized in the high Tc cuprate superconductors. Hence, apart from its fundamental interest, there is also a strong practical incentive to fully explore the physics we will consider here. However, originally research in this ﬁeld had been mostly driven by the quest to understand the eﬀect of magnetic impurities in superconductors. It is clear that an imbalance n↑ = n↓ between the two spin populations, as created by a diﬀerence between spin-up chemical potential μ↑ and spin-down chemical potential μ↓ is unfavorable to the BCS state since the formation of Cooper pairs imply naturally

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n↑ = n↓ . In the following, we will consider the diﬀerence μ↑ − μ↓ ≡ 2μ∗ as our primary external constraint on the system, because it is easier to handle. It is then a matter of standard thermodynamic translation in terms of conjugate variable to obtain the eﬀect of an imbalance in the spin populations, which is the natural variable of cold gases. Qualitatively we can think of μ∗ as an eﬀective magnetic ﬁeld and its unfavorable eﬀect on pairing implies that, beyond some critical chemical potential diﬀerence μ∗c , the BCS state will be destroyed in favor of the normal state. In order to ﬁnd the dependence on temperature μ∗c (T ) of this critical “ﬁeld”, it is natural to generalize the standard BCS equation which gives the critical temperature to the case where μ∗ = 0. One ﬁnds easily (1)

1 = N0 V

0

ωc

1 − f (Ek,↑ ) − f (Ek,↓ ) , 2Ek

where f (E) = [exp[E/T ] − 1]−1 is the Fermi distribution, ωc the standard cut-oﬀ of BCS theory, N0 the single-spin density of states at the Fermi surface and V the attractive interaction, responsible for pairing. As usual the standard single-particle excitation energy Ek = ξk2 + Δ2 is related to the kinetic energy k = h ¯ 2 k 2 /2m and ξk = k − μ is just this kinetic energy measured from the averaged chemical potential μ = (μ↑ + μ↓ )/2, while the gap Δ is zero at the critical ﬁeld we are looking for. The eﬀect of μ∗ enters this equation only because the energies entering the Fermi distributions are the up-spin and down-spin single-particle excitation energies Ek,↑,↓ = ±μ∗ + Ek . It is worth pointing out that the transition corresponding to this equation is just a natural continuation of the standard second-order phase transition from the normal phase toward the BCS phase at μ∗ = 0. It corresponds to an absolute instability of the normal phase with respect to the superconducting one, and is often called a spinodal instability. There is no supercooling eﬀect at this transition, and the normal state cannot possibly escape to go to the superﬂuid one. The result is shown in ﬁg. 1. At T = 0 one ﬁnds the simple result μ∗c = Δ0 /2, where Δ0 is the gap at zero temperature when the spin populations are equal. However the result displays a rather strange re-entrant behaviour, with the critical ﬁeld at intermediate temperature being larger than the one obtained at T = 0. However it was realized quite early by Clogston and Chandrasekhar [3, 4], independently, that there is, at T = 0, a ﬁrst-order transition from the normal to the standard BCS superﬂuid state occurring at a higher ﬁeld. The location of this transition is easy to obtain. At zero temperature, for ﬁxed chemical potentials μ↑ and μ↓ , the stable phase is given by the minimum of the thermodynamic potential (2)

G = E − μ↑ n↑ − μ↓ n↓ μ↑ − μ↓ μ↑ + μ↓ (n↑ + n↓ ) − (n↑ − n↓ ) = E − μ∗ (n↑ − n↓ ), =E− 2 2

where in the last equality we have just used the deﬁnition of μ∗ and dropped for clarity the term corresponding to the total spin population since it does not play here any role. Now we have in the standard BCS state n↑ = n↓ , while its energy E is given in terms of

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Fig. 1. – Temperature of the spinodal transition, compared to the critical temperature Tc0 for equal populations, as a function of the chemical potential diﬀerence μ∗ expressed in terms of the zero temperature gap Δ0 for equal populations.

the zero-temperature gap Δ0 by the standard result E = −(1/2)N0 Δ20 compared to the normal-state energy, which leads to (3)

1 Gs = − N0 Δ20 . 2

On the other hand, we have taken by deﬁnition the energy of the normal state equal to zero for equal populations and, in contrast with the BCS state, the normal state has a non-zero susceptibity leading to n↑ − n↓ = 2N0 μ∗ . As a result we have E = N0 μ∗2 and (4)

Gn = −N0 μ∗2 .

Clearly, for μ∗ = 0, the BCS state has the lower G while for large μ∗ this will be the normal state, as expected. The transition occurs for Gn = Gs , which leads, for the critical μ∗ , to (5)

Δ0 μ∗c = √ . 2

Here the transition is ﬁrst order since the gap, which is the OP in the present case, goes discontinously from zero to √ Δ0 , when one goes from the normal to the superﬂuid state. ∗ Since the above μc = Δ0 / 2 is higher than the one, equal to Δ0 /2, that we have found from eq. (1),√ it is clear that, coming from the normal state with a high μ∗ , the transition ∗ at μc = Δ0 / 2 will occur ﬁrst. However, supercooling below this transition is possible, since it is ﬁrst order. This may occur down to Δ0 /2 where the normal state becomes absolutely unstable with respect to the BCS state.

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Fig. 2. – Temperature of the phase transition between normal and BCS state. The dashed line gives the location of the ﬁrst-order transition corresponding to the Clogston-Chandrasekhar limit. The full line is the same as in ﬁg. 1. Above the tricritical point (TCP) it gives the transition which is second order. Below the TCP, it corresponds to the position of the spinodal line where the normal state is completely unstable with respect to the transition toward the BCS state.

The resulting transition line is given now in ﬁg. 2. When the T = 0 ﬁrst-order transition we have just found is followed at non-zero temperature, one ﬁnds that this line meets the second-order transition line resulting from eq. (1) for some intermediate temperature at the so-called tricritical point (TCP). Hence, at this stage, we have reached the conclusion that the transition from normal to superﬂuid is second order above the TCP, but becomes ﬁrst order at lower temperatures. As we will see, the actual situations is much more complicated, but this TCP plays an important role in our understanding, and we will consider it again below. 3. – The Fulde-Ferrell-Larkin-Ovchinnikov phases Not much later, the independent work of Fulde and Ferrell [5], and of Larkin and Ovchinnikov [6] showed that the actual transition is more complex. Indeed they considered the possibility for the Cooper pairs to have a non-zero total momentum q. This is in contrast with the standard BCS phase where it is energetically more favourable to have all the pairs with a total momentum q = 0. They found that, at T = 0, for high eﬀective ﬁeld μ∗ , the transition toward the superﬂuid state toward an order parameter with q = 0, is energetically more advantageous, leading in this way to an extension of the superﬂuid stability domain in the presence of an eﬀective ﬁeld μ∗ .

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Fig. 3. – Fermi surfaces of up and down spins when their chemical potentials are diﬀerent.

This result is quite remarkable since it corresponds, as we will see below in more details, to a spontaneous breaking of translational invariance produced by the ﬁeld μ∗ . This is quite similar to what occurs in a type-II superconductor in the presence of a magnetic ﬁeld B, as we have seen above. The superﬂuid phases having such an order parameter with q = 0 are called FFLO or LOFF phases. However, despite many experimental eﬀorts and a good deal of claims, these phases have not yet been observed in standard superconductors, at least very clearly. Nevertheless, the existence of such an eﬀect seems quite real. This is perhaps most convincingly shown by experiments where a nearby magnetic material was responsible for the eﬀective ﬁeld μ∗ . This was felt by a superconductor, due to the proximity eﬀect. The existence of a resulting change of sign of the OP could be proved by the study of the characteristics of the associated tunnel junction [7]. Qualitatively one can understand in the following way the energetical advantage of having Cooper pairs with a total momentum q = 0. There is always a loss in kinetic energy for electrons in forming Cooper pairs, compared to the kinetic energy of a simple T = 0 Fermi sea, because one has to excite electrons away from the Fermi surface in order to build up the Cooper pair wave function. Naturally this loss is more than compensated by the gain in attractive interaction energy obtained by this formation. However, in order to minimize this kinetic energy loss, it is better to have the electrons as much as possible near the Fermi surface. When the up- and down-spin Fermi surfaces are identical, this requirement is completely compatible with the standard formation of k↑ , −k↓ pairs. However, when the up- and down-spin chemical potentials are diﬀerent, the Fermi surfaces are also diﬀerent and it is no longer possible to take each electron in the vicinity of its Fermi surface while having at the same time a zero total momentum. As shown in ﬁg. 3, taking each electron in the vicinity of its Fermi surface implies a non-zero total momentum k↑ + k↓ = q. Hence taking Cooper pairs with non-zero total momentum allows to better cope with the requirement of minimizing the kinetic energy. However, although this argument shows why pairs with q = 0 are coming in, it does not prove that this is energetically favourable globally. Indeed, since the total

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momentum is conserved by the interaction, we have to choose a speciﬁc total momentum q, independent of k. But, as shown in ﬁg. 3, if a given choice of q allows to take both electrons in the vicinity of their respective Fermi surface for some wave vectors k, this is not true for all wave vectors (since one Fermi surface is not related to the other by a simple translation q): in ﬁg. 3 this would not work at all on the opposite sides of the Fermi surfaces. Hence a quantitative calculation is necessary to obtain the ﬁnal answer. The ﬁrst step was done by Fulde and Ferrell [5], and Larkin and Ovchinnikov [6]. Looking for a second-order phase transition, they found indeed that taking q = 0 is globally favorable. Speciﬁcally they obtained that at T = 0 the transition occurs for a critical ﬁeld μ∗ = 0.754 Δ0 with a corresponding wave vector q = 2.40 μ∗ /vF , where vF is the Fermi velocity (in the weak-coupling regime, the two Fermi surfaces are only slightly diﬀerent and, to lowest order, they have the same vF ). This value of q implies that the order parameter has spatial variations with wavelength of order of the size of a Cooper pair. Nevertheless we see that the resulting critical ﬁeld is only barely above the Clogston-Chandrasekhar result. In this respect it is worth noting that this situation is much improved when one goes to lower spatial dimensions, which is relevant for the planar structure of high-Tc superconductors and could also be realized with cold gases. Although the modulus of q is ﬁxed by the energy minimization, its direction is not and we have naturally a degeneracy with respect to this direction, linked to the spontaneous breaking of rotational invariance, but also with respect to any superposition of similar plane waves. The work of Fulde and Ferrell [5] concentrated on the case where a single wave vector q is chosen. This corresponds to an order parameter Δ(r) which has a single plane-wave form Δ(r) ∼ exp[iq · r]. Larkin and Ovchinnikov [6] went further by noticing that, just below the transition, in a standard Ginzburg-Landau analysis, the fourth-order term would produce couplings between diﬀerent plane waves which would lift the degeneracy and select a speciﬁc order parameter. In order to be slightly more speciﬁc, let us ﬁrst recall the standard expression of the free energy F in terms of an expansion in powers of the order parameter Δ, assumed ﬁrst to be space independent (6)

F = a0 Δ2 + a2 Δ4 ,

where the normal state free energy is taken to be zero. The coeﬃcient a2 is always positive, while we have a0 > 0 above and a0 < 0 below the critical temperature. In a T = 0 analysis, the free energy is merely the energy, and the role of the temperature is played by the eﬀective ﬁeld μ∗ . Now, because of the degeneracy with respect to the momentum direction, the order parameter can be any superposition of plane waves (7)

Δ(r) =

Δq eiq·r ,

q

provided the modulus q is ﬁxed to the above-mentioned value. The generalization to this

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case of the above Ginzburg-Landau expansion for the free energy is (8) F = a0 |Δq |2 + J(q1 , q2 , q3 , q4 )Δq1 Δ∗q2 Δq3 Δ∗q4 , q

q1 +q3 =q2 +q4

where a0 depends only on the modulus of q by rotational invariance, but we have not written it explicitely since this modulus is ﬁxed. On the other hand J(q1 , q2 , q3 , q4 ) depends on the relative orientation of the various wave vectors and minimization will produce a selection among the possible orientations. Indeed a0 as well as J(q1 , q2 , q3 , q4 ) can be explicitely calculated within BCS theory. But we see that this problem is immediately extremely complicated, since we want to minimize the free energy with respect to all the possible Δq , which means that we have a functional space to explore. Larkin and Ovchinnikov did simplify it somewhat by the following reasonable assumption. The superposition of the various plane waves will give rise, by constructive interference, to maxima of Δ(r) (correspondingly at these points the diﬀerence in spin populations will be minimal). Larkin and Ovchinnikov assumed that these would be regularly arranged, as in a cristalline lattice. They selected the subset of wave vectors qi giving rise to such a structure. With this restriction they could perform a complete exploration and came to the conclusion that the best structure was obtained by taking just the equal superposition of two plane waves with opposite wave vectors. In other words, the best order parameter is given by (9)

Δ(r) ∼ cos(q · r),

where naturally a degeneracy with respect to global rotation is left, but no longer with respect to any superposition of plane waves. Let us stress again that this GinzburgLandau analysis assumes the transition to be second order. The transition line is easy to follow at non-zero temperature because all possible FFLO phases have the same critical temperature. When this is done one ﬁnds, as shown in ﬁg. 4, that this line follows rather closely the Clogston-Chandrasekhar line and meets it at the tricritical point, which we have already considered above. This makes it clearly worthwhile to consider more closely the vicinity of this tricritical point, which will give us a much better understanding of how the various transitions we have already discussed are related. I will sketch in the next section the results of the analytical study which has been done with C. Mora [8]. 4. – Vicinity of the tricritical point This domain is particularly interesting because we will be able to make a generalized Ginzburg-Landau analysis for the following reasons. First, since we know that, at temperatures above the TCP, the transition is just the standard second-order BCS transition, we know by continuity that the order parameter will be small anyway in the vicinity of the TCP. This is true not only naturally if the transition is second order, but also if it is ﬁrst order. This point is particularly interesting because we will be able to

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Fig. 4. – Temperature of the phase transition between normal and BCS state. The part above the TCP is the same as in ﬁg. 2, and below the TCP the dashed line is again the location of the ﬁrst-order Clogston-Chandrasekhar limit. The full line below the TCP is the FFLO second-order phase transition.

study the possibility of a ﬁrst-order transition, whereas this is in general quite diﬃcult since the order parameter is not small by deﬁnition and no expansion in its powers is accordingly possible. The second reason has to do with the spatial dependence of the order parameter. We know that, at temperatures above the TCP, we have the standard uniform BCS phase. Again by continuity we know that the spatial variation of the order parameter (which occurs in the FFLO phases) will have only a long wavelength and a small wave vector, which will allow us to perform an expansion in powers of this wave vector q. It turns out that an expansion up to sixth order in powers of the couple of variable (Δ, q) is enough. In order to clarify the fairly complex situation we want to consider, let us ﬁrst display the variation of the free energy for a homogeneous order parameter, when one has to go up to the sixth-order term in the expansion which is not the standard Ginzburg-Landau expansion we have seen above. In this case the expression of the free energy is (10)

F = a0 Δ2 + a2 Δ4 + a4 Δ6 ,

where again all the coeﬃcients can be calculated explicitely within BCS theory. In this expression a4 is always positive in the vicinity of the TCP, which is the basic reason why we can stop the expansion at order six. As above in eq. (6) the coeﬃcient a0 is positive above the second-order transition and negative below it. The new point is the behaviour of a2 . Above the TCP this coeﬃcient is positive as was the case in eq. (6). However it decreases when we come near the TCP and it changes sign when we cross the TCP.

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Fig. 5. – Free energy F (Δ) as a function of Δ given by eq. (10) in the cases where a2 is positive (left panel) or negative (right panel), for temperatures above, below or at the corresponding critical temperature Tc .

This is actually the essential property which gives rise to the TCP: at this point we have a2 = 0. In ﬁg. 5 we show F (Δ) as given by eq. (10). On the left panel we see the case where a2 > 0, for decreasing values of a0 . Qualitatively there is no diﬀerence with what one obtains from eq. (6), because the a4 term just reinforces the eﬀect of the a2 term, and in particular the transition occurs for a0 = 0 and is second order. On the other hand, we see on the right panel that the situation is qualitatively modiﬁed when a2 < 0. Indeed this introduces a downward bending of the graph and as a result, when a0 is lowered, a minimum develops for a non-zero value of Δ. When the value of F (Δ) reaches zero at this minimum, due to the lowering of a0 , there is a transition toward the superﬂuid phase, which is a ﬁrst-order transition since the value of Δ will be non-zero. Accordingly, we see that, because of the change of sign of a2 , the standard second-order phase transition is superseded by a ﬁrst-order transition. At this stage it is worth making a side remark on the curve for T = Tc on the right panel. It has two minima, one for Δ = 0 which corresponds to the normal state, and another one at non-zero Δ which corresponds to the superﬂuid state. Naturally we ﬁnd in between these two minima a maximum of the free energy. Now the gap equation of BCS theory can be obtained quite generally by writing that the free energy is extremal with respect to the order parameter Δ. This means that, in the present situation, this gap equation will have three solutions corresponding to the two minima and to the maximum. In the course of his study of the generalization at non zero temperature of the ClogstonChandrasekhar limit, Sarma [9] found indeed three solutions (we will see below that the Clogston-Chandrasekhar limit ﬁts indeed in our framework). He studied in particular the one corresponding to the maximum, which is often referred to as the Sarma phase in the recent literature. However this phase does not appear in the ﬁnal thermodynamic results of Sarma’s paper. This is expected, since it is not a physically acceptable solution because it corresponds to a maximum, and not a minimum of the free energy. This

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is naturally the situation in the weak-coupling BCS limit, and the situation might be diﬀerent in cold gases away from weak coupling. However, it is clearly necessary to check in these cases that the Sarma phase found in the course of calculations corresponds to a minimum, and not a maximum, of the free energy. Let us come back now to our problem and take into account that, having in mind FFLO phases, we want to consider space-dependent order parameters which we can always Fourier expand as in eq. (7). We restrict ourselves to the interesting domain just below the TCP. Since the resulting expression for the free energy is fairly complicated, we write it with reduced units (but we do not change the notations for simplicity). Without entering in the details, we set a0 ∼ (a22 /a4 )A0 , and q 2 as well as Δ2 are also expressed in units proportional to a2 /a4 . The reduced coeﬃcient A0 (μ∗ , T ), obtained from BCS theory, is an increasing function of the eﬀective ﬁeld μ∗ and temperature T in the domain we are interested in. This reduced free energy reads (the normal-state free energy being taken as zero) (11)

1 2 1 4 F = |Δq | A0 − q + q − 3 5 q

1 1 2 ∗ ∗ 2 2 2 − Δq1 Δq2 Δq3 Δq4 1 − q1 + q2 + q3 + q4 + q1 · q3 + q2 · q4 + 4 3 1 Δq1 Δ∗q2 Δq3 Δ∗q4 Δq5 Δ∗q6 . + 8

2

In this expression all the transitions we have discussed previously are present. Indeed the standard BCS spinodal transition corresponds to a uniform order parameter, so we have q = 0, and is a second-order transition, that is Δ = 0. Accordingly, its location in the (μ∗ , T )-plane is given by A0 = 0. However we know that a ﬁrst-order transition to a uniform order parameter with Δ = 0, as found by Clogston-Chandrasekhar, is more favorable. And indeed we ﬁnd that, for q = 0, when A0 = 1/8, the minimum found at Δ = 1 has already a free energy F = 0 (this corresponds to the minimum of F (Δ) at Tc on the right panel of ﬁg. 5). The fact that the corresponding value 1/8 of A0 is larger than zero implies that this transition occurs (for ﬁxed μ∗ ) at higher temperature than the standard BCS transition, so that it supersedes it. Now we can consider what occurs when we take into account the possibility of an order parameter with q = 0. We look ﬁrst for a second-order transition, that is Δ = 0. We see immediately from the ﬁrst term in eq. (11), of order Δ2 , that it is indeed better to take q 2 = 5/6 which gives the minimum of the q-dependent coeﬃcient. This leads to A0 = 5/36, which is even higher than the Clogston-Chandrasekhar value A0 = 1/8. Therefore this FFLO transition overtakes the Clogston-Chandrasekhar one. It is interesting to note, at this stage, that there is no barrier in free energy between q = 0 and q 2 = 5/6, as it is obvious from the ﬁrst term of eq. (11). This implies that the transition to a q = 0 order parameter has a spinodal character. It is impossible to escape it, because it corresponds to an absolute instability. In this weak-coupling limit it is not possible to believe that the FFLO phases will be “missed” by the system. This is directly linked to

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the fact that the coeﬃcient of the q 2 term is negative. If we do not take reduced units, this coeﬃcient is actually found proportional to a2 . Above the TCP it will be positive and we are back to the standard BCS transition to a uniform order parameter. On the other hand, below the TCP it is negative and we ﬁnd FFLO phases. In other words a characteristic property of the TCP is also the fact that this coeﬃcient of the q 2 term is equal to zero at the TCP. Finally, when we put together all the results found above, we may naturally wonder if it is not possible to ﬁnd an even better solution by looking for a ﬁrst-order transition Δ = 0 at q = 0. This is indeed what we have found [8]. For q 2 = 0.68 and Δ = 0.27, one ﬁnds a minimum equal to zero in the free energy if one takes A0 = 5/36 + 2.10−3 , which is slightly higher than the FFLO value. Hence this ﬁrst-order transition supersedes the standard FFLO second-order phase transition. However, in contrast with this standard FFLO transition, this result is not obtained for any superposition of plane waves. We have in eq. (11) to take advantage of the speciﬁc dependence of the fourth-order term on the directions of the various qi , in much the same way as in the Larkin-Ovchinnikov analysis. We ﬁnd that the above result is obtained for an order parameter which has the same spatial dependence, namely Δ(r) ∼ cos(q · r), as the one obtained by Larkin and Ovchinnikov. The fact that the result for A0 is barely higher than the standard FFLO one is somewhat puzzling (which implies that the transition lines are very close; in ﬁg. 4 this ﬁrst-order transition line is just above the FFLO one). However, we note that the value of Δ itself is not particularly small, compared to the Clogston-Chandrasekhar value. There is still a strange feature in our result. We have found that the transition toward Δ(r) ∼ cos(q · r) is ﬁrst order in the vicinity of the TCP, while Larkin and Ovchinnikov have a second-order phase transition at T = 0. This implies that, as the temperature is lowered, some change must occur. In this way we have been led to investigate the problem away from the TCP [10]. This is a much more diﬃcult problem since no expansion in powers of Δ is any longer possible, that is one needs to go to all orders in Δ to ﬁnd the actual solution. We have used a method which mixes analytical and numerical ingredients. The ﬁnal result is the following. When the temperature is lowered, the transition stays ﬁrst order, that is the qualitative situation is analogous to what we have found in the vicinity of the TCP. However, the order parameter changes. At some temperature the best solution switches to a superposition of two cosines along orthogonal directions Δ(r) ∼ i=x,y cos(qi · r) and at even lower temperatures it goes to a superposition of three cosines along orthogonal directions Δ(r) ∼ i=x,y,z cos(qi · r). This last form is the actual solution at T = 0. It overtakes the solution found by Larkin and Ovchinnikov. Indeed this solution was beyond their scope since they had restricted their investigation to a second-order phase transition. 5. – Collective mode in the BEC-BCS crossover Let us switch now to our second topic. Since this problem has been in some respect investigated in the literature and that the details of our work, done in collaboration with M.Yu. Kagan and S. Stringari, will be published quite soon [11], I will only stress here

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the most interesting and new features. We will consider here only the collective mode in a homogeneous superﬂuid. For trapped gas, the same physics would correspond to the well-known collective modes which have been very much investigated experimentally and theoretically. In the low-frequency limit this mode reduces to sound propagation with a linear dispersion relation, but we are mostly interested in what happens at higher frequency in the non-linear regime. In particular we will consider the merging with the pair-breaking continuum, which in the BEC limit reduces to molecular dissociation continuum. Our considerations will be restricted to T = 0. In this BEC limit this mode reduces to the well-known Bogoliubov mode with dispersion relation ω 2 = c2s q 2 + (q 2 /2Mm )2 . The BCS limit of this mode is less standard, since in superconductors it is pushed up to plasma frequency, because electrons are charged particles, and it becomes physically irrelevant. In superﬂuid 3 He, which is a neutral BCS superﬂuid, the strong hard-core repulsion produces a very weak compressibility and a very high sound velocity, which dominates the physics of this mode. So it seems that ultracold fermionic gases are the ﬁrst systems where the pure Bogoliubov-Anderson (as it is called in this limit) can be observed. For lack of a much better theory, this collective mode will be investigated within dynamical self-consistent BCS theory. While it is known that this theory gives a proper description both in the BCS and in the BEC limit, it must be considered in between as an interpolation model. However, BCS theory is a very coherent theoretical framework, and one may expect that it gives qualitatively correct results. Moreover when we ﬁnd a general physical reason supporting our ﬁndings, we will gain conﬁdence in their validity. Let us ﬁrst consider, in the (q, ω)-plane, the domain corresponding to the pairbreaking continuum. The threshold ωth for pair-breaking is given by (12)

ωth = 2Δ for μ > 0 and q ≤ 2 2mμ , ωth = 2 (q 2 /8m − μ)2 + Δ2 otherwise

and is shown in ﬁg. 6 for various values of 1/kF a. In the BCS limit one recovers the √ well-known threshold ωth = 2Δ when the wave vector is less than 2 2mμ. In the BEC limit, one has Δ/μ → 0− which gives ωth = 2|μ| + q 2 /4m. This is just the energy necessary to break a molecule with binding energy b = 2|μ| = 1/ma2 into two fermions with total momentum q, leading to an additional kinetic energy (1/2)q 2 /(2m), just as for a molecule of mass 2m. We display ﬁrst in ﬁg. 7 the result for the collective mode dispersion relation on the BCS side for two values of 1/kF a. It shows an interesting anticrossing behaviour. This is as if we had a coupled two-level system. One of the states is just the collective mode. The other one is the pair breaking continuum. Naturally one cannot consider this one as a single state, but the singularity in the BCS density of states at ω = 2Δ means that there is an accumulation of states near this energy, so the physical situation is not so diﬀerent. Looking at the ﬁgure, it seems that the collective mode merges at some point with the threshold 2Δ. Actually this is not correct as shown by detailed analysis. The

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711

Fig. 6. – Threshold ωth as a function of wave vector q for, from bottom to top, 1/kF a = −1, −0.5, 0 (unitarity), 0.553 (μ = 0), 1 and 2.

Fig. 7. – Dispersion relation ω/EF of the collective mode as a function of q/kF for 1/kF a = −1 (lower thick line) and −0.5 (upper thick line). The location of the threshold for pair breaking is given in each case by the thinner line.

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Fig. 8. – Dispersion relation ω/EF of the collective mode as a function of q/kF at unitarity (thick line). The location of the threshold for pair breaking is given by the thinner line. The collective mode merges into the continuum for q/kF = 1.76, as can be seen from ﬁg. 9.

collective mode approaches the threshold exponentially and merges with the continuum only at higher wave vector, as indicated by the dots near the right side of the ﬁgure. We consider next the dispersion relation at unitarity, which is found in ﬁg. 8. Qualitatively the result is rather similar to the one found on the BCS side. However, the interesting point here is the very wide range of quasi-linear behaviour for this dispersion relation. This may explain why hydrodynamics works so well in traps around unitarity at the fairly high frequencies produced in radial oscillations, since T = 0 hydrodynamics is valid as long as the dispersion relation is linear. We turn now to the speciﬁc location of the wave vector qm where the collective mode merges with the pair-breaking continuum. A natural expectation would be that qm → ∞ as one approaches unitarity. This would allow to recover on the BEC side a collective mode which never meets the continuum, as is the case for the Bogoliubov mode. The fact that things are more complicated is already plain from ﬁg. 8 where one sees that, at unitarity, merging occurs still at a ﬁnite value of qm . The whole picture is given in ﬁg. 9. Surprisingly a higher-frequency branch of the collective mode appears at unitarity, which merges with the continuum at lower frequency, as is indicated schematically in the insert of ﬁg. 9. Hence on the BEC side, that is for a > 0, there are two merging points qm . The evolution of their position with 1/kF a is indicated in the main ﬁgure. We see that, when 1/kF a increases, they come closer until they become equal. At this stage the collective mode dispersion relation only touches the continuum. For larger

Introduction to FFLO phases and collective mode in the BEC-BCS crossover

713

Fig. 9. – Wave vector qm , in units of kF , at which the collective mode dispersion relation meets the excitation continuum, as a function of 1/kF a (note that the qm scale is logarithmic). The √ dashed line corresponds to q = 2 2mμ. It goes to q/kF = 2 when 1/a → −∞. The insert shows schematically the two branches of the collective mode dispersion relation on the a > 0 side of unitarity.

Fig. 10. – Dispersion relation ω/EF of the collective mode as a function of q/kF for 1/kF a = 1 (thick line). The threshold for pair breaking is given by the thinner line. The dashed line indicates the result obtained from the Bogoliubov formula.

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1/kF a one recovers a single dispersion relation located below the continuum, that is the same qualitative situation as the standard Bogoliubov mode. The interpretation of this strange behaviour is the following. Physically the Bogoliubov mode at high frequency corresponds to kick a molecule out of the condensate. However, it has been pointed out [12] that, while molecular formation goes unhindered at high momentum, it becomes restricted by the eﬀect on the molecular state of Pauli exclusion by the Fermi sea made of the other fermions. Hence, near unitarity, molecules cannot form at lower momentum and the collective mode cannot exist. We note that, since these arguments do not rely speciﬁcally on BCS theory, we expect this behaviour to be qualitatively present in the exact theory. Finally in ﬁg. 10 we compare in the BEC regime our result for the collective mode dispersion relation with the standard Bogoliubov result. We see that, while they agree perfectly well at low frequency (which is quite natural, since the sound velocity is the same in the two results), at higher frequency the Bogoliubov result is slightly higher than the collective mode we ﬁnd. This can be understood qualitatively because, at higher momentum q ∼ 1/a, the collective mode will feel the internal structure of the molecules whose size is of order a. Hence the fact that our molecules are composite bosons will appear. Since Bogoliubov theory deals only with elementary bosons, it is quite natural that it does not lead to exactly the same result in this high-q regime. Again, since this physical argument does not make use speciﬁcally of BCS theory, we expect this result to hold also in the exact theory of this regime. ∗ ∗ ∗ We are grateful to C. Mora for providing some of the ﬁgures used in this paper. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Casalbuoni R. and Nardulli G., Rev. Mod. Phys., 76 (2004) 263. Combescot R., Europhys. Lett., 55 (2001) 150. Clogston A. M., Phys. Rev. Lett., 9 (1962) 266. Chandrasekhar B. S., Appl. Phys. Lett., 1 (1962) 7. Fulde P. and Ferrell R. A., Phys. Rev., 135 (1964) A550. Larkin A. I. and Ovchinnikov Y. N., Zh. Exp. Teor. Fiz., 47 (1964) 1136 (Sov. Phys. JETP, 20 (1965) 762). Kontos T., Aprili M., Lesueur J. and Grison X., Phys. Rev. Lett., 86 (2001) 304. Combescot R. and Mora C., Eur. Phys. J. B, 28 (2002) 397. Sarma G., J. Phys. Chem. Solids, 24 (1963) 1029. Combescot R. and Mora C., Phys. Rev. B, 71 (2005) 144517; Mora C. and Combescot R., Phys. Rev. B, 71 (2005) 214504. Combescot R., Kagan M. Yu. and Stringari S., Phys. Rev. A, 74 (2006) 042717. Combescot R., Leyronas X. and Kagan M. Yu., Phys. Rev. A, 73 (2006) 023618.

Strongly correlated quantum phases of ultracold atoms in optical lattices I. Bloch Institut f¨ ur Physik, Johannes Gutenberg-Universit¨ at - 55099 Mainz, Germany

We review several experimental aspects of ultracold bosonic and fermionic quantum gases in optical lattices. After introducing fundamental aspects of optical lattices, we use the superﬂuid-Mott insulator transition of ultracold bosonic quantum gases in optical lattices, to highlight the physics of strongly correlated quantum systems. We discuss the coherence properties and recent measurements of the shell structure of in the Mott insulating phase. Furthermore, we show how quantum noise correlations can be used to detect quantum phases of strongly correlated bosonic and fermionic quantum gases in optical lattices. 1. – Introduction Ultracold quantum gases in optical lattices form almost ideal conditions to analyze the physics of strongly correlated quantum phases in periodic potentials. Such strongly correlated quantum phases are of fundamental interest in condensed-matter physics, as they lie at the heart of topical quantum materials, such as high-Tc superconductors and quantum magnets, which pose a challenge to our basic understanding of interacting many-body systems. Quite generally, such strongly interacting quantum phases arise, when the interaction energy between two particles dominates over the kinetic energy of the two particles. Such a regime can either be achieved by increasing the interaction strength between the atoms via Feshbach resonances (see, e.g., articles of W. Ketterle and T. Esslinger in this volume), or by decreasing the kinetic energy, such that eventually the interaction energy is the largest energy scale in the system. The latter can for example simply be achieved by increasing the optical lattice depth. c Societ` a Italiana di Fisica

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This article tries to give an introduction into the ﬁeld of optical lattices and the physics of strongly interacting quantum phases. A prominent example hereof is the superﬂuid to Mott insulator transition [1-5], which transforms a weakly interacting quantum gas into a strongly correlated many-body system. Dominating interactions between the particles are in fact crucial for the Mott insulator transition and also for the realization of controlled interaction-based quantum gates [6-8] of which several have been successfully realized experimentally [9-11], which will also be discussed throughout the text. The article is structured as follows: in sect. 2 the basics of trapping neutral atoms in optical lattice potentials is discussed; in sect. 3 we turn to a discussion of the superﬂuid to Mott insulator transition; sect. 4 discusses the inﬂuence of the interactions between the particles on the stability of the coherent matter wave ﬁeld of a BEC, leading to a pronounced series of collapses and revivals of its macroscopic wave function; in sect. 5 quantum gate arrays based on controlled collisions between neutral atoms are discussed and sect. 6 explains how spin changing collisions can be used to generate robust entangled states; ﬁnally sect. 7 highlights how Hanbury-Brown and Twiss type noise correlations in expanding atom clouds can be used to read out the ﬁnal quantum states of an opticallattice–based quantum simulator. 2. – Optical lattices . 2 1. Optical dipole force. – In the interaction of atoms with coherent light ﬁelds, two fundamental forces arise [12, 13]. The so-called Doppler force is dissipative in nature an can be used to eﬃciently laser cool a gas of atoms and relies on the radiation pressure together with spontaneous emission. The so-called dipole force on the other hand creates a purely conservative potential in which the atoms can move. No cooling can be realized with this dipole force, however if the atoms are cold enough initially, they may be trapped in such a purely optical potential [14, 15]. How does this dipole force arise? We may grasp the essential points through a simple classical model, in which we view the electron as harmonically bound to the nucleus with oscillation frequency ω0 . An external oscillating electric ﬁeld of a laser E with frequency ωL can now induce an oscillation of the electron resulting in an oscillating dipole moment d of the atom. Such an oscillating dipole moment will be in phase with the driving oscillating electric ﬁeld, for frequencies much lower than an atomic resonance frequency and 180◦ out of phase, for frequencies much larger than the atomic resonance frequency. The induced dipole moment again interacts with the external oscillating electric ﬁeld, resulting in a dipole potential Vdip experienced by the atom [15-19]: (1a)

1 Vdip = − dE, 2

where · denotes a time average over fast oscillating terms at optical frequencies. From eq. (1) it becomes immediately clear that for a red detuning (ωL < ω0 ), where d is in phase with E, the potential is attractive, whereas for a blue detuning (ωL > ω0 ), where d is in 180◦ out of phase with E, the potential is repulsive. By relating the dipole moment

Strongly correlated quantum phases of ultracold atoms in optical lattices (a)

717

(b)

(c)

Fig. 1. – (a) Gaussian laser beam together with corresponding trapping potential for a red detuned laser beam. (b) A red detuned laser beams leads to an attractive dipole potential, whereas a blue detuned laser beam leads to a repulsive potential (c).

to the polarizability α(ωL ) of an atom and expressing the electric ﬁeld amplitude E0 via the intensity of the laser ﬁeld I, one obtains for the dipole potential: (2a)

Vdip (r) = −

1 Re(α)I(r). 2 0 c

A spatially dependent intensity proﬁle I(r) can therefore create a trapping potential for neutral atoms. For a two level atom a more useful form of the dipole potential may be derived within the rotating-wave approximation, which is a reasonable approximation provided that the detuning Δ = ωL − ω0 of the laser ﬁeld ωL from an atomic transition frequency ω0 is small compared to the transitions frequency itself Δ ω0 . Here one obtains [15] (3)

Vdip (r) =

3πc2 Γ I(r), 2ω03 Δ

with Γ being the decay rate of the excited state. Here a red detuned laser beam (ωL < ω0 ) leads to an attractive dipole potential and a blue detuned laser beam (ωL > ω0 ) leads to a repulsive dipole potential. By simply focussing a Gaussian laser beam, this can be used to attract or repel atoms from an intensity maximum in space (see ﬁg. 1). For such a focussed Gaussian laser beam the intensity proﬁle I(r, z) is given by (4)

I(r, z) =

2 2 2P e−2r /w (z) , 2 πw (z)

2 ) is the 1/e2 radius depending on the z-coordinate, zR = where w(z) = w0 (1 + z 2 /zR πw2 /λ is the Rayleigh length and P is the total power of the laser beam [20]. Around

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the intensity maximum a potential depth minimum occurs for a red detuned laser beam, leading to an approximately harmonic potential of the form (5)

Vdip (r, z) ≈ −V0

1−2

r w0

2 −

z zR

2 ! .

This harmonic conﬁnement is characterized by radial ωr and axial ωax trapping fre2 quencies ωr = (4V0 /mw02 )1/2 and ωz = (2V0 /mzR ). Great care has to be taken to minimize spontaneous scattering events, as they lead to heating and decoherence of the trapped ultracold atom samples. For a two-level atom, the scattering rate Γsc (r) can be estimated [15] through 3πc2 Γsc (r) = 2¯hω03

(6)

Γ Δ

2 I(r).

From eqs. (3), (6) it can be seen that the ratio of scattering rate to optical potential depth can always be minimized by increasing the detuning of the laser ﬁeld. In practice however, such an approach is limited by the maximum available laser power. For experiments with ultracold quantum gases of alkali atoms, the detuning is typically chosen to be large compared to the excited state hyperﬁne structure splitting and in most cases even large compared to the ﬁne structure splitting in order to suﬃciently suppress spontaneous scattering events. Typical detunings range from several tens of nm to optical trapping in CO2 laser ﬁelds. A laser trap formed by a CO2 laser ﬁelds can be considered as a quasi-electrostatic trap, where the detuning is much larger than the optical resonance frequency of an atom [21-23]. One ﬁnal comment should be made about state-dependent optical potentials. For a typical multi-level alkali atom, the dipole potential will both depend on the internal magnetic substate mF of a hyperﬁne ground state with angular momentum F , as well as on the polarization of the light ﬁeld P = +1, −1, 0 (circular σ ± and linear polarization). One can then express the lattice potential depth through [24, 15]:

(7)

Vdip (r) =

πc2 Γ 2ω03

2 + P gF m F 1 − P gF m F + Δ2,F Δ1,F

I(r).

Here gF is the Land´e factor and Δ2,F , Δ1,F refer to the detuning relative to the transition between the ground state with hyperﬁne angular momentum F and the center of the excited state hyperﬁne manifold on the D2 and D1 transition, respectively. For large detunings relative to the ﬁne structure splitting ΔF S , the optical potentials become almost spin independent again. For detunings of the laser frequency in between the ﬁne structure splitting, very special spin-dependent optical potentials can be created that will be discussed below.

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Fig. 2. – One-dimensional optical-lattice potential. By interfering two counterpropagating Gaussian laser beams, a periodic intensity proﬁle is created due to the interference of the two laser ﬁelds.

. 2 2. Optical lattice potentials. – A periodic potential can simply be formed by overlapping two counterpropagating laser beams. Due to the interference between the two laser beams an optical standing wave with period λ/2 is formed, in which the atoms can be trapped. By interfering more laser beams, one can obtain one-, two- and threedimensional periodic potentials [25], which in their simplest form will be discussed below. Note that by choosing to let two laser beams interfere under an angle less than 180◦ , one can also realize periodic potentials with a larger period. . 2 2.1. 1D lattice potentials. The simplest possible periodic optical potentials is formed by overlapping two counterpropagating focussed Gaussian laser beams (see ﬁg. 2) which results in a trapping potential of the form (8)

V (r, z) = −Vlat · e−2r

2

/w2 (z)

r2 · sin2 (kx) ≈ −Vlat · 1 − 2 2 · sin2 (kz) , w (z)

where w0 denotes the beam waist, k = 2π/λ is the wave vector of the laser light and Vlat is the maximum depth of the lattice potential. Note that due to the interference of the two laser beams Vlat is four times larger than V0 if the laser power and beam parameters of the two interfering lasers are equal. . 2 2.2. 2D lattice potentials. Periodic potentials in two dimensions can be formed by overlapping two optical standing waves along diﬀerent directions. In the simplest form one chooses two orthogonal directions and obtains at the center of the trap an optical potential of the form (neglecting the harmonic conﬁnement due to the Gaussian beam proﬁle of the laser beams): (9)

V (y, z) = −Vlat cos2 (kx) + cos2 (ky) + 2e1 · e2 cos φ cos(kx) cos(ky) .

Here e1,2 denote the polarization vectors of the laser ﬁelds, each forming one standing wave and φ is the time between them. If the polarization vectors are chosen not to be orthogonal to each other, then the resulting potential will not only be the sum of the potentials created by each standing wave, but will be modiﬁed according to the time phase φ used (see ﬁg. 3). In such a case it is absolutely essential to stabilize the time phase

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(a)

(b)

Fig. 3. – 2D Optical lattice potentials for a lattice with (a) orthogonal polarization vectors and (b) parallel polarization vectors and a time phase of φ = 0.

between the two standing waves [26], as small vibrations will usually lead to ﬂuctuations of the time phase, resulting in severe heating and decoherence eﬀects of the ultracold atom samples. In such a two-dimensional optical lattice potential, the atoms are conﬁned to arrays of tightly conﬁning one-dimensional tubes (see ﬁg. 4a). For typical experimental parameters the harmonic trapping frequencies along the tube are very weak and on the order of 10– 200 Hz, while in the radial direction the trapping frequencies can become as high as up to 100 kHz, thus allowing the atoms to eﬀectively move only along the tube for deep lattice depths [27-31]. . 2 2.3. 3D lattice potentials. For the creation of a three-dimensional lattice potential, three orthogonal optical standing waves have to be overlapped. Here we only consider the case of independent standing waves, with no cross interference between laser beams of diﬀerent standing waves. This can for example be realized by choosing orthogonal polarization vectors between diﬀerent standing wave light ﬁelds and also by using diﬀerent wavelengths for the three standing waves. In this case the resulting optical potential is simply given by the sum of three standing waves: (10)

2

2 +z 2 )/wx

2

+y 2 )/wz2

V (r) = −Vx e−2(y −Vz e−2(x

2

sin2 (kx) − Vy e−2(x

+z 2 )/wy2

sin2 (ky)

sin2 (kz).

Here Vx,y,z are the potential depths of the individual standing waves along the diﬀerent directions. In the center of the trap, for distances much smaller than the beam waist, the trapping potential can be approximated as the sum of a homogeneous periodic lattice potential and an additional external harmonic conﬁnement due to the Gaussian laser beam proﬁles: (11)

V (r) ≈ Vx · sin2 (kx) + Vy · sin2 (ky) + Vz · sin2 (kz) +

m 2 2 ωx x + ωy2 y 2 + ωz2 z 2 , 2

where ωx,y,z are the eﬀective trapping frequencies of he external harmonic conﬁnement.

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(a)

(b)

Fig. 4. – Two Dimensional (a) and three-dimensional (b) optical-lattice potentials formed by superimposing two or three orthogonal standing waves. For a two-dimensional optical lattice, the atoms are conﬁned to an array of tightly conﬁning one-dimensional potential tubes, whereas in the three-dimensional case the optical lattice an be approximated by a three-dimensional simple cubic array of tightly conﬁning harmonic-oscillator potentials at each lattice site.

They can again be approximated by (12)

ωx2 =

4 m

Vy Vz + 2 2 wy wz

;

2 = (cycl. perm.). ωy,z

In addition to this harmonic conﬁnement due to the gaussian laser beam proﬁles, a conﬁnement due to a magnetic trapping typically exists, which has to be taken into account as well for the total harmonic conﬁnement of the atom cloud. For suﬃciently deep optical-lattice potentials, the conﬁnement on a single lattice site is also approximately harmonic. Here the atoms are very tightly conﬁned with typical trapping frequencies ωlat of up to 100 kHz. One can estimate the trapping frequencies at a single lattice site through a Taylor expansion of the sinusoidally varying lattice potential at a lattice site and obtains # (13)

ωlat ≈

Vlat ¯h2 k 4 . Er m 2

722

I. Bloch (a)

mirror

(b)

'x

angle T lin atom cloud

lin

O/4

EOM

Fig. 5. – Schematic experimental setup. A one-dimensional optical standing wave laser ﬁeld is formed by two counterpropagating laser beams with linear polarizations. The polarization angle of the returning laser beam can be adjusted through an electro-optical modulator. The dashed lines indicate the principal axes of the wave plate and the EOM.

Here Er = h ¯ 2 k 2 /2m is the so-called recoil energy, which is a natural measure of energy scales in optical-lattice potentials. . 2 3. Spin-dependent optical-lattice potentials. – In order to realize a spin-dependent lattice potential, a standing-wave conﬁguration formed by two counterpropagating laser beams with linear polarization vectors enclosing an angle θ has been proposed [24, 7, 6]. Such a standing-wave light ﬁeld can be decomposed into a superposition of a σ + and σ − polarized standing-wave laser ﬁeld, giving rise to lattice potentials V+ (x, θ) = V0 cos2 (kx + θ/2) and V− (x, θ) = V0 cos2 (kx − θ/2). Here k is the wave vector of the laser light used for the standing wave and V0 is the potential depth of the lattice. By changing the polarization angle θ one can thereby control the separation between the two potentials Δx = θ/180◦ · λx /2. When increasing θ, both potentials shift in opposite directions and overlap again when θ = n · 180◦ , with n being an integer. For a spin-dependent transport, two internal spin states of the atom should be used, where one spin state dominantly experiences the V+ (x, θ) potential and the other spin state mainly experiences the V− (x, θ) dipole force potential. Such a situation can be realized in rubidium by tuning the wavelength of the optical lattice laser to a value of λx ≈ 785 nm between the ﬁne-structure splitting of the rubidium D1 and D2 transition. Then the dipole potential experienced by an atom in, e.g., the |1 ≡ |F = 2, mF = −2 state is given by V1 (x, θ) = V− (x, θ) and that for an atom in the |0 ≡ |F = 1, mF = −1 state is given by V0 (x, θ) = 3/4V+ (x, θ) + 1/4V− (x, θ). √If an atom is now ﬁrst placed in a coherent superposition of both internal states 1/ 2(|0 + i|1) and the polarization angle θ is continuously increased, the spatial wave packet of the atom is split with both components moving in opposite directions. This can be used to coherently move atoms . across lattices and realize quantum gates between them (see subsect. 2 3). The polarization angle θ —and therefore the relative position of the two standing-wave . potentials (see subsect. 2 3)— can be dynamically controlled through the use of a quarter wave plate and an electro-optical modulator (EOM) that enable one to dynamically rotate the polarization vector of the retro-reﬂected laser beam through an angle θ by applying an appropriate voltage to the EOM (see ﬁg. 5) [32].

723

Strongly correlated quantum phases of ultracold atoms in optical lattices

(a)

(b) 10-1

0.7

0.5

J (Er)

U (Er)

0.6

0.4 0.3

10-2

0.2 10-3

0.1 0

5

10

15

20

25

30

0

5

10

15

20

25

30

V (Er)

V (Er)

Fig. 6. – Onsite interaction matrix element U for 87 Rb (a) and tunnel matrix element J (b) vs. lattice depth. All values are given in units of the recoil energy Er .

3. – Bose-Hubbard model of interacting bosons in optical lattices The behavior of bosonic atoms with repulsive interactions in a periodic potential is fully captured by the Bose-Hubbard Hamiltonian of solid-state physics [1, 2], which in the homogeneous case can be expressed through (14)

H = −J

i,j

1 a ˆ†i a ˆj + U n ˆ i (ˆ ni − 1). 2 i

Here a ˆ†i and a ˆi describe the creation and annihilation operators for a boson on the i-th lattice site and n ˆ i counts the number of bosons on the i-th lattice site. The tunnel coupling between neighboring potential wells is characterized by the tunnel matrix element (15) J = − d3 xw(x − xi )(−¯h2 ∇2 /2m + Vlat (x))w(x − xj ), where w(x − xi ) is a single particle Wannier function localized to the i-th lattice site and Vlat (x) indicates the optical lattice potential. The repulsion between two atoms on a single lattice site is quantiﬁed by the on-site matrix element U (16) U = (4π¯h2 a/m) |w(x)|4 d3 x, with a being the scattering length of an atom. Due to the short range of the interactions compared to the lattice spacing, the interaction energy is well described by the second term of eq. (14) which characterizes a purely on-site interaction. Both the tunneling matrix element J and the onsite interaction matrix element can be calculated from a band structure calculation (see ﬁg. 6). The tunnel matrix element J is related to the width of the lowest Bloch band through (17)

4J = |E0 (q = π/a) − E0 (q = 0)|,

724

I. Bloch

1000

U/J

100

10

1 0

5

10

15

20

25

30

V (Er) Fig. 7. – U/J vs. optical-lattice potential depth for 87 Rb. By increasing the lattice depth one can tune the ratio U/J, which determines whether the system is strongly or weakly interacting.

where a is the lattice period, such that q = π/a corresponds to the quasimomentum at the border of the ﬁrst Brillouin zone. Care has to be taken to evaluate the tunnel matrix element through eq. (15), when the wannier function is approximated by the Gaussian ground-state wave function on a single lattice site. This usually results in a severe underestimation of the tunnnel coupling between lattice sites. The interaction matrix element can be evaluated through the Wannier function with the help of eq. (16). In this case, however, the approximation of the wannier function through the Gaussian ground-state wave function yields a very good approximation. Recently [33] has carried out a more sophisticated approximation of the tunnel matrix element and the onsite interaction matrix element, he ﬁnds (in units of the recoil energy) (18)

J≈

8 π

Vlat Er

3/4

√ e−2 Vlat /Er

and for the interaction matrix element (19)

√ a Vlat U ≈ 4 2π . λ Er

The ratio U/J is crucial for determining, whether one is in a strongly interacting or a weakly interacting regime. It can tuned continuously by simply changing the lattice potential depth (see ﬁg. 7). . 3 1. Ground states of the Bose-Hubbard Hamiltonian. – The Bose-Hubbard Hamiltonian of eq. (14) has two distinct ground states depending on the strength of the interactions U relative to the tunnel-coupling J. In order to gain insight into the two

Strongly correlated quantum phases of ultracold atoms in optical lattices

725

Fig. 8. – Ground state of two interacting particles in a double well. For interaction energies U smaller than the tunnel coupling J the ground state of the two-body system is realized by the “superﬂuid” state a. If on the other hand U is much larger than J, then the ground state of the two-body system is the Mott insulating state b.

limiting ground-states, let us ﬁrst consider the case of a double-well system with only two interacting neutral atoms. . 3 2. Double-well case. – In the double-well system √ the two lowest-lying states for noninteracting√particles are the symmetric |ϕS = 1/ 2(|ϕL +|ϕR ) and the anti-symmetric |ϕA = 1/ 2(|ϕL − |ϕR ) states, where |ϕL and |ϕR are the ground states of the leftand right-hand side of the double-well potential. The energy diﬀerence between |ϕS and |ϕA will be named 2J, which characterizes the tunnel coupling between the two wells and depends strongly on the barrier height between the two potentials. In case of no interactions, the ground state of the two-body system is realized when each atom is in the symmetric ground state of the double-well system (see ﬁg. 8a). Such a situation yields an average occupation of one atom per site with the single-site manybody state actually being in a superposition of zero, one and two atoms. Let us now consider the eﬀects due to a repulsive interaction between the atoms. If both atoms are again in the symmetric ground state of the double well, the total energy of such a state will increase due to the repulsive interactions between the atoms. This higher energy cost is a direct consequence of having contributions where both atoms occupy the same site of the double well. This leads to an interaction energy of 1/2U for this state. If this energy cost is much greater than the splitting 2J between the symmetric and anti-symmetric ground states of the noninteracting system, the system can minimize its energy when each atom is in√a superposition of the symmetric and antisymmetric ground state of the double well √ 1/ 2(|ϕS ± |ϕA ). The resulting many-body state can then be written as |Ψ = 1/ 2(|ϕL ⊗ |ϕR + |ϕR ⊗ |ϕL ). Here exactly one atom occupies the left and right site of the double well. Now the interaction energy vanishes because both atoms never occupy the same lattice site. The system will choose this new “Mott insulating” ground state when the energy costs of populating the antisymmetric state of

726

I. Bloch

the double-well system are outweighed by the energy reduction in the interaction energy. It is important to note that precisely the atom number ﬂuctuations due to the delocalized single-particle wave functions make the “superﬂuid” state unfavorable for large U . Such a change can be induced by adiabatically increasing the barrier height in the double well system, such that J decreases exponentially and the energy cost for populating the antisymmetric state becomes smaller and smaller. Eventually it will then be favorable for the system to change from the “superﬂuid” ground state, where for U/J → ∞ each atom is delocalized over the two wells, to the “Mott insulating” state, where each atom is localized to a single lattice site. . 3 3. Multiple-well case. – The above ideas can be readily extended to the multiplewell case of the periodic potential of an optical lattice. For U/J 1 the tunnelling term dominates the Hamiltonian and the ground state of the many-body system with N atoms is given by a product of identical single-particle Bloch waves, where each atom is spread out over the entire lattice with M lattice sites:

(20)

|ΨSF U/J≈0 ∝

M

N a ˆ†i

|0.

i=1

Since the many-body state is a product over identical single-particle states, a macroscopic wave function can be used to describe the system. Here the single-site many-body wave function |φi is almost equivalent to a coherent state. The atom number per lattice site then remains uncertain and follows a Poissonian distribution with a variance given by the average number of atoms on this lattice site Var(ni ) = ˆ ni . The non-vanishing expectation value of ψi = φi |ˆ ai |φi then characterizes the coherent matter wave ﬁeld on the i-th lattice site. This matter wave ﬁeld has a ﬁxed phase relative to all other coherent matter wave ﬁelds on diﬀerent lattice sites. If, on the other hand, interactions dominate the behavior of the Hamiltonian, such that U/J 1, then ﬂuctuations in the atom number on a single lattice site become energetically costly and the ground state of the system will instead consist of localized atomic wave functions that minimize the interaction energy. The many-body ground state is then a product of local Fock states for each lattice site. In this limit the ground state of the many-body system for a commensurate ﬁlling of n atoms per lattice site is given by

(21)

|ΨM I J≈0 ∝

M "

(ˆ a†i )n |0.

i=1

Under such a situation the atom number on each lattice site is exactly determined but the phase of the coherent matter wave ﬁeld on a lattice site has obtained a maximum uncertainty. This is characterized by a vanishing of the matter wave ﬁeld on the i-th lattice site ψi = φi |ˆ ai |φi ≈ 0.

Strongly correlated quantum phases of ultracold atoms in optical lattices a

b

c

d

e

f

g

h

727

Fig. 9. – Absorption images of multiple matter wave interference patterns after releasing the atoms from an optical-lattice potential with a potential depth between (a) 7 Er and (h) 20 Er . The ballistic expansion time was 15 ms.

In this regime of strong correlations, the interactions between the atoms dominate the behavior of the system and the many-body state is not amenable anymore to a description as a macroscopic matter wave, nor can the system be treated by the theories for a weakly interacting Bose gas of Gross, Pitaevskii and Bogoliubov [34, 35]. For a 3D system, the transition to a Mott insulator occurs around U/J ≈ z · 5.6 [36, 2, 37, 33], where z is the number of next neighbors to a lattice site (for a simple cubic crystal z = 6). . 3 4. Superﬂuid to Mott insulator transition. – In the experiment the crucial parameter U/J that characterizes the strength of the interactions relative to the tunnel coupling between neighboring sites can be varied by simply changing the potential depth of the optical lattice potential. By increasing the lattice potential depth, U increases almost linearly due to the tighter localization of the atomic wave packets on each lattice site and J decreases exponentially due the decreasing tunnel coupling. The ratio U/J can therefore be varied over a large range from U/J ≈ 0 up to values in our case of U/J ≈ 2000. In the superﬂuid regime [38] phase coherence of the matter wave ﬁeld across the lattice characterizes the many-body state. This can be observed by suddenly turning oﬀ all trapping ﬁelds, such that the individual matter wave ﬁelds on diﬀerent lattice sites expand and interfere with each other. After a ﬁxed time of ﬂight period the atomic density distribution can then be measured by absorption imaging. Such an image directly reveals the momentum distribution of the trapped atoms. In ﬁg. 9b an interference pattern can be seen after releasing the atoms from a three-dimensional lattice potential. If on the other hand the optical lattice potential depth is increased such that the system is very deep in the Mott insulating regime (U/J → ∞), phase coherence is lost between the matter wave ﬁelds on neighboring lattice sites due to the formation of Fock states [2, 1, 39, 33]. In this case no interference pattern can be seen in the time-of-ﬂight images (see ﬁg. 9h) [3-5]. For a Mott insulator at ﬁnite U/J one expects a residual visibility in the interference pattern [40, 41], which can be caused, on the one hand, by

728

Visibility V

I. Bloch

100 4

10-1 10-2

100

101 U/zt

102

Fig. 10. – Visibility of the interference pattern vs. U/z t, the characteristic ratio of interaction to kinetic energy. The data is shown for two atom numbers 5.9 × 105 atoms (black circles), and 3.6 × 105 atoms (grey circles). The former curve has been oﬀset vertically for clarity. The lines are ﬁts to the data in the range 14–25ER , assuming a coherent particle hole admixture as in eq. (22) (see ref. [42]).

the residual superﬂuid shells and, on the other hand, through an admixture of coherent particle hole pairs to the ideal MI ground state. A more detailed picture for the residual short-range coherence features beyond the SFMI transition is obtained by considering perturbations deep in the Mott insulating regime at J = 0. There, the ﬁrst-order correlation function G(1) (R) describing the coherence properties vanishes beyond R = 0 and the momentum distribution is a structureless Gaussian, reﬂecting the Fourier transform of the Wannier wave function. With increasing tunneling J, the Mott state at J/U → 0 is modiﬁed by a coherent admixture of particlehole pairs. However due the presence of a gapped excitation spectrum, such particlehole pairs cannot spread out and are rather tightly bound to close distances. They do, however, give rise to a signiﬁcant degree of short-range coherence (see ﬁg. 10). Using ﬁrst-order perturbation theory with the tunneling operator as a perturbation on the dominating interaction term, one ﬁnds that the amplitude of the coherent particle-hole admixtures in a Mott insulating state is proportional to J/U : (22)

|ΨU/J ≈ |ΨU/J→∞ +

J † a ˆi a ˆj |ΨU/J→∞ . U i,j

Within a local density approximation, the inhomogeneous situation in a harmonic trap is described by a spatially varying chemical potential μR = μ(0) − R with R = 0 at the trap center. Assuming, e.g., that the chemical potential μ(0) at trap center falls into the n ¯ = 2 “Mott-lobe”, one obtains a series of MI domains separated by a SF by

729

Strongly correlated quantum phases of ultracold atoms in optical lattices

a z

d

2

1

y

x

20 Slice Fraction (%)

Number of atoms in plane (x1000)

3

15

c

d

e

f

10 5 0

Number of atoms in plane (x1000)

3

b

n=1 n=2

2 1

Slice Fraction (%)

0 15 10 5 0 0 -20

-10

0

z Position (μm)

10

-20

-10

0

10

z Position (μm)

20 -20

-10

0

10

20

z Position (μm)

Fig. 11. – Integrated distribution of a superﬂuid (a) and a Mott insulating state (b) calculated for a lattice with harmonic conﬁnement. Grey solid lines denote the total density proﬁles, blue (red) lines the density proﬁles from singly (doubly) occupied sites. A vertical magnetic-ﬁeld gradient is applied which creates almost horizontal surfaces of equal Zeeman shift over the cloud (dashed lines in inset). A slice of atoms can be transferred to a diﬀerent hyperﬁne state by using microwave radiation only resonant on one speciﬁc surface (coloured areas in insets). Spin changing collisions can then be used to separate singly (blue) and doubly occupied sites (red) in that plane into diﬀerent hyperﬁne states. Experimental data: (c) 1.0 × 105 atoms in the superﬂuid regime (V0 = 3Er ), (d) 1.0 × 105 atoms in the Mott regime (V0 = 22Er ), (e) 2.0 × 105 atoms, (f) 3.5 × 105 atoms. The grey data points denote the total density distribution and the red points the distribution of doubly occupied sites. The blue points show the distribution of sites with occupations other than n = 2. The solid lines are ﬁts to an integrated Thomas-Fermi distribution in (c), and an integrated shell distribution for (d) to (f). The n = 2 data points are oﬀset vertically for clarity.

moving to the boundary of the trap where μR vanishes. In this manner, all the diﬀerent phases which exist for given J/U below μ(0) are present simultaneously! The SF phase has a ﬁnite compressiblity κ = ∂n/∂μ and a gapless excitation spectrum of the form ω(q) = cq because there is a ﬁnite superﬂuid density ns . By contrast, in the MI-phase both ns and κ vanish identically. As predicted by [2], the incompressibility of the MI phase allows to distinguish it from the SF by observing the local density distribution in a trap. Since κ = ∂n/∂μ = 0 in the MI, the density stays constant in the Mott phases, even though the external trapping potential is rising. The existence of such wedding-cake–like density proﬁles of a Mott insulator has been supported by accurate Monte Carlo [43, 44] and DMRG [45] calculations in one, two, and three dimensions. Very recently in-trap density proﬁles have been detected experimentally by [46] and [47]. In the latter case it has been possible to directly observe the wedding cake density proﬁles and thus conﬁrm the incompressibiltiy of the Mott insulating regions of the atomic gas in the trapping potential (see ﬁg. 11). It should be

730

I. Bloch

Fig. 12. – (a) Excitation gap in the Mott insulator phase with exactly n = 1 atom on each lattice site. (b) If a correct potential gradient is added, atoms can tunnel again.

noted that the in-trap density proﬁles can be used as a sensitive thermometer for the strongly interacting quantum gas. Already for small temperatures around T ≈ 0.2U/kB the wedding cake proﬁles become completely washed out. Close to the transition point, higher-order perturbation theory or a Greens function analysis can account for coherence beyond nearest neighbors and the complete liberation of the particle-hole pairs, which eventually leads to the formation of long-range coherence in the superﬂuid regime. The coherent particle-hole admixture and its consequence on the short-range coherence of the system have been investigated theoretically and experimentally in [42, 48, 49]. It has been demonstrated that through a quantitative analysis of the interference pattern, one can even observe traces of the shell structure formation in the Mott insulating regime. In addition to the fundamentally diﬀerent momentum distributions in the superﬂuid and Mott insulating regime, the excitation spectrum is markedly diﬀerent as well in both cases. Whereas the excitation spectrum in the superﬂuid regime is gapless, it is gapped in the Mott insulating regime. This energy gap of order U (deep in the MI regime) can be attributed to the now localized atomic wave functions of the atoms [1, 2, 37, 33]. Let us consider for example a Mott insulating state with exactly one atom per lattice site. The lowest-lying excitation to such a state is determined by removing an atom from a lattice site and placing it into the neighboring lattice site (see ﬁg. 12a). Due to the onsite repulsion between the atoms, however, such an excitation costs energy U which is usually not available to the system. Therefore these are only allowed in virtual processes and an atom in general has to remain immobile at its original position. If one adds a potential gradient such that the energy diﬀerence between neighboring lattice sites ΔE exactly matches the onsite energy cost U , then such an excitation becomes energetically possible and one is able to resonantly perturb the system (see ﬁg. 12b and ﬁgs. 13). It has been possible to measure this change in the excitation spectrum by applying varying magnetic-ﬁeld gradients to the system for diﬀerent lattice potential depths and detecting the response of the system to such perturbations [3, 50, 51]. Recently, the excitation spectrum has also been probed via an intensity modulation of the lattice potential, which is eﬀectively a Bragg spectroscopy for momentum transfer

731

Strongly correlated quantum phases of ultracold atoms in optical lattices 50

40

a

b

40 Interference Peak Width (Pm)

30 30 20

20 10

10 0

1

2

3

4

0

5

1

2

3

4

5

1

2

3

4

5

36 32

28

c

d

26

28

24

24

22

20

20

16

18 0

1

2

3

4

5

0 'E/h (kHz)

Fig. 13. – Probing the excitation probability vs. an applied vertical potential gradient. Width of interference peaks after adiabatic rampdown vs. the energy diﬀerence between neighboring lattice sites E, due to the potential gradient at diﬀerent lattice depths (a) Vmax = 10Er , (b) Vmax = 13Er , (c) Vmax = 16Er and (d) Vmax = 20Er . The emergence of an energy gap can be observed as the system is converted into a Mott insulator. (Adapted from [3].)

q = 0 [5]. Here a similar behavior was predicted and observed: in the superﬂuid a broad excitation spectrum was detected [5,52], whereas the Mott insulating state display sharp resonances at integer multiples of the onsite interaction matrix element U [5]. Using Bragg spectroscopy it should also be possible to probe the excitation spectrum of a Mott insulator in more detail at various momentum transfers, which has been suggested and evaluated recently by [53]. 4. – Collapse and revival of a macroscopic quantum ﬁeld A long-standing question in interacting macroscopic quantum systems has been directed towards the problem of what happens to an initially well-deﬁned relative phase between two macroscopic quantum systems after they have been isolated from each other [54-58]. Equivalently one may ask: how do the individual macroscopic quantum ﬁelds evolve after they have been isolated from each other? Such a situation can be realized for example with a Bose-Einstein condensate that is split into two parts, such that a constant relative phase is initially established between the two subsystems BEC1 and BEC2 (see ﬁg. 14). Whenever a condensate is split into two parts such that a ﬁxed relative phase is established between those two parts, the many-body state in each of the BECs is in a

732

I. Bloch

Fig. 14. – A Bose-Einstein condensate is split into two parts with an initially constant phase between the two subsystems BEC1 and BEC2.

superposition of diﬀerent atom number states. Let us now consider the case of repulsive interactions between the atoms and determine how such superpositions of atom number states evolve over time, taking into account the collisions between the atoms. Let us ﬁrst assume that all atoms in a subsystem occupy the ground state of its external conﬁning potential. If the interaction energy is then small compared to the vibrational spacing in this potential well, the Hamiltonian governing the behavior of the atoms is given by (23)

H=

1 Un ˆ (ˆ n − 1). 2

The eigenstates of the above Hamiltonian are Fock states |n in the atom number, with eigenenergies En = U n(n − 1)/2. The evolution with time of such an n-particle state is then simply given by |n(t) = |n(0) × exp[−iEn t/¯h]. If the atoms in such a subsystem are brought into a superposition of atom number states |n, which always occurs whenever a ﬁxed relative phase persists between the two subsystems, each subsystem is in a superposition of eigenstates |n which results in a dynamical evolution of this state over time. Let us consider for example a coherent state n |α = exp[−|α|2 /2] n √αn! |n in each subsystem [59]. Here α is the amplitude of the coherent state with |α|2 corresponding to the average atom number in the subsystem. The evolution with time of such a coherent state can be evaluated by taking into account the time evolution of the diﬀerent Fock states forming the coherent state: (24)

|α(t) = e−|α|

2

/2

αn 1 √ e−i 2 U n(n−1)t/¯h |n. n! n

The coherent matter wave ﬁeld ψ in each of the subsystems can then simply be evaluated through ψ = α(t)|ˆ a|α(t), which exhibits an intriguing dynamical evolution [54-63].

Strongly correlated quantum phases of ultracold atoms in optical lattices

3

a

b

733

c

0 -3 -3

0

3

d

e

Fig. 15. – Calculated quantum dynamics of an initially slightly number squeezed state with an average number of three atoms. Such a state can be parameterized as |α(t) ∝ P n(n−1) αn √ g exp[−i 12 U n(n − 1)t/¯ h]|n, where 0 < g < 1 characterizes the sub-Poissonian n n! character of the many-body state. In the graph g = 0.8. The dynamical evolution of the quantum state is caused by the coherent cold collisions between the atoms. The graphs show the overlap of the dynamically evolved input state with an arbitrary coherent state of amplitude β. Evolution times are (a) 0 h/U ; (b) 1/8 h/U ; (c) 1/4 h/U ; (d) 3/8 h/U ; (e) 1/2 h/U .

At ﬁrst, the diﬀerent phase evolutions of the atom number states lead to a collapse of ψ. However, at integer multiples in time of h/U all phase factors in the above equation rephase modulo 2π and thus lead to a revival of the initial coherent state (see also ﬁg. 15). The collapse and revival of the coherent matter wave ﬁeld of a BEC is reminiscent to the collapse and revival of the Rabi oscillations in the interaction of a single atom with a single-mode electromagnetic ﬁeld in cavity quantum electrodynamics [64, 65]. There, the nonlinear atom-ﬁeld interaction induces the collapse and revival of the Rabi oscillations whereas here the nonlinearity due to the interactions between the atoms themselves leads to the series of collapse and revivals of the matter wave ﬁeld. It should be pointed out that such a behavior has also been theoretically predicted to occur for a coherent light ﬁeld propagating in a nonlinear medium [61] but to our knowledge has never been observed experimentally. In order to realize a coherent state in a potential well, one can again use the opticallattice potential and ramp it to a potential depth VA , which is still completely in the superﬂuid regime. Then, for low lattice depths, the many-body state in each potential well is almost equal to that of a coherent state with a corresponding average atom number. In such a situation the phase of the matter wave ﬁeld on the i-th lattice site is ﬁxed relative to the matter wave ﬁelds on the other lattice sites. Then, in order to

734

I. Bloch

c

e

f

g

d

1

0

Column density (a.u.)

b

600 μm

a

Fig. 16. – Dynamical evolution of the multiple matter wave interference pattern after jumping from a potential depth VA = 8 Er to a potential depth VB = 22 Er and a subsequent hold time τ . Hold times τ : (a) 0 μs; (b) 100 μs; (c) 150 μs; (d) 250 μs; (e) 350 μs; (f) 400 μs and (g) 550 μs.

isolate the wells from each other, one rapidly increases the lattice potential depth to VB with negligible tunnel coupling on a timescale that is fast compared to the tunnelling time h/J in the system. Thereby the atoms do not have time to redistribute themselves during the ramp-up of the optical potential and we preserve the initial atom number distribution on each lattice site. On the other hand, the time scale is slow compared to the oscillation frequencies on each lattice site such that no vibrational excitations are created in the ramp-up process and all atoms remain in the vibrational ground state of each well. Using this method one can freeze out the atom number distribution at a potential depth VA and the dynamics of each of the matter wave ﬁelds on diﬀerent lattice sites is now governed by the Hamiltonian of eq. (23). The dynamical evolution of the matter wave ﬁelds can subsequently be followed by holding the atoms at the lattice potential depth VB for a variable hold time and then releasing them suddenly from the combined optical and magnetic trapping potentials. After a suitable time-of-ﬂight period one then takes absorption images of the multiple matter wave interference pattern (see ﬁg. 16). Initially, directly after ramping up the lattice potential, the interference pattern is clearly visible, however after a time of ≈ 250 μs the interference pattern is completely lost. Here the vanishing of the interference pattern is caused by the collapse of the matter wave ﬁelds on each lattice site. But after a total hold time of 550 μs the original interference pattern is regained again, showing that the matter wave ﬁelds have revived. It is important to note that the atom number statistics in each of the wells remains constant throughout the dynamical evolution time. This is fundamentally diﬀerent from the vanishing of the interference pattern in the Mott insulator case, where the atom number distribution changes, but no further dynamical evolution occurs.

Strongly correlated quantum phases of ultracold atoms in optical lattices

735

5. – Quantum gate arrays via controlled collisions . 5 1. Spin-dependent transport. – So far the optical potentials used have been mostly independent of the internal ground state of the atom. However, it has been suggested that by using spin-dependent periodic potentials one could bring atoms on diﬀerent lattice sites into contact and thereby realize fundamental quantum gates [7,66-68], create largescale entanglement [6, 69], excite spin waves [70], study quantum random walks [71] or form a universal quantum simulator to simulate fundamental condensed-matter physics Hamiltonians [72]. We show how the wave packet of an atom that is initially localized to a single-lattice site can be split and delocalized in a controlled and coherent way over a deﬁned number of lattice sites. For this, the atoms are loaded into a three-dimensional lattice potential, on which along one dimension a spin dependent transport is implemented. An experimental setup similar to ﬁg. 5 is used. In such a setup the polarization angle θ is initially set to a linlin polarization conﬁguration. Shortly before moving the atoms along this standing wave direction, the lattice potentials are adiabatically turned oﬀ along the y- and z-direction. This is done in order to reduce the interaction energy, which strongly depends on the conﬁnement of the atoms at a single lattice site. One can thereby study the transport process itself, without having to take into account the phase shifts in the many-body state that result from a coherent collisional interaction between atoms. During the shifting process of the atoms it is crucial to avoid unwanted vibrational excitations, especially if the shifting process would be repeated frequently. Therefore care has to be taken not to expose the atoms to high accelerations, which could excite them to higher vibrational bands [32, 66, 73, 74]. In order to verify the coherence of the spin-dependent transport, the interferometer sequence of ﬁg. 17 has been used. Let us ﬁrst consider the case of a single atom being initially localized to the j-th lattice site. First, the atom is placed in a coherent superposition of the two internal states |0j and |1j with a π/2 microwave pulse (here the index denotes the position in the lattice). Then the polarization angle θ is rotated to 180◦ , such that the spatial wave packet of an atom in the |0 and the |1 state are transported in √ opposite directions. The ﬁnal state after such a movement process is then given by 1/ 2(|0j + i exp[iβ]|1j+1 ), where the wave function of an atom has been delocalized over the j-th and the (j + 1)-th lattice site. The phase β between the two wave packets depends on the accumulated kinetic and potential energy phases in the transport process and in general will be nonzero. In order to reveal the coherence between the two wave packets, a ﬁnal π/2 microwave pulse is applied, which erases the which-way information encoded in the hyperﬁne states. The atoms are then released from the conﬁning potential by suddenly turning oﬀ the standing-wave optical potential and the momentum distribution of the trapped atoms in the |1 state is subsequently observed after a time-of-ﬂight period. As a result of the above sequence, the spatial wave packet of an atom in the |0 (|1) state is delocalized over two lattice sites resulting in a double slit momentum distribution w(p) ∝ exp[−p2 /(¯ h/σx )2 ] · cos2 (p δx0 /2¯ h + β/2) (see ﬁg. 18a and ﬁg. 19) where δx0 denotes the separation between

736

I. Bloch

Fig. 17. – General interferometer sequence used to delocalize an atom over an arbitrary number of lattice sites. Initially an atom is localized to the j-th lattice site. The graph on the left indicates the EOM voltage and the sequence of π/2 and π microwave pulses that are applied over time (see text).

Fig. 18. – Observed interference patterns in state |1 after initially localized atoms have been delocalized over (a) two, (b) three, (c) four, (d) ﬁve, (e) six and (f) seven lattice sites using the interferometer sequence of ﬁg. 17. The time of ﬂight period before taking the images was 14 ms and the horizontal size of each image is 880 μm.

Strongly correlated quantum phases of ultracold atoms in optical lattices

737

Column Density (a.u.)

1.0 0.8 0.6 0.4 0.2 0 0

100

200

300

400

500

600

700

x (Pm) Fig. 19. – Proﬁle of the interference pattern obtained after delocalizing atoms over three lattice sites with a π/2-π-π/2 microwave pulse sequence. The solid line is a ﬁt to the interference pattern with a sinusoidal modulation, a ﬁnite visibility (≈60%) and a Gaussian envelope. The time of ﬂight period was 15 ms.

the two wave packets and σx is the spatial extension of the Gaussian ground-state wave function on each lattice site. In order to increase the separation between the two wave packets further, one could increase the polarization angle θ to further integer multiples of 180◦ . In practice, such an approach is however limited by the ﬁnite maximum voltage that can be applied to the EOM, however the use of spin-echo π-pulses, allows one to reverse the roles of both spin states and implement a transport over almost arbitrary distances. With increasing separation between the two wave packets the fringe spacing of the interference pattern further decreases (see ﬁg. 18). Through these experiments it has become possible to demonstrate the coherent spin-dependent transport of neutral atoms in optical lattices, thereby showing an essential level of coherent control for many future applications. The method demonstrated, e.g., also provides a simple way to continuously tune the interspecies interactions by controlling the overlap of the two ground-state wave functions for the two spin states. Furthermore, if such a transport is carried out in a three-dimensional lattice, where the on-site interaction energy between atoms is large, one can induce interactions between almost any two atoms on diﬀerent lattice sites in a . controlled way as will be shown in the next subsect. 5 2. . 5 2. Controlled collisions. – In order to realize a controlled interaction between the particles on diﬀerent lattice sites in a 3D Mott insulating quantum register, the above spin-dependent transport sequence can be used. This leads to collisions between neighboring atoms and can be described through an ensemble of quantum gates acting in parallel [6, 66]. Alternatively, these quantum gates can be described as a controllable

738

I. Bloch (a)

(b) Shift

time

π/2 ϕ

ϕ

ϕ π/2

j

j+1

j+2

Lattice Site

Fig. 20. – (a) Controlled interactions between atoms on diﬀerent lattice sites can be realized with the help of spin-dependent lattice potentials. In such spin-dependent potentials, atoms in a, let us say, blue internal state experience a diﬀerent lattice potential than atoms in a red internal state. These lattices can be moved relative to each other such that two initially separated atoms can be brought into controlled contact with each other. (b) This can be extendended to form a massively parallel quantum gate array. Consider a string of atoms on diﬀerent lattice sites. First the atoms are placed in a coherent superposition of the two internal states (red and blue). Then spin dependent potentials are used to split each atom such that it simultaneously moves to the right and to the left and is brought into contact with the neighboring atoms. There both atoms interact and a controlled phase shift j is introduced. After such a controlled collision the atoms are again moved back to their original lattice sites.

quantum Ising interaction [69]: (25)

Hint ∝ g(t)

1 + σz(j) 1 − σz(j+1) j

2

2

.

(j)

Here g(t) denotes the time dependent coupling constant and σz is the Pauli Spin operator t acting on an atom at the j-th lattice site. For an interaction phase of ϕ = 2π × 0 hold g(t) dt/h = (2n+1)π one obtains a maximally entangled cluster state, whereas for ϕ = n2π one obtains a disentangled state [69]. Here thold denotes the time for which the atoms are held together at a common site and n is an integer. Let us point out that the creation of such highly entangled states can be achieved in a single lattice shift operational sequence described above and depicted in ﬁg. 20, independent of the number of atoms to be entangled [6, 69]. A π/2 pulse ﬁrst allows one to place the atom in a coherent superposition of the two states |0 and |1. After creating such a coherent superposition, a spin-dependent transfer is used to split and move the spatial wave function of the atom over half a lattice spacing in two opposite directions depending on its internal state (see ﬁg. 20). Atoms on neighboring sites interact for a variable amount of time thold that leads to a controlled conditional phase shift of the corresponding many-body state. After half of the hold time, a microwave π pulse is furthermore applied. This spin-echo type pulse is mainly used to

Strongly correlated quantum phases of ultracold atoms in optical lattices

739

0.6 0.5

Visibility

0.4 0.3 0.2 0.1 0 0

100

200 300 400 Hold Time (μs)

500

Fig. 21. – Visibility of Ramsey fringes vs. hold times on neighboring lattice sites for the experimental sequence of ﬁg. 20. The solid line is a sinusoidal ﬁt to the data including an oﬀset and a ﬁnite amplitude. Such a sinusoidal behavior of the visibility vs. the collisional phase shift (determined by the hold time thold ) is expected for a Mott insulating state with an occupancy of n = 1 atom per lattice site.

cancel unwanted single-particle phase shifts, e.g., due to inhomogeneities in the trapping potentials. It does not however aﬀect the non-trivial and crucial collisional phase shift due to the interactions between the atoms. After such a controlled collision, the atoms are moved back to their original site. Then a ﬁnal π/2 microwave pulse with variable phase is applied and the atom number in state relative to the total atom number is recorded. For short hold times, where no signiﬁcant collisional phase shift is acquired, a Ramsey fringe with a high visibility of approx. 50% is recorded (see ﬁg. 21). For longer hold times one notices a strong reduction in the visibility of the Ramsey fringe, with an almost vanishing visibility of approximately 5% for a hold time of 210 μs. This hold time corresponds to an acquired collisional phase shift of ϕ = π for which a minimum visibility is expected if the system is becoming entangled. For a two-particle system this can be understood by observing the resulting Bell state: (26)

√

α 1/ 2 |0j |+α j+1 + |1j |−j+1 ,

after the ﬁnal π/2 pulse of the Ramsey sequence has been applied to the atoms. Here α |+α j+1 and |−j+1 represent two orthogonal superposition states of |0 and |1 for which α 2 |1|+ | + |1|−α |2 = 0.5. A measurement of atoms in state |1 therefore becomes independent of the phase corresponding to a vanishing Ramsey fringe. This indicates that no single-particle operation can place all atoms in either spin-state when a maximally entangled state has been created. The disappearance of the Ramsey fringe has been shown to occur not only for a two-particle system, but is a general feature for an arbitrary N -particle array of atoms that have been highly entangled with the above experimental sequence [75, 66]. For longer hold times, however, the visibility of the Ramsey fringe increases again reaching a maximum of 55% for a hold time of 450 μs (see ﬁg. 21).

740

I. Bloch

Here the system becomes disentangled again, as the collisional phase shift is close to ϕ = 2π and the Ramsey fringe is restored with maximum visibility. The timescale of the observed collisional phase evolution is in good agreement with the measurements on the Mott insulator transition of the previous section and ab initio calculations of the onsite matrix element U . In a one-dimensional lattice shift the system is very susceptible to vacant lattice sites, as a defect will immediately limit the size of the cluster. However, the scheme can be extended to two- or three dimensions by using two additional lattice shift operations along the remaining orthogonal lattice axes. As long as the ﬁlling factor of lattice sites would exceed the percolation threshold (31% for a 3D simple cubic lattice system) a large entangled cluster should be formed, making maximum entanglement of literally 100000 of atoms possible in only three operational steps. In addition, novel ﬁltering schemes have been proposed, which would allow one create a Mott state with high ﬁdelity through a coherent coupling to a BEC reservoir [76]. . 5 3. Using controlled collisional quantum gates. – It has been proposed to use such controlled interactions of Ising or Heisenberg type to simulate the behavior of quantum magnets [70] and other quantum system [72], to realize quantum gates between diﬀerent atoms [7, 6, 66-68] or to generate highly entangled cluster states |φc [66]. In a onedimensional spin chain of length N , such a cluster state can be written to be of general form [69]

(27)

|φc =

1 2N/2

N D

|0i σzi+1 + |1i .

i=1

Up to local unitary √ transformations, for N = 2 the cluster state simply corresponds to √ the Bell state 1/ 2(|0|0 + |1|1) and for N = 3 to the three particle GHZ state 1/ 2(|0|0|0 + |1|1|1) [77], however for larger N no such simple identiﬁcation can be made. Cluster states provide a high degree of persistent entanglement and a new form of quantum computing has been proposed based on such a state —the so-called one-way quantum computer [67, 78]. At its heart lies the idea of ﬁrst generating a massively entangled 2D cluster state. Then only single-particle operations have to be implemented on this cluster state to realize quantum logical operations in the system and to perform a quantum computation [67, 78]. 6. – Entanglement generation via spin-changing collisions In one of the ﬁnal section I would like to outline a rather novel method for the creation of entanglement based on coherent spin-changing collisions between neutral atoms. In a usual collision event between two atoms in spin states m1 , m2 , both particles retain their spin orientation. However, it is also possible that when two atoms collide, their spins

Strongly correlated quantum phases of ultracold atoms in optical lattices

741

~ω

ΩS (m1=0,m2=0)

(m3=+1,m4=-1)

Fig. 22. – Two-atoms in the vibrational ground state of a single lattice site can undergo a spin changing collision, in which the spin state of the individual atoms is changed, however, the total magnetization is conserved. Such coherent spin changing collisions can be described as coherent coupling between the initial and ﬁnal state, leading to Rabi-type oscillations between the two two-particle states [11].

are ﬂipped and they exit the collision event in spin states m3 , m4 . Since total angular momentum has to be conserved in such a collision event, we ﬁnd that m1 +m2 = m3 +m4 . Such spin-changing collisions are the basis for spinor dynamics in multi-state BoseEinstein condensates trapped in optical traps and have become a focus of research in recent years [79,80]. How can such spin-changing collisions be used to create an entangled state? Let us for example consider two 87 Rb atoms, both in the (f = 1, mf = 0) internal hyperﬁne state and trapped in the vibrational ground state at the single site of an opticallattice potential. If two such atoms undergo a spin-changing √ collision event, they can transform from spin states |0, 0 into the triplet spin state 1/ 2(| + 1, −1 + | − 1, +1), which forms an entangled Bell state. From this state the atoms could collide again back into the original initial state |0, 0. This coherent collisional process can in fact be described by a coherent coupling with resonant coupling strength ΩS between the √ initial and ﬁnal two particle states |0, 0 ↔ 1/ 2(| + 1, −1 + | − 1, +1) [11] (see ﬁg. 22). Such a coherently coupled two-level system formed out of two-particle state will then exhibit Rabi-type oscillations between the initial and ﬁnal states. High contrast Rabitype oscillations of this type have indeed only recently been observed [11] and can be seen in ﬁg. 23. If such spin-changing collisions are carried out on resonance, atoms initially prepared in state |0, 0 will have completely converted in Bell pairs after a time tπ = π/ΩS . Such Bell pairs are known to be very robust against decoherence processes for example due to global magnetic-ﬁeld ﬂuctuations and have been observed to possess lifetimes of several seconds [81] in ion-trap experiments. They are therefore promising candidates for the creation of robust multiparticle entanglement or Einstein-Podolsky-Rosen entangled states [82, 83].

742

I. Bloch

Relative spinpopulation

0.8

0.6

0.4

0.2 20

40

60

80

time (ms) Fig. 23. – Rabi-type oscillations in an array of doubly occupied lattice sites, showing the coherent √ collisional coupling between states |0, 0 ↔ 1/ 2(| + 1, −1 + | − 1, +1).

7. – Quantum noise correlations For now almost 10 years, absorption imaging of released ultracold quantum gases has been a standard detection method for revealing information on the macroscopic quantum state of the atoms in the trapping potential. For strongly correlated quantum states in optical lattices, however, the average signal in the momentum distribution that one usually observes, e.g. for a Mott insulating state of matter, is a featureless Gaussian wave packet. From this Gaussian wave packet one cannot deduce anything about the strongly correlated quantum states in the lattice potential apart from the fact that phase coherence has been lost. Recently, however, the widespread interest in strongly correlated quantum gases in optical lattices as quantum simulators has lead to the prediction of fascinating new quantum phases for ultracold atoms, e.g. with antiferromagnetic structure, spin waves or charge density waves. So far it has not been clear how one could detect those states. Recently a theoretical proposal by Altman et al. [84] has shown that noise correlation interferometry could be a powerful tool to directly visualize such quantum states. Noise correlation in expanding ultracold atom clouds can in fact be seen as a powerful way to read out the quantum states of an optical lattice-based quantum simulator. The basic eﬀect relies on fundamental Hanbury-Brown–Twiss correlations [85-88] in the ﬂuctuation signal of an atomic cloud. For bosons, e.g., a bunching eﬀect of the ﬂuctuations is predicted to occur at special momenta of the expanding cloud, which directly reﬂect the ordering of the atoms in the lattice. Such bunching eﬀects in momentum space can be directly revealed as spatial correlations in the expanding atom cloud. Our goal therefore is to reveal correlations in the ﬂuctuations of the expanding atomic gas after it

Strongly correlated quantum phases of ultracold atoms in optical lattices

Path A

743

Path B

click!

click!

r

Fig. 24. – Hanbury-Brown–Twiss correlations in expanding quantum gases from an optical lattice. For bosonic particles that are detected at distances r (e.g. on a CCD camera), an enhanced detection probability exists due to the two indistinguishable paths the particles can take to the detector. This leads to enhanced ﬂuctuations at special detection distances r, depending on the ordering of the atoms in the lattice. Detection of the noise correlation can therefore yield novel information on the quantum phases in an optical lattice.

has been released from the trap. Such correlations in the expanding cloud at distance d can be quantiﬁed through the second-order correlation function (28)

n(x + d/2) n(x − d/2)d2 x C(d) = . n(x + d/2)n(x − d/2)d2 x

Here n(x) is the density distribution of a single expanding atom cloud and the angle brackets · denote a statistical averaging over several individual images taken for diﬀerent experimental runs. Spatial correlations in the noise of expanding atom clouds arise here due to a fundamental indistinguishability of the particles as is well known from the foundational experiments in quantum optics of Hanbury Brown and Twiss [85-88]. Let us for simplicity consider two detectors spaced at a distance d below our trapped atoms and furthermore restrict the discussion two only two atoms trapped in the lattice potential (see ﬁg. 24). As the trapping potential is removed and the particles propagate to the detectors, there are two possibilities for the particles to reach these detectors, such that one particle is detected at each detector. First, the particles can propagate along path A in ﬁg. 24 to achieve this. However, another propagation path exists, which is equally probable, path B in ﬁg. 24. If we fundamentally cannot distinguish which way the particles have been propagating to our detectors, we have to form the sum for bosons or diﬀerence for fermions of the two propagation amplitudes and square the resulting value to obtain

744

I. Bloch

a

x10-4

c

6 4 2

2hk

-2

b

6

d

0.2

4 2

0.1

0 0

-400 -200

0

200 400

-400 -200

0

x (μm)

x (μm)

200 400

-2

Corr. Amp. (x10-4)

Column Density (a.u.)

0

Fig. 25. – Single-shot absorption image including quantum ﬂuctuations and associated spatial correlation function. (a) 2D column density distribution of a Mott insulating atomic cloud containing 6 × 105 atoms, released from a 3D optical lattice potential with a lattice depth of 50 Er . The white bars indicate the reciprocal lattice scale l deﬁned in eq. (29). (b) Horizontal cut (black line) through the centre of the image in a and Gaussian ﬁt (red line) to the average over 43 independent images each one similar to (a). (c) Spatial noise correlation function obtained by analyzing the same set of images, which shows a regular pattern revealing the lattice order of the particles in the trap. (d) Horizontal proﬁle through centre of pattern, containing the peaks separated by integer multiples of l. The width of the individual peaks is determined by the optical resolution of our imaging system.

the two particle detection probability at the detectors. As one increases the separation between the detectors, the phase diﬀerence between the two propagation paths increases, leading to constructive and destructive interference eﬀects in the two-particle detection probability. The length scale of this modulation in the two-particle detection probability of the expanding atom clouds depends on the original separation of the trapped particles at a distance alat and is given by the characteristic length scale (29)

l=

h t, malat

where t is time of ﬂight. Such Hanbury-Brown–Twiss correlations in the shot noise of an expanding atom cloud from a Mott insulating state of matter have recently been observed for 3D and 2D Mott insulating states (see ﬁg. 25) [89, 90]. Similar pair correlations at the shot noise level

0.2

a

c

b

d

4 2 0

0.1

-2 0

-4 -400

-200

0

x (m)

200

400

-200

0

200

C(d) - 1 (x10 - 4/px)

745

Strongly correlated quantum phases of ultracold atoms in optical lattices

x (m)

Fig. 26. – Noise correlations of a band insulating Fermi gas. Instead of the correlation “bunching” peaks observed in ﬁg. 25 the fermionic quantum gas shows an HBT type antibunching eﬀect, with dips in the observed correlation function [92].

have also been obtained in the group of D. Jin [91] for dissociated molecular fragments. Here the correlations however arise due to the dissociation event and not due to indistinguishable pathways of the particles. Very recently, our group has been able to analyze the quantum noise correlation of a band insulating single-component Fermi gas in an optical lattice [92]. Although the density distribution is there almost identical to the one of a bosonic Mott insulating cloud for J → 0, a profound diﬀerence can be observed in both cases. Whereas the bosonic particles exhibit an HBT-type bunching, the fermionic particles exhibit an antibunching, which can be seen as a decreased detection probability for ﬁnding two particles at integer multiples of l (see ﬁg. 26). Such a quantum statistical antibunching eﬀect for Fermions has been observed for electrons in semiconductors [93,94], as well as for free electrons [95] and neutrons [96]. Together with the experiments performed by [97] (see article of W. Vassen in this volume), our experiments are the ﬁrst demonstration of antibunching with atoms [92].

746

I. Bloch

8. – Outlook Ultracold atoms in optical lattices have proven to be versatile model system for the investigation of strongly correlated quantum physics with ultracold atoms. For the future, the focus of research will shift towards the realization of quantum magnetism with ultracold atoms. Although no long-range direct interaction exists between the particles on diﬀerent lattice sites, superexchange processes can mediate eﬀective long-ranged spinspin interaction between the particles [98]. Such superexchange processes fundamentally rely on second-order tunnelling events, which lead to eﬀective spin-spin coupling strengths on the order of J 2 /U . To observe these phases, the temperature of the many-body system will have to be below this superexchange-interaction energy scale, which presents a challenge to current experiments, however does not seem out of reach [99]. If such systems for fermionic particles could eventually be doped or analyzed at non-integer ﬁlling, one could hope to solve one of the long-standing problems of condensed matter physics, namely whether the fermionic Hubbard model with repulsive interactions contains a superconducting phase [100, 101] and what the mechanism behind such a superconducting phase could be. ∗ ∗ ∗ The author would like to acknowledge funding through a EU Marie-Curie-Excellence grant (QUASICOMBS), EU-STREP (OLAQUI), the DFG and AFOSR.

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Finite-temperature eﬀects in ultracold Fermi gases K. Levin and Q. Chen James Franck Institute and Department of Physics, University of Chicago Chicago, Illinois 60637, USA

1. – BCS-BEC crossover theory and the physical eﬀects of temperature The study of ultracold trapped fermionic gases is a rapidly exploding subject [1-10] which is deﬁning new directions in condensed-matter and atomic physics. It has also captured the attention of physicists who study color superconducting quark matter as well as nuclear matter [11-13]. Indeed, it is hard, in recent times, to ﬁnd a subﬁeld of physics which appeals this broadly to the research community. As we come to understand the gases, experimentalists will move to address fermionic atoms in optical lattices; this will provide important insight to condensed-matter physicists as analogue systems for “solving” intractable many body problems. What makes these gases (and lattices) so important is their remarkable tunability and controllability. Using a Feshbach resonance, one can tune the attractive two-body interaction from weak to strong, and thereby make a smooth crossover from a BCS superﬂuid to a Bose-Einstein condensation (BEC) [14, 15]. Not only does this allow high transition temperatures Tc (relative to the Fermi energy EF ) but it may also provide insights into the high-temperature cuprate superconductivity [16-19]. Furthermore, one can tune the population of the two spin states, essentially at will, in this way, allowing exploration [20-22] of exotic polarized phases such as the FFLO [23-25] superﬂuids, of interest to condensed matter, nuclear and particle physicists. One will be able to tune the lattice parameters such as bandwidth, on-site repulsion, even random disorder, etc. and thereby study the famous, and as yet, unsolved Hubbard model Hamiltonian. This paper will concentrate on those issues relating to the eﬀects which are of current interest to experimentalists. In particular, we will study BCS-BEC crossover in atomic c Societ` a Italiana di Fisica

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Fermi gases, looking at a wide range of diﬀerent experiments. Our group has been principally interested in the eﬀects of ﬁnite temperature [26, 17, 18] and in this review, we will discuss how temperature T enters into the standard crossover theory, and how temperature can be measured, and how temperature changes the character of the gas from a superﬂuid at low T to an unusual (but strongly interacting) normal ﬂuid at moderate T , and to ultimately a Fermi gas at high T . Temperature, of course, plays an important role in these Fermi-gas experiments. As we shall see in this paper, the earliest evidence for superﬂuidity is generally based [1-8] on a comparison of the behavior of the gas at two diﬀerent temperatures, presumably, one above and one below Tc . Experimentally, one is never strictly at T = 0, and as a result there are thermal excitations of the gas which need to be characterized both theoretically and experimentally. Before we begin with the ultracold gases, it is useful to discuss to what extent the BCS-BEC crossover scenario relates to the high-temperature copper oxide superconductors [17, 18]. While there is much controversy in the ﬁeld of high-Tc superconductivity, there is a school of thought [17,27-31,19] which argues that these systems are somewhere intermediate between BCS and BEC. These arguments are based on the observation that Tc is anomalously high, and therefore the attractive interaction is presumably stronger than in conventional superconductors. These are quasi-two-dimensional (2D) materials which means that they have a tendency to exhibit “pre-formed” pairs, that is, pairing in advance of Bose condensation. This is often referred to as ﬂuctuation eﬀects, but preformed pairs are naturally associated with a more BEC-like scenario. Importantly these pairs can be used to explain the widespread and anomalous “pseudogap” eﬀects [16, 32] which are the focus [33] of most of the current attention in the ﬁeld. The existence of “pre-formed” pairs means that a characteristic (pseudogap) energy must be supplied in the normal state to break the pairs and create fermionic excitations. Hence we say that there is a gap or, more precisely, a pseudogap in the fermionic excitation spectrum. The case that the crossover scenario is relevant to the cuprates was made quite eloquently by A. J. Leggett in a recent status report [19] on high-Tc superconductivity. In this article he summarized the eight salient “facts” about the cuprates. It is worth quoting one here, which relates to their anomalously short coherence length. “The size of the pairs is somewhere in the range 10-30 ˚ A —from measurements of the upper critical ﬁeld, Fermi velocity and Tc . This means that the pair size is only moderately greater than the inter-conduction electron in-plane spacing, putting us in the intermediate regime of the so-called Bose-Einstein condensate to BCS superconductor (BEC-BCS) crossover, and leading us to expect very large eﬀects of ﬂuctuations (they are indeed found).” The ﬁeld of BCS-BEC crossover is built around early observations by Eagles [15] and Leggett [14] that the BCS ground state, proposed by Bardeen, Cooper, and Schrieﬀer in 1957, is much more general than was originally thought. If one increases the strength of the attraction and self-consistently solves for the fermionic chemical potential μ, this wave function will correspond to a more BEC-like form of superﬂuidity. Knowing the ground state what is the nature of the superﬂuidity at ﬁnite T ? That is the central question we will address in this paper.

753

Finite-temperature effects in ultracold Fermi gases

'

'7

' sc

7c

T*

Fig. 1. – Contrasting behavior of the excitation gap Δ(T ) and superﬂuid order parameter Δsc (T ) vs. temperature. The height of the shaded region roughlvy reﬂects the density of non-condensed pairs at each temperature.

Even without a detailed theoretical framework, we can make three important observations. – As we go from BCS to BEC, pairs will form above Tc without condensation. In the normal state, it pays in general to take advantage of the attractive pairing interaction. Only in the extreme BCS limit do pairs form exactly at Tc . – The fundamental statistical entities in these superﬂuids are fermions. We can think of pairs of fermions as a form of “boson”, although the statistics are not precisely the same. We measure these “bosonic” or pair degrees of freedom indirectly via the fermionic gap parameter Δ(T ). In the fermionic regime this parameter is the minimum energy which must be supplied to create fermionic excitations. It tells us about bosons indirectly through the binding together of two fermions. – In general there will be two types of excitations in these BCS-BEC crossover systems. Only in strict BCS theory are the excitations of solely fermionic character, and only in the strict BEC limit will the excitations be solely of bosonic character. More generally in the intermediate case (often called the “unitary” regime) the excitations consist of a mix of both fermions and bosons. These observations are illustrated by ﬁgs. 1 and 2. In ﬁg. 1 we schematically plot the gap parameter Δ(T ) as a function of T , along with the superﬂuid order parameter Δsc (T ). The former, which represents the “bosonic” degrees of freedom, shows that pairs continuously form once temperature is less than a crossover temperature T ∗ , while the order parameter turns on as in a second-order phase transition at Tc . The height of the shaded region reﬂects the number of non-condensed pairs. This number increases monotonically with decreasing T , until Tc is reached. As T further decreases below Tc the number of non-condensed pairs begins to decrease monotonically due to the condensation of zero momentum pairs. In ﬁg. 2 we present a schematic plot of the excitation type, which shows that between BCS and BEC (i.e., in the unitary regime) there will be a mix of fermions and bosons.

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BCS

Unitary

BEC

Fig. 2. – Schematic illustration of excitations in the BCS, unitary and BEC regimes. The black discs represent fermionic excitations. Pair excitations become progressively dominant as the system evolves from the BCS to BEC regime.

These bosons and fermions are not separate ﬂuids, but rather they are strongly interconnected. Indeed, the gap in the fermionic spectrum (related to Δ) is a measure of the number of bosons in the system. 2. – Theory outline In this paper we will presume that the ground state is given by the standard BCS-like wave function, (2.1)

Ψ0 = Πk uk + vk c†k,↑ c†−k,↓ |0,

where c†k,σ and ck,σ are the creation and annihilation operators for fermions of momentum k and spin σ =↑, ↓. The variational parameters vk and uk are usually represented by the two more directly accessible parameters Δsc (T = 0) and μ, which characterize the fermionic system. Here Δsc (T = 0) is the zero-temperature superconducting order parameter. These fermionic parameters are uniquely determined in terms of the attractive interaction U and the fermionic density n. The variationally determined self-consistency conditions are given by two BCS-like equations which we refer to as the “gap” and “number” equations, respectively. At T = 0, the eﬀects of BEC-BCS crossover are most directly reﬂected in the behavior of the fermionic chemical potential μ, which is associated with the number equation. We plot the behavior of μ in ﬁg. 3, which indicates the BCS and BEC regimes. In the weakcoupling regime μ = EF and ordinary BCS theory results. With increasing |U |, μ begins to decrease, eventually crossing zero and then ultimately becoming negative in the BEC regime. We generally view μ = 0 as a crossing point. For positive μ the system has a remnant of a Fermi surface, and we say that it is “fermionic”. For negative μ, the Fermi surface is gone and the material is “bosonic”. One can debate whether other ground states ought to be considered. Indeed the work of the Camerino group [34, 35] is based on a ﬁnite-T approach ﬁrst introduced by Nozieres and Schmitt-Rink (NSR) [36, 31]. This leads to a diﬀerent, and not as readily characterized ground state. We list some arguments in support of the ground state in

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Finite-temperature effects in ultracold Fermi gases

1

1

PG Unitary

0

-1 0

1 U/Uc

2

BEC

PG

BCS

BEC P/EF

P/EF

BCS

Unitary 0

-1 -3

-2

-1

0 1/kFa

1

2

3

Fig. 3. – Typical behavior of the chemical potential μ at T = 0 in the three regimes, as a function of the interaction strength U/Uc , or, equivalently, 1/kF a. As U/Uc increases from 0, the chemical potential μ starts to decrease and then becomes negative. The character of the system changes from fermionic (μ > 0) to bosonic (μ < 0). The pseudogap (PG) or unitary regime corresponds to non-Fermi-liquid–based superconductivity, and Uc (< 0) corresponds to critical coupling for forming a two-fermion bound state in vacuum [17], i.e., the unitary point where the two-body s-wave scattering length a diverges.

eq. (2.1). i) This is the basis for the widely studied Bogoliubov-de Gennes approach, which can be applied [37-40] to the BCS-BEC crossover problem at T = 0. ii) At arbitrarily strong coupling (and T = 0), this ground state can be shown to coincide with a Gross-Pitaevskii (GP) description [41] of the boson system. iii) This ground state is the basis for the rapidly proliferating theoretical literature [40, 42-47] on spin-polarized Fermi superﬂuids. In addition to our own work, there have been some additional studies which include the eﬀects of temperature [48-50], albeit at a lower-order mean-ﬁeld theory than considered here. We begin at the more physical level by stressing the analogy between condensation in this composite boson or fermionic superﬂuid and condensation in a gas of ideal point bosons. Our microscopic theory treats self-consistently two-particle and one-particle Green’s functions on an equal footing. Because the physics is so simple and clear, we can fairly readily anticipate the form of the central equations of this BCS-BEC generalization of BCS theory. It is important to stress, however, that these equations can be derived more rigorously from a truncated series of equations of motion for the appropriate Green’s functions [16, 51]. There are three principal equations which govern Bose condensation: the vanishing of the bosonic chemical potential at all T ≤ Tc is the ﬁrst. We will refer to this condition as the “BEC condition”. It is related to the usual Thouless criterion for superconductivity, but the latter is generally associated only with the temperature Tc . The second equation is the boson number equation. All “bosons” must be accounted for as either condensed or non-condensed. The third equation is the number of non-condensed “bosons”, which are created by thermal excitations. This is determined simply by inserting the known

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Table I. – BCS theory by way of BEC analogy. Here we compare condensation in composite and point bosons; μB is the bosonic chemical potential, N0 and NT are the number of condensed and non-condensed bosons, respectively. We deﬁne μpair as the chemical potential for the noncondensed pairs. Here Δ(T ) is the total fermionic gap which contains contributions from the non-condensed (Δ2pg ) and condensed terms (Δ2sc ). In the strict BCS limit Δpg = 0, so that the order parameter and gap are identical.

Pair chemical potential Total “number” of pairs Non-condensed pairs

Composite bosons

Point bosons

μpair = 0, T ≤ Tc Leads to BCS gap equation for Δ(T )

μB = 0, T ≤ Tc

Δ2 (T ) = Δ2sc (T ) + Δ2pg (T ) P ZΔ2pg = q =0 b(Ωq )

N = N0 + N T P NT = q =0 b(Ωq )

excitation spectrum of the excited pairs or bosons, into the Bose distribution function b(x). With this equation, and the second equation, one can then deduce the number of condensed bosons. These three central equations for bosons are indicated in table I, on the far right, for true point bosons, and in the second column for the composite bosons which appear in fermionic superﬂuids. For these composite bosons the quantity which provides a measure of the “number” of bosons (N ) is given by Δ2 (T ) (up to a constant coeﬃcient, Z). How does one quantitatively establish the appropriate “boson number” for the fermionic case? This is determined via the self-consistent gap equation for Δ(T ), which, in turn, is determined using the ﬁrst condition: that the pair chemical potential μpair is zero at and below Tc . How does one compute the number of excited pairs? Once the gap equation is interpreted in terms of the appropriate non-condensed pair propagator (see below), then one knows the related excitation spectrum Ωq of this propagator. The quantity Z which appears in the last equation of table I (for the composite bosons) gives the relation between the gap associated with non-condensed pairs (Δ2pg ) and the number of all pairs [ b(Ωq )]. It can be readily calculated in this theory; once one has the non-condensed pair propagator, Z appears as the inverse residue. (Deep in the BEC regime [52], Z is relatively simple to compute, for here the boson number density approaches the asymptote n/2.) More precisely, the total number of bosons in the present case has to be determined self-consistently through the gap equation for Δ(T ). It also involves the fermion number equation through the related fermionic chemical potential. To be consistent with the ground-state variational conditions, the vanishing of the pair chemical potential is associated with a particular form for the pair propagator involving dressed Green’s functions. These, in turn, determine the fermionic chemical potential through the fermion number equation. . 2 1. Microscopic T -matrix scheme. – Next, we implement this picture microscopically via a T -matrix approximation. We include spin indices throughout so that it will be clear

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Finite-temperature effects in ultracold Fermi gases

how to apply this scheme to spin-polarized superﬂuids. This means that we consider the coupled equations between the particles (with propagator G) and the pairs (with propagator t(P )) and drop all higher-order terms. This theory does not include direct “boson-boson” interactions, although the pairs do interact indirectly via the fermions, in an averaged or mean-ﬁeld sense. Here, for all T ≤ Tc , the BEC condition is interpreted as requiring that the pair chemical potential μpair associated with the non-condensed pairs, vanish. Within a T -matrix scheme, the pair propagator is given by −1 t−1 + χ(P ), pg (P ) = U

(2.2)

where χ is the pair susceptibility. The function χ(P ) is, in many ways, the most fundamental quantity we introduce. We will show that one obtains consistent answers between T -matrix–based approaches and the BCS-Leggett ground-state equations, provided the components of the pair susceptibility, deﬁned by (2.3)

χ(P ) =

1 χ↑↓ (P ) + χ↓↑ (P ) 2

are given by the product of one dressed and one bare Green’s function (2.4)

χ↑↓ (P ) =

G0↑ (P − K)G↓ (K),

χ↓↑ (P ) =

K

G0↓ (P − K)G↑ (K),

K

where P = (iΩl , p), and G and G0 are the full and bare Green’s functions respectively. Here G−1 ¯ 2 k 2 /2m is the kinetic energy of fermions, 0,σ (K) = iωn −ξk,σ , ξk,σ = k −μσ , k = h and μσ is the fermionic chemical potential for spin σ =↑, ↓. Throughout this paper, we take ¯h = 1, kB = 1, and use the four-vector notation K ≡ (iωn , k), P ≡ (iΩl , q), K ≡T n k , etc., where ωn = (2n + 1)πT and Ωl = 2lπT are the standard odd and even Matsubara frequencies [53] (where n and l are integers). We now evaluate the BEC condition (2.5)

−1 t−1 + χ(0). pg (0) = 0 = U

The one-particle Green’s function for fermions with spin σ is (2.6)

−1 G−1 σ (K) = G0σ (K) − Σσ (K) = iωn − ξkσ − Σσ (K),

where σ ¯ ≡ −σ, and the self-energy Σσ is of the BCS-like form

(2.7)

Σσ (K) = −Δ2 G0¯σ (−K) =

Δ2 . iω + ξk¯σ

It should be noted that we use a contact potential so that the symmetry factor ϕk associated with the pairing interaction is trivially ϕk = 1. For a non-trivial ϕk , one

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only needs to replace Δ with Δϕk in eq. (2.7). We will see below how this form for the self-energy very naturally arises (below Tc ) in a T -matrix approach. Thus G−1 σ (K) = iω − ξkσ −

(2.8)

Δ2 . iω + ξk¯σ

Now we are in position to calculate the pair susceptibility at P = 0 (2.9)

χ(0) = χ↑↓ (0) = χ↓↑ (0) = −

K

1 . (iωn − Ek↓ )(iωn + Ek↑ )

Substituting this expression into our BEC condition eq. (2.5), we obtain the gap equation (2.10)

1 1 − f (Ek↓ ) − f (Ek↑ ) 1 1 − 2f¯(Ek ) 0= + + = U 2Ek U 2Ek k

k

after carrying out the Matsubara summation. Here μ = (μ↑ + μ↓ )/2 and h = (μ↑ − μ↓ )/2, 2 Ek = ξk + Δ2 , Ek↑ = −h + Ek and Ek↓ = h + Ek , where ξk = k − μ. In addition, we deﬁne the average f¯(x) ≡ [f (x + h) + f (x − h)]/2, where f (x) is the Fermi distribution function. The coupling constant U can be replaced in favor of the dimensionless parameter, 1/kF a, via the relationship m/(4πa) = 1/U + k (2 k )−1 , where a is the two-body swave scattering length, and kF is the non-interacting Fermi wave vector for the same total number density in the absence of population imbalance. Therefore the gap equation can be rewritten as 1 − 2f¯(Ek ) m 1 = (2.11) − − . 4πa 2Ek 2 k k

Here the “unitary scattering” limit corresponds to resonant scattering where a → ∞. This scattering length is tunable by magnetic-ﬁeld application and we say that we are on the BCS or BEC side of resonance, depending on whether the ﬁelds are higher or lower than the resonant ﬁeld, or alternatively whether a is negative or positive, respectively. Finally, in terms of Green’s functions, we readily arrive at the number equations: nσ = K Gσ (K), which are consistent with their ground-state counterparts (2.12)

nσ =

f (Ekσ )u2k + f (Ek¯σ )vk2 , k

where the coherence factors u2k , vk2 = (1 ± ξk /Ek )/2. Next we use this T -matrix scheme to derive eq. (2.7) and separate the contribution from condensed and non-condensed pairs. The diagrammatic representation of our T -matrix scheme is shown in ﬁg. 4. The ﬁrst line indicates tpg , and the second the total self-energy. One can see throughout the combination of one dressed and one bare Green’s

759

Finite-temperature effects in ultracold Fermi gases

tpg

Σpg

Σsc tpg

Σ

tsc

Fig. 4. – T -matrix and self-energy diagrams for the present T -matrix scheme. The self-energy comes from contributions of both condensed (Σsc ) and non-condensed (Σpg ) pairs. Note that there is one dressed and full Green’s function in the T -matrix. Here tpg represents the propagator for the non-condensed pairs.

function, as represented by the thick and thin lines. The self-energy consists of two contributions from the non-condensed pairs or pseudogap (pg) and from the condensate (sc). There are, analogously, two contributions in the full T -matrix (2.13)

t = tpg + tsc , U , 1 + U χ(P )

(2.14)

tpg (P ) =

(2.15)

tsc (P ) = −

P = 0,

Δ2sc δ(P ), T

where we write Δsc = −U k c−k↓ ck↑ . Similarly, we have for the fermion self-energy pg t(P )G0,¯σ (P − K). (2.16) Σσ (K) = Σsc σ (K) + Σσ (K) = P

We can see at once that (2.17)

Σsc σ (K) =

tsc (P )G0,¯σ (P − K) = −G0,¯σ (−K)Δ2sc .

P

The vanishing of the pair chemical potential implies that (2.18)

−1 t−1 + χ(0) = 0, pg (0) = U

T ≤ Tc .

Moreover, a vanishing chemical potential means that tpg (P ) diverges P = 0. Thus, we may approximate [54] eq. (2.16) to yield (2.19)

Σσ (K) ≈ −G0,¯σ (−K)Δ2 ,

where (2.20)

Δ2 (T ) ≡ Δ2sc (T ) + Δ2pg (T ),

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Importantly, we are led to identify the quantity Δpg Δ2pg ≡ −

(2.21)

tpg (P ).

P =0

Note that in the normal state (where μpair is non-zero) eq. (2.19) is no longer a good approximation. We now have a closed set of equations for addressing the ordered phase. This approach can be readily generalized [55] to treat more exotic polarized phases such as the LOFF state [23,25,24]. We can similarly extend this approach to temperatures somewhat above Tc , by self-consistently including a non-zero pair chemical potential. This is a necessary step in addressing a trap as well [56]. Additionally, the propagator for non-condensed pairs can now be quantiﬁed, using the self-consistently determined pair susceptibility. At small four-vector P , we may expand the inverse of tpg , after analytical continuation (iΩl → Ω + i0+ ), to obtain

p2 2 t−1 ≈ a Ω + Z Ω − + μ + iΓ , 1 pair P pg 2M ∗

(2.22)

where the imaginary part ΓP → 0 rapidly as p → 0 below Tc . Because we are interested in the moderate- and strong-coupling cases, we drop the a1 Ω2 term in eq. (2.22), and hence (2.23)

tpg (P ) =

Z −1 , Ω − Ωp + μpair + iΓP

where we associate Ωp ≈

(2.24)

p2 . 2M ∗

This establishes a quadratic pair dispersion and deﬁnes the eﬀective pair mass, M ∗ . They can be calculated via a small p expansion of χ(P ), (2.25)

Z=

/ ∂χ // , ∂Ω /Ω=0,p=0

/ 1 1 ∂ 2 χ // = − . 2M ∗ 6Z ∂p2 /Ω=0,p=0

Finally, one can rewrite eq. (2.21) as (2.26)

Δ2pg (T ) = Z −1

b(Ωp ).

p

We now return to the strong analogies between our T -matrix approach and Bose condensation of point bosons, as summarized in table I. We have three central equations.

Finite-temperature effects in ultracold Fermi gases

761

1) The pair chemical potential must vanish at and below Tc , (2.27)

μpair = 0,

(T ≤ Tc ).

Importantly this condition leads to the mean-ﬁeld gap equation derived in eq. (2.10). 2) There must be a conservation of the total number of (composite) “bosons” in the system. For this condition, our central equation is eq. (2.20). Here it is understood that the number of “bosons” is eﬀectively represented by the parameter Δ2 (T ). Unlike the point boson case, the “total boson number” is temperature dependent and has to be self-consistently determined. 3) The number of non-condensed pairs is readily computed in terms of the pair dispersion, just as in conventional BEC. For this condition our central equation is eq. (2.26). Then, just as in conventional BEC, the number of condensed bosons (proportional to Δ2sc ) is determined by the diﬀerence between Δ2 (T ) and Δ2pg (T ). This, in turn, determines Tc as the lowest temperature(s) in the normal state at which non-condensed pairs exhaust the total weight of Δ2 so that Δ2pg = Δ2 . Solving for the “transition temperature” in the absence of pseudogap eﬀects [48-50] leads to the quantity TcM F . More precisely, TcM F should be thought of as the temperature at which the excitation gap Δ(T ) vanishes. This provides a reasonable estimate, for the pairing onset temperature T ∗ (when a stable superﬂuid phase exists). This is distinguished from the transition temperature. We note that T ∗ represents a smooth crossover rather than an abrupt phase transition. It should be stressed that the dispersion relation for the non-condensed pairs is quadratic. While one will always ﬁnd a linear dispersion in the collective mode spectrum [57], within the present class of BCS-BEC crossover theories, the restriction to a T -matrix scheme means that there is no feedback from the collective modes onto the pair excitation spectrum. In eﬀect, the T -matrix approximation does not incorporate pair-pair interactions at a level needed to arrive at this expected linear dispersion in the pair excitation spectrum. Nevertheless, this level of approximation is consistent with the underlying ground-state wave function. 3. – Behavior of Tc and trap eﬀects Before turning to experiment, it is important to discuss the behavior of Tc which is plotted as a function of scattering length in the left panel of ﬁg. 5 for the homogeneous case, presuming s-wave pairing. Starting from the BCS regime this ﬁgure shows that Tc initially increases as the interaction strength increases. However, this increase competes with the opening of a pseudogap or excitation gap Δ(Tc ). Technically, the pairs become eﬀectively heavier before they form true bound states. Eventually Tc reaches a maximum (very near unitarity) and then decreases slightly until ﬁeld strengths corresponding to the point where μ becomes zero. At this ﬁeld value (essentially where Tc is minimum), the system becomes a “bosonic” superﬂuid, and beyond this point Tc increases slightly to reach the asymptote 0.218EF corresponding to an ideal Bose gas.

762

K. Levin and Q. Chen 0.5

0.3

0.4

0.2 Tc/TF

Tc /TF

0.3

0.2

0.1 0.1

0

-2

0

2 1/kFa

4

0

-2

0

2

4

1/kFa

Fig. 5. – Typical behavior of Tc as a function of 1/kF a in a homogeneous system (left panel) and in a trapped Fermi gas (right panel). Tc follows the BCS predictions and approaches the BEC asymptote 0.218TF and 0.518TF in the homogeneous and trapped cases, respectively. In contrast to the homogeneous case, the BEC asymptote in a trap is much higher due to a compressed proﬁle for trapped bosons. In the homogeneous case, Tc reaches a maximum around 1/kF a = 0 and a minimum around where μ = 0. In the trapped case, this maximum/minimum behavior is washed out largely by the shrinking cloud size as 1/kF a increases.

Trap eﬀects change these results only quantitatively as seen in the right panel of ﬁg. 5. Here the maximum in Tc may no longer be visible. The calculated value of Tc (∼ 0.3TF ) at unitarity is in good agreement with experiment [8,58] and other theoretical estimates [59]. To treat these trap eﬀects, one introduces the local density approximation (LDA) in which Tc is computed under the presumption that the chemical potential μ → μ − V (r). Here we consider a spherical trap with V (r) = 12 mω 2 r2 . The Fermi energy EF is determined by the total atom number N via EF ≡ kB TF = h ¯ ω(3N )1/3 ≡ ¯h2 kF2 /2m, where kF is the Fermi wave vector at the center of the trap. It can be seen that the homogeneous curve is eﬀectively multiplied by an “envelope” curve when a trap is present. This envelope, with a higher BEC asymptote, reﬂects the fact that the particle density at the center of the trap is higher in the bosonic, relative to the fermionic case. In this way Tc is relatively higher in the BEC regime, as compared to its counterpart in the homogeneous case. Figure 6 is a central one, for it prepares us for understanding various experiments. It presents a plot of the position-dependent excitation gap Δ(r) and particle density n(r) proﬁle over the extent of the trap. An important point needs to be made: because the gap is largest at the center of the trap, bosonic excitations will be dominant there. At the edge of the trap, by contrast, where fermions are only weakly bound (since Δ(r) is small), the excitations will be primarily fermionic. We will see the implications of these observations as we examine thermodynamic [60] and radio frequency (RF) spectra data [61] in the ultracold gases. One should appreciate that temperature is not straightforward to measure in these cold gases; it is most readily measured at the Fermi gas endpoint [3] or in the deep BEC

763

Finite-temperature effects in ultracold Fermi gases 1

1/kFa = 0

n(r)/2n(0), 'EF

0.8

'EF

0.6

T/TF = 1/4

n(r)/n(0) u

0.4

0.2

0

0

0.2

0.6

0.4

0.8

1

r/RTF

Fig. 6. – Typical spatial proﬁle for density n(r) and fermionic excitation gap Δ(r) of a Fermi gas in a trap. The curves are computed at T = TF /4 and at unitarity, where 1/kF a = 0. Here RTF is the Thomas-Fermi radius, which gives the cloud size at T = 0 in the non-interacting limit.

regime [62]. At unitarity, the physical temperature can be extracted using phenomenological ﬁts to the particle density proﬁles based on the universality hypothesis [63, 58, 64] with proper re-calibration [8, 65] below Tc . For a more general magnetic ﬁeld, one has to resort to adiabatic sweep thermometry [60, 66]. Here, the magnetic ﬁeld of interest is accessed via a slow, adiabatic, or isentropic, magnetic-ﬁeld sweep starting from either the BCS or BEC endpoints, where the temperature (and the entropy [67]) are known. A ﬁnite-temperature theory of BCS-BEC crossover is required to calculate the entropy [60] at general magnetic ﬁelds. In this way, the physical temperature can be associated with the measured endpoint temperature. Indeed, we will see below that the temperature which appears in the measured superﬂuid phase diagram [3] or in the RF pairing gap experiments [62] is given in terms of the endpoint temperature. 4. – Experimental evidence for a pseudogap in cold gases Our ﬁnite temperature generalization of the BCS-like ground state has introduced the concept of a “pseudogap”. This pseudogap in the fermionic spectrum should be viewed as synonymous with the concept of non-condensed pairs, or with pairs which have a ﬁnite center-of-mass momentum. They are important both above and below Tc . In this section we want to explore the evidence for these non-condensed pairs using three diﬀerent experiments: density proﬁles, normal-state thermodynamics and RF pairing gap spectroscopy. In this section we will consider the case of an unpolarized gas. In ﬁg. 7 we plot a decomposition of the particle density proﬁles [65] for various temperatures above and below Tc . The various color codes (or gray scales) indicate the condensate along with the non-condensed pairs and the fermions. This decomposition is based on the superﬂuid density so that all atoms participate in the condensation at T = 0.

764

K. Levin and Q. Chen

0.06 0.04

Density Profiles

0.02

n np ns nQP

Unitary

T/Tc = 1.5

0.5

1.0

0.25

0.75

0

0 0.04 0.02 0 0.04 0.02 0

0 0.5 1 1.5 1 0.5 0 0.5 1 1.5 0 0.5 1 1.5 1 0.5 0 0.5 1 1.5 r/RTF r/RTF

Fig. 7. – Decomposition of density proﬁles at various temperatures at unitarity. Here green refers to the condensate, red to the non-condensed pairs and blue to the excited fermionic states. Tc = 0.27TF , and RTF is the Thomas-Fermi radius. The presence of non-condensed pairs is essential [65] for explaining why there are no sharp features in these proﬁles, associated with the interface of the normal and superﬂuid regions. Here ns , np , and nQP denote density of superﬂuid, incoherent pairs, and fermioinic quasiparticle, respectively.

An important observation should be noted. The non-condensed pairs are responsible for smoothing out what otherwise would be a discontinuity between the fermionic and condensate contributions. This leads to a featureless proﬁle, in agreement with experiment [68,69]. Indeed, these experimental observations presented a challenge for previous theories [70,63] which ignored non-condensed pairs, and therefore predicted an eﬀectively bimodal proﬁle with a kink at the edge of the superﬂuid core. One can see from the ﬁgure that even at Tc , the system is diﬀerent from a Fermi gas. That is, non-condensed pairs are present in the central region of the trap when the condensate is gone. Even at T /Tc = 1.5 there is a considerable fraction of non-condensed pairs. It is not until around T ∗ = 2Tc for this unitary case, that non-condensed pairs have ﬁnally disappeared. We next turn to a detailed comparison of theory and experiment for thermodynamics. Figure 8 presents a plot of energy E as a function of T comparing the unitary and non-interacting regimes. The solid curves are theoretical while the data points are measured in 6 Li [8]. There has been a recalibration of the experimental temperature scale in order to plot theory and experiment in the same ﬁgure. The latter was determined via Thomas-Fermi ﬁts to the density proﬁles. To arrive at the calibration, we applied the same ﬁts to the theoretically produced density proﬁles, examples of which appear in ﬁg. 7. Good agreement between theory and experiment is apparent in ﬁg. 8. In the ﬁgure, the temperature dependence of E reﬂects primarily fermionic

765

Finite-temperature effects in ultracold Fermi gases

Theory, noninteracting Theory, unitary Noninteracting Unitary

E/EF

4

2

0

Tc = 0.29 0

0.5

1

1.5

T/TF Fig. 8. – Pseudogap eﬀects as apparent from thermodynamics. From ref. [8]. The fact that the experimental data (symbols) (and the two theoretical curves) for non-interacting and unitary Fermi gases do not merge until higher T ∗ > Tc is consistent with the presence of a normal-state pseudogap.

excitations at the edge of the trap, although there is a small bosonic contribution as well. Importantly one can see the eﬀect of a pseudogap in the unitary case. The temperature T ∗ is visible from the plots as that at which the non-interacting and unitary curves merge. This corresponds roughly to T ∗ ≈ 2Tc . Measurements [62] of the excitation gap Δ can be made more directly, and, in this way one can further probe the existence of a pseudogap. This pairing gap spectroscopy is based on using a third atomic level, called |3, which does not participate in the superﬂuid pairing. Under application of RF ﬁelds, one component of the Cooper pairs, called |2, is excited to state |3. If there is no gap Δ then the energy it takes to excite |2 to |3 is the atomic level splitting ω23 . In the presence of pairing (either above or below Tc ) an extra energy Δ must be input to excite the state |2, as a result of the breaking of the pairs. Figure 9 shows a plot of the spectra for 6 Li near unitarity for four diﬀerent temperatures, which we discuss in more detail below. In general for this case, as well as for the BCS and BEC limits, there are two peak structures which appear in the data and in the theory [71, 61]: the sharp peak at ω23 ≡ 0 which is associated with “free” fermions at the trap edge and the broader peak which reﬂects the presence of paired atoms; more precisely, this broad peak derives from the distribution of Δ in the trap. At high T (compared to Δ), only the sharp feature is present, whereas at low T only the broad feature remains. The sharpness of the free-atom peak can be understood as coming from a large phase space contribution associated with the 2 → 3 excitations [61]. These data alone do not directly indicate the presence of superﬂuidity, but rather they provide strong evidence for pairing.

766

K. Levin and Q. Chen

Fig. 9. – Experimental RF Spectra for 6 Li at unitarity at 834 G. The temperatures labeled in the ﬁgure were computed theoretically at unitarity based on adiabatic sweeps from BEC. The two top curves, thus, correspond to the normal phase, thereby, indicating pseudogap eﬀects. Here EF = 2.5 μK, or 52 kHz. From ref. [62].

It is interesting to return to discuss the temperatures in the various panels. What is measured experimentally are temperatures T which correspond to the temperature at the start of an adiabatic sweep from the BEC limit to unitarity. Here ﬁts to the BEC-like proﬁles are used to deduce T from the shape of the Gaussian tails in the trap. Based on knowledge [60] about thermodynamics (energy E in the previous ﬁgure or, equivalently, entropy S), and given T , one can then compute the ﬁnal temperature in the unitary regime, assuming S is constant. Indeed, this adiabaticity has been conﬁrmed experimentally in related work [69]. We ﬁnd that the four temperatures are as indicated in the ﬁgures. Importantly, one can conclude that the ﬁrst two cases correspond to a normal state, albeit not far above Tc . In this way, these ﬁgures suggest that a pseudogap is present as reﬂected by the broad shoulder above the narrow free-atom peak. 5. – Establishing superﬂuidity in cold Fermi gases From the time of the earliest discoveries [72, 1-8] there was a drive to establish the existence of superﬂuidity which is more diﬃcult on the BCS than on the BEC side [2, 1] of the resonance. There has been a sequence of experiments which have eﬀectively made this case, beginning ﬁrst with fast-sweep experiments [3, 4], then thermodynamical measurements [8], and ﬁnally detection of quantized vortices [5]. We discuss the ﬁrst two methodologies here in the context of our theoretical framework. We limit our discussion ﬁrst to the case of unpolarized superﬂuids. The left panel in ﬁg. 10 is a plot of the ﬁrst phase diagram representing the condensate fraction vs. 1/kF0 a, as obtained in ref. [3] for 40 K. Subsequently, similar studies [4] were undertaken for 6 Li. The ﬁgure is based on starting the system oﬀ in the free Fermi gas regime where it can be associated with an initial known temperature (T /TF )0 , and then

767

Finite-temperature effects in ultracold Fermi gases 0.25

0.25

(T/TF)0

0.15 0.10 0.05

0.15 0.1

0.05 0

0.5

0

1/k0Fa

-0.5

-1.0

-1.5

0 0.01 0.025 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.2

(T/TF)0

-0.020 0.010 0.025 0.050 0.075 0.100 0.125 0.150 0.175

0.20

0 1.0

NS /N =

N0/N =

1

0.5

0

-0.5 1/kF0 a

-1

-1.5

Fig. 10. – Earliest evidence for superﬂuidity: phase diagram of 40 K as a function of (T /TF )0 and 1/kF0 a. This compares both experiment (left panel) and theoretical computations (right panel). In the left panel, a contour plot of the measured condensate fraction N0 /N as a function of 1/kF0 a and eﬀective temperature (T /TF )0 is compared with theoretically calculated contour lines at Ns /N = 0 (at the superﬂuid transition, black curve) and 0.01 (red curve). The experimental data have an overall systematic uncertainty of approximately 0.1 in 1/kF0 a. The experimental contour at N0 /N = 0.01 and the theoretical line for Ns /N = 0.01 are in good agreement. The 0 dashed line represents the naive BCS result Tc /TF0 ≈ 0.615eπ/2kF a . The right panel represents a more complete theoretically computed equilibrium phase diagram, with contour lines for Ns /N . Here all temperatures are measured in the Fermi-gas regime. From ref. [73].

adiabatically sweeping closer to unitarity. (Here TF = TF0 , kF = kF0 and (T /TF )0 are all measured in the non-interacting Fermi gas limit.) Once the near-unitary gas (of ﬁxed entropy) is obtained, a fast sweep is made to the BEC regime, where the condensate fraction can be read oﬀ from a bi-modal proﬁle. The presumption here, for which there is considerable experimental support [3, 4], is that, even if the condensate fraction is not conserved upon a fast sweep to BEC, the presence or absence of a condensate will be preserved. The time frame for the sweep will not allow a condensate to form in the BEC if there were none present near unitarity, nor will it allow a condensate to disappear if it was present initially. The lines drawn on top of the experimental contour plots are the calculated [73] condensate fraction contours as a function of the adiabatic sweep-projected temperatures (T /TF )0 for a 0% and 1% condensate fraction. These essentially correspond to the normal-superﬂuid phase boundary which is expected to be rather well measured in these sweep experiments. The ﬁgure to the right presents a full plot of the theoretical condensation fraction, importantly, measured with respect to the adiabatic sweep-projected temperatures (T /TF )0 . This, then, is the theoretical phase diagram. For the 1% case, the overall trends yield good agreement between theory and experiment, except for the small “overshoot” (of unknown origin) which appears in the BEC side of the data. The second-generation experiments which helped to establish superﬂuidity were based on thermodynamical measurements. In ﬁg. 11 we show data from ref. [8]. What is plotted is the measured energy as a function of temperature on a log-log scale. This temperature represents a theoretical recalibration of an eﬀective proﬁle-measured temperature T˜.

768

K. Levin and Q. Chen

1

Noninteracting Unitary Noninteracting Unitary Power law fit

T /TF = 0.27

0.8

T/TF

E(theat)/E0 -1

10

0.1

Tc

0.2 0

0.01

0.1

T/TF

0

0.2

~ 1+E T

0.8

1

Fig. 11. – Evidence for a superﬂuid phase via plots of the energy E vs. physical temperature T . The upper curve (long dashed line) and data points correspond to the BCS or essentially free Fermi gas case, and the lower curve and data correspond to unitarity. The latter provide indications for a phase transition via a slope change. The thin dashed line represents a power law ﬁt to the unitary data below the transition. The inset shows how temperature must be recalibrated below Tc . Here β is a parameter which characterizes the interaction energy at unitarity for T = 0, and T˜ is an empirical temperature scale. From ref. [8].

The key feature here is that the data (indicated by the points) show an abrupt change at a temperature one can call Tc . This abrupt change occurs for the unitary scattering case. No such feature is seen for the non-interacting Fermi gas, also plotted in the ﬁgure. Furthermore, this abrupt change in E vs. T is seen even in the raw data points (not shown here), without appealing to a theoretical re-calibration. All of this is very suggestive of a speciﬁc-heat discontinuity, which is to be associated with a phase change —presumably to a superﬂuid phase. The inset in the ﬁgure shows how the eﬀective temperature T˜ which is obtained from a Thomas-Fermi ﬁt to the calculated density proﬁle compares with the physical temperature T . The inset shows that a recalibration is necessary only below Tc , to account for the presence of a condensate. We stress that the observation of a phase change made by the Duke group [8] is not dependent on this recalibration. Hence these experiments provide good evidence for a transition between a normal and superﬂuid phase. The last-generation experiment to make the case for superﬂuidity was the observation of quantized vortices by the MIT group [5]. These very convincing experiments will be discussed elsewhere in this lecture series.

Finite-temperature effects in ultracold Fermi gases

769

6. – Fermi gases with imbalanced spin population The latest excitement in the ﬁeld of trapped fermions pertains to gases with deliberately imbalanced spin populations [20-22]. In large part this is motivated by interest from theorists in other disciplines such as dense QCD and (isospin asymmetric) nuclear matter [11-13, 74]. From the condensed-matter viewpoint there has been an underlying interest in exotic and intriguingly elusive phases associated with Zeeman eﬀects in superconductors, such as that proposed by Larkin and Ovchinnikov and by Fulde and Ferrell (LOFF). In the LOFF state [23] the condensate has a net momentum of a ﬁxed q or of +q and −q. Even more elaborate crystalline lattices of various qi have also been contemplated [24]. An additional, and very important motivation for these studies is associated with the recent claims [22] that when there is a population imbalance, the density proﬁles will indicate whether or not superﬂuidity is present and they, moreover, provide an internal mechanism for thermometry. Thus, because these experiments are claimed to identify Tc itself, and because they pertain to thermometry, it should be clear that a theoretical analysis of these experiments requires an understanding of the eﬀects of T = 0. We begin our discussion by summarizing some key experimental observations [20-22]. In a trapped cloud there appears to be a superﬂuid core, which, at the lowest temperatures, is unpolarized. Outside of this core there is a normal region, in which both spin components are present and this carries a signiﬁcant fraction of the polarization. Beyond this “mixed, normal region” there is a free Fermi gas composed of the majority spin species, which carries additional polarization. At suﬃently low temperatures, there appears to be a form of phase separation in which the superﬂuid and normal phases are associated with zero and ﬁnite polarization, respectively. We turn now to a theoretical understanding [56,75] of these experiments at T = 0 ﬁrst for the case of a homogeneous gas, and then later in the trapped conﬁguration. There has been some work along these same lines elsewhere in the literature [49], although without incorporating non-condensed pair eﬀects. We focus on the Sarma or breached pair phase [76]. The ﬁgures we present do not include the more exotic LOFF or phaseseparated states. The formal structure for addressing the former [55] is very similar to that of the Sarma state; we will defer a brief discussion of these until the end. To help with the clarity of the presentation, we state our major conclusions for the homogeneous and trapped conﬁgurations in the unitary regime, at the outset. – In the homogeneous case temperature serves to stabilize the polarized superﬂuid phase. Such a phase is unstable in the ground state. – In a trap at low T , the superﬂuid core remains unpolarized. Except at very low T , spin polarization will be mostly carried by fermions coexisting with strongly interacting, thermally excited non-condensed pairs which appear outside the core. Thus, pseudogap eﬀects, which have been emphasized throughout, are very important here.

770

K. Levin and Q. Chen 0.3 p = 0.05

p = 0.1

Tc /TF

0.2

0.1

(b)

(a) 0 p = 0.3

p = 0.16

Tc /TF

0.2

0.1

(d)

(c) 0 -3

-2

-1

0

1/kFa

1

2

-3

-2

-1

0

1

2

3

1/kFa

Fig. 12. – Typical behavior of Tc as a function of 1/kF a for spin polarizations p = 0.05, 0.1, 0.16, and 0.3. The Tc curve splits into two disconnected curves around p = 0.14. This ﬁgure should be compared with left panel of ﬁg. 5 for the unpolarized case. From ref. [75].

– With increasing T , polarization tends to continuously penetrate into the superﬂuid core of the trap, until at, or even below Tc , the polarization is uniformly distributed throughout the cloud. – This superﬂuid Sarma or breached pair phase is limited in the amount of polarization it can accommodate, especially near unitarity. This applies to both traps and homogeneous systems. Figure 12 presents a plot of Tc for a homogeneous system as a function of 1/kF a for various polarizations. This ﬁgure should be compared with the left panel of ﬁg. 5. With this comparison, one sees at once that there can be no superﬂuidity in the deep BCS regime, once the polarization is diﬀerent from zero. And when superﬂuidity ﬁrst appears on the BCS side of resonance it is associated with two Tc ’s at given 1/kF a. The larger the polarization p, the harder it is for a homogeneous system to support superﬂuidity, except in the BEC regime. To understand the meaning of these two Tc ’s we plot the superﬂuid density ns (T ) as a function of T in ﬁg. 13, for several diﬀerent values of 1/kF a. If one focuses on the unitary case, for deﬁniteness, one can see that ns vanishes at two diﬀerent temperatures. The lower Tc corresponds to the onset of superﬂuidity. At temperatures below this, the (breached pair or Sarma) state is unstable. Similarly, at the upper Tc , superﬂuidity disappears in the usual way; it is destroyed by thermal excitations. For positive values of 1/kF a, the ﬁgure shows that the ns (T ) curves stop abruptly. This is indicated by the dotted segments of the curves which represent thermodynamically unstable solutions.

771

Finite-temperature effects in ultracold Fermi gases 1 1/kFa=1.5

0.8 1.0

ns /n

0.6

0.5

0.4 0.2

0.2 -0.5 0

0

0

0.2

0.1

T/TF

Fig. 13. – Normalized superﬂuid density ns /n as a function of T /TF at p = 0.1 for various 1/kF a from BCS to BEC. The dotted (segments of the) curves represent unstable solutions. The fact that at resonance and on the BCS side there are 2 Tc ’s is consistent with the previous ﬁgure. From ref. [75].

The fact that Tc may be double-valued could have been anticipated in a simpler set of calculations performed at the strict mean-ﬁeld level, and discussed elsewhere in the literature [74]. Here one solves eqs. (2.11) and (2.12) only, without imposing (2.20) and (2.21). In ﬁg. 14, we present a plot of TcMF as a function of 1/kF a for a range of p. 0.4 1/kFa = 0.2

0.6

-0.5 -0.65

0 0.6

0.4

0.7

0.2

0.9

0.2

0.2 0

0.99

0.4

-0.3

0.5

T/TF 0.3

TcMF/TF

0.3

'/EF

-0.1

p = 0.01

0 -3

5

.0 0.02 0

-2

0.1

0.2

0.1

-1

0

1

1/kFa

Fig. 14. – Mean-ﬁeld behavior of TcMF as a function of 1/kF a for diﬀerent spin polarizations p. Shown in the inset is the pairing gap Δ(T ) at diﬀerent 1/kF a for p = 0.3 which can vanish at h2 kF2 /2m is the non-interacting Fermi energy in two distinct temperatures. Here EF ≡ kB TF ≡ ¯ MF the absence of polarization. The quantity Tc represents the pairing onset temperature T ∗ .

772

K. Levin and Q. Chen

Gaps, Gn

0.8

T/TF = 0.1

0.14

0.28

0.26

'

'sc 0.4

Gnu

Densities

0

Tc/TF = 0.27

0.02

0.01

0

0

np

nn

0.5

1

r/RTF

0

0.5

r/RTF

1

0

0.5

r/RTF

1

0

0.5

1

1.5

r/RTF

Fig. 15. – Spatial proﬁles of the gap (Δ), order parameter (Δsc ), and the density of the up (n↑ ) and down (n↓ ) spin components and their diﬀerence (δn) for a unitary gas in a trap at T /TF = 0.1, 0.14, 0.26, and 0.28, from left to right. Except at very low T , most of the polarization is carried by fermions co-existing with non-condensed pairs, i.e., in the pseudogap regime where Δsc = 0, but Δ = 0. Here Tc /TF ≈ 0.27.

This quantity can be viewed as the pairing onset temperature T ∗ , when there is a stable superﬂuid phase. In the inset of ﬁg. 14 we plot Δ(T ) at diﬀerent 1/kF a for p = 0.3. For p < 0.9 and suﬃciently low TcMF , we ﬁnd that there are two TcMF values. This structure implies that Δ is non-monotonic [74] in T , as indicated by the bottom curve in the inset of ﬁg. 14. The two zeroes of Δ represent the two values of TcMF . In contrast to the more conventional behavior (shown in the top curve for stronger pairing interaction), Δ increases initially with T at low temperature when 1/kF a is suﬃciently small. We summarize by noting that these results indicate that temperature enables pairing associated with the breached pair or Sarma state. This was also inferred in refs. [49] and [74]. In general superﬂuids, one would argue that these two eﬀects compete. We may view, then, this unusual polarized phase as an “intermediate temperature superﬂuid”. We now turn to the behavior of these superﬂuids in a trap. The same calculations are applied to the Sarma or breached pair state using the LDA to incorporate trap eﬀects. Figure 15 shows the resulting behavior at unitarity for polarization of 15% and for various temperatures from below to just above Tc . The upper panels plot the order parameter Δsc and the (total) gap parameter Δ. Superposed on these plots is the polarization δn. The lower panels present the density proﬁles for each spin state. Several important features can be gleaned from the upper panels. At the lowest temperatures the bulk of the polarization is in a region where Δsc = 0, but Δ = 0; thus polarization is excluded at low T from the superﬂuid core. Moreover, it can be seen that an excitation gap Δ is present throughout most of the cloud. Whenever Δ = 0 one can infer that both spin states are present. For non-zero Δ, the particle proﬁles are necessarily diﬀerent from those of a non-interacting gas. The bulk of the polarization appears in the “normal, mixed region”, and within this portion of the trap there are strong interactions between

773

Finite-temperature effects in ultracold Fermi gases 6

0.03 T/TF ≤ 0.05

Experiment

0.02

n↑ Atomic Density (kF3 )

2

n↓

11

Atomic Density (10 /cm3)

4

0

3

δn

2

0.01

0 0

500

Δ=0

0.01

0

Axial Position (μm)

n↓

Δsc = 0

0.015

0.005

-500

n↑

0

1

-1

T/TF = 0.06 p = 0.35

Theory

p = 0.35

-0.005

δn

-1

0

1

r/RTF

Fig. 16. – Comparison of theoretically computed density distributions of the up- and downspin components (upper row) and their diﬀerence (lower row) at unitarity with the Rice group data. The central issue here is not the quantitative comparison, but rather that (as seen in the theoretical curves) polarization is conﬁned to outside the superﬂuid core. In the Rice data, n↓ has a narrower distribution due to non-LDA eﬀects. Data from ref. [77].

the two spin states. Only at the very edge of the trap is there an exclusively majority component (and here Δ = 0). This region is occupied by a non-interacting Fermi gas and can thereby be used to set the temperature scale for the trapped cloud. We end this section with some comparisons with experiment. We address qualitative eﬀects in ﬁg. 16 and semi-quantitative eﬀects in ﬁg. 17. The former is for the unitary case and the latter is in the BEC regime. The left panels in ﬁg. 16 show data from the Rice group, ref. [77]. The upper ﬁgure plots the density proﬁles for each spin state and the lower panel, their diﬀerence. The unpolarized core is evident, as is the sharp edge beyond which polarization abruptly appears. This behavior has been interpreted [21, 77] as suggesting phase separation. Because the aspect ratios for the trap are not maintained in the proﬁles, it has been argued [21, 77] that the LDA scheme may not be appropriate for addressing experiments on these highly anisotropic traps. In the experimental data, non-LDA eﬀects lead to a narrower distribution for n↓ along the axial direction while it is broadened in the radial direction. Despite this caveat, we plot our counterpart theoretical results at T = 0.06TF (right panels) for qualitative comparison. This plot is designed primarily to introduce theoretical observations (which can be superposed, in eﬀect, on the experimental plot) concerning where the superﬂuid phase resides in the trap. This knowledge cannot be directly gleaned from these particular experiments. These low-T theoretical results are rather striking, for they make it clear that when the order parameter is present the po-

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K. Levin and Q. Chen 0.03

Theory 0.02

nn

0.01

T/TF = 0.4

np

0 0.01 0

Gn

0

0.5

0.04

1

1.5

np

0.004

0.03

nn

0.02 0.01

np

0 0.3

0.4

0.5

r/RTF

T/TF = 0.1

0

Gn

0.01 0

0

0.5

1

1.5

r/RTF

Fig. 17. – Comparison of theoretically computed density distributions of the up- and down-spin components and their diﬀerence δn at diﬀerent temperatures in the BEC regime with MIT data from ref. [22], for p = 0.6. Upper/lower panels are above/below Tc . The arrow in the bottom theoretical curve for δn indicates the condensate edge which is reﬂected in features in both n↑ and n↓ , much as in the data.

larization is excluded. Rather, the polarization appears in the Fermi-gas region outside the condensate core. At higher T and lower p (e.g., T = 0.1TF , p = 0.15), the polarization is carried largely within the “pseudogap” regime, where there are strong pairing correlations, but no long-range order. We turn ﬁnally to a comparison of data from the MIT group [22], and in the BEC regime. The experimental data (for polarizations near 0.6) are plotted on the two right panels which show the particle density proﬁles for each spin state and their diﬀerence δn. The upper panels correspond to temperatures which are believed to be in the normal state, whereas, the lower panels are for the superﬂuid phase. The lightly dotted curves in the experiment represent extrapolations of the Thomas-Fermi ﬁts to the curves at large radii. In the superﬂuid phase, the data show that the polarization at the core center is considerably smaller than it is for the normal state. One apparent diﬀerence between above and below Tc experiments, is that at low T i) the minority component has contracted into the center of the trap. ii) Another signature is a kink in the majority proﬁle. Yet another signature is that iii) there is a clear bi-modality in the minority component.

Finite-temperature effects in ultracold Fermi gases

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This behavior can be compared with the theoretical results plotted in the left two panels above and below Tc for a polarization p = 0.6. The kink in the majority component at low T can be clearly attributed to the edge of the condensate as denoted by the vertical arrow (where Δsc = 0). The bi-modality in the minority component is ampliﬁed in the lower inset to make it more evident. These theoretical plots, at a qualitative level, exhibit the three central features of the data noted above. 7. – Conclusions Throughout this paper we have stressed that temperature is important in understanding and characterizing the ultracold Fermi gases. Experimentally, one is never strictly at T = 0; as a result, there are thermal excitations of the gas which need to be characterized both theoretically and experimentally. At a formal level, we have discussed how temperature enters into the standard BCS-BEC crossover theory by leading to a distinction between the superconducting order parameter Δsc and the gap parameter Δ. This diﬀerence reﬂects the existence of pre-formed pairs above Tc and non-condensed pairs below. At a more physical level, we have shown how temperature changes the character of the gas from a superﬂuid at low T to an unusual (but strongly interacting) normal ﬂuid at moderate T , and to ultimately a free Fermi gas at high T . This strongly interacting normal ﬂuid is most interesting, for unlike the Landau Fermi liquid (or gas) which exists just above Tc in a strictly BCS superﬂuid, here there is a normal-state excitation gap or pseudogap. By looking at three diﬀerent experiments, we have provided evidence for this normal-state excitation gap (in RF pairing gap spectroscopy [62] and in thermodynamics [8] and for the below-Tc counterpart (non-condensed pairs), via the shape of the particle density proﬁles [68, 69]. In this paper we have discussed how temperature can be measured through adiabatic sweep thermometry [3] and through Thomas-Fermi ﬁts to density proﬁles [68,8]. We have shown how temperature played an important role in establishing the ﬁrst-generation and earliest evidence for superﬂuidity —based on fast-sweep experiments which yield [3] a phase diagram in the temperature vs. 1/kF a plane. It also played an important role in the second-generation indications for superﬂuidity associated with thermodynamical measurements [8]. Finally, we have addressed a new and exciting class of experiments [20-22] involving spin population imbalance and shown that here too temperature plays a critical role. From an experimental point of view, the most exciting features of these experiments are that they show how the density proﬁles can be used to establish the transition temperature Tc . Moreover, through the wings of the proﬁles, they provide a theory-independent mechanism for thermometry. From our theoretical perspective, which focuses on temperature [75,56], what is also exciting is that i) the stability and character of these polarized superﬂuids is very sensitive to temperature and ii) even in the normal state one sees strong interactions between the two spin components, which we associate with the ﬁnite-T pseudogap eﬀects we have been discussing in this paper. As stated in ref. [22], “Already at high temperatures, above the phase transition, the larger [majority] cloud’s proﬁle is strongly

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deformed in the presence of the smaller [minority] cloud, a direct signature of interaction.” Our discussion, thus far, has not addressed phase separation [42] or the exotic LOFF states [23, 25] associated with polarized superﬂuids. The latter, which correspond to a condensate with a ﬁnite momentum q, may well be ground states of these spin-polarized gases for some range of scattering lengths a, since the polarized BCS like phase (with q = 0) is only stable at intermediate temperatures. These LOFF phases are discussed elsewhere in this volume. There are arguments in the literature that, in some more general form, they play an important role near unitarity [48, 40] in trapped gases. At a theoretical level, LOFF states may be included [55] following the formalism we have outlined in this paper. Underlying the interest in this general ﬁeld of ultracold gases is the possibility that they may shed light on the high-temperature cuprate superconductors either in the context [17, 19] of the BCS-BEC crossover scenario or in the context of optical lattices and Hubbard model simulations. For the former, one can return to the question of where they would ﬁt on a phase diagram plot (such as that presented in ﬁg. 5 which addresses Tc vs. 1/kF a). When the calculations are properly redone for d-wave pairing on nearly two-dimensional lattices, the values and shape of the Tc curve are in quite good agreement with experiment [16-18]. Moreover, Tc vanishes well before the BEC limit is reached [26]; one can, then, deduce that in this scenario the cuprates are close to the unitary regime, just as are the ultracold gases. Whether or not this crossover picture turns out to be correct for high Tc , another very important rationale for its study is the possibility of generalizing what is undoubtedly the most successful theory in condensed-matter physics: BCS theory. We now understand from the ultracold gases that the nature-made superconductors to which the original theory has been applied, are only a very special case of a much more general class of superﬂuids. And it is most ﬁtting on this ﬁftieth anniversary of BCS that we pay homage to this most remarkable of theories by recognizing its even greater generality. ∗ ∗ ∗ We wish to warmly thank all our past collaborators who have contributed to the work and ﬁgures presented here: J. Stajic, Yan He and Chih-Chun Chien, as well as J. Thomas, J. Kinast and A. Turlapov, as well as M. Holland, M. Chiofalo and J. Milstein, as well as D. Jin, C. Regal and M. Greiner. This work was supported by NSF PHY-0555325 and NSF-MRSEC Grant No. DMR-0213745. REFERENCES [1] Greiner M., Regal C. A. and Jin D. S., Nature (London), 426 (2003) 537. [2] Jochim S., Bartenstein M., Altmeyer A., Hendl G., Riedl S., Chin C., Denschlag J. H. and Grimm R., Science, 302 (2003) 2101. [3] Regal C. A., Greiner M. and Jin D. S., Phys. Rev. Lett., 92 (2004) 040403. [4] Zwierlein M. W., Stan C. A., Schunck C. H., Raupach S. M. F., Kerman A. J. and Ketterle W., Phys. Rev. Lett., 92 (2004) 120403.

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[5] Zwierlein M. W., Abo-Shaeer J. R., Schirotzek A. and Ketterle W., Nature (London), 435 (2005) 170404. [6] Kinast J., Hemmer S. L., Gehm M. E., Turlapov A. and Thomas J. E., Phys. Rev. Lett., 92 (2004) 150402. [7] Bartenstein M., Altmeyer A., Riedl S., Jochim S., Chin C., Denschlag J. H. and Grimm R., Phys. Rev. Lett., 92 (2004) 203201. [8] Kinast J., Turlapov A., Thomas J. E., Chen Q. J., Stajic J. and Levin K., Science, 307 (2005) 1296. [9] Bourdel T., Khaykovich L., Cubizolles J., Zhang J., Chevy F., Teichmann M., Tarruell L., Kokkelmans S. J. and Salomon C., Phys. Rev. Lett., 93 (2004) 050401. [10] Partridge G. B., Strecker K. E., Kamar R. I., Jack M. W. and Hulet R. G., Phys. Rev. Lett., 95 (2005) 020404. [11] Gubankova E., Schmitt A. and Wilczek F., Phys. Rev. B, 74 (2006) 064505. [12] Liu W. V. and Wilczek F., Phys. Rev. Lett., 90 (2003) 047002. [13] Forbes M. M., Gubankova E., Liu W. V. and Wilczek F., Phys. Rev. Lett., 94 (2005) 017001. [14] Leggett A. J., Diatomic molecules and Cooper pairs, in Modern Trends in the Theory of Condensed Matter, edited by Pekalski A. and Przystawa J. (Springer-Verlag, Berlin) 1980, pp. 13-27. [15] Eagles D. M., Phys. Rev., 186 (1969) 456. ´ B. and Levin K., Phys. Rev. Lett., 81 (1998) 4708. [16] Chen Q. J., Kosztin I., Janko [17] Chen Q. J., Stajic J., Tan S. N. and Levin K., Phys. Rep., 412 (2005) 1. [18] Chen Q. J., Stajic J. and Levin K., Low Temp. Phys., 32 (2006) 406 (Fiz. Nizk. Temp., 32 (2006) 538). [19] Leggett A. J., Nature Phys., 2 (2006) 134. [20] Zwierlein M. W., Schirotzek A., Schunck C. H. and Ketterle W., Science, 311 (2006) 492. [21] Partridge G. B., Li W., Kamar R. I., Liao Y. A. and Hulet R. G., Science, 311 (2006) 503. [22] Zwierlein M. W., Schunck C. H., Schirotzek A. and Ketterle W., Nature (London), 442 (2006) 54. [23] Fulde P. and Ferrell R. A., Phys. Rev., 135 (1964) A550; Larkin A. I. and Ovchinnikov Y. N., Zh. Eksp. Teor. Fiz., 47 (1964) 1136 (Sov. Phys. JETP, 20 (1965) 762). [24] Casalbuoni R. and Nardulli G., Rev. Mod. Phys., 76 (2004) 263. [25] Mora C. and Combescot R., Phy. Rev. B, 71 (2006) 214504. ´ B. and Levin K., Phys. Rev. B, 59 (1999) 7083. [26] Chen Q. J., Kosztin I., Janko [27] Pieri P. and Strinati G. C., Phys. Rev. B., 71 (2005) 094520. [28] Friedberg R. and Lee T. D., Phys. Lett. A, 138 (1989) 423. [29] Deutscher G., Nature (London), 397 (1999) 410. [30] Uemura Y. J., Physica C, 282-287 (1997) 194. [31] Randeria M., Crossover from BCS theory to Bose-Einstein condensation, in BoseEinstein Condensation, edited by Griffin A., Snoke D. and Stringari S. (Cambridge University Press, Cambridge) 1995, pp. 355-92. [32] Stajic J., Milstein J. N., Chen Q. J., Chiofalo M. L., Holland M. J. and Levin K., Phys. Rev. A, 69 (2004) 063610. [33] Lee P. A., Nagaosa N. and Wen X. G., Rev. Mod. Phys., 78 (2006) 17. [34] Pieri P., Pisani L. and Strinati G. C., Phys. Rev. Lett., 92 (2004) 110401. [35] Perali A., Pieri P. and Strinati G. C., Phys. Rev. Lett., 93 (2004) 100404. `res P. and Schmitt-Rink S., J. Low Temp. Phys., 59 (1985) 195. [36] Nozie

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Chien C.-C., He Y., Chen Q. J. and Levin K., Phy. Rev. A, 73 (2006) 041603(R). Machida M. and Koyama T., Phys. Rev. Lett., 94 (2005) 140401. Sensarma R., Randeria M. and Ho T.-L., Phy. Rev. Lett., 96 (2006) 090403. Kinnunen J., Jensen L. M. and Torma P., Phy. Rev. Lett., 96 (2006) 110403. Pieri P. and Strinati G. C., Phys. Rev. Lett., 91 (2003) 030401. Sheehy D. E. and Radzihovsky L., Phys. Rev. Lett., 96 (2006) 060401. De Silva T. N. and Mueller E. J., Phys. Rev. A, 73 (2006) 051602(R). Haque M. and Stoof H. T. C., Phys. Rev. A, 74 (2006) 011602. He L. Y., Jin M. and Zhuang P. F., e-print cond-mat/0606322. Pao C. H., Wu S. T. and Yip S. K., Phys. Rev. B, 73 (2006) 132506. Pieri P. and Strinati G. C., Phys. Rev. Lett., 96 (2006) 150404. Machida K., Mizushima T. and Ichioka M., Phys. Rev. Lett., 97 (2006) 120407. Yi W. and Duan L. M., Phys. Rev. A, 73 (2006) 031604(R). Gubbels K. B., Romans M. W. J. and Stoof H. T. C., e-print cond-mat/0606330. Chen Q. J., Generalization of BCS theory to short coherence length superconductors: A BCS–Bose-Einstein crossover scenario, Ph.D dissertation, University of Chicago (2000). Stajic J., Chen Q. J. and Levin K., Phys. Rev. A, 71 (2005) 033601. Fetter A. L. and Walecka J. D., Quantum Theory of Many-Particle Systems (McGrawHill, San Francisco) 1971. ´ B. and Levin K., Physica C, 321 (1999) 113. Maly J., Janko Chen Q. J., Yan H., Chien C.-C. and Levin K., preprint, cond-mat/0608662. Chien C.-C., Chen Q. J., He Y. and Levin K., Phys. Rev. A, 74 (2006) 021602(R). Kosztin I., Chen Q. J., Kao Y. J. and Levin K., Phys. Rev. B, 61 (2000) 11662. Kinast J., Turlapov A. and Thomas J. E., Phys. Rev. Lett., 94 (2005) 170404. Perali A., Pieri P., Pisani L. and Strinati G. C., Phys. Rev. Lett., 92 (2004) 220404. Chen Q. J., Stajic J. and Levin K., Phys. Rev. Lett., 95 (2005) 260405. He Y., Chen Q. J. and Levin K., Phys. Rev. A, 72 (2005) 011602(R). Chin C., Bartenstein M., Altmeyer A., Riedl S., Jochim S., Hecker-Denschlag J. and Grimm R., Science, 305 (2004) 1128. Ho T.-L., Phys. Rev. Lett., 92 (2004) 090402. Thomas J. E., Kinast J. and Turlapov A., Phys. Rev. Lett., 95 (2005) 120402. Stajic J., Chen Q. J. and Levin K., Phys. Rev. Lett., 94 (2005) 060401. Williams J. E., Nygaard N. and Clark C. W., New J. Phys., 6 (2004) 123. Carr L. D., Shlyapnikov G. V. and Castin Y., Phys. Rev. Lett., 92 (2004) 150404. O’Hara K. M., Hemmer S. L., Gehm M. E., Granade S. R. and Thomas J. E., Science, 298 (2002) 2179. Bartenstein M., Altmeyer A., Riedl S., Jochim S., Chin C., Denschlag J. H. and Grimm R., Phys. Rev. Lett., 92 (2004) 120401. Chiofalo M. L., Kokkelmans S. J. J. M. F., Milstein J. N. and Holland M. J., Phys. Rev. Lett., 88 (2002) 090402. ¨ rma ¨ P., Science, 305 (2004) 1131. Kinnunen J., Rodriguez M. and To Kinast J., Turlapov A. and Thomas J. E., Phys. Rev. A, 70 (2004) 051401(R). Chen Q. J., Regal C. A., Greiner M., Jin D. S. and Levin K., Phy. Rev. A, 73 (2006) 041601(R). Sedrakian A. and Lombardo U., Phys. Rev. Lett., 84 (2000) 602. Chien C.-C., Chen Q. J., He Y. and Levin K., Phy. Rev. Lett., 97 (2006) 090402. Sarma G., J. Phys. Chem. Solids, 24 (1963) 1029. Partridge G. B., Li W. H., Liao Y. A., Hulet R. G., Haque M. and Stoof H. T. C., e-print cond-mat/0608455.

[52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77]

Normal-superﬂuid transition temperature in the unitary Fermi gas N. V. Prokof’ev Physics Department, University of Massachusetts - Amherst, MA 01003, USA BEC-INFM, Dipartimento di Fisica, Universit` a di Trento Via Sommarive 14, I-38050 Povo, TN, Italy

1. – Introduction In this lecture I review work done in collaboration with E. Burovski and Boris Svistunov from the University of Massachusetts, Amherst, and Matthias Troyer from ETH, Z¨ urich. It is based on results published in refs. [1, 2]. I will focus on the transition temperature from normal to superﬂuid state and the normal-state thermodynamics in a system of fermions with attractive short-range potential, ﬁne tuned to have an inﬁnite s-wave scattering length, as → ∞. More speciﬁcally, if r0 is the range of the interaction potential, then we are interested in the unitary limit at vanishing particle density, nr03 → 0. For the system of ultracold fermions this limit corresponds to the case of broad Feshbach resonance. Since neither as nor r0 can appear in the ﬁnal answers, the unitary point has a remarkable feature: using Fermi momentum kF = (3π 2 n)1/3 and Fermi energy F = kF2 /2m to deﬁne units for distance and energy, one obtains a universal description of all physical properties irrespective of the microscopic origin of the interacting Fermi system. The unitary limit was intensively studied both theoretically and experimentally in recent years [3-6]. Most theoretical work is based on the mean-ﬁeld and variational treatments which have no small parameters to control approximations made. As a consequence, there is a lot of scatter in the ﬁnal results. Unfortunately, the analysis of experimental data also often involves mean-ﬁeld approximations, largely due to lack of c Societ` a Italiana di Fisica

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reliable thermometry. So far, with the exception of one study performed by the Seattle group [7], numerical simulations were not done in the universal limit of the problem [8-10]. Our method of solving the problem is numerical. It is based on the Diagrammatic Monte Carlo (MC) technique [11,12] which sums all Feynman diagrams for the interacting system of N fermions in a ﬁnite volume V = L3 with the Hamiltonian (1)

H=

(kiσ ) +

V (ri↑ − rj↓ ).

i<j

i,σ=↑↓

In the experimental system, the single-particle dispersion law is parabolic (k) = k 2 /2m, and the interaction potential has a ﬁnite range r0 so that the Fourier transform of V (r) goes to zero for large momenta k k0 ∼ 1/r0 . This provides a natural ultraviolet regularization of the problem. Alternatively, one may look at the model with purely contact interactions V (r) = U δ(r) and remove ultraviolet divergences by modifying the ∞ dispersion relation (k) for large momenta k k0 so that the integral k0 dk/ (k) is convergent. Due to universality of the unitary point, either way of looking at the problem will result in identical ﬁnal answers in the nr03 → 0 limit. However, for numerical reasons (to avoid the sign problem) it is crucial to have contact interactions. Though our numerical scheme can work with arbitrary dispersion relation and can be formulated directly in the space-time continuum, we choose to work with the lattice tight-binding model (2)

(k) = −t

eikg + zt,

g

where g enumerates z = 6 nearest-neighbor displacement vectors on a simple cubic lattice with the unit length |g| = a. In the ka 1 limit we recover the parabolic dispersion relation (k) → k 2 /2m with m = 1/2ta2 . In what follows we use units such that t = 1 and a = 1; the Fermi momentum and Fermi energy for the lattice system with ﬁlling factor ν = na3 are identical to those of a continuum gas with the same eﬀective mass and number density n. Our motivation for considering a lattice model was based solely on the possibility of comparing to available numerical data for the superﬂuid-normal transition at large ﬁlling factors [10] and exact diagonalization studies [13]. 2. – Numerical method The starting point is a series of Feynman space-time diagrams for the partition function Z, depicted in ﬁg. 1. They consist of four-point vertices with two incoming (spin up and spin down) and two outgoing fermionic lines placed at diﬀerent space-time points (r1 τ1 , . . . , rp τp ). In a particular Feynman diagram one also speciﬁes how incoming and outgoing lines connect to each other. The corresponding diagram contribution to Z is given by the product of single-particle Matsubara propagators for the non-interacting system, Gσ (i, j) = Gσ (ri − rj , τi − τj ; β, μ), at temperature T = 1/β and chemical

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Normal-superfluid transition temperature in the unitary Fermi gas

Z = 1+

+

−

−

= 1+

+

+

+ ...

+ ...

Fig. 1. – Diagrammatic series for the partition function in terms of Feynman diagrams. First we show various diagrams with explicit connections (by single-particle propagators) between vertices. The last line represents vertex conﬁgurations for diagrams summed over all possible connections.

potential μ, a factor (−U )p , and a sign factor (−1)P↑ +P↓ (3)

(−U )n (−1)P↑ +P↓ G↑ (1, i1 ) . . . G↑ (p, ip ) G↓ (1, j1 ) . . . G↓ (p, jp ),

where p is the diagram order and P↑ (P↓ ) is the permutation of the i1 , . . . , ip (j1 , . . . , jp ) set. Because of the sign problem [14], the Monte Carlo summation of individual Feynman diagrams with the weights deﬁned by eq. (3) is not possible. However, the sum of all diagrams of a given order with the ﬁxed space-time positions of all vertices is always sign positive. Indeed, the sum over p! connections between the incoming and outgoing lines for one spin direction forms a determinant built on single-particle propagators (4)

det Aσ (Sp ),

Aσij = Gσ (i, j),

i, j = 1, . . . , p,

where a short-hand notation (5)

Sp = {(rj , τj ), j = 1, . . . , p},

was introduced to specify the vertex conﬁguration used for the construction of the A-matrix. For equal number of spin-up and spin-down particles and contact interaction an identical determinant is formed by the sum over all p! connections for the opposite spin component, and we arrive at the following formula for the partition function:

(6)

Z=

∞ p=0

(−U )p

Sp

det2 A(Sp ).

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N. V. Prokof’ev d

G2 =

+

=

+ +

+ ... + ...

Fig. 2. – Diagrammatic series for the correlation function (8). Diamonds represent the two-point creation/annihilation operators P and P † .

I suppress the spin index since the determinant can be calculated for any of the spin components. The sum over Sp is understood as (7)

Sp

≡

dr1 . . . drp

dτ1 . . . dτp . 0<τ1 <τ2 <...<τp <β

For the lattice model an integration over space should be replaced with the sum over lattice sites. Unlike eq. (3), the sum over vertex conﬁgurations in eq. (6) is suitable for the systematic error-free Monte Carlo sampling [15, 16]. For future use, I denote the set of all possible vertex conﬁgurations (5) by S(Z) , i.e., S(Z) = {p, Sp }. To determine the critical point of the superﬂuid-normal transition and to introduce the worm algorithm for eﬃcient Monte Carlo sampling of the conﬁguration space, we consider the pair correlation function (8)

$ % g2 (rτ ; r τ ) , G2 (rτ ; r τ ) = Tτ P (r, τ )P † (r , τ ) ≡ Z

where (9)

g2 (rτ ; r τ ) = Tr Tτ P (r, τ )P † (r , τ ) e−βH ,

and P (r, τ ) and P † (r , τ ) are the pair annihilation and creation operators in the Heisenberg picture, respectively: P (r, τ ) = cr↑ (τ )cr↓ (τ ). The non-zero asymptotic value of G2 (rτ ; r τ ) as |r − r | → ∞ is proportional to the condensate density. Feynman diagrams for g2 (rτ ; r τ ) are similar to those for Z, but contain two extra elements: a pair of two-point vertices with two incoming (outgoing) ends which represent P (r, τ ) (P † (r , τ )), see ﬁg. 2. The vertex conﬁgurations for the correlation function (8) slightly diﬀer from those for the partition function (5) by the presence of the two extra elements: the conﬁguration space for eq. (8) is S(G) = {p, S˜p }, with (10)

S˜p = {P (r, τ ), P † (r , τ ), (rj , τj ), j = 1, . . . , p}.

Normal-superfluid transition temperature in the unitary Fermi gas

783

The diagrammatic expansion for g2 (rτ ; r τ ) is similar to eq. (6) (11)

g2 (rτ ; r τ ) =

∞

(−U )p

p=0

: S˜p , det2 A

S˜p

: S˜p ) is a (p + 1) × (p + 1) matrix which diﬀers from that deﬁned in eq. (4) only where A( :ij = G(ri − r, τi − τ ) by that it has an extra row i0 and an extra column j0 , such that A 0 :i j = G(r − rj , τ − τj ). and A 0 A detailed description of the algorithm we use to sum the series (6) and (11) can be found in ref. [2]. Here I brieﬂy mention that the generic strategy of the worm algorithm [17] is to update the relevant conﬁguration space of the model (which is the unity of S(Z) and S(G) conﬁgurations) through local moves of the pair creation and annihilation operators. These moves switch between the S(Z) and S(G) diagrams; in the former we collect statistics for the system density (the simulation is done in the grand-canonical ensemble), kinetic and potential energy, in the latter we evaluate the space-time integral of the pair correlation function (12)

K = (V β)−2

V

dr dr

β

dτ dτ G2 (r − r , τ − τ ),

0

i.e. its zero-momentum, zero-frequency component. 3. – Extrapolation towards continuum system in the thermodynamic limit The MC setup discussed in previous sections works in the grand-canonical ensemble with external parameters {L, T, μ}. In order to extract the critical temperature of a continuum gas from MC data, one has to perform the two-step extrapolation. i) Upon taking the limit of L → ∞, one obtains Tc (μ), the critical temperature of a lattice system at a given chemical potential, and translates it into Tc (ν) by extrapolating the measured ﬁlling factor to the inﬁnite system size: ν ≡ ν(μ, T = Tc (μ), L → ∞). ii) The extrapolation towards the continuum limit is then done by taking the limit of ν → 0. The ﬁrst step is based on the ﬁnite-size scaling technique [18, 14]. We expect the superﬂuid-normal transition to be in the universality class of the three-dimensional XY model. This simpliﬁes the analysis considerably, since we can utilize well-known properties and critical exponents for this universality class [19]. At the critical point the correlation function (8) decays at large distances as a power law: G2 (t, τ ) ∝ r−(1+η) , where η = 0.0380 is the anomalous dimension. This power law translates into the K ∝ 1/L(1+η) decay of the integrated correlation function eq. (12). The ﬁnite-size technique is then to look at the rescaled quantity (13)

R(L, T ) = L1+η K(L, T ),

which is supposed to become size-independent at the critical point, diverge in the superﬂuid phase and go to zero in the normal state. This behavior implies that if one considers

784

N. V. Prokof’ev

=

+

=

+

+

+ ...

Fig. 3. – Ladder diagrams for the eﬀective vertex Γ(ω, p). Small dots represent the bare Hubbard interaction −U , and lines are for the single-particle propagators.

a series of system sizes L1 < L2 < L3 . . . , then the crossing of R(Li , T ) and R(Lj , T ) curves can be used to obtain an estimate TLi ,Lj for the critical temperature [18]. Indeed, for temperatures in the vicinity of the critical point the correlation length diverges as ξcorr ∝ |t|−νξ , where t = (Tc − T )/Tc , and νξ = 0.6717 for the U (1) universality class. Finite-size scaling of quantities similar to R(L, T ) in the vicinity of the transition point is described by the following formula: (14)

R = f (x)(1 + cL−ω + . . .),

where x = (L/ξcorr )1/νξ ∼ tL1/νξ is the dimensionless scaling variable, f (x) is the universal scaling function analytic at x = 0, c is a non-universal constant, ω ≈ 0.80 is the critical exponent of the leading irrelevant ﬁeld [20], and dots represent higher-order corrections to scaling. If the irrelevant ﬁeld corrections were not present, all R(Li , T ) curves would intersect at a unique point, Tc (μ). Expanding eq. (14) up to terms linear in t, one obtains for the crossing of two R-curves (15)

TLi ,Lj − Tc (μ) =

const 1/νξ +ω

Lj

ω

(Lj /Li ) − 1 1/νξ

1 − (Li /Lj )

.

To employ eq. (15) one performs a linear ﬁt of the sequence of TLi ,Lj against the righthand side of eq. (15) for several pairs of system sizes. The intercept of the best-ﬁt line yields the thermodynamic-limit critical temperature Tc (μ). Once we determine the critical temperature for a given value of the chemical potential, or ﬁlling factor (obtained by extrapolating the ν(L, T, μ) data to the thermodynamic limit), we study the ﬁnite-density correction to the universal ratio Tc (ν)/ F (ν) for the continuous low-density system. The theoretical expectation is that this correction vanishes as ν 1/3 . Indeed, consider the process of two particles scattering oﬀ each other at low energies. The standard solution for the eﬀective four-point vertex (given by the sum of ladder diagrams depicted in ﬁg. 3) is (16)

Γ−1 (ω, p) = U −1 + Π(ω, p),

Normal-superfluid transition temperature in the unitary Fermi gas

785

where Π(ξ, p) is the polarization operator (the integration is over the Brillouin zone) (17)

Π(ω, p) = BZ

dk 1 . (2π)3 ω + p/2+k + p/2−k

The position of the resonance (as → ∞) is deﬁned by the divergence of Γ(0, 0), which for the contact interaction happens at U =−

(18)

BZ

dk 1 (2π)3 2 k

−1 = −7.914.

At the resonance point, the dependence of Γ on frequency and momentum starts from the universal leading term Γcont and small lattice correction (19)

−1

Γ

(ω, p) = BZ

1 dk 1 −1 − , = Γ−1 cont + δΓ (2π)3 ω + p/2+k + p/2−k 2 k

where (20)

Γ−1 cont (ω, p)

m3/2 =− 4π

ω+

p2 , 4m

and (21)

p2 ω δΓ−1 (ω, p) ≈ A + B, 4 16 1 dk 1 dk A= − , 3 ε2 (2π)3 (k 2 /2m)2 (2π) BZ k 1/m dk (∂ 2 εk /∂kx ∂kx ) dk B= − . 3 2 2 3 (2π) (k /2m) ε2k BZ (2π)

At small ﬁlling factor the typical values of ω and p are set by the Fermi energy and Fermi 1/3 momentum, i.e. we have Γ−1 , and δΓ−1 ∼ kF2 ∼ ν 2/3 . Hence, the leading cont ∼ kF ∼ ν lattice correction is of the form (22)

δΓ ∼ ν 1/3 . Γ

Incidentally, eq. (21) hints into an intriguing possibility of completely suppressing the leading-order correction by tuning the single-particle spectrum k so that A = B = 0. We did not explore this possibility in the present study.

786

N. V. Prokof’ev

R(L,T)

0.17

0.20

0.16

0.15

0.15

0.05

0.10

0.15

0.10

3

30 fermions on a 6 lattice 3 80 fermions on a 8 lattice 3 240 fermions on a 12 lattice 0.05 3.50

3.75

4.00

4.25

4.50

4.75

E

Fig. 4. – A typical crossing of R(L, T ) curves. The error bars are 2σ, and solid lines are the linear ﬁts to MC points. The dependence of the ﬁlling factor on L is shown in the inset. ν = 0.148(1). From this plot and eq. (15) one obtains 1/Tc (ν = 0.148(1)) = 4.41(5).

4. – Simulation results We performed simulations for ﬁlling factors ranging from 0.95 down to 0.06 with up to about 300 fermions on lattices with up to 163 sites. The typical rank of determinants involved in computations of acceptance ratios was up to M ∼ 5000. We validated our numerical procedures by comparing results against the exact diagonalization data for a 4 × 4 cluster [13], and other simulations of the critical temperature at quarter ﬁlling ν = 0.5 [10,21] and ν = 0.25 [21]. In all cases we obtained agreement within the statistical error bars of a few percent. In ﬁg. 4 we show a typical example of the ﬁnite-size analysis. Despite the fact that numerically accessible system sizes are quite small, we were able to pinpoint the phase transition temperature to within a few percent accuracy. The dependence of Tc / F on the ﬁlling factor is shown in ﬁg. 5. It is clearly seen that the presence of the lattice suppresses the critical temperature considerably, nearly by a factor of 4, depending on the ﬁlling factor. Strong dependence of Tc on ν is in apparent contradiction with ref. [7], which claims weak or no ν-dependence. This disagreement might be due to the diﬀerence in the single-particle spectra εk used: ref. [7] employs the parabolic spectrum with the spherically symmetric cutoﬀ, while we use the tight-binding dispersion law. Indeed, eq. (21) indicates that a particular choice of εk does inﬂuence lattice corrections, which may even have diﬀerent signs for diﬀerent εk . The predicted ∼ ν 1/3 scaling (22) sets in at about ν ≈ 0.5. We thus use a linear ﬁt Tc (ν)/εF (ν) = Tc /εF − const · ν 1/3 to eliminate lattice corrections in the ﬁnal result. This

787

Normal-superfluid transition temperature in the unitary Fermi gas

Tc/HF 0.150

0.125

0.100

A. Sewer et al., 2002

0.075

0.050 T.A. Maier et al., 2006 0.025 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Q

1/3

Fig. 5. – Critical temperature dependence on the ﬁlling factor (circles). The ν = 1 point corresponds to half-ﬁlling. The error bars are one standard deviation. We also show results of refs. [10, 21] at quarter-ﬁlling and ν = 0.25.

ﬁtting procedure results in the best-ﬁt line given by 0.152(7) − 0.13(2)ν 1/3 . We further analyze the ﬁt residues in order to estimate the eﬀect of sub-leading lattice corrections which are expected to be proportional to ν 2/3 . From ﬁg. 6 we conclude that sub-leading corrections, if any, are smaller than the uncertainty of the ν 1/3 ﬁt. This analysis yields the ﬁnal result (23)

Tc = 0.152(7) εF ,

for the continuum system, which is noticeably below the transition temperature in the BEC limit TBEC = 0.218εF . Various approximate analytical treatments led in the past to Tc being either above or below TBEC , and here we deﬁnitively show that it is below. There have been several numerical attempts to determine the universal ratio Tc /εF for the resonant gas. The simulations of ref. [8] yield Tc /εF = 0.05, but at the value of the scattering length which has not been determined precisely. This result most probably corresponds to a deep BCS regime, where the transition temperature is exponentially suppressed. Lee and Sch¨ afer [9] report an upper limit for Tc /εF < 0.14, based on the study of the caloric curve, E(T ), for temperatures T > 0.14εF and ﬁlling factors ν ≥ 0.5. In this parameter range the speciﬁc heat shows no sign of the anomalous behavior characteristic of the transition point. This upper limit is, of course, consistent with Tc (ν = 0.5)/εF ≈ 0.054, see ﬁg. 5. The Seattle group has performed simulations of the caloric curve and condensate fraction, n0 , of the unitary gas at much smaller ﬁlling factors [7]. Using visual inspection of the caloric-curve shape, the critical temperature was estimated to be Tc /εF = 0.22(3).

788

N. V. Prokof’ev

0.0150 0.0125

fit residues

0.0100 0.0075 0.0050 0.0025 0.0000 -0.0025 0.0

0.2

0.4

0.6

0.8

Q

2/3

Fig. 6. – Linear ﬁt residues plotted vs. ν 2/3 (circles). Zero level is shown by the horizontal line, the dashed lines are linear ﬁts to the data points for ν < 0.5 and ν < 0.35, respectively.

The authors performed neither the ﬁnite-size analysis nor the ν → 0 extrapolation to justify this result as an outcome of the mathematically rigorous procedure relating raw data to Tc . The overall shape of the E(T ) curve seems to be little aﬀected by the ﬁnite volume of the system. This is hardly surprising since, even in the thermodynamic limit, E(T ) and its derivative dE/dT are continuous at the transition point. These properties also make E(T ) not particularly suitable for reliable estimates of critical parameters. On the other hand, the condensate fraction which has singular properties at Tc does show sizable ﬁnite-size corrections, see ﬁg. 1 of ref. [7]. At this point we note that the scaling of n0 with the system size is identical to that of K(L, T ). One may wonder what the data of ref. [7] would predict if they were analyzed using ﬁnite-size scaling techniques. To answer this question we digitized the data of Seattle’s group for the condensate fraction (using ﬁg. 1 in ref. [7]) and plotted them as n0 L1+η vs. T /εF (the Fermi energy was computed using particle number and system size information provided in the text). The result is shown in ﬁg. 7. The intersection of scaled curves turns out to be inconsistent with the result Tc /εF = 0.22(3) suggested by the authors, in fact, the numeric uncertainties are too large to claim any reliable estimate. We have also determined other universal ratios for the continuous unitary Fermi gas. The ﬁlling factor dependence of the system energy and chemical potential at the critical point is shown in ﬁg. 8. We used linear ﬁts to extrapolate to the zero-density limit with the ﬁnal result (24)

E/(N εF ) = 0.31(1)

(T = Tc ),

(25)

μ/εF = 0.493(14)

(T = Tc ).

Normal-superfluid transition temperature in the unitary Fermi gas

789

Fig. 7. – Finite-size scaling of the condensate fraction data from ref. [7]. Raw data points are rescaled similar to eq. (13) by the L1+η factor. Shaded vertical strips represent results for Tc /εF of this work and ref. [7], respectively, solid lines are drawn to guide the eye.

Fig. 8. – Energy (left-hand panel) and chemical-potential (right-hand panel) dependence on the ﬁlling factor along the critical isotherm T = Tc (ν). Dots are the MC results, dashed lines are the linear ﬁts.

790

N. V. Prokof’ev

The numerical values of other thermodynamic functions at criticality can be easily restored using special properties of the unitary gas thermodynamics [22,23]. In short, if one writes the free energy in terms of the universal function of the dimensionless temperature variable x = T /εF (26)

F = N εF fF (x),

then it immediately follows that all other thermodynamic functions obey similar scaling relations: (27)

E = N εF fE (x),

energy,

(28)

μ = εF fμ (x),

chemical potential,

(29)

P = n εF fP (x),

pressure,

(30)

S = N fS (x),

entropy.

Moreover, all scaling functions are related to each other, e.g. (31) (32) (33)

fP = (2/3)fE , (5/3)fE − fμ , x fF = fμ − (2/3)fE , fS =

and allow one to express fS and fF functions through numerically available functions fE and fμ . Another useful relation is (the prime stands for the derivative) (34)

fF =

fμ − (5/3)fE , x

which allows to extract fF directly from fE and fμ , and thus provides a simple check for the data consistency: The result (33) for the fF curve should be consistent with the derivative deduced from (34). Unfortunately, the value of entropy cannot be accurately deduced this way since (5/3)fE and fμ almost compensate each other within the error bars. Apart from critical parameters, we have also simulated properties of the normal state since they are important for the proper analysis of the experimental system conﬁned in the external potential. Indeed, when system parameters in the middle of the trap correspond to the critical point, the rest of the system is in the normal state. Moreover, it is often convenient to relate the critical temperature to the total number of particles in the system and trap parameters, which, in turn, requires precise knowledge of the equation of state. In ﬁg. 9 we show results for the temperature dependence of energy and chemical potential. Each point was obtained using data analysis similar to that depicted in ﬁg. 8. In the high-temperature region we simulated up to 80 fermions on lattices with up to 323 sites. In this region, the condition ν 1 is necessary but not

791

Normal-superfluid transition temperature in the unitary Fermi gas

E/NHF Boltzmann gas

Non-interacting Fermi gas

1.5

1.0

0.5 T-L. Ho and E. Mueller, 2004

PHF

Boltzmann gas

0.50 0.25 0.00 -0.25 -0.50 T-L. Ho and E. Mueller, 2004

0.0

0.2

0.4

0.6

0.8

1.0

1.2

T/HF

Fig. 9. – Temperature dependence of energy per particle (upper panel) and chemical potential (lower panel) in the normal state. Circles are the MC results, dotted lines and dashed lines correspond to the Boltzmann and non-interacting Fermi gases, respectively, the dot-dashed lines are the asymptotic prediction of ref. [23] (plus the ﬁrst virial Fermi correction), triangles are the MC results of ref. [7], and the stars denote the ground-state ﬁxed-node MC results [24].

suﬃcient for extrapolation to the continuum limit, for it is crucial to keep temperature much smaller than the bandwidth: T 6t. Using eqs. (33) and (34) we performed a self-consistency check shown in ﬁg. 10 where results for the free energy are plotted along with the free-energy derivatives. At temperatures above εF our results for both energy and chemical potential approach the virial expansion of ref. [23]. For T /εF 0.5 we ﬁnd reasonable agreement with the Monte Carlo data of ref. [7]. Though we do not have data points for the superﬂuid state T < Tc there is still a reasonable agreement even at Tc with the T → 0 ﬁxed-node MC values [24], apparently, due to weak dependence of energy and chemical potential on temperature. It is also well established that the T = 0 simulations are consistent with the experimental results [5, 6], apart from the gap values.

792

N. V. Prokof’ev

F/NHF 0.0

-0.5

Boltzmann gas

J. Carlson et al. 2003, G.E. Astrakharchik et al. 2004

-1.0 T-L. Ho and E. Mueller, 2004

-1.5

-2.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

T/HF

Fig. 10. – Free energy vs. temperature. Dots are the MC data, and dashes represent the error bar range for the free-energy derivative calculated using eq. (34). Triangles are MC results of ref. [7], the star denotes the ground-state ﬁxed-node MC result [24], the dotted line shows the Boltzmann gas curve, and the dashed is the asymptotic prediction of ref. [23].

5. – Trapped Fermi gas As mentioned above, in the experimental realization of the ultracold resonant Fermi gas one has to deal with the parabolic trapping potential, Vext (r). The standard procedure, especially in a system with a short “healing length” (which is just the interatomic distance in the present case) is to use the local density approximation (LDA), i.e. to replace the chemical potential with its coordinate-dependent counterpart μ(r) = μ−Vext (r). Likewise, other thermodynamic functions are to be understood as local, coordinatedependent quantities. Obviously, the data presented in the previous section for the system thermodynamics in the normal state are suﬃcient to relate physics happening in the trap center to parameters characterizing the system as a whole. For example, consider the case of N particles in a cigar-shaped parabolic trap, characterized by the axial and radial frequencies ω and ω⊥ . The characteristic energy in this case is EF = (3N ω2 ω⊥ )1/3 , which would coincide with the Fermi energy for a non-interacting gas in the trap center. Note that we denote the Fermi energy in the trap by a capital EF in order to avoid confusion with the uniform system Fermi energy εF . To ﬁnd the ratio Tc /EF one has to know the total number of particles, which is an integral of the system density over volume. In the middle of the trap n and μ are determined from the universal relations at the critical point, and for the rest one can use LDA and the known equation of state

Normal-superfluid transition temperature in the unitary Fermi gas

793

to complete the calculation. The ﬁnal result is (35)

Tc /EF = 0.20(2).

Since numerical data are available at a discrete set of points, the actual calculation starts with the interpolation procedure which produces continuous functional behavior for thermodynamic functions, consistent with the discrete set of simulated points. We use a piecewise-cubic ansatz with a smooth crossover to the virial expansion for the free energy. Temperature dependence of both energy and entropy are then deduced using numerical integration of the free-energy curve. We quote here a conservative error bar in eq. (35) which incorporates both the uncertainty of the critical temperature and systematic uncertainties which stem from restoring the continuous functional dependence of the chemical potential from the discrete set of Monte Carlo points with ﬁnite error bars. Experimentally, the temperature of the strongly interacting Fermi gas is not measured directly. On the contrary, the thermometry of the weakly interacting Fermi gases is well established. In the adiabatic experiments one starts from the non-interacting gas at some temperature (in units of Fermi energy) T (in) /EF , and slowly ramps magnetic ﬁeld towards the Feshbach resonance, thus adiabatically connecting the system at unitarity to

Fig. 11. – Axial density proﬁles: experimental data (dots) are from ﬁg. 3 of the most recent Rice group paper [6]. The full line is calculated using the LDA approach for T /EF = 0.03, the dashed line corresponds to T /EF = 0.16, and the dot-dashed line is for T /EF = 0.22.

794

N. V. Prokof’ev

a non-interacting one. Assuming entropy conservation during the magnetic-ﬁeld ramp, one can use Monte Carlo data for the entropy (integrated over the trap) to relate the initial and ﬁnal temperatures. Since our results for the entropy still have very large error bars at the critical point, we cannot use this protocol for accurate estimates of T (in) /EF . An alternative thermometry can be built on recent advances in the experimental technique developed by Innsbruck and Rice groups [6] with direct imaging of the in situ density proﬁles. Such density proﬁles can be directly ﬁt to the LDA proﬁles which give the shape of the cloud depending on T /EF . As an illustrative example of such a ﬁt, we have analyzed the experimental density proﬁle measured by the Rice group, see ﬁg. 11. From this analysis we deduce an upper bound on the sample temperature as T < 0.1EF which is consistent with measurements of the condensate fraction. Since in Rice experiments the gas is highly degenerate, we can only put an upper limit on temperature. 6. – Conclusions We have developed a worm-type scheme within the systematic-error–free determinant diagrammatic Monte Carlo approach for fermions. We applied it to the Fermi-Hubbard model with attractive interaction and equal number of spin-up and spin-down particles. At ﬁnite densities, the model describes ultracold atoms in optical lattice. In the limit of vanishing ﬁlling factor, ν → 0, and ﬁne-tuned (to the resonance in the two-particle s-wave channel) on-site attraction, a universal regime sets in, which is identical to the BCS-BEC crossover in the continuous space. In the present work, we conﬁned ourselves to a special value of the on-site interaction, U = U∗ ≈ −7.914t, corresponding to the divergent scattering length. At U = U∗ and ν → 0, the system reproduces the unitary point of the BCS-BEC crossover. The unitary regime is scale-invariant: all thermodynamic potentials are expressed in terms of dimensionless scaling functions of the dimensionless ratio T /εF (temperature in units of Fermi energy). For the critical temperature of the superﬂuid-normal transition in the uniform case we found Tc /εF = 0.152(7). Our results for system thermodynamics in the normal state form basis for an unbiased thermometry of trapped fermionic gases in the unitary regime: In particular, we found (within the local density approximation) that for the parabolic conﬁnement, the critical temperature in units of the characteristic trap energy EF is T /EF = 0.20(2). ∗ ∗ ∗ We would like to thank A. Bulgac, P. Magierski, and J. Drut, who kindly provided us with their numeric data for energy and chemical potential. We are also indebted to W. Li and R. Hulet for providing us with their unpublished experimental data. This research was enabled by computational resources of the Center for Computational Sciences and in part supported by the Laboratory Research and Development program at Oak Ridge National Laboratory. Part of the simulations were performed on the “Hreidar” cluster of ETH Z¨ urich. We also acknowledge partial support by NSF grants Nos. PHY-0426881 and PHY-0456261.

Normal-superfluid transition temperature in the unitary Fermi gas

795

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Introduction to color superconductivity G. Nardulli Dipartimento di Fisica, Universit` a di Bari - I-70126 Bari, Italy INFN, Sezione di Bari - I-70126 Bari, Italy

1. – Nuclear matter and QCD At small nuclear density and low temperature the relevant degrees of freedom of nuclear matter are the hadrons: nucleons, hyperons, mesons. At high density and/or high temperature the fundamental ﬁelds are those of quarks, spin-(1/2) particles carrying electric charge and a quantum number called color, and those of gluons, massless spin-1 particles carrying color in eight possible diﬀerent ways. The corresponding theory is called Quantum ChromoDynamics (QCD). It resembles Quantum Electrodynamics (QED), where the role of the fundamental fermion is played by the electron and that of the eight gluons by the photon (carrying no charge and therefore unique). Similarly to QED, QCD is a gauge theory based on a local unitary gauge group. For QCD it is SU (3), while QED is based on the Abelian group U (1). Quarks belong to the fundamental and gluons to the adjoint representation of SU (3), which is why each quark has three and each gluon eight possible colors. Besides color, quarks carry another quantum number, ﬂavor, that aﬀects their mass. Six diﬀerent ﬂavors are known, but we will consider here only the lightest quarks, called up, down and strange (with mu md ms ). A peculiar property of QCD is Asymptotic Freedom (AF), which means that at large momenta the eﬀective QCD coupling constant becomes smaller and smaller with increasing momenta. In the ordinary world, i.e. at zero temperature and zero density, the study of the QCD degrees of freedom is made diﬃcult by the fact that quarks and gluons are conﬁned inside hadrons. Studying the QCD vacuum state under extreme conditions can therefore help our understanding of strong interactions, because their description is simpler in these regimes, due to the AF property of QCD. In particular one might perform the study in c Societ` a Italiana di Fisica

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two very diﬀerent regimes: high temperature and small density or small temperature and high density. The ﬁrst regime is presently studied at the Relativistic Heavy Ion Collider at Brookhaven, USA, and will be investigated in the near future at the Large Hadron Collider at CERN. The other regime is not accessible for the present at the accelerators, but can be indirectly investigated by the study of the properties of compact stars such as pulsars, where both conditions, small temperatures and high densities, are likely to occur. As we noted, quarks are spin-(1/2) fermions with three colors, conventionally called red, green, blue (r, g, b) and, if light, in three ﬂavors (u, d, s). Since they belong to the representation 3 of the color symmetry group, when two quarks scatter they are either in a sextet (color symmetric) or in antitriplet (color antisymmetric) state. QCD interactions are mediated by one gluon exchange (in the perturbative regime). In the antitriplet channel this interaction is attractive. As a consequence of the Cooper’s theorem, the perturbative vacuum is unstable and one expects the formation of Cooper pairs of quarks (diquarks) and color superconductivity (this is so because the condensate is colored, not white as in the quark-antiquark channel, relevant at zero density). Moreover, the two quarks scatter preferably in the spin-0 channel, since this corresponds to a larger phase space. At extreme densities, considering QCD with 3 ﬂavors, the attractive interaction in color antisymmetric channel, together with S = 0, implies, by the Pauli principle, antisymmetry in ﬂavor. This is the basic idea of the models that we discuss in the next section. 2. – The true vacuum Since the naive vacuum is unstable, we should try to ﬁnd the true superconducting vacuum state. To do that, we have to show that the formation of a quark-quark condensate occurs because of the color interaction. The interaction term is fundamentally provided by gluon exchanges and to compute it and derive the gap equation one can follow diﬀerent approaches, such as, for example, the one of Bailin and Love in [1] based on the resummation of bubble diagrams (for previous work, see references in [1]). For this method and similar approaches, we refer to the reviews [2]. For pedagogical purposes it is however suﬃcient to consider a Nambu-Jona Lasinio [3] (NJL) interaction (4-fermion interaction) that mimics the QCD interaction. In this approximation the Hamiltonian is (1)

H=

¯ ∂/ + μγ0 )ψ + Gψγ ¯ μ T A ψ ψγ ¯ μT Aψ . d3 x ψ(i

Here G is a coupling constant with dimensions (mass)−2 and T A are SU (3) generators. Notice that a form factor is included in the deﬁnition, to take into account the asymptotic freedom property of QCD. It would act as an ultraviolet cutoﬀ. In order to obtain the gap equation, we make a mean-ﬁeld ansatz (2)

% $ T T Cψβj → ψαi Cψβj . ψαi

Introduction to color superconductivity

799

C is the charge conjugation matrix. Here and below Greek indices α, β are color indices, Latin indices i, j are ﬂavor indices. Let us now show that one expects for the vacuum expectation value of the quark bilinear the result (3)

$

% ΔIK T ψαi Cψβj = αβI ijK 2G

(sum over repeated indices). It is useful to note that, by evaluating (1) in mean-ﬁeld approximation, i.e. substituting to a pair of fermion ﬁelds in (1) their mean value given by (3) and (2), the interaction term takes the form (4)

∼ Δψψ + h.c.,

i.e. the fermions acquire a Majorana mass. To prove (3), we note that, as we have already discussed, QCD interaction favors antisymmetry in color; the pair has zero angular momentum (in this way the entire Fermi surface is available and the eﬀect becomes macroscopic); if it has also spin zero, the fermion pair must be in an antisymmetric state of ﬂavor to produce an antisymmetric wave function. Neglecting the sextet contribution, which is small, leaves us with (3). Clearly the form of the matrix ΔIK corresponding to the true vacuum state depends on dynamical eﬀects. In the literature various cases have been discussed: 1) 2 ﬂavor SuperConductivity (2SC): ΔIK = ΔδI3 δK3 , see ref. [4]; 2) Color-Flavor-Locking (CFL): ΔIK = δIK Δ, see ref. [5]; 3) gapless Color-Flavor-Locking (gCFL): ΔIK = δIK ΔK , see refs. [6, 7]; 4) 2 ﬂavor LOFF phase: ΔIK = δI3 δK3 Δ(r), see refs. [8-11]; 5) 3 ﬂavor LOFF phase: ΔIK = δIK ΔK (r), see ref. [12]. It is an incomplete list, for other results see [13, 14] and references in [2]. In the case of the LOFF phases the r dependence of the condensates produces an inhomogeneous type of superconductivity. Various r dependences have been discussed, e.g. the single plane wave or the sum of several plane waves (see below). In the cases of the gCFL phase and the 3 ﬂavor LOFF phase, one has to take into account the diﬀerences of quark masses (as we noted above, ms is signiﬁcantly larger that the up and down masses) and the diﬀerences of quark chemical potentials (μu , μd , μs are in general diﬀerent). This introduces more parameters in the description, but their number is reduced by imposing that the quark matter is electrically and color neutral (such constraint is automatically implemented in the CFL model). We will concentrate our attention on the above-mentioned models.

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3. – Homogeneous color superconductivity To show that color superconductivity can exist, let us consider the ﬁrst two cases (2SC and CFL). The 2SC model corresponds to the presence in the condensate of only the u and d ﬂavors: this means that this ansatz may be relevant in the situation characterized by an intermediate chemical potential mu , md μ ms . Therefore one can imagine of having taken the limit ms → ∞ and the theory is eﬀectively a two-ﬂavor theory. The CFL model holds in the other limit mu , md , ms μ. It takes its name from the property 3

(5)

αβK ijK = δαi δβj − δαj δβi ,

K=1

by which the color and ﬂavor indices are locked. Let us now introduce a, a† , b, . . . annihilation/creation operators of particles and holes (6) ψαi = uk ak eikx + b†k e−ikx . αi

k

The existence of the condensate gives rise to the following Hamiltonian: |k − μ|a†k ak + (k + μ)b†k bk + H= (7) k

+

Δ k

2

e−iΦ ak a−k +

Δ +iΦ † † e bk b−k + h.c., 2

where we have omitted for simplicity any color-ﬂavor indices as we wish to stress the mechanism that produces the condensate. Let us now perform a unitary Bogoliubov transformation. To do that, we introduce annihilation and creation operators for quasiparticle and quasiholes yk = cos θak − eiΦ sin θa†−k ,

(8)

zk = cos ϕbk − eiΦ sin ϕb†−k . By an appropriate choice of the parameters of the transformation, we can transform the original Hamiltonian in a new Hamiltonian describing a gas of non-interacting quasiparticles (9) H= y (k)yk† yk + z (k)zk† zk , with (10)

y (k) =

(k − μ)2 + Δ2 ,

These equations show two eﬀects:

z (k) =

(k + μ)2 + Δ2 .

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Introduction to color superconductivity

1) The quasiparticles and quasiholes are free. 2) In the dispersion laws of the quasiparticles a mass term proportional to Δ (the gap parameter) appears. To obtain (9), i.e. a free Hamiltonian, the parameters have to be chosen as follows: cos 2θ =

(11)

cos 2ϕ =

|k − μ| |k − μ|2 + Δ2 k+μ

,

|(k + μ)2 + Δ2

.

We note that, diﬀerently from the original ak , bk operators that annihilate the false vacuum, the quasiparticles annihilation operators destroy the true vacuum yk |0 >= 0,

(12)

zk |0 >= 0.

We have still to prove that Δ = 0. We do that and get an equation for Δ by substituting for yk , zk in eq. (3). We get in this way an integral equation that has the following schematic form (gap equation):

(13)

Δ=C

d k 3

Δ (k − μ)2 + Δ2

+

Δ

(k + μ)2 + Δ2

.

We see immediately that the origin of the instability of the false vacuum lies in the ﬁrst of the two terms in the r.h.s. of (13): If Δ = 0 there is no compensation for the divergence at k = μ. To be more quantitative, let us consider the two models in more detail. In the CFL model (ms = mu = md = 0) all the 3 × 3 = 9 quarks acquire a Majorana mass. The CFL condition gives two diﬀerent sets of eigenvalues. The ﬁrst one comprises 8 degenerate masses Δ1 = Δ2 = . . . = Δ8 and the second set the non-degenerate mass Δ0 . The actual values of the gaps depend on the model and the approximations involved; typical values are (14)

Δ1 = . . . = Δ8 = Δ ≈ 20–100 MeV,

Δ0 ≈ −2Δ ,

for μ ≈ 400–500 MeV. Let us observe that, since we start with a massless theory, the QCD Hamiltonian has, besides the local color symmetry SU (3), also a global chiral symmetry SU (3)L ×SU (3)R . The pairing occurs for left-handed and right-handed quarks separately. Therefore these condensates break SU (3)c ⊗SU (3)L ⊗SU (3)R ; ﬁnally, since the pair has baryonic number 2/3 (each quark has baryonic number equal to 1/3 since the nucleon is made up by three quarks), also U (1)B is broken. It should be observed, however, that a diagonal SU (3)c+L+R symmetry remains unbroken.

802

G. Nardulli

In the 2SC model (ms μ mu , md ) there are 4 massless quarks (the up and down quarks with colors 1 and 2), while the strange quark with any color and the u and d quarks with color 3 remain ungapped. Numerical results for this case are qualitatively in agreement with the results (14) of the CFL model. It is interesting to observe that the spontaneous breaking of global symmetries implies the existence of Nambu-Goldstone bosons; for internal symmetries there are as many NGBs as there are broken generators Ga . As they are massless, they are the lowest-energy quasiparticles of the eﬀective theory. On the other hand, for broken local gauge symmetries, by the Higgs-Anderson mechanism the gauge bosons acquire masses and there are no NGBs. In the CFL model the diquark condensate breaks SU (3)L × SU (3)R × SU (3)c × U (1)B to SU (3)c+L+R × Z2 . All the 9 quarks are massive and they belong to a SU (3)c+L+R singlet and a SU (3)c+L+R octet. All the 8 gluons are massive and are degenerate. There are 8 + 1 Nambu-Goldstone bosons. In the 2SC model the condensate breaks SU (3)c ⊗ SU (2)L ⊗ SU (2)R × U (1)B down to SU (2)c × SU (2)L × SU (2)R . While the chiral group is unbroken, SU (3)c is broken to SU (2)c . Therefore 3 gluons remain massless and 5 acquire a mass. As to the other quasiparticles, there is one would-be NGB associated to the breaking of the axial symmetry; moreover of the 6 quarks (2 ﬂavors in 3 colors) 4 are massive and 2 are massless. One can show that, for arbitrarily large μ, the CFL model is favored [15]; however, for intermediate μ other phases should be considered. At lower densities, in a range presumably more relevant for the study of compact stars, taking into account strange quark mass eﬀects implies that a new phase, called gCFL (gapless CFL) phase, is favored. Let us therefore consider now this new phase. The Lagrangian for gluons and ungapped quarks with mu = md = 0 and ms = 0 can be written as follows (color, ﬂavor and spin indices suppressed): L = ψ¯ (i D / − M + μ γ0 ) ψ,

(15)

where M = diag(0, 0, ms ) is the mass matrix in ﬂavor space and the matrix of chemical potential is given by

2 αβ αβ αβ √ μ (16) μαβ = (μ δ − μ Q ) δ + δ T + T μ b ij Q ij ij 3 3 8 8 ij 3 (i, j = 1, 3 ﬂavor indices; α, β = 1, 3 color indices). Moreover T3 = 12 diag(1, −1, 0), 1 T 8 = 2√ diag(1, 1, −2) in color space and Q = diag(2/3, −1/3, −1/3) in ﬂavor space; μQ 3 is the electrostatic chemical potential; μ3 , μ8 are the color chemical potentials associated respectively to the color charges T3 and T8 ; μb is the quark chemical potential. Working in the mean-ﬁeld approximation and neglecting the antiquark (hole) contribution, one can derive the quasiparticle dispersion law, whose knowledge allows the evaluation of the grand potential. At zero temperature it is given by (17)

Ω=−

1 2π 2

Λ

dp p2 0

9 j=1

| j (p)| +

μ4Q 1 2 Δ1 + Δ22 + Δ23 − , G 12π 2

803

Introduction to color superconductivity

where Λ is the ultraviolet cutoﬀ, j (p) are the quasiparticle dispersion laws and G is the Nambu-Jona Lasinio coupling constant. In any realistic astrophysical application of these models one has to impose electric neutrality (the global electric charge must vanish) and color neutrality (the color charges must vanish as well). In the absence of these conditions, electric and color forces would destroy quark matter stability. In order to enforce electrical and color neutrality, one has to minimize the grand potential with respect to μQ , μ3 and μ8 . Including the stationary conditions with respect to the gap parameters Δ1 , Δ2 , Δ3 , one ends up with a system of six equations which must be solved simultaneously. Once this system of equations is solved, one may express Δ1 , Δ2 , Δ3 and the chemical potentials μQ , μ3 and μ8 as functions of m2s /μb . The superconductive state is gapless because the equation j (p) = 0 has solutions for ﬁnite p; in other terms, diﬀerently from the CFL or the 2SC models, where the dispersion laws have the form (10), in the gCFL model there are gapless modes (this happens also in the similar model with two ﬂavors, the gapless 2SC model [14]). This result can be understood as follows. The chemical potentials of the various quarks are in general diﬀerent, so that in general one expects δμ = 0. It follows that the generic quasiparticle dispersion law has the form (18)

(p) = ±δμ +

(p − μ)2 + Δ2 ,

which shows that, for appropriate values of δμ there are gapless modes. Before being accepted, the tentative ground state must pass a stability test. For gluons this means that the Meissner masses, obtained by the eigenvalues of the polarization tensor, must be real. In the CFL phase, with ms = 0 the Meissner masses are degenerate with the value

2 1 μ2 g 2 11 ln 2 + − − (19) m2M = b 2 > 0. π 36 27 2 However, for a non-zero strange quark in the gCFL phase the masses are in general diﬀerent, due to symmetry breaking, and some of them are actually imaginary [7]. This means that the gCFL state cannot be the true vacuum state, even if its free energy is smaller than that of the normal state. 4. – Color superconductivity and compact stars Color superconductivity might be realized in compact stars. This follows from the following considerations. The BCS critical temperature is given by (20)

Tc = 0.57ΔBCS

and in QCD ΔBCS is expected to range between 20 and 100 MeV, see eq. (14). This result must be evaluated in the context of the thermal history of pulsars. These compact stars are formed after a supernova explosion. The temperature at the interior of the supernova

804

G. Nardulli

is about 1011 K, corresponding to 10 MeV. Then the star cools very rapidly by neutrino emission with the temperature going down to ∼ 109 –1010 K in about one day. Neutrino emission is then the dominating cooling process for ∼ 103 years. When the star reaches the temperature of ∼ 106 K, it cools down due to X-ray and photon emission, so that in a few million years it reaches a surface temperature ∼ 105 K. Therefore for the largest part of its existence a neutron star has T < Tc , which implies the possibility of color superconductive states if μ is large enough. Since the star temperature is much smaller than the typical BCS energy gap (Tn.s. /ΔBCS ≈ 10−6 –10−7 ), one can assume that the compact star is eﬀectively at zero temperature. We have shown that QCD favors the formation of BCS condensates in idealized cases, e.g. two or three massless ﬂavors of quarks. However, in realistic cases the three quarks have diﬀerent masses and, as a consequence, diﬀerent Fermi momenta. Let us estimate the order of magnitude of the scales involved in the description of a neutron star with a quark core. We begin with the simple case of a free gas of three ﬂavor quarks, assuming that the up and down quarks are massless and the strange quark has mass ms [16]. We also assume that the weak interactions are in equilibrium. One gets for quark chemical potentials and Fermi momenta the results 2 μu = μ − μe , 3 1 μd = μ + μe , 3 1 μs = μ + μe , 3

(21)

puF = μu , pdF = μd , psF =

μ2s − Ms2 ,

where μ is the average chemical potential (22)

μ=

1 (μu + μd + μs ) 3

and μe the chemical potential of the electrons. Notice that (23) μi Ni + μe Ne = μNq − μe Q , i=u,d,s

where (24)

Nq =

Ni ,

Q=

i=u,d,s

2 1 Nu − (Nd + Ns ) − Ne . 3 3

This result reﬂects the fact that the u quark has electric charge +2/3e, the d and s charge −1/3e, and the electron −e. The chemical potential for the electrons is ﬁxed by requiring electrical neutrality. This corresponds to the following condition for the grand potential Ω at zero temperature: (25)

Q=

∂Ω = 0. ∂μe

805

Introduction to color superconductivity

For each fermionic species, omitting the volume factor, Ω is given by 1 Ω= 2 π

(26)

pF

p2 (E(p) − μ)dp .

0

In our case we get (27)

piF μe 3 1 Ω= 2 p2 (Ei (p) − μi )dp + 2 p2 (p − μe )dp , π π 0 0 i=u,d,s

where (28)

Eu,d (p) = p,

Es (p) =

p2 + m2s .

An analytical expression can be obtained by performing an expansion up to the order m4s /μ4 . One gets μe ≈

(29)

m2s 4μ

and (30)

Ω≈−

3 4 3 7 − 12 log(ms /2μ) 4 μ + 2 m2s μ2 − ms . 2 4π 4π 32π 2

The baryon density is obtained as (31)

ρB = −

1 i 3 1 ∂Ω = (pF ) . 3 ∂μ 3π 2 i=u,d,s

With the same approximation as before one ﬁnds (32)

2 μ3 1 ms . ρB ≈ 2 1 − π 2 μ

We note that densities in the core are of the order of 1015 g/cm3 , corresponding to a chemical potential of the order of 400 MeV. Let us discuss the range of values μ ∼ 400 MeV of the average chemical potential, with a strange mass of the order of 200–300 MeV (this is the eﬀective density-dependent strange-quark mass). With ms = 300 MeV, one ﬁnds μe = 56 MeV with Fermi momenta (33)

puF = 365 MeV,

pdF = 418 MeV,

psF = 290 MeV,

806

G. Nardulli

and a baryon density about 4.4 times the nuclear matter density. With ms = 200 MeV the result is μe = 25 MeV and (34)

puF = 384 MeV,

pdF = 408 MeV,

psF = 357 MeV,

and a baryon density about 5.1 times the nuclear matter density. To go to baryon densities relevant to the central core of the star, i.e. densities from 6 to 8 times the nuclear matter density, one needs to go to higher values of μ and lower values of ms , where the diﬀerence among the Fermi momenta is lower. This can be seen using the approximate expression for μe (35)

puF ≈ μ −

m2s , 6μ

pdF ≈ μ +

m2s , 12μ

psF ≈ μ −

5m2s , 12μ

with (36)

pdF − puF ≈ puF − psF ≈

m2s . 4μ

In conclusion the conditions for the formation of color condensates can be indeed realized in compact stars, more exactly in their cores. The precise exact nature of the condensed phase is, however, still matter of debate. In fact, it is quite likely that the CFL is not favored, due to the non-vanishing strange-quark mass. Since the gCFL phase is unstable, one has to search for an alternative. A possibility is oﬀered by the inhomogeneous color superconductivity that we now discuss. 5. – Inhomogeneous color superconductivity: LOFF phase with two ﬂavors The relevance of the Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) phase [8-12] follows from the fact that, for appropriate values of δμ, pairing with non-vanishing total momentum, p1 + p2 = 2q = 0, is favored. It is a well-known fact that, if fermions of diﬀerent chemical potentials interact in an attractive channel, pairing √ occurs only if the diﬀerence in chemical potentials δμ does not exceed the limit ΔBCS / 2, called the Clogston-Chandrasekhar limit. For δμ much larger than the Clogston-Chandrasekhar limit, the favored state is the normal phase, but in a window of intermediate values of δμ pairing with non-vanishing total momentum is favored, as ﬁrst shown in the context of condensed matter in [8]. The LOFF phase is characterized by gap parameters not uniform in space that, in the simplest case, have the form of a single plane wave: Δ(r) = Δ exp[2iq · r]. Both cases of two and three ﬂavours have been discussed in the literature. Let us begin with the case of 2 ﬂavors, i.e. let us assume that the strange quark is decoupled due to its (rather) large mass. In nature ﬂavor symmetry is broken not only explicitly by quark mass terms, but also by weak interactions. Therefore in the applications of the color superconductivity one has to take into account this symmetry

807

Introduction to color superconductivity

breaking; for example in compact stars, considering only two ﬂavors, isospin is broken by δμ = μu − μd = 0, due to the process d → u e− ν¯e .

(37)

At the equilibrium (including the reaction u e− → d νe ) one has δμ = −μe . To simplify the problem, I will consider the case of two massless quarks with chemical potentials μu and μu given by (38)

μu = μ + δμ,

μd = μ − δμ,

where up and down refer to ﬂavor. The condensate has the form $

(39)

% ψiα ψjβ ∝ αβ3 ij .

√ Let ΔBCS be the value of the homogeneous BCS condensate. For δμ < ΔBCS / √2, the energetically favored state is the homogeneous one. To show that for δμ > ΔBCS / 2 the Cooper pair might prefer to have non-zero total momentum, we consider a four-fermion interaction modelled on one-gluon exchange, that is 3 ¯ μ a ¯ μ a λ ψ ψγ λ ψ , L = − g ψγ 8

(40)

where λa are Gell-Mann matrices. In the mean-ﬁeld approximation it reduces to

1 L = − αβ3 ij ψiα ψjβ Δe2iq·r + c.c. + (L → R) , 2

(41)

where we have deﬁned $ % 1 ΓS e2iq·r = − αβ3 ij ψαi ψβj , 2

(42) and (43)

Δ = gΓS .

The gap equation has the form (44)

Δ=i

gρ Δ 2

dv 4π

+δ

−δ

dξ

1 dE , 2π (E − μ ¯ + i sign E)2 − ξ 2 − Δ2

where ρ = 4μ2 /π 2 is the density of states, δ the ultraviolet cutoﬀ and ξ the component of the quark momentum parallel to the Fermi velocity v, measured from the Fermi surface. Moreover, (45)

μ ¯ = δμ − v · q .

808

G. Nardulli

Performing the integration over the energy, we get (46)

1=

gρ 2

dv 4π

δ

0

dξ ξ 2 + Δ2

θ( − |¯ μ|) ,

where (47)

θ( − |¯ μ|) = 1 − θ(− − μ ¯) − θ(− + μ ¯) .

It can be shown that there is a ﬁrst-order transition in δμ, between the homogenous state and the normal state and a second-order transition between the √ LOFF state and the normal one. The ﬁrst-order transition occurs near δμ1 ∼ ΔBCS / 2, while the second-order phase transition is at the critical point δμ2 = 0.754ΔBCS (see [8]). Near δμ2 one has (48)

ΔLOFF =

1.757 δμ2 (δμ2 − δμ) = 1.15 ΔBCS

δμ2 − δμ , ΔBCS

which is non-vanishing, and the grand potential is given by (49)

ΩLOFF − Ωnormal = −0.439 ρ(δμ − δμ2 )2 ,

showing that the LOFF phase is indeed favored. In the more general case, one can assume the ansatz

(50)

Δ(r) = Δ

P

e2iqm ·r

m=1

(the previous case corresponds to P = 1). Only approximate solutions can be given for the gap (50) either near the second-order phase transition, by the Ginzburg-Landau method, or by an eﬀective gap equation that captures the main aspects of the structure represented by eq. (50). . 5 1. Gap equation in the Ginzburg-Landau approximation. – In the Ginzburg-Landau approximation the gap equation for condensate (50) is as follows: (51)

∂Ω = 0, ∂Δ

with (52)

Ω α β γ = P Δ2 + Δ4 + Δ6 ρ 2 4 6

809

Introduction to color superconductivity

and (53)

2 (1 − Π(q)) , Π(q) ≡ Π(q, q) , gρ P 2 β=− J(qk , q , qm , qn )δqk −q +qm −qn , gρ

α=

k, ,m,n=1

γ=−

2 gρ

P

K(qk , q , qm , qj , qi , qn )δqk −q +qm −qj +qi −qn .

k, ,m,j,i,n=1

Here δ(qk − qn ) means the Kronecker delta: δn,k and the following deﬁnitions have been adopted: (54)

(55)

(56)

Π(q1 , q2 ) = +

igρ 2

dw ˆ 4π

+δ

−δ

dξ

2 dE " fi (E, δμ, {q}) , 2π i=1

+∞

−∞

4 dE " fi (E, δμ, {q}) , −δ −∞ 2π i=1 +∞ 6 dE " igρ dw ˆ +δ K(q1 , q2 , q3 , q4 , q5 , q6 ) = + dξ fi (E, δμ, {q}). 2 4π −δ −∞ 2π i=1

J(q1 , q2 , q3 , q4 ) = +

igρ 2

dw ˆ 4π

+δ

+∞

dξ

We have put w ≡ vF w ˆ and (57)

fi (E, δμ, {q}) =

1 E + i sign E − δμ + (−1)i [ξ − 2

i

k k=1 (−1) w

· qk ]

;

M moreover, the condition k=1 (−1)k qk = 0 holds, with M = 2, 4, 6 respectively for Π, J and K. The critical value δμ2 is obtained by the condition that, at δμ = δμ2 , α vanishes. This approximation is adequate to deal with most of the structures that can be obtained summing several plane waves. This analysis has been performed by Bowers and Rajagopal in [9]. The most interesting case is oﬀered by the cubic structures. They are formed either by six plane waves pointing to the six faces of a cube (the so-called body-centered cube, b.c.c.) or by eight plane waves pointing to the vertices of a cube (face-centered cube, f.c.c.). In the former case the grand potential turns out to be bounded from below and smaller than all the other crystalline structures. In the latter case the grand potential is unbounded from below, because the coeﬃcient γ is negative. This represents a problem, because it shows that the analysis is incomplete and higher-order terms, e.g. ∼ Δ8 , should be included. . 5 2. Eﬀective gap equation. – If one is far away from the second-order phase transition, the Ginzburg-Landau approach is not valid. One can use in this case an eﬀective gap equation obtained by an appropriate average of the original Lagrangian. The smoothing procedure is described in detail in [11] and we can recall here only the main points. The

810

G. Nardulli

method employed uses eﬀective quark ﬁelds where the large part of the quark momentum μv has been extracted by a factor exp[iμv] (v the Fermi velocity). Therefore in the condensate term of the Lagrangian, one has a factor (58)

exp[i(μu vu + μd vd + 2qm ) · r] .

m

Next we multiply the Lagrangian by some appropriate function g(r) and make an average over a crystal cell. In the gap equation the relevant integration momenta are small. Therefore, one can assume that the ﬁelds are almost constant in the averaging procedure. Averaging the factor (58) produces as a result that the two velocities are antiparallel up to terms of the order of δμ/μ; moreover the term (58) is substituted by its average

(59)

P

ΔE (v, 0 ) =

Δeﬀ (v · nm , 0 ) ,

m=1

where 0 is the quasiparticle energy and nm = qm /q. It can be shown that this procedure is valid for Δ not too small, i.e. far away from the second-order phase transition. The exact form of the function Δeﬀ depends on the function g(r) chosen for the average. In particular g(r) can be chosen in such a way that

(60)

Δeﬀ = Δθ(Eu )θ(Ed ) =

Δ, 0,

for (ξ, v) ∈ PR, elsewhere,

where for each plane wave of wave number nm , one has (61)

Eu,d = ±δμ ∓ qnm · v +

ξ 2 + Δ2

and the pairing region (PR) is deﬁned by the condition Eu > 0, Ed > 0. This procedure simpliﬁes the gap equation that assumes the form

(62)

PΔ = i

gρ 2

dv 4π

ΔE (v, 0 ) d2 . 2π 20 − 2 − Δ2E (v, 0 )

The energy integration is performed by the residue theorem and the phase space is divided into diﬀerent regions according to the pole positions, deﬁned by (63)

=

ξ 2 + Δ2E (v, ) .

811

Introduction to color superconductivity

Therefore we get

(64)

P 2δ ΔE (v, ) dv P Δ ln dξ = 2 Δ0 4π ξ + Δ2E (v, ) Pk k=1 P kΔ dv dξ = , 2 4π ξ + k 2 Δ2 Pk k=1

where the regions Pk are deﬁned as follows: Pk = {(v, ξ) | ΔE (v, ) = kΔ}

(65)

and we have made use of the equation 2δ 2 = ln , gρ ΔBCS

(66)

relating the BCS gap ΔBCS to the four-fermion coupling g and the density of states. The ﬁrst term in the sum, corresponding to the region P1 , has P equal contributions with a dispersion rule equal to the Fulde-Ferrel P = 1 case. This can be interpreted as a contribution from P non-interacting plane waves. In the other regions the diﬀerent plane waves have an overlap. The free energy Ω is obtained by integrating in Δ the gap equation. At ﬁxed δμ, Ω is a function of Δ and q, therefore the energetically favored state satisﬁes the conditions ∂Ω = 0, ∂Δ

(67)

∂Ω = 0, ∂q

and must be the absolute minimum. The result of this analysis is that the body-centered cube (b.c.c.) is the favored structure up to δμ ≈ 0.95ΔBCS . For larger values of δμ < 1.32ΔBCS the favored structure is the face-centered cube (f.c.c.). 6. – LOFF phase of QCD with three ﬂavors in the Ginzburg-Landau approximation We shall describe in this section some results for the LOFF phase of QCD with three ﬂavours, obtained in the Ginzburg-Landau (GL) approximation [12]. We shall limit our presentation to the case of space modulation given by a single plane wave. The free energy per unit volume Ω in the GL limit is

(68)

Ω = Ωn +

3 I=1

⎛

⎞ α β β IJ ⎝ I Δ2I + I Δ4I + Δ2I Δ2J ⎠ + O(Δ6 ), 2 4 4 J =I

812

G. Nardulli

where Ωn refers to the normal phase Ωn = −

(69)

μ4 3 4 μu + μ4d + μ4s − e 2 , 2 12π 12π

and one has assumed for the condensate the pairing ansatz ψiα C γ5 ψβj ∝

(70)

3

ΔI (r) αβI ijI ,

I=1

with ΔI (r) = ΔI exp [2iqI · r] .

(71)

In (69) μj are chemical potentials for quarks or for the electron, while the coeﬃcients of (68) can be found in [12]. One works in the approximation of vanishing color chemical potentials μ3 = μ8 = 0, which is valid for the present case, since they vanish in the normal phase and we work near the second-order phase transition. β-equilibrium is imposed together with the electric-neutrality condition −

(72)

∂Ω = 0. ∂μe

These conditions, together with the gap equations, give, for each value of the strangequark mass, the electron chemical potential μe and the gap parameters ΔI . Moreover one determines qI by solving, together with the gap equation and eq. (72), also (73)

3 ∂αI ∂βIJ ∂Ω = ΔI + ΔI Δ2J , 0= ∂qI ∂qI ∂qI

I = 1, 2, 3.

J=1

Condition (72) gives (74)

μe ≈

m2s , 4μ

which is valid up to terms of the order (1/μ). This result is identical to the free-fermion case, which was expected since one works near the transition point between the LOFF and the normal phase. It follows that (75)

δμdu = δμus ≡ δμ

and (76)

δμds = 2δμ .

813

Introduction to color superconductivity

To evaluate (73), it is suﬃcient to work at the O(Δ2 ), which leads to q = 1.1997|δμ|. As to the orientation of qj , the results obtained in ref. [12] indicate that the favored solution has Δ1 = 0 and therefore q1 = 0. Furthermore q2 = q3 and Δ2 = Δ3 . These results are consequences of the GL limit. In fact, as shown by eqs. (75) and (76), the surface separation of d and s quarks is larger, which implies that Δ1 pairing is disfavored. On the other hand, the surface separations of d-u and u-s quarks are equal, which implies that Δ2 and Δ3 must be almost equal. Finally q2 , q3 are parallel because the pairing region on the u-quark surface at the GL point is formed by two distinct rings (in the northern and southern emisphere, respectively), while for antiparallel q2 , q3 the two rings overlap, which reduces the phase space available for pairing. The results one obtains can be summarized as follows. Leaving the strange-quark mass as a parameter (though a complete calculation would need its determination, considering also the possibility of condensation in the q q¯ channel) the free energy can be evaluated as a function of the variable m2s /μ. One ﬁnds an enlargement of the range where color superconductivity is allowed, with the LOFF state having free energy lower than the normal and the gCFL phases. For example, for μ = 500 MeV, and with the coupling ﬁxed by the value Δ0 = 25 MeV of the homogeneous gap for two ﬂavors, at about m2s /μ = 150 MeV the LOFF phase has a free energy lower than the normal one. This point corresponds to a second-order transition. Then the LOFF state is energetically favored till the point where it meets the gCFL line at about m2s /μ = 128 MeV. This is a ﬁrst-order transition since the gaps are diﬀerent in the two phases. 7. – Stability of the LOFF phase of QCD with three ﬂavors In [17] the gluon Meissner masses in the three-ﬂavor LOFF phase of QCD were computed using an expansion in 1/μ and the GL approximation. At the order of g 2 there is a contribution g 2 μ2 /(2π 2 ) identical for all the eight gluons; this result for the LOFF phase is identical to those of the normal or the CFL case as this term is independent of the gap parameters, see (19). Another contribution to the polarization tensor is given by μ ν iΠμν ab (x, y) = − Tr[ i S(x, y) i Ha i S(y, x) i Hb ],

(77)

where the trace is over all the internal indices; S(x, y) is the quark propagator, and Haμ is the vertex. The quark propagator S has components S ij (i, j = 1, 2), where each S ij is a 9 × 9 matrix in the color-ﬂavor space. At the fourth order in Δ one has

S 11 = S011 + S011 Δ S022 Δ S011 + S011 Δ S022 Δ S011 ,

S 21 = S022 Δ S011 + S011 Δ S022 Δ S011 ,

(78) (79)

where S0ij is the 18 × 18 matrix (80)

S0 =

11 [S0 ]AB 0

0 [S022 ]AB

= δAB

−1 ¯A ) (p0 − ξ + μ 0

0 ¯A ) (p0 + ξ − μ

−1

,

814

G. Nardulli

A, B = 1, . . . , 9 are indices in the basis A = (ur , dg , sb , dr , ug , sr , ub , sg , db ), Δ and Δ are 9 × 9 matrices containing the gap parameters ΔI , p0 is the energy, ξ = |p| − μ, μ ¯A = μA − μ. S 12 and S 22 are obtained by the changes 11 ↔ 22 and Δ ↔ Δ . In the region where the use of the GL expansion is justiﬁed, i.e. 128 MeV < m2s /μ < 150 MeV, all the squared gluon Meissner masses were found positive and therefore the LOFF phase of three-ﬂavor QCD is free from the chromomagnetic instability. 8. – Neutrino emission by pulsars and the LOFF state As noted above, neutrino emission due to direct Urca processes, when kinematically allowed, is the most important cooling mechanism for a young neutron star. When the temperature of the compact star is of the order of ∼ 1011 K, neutrinos are able to escape, which produces a decrease in temperature. For smaller temperatures, e.g. below 109 K, the direct nuclear Urca processes n → p+e+ ν¯e and e− +p → n+νe , would produce rapid cooling. However, they are not kinematically allowed, because energy and momentum cannot be simultaneously conserved. Therefore only modiﬁed Urca processes, involving another spectator particle, can take place. The resulting cooling is less rapid because neutrino emission rates turn out to be εν ∼ T 8 , much smaller than the emission rate εν ∼ T 6 due to direct Urca processes. These considerations apply to stars containing only nuclear matter. If hadronic densities in the core of neutron stars are suﬃciently large, as we have seen, the central region of the star may comprise deconﬁned quark matter in a color condensed state. Therefore, direct Urca processes involving quarks, i.e. the processes d → u + e− + ν¯e and u + e− → d + νe , may take place and contribute to the cooling of the star. In the color-ﬂavor locked phase, in which light quarks of any color form Cooper pairs with zero total momentum, and all fermionic excitations are gapped, the corresponding neutrino emissivity and the speciﬁc heat C are suppressed by a factor e−Δ/T and the cooling is distinctly less rapid compared to quarks in the normal phase. However, at densities relevant for compact stars, the quark number chemical potential μ should be of the order of 500 MeV and eﬀects due to the strange-quark mass ms are relevant. In [18] a calculation of neutrino emission rate and the speciﬁc heat of quark matter in the LOFF superconductive phase with three ﬂavors was performed. It was shown that, due to the existence of gapless modes in the LOFF phase, a neutron star with a quark LOFF core cools faster than a star made of nuclear matter only. This follows from the fact that in the LOFF phase neutrino emissivity and quark speciﬁc heat are parametrically similar to the case of unpaired quark matter (εν ∼ T 6 and C ∼ T , respectively). Therefore, the cooling is similar to that of a star comprising unpaired quark matter. These results are still preliminary, since the simple ansatz of a single plane wave should be substituted with a more complex behavior and if a more complete calculation would conﬁrm the results of [18] remains to be seen.

815

Introduction to color superconductivity

∗ ∗ ∗ I would like to thank R. Anglani, R. Casalbuoni, M. Ciminale, R. Gatto, N. Ippolito, M. Mannarelli and M. Ruggieri for a most fruitful collaboration on the themes covered by this review. REFERENCES [1] Bailin D. and Love A., Phys. Rep., 107 (1984) 325. [2] Rajagopal K. and Wilczek F., in Handbook of QCD, edited by Shifman M. (World Scientiﬁc) 2001, pp. 2061 (arXiv:hep-ph/0011333); Nardulli G., Riv. Nuovo Cimento, 25, No. 3 (2002) (arXiv:hep-ph/0202037); Schafer T., arXiv:hep-ph/0304281. [3] Nambu Y. and Jona Lasinio G., Phys. Rev., 122 (1961) 345; 124 (1961) 246. [4] Alford M., Rajagopal K. and Wilczek F., Phys. Lett. B, 422 (1998) 247 (arXiv:hepph/9711395). [5] Alford M., Rajagopal K. and Wilczek F., Nucl. Phys. B, 537 (1999) 443 (arXiv:hepph/9804403). [6] Alford M., Kouvaris C. and Rajagopal K., Phys. Rev. Lett., 92 (2004) 222001 (arXiv:hep-ph/0311286); Phys. Rev. D, 71 (2005) 054009 (arXiv:hep-ph/0406137). [7] Casalbuoni R., Gatto R., Mannarelli M., Nardulli G. and Ruggieri M., Phys. Lett. B, 605 (2005) 362; 615 (2005) 297(E) (arXiv:hep-ph/0410401); Fukushima K., Phys. Rev. D, 72 (2005) 074002 (arXiv:hep-ph/0506080). [8] Larkin A. I. and Ovchinnikov Yu. N., Zh. Eksp. Teor. Fiz., 47 (1964) 1136 (Sov. Phys. JETP, 20 (1965) 762); Fulde P. and Ferrell R. A., Phys. Rev. A, 135 (1964) 550. [9] Alford M., Bowers J. A. and Rajagopal K., Phys. Rev. D, 63 (2001) 074016 (arXiv:hep-ph/0008208); Bowers J. A. and Rajagopal K., Phys. Rev. D, 66 (2002) 065002 (arXiv:hep-ph/0204079). [10] Casalbuoni R. and Nardulli G., Rev. Mod. Phys., 76 (2004) 263 (arXiv:hepph/0305069). [11] Casalbuoni R., Ciminale M., Mannarelli M., Nardulli G., Ruggieri M. and Gatto R., Phys. Rev. D, 70 (2004) 054004 (arXiv:hep-ph/0404090). [12] Casalbuoni R., Gatto R., Ippolito N., Nardulli G. and Ruggieri M., Phys. Lett. B, 627 (2005) 89; 634 (2006) 565(E) (arXiv:hep-ph/0507247); Mannarelli M., Rajagopal K. and Sharma R., Phys. Rev. D, 73 (2006) 114012 (arXiv:hep-ph/0603076); Rajagopal K. and Sharma R. arXiv:hep-ph/0605316. [13] Rapp R., Schafer T., Shuryak E. V. and Velkovsky M., Phys. Rev. Lett., 81 (1998) 53 (arXiv:hep-ph/9711396). [14] Shovkovy I. and Huang M., Phys. Lett. B, 564 (2003) 205 (arXiv:hep-ph/0302142); Phys. Rev. D, 70 (2004) 051501 (arXiv:hep-ph/0407049); Phys. Rev. D, 70 (2004) 094030 (arXiv:hep-ph/0408268). [15] Rajagopal K. and Shuster E., Phys. Rev. D, 62 (2000) 085007 (arXiv:hep-ph/0004074). [16] Alford M. G., Berges J. and Rajagopal K., Nucl. Phys. B, 571 (2000) 269. [17] Ciminale M., Nardulli G., Ruggieri M. and Gatto R., Phys. Lett. B, 636 (2006) 317 (arXiv:hep-ph/0602180). [18] Anglani R., Mannarelli M., Nardulli G. and Ruggieri M., arXiv:hep-ph/0607341.

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Production of a degenerate Fermi gas of metastable helium-3 atoms W. Vassen, T. Jeltes, J. M. McNamara and A. S. Tychkov Laser Centre Vrije Universiteit - De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands

1. – Introduction Most of this book discusses Fermi gases composed of alkali atoms. Since the ﬁrst realization of a degenerate 40 K Fermi gas in 1999, 6 Li has been the only other fermionic atom to be cooled below the Fermi temperature. In this contribution we will discuss our recent results on cooling fermionic 3 He below the Fermi temperature. This has not been performed with ground-state 3 He, but with 3 He in the metastable 2 3 S1 state, which has an internal energy of 20 eV. Helium atoms in this metastable state are denoted by He∗ and have a lifetime of ∼ 8000 s, which is inﬁnite for all practical purposes. He∗ atoms can be eﬃciently cooled with laser light at 1083 nm and samples of ∼ 109 atoms of either isotope can be conﬁned in a magneto-optical trap (MOT) at temperatures around 1 mK. The density in a He∗ MOT is limited to ∼ 4 × 109 cm−3 due to large losses associated with Penning ionization (PI) and associative ionization (AI) (1)

He∗ + He∗ → He + He+ + e− − He∗ + He∗ → He+ 2 +e

(PI), (AI).

These loss processes (summarized as Penning ionization in the rest of this contribution) are discussed in many studies of ultracold metastable helium and are both a strength and a weakness of this atom in cold-atom experiments. To begin with the weak side, the rate constant for ionizing collisions in a typical MOT is ∼ 5 × 10−9 cm3 /s and it is this rate that limits the achievable densities in a MOT and forces He∗ MOTs to have a large diameter (up to 1 cm) in order to accumulate a large number of atoms. When the light c Societ` a Italiana di Fisica

817

818

W. Vassen, T. Jeltes, J. M. McNamara and A. S. Tychkov

is switched oﬀ this loss rate is reduced to ∼ 10−10 cm3 /s [1], still far too large for further cooling processes lasting seconds. It is only when the atoms are fully spin-polarized that the rate constant drops by four orders of magnitude [2] and becomes acceptable. Before our work on cooling and trapping 3 He∗ [1,3,4] this suppression of Penning ionization had only been demonstrated for the bosonic isotope 4 He, which, due to this reduced loss rate constant, could be brought to Bose-Einstein condensation (BEC) [5-7]. The reduction in loss rate can be simply understood by considering the angular momentum quantum numbers (for 4 He∗ ) in more detail (2)

He∗ (J = 1) + He∗ (J = 1) → He(J = 0) + He+ (J = 1/2) + e− (J = 1/2) .

The 2 3 S1 state has a total angular momentum (spin) J = 1. In a collision between two He∗ atoms a total angular momentum of 0, 1 and 2 is therefore possible, while for the reaction products the maximum combined angular momentum is 1. Hence, Penning ionization is forbidden (as long as spin is conserved) in either of the two cases in which the atoms are spin-polarized in either of the fully stretched magnetic substates (m = +1 or m = −1) for which J = 2. Experimental studies have to date concentrated on the m = +1 state which is magneticallly trappable. Only due to the spin-dipole interaction Penning ionization may occur, and the rate constant for that process is ∼ 10−14 cm3 /s [2]. Consequently, a mixture containing m = +1 atoms will show strong Penning ionization. In contrast to the 4 He case, the fermionic isotope 3 He has nuclear spin I = 1/2, which makes the level structure more complex. Due to this nuclear spin, the metastable 2 3 S1 state is split by the hyperﬁne interaction into an F = 3/2 and F = 1/2 state (splitting 6740 MHz) with the F = 3/2 state having the lowest energy (see ﬁg. 1). When we consider magnetic trapping, three magnetic substates are in principle trappable, i.e., |F, MF = |1/2, +1/2, |3/2, +1/2 and |3/2, +3/2. In a magnetic trap we may trap atoms in any of these states, although PI will strongly reduce the lifetime of any mixture of these spin states. The Pauli exclusion principle, however, suppresses collisions between identical fermions and PI should be suppressed in a cloud of 3 He∗ atoms in a single spin state, though this would come at the cost of rethermalizing collisions and our ability to evaporatively cool such a sample. Sympathetic cooling overcomes this limitation by introducing a second component to the gas; thermalization between the two components then allows the mixture as a whole to be cooled. As sympathetic cooling usually works best in a mixture with bosonic atoms, we need to consider bosons that do not Penning ionize in collisions with 3 He∗ . An obvious choice for us was 4 He∗ as the experimental set-up for evaporative cooling was available. However, we cannot use all three trappable 3 He∗ states now. Although there have been no theoretical predictions, we may expect that only in collisions between the |3/2, +3/2 state of 3 He∗ and the |J, mJ = |1, +1 state of 4 He∗ the same reduction of Penning ionization as in a pure gas of |1, +1 atoms of 4 He∗ will occur because of conservation of angular momentum as above. Of course this may have considerable consequences for further studies of ultracold 3 He∗ clouds and 3 He∗ /4 He∗ mixtures. For the mixtures only a single spin combination is expected to be

Production of a degenerate Fermi gas of metastable helium-3 atoms

819

Fig. 1. – 4 He and 3 He energy levels relevant for laser cooling and probing at 1083 nm.

stable, i.e., 3 He∗ in the |3/2, +3/2 state and 4 He in the |1, +1 state. For an ultracold 3 He∗ gas there are more options, however no mixtures of diﬀerent magnetic substates. This hampers studies where one controls the scattering length via Feshbach resonances. Although no studies of possible Feshbach resonances have so far been performed, it may be expected that even if there were magnetic ﬁelds at which they existed, the required mixture of diﬀerent hyperﬁne states (|1/2, +1/2 and |3/2, +1/2 for instance) would be very unstable. One should not draw the conclusion that only the 3 He∗ |3/2, +3/24 He |1, +1 system would be suitable for producing and studying an ultracold gas of 3 He∗ atoms. Any bosonic alkali atom in a fully stretched hyperﬁne state may show the suppression of Penning ionization in collisions with 3 He∗ atoms in the |3/2, +3/2 state. It remains to be seen, of course, if this suppression will be as large as observed in collisions with 4 He |1, +1 atoms. From an experimentalists point of view, the Penning ionization process provides new and exciting opportunities to investigate cold clouds of He∗ atoms. First, as the dominant loss process produces ions, we have a direct and “nondestructive” detection method. Ions

820

W. Vassen, T. Jeltes, J. M. McNamara and A. S. Tychkov

can be eﬃciently detected with an electron multiplier: when an ion hits a microchannel plate (MCP) detector with suﬃcient kinetic energy, an electron is released which can be ampliﬁed and detected. When the count rate is not too high, one can even count individual ions. Second, in the same way as ions can be detected, also neutral He∗ atoms can be detected: the 20 eV internal energy far exceeds the electron work function of almost all materials and the released electron can be counted by a similar MCP detector. The application of He+ and He∗ detectors allows alternative means of observation of an ultracold cloud of metastable helium, besides the standard method of absorption imaging. The ion detector is eﬀectively a density probe (see eq. (1)) while the He∗ detector measures the arrival time (and position in the case of a position-sensitive MCP) of atoms after release or escape from the trap. 2. – Relevant atomic physics In contrast to all alkali cold atom research, experiments on metastable helium are performed with atoms in the highly excited metastable 2 3 S1 state which lies 20 eV above the 1 1 S0 ground state. The ionization limit is at 25 eV. Collisions between He∗ atoms therefore contain 40 eV internal energy which is more than enough to ionize one of the two collision partners, while the other is de-excited (see eq. (1)). Atoms in the 2 3 S1 state can be laser cooled by applying radiation at 1083 nm to excite the 2 3 P state. In ﬁg. 1 the energy level structure relevant for laser cooling is summarized. For 4 He the D2 transition, 2 3 S1 → 2 3 P2 , that is commonly used for eﬃcient cooling and trapping in a MOT, is indicated, whilst for 3 He cooling is performed on the C3 (F = 3/2 → 5/2) transition to prevent optical pumping to the F = 1/2 hyperﬁne state. The more than 33 GHz frequency diﬀerence between the two laser cooling transitions implies that the laser radiation for cooling both isotopes cannot come from one laser. Moreover, there is a near coincidence between the 4 He laser cooling transition and the C9 transition of 3 He, which may cause unwanted optical pumping. However, when only cooling 3 He∗ in the F = 3/2 state, no repumper is required [3]. The Zeeman splitting of the metastable states involved in magnetic trapping and evaporative cooling is shown in ﬁg. 2. The fully stretched states of both isotopes show the same magnetic-ﬁeld dependence. For sympathetic cooling of 3 He∗ by 4 He∗ , evaporative cooling of 4 He∗ is performed on the MJ = 1 → 0 transition. The corresponding RF radiation will then not aﬀect the 3 He∗ atoms (at the same temperature) as these are resonant at lower frequency. Due to the large hyperﬁne and ﬁne-structure splittings the magneticﬁeld dependence is linear in magnetic traps typically used in BEC experiments of 4 He∗ . 3. – Molecular and collision physics Collisions between two He∗ atoms are described with interatomic interaction potentials that can be calculated ab initio in the Born-Oppenheimer approximation, where the total electronic spin S is a good quantum number. The possible values S = 0, 1, and 2 correspond to a singlet, triplet and quintet potential (see ﬁg. 3). In a MOT the

821

Production of a degenerate Fermi gas of metastable helium-3 atoms

E

3He*

4He*

mF

mJ

+3/2

+1

ωrf +1/2 B

0 -1/2

-3/2

-1

Fig. 2. – Magnetic-ﬁeld dependence of the 3 He∗ F = 3/2 and 4 He∗ J = 1 magnetic substates. Evaporative cooling, at RF frequency ωrf , is performed on the MJ = +1 → MJ = 0 transition.

atoms are unpolarized and collisions take place in all three potentials. In a magnetic trap with atoms in the fully stretched state, collisions take place in only the quintet potential and the binding energy of the least bound state determines the scattering length. The quantum chemistry calculations of the quintet potential have recently become so good that the scattering length for collisions between spin-polarized 4 He∗ atoms can be

0.04 0.03 Energy (Eh)

0.02 0.01 5 + Σg

0.00 −0.01

3 + Σu

−0.02 −0.03

4

6

1 + Σg

8 10 12 Internuclear distance R (a0)

Fig. 3. – Short-range potentials in atomic units.

14

822

W. Vassen, T. Jeltes, J. M. McNamara and A. S. Tychkov

calculated with an accuracy of 3%. The calculated value is a44 = +7.64(20) nm [8], which agrees very well with an experimental value a44 = +7.512(5) nm recently obtained in Paris [9]. The value for a44 is relatively large, about 50% larger than corresponding numbers for Rb and Na. This means that evaporative cooling of 4 He∗ proceeds eﬃciently and short RF ramps can be used. The accuracy of the theoretical calculations suggests that similar accuracy can be obtained when calculating the molecular potentials between two spin-polarized 3 He∗ atoms, and, more importantly, for one spin-polarized 3 He∗ atom and one spin-polarized 4 He∗ atom. These calculations have recently been performed as well. A value a34 = +28.8+3.9 −3.3 nm [10] results, which is positive and exceptionally large. The large value implies that sympathetic cooling of 3 He∗ by 4 He∗ in a magnetic trap should proceed very eﬃciently. Moreover, an ultracold mixture of spin-polarized 3 He∗ and 4 He∗ atoms should be stable as the interaction is repulsive. Of course, what the two-body and three-body loss rate constants will be for such a large heteronuclear scattering length remains to be seen. Calculations have not yet been performed, but an order of magnitude estimate of the three-body loss rate constant K3BBF for collisions between two 4 He∗ bosons and one 3 He∗ fermion, which is expected to be the largest, gives K3BBF ≈ 1 × 10−24 cm6 /s [4, 11]. This value is 1–3 orders of magnitude larger than the three-body loss rate of ultracold 4 He∗ alone. 4. – MOT results We load our MOT with either 4 He∗ , 3 He∗ , or an arbitrary mixture of the two. We start with a DC discharge to populate the metastable state. The discharge is maintained inside our source chamber. The helium mixture is prepared outside the vacuum chamber and coupled via a needle valve into a discharge tube which is positioned 1–2 cm in front of a skimmer with a 1 mm hole. The diverging He∗ beam is collimated over a length of 20 cm by applying transversal laser cooling in two dimensions to increase the beam intensity [12], and deﬂected over an angle of 1 degree to purify the He∗ beam. The latter is important to prevent ground-state atoms, VUV photons and singlet metastables (from the source) reaching the ultrahigh vacuum (UHV) chamber in which the ﬁnal cooling and trapping experiments take place. After the collimation/defection zone the He∗ beam enters a 2 m Zeeman slower which slows the atoms to a velocity of ∼ 70 m/s, which is low enough to capture them in a MOT. The MOT conﬁguration is standard except that very large MOT beams have to be used in order to trap as many atoms as possible. Slowing 4 He∗ atoms from a longitudinal velocity of up to 1050 m/s (∼ 30% of the atoms in the atomic beam) we trap typically 2 × 109 atoms at a temperature of 1 mK. Alternatively, using a pure 3 He∗ source, we trap up to 1 × 109 3 He∗ atoms in the F = 3/2 state at the same temperature. The number of trapped 3 He∗ atoms is quite large considering that only a few percent of the 3 He∗ atoms can be slowed due to the larger average velocity of 3 He∗ atoms leaving the source; we, however, increased the loading time and used larger laser power (compared to loading the 4 He∗ MOT). To trap a mixture of 4 He∗ and 3 He∗ atoms we use two lasers. Our 4 He laser is a 250 mW LNA laser while our 3 He laser is a 1 W ytterbium ﬁber laser. The output of both

Production of a degenerate Fermi gas of metastable helium-3 atoms

823

lasers is split, sent through various acousto-optic modulators (AOMs) and overlapped on several beam splitters to provide the collimation/deﬂection beam at zero detuning, the Zeeman slower beam at −250 MHz detuning and the MOT beams at −40 MHz detuning. A complication in magneto-optical trapping is the optical pumping that occurs in 3 He∗ due to the 4 He laser beams [3]. We found that optimal trapping is obtained using a repumper beam close to the C2 transition, simply produced by double passing our Zeeman slower AOM. We trap a mixture of up to 7 × 108 3 He∗ and 1.5 × 109 4 He∗ atoms simultaneously at a temperature of ≈ 1 mK with this setup [7]. . 4 1. Homonuclear and heteronuclear collisions in a MOT . – From studies of the ion production in combination with absorption imaging, it is possible to extract two-body loss rate constants in an unpolarized cloud of He∗ atoms at temperatures of ∼ 1 mK. Two diﬀerent loss rates may be studied. In the presence of MOT light, losses are dominated by light-assisted collisions, while in the dark all atoms are in the metastable ground state and loss rates are much smaller. Experimentally these losses are studied using an MCP detector that attracts all ions produced by PI/AI, a second MCP detector that measures the temperature, and absorption imaging to determine the number of metastables in the MOT as well as the density proﬁle of the cloud. The loss rate for collisions in the dark is determined by measuring the reduction in the ion signal when the MOT light is switched oﬀ for a very short time, short enough not to disturb the density proﬁle in the MOT. The goal of the research, which was performed in a setup similar to the setup used for magnetic trapping and cooling towards degeneracy, was to investigate the diﬀerence in PI/AI loss rate constant for homonuclear bosonic 4 He∗ -4 He∗ collisions, fermionic 3 He∗ -3 He∗ collisions and heteronuclear 4 He∗ -3 He∗ collisions, due to quantum statistics. For collisions in the presence of MOT light the diﬀerence between the homonuclear loss rate constant for 4 He∗ and 3 He∗ turned out to be small as many partial waves contribute to the loss rate constant. For our typical MOT parameters the two-body loss rate coeﬃcient were 3.3(7) × 10−9 cm3 /s for 4 He∗ and 5.5(8) × 10−9 cm3 /s for 3 He∗ [1]. These numbers are not directly comparable to other experimental values as they strongly depend on the MOT laser intensity and detuning. Also in the case of collisions in the dark we did not ﬁnd large diﬀerences in the Penning ionization rate despite the fact that quantum statistics should play a prominent role now as only the lowest partial waves contribute. However, in the dark (when all atoms have decayed to the metastable state), the atoms are distributed over all magnetic substates and the diﬀerence between the 4 He∗ and 3 He∗ loss rate constant is not very large. We both measured (at the temperature of the MOT), and calculated (for a large range of temperatures) these loss rate constants. In the calculations we included quantum threshold behavior, applied Wigner’s spin conservation rule and incorporated the required quantum-statistical symmetry requirements in a single channel model [1,13]. In ﬁg. 4 the calculated loss rate constants for both the homonuclear cases and the heteronuclear case are plotted as a function of temperature. Our measurements are also included and fall on the theoretical curves. The error in the measurements in the homonuclear case are primarily due to the measurement of the density proﬁle of the single-isotope MOT. The

824

W. Vassen, T. Jeltes, J. M. McNamara and A. S. Tychkov

th KSS 1010 cm3 s1

5.0 4.0 3.0 2.0 1.5 1.0 0.7 0.5 0.01

0.1

1 Temperature mK

10

100

Fig. 4. – Theoretical loss rate coeﬃcients (in the dark) as a function of temperature and measured values; 3 He-3 He: dashed curve and experimental point (diamond), 4 He-4 He: dotted curve and experimental point (triangle), 3 He-4 He: solid curve and experimental point (square).

error in the heteronuclear case is (only slightly) larger. This can be understood from the fact that the heteronuclear loss rate in a two-isotope MOT (TIMOT) is measured from a total loss rate that is a combination of homonuclear and heteronuclear losses. As a bonus we have deduced the fraction of triplet metastables in our beam leaving the DC discharge. This fraction turns out to be 0.01% [1]. We have deduced this number by measuring the linear losses from our 4 He∗ MOT due to collisions with fast ground state 4 He atoms from the source (we did not use a deﬂection zone in this experiment). From the known total cross-section a ground-state beam intensity of 4 × 1018 s−1 can be calculated, while the intensity of the 4 He∗ beam is 4 × 1014 s−1 for the discharge conditions of our source. 5. – Magnetic trapping and one-dimensional Doppler cooling In order to proceed towards evaporative and sympathetic cooling, the atoms have to be transferred from the MOT to a magnetic trap. Our trap is a cloverleaf trap with bias compensation. Its geometry, including the MOT laser beams, is shown in ﬁg. 5. The coils are positioned in re-entrant windows that are separated by 3 cm. The vertical direction is used for absorption imaging as well as for time-of-ﬂight (TOF) measurement (using an MCP positioned 17 cm below the trap) of the velocity distribution and atom number. To measure an absorption image, the MCP, which is mounted on a translation stage, is moved out of the vertical laser beam. To load the magnetic trap all currents and laser beams that produce the MOT are switched oﬀ. Next a weak axial magnetic ﬁeld is applied and the atoms are spin-polarized

Production of a degenerate Fermi gas of metastable helium-3 atoms

825

Fig. 5. – Experimental setup for magneto-optical trapping, magnetic trappping and detection on a microchannel plate detector.

by a short laser pulse along the ﬁeld direction. The currents for the cloverleaf trap and the one-dimensional (1D) laser cooling beam are then switched on simultaneously; ∼ 60% of the 4 He∗ atoms are transferred to the magnetic trap and a temperature of 0.15 mK is reached. 1D Doppler cooling turned out to be very eﬃcient, both for 4 He∗ and 3 He∗ [7]. The cooling mechanism in this case relies on standard and fast Doppler cooling along the axial (magnetic ﬁeld) direction, simply implemented by retroreﬂecting a very weak circularly polarized laser beam, about one linewidth red-detuned from resonance at the center of the trap. The bias magnetic ﬁeld was 24 G to prevent optical pumping to unwanted magnetic substates. Cooling in the radial direction is much slower and relies on absorption of the red-detuned ﬂuorescence photons; optimum results (low temperature and no atoms lost during Doppler cooling) were obtained for 2 s cooling. For the 1D Doppler cooling to work, the optical density has to be large; we found that it only works well for more than 108 atoms in the trap. The resulting phase space density of our 4 He∗ atoms was ≈ 10−4 , a factor 600 higher than without 1D Doppler cooling. The lifetime of the atoms in our trap is fully determined by collisions with background gas. Typically a lifetime of 2 minutes is measured. No indications of two-body losses are found. To increase the collision rate the mixture (or pure cloud) is adiabatically compressed by reducing the bias magnetic ﬁeld to a value ∼ 3 G. The trap frequencies are then 273 (237) Hz in radial direction and 54 (47) Hz in axial direction for 3 He (4 He). The trap thus has a cigar shape.

826

W. Vassen, T. Jeltes, J. M. McNamara and A. S. Tychkov

In the following we ﬁrst discuss our results for evaporative cooling and BEC of 4 He∗ as these are crucial to the success of sympathetically cooling 3 He∗ to quantum degeneracy, which will be discussed subsequently. 6. – Bose-Einstein condensation of helium-4 The ultracold 4 He∗ cloud at a phase space density of ≈ 10−4 is further cooled by RF-induced evaporative cooling. For this purpose an RF coil consisting of two windings (diameter 4 cm) is positioned 7 cm from the trap center just behind the Zeeman slower inside the vacuum. A single exponential ramp, starting at 50 MHz and cut at ≈ 8.5 MHz, cools the atoms in 12 s to BEC [7]. The temperature at the onset of BEC is ≈ 2 μK. We can also reach BEC in a much shorter ramp, as short as 2 s, however at the cost of atom number. The beauty of our experiment is that we have observed BEC by applying three diﬀerent detection techniques [7] (see ﬁg. 6). The most sensitive technique is a TOF analysis of the expanding cloud after release from the trap, measured on the He∗ MCP mounted below the trap. A TOF spectrum is shown in ﬁg. 6a. It clearly shows the typical double structure: a broad thermal velocity distribution together with the inverted parabola of a BEC in the Thomas-Fermi limit. When we move the MCP detector away in horizontal direction, we measure the absorption image shown in ﬁg. 6b. This shows the typical signature of BEC in expansion: a round thermal cloud and the elliptical shape of the condensate. In situ detection of the sudden appearance of a condensate is possible by observing the ion production in the cloud during evaporative cooling. When a BEC forms, the density in the center of the trap suddenly increases, giving rise to increased two- and three-body PI losses. We see this eﬀect clearly on the second MCP detector that attracts all ions (ﬁg. 6c). Analysis of the TOF spectra (on CCD and MCP detectors) allows us to measure the expansion of the condensate in all three spatial dimensions and determine the number of atoms in the condensate. Due, in the main, to detector saturation we can only say that our largest BECs contain between 1.5 × 107 and 4 × 107 atoms [7]. These are large numbers, comparable to some of the largest condensates produced to date, and at least one order of magnitude larger than other BECs of He∗ . In BEC decay experiments [7], we were able to detect a condensate up to 75 s after it was formed, due to the wide range and high sensitivity of the MCP detector. 7. – Fermi degeneracy of helium-3 We discussed in sect. 4 that we can trap a mixture of up to 7× 108 3 He∗ and 1.5 × 109 4 He∗ atoms simultaneously at a temperature of ≈ 1 mK in our TIMOT. Given the size of our 4 He∗ reservoir, it is impossible to cool so many fermions to quantum degeneracy by sympathetic cooling. Our BEC experiments show that we cross the BEC threshold with typically ≈ 5 × 107 bosons. We therefore reduce the number of fermions in our TIMOT to ≈ 107 by changing the composition of the isotopic mixture in our helium reservoir to 90% 4 He and 10% 3 He or reducing the loading time of the 3 He∗ MOT. The number of

827

Production of a degenerate Fermi gas of metastable helium-3 atoms

He flux arb. units

100

a

80 60 40 20

170

0.30

190 200 Time ms

210

b

0.25 Axial cm

180

0.20 0.15 0.10 0.05 0.00 0.0

0.1

0.2 0.3 Radial cm

0.4

0.5

1.0 Time s

0.5

0.0

Ion flux 105 s1

c 1.5 1.0 0.5 0.0

2.0

1.5

Fig. 6. – Observation of BEC: a) on the He∗ microchannel plate detector, the dashed ﬁt shows the condensed fraction and the dashed-dotted ﬁt the thermal component; b) on a CCD camera, after 19 ms expansion a round thermal cloud surrounding an elliptical condensate is visible; c) on the ion microchannel plate detector, the condensate starts to grow at t = −0.95 s, at t = −0.45 s the RF ramp ends and at t = 0 the trap is switched oﬀ.

828

W. Vassen, T. Jeltes, J. M. McNamara and A. S. Tychkov

Fig. 7. – Potential energy of the diﬀerent magnetic substates of both 4 He∗ and 3 He∗ in the cloverleaf magnetic trap. In the center of the trap the m = +1 4 He atoms are (at T = 0) removed with RF at a frequency of 8.4 MHz, while the m = +3/2 3 He∗ atoms can be removed at a frequency of 5.6 MHz.

He∗ atoms should not be chosen too low as then the 1D Doppler cooling does not work as well as at higher number due to loss of optical density. There is clearly a compromise here which still needs optimization. After 1D Doppler cooling and compression we have a mixture in thermal equilibrium and a collision rate suitable for eﬃcient evaporative cooling. We then perform RF-induced evaporative cooling of 4 He∗ and sympathetically cool 3 He∗ in the |3, 2, +3/2 state. Measuring the number of 4 He∗ atoms and 3 He∗ atoms during evaporative cooling, we see the typical reduction in the number of 4 He∗ atoms while the number of 3 He∗ atoms stays roughly constant. This shows that sympathetic cooling works eﬃciently, as expected from the large value of the heteronuclear scattering length. We have not (yet) observed the 3 He∗ atoms by absorption imaging, instead we rely upon TOF measurements in our determination of temperature and number of atoms. Of course an MCP does not discriminate between the two isotopes so, in order to follow the sympathetic cooling process, we measure the number of atoms from one isotope after removing the other from the trap. At high temperature (T > 20 μK) we use resonant laser light for this purpose. At these temperatures the number of 4 He∗ atoms is much larger than the number of 3 He∗ atoms so the 4 He∗ number of atoms and temperature are not very diﬃcult to measure. On the other hand, it is very diﬃcult to measure the number of 3 He∗ at these “high” temperatures. To measure a 3 He∗ TOF we need to push a much larger number of 4 He∗ away, while we know that 4 He∗ light aﬀects the 3 He∗ cloud by oﬀ-resonant excitation of the C9 transition (see ﬁg. 1). This “pushing” technique therefore is not very accurate. Luckily, when the temperature has decreased below ≈ 20 μK we can selectively remove either 4 He∗ or 3 He∗ by applying a short RF ramp. This is illustrated in ﬁg. 7. When we want to measure either 4 He∗ or 3 He∗ we generate a short RF sweep and cut through the distribution of the other isotope. The frequency span of this ramp is, at temperatures below 20 μK, small enough not to disturb 3

829

Production of a degenerate Fermi gas of metastable helium-3 atoms

MCP Voltage mV

250 200 150 100 50 170

180 190 Time ms

200

210

Fig. 8. – Time-of-ﬂight signal of a degenerate Fermi gas of 3 He∗ atoms together with a ﬁt to a Fermi-Dirac velocity distribution. The ﬁt shows that in this case the number of 3 He∗ atoms is 2.1 × 106 , the temperature 0.8 μK and the degeneracy parameter T /TF = 0.45.

the isotope that we want to study. We have noted that 3 He∗ is eﬃciently removed despite the fact that two RF photons have to be absorbed to remove it from the trap. We are thus able to measure the mixture by simply releasing it from the trap or by measuring each isotope separately. Applying a magnetic-ﬁeld gradient, it is possible to separate both isotopes in time as well. Our ﬁrst experiments aimed at producing a degenerate Fermi gas (DFG), so we applied a single RF ramp with end frequency below 8.4 MHz. This cools and eﬀectively removes all 4 He∗ atoms from the trap, while the 3 He∗ is sympathetically cooled and remains in the trap. A “typical” TOF spectrum obtained in this way is shown in ﬁg. 8. A perfect ﬁt is obtained when we ﬁt this spectrum with a Fermi-Dirac TOF function; a Maxwell-Boltzmann classical TOF function clearly does a less good job. This TOF corresponds to a degenerate Fermi gas with a temperature of 0.8 μK and degeneracy parameter T /TF = 0.45, with N3 = 2.1 × 106 3 He∗ atoms in the |3/2, +3/2 state. To calculate the Fermi temperature we used the standard formula (3)

kB TF = h(6N3 νr2 νa )1/3 ,

with kB Boltzmann’s constant, h Planck’s constant and νa and νr the axial and radial trap frequencies. In alkali absorption spectra on a CCD camera the TOF distribution in two dimensions is observed and the signal is integrated over the third (line-of-sight) dimension. This already makes it diﬃcult to see the diﬀerence between a Maxwell-Boltzmann distribution and a Fermi-Dirac distribution. In our case we eﬀectively integrate over two dimensions (the surface of the MCP detector) and only the expansion in the radial

830

W. Vassen, T. Jeltes, J. M. McNamara and A. S. Tychkov

a

1.10

Fitted temperature zK

1.05 1.00 0.95

Σ0

0.90 0.0

1.0 2.0 1.5 0.90 5.25 b 0.85 c 5.00 0.80 4.75 0.75 4.50 0.70 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Fraction of Σ0 removed 0.5

Fig. 9. – Temperatures obtained by repeatedly ﬁtting, with a classical Maxwell-Boltzmann distribution, three time-of-ﬂight spectra from which increasingly large central fractions have been removed: (a) a degenerate 3 He∗ Fermi gas at T /TF = 0.5; (b) a cloud of 4 He∗ with T TC ; and (c) a cloud of 4 He∗ atoms just above TC .

direction is measured. This makes the small diﬀerence between these two velocity distributions even smaller. However, our TOF spectra show a very good signal-to-noise ratio and we can ﬁt far into the wings of the distribution. To show that indeed the FermiDirac velocity distribution is best in ﬁtting a degenerate Fermi gas and to compare these results also with similar results for bosons close to BEC we measured four TOF spectra, two (for 3 He∗ and 4 He∗ ) close to degeneracy and two (for 3 He∗ and 4 He∗ ) far above. We ﬁtted these four TOF’s with a gaussian (for a Maxwell-Boltzmann distribution) and extracted the temperature and rms width σ0 . Then, as described by Schreck [14], we removed increasingly larger central fractions of the TOF spectrum and ﬁtted it again. The temperature we extract as a function of the fraction of σ0 removed is plotted in ﬁg. 9, in ﬁg. 9a for 3 He∗ below TF , in ﬁg. 9b for 4 He∗ far above TC (a similar spectrum is obtained for 3 He∗ above TF ) and in ﬁg. 9c for 4 He∗ just above TC . For a cloud displaying Fermi-Dirac statistics the population of low-energy states is suppressed due to the Pauli exclusion principle and ﬁtting a Gaussian to the whole TOF will lead to an overestimation of the temperature. By ﬁtting only the wings this overestimation is reduced and

831

Production of a degenerate Fermi gas of metastable helium-3 atoms

MCP Voltage mV

250 200 150 100 50 170

180 190 200 Time of flight ms

210

Fig. 10. – Time-of-ﬂight spectrum for a degenerate mixture of 4.2 × 105 (T /TF = 0.5) 3 He∗ and 1 × 105 4 He∗ atoms. The dashed curve is a Fermi-Dirac velocity distribution ﬁtted to the wings while the dashed-dotted curve is a ﬁt of a pure Bose-Einstein condensate to the central part.

the extracted temperature will fall as a function of the fraction of σ0 removed in the ﬁt. Figure 9a clearly shows this. The opposite behaviour was observed for the bosons where the low-energy states show enhanced population due to Bose enhancement (ﬁg. 9c). A thermal cloud should see no dependence on the fraction of σ0 removed and this is shown clearly in ﬁg. 9b, which was recorded at a temperature of ∼ 5 μK, far above TC (and TF ). From ﬁg. 9a we again recover T /TF = 0.5 when we ﬁt only the wings removing the central part up to 1.75σ0 . The large value for the theoretical heteronuclear scattering length of +29 nm suggests that thermalization should be very eﬃcient. Indeed, we can produce a degenerate Fermi gas with an RF ramp of 2.5 s, albeit with a slightly larger degeneracy parameter. For such a ramp we measure T /TF = 0.75, with N3 = 4 × 106 . In this way we thus produce larger clouds of degenerate 3 He∗ gas. 8. – Mixtures It is straightforward to continue along the lines described in the previous section to realize a quantum degenerate mixture of 4 He∗ in the |1, +1 state and 3 He∗ in the |3/2, +3/2 state. The only diﬀerence is the end frequency of the RF ramp which now is to be chosen slightly above the bottom of the 4 He∗ trap, i.e., slightly above 8.4 MHz. This results in a pure BEC, sitting (presumably) in the center of a degenerate Fermi gas. An observed TOF for this situation is shown in ﬁg. 10. This speciﬁc mixture contained N3 = 4.2 × 105 3 He∗ fermions and a condensate with N4 = 1 × 105 4 He∗ atoms at a temperature corresponding to T /TF = 0.5. The two theoretical curves represent the Fermi-Dirac velocity distribution, ﬁtted to the wings, and the inverted parabola of a pure

832

W. Vassen, T. Jeltes, J. M. McNamara and A. S. Tychkov

Fig. 11. – T = 0 calculation of the spatial distribution of a Bose-Einstein condensate of 105 4 He∗ atoms in the |1, +1 state (left) and a degenerate Fermi gas of 106 3 He∗ atoms in the |3/2, +3/2 state (right) in our cloverleaf magnetic trap. The strong repulsion due to the large and positive 3-4 scattering length causes the fermions to be repelled out of the core of bosons.

BEC in the Thomas-Fermi limit. These experiments are still in a preliminary phase but we may already conclude from our observations that the lifetime of a condensate (τC ) in the presence of a DFG of 3 He∗ is signiﬁcantly shorter than in case of a pure condensate. We ﬁnd τC = 1 s for a pure condensate [7] and τC ∼ 10 ms for the mixture. This can be understood when indeed, as suggested in sect. 3, the three-body loss rate K3BBF is large. 9. – Prospects Now that we have seen that we can produce a degenerate Fermi gas of metastable 3 He atoms as well as a degenerate boson-fermion mixture of metastable atoms, it is interesting to discuss what these gases may be good for. We have a system with both a naturally large and positive scattering length and this oﬀers some interesting possibilities. It may be possible to observe phase separation [15] in this system. For the large scattering lengths of this system our mixture may already exhibit phase separation, although we have not demonstrated this. A T = 0 calculation based on the Gross-Pitaevskii equation for the bosons and the Thomas-Fermi equation for the fermions [16], shown in ﬁg. 11, shows that the fermions are expected to be expelled from the hard core of bosons already at the densities calculated for observed particle numbers at degeneracy. As discussed in sect. 3, it will be very diﬃcult to make spin mixtures of ultracold 3 He∗ atoms. This hampers observation of BCS-like phenomena observed for the Li and K systems. However, as the 3-4 scattering is so large, we need only a very small tuning of the scattering lengths to open up the p-wave window for Cooper pairing of |3/2, +3/2 3 He∗ atoms. This pairing is mediated by density ﬂuctuations in the 4 He∗ cloud in a boson-fermion mixture [17]. It seems not entirely impossible to use an optical Feshbach resonance for this purpose. Also loading such a mixture in an optical lattice oﬀers many interesting new possibilities [18]. First, this oﬀers a new system for the observation of exotic new phases and secondly a boson-fermion mixture in an optical lattice will be a naturally clean system with at most

Production of a degenerate Fermi gas of metastable helium-3 atoms

833

one boson per lattice site due to the fast Penning ionization that will occur when two 4 He∗ atoms or a 4 He∗ and a 3 He∗ atom occupy the same lattice site. Another important application of ultracold 4 He∗ and 3 He∗ will be in metrology. A source of ultracold metastable helium atoms provides an almost ideal opportunity to measure with high spectral resolution narrow transitions from the 2 3 S1 state. Of particular interest are the electric-dipole–allowed 1083 nm line (to 2 3 P ) with a natural linewidth of 1.6 MHz and the magnetic-dipole–allowed 1557 nm line (to 2 1 S), with a natural linewidth of 8 Hz [19, 20]. Observation of these transitions in a 1D optical lattice will allow virtually Doppler-free excitation oﬀering a resolution below 1 kHz. This allows tests of bound state QED in two-electron systems, and a measurement of diﬀerences in nuclear charge radii as well as nuclear masses. The latter will be possible if we measure the transition isotope shift of the 1557 nm line, which at present can be calculated from theory with an accuracy better than 1 kHz [20]. Perhaps the most promising experiment that can be performed with an ultracold cloud of 3 He∗ atoms is the analogue of the recently realized Hanbury-Brown and Twiss experiment for 4 He∗ atoms in Orsay [21]. For 4 He∗ bunching was observed in an ultracold gas of bosons, close to BEC, while for a BEC this bunching was absent, just as expected for a coherent source of atoms. This in complete analogy to the classical light source and the laser. We have the perfect source to repeat this experiment with fermions and to observe anti-bunching for an ultracold gas of 3 He∗ . Actually, on July 14, two weeks after the end of the School, we indeed observed antibunching in our setup, in a collaboration with the Orsay group, who brought their position-sensitive MCP detector to Amsterdam [22]. All these proposed experiments will proﬁt from the possibility to observe an ultracold cloud of metastable atoms using the three detection techniques: absorption imaging (so far only used on the bosons), TOF measurements using an MCP and ion detection (also using an MCP detector). The last detection technique was so far only used for the bosons as well.

REFERENCES [1] Stas R. J. W., McNamara J. M., Hogervorst W. and Vassen W., Phys. Rev. A, 73 (2006) 032713. [2] Shlyapnikov G. V., Walraven W. T. M., Rahmanov U. M. and Reynolds M. W., Phys. Rev. Lett., 73 (1994) 3247. [3] Stas R. J. W., McNamara J. M., Hogervorst W. and Vassen W., Phys. Rev. Lett., 93 (2004) 053001. [4] McNamara J. M., Jeltes T., Tychkov A. S., Hogervorst W. and Vassen W., Phys. Rev. Lett., 97 (2006) 080404. [5] Robert A., Sirjean O., Browaeys A., Poupard J., Nowak S., Boiron D., Westbrook C. and Aspect A., Science, 292 (2001) 461. ´ nard J., Wang J., Barrelet C. J., Perales F., Rasel [6] Pereira Dos Santos F., Leo E., Unnikrishnan C. S., Leduc M. and Cohen-Tannoudji C., Phys. Rev. Lett., 86 (2001) 003459.

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[7] Tychkov A. S., Jeltes T., McNamara J. M., Tol P. J. J., Herschbach N., Hogervorst W. and Vassen W., Phys. Rev. A, 73 (2006) 031603(R). [8] Przybytek M. and Jeziorski B., J. Chem. Phys., 123 (2005) 134315. ´ J., Rapol U. D., Leduc M. and Cohen[9] Moal S., Portier M., Kim J., Dugue Tannoudji C., Phys. Rev. Lett., 97 (2006) 023203. [10] Przybytek M. and Jeziorski B., private communication (2005). [11] D’Incao J. P., Suno H. and Esry B. D., Phys. Rev. Lett., 93 (2004) 123201. [12] Rooijakkers W., Hogervorst W. and Vassen W., Opt. Commun., 123 (1996) 321. [13] McNamara J. M., Stas R. J. W., Hogervorst W. and Vassen W., to be published. [14] Schreck F., PhD Thesis, Universit´e Paris VI (2002). [15] Molmer K., Phys. Rev. Lett., 80 (1998) 1804. [16] Marten F., Bachelor thesis, Vrije Universiteit, Amsterdam (2005). [17] Efremov D. V. and Viverit L., Phys. Rev. B, 65 (2002) 134519. [18] Lewenstein M., Santos L., Baranov M. A. and Fehrmann H., Phys. Rev. Lett., 92 (2004) 050401. [19] Baklanov E. V. and Denisov A. V., Quantum Electron., 27 (1997) 463. [20] van Leeuwen K. A. H. and Vassen W., Europhys. Lett., 76 (2006) 409. [21] Schellekens M., Hoppeler R., Perrin A., Gomes J. V., Boiron D., Aspect A. and Westbrook C., Science, 310 (2005) 648. [22] Westbrook C. I., Schellekens M., Perrin A., Krachmalnicoff V., Viana Gomes J., Trebbia J.-B., Esteve J., Chang H., Bouchoule I., Boiron D., Aspect A., Jeltes T., McNamara J., Hogervorst W. and Vassen W., arXiv:quant-ph/0609019 (2006).

Excited states on optical lattices: Atomic lattice excitons A. Kantian Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences A-6020 Innsbruck, Austria Institute for Theoretical Physics, University of Innsbruck - A-6020 Innsbruck, Austria

1. – Introduction Cold fermionic atoms in optical lattices [1-3] oﬀer novel opportunities for the study of excited many-body states. Recent experiments with cold lattice-conﬁned bosonic atoms suggest long lifetimes for states that are very diﬀerent from the ground state of the Hubbard-type Hamiltonian that describes these systems [4]. A particular excited state that is intensely studied in condensed-matter physics is that of excitons in semiconductors, i.e. bound states of electrons and holes. In such solid-state materials, these are created by transfer of electrons from the valence band of the solid into the conduction band. These electrons then bind via an eﬀective Coulomb attraction with the holes they leave behind to form excitons. They can thus be regarded as the elementary quanta of electronic excitation in a crystal. The ensemble of these quasiparticles that makes up the state has properties that are diﬀerent from the underlying fermions. In particular, they obey diﬀerent exchange statistics, and are predicted to condense, akin to a molecular BEC [5], at suﬃciently low temperatures. Furthermore, in certain regimes, two excitons may form a bound state, creating a biexciton, which should also be capable of condensation (cf. [6, 7] for a general introduction and survey to exciton and biexciton physics). The intention of this article is to show how excited states of the exciton type can be created with cold fermionic atoms conﬁned in optical lattices. In sect. 2 ﬁrst the single atomic lattice exciton (ALE) is described, then the question of exciton-exciton interaction c Societ` a Italiana di Fisica

835

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is brieﬂy addressed, after which the exciton condensate is described using BCS meanﬁeld theory. In sect. 3 two potential approaches to experimentally probe a condensate of ALEs are described in order to obtain the condensate fraction and pairing correlations, respectively. Finally, sect. 4 brieﬂy summarizes a protocol by which ultracold ALEs may be prepared in an optical lattice, in a situation that is favorable for the production of an exciton condensate. The realization of excitons proposed here has a number of unique properties, which in particular should facilitate the condensate formation. The ALEs are only very weakly coupled to their environment, so that in contrast to the usual situation in solid-state physics, dissipation can be neglected on typical experimental time scales. They are expected to exhibit long lifetimes [4], with a very low recombination rate as the interaction between particle and hole is typically much smaller than the band gap, and it therefore takes high-order scattering processes to relax a conduction band atom into a lower band. As the atoms are neutral overall, the issue of spontaneous radiative relaxation into a lower band can also safely be neglected. Finally, the properties of cold atomic gases on optical lattices oﬀer a wide range of control over parameters and possibilities for measurement. Interaction strength, hopping rates, the density of particles and holes: all of these crucial properties can be manipulated in a manner diﬃcult to achieve in solid-state materials [1, 3]. Many properties that cannot be probed in semiconductors can also be measured directly, such as the density-density correlations.

2. – Description of atomic lattice excitons . 2 1. The qualitative picture. – Degenerate gases of fermionic atoms in optical lattices have already been realised in several experiments [8-11]. At unit ﬁlling and T ≈ 0, the ground state of the lattice-conﬁned atoms is that of the ﬁlled lowest Bloch band. As an excited state above this ground state, we consider now a situation where the lowest Bloch band (the valence band ) is partially emptied of atoms, which are transferred into the ﬁrst excited Bloch band (the conduction band ), as shown in ﬁg. 1(a). In addition, we assume a repulsive interaction between atoms in diﬀerent bands, which could, for example, be generated by transferring the conduction band atoms into a diﬀerent internal state and exploiting s-wave scattering. Qualitatively, a bound particle-hole state —the ALE— is formed because the interband repulsion makes it energetically favorable for the conduction band atom to tunnel to an occupied site if a valence band atom tunnels back, which is pictured schematically in ﬁg. 1(a). This ensures that the atom and hole tunnel together. . 2 2. An eﬀective Hamiltonian description. – Analogous to the case of excitons in semiconductors [6], a formal description of the interacting populations of conduction band atoms and valence band holes can be obtained through an eﬀective Hamiltonian. At low temperatures, the ﬁeld operator of both the conduction and valence band atoms

837 E/(2J)

atomic lattice excitons

(a)

(b) scattering continuum

bound particle-hole pair: exciton

Forbiddden by on-site repulsion

-1

- 0.5

0

Ka/

0.5

1

Fig. 1. – (a) Set-up illustration for the atomic lattice exciton. (b) Eigenvalues over quasimomentum for a single particle-hole pair with Jp = J = J in 1D.

is expanded in the localized Wannier modes of the respective bands (cf. [1]) (1)

ψ {c,v} (r) =

c{c,v} w{c,v} (r − ax), x

x {c,v}

where x := (x1 , . . . , xD ) ∈ ZD , a is the lattice spacing and w{c,v} (r − ax) and cx are the Wannier modes and annihilation operators in conduction band and valence band at site x, respectively. Inserting this decomposition into the two-species Hamiltonian in second quantization, assuming positive s-wave scattering length, and replacing valence band creation operators with hole annihilation operators d†x := cvx , we arrive at the particle-hole Hubbard-Hamiltonian with on-site particle-hole attraction (2)

H = −Jp

xx

c†x cx − Jh

yy

d†y dy − U

c†x d†x dx cx .

x

Here, cx := ccx , Jp and Jh are the hopping rates for particles and holes respectively, · denotes summation over nearest neighbors and U = 4π2 as /matom > 0 gives the strength of the on-site attraction. Interband repulsion between the atoms has thus been rewritten as atom-hole attraction(1 ). The assumption underlying the derivation of Hamiltonian (2) is that the depth of the optical lattice is such that in the Wannier basis only nearest-neighbour tunneling has a ﬁnite amplitude. Further, it is assumed that the hopping rates, the interband repulsion and the thermal energy are smaller than the band gap. All these requirements can be fullﬁlled in current experiments, and (1 ) The physics of excitons is signiﬁcantly modiﬁed when terms of form ccdd and c† c† d† d† can no longer be neglected in the derivation of eq. (2), as they have been there. They correspond to annihilation and creation of two particle-hole pairs into/out of the exciton vacuum. These terms must be kept to describe a semiconductor in which the gap between valence and conduction band becomes smaller than the interaction strength. In the resultant state, called excitonium, the ground state is rearranged, as stable excitons form spontaneously (cf. [7], chapt. 10 and [12] for more background). Creating an analogous set-up with ALEs, while interesting, would require deliberate engineering, and in particular long-range interactions of some sort.

838

A. Kantian

consequently these Hubbard-type Hamiltonians have shown to be a good match to the physics of lattice-conﬁned atoms [2, 8]. . 2 3. The single exciton: analytical solution for Jp = Jh . – To obtain the description for a single ALE, the ansatz for the exciton creation operator with quantum numbers α is made as A†α |v :=

(3)

† † φα x,y cx dy |v,

xy

with φα x,y , the wave function of the exciton, being determined by the eigenvalue equation † HA†α |v = εex α Aα |v.

(4)

We constrain this equation ﬁrst to the special case of Jp = Jh =: J, which can be treated analytically by introducing center-of-mass (COM) and relative coordinates R = (x + y)/2, r = x − y and making a product-ansatz φ = eiK(x+y)/2 ρx−y . Factoring out the COM motion, one is left with an equation for the relative wave function ρr for each COM quasimomentum K. Each of these equations has a single bound-state solution, the exciton solution (5)

ρr = C BZ

dD k eikr , D (2π)D εex K + d=1 2JKd cos(kd )

where ed is the unit vector in direction d, JKd := 2J cos(Kd /2) is the hopping rate in direction d in relative coordinates, C is the normalization constant, and BZ denotes the ﬁrst Brillouin zone [−π, π]D . The energy εex K of the exciton solution is determined by solving the implicit equation CU −1 = ρ0 (cf. [4, 13] for details). Overall, in contrast to free space, the δ-function pseudopotential and periodic potential together create a single bound-state solution for each value of the COM q.m. K, with all other solutions describing unbound states, as summarized in ﬁg. 1(b). Taken together, the bound-state solutions describe the dispersion relation of the lattice exciton. ex In one dimension εex U 2 + (2JK )2 , and thus can be written explicitly as ε = − K K bound states exists for arbitrarily small positive values of U . Calculating εex K numerically in higher dimensions, we still ﬁnd bound-state solutions for any positive value of U in 2D. In 3D, however, there is a critical threshold of (U/J)crit ≈ 7.95, that must be matched or exceeded for an exciton state to form. Generally, Jp = Jh will be the case usually encountered in an experiment for the suggested implementations, as upper and lower Bloch band will always diﬀer in width. We do not expect the critical U -value for the appearance of the 3D bound state to shift to larger values, as for example the extreme limit of one hopping rate being zero can again be solved analytically, yielding in 3D half the critical value obtained in the case of equal hopping rates, i.e. (U/J)crit ≈ 3.97.

atomic lattice excitons

839

. 2 4. Interaction of two excitons: stability of the single-exciton picture. – The eﬀective interaction of two excitons will strongly inﬂuence the overall physical behavior of the many-exciton system. In particular, the question of possible bound states of two ALEs, the so-called biexcitons, has to be addressed. If such a state exists, and has a binding energy comparable to or larger than any remaining thermal energy, the presence of these molecular modes at lower energy can make the system of excitons unstable. We believe it to be unlikely that such bound states exist for the Hamiltonian (2), which contains only on-site attraction(2 ). In the case of (Jh /U, Jp /U ) 1, where the exciton can in a ﬁrst approximation be regarded as a hard-core boson, this can be made explicit. Second-order perturbation theory [14] yields a dimension-independent hopping rate of J = 2Jp Jh /U and a weak nearest-neighbor repulsion of U = 2(Jp2 + Jh2 )/U (cf. [15,16]), which precludes bound states in this limit(3 ). To gain additional insight beyond the tightbinding case, we have performed small-scale exact diagonalizations for two interacting excitons in 1D (2 atoms, 2 holes on up to 22 lattice sites). The system parameters were ranging from U/J = 1 to U/J = 25, and strong imbalances in the hopping rates of holes and particles were considered as well (up to Jp /Jh = 40). The results show only eﬀective repulsion between the two excitons and a complete absence of bound states of four particles. As is well known that with increasing spatial dimensionality it becomes generally more unlikely for bound states to form in scattering problems, we expect that this result will hold for arbitrary values of (Jh /U, Jp /U ) in 2D and 3D. . 2 5. Lattice excitons at zero temperature: the exciton condensate. – A substantial proportion of the fundamental interest in semiconductor excitons has stemmed from their predicted ability to undergo condensation at comparatively high temperatures, around 10–20 K. We believe that such a condensate can be achieved with the ALEs proposed here. As we assume absence of biexciton modes for now, the ALEs can be considered as behaving akin to a ensemble of repulsively interacting lattice “molecules”. A general treatment of an exciton condensate starts from the assumption that the lowest energy mode of the exciton, i.e. the COM mode with K = 0 attains macroscopic occupation at zero temperature. Writing its creation operator in quasimomentum basis, A†0 = k∈BZ ρˆk c†k d†−k , suggests that this can be seen as pairing of fermions at opposite (2 ) Once longer-range interactions are introduced, there will be biexciton states in some regime, however, and will have to be taken into account. (3 ) The eﬀective range of validity around (Jh /U, Jp /U ) = 0 for such a bosonization procedure is diﬃcult to determine. Fermionic exchange becomes relevant immediately with ﬁrst order of (Jh /U, Jp /U ), as the exciton acquires ﬁnite size. The commutator relation of the exciton operator shows this, [AK , A†K ] = δKK −DKK where the operator DKK scales as O(nex a3ex ) and contains the fermionic exchange eﬀects, aex being the radius of the exciton and nex their density. Thus, only in the extreme tight-binding/low-density limit can we think of the excitons as obeying true Bose statistics. In applying general bosonization procedures [17] for non-perturbative values of (Jh /U, Jp /U ), we have consequently encountered signiﬁcant discrepancies to exact small-scale calculations already (cf. [18]).

840

A. Kantian

quasimomenta. The ground state is described by the coherent state (6)

|Ψ =

" k

u(ˆ ρk ) + v(ˆ ρk )c†k d†−k |0,

|u(ˆ ρk )|2 + |v(ˆ ρk )|2 = 1.

Following the established mean-ﬁeld approximation for lattice fermions [15,19], the pair ing amplitude ΔP := U k∈BZ uk vk is introduced to describe anomalous pairing and to act as the order parameter of the condensate phase. Using it, the self-consistent equations that determine uk := u(ˆ ρk ) and vk := v(ˆ ρk ) are rewritten in terms of ΔP and μex , the chemical potential of the excitons. It is introduced to ﬁx the average exciton density nex , as the particle number is no longer conserved in the mean-ﬁeld approach. This yields the standard equations(4 ) (7)

U −1 =

1 2

BZ

dD k 1 , (2π)D EK

nex =

1 2

BZ

dD k (2π)D

1−

ξK EK

,

where (8)

ξk = −(Jp + Jh )

D

(cos(kd )) − μex − U nex ,

Ek =

ξk2 + Δ2P

d=1

and Ek is the new quasiparticle dispersion. As discussed in [15, 19], these equations provide a good qualitative interpolation between the limits of low density (nex 1) and high density (nex ≈ 1/2), where they become exact. Notice that in the strong-coupling regime the ordinary Hartree term U nex in eq. (8) cannot be discarded as is done in treatments of the weak-coupling case. It provides for the renormalization of the chemical potential, otherwise the approach cannot adequately capture the density dependence of ΔP and μex . No Fock term appears in this model, as there are no interactions between atoms in the same band. For longer-range interactions it would be non-vanishing, and would have to be included as well. 3. – Probing atomic lattice excitons In order to demonstrate particle-hole pairing and in particular to detect the macroscopic occupation of the exciton ground state, it is possible to employ a range of measuring techniques that have been developed for cold-atoms experiments. At zero temperature, with no thermal excitation of collective exciton modes with K = 0, the pairing amplitude ΔP is directly proportional to the condensate fraction. It can be probed by (4 ) In an exact treatment, uk and vk must be determined from the ground state of the trans√ formed system Hamiltonian D† HD, where D := exp[i nex (A†0 − A0 )], under the constraint †

D cx dy D = 0, ∀ x, y [20]. The mean-ﬁeld approximation then is equivalent to diagonalizing D† HD while neglecting all quartic terms and evaluating the constraint just over the resultant ground state (cf. [7], chapt. 2 for details).

841

atomic lattice excitons

the application of RF spectroscopy, following the example of [21]. A RF pulse is used to couple the conduction band atoms, which are at energy ωc , to another internal state with energy ω3 . This state might be chosen such that it does not scatter oﬀ either conduction or valence atoms. It is further assumed that this state is still lattice conﬁned, with (3) (3) (3) single-particle energy ξk = εk − μ3,eﬀ , where εk is the dispersion relation of the state on the lattice and μ3,eﬀ is the eﬀective chemical potential, i.e. the chemical potential plus any mean-ﬁeld shift from interactions. If the RF pulse with frequency ωRF has an amplitude Ω that is varying slowly on length scales of the lattice size, the transfer rate I(δ) into the new internal state —which is the spectrum— is given by [22] (9)

I(δ) = −2π|Ω|2

(3) ˜ + Ek , vk2 δ ξk + Δ

˜ = μ3 − μc − δ. Δ

k

Here, δ = ωRF − (ω3 − ωc ) denotes the detuning of the RF pulse from the transition frequency between the two internal states. μc is the chemical potential of the conduction band atoms. The spectrum described by eq. (9) has a gap, given by the minimum value of the (3) δ-function argument. Its value is δgap = ε0 + E0 − μc , assuming that the additional internal state is initially unpopulated and does not scatter oﬀ the other states, i.e. μ3 = 0. If nex is independently determined, ΔP as well as the mean-ﬁeld shift can be determined self-consistently from the gap. Another measurement scheme is based on the exploitation of atom shot noise as proposed in [23] and demonstrated, e.g., in [24] and [25]. This allows to detect the atom-hole pairing via the second-order correlation function that can be obtained from the ﬂuctuations in the density proﬁle of the atomic gas. From the assumption of paring of particles and holes at opposite quasimomenta, we expect pronounced anticorrelation of conduction and valence band atoms at equal quasimomentum. To obtain the density proﬁles experimentally, and from them the ﬂuctuations, the Brillouin zones need to be resolved, i.e. lattice quasimomentum needs to be mapped to real space position on the detector. This is achieved by ramping down the lattice suﬃciently slowly to keep the atoms within their respective bands whilst preserving their quasimomentum (cf. [8]). Valence band atoms then occupy the ﬁrst, and conduction band atoms the second Brillouin zone (cf. ﬁg. 2(a)). The density ﬂuctuations at equal quasimomenta are anticorrelated according to the connected correlation function

(10)

(cck )† cck (cvk )† cvk

−

(cck )† cck

−

(cvk )† cvk

=

−u2k vk2 0

k = k k = k .

An example plot of this correlation function is shown in ﬁg. 2(b). Considering that the correlation at k = k equals u2k vk2 in the mean-ﬁeld theory, this approach provides an alternative for determining the condensate fraction via the deﬁnition of ΔP .

842

A. Kantian

(a)

(b) Second Brillouin Zone

First Brillouin Zone

Fig. 2. – (a) Column-integrated density proﬁle of the valence band and conduction band atoms (in the ﬁrst and second Brillouin zone, respectively) for the condensate ground state. (b) Columnintegrated connected correlation function between conduction and valence band atoms, for quasimomenta k = k , calculated along the dashed lines of panel (a).

4. – Exciton formation on an optical lattice In semiconductors, excitons can be created directly via a laser pulse tuned at the band gap. Provided that recombination times are considerably longer than the time scales required for thermal equilibriation, they can dissipate energy via interactions with lattice phonons until they reach the ground state. When below critical temperature, it is thus assumed that they would attain the desired condensate state automatically, though with a lifetime that would be fundamentally limited by the exciton lifetime (cf. [7], p. 57). In contrast, atoms on optical lattices do not couple to dissipative channels, and thus an excited state one is interested in must be prepared directly. There are no internal dynamics to disperse any excess energy that may have entered the system during the preparation stage. One such preparation scheme we are investigating can be described as melting [26]. It starts out with a deep optical lattice and the valence band completely ﬁlled with one type of fermionic atom, whereas the conduction band is ﬁlled with a number of atoms in a diﬀerent internal state that corresponds to the ﬁnal number of excitons desired. Using a Feshbach resonance to adiabatically ramp up the s-wave scattering length between the two states, the energies of valence atoms on lattice sites with an additional conduction band atom on top diﬀer from singly occupied sites by the interaction energy. Valence atoms on doubly occupied sites can thus be selectively transfered into a diﬀerent, separately trapped internal state via a Raman process [27], leaving just a conduction band atom and a valence hole behind. Finally, by ramping down the lattice suﬃciently slowly to the desired ﬁnal height, we end up with a state of lattice excitons that will be close to the ground state of the excitons. For more details, see [13].

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atomic lattice excitons

5. – Summary and outlook We have discussed how a particular class of non-equilibrium many-body states, i.e. excitons, can be engineered and probed using cold fermionic gases on optical lattices. Speciﬁcally, we have argued that the predicted condensation of the constituent quasiparticles of these states should be achievable in a clean and controlled manner. While we have focused on the case of on-site attraction as the most immediately realizable case, the extension to longer-rage interactions is straightforward. These might be implemented using polar molecules [28]. One possible extension of this work could be the investigation of biexcitons on a lattice, —which have been observed in semiconductors [7]— now employing two internal states of atoms in each band. The creation of a biexciton condensate would then be the next logical step. While currently being outside the experimental capabilities in solid-state materials, this might be achievable in optical lattices. Though the hopping rates of any such four-body state could be quite low as compared to the overall lifetime of experiment, given the currently achievable single-atom tunneling rates, there is still room for improvement. Another possibility might be to take the idea of quantum simulation a step further by introducing a bosonic species of atoms to act as a sympathetic coolant/phonon reservoir [29]. The eﬀects of cooling, decoherence and recombination on excitons could be studied in this way, with all the advantages of parameter control that cold atoms can oﬀer. ∗ ∗ ∗ The author wants to thank all the researchers involved in this project on lattice excitons, whose contributions, guidance and ideas have been invaluable: A. Daley, P. ¨ rma ¨ and P. Zoller. Further acknowledgement for many insightful discussions is due To ¨chler, M. Combescot and M. Szymanska. This work was supported to H.-P. Bu by the European commission through the integrated project FET/QIPC “SCALA”. REFERENCES [1] Jaksch D. and Zoller P., Ann. Phys. (N.Y.), 315 (2005) 52. [2] Bloch I., Nature Phys., 1 (2005) 1. [3] Lewenstein M., Sanpera A., Ahufinger V., Damski B., Sen De A. and Sen U., cond-mat/0606771. [4] Winkler K., Thalhammer G., Lang F., Grimm R., Hecker Denschlag J., Daley ¨chler H. P. and Zoller P., Nature, 441 (2006) 7095. A. J., Kantian A., Bu [5] Greiner M., Regal C. A. and Jin D. S., Nature, 426 (2003) 537. [6] Hanamura E. and Haug H., Phys. Rep., 33 (1977) 209. [7] Moskalenko S. A. and Snoke D. W., Bose-Einstein Condensation of Excitons and Biexcitons (Cambridge University Press) 2000. ¨ hl M., Moritz H., Sto ¨ ferle T., Gu ¨nter K. and Esslinger T., Phys. Rev. Lett., [8] Ko 94 (2005) 080403. ¨nter K., Sto ¨ ferle T., Moritz H., Ko ¨ hl M. and Esslinger T., Phys. Rev. Lett., [9] Gu 96 (2006) 180402.

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[10] Ospelkaus S., Ospelkaus C., Wille O., Succo M., Ernst P., Sengstock K. and Bongs K., Phys. Rev. Lett., 96 (2006) 180403. [11] Chin J. K., Miller D. E., Liu Y., Stan C., Setiawan W., Sanner C., Xu K. and Ketterle W., cond-mat/0607004. [12] Guseinov R. R. and Keldysh L. V., Sov. Phys. JETP, 36 (1973) 6. [13] Kantian A. et al., in preparation. [14] Cohen-Tannoudji C., Dupont-Roc J. and Grynberg G., Atom-Photon Interactions — Basic Processes and Applications (Wiley Interscience) 1998. `res P. and Schmitt-Rink S., J. Low Temp. Phys., 59 (1985) 3/4. [15] Nozie [16] Lee C., Phys. Rev. Lett., 93 (2004) 120406. [17] Girardeau M. D., J. Mat. Phys., 16 (1975) 9. [18] Combescot M. and Betbeder-Matibet O., Europhys. Lett., 58 (2002) 87. `res P., J. Phys. (Paris), 43 (1982) 1069. [19] Comte C. and Nozie [20] Keldysh L. V. and Kozlov A. N., Sov. Phys. JETP, 27 (1968) 3. ¨ rma ¨ P. and Zoller P., Phys. Rev. Lett., 85 (2000) 487. [21] To ¨ rma ¨ P., private communication. [22] To [23] Altman E., Demler E. and Lukin M. D., Phys. Rev. A, 70 (2004) 013603. ¨ lling S., Gerbier F., Widera A., Mandel O., Gericke T. and Bloch I., Nature, [24] Fo 434 (2005) 481. [25] Greiner M., Regal C. A., Stewart J. T. and Jin D. S., Phys. Rev. Lett., 94 (2005) 110401. [26] Jaksch D., Venturi V., Cirac J. I., Williams C. J. and Zoller P., Phys. Rev. Lett., 89 (2002) 040402. [27] Rabl P., Daley A. J., Fedichev P. O., Cirac J. I. and Zoller P., Phys. Rev. Lett., 91 (2003) 110403. [28] Micheli A., Brennen G. K. and Zoller P., Nature Phys., 2 (2006) 341. [29] Griessner A., Daley A. J., Clark S. R., Jaksch D. and Zoller P., condmat/0607254.

Expansion of a lithium gas in the BEC-BCS crossover L. Tarruell, M. Teichmann, J. McKeever, T. Bourdel, J. Cubizolles, N. Navon, F. Chevy and C. Salomon ´ Laboratoire Kastler Brossel, Ecole Normale Sup´erieure - 24 rue Lhomond, 75005 Paris, France

L. Khaykovich Department of Physics, Bar Ilan University - 52900 Ramat Gan, Israel

J. Zhang SKLQOQOD, Institute of Opto-Electronics, Shanxi University - Taiyuan 030006, PRC

We present an experimental study of the time-of-ﬂight properties of a gas of ultracold fermions in the BEC-BCS crossover. Since interactions can be tuned by changing the value of the magnetic ﬁeld, we are able to probe both non-interacting and strongly interacting behaviors. These measurements allow us to characterize the momentum distribution of the system as well as its equation of state. We also demonstrate the breakdown of superﬂuid hydrodynamics in the weakly attractive region of the phase diagram, probably caused by pair breaking taking place during the expansion. 1. – Introduction Feshbach resonances in ultracold atomic gases oﬀer the unique possibility of tuning interactions between particles, thus allowing one to study both strongly and weakly interacting many-body systems with the same experimental apparatus. A recent major achievement was the experimental exploration of the BEC-BCS crossover [1-6], a scenario proposed initially by Eagles, Leggett, Nozi`eres and Schmitt-Rink to bridge the gap between the Bardeen-Cooper-Schrieﬀer (BCS) mechanism for superconductivity in metals, and the Bose-Einstein condensation of strongly bound pairs [7-9]. Here, we present a c Societ` a Italiana di Fisica

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H Y X Z

Fig. 1. – Ioﬀe-Pritchard trap and crossed dipole trap used for the experiments. The crossed geometry allows us to change the aspect ratio of the trap.

study of the crossover using time-of-ﬂight measurements. This technique gives access to a wide range of physical properties of the system and has been successfully used in diﬀerent ﬁelds of physics. The observation of elliptic ﬂows was for instance used to demonstrate the existence of quark-gluon plasmas in heavy-ion collisions [10]. In cold atoms, the ellipticity inversion after free ﬂight is a signature of Bose-Einstein condensation [11,12]. In an optical lattice the occurrence of interference peaks can be used as the signature of the superﬂuid to insulator transition [13] and, with fermions, it can be used to image the Fermi surface [14]. Two series of time-of-ﬂight measurements are presented: expansion of the gas without interactions, which gives access to the momentum distribution, a fundamental element in the BEC-BCS crossover, or with interactions, which allows us to characterize the equation of state of the system, and probe the validity of superﬂuid hydrodynamics. 2. – Experimental method In a magnetic trap, a spin-polarized gas of N = 106 6 Li atoms in |F = 3/2, mF = +3/2 is sympathetically cooled by collisions with 7 Li in |F = 2, mF = +2 to a temperature of 10 μK. This corresponds to a degeneracy of T /TF ∼ 1, where TF = ¯ ω (6N )1/3 /kB is the Fermi temperature of the gas. The magnetic trap frequencies are 4.3 kHz (76 Hz) in the radial (axial) direction, and ω ¯ = (ωx ωy ωz )1/3 is the mean frequency of the trap. Since there are no thermalizing collisions between the atoms in a polarized Fermi gas, the transfer into our ﬁnal crossed dipole trap, which has a very diﬀerent geometry (ﬁg. 1), is done in two steps. We ﬁrst transfer the atoms into a mode-matched horizontal single beam Yb:YAG dipole trap, with a waist of ∼ 23 μm. At maximum optical power (2.8 W), the trap depth is ∼ 143 μK (15 TF ), and the trap oscillation frequencies are 6.2(1) kHz (63(1) Hz) in the radial (axial) direction. The atoms are transferred in their absolute ground state |F = 1/2, mF = +1/2 by an RF pulse. We then sweep the magnetic ﬁeld to 273 G and drive a Zeeman transition between |F = 1/2, mF = +1/2 and

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Scattering length [a 0]

4000 3000 2000 1000 0 -1000 -2000 -3000 -4000 0

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Fig. 2. – 6 Li Feshbach resonance between |F = 1/2, mF = +1/2 and |F = 1/2, mF = −1/2. The broad Feshbach resonance is located at 834 G. The balanced mixture is prepared at 273 G.

|F = 1/2, mF = −1/2 to prepare a balanced mixture of the two states (better than 5%). At this magnetic ﬁeld, the scattering length between both states is −280 a0 (ﬁg. 2). After 100 ms the mixture has lost its coherence, initiating collisions in the gas. During the thermalization process about half of the atoms are lost. We then perform a ﬁnal evaporative cooling stage by lowering the trap depth to ∼ 36 μK. At this point, we ramp up a vertical Nd:YAG laser beam (power 126 mW and waist ∼ 25 μm), obtaining our ﬁnal crossed dipole trap conﬁguration (ﬁg. 1). The measured degeneracy is T /TF 0.15. The magnetic ﬁeld is then increased to 828 G (in the vicinity of the Feshbach resonance, see ﬁg. 2), where we let the gas thermalize for 200 ms before performing subsequent experiments. 3. – Momentum distribution In standard BCS theory, the ground state of an homogeneous system is described by a pair condensate characterized by the many-body wave function |ψ =

" k

(uk + vk a†k,↑ a†−k,↓ )|0,

where |0 is the vacuum and a†k,σ is the creation operator of a fermion with momentum k and spin σ. In this expression, |vk |2 can be interpreted as the occupation probability in momentum space, and is displayed in ﬁg. 3a for several values of the interaction parameter 1/kF a, where kF is the Fermi wave vector of the non-interacting Fermi gas (EF = 2 kF2 /2m). One eﬀect of the pairing of the atoms is to broaden the momentum distribution. In the BCS limit (1/kF a → −∞), the broadening with respect to the momentum distribution of an ideal Fermi gas is very small, of the order of the inverse of the coherence length ξ. In the unitary limit (1/kF a → 0) it is expected to be of the order of kF . In the BEC limit (1/kF a → ∞) we have molecules of size a so the momentum distribution, which is given by the Fourier transform of the molecular wave function, has a width /a.

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Fig. 3. – (a) Momentum distribution of a uniform Fermi gas for 1/kF a = −1 (solid line), 1/kF a = 0 (dotted line) and 1/kF a = 1 (dashed line) calculated from mean ﬁeld BCS theory at T = 0 [15]. The results obtained from quantum Monte Carlo simulations [16] show that BCS theory slightly underestimates the broadening. (b) Measured momentum distribution of a trapped Fermi gas on the BCS side of the resonance (1/kF0 a = −0.42). (c) Unitary limit (1/kF0 a = 0). (d) BEC side of the resonance (1/kF0 a = 0.38). The solid lines in (b), (c) and (d) are the predictions of BCS mean-ﬁeld theory taking into account the trapping potential with a local density approximation [15]. kF0 is deﬁned in the text.

In a ﬁrst series of expansion experiments, we have measured the momentum distribution of a trapped Fermi gas in the BEC-BCS crossover. Similar experiments have been performed at JILA on 40 K [17]. In order to measure the momentum distribution of the atoms, the gas must expand freely, without any interatomic interactions. To achieve this, we quickly switch oﬀ the magnetic ﬁeld so that the scattering length is brought to zero (see ﬁg. 2) [18]. We prepare N = 3×104 atoms at 828 G in the crossed dipole trap with frequencies ωx = 2π×2.78 kHz, ωy = 2π × 1.23 kHz and ωz = 2π × 3.04 kHz. The magnetic ﬁeld is adiabatically swept in 50 ms to diﬀerent values in the crossover region. Then, we simultaneously switch oﬀ both dipole trap beams and the magnetic ﬁeld (with a linear ramp of 296 G/μs). After

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1 ms of free expansion, the atoms are detected by absorption imaging. The measured density proﬁles give directly the momentum distribution of the gas integrated along the imaging direction. In ﬁg. 3, we show the measured momentum distributions for three diﬀerent interaction parameters, corresponding to the BCS side of the resonance, the unitary limit and the BEC side of the resonance. Together with our data, we have plotted the predictions of mean ﬁeld BCS theory at T = 0, taking into account the trapping potential with a local density approximation [15]. kF0 is now the Fermi wave vector calculated at the center of the harmonic trap for an ideal gas. Some precautions need to be taken concerning this type of measurements due to possible density-dependent losses during the magnetic ﬁeld switch-oﬀ. If the magnetic ﬁeld is not turned oﬀ fast enough, some atoms can be bound into molecules while the Feshbach resonance is crossed. The molecules are not detected with the imaging light and therefore will appear as a loss of the total number of atoms. Even if, as in our case, the Feshbach resonance is crossed in only 1 μs, this time may not be small compared to the typical many-body timescale (/EF ∼ 1.3 μs for ﬁg. 3 data). To investigate quantitatively this eﬀect, we have performed an additional experiment in a more tightly conﬁning trap. We prepare a gas of 5.9 × 104 atoms at 828 G in a trap with frequencies ωx = 2π × 1.9 kHz, ωy = 2π × 3.6 kHz and ωz = 2π × 4.1 kHz. The total peak density in the trap is 1.3 × 1014 atoms/cm3 . We let the gas expand at high ﬁeld for a variable time tB , then switch oﬀ B and detect the atoms after 0.5 ms of additional free expansion. Assuming hydrodynamic expansion at unitarity we calculate the density after tB [19] and obtain the fraction of atoms detected as a function of the density of the gas when the resonance is crossed. For instance, we detect 60% fewer atoms for tB = 0 compared to tB = 0.5 ms, where the density is a factor 103 lower. The results are nicely ﬁtted by a Landau-Zener model: n(tB ) Ndetected /Ntotal = exp −A , 2B˙ where n(tB ) is the total density at tB , B˙ the sweep rate and A the coupling constant between the atoms and the molecules. We determine A 5 × 10−12 G m3 /s. Our result is ﬁve times smaller than the MIT value A 24 × 10−12 G m3 /s [20], measured at a total peak density of 1013 atoms/cm3 (one order of magnitude smaller than in our experiment). The theoretical prediction, assuming only two-body collisions, is A = 19×10−12 G m3 /s [21]. The diﬀerence between our measurement and theory may suggest that many-body eﬀects are important in our case. Finally, using our value of A the model predicts an atom number loss of about 27% for the momentum distribution measurements of ﬁg. 3. This loss is comparable to our shot-to-shot ﬂuctuations in atom number and therefore was not unambiguously observed. In conclusion, we have performed a measurement of the momentum distribution of a trapped Fermi gas. The results are found in reasonable agreement with BCS theory despite the fact that it is not expected to be quantitatively correct in the strongly interacting regime. In future work, experiments at lower density will be performed, in

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order to avoid the observed loss eﬀect. This should allow us to distinguish between BCS and more exact theories [16]. It would also be interesting to perform measurements at diﬀerent temperatures as in ref. [22]. 4. – Release energy In a second series of experiments, we have performed expansions at constant magnetic ﬁeld, thus keeping the interactions present during the time of ﬂight. The analysis of size measurements across the BEC-BCS crossover yields valuable information on the inﬂuence of interactions on the properties of the system. In particular, we have measured the release energy of the gas in the BEC-BCS crossover [3]. On resonance (1/kF0 a = 0), the gas reaches a universal behavior [23]. The chemical potential μ is proportional to the Fermi energy μ = (1 + β)EF . We have determined the universal scaling parameter β from our release energy measurement. The starting point for the experiment is a nearly pure molecular condensate of 7× 104 atoms at 770 G, in an optical trap with frequencies ωx = 2π × 830 Hz, ωy = 2π × 2.4 kHz, and ωz = 2π × 2.5 kHz. We slowly sweep the magnetic ﬁeld at a rate of 2 G/ms to various values across the Feshbach resonance. We detect the integrated density proﬁle after a time-of-ﬂight expansion of 1.4 ms in several stages: 1 ms of expansion at high magnetic ﬁeld, followed by a fast ramp of 100 G in 50 μs in order to dissociate the molecules and, after the fast switch-oﬀ of the magnetic ﬁeld, 350 μs of ballistic expansion. Figure 4 presents the gas energy released after expansion, which is calculated from Gaussian ﬁts to the optical density after time of ﬂight: Erel = m(2σy2 + σx2 )/2τ 2 , where σi is the rms width along i and τ is the time of ﬂight. We assume that the size σz (which is not observed) is equal to σy . Note that both in the weakly interacting case

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Table I. – List of the recent experimental measurements and theoretical predictions of the universal scaling parameter β. β This work ENS 2004 [3] Innsbruck [24] Duke [25] Rice [26]

−0.59(15) −0.64(15) −0.73+12 −0.09 −0.49(4) −0.54(5)

JILA [27]

+0.05 −0.54−0.12

Theoretical predictions at T = 0

BCS theory [7-9] Astrakharchik et al. [28] Carlson et al. [29, 30] Perali et al. [31] Pad´e approximation [23, 32] Steel [33] Haussmann et al. [34]

−0.41 −0.58(1) −0.58(1) −0.545 −0.67 −0.56 −0.64

Theoretical predictions at T = Tc

Bulgac et al. [35] Burovski et al. [36]

−0.55 −0.507(14)

Experimental results on 6 Li at ﬁnite T

Experimental result on extrapolation to T = 0

40

K

and unitarity limit the density has a Thomas-Fermi proﬁle and the release energy can be calculated from the exact proﬁles. However, we have chosen this Gaussian shape to describe the whole crossover region with a single ﬁt function. This leads to a rescaling of the release energy. In particular, the ideal Fermi gas release energy in a harmonic trap is Erel = 3/8EF , but when using the Gaussian ﬁt to the Thomas-Fermi proﬁle we get instead Erel = 0.46EF as shown in ﬁg. 4. The release energy in the BEC-BCS crossover varies smoothly. It presents a plateau for −1/kF a ≤ −0.5, (BEC side) and then increases monotonically towards that of a weakly interacting Fermi gas. On resonance, the release energy scales as √ 0 0 Erel = 1 + β Erel , where Erel is the release energy of the non-interacting Fermi gas. The square root comes from the average over the trap. At 834 G, we get β = −0.59(15). This value is slightly diﬀerent from our previous determination β = −0.64(15), where the resonance was assumed to be located at 820 G instead of 834 G [3]. Our result agrees with other measurements performed on 6 Li and with some theoretical predictions (see table I). Remarkably, the recent 40 K measurement at JILA is also in very good agreement, thus proving the universality of the unitarity regime. 5. – Ellipticity Nontrivial information can be extracted from the measurement of the aspect ratio of the cloud after expansion. For instance, in the ﬁrst days of gaseous Bose-Einstein

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condensates, the onset of condensation was characterized by an ellipticity inversion after time of ﬂight, a dramatic eﬀect compared to the isotropic expansion of a non condensed Boltzmann gas. In the case of strongly interacting Fermi gases, ellipticity measurements can be used as probes for the hydrodynamic behavior of the system, and constitute an indirect signature of the appearance (or breakdown) of superﬂuidity. We have studied the ellipticity of the cloud as a function of the magnetic ﬁeld for diﬀerent temperatures. As before, the density proﬁles are ﬁtted with Gaussians, and the ellipticity is deﬁned as η = σy /σx . We prepare N = 3 × 104 atoms at 828 G in a crossed dipole trap. The magnetic ﬁeld is adiabatically swept in 50 ms to diﬀerent values in the crossover region. Then, we switch oﬀ both dipole trap beams and let the gas expand for 0.5 ms in the presence of the magnetic ﬁeld. After 0.5 ms of additional expansion at B = 0, the atoms are detected by absorption imaging. Figures 5a and b show the measured value of the ellipticity as a function of the magnetic ﬁeld for two diﬀerent samples, which are at diﬀerent temperatures. Together with the experimental results we have plotted the expected anisotropy from superﬂuid hydrodynamics [19]. For this, we have extracted from the quantum Monte Carlo simulation of ref. [28] the value of the ∂μ polytropic exponent γ, deﬁned as γ = nμ ∂n . The ﬁrst series of measurements is done in a trap with frequencies ωx = 2π ×1.39 kHz, ωy = 2π×3.09 kHz, ωz = 2π×3.38 kHz and trap depth ∼ 1.8 TF . The measured ellipticity (ﬁg. 5a) is in good agreement with the hydrodynamic prediction on the BEC side, at resonance and on the BCS side until 1/kF0 a = −0.15. It then decreases monotonically to 1.1 at 1/kF0 a = −0.5. For the second series of experiments we prepare a colder sample in a trap with frequencies ωx = 2π × 1.24 kHz, ωy = 2π × 2.76 kHz, ωz = 2π × 3.03 kHz and trap depth ∼ 1.6 TF . In this case the behavior of the anisotropy is very diﬀerent (ﬁg. 5b). We

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observe a plateau until 1/kF0 a = −0.33, in good agreement with the hydrodynamic prediction, and at this critical magnetic ﬁeld there is a sharp decrease of η to a value close to 1.2. This sharp transition seems analogous to the sudden increase of the damping of the breathing mode observed in Innsbruck [37]. In a third experiment, we measure the ellipticity at unitarity as a function of trap depth (hence of the gas temperature). Below a critical trapping laser intensity, the ellipticity jumps from a low value (1.1) to the hydrodynamic prediction 1.45. In all cases, the decrease of the anisotropy indicates a breakdown of superﬂuid hydrodynamics in the weakly attractive part of the phase diagram or at higher temperature. A ﬁrst possibility would be that the gas crosses the critical temperature in the trap. However, we know from the MIT experiment [38] that pair breaking can occur during the expansion. During the time of ﬂight, both the density and kF decrease. On the BEC side of resonance, the binding energy of the molecules (−2 /ma2 ) does not depend on the density and the pairs are very robust. By contrast, on the BCS side of resonance the generalized Cooper pairs become fragile as the gap decreases with 1/kF a and they can be broken during the expansion. Our experiments use the ellipticity of the cloud as a probe and are complementary to the MIT approach, where the breakdown of superﬂuidity was characterized by the disappearance of vortices during the expansion of the gas. We are planning additional experiments in order to investigate whether the breakdown of superﬂuidity occurs in the trap or during the expansion. 6. – Conclusion The results presented here constitute a ﬁrst step in the understanding of the free ﬂight properties of strongly correlated fermionic systems. In future work, we will investigate more thoroughly the pair breaking mechanism taking place during the expansion in the BCS part of the phase diagram. We point out the need for a dynamic model of the expanding gas at ﬁnite temperature. ∗ ∗ ∗ We gratefully acknowledge support by the IFRAF institute and the ACI Nanosciences 2004 NR 2019. We thank the ENS ultracold atoms group, S. Stringari, R. Combescot, D. Petrov and G. Shlyapnikov for stimulating discussions. Laboratoire Kastler Brossel is a research unit No. 8552 of CNRS, ENS, and Universit´e Paris 6.

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[34] Haussmann R., Rantner W., Cerrito S. and Zwerger W., e-print cond-mat/0608282. [35] Bulgac A., Drut J. E. and Magierski P., Phys. Rev. Lett., 96 (2006) 090404. [36] Burovski E., Prokof’ev N., Svistunov B. and Troyer M., Phys. Rev. Lett., 96 (2006) 160402. [37] Bartenstein M., Altmeyer A., Riedl S., Jochim S., Chin C., Hecker Denschlag J. and Grimm R., Phys. Rev. Lett., 92 (2004) 203201. [38] Schunck C. H., Zwierlein M. W., Schirotzek A. and Ketterle W., Phys. Rev. Lett., 98 (2007) 050404.

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Quantum degenerate gases and the mixtures of ytterbium atoms T. Fukuhara, S. Sugawa, Y. Takasu Department of Physics, Graduate School of Science, Kyoto University - Kyoto 606-8502, Japan

Y. Takahashi Department of Physics, Graduate School of Science, Kyoto University - Kyoto 606-8502, Japan CREST, Japan Science and Technology Agency - Kawaguchi, Saitama 332-0012, Japan

1. – Introduction Since the realization of quantum degenerate gases, such as Bose-Einstein condensation (BEC) [1] and Fermi degeneracy [2], they have been applied to extensive investigations, and these investigations have yielded signiﬁcant results. One of the most exciting current directions in such a ﬁeld is the study of degenerate Fermi gases and quantum degenerate mixtures, where ytterbium (Yb) atoms are attractive due to many unique features. One is the existence of various stable isotopes (table I), which allows us to study a variety of quantum degenerate gases and their mixtures. Especially, in studying a Fermi gas, the existence of two stable fermionic isotopes, whose natural abundance is more than ten percent, is a unique advantage. Other unique properties of Yb come from the two valence electrons, which make it similar to the alkaline-earth atoms such as Ca and Sr, see ﬁg. 1. First, no electron spin in the ground state is expected to suppress heating and atom loss caused by spin relaxation. Second, an ultranarrow transition of 1 S0 -3 P0 (the natural linewidth ∼ 10 mHz [3]), which is an excellent candidate for an atomic clock [3, 4] and whose optical frequency was measured with an uncertainty of 4.4 kHz [5], enables us to detect a small energy diﬀerence such as the pairing gap of Fermi superﬂuidity. Third, the metastable 3 P2 state has a large magnetic dipole moment of 3μB (μB is the Bohr magnec Societ` a Italiana di Fisica

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T. Fukuhara, S. Sugawa, Y. Takasu and Y. Takahashi

Table I. – Natural abundance and nuclear spin of Yb isotopes. Isotope

Natural abundance (%)

Nuclear spin

168 170 171 172 173 174 176

0.13 3.05 14.3 21.9 16.12 31.8 12.7

0 0 1/2 0 5/2 0 0

ton), resulting in a factor of 9 larger magnetic dipole-dipole interaction than that of alkali atoms, and thus studies of anisotropic superﬂuidity can be expected. A large mass is also an important aspect of Yb atoms, which leads to possibilities for interesting heteronuclear molecules with a large mass diﬀerence [6]. These unique properties make production of degenerate Fermi gases and degenerate mixtures of Yb atoms quite appealing. Our group realized BEC of 174 Yb by all-optical means in 2003 [7]. In this article, we report recent experimental results concerning the production of degenerate Fermi gases and BEC and also degenerate mixture of Yb isotopes. First, we introduce our experimental procedure for creating a degenerate gas of Yb isotopes (sect. 2), and then present our recent experiments concerning four Yb isotopes; 173 Yb (sect. 3), 171 Yb (sect. 4), 170 Yb (sect. 5), and 176 Yb (sect. 6). Section 7 gives conclusions and future prospects for degenerate Yb atoms.

6s6p 1P1

25068cm-1

5.5ns 3P 2

6s6p Zeeman slower Probe 399nm

3P 1 3P 0

19710cm-1 17992cm-1

875ns

17288cm-1

FORT 532 nm MOT 556nm

6s2

1S 0

0 cm-1

Fig. 1. – Energy levels of Yb atoms. The wavelengths of the lasers for cooling, trapping, and probing are also represented.

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2. – Cooling and trapping of Yb atoms One of the advantages of working with Yb atoms is that diﬀerent isotopes can be cooled in the same experimental setup with only a slight change. In this section, we present common procedure for cooling Yb isotopes to quantum degeneracy. First, Yb atoms in a thermal beam generated by an oven at 375 ◦ C are decelerated by a Zeeman slower with a strong transition (1 S0 -1 P1 ; the wavelength of 399 nm and the linewidth of 29 MHz) and then loaded into a magneto-optical trap (MOT) with an intercombination transition (1 S0 -3 P1 ; the wavelength of 556 nm and the linewidth of 182 kHz) [8], as shown in ﬁg. 1. The isotope shifts of these transitions are typically much larger than their linewidth, and thus we can select particular isotopes by tuning cooling and trapping laser frequency to its near resonance. The MOT beam is generated by a dye laser whose frequency is narrowed to less than 100 kHz and stabilized by an ultralow expansion cavity, whose stability is typically less than 20 Hz/s. The intensity of each MOT beam is typically 60Is,3 P1 (Is,3 P1 = 0.14 mW/cm2 is the saturation intensity of the 1 S0 -3 P1 transition) and the detuning is 7Γ (Γ is the linewidth of the 1 S0 -3 P1 transition). This scheme enables us to prepare 2 × 107 atoms in the MOT. Laser-cooled Yb atoms are transferred to a crossed far-oﬀ-resonance trap (FORT) [9] with horizontal and vertical beams which are left on throughout the laser cooling process. The beams are independently produced by two 10 W diode-pumped solid-state lasers at 532 nm. The power of each beam is up to 6 W and the 1/e2 beam radii at the crossed point are 15 μm (horizontal beam) and 24 ∼ 86 μm (vertical beam). The radius and the power of the vertical beam are slightly changed for each experiment while the trap depth is kept to similar values, 17 ∼ 30 μK. Although the crossed FORT with almost equal trap depths leads to an extremely high density of more than 1014 cm−3 , rapid atom loss due to three-body recombination prevents eﬃcient evaporative cooling. For this reason, we use greatly asymmetrical potential depths at the initial stage of evaporation, typically 620 μK for the horizontal FORT and 20 μK for the vertical FORT. Thus, 2 × 106 atoms are trapped mainly in the horizontal FORT, whose radial and axial trapping frequencies at full power are 3.6 kHz and 30 Hz, respectively. The radial trapping frequency is measured by parametric resonance methods [10]. Forced evaporative cooling is carried out by ramping down the intensity of the horizontal beam, while keeping the vertical FORT power constant. When the potential depth of the horizontal FORT is reduced to approximately the same as that of the vertical FORT, almost all the atoms are trapped in the crossed region. Further decrease of the horizontal FORT potential leads to evaporation in the crossed region. Finally, the potential depth along the vertical direction, which is considerably aﬀected by gravity at this stage of evaporation, is reduced below a few μK, where degenerate gases appear. We measure atom numbers and temperatures of atom clouds using absorption imaging technique. After evaporation, the trapping beams are turned oﬀ within a ten hundred ns and at a time-of-ﬂight (TOF) time later the released gas is illuminated by 100 μs probe beam pulse resonant with the strong 1 S0 -1 P1 transition. In the case of 174 Yb, we obtain nearly 105 condensed atoms in 10 s of MOT loading and 6 s of evaporation.

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3. – Fermi degeneracy of

173

Yb

In this section, we present all-optical formation of degenerate fermionic 173 Yb gases [11]. The basic experimental scheme is described in sect. 2, but we should care about the fact that fermionic 173 Yb atoms have nuclear spin in the ground state and there exists hyperﬁne structure in the excited 1 P1 and 3 P1 states. Atoms are pre-cooled by the Zeeman slower with the 1 S0 -1 P1 (Fg = 5/2 → Fe = 7/2) transition and the MOT with the 1 S0 -3 P1 (Fg = 5/2 → Fe = 7/2) transition. The loading time for the MOT is typically 30 seconds, which leads to 2 × 107 173 Yb atoms in the MOT. The intensity of the MOT beam is decreased to eﬃciently transfer the atoms from the MOT to the FORT. In the case of 174 Yb, which has no electronic and nuclear spin in the ground state, the intensity is decreased down to 5Is,3 P1 in our system, which results in the temperature for the MOT clouds of 50 μK. However, 173 Yb has nuclear spin I = 5/2 and the eﬀective radiation pressure of the MOT beam is smaller, due to the optical pumping eﬀect [9]. For this reason, in this experiment of 173 Yb, the intensity is lowered to 25Is,3 P1 . The temperature of 173 Yb in the MOT is 15 μK, which is low compared with 174 Yb MOT because of the sub-Doppler cooling mechanism [8, 12]. The laser-cooled atoms are loaded into the crossed FORT, whose 1/e2 beam radii at the crossed point are 15 μm (horizontal beam) and 24 μm (vertical beam, 700 mW). 2 × 106 atoms at 100 μK are trapped mainly in the horizontal FORT. The Fermi temperature of the trapped atoms is approximately 5 μK, where we simply assume that six spin components are equally distributed. The probing beam is a linearly polarized light pulse resonant with 1 S0 -1 P1 (Fg = 5/2 → Fe = 7/2) transition propagating along the direction of 0.9 G magnetic ﬁeld. The intensity of the beam is 0.02Is,1 P1 , where Is,1 P1 is the saturation intensity of the 1 S0 -1 P1 transition. The pulse duration of 100 μs is much longer than the absorption cycle of 550 ns at this intensity. It should be noted that transition rates are diﬀerent for each spin component and thus a total absorption rate generally depends on the distribution of the spin components. However, it turned out from numerical calculations based on the rate equation that, in the case of our condition, the total absorption rate is the same within 2% for any initial spin distribution because of the rapid redistribution due to the optical pumping eﬀect. This enables us to determine the total number of atoms exactly, without knowing the population of each component. To perform eﬃcient evaporation, a large elastic collision rate is necessary. When temperatures are much less than the threshold energy Eth (l) for a given partial wave l (and there is not a shape resonance [13]), the collisions associated with a partial wave higher than l are negligible. This threshold energy can be approximately determined by the centrifugal barrier and the van der Waals potential of C6 /R6 , (1)

Eth (l) = 2

¯ 2 l(l + 1) h 6μ

3/2

−1/2

C6

,

where μ is the reduced mass. In the case of fermionic atoms, we have only to consider the s-wave scattering between non-identical fermions at temperatures much less than

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the p-wave threshold energy Eth (l = 1) because there is no s-wave scattering between identical atoms from Fermi-Dirac statistics. We carried out the cross-dimensional rethermalization technique to deduce the s-wave scattering length [14]. Assuming the coeﬃcient of the van der Waals C6 = 2000 a.u. [15], the p-wave threshold energy is 43 μK. Therefore we measure a rethermalizing process at 6 μK, where the s-wave scattering is dominant unless there exists a p-wave shape resonance. Typically 1.2 × 104 atoms at 6 μK are prepared in the crossed FORT and then a 40% intensity modulation of 800 Hz, twice the trapping frequency of one horizontal direction, is applied to the vertical FORT during 5 ms, which causes heating along the horizontal direction. Following the modulation, thermal relaxation is observed as the time evolution of temperatures in the horizontal and vertical directions. An exponential ﬁt to the data extracts a rethermalization time τ = 4 ± 1 ms. To conﬁrm that the relaxation is not due to anharmonic mixing, we have also applied this technique with the same conﬁguration for 176 Yb atoms, whose scattering length is not large [16]. The result for the 176 Yb atoms is that the rethermalization time is observed to be longer than 50 ms, which means that the relaxation time due to the anharmonicity is not shorter than 50 ms, therefore the anharmonicity eﬀect is negligible. In the case of 173 Yb atoms, if we assume that the atoms are equally populated among the six spin states, the rethermalization time τ and the elastic σ are related by the following equation: ατ = 2collision cross-section 5 3 3 ¯ σ¯ v , where n ¯ = n (r)d r/ n(r)d r is the average density and v¯ = 4 (kB T )/(πm) 6n is the mean relative velocity with kB the Boltzmann constant and m the ytterbium mass. The factor 5/6 comes from the fact that there are no s-wave collisions between identical spin components. The constant α represents how many elastic collisions lead to cross-dimensional rethermalization and turned out to be about 2.7 by the Monte Carlo simulation [14]. From this analysis, we have deduced the absolute value of the scattering length |as | = 6 ± 2 nm, whose error is mainly caused by the uncertainty in the time τ and the density n ¯ . This large scattering length allows us to cool the atoms to the quantum degenerate regime. It is noted that the scattering length would be larger than this value if six spin components are not equally distributed. After 2 s of forced evaporative cooling, the potential depth of the horizontal FORT is reduced to 30 μK, and typically 8 × 104 atoms are trapped in the crossed region. The temperature is about 4TF , where the Fermi temperature TF is 1.4 μK at this reduced FORT power. Further decrease of the horizontal FORT power leads to the evaporation in the crossed region, and ﬁnally the potential depth along the vertical direction is reduced below 2 μK. A few thousand atoms remain in the trap, whose mean trapping frequency is 450 Hz, and the temperatures are approximately half of TF 300 nK. An eﬀect of quantum degeneracy is already observed as a decrease in cooling efﬁciency [2]. Figure 2 shows the number of atoms N and the temperature renormalized by the Fermi temperature T¯ = T /TF at each stage of the evaporation in the crossed region. The line represents the trajectory of evaporation with cooling eﬃciency d(ln T¯ ) γ = 3 d(ln N ) = 2.4, which is extracted from a ﬁt to the data above the Fermi temperature. Below the Fermi temperature, the eﬃciency falls due to the Fermi pressure and the Pauli blocking [2].

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T. Fukuhara, S. Sugawa, Y. Takasu and Y. Takahashi

Fig. 2. – Trajectory of evaporative cooling in the crossed FORT. The data of evaporation are shown as solid circles. The solid line shows the evaporation trajectory with γ = 2.4. Although the evaporation eﬃciency is nearly constant above the Fermi temperature TF , it decreases below TF because of the Fermi pressure and Pauli blocking.

Distribution (arb. units)

In order to enter further degenerate regime, we have increased the volume of the optical trap by changing the beam waist and the power of the vertical beam to 86 μm and 4.7 W, respectively. Owing to this change, we can obtain approximately a factor of 5 more atoms at half the Fermi temperature, which enables us to do further cooling. After 4.5 s of evaporation, 1 × 104 atoms are cooled down to ∼ 75 nK, corresponding to T /TF = 0.37 ± 0.06. This error comes from an uncertainty of ±8% in the temperature and ±6% in the trapping frequency. The temperature of atom clouds is determined by a Fermi-Dirac ﬁt to the data (ﬁg. 3) [17]. We also extract T /TF = 0.36 ± 0.04 from the fugacity [17] and these values are consistent.

-5

0 Velocity (μm/ms)

5

Fig. 3. – Velocity distribution after 4.5 s of evaporation. The data are averaged over ﬁve measurements. Solid line shows the Fermi-Dirac distribution with T /TF = 0.37. Two dashed lines correspond to the Fermi velocity vF = 4.4 μm/ms.

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It is worth noting that, in this experiment, we can produce degenerate Fermi gases without preparing the two-spin mixture which is stable for spin relaxation. From this result, we deduce that 173 Yb is less aﬀected by spin relaxation because it has no electron spin. Recently, physics of a more-than-two component Fermi gas is attracting theoretical interests [18]. To our knowledge, no stable three- (or more) component degenerate Fermi gas has been realized so far. The obtained degenerate gases of 173 Yb may have potential for this new physics. 4. – Sympathetic cooling of

171

Yb with

174

Yb

Diﬀerently from 173 Yb atoms, the elastic collision rate of 171 Yb is not large, and thus thermalization process takes much longer time, which prevents evaporation in a two-spin mixture toward quantum degeneracy. To overcome this problem, we applied sympathetic cooling with bosonic 174 Yb isotope [19]. The experimental sequence is slightly modiﬁed for simultaneous trapping of two isotopes in the MOT, which requires bichromatic beams whose frequencies are near resonant to each isotope. For this purpose, we use frequency sidebands generated by an electrooptic modulator at the frequency corresponding to the isotope shift. Each isotope is independently loaded into the MOT; ﬁrst the frequency of the Zeeman slower laser is tuned to the resonance of one isotope (in this experiment, 171 Yb), and then is changed to the other isotope (174 Yb). The FORT beams produce the same conservative potential for both isotopes because the isotope shift is negligibly small compared with the detuning of the FORT laser, and thus both isotopes are simultaneously trapped in the FORT. We can use isotope-selective absorption imaging because the isotope shift of the 1 S0 -1 P1 transition for the imaging is much larger than the linewidth. The fermionic 171 Yb isotope is sympathetically cooled through elastic 171 Yb-174 Yb collisions. The collision rate was found to be large enough to perform eﬃcient sympathetic cooling, which results in cooling of 171 Yb atoms. The lowest temperature obtained so far was 0.9 of the Fermi temperature. By increasing the number of 174 Yb atoms, we expect to obtain further low temperature of 171 Yb atoms with this method. 5. – Bose-Einstein condensation of

170

Yb

The natural abundance of 170 Yb atoms is relatively small (3.05%). Therefore, we need a longer loading time of 120 s for the MOT to obtain 2× 107 atoms. Once we collect enough atoms in the FORT by transfer from the MOT, whether the atoms can be cooled to quantum degeneracy depends only on their collisional properties. The elastic collision cross-section of 170 Yb atoms was found to be smaller than that of 174 Yb atoms, whose BEC was previously realized [7], and thus we performed the evaporative cooling that was twice as long as that for 174 Yb, which has resulted in Bose-Einstein condensation of 1 × 104 170 Yb atoms. We have observed bimodal momentum distribution in the absorption image (ﬁg. 4) and anisotropic expansion of atoms released from the trap. From the free expansion of the condensate [20], we have extracted the scattering length

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Integrated optical density (arb. units)

T. Fukuhara, S. Sugawa, Y. Takasu and Y. Takahashi

(a)

(b)

(c) -100

-50

0

50

100

Position (μm) Fig. 4. – Density distributions integrated over the vertical direction of 170 Yb atoms after 10 ms of free expansion: (a) thermal cloud, T = 350 nK; (b) bimodal distribution, T = 200 nK; (c) nearly pure condensate with 1 × 104 atoms. The solid lines show the Gaussian ﬁts to the thermal component.

as,170 = 2.9 ± 1.1 nm. This realization of BEC in of a BEC-BEC mixture with 170 Yb-174 Yb. 6. – BEC-BEC mixture of

174

Yb and

176

170

Yb opens up possibility for a study

Yb

The sympathetic cooling technique has been applied to a 174 Yb-176 Yb Bose-Bose mixture because the elastic collision cross-section of 176 Yb is not large [16]. From the thermalization process between the two isotopes, we found that the interspecies elastic collision cross-section is large. Owing to the rapid thermalization, sympathetic cooling works eﬃciently, which results in the phase space densities of higher than one for both isotopes. We have observed almost pure BEC of 1 × 104 174 Yb atoms and the bimodal distribution of 1 × 104 176 Yb atoms (ﬁg. 5), which means that a quantum degenerate mixture was achieved. The number of condensed atoms of 176 Yb is ∼ 1000. We performed further cooling of 176 Yb atoms to obtain a large condensation fraction. However, the number of condensate atoms did not increase. One possibility to explain this result is that the scattering length as of 176 Yb is negative because BEC with attractive interaction becomes unstable when the number of condensed atoms exceeds the critical value Ncr 0.5aho /|as |, where aho = ¯ h/(m¯ ω ) is the harmonic oscillator length with the geometrical average ω ¯ of the trap frequencies [21]. If as = −1 nm, the critical value Ncr will be ∼ 300 in our system, which is consistent with the experimental result. In order to conﬁrm this possibility, we hold the 174 Yb-176 Yb mixture in the optical trap with constant trap depths after evaporation and measure their atom loss. We observed a rapid decay of

Integrated optical density (arb. units)

Quantum degenerate gases and the mixtures of ytterbium atoms

-50

0 Position (μm)

865

50

Fig. 5. – Density distribution integrated over the vertical direction of 176 Yb atoms after 8 ms of free expansion. The data are averaged over ﬁve measurements. The solid line shows the Bose-Einstein distribution ﬁt and the dashed line shows the Gaussian ﬁt to thermal component, yielding a temperature of T ∼ 90 nK.

176

Yb atoms compared to 174 Yb atoms. This supports the negative scattering length of Yb. In addition, recently our group performed photoassociation experiment and the result is consistent with a small negative scattering length of 176 Yb. 176

7. – Prospects We studied cooling of Yb isotopes to quantum degeneracy. We have produced degenerate Fermi gases of 173 Yb and BEC of 170 Yb, following the realization of BEC in 174 Yb. A degenerate mixture of 174 Yb and 176 Yb has also been obtained and it is suggested that the scattering length of 176 Yb is negative. In addition, fermionic 171 Yb atoms have been cooled to T /TF = 0.9 by sympathetic cooling with 174 Yb. The realization of a degenerate 173 Yb gas can be considered as a ﬁrst step toward studying a more-than-two component degenerate Fermi gas. It is also interesting to investigate degenerate Fermi gases using the ultra-narrow optical transition. Furthermore, the results obtained so far would promise the production of degenerate Bose-Fermi mixtures such as 174 Yb-171 Yb and 174 Yb-173 Yb, and also another Bose-Bose mixture of 170 Yb-174 Yb. ∗ ∗ ∗ This work was partially supported by Grant-in-Aid for Scientiﬁc Research of JSPS (18043013, 18204035), SCOPE-S, and 21st Century COE “Center for Diversity and Universality in Physics” from MEXT of Japan. T. Fukuhara would like to acknowledge support from Yoshida Scholarship Foundation, and Y. Takasu from JSPS.

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REFERENCES [1] Anderson M. H., Ensher J. R., Matthews M. R., Wieman C. E. and Cornell E. A., Science, 269 (1995) 198; Davis K. B., Mewes M.-O., Andrews M. R., van Druten N. J., Durfee D. S., Kurn D. M. and Ketterle W., Phys. Rev. Lett., 75 (1995) 3969; Bradley C. C., Sackett C. A., Tollett J. J. and Hulet R. G., Phys. Rev. Lett., 75 (1995) 1687. [2] DeMarco B. and Jin D. S., Science, 285 (1999) 1703. [3] Porsev S. G. and Derevianko A., Phys. Rev. A, 69 (2004) 042506. [4] Takamoto M., Hong F.-L., Higashi R. and Katori H., Nature, 435 (2005) 321. [5] Hoyt C. W., Barber Z. W., Oates C. W., Fortier T. M., Diddams S. A. and Hollberg L., Phys. Rev. Lett., 95 (2005) 083003. [6] Petrov D. S., Salomon C. and Shlyapnikov G. V., J. Phys. B, 38 (2005) S645; see also the lecture note of G. Shlyapnikov in this volume, p. 385. [7] Takasu Y., Maki K., Komori K., Takano T., Honda K., Kumakura M., Yabuzaki T. and Takahashi Y., Phys. Rev. Lett., 91 (2003) 040404. [8] Kuwamoto T., Honda K., Takahashi Y. and Yabuzaki T., Phys. Rev. A, 60 (1999) R745. [9] Takasu Y., Honda K., Komori K., Kuwamoto T., Kumakura M., Takahashi Y. and Yabuzaki T., Phys. Rev. Lett., 90 (2003) 023003. ¨nsch T. W., Phys. Rev. A, 57 [10] Friebel S., D’Andrea C., Walz J., Weitz M. and Ha (1998) R20. [11] Fukuhara T., Takasu Y., Kumakura M. and Takahashi Y., Phys. Rev. Lett., 98 (2007) 030401. [12] Maruyama R., Wynar R. H., Romalis M. V., Andalkar A., Swallows M. D., Pearson C. E. and Fortson E. N., Phys. Rev. A, 68 (2003) 011403(R). [13] Boesten H. M. J. M., Tsai C. C., Verhaar B. J. and Heinzen D. J., Phys. Rev. Lett., 77 (1996) 5194; Burke J. P., Greene C. H., Bohn J. L., Wang H., Gould P. L. and Stwalley W. C., Phys. Rev. A, 60 (1999) 4417; Tojo S., Kitagawa M., Enomoto K., Kato Y., Takasu Y., Kumakura M. and Takahashi Y., Phys. Rev. Lett., 96 (2006) 153201. [14] Monroe C. R., Cornell E. A., Sackett C. A., Myatt C. J. and Wieman C. E., Phys. Rev. Lett., 70 (1993) 414. [15] From recent photoassociation spectroscopy experiment, the value of C6 is estimated to be approximately 2000 a.u. (K. Enomoto et al., in preparation). [16] Takasu Y., Fukuhara T., Kitagawa M., Kumakura M. and Takahashi Y., Laser Phys., 16 (2006) 713. [17] DeMarco B., Ph.D. Thesis (University of Colorado, Boulder) 2001. [18] Honerkamp C. and Hofstetter W., Phys. Rev. Lett., 92 (2004) 170403; Bedaque P. F. and D’Incao J. P., cond-mat/0602525; Paananen T., Martikainen J.-P. and ¨ rma ¨ P., cond-mat/0603498. To [19] Honda K., Takasu Y., Kuwamoto T., Kumakura M., Takahashi Y. and Yabuzaki T., Phys. Rev. A, 66 (2002) 021401(R). [20] Castin Y. and Dum R., Phys. Rev. Lett., 77 (1996) 5315. [21] Bradley C. C., Sackett C. A. and Hulet R. G., Phys. Rev. Lett., 78 (1997) 985, and references therein.

Ultracold fermions in a 1D optical lattice F. Ferlaino(∗ ), G. Modugno, G. Roati and M. Inguscio LENS and Dipartimento di Fisica, Universit` a di Firenze Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy

1. – Introduction In the last years, ultracold atomic physics has been successfully extended into the ﬁeld of degenerate Fermi gases, and optically produced lattices. A growing attention is now addressed to the possibility to combine these two directions toward the production of fermionic systems with tunable interactions in periodic potentials. Optical lattices have already been extensively studied in combination with BoseEinstein condensates giving rise to spectacular eﬀects at the intersection between diﬀerent areas of physics. Notable examples are: macroscopic tunneling of condensed atoms through the lattice [1-3], Bloch oscillations [4, 5], superﬂuid-to-Mott-insulator transition [6], system with reduced dimensionality [7, 8]. Fermions in periodic potential promise to be an even richer system. Recent experimental investigations have been performed on Fermi gases [9-12], fermionic molecules [13], fermion-composed pairs [14, 15] and Bose-Fermi mixtures [16, 17] in a 1D and 3D optical lattice. These systems realize a new laboratory to study a variety of interdisciplinary phenomena, ranging from interferometry to solid-state physics and superﬂuidity. The tight resemblance between fermionic atoms in a lattice and electrons in a crystal naturally induces to extend solid-state eﬀects to the ﬁeld of atomic gases with crucial advantage: a Fermi gas in an optical lattice is a perfect non-interacting system(1 ). This peculiarity has (∗ ) Also at Institut of Experimental Physics and Center for Quantum Physics, University of Innsbruck, 6020 Innsbruck, Austria. (1 ) Ultracold fermionic atoms prepared in just one internal state are non-interacting: s-wave c Societ` a Italiana di Fisica

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allowed investigations on single-particle coherence eﬀects and transport phenomena which are usually overwhelmed by interaction, leading to the observation of spatially resolved Bloch oscillations [11], band insulator [12] and localization [16]. Controlled collisional mechanisms have also been added to explore the conduction mechanisms from zerointeraction limit, to the collisionless and hydrodynamics regime revealing the insulatorto-conductor crossover [18]. On the other hand, fermions can also act as impurities for a Bose gas in a lattice leading to a decrease of the phase space density of a condensate [17] and to a modiﬁcation of in the superﬂuid-to-Mott-insulator transition [19]. One of the most exciting perspectives is the employment of an optical lattice as a probe for the fermionic superﬂuidity in a strong interacting two-components Fermi gas [20, 21]. Recently the group of Wolfgang Ketterle has obtained the ﬁrst evidence of superﬂuidity in a 3D optical lattice [15] by observing the appearance of distinct interference peaks. 2. – Geometrical localization in a 1D optical lattice A single particle in an inﬁnite periodic potential is described by the well-known Bloch function. This wave function exhibits a periodicity equal to the one of the lattice and spreads along all the lattice due to the translational invariance of the system. As soon as this invariance is destroyed or modiﬁed, diﬀerent localization eﬀects appear depending on the perturbation applied. The translational properties can be changed by adding an external force, a disorder in the periodic potential or by introducing some imperfections. In the domain of ultracold atomic gases, the simpler localization eﬀects can be produced using an extra potential, since one can easily manipulate the gases by applying magnetic ﬁelds. A notable example is provided by the linear potential where delocalized Bloch functions are mapped into localized Wannier-Stark states. Another example of great interest in the atomic physics ﬁeld is the case of a combined harmonic (i.e. quadratic) and periodic potential (CHP). This is the most common conﬁguration encountered in experiments, where additional conﬁning potentials are usually present. The CHP potential results in a drastic modiﬁcation of the energy spectrum with respect to the usual band structure encountered in solid-state physics. Theoretical investigations on this topics have been recently reported [18, 22-24]. . 2 1. The “bent-tube” spectrum. – Here we discuss the energy spectrum of a single particle conﬁned in the combined harmonic and periodic potential. The quadratic conﬁnement exhibits a cylindric geometry and the one-dimensional optical lattice is superimposed along the weak axis of the parabolic potential (we label the axial direction as x). The full 3D Hamiltonian is thus (1)

H=

2

pz + p2y p2x 1 1 + mωa2 x2 + sEr sin2 (kx) + + mωr2 (z 2 + y 2 ) , 2m 2 2m 2

collisions are forbidden by the Pauli exclusion principle, while high-order partial waves are thermally suppressed. An optical lattice is a perfect periodic potential free from impurities or phononic excitations.

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Ultracold fermions in a 1D optical lattice

where m is the atomic mass, ωa,r the axial and radial frequency of the harmonic trap, s the lattice depth in units of the recoil energy Er = h ¯ 2 k 2 /2m, with k the wave number. The 3D problem can be reduced to a 1D problem since the Hamiltonian of eq. (1) is decoupled in the three directions. We focus on the stationary single-particle Schr¨ odinger equation along the lattice direction (2)

¯ 2 ∂2 h 1 2 2 2 − + mω x + sEr sin (kx) ψ = En ψ. 2m ∂x2 2

Unlike the linear potential, the presence of a harmonic potential prevents analytic solution of eq. (2). Just in tight binding and single-band approximations an analytic solution can be recovered [22, 23]. Nevertheless, these approximations constrain the analysis to deep optical lattices and to systems with such a high phase space density to ensure the occupancy of a single band. The eigenvalue problem (2) can be solved numerically for the typical range of experimental parameters. The energy spectrum in x-space of the ﬁrst 1000 eigenfunctions is shown in ﬁg. 1. Each line corresponds to a density plot of the wave function. This spectrum belongs to a new class of energy diagrams which diﬀers drastically from the one of a pure harmonic trap or a pure optical lattice. Three major features can be identiﬁed: the shell structure, a spatial energy gap, and the appearance of localized states. – Shell structure: Since the harmonic potential destroys the translational invariance of the system, the quasi-momentum is no more a good quantum number and the energy spectrum should now be calculated in real x-space (see ﬁg. 1). This spectrum clearly exhibits a shell-like structure. The shells can be regarded as energy bands separated by a forbidden energy region. – Energy gap: For a given x-position, not all the energies are allowed because a gap opens in the presence of a periodic conﬁnement. In analogy with the Bloch picture, one can introduce a bandwidth Ebw which deﬁnes the interval of accessible energy. For deep lattices the width of the ﬁrst energy band scales as (3)

Ebw

8Er s3/4 e−2 √ = π

√

s

.

We also found a correspondence between the forbidden energy interval Egap in xspace and the usual band gap in the quasi-momentum space. Nevertheless, one can no more refer to an absolute energy gap in the bent-tube spectrum but has to consider a spatial energy gap. – Localized and delocalized states: In each energy shell one can recognize two distinct classes of solutions. For low energies (E < Ebw ), the eigenstates spread symmetrically around the potential minimum. They are delocalized along the lattice with an extension which exceeds the harmonic oscillator length. These delocalized states

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Fig. 1. – Spectrum of the Hamiltonian: representation of the 1D spectrum of the single-particle Schr¨ odinger equation for a combined periodic and parabolic potential. Each line represents one eigenstate of the system. The vertical position corresponds to the energy of the eigenstate. The parameters of the potential are s = 3, λ = 830 nm, ω = 2π × 16 s−1 and the mass is that of 87 Rb.

are the analogous of Bloch states in a pure periodic potential. The density distributions |ψ|2 of the ﬁrst two delocalized functions are shown in ﬁg. 2a)-b). The shape of the wave functions is similar to the corresponding eigenstates of a pure harmonic oscillator while the fast modulation is due to the optical potential. For energy larger than Ebw , a second class of solutions appears. These eigenstates are no more symmetric with respect to the center of the trap and become localized on both sides of the potential. We interpret this class of solutions as localized states similarly to the Wannier-Stark states found in the presence of a linear force. Typical localized states are shown in ﬁg. 2c)-d). These states spread over few lattice sites depending on the lattice height s, on the harmonic conﬁnement ωa,r , and on the energy level. A particle has now the highest probability to occupy a space region on one side of the potential while the occupation probability decreases to zero around the trap minimum. Going further in energy, the corresponding eigenfunctions are again centered around the trap minimum. These states correspond to delocalized particles in the second band. By further increasing the energy, localized states in the second band also appear.

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density (arb.units)

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Fig. 2. – Density distribution of a particle trapped in the CHP potential with s = 3. For E < Ebw , the solutions of eq. (2) are delocalized functions spreading symmetrically around the trap minimum. The density distributions |ψ|2 of eigenstates relative to the ﬁrst (a) and second (b) energy level are reported. For E > Ebw , the solutions are states localized on one side of the potential, as shown by the density distribution of the 121st (c), 122nd (d) eigenfunction.

. 2 2. Localization and addressability. – The degree of localization of particles is a crucial point connected to the possibility of addressing particles within a single lattice site. The extension of localized states depends on the local gradient provided by the quadratic potential, on the lattice depth and on the energy level. In ﬁg. 3a) we show the energy spectrum for a potential depth of s = 0.3, and 9. Already for moderate heights of the periodic potential, a substantial localization takes place. This behavior is quantiﬁed in ﬁg. 3b), where we plot the extension of the localized state as a function of the lattice height for typical trap parameters. For small lattice depths, these states are localized in space over several lattice sites (the lattice spacing is λ2 0.4 μm). Increasing the depth of the optical lattice, the extension of the localized state shrinks and the smallest possible extension is given by the ground state in each lattice site. Contrary to WannierStark functions in linearly tilted lattice, here the degree of localization for a given s and ωa,r is not constant and changes with the energy level considered. To clarify this point, we also plot the function extension for diﬀerent axial energies (Ea /kB = 100, 200, and 300 nK). We found the localization to be enhanced at high energies. As we will discuss in sect. 3, this counterintuitive result can be directly observed in a sample of fermions

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s=0.3

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lattice height (Er) Fig. 3. – (a) Energy spectrum for s = 0.3 and s = 9. The representation as well as the parameters of the potential are the same as in ﬁg. 1. (b) Extension of the localized states in the ﬁrst band as a function of the lattice height for three energies (kB × 100 nK, kB × 200 nK, and kB × 300 nK).

where the highest axial energy can be chosen by tuning the Fermi energy EF with respect to Ebw . Note that for small s, inter-band transitions reduce the degree of localization of our states since tunneling processes between bands are strongly enhanced. The wave function has then substantially contributions of both localized states within the ﬁrst band and delocalized within the second one. . 2 3. Semi-classical motion in the CHP potential. – The dynamics of particles in the CHP potential can be investigated either studying the time evolution of eq. (2) or using a semi-classical model [10]. While the single-particle quantum problem is powerful in understanding the properties of the CHP potential and the static behavior of an atomic gas, the semi-classical model turns out to be more appropriate in describing the dynamics of the system and is able to reproduce rather closely the motion observed in our experiments (see sect. 4). This approach has already been successfully applied to study the motion of an electron in a lattice subjected to a constant force. In particular, Bloch oscillations in quasi-momentum space have been predicted. The crucial idea beyond this model is that the periodic potential aﬀects the dynamics by increasing the particle inertia and the motion is scaled thanks to a global renormalized mass (m → m∗ ). In the last years, big eﬀorts have been spent trying to apply this model to the case of harmonic forces. A renormalized mass can be introduced which is now energy dependent p2x (m∗ → m∗ (E)). Consequently, the energy dispersion is also modiﬁed ( 2m → ε(px )) and can be calculated within the tight-binding approximation, a linearizing process, and/or numerical models. A detailed description can be found in refs. [10, 25]. In particular we are interested on the eﬀect of a sudden small displacement of the harmonic trap

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Fig. 4. – (a) Phase trajectories for a trapped 1D Fermi gas in a lattice at T = 0. The dashed orbit corresponds to the isoenergetic p curve with energy E = Ebw . The ordinate and abscissa are h and x∗ ≡ mωx2 x2 /2Ebw . (b) Frequency dispersion of particles moving in units of p∗ ≡ px d/2¯ along closed orbits corresponding to the oscillation of delocalized atoms in the CHP potential. Figure from [25].

center. The physics becomes particulary clear in the semiclassical phase space, as shown in ﬁg. 4a). The lines are iso-energetic single-particle orbits and we can again distinguish between two diﬀerent classes of orbits, closed and open, separated by the dashed orbit with energy Ebw . Closed orbits, appearing for E < Ebw , belong to delocalized state. Particles stored in these orbits can fully convert their potential energy into kinetic energy and their motion proceeds as an oscillation around the trap minimum. The singleparticle oscillation frequency depends strongly on energy and vanishing for E = Ebw . The frequency dispersion for closed orbit is reported in ﬁg. 4b). Localized particles occupying open orbits (E > Ebw ) can instead just oscillate on one side of the potential. This is the equivalent of Bloch oscillations encountered in linearly tilted lattices. 3. – Localized fermions and bosons The single-particle energy spectrum described above has shown the existence of localized and delocalized states. This picture can be generalized to describe ultracold atomic gases with negligible inter-particle interaction such as Fermi gases. To go from a single-particle and a many-particle description one needs to introduce quantum statistic in the problem, which also sets new energy scale in addition to the ones ﬁxed by the lattice (Ebw , EGap ). In a spin-polarized Fermi gas, atoms can be regarded as a collection of non-interacting single particles fulﬁlling the Pauli exclusion principle. At zero temperature, the Pauli principle enforces fermions to occupy one by one the energy levels until the Fermi energy EF is reached. In a sample of bosons above the condensation threshold, atoms arrange themselves according to the Bolzmann statistics. The relevant energy scale is then the thermal energy kB T . The situation becomes more challenging dealing with a Bose-

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Einstein condensate. Condensed atoms are no more well described by a single-particle picture, since the inter-particle interaction modiﬁes substantially the eigenvalue problem. The energy spectrum discussed above can then just be regarded as a qualitative guideline. The energy scale is now determined by the chemical potential and condensed bosons are expected to occupy just a narrow region at the bottom of the ﬁrst energy band, e.g. delocalized states. For these states the harmonic potential slightly deforms the lattice and the atoms can be described within the semi-classical approach as delocalized Bloch states of mass m∗ which takes into account diﬀerent inertia of condensed atoms [3,26-28]. This description is valid until the harmonic potential energy is small compared with the chemical potential, otherwise energetic and dynamical instabilities appear in the system, as observed in [29, 30]. . 3 1. K-Rb mixture in a combined harmonic and periodic potential. – We now describe a series of experiments in which we have studied the properties of fermions and bosons in the CHP discussed above. The experiments reported here are performed either with a sample of fermionic 40 K atoms, with one of bosonic 87 Rb atoms, or a mixture of both. We initially prepare a 87 Rb-40 K Bose-Fermi mixture in a magnetic trap via sympathetic and evaporating cooling, and we then superimpose a one-dimensional optical lattice. A detailed description of our experimental procedure can be found in [9, 31]. After the laser cooling phase (pre-cooling stage), the atomic sample is initially loaded into a harmonic trap with cylindrical symmetry along the axial direction (x-axis). Trapped Rb atoms are selectively evaporated using radio-frequency sweeps while K atoms are sympathetically cooled through elastic interspecies collisions. Both species are trapped in their doubly polarized spin states (|F = 9/2, mF = 9/2 for K and |2, 2 for Rb). In these states, the two samples experience the same magnetic potential, with axial and radial harmonic frequencies ωa = 2π × 24(16.3) s−1 and ωr = 2π × 275(187) s−1 for K (Rb). The evaporation procedure can be optimized to reach the desired temperature and atom number. The bosons can be completely removed forcing the evaporation below the trap bottom or prepared at temperatures as low as 100 nK (below this temperature it is diﬃcult to prevent condensation), with atom numbers ranging from 104 to 105 . The typical sample of spin-polarized fermions is composed of 104 -105 atoms at temperatures that can be adjusted between 0.2TF and 3TF . The Fermi temperature for 104 atoms is TF = 200 nK. Notice that at such low temperatures, fermions prepared in a single spin state are identical and cannot collide between themselves. We then superimpose to the magnetically trapped sample a 1D optical lattice along the weak x axis of the magnetic trap. The optical lattice is created by a laser beam in standing-wave conﬁguration. The laser wavelength λ can be chosen in the range 790–870 nm which is red-detuned from all the K and Rb optical transitions. Most of the experiments have been performed at λ = 863 nm, where the recoil energies are Er = 146 nK and Er = 317 nK for Rb and K, respectively. The lattice height in units of the recoil energy is calibrated by means of Bragg spectroscopy on the condensate and can be adjusted in the range s = 0–10. The lattice is switched on adiabatically about 500 ms before the end of the evaporation, to allow for thermalization in the combined potential.

Ultracold fermions in a 1D optical lattice

875

The dipolar oscillations in the CHP potential are excited by displacing axially the harmonic trap minimum. The minimum kick we give to the atoms amounts to a displacement of the trap of about 10 μm. After a variable holding time in the trap, the atoms are released from the potentials and imaged after a ballistic expansion. . 3 2. Radio-frequency spectroscopy of localized states. – The bent-tube energy spectrum suggests the possibility to address selectively localized and delocalized states. We discuss here a band spectroscopy technique selective on the energy levels which allows a clear identiﬁcation of the two classes of states. To populate a high number of localized states, the average axial energy of the atomic sample has to exceed the bandwidth (E > Ebw ). This condition can be easily achieved in a thermal sample of bosonic atoms, where the atomic temperature can be chosen by controlling the evaporative cooling procedure. We concentrate our eﬀort on a bosonic thermal cloud, although the spectral features are equally valid for fermions. We prepare a sample of 87 Rb atoms in the CHP potential at a temperature ranging from 500 to 600 nK. We then superimpose adiabatically a 1D optical lattice along the axial direction. The optical depth can be adjusted in the range 0 < s < 10. These conditions ensure large population in localized states. Our spectroscopic technique is based on the selective removal of delocalized bosons from the trap. After the preparation of the atomic cloud in the CHP potential, a radio-frequency (rf) ﬁeld is applied in order to induce spin-ﬂip transitions from trapped to untrapped Zeeman states. All atoms whose wave function has a spatial overlap with the magnetic ﬁeld shell are removed from the trap. The magnetic ﬁeld shell is deﬁned by the following resonance condition: (4)

hν = μB B(r)/2,

where B is the magnetic ﬁeld of the trap and μB the Bohr magneton. A periodic sweep (1 kHz) of the radio frequency within an interval Δν = νup − νlow = 3 kHz deﬁnes the spatial region in which the atoms are removed from the potential. After 100 ms we switch oﬀ the r.f. ﬁeld together with the optical and the magnetic potential. The remaining atoms are imaged after 1.5 ms of free expansion. In ﬁg. 5, we report a sketch of the experimental technique (a) and series of absorption images taken for diﬀerent values of the r.f. knife νc (b)-(i). Starting from below, when the resonant condition (4) is satisﬁed, the r.f. signal starts to remove the less energetic atoms in the CHP potential. These atoms are delocalized on the bottom of the energy spectrum occupying the center of the trap (see ﬁg. 1). The remaining atoms occupy just the edge of the potential and we clearly observe the appearance of two clouds on the side of the initial atomic distribution (ﬁg. 5b)-g)). For hνc ∼ Ebw /2, we are able to transfer all the delocalized atoms into untrapped Zeeman sublevels (ﬁg. 5d)). By increasing the r.f. frequency, the hole in the spatial distribution deepens, and localized states are also removed. When νc > Ebw /2, atoms in the trap center are no longer removed from the potential and we observe three clouds. The central cloud corresponds to delocalized atoms in the bottom of the ﬁrst band while the lateral cloud are due to particles localized

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Fig. 5. – Band spectroscopy of localized and delocalized bosons for s = 9. (a) Sketch of the spectroscopy technique: the blue region indicates the occupied levels while the gray one corresponds to the energy levels depleted by the radio-frequency signal within Δν. (b)-(i) Absorption images of a thermal cloud of Rb atoms after 1.5 ms of time ﬂight for diﬀerent values of the r.f. signal. The frequencies indicate the upper value νup of the r.f. ﬁeld, the frequency interval of the r.f. ﬁeld is 3 kHz.

in the higher energy state (ﬁg. 5h)). By further increasing the frequencies, the side peaks disappear and the cloud is again unaﬀected by the r.f. knife. This spectroscopy technique allows ﬁrst direct observation of the energy distribution in the CHP potential revealing the existence of localized and delocalized states. These result are also promising to address and control particles within a deﬁned spatial region. 4. – Dynamical response: localized vs. delocalized atoms So far we have discussed the stationary states of atomic gases trapped in a CHP potentials. This study has revealed the existence of localized and delocalized states in the energy distribution of an atomic sample. These two classes of states also exhibit a diﬀerent dynamical response when subjected to external perturbations. In this section, we analyze the behavior of the system when a sloshing mode is excited by a sudden displacement of the harmonic trap minimum. In a pure harmonic potential the center of mass of an atomic cloud oscillates with the trap frequency irrespective of temperature, inter-particle interactions, and quantum statistics (Kohn’s theorem). Adding periodic potential, this result is no longer true and the particle motion is strongly perturbed also in the absence of inter-particle collisions. We focus here on a spin-polarized Fermi gas where the atomic collisions are suppressed by the quantum statistics. . 4 1. Transport of a Fermi gas. – Let us consider the center-of-mass evolution of a spin-polarized Fermi gas in the CHP potential when a sloshing mode is excited. We

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Fig. 6. – (a)-(b): Dipole oscillations of a Bose-Einstein condensate of 87 Rb atoms (a) and a Fermi gas of 40 K atoms in the presence (ﬁlled circles and full line) and in the absence (triangles and gray line) of a lattice with height s = 2. The Fermi gas is prepared at T = 0.3 TF with EF = 0.5Er . (c)-(d): Relative frequency of a condensate (c) and a Fermi gas (d) as a function of the lattice height. The continuous lines are the theoretical predictions using the semi-classical model.

perturb the system by suddenly displacing the harmonic trap minimum along the axial direction [32]. The typical displacement is xd = 15 μm, which is much smaller than the 1/e2 axial radius of the cloud (110 μm). After a variable hold time in the trap the atoms are released from the combined potential. We detect the center-of-mass position by absorption images taken after 8 ms of ballistic expansion. The fraction of fermions in localized and delocalized states clearly depends on the position of the Fermi energy relative to the ﬁrst band of the energy spectrum. We can tune the ratio EF /Ebw , for example by varying the depth of the lattice at ﬁxed atom number, as illustrated in ﬁg. 3a). We perform a ﬁrst set of measurements with Ebw ∼ EF where most fermions are delocalized at the bottom of the trap. The centerof-mass motion of a Fermi gas together with the one of a Bose-Einstein condensate is shown in ﬁg. 6a)-b). While a Bose-Einstein condensate exhibits a collective oscillation with reduced frequency [2], the Fermi gas motion proceed also with strong damping [10]. The reduced oscillation frequency observed in both samples can be explained within the semiclassical model [10, 26]. A delocalized particle obeys to the band-like velocity dispersion law where a renormalized mass m∗ is introduced to consider the modiﬁcation

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of the atom inertia due to the lattice. The oscillation frequency ωd is shifted according to the following law: m ωa , (5) ωd = m∗ where m is the atomic mass and ωa the axial magnetic frequency. In ﬁg. 6c)-d), we report the oscillation frequency as a function of the lattice height for both a condensed and fermionic sample. The experimental data are well ﬁtted by the theoretical curve calculated for particles moving at the bottom of the band with a re-normalized mass (solid line) [10, 26, 2, 3]. While the renormalized mass theory is based on the singleparticle coherence induced by the lattice, one can also describe the condensate motion within a Josephson-like description exploiting the macroscopic quantum coherence of the system. This alternative picture is based on a set of Gross-Pitaevski equations coupled one to the other by the tunneling rate K. These two equivalent descriptions are linked by the following relation: λ2 K 2¯h2 m (6) ωd = ω , where K = . a λ 2 m∗ 2¯h2 Despite the analogy in frequency shifts observed for bosons and fermions, only the fermions motion is damped. The damping observed in ﬁg. 6b) can be qualitatively under. stood using the semiclassical picture for delocalized states summarized in subsect. 2 3. Fermions occupy one by one the energy level of the CHP spectrum. Each energy level corresponds to a diﬀerent orbit in the phase space and then to a diﬀerent oscillation frequency. The atoms with lower energy start to participate in phase to the collective dipolar oscillations, but since there is a frequency dispersion due to the non-parabolic shape on the band, as summarized in ﬁg. 4, they get dephased in a few oscillations, causing a damping of the macroscopic motion. According to our parameters, the Fermi gas is prepared at a temperature T = 0.3 TF and E/Ebw = 0.3, yielding a maximum frequency shift ω(E)/ωa of 10%. Note that the dephasing also aﬀects the motion of condensed atoms nevertheless their narrower energy distribution both in momentum and coordinate space makes this eﬀect negligible, and the observed motion is undamped. To summarize, we have observed that delocalized fermions (E < Ebw ) can symmetrically oscillate around the trap minimum with a reduced frequency and a damping due to the dephasing of particles in diﬀerent orbits. When E > Ebw also localized states start to be populated. The number of localized fermions can be enhanced by increasing the depth of the periodic potential. The dynamics of localized particles strongly diﬀers from the previous one. For s = 8, the Fermi energy lies in the ﬁrst bandgap: both classes of states, delocalized across the trap center and localized on the sides, will be occupied. The center-of-mass motion is shown in ﬁg. 7b). We again observe a reduced oscillation frequency and a damping ascribing again to the delocalized fraction. Nevertheless the most striking eﬀects are the appearance of an oﬀset and the strong decrease of the oscillation amplitude. Localized fermions no longer oscillate around the trap minimum but move on

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Fig. 7. – Dynamical response of a Fermi gas in the regime EF > Ebw , where E = 0.3EF , and Ebw = 0.12Er with EF /kB = 430 nK, Er /kB = 415 nK and s = 8. (a) Axial distribution of the Fermi gas showing the localized and delocalized component. (b) Evolution of the center-of-mass position of a Fermi gas in the harmonic trap (gray triangles) and combined with an optical lattice (black circles). The position of the cloud is detected after 8 ms of ballistic expansion.

one edge of the potential. This is a direct consequence of the energy gap which conﬁnes the motion in a small region ranging over few lattice sites (see ﬁg. 7b)). Since the system is free from collisions, fermions cannot change their energy level and transitions from localized to delocalized states at the center of the trap are forbidden. This explains the long-lived oﬀset observed. Observing the atomic distribution, it is possible to distinguish two diﬀerent components: one composed of localized fermions whose position is frozen during the time evolution and the other oscillating in the combined trap (delocalized fermions), as illustrated in ﬁg. 7a). By tuning the lattice depth, one can thus observe a crossover from a conducting to an insulating behavior of the system. 5. – Conduction of a Fermi gas So far, we have investigated the dynamics of a non-interacting atomic Fermi gas in a CHP potential. Non-interacting fermions in localized states behave as an insulator when subjected to an external driven force. The next step is to consider the eﬀect of atomic collisions on the transport phenomena described above. These collisions are normally present in Bose-Fermi mixtures, in sample of non-identical fermions (e.g., fermions in diﬀerent internal states) and in bosonic gases. At low temperatures the collisional rate Γ ≈ 8πa2 nv depends just on the mean atomic density n, the thermal velocity v and the scattering length a. We ﬁnd that collisions aﬀect dramatically the transport of particles through the lattice. In particular scattering mechanisms destroy the localization and drive the system from an insulating to a conducting behavior [16]. This phenomenology provides a straightforward link with the well-known theory of conduction in metals. In the absence of interactions, electrons in a tilted lattice are expected not to change their quantum

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state and the whole system behaves like an insulator for DC currents. This localization is rapidly destroyed by collisions and a macroscopic current is established. Small collisional rates favor a current through the lattice, whereas in the regime of high collisional rate the current is hindered by collisions and the conductivity decreases linearly by increasing collisional rates. The latter regime is the usual one for real solids where electrons strongly interact with lattice phonons and impurities. . 5 1. From an insulating to a conducting Fermi gas: admixture of bosons. – We here discuss an experiment in which we have studied how the transport of a Fermi gas is modiﬁed by the presence of bosonic atoms [16]. This combination allows to explore a wide range of collisional rates, since one can keep the number of fermions ﬁxed while the boson number can be varied by more than one order of magnitude. Rubidium and potassium atoms experience strong mutual attractive interactions [33-35]. The interspecies triplet scattering length aRb,k has been measured through the Feshbach spectroscopy technique to be aRb,K = (−215 ± 10)a0 [36]. In this set of measurements, the number of bosons in the mixture has been chosen by changing the parameters of the radio-frequency evaporation ramp. We can easily either remove all the bosons by sweeping the radio-frequency below the trap bottom or produce a mixture with NRb,K = 104 –105 . The working temperature ranges from TF to 1.5TF , where TF = 300–400 nK. This temperature scale ensures that E > Ebw corresponding to a high occupation of localized states. The motion is again excited by suddenly displacing the magnetic trap along the axial direction. As show in ﬁg. 8a), collisions modify drastically the dynamical behavior of the Fermi gas. We observe an exponential decay of the center-of-mass position to the trap minimum. In particular, the oﬀset due to localized fermions washes out and the system relaxes towards the equilibrium position of the combined potential. During interspecies collisions the momentum of fermions is no more conserved and fermions can hop between diﬀerent localized states until the minimum energy conﬁguration is reached. Using the solid-state picture, we interpret this macroscopic transport as a DC current induced by collisions. Here bosons act as impurities in the crystal and the relaxation time τ (e.g., the decay time) towards the trap center corresponds somehow to the resistivity of a crystal. To better understand the role of collisions, we have repeated the measurement with diﬀerent numbers of bosons (e.g. diﬀerent collisional rates ΓRb,K ). In ﬁg. 8b) we report τ for a fermionic cloud as a function of the interspecies collisional rate. The number of bosons ranges from 2.5 × 104 to 3 × 105 corresponding to a change in ΓRb,K between 40 s−1 and 550 s−1 . The average collisional rate is calculated considering the spatial overlap of both clouds in the combined potential. Figure 8b) shows the crossover from an insulator to an Ohmic conductor, where conduction is hindered by collisions. In the regime of small ΓRb,K , the resistivity decreases by increasing collisional rates. That is what expected if the collisions assist the hopping between diﬀerent localized states. This regime of anomalous conduction was accessed so far only in semiconductor superlattices [37, 38]. In the high collisional rates regime, the system exhibits an Ohmic behavior and the resistivity increases by increasing the number of impurities (e.g. bosons). Collisions with

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Collisional rate (s )

Fig. 8. – Crossover from an insulator to an Ohmic conductor by tuning the collisional rate in a Bose-Fermi mixture. (a) Center-of-mass motion of a cloud of fermions. A pure fermionic sample (triangles and gray line) does not move toward the trap center, whereas an identical sample with an admixture of bosons reveals a current through the potential (circles). The data are ﬁtted with an exponential decay (black line). The expansion time of the cloud is 8 ms and the lattice height for K (Rb) atoms is s = 3 (9). The temperature and the atom number of the fermions are T = 300 nK and N = 5 × 104 , the number of admixed bosons is Nb = 1 × 105 . (b) Oﬀset decay time τ of a cloud of fermions in a mixture with bosons in dependence on the collisional rate (dots). The number of bosons was changed over more than one order of magnitude. The decay time is the analogous to the resistivity of the sample.

bosons impede indeed the current through the potential like in an electric conductor. Note that the number of bosons is now much higher than the number of fermions and the bosons can be regarded as a thermal bath for the fermions. . 5 2. Two-component Fermi gas. – The introduction of collisional channels to a noninteracting system yields a qualitative change in their physical behavior. A part from the already discussed Bose-Fermi case, another remarkable example is provided by twocomponent Fermi gases. Recently this system has been the subject of large interest, being the starting point for the production of fermionic superﬂuids, of molecular Bose-Einstein condensates and strong interacting systems in the BCS-BEC crossover. We will now discuss how the dynamics on spin-polarized fermions change when a second fermionic component is added. We can produce a two-component Fermi gas in the CHP potential by transferring a fraction of fermions in the subsequent Zeeman state by means of radiofrequency sweep. As already described, we prepare a pure sample of 40 K atoms in their stretched state (F = 9/2; mF = 9/2) by evaporating the whole rubidium component. We then transfer one third of the atoms in the (F = 9/2; mF = 7/2) state using a radiofrequency signal. We restrict our attention to the case of cold sample (T = 0.5TF ) and small lattice depth (s = 0.6) where the fermions are mostly delocalized around the center band. In this situation the atomic energy is lower than the bandwidth Ebw (E < Ebw ). In the absence of interactions (e.g. one-component Fermi gas), the dipolar oscillation in the

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Position (μm)

60

55

50

45

40 0

50

100

150

200

250

Time(ms) Fig. 9. – Dipole oscillations of a Fermi gas in the CHP potential with s = 0.6 in the absence (triangles) and in the presence (ﬁlled circles) of a second distinguishable Fermi gas. The temperature and the atom number of the fermions in mF = 9/2 are T = 0.5TF and N9/2 = 2 × 104 , the number of admixed fermions in mF = 7/2 is N7/2 = 104 . The damping is due to collisions between the two components.

CHP potential proceed essentially undamped with a slightly reduced frequency. When a fermionic fraction is transferred in another Zeeman state, collisions between non-identical fermions are switched on and the center-of-mass motion results in a damped oscillation, as shown in ﬁg. 9. In the E < Ebw limit, the main contribution of intercomponent collisions to the dynamics consists thus in a dephasing of the center-of-mass motion. A pair of colliding fermions can change their momentum during the scattering and both atoms have a chance of being dephased after the collision. The number of dephased atoms clearly increase with time, giving rise to an exponential decay of the dipole oscillation. Note that a damping due to collisions between the two components is present already in the harmonic trap, because of their diﬀerent oscillations frequencies, but it almost doubles in the presence of the lattice [39]. 6. – Conclusion and outlook One of the main topics here discussed is whether a harmonic conﬁned Fermi gas is able to perform center-of-mass oscillations in the presence of a one-dimensional optical lattice. The combined potential plays the twofold role of modifying the energy dispersion of the system and of breaking the translational symmetry. In the absence of interaction, oscillatory motions are strongly inhibited and, for high lattice depths fermions exhibit an overdamped motion centered on one side of the potential. The appearance of such oﬀset shows the insulating behavior of the system. When collisions are added, the center-ofmass oscillations are destroyed by scattering processes and the global motion results in an exponential decay of the oﬀset towards the potential minimum. This behavior diﬀers

Ultracold fermions in a 1D optical lattice

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from the one observed in a pure harmonic trap or periodic potential and opens the intriguing perspective to combine the CHP potential with a strongly interacting Fermi gas. We have shown that already in the collisionless regime the motion of a two-component Fermi gas is remarkably modiﬁed by the presence of a lattice. The observed damping is expected to be even enhanced in the hydrodynamics regime due to umklapp collisions, where colliding particles exchange momentum with the lattice (Δk = ±2¯hk) [40]. The subjects here discussed are even more relevant if we consider the case of a two-component Fermi gas in the superﬂuid phase. This immediately rises the question whether the CHP potential can provide an unambiguous signature for superﬂuidity. Recent theoretical investigations predict a marked change in the center-of-mass dynamics with the occurrence of persistent (undamped) Josephson-like oscillations [40, 21, 41] and with a change in the BCS transition temperature [20]. ∗ ∗ ∗ We would like to thank the former team members who have contributed to the research work reported here: E. de Mirandes, H. Ott and R. Heidemann. We would like ´ and the LENS quantum gas group for stimulating to thank S. Stringari, L. Pezze discussions. This work was supported by MIUR, by EU under contracts HPRICT 199900111 and MEIF-CT-2004-009939, and by Ente CRF, Firenze. REFERENCES [1] Hensinger W. K., Haffner H., Browaeys A., Heckenberg N. R., Helmerson K., McKenzie C., Milburn G. J., Phillips W. D., Rolston S. L., Rubinsztein-Dunlop H. and Upcroft B., Nature, 412 (2001) 52. [2] Cataliotti F. S., Burger S., Fort C., Maddaloni P., Minardi F., Trombettoni A., Smerzi A. and Inguscio M., Science, 293 (2001) 843. [3] Fort C., Cataliotti F. S., Fallani L., Ferlaino F., Maddaloni P. and Inguscio M., Phys. Rev. Lett., 90 (2003) 140405. ¨ller J. H., Cristiani M., Ciampini D. and Arimondo E., Phys. Rev. [4] Morsch O., Mu Lett, 87 (2001) 140402. ¨ffner H., McKenzie C., Browaeys A., Cho [5] Denschlag J. H., Simsarian J. E., Ha D., Helmerson K., Rolston S. L. and Phillips W. D., J. Phys. B, At. Mol. Opt. Phys., 35 (2002) 3095. ¨nsch T. W. and Bloch I., Nature, 415 [6] Greiner M., Mandel O., Esslinger T., H a (2002) 39. ¨nsch T. W. and Esslinger T., Phys. Rev. Lett, [7] Greiner M., Bloch I., Mandel O., H a 87 (2001) 160405. [8] Burger S., Cataliotti F. S., Fort C., Maddaloni P., Minardi F. and Inguscio M., Europhys. Lett., 57 (2002) 1. [9] Modugno G., Ferlaino F., Heidemann R., Roati G. and Inguscio M., Phys. Rev. A, Rapid Commun., 68 (2003) 011601(R). ´ L., Pitaevskii L., Smerzi A., Stringari S., Modugno G., de Mirandes E., [10] Pezze Ferlaino F., Ott H., Roati G. and Inguscio M., Phys. Rev. Lett., 93 (2004) 120401. [11] Roati G., de Mirandes E., Ferlaino F., Ott H., Modugno G. and Inguscio M., Phys. Rev. Lett., 92 (2004) 230402.

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¨ hl M., Moritz H., Sto ¨ ferle T., Gu ¨nter K. and Esslinger T., Phys. Rev. Lett, [12] Ko 94 (2005) 080403. [13] Ospelkaus C., Ospelkaus S., Humbert L., Ernst P., Sengstock K. and Bongs K., Phys. Rev. Lett, 97 (2006) 120402. ¨ ferle T., Moritz H., Gu ¨nter K., Ko ¨ hl M. and Esslinger T., Phys. Rev. Lett, [14] Sto 96 (2006) 030401. [15] Chin J. K., Miller D. E., Liu Y., Stan C., Setiawan W., Sanner C., Xu K. and Ketterle W., cond-mat, 0607004 (2006). [16] Ott H., de Mirandes E., Ferlaino F., Roati G., Modugno G. and Inguscio M., Phys. Rev. Lett., 92 (2004) 160601. ¨nter K., Sto ¨ ferle T., Moritz H., Ko ¨ hl M. and Esslinge T., Phys. Rev. Lett, 96 [17] Gu (2006) 180402. ¨rck V., Modugno G. and [18] Ott H., de Mirandes E., Ferlaino F., Roati G., T u Inguscio M., Phys. Rev. Lett., 93 (2004) 120407. [19] Ospelkaus S., Ospelkaus C., Wille O., Succo M., Ernst P., Sengstock K. and Bongs K., Phys. Rev. Lett, 96 (2006) 180403. [20] Orso G. and Shlyapnikov G. V., Phys. Rev. Lett., 95 (2005) 260402. [21] Wouters M., Tempere J. and Devreese J. T., Phys. Rev. A, 70 (2004) 013616. [22] Rigol M. and Muramatsu A., Phys. Rev. A, 70 (2004) 043627. [23] Hooley C. and Quintanilla J., Phys. Rev. Lett., 93 (2004) 080404. [24] Ruuska V. and Torma P., New J. Phys., 6 (2004) 59. ´ L., Dynamical Behaviour of a Trapped Ideal Fermi Gas in an Optical Lattice, [25] Pezze Diploma Thesis, University of Trento (2003). ¨ mer M., Pitaevskii L. and Stringari S., Phys. Rev. Lett., 88 (2002) 180404. [26] Kra ¨mer M., Menotti C., Pitaevskii L. and Stringari S., Eur. Phys. J. D, 27 (2003) [27] Kra 247. [28] Fallani L., Cataliotti F. S., Catani J., Fort C., Modugno M., Zawada M. and Inguscio M., Phys. Rev. Lett, 91 (2003) 240405. [29] Fallani L., De Sarlo L., Lye J., Modugno M., Saers R., Fort C. and Inguscio M., Phys. Rev. Lett, 93 (2004) 140406. [30] Cataliotti F., Fallani L., Ferlaino F., Fort C., Maddaloni P. and Inguscio M., New J. Phys., 5 (2003) 71. [31] Roati G., Riboli F., Modugno G. and Inguscio M., Phys. Rev. Lett., 89 (2002) 150403. [32] Burger S., Cataliotti F. S., Fort C., Minardi F., Inguscio M., Chiofalo M. L. and Tosi M. P.,, Phys. Rev. Lett., 86 (2003) 4447. [33] Modugno G., Roati G., Riboli F., Ferlaino F., Brecha R. and Inguscio M., Science, 297 (2003) 2240. [34] Ferlaino F., de Mirandes E., Roati G., Modugno G. and Inguscio M., Phys. Rev. Lett, 92 (2004) 140405. [35] Ospelkaus C., Ospelkaus S., Sengstock K. and Bongs K., Phys. Rev. Lett, 96 (2006) 020401. [36] Ferlaino F., D’Errico C., Roati G., Zaccanti M., Inguscio M. and Modugno G., Phys. Rev. A, 73 (2006) 040702(R). [37] Esaki L. and Chang L. L., Phys. Rev. Lett., 33 (1974) 495. [38] Rauch C., Strasser G., Unterrainer K., Boxleitner W., Gornik E. and Wacker A., Phys. Rev. Lett., 81 (1998) 3495. [39] Ferlaino F., Brecha R., Hannaford P., Riboli F., Roati G., Modugno G. and Inguscio M., J. Opt. B: Quantum Semiclass. Opt., 5 (2003) S3. [40] Orso G., Pitaevskii L. P. and Stringari S., Phys. Rev. Lett., 93 (2004) 020404. [41] Pitaevskii L. P., Stringari S. and Orso G., Phys. Rev. A, 71 (2005) 053602.

International School of Physics “Enrico Fermi” Villa Monastero, Varenna Course CLXIV 20–30 June 2006 “Ultra-cold Fermi Gases” Directors

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Francesca FERLAINO LENS Via Nello Carrara 1 50019 Sesto Fiorentino FI Italy Tel.: +39 055 4572460 Fax: +39 055 4572451 [email protected]

Wolfgang KETTERLE MIT and Department of Physics Massachusetts Institute of Technology 77 Massachusetts Ave. Cambridge, MA 02139-4307 USA Tel.: +1 617 2536815 Fax: +1 617 2534876 [email protected]

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c Societ` a Italiana di Fisica

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Eugene DEMLER Lyman Laboratory Department of Physics Harvard University Harvard, MA 02138 USA Tel.: +1 617 4961045 [email protected] Antoine GEORGES Centre de Physique Th´eorique (CPHT) Ecole Polytechnique 91128 Palaiseau France Tel.: +33 1 69334092 [email protected] Rudi GRIMM Institute of Quantum Optics and Quantum Information (IQOQI) 6020 Innsbruck Austria Tel.: +43 512 5076300 [email protected] Murray HOLLAND JILA National Institute of Standards and Technology and Department of Physics University of Colorado Boulder, CO 80309-0440 USA Tel.: +1 303 4924172 [email protected] Deborah JIN JILA University of Colorado Boulder, CO 80309-0440 USA Tel.: +1 303 4920256 [email protected]

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Martin TEICHMANN Laboratoire Kastler Brossel 24 rue Lhomond 75231 Paris France Tel.: +33 1 44322019 Fax: +33 1 44323434 [email protected]

Gergely SZIRMAI Departement of Physics of Complex Systems Hungarian Academy of Sciences E¨ otv¨ os University P´ azm´any P´eter s´et´any 1/A 1118 Budapest Hungary Tel.: +36 1 209 0555 6519 Fax: +36 1 3722866 [email protected]

Kristan TEMME Institut f¨ ur Theoretische Physik Universit¨ at Heidelberg Philosophenweg 16 69120 Heidelberg Germany Tel.: +49 6221 549406 [email protected]

Matthias TAGLIEBER Max-Planck Institute for Quantum Optics Hans-Kopfermann-Str. 1 85748 Garching Germany Tel.: +49 89 21803704 Fax: +49 89 285192 [email protected]

Leticia TARRUELL Laboratoire Kastler Brossel 24 rue Lhomond 75231 Paris France Tel.: +33 1 44323303 Fax : +33 1 44323434 [email protected]

Tobias TIECKE Van der Waals-Zeeman Institute University of Amsterdam Valckenierstraat 65 1018 XE Amsterdam The Netherlands Tel.: +31 20 5256365 Fax: +31 20 5255788 [email protected]

Gael VAROQUAUX Institut d’Optique Groupe d’Optique Atomique Centre Universitaire d’Orsay Bat. 503 9140 Orsay France Tel.: +33 1 69358830 Fax: +33 1 69358700 [email protected]

897

Elenco dei partecipanti Arne-Christian VOIGT Max-Planck Institute for Quantum Optics Ludwig-Maximilians-University M¨ unich Schellingstr. 4/III ¨nich 80799 Mu Germany Tel.: +49 89 21803704 Fax: +49 89 285192 [email protected]

Eric WILLE Institut f¨ ur Quantenoptik und Quanteninformation ¨ Osterreichische Akademie der Wissenschaften ICT-Geb¨aude, Technikerstraße 21a 6020 Innsbruck Austria Tel.: +43 512 5074762 Fax: +43 512 5079816 [email protected]

Thomas VOLZ Max-Planck Institute for Quantum Optics Hans-Kopfermann-Str. 1 85748 Garching Germany Tel.: +49 89 32905241 Fax +49 89 32905395 [email protected]

Patrick WINDPASSINGER Niels Bohr Institut University of Copenhagen Danish Quantum Optics Center (QUANTOP) Blegdamsvej 17 2100 Copenhagen Denmark Tel.: +45 35 325395 [email protected]

Cheng-Ching WANG Department of Physics The University of Texas at Austin 1 University Station C1600 Austin, TX 78712-0264 USA Tel.: +1 512 4716164 [email protected]

F´elix WERNER Laboratoire Kastler Brossel 24 rue Lhomond 75231 Paris France Tel.: +33 1 44323424 Fax: +33 1 44323434 [email protected]

Matthew WRIGHT Institut f¨ ur Experimentalphysik Universit¨ at Innsbruck Technikerstrasse 25 6030 Innsbruck Austria Tel.: +43 512 5076379 Fax: +43 512 5072951 [email protected] Matteo ZACCANTI LENS e Universit` a di Firenze Via Nello Carrara 1 50019 Sesto Fiorentino FI Italy Tel.: +39 055 4572460 Fax: +39 055 4572451 [email protected]

898

Elenco dei partecipanti

Alessandro ZENESINI Dipartimento di Fisica Universit` a di Pisa L.go B. Pontecorvo 3 56127 Pisa Italy Tel.: +39 050 2214571 Fax: +39 050 2214333 [email protected]

Observers

Oleksandr ZOZULYA ITFA University of Amsterdam J/K 2.47 Valckenierstraat 65 1018 XE Amsterdam The Netherlands Tel.: +31 20 5255736 Fax: +31 20 5255778 [email protected]

Johannes HECKER DENSCHLAG Institut f¨ ur Experimentalphysik Universit¨ at Innsbruck Technikerstrasse 25 6020 Innsbruck Austria Tel.: +43 512 5076340 Fax: +43 512 5072921 [email protected]

´ Andras CSORDAS Research Group for Statistical Physics Hungarian Academy of Sciences P´ azm´any P´eter s´et´any 1/A 1117 Budapest Hungary Tel.: +36-1-209-0555, Extension: 6544 Fax: +36-1-372-2866 [email protected]

Jiannis K. PACHOS DAMTP, CMS Cambridge University Wilberforce Rd. Cambridge CB3 0WA United Kingdom Tel.: +44 122 3336188 [email protected]

PROCEEDINGS OF THE INTERNATIONAL SCHOOL OF PHYSICS “ENRICO FERMI”

Course I (1953) Questioni relative alla rivelazione delle particelle elementari, con particolare riguardo alla radiazione cosmica edited by G. Puppi Course II (1954) Questioni relative alla rivelazione delle particelle elementari, e alle loro interazioni con particolare riguardo alle particelle artiﬁcialmente prodotte ed accelerate edited by G. Puppi Course III (1955) Questioni di struttura nucleare e dei processi nucleari alle basse energie edited by C. Salvetti Course IV (1956) Propriet` a magnetiche della materia edited by L. Giulotto Course V (1957) Fisica dello stato solido edited by F. Fumi Course VI (1958) Fisica del plasma e relative applicazioni astroﬁsiche edited by G. Righini Course VII (1958) Teoria della informazione edited by E. R. Caianiello

Course XIII (1959) Physics of Plasma: Experiments and Techniques ´n edited by H. Alfve Course XIV (1960) Ergodic Theories edited by P. Caldirola Course XV (1960) Nuclear Spectroscopy edited by G. Racah Course XVI (1960) Physicomathematical Aspects of Biology edited by N. Rashevsky Course XVII (1960) Topics of Radiofrequency Spectroscopy edited by A. Gozzini Course XVIII (1960) Physics of Solids (Radiation Damage in Solids) edited by D. S. Billington Course XIX (1961) Cosmic Rays, Solar Particles and Space Research edited by B. Peters Course XX (1961) Evidence for Gravitational Theories edited by C. Møller

Course VIII (1958) Problemi matematici della teoria quantistica delle particelle e dei campi edited by A. Borsellino

Course XXI (1961) Liquid Helium edited by G. Careri

Course IX (1958) Fisica dei pioni edited by B. Touschek

Course XXII (1961) Semiconductors edited by R. A. Smith

Course X (1959) Thermodynamics of Irreversible Processes edited by S. R. de Groot

Course XXIII (1961) Nuclear Physics edited by V. F. Weisskopf

Course XI (1959) Weak Interactions edited by L. A. Radicati

Course XXIV (1962) Space Exploration and the Solar System edited by B. Rossi

Course XII (1959) Solar Radioastronomy edited by G. Righini

Course XXV (1962) Advanced Plasma Theory edited by M. N. Rosenbluth

Course XXVI (1962) Selected Topics on Elementary Particle Physics edited by M. Conversi Course XXVII (1962) Dispersion and Absorption of Sound by Molecular Processes edited by D. Sette Course XXVIII (1962) Star Evolution edited by L. Gratton Course XXIX (1963) Dispersion Relations and their Connection with Casuality edited by E. P. Wigner Course XXX (1963) Radiation Dosimetry edited by F. W. Spiers and G. W. Reed Course XXXI (1963) Quantum Electronics and Coherent Light edited by C. H. Townes and P. A. Miles Course XXXII (1964) Weak Interactions and High-Energy Neutrino Physics edited by T. D. Lee Course XXXIII (1964) Strong Interactions edited by L. W. Alvarez Course XXXIV (1965) The Optical Properties of Solids edited by J. Tauc Course XXXV (1965) High-Energy Astrophysics edited by L. Gratton

Course XLI (1967) Selected Topics in Particle Physics edited by J. Steinberger Course XLII (1967) Quantum Optics edited by R. J. Glauber Course XLIII (1968) Processing of Optical Data by Organisms and by Machines edited by W. Reichardt Course XLIV (1968) Molecular Beams and Reaction Kinetics edited by Ch. Schlier Course XLV (1968) Local Quantum Theory edited by R. Jost Course XLVI (1969) Physics with Intersecting Storage Rings edited by B. Touschek Course XLVII (1969) General Relativity and Cosmology edited by R. K. Sachs Course XLVIII (1969) Physics of High Energy Density edited by P. Caldirola and H. Knoepfel Course IL (1970) Foundations of Quantum Mechanics edited by B. d’Espagnat Course L (1970) Mantle and Core in Planetary Physics edited by J. Coulomb and M. Caputo Course LI (1970) Critical Phenomena edited by M. S. Green

Course XXXVI (1965) Many-body Description of Nuclear Structure and Reactions edited by C. L. Bloch

Course LII (1971) Atomic Structure and Properties of Solids edited by E. Burstein

Course XXXVII (1966) Theory of Magnetism in Transition Metals edited by W. Marshall

Course LIII (1971) Developments and Borderlines of Nuclear Physics edited by H. Morinaga

Course XXXVIII (1966) Interaction of High-Energy Particles with Nuclei edited by T. E. O. Ericson

Course LIV (1971) Developments in High-Energy Physics edited by R. R. Gatto

Course XXXIX (1966) Plasma Astrophysics edited by P. A. Sturrock

Course LV (1972) Lattice Dynamics and Forces edited by S. Califano

Course XL (1967) Nuclear Structure and Nuclear Reactions edited by M. Jean and R. A. Ricci

Course LVI (1972) Experimental Gravitation edited by B. Bertotti

Intermolecular

Course LVII (1972) History of 20th Century Physics edited by C. Weiner

Course LXXII (1977) Problems in the Foundations of Physics edited by G. Toraldo di Francia

Course LVIII (1973) Dynamics Aspects of Surface Physics edited by F. O. Goodman

Course LXXIII (1978) Early Solar System Processes and the Present Solar System edited by D. Lal

Course LIX (1973) Local Properties at Phase Transitions ¨ller and A. Rigamonti edited by K. A. Mu Course LX (1973) C*-Algebras and their Applications to Statistical Mechanics and Quantum Field Theory edited by D. Kastler

Course LXXIV (1978) Development of High-Power Lasers and their Applications edited by C. Pellegrini Course LXXV (1978) Intermolecular Spectroscopy and Dynamical Properties of Dense Systems edited by J. Van Kranendonk

Course LXI (1974) Atomic Structure and Mechanical Properties of Metals edited by G. Caglioti

Course LXXVI (1979) Medical Physics edited by J. R. Greening

Course LXII (1974) Nuclear Spectroscopy and Nuclear Reactions with Heavy Ions edited by H. Faraggi and R. A. Ricci

Course LXXVII (1979) Nuclear Structure and Heavy-Ion Collisions edited by R. A. Broglia, R. A. Ricci and C. H. Dasso

Course LXIII (1974) New Directions in Physical Acoustics edited by D. Sette

Course LXXVIII (1979) Physics of the Earth’s Interior edited by A. M. Dziewonski and E. Boschi

Course LXIV (1975) Nonlinear Spectroscopy edited by N. Bloembergen

Course LXXIX (1980) From Nuclei to Particles edited by A. Molinari

Course LXV (1975) Physics and Astrophysics of Neutron Stars and Black Hole edited by R. Giacconi and R. Ruffini

Course LXXX (1980) Topics in Ocean Physics edited by A. R. Osborne and P. Malanotte Rizzoli

Course LXVI (1975) Health and Medical Physics edited by J. Baarli

Course LXXXI (1980) Theory of Fundamental Interactions edited by G. Costa and R. R. Gatto

Course LXVII (1976) Isolated Gravitating Systems in General Relativity edited by J. Ehlers

Course LXXXII (1981) Mechanical and Thermal Behaviour of Metallic Materials edited by G. Caglioti and A. Ferro Milone

Course LXVIII (1976) Metrology and Fundamental Constants edited by A. Ferro Milone, P. Giacomo and S. Leschiutta

Course LXXXIII (1981) Positrons in Solids edited by W. Brandt and A. Dupasquier

Course LXIX (1976) Elementary Modes of Excitation in Nuclei edited by A. Bohr and R. A. Broglia

Course LXXXIV (1981) Data Acquisition in High-Energy Physics edited by G. Bologna and M. Vincelli

Course LXX (1977) Physics of Magnetic Garnets edited by A. Paoletti

Course LXXXV (1982) Earthquakes: Observation, Theory and Interpretation edited by H. Kanamori and E. Boschi

Course LXXI (1977) Weak Interactions edited by M. Baldo Ceolin

Course LXXXVI (1982) Gamow Cosmology edited by F. Melchiorri and R. Ruffini

Course LXXXVII (1982) Nuclear Structure and Heavy-Ion Dynamics edited by L. Moretto and R. A. Ricci

Course CII (1986) Accelerated Life Testing and Experts’ Opinions in Reliability edited by C. A. Clarotti and D. V. Lindley

Course LXXXVIII (1983) Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics edited by M. Ghil, R. Benzi and G. Parisi

Course CIII (1987) Trends in Nuclear Physics edited by P. Kienle, R. A. Ricci and A. Rubbino

Course LXXXIX (1983) Highlights of Condensed Matter Theory edited by F. Bassani, F. Fumi and M. P. Tosi

Course CIV (1987) Frontiers and Borderlines in ManyParticle Physics edited by R. A. Broglia and J. R. Schrieffer

Course XC (1983) Physics of Amphiphiles: Micelles, Vesicles and Microemulsions edited by V. Degiorgio and M. Corti Course XCI (1984) From Nuclei to Stars edited by A. Molinari and R. A. Ricci Course XCII (1984) Elementary Particles edited by N. Cabibbo Course XCIII (1984) Frontiers in Physical Acoustics edited by D. Sette Course XCIV (1984) Theory of Reliability edited by A. Serra and R. E. Barlow Course XCV (1985) Solar-Terrestrial Relationship and the Earth Environment in the Last Millennia edited by G. Cini Castagnoli

Course CV (1987) Confrontation between Theories and Observations in Cosmology: Present Status and Future Programmes edited by J. Audouze and F. Melchiorri Course CVI (1988) Current Trends in the Physics of Materials edited by G. F. Chiarotti, F. Fumi and M. Tosi Course CVII (1988) The Chemical Physics of Atomic and Molecular Clusters edited by G. Scoles Course CVIII (1988) Photoemission and Absorption Spectroscopy of Solids and Interfaces with Synchrotron Radiation edited by M. Campagna and R. Rosei

Course XCVI (1985) Excited-State Spectroscopy in Solids edited by U. M. Grassano and N. Terzi

Course CIX (1988) Nonlinear Topics in Ocean Physics edited by A. R. Osborne

Course XCVII (1985) Molecular-Dynamics Simulations of Statistical-Mechanical Systems edited by G. Ciccotti and W. G. Hoover

Course CX (1989) Metrology at the Frontiers of Physics and Technology edited by L. Crovini and T. J. Quinn

Course XCVIII (1985) The Evolution of Small Bodies in the Solar System ˇ Kresa `k edited by M. Fulchignoni and L.

Course CXI (1989) Solid-State Astrophysics edited by E. Bussoletti and G. Strazzulla

Course XCIX (1986) Synergetics and Dynamic Instabilities edited by G. Caglioti and H. Haken

Course CXII (1989) Nuclear Collisions from the Mean-Field into the Fragmentation Regime edited by C. Detraz and P. Kienle

Course C (1986) The Physics of NMR Spectroscopy in Biology and Medicine edited by B. Maraviglia

Course CXIII (1989) High-Pressure Equation of State: Theory and Applications edited by S. Eliezer and R. A. Ricci

Course CI (1986) Evolution of Interstellar Dust and Related Topics edited by A. Bonetti and J. M. Greenberg

Course CXIV (1990) Industrial and Technological Applications of Neutrons edited by M. Fontana and F. Rustichelli

Course CXV (1990) The Use of EOS for Studies of Atmospheric Physics edited by J. C. Gille and G. Visconti

Course CXXIX1 (1994) Observation, Prediction and Simulation of Phase Transitions in Complex Fluids edited by M. Baus, L. F. Rull and J. P. Ryckaert

Course CXVI (1990) Status and Perspectives of Nuclear Energy: Fission and Fusion edited by R. A. Ricci, C. Salvetti and E. Sindoni

Course CXXX (1995) Selected Topics in Nonperturbative QCD edited by A. Di Giacomo and D. Diakonov

Course CXVII (1991) Semiconductor Superlattices and Interfaces edited by A. Stella Course CXVIII (1991) Laser Manipulation of Atoms and Ions edited by E. Arimondo, W. D. Phillips and F. Strumia

Course CXXXI (1995) Coherent and Collective Interactions of Particles and Radiation Beams edited by A. Aspect, W. Barletta and R. Bonifacio Course CXXXII (1995) Dark Matter in the Universe edited by S. Bonometto and J. Primack

Course CXIX (1991) Quantum Chaos edited by G. Casati, I. Guarneri and U. Smilansky

Course CXXXIII (1996) Past and Present Variability of the SolarTerrestrial System: Measurement, Data Analysis and Theoretical Models edited by G. Cini Castagnoli and A. Provenzale

Course CXX (1992) Frontiers in Laser Spectroscopy ¨nsch and M. Inguscio edited by T. W. Ha

Course CXXXIV (1996) The Physics of Complex Systems edited by F. Mallamace and H. E. Stanley

Course CXXI (1992) Perspectives in Many-Particle Physics edited by R. A. Broglia, J. R. Schrieffer and P. F. Bortignon

Course CXXXV (1996) The Physics of Diamond edited by A. Paoletti and A. Tucciarone

Course CXXII (1992) Galaxy Formation edited by J. Silk and N. Vittorio

Course CXXXVI (1997) Models and Phenomenology for Conventional and High-Temperature Superconductivity edited by G. Iadonisi, J. R. Schrieffer and M. L. Chiofalo

Course CXXIII (1992) Nuclear Magnetic Double Resonsonance edited by B. Maraviglia Course CXXIV (1993) Diagnostic Tools in Atmospheric Physics edited by G. Fiocco and G. Visconti Course CXXV (1993) Positron Spectroscopy of Solids edited by A. Dupasquier and A. P. Mills jr. Course CXXVI (1993) Nonlinear Optical Materials: Principles and Applications edited by V. Degiorgio and C. Flytzanis Course CXXVII (1994) Quantum Groups and their Applications in Physics edited by L. Castellani and J. Wess Course CXXVIII (1994) Biomedical Applications of Synchrotron Radiation edited by E. Burattini and A. Balerna 1 This

Course CXXXVII (1997) Heavy Flavour Physics: a Probe of Nature’s Grand Design edited by I. Bigi and L. Moroni Course CXXXVIII (1997) Unfolding the Matter of Nuclei edited by A. Molinari and R. A. Ricci Course CXXXIX (1998) Magnetic Resonance and Brain Function: Approaches from Physics edited by B. Maraviglia Course CXL (1998) Bose-Einstein Condensation in Atomic Gases edited by M. Inguscio, S. Stringari and C. E. Wieman Course CXLI (1998) Silicon-Based Microphotonics: from Basics to Applications edited by O. Bisi, S. U. Campisano, L. Pavesi and F. Priolo

course belongs to the NATO ASI Series C, Vol. 460 (Kluwer Academic Publishers).

Course CXLII (1999) Plasmas in the Universe edited by B. Coppi, A. Ferrari and E. Sindoni

Course CLIV (2003) Physics Methods in Archaeometry edited by M. Martini, M. Milazzo and M. Piacentini

Course CXLIII (1999) New Directions in Quantum Chaos edited by G. Casati, I. Guarneri and U. Smilansky

Course CLV (2003) The Physics of Complex Systems (New Advances and Perspectives) edited by F. Mallamace and H. E. Stanley

Course CXLIV (2000) Nanometer Scale Science and Technology edited by M. Allegrini, N. Garc´ıa and O. Marti

Course CLVI (2003) Research on Physics Education edited by E.F. Redish and M. Vicentini

Course CXLV (2000) Protein Folding, Evolution and Design edited by R. A. Broglia, E. I. Shakhnovich and G. Tiana Course CXLVI (2000) Recent Advances in Metrology and Fundamental Constants edited by T. J. Quinn, S. Leschiutta and P. Tavella Course CXLVII (2001) High Pressure Phenomena edited by R. J. Hemley, G. L. Chiarotti, M. Bernasconi and L. Ulivi Course CXLVIII (2001) Experimental Quantum Computation and Information edited by F. De Martini and C. Monroe Course CXLIX (2001) Organic Nanostructures: Science and Applications edited by V. M. Agranovich and G. C. La Rocca Course CL (2002) Electron and Photon Conﬁnement in Semiconductor Nanostructures ´dran, A. Quatedited by B. Deveaud-Ple tropani and P. Schwendimann Course CLI (2002) Quantum Phenomena in Mesoscopic Systems edited by B. Altshuler, A. Tagliacozzo and V. Tognetti Course CLII (2002) Neutrino Physics edited by E. Bellotti, Y. Declais and P. Strolin Course CLIII (2002) From Nuclei and their Constituents to Stars edited by A. Molinari, L. Riccati, W. M. Alberico and M. Morando

Course CLVII (2003) The Electron Liquid Model in Condensed Matter Physics edited by G. F. Giuliani and G. Vignale Course CLVIII (2004) Hadron Physics edited by T. Bressani, U. Wiedner and A. Filippi Course CLIX (2004) Background Microwave Radiation and Intracluster Cosmology edited by F. Melchiorri and Y. Rephaeli Course CLX (2004) From Nanostructures to Nanosensing Applications edited by A. D’Amico, G. Balestrino and A. Paoletti Course CLXI (2005) Polarons in Bulk Materials and Systems with Reduced Dimensionality edited by G. Iadonisi, J. Ranninger and G. De Filippis Course CLXII (2005) Quantum Computers, Algorithms and Chaos edited by G. Casati, D. L. Shepelyansky, P. Zoller and G. Benenti Course CLXIII (2005) CP Violation: From Quarks to Leptons edited by M. Giorgi, I. Mannelli, A. I. Sanda, F. Costantini and M. S. Sozzi Course CLXIV (2006) Ultra-Cold Fermi Gases edited by M. Inguscio, W. Ketterle and C. Salomon Course CLXV (2006) Protein Folding and Drug Design edited by R. A. Broglia, L. Serrano and G. Tiana Course CLXVI (2006) Metrology and Fundamental Constants ¨nsch, S. Leschiutta, A. edited by T. W. Ha J. Wallard and M. L. Rastello