Advances in
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS V O L U M E 54
Editors PAUL R. B ERMAN
University of Michigan Ann Arbor, Michigan C HUN C. L IN
University of Wisconsin Madison, Wisconsin E NNIO A RIMONDO
University of Pisa Pisa, Italy
Editorial Board C. J OACHAIN
Université Libre de Bruxelles Brussels, Belgium M. G AVRILA
F.O.M. Insituut voor Atoom- en Molecuulfysica Amsterdam, The Netherlands M. I NOKUTI
Argonne National Laboratory Argonne, Illinois
ADVANCES IN
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by
P.R. Berman PHYSICS DEPARTMENT UNIVERSITY OF MICHIGAN ANN ARBOR , MI , USA
C.C. Lin DEPARTMENT OF PHYSICS UNIVERSITY OF WISCONSIN MADISON , WI , USA
and
E. Arimondo PHYSICS DEPARTMENT UNIVERSITY OF PISA PISA , ITALY
Volume 54
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK OXFORD • PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE SYDNEY • TOKYO Academic Press is an imprint of Elsevier
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ISBN-13: 978-0-12-003854-1 ISBN-10: 0-12-003854-4 ISSN: 1049-250X
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07 08 09 10 11
10 9 8 7 6 5 4 3 2 1
IN MEMORIAM HERBERT WALTHER (1935–2006) It is with great sadness that the Editors report the passing of Professor Herbert Walther on July 22, 2006. Professor Walther expanded the horizons of our understanding of the interaction of radiation with matter. The physics community will sorely miss his presence. Professor Walther assumed co-editorship of the Advances series starting with Volume 33 and continued in that role through Volume 51, which was a special issue, edited by Henry Stroke, to honor his co-editor Benjamin Bederson. Volume 53, edited by Gerhard Rempe and Marlan Scully, was published as a tribute to Professor Walther.
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Contents C ONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P REFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Experimental Realization of the BCS-BEC Crossover with a Fermi Gas of Atoms C.A. Regal and D.S. Jin 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BCS-BEC Crossover Physics . . . . . . . . . . . . . . . . . . . . . . . Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . Cooling a Fermi Gas and Measuring its Temperature . . . . . . . . . . Elastic Scattering near Feshbach Resonances between Fermionic Atoms Creating Molecules from a Fermi Gas of Atoms . . . . . . . . . . . . Inelastic Collisions near a Fermionic Feshbach Resonance . . . . . . . Creating Condensates from a Fermi Gas of Atoms . . . . . . . . . . . The Momentum Distribution of a Fermi Gas in the Crossover . . . . . Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 10 18 24 36 42 51 56 64 71 72 72
Deterministic Atom–Light Quantum Interface Jacob Sherson, Brian Julsgaard and Eugene S. Polzik 1. 2. 3. 4. 5. 6. 7. 8. A. B. 9.
Introduction . . . . . . . . . . . Atom–Light Interaction . . . . . Quantum Information Protocols Experimental Methods . . . . . Experimental Results . . . . . . Conclusions . . . . . . . . . . . Acknowledgements . . . . . . . Appendices . . . . . . . . . . . Effect of Atomic Motion . . . . Technical Details . . . . . . . . References . . . . . . . . . . . .
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Cold Rydberg Atoms J.-H. Choi, B. Knuffman, T. Cubel Liebisch, A. Reinhard and G. Raithel 1. 2. 3. 4. 5. 6. 7.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preparation and Analysis of Cold Rydberg-Atom Clouds . . . . . . . Collision-Induced Rydberg-Atom Gas Dynamics . . . . . . . . . . . . Towards Coherent Control of Rydberg-Atom Interactions . . . . . . . Rydberg-Atom Trapping . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Realization of Rydberg-Atom Trapping . . . . . . . . . Landau Quantization and State Mixing in Cold, Strongly Magnetized Rydberg Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132 135 149 159 176 186 192 196 197 198
Non-Perturbative Quantal Methods for Electron–Atom Scattering Processes D.C. Griffin and M.S. Pindzola 1. 2. 3. 4. 5. 6. 7. 8.
Introduction . . . . . . . . . . . . . . . . . . . . . . The Configuration-Average Distorted-Wave Method The R-Matrix with Pseudo-States Method . . . . . . The Time-Dependent Close-Coupling Method . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
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R-Matrix Theory of Atomic, Molecular and Optical Processes P.G. Burke, C.J. Noble and V.M. Burke 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Introduction . . . . . . . . . . . . . . . . . . . . . . Electron Atom Scattering at Low Energies . . . . . Electron Scattering at Intermediate Energies . . . . Atomic Photoionization and Photorecombination . . Electron Molecule Scattering . . . . . . . . . . . . . Positron Atom Scattering . . . . . . . . . . . . . . . Atomic and Molecular Multiphoton Processes . . . Electron Energy Loss from Transition Metal Oxides Conclusions . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
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Electron-Impact Excitation of Rare-Gas Atoms from the Ground Level and Metastable Levels John B. Boffard, R.O. Jung, L.W. Anderson and C.C. Lin 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . Background: Excitation of Helium and the Multipole Field Picture Argon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Krypton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison to Theoretical Calculations . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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320 321 325 342 348 372 384 397 406 410 418 418
Internal Rotation in Symmetric Tops I. Ozier and N. Moazzen-Ahmadi 1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . Spectroscopy from 50 kHz to 1000 cm−1 Discussion . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .
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424 436 449 498 505 506
Introduction . . . . . . . . . . . . . . . . . . . . . . . Tunnel Ionization and Electron Re-collision . . . . . . Producing and Measuring Attosecond Optical Pulses . Measuring an Attosecond Electron Pulse . . . . . . . Attosecond Imaging . . . . . . . . . . . . . . . . . . . Imaging Electrons and their Dynamics . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Attosecond and Angstrom Science Hiromichi Niikura and P.B. Corkum 1. 2. 3. 4. 5. 6. 7. 8.
Atomic Processing of Optically Carried RF Signals Jean-Louis Le Gouët, Fabien Bretenaker and Ivan Lorgeré
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2. 3. 4. 5. 6. 7. 8. 9. 10.
Radio Frequency Spectral Analyzers . . . . . . . . . . . Spectrum Photography Architecture . . . . . . . . . . . Frequency Selective Materials as Programmable Filters Rainbow Analyzer . . . . . . . . . . . . . . . . . . . . . Photon Echo Chirp Transform Spectrum Analyzer . . . Frequency Agile Laser Technology . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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I NDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C ONTENTS OF VOLUMES IN T HIS S ERIAL . . . . . . . . . . . . . . .
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Controlling Optical Chaos, Spatio-Temporal Dynamics, and Patterns Lucas Illing, Daniel J. Gauthier and Rajarshi Roy 1. 2. 3. 4. 5. 6. 7. 8. 9.
Introduction . . . . . . . . . . . . . Recent Examples . . . . . . . . . . Control . . . . . . . . . . . . . . . . Synchronization . . . . . . . . . . . Communication . . . . . . . . . . . Spatio-Temporal Chaos and Patterns Outlook . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . References . . . . . . . . . . . . . .
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CONTRIBUTORS
Numbers in parentheses indicate the pages on which the author’s contributions begin.
C.A. R EGAL (1), JILA, National Institute of Standards and Technology Quantum Physics Division and University of Colorado, and Department of Physics, University of Colorado, Boulder, CO 80309-0440, USA D.S. J IN (1), JILA, National Institute of Standards and Technology Quantum Physics Division and University of Colorado and Department of Physics, University of Colorado, Boulder, CO 80309-0440, USA JACOB S HERSON (81), QUANTOP, Danish Research Foundation Center for Quantum Optics, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark; Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, bygning 1520, DK-8000 Aarhus C, Denmark B RIAN J ULSGAARD (81), QUANTOP, Danish Research Foundation Center for Quantum Optics, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark E UGENE S. P OLZIK (81), Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, bygning 520, DK-8000 Aarhus C, Denmark J.-H. C HOI (131), FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA B. K NUFFMAN (131), FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA T. C UBEL L IEBISCH (131), FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA A. R EINHARD (131), FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA G. R AITHEL (131), FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA D.C. G RIFFIN (203), Department of Physics, Rollins College, Winter Park, FL 32789, USA M.S. P INDZOLA (203), Department of Physics, Auburn University, Auburn, AL 36849, USA xi
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P.G. B URKE (237), Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 1NN, UK C.J. N OBLE (237), CSE Department, CCLRC Daresbury Laboratory, Warrington WA4 4AD, UK V.M. B URKE (237), CSE Department, CCLRC Daresbury Laboratory, Warrington WA4 4AD, UK J OHN B. B OFFARD (319), Department of Physics, University of Wisconsin, Madison, WI 53706, USA R.O. J UNG (319), Department of Physics, University of Wisconsin, Madison, WI 53706, USA L.W. A NDERSON (319), Department of Physics, University of Wisconsin, Madison, WI 53706, USA C.C. L IN (319), Department of Physics, University of Wisconsin, Madison, WI 53706, USA I. O ZIER (423), Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada V6T 1Z1 N. M OAZZEN -A HMADI (423), Department of Physics and Astronomy, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 H IROMICHI N IIKURA (511), National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario, Canada K1A0R6; PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho, Kawaguchi-city, Saitama, Japan P.B. C ORKUM (511), National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario, Canada K1A0R6 J EAN -L OUIS L E G OUËT (549), Laboratoire Aimé Cotton, CNRS UPR3321, bâtiment 505, campus universitaire, 91405 Orsay Cedex, France FABIEN B RETENAKER (549), Laboratoire Aimé Cotton, CNRS UPR3321, bâtiment 505, campus universitaire, 91405 Orsay Cedex, France I VAN L ORGERÉ (549), Laboratoire Aimé Cotton, CNRS UPR3321, bâtiment 505, campus universitaire, 91405 Orsay Cedex, France L UCAS I LLING (615), Department of Physics and Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708, USA DANIEL J. G AUTHIER (615), Department of Physics and Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708, USA R AJARSHI ROY (615), Department of Physics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
PREFACE Volume 54 of the Advances Series contains ten contributions, covering a diversity of subject areas in atomic, molecular and optical physics. The chapter by Regal and Jin reviews the properties of a Fermi degenerate gas of cold potassium atoms in the crossover regime between the Bose–Einstein condensation of molecules and the condensation of fermionic atom pairs. The transition between the two regions can be probed by varying an external magnetic field. Sherson, Julsgaard, and Polzik explore the manner in which light and atoms can be entangled, with applications to quantum information processing and communication. They report on the result of recent experiments involving the state entanglement of distant ensembles and a method for storing the quantum state of an optical field in an atomic ensemble. Recent developments in cold Rydberg atom physics are reviewed in the chapter by Choi, Knuffman, Cubel Liebisch, Reinhard, and Raithel. Fascinating experiments are described in which cold, highly excited atoms (“Rydberg” atoms) and cold plasmas are generated. Evidence for a collective excitation of Rydberg matter is also presented. Griffin and Pindzola offer an account of non-perturbative quantal methods for electron–atom scattering processes. Included in the discussion are the R-matrix with pseudo-states method and the time-dependent close-coupling method. An extensive review of the R-matrix theory of atomic, molecular, and optical processes is given by Burke, Noble, and Burke. They present a systematic development of the R-matrix method and its applications to various processes such as electron–atom scattering, atomic photoionization, electron–molecule scattering, positron–atom scattering, and atomic/molecular multiphoton processes. Electron impact excitation of raregas atoms from both their ground and metastable states is discussed in the chapter by Boffard, Jung, Anderson, and Lin. Excitation cross sections measured by the optical method are reviewed with emphasis on the physical interpretation in terms of electronic structure of the target atoms. Ozier and Moazzen-Ahmadi explore internal rotation of symmetric top molecules. Developments of new experimental methods based on high-resolution torsional, vibrational, and molecular beam spectroscopy allow accurate determination of internal barriers for these symmetric molecules. The subject of attosecond and angstrom science is reviewed by Niikura and Corkum. The underlying physical mechanisms allowing one to generate attosecond radiation pulses are described and the technology needed for the preparation of such pulses is discussed. Le Gouët, Bretenaker, and Lorgeré describe how rare earth ions embedded in crystals can be used for processing optically carried broadband radio-frequency signals. Methods for reaching tens of gigahertz xiii
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instantaneous bandwidth with submegahertz resolution using such devices are analyzed in detail and demonstrated experimentally. Finally, in the chapter by Illing, Gauthier, and Roy, it is shown that small perturbations applied to optical systems can be used to suppress or control optical chaos, spatio-temporal dynamics, and patterns. Applications of these techniques to communications, laser stabilization, and improving the sensitivity of low-light optical switches are explored. The Editors would like to thank all the contributing authors for their contributions and for their cooperation in assembling this volume. They would also like to express their appreciation to Dr. Anita Koch at Elsevier for her invaluable assistance. Ennio Arimondo
Paul Berman
Chun Lin
ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54
EXPERIMENTAL REALIZATION OF THE BCS-BEC CROSSOVER WITH A FERMI GAS OF ATOMS C.A. REGAL and D.S. JIN JILA, National Institute of Standards and Technology Quantum Physics Division and University of Colorado, and Department of Physics, University of Colorado, Boulder, CO 80309-0440, USA 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. BCS-BEC Crossover Physics . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Pairing in a Fermi Gas of Atoms . . . . . . . . . . . . . . . . . . . . 2.2. Varying Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Simple Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Beyond T = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Modern Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. A Specific Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Cooling a Fermi Gas and Measuring its Temperature . . . . . . . . . . . 4.1. Cooling 40 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Measuring the Temperature of a Fermi Gas . . . . . . . . . . . . . . 5. Elastic Scattering near Feshbach Resonances between Fermionic Atoms 5.1. Measuring the Elastic Collision Cross Section . . . . . . . . . . . . 5.2. Anisotropic Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Measuring the Mean-Field Interaction Energy . . . . . . . . . . . . 5.4. 40 K Feshbach Resonance Summary . . . . . . . . . . . . . . . . . . 6. Creating Molecules from a Fermi Gas of Atoms . . . . . . . . . . . . . . 6.1. Magnetic-Field Association . . . . . . . . . . . . . . . . . . . . . . 6.2. Rf Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Understanding Molecule Conversion Efficiency . . . . . . . . . . . 6.4. A Precise Measurement of B0 . . . . . . . . . . . . . . . . . . . . . 7. Inelastic Collisions near a Fermionic Feshbach Resonance . . . . . . . . 7.1. Expected Inelastic Decay Processes . . . . . . . . . . . . . . . . . . 7.2. Lifetime of Feshbach Molecules . . . . . . . . . . . . . . . . . . . . 7.3. Three-Body Recombination . . . . . . . . . . . . . . . . . . . . . . 7.4. Comparison of 40 K and 6 Li . . . . . . . . . . . . . . . . . . . . . . 8. Creating Condensates from a Fermi Gas of Atoms . . . . . . . . . . . . . 8.1. Emergence of a Molecular Condensate from a Fermi Gas of Atoms 8.2. Observing Condensates in the Crossover . . . . . . . . . . . . . . . 1
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8.3. Measurement of a Phase Diagram . . . . . . . . . . . . . 9. The Momentum Distribution of a Fermi Gas in the Crossover 9.1. Measuring the Momentum Distribution . . . . . . . . . . 9.2. Extracting the Kinetic Energy . . . . . . . . . . . . . . . 9.3. Comparing the Kinetic Energy to Theory . . . . . . . . . 9.4. Temperature Dependence . . . . . . . . . . . . . . . . . . 10. Conclusions and Future Directions . . . . . . . . . . . . . . . 11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 12. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract Ultracold atomic gases have proven to be remarkable model systems for exploring quantum mechanical phenomena. Experimental work on gases of fermionic atoms in particular has seen astounding recent progress. In the short span of time between 2001 and 2004 accessible Fermi gas systems evolved from normal Fermi liquids at moderate temperatures to superfluids in the BCS-BEC crossover. This was made possible by unique control over interparticle interactions using Feshbach resonances in 6 Li and 40 K gases. In this chapter we present the story of the experimental realization of BCS-BEC crossover physics from the point of view of studies using 40 K at JILA. We start with some historical context and an introduction to the theory of the BCS-BEC crossover and Feshbach resonances. We then present studies of a normal 40 K Fermi gas at a Feshbach resonance and the work required to cool the gas to temperatures where superfluidity in the crossover is predicted. These studies culminated in the first observation of a phase transition in the BCS-BEC crossover regime, a task accomplished through detection of condensation of fermionic atom pairs. We also discuss subsequent work that confirmed the crossover nature of the pairs in these condensates.
1. Introduction 1.1. H ISTORICAL P ERSPECTIVE 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 (Onnes, 1911). 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 (Allen and Misener, 1938; Kapitza, 1938). Much later, 3 He, the fermionic helium isotope, was also found to be superfluid at yet a much colder temperature than 4 He
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F IG . 1. 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. (Holland et al., 2001).)
(Osheroff et al., 1972). Relatively recently in 1986, high-temperature superconductors in Copper-oxide materials further enlarged the list of superconducting materials (Bednorz and Mueller, 1986). These “super” systems, which we will refer to in general as superfluids, are listed in Fig. 1, but they are only classic examples. There are many other physical systems that display superfluid properties from astrophysical phenomena such as neutron stars, to excitons in semiconductors, to atomic nuclei (Snoke and Baym, 1995). Although the physical properties of these systems vary widely, they are all linked by their 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 super-systems are superconductors, and if a robust room-temperature superconductor were created it would be an amazing discovery. Some of the first attempts to understand the phenomenon of superfluidity were in the context of Bose–Einstein condensation (BEC) of an ensemble of bosonic particles (Randeria, 1995). BEC is a consequence of the quantum statistics of bosons, which are particles with integer spin, and it results in a macroscopic occupation of a single quantum state (Fig. 2) (Bose, 1924; Einstein, 1925). Fritz London proposed in 1938 that superfluid 4 He was a consequence of Bose–Einstein condensation of bosonic 4 He (London, 1938). (4 He behaves as a boson because it is made up of an even number of 1/2 integer spin fermions—electrons, protons, and neutrons.) Physicists such as Blatt et al. pushed a similar idea in the context of superconductors in proposing that “at low temper-
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F IG . 2. Quantum statistics: Bosons versus fermions with weak interactions at T = 0. Bosons form a BEC in which all of the bosons macroscopically occupy a single quantum state. Due to the Pauli exclusion principle, fermions form a Fermi sea in which each energy state up to the Fermi energy is occupied.
ature, charge carrying bosons occur, e.g., because of the interaction of electrons with lattice vibrations” (Blatt and Butler, 1954). For the case of tightly bound bosons, such as 4 He, London’s hypothesis turned out to be correct. However, the very strong interactions in 4 He made it difficult to verify the existence of condensation (Sokol, 1995), and for many years 4 He studies did not mention BEC. In the case of superconductors, discussion of BEC was overshadowed by the amazing success in 1957 of the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity (Bardeen et al., 1957a, 1957b). In 1956 Cooper found that a pair of fermions in the presence of a filled Fermi sea (Fig. 2) will form a bound pair with an arbitrarily small attractive interaction (Cooper, 1956). The BCS theory solved this problem in the case where many pairs can form in the Fermi sea. 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. Figure 1 sorts the classic superfluid systems according to the strength of the interaction between the fermions. 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 right side of Fig. 1. The theory perfectly described conventional superconductors for which the attraction between fermions is ∼10,000 times less than the Fermi energy, EF . The BCS ground state was also able to accurately describe superfluid 3 He and many (although arguably not all (Chen et al., 2005b)) aspects of high-Tc superconductors. The theory in its original form, however, did not at all apply to the case of the tightly bound boson, 4 He. In 1995 a completely new system joined 4 He on the left side of Fig. 1. Here the composite bosons were alkali atoms, such as 87 Rb, that had been cooled as a gas down to nanoKelvin temperatures via laser cooling and evaporative cooling (Phillips, 1998; Ketterle et al., 1999; Cornell et al., 1999). At
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F IG . 3. Bose–Einstein condensation in a dilute gas of 87 Rb atoms. (a) Phase space density criterion for condensation. (b) Momentum 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. (Anderson et al., 1995).)
2π h¯ 2 these temperatures the thermal deBroglie wavelength λdeBroglie = mkb T of the particles becomes on the order of the interparticle spacing in the gas and a Bose–Einstein condensate is formed (Fig. 3(a)) (Anderson et al., 1995; Davis et al., 1995). In contrast to 4 He the alkali BEC that was created was weakly interacting, making the condensation stunningly visible, as shown in Fig. 3(b). Experiments also verified the superfluid nature of these gases (Matthews et al., 1999; Raman et al., 1999). In this way both the BEC and superfluid properties could be clearly seen in one system and understood extremely well theoretically (see, e.g., Ref. (Dalfovo et al., 1999)). However, the fact that the dilute gas BEC was found to be a superfluid was not at all a surprise. Although a long and complicated history was required, physicists now understand the basic connection between BEC and superfluidity (see, e.g., Ref. (Tilley, 1986)). It is commonly accepted that superfluidity is always intimately connected to the macroscopic occupation of a quantum state. Besides providing the first clear evidence for BEC, ultracold alkali gases opened a world of possibilities for studying superfluid systems. Many of the initial experiments with alkali BEC could be perfectly described by existing theories. However, recent work in the field of BEC has developed techniques to reach a regime that is more relevant for the outstanding theoretical questions in condensed matter physics, which are most commonly in strongly correlated systems. For example, experiments achieved BEC with much stronger interatomic interactions than typical alkali gases; furthermore, these interactions could even be controllably tuned (Inouye et al., 1998; Roberts et al., 1998). A phase transition to the highly-correlated Mott insulator state was observed through studies of quantum gases in optical lattice potentials (Greiner et al., 2002). These bosonic systems require theory that goes beyond mean-field interactions; yet they have a controllability rarely found in solid state materials.
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At the same time as the creation of the first strongly interacting Bose gases, the techniques used to create an alkali BEC were applied to the other class of quantum particles, fermions. 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. Due to the quantum statistics of fermions, the s-wave collisions required for evaporative cooling are not present at ultracold temperatures in a gas of spin-polarized, identical fermions. The solution to this problem was to introduce a second particle for the evaporative cooling, either another state of the fermionic atom or another species entirely. The first gas of fermionic atoms to enter the quantum degenerate regime was created at JILA in 1999 using 40 K (DeMarco and Jin, 1999). The observation in these experiments was not a phase transition, as in the Bose gas, but rather the presence of more and 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 (Truscott et al., 2001; Schreck et al., 2001; Roati et al., 2002; Granade et al., 2002; Hadzibabic et al., 2002; Goldwin et al., 2004; Bartenstein et al., 2004b; Köhl et al., 2005; Ospelkaus et al., 2006). The next goal after the creation of a normal Fermi gas of atoms was to form a superfluid in a paired Fermi gas. In conventional superconductors swave pairing occurs between spin up and spin down electrons. The hope was that s-wave pairing could similarly occur with the creation of a two-component atomic gas with an equal Fermi energy for each component. Indeed, such a two-component gas can be realized using an equal mixture of alkali atoms in two different hyperfine spin states. The simplistic view was that a BCS state would appear if the temperature of this two-component gas were cold enough and the interaction between fermions attractive and large enough. However, for typical interactions 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 large compared to typical values (|a| ≈ 2000a0 ), as well as attractive, bringing the BCS transition temperature closer to realistic temperatures (Stoof et al., 1996; Combescot, 1999). It was then recognized that a type of scattering resonance, known as a Feshbach resonance, could allow arbitrary changes in the interaction strength. Theories were developed that explicitly treated the case where the interactions were enhanced by a Feshbach resonance (Timmermans et al., 2001; Holland et al., 2001; Ohashi and Griffin, 2002). In these proposals, however, increasing the interactions beyond the perturbative limit of BCS theory meant that the physical system would not be simply a BCS state, but rather something much more interesting. It was predicted that a
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Feshbach resonance would allow the realization of a system with an excitation gap on the order of the Fermi energy and provide the ability to continually tune within this region (shaded box in Fig. 1) (Holland et al., 2001). This system, if achievable, would be the experimental realization of a theoretical topic that dated back to the late 1960s, the BCS-BEC crossover. 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 (Eagles, 1969; Leggett, 1980). 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, i.e., the fundamental physics behind all of the systems in Fig. 1 (Eagles, 1969; Leggett, 1980; Nozieres and Schmitt-Rink, 1985; Randeria et al., 1990; Dreschler and Zwerger, 1992; Haussman, 1994). After nearly a century of 4 He and superconductors being considered separate entities, an experimental realization of a superfluid in the BCS-BEC crossover regime would provide a physical link between the two. 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, which are both characteristics expected to be found in a Fermi gas in the crossover (Randeria, 1995; Chen et al., 2005b). Thus, starting in about 2001 a major goal in dilute Fermi gas experiments was to achieve a superfluid Fermi gas at a Feshbach resonance, often referred to as a “resonance superfluid” (Holland et al., 2001). However, achievement of this experimental goal was a number of years and many steps away. The existence of Feshbach resonances had been predicted by atomic physicists both in the 6 Li and in the 40 K systems (Houbiers et al., 1998; Bohn, 2000), and the first step was to locate these resonances (Loftus et al., 2002; O’Hara et al., 2002b; Dieckmann et al., 2002; Regal et al., 2003b). Subsequent experimental studies appeared 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 (O’Hara et al., 2002a; Regal and Jin, 2003; Gupta et al., 2003; Gehm et al., 2003; Bourdel et al., 2003). Then Fermi gases were reversibly converted to gases of diatomic molecules using Feshbach resonances (Regal et al., 2003a; Strecker et al., 2003; Cubizolles et al., 2003; Jochim et al., 2003a). The observation that these molecules were surprisingly long-lived created many opportunities for further study (Strecker et al., 2003; Jochim et al., 2003a; Cubizolles et al., 2003; Regal et al., 2004a). Condensates of diatomic molecules in the BEC limit were achieved (Greiner et al., 2003; Jochim et al., 2003b; Zwierlein et al., 2003; Bourdel et al., 2004; Partridge et al., 2005); then these
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condensates were found to exist in the crossover regime (Regal et al., 2004b; Zwierlein et al., 2004), signalling the existence of a phase transition in the BCSBEC crossover regime. Collective excitations (Bartenstein et al., 2004a; Kinast et al., 2004a, 2004b) and thermodynamic properties (Bartenstein et al., 2004b; Bourdel et al., 2004; Regal et al., 2005; Kinast et al., 2005b) were also measured, and the nature of the pairs was probed in a variety of ways (Chin et al., 2004; Greiner et al., 2005a; Partridge et al., 2005). Most recently a vortex lattice was even created in the crossover (Zwierlein et al., 2005a). Developing these 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 condensed matter physics. Some 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 dilute 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. The power to test these many-body theories comes from the fact that by using ultracold, dilute gases one can create a very clean strongly interacting Fermi system. 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, uniting the basic physics surrounding “super” systems. On the other hand, the complicated materials physics involved in, for example, high-Tc superconductors, 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. 1.2. C ONTENTS In this chapter we will present some of the first experimental work studying fermions in the BCS-BEC crossover in a dilute atomic system. We and our co-workers
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performed these experiments at JILA between 2001 and 2005 using 40 K atoms. As outlined above this was an exciting time in the field of Fermi gases. In 2001 in our lab at JILA the technology existed to create two-component Fermi gases at temperatures around 0.25TF (DeMarco, 2001). Predictions had been made for the existence of a superfluid state near a Feshbach resonance for a Fermi gas on the order of this temperature (Timmermans et al., 2001; Holland et al., 2001; Ohashi and Griffin, 2002). Yet in the early days of this work many physicists were skeptical about the feasibility of experimentally realizing such a state. The Feshbach resonances that had been observed in Bose gases were associated with extremely fast inelastic decay of the trapped gas (Inouye et al., 1998; Stenger et al., 1999; Roberts et al., 2000). 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 (Fedichev et al., 1996; Nielsen and Macek, 1999; Esry et al., 1999; Braaten et al., 2001). Carl Wieman’s group at JILA produced the only experiments studying BEC near a Feshbach resonance over long time scales (Roberts et al., 1998). In this group 85 Rb BECs were studied at very low densities, where two-body elastic collisions dominate over threebody collisions. For two-component Fermi gases three-body decay processes were expected to be suppressed compared to the Bose case (Esry et al., 2001; Petrov, 2003), due to the Pauli exclusion principle. Still there was a fair amount of contention about the degree of this suppression. Further difficulties stemmed from the fact that not all researchers in the atomic physics community were familiar with theories from condensed matter physics such as BCS-BEC crossover theory. Even when these theories were understood, there was significant confusion about the relation between classic crossover theory and the two-body physics of the Feshbach resonances in 6 Li and 40 K. An additional challenge was creating a sufficiently cold Fermi gas near a Feshbach resonance. To be certain of achieving the predicted phase transition, temperatures well below the predicted Tc would have to be achieved. A technical challenge was that the states required to access 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. Much of the work to create the results presented in this chapter involved overcoming these difficulties and sources of confusion, and as we shall see there are now clear answers to many of these questions. These answers were found through careful studies of the normal state of a Fermi gas near the Feshbach resonance and significant work to cool a 40 K gas to the coldest temperatures possible. In late 2003 we observed a phase transition in the crossover and since then have been able to study BCS-BEC crossover physics with a Fermi gas of atoms.
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In the first few sections we discuss theory that is necessary to understand experiments presented in later sections. The goal is not to rigorously derive modern theories, but rather to convey the mindset that many experimentalists currently use to think about the crossover problem with atomic Fermi gases. Section 2 presents the ideas of BCS-BEC crossover physics through simplified theory. Section 3 introduces Feshbach resonances and discusses how well the Feshbach resonance systems reproduce conditions for the classic “universal” BCS-BEC crossover problem. The main body of our chapter presents experiments using an ultracold gas of 40 K atoms. Section 4 describes cooling methods and temperature measurement techniques. Section 5 contains experiments that probe the presence of Feshbach resonances in the 40 K system and study their ability to tune atomic scattering properties. Section 6 introduces the creation of molecules from a Fermi gas of atoms, which is the analog of the BCS-BEC crossover in the normal state. Section 7 concentrates on the stability of fermionic atoms and pairs against inelastic processes in the presence of a Feshbach resonance; this is a subject of practical importance to the ability to study the BCS-BEC crossover with a Fermi gas of atoms. Section 8 describes the first experiments to observe a phase transition in a Fermi gas of atoms in the BCS-BEC crossover regime. Section 9 focuses on a measurement of the momentum distribution of the pairs in the crossover; this measurement is an important probe of the nature of the pairs in the crossover. Some of the experiments we discuss in this chapter were also performed in 6 Li gases in the groups of R. Grimm, R. Hulet, W. Ketterle, C. Salomon, or J. Thomas. In addition the experiments presented in this chapter represent only a fraction of the experimental work in atomic physics on the BCS-BEC crossover. There are numerous experiments using 6 Li gases not mentioned here that have studied fascinating aspects of BCS-BEC crossover physics.
2. BCS-BEC Crossover Physics To understand the experimental work presented in this chapter it is helpful to be familiar with BCS-BEC crossover physics. In this section we present the theory of the BCS-BEC crossover, first from a purely qualitative point of view and then from a slightly more quantitative perspective. In this quantitative perspective it is not our goal to present the most sophisticated theory, but rather a theory that introduces important parameters and illustrates the key differences between the crossover problem and the BCS limit. Lastly we discuss some of the unanswered questions related to the crossover problem; this conveys a sense of the importance of experimental studies of crossover physics.
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F IG . 4. Cartoon illustration of the continuum of pairing in the BCS-BEC crossover.
2.1. PAIRING IN A F ERMI G AS OF ATOMS As we have seen, superfluids are fundamentally associated with the quantum properties of bosons. Since all visible matter is made up of fermions, creating a superfluid most often requires pairing of fermions. The simplest (although historically not the most famous) way to imagine pairing fermions is to create a two-body bound state with two fermions. Two half-integer spin fermions when paired will produce an integer spin particle, which is a composite boson. In the case of experiments discussed in this chapter 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 left side of Fig. 4 represents a superfluid containing these type of pairs. The two shades of gray represent fermions in two different spin states; two states are required if the fermions are to pair via s-wave (l = 0) interactions. In other pairing mechanisms, such as Cooper pairing, the underlying fermionic nature of the system is much more apparent. Cooper considered the problem of two fermions with equal and opposite momentum outside a perfect Fermi sea (Cooper, 1956). The energy of the two fermions turns out to be less than the expected value of 2EF for arbitrarily weak attractive interactions. This result is in surprising contrast to the result of the problem of two fermions in vacuum; in this case there will not be a 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 (de Gennes, 1966). Considering only one pair of electrons as free to pair on top of a static Fermi sea is not a sufficient solution to the pairing problem in a Fermi sea. All fermions should be allowed to participate in the pairing, and we expect that pairs should form until an equilibrium point is reached. At this equilibrium point the remain-
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ing ensemble of fermions is disturbed enough from a Fermi sea configuration to no longer lead to a bound state at the given interaction strength (Tinkham, 1980). The BCS state was an approximate solution to this many-body problem. A description of the full BCS theory is beyond the scope of this current discussion, but is presented in the original papers (Bardeen et al., 1957a, 1957b) and discussed in numerous textbooks, for example, Refs. (de Gennes, 1966; Tinkham, 1980). Qualitatively the BCS state consists of loose correlations between fermions across the Fermi surface in momentum space (Fig. 4 right side). Spatially the pairs are highly overlapping and cannot simply be considered to be composite bosons. In the BCS limit the momentum distribution only deviates from that of a normal Fermi sea in an exponentially small region near the Fermi surface. It is interesting to consider what happens if diatomic molecules become more and more weakly bound, to the point where the binding energy of the molecules, Eb , becomes less than the Fermi energy, EF . One could also consider increasing the interaction energy of a Cooper paired state until it is close to 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 considered the cusp of the crossover problem, and pairing in such a state is represented in the middle of Fig. 4. 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.
2.2. VARYING I NTERACTIONS It is instructive to consider a physical situation that will allow the realization pairing throughout the crossover (Fig. 4) (Leggett, 1980). Suppose we start with an attractive potential between two atomic fermions in vacuum, such as the square well potential with characteristic range r0 shown in Fig. 5. If this potential is very shallow there is a weak attractive interaction between the fermions. If we make this potential deeper the interaction between fermions becomes stronger, and for a strong enough attraction a bound molecular state will appear. This molecule will become more and more deeply bound as the potential becomes deeper. The interaction in this system can be characterized by the s-wave scattering length a. The quantity a comes out of studying two-body, low-energy scattering and is related to the s-wave collision cross section through σ = 4πa 2 . The top of Fig. 5 shows a pictorial representation of a. Just before the bound state appears
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F IG . 5. Scattering wave-functions in the presence of an attractive potential (right) and a more deeply attractive potential (left), in a regime where a weakly bound state of the potential (dashed line) is near threshold. r here describes the relative position of two distinguishable fermions. As in the bottom plot the scattering length diverges as the bound state moves through threshold.
a is large and negative, corresponding to a strong attractive interaction. As the bound state comes through threshold a diverges and then becomes large and positive, corresponding to a strong repulsive interaction. When a is much larger than r0 , the interaction is independent of the exact form of the potential, and a > 0 is universally related to the binding energy of the two-body bound state through h¯ 2 Eb = ma 2 , where m is the mass of a fermion (see, e.g., Ref. (Sakurai, 1994)). Now if we consider an ensemble of many fermions under the situation in Fig. 5, we have a system that can be tuned from BCS to BEC simply by tweaking the attractive potential. To the far right we have a small negative a and thus the BCS limit. In the opposite limit we have an ensemble of diatomic molecules; hence the BEC limit. It is important to note that although the interaction between fermions in pairs is strongest in the BEC limit, from the point of view of collisions in the gas, the BEC limit is actually weakly interacting because the molecule–molecule
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interaction is weak.1 The most strongly interacting gas from the point of view of collisions occurs near the divergence of a. Here many-body calculations are difficult because there is no small expansion parameter. The precise point at which a diverges is known as unitarity. Here the only length scale in the problem is 1/kF , giving this point many unique properties (Baker, 1999; Heiselberg, 2001; Kinast et al., 2005a; Thomas et al., 2005).
2.3. S IMPLE T HEORY BCS theory was originally applied in the limit where the interaction energy is extremely small compared to the Fermi energy. In this case the chemical potential, μ, can be fixed at EF , and many calculations become reasonably simple. Leggett pointed out that if the BCS gap equation is examined allowing μ to vary, the gap equation actually becomes precisely the Schrödinger equation for a diatomic molecule in the limit where μ dominates (Leggett, 1980). Below we briefly illustrate the key steps in an application of BCS theory to the entire crossover. This gives qualitatively correct results for the entire spectrum of pairing. The structure of the theory below originates in the work of Nozieres and Schmitt-Rink (NSR) in Ref. (Nozieres and Schmitt-Rink, 1985) and of Randeria et al. in Ref. (Randeria et al., 1990). We will consider a homogeneous Fermi system in three dimensions in an equal mixture of two different states at T = 0. Application of usual BCS theory results in the gap equation Δk = −
k
Ukk
k , 2Ek
(1)
2 2 where Ek = ξk2 + Δ2 , ξk = k − μ, and k = h¯2mk . Ukk < 0 is the attractive interaction for scattering of fermions with momenta k and −k to k and −k. We also obtain the number equation ξk Ntot = (2) 1− , Ek k
where Ntot is the total number of fermions in both states. To solve Eq. (1) in the BCS limit the standard approach is to assume that the potential is constant at a value U < 0, which implies that the gap is constant as
1 In the limit a r the molecule–molecule scattering length a mm is predicted to be 0.6a, where a 0 is the scattering length for atoms scattering above threshold (Petrov et al., 2004).
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well, i.e., Δk = Δ. In this case the gap equation (1) becomes 1 1 . − = U 2Ek
15
(3)
k
One will find, however, that this equation diverges. For BCS superconductors this problem is resolved because the interaction is limited to within the Debye energy, hω ¯ D , of EF . This is due to the nature of the phonon-mediated interaction between the electrons that gives rise to the attractive interaction. Further simplifications in the BCS limit are that μ = EF and that, since hω ¯ D EF , the density of states is constant at the value N (ξ = 0). The gap equation then becomes 1 = − N (0)U
h ¯ ωD
−h¯ ωD
dξ
2 Δ2 + ξ 2
.
(4)
Solving Eq. (4) produces the BCS result ≈ 2h¯ ωD e−1/N(0)|U | . To extend this calculation to the crossover in atomic systems we can no longer apply the h¯ ωD cutoff. The solution to the divergence problem in this case is nontrivial and requires a renormalization procedure, the full description of which we will not present here (but see, for example, Randeria (1995) and references therein). The result of such a procedure is a “renormalized” gap equation 1 1 1 m − = , − (5) V 2Ek 2k 4π h¯ 2 a k where the interaction is now described by the s-wave scattering length a instead of U and V is the volume of the system. Furthermore, in the crossover we cannot assume μ = EF ; instead we must solve the gap equation (5) and number equation (2) simultaneously for μ and the gap parameter Δ. We will solve for these √ parameters as a function of the dimensionless parameter kF a, where kF = 2mEF /h¯ . As pointed out in a useful paper by M. Marini et al. this can actually be done analytically using elliptic integrals (Marini et al., 1998). The solid lines in Fig. 6 show the result of this calculation of Δ and μ. We also plot the values of Δ and μ as they would be 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. In typical crossover experiments with 40 K or 6 Li, this regime corresponds to varying a from −2000a0 through ∞ and to 2000a0 , where a0 is the Bohr radius (0.0529 nm). 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 −Eb /2 = −( kF1 a )2 EF in the BEC limit. /EF −π 16 1 is e kF |a| in the BCS limit and 3π kF a in the BEC limit (Marini et al., 1998). Although referred to as the gap parameter, Δ only has meaning as the excitation gap, i.e., the smallest possible energy that can create a hole (remove a fermion) in
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F IG . 6. Gap parameter, Δ, and chemical potential, μ, of a homogeneous Fermi gas at T = 0 as determined through 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 < 1/kF a < 1.
the superfluid, in the BCS limit. In general the excitation energy is
2 h¯ 2 k 2 Egap = min Ek = min − μ + Δ2 2m (Randeria, 1995). This is Δ when μ is positive (as in the BCS limit), but becomes μ2 + Δ2 when μ is negative. 2.4. B EYOND T = 0 The phase transition temperature, Tc , is an important parameter for any superfluid system. In the BCS-BEC crossover the transition temperature increases as the interaction is increased, i.e., it is lowest in the perturbative BCS regime and highest in the BEC limit (Fig. 1). In a homogeneous system, in the BCS limit −π
Tc /TF = π8 eγ −2 e 2kF |a| where γ = 0.58 (Haussman, 1994), and in the BEC limit Tc /TF = 0.22 (Pethick and Smith, 2002). Note that BCS transition temperatures can be extremely small due to the exponential dependence upon 1/kF a. For example, at a typical interaction strength for alkali gases (a = −100a0 ) and a typical kF (1/kF = 2000a0 ) the BCS transition temperature would be ∼10−14 TF , which is a completely inaccessible temperature in contemporary atomic systems. Still, at 1/kF a = −1 where BCS theory nearly applies, the transition temperature is on the order of 0.1TF , which is accessible in current atomic systems. In the BCS limit pairing and the phase transition to a superfluid state occur at the same temperature. However, in the BEC limit this is not the case; because
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EXPERIMENTAL REALIZATION OF BCS-BEC CROSSOVER
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the constituent fermions are very tightly bound, pairs can form far above Tc . It is natural to expect that there would be a crossover between these two behaviors in the BCS-BEC crossover, i.e., at the cusp of the crossover the pairing temperature, T ∗ , would be distinct from Tc , yet not far from it. In a simple picture T ∗ should be related to a pair dissociation temperature, which in the case of molecules is ∼Eb /kb (Randeria, 1995). It is important to differentiate between the superconducting order parameter, which exists below Tc , and the pairing gap, which exists below T ∗ (Chen et al., 2005b). The pairing gap is associated with so-called pre-formed pairs, which are pairs that exist at temperatures above the critical temperature for condensation, Tc . Aspects of the pseudogap observed in high-Tc superconductors may be a manifestation of pre-formed pairs (Chen et al., 2005b).
2.5. M ODERN C HALLENGES The discussions and calculations above, while providing an introduction to basic crossover theory, are far from the state-of-the-art for theory in the field. There are noticeable problems with the NSR theory presented in Section 2.3. For example, the result for the chemical potential at unitarity (1/kF a = 0) is significantly different from the result of more accurate calculations using full Monte Carlo simulations (Carlson et al., 2003; Astrakharchik et al., 2004). The chemical potential is often written as μ = (1 + β)EF , and the NSR theory produced β = −0.41 at unitarity, while the Monte Carlo simulation of Ref. (Astrakharchik et al., 2004) finds β = −0.58 ± 0.01 at unitarity. As another example, extension of the NSR theory predicts that amm = 2aaa , while a full 4-body calculation in the BEC limit yields amm = 0.6aaa (Petrov et al., 2004). (Experiments at ENS have shown that amm = 0.6+0.3 −0.2 aaa (Bourdel et al., 2004), in agreement with the full calculation but not the NSR theory.) Both of these problems point to the fact that the NSR ground state, which only includes two-particle correlations, is not sufficient to accurately describe the system. Thus, it is clear that adding higher-order correlations to BCS-BEC theory is necessary, yet not a simple task (Holland et al., 2005). An even greater challenge is to extend accurate theories to nonzero temperatures where predictions can be made about the critical temperatures and the role of pre-formed pairs. Furthermore, all of the calculations considered thus far are carried out for a homogeneous Fermi system. However, the experiments with ultracold gases take place in traps (most often harmonic traps). The use of an inhomogeneous density gas can lead to qualitative changes in the system that must be accounted for in theory: Strong interactions can modify the density of the trapped gas (Vichi and Stringari, 1999), and signals can become blurred as the density, and hence the gap, varies across the sample (Chin et al., 2004).
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3. Feshbach Resonances In the previous section we determined that varying the fermion–fermion interaction is the key to accessing BCS-BEC crossover physics. We also observed that to arbitrarily vary the interactions we could imagine using a variable attractive potential with a bound state near threshold. Amazingly, atomic systems can achieve exactly such a physical situation. The attractive potential is provided by the van der Waals interaction between two atoms, and the knob to tune this potential is a homogeneous magnetic field. The sensitive magneticfield dependence of the potential can be provided through a scattering resonance known as a Feshbach resonance (Fano, 1961; Feshbach, 1962; Stwalley, 1976; Tiesinga et al., 1993). The goal of this section will be to understand exactly how a Feshbach resonance allows us to arbitrarily tune the interaction using a magnetic field. In this section we will be discussing only two-body physics, namely the problem of two fermions scattering in vacuum. We will find that the Feshbach resonance used for the experiments in this chapter approximates well the classic two-body system required for study of the BCS-BEC crossover problem of Section 2.
3.1. D ESCRIPTION Calculating the interaction between two ground state alkali atoms is a nontrivial problem that has been studied extensively in atomic physics (Burke, 1999). The result of studies of this problem show that for S-state atoms the interatomic potential is repulsive for very small r and has a weak attractive tail that goes as −C6 /r 6 as r → ∞. This weak attractive tail is a result of the interaction between mutually induced dipole moments of the atoms, which is known as the van der Waals interaction. The interatomic potentials are deep enough to contain a large number of bound vibrational states. A Feshbach resonance occurs when one of these bound states (often called the bare molecule state) coincides with the collision energy of two free atoms in a different scattering channel. Such a situation is depicted in Fig. 7(a). The interatomic potential of the two free atoms is often referred to as the open channel, while the potential containing the bare molecule state is referred to as the closed channel. When the closed and open channels describe atoms in different magnetic sublevels, they can be shifted with respect to each other through the Zeeman effect using an external magnetic field (Fig. 7(b)). Typically the effect of the coupling between the closed and open channels is small, but at a Feshbach resonance when the open-channel dissociation threshold is nearly degenerate with the bare molecule state, the effect of the coupling can be significantly enhanced. This coupling changes the effective interatomic potential, which we will refer to as the multichannel potential (Szyma´nska et al., 2005).
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F IG . 7. (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 magnetic field than the open channel threshold. This can lead to a crossing of the two levels.
A bound state will be added to this multichannel potential at a magnetic field value near (but not exactly at) the magnetic field position of the crossing of the bare molecule and open channel threshold. When we use the term “molecule” in this chapter we are always referring to this additional bound state of the multichannel potential, a so-called dressed molecule. The wave-function of these molecules is generally a linear superposition of open and closed channel wave-functions. As we will see in the next section, the open-channel component dominates for the molecules we study. As the magnetic field is tuned this multichannel bound state moves through threshold, and the scattering length between atoms in the open channel diverges. The scattering length near a Feshbach resonance varies with the magnetic field, B, according to the following equation (Burke, 1999): w . a(B) = abg 1 − (6) B − B0 Here abg is the triplet background (nonresonant) scattering length for atoms scattering in the open channel, B0 is the magnetic field position at which the molecular bound state of the coupled system 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 8 shows how a diverges according to Eq. (6) for the 40 K resonance described in the next section. 3.2. A S PECIFIC E XAMPLE To further illustrate Feshbach resonance physics it is useful to consider a specific atomic example. Here we will discuss the case of the 40 K resonance that is used for most of our BCS-BEC crossover studies. The open channel in this case describes the scattering of |f, mf = |9/2, −9/2 and |9/2, −7/2 atoms, which are
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F IG . 8. Behavior of the scattering length at a Feshbach resonance in 40 K between the mf = −7/2, −9/2 spin states.
F IG . 9. 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 the hyperfine structure of 40 K is inverted.
the two lowest energy states in the 40 K system (see Fig. 9). f describes the total atomic angular momentum and mf is the magnetic quantum number. For this particular problem the open channel couples to only one closed channel, |f, mf = |9/2, −9/2 and |7/2, −7/2 atoms. We will compare the results of this example to the requirements for studying the classic BCS-BEC crossover problem of Section 2. In the classic problem instead of two coupled channels a single variable depth potential with r0 a is considered (where r0 is the range of the potential). To calculate exactly the properties of our 40 K resonance we would need to carry out a full coupled channels calculation using realistic potassium potentials. The description of such a calculation is beyond the scope of this chapter but is de-
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EXPERIMENTAL REALIZATION OF BCS-BEC CROSSOVER
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scribed nicely in, for example, Ref. (Burke, 1999).2 Instead, for the demonstrative purposes of this section, we will examine the results of a simpler technique that is derived from K. Góral et al. in Ref. (Góral et al., 2004) and applied to the case of 40 K by M. Szyma´nska et al. in Ref. (Szyma´nska et al., 2005). We will solve the coupled Schrödinger equations that describe our two-channel system with a few simplifying assumptions. This technique reproduces the important physics of our 40 K resonance using a small set of experimentally measurable parameters. The main approximation is the so-called “pole approximation” described in Ref. (Góral et al., 2004) that holds when the open channel is strongly coupled to only one bare molecule state. Our main goal will be to use the simplified two-channel calculation to determine the binding energy of the molecular state in the multichannel potential as a function of the magnetic field. The Hamiltonian for our two-channel system is
2 − h¯m ∇ 2 + Voc W H = (7) , 2 W − h¯m ∇ 2 + Vcc where Voc is the uncoupled open channel potential, Vcc is the uncoupled closed channel potential, the potential W describes the coupling between the open and closed channel, and m is the mass of 40 K. It can be shown that the solution to this problem depends on only a few accessible parameters of the 40 K system (Góral et al., 2004). For the 40 K resonance we are considering, these parameters include the background scattering length abg = 174a0 (Loftus et al., 2002), the van der Waals coefficient C6 = 3897 a.u. (Derevianko et al., 1999), the resonance width w = 7.8 G (Regal et al., 2003b), and the resonance position B0 = 202.1 G (Regal et al., 2004b). Also useful is the binding energy of the first bound state in Voc , E−1 = 8.75 MHz (which can be attained from abg and C6 ) (Góral et al., 2004; Szyma´nska et al., 2005). Finally, we need the difference in magnetic moment (change in energy with magnetic field) of the open channel threshold with respect to the closed channel bare molecule. In our simplified calculation we will assume this to be the linear value that best approximates the result of a full coupled channels calculation, μco = 1.679μB = h × 2.35 MHz/G (Szyma´nska et al., 2005; Szyma´nska, 2005).3 These parameters, which in the end come from experimentally measured values, of course all have uncertainties, which we will ignore for now. The solution to the coupled Schrödinger equations based on the Hamiltonian above using the pole approximation is outlined in Ref. (Góral et al., 2004). The 2 We have compared some of the data presented in this chapter to full coupled channels calculations carried out by Chris Ticknor in John Bohn’s group at JILA. See Sections 5 and 6. 3 For 40 K µ from the result of the full calculation comes quite close to the difference in magnetic co moments of the closed and open channel thresholds, which is μco = h × 2.49 MHz/G at 200 G according to the Breit–Rabi formula (Corney, 1977).
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F IG . 10. Multichannel bound state of a Feshbach resonance in 40 K determined through the simplified calculation of Refs. (Góral et al., 2004; Szyma´nska et al., 2005) described in the text.
calculation is not computationally intensive and, after inserting the parameters above, provides the multichannel binding energy Eb , which we plot as the solid line in Fig. 10. Here Eb is plotted with respect to the energy of the open channel threshold, i.e., the open channel threshold is zero for all values of B. The dotted line shows the movement of the bare molecule energy with magnetic field, and the flat dashed line is the value of E−1 . The multichannel bound state is the dressed state of the avoided crossing of these two levels. The bare molecule state crosses threshold about 9 G higher than the position where the multichannel bound state comes through threshold, B0 . The difference between these two crossings is proportional to the resonance width parameter w and related to the interchannel coupling parameter in the Hamiltonian, W (Góral et al., 2004). Note that for 40 K the multichannel bound state at low field adiabatically connects to the highest-lying vibrational state of the open channel, rather than the bare molecule of the closed channel. The physics we study in this chapter actually occurs over a very small region near threshold in Fig. 10. The solid line in Fig. 11 shows a closeup of the multichannel bound state near threshold. Within a few gauss range of the Feshbach h¯ 2 resonance position the result obeys the expectation for a r0 , Eb = ma 2 (where a is determined through Eq. (6)), which is shown by the dashed line in Fig. 11. Farther away there is a clear deviation from quadratic behavior in 1/a. This behavior can be estimated by subtracting the range of the van der Waals potential r0 h¯ 2 from a in the calculation of the binding energy, Eb = m(a−r 2 (Góral et al., 2004; 0) Szyma´nska et al., 2005; Chin, 2005). The range of the van der Waals potential is given by (Gribakin and Flambaum, 1993) 1 (3/4) mC6 1/4 . r0 = √ (8) 8 (5/4) h¯ 2
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F IG . 11. Binding energy near the Feshbach resonance peak using three calculations of varying degrees of approximation. These results agree with those in Ref. (Szyma´nska et al., 2005).
F IG . 12. Binding energy in the magnetic-field regime used for the BCS-BEC crossover studies of this chapter.
The value of r0 is ∼60a0 for 40 K (and ∼30a0 for 6 Li). In Fig. 12 we examine the binding energy even closer to threshold. Here the h¯ 2 result of the simplified two-channel calculation and Eb = m(a−r (solid line) )2 2
0
h¯ are indistinguishable, while the Eb = ma 2 prediction (dashed line) only deviates slightly. We also plot the closed channel contribution to the molecule b wave-function, which is given by dE dB /μco . The Feshbach molecules that we will be interested in for the BCS-BEC crossover studies will have binding energies on order of or smaller than EF . A typical value of EF in our experiments (h × 15 kHz) is shown by the horizontal dotted line in Fig. 12. At this point the closed channel fraction according to our two-channel calculation is only 2%; this result is in agreement with full coupled channels calculations (Ticknor, 2005;
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Szyma´nska, 2005). This open-channel dominance means that, for our crossover studies, the two coupled channels problem of the Feshbach resonance can be approximated by small effective changes to the open channel potential. A resonance for which this is true is often referred to as a “broad resonance”, while the limit in which the closed channel dominates for Eb ≈ EF is referred to as a “narrow resonance” (Diener and Ho, 2004; Bruun and Pethick, 2004; h¯ 2 De Palo et al., 2004). In the case in which Eb ≈ m(a−r 2 we can derive a simple 0) equation that describes this criterion for a broad resonance in a Fermi system, w 2 μ2co 4EF
h¯ 2 2 mabg
1.
(9)
For the 40 K resonance we have been considering the left side of this expression has the numerical value of 43, and for the 6 Li resonance at B0 = 834 G it is over 10,000 (Chin, 2005; Bartenstein et al., 2005), indicating both Feshbach resonances used for BCS-BEC crossover studies thus far can be considered broad. In conclusion, we have seen how in atomic systems a Feshbach resonance can be used to add an additional bound state to an interatomic potential, leading to a divergence in the zero-energy scattering cross section for atoms colliding through the open channel. The Feshbach resonances that have so far been used for BCS-BEC crossover studies are broad resonances. These resonances can be approximated by a single channel problem and display universal properties, i.e., they have no dependence on the details of the atomic structure, but rather only on the parameters a and kF . For the 40 K resonance there are slight deviations from the a r0 limit that must be taken into account for precise measurements (Fig. 12). Still the physics of a 40 K gas at this resonance should basically reproduce the classic BCS-BEC crossover scenario envisioned by Leggett (Leggett, 1980) and described in Section 2.2.
4. Cooling a Fermi Gas and Measuring its Temperature To access the superfluid state of a Fermi gas in the crossover, the gas must be cooled below the critical temperature, Tc . To achieve such a temperature with a trapped gas of fermions was one of our largest challenges. In this section we describe how we cool 40 K and assess the success of the cooling through temperature measurements. 4.1. C OOLING 40 K The apparatus used to cool 40 K for these BCS-BEC crossover experiments employs the strategy used for some of the first experiments with 87 Rb BEC
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(Anderson et al., 1995; Myatt, 1997; Ensher, 1998). We perform the “usual” combination of trapping and cooling in a magneto-optical trap (MOT) followed by evaporative cooling (Phillips, 1998; Ketterle et al., 1999; Cornell et al., 1999). 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 a ultra high vacuum region for evaporative cooling (DeMarco, 2001). The major difference compared to the 87 Rb experiments stems from the fact that elastic collisions between identical fermions are suppressed at ultracold temperatures. This is because quantum statistics require antisymmetry of the total wave-function for two colliding fermions, which 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 (DeMarco et al., 1999). 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 9 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 both be held 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 could provide 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 (DeMarco and Jin, 1999; DeMarco, 2001). 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. One of the first steps in accessing the BCS-BEC crossover was to create a degenerate Fermi gas in an equal mixture of the f = 9/2 Zeeman states between which a Feshbach resonance was predicted, mf = −9/2 and −7/2 (Fig. 9). To accomplish this we needed to trap the high field seeking spin states of the f = 9/2 manifold; these states cannot be trapped in a magnetic trap. Thus, we utilized a far-off-resonance optical dipole trap (FORT) to confine the required spin states for the crossover experiments. Such a trap can be formed at the focus of a Gaussian laser beam whose optical frequency is far detuned from the 40 K transitions. We found that the best way to realize a cold gas in an optical dipole trap was to load a relatively hot gas of fermions after some evaporation in the magnetic trap. Typically we load the optical trap when the sample in the magnetic trap reaches T /TF ≈ 3 (T ≈ 5 µK with a few 107 atoms). After transfer to the mf = −9/2, −7/2 spin states the gas is evaporated in the optical trap simply by lowering the depth of the trap and allowing the hottest atoms to spill out. This evaporation and the subsequent temperature measurements were typically performed at a magnetic field of ∼235 G. At this field the scattering length is
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near the background (nonresonant) value of 174a0 , and from this field Feshbach resonances can easily be accessed.4 Evaporative cooling in the optical trap actually allowed us to achieve colder temperatures than previous records. We could cool to T /TF ≈ 0.1 (and possibly colder) with up to 106 atoms in each spin state. At these temperatures another impediment to cooling fermions appears—Pauli blocking of collisions (DeMarco et al., 2001). In a degenerate Fermi sea the Pauli exclusion principle forbids collisions for which the final state would place fermions in already occupied levels. This results in a suppression of the collision rate compared to the classical expectation and may make evaporative cooling more difficult in an all-fermion system. Many recent experiments have used a bosonic atom as their “second particle” as a possible approach to avoiding this problem (Truscott et al., 2001; Schreck et al., 2001; Roati et al., 2002; Hadzibabic et al., 2002; Goldwin et al., 2004).
4.2. M EASURING THE T EMPERATURE OF A F ERMI G AS In atomic gas experiments the standard technique for measuring the temperature of a gas is to suddenly turn off the trapping potential confining the atoms and then wait for an “expansion time” t. A two-dimensional image of the atomic distribution then reveals the velocity, or momentum, distribution of the gas (Ketterle et al., 1999). This technique is often referred to as time-of-flight imaging. For a classical gas the velocity spread is directly related to the temperature of the gas through the Maxwell–Boltzmann distribution. In the quantum regime measuring temperature from such distributions becomes trickier. In this section we will describe techniques for measuring the temperature of an ideal Fermi gas using images of gas distributions. 4.2.1. An Optically Trapped Fermi Gas Before presenting temperature measurements, it is useful to describe the expected distribution of an optically trapped Fermi gas. First, we analyze the optical dipole potential created from a single Gaussian laser beam using a harmonic approximation; this is the potential used for most of the BCS-BEC crossover experiments presented in this chapter. Then we present the most relevant formulas for describing the spatial and momentum distributions of a Fermi gas in such a harmonic trap. 4 The background scattering length is reasonably large compared to many other alkali atoms, resulting in nice evaporative cooling. Yet the interaction it provides still allows us to approximate the gas as an ideal Fermi gas for the purpose of temperature measurement.
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F IG . 13. Potential for potassium atoms created by a single 1064 nm Gaussian laser beam with w = 15 µm and a power of 10 mW. The y-direction includes the effect of gravity. Note that the horizontal scale in the third graph (z-direction) is a factor of 100 larger than the scale in the first two graphs. The dashed line in the left-most graph shows the parabola that approximates the potential for small x.
An optical dipole trap results from the interaction between a light field and the oscillating atomic electric dipole moment that is induced by the light field. This effect, known as the ac Stark shift, forms the conservative part of the interaction of atoms with light. The shift is proportional to the light intensity, and the sign depends upon the sign of the detuning of the light compared to the atomic resonance. When the light is detuned red of the atomic transition and the light intensity varies in space, an attractive potential well can be formed. While the photon-scattering rate decreases quadratically with the frequency detuning between the light and the atomic transition, the ac Stark shift varies linearly with this detuning. Thus, a trap with a large detuning results in a significant dipole potential at a low photonscattering rate. This allows use of the dipole potential for atom traps with long storage times, as was first realized in Ref. (Chu et al., 1986). The simplest optical dipole trap is formed by the focus of a single red-detuned Gaussian laser beam. The trap potential from the beam will be proportional to the laser intensity, which is given by 2
I (r, z) =
2
−2r /w Ipk 1+(z/zr )2 e 1 + (z/zR )2
(10)
for a beam pointing along zˆ . Here w is the beam waist (1/e2 radius), zR = πw2 /λ 2p is the Rayleigh range, Ipk = πw 2 is the peak intensity, and p is the total optical power. The strength of the potential created can easily be calculated from atomic properties (Grimm et al., 2000). Figure 13 shows cross sections of the potential for a typical trap that would hold a degenerate Fermi gas in our system, where we have included the effect of gravity in the y-direction. Gravity effectively
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lowers the trap depth in the y-direction, making evaporation in shallow traps onedimensional. At the bottom of the trap we can approximate the Gaussian potential as a parabolic potential (dashed line in Fig. 13). In this case we have a harmonic trap with oscillator frequencies given by 4U0 4U0 ωr = 2πνr = (11) , ωz = 2πνz = , 2 mw2 mzR where U0 is the depth of the potential. In shallow traps the harmonic approximation breaks down since atoms sample the potential near the top of the well where the Gaussian rises less steeply than the parabola. We account for this effect in experiments by modelling the real trap as a harmonic trap with a lower effective frequency. We determine the effective frequency experimentally through observation of the Fermi gas excitation frequencies in the weakly interacting regime. We now consider a Fermi gas in a harmonic potential described by the Hamiltonian for a particle in a harmonic well H =
1 p2 + mωr2 ρ 2 , 2m 2
(12)
where p 2 = px2 + py2 + pz2 and ρ 2 = x 2 + y 2 + λ2 z2 (λ = ωz /ωr ). The density 2
, where ω¯ = (ωr2 ωz )1/3 (see, e.g., Ref. (Pethick and of states is g() = 2(hω) ¯¯ 3 Smith, 2002)). The statistics of the Fermi gas are described by the Fermi–Dirac distribution function 1 , f () = (13) e kb T /ζ + 1 where ζ = eμ/kb T is the fugacity. We can now calculate the Fermi energy, which is defined as the energy of the highest occupied level of the potential at T = 0. We simply equate the integral over all states up to EF to the number of particles in one fermion spin state, N , to find EF h¯ ω¯ = (6N )1/3 . TF = (14) kb kb The temperature compared to TF describes the degeneracy of the Fermi gas and in the classical limit is related to peak phase space density through PSDpk = (T /TF )−3 /6. Also using the distribution function and the density of states we can obtain thermodynamic quantities such as the energy per particle, which will become important in Section 9. ∞ g()f () d Li4 (−ζ ) U = 3kb T = 0 ∞ . E= (15) N Li3 (−ζ ) 0 g()f () d
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F IG . 14. Energy of an ideal, trapped Fermi gas.
∞ k n The function Lin (x) = k=1 x /k appears often in the analysis of a harmonically trapped Fermi gas. Lin is the Poly-Logarithmic function of order n, sometimes written gn . Figure 14 plots the result of Eq. (15). In the classical regime the energy is proportional to the temperature, while in the Fermi gas limit the energy asymptotes to 34 EF ( 38 EF kinetic energy and 38 EF potential energy). Distribution functions in position and momentum can be determined through standard statistical mechanics techniques and the Thomas–Fermi approximation, which holds when many oscillator states are occupied (see Refs. (Butts and Rokhsar, 1997; DeMarco, 2001)). Table I shows the resulting Fermi–Dirac distribution functions in position and momentum space. The full expression is shown first, but it is also useful to understand the classical and T = 0 limits. In the classical limit the distribution is Gaussian, and at T = 0 the distributions are only nonzero for values less than the Fermi radius rF or Fermi momentum pF = h¯ kF . h2 k 2
Note that kF is defined through EF = ¯2mF and is related to the inverse distance between fermions at the center of the trap; at T = 0 the number density of one fermion spin state at the center of the trap is written in terms of kF as npk =
kF3 . 6π 2
4.2.2. Measuring Temperature from the Momentum Distribution In the experiment we access the distribution of the Fermi gas through absorption images of an expanded gas. Absorption images are taken by illuminating the atoms with a resonant laser beam pulse and imaging the shadow cast by the atoms onto a CCD camera. These images effectively integrate through one dimension to give a two-dimensional image (for example, Fig. 15(a)). The appropriate function for this distribution is n(ρ) or Π(p) integrated over one dimension (here x) (DeMarco, 2001). Writing the result in terms of the experimentally observed optical depth (OD), we find for the Fermi–Dirac distribution
ODFD (y, z) = ODpk Li2 −ζ e
−
y2 2σy2
e
−
z2 2σz2
/Li2 (−ζ ).
(16)
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Table I Distribution functions for a harmonically trapped Fermi gas Validity
Spatial distribution where σr2 =
all T /TF
n(ρ) =
kb T 2E and rF2 = F2 mωr2 mωr
2 2 λN Li3/2 (−ζ e−ρ /2σr )/Li3 (−ζ ) (2π )3/2 σr3 2 2 λN nc (ρ) = e−ρ /2σr (2π )3/2 σr3 8 1 − ρ 2 3/2 for ρ < r , 0 otherwise n0 (ρ) = λN F rF2 rF3 π 2
T /TF 1 T =0 Validity
2 = 2mE Momentum distribution where σp2 = mkb T and pF F
all T /TF
Π(p) =
2 2 N Li3/2 (−ζ e−p /2σp )/Li3 (−ζ ) (2π )3/2 σp3 2 2 N Πc (p) = e−p /2σp (2π )3/2 σp3 2 3/2 Π0 (p) = N3 82 1 − p2 for p < pF , 0 otherwise p p π
T /TF 1 T =0
F
F
F IG . 15. Nonclassical 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. Here the integration is along the z-direction. (b) Azimuthally averaged profile of the absorption image.
In the classical limit this equation becomes a two-dimensional Gaussian function: ODgauss (y, z) = ODpk e
−
y2 2σy2
e
−
z2 2σz2
(17)
.
These forms are applicable for both the spatial and momentum profiles, and for kb T 2 2 arbitrary expansion times through the relations σy2 = mω 2 [1 + (ωr t) ] and σz = kb T [1 + (ωz t)2 ], mωz2
r
where t is the expansion time.
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F IG . 16. Theoretical cross section of a harmonically trapped Fermi gas momentum distribution integrated along one dimension. TF is held fixed as T /TF varies.
Examples of cross sections of ODFD for various degeneracies, and constant TF , are shown in Fig. 16. As the gas is cooled the overall distribution becomes narrower, while the shape of the distribution becomes flatter. Note that as the temperature is lowered far below TF the changes in the distribution become small compared to TF . Still, down to T /TF ≈ 0.1 the temperature can be determined from least-squared fits to such distributions. Figure 15(a) is a sample absorption image of an expanded Fermi gas. The points in Fig. 15(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 Fermi–Dirac distribution (Eq. (16)), which reveals that the gas is at a temperature of 0.1TF . For comparison we plot as a dashed line the result of a fit appropriate for a classical distribution (Eq. (17)). Clearly the experimental distribution is flatter than a classical distribution. To evaluate this thermometry, we can 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. σy and σz tell us the temperature; 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 calculated through σy2 mωr2 T . = (18) TF h¯ ω(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 calculated from the total absorption in the image. Figure 17(a) shows a comparison of ζ with T /TF from Eq. (18). The line shows the expected relationship for an ideal Fermi gas. In Fig. 17(b) we convert ζ to T /TF for a more direct comparison. In general we find the two values agree, indicating the fits are extracting the correct information. Note that the noise in the points becomes large
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F IG . 17. Analysis of fits to Eq. (16) for expansion images of an optically trapped gas with an equal mixture of mf = −9/2, −7/2 atoms (Regal and Jin, 2003). For these data the integration was in the x-direction, and T is extracted from σy .
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 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. 4.2.3. Measuring Temperature Using an Impurity Spin State A second technique for measuring temperature that we have explored is impurity spin-state thermometry. Eric Cornell proposed this idea as a method to check the Fermi–Dirac surface-fit technique outlined in the previous section. A check is especially necessary for the coldest temperature gases at 0.1TF and below, because of the decrease in the sensitivity of the Fermi–Dirac fits at these temperatures. We have not done extensive work using the impurity thermometer; however, as we will see here this thermometer works quite well and has unexplored potential, in particular as a technique that could measure temperatures less than 0.1TF . 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. 18). 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 nondegenerate. Provided all of the spin states in the system are in thermal equilibrium, the temperature of the sysmσ 2
tem will be k tim2 , where σim is simply determined from a fit of the impurity gas b momentum distribution to a Gaussian distribution (Eq. (17)).
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F IG . 18. Measurement of T through an embedded impurity spin state. All three components are overlapped in the optical 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 good signal to noise 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 the T /TF be large enough that the energy of the impurity gas changes significantly with temperature. Figure 14 illustrates the range of T /TF for which this is the case. 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. 18). For this experiment the gas was prepared at a low magnetic field of a few gauss where the three-state mixture is fully stable. Here 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) (DeMarco, 2001). For analysis of the impurity spin-state data in the most general case where the impurity is not fully classical, it is useful to introduce the variable Tgauss . For mσ 2
a momentum space distribution Tgauss is defined as k gauss 2 , where σgauss is the bt result from a least-squared fit to the Gaussian distribution of Eq. (17). As we saw in Fig. 15 the Fermi distributions will not be well fit by a Gaussian, but the result is a well-defined quantity. Figure 19 shows the result of Tgauss /TF as a function of degeneracy, as determined through least-squared fits to theoretical distributions. Tgauss provides much the same information as the Fermi gas energy E (Fig. 14). However, it is more useful for our current purposes because it can be extracted
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F IG . 19. Dimensionless plot of the variable Tgauss (defined in the text) versus temperature.
F IG . 20. Impurity spin-state thermometry. Here we plot Tgauss for all of the states in the gas for multiple experiment iterations in which T was held fixed. The mf = 9/2 points have an average degeneracy of T /TF = 0.13.
from real images with better signal to noise than E, and it is convenient because in the classical limit it becomes precisely the real temperature of the system. In Fig. 20 we plot Tgauss as a function of N for a cold gas of atoms distributed among the three states mf = 9/2, 7/2, and 5/2. N is the measured number of atoms in the spin state from which Tgauss is extracted. The seven sets of points come from multiple iterations of the experiment for which the temperature was expected to be constant. Since the trap strength was held constant, N uniquely defines EF , and given that the temperature T is constant for all points on the plot, we can translate the dimensionless theoretical result of Fig. 19 to the situation of Fig. 20. The line in Fig. 20 shows the best-fit curve to the data, in which the only free parameter is the real temperature T . Note T is the value of Tgauss as N goes to zero. If the gas were fully classical Tgauss would be constant as a function of N ;
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F IG . 21. 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.
it is a manifestation of Pauli blocking that different components of an equilibrium gas can have the same temperature but different energy (DeMarco et al., 2001). Figure 21 shows the results of four experiments like the one shown in Fig. 20. The temperature result from the impurity measurement is compared to the result from the surface fits described in Section 4.2.2 (applied to the mf = 9/2 cloud). We see that both methods agree to within the uncertainty for clouds in the T /TF = 0.1–0.2 range. 4.2.4. Temperature in the BCS-BEC Crossover Our focus thus far has been on measuring the temperature of an ideal, normal Fermi gas. For the BCS-BEC crossover experiments we would like to know the temperature of our gas at any interaction strength. To this point there have been no experiments that have directly measured the temperature of a gas throughout the entire crossover. The main technique that has been used thus far is to instead measure the entropy of the system, which, through theory, can be translated to a temperature at any point in the crossover. The entropy, S, can be determined through temperature measurements in the weakly interacting regime. Then if experiments are performed using adiabatic magnetic-field ramps to the crossover regime, the entropy will be held constant at the weakly interacting value. As we have seen in 40 K it is convenient to measure the temperature of the weakly interacting gas above the Feshbach resonance with a Fermi gas, while in most of the 6 Li experiments the temperature is measured through the condensate fraction close to the BEC limit. In either case, the limitation of this technique is that it relies upon theory to convert between entropy and temperature in the crossover.
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Some initial work on this theoretical problem can be found in Ref. (Chen et al., 2005a). An alternative temperature measurement in the crossover was applied in the group of J. Thomas where they measured the temperature at unitarity by using fits to the momentum distribution in the hydrodynamic limit (Kinast et al., 2005b); here the distribution is simply a rescaled Fermi distribution and thus amenable to the same fitting procedure as the weakly interacting regime (Jackson et al., 2004). The impurity spin state thermometer described in Section 4.2.3 is another possible, but as of yet unexplored, direct thermometer in the crossover regime.
5. Elastic Scattering near Feshbach Resonances between Fermionic Atoms Some of the first signatures of the presence of fermionic Feshbach resonances were the observation of magnetic-field-dependent changes in the elastic-scattering properties of a normal Fermi gas (Loftus et al., 2002; O’Hara et al., 2002a; 2002b; Regal et al., 2003b; Regal and Jin, 2003; Gupta et al., 2003; Gehm et al., 2003; Bourdel et al., 2003). Here we present three different techniques that we used to probe changes in scattering properties at a fermionic Feshbach resonance. Two techniques probe the collision cross section, which tells us the magnitude of a. In the third measurement we observe the sign of a change and see evidence for unitarity-limited interactions.
5.1. M EASURING THE E LASTIC C OLLISION C ROSS S ECTION The first Feshbach resonance we searched for experimentally is the resonance described in Section 3 that occurs between the mf = −9/2 and mf = −7/2 spin states. The original theoretical prediction for the location of this resonance was B0 = 196+9 −21 G, based on available potassium potentials (Bohn, 2000). We first experimentally measured the position of this resonance using the technique of cross-dimensional rethermalization, which measures the collision cross section (Monroe et al., 1993). This was a technique that had provided much information about a Feshbach resonance in the bosonic 85 Rb gas (Roberts et al., 1998, 2001a). 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 could selectively modulate one radial direction because for this measurement our optical trap was not cylindrically symmetric (νx = 1.7νy ).) 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
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F IG . 22. 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 (Regal et al., 2003b). In between the peak and dip in σ the interaction is attractive; everywhere else it is repulsive.
√ collision cross section through 1/τ = 2nσ v/α. v = 4 kb T /πm is the mean relative speed between colliding fermions and n = N1tot n7 (r)n9 (r) d 3 r is the density weighted density. α is the calculated number of binary s-wave collisions required for rethermalization (DeMarco et al., 1999). Figure 22 plots the result of this measurement as a function of the magnetic field B. The magnetic field was calibrated through radio-frequency (rf) 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. 22). This allows a measurement of the magnetic-field width of the resonance w, which as we saw in Section 3 describes the coupling strength. The line in Fig. 22 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 (Regal et al., 2003b; Burke et al., 1999). 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.
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5.2. A NISOTROPIC E XPANSION A disadvantage of the cross-dimensional rethermalization method is that it only provides a valid measurement of σ in the so-called collisionless regime. A trapped gas is considered collisionless if the trap oscillator period 1/ν is much shorter than the mean time between collisions in the gas, 1/Γ . In the opposite limit where Γ ν the gas is collisionally hydrodynamic and the cross-dimensional rethermalization time will be determined by the oscillator period 1/ν instead of the mean time between collisions. At the peak of the Feshbach resonance the fermion–fermion interactions can easily become strong enough to make the gas collisionally hydrodynamic. In this regime a technique more suited to measuring changes in the elastic cross section is anisotropic expansion. In the hydrodynamic limit collisions during the expansion transfer kinetic energy from the elongated axial (z) cloud dimension into the radial (r) direction. This changes the aspect ratio of the expanded cloud (σz /σr ) compared to the collisionless expectation. This effect was first observed in a 6 Li Fermi gas in Ref. (O’Hara et al., 2002a). Figure 23 presents a measurement of anisotropic expansion in a 40 K gas at T /TF = 0.34. Here we enhance interactions between the mf = −9/2, −5/2 spin states using a Feshbach resonance between these states at ∼224 G. At the position of the Feshbach resonance where σ is large, the aspect ratio σz /σy decreases. As the gas becomes collisionless away from the resonance the aspect ratio smoothly evolves to the collisionless value. The key to observing anisotropic expansion in these systems is that during the beginning of the expansion the magnetic field must remain at the value near the Feshbach resonance. In these experiments the magnetic field remained high for 5 ms of expansion and an absorption image was taken after a total expansion time of 20 ms. In this measurement the mf =
F IG . 23. Anisotropic expansion of a strongly interacting Fermi gas near a Feshbach resonance between the mf = −5/2, −9/2 spin states (Regal and Jin, 2003).
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−9/2, −5/2 gas is 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 nonequilibrium 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 = 2nσ v in the gas, using an elastic 2 and |a | = 2000a (as was measured collision cross section given by σ = 4πa59 59 0 near the resonance peak (Regal and Jin, 2003)) 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 (Menotti et al., 2002; Kagan et al., 1997). In ultracold gas experiments anisotropic expansion is often associated with superfluidity of a BEC. This is because a typical ultracold Bose gas is collisionless, while below the superfluid transition the gas obeys superfluid hydrodynamic equations. Near a Feshbach resonance, however, where the gas can be collisionally hydrodynamic, anisotropic expansion can be observed both above and below the superfluid transition temperature. Still anisotropic expansion has been put forth as a possible signature of superfluidity in the BCS-BEC crossover regime (Menotti et al., 2002; Kagan et al., 1997). In pursuing this signature it is important to carefully distinguish between collisional and superfluid hydrodynamics and take into account that changes during expansion will affect the many-body state (Gupta et al., 2004). Such analysis is possible and has been considered in Refs. (O’Hara et al., 2002a; Bourdel et al., 2004).
5.3. M EASURING THE M EAN -F IELD I NTERACTION E NERGY Measurements of the collision cross section are useful for detecting the strength of the interaction 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 (Inouye et al., 1998; Cornish et al., 2000), while attractive interactions cause condensates with large atom number to become mechanically unstable (Gerton et al., 2000; Roberts et al., 2001b). For an atomic Fermi gas the mean-field interaction energy has a smaller impact on the thermodynamics. Here we discuss a novel spectroscopic technique that measures the mean-field energy of a two-component Fermi gas directly (Regal and Jin, 2003; Gupta et al., 2003).
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F IG . 24. Radio-frequency spectra. (a) Transition of interest. (b) rf line-shapes with (solid line) and without (dotted line) interactions (Regal and Jin, 2003).
In this measurement we again used the Feshbach resonance between the mf = −5/2, −9/2 spin states. At magnetic fields near the resonance peak, the meanfield energy in the Fermi gas was measured using rf spectroscopy (Fig. 24(a)). First, optically trapped atoms were evaporatively cooled in a 72/28 mixture of the mf = −9/2 and mf = −7/2 spin states. After the evaporation the optical trap was recompressed to achieve a larger density, and the magnetic field was ramped to the desired value near the 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 rf π-pulse. The fraction of mf = −7/2 atoms remaining after the pulse was measured as a function of the rf 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 nonlinear Zeeman shifts. Sample rf absorption spectra are shown in Fig. 24(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 rf line-shape had a Fourier width defined by the rf pulse duration. At the Feshbach resonance we observed two changes to the rf spectra. First, the frequency for maximum transfer was shifted relative to the expected value from a magnetic field calibration. Second, the maximum transfer was reduced and the measured line-shape is 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 ν =
2h¯ n9 (a59 − a79 ), m
(19)
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F IG . 25. Scattering length as measured through the mean-field interaction (Regal and Jin, 2003). 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).
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 (Harber et al., 2002). Here we have ignored a nonresonant 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 line-shape 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 ν as a function of B near the Feshbach resonance peak. The rf frequency for maximum transfer was obtained from Lorentzian fits to spectra like those shown in Fig. 24(b). The expected resonance frequency was then subtracted to yield ν. The scattering length a59 was obtained using Eq. (19) with n9 = 0.5npk and a79 = 174a0 , where npk is the peak density of the mf = −9/2 gas (Loftus et al., 2002). 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. 25. The solid line shows a fit to the expected form for a Feshbach resonance, Eq. (6). 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. When B is tuned very close to the Feshbach resonance peak, we expect the measured a59 to have a maximum value of on the order of 1/kF due to the unitarity limit. This saturation can be seen in the data shown in Fig. 25. Two points
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Table II Observed Feshbach resonances in 40 K Open channel |f, mf
l B0 (G)
|9/2, −9/2 + |9/2, −7/2 |9/2, −9/2 + |9/2, −5/2 |9/2, −7/2 + |9/2, −5/2 |9/2, −7/2
s s s p
w (G)
202.10 ± 0.07 7.8 ± 0.6 224.21 ± 0.05 9.7 ± 0.6 ∼174 ∼7 ∼198.8
Reference (Regal et al., 2003b, 2004b) (Regal and Jin, 2003) unpublished (Regal et al., 2003b; Ticknor et al., 2004)
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 unitaritylimited point on the attractive interaction side of the resonance (higher B) has an effective scattering length of ∼ 2/kF . (Here h¯ kF is the Fermi momentum for the mf = −9/2 gas.)
5.4.
40 K
F ESHBACH R ESONANCE S UMMARY
Table II lists the Feshbach resonances we have studied experimentally in 40 K. All of these resonances were originally located by measuring scattering properties using the techniques described above. We include the states between which the resonance occurs, the partial wave of the resonant collision l, our most precise measurement of the resonance position B0 , and the resonance width w.
6. Creating Molecules from a Fermi Gas of Atoms After locating Feshbach resonances in our 40 K system, we wanted 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 crossover problem. We were motivated to believe that it would be possible to create Feshbach molecules by experiments carried out in the Wieman group at JILA (Donley et al., 2002; Claussen et al., 2002). By pulsing the magnetic field quickly towards the Feshbach resonance they were able to observe coherent oscillations between atoms and Feshbach molecules 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. Another of our contributions to the study of Feshbach molecules was a spectroscopic detection technique that firmly
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F IG . 26. Creating molecules using magnetic-field ramps across a Feshbach resonance.
established that Feshbach molecules had been created. 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 (Hodby et al., 2005). Lastly, we describe how the dissociation of molecules in a sample at low density provided our most precise measurement of the Feshbach resonance position. 6.1. M AGNETIC -F IELD A SSOCIATION Figure 26 shows the behavior of the bound molecular state at a Feshbach resonance presented in Section 3. Given this picture, one would expect that atoms could be converted to molecules simply by ramping the magnetic field in time across the Feshbach resonance position B0 (Timmermans et al., 1999; Abeelen and Verhaar, 1999; Mies et al., 2000). The only requirement to 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 (twobody adiabatic). To a very good approximation the Feshbach molecules would have twice the polarizability of the atoms (Ratcliff et al., 1987) 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 introduced in the previous section. 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 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. 27 we present the observed total atom number in
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F IG . 27. Creation of molecules as seen through atom loss (Regal et al., 2003a). 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 (Regal and Jin, 2003).
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 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. 6.2. R F S PECTROSCOPY While suggestive of molecule creation, the measured atom loss was not conclusive proof for the existence of Feshbach molecules. We thus employed a spectroscopic technique to probe the molecules. First, we created the molecules with a magneticfield ramp across the Feshbach resonance and stopped at a magnetic field Bhold . At Bhold , a 13 µs radio frequency (rf) pulse was applied to the cloud; the rf frequency was chosen so that the photon energy was near the energy splitting between the mf = −5/2 and mf = −7/2 atom states (see Fig. 24(a)). 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 magneticfield gradient during free expansion (Stern–Gerlach imaging) or by leaving the magnetic field high (215 G) and taking advantage of nonlinear Zeeman shifts. Figure 28(a) shows a sample rf spectrum at Bhold = 222.49 G; the resulting number of atoms in the mf = −7/2 state is plotted as a function of the fre-
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F IG . 28. Rf spectrum for an atom/Feshbach molecule mixture (Regal et al., 2003a). (a) Transfer to the mf = −7/2 states as a function of rf 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.
quency of the rf pulse. 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 rf pulse. Nearby is a broader, asymmetric peak shifted lower in frequency. Here we find that after the rf pulse the total number of observed atoms (mf = −5/2 + −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. 28(b)). The asymmetric peak corresponds to the dissociation of molecules into free mf = 7/2 and mf = −9/2 atoms. Since the applied rf pulse stimulates a transition to a lower energy Zeeman state, we expect hνrf = hν0 − Eb − E, where Eb is the binding energy of the molecule, ν0 is the atom–atom transition frequency for noninteracting atoms, and we have ignored mean-field interaction energy shifts. The remaining energy, E, must by imparted to the dissociating atom pair as kinetic energy. Two separate linear fits are applied to the kinetic energy data in
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F IG . 29. The frequency shift (ν) from the expected mf = −5/2 → −7/2 transition plotted versus 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 (Regal et al., 2003a).
Fig. 28(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, E, beyond the binding energy when the molecule is dissociated. The observed line-shape of the asymmetric peak in Fig. 28(a) should depend upon the Franck–Condon factor, which gives the overlap of the molecular wavefunction with the atomic wave-function. C. Ticknor and J. Bohn calculated this multichannel Franck–Condon overlap as a function of energy. The resulting transition rate, convolved with the frequency width of the applied rf and scaled vertically, is shown as the solid line in Fig. 28(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. (Chin and Julienne, 2005). In Fig. 29 we plot the magnetic-field dependence of the frequency shift ν = νrf − ν0 , which to first order should correspond to the molecular binding energy. While ν could in principle be obtained directly from the transfer spectrum (Fig. 28(a)), we use the appearance of the threshold in the energy of the mf = −7/2 cloud, as it is more clear (Fig. 28(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 C. Ticknor and
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F IG . 30. Dissociation of molecules with radio frequencies (Greiner et al., 2005b). (a) The atoms resulting from the dissociation have equal and opposite momenta. (b) Absorption image of a dissociated molecular cloud.
J. Bohn. This measurement of Eb accentuates 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 compared to the molecules chemists are accustomed to studying. The excellent agreement with theory in Fig. 29 left no doubt that efficient creation of Feshbach molecules is possible. In addition, our rf spectroscopy technique was extended for a variety of other measurements in paired systems. It was proposed that rf spectroscopy could be used to measure the excitation gap in a superfluid Fermi gas (Petrosyan, 1999; Törmä and Zoller, 2000); such a measurement is published in Ref. (Chin et al., 2004). Our rf spectroscopy technique was also extended to detect confinement induced molecules in a one-dimensional Fermi gas (Moritz et al., 2005). Molecule dissociation via rf spectroscopy has proven useful for giving atoms in a molecule a large relative momentum, as in Ref. (Greiner et al., 2005b). Dissociating the molecules far above threshold produces a fun absorption image. The dissociated atoms fly out in a spherical shell, and the resulting absorption image is a ring structure (Fig. 30).
6.3. U NDERSTANDING M OLECULE C ONVERSION E FFICIENCY While using magnetic-field ramps to create molecules was very successful, there were many outstanding questions about the physics of the process. For example, what parameters define the conversion efficiency from atoms to molecules? The first important parameter turns out to be the rate of the magnetic-field ramp across the resonance. If the ramp is too fast no molecules will be created 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. This effect is shown in Fig. 31 where we observe molecule creation through atom loss as a function of the inverse magnetic field ramp rate across the resonance, (dB/dt)−1 .
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F IG . 31. Time scale for two-body adiabaticity at a Feshbach resonance in 40 K (Regal et al., 2003a).
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 , (20) where f is the fraction of atoms converted to molecules, fm is the maximum fraction of atoms that can be converted to molecules, and δLZ = β(dB/dt)−1 is the Landau–Zener parameter (Mies et al., 2000; Góral et al., 2004). We can fit Fig. 31 with β as the fitting parameter to find in this case β ≈ 20 µs/G (Regal et al., 2003a; Hodby et al., 2005). Reference (Góral et al., 2004) predicts that β = αnwabg , where n is the atomic gas density, w is the width of the Feshbach resonance, abg is the background scattering length, and α is a proportionality constant. A study by Hodby et al. verified the linear dependence of β upon the density n, but the proportionality constant α for this expression is still under investigation (Hodby et al., 2005). Notice in Fig. 31 that even at rates a few times slower than β not all the atoms are converted to molecules. One would expect that if the atom–molecule system were in chemical equilibrium and the temperature of the molecular sample were much less than Eb /kb then 100% of the atoms should be converted to molecules. An important point to recognize for all of the experiments in this section is that we are not operating in chemical equilibrium. At the final magnetic-field value in these experiments the time scale for chemical equilibrium is significantly longer than time scales in Fig. 31, and for many of the experiments described in this chapter we work on time scales intermediate between the time scale of β and the chemical equilibrium time scale. Thus, the observed saturation in molecule conversion is important to understand. Reference (Hodby et al., 2005) studies this phenomenon both for a bosonic gas of 85 Rb and our fermionic gas.
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F IG . 32. Dependence of molecule conversion on initial T /TF of a two-component Fermi gas (Hodby et al., 2005).
Eric Cornell suggested that the saturation in molecule conversion was likely related to phase space density (or T /TF in our case) based on intuitive arguments. An adiabatic process smoothly alters the wave function describing atom pairs but does not change the occupation of states in phase space. Thus to form a molecule a pair of atoms must initially be sufficiently close in phase space that their combined wave-function can evolve smoothly into the Feshbach molecule state as the resonance is crossed. In other words one would expect a molecule to form if a pair of atoms has a relative position rrel and relative velocity vrel such that |δrrel mδvrel | < γ h,
(21)
where γ will be an experimentally determined constant. Figure 32 shows the result of a measurement of the saturation in molecule conversion (fm ) for a 40 K gas as a function of T /TF , which is monotonically related to the phase space density of the gas. We see that indeed the conversion fraction does increase as T /TF decreases (phase space density increases) with a maximum conversion at our lowest temperatures of ∼90%. The work in 85 Rb found a similar dependence. To quantify the conversion expected for the many-body problem given the two-body criterion above (Eq. (21)) an algorithm described in Ref. (Hodby et al., 2005) was developed. The line in Fig. 32 is the result of this algorithm for the best fit value of γ . We found that for the fermion data γ = 0.38 ± 0.04 and for the boson data γ = 0.44 ± 0.03, indicating that a similar process is at work in both the Fermi and Bose cases.
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F IG . 33. Determination of the position of the Feshbach resonance via molecule dissociation in a low-density Fermi gas (Regal et al., 2004b). 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.
6.4. A P RECISE M EASUREMENT OF B0 A precise determination of the magnetic-field location of the Feshbach resonance B0 is an essential ingredient for exploring the BCS-BEC crossover regime. Knowledge of the position and the width of the resonance allows a precise calculation of the interaction strength at a particular magnetic field (see Section 3). As we saw in Section 5, B0 can be measured via scattering properties of the resonance, but our most precise measurement of B0 actually comes from the study of Feshbach molecules. In particular we looked for dissociation of Feshbach molecules in a low density gas as a function of magnetic field. 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. Figure 33 shows the result of such a measurement. Molecules created by a slow magnetic-field ramp across the resonance were dissociated by raising the magnetic field to a value Bprobe near the Feshbach resonance (inset to Fig. 33). Note that to avoid many-body effects, we dissociated the molecules after allowing the gas to expand from the trap to much lower density. This plan also allowed us to be certain we would not create molecules in the ramp of the magnetic field to near zero field for imaging. The measured number of atoms increases sharply at B0 = 202.10 ± 0.07 G. This measurement of the resonance position agreed well with previous less precise results (Regal et al., 2003b, 2004a).5
5 Note that this measurement of the resonance position is quite similar to the molecule association result of Fig. 27. However, the disadvantages of the association method are: (1) The molecule creation must take place in a high density sample; hence many-body effects may play a role. (2) The details of the magnetic-field ramp to go to low field for imaging, in particular its initial rate, are crucial.
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7. Inelastic Collisions near a Fermionic Feshbach Resonance The elastic collisions between atoms discussed in Section 5 are often referred to as good collisions. These collisions allow rethermalization in the gas but do not change the internal state of the atoms or molecules. In atomic gas experiments a constant concern is inelastic collisions between atoms, often referred to as bad collisions. In these collisions the products often are particles in lowerenergy internal states. The difference in energy between the incoming particles and the products of the collision must be carried away in the form of kinetic energy. When this energy difference is large compared to other energy scales such as the trap depth particles can be lost from the trap and significant heating of the sample can occur. Near a Feshbach resonance inelastic collisions can be enhanced along with the elastic collisions. To accomplish the work in this chapter we spent a large fraction of our time understanding the inelastic processes near 40 K Feshbach resonances and designing experiments that minimize the effect of inelastic collisions. In this section we will discuss the inelastic collisions near a 40 K Feshbach resonance and present measurements of relevant inelastic collision rates. We observe clear evidence of inelastic processes near the fermionic Feshbach resonance but find that despite these inelastic processes the lifetime of the sample is long enough to study BCS-BEC crossover physics.
7.1. E XPECTED I NELASTIC D ECAY P ROCESSES Let us first consider the stability of free fermionic atoms on the BCS side of a fermionic Feshbach resonance. In particular consider the Feshbach resonance between the |f, mf = |9/2, −9/2 and |9/2, −7/2 states that is used for many of the experiments in 40 K. Since these are the two lowest energy states of 40 K the only inelastic collision involving two of these fermions that is energetically favorable is |9/2, −9/2 + |9/2, −7/2 → |9/2, −9/2 + |9/2, −9/2. This process however is forbidden due to the fermionic nature of the particles (Bohn, 2000). Thus, any inelastic collision with these states must involve at least three fermions. A three-body inelastic collision in a two-component Fermi gas with components X and Y would take the form X + X + Y → X + (XY )− .
(22)
Here the subscript − represents a lower-energy molecular state. Such lowerenergy molecular states are always present in these atomic gas systems as there are many vibrational levels of the interatomic potential. To conserve energy
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F IG . 34. Particles involved in inelastic collisions in a Feshbach molecule/atom mixture. (a) Eq. (23); (b) Eq. (24).
and momentum in the collision the products X and (XY )− carry away the binding energy of the (XY )− molecule in the form of relative kinetic energy. Theory predicts that this three-body collision process will be suppressed for swave interactions between fermions because it requires that two identical fermions approach each other (Esry et al., 2001; Petrov, 2003; Petrov et al., 2004; D’Incao and Esry, 2005). However, while the rate of this inelastic collision process is suppressed, it is not forbidden, making it an important process near a Feshbach resonance (Regal et al., 2003b; Suno et al., 2003). As we cross the Feshbach resonance to the BEC side with a cold 40 K Fermi gas we want to consider the stability of a mixture of fermionic atoms and Feshbach molecules. An isolated Feshbach molecule for the mf = −7/2, −9/2 40 K resonance will be stable for the same reason as the two fermion mixture is stable. Note that the case in which the two atoms in the molecule are not in the lowest energy internal states, such as molecules created using 85 Rb, is quite different (Donley et al., 2002; Hodby et al., 2005). These 85 Rb Feshbach molecules will spontaneously dissociate as observed in Ref. (Thompson et al., 2005). For our 40 K molecules however we again expect that any decay processes will require more than two fermions, for example, (Petrov et al., 2004; D’Incao and Esry, 2006) X + XY → X + (XY )− ,
(23)
XY + XY → XY + (XY )− .
(24)
The first process is reminiscent of Eq. (22) above. These processes are often referred to as collisional quenching of vibrations (Balakrishnan et al., 1998; Forrey et al., 1999; Soldán et al., 2002). Again, we expect some suppression of these decay channels due to Fermi statistics since two identical fermions must approach each other, as shown schematically in Fig. 34.
7.2. L IFETIME OF F ESHBACH M OLECULES In this section we present experimental data on the stability of a mixture of atoms and Feshbach molecules. To obtain these data we created a molecule sample at
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F IG . 35. Feshbach molecule loss rate as a function of the atom–atom scattering length near a Feshbach resonance in 40 K (Regal et al., 2004a). N here 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 expected from two-body theory is larger than the interparticle spacing.
the mf = −7/2, −9/2 Feshbach resonance in which typically 50% of 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 (Regal et al., 2004a). Figure 35 shows the result of this measurement at a variety of magnetic fields on the BEC side of the Feshbach resonance. The plot shows N˙ /N versus 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. Physically this effect is partially related to the overlap between the wave-functions of the XY molecule and the (XY )− molecule. As the Feshbach resonance is approached the XY molecules become extremely weakly bound and quite large; hence, they have less overlap with the small (XY )− molecules. A scaling law for the dependence of the molecule decay rate upon the atom– atom scattering length a was found in (Petrov et al., 2004) and later in (D’Incao and Esry, 2005) for both the processes of Eqs. (23) and (24) schematized in Fig. 34. The scaling law is found by solving the full few-body problem in the limit where the molecules are smaller than the interparticle spacing, yet a r0 . Physical effects important to the result are the Fermi statistics and the wave-function overlap. The prediction for Eq. (23) (atom–molecule collisions) is that the decay rate should scale with a −3.33 and for Eq. (24) (molecule–molecule collisions) with a −2.55 . Since our measurement was carried out with thermal molecules the density of the molecule gas remains approximately constant over the a = 1000a0 to 3000a0 range. (The peak atom density in one spin state in the weakly interacting regime was n0pk = 7.5 × 1012 cm−3 .) Thus, we can measure the power law by fitting the data in Fig. 35 to the functional form Ca −p , where C and p are constants. We fit
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only points for which the interatomic spacing at the peak of the cloud is larger than the expected size of a two-body molecule, a/2. We find 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 (Zhang et al., 2005). In general we find that the lifetime of the molecules is surprisingly long near the Feshbach resonance. The molecule lifetime for magnetic fields at which a > 3000a0 is greater than 100 ms. This is much longer than lifetimes observed in bosonic systems for similar densities and internal states (Xu et al., 2003; Dürr et al., 2003). 100 ms is actually a long time compared to many other time scales in our Fermi gas—for example, the time scale for two-body adiabaticity, the mean time between elastic collisions, and the radial trap period. This comparison suggested that it would indeed be possible to study BCS-BEC crossover physics using atomic 40 K gases. 7.3. T HREE -B ODY R ECOMBINATION We also observe inelastic decay of fermionic atoms on the BCS side of the Feshbach resonance, where the decay is due to Eq. (22) (Regal et al., 2003b). These collisions cause both particle loss and heating. The heating can result from a combination of two processes: First, the density dependence of the three-body process results in preferential loss in high density regions of the cloud (Weber et al., 2003). Second, the products of the inelastic collision, which have kinetic energies on order the binding energy of the XY− molecules, can collide with other particles on the way out of the sample in a hydrodynamic gas causing transfer of energy to the gas. We have found that one relevant measure of the effect of inelastic decay processes on our ability to study BCS-BEC crossover physics is the heating of our Fermi gas during magnetic-field ramps that are sufficiently slow to be adiabatic compared to many-body time scales (see Section 8). We performed an experiment in which we approached the Feshbach resonance at rate of (6 ms/G)−1 , waited 1 ms, and then ramped back at the same slow rate to the weakly interacting regime. The result of this experiment is shown in Fig. 36. If we start with a cloud initially at T /TF = 0.10, T /TF upon return increases by less than 10% for a ramp to 1/kF0 a = 0 (yet by 80% for a ramp to 1/kF0 a = 0.5). (kF0 is the Fermi wave-vector measured in the weakly interacting regime.) For this measurement the peak density in one spin state was npk = 1.2 × 1013 cm−3 . 7.4. C OMPARISON OF 40 K AND 6 L I In both 40 K and 6 Li gases the scaling law of the decay rate as a function of molecule size is the same (Regal et al., 2004a; Zhang et al., 2005). However,
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F IG . 36. T /TF measured in the weakly interacting regime as a function of the final magnetic field in an adiabatic ramp towards the Feshbach resonance and back (Regal et al., 2005). The magnetic field is represented through the dimensionless parameter 1/kF0 a.
comparison of the absolute decay rates for similar densities and a ≈ 5000a0 shows that the inelastic decay of 40 K occurs more than two orders of magnitude faster than at the 6 Li Feshbach resonance at 834 G (Bartenstein et al., 2004b; Zhang et al., 2005; Bartenstein et al., 2005). This difference has so far not been fully explained theoretically. However, it must be related in some way to the difference in the full three-body potentials for 40 K versus 6 Li atoms. This difference in lifetime between the two atomic species affects how experiments in the BCSBEC crossover are approached in 40 K versus 6 Li. For example, in our experiments we obtain the coldest gases in 40 K by evaporating a Fermi gas in the weakly interacting regime where inelastic decay is negligible. Experimenters using 6 Li, however, obtain their coldest gases through evaporation near the Feshbach resonance or slightly to the molecular side of the resonance (Chin and Grimm, 2004; Jochim et al., 2003b). In our discussions thus far we have only considered inelastic collisions in which the final state is a more deeply bound molecule. In the 6 Li systems experimenters make use of inelastic collisions that interconvert atoms on the repulsive side of the Feshbach resonance and Feshbach molecules (Chin and Grimm, 2004; Jochim et al., 2003b). The rate of these inelastic collisions in the 6 Li system is faster than inelastic decay to more deeply bound molecules, and the binding energy h¯ 2 of the Feshbach molecules Eb ≈ ma 2 is small enough to not cause significant problems in the cooling process. Hence, they typically start with a hot gas of atoms on the BEC side of the resonance and after evaporative cooling observe a pure sample of Feshbach molecules (Jochim et al., 2003a, 2003b). Thus, in contrast to the 40 K experiments in this chapter, 6 Li experiments can operate with chemical equilibrium between atoms and Feshbach molecules.
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8. Creating Condensates from a Fermi Gas of Atoms So far we have considered the normal state of the Feshbach resonance/Fermi gas system and found that it has all the elements necessary to study BCS-BEC crossover physics. However, the true test of whether we could access BCS-BEC crossover physics with our atomic gas would be to observe a phase transition. The phase transition could be distinguished through observation of the onset of either superfluid behavior or condensation. Due to the linked nature of these phenomena in massive systems one would necessarily imply the other. Just as with alkali BEC in 87 Rb and 23 Na (Fig. 3), the observable of choice for the first experiments to observe this phase transition in the BCS-BEC crossover was condensation. In this section we will discuss how we were able to show condensation of fermionic atom pairs in the BCS-BEC crossover regime. This demonstration relied heavily upon our previous knowledge of the normal state of a Fermi gas at a Feshbach resonance. First we present our observation of condensation of Feshbach molecules to create one of the first molecular BECs (Greiner et al., 2003). This work led the way to observation of condensation of fermionic atom pairs in the crossover regime. Then we present a phase diagram of the BCS-BEC crossover regime attained through measurements of condensate fraction. 8.1. E MERGENCE OF A M OLECULAR C ONDENSATE FROM A F ERMI G AS OF ATOMS We have seen that a Feshbach resonance can be used to create a large number of ultracold molecules starting with a Fermi gas of atoms. After observing that these molecules can be long lived, the creation of a BEC from these bosonic Feshbach molecules was an obvious goal. Previously we had created molecules by applying a magnetic-field ramp just slow enough to be two-body adiabatic; in the experiments here the idea is to apply a magnetic-field ramp that is not only two-body adiabatic, but also slow with respect to the many-body physics timescale (manybody adiabatic). With such a magnetic-field ramp across the Feshbach resonance the entropy of the original quantum Fermi gas should be conserved (Cubizolles et al., 2003; Carr et al., 2004). For an initial atom gas with a sufficiently low T /TF the result should be a low entropy sample of bosonic molecules, which for a low enough entropy is a BEC. To pursue this idea experimentally we again 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 is that the
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lifetime of the Feshbach molecules can be much longer than the typical collision time in the gas and longer than the radial trapping period (see Section 7). The relatively long molecule lifetime near the Feshbach resonance allows the atom/molecule mixture to achieve thermal equilibrium during the magnetic-field ramp. Note however that since the optical trap is strongly anisotropic (νr /νz ≈ 80) we may attain 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 both the molecules and the residual atoms using time-of-flight absorption imaging. After typically 10 to 20 ms of expansion we applied a radio frequency (rf) pulse that dissociated the molecules into free atoms in the mf = −5/2 and mf = −9/2 spin states (Regal et al., 2003a). Immediately after this rf dissociation pulse we took a spin-selective absorption image. The rf pulse had a duration of 140 µs and was detuned 50 kHz beyond the molecule dissociation threshold where it did not affect the residual unpaired atoms in the mf = −7/2 state. We selectively detected the expanded molecule cloud by imaging atoms transferred into the previously unoccupied mf = −5/2 state by the rf dissociation pulse. Alternatively we could image only the expanded atom cloud by detecting atoms in the mf = −7/2 spin state. Close to the Feshbach resonance, the atoms and molecules are strongly interacting with effectively repulsive interactions. The scattering length for atom– molecule and molecule–molecule collisions close to the Feshbach resonance was calculated by Petrov et al. to be 1.2a and 0.6a, respectively, where a is the atom– atom scattering length (Petrov et al., 2004). During the initial stage of expansion the positive interaction energy is converted into additional kinetic energy of the expanding cloud. Therefore the measured momentum distribution is very different from the original momentum distribution of the trapped gas. In order to reduce the effect of these interactions on the molecule time-of-flight images we used the magnetic-field Feshbach resonance to control the interparticle interaction strength during expansion. We could significantly reduce the momentum kick due to the interaction energy by rapidly 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 is reduced to ∼500a0 . We found that this magnetic-field jump resulted in a loss of typically 50% of the molecules, which we attribute to the reduced molecule lifetime away from the Feshbach resonance. To attempt to observe condensation of molecules we monitored the molecule momentum distribution while varying the temperature of the initial weakly interacting Fermi gas, (T /TF )0 . Below (T /TF )0 of 0.17 we observed the sudden onset of a pronounced bimodal momentum distribution. Figure 37 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 mo-
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F IG . 37. 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 (Greiner et al., 2003). In the right sample the molecules form a Bose–Einstein condensate. The lines illustrate the result of bimodal surface fits.
F IG . 38. Molecular condensate fraction N0 /N versus the scaled temperature T /Tc (Greiner et al., 2003). The temperature of the molecules is varied by changing the initial temperature of the fermionic atoms prior to the formation of the molecules, yet measured through the momentum distribution of the molecular thermal gas.
mentum distribution is a striking indication that the cloud of weakly bound molecules has undergone a phase transition to a BEC (Anderson et al., 1995; Davis et al., 1995). To obtain thermodynamic information about the molecule cloud we fit the momentum distribution with a two-component fit. The fit function is the sum of an inverted parabola describing the Thomas–Fermi momentum distribution of a bosonic condensate and a Gaussian momentum distribution describing the noncondensed component of the molecule cloud. In Fig. 38 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
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F IG . 39. Time scale for many-body adiabaticity (Greiner et al., 2003). We plot the fraction of condensed molecules versus the time in which the magnetic field is ramped across the Feshbach resonance from 202.78 to 201.54 G.
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. From Fig. 38 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 (Giorgini et al., 1996). As expected we found that the creation of a BEC of molecules requires that the Feshbach resonance be traversed sufficiently slowly to be many-body adiabatic. We expect that this many-body time scale to be determined by the time it takes atoms to collide and move in the trap. In Fig. 39 the measured condensate fraction is plotted versus the ramp time across the Feshbach resonance starting with a Fermi gas at a temperature ∼0.1TF . Our fastest ramps resulted in a much smaller condensate fraction while the largest condensate fraction appeared for a B-field ramps slower than ∼3 ms/G. In conclusion, we have discussed the creation of a BEC of weakly bound molecules starting with a gas of ultracold fermionic atoms. With a relatively slow ramp of an applied magnetic field that converts most of the fermionic atoms into bosonic molecules and with an initial atomic gas below T /TF = 0.17, we observe a molecular condensate in time-of-flight absorption images. Our experiment approaches the BEC limit, in which superfluidity occurs due to BEC of essentially local pairs whose binding energy is larger than the Fermi energy. Strikingly, our 40 K molecular condensate is not formed by any active cooling of the molecules, but rather merely by traversing the BCS-BEC crossover regime. At the same time as these experiments in 40 K, experiments using 6 Li created a similar BEC of Feshbach molecules. Their approach however was direct evaporative cooling of the Feshbach molecules (Jochim et al., 2003b; Zwierlein et al., 2003).
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8.2. O BSERVING C ONDENSATES IN THE C ROSSOVER To create a molecular condensate 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 required a probe of the momentum distribution of pairs in the crossover. In the BEC limit the momentum distribution of the molecules could be measured using standard time-of-flight absorption imaging. However, this method is problematic in the crossover because the pairs depend on many-body effects and are not bound throughout expansion of the gas. Second, to prove observation of condensation in the BCS-BEC crossover regime 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 effects 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. 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, where time-of-flight imaging could be used to measure the momentum distribution of the weakly bound molecules. The projecting magneticfield ramp was completed on a timescale that allowed molecule formation but was still too brief for particles to collide or move significantly in the trap. This is possible due to the clear separation of the two-body and many-body time scales. The timescale for many-body adiabaticity in Fig. 39 is two orders of magnitude longer than the timescale for two-body adiabaticity shown in Fig. 31. The key to the second problem, verifying condensation of many-body pairs, came from careful understanding of the two-body physics. As discussed in Section 6 we were able to precisely measure the magnetic-field position above which a two-body bound state no longer exists, B0 (Fig. 33). If we observed condensation of fermionic atom pairs at B > B0 we could be assured that these were pairs that were the result of many-body effects. To perform experiments making use of these ideas, we continued with the same experimental setup as the last section where we discussed the creation of molecular condensates. We initially prepared the ultracold two-component atom gas at a magnetic field far above the Feshbach resonance. Here the gas is not strongly interacting, and we measured (T /TF )0 through surface fits to time-of-flight images of the Fermi gas (Section 4). The field was then slowly lowered at the many-body adiabatic rate of 10 ms/G to a value Bhold near the resonance. Whereas before we
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F IG . 40. A typical magnetic-field ramp used to measure the fraction of condensed fermionic atom pairs (Regal et al., 2004b). The system is first prepared by a slow magnetic-field ramp towards the resonance to a variable position Bhold , indicated by the two-sided arrow. After a time thold the optical trap is turned off at t = 0 and the magnetic field is quickly lowered by ∼10 G. After free expansion, the molecules are imaged on the BEC side of the resonance (◦).
had considered only values of Bhold below B0 , on the BEC side, now we explore the behavior of the sample when ramping slowly to values of Bhold on either side of the Feshbach resonance. To probe the system we projected the fermionic atoms pairwise onto molecules and measured the momentum distribution of the resulting molecular gas. This projection was accomplished by rapidly lowering the magnetic field by ∼10 G at a rate of (40 µs/G)−1 while simultaneously releasing the gas from the trap (Fig. 40). This puts the gas far on the BEC side of the resonance, where it is weakly interacting. After a total of typically 17 ms of expansion the molecules were selectively detected using rf 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 noncondensed molecules. Figures 41 and 42 present the main result of this section. In Fig. 41(a) we plot the measured condensate fraction N0 /N as a function of the magnetic-field detuning from the resonance, B = Bhold − B0 . The data in Fig. 41(a) was taken for a Fermi gas initially at T /TF = 0.08 and for two different wait times at Bhold . Condensation is observed on both the BCS (B > 0) and BEC (B < 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. 41(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 41 compares the condensate fraction measured using the 40 µs/G (circles) rate to that using a ramp that was ∼7 times faster (open squares). We found that the measured con-
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F IG . 41. Measured condensate fraction as a function of detuning from the Feshbach resonance B = Bhold − B0 (Regal et al., 2004b). (a) Data for thold = 2 ms (") and thold = 30 ms (P) with an initial cloud at T /TF = 0.08. (b) Data for two different projection magnetic-field ramp rates: 40 µs/G (circles) and ∼6 µs/G (squares). The dashed lines reflect the uncertainty in the Feshbach resonance position.
densate 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 (Zwierlein et al., 2004). They also monitored the delayed response of the many-body system after modulating the interaction strength (Zwierlein et al., 2005b). 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 been a number of theoretical papers on the subject of the pairwise projection technique for measuring condensate fraction in the crossover (Avdeenkov and Bohn, 2005; Altman and Vishwanath, 2005; Perali et al., 2005). 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 introduced a method for probing the momentum distribution of fermionic atom pairs and employed this technique to observe condensation near a Feshbach resonance. By projecting the system onto a molecule gas, we observe condensation of fermionic pairs as a function of the magnetic-field detuning from the resonance as shown in Fig. 42. While Fig. 42 is
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F IG . 42. 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 B = 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. 3D artistry is courtesy of Markus Greiner.
reminiscent of Fig. 3 where condensation was observed a function of T /Tc , note that the condensate here actually appears as a function of interaction strength, not temperature.
8.3. M EASUREMENT OF A P HASE D IAGRAM In addition to varying B and measuring the condensate fraction, we can also vary the initial temperature of the Fermi gas. Figure 43 is a phase diagram created from data varying both B and (T /TF )0 . B is converted to the dimensionless parameter 1/kF0 a, where a is calculated directly from B through Eq. (6) and kF0 is extracted from the weakly interacting Fermi gas. The shades of gray represent the measured condensate fraction using the projection technique. The boundary between the black and light gray 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 B decreases, as expected based upon BCS-BEC crossover theories (Section 2). These 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. 43 always remains well below one (Greiner et al., 2003). This is not observed in the case of 6 Li experiments
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F IG . 43. Transition to condensation as a function of both interaction and T /TF (Regal et al., 2004b). The data for this phase diagram are collected with the same procedure as shown in Fig. 40 with thold ≈ 2 ms. The contour plot is obtained using a Renka–Cline interpolation of approximately 200 distinct data points.
(Zwierlein et al., 2004), suggesting that technical issues particular to 40 K may play a role. As part of our probing procedure the magnetic field is set well below the Feshbach resonance where the molecule lifetime is only on the order of milliseconds (Fig. 35) (Regal et al., 2004a; Petrov et al., 2004). This results in a measured loss of 50% of the molecules and may also reduce the measured condensate fraction. Further, temperature measurements using the surface-fitting procedure become less accurate below T /TF = 0.1 (see Section 4). Further, it is expected that the isentropic ramp from a weakly interacting Fermi gas to the crossover results in a larger value of T /TF than (T /TF )0 . The extent of this adiabatic heating is currently under theoretical investigation (Chen et al., 2005a). Density-dependent loss processes could also play a role in heating of the sample, especially as the BEC limit is approached (see Section 7). Whether any or all of these processes play a role in the small condensate fraction is a subject of current study in our group.
9. The Momentum Distribution of a Fermi Gas in the Crossover The measurements of condensation in the crossover described in the last section probed the phase coherence between fermionic atom pairs. As discussed in Sec-
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tion 2, it is only in the BCS limit that the pairs are always coherent, and when the interaction becomes large so-called preformed pairs are predicted to exist above the phase transition temperature Tc (Randeria, 1995; Chen et al., 2005b). To verify such theories it is important that techniques for detecting pairing be developed alongside studies probing the phase transition. There have been a number of probing techniques used to detect pairing including rf spectroscopy (Chin et al., 2004), magnetic-field modulation (Greiner et al., 2005a), spectroscopic probes (Partridge et al., 2005), and measurements of the atomic momentum distribution (Bourdel et al., 2003; Regal et al., 2005). This section focuses on measurements of the atomic momentum distribution in the BCS-BEC crossover using 40 K (Regal et al., 2005). The classic characteristic change of the momentum distribution of a superfluid Fermi system is a broadening of the Fermi surface compared to the ideal Fermi sea (see, for example, de Gennes (1966)). Figure 44 (inset) shows the expected momentum distribution of a homogeneous, zero-temperature Fermi system. In the BCS limit (1/kF a → −∞) the amount of broadening is small and associated with . As the interaction increases this effect grows until at unitarity (1/kF a = 0) the effect is on order of EF , and in the BEC limit (1/kF a → ∞) the momentum distribution becomes the square of the Fourier transform of the molecule wave-function (Nozieres and Schmitt-Rink, 1985). This kinetic energy increase can be interpreted as a cost of pairing. Amazingly, the total energy of the system is lowered despite this large kinetic energy increase. Here we will discuss the atom momentum distribution of a trapped Fermi gas. The case of an inhomogeneous trapped gas is more complicated than the homogeneous case. Nonetheless, in the strongly interacting regime the momentum distribution probes the pair wave-function and consequently the nature of the pairs: Small, tightly bound pairs will broaden the momentum distribution more than large, weakly bound pairs. Similarly a fully paired gas will broaden the distribution more than a gas with a small fraction of paired atoms. It is expected that this measurement will probe the pairs independent of whether they have formed a condensate. Throughout this section we will compare our data to theory of the momentum distribution of a trapped gas. This theoretical work is the result of a collaboration between our group, Murray Holland of JILA, and Stefano Giorgini, a JILA visiting fellow from Trento, Italy.
9.1. M EASURING THE M OMENTUM D ISTRIBUTION We found that the momentum distribution of a Fermi gas in the crossover could actually be measured using the standard technique of time-of-flight expansion followed by absorption imaging (Anderson et al., 1995). The key to measuring the atom momentum distribution is that the gas must expand freely without any interatomic interactions; to achieve this we quickly changed the scattering length
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F IG . 44. Theoretical column integrated momentum distributions of a trapped Fermi gas n(k) calculated from a mean-field theory at T = 0 (Viverit et al., 2004; Regal et al., 2005). The normalization is given by 2π n(k)k dk = 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.
to zero for the expansion. This 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 (Bourdel et al., 2003). In this work we carried out measurements down to T /TF ≈ 0.1, where pairing becomes a significant effect and condensates have been observed (Regal et al., 2004b; Zwierlein et al., 2004). To understand what we expect for our trapped atomic 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 44 shows a calculation of an integrated column density from the result of a mean-field calculation at T = 0 as described in Ref. (Viverit et al., 2004). First we will discuss the atomic momentum distribution measured with a low temperature Fermi gas. We started with a weakly interacting mf = −7/2, −9/2 gas at T = 0.12TF in a trap with a radial frequency of νr = 280 Hz and νr /νz = 14.6 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 6 For this measurement we introduced the use of a crossed dipole trap configuration. One beam is oriented parallel to the force of gravity (y) ˆ with a waist of wy = 200 µm, and the second beam is perpendicular to the first (ˆz) and has wz = 15 µm.
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F IG . 45. 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. 44 (Regal et al., 2005). The curves, in order of decreasing peak OD, correspond to 1/kF0 a = −71, −0.66, 0, and 0.59. 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 k 3 . The lines are the results of a fit to Eq. (16).
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 typical many-body timescales as determined by EhF = 90 µs. The cloud was allowed to freely expand for 12.2 ms, and then an absorption image is taken. The imaging beam propagated along zˆ and selectively probed the mf = −9/2 state. Samples of these absorption images, azimuthally averaged, are shown in Fig. 45 for various values of 1/kF0 a. We observe a dramatic change in the distribution that is qualitatively very similar to the prediction in Fig. 44. Some precautions need to be taken in the quantitative comparison of Figs. 44 and 45. First, the magneticfield ramp to the Feshbach resonance, while adiabatic with respect to most time scales, is not fully adiabatic with respect to the axial trap period. Second, in the experiment an adiabatic field ramp keeps the entropy of the gas, not T /TF , constant. However, we expect the resulting change in T /TF to have a minimal effect on the distribution for 1/kF0 a < 0 (Chen et al., 2005a). Third, the theory assumes T = 0 and does not include the Hartree term, thus underestimating the broadening on the BCS side compared to a full theory (Astrakharchik et al., 2004).
9.2. E XTRACTING THE K INETIC E NERGY It is natural now to consider extracting the kinetic energy from the momentum distribution. While the momentum distribution should be universal for small momenta, for large momenta it is influenced by details of the interatomic scattering
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potential. In the extreme case of a delta potential, which we used for the calculation in Fig. 44, the momentum distribution has a tail with a 1/k 4 dependence, giving rise to a divergence of the kinetic energy. In the experiment we avoid a dependence of the measured kinetic energy on details of the interatomic potential because the magnetic-field ramp 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 (Gribakin and Flambaum, 1993). Thus, the results presented here represent a universal quantity, independent of the details of the interatomic potential. Although universal in this sense, the measured kinetic energy is intrinsically dependent on the dynamics of the magnetic-field ramp, with faster ramps corresponding to higher measured energies. To exactly obtain the kinetic energy from the experimental data we would need 3 to take the second moment of the distribution, which is proportional to k OD/ kOD. As illustrated in Fig. 45 (inset) this is difficult because of the decreased signal-to-noise ratio for large k. Thus, our approach will be to apply a 2D surface fit to the image and extract an energy from the fitted function. In the limit of weak interactions the appropriate function is that for an ideal, harmonically trapped Fermi gas (Eq. (16)). Using the results of a fit to this function the kinetic energy per particle is given by Ekin =
3 mσx σy Li4 (−ζ ) , 2 t 2 Li3 (−ζ )
(25)
where t is the expansion time. Although Eq. (16) is not an accurate theoretical description of the Fermi gas in the crossover, empirically we find it to be a fitting function that describes the data reasonably well throughout the crossover, as illustrated in Fig. 45 (inset). Figure 46 shows the result of extracting Ekin as a function of 1/kF0 a; we see that Ekin more than doubles between the noninteracting regime and unitarity.
9.3. C OMPARING THE K INETIC E NERGY TO T HEORY As mentioned earlier Ekin of a trapped gas is affected both by the broadening due to pairing (Fig. 44 (inset)) and by changes in the trapped gas density profile. In 0 the BCS limit, the broadening due to pairing scales with e−π/2kF |a| and is thus exponentially small compared to density profile changes, which scale linearly with 0 using a mean-field calculation in kF0 |a|. In this limit we can calculate Ekin /Ekin the normal state (Vichi and Stringari, 1999). S. Giorgini found that to lowest order 0 = 2048 k 0 |a| + 1. We plot this result in Fig. 46 (inset) and in kF0 |a|, Ekin /Ekin 945π 2 F find good agreement for the weakly interacting regime (1/kF0 a < −1). However, caution must be taken with the apparent agreement for −0.5 < 1/kF0 a < −1. The agreement could be explained by the lack of pairing in this theory being
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F IG . 46. The measured energy Ekin of a Fermi gas at (T /TF )0 = 0.12 in the crossover nor0 = 0.25k µK (Regal et al., 2005). The dash–dot line is the expected energy ratio malized to Ekin b from a calculation only 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.
compensated by the theory’s overestimation of the density profile change when |a| becomes larger than 1/kF . In the crossover regime where the pairs are more tightly bound, pairing provides a significant contribution to the change in the momentum distribution. At unitarity a full Monte Carlo calculation predicts the radius of the Fermi gas density profile will become (1 + β)1/4 R0 = 0.81R0 , where R0 is the Thomas–Fermi radius of a noninteracting Fermi gas (Astrakharchik et al., 2004). Just this rescal0 = 1.54. Thus, at unitarity, pairing effects on the ing would result in Ekin /Ekin momentum distribution must account for a large fraction of the measured value 0 = 2.3 ± 0.3 (Fig. 46). of Ekin /Ekin In the BEC limit and at T = 0 we expect the measured energy would be that of an isolated diatomic molecule after dissociation by the magnetic-field ramp. Provided the scattering length associated with the initial molecular state, a(t = 0), is much larger than r0 ≈ 60a0 , the wave function for the molecule is given by ψ = Ae−r/a(t=0) /r where r is the internuclear separation and A is a normalization constant. M. Holland and S. Giorgini calculated the measured energy from the solution of the Schrödinger equation with a time-dependent boundary condi1 tion on the two-particle wave-function d log(rψ) = − a(t) , where a(t) is the dr r=0 scattering length fixed by the magnetic field at time t. In Fig. 46 we show the result of this calculation for a pure gas of molecules with zero center of mass momentum and a (2 µs/G)−1 ramp rate. We find reasonable agreement considering that there is a large systematic uncertainty in the theory prediction resulting from the experimental uncertainty in the magnetic-field ramp rate and also that this two-body theory should match the data only in the BEC limit (1/kF0 a 1).
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0 normalized to the Fermi energy at F IG . 47. Temperature dependence of Ekin = Ekin − Ekin 0 a = 0. (T /TF ) is the temperature of the noninteracting gas.
A greater theoretical challenge is 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. (Chiofalo et al., 2005) has addressed this calculation using the NSR ground state (Nozieres and Schmitt-Rink, 1985). In the strongly interacting regime the result does not accurately reproduce the measured kinetic energy; this suggests that more sophisticated crossover theories are necessary.
9.4. T EMPERATURE D EPENDENCE We have also studied the dependence of the momentum distribution on (T /TF )0 . To vary the temperature of our gas, we typically recompress the optical dipole trap after evaporation and heat the cloud through resonant modulation of the optical trap strength. The experimental sequence for measuring the momentum distribution was the same as before except the ramp rate to a = 0 for expansion was ∼(8 µs/G)−1 . Figure 47 shows the measured kinetic energy change 0 . Ideally for this measurement we would have liked to Ekin = Ekin − Ekin have varied the temperature while maintaining a fixed density. However, for these data the density does vary some. For example, for the coldest dataset (circles) the peak density, for atoms in one spin state, in the weakly interacting regime is n0pk = 1.4 × 1013 cm−3 and EF0 = 0.56 µK. For the hottest dataset (stars) n0pk decreases to 6 × 1012 cm−3 and EF0 = 0.79 µK. On the BEC side of the resonance (1/kF a > 0), Ekin decreases dramatically with (T /TF )0 . Because Ekin should be proportional to the molecule fraction, this result is closely related to our recent observation that the molecule conversion efficiency scales with T /TF (Hodby et al., 2005). In the strongly interacting
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regime we also observe a decrease in Ekin with increasing (T /TF )0 . However, the decrease is not as steep as would be expected if the phase transition, which occurs at (T /TF )0 ≈ 0.15 as measured through N0 /N (Fig. 43), were responsible for the change in the momentum distribution. Instead the temperature dependence of Ekin is consistent with the expectation that the changes in the kinetic energy are caused by pairing and not condensation (Regal et al., 2004b; Chin et al., 2004; Chen et al., 2005b).
10. Conclusions and Future Directions In this chapter we have presented the story of the realization of BCS-BEC crossover physics with a gas of 40 K atoms. Experiments with 40 K and 6 Li have shown that a Fermi gas at a broad Feshbach resonance crosses a phase transition to a superfluid state and displays properties of the classic BCS-BEC crossover problem. With this system we can study the evolution from fermionic superfluidity of pairs nearly described by BCS theory to BEC of diatomic molecules. This new system provides a physical link between two descriptions of superfluid systems, BCS and BEC, that were historically thought to be distinct. The fermionic superfluids created in these gases have extremely high transition temperatures, Tc ≈ 0.1TF . In these model systems the absolute value of Tc ≈ 100 nK is very cold. However, for typical values of TF in metals the corresponding transition temperature would be above room temperature. Fully understanding this model system will perhaps contribute to the efforts to create higher transition temperature superconductors in real materials. These atomic gas systems also provide the opportunity to study aspects of quantum systems that are not typically accessible in solid state materials. For example, experiments such as those in Section 8 utilize dynamics for measurements. The ability to study the physics of the BCS-BEC crossover in real time may provide new insight into this many-body quantum system. In the immediate future there are a plethora of experiments that could be done to study BCS-BEC crossover physics with this new system. This list includes both more precise versions of previous measurements as well as experiments designed to study entirely new phenomena. More precise experiments will be vital to rigorous comparison with sophisticated crossover theories. New phenomena that could be studied include, for example, higher partial wave pairing (Regal et al., 2003b; Zhang et al., 2005; Schunck et al., 2005). This would be an important contribution due to the relevance to cuprate superconductors, which have been shown to have d-wave symmetry. Predictions have been made for the behavior of an atomic Fermi gas in the BCS-BEC crossover with unequal particle number in each spin state of the two-component gas (Mizushima et al., 2005; Sheehy and
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Radzihovsky, 2006). Such a “magnetized” system has been the topic of a number of recent experimental pursuits (Zwierlein et al., 2006). Further, quantum Fermi gases have now been studied in optical lattice potentials (Ott et al., 2004; Köhl et al., 2005); studying crossover physics in the presence of such a lattice could more closely mimic conditions in real solids.
11. Acknowledgements We are grateful to many different people for their contributions to this work. We thank our experimental co-workers Markus Greiner and Jayson Stewart, and we are indebted to Eric Cornell, Carl Wieman, and the rest of the BEC collaboration at JILA for stimulating discussions. We have benefited from collaboration with many theorists, including Murray Holland, Stefano Giorgini, Marilu Chiofalo, John Bohn, Chris Ticknor, Alexander Avdeenkov, Marzena Szyma´nska, Paul Julienne, Chris Greene, Brett Esry, Leo Radzihovsky, Kathy Levin, and Qijin Chen. Funding for this work was provided by NSF, NIST, ONR, NASA, and the Hertz foundation.
12. References Abeelen, F.A., Verhaar, B.J. (1999). Time-dependent Feshbach resonance scattering and anomalous decay of a Na Bose–Einstein condensate. Phys. Rev. Lett. 83, 1550. Allen, J.F., Misener, A.D. (1938). Flow of liquid helium II. Nature 141, 75. Altman, E., Vishwanath, A. (2005). Dynamic projection on Feshbach molecules: A probe of pairing and phase fluctuations. Phys. Rev. Lett. 95 (11), 110404. Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A. (1995). Observation of Bose–Einstein condensation in a dilute atomic vapor. Science 269, 198. Astrakharchik, G.E., Boronat, J., Casulleras, J., Giorgini, S. (2004). Equation of state of a Fermi gas in the BEC-BCS crossover: A quantum Monte Carlo study. Phys. Rev. Lett. 93 (20), 2004040. Avdeenkov, A.V., Bohn, J.L. (2005). Pair wave functions in atomic Fermi condensates. Phys. Rev. A 71, 023609. Baker, G.A. (1999). Neutron matter model. Phys. Rev. C 60, 054311. Balakrishnan, N., Forrey, R.C., Dalgarno, A. (1998). Quenching of H2 vibrations in ultracold 3 He and 4 He collisions. Phys. Rev. Lett. 80 (15), 3224. Bardeen, J., Cooper, L.N., Schrieffer, J.R. (1957a). Microscopic theory of superconductivity. Phys. Rev. 106, 162. Bardeen, J., Cooper, L.N., Schrieffer, J.R. (1957b). Theory of superconductivity. Phys. Rev. 108, 1175. Bartenstein, M., Altmeyer, A., Riedl, S., Jochim, S., Chin, C., Hecker Denschlag, J., Grimm, R. (2004a). Collective excitations of a degenerate gas at the BEC-BCS crossover. Phys. Rev. Lett. 92, 203201. Bartenstein, M., Altmeyer, A., Riedl, S., Jochim, S., Chin, C., Hecker Denschlag, J., Grimm, R. (2004b). Crossover from a molecular Bose–Einstein condensate to a degenerate Fermi gas. Phys. Rev. Lett. 92, 120401.
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54
DETERMINISTIC ATOM–LIGHT QUANTUM INTERFACE JACOB SHERSON1,2 , BRIAN JULSGAARD1,* and EUGENE S. POLZIK1 1 QUANTOP, Danish Research Foundation Center for Quantum Optics, Niels Bohr Institute,
Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark 2 Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, bygning 1520,
DK-8000 Aarhus C, Denmark 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 2. Atom–Light Interaction . . . . . . . . . . . . . . . . 2.1. Atomic Spin Operators . . . . . . . . . . . . . 2.2. Polarization States of Light . . . . . . . . . . . 2.3. Off-Resonant Coupling . . . . . . . . . . . . . 2.4. Propagation Equations . . . . . . . . . . . . . . 2.5. The Rotating Frame . . . . . . . . . . . . . . . 2.6. Two Oppositely Oriented Spin Samples . . . . 3. Quantum Information Protocols . . . . . . . . . . . . 3.1. Entanglement—Two Mode Squeezing Protocol 3.2. Quantum Memory . . . . . . . . . . . . . . . . 4. Experimental Methods . . . . . . . . . . . . . . . . . 4.1. Paraffin Coated Vapor Cells . . . . . . . . . . . 4.2. Detection of Polarization States . . . . . . . . . 4.3. Magnetic Fields . . . . . . . . . . . . . . . . . 5. Experimental Results . . . . . . . . . . . . . . . . . 5.1. Projection Noise Level . . . . . . . . . . . . . . 5.2. Decoherence . . . . . . . . . . . . . . . . . . . 5.3. Entanglement Results . . . . . . . . . . . . . . 5.4. Quantum Memory Results . . . . . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . 7. Acknowledgements . . . . . . . . . . . . . . . . . . 8. Appendices . . . . . . . . . . . . . . . . . . . . . . . A. Effect of Atomic Motion . . . . . . . . . . . . . . . . A.1. Modeling Atomic Motion . . . . . . . . . . . . A.2. Atomic Motion as a Source of Decoherence . . B. Technical Details . . . . . . . . . . . . . . . . . . . . B.1. Light Polarization and Stark Shifts . . . . . . . B.2. Influence of Laser Noise . . . . . . . . . . . . . 9. References . . . . . . . . . . . . . . . . . . . . . . .
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* Present address: Lund Institute of Technology, Box 118, S-221 00 Lund, Sweden.
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© 2007 Elsevier Inc. All rights reserved ISSN 1049-250X DOI: 10.1016/S1049-250X(06)54002-9
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Abstract The notion of an atom–light quantum interface has been developed in the past decade, to a large extent due to demands within the new field of quantum information processing and communication. A promising type of such interface using large atomic ensembles has emerged in the past several years. In this chapter we review this area of research with a special emphasis on deterministic high fidelity quantum information protocols. Two recent experiments, entanglement of distant atomic objects and quantum memory for light are described in detail.
1. Introduction Interaction of light with atoms has always been one of the most exciting areas in AMO physics. A complete quantum-mechanical approach to atom–light interactions allows one to consider aspects such as quantum state transfer between atoms and light and generation of entangled states of light and atoms. Entangled or non-separable states form the basis for Quantum Information Processing and Communication where information is encoded, processed and transferred in quantum states of light and matter (for a collection of review articles on QIPC see, for example, Zoller et al., 2005). Processing information encoded in quantum states provides parallelism and unconditional security, the two most promising properties within this field. State transfer between matter, where quantum information is processed and stored, and light, the prime long distance carrier of information, is crucial for quantum networks and other applications. A well-designed interaction between two or more quantum systems is an efficient tool for creation of a desired joint entangled state of the systems. Such interaction alone, or combined with a projective measurement on one of the systems, can lead to the creation of a non-trivial quantum state of the remaining system(s). In this review we describe recent progress in the development of a tool box for deterministic quantum state generation of light and atomic ensembles via the well known dipole interaction. This tool box forms the basis for a quantum atom–light interface capable of generating entangled states of atoms and light on demand, performing efficient quantum state exchange between light and atoms— quantum memory for light, and long distance transfer of a quantum state of atoms via teleportation and other quantum transfer protocols. A key to successful quantum state engineering is a favorable balance between efficient unitary interaction between the systems in question (light and atoms within the context of this chapter) and decoherence caused by interaction with the environment. Historically the first setting considered most promising for the light– atoms interaction was Cavity Quantum Electrodynamics (Cavity QED) where the Jaynes–Cummings type of interaction could be realized. Driven by the interest in single photon–single atom interactions (Cirac et al., 1997), this field of research has achieved spectacular successes in both optical (McKeever et al., 2004;
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Kuhn et al., 2002) and microwave (Raimond et al., 2001) domains. Through the dramatic improvements in cavity quality factors and with the use of cold atoms, coupling of single atoms to single photon cavity modes has been demonstrated. However, formidable technical challenges of the strong coupling cavity QED regime promoted searches for alternative routes to atom–light quantum interfaces. One of the most successful of these emerged in the second half of the 90s with the recognition of the fact that the use of collective quantum states of large atomic ensembles instead of single atoms can provide efficient “strong” coupling to light outside of the cavity QED regime. An intuitive explanation goes as follows. In order to be efficiently coupled to a single atom, light has to be focused and shaped to match the atomic dipole pattern. Since this is very difficult in free space, a cavity enhancing interaction of light with a specific mode is necessary. On the other hand, a collection of atoms with transverse dimensions large compared to the wavelength of light collectively couples very efficiently to any spatial light mode which matches its shape. In classical electrodynamics this is manifested by large absorption and dispersion that characterizes an atomic ensemble interacting with nearly resonant light. Luckily, the same scaling persists for some collective quantum properties of light and atoms. The first approach to coupling quantum features of light to an atomic ensemble was based on a straightforward assumption that if the input light is completely absorbed by atoms, its quantum features should be to a certain extent transferred to the atoms. The proposal of Kuzmich et al. (1997) considered a V-type atomic level scheme where excitation in the two arms of the V-transition is carried out by quantum correlated—entangled—light modes. Absorption of these modes leads to the creation of entanglement in the excited state of atoms. If excitation, for example, is performed with squeezed light, the excited state of atoms becomes spin squeezed. In case of cw excitation spontaneous emission was shown to limit the degree of spin squeezing of atoms to 50%. The experimental implementation of this proposal has been carried out for cold cesium atoms excited by squeezed light resulting in the first demonstration of a macroscopic (106 atoms) ensemble of entangled atoms (Hald et al., 1999). Efficient coupling at the quantum level also requires that no other modes of light but the desired input mode couple to the atoms. In classical physics other modes are of no interest provided they do not contain any light. In quantum physics any empty mode is in a vacuum state. Coupling to such modes leads to spontaneous emission of atoms, the major source of decoherence which a quantum state engineer has to fight. In the Cavity QED setting interaction with vacuum modes of the field is made small compared to coupling to the relevant input mode by the strong coupling provided by the cavity. How can this type of decoherence be suppressed for a large atomic ensemble in the absence of any cavity? The solution is to use for coupling with light two substates of the atomic ground state rather than a ground and an excited state as in the standard Jaynes–Cummings
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approach. The advantages of using ground state levels are multiple. Their spontaneous decay is negligible leading to long coherence times. Their energy spacing is in the radio-frequency or microwave domain which means that with the size of the ensemble smaller than the radio—or microwave—wavelength the position of each atom is irrelevant and, hence, collective coupling insensitive to atomic motion can be achieved. In order to circumvent limitations due to spontaneous emission, the idea of complete absorption has been taken further by Kozhekin et al. (2000) where a driven Raman transition involving a weak quantum mode in one arm has been shown to be capable of faithful transfer of the light quantum state onto ground state collective atomic coherence. In a parallel development, a celebrated electro-magnetically induced transparency process utilizing carefully timed Raman pulses has been proposed for quantum storage of light (Fleischhauer and Lukin, 2002). The first experiment testing these ideas has been recently carried out (Eisaman et al., 2005). In all of the above approaches to light–atom quantum interfaces the tool box has included light–atoms interactions only. A significant next step allowing implementation of atomic entanglement and quantum memory has been made by adding a quantum measurement and feedback to the picture. The two experiments which are central to this chapter, the deterministic entanglement of distant atomic objects (Julsgaard et al., 2001) and the deterministic quantum memory for light (Julsgaard et al., 2004a), have been carried out following essentially the same scenario. A pulse of light interacts with atoms via a quantum non-demolition (QND) Hamiltonian, a projective homodyning measurement is performed on the transmitted light, and the feedback conditioned on the result of the measurement is applied to atoms (Figs. 1 and 2). Early proposals for using the QND interaction for atomic quantum state engineering have been published by Sanders and Milburn (1989), Corney and Milburn (1998), and Wiseman (1998). In particular, spin squeezing in an atomic ensemble generated by a QND measurement has been studied theoretically in Kuzmich et al. (1998), Takahashi et al. (1999), Madsen and Mølmer (2004), and experimentally demonstrated in Kuzmich et al. (2000) and Geremia et al. (2004). Quantum feedback has been explored in a series of papers by Wiseman and co-authors. In particular, quantum feedback in relation to spin squeezing has been analyzed in Thomsen et al. (2002). To complete this brief introduction into atom–light quantum interface research, we wish to draw a line between the deterministic interface reviewed in this chapter and probabilistic schemes. The latter are based on generation of entanglement between light and atoms conditioned on a random event of detection of a photon in a certain mode. Spectacular experimental progress along these lines has been achieved for schemes involving a single ion (Blinov et al., 2004; Polzik, 2004), single atom (Volz et al., 2006), and atomic ensembles (Kuzmich
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et al., 2003; van der Wal et al., 2003; Chaneliere et al., 2005; Chou et al., 2005; Eisaman et al., 2005). The chapter is organized as follows. In Section 2 we introduce quantum variables for light—Stokes operators, and for atoms—collective spin operators. An effective Hamiltonian and equations of motion for the atomic and light variables are then derived. We then expand the theory to two atomic ensembles with an addition of a bias magnetic field. In Section 3 we formulate the equations of motion in the language of canonical variables for light and atoms. Two main quantum information protocols are then described theoretically: entanglement of two atomic ensembles and quantum memory for light. In both cases the treatment includes a simple decoherence model. The quantum memory protocol which we dubbed “the direct mapping protocol” represents an alternative to the teleportation method for a high-fidelity quantum state transfer from one system to another. Unlike teleportation it involves only two objects: one carrying the input state and another the target on which the input state is transferred. Section 4 describes experimental methods and the key elements of the setup including atomic cells, magnetic fields and detection of light. Application of the magneto-optical resonance method for characterization of the collective atomic spin states is outlined. Magnetic feedback used for the spin manipulation is discussed. Section 5 contains the main experimental results on atomic entanglement and quantum memory for light. The determination of the benchmark atomic quantum noise level—the projection noise—is described in detail. The effects leading to decoherence of generated atomic entangled states are summarized. In the appendices we discuss various technical aspects, such as the effect of atomic motion on quantum state generation and decoherence, the influence of polarization of light, and of technical noise of the probe laser. We conclude with a brief summary and outlook.
2. Atom–Light Interaction In this section we introduce the quantum variables for light and atoms and describe the off-resonant dipole interaction between the ground state 6S1/2 and the excited state 6P3/2 in cesium. We use spin operators for atoms and Stokes operators for light as a convenient way to describe the interaction, and present an effective Hamiltonian which describes the dynamics of the ground state spin and the light. For simplicity the Hamiltonian is specific to the cesium ground state, although, the same procedure can be applied for any other atom with a magnetically non-degenerate ground state. With this as a starting point we derive equations of motion for the light and atomic operators of a single atomic ensemble.
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F IG . 1. Schematic layout illustrating main components of the experiments. Two clouds of Cs atoms contained in paraffin coated glass cells are placed inside the magnetic shields. A light pulse consisting of orthogonally polarized strong classical and weak quantum fields passes through the cells and is detected by the detector on the right. Electronic feedback is applied to the rf magnetic coils surrounding cells to rotate the collective atomic spin. The bias magnetic field is applied to atoms in both cells. Atoms in the two cells are optically pumped as shown in the figure.
A significant next step for quantum information protocols discussed in this chapter is the transition from a single atomic ensemble to two oppositely oriented ensembles accompanied by the addition of a constant magnetic field (Fig. 1). If this field is oriented along the mean atomic spin direction it allows for achieving entanglement between two ensembles with a single pulse of light instead of two pulses required in the absence of the field. An even more important advantage is brought about by the possibility to make measurements at a rather high Zeeman frequency, thus achieving quantum limits of sensitivity with macroscopic numbers of atoms via spectral filtering of classical noise. 2.1. ATOMIC S PIN O PERATORS The ground states of cesium are characterized by its outermost electron which is in the 6S1/2 state, i.e. the orbital angular momentum L is zero. The electron spin S and thus the total electronic angular momentum J has quantum number S = J = 1/2. The nuclear spin I of cesium-133 has I = 7/2, and the coupling between the nucleus and the electron gives rise to the total angular momentum F = I + J with quantum numbers F = 3 and F = 4. It is indeed the total angular momentum F which interests us in this work since F and the magnetic quantum numbers mF define the energy levels of the ground states in the limit of low magnetic field discussed here. Furthermore, we restrict ourselves to one hyperfine level, F = 4, which is possible experimentally since the hyperfine splitting νhfs = 9.1926 GHz is large compared to typical resolutions
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of our laser systems. We choose to denote the total angular momentum of a single atom by j and for a collection of atoms (in the F = 4 state) we denote the collective total angular momentum by J, i.e. J=
N
j(i) ,
(1)
i=1
where N is the number of atoms in the F = 4 state and j(i) is the total angular momentum of the ith atom. The reason for using J and not F is conventional, and indeed, we wish to think about our spins more abstractly than just the properties of some atoms. Many of the results should be applicable in a broader sense than to a collection of cesium atoms. In our experiments the number of atoms N is of order 1012 and we will almost always aim at having all atoms polarized along one direction which we denote as the x-axis. With the x-axis as quantization axis we have mF = 4 for all atoms to a high degree of accuracy, and the collective spin Jˆx is really a macroscopic entity. With this experimental choice, we may treat the x-component of the collective spin as a classical c-number, i.e. we replace the operator Jˆx by the number Jx . The transverse spin components Jˆy and Jˆz maintain their quantum nature. They typically have zero or a small mean value. The quantum fluctuations are governed by the commutation relation and the Heisenberg uncertainty relation (with h¯ = 1) Jˆy , Jˆz = iJx (2) J2 ⇒ Var Jˆy · Var Jˆz x . 4
(3)
With 1012 atoms the quantum uncertainty of the angle of the collective spin direction is of order 10−6 .
2.2. P OLARIZATION S TATES OF L IGHT All our experiments involve narrow-band light interacting with atomic spin states, and it turns out that the polarization states of the light form a convenient language to describe the light degrees of freedom. Consider a pulse of light, or a collection of photons, propagating in the z-direction. The polarization state is well described by the Stokes operators 1 1 Sˆx = nˆ ph (x) − nˆ ph (y) = aˆ x† aˆ x − aˆ y† aˆ y , 2 2 1 1 Sˆy = nˆ ph (+45◦ ) − nˆ ph (−45◦ ) = aˆ x† aˆ y + aˆ y† aˆ x , 2 2 1 1 † aˆ x aˆ y − aˆ y† aˆ x , Sˆz = nˆ ph (σ+ ) − nˆ ph (σ− ) = 2 2i
(4)
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where nˆ ph (x) is the number of photons in the pulse with x-polarization, and so on. The Stokes operators are dimensionless as written here, they count photons. At our convenience we later break these up into temporal or spatial slices. We make the following assumption central to the experiments described below. We assume that light consists of a strong component linearly polarized along the x-direction and a much weaker component polarized in the y-plane (Fig. 1). This means that we can treat the x-mode operators aˆ x , aˆ x† , and hence Sˆx → Sx as a cnumber. Note, this is very similar to the approximation of a well polarized sample of spins in the previous section. Specifically we find (with aˆ x† = aˆ x = Ax being real) that Sˆy = Ax · (aˆ y + aˆ y† )/2 and Sˆz = Ax · (aˆ y − aˆ y† )/2i. We see that in our approximation the quantum properties of Sˆy and Sˆz are solely encoded in the y-polarized part of the light. It can be shown that the Stokes vector satisfies angular commutation relations Sˆy , Sˆz = iSx (5) S2 ⇒ Var Sˆy · Var Sˆz x 4
(pulse of light).
(6)
2.3. O FF -R ESONANT C OUPLING We consider the special case with a propagating beam of light coupled offresonantly to the 6S1/2,F =4 → 6P3/2,F =3,4,5 transitions in cesium. Then absorption effects can be neglected and all the dynamics are of a dispersive nature. Furthermore, the optically excited states can be adiabatically eliminated and an effective Hamiltonian describing the light interacting with only ground state degrees of freedom is obtained (Julsgaard, 2003; Hammerer et al., 2005a)1 . h¯ cγ λ2 eff =− Hˆ int 8AΔ 2π
L
ˆ t) + a1 · Sˆz (z, t)jˆz (z, t) a0 · φ(z,
0
ˆ t)jˆz2 (z, t) − Sˆ− (z, t)jˆ+2 (z, t) − Sˆ+ (z, t)jˆ−2 (z, t) + a2 φ(z, × ρA dz.
(7)
Here we have assumed a one-dimensional theory for the light which is sufficient for a beam cross section A that is much larger than the squared wavelength λ2 . The small letter spin operators jˆ(z, t) are dimensionless and refer to single atoms ˆ t) are taken to count the number at position z at time t. The Stokes operators S(z, of photons per unit length at position z and time t. The integration then runs over 1 Julsgaard (2003) contains a factor of two error which has been corrected in Eq. (7).
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the entire sample of length L with atomic density ρ. The factor γ in front of Eq. (7) is the natural FWHM line width of the optical transition 6S1/2 → 6P3/2 and Δ is the detuning from the F = 4 to F = 5 transition with red being positive. ˆ t) is the photon flux per unit length, Sˆ+ = Sˆx + As for the operators, φ(z, † † aˆ − and Sˆ− = Sˆx − i Sˆy = −aˆ − aˆ + are raising and lowering operators i Sˆy = −aˆ + converting σ+ -photons into σ− -photons or vice versa, and jˆ± = jˆx ± i jˆy are the usual raising and lowering operators for the spin. The parameters a0 , a1 , and a2 for the F = 4 ground state in cesium are given by 1 7 1 + + 8 → 4 (F = 4), a0 = 4 1 − 35 /Δ 1 − Δ45 /Δ 1 35 21 − − + 176 → 1, a1 = (8) 120 1 − Δ35 /Δ 1 − Δ45 /Δ 1 21 5 − + 16 → 0, a2 = 240 1 − Δ35 /Δ 1 − Δ45 /Δ where the limit is calculated for very large values of the detuning. The detunings Δ35 /2π = 452.2 MHz and Δ45 /2π = 251.0 MHz are given by the splitting in the excited state. Let us comment on the different terms in the Hamiltonian (7). The first term containing a0 just gives a Stark shift to all atoms independent of ˆ t). The second term the internal state but proportional to the photon density φ(z, containing a1 rotates the Stokes vector S and the spin vector J around the z-axis, known as Faraday rotation—we look more closely into this below. The last terms proportional to a2 are higher order couplings between the light and the atoms and since a2 is small for a sufficiently large detuning these can normally be neglected. All these terms conserve individually the z-projection of the total angular momentum of light and atoms, e.g., the Sˆ− jˆ+2 term can change a σ+ photon into a σ− photon (changing the light angular momentum along z by −2h¯ while the atoms receive 2h¯ mediated by the atomic raising operator jˆ+2 . The total angular momentum must have its z-projection invariant since the physical system is axially symmetric around the direction of light propagation (the z-axis). For us the term proportional to a1 is useful and relevant. This term represents the QND interaction. The higher order terms proportional to a2 create some minor problems which can be minimized with large detuning (this will be discussed further in Appendices B.1 and B.2). For a detailed treatment of the higher order—atomic alignment—effects we refer to Kupriyanov et al. (2005). The zeroth order term proportional to a0 produces an overall phase shift and can be omitted.
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2.4. P ROPAGATION E QUATIONS The Hamiltonian (7) is a very convenient starting point for many calculations and we now show the procedure to derive equations of motion. For the spin operators we need the Heisenberg evolution ∂ jˆi /∂t = i1h¯ [jˆi , Hˆ ] and for the Stokes operators the Maxwell–Bloch equations (∂/∂t + c · ∂/∂z)Sˆi (z, t) = i1h¯ [Sˆi (z, t), Hˆ int ], see Julsgaard (2003). For the latter we neglect retardation effects, i.e. we do not calculate dynamics on the time scale L/c of propagation across the sample. This is equivalent to setting the speed of light to infinity and hence leaving out the ∂/∂t term. If we consider only the term proportional to a1 and neglect the other we find ∂ jˆx (z, t) cγ λ2 ˆ =+ a1 Sz (z, t)jy (z, t), ∂t 8AΔ 2π ∂ jˆy (z, t) cγ λ2 ˆ =− a1 Sz (z, t)jx (z, t), ∂t 8AΔ 2π ∂ jˆz (z, t) = 0. ∂t
(9)
∂ ˆ γρ λ2 ˆ Sx (z, t) = + a1 Sy (z, t)jˆz (z, t), ∂z 8Δ 2π γρ λ2 ˆ ∂ ˆ Sy (z, t) = − a1 Sx (z, t)jˆz (z, t), ∂z 8Δ 2π ∂ ˆ Sz (z, t) = 0. ∂z
(10)
and
We observe that jˆz (z, t) and Sˆz (z, t) are individually conserved during the interaction which is also apparent from the Hamiltonian (7) since these operators commute with the a1 -term. The effect of the interaction is that the spin rotates around the z-axis with an amount proportional to Sˆz , and the Stokes vector rotates around the z-axis proportionally with jˆz . Below we assume that these rotations are small and that the dominant classical (mean) polarization vector of light and the orientation vector of the collective atomic spin stay oriented along the x-direction after the interaction. This turns out to be a very good approximation for experimentally attainable values of the interaction strength. Under this assumption the first line of the systems (9) and (10), respectively, can be omitted. Furthermore, given the QND structure of the remaining equations, we can sum over the individual atomic spins and obtain the same equation for the collective spin variables (1). In our continuous notation L we have Jˆi (t) = 0 jˆi (z, t)ρA dz. As for the light operators we concentrate on the in- and out-going parts only. Hence we define Sˆiin = cSˆi (z = 0, t) and
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Sˆiout = cSˆi (z = L, t). The multiplication by the speed of light c turns the normalization into photons per unit time instead of per unit length. With the assumption of small rotation angles, integrating Eqs. (9) and (10) over space from z = 0 to z = L leads to the following very important equations: Sˆyout (t) = Sˆyin (t) + aSx Jˆz (t),
(11)
Sˆzout (t)
(12)
=
Sˆzin (t),
∂ ˆ Jy (t) = aJx Sˆzin (t), ∂t ∂ ˆ Jz (t) = 0, ∂t
(13) (14)
2
γ λ where a = − 8AΔ 2π a1 . In and out refer to light before and after passing the atomic sample, respectively. We note from Eqs. (11) and (14) that in the case of a large interaction strength (i.e. if aSx Jˆz dominates Sˆyin ) a measurement on Sˆyout amounts to a QND measurement of Jˆz . Using off-resonant light for QND measurements of spins has also been discussed in Kuzmich et al. (1998) and Takahashi et al. (1999). Equation (13) implies that a part of the state of light is also mapped onto the atoms—we denote this as back action. This opens up the possibility of using this sort of system for quantum memory which will be the topic of Sections 3.2 and 5.4.
2.5. T HE ROTATING F RAME In the experiment a constant and homogeneous magnetic field is added in the x-direction. We discuss the experimental reason for this below. For our modeling, the magnetic field adds a term Hˆ B = Ω Jˆx to the Hamiltonian. This makes the transverse spin components precess at the Larmor frequency Ω depending on the strength of the field. Introducing the rotating frame coordinates: ˆ ˆ Jy Jy cos(Ωt) sin(Ωt) (15) = − sin(Ωt) cos(Ωt) Jˆz Jˆ z
we can easily show that Eqs. (11)–(14) transform into: Sˆyout (t) = Sˆyin (t) + aSx Jˆy (t) sin(Ωt) + Jˆz (t) cos(Ωt) , Sˆzout (t)
=
Sˆzin (t),
∂ ˆ J (t) = aJx Sˆzin (t) cos(Ωt), ∂t y ∂ ˆ J (t) = aJx Sˆzin (t) sin(Ωt). ∂t z
(16) (17) (18) (19)
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Thus, the atomic imprint on the light is encoded at the Ω-sideband instead of at the carrier frequency. The primary motivation for adding the magnetic field is the fact that lasers are generally much more quiet at high sideband frequencies than at the carrier frequency. A measurement without a magnetic field is a DC measurement for which the technical noise dominates the subtle quantum signal. Also, as the measurement time is longer than 1/Ω, Eq. (16) enables us to access both Jy and Jz at the same time. We are of course not allowed to perform nondestructive measurements on these two operators simultaneously since they are non-commuting. This is also reflected by the fact that neither Jˆy nor Jˆz are constant in Eqs. (18) and (19). Below we consider two atomic samples and we see that a QND-type interaction can be regained in this setting. 2.6. T WO O PPOSITELY O RIENTED S PIN S AMPLES Inspired by the above we now assume that we have two atomic samples with oriented spins such that Jx1 = −Jx2 ≡ Jx . We re-express the equations of motion (16)–(19) for two samples in a way which is much more convenient for the understanding of our entanglement creation and verification procedure. For two atomic samples we write equations of motion: (t) + Jˆy2 (t) sin(Ωt) Sˆyout (t) = Sˆyin (t) + aSx Jˆy1 (t) + Jˆz2 (t) cos(Ωt) , + Jˆz1 (20) ∂ ˆ (21) (t) = a(Jx1 + Jx2 )Sˆzin (t) cos(Ωt) = 0, J (t) + Jˆy2 ∂t y1 ∂ ˆ (t) = a(Jx1 + Jx2 )Sˆzin (t) sin(Ωt) = 0. J (t) + Jˆz2 (22) ∂t z1 (t)+ Jˆ (t) and Jˆ (t)+ Jˆ (t) have zero time derivative The fact that the sums Jˆy1 y2 z1 z2 relies on the assumption of opposite spins of equal magnitude. The constancy of these terms together with Eq. (20) allows us to perform QND measurements on the two sums. We note that each of the sums can be accessed by considering the two operators T
Sˆyout cos(Ωt) dt =
T
0
0
T
T
0
Sˆyout sin(Ωt) dt =
0
aSx T ˆ (t) , Jz1 (t) + Jˆz2 Sˆyin cos(Ωt) dt + 2
(23)
aSx T ˆ (t) . Jy1 (t) + Jˆy2 Sˆyin sin(Ωt) dt + 2
(24)
T T We have used the fact that 0 cos2 (Ωt) dt ≈ 0 sin2 (Ωt) dt ≈ T /2 and that T 0 cos(Ωt) sin(Ωt) dt ≈ 0. Each of the operators on the left-hand side can
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be measured simultaneously by making an Sˆy -measurement and multiplying the photo-current by cos(Ωt) or sin(Ωt) followed by integration over the duration T . (t) + Jˆ (t) and Jˆ (t) + Jˆ (t) enThe possibility to gain information about Jˆy1 y2 z1 z2 ables us to generate entangled states, the topic of Sections 3.1 and 5.3. At the same time we must lose information about some other physical variable. This is indeed (t) − Jˆ (t) and Jˆ (t) − Jˆ (t), true, the conjugate variables to these sums are Jˆz2 z1 y1 y2 respectively. These have the time evolution ∂ ˆ (25) (t) = 2aJx Sˆzin (t) cos(Ωt), Jy1 (t) − Jˆy2 ∂t ∂ ˆ (t) = 2aJx Sˆzin (t) sin(Ωt). (26) Jz1 (t) − Jˆz2 ∂t We see how noise from the input Sˆz -variable is piling up in the difference components while we are allowed to learn about the sum components via Sˆy measurements. The equations above clearly describe the physical ingredients for light and atoms. However, when transfer of a quantum state between two very different systems—light and atoms—is concerned, an isomorphic formulation, such as rendered by canonical variables, is desirable. In Section 3 we will re-write the theory in this more convenient language.
3. Quantum Information Protocols In this section we introduce in detail the two main protocols of this chapter: deterministic generation of entanglement of two macroscopic objects and deterministic quantum state mapping from one system onto another—the so-called direct mapping protocol. The latter is then applied as the quantum memory for light protocol. In order to simplify the description of the protocols we re-write equations of motion of the previous section in the language of canonical variables. This also allows for unified treatment of a single ensemble without- and two ensembles, with a constant magnetic field. For a single sample we define: Jˆy Xˆ As = √ , Jx 1 Xˆ Ls = √ Sx T
Jˆz PˆAs = √ , Jx T
Sˆy (t) dt,
0
1 PˆLs = √ Sx T
T
(27) Sˆz (t) dt.
0
For two samples we get two sets of canonical variables by defining the atomic operators: Xˆ A1 =
− Jˆ Jˆy1 y2 , √ 2Jx
(28a)
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+ Jˆ Jˆz1 √ z2 , 2Jx
Jˆ − Jˆz2 , Xˆ A2 = − z1 √ 2Jx
PˆA2 =
+ Jˆ Jˆy1 y2 √ 2Jx
[3 (28b) (28c) (28d)
and the light operators: 2 Sx T
Xˆ L1 =
2 Sx T 2 Sx T
T
Sˆz (t) cos(Ωt) dt,
(29b)
T
Sˆy (t) sin(Ωt) dt,
(29c)
Sˆz (t) sin(Ωt) dt.
(29d)
0
PˆL2 =
(29a)
0
Xˆ L2 =
Sˆy (t) cos(Ωt) dt,
0
PˆL1 =
T
2 Sx T
T 0
ˆ Pˆ operators satisfy the usual commutation relation, e.g., Each pair of X, [Xˆ L1 , PˆL1 ] = i. Equations (11)–(14) and (20)–(22) now translate into out in in = Xˆ Li + κ PˆAi , Xˆ Li
(30a)
out in = PˆLi , PˆLi
(30b)
out Xˆ Ai
=
in Xˆ Ai
out in PˆAi = PˆAi ,
in + κ PˆLi ,
(30c) (30d)
where we recall that i = 1, 2, s refer to the definitions above and not the two samples. Note, that in the case of two samples the two sets of interacting light and atomic operators are decoupled. The √parameter describing the strength of light/matter-interactions is given by κ = a Jx Sx T . The limit to strong coupling is reached around κ ≈ 1. This set of equations represents the starting point for numerous applications, which in the context of this paper are implemented in the double sample setup.
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3.1. E NTANGLEMENT—T WO M ODE S QUEEZING P ROTOCOL It follows from Eqs. (30a) and (30d) that a sufficiently precise measurement of Xˆ L of a light pulse transmitted through two atomic ensembles renders knowledge about PˆA . The measurement projects PˆA into a two-mode squeezed or Einstein– Podolsky–Rosen entangled state (Julsgaard et al., 2001). The necessary and sufficient condition for such an entangled state has been derived by Duan et al. (2000). Demonstration of entanglement is thus reduced to the fulfillment of this criterion. The criterion, which is introduced shortly, utilizes the vacuum state noise of a canonical variable, which for atomic ensembles reduces to the coherent spin state fluctuations. Hence, these fluctuations, which are also called projection noise, form an extremely important benchmark in experiments with atomic ensembles. 3.1.1. The Coherent Spin State To create entangled or squeezed states one has to generate states which exhibit less fluctuations than all equivalent classical states. The boundary occurs at the coherent spin state (CSS) in which all spins are independent realizations of a single spin oriented along a specific direction. The characteristics of this state are discussed in more detail in Section 5.1 so for now we need only to quantify the role of the coherent spin state as a boundary between classical and purely quantum mechanical states. From the canonical atomic operators a Heisenberg uncertainty relation can be formed: |[Xˆ Ai , PˆAi ]|2 1 = . Var Xˆ Ai Var PˆAi (31) 4 4 For the coherent spin state the two variances are both equal to one half, thus confirming that it is the classical state with the least possible noise. For a state to be squeezed, it has to have less noise in one of the quadratures. Since the Heisenberg uncertainty relation still has to be fulfilled it follows that the other quadrature has to exhibit excessive fluctuations, it is anti-squeezed. Entanglement is the non-local interconnection of two systems, such that it is impossible to write the total density matrix as a product of density matrices for each system. It has been shown that for effectively continuous variable systems such as the ones described so far in this paper the necessary and sufficient criterion for entanglement is (Duan et al., 2000) Var PˆA1 + Var PˆA2 < 1. (32) 3.1.2. Entanglement Generation and Verification We now turn to the actual understanding of entanglement generation and verification. We concentrate here on generation of an entangled state conditioned on the
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result of a measurement. Generation of an unconditional entangled state with the help of feedback is described in the experimental section below. For generation of conditional entanglement we perform the following steps: First the atoms are prepared in the oppositely oriented coherent states corresponding to creating the vacuum states of the two modes (Xˆ A1 , PˆA1 ) and (Xˆ A2 , PˆA2 ). Next a pulse of light called the entangling pulse is sent through the atoms and we measure the two opout and X ˆ out with outcomes A1 and B1 , respectively. These results bear erators Xˆ L1 L2 information about the atomic operators PˆA1 and PˆA2 and hence we reduce variances Var(PˆA1 ) and Var(PˆA2 ). To prove that we have an entangled state we must confirm that the variances of PˆA1 and PˆA2 fulfill the criterion (32). That is we need to know the mean values of PˆA1 and PˆA2 with a total precision better than unity. To demonstrate that, we send a second verifying pulse through the atomic out and X ˆ out with outcomes A2 and B2 . Now it is a samples again measuring Xˆ L1 L2 matter of comparing A1 with A2 and B1 with B2 . If the results are sufficiently close the state created by the first pulse was entangled. Let us be more quantitative. The interaction (30a) mapping the atomic operators PˆAi to field operators Xˆ Li is very useful for large κ, and useless if κ 1. We will describe in detail the role of κ for all values. To this end we first describe the natural way to determine κ experimentally. If we repeatedly perform the first two steps of the measurement cycle, i.e. prepare coherent states of the atomic spins, send in the first measurement pulse, and record outcomes A1 and B1 , we may deduce the statistical properties of the measurements. Theoretically we expect from (30a) 1 κ2 + . (33) 2 2 The first term in the variances is the shot noise (SN) of light. This can be measured in the absence of the interaction where κ = 0. The quantum nature of the shot noise level is confirmed by checking the linear scaling with the photon number of the pulse. The second term arises from the projection noise (PN) of atoms. Hence, we may calibrate κ 2 to be the ratio κ 2 = PN/SN of atomic projection noise to shot noise of light. Theoretically κ 2 has a linear scaling κ 2 = aJx Sx T with a macroscopic spin Jx that must be confirmed in the experiment (see Section 5.1). Next we describe how to deduce the statistical properties of the state created by the entangling pulse. Based on the measurement results A1 and B1 of this pulse we must predict the mean value of the second measurement outcome. If κ → ∞ in we ought to trust the first measurement completely since the initial noise of Xˆ Li is negligible, i.e. A2 = A1 and B2 = B1 . On the other hand, if κ = 0 we know that atoms must still be in the vacuum state such that A2 = B2 = 0. It is natural to take in general A2 = αA1 and B2 = αB1 . We need not know a theoretical value for α. The actual value can be deduced from the data. If we (i) (i) (i) (i) repeat the measurement cycle N times with outcomes A1 , B1 , A2 , and B2 , A1 = B1 = 0 and
Var(A1 ) = Var(B1 ) =
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the correct α is found by minimizing the conditional variance Var(A2 |A1 ) + Var(B2 |B1 ) (i) 1 (i) (i) 2 (i) 2 A2 − αA1 . + B2 − αB1 N −1 N
= min α
(34)
i
In order to deduce whether we fulfill the entanglement criterion (32) we compare the above to our expectation from (30a). For the verifying pulse we get 2 out out 2 in,2nd = Xˆ + κ Pˆ ent − Pˆ ent Xˆ − Xˆ Li
Li
Ai
Li
ent 1 , = + κ 2 Var PˆAi 2
Ai
(35)
in,2nd refers to the incoming light of the verifying pulse which has zero where Xˆ Li ent ˆ mean. PAi refers to the atoms after being entangled. We see that the practical entanglement criterion becomes ent ent Var(A2 |A1 ) + Var(B2 |B1 ) = 1 + κ 2 Var PˆA1 + Var PˆA2
< 1 + κ 2 = Var(A1 ) + Var(B1 ).
(36)
Simply stated, we must predict the outcomes A2 and B2 with a precision better than the statistical spreading of the outcomes A1 and B1 with the additional constraint that A1 and B1 are outcomes of quantum noise limited measurements. 3.1.3. Theoretical Entanglement Modeling We described above the experimental procedure for generating and verifying the entangled states. Here we present a simple way to derive what we expect for the ent ). mean values (i.e. the α-parameter) and for the variances Var(PˆAi We calculate directly the expected conditional variance of A2 based on A1 : out,2nd out,1st 2 − α Xˆ L1 Xˆ L1 in in,2nd in,1st ent 2 − α Xˆ L1 + κ PˆA1 − α PˆA1 = Xˆ L1 1 = 1 + α 2 + κ 2 (1 − α)2 . (37) 2 In the second step we assume that a perfect QND measurement without any dein = Pˆ ent . By taking the derivative with respect to coherence is performed, i.e. PˆA1 A1 α we obtain the theoretical minimum κ2 1 + κ2 ent ent 1 + Var PˆA2 = ⇒ Var PˆA1 1 + κ2
Var(A2 |A1 ) + Var(B2 |B1 ) = 1 +
(38)
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obtained with the α-parameter α=
κ2 . 1 + κ2
(39)
It is interesting that, in principle, any value of κ leads to the creation of entanglement. The reason for this is our prior knowledge of the input state. The atoms are in a coherent state which is as well defined in terms of variances as is possible for separable states. We need only an “infinitesimal” extra knowledge about the spin state to go into the entangled regime. It is also interesting to see what happens to the conjugate variables Xˆ Ai in the entangling process. This is governed by Eq. (30c). We do not perform measurein so all we know is that both X ˆ in and Pˆ in are in ments of the light operator PˆLi Li Ai ent ) = (1 + κ 2 )/2 and we preserve the minimum the vacuum state. Hence Var(Xˆ Ai ent ) Var(Pˆ ent ) = 1/4. uncertainty relation Var(Xˆ Ai Ai 3.1.4. Entanglement Model with Decoherence Practically our spin states decohere between the light pulses and also in the presence of the light. We model this decoherence naively by attributing the entire effect to the time interval between the two pulses, i.e. we assume that there is no decoherence in the presence of the light but instead a larger decoherence between the pulses. We may then perform an analysis in complete analogy with the above ent = β Pˆ in + 1 − β 2 Vˆ where Vˆ is a vacuum with the only difference that PˆA1 p p A1 operator admixed such that β = 0 corresponds to a complete decay to the vacuum state and β = 1 corresponds to no decoherence. Completing the analysis we find the theoretical conditional variances 1 + (1 − β 2 )κ 2 1 + κ2 ent 1 + (1 − β 2 )κ 2 ent + Var PˆA2 = ⇒ Var PˆA1 1 + κ2
Var(A2 |A1 ) + Var(B2 |B1 ) = 1 + κ 2
(40)
obtained with α-parameter α=
βκ 2 . 1 + κ2
(41)
In the limit β → 1 these results agree with (38) and (39). For β → 0 we have α → 0 (outcomes A1 and B1 are useless) and the variance approaches that of the vacuum state which is a separable state. Note that in the analysis above, the decoherence is assumed to be towards the coherent spin state. To model decoherence towards, e.g., a thermal state, the more sophisticated methods discussed in Section 3.1.5 have to be employed.
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3.1.5. Gaussian State Modeling Following the extensive study of the evolution of Gaussian states during arbitrary interactions and measurements (Giedke and Cirac, 2002; Eisert and Plenio, 2003) the development of spin squeezing in a single atomic sample and entanglement in two samples were treated in detail theoretically in Madsen and Mølmer (2004) and Sherson and Mølmer (2005). Arbitrary Gaussian states of n canonical modes are fully characterized by a 2n×1 vector, v, describing the mean values and a 2n×2n matrix, γ , describing the correlations within the atomic and light systems and the cross-correlations between these. In this formalism an interaction between light and atoms is governed by: v → Sv and γ → Sγ S T . For the Faraday interaction, the coefficients of the matrix S are easily determined from Eqs. (30). Decoherence is easily included but most importantly, there is an explicit expression for the state of the remaining modes after an arbitrary homodyne measurement on a number of the modes is performed. If the incoming pulse of light is split into a large number of segments and the interaction with atoms and subsequent measurement of each segment is treated sequentially differential equations for v(t) and γ (t) can be obtained in the limit of infinitesimal segment durations. The former is a stochastic differential equation determined by the outcome of the measurement, whereas the latter is deterministic, although non-linear because of the measurement dynamics. In Madsen and Mølmer (2004) and Sherson and Mølmer (2005) such differential equations are solved giving the time resolved dynamics in the presence of, e.g., atomic decoherence and light losses. In this way, e.g., analytic expressions for the optimal degree of spin squeezing and the degree of entanglement at arbitrary rotation frequencies (the verification and generalization of Eq. (38)) can be obtained easily.
3.2. Q UANTUM M EMORY For complete quantum memory we require (1) that the light state to be stored is supplied by a third party in an unknown state, (2) that this state is mapped onto an atomic state with a fidelity higher than the best classical fidelity, and finally (3) that the stored state can be retrieved from memory. As is described in Section 5.4 the first two criteria have been met experimentally in Julsgaard et al. (2004a) whereas the last one still remains an unsolved experimental challenge. A recently developed experimentally feasible protocol for retrieval now exists and is discussed in Section 3.2.3. 3.2.1. Direct Mapping Protocol In Eqs. (30a)–(30c) one of the light variables is mapped onto one of the atomic variables. This represents a natural starting point for a quantum memory protocol
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in which the entire light mode described by the two non-commuting variables Xˆ in and Pˆ in is faithfully stored. In the so-called “direct mapping protocol” of Julsgaard et al. (2004a) the mapping is completed by measuring the remaining light quadrature Xˆ Lout = Xˆ Lin + κ PˆAin and feeding the result back into the atomic Xˆ A with a gain of g: out in Xˆ A = Xˆ A + κ PˆLin ,
(42a)
PˆAout = PˆAout − g Xˆ Lout = PˆAin (1 − κg) − g Xˆ Lin .
(42b)
If κ = g = 1 and the initial atomic state is assumed to be a coherent state with zero mean value the mean values of both light variables are stored faithfully in the atoms. Although the initial atomic state has zero mean, it is a quantum mechanically fluctuating state, and any uncanceled atomic part increases the noise of the final state and thus degrades the mapping performance. Although this protocol works for any state, in the following we discuss storage of coherent states of light, i.e. vacuum states which are displaced by an unknown amount in phase space. For in contribution limthe storage of an arbitrary coherent light state the remaining Xˆ A its the storage fidelity to 82%. This can be remedied by initially squeezing the atomic state, in which case 100% fidelity can be reached in the limit of infinite squeezing. 3.2.2. Mapping with Decoherence Just as in the case of entanglement generation the spin states decohere. Again we can model this by a beam splitter type admixture of vacuum components right after the passage of the first light pulse. We can furthermore model light damping (e.g., reflection losses) in a similar way to obtain: in out → β Xˆ A + κ PˆLin + 1 − β 2 VXA , Xˆ A (43a) PˆAout → β − gκ ζ PˆAin − g ζ Xˆ Lin + 1 − β 2 VP A − g 1 − ζ VXL . (43b) Here 1 − ζ is the fraction of the light lost after the interaction with the atoms. √ = βκ and gF = g ζ , We see that PˆLin and Xˆ Lin are mapped with gains gBA respectively. The variances can easily be calculated to be: 2 , Var X out = 1 + gBA Var P out = 1 +
gF2 ζ
+
(44a) 2 gF2 gBA β2
− 2gF gBA .
(44b)
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3.2.3. Quantum Memory Retrieval As mentioned in Julsgaard et al. (2004a) the stored state can in principle be retrieved by inverting the roles of light and atoms in the direct mapping protocol. This would involve first an interaction between a read-out light beam with the atomic sample acting as a storage medium. According to Eq. (30a) this would map PˆA onto the light. Next Xˆ A has to be measured and feedback applied to the read-out beam according to the result of the measurement. However, since the atomic measurement requires a certain time during which the read-out pulse propagates at the speed of light, the feedback is only practical for pulse durations shorter than a microsecond. In the experiments of Julsgaard et al. (2004a) pulses of millisecond duration (∼ 300 km) are required in order to obtain a sufficiently high interaction strength, and the inverse direct mapping protocol is thus infeasible for this experimental realization. Several years ago a retrieval scheme, which did not involve measurements, but instead used two orthogonal passages of the read-out pulse was proposed (Kuzmich and Polzik, 2003). In each passage one of the atomic quadratures is mapped onto the light pulse, and in this way retrieval fidelities of up to 82% can be achieved with coherent readout light and up to 100% with perfectly squeezed readout light. Unfortunately in order to preserve the QND nature of each of the two interactions the light has to pass entirely through the atomic medium before proceeding to the second passage. This again renders the protocol inapplicable to all setups requiring “long” pulses. Recently this problem was eliminated by solving the complex dynamics arising when a light beam passes through the atomic medium along two orthogonal directions simultaneously (Sherson et al., 2005b). This protocol works with arbitrary pulse durations and the fidelity of this two-pass protocol has been calculated both for a coherent input state and for a light qubit. It was also shown that if the light is reflected back after the second passage, thus completing four passages, and a time dependent interaction strength is applied perfect retrieval can be achieved without requiring squeezed initial states. In a related proposal (Fiurášek et al., 2005) retrieval is achieved by sending two different beams through the atomic samples simultaneously, each in a direction orthogonal to the other. The fidelity of retrieval for coherent states including realistic light losses is calculated and shown to exceed the classical bound. 3.2.4. Alternative Quantum Memory Proposals In Kuzmich and Polzik (2003) various protocols which can be implemented using the QND-Faraday interaction are described in detail, including atom-atom teleportation. Instead of reviewing these, we would like to mention some recent theoretical proposals for improving the quantum memory performance. All of these break with the simple interaction scheme either by introducing a non-QND
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interaction (as was also done in Sherson et al., 2005b; Fiurášek et al., 2005) or by exploiting the fact that the atoms involved are not simple spin-1/2 atoms and thus contain more than two magnetic sublevels. In Opatrny and Fiurasek (2005) it is proposed to exploit additional atomic coherences to enhance the capacity of an atomic quantum memory. Remember that during the usual QND Faraday interaction m = 1 atomic coherences are coupled to the sideband of the light at the Larmor frequency. If, in addition, circularly polarized classical fields are present during the interaction, coupling between m = 2 coherences and a light sideband at twice the Larmor frequency is also created. In this way an additional quantum channel is added to the quantum memory, thus enhancing its capacity. By appropriately tuning the detuning of the additional coupling field two-mode squeezer and beam splitter Hamiltonians can also be realized. With the latter, a quantum memory is realized in a single passage without a requirement of measurements or prior squeezing of atoms or light. In Opatrny (2005) the author proposes to create a single cell atomic memory by optically pumping the sample into an incoherent mixture of the two extreme magnetic sublevels (m = −4 and m = 4 for the F = 4 hyperfine level of cesium). This is done with suitable linearly polarized light. A light pulse sent through the atomic sample interacts with both extreme coherences simultaneously, creating the usual QND–Faraday interaction. Apart from the prospect of making the memory more compact, this proposal would also substantially decrease the reflection losses associated with the many air-glass transitions in a two-cell setup. A main problem in this approach is that in the presence of the probe light the two coherences experience different AC-Stark shifts and thus different phase evolutions. For relevant pulse durations this effect is in fact significant. The author proposes to solve this by either decreasing the bias magnetic field or by introducing an additional AC-Stark shift by adding a light field with appropriate polarization and detuning. An atomic memory could of course also be implemented using light–atom teleportation. This could be achieved by first sending an auxiliary pulse through the two oppositely oriented atomic samples. This would entangle the light and atomic systems via the QND–Faraday interaction. After this, the auxiliary light beam should be mixed on a 50/50 beam splitter with the quantum light signal to be teleported. Measurements of Sy and Sz in the two output ports respectively and subsequent feedback to the atomic system would complete the teleportation. Unfortunately the achievable fidelity is limited to 67% with the usual QNDinteraction. It was therefore proposed in Hammerer et al. (2005a) to replace the two atomic samples in the protocol just sketched by a single sample, still with a constant bias magnetic field. This changes the interaction dramatically. Both atomic quadratures are transferred to the light, albeit at the cost of additional new modes of light. The resulting multi-mode light–atom entanglement enables teleportation with up to 77% fidelity.
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F IG . 2. (a) A photographic view of a typical experimental setup. Atomic vapor cells are placed inside the cylindrical magnetic shields. The pumping beams are indicated with dashed arrows and the path of the quantum probe field is marked with the solid arrows. (b) A schematic view of the setup. The probe pulses reach a detection system measuring Sˆy (t). The photo current is sent to a lock-in amplifier which singles out the sin(Ωt) and cos(Ωt) parts as we discuss in Section 4.2. These are integrated and stored in a PC. Typically, the pulse sequence consists of (1) a pumping pulse, (2) and (3) two laser pulses for quantum manipulation and detection of the atomic states. A very small portion of the probe field is sent through each sample in the x-direction to measure the magnitude Jx of the macroscopic spins by Faraday rotation measurements.
4. Experimental Methods In this section we describe the typical setup for our experiments. It is centered around two glass cells filled with Cs vapor at room temperature placed in two separate magnetically insulating shields with a bias magnetic field inside. Additional coils are used to apply an rf magnetic field with the frequency equal to the Larmor frequency of the bias field. Atoms in the cells are optically pumped. The setup, therefore, is similar to that of a classical magneto-optical resonance experiment. The typical experimental setup is shown in Fig. 2. A Verdi V8 pumped Ti:sapphire laser delivers what we call the probe field. This is used for quantum manipulation of the atomic samples and also for detecting the macroscopic spin Jx . Diode lasers provide optical pumping fields for creating highly polarized spin states. The cw lasers are modulated by AOMs or EOMs to achieve the desired pulse shapes. Atoms are contained in vapor cells and placed in stable magnetic fields. The magnetic field homogeneity must be of order 10−3 across the vapor cell volume.
4.1. PARAFFIN C OATED VAPOR C ELLS In our experiments the atomic samples are contained in a paraffin coated vapor cell. The coating prevents depolarization of the spin state when atoms hit the
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F IG . 3. The setup for measuring Sˆy . A strong classical pulse is mixed on a polarizing beam splitter with the quantum field which can be in a vacuum state, as in the entanglement experiment. Measurement is performed in the 45◦ - and −45◦ -basis by two detectors. Half of the difference of the two photo currents is Sˆy . An optional λ/4-retardation plate turns this measurement into the measurement of Sˆz component.
walls. We have measured spin coherence times exceeding 40 ms and spin polarizations exceeding 99% by the methods described in Section 4.3.1. The atomic density and thereby the macroscopic spin Jx can be controlled by heating or cooling the vapor cell. In order to achieve a stable vapor density, temperature gradients across the cell should be avoided. Temperature control by air flow is a convenient solution. Metal heating/cooling elements cause severe problems since the atoms are disturbed by random magnetic fields created by thermal currents even if aluminum is used, see Julsgaard (2003). Further information on paraffin coated cells can be found in Bouchiat and Brossel (1966) and Alexandrov et al. (1996, 2002).
4.2. D ETECTION OF P OLARIZATION S TATES The Stokes parameters are measured with low noise photo detectors. We use high quantum efficiency photo diodes and home made amplifiers characterized by negligible electronic noise compared to the shot noise of light at optical power higher than 1 mW. In Fig. 3 we depict how the Sˆy -component of light is measured. The differential photo current i(t) from the two detectors corresponds to a realization of the measurement of Sˆy . By passing i(t) through a lock-in amplifier we can detect the sine and cosine components at the Larmor precession frequency Ω. Practically, the current i(t) is multiplied by cos(Ωt) and sin(Ωt) and integrated over time for this purpose. According to (29a)–(29d), with appropriate scaling, this exactly corresponds to measuring the Xˆ L1 and Xˆ L2 components of light. For Sˆz -detection we would measure PˆL1 and PˆL2 .
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4.3. M AGNETIC F IELDS As was discussed in Section 2.5, external magnetic fields are added to control the dynamics of the atomic samples. The interaction of atoms with a magnetic field is governed by the Hamiltonian Hˆ mag = gF μB J · B + O B 2 . (45) We stress here that J is the total angular momentum of the atom including the nuclear spin. For the F = 4 ground state of cesium, gF ≈ 1/4. The second term O(B 2 ) reminds us that the above linear equation is only approximately true. When the magnetic energy becomes comparable to the hyperfine splitting of the ground state the response is non-linear. We comment on this below. Adding a constant bias magnetic field Bx in the x-direction leads to the equations of motion (15) with Larmor precession at frequency ΩL = gF μB Bx /h¯ . If we furthermore add an RF magnetic field at frequency Ω along the y-direction such that Bext = Bx ex + Bc cos(Ωt + φ) + Bs sin(Ωt + φ) ey (46) with constants Bc and Bs we may derive for the rotating frame coordinates Jˆy and Jˆz of (15) that ∂ Jˆy (t) ∂t
= −ωs sin(ΩL t) sin(Ωt + φ)Jx ,
(47) ˆ ∂ Jz (t) = −ωc cos(ΩL t) cos(Ωt + φ)Jx , ∂t with ωc,s = gF μB Bc,s /h¯ . Choosing the phase and the frequency of the RF-drive such that φ = 0 and Ω = ΩL we obtain: ∂ Jˆy (t)
∂ Jˆz (t) ω s Jx ω c Jx (48) , =− . ∂t 2 ∂t 2 These equations are valid for interaction times T such that ωc T , ωs T 1 ΩT . We see that with pulses of RF-magnetic fields we are able to change the spin components Jˆy and Jˆz by an amount controlled by the sine and cosine components Bs and Bc . This has several experimental applications, which are discussed below. Jx (t) = Jx (0),
=−
4.3.1. Characterizing the Spin State with the Magneto-Optical Resonance Method Equations (20) and (47) describe the Magneto-Optical Resonance method (MORS) which we use extensively for the spin state characterization. Application
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of MORS to our experiments is described in detail in Julsgaard et al. (2004b). Within this method the RF frequency is scanned around ΩL and the oscillating transverse spin components are probed via their introduction of oscillating polarization rotation of the optical probe through the Faraday interaction. In order to quantitatively explain the MORS signal as the RF-frequency is scanned across ΩL , we need to return to the second order term mentioned in Eq. (45). The single atom transverse spin components jˆy and jˆz can be expressed in terms of coherences σˆ m,m±1 in the following way: 1 jˆy = F (F + 1) − m(m + 1)(σˆ m+1,m + σˆ m,m+1 ), 2 m (49) 1 jˆz = F (F + 1) − m(m + 1)(σˆ m+1,m − σˆ m,m+1 ). 2i m In the absence of the second order term in (45) the energy separation h¯ ΩL between states |m and |m + 1 is the same for all m and all terms σˆ m+1,m have the same resonant frequency. The second order term in (45), however, makes the frequency of the coherences σˆ m+1,m slightly different for different values of m. It can be shown, that the quadratic Zeeman frequency difference ωQZ between σˆ m,m+1 and σˆ m−1,m is ωQZ = 2ΩL /ωhfs where ωhfs = 2π · 9.1926 GHz is the hyperfine splitting of the Cesium ground state. We typically have ΩL = 2π · 322 kHz and the effect is small but detectable. In the special case that the amplitude and frequency of the driving RF-field vary slowly compared to the spin coherence time, the off-diagonal coherences follow the diagonal populations adiabatically and we may write e.g. Jˆy as:
F −1 iΩt [F (F + 1) − m(m + 1)] · e [σˆ m+1,m+1 − σˆ m,m ] Jˆy = Re const i(Ωm+1,m − Ω) − Γm+1,m /2 m=−F (50) where Γm+1,m are the FWHM line-widths giving an exponential e−Γ t/2 decay of each coherence. The Larmor frequency ΩL has been replaced by the individual coherence evolution frequencies Ωm+1,m . For Jˆz we have to take the imaginary part instead of the real part in Eq. (50). We see that two adjacent magnetic sublevels act as a single two level system with the usual Lorentzian response to a driving RF field. Scanning the RF frequency we get eight Lorentzian peaks, the magnitudes of which will depend on the populations of the magnetic sublevels. The MORS signal is proportional to the square of the term in the square brackets in Eq. (50). An example of such a signal can be seen in Fig. 4. The application of this method is twofold. First, we gain information on the distribution of population among the different ground state magnetic sublevels. From this we infer that we are able to optically pump the atoms to such an extent that only the outermost coherence (mF = 4 ↔ mF = 3) becomes significant. Second, measuring
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F IG . 4. MORS signal, in arbitrary units (a.u.) of a poorly oriented sample. Each peak corresponds to a mF ↔ mF + 1 Zeeman resonance. The height of each resonance is proportional to the population difference of the two relevant sublevels.
the widths of the resonances under different experimental conditions allows us to quantify the effect of different decoherence mechanisms as discussed in Section 5.2. A second implication of the second order term is that the equations of motion ∂ jˆy /∂t = −Ω jˆz and ∂ jˆz /∂t = Ω jˆy describing Larmor precession are too simple and must, in principle, be generalized. However, this is not necessary for our case with highly polarized samples. We have almost all the population in the mF = 4 and mF = −4 states for the two samples, respectively. Then there is effectively only one non-vanishing frequency component in the sums (49). For the jx = 4 sample the single term is σˆ 3,4 and for the jx = −4 sample the only term is σˆ −3,−4 .
4.3.2. Manipulating the Spin State Of course, as suggested by Eq. (48), an appropriate choice of phase, strength and envelope function of the RF-field allows us to create an arbitrary mean value of either of the spins or any combination of these. This has two applications. First, it enables us to create a large classical mean value and observe the evolution of this state, thus constituting another mechanism for studying decoherence and calibrating the system as is discussed briefly at the end of Appendix B.1. Secondly, we can actively feed back the result of a quantum probing of the spin state, thus creating a particular desired state. This is the keystone element for creation of deterministic entanglement, discussed in Section 5.3.2, and for the quantum mapping experiment discussed in Section 5.4.
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5. Experimental Results 5.1. P ROJECTION N OISE L EVEL Since the Heisenberg uncertainty relation sets the starting point of all our calculations, one of the most important tasks in our experiments is the achievement of quantum noise limited performance. Practically, it is also one of the most difficult tasks. When we detect polarization states of light we observe noise in the signals. After the light has passed the atomic samples, there is a contribution to this noise from the light itself and from the atomic spins. The noise contribution from atoms in the minimum uncertainty state (the coherent spin state) is called projection noise. We discussed the ratio of the projection noise to the quantum noise of light (shot noise) already in Section 3.1. We found that theoretically this ratio should be κ 2 = a 2 Jx Sx T .
(51)
κ2
came from the more general interaction equations (30a)–(30d). In The ratio the present section we discuss how to determine this projection noise level experimentally and how to predict the noise level from independent measurements. 5.1.1. Measuring the Macroscopic Spin The ratio of projection noise to shot noise is proportional to the macroscopic spin Jx . This linearity is the fingerprint of quantum noise and is essential to establish experimentally. To this end we need a good measure of Jx . We measure Jx by detecting polarization rotation of a linearly polarized laser field propagating through the atomic samples along the x-direction. To see what happens in this setting we consider Eqs. (9) and (10). These equations assume propagation along the z-direction so we assume the spin to be polarized along the z-direction in the following. For linearly polarized light we have Sˆz = 0 and the effect on the transverse spin components jˆx , jˆy is negligible. It can also be shown easily that the a0 and a2 terms of the Hamiltonian (7) play no role in this calculation. We are left with the evolution of Sˆx and Sˆy according to (10) and after integration over the sample we find Sxout = Sxin cos(2θF ) − Syin sin(2θF ),
(52)
Syout = Sxin sin(2θF ) + Syin cos(2θF ), where “in” refers to the polarization state before the sample at z = 0 and “out” refers to the state after the sample at z = L. The angle θF is given by (in radians) θF = −
a1 γ λ2 ρL · jˆz . 32πΔ
(53)
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If a linearly polarized beam of light is rotated by the angle θ , the Stokes vector is rotated by 2θ. Thus, in the above, θF is the polarization rotation caused by the spin orientation along the direction of light propagation. We note that the angle θF depends on the density ρ of atoms times the length L that the light traverses. We wish to re-express this in terms of the macroscopic spin size Jz = Nat jˆz of the entire sample (remember we have the spins polarized along z in this discussion). To this end we observe that Nat = ρV ≡ ρAcell L where V is the vapor cell volume and Acell is the area of the vapor cell transverse to the beam direction. This will conveniently be an effective area for cells that are not exactly box like. Returning to the usual convention of spin polarization along the x-axis we then rewrite (53) as θF = −
a 1 γ λ2 Jx . 32πAcell Δ
(54)
5.1.2. Predicting the Projection Noise Level Now, let us return to the predicted ratio of projection to shot noise (51). This prediction relies on Eqs. (11)–(14) which are derived under the assumption that all atoms in the sample interact with the laser beam which has a cross sectional area A. In experiments the laser beam does not intersect all the atoms. In Appendix A we show that the random motion of atoms in and out of the beam modifies the expected variance of the transverse spin components Jˆy and Jˆz by statistical effects from the usual Jx /2 to p 2 (1 + σ 2 )Jx /2 where p = A/Acell is the mean time of an atom inside the laser beam and σ 2 is the relative variance of p. We furthermore present a simple model for p and σ and show that the atomic motion acts as an effective source of decoherence between two probe pulses. We incorporate atomic motion into Eq. (51) by replacing A with Acell in the factor a and multiplying the whole expression by 1 + σ 2 . We then find 2 γ λ2 a 1 Jx Sx T 1 + σ 2 κ 2 = a 2 Jx Sx T · p 2 1 + σ 2 = 8Acell Δ 2π =
(1 + σ 2 )γ λ3 a1 P · T · θF . 32π 2 Acell Δh¯ c
(55)
In the last step we replaced Sx = φ/2 = P /2h¯ ω = P λ/4π hc ¯ where P is the optical power. We also inserted Eq. (54) to express κ 2 as a function of θF . However, we must remember that the area Acell in (54) refers to the transverse area for a beam propagating in the x-direction while the Acell from the relation p = A/Acell refers to the transverse area for a beam along z. Hence, the last step above is valid for a vapor cell of cubic symmetry only, but it can still be an irregularly shaped cell. In other cases the generalization is straightforward. We have reached an expression for κ 2 in terms of convenient parameters from an
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F IG . 5. Measured atomic noise relative to shot noise of light. The linearity is a clear signature of 2 = 0.104(2) · θ should be compared to the theoretical the projection noise limitation. The slope κexp F 2 = 0.140 · θ from Eq. (56) with σ 2 = 0. value of κth F
F IG . 6. Slope of measured κ 2 vs. θ normalized to the experimentally predicted level (without the 2 /κ 2 = 1 + 0.47(13)/T [ms]. factor 1 + σ 2 ) vs. Tprobe . The fit gives κexp th
experimental point of view. With γ /2π = 5.21 MHz and λ = 852.3 nm together with hc ¯ we reach our final theoretical estimate for the projection to shot noise ratio expressed in convenient units: 2 = κth
56.4 · P [mW] · T [ms] · θF [deg] · a1 (Δ) · (1 + σ 2 ) , Acell [cm2 ] · Δ[MHz]
(56)
where a1 (Δ) was defined in Eq. (8). 5.1.3. Experimental Investigation Turning to experiment, in Fig. 5 we see an example where the measured noise relative to the shot noise of light is plotted. The data are clearly linear. With
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Δ/2π = 700 MHz, T = 2.0 ms, P = 4.5 mW, and σ 2 = 0 for the moment, we predict a linear slope of 0.140 which is somewhat higher than the measured value of 0.104. Including the σ 2 from atomic motion makes the discrepancy slightly worse but given the simplicity of the theoretical model we consider the agreement satisfactory. To test the scaling properties predicted in the atomic motion calculations, we fix the power P , detuning Δ and macroscopic spin size J but vary the probe duration T . The measured noise is plotted in Fig. 6 relative to the prediction (56) with σ 2 = 0. We see that as T is increased we do see a lower and lower noise level which corresponds to decreasing σ 2 . The solid line in the figure represents a fit where σ 2 = (0.47±0.13)/T [ms]. To compare this to the simple model described in (A.4) we estimate our beam diameter to be 1.6 cm which gives A ≈ 2.0 cm; moreover, L = 3.0 cm, v0 = 13.7 cm/ms (cesium at room temperature). For T = 1 ms we get the prediction σ 2 = 0.44. This is in very good agreement with the measured data, but this agreement must be viewed as fortuitous. As mentioned in Appendix A.1, numerical simulations of atomic motion have shown that the variance estimate (A.4) is almost four times too high. The high experimental value must be attributed to the additional Doppler broadening effect. We also note the relatively high uncertainty of 0.13. But all together, however, we have a qualitative understanding of the physics and a quantitative agreement within a few tens of percent. 5.1.4. Thermal Spin Noise Another issue concerning the projection noise level is the question of thermal spin noise. For the establishment of the correct noise level we must be in the CSS with high precision. For the CSS the spin is completely polarized along the x-direction and Var(jy ) = Var(jz ) = F /2 = 2 for the F = 4 ground state. As a very different example we may consider a completely unpolarized sample. We then have by symmetry Var(jx ) = Var(jy ) = Var(jz ) = (jx2 + jy2 + jz2 )/3 = F (F + 1)/3 = 20/3. This is a factor of 10/3 higher than the CSS noise and, even for fairly good polarization, the thermal noise may be significant. In our experiments with quantum information protocols we exceed a spin polarization of 99% which means that the thermal noise must be very small compared to the true projection noise. The degree of spin polarization has been measured independently as described in Section 4.3.1 with methods similar to those in Julsgaard et al. (2004b). A nice illustration of the fact that we get lower noise for the CSS than in the unpolarized case is given in Fig. 7. Experimentally, we perform measurements on very poor vapor cells where the macroscopic spin life time is small. We optically pump the sample and wait for some variable delay time before probing the spin noise. For long times the spins reach thermal equilibrium, where the noise of each atom in F = 4 contributes 20/3. The fraction of atoms in F = 4 is 9/16,
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F IG . 7. Coherent state noise compared to the completely unpolarized spin noise. The data is taken with a vapor cell in which the spin life time is very short. The noise level increases on a time scale of roughly 8 ms to the thermal equilibrium level. The increase in noise is consistent with predictions for the coherent and unpolarized spin states.
whereas the remaining 7/16 are in the F = 3 state and do not contribute because of the large detuning. Initially, all atoms are in F = 4 in the CSS and they each contribute the value 2 to the noise. Hence the measured noise must be on the form 20 9 Measured noise ∝ 2 · exp(−Γ t) + (57) · 1 − exp(−Γ t) . 3 16 The predicted ratio of final to initial noise is thus 15/8 ≈ 1.88. Experimentally we find the ratio 2.05 ± 0.09 which is consistent. To sum up, there is strong evidence that we really do create the CSS with the correct minimum uncertainty noise. 5.1.5. Concluding Remarks on the Projection Noise Level Let us sum up the discussion of the projection noise level. To reach the quantum noise limited performance one should first observe the atomic noise grow linearly with the macroscopic spin size Jx . An experimental example of this was shown in Fig. 5. The linearity of the noise basically arises from the fact that different atoms yield independent measurements when their spin state is detected. Technical noise sources from e.g. external electromagnetic fields couple to all atoms and the effect on the noise variance would be quadratic. However, linearity alone is not enough. An ensemble of independent and unpolarized atoms would also show a linear increase in the spin noise with an increase in the number of atoms. Since unpolarized atoms have larger noise variance than the 100% polarized atomic sample, we must know independently that the spin orientation is high. In our experiments the spin samples are polarized better than 99%.
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One may argue that the small fraction of atoms that are not in the completely polarized state could, in principle, form exotic-multi particle states with a very high variance of the detected spin noise. The results discussed in Section 5.1.4 prove that this is not the case. Finally, as derived in Appendix A the atomic motion leads to an increased ratio of atomic to shot noise. Generally a large ratio of atomic projection noise to shot noise is good for the quantum information protocols. However, as discussed in Appendix A.2, we do not gain anything by the increase of the atomic noise caused by atomic motion since there is an acco
