Advances in
ATOMZC, MOLECULAR, AND OPTZCAL PHYSICS VOLUME 41
Editors BENJAMIN BEDERSON New York University New York, New York HERBERT WALTHER Max-Planck-Institutfr Quantenoptik Garching bei Munchen Germany
Editorial Board P. R. BERMAN University of Michigan Ann Arbor, Michigan M. GAVRILA E O.M. Instituut voor Atoom-en Molecuulfyica Amsterdam The Netherlands M. INOKUTI Argonne National Laboratory Argonne, Illinois W. D. PHILLIPS National Institute for Standards and Technology Gaithersburg, Maryland
Founding Editor SIRDAVIDR. BATES
Supplements 1. Atoms in Intense Laser Fields, Mihai Gavrila, Ed. 2. Cavity Quantum Electrodynamics, Paul R. Berman, Ed. 3. Cross Section Data, Mitio Inokuti, Ed.
ADVANCES IN
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by
Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK dr
Herbert Walther UNIVERSITY OF MUNICH AND
mR QUANTENOPTIK
MAX-PLANK-INSTITUT MUNICH, GERMANY
Volume 41
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This book is printed on acid-free paper. @ Copyright 0 1999 by Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. The appearance of code at the bottom of the first page of a chapter in this book indicatesthe Publisher’s consent that copies of the chapter may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts01923), for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for chapters are as shown on the title pages: if no fee code appears on the chapter title page, the copy fee is the same as for current chapters. 1049-250x199 $30.00 Academic Press 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA http:llwww.apnet.com Academic Press 24-28 Oval Road, London NW17DX, UK http://www.hbuk.co.uk. lap1 International Standard Serial Number: 1049-25OX International Standard Book Number: 0-12-003841-2 Printed in the United States of America 98990001 0 2 M V 9 8 7 6 5 4 3 2 1
Contents CONTRIBUTORS
.........................................................
vii
Two-Photon Entanglement and Quantum Reality Yanhua Shih I. Introduction ......................................................
11. “Ghost” Image and Interference .................................... 111. Experimental Testing of Bell’s Inequalities ...........................
IV. V. VI. VII. VIII.
Why Two-Photon But Not Two Photons? ............................ Conclusion ....................................................... Acknowledgments ................................................ Notes ............................................................ References .......................................................
2 5 14
22 35 35 36 36
Quantum Chaos with Cold Atoms Mark G. Raizen I. Introduction ...................................................... 11. Two-Level Atoms in a Standing-Wave Potential ...................... 111. Experimental Method ............................................. IV. Single Pulse Interaction ............................................ V. KickedRotor ..................................................... VI. The Modulated Standing Wave ..................................... VII. Conclusion and Future Directions ................................... VIII. Acknowledgments ................................................ IX. References ........................................................
43 45 49 54 59
72 78 79 79
Study of the Spatial and Temporal Coherence of High-Order Harmonics Pascal Saligres, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein I. Introduction ...................................................... 11. Theory of Harmonic Generation in Macroscopic Media
...............
111. Phase Matching ................................................... IV. Spatial Coherence ................................................. V. Temporal and Spectral Coherence ................................... VI. Future Applications ............................................... VII. Conclusion .......................................................
84 91 99 106 116 131 136
vi
Contents
VIII . Acknowledgments ................................................ IX. References .......................................................
137 137
Atom Optics in Quantized Light Fields Matthias Freyburger, Alois M . Herkommer. Daniel S . Krahmer, Erwin Mayr, and Wolfgang P. Schleich I. Introduction ...................................................... 11. Ante ............................................................. 111. Atomic Deflection by a Resonant Quantum Field ..................... IV. Atom Optics in Nonresonant Fields ................................. V. The Bragg Regime ................................................ VI . Conclusion ....................................................... VII. Acknowledgments ................................................ VIII . References .......................................................
142 145 149 162 170 175 176 176
Atom Waveguides Victor I. Balykin I. Introduction ...................................................... 11. Guiding of Atoms with Static Electrical and Magnetic Fields .......... 111. Evanescent Light Wave ............................................ IV. Guiding Atoms with Evanescent Wave .............................. V. Atom Waveguide with Propagating Laser Fields ...................... VI . Experiments with Atom Guiding .................................... VII . Acknowledgments ................................................ VIII . References .......................................................
182 184 187 213 236 250 257 257
Atomic Matter Wave Amplification by Optical Pumping Ulf Janicke and Martin Wilkens I. Introduction ...................................................... I1. Model of an Atom Laser ........................................... III. Master Equation .................................................. IV. Photon Reabsorption .............................................. v. summary ........................................................ VI . Acknowledgments ................................................ VII . Appendix A: N-Atom Master Equation .............................. VIII . References .......................................................
262 264 272 278 291 294 294 303
........................................................ ....................................
305 31 1
SUBJECTINDEX CONTENTS OF VOLUMES IN THIS SERIES
Contributors Numbers in parentheses indicate pages on which the author’s contributions begin. PHILIPPE ANTOINE (83), CEAIDSMIDRECAMISPAM, Centre d’Etudes de Saclay, F-91191 Gif-sur-Yvette,France VICTOR I. BALYKIN (181), Institute of Laser Science, University of ElectroCommunications,Tokyo, Japan and Institute of Spectroscopy,Russian Academy of Sciences, Troitsk, Moscow region, 142092, Russia MATTHIAS FREYBURGER (143), Abteilung fur Quantenphysik, Universitat Ulm, 89069 Ulm, Germany ALOISM. HERKOMMER ( 143), Abteilung fur Quantenphysik, Universitat Ulm, 89069 Ulm, Germany ANNEL’HUILLIER (83), CEA /DSM/DRECAM /SPAM, Centre d’Etudes de Saclay, F-9 1191 Gif-sur-Yvette,France ULFJANICKE (261), Daisendorferstr. 14a, 88709 Meersburg, Germany DANIELS. KRLHMER(143), Abteilung fur Quantenphysik, Universitat Ulm, 89069 Ulm, Germany MACIEJLEWINSTEIN (83), CEA/DSM/DRECAM/SPAM, Centre d’Etudes de Saclay, F-9 1191 Gif-sur-Yvette,France ERWINMAYR(143), Abteilung fur Quantenphysik, Universitat Ulm, 89069 Ulm, Germany MARKG. RAIZEN(43), Department of Physics, The University of Texas at Austin, Austin, Texas 78712-1081 PASCAL SALIQRES (83), CEA/DSM/DRECAM/SPAM, Centre d’Etudes de Saclay, F-91191 Gif-sur-Yvette,France WOLFGANG P. SCHLEICH (143), Abteilung fur Quantenphysik, Universitat Ulm, 89069 Ulm,Germany vii
...
Vlll
Contributors
YANHUA SHIH(l), Department of Physics, University of Maryland at Baltimore County, Baltimore, Maryland 21250 MARTIN WILKENS (261), Institut fur Physik, Universitat Potsdam, 14469 Potsdam, Germany
Advances in
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 41
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ADVANCES IN ATOMIC, MOLECULAR, AND OF‘TICALPHYSICS, VOL. 41
TWO-PHOTON ENTNGLEMENT AND QUANTUM REALITY YANHUA SHIH Department of Physics, University of Maryland at Baltimore County, Baltimore, Maryland
I. Introduction ............. 11. “Ghost” Image and Interference .................................... A. “Ghost” Image Experiment . .
III. Experimental Testing of Bell’s Inequalities ...........................
IV.
V. VI. VII. VIII.
A. Double Entanglement of Type-I1 SPDC B. Experiment One: Bell’s Inequality for C. Experiment Two: Bell’s Inequality for Why Two-Photon But Not Two Photons? ............................ A. Is Two-Photon Interference the Interference of Two Photons? ......... B. Entangled State and Two-Photon Wavepacket ....... C. Experiment One: Two-Photon Interference ........................ D. Two-Photon Wavepacket in Bell’s Inequality Measurement .... E. Experiment Two: Single-Photon Measurement of a Tw Conclusion .................................................... Acknowledgments Notes ......................................................... References ..................................................... Appendix A: The Two-Photon State Appendix B: The Biphoton Wavefunction ............................
2 5 6 8 14 15 17 19 22 23 26 28 30 32 35 35 36 36 38 39
Abstract: One of the most surprising consequences of quantum mechanics is the entanglement of two or more distant particles. In 1935, Einstein-Podolsky-Rosen suggested the first classic two-particle entangled state, and proposed a gedunkenexperiment. What was surprising about the EPR state and the outcome of the EPR gedunkenexperiment is the following: The value of an observable for neither single particle is determined. However, if one of the particles is measured to have a certain value for that observable, the other one is 100% determined. A simple yet fundamental question was then asked by EPR: “Does a single particle have definite value for an observable, in the course of its travel, regardless of whether we measure it or not?” Quantum mechanics answers: “No.” EPR thought: “It should!” In 1964, J. S. Bell proofed a theorem to show that an inequality must be obeyed by any theories that subject to Einstein’s local realism. It is this work that made possible the real-life experimental testing. The progression from gedunken to real experiment in recent years has been greatly aided by the use of Spontaneous Parametric Down Conversion (SPDC). The distinctiveentanglementquantum 1
Copyright 0 1999 by Academic Press All rights of reproductionin any form reserved. 1049-25OX/W$30.00 ISBN 0-12-003841-2/ISSN
2
Yanhua Shih
nature of the resulting two-photon state of SPDC has allowed us to demonstrate the “spooky” EPR phenomenon as well as the violation of Bell’s inequalities. In addition to reviewing several recent experiments, we introduce a new concept of “biphoton” in this chapter, which may be considered as a different approach to challenge the EPR puzzle.
I. Introduction One of the most surprising consequences of quantum mechanics has been the entanglement of two or more distant particles. The two-particle entangled state was mathematically formulated by Schrodinger (1935). Consider a pure state for a system composed of two spatially separated subsystems, ij =
IWWl9
IW
=
c a. b
c(a7
b) l a ) Ib)
(1)
where { I a ) } and {I b ) } are two sets of orthogonal vectors for subsystems 1 and 2, respectively, and i j is the density matrix. If c(a, b) does not factor into a product of the formf ( a ) X g(b),then it follows that the state does not factor into a product state for subsystems 1 and 2: ij #
61 63
lj2
The state was defined by Schrodinger as the entangled state. The first classic example of a two-particle entangled state was suggested by Einstein, Podolsky, and Rosen (1935) in their famous gedankenexperiment:
where a and b are the momentum or the position of particles 1 and 2, respectively, and co is a constant. A surprising feature of the EPR state is the following: the value of an observable (momentum or position) for neither single subsystem is determinate. However, if one of the subsystems is measured to be at a certain value f o r that observable, the other one is 100% determined. This point can be easily seen from the delta function in Eq. (2). A simple yet fundamental question naturally followed, as EPR asked 60 years ago: “Does a single particle have definite momentum in the state of Eq. (2) in the course of its travel, regardless of whether we measure it or not?” Quantum mechanics answers “No!” The memorable quote from Wheeler (1983) “No elementary quantum phenomenon is a phenomenon until it is a recorded phenomenon” summarizes what Copenhagen has been trying to tell us. By 1927, most physicists accepted the Copenhagen interpretation as the standard view of quantum formalism. Einstein, however, refused to compromise. As Pais (1982) recalled vividly: around 1950 during a walk, Einstein suddenly stopped and “asked me if I really believed that the moon exists only if I look at it.”
TWO-PHOTON ENTANGLEMENT AND QUANTUM REALITY
3
Einstein, Podolsky, and Rosen published their famous paper in 1935: “Can quantum-mechanical description of physical reality be considered complete?” In this paper EPR suggested the classic EPR state, Eq. (2), for a gedunkenexperiment, and then give their criteria: Locality: There is no action-at-a-distance; Reality: “If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this quantity.” According to EPR, because we can predict with certainty the outcome result of measuring the momentum of particle 1 by measuring the momentum of particle 2, and the measurement of particle 2 cannot cause any disturbance to particle 1, if the measurements are space-like separated events, the momentum of particle 1 must be an element of physical reality. A similar argument shows that the position of particle 1 must be physical reality too. However, this is not allowed by quantum mechanics. Now consider the following. Completeness: “Every element of the physical reality must have a counterpart in the complete theory.” This leads to the question: “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” The state of the signal-idler photon pair of the spontaneous parametric down conversion (SPDC) is a typical entangled EPR state. SPDC is a nonlinear optical process from which a pair of signal-idler photon is generated when a pump laser beam is incident onto an optical nonlinear crystal. Quantum mechanically, the state can be calculated by first-order perturbation theory, see for example, Appendix A, note 1.
IW =
c s, i
S(w,
+
wi - w,,P(k,
+ k; -
k,)af(w(k,))at(w(k;)) 10)
(3)
where w j , kj ( j = s, i, p) are the frequency and wavevectors of the signal (s), idler (i), and pump (p) respectively; w,, and k,, can be considered as constants; usually a single mode laser is used for pump; and uf and ut are creation operators for signal and idler photons, respectively.Equation (3) tells us that there are two eigenmodes excited together. The signal or idler photon could be in any modes of its superposition (uncertain); however, if one is known to be in a certain mode the other one is determined with certainty. (1) Do we have such a state? (2) How special is it physically? In this chapter, we review a “ghost image” experiment (Pittman et ul., 1995) and a “ghost interference’’ experiment (Strekalov et ul., 1995) in Section I1 to answer these questions and to show the striking EPR phenomenon. Another example of an entangled two-particle system suggested by Bohm (1951) is a singlet state of two spin 1/2 particles:
4
Yanhua Shih
where the kets I +) and I -) represent states of spin up and down, respectively, along an arbitrary direction ii. Again for this state, the spin for neither particle is determined; however, if one particle is measured to be spin up along a certain direction, the other one must be spin down along that direction. Does a single particle in the Bohm state have a definite spin in the course of its travel, regardless of whether we measure it or not? No! The spin for neither particle is defined in Eq. (4).It does not make sense to EPR in the first place: According to EPR, because we can predict with certainty the outcome result of measuring any components of the spin of particle 1 by measuring some component of the spin of particle 2, and the measurement of particle 2 cannot cause any disturbance to particle 1, if the measurements are space-like separated events, the chosen spin component of particle 1 must be an element of physical reality. Following this argument, all of the components of spin of particle 1 must be physical realities associated with it. However, as is well known, this is not allowed by quantum mechanics. Is it possible to have a “better” theory, which provides correct predictions like quantum mechanics and at the same time respects its description of physical reality by EPR as “complete”? It was Bohm who first attempted a version of a so-called “hidden variable theory,” which seemed to satisfy these requirements (Bohm, 1952a,b, 1957). The hidden variable theory was successfully applied to many different quantum phenomena until 1964, when Bell proofed a theorem to show that an inequality that is violated by certain quantum mechanical statistical predictions can be used to distinguish local hidden variable theory from quantum mechanics (Bell, 1964 and 1987). Since then, the testing of Bell’s inequalities has become a key instrument for the study of fundamental problems of quantum theory. However, as it is not the purpose of this chapter to discuss the details about Bell’s inequality, see Bell (1964, 1987) and note 2 for additional reading. The experimental testing of Bell’s inequality started from the early 1970s (note 2). Most of the historical experiments employed two-photon sources of atomic cascade decay. The two-photon state of atomic cascade decay is similar to that of Eq. (3), except that the momentum S function, S(p, + p2), is not strictly true, because of the recoil of the atom, that is, the momenta of the pair are not necessarily to be exactly in opposite directions. Thus if particle 1 is measured at a certain direction, particle 2 could be in any direction of a large solid angle. This so-called “collection efficiency loophole” has been criticized by many serious physicists and philosophers. Since SPDC was introduced to the experimentaltesting of Bell’s inequality (Alley and Shih, 1986; Shih and Alley, 1987), the “collection efficiency loophole” has never been a problem. In addition, unlike the atomic cascade decay experiments, there is no need to “subtract” “noise” any more, which definitely influences the credence of the “violation” of Bell’s inequalities. In Section 111, we will review the violations of Bell’s inequality in two types of experimentsby using a two-photon source of SPDC (Kwiat er al., 1995;Strekalov et al., 1996). One of the experiments held a “world record,” which violated a
TWO-PHOTON ENTANGLEMENTAND QUANTUM REALITY
5
Bell’s inequality with more than 100 standard deviations. Once again, there was no “subtraction” of “noise” in these experiments. The important physics we want to emphasize here is that the “click-click” detection events are ensured space-like separated events in all our measurements, by using short coincidence time windows, which only accept that detection events happened at a time interval shorter than the optical distance between the two detectors. (See locality criterion of EPR.) Notice that we are talking about two-photon. Why two-photon but not two photons? What is the difference between two-photon and two photons? If “twophoton” is not “two photons,” then what is it? Do we have single particle reality in an entangled two-particle system? What information is available for “a single photon” in a two-photon measurement?These questions will be examined in Section IV and two experiments will be reviewed for this purpose (see notes 3 and 4). We must find a way out from the 60-year-old EPR puzzle, and hope that these questions and answers as well as the experiments themselves enlighten a better understanding of the quantum world. Recently, Greenberger-Horne-Zeilingerdemonstrated Bell’s theorem in a new way, by analyzing a three or more than three multiparticle entangled system (Greenberger et af., 1990). Unlike Bell’s original theorem, GHZ’s demonstration of the incompatibility of quantum mechanics with EPR local realism considers only “perfect” correlations rather than statistical correlations and as such it completely dispenses with inequalities. GHZ’s incompatibility is stronger than the one previously revealed for two-particle systems. The testing of GHZ theorem is our current and near future experimental goal. Theoretically, we have demonstrated the possibility of producing an entangled three-photon GHZ state in nonlinear optical processes (Keller et af., 1998). It is also important to point out that the three-photon wavepacket, or triphoton, is a crucial subject for the understanding of three-particle physics.
11. “Ghost” Image and Interference We review two experiments in this section (Pittman et af.,1995; Strekalov et al., 1995). The first experiment is a so-called two-photon “ghost” imaging experiment in which the signal-idler pair, generated in SPDC, is propagated to different directions and detected by two distant photon counting detectors. An aperture (mask) placed in front of one of the detectors is illuminated by the signal beam through a convex lens. Surprisingly, an image of this aperture is observed by scanning the other detector in the transverse plane of the idler beam, provided that the detectors catch the signal-idler twin and that if the two detectors and the convex lens are in the correct positions, that is, they satisfy the Gaussian thin lens equation. The second experiment demonstrates a “ghost” interference. The experimental setup is similar to the image experiment, except that a Young’s double-slit,
6
Yanhua Shih
rather than an aperture, is inserted into the path of the signal beam. There is no interference pattern behind the double-slit. However, an interference pattern is observed in the idler beam if the detectors catch the signal-idler twin. The interference pattern is definitely not the “hidden” pattern behind the double-slit, because the period of the interference pattern is not a function of the distance between the slit and the signal detector, but rather a function of a distance from the double-slit going backwards to the nonlinear crystal of SPDC and then to the idler detector along the “empty” idler beam.
A. “GHOST”IMAGEEXPERIMENT The experimental setup is shown in Fig. 1. The 35 1.1 nm line of an argon ion laser is used to pump a BBO (P-BaB,O,) crystal, which is cut at a degenerate type-I1 phase matching angle (note 6) to produce a pair of orthogonally polarized signal (e-ray of the BBO) and idler (0-ray of the BBO) photon. The pair emerges from the crystal nearly collinear, with w, = wi w,,/2, where wj ( j = s, i, p ) are the frequencies of the signal, idler, and pump, respectively. The pump is then separated from the down conversion by a UV grade fused silica dispersion prism and the remaining signal and idler beams are sent in different directions by a polarization beam-splitting Thompson prism. The signal beam passes through a convex lens with a 400 mm focal length and illuminates a chosen aperture (mask). As an
-
polarizing idler beam splitter
X-Y scanning fiber
FIG. 1. Schematic set-up of the two-photon “ghost” image experiment.
TWO-PHOTON ENTANGLEMENT AND QUANTUM REALITY
7
FIG.2. (a) A reproduction of the actual aperture “UMBC” placed in the signal beam. (b) The image of “UMBC”:coincidence counts as a function of the fiber tip’s transverse plane coordinates. The scanning step size is 0.25 mm. The data show a “slice” at the half-maximum value.
example, we have used letters “UMBC” for the object mask. Behind the aperture is the detector package D which consists of a 25 mm focal length collection lens in whose focal spot is a 0.8 mm diameter dry-ice-cooled avalanche photodiode. The idler beam is met by detector package D,, which consists of a 0.5 mm diameter multimode fiber whose output is mated with another dry-ice-cooled avalanche photodiode. The input tip of the fiber is scanned in the transverse plane by two encoder drivers. The output pulses of each detector, which are operating in the Geiger mode, are sent to a coincidence counting circuit with a 1.8 ns acceptance time window for the signal-idler twin detection. Both detectors are preceded by 83 nm bandwidth spectral filters centered at the degenerate wavelength, 702 nm. By recording the coincidence counts as a function of the fiber tip’s transverse plane coordinates, we see the image of the chosen aperture (for example “UMBC”), as is reported in Fig. 2. It is interesting to note that whereas the size of the “UMBC” aperture inserted in the signal beam is only about 3.5 mm X 7 mm, the observed image measures 7 mm X 14 mm. We have therefore managed linear magnification by a factor of 2. Despite the completely different physical situation, the remarkable feature here is that the relationship between the focal length of the lens$ the aperture’s optical distance from the lens So,and the image’s optical
8
Yanhua Shih
distance from the lens (from lens back through beamsplitter to BBO then along the idler beam to the image) Si, satisfies the Gaussian thin lense equation: -1- + -1 = - 1
f
si
so
In this experiment, we chose So = 600 mm, and the twice-magnified clear image was found when the fiber tip was in the plane of Si = 1200 mm. To understand this unusual phenomenon, we examine the quantum nature of the two-photon state, Eq. (3), of SPDC, entangled by means of two S functions, which is usually called phase matching conditions (note 1): w,
+ wi = up,
k,
+ ki = k,
(6)
where kj( j= s, i, p ) is the wavevector of the signal, idler, and pump, respectively. The spatial correlation of the signal-idler pair, which encourages two dimensional correlation applications, is the result of the transverse components of the wavevector phase-matching condition:
k, sin a, = ki sin ai
(7)
where a, and a i are the scattering angles inside the crystal. Upon exiting the crystal, Snell’s law thus provides: w, sin
p,
= wi
sin pi
(8)
where p, and pi are the exit angles of the signal and idler with respect to k, direction. Therefore, in the near degenerate case, the signal-idler pair are emitted at roughly equal, yet opposite, angles relative to the pump, and the measurement of the momentum (vector) of the signal photon determines the momentum (vector) of the idler photon with unit probability and vice versa. This then allows for a simple explanation of the experiment in terms of “usual” geometrical optics in the following manner: We envision the crystal as a “hinge point” and “unfold” the schematic of Fig. 1 into that shown in Fig. 3. Because of the equal-angle requirement of Eq. (8), we see that all the signal-idler pairs that result in a coincidence detection can be represented by straight lines (but keep in mind the different propagation directions) and therefore the image is well produced in coincidences when the aperture, lens, and fiber tip are located according to Eq. (5). In other words, the image is exactly the same as one would observe on a screen placed at the fiber tip if detector D ,were replaced by a point-like light source and the BBO crystal by a reflecting mirror (note 7).
B. “GHOST”INTERFERENCE-DIFFRACTION The schematic experimental set-up is illustrated in Fig. 4. It is similar to the “ghost image” experiment except that after the separation of signal and idler, the
TWO-PHOTON ENTANGLEMENTAND QUANTUM REALITY
-S
-
6Wmm
9
-
S’ 1200mm
FIG. 3. A conceptual “unfolded” version of the schematic shown in Fig. I , which is helpful for understanding the physics. Although the placement of the lens and the detectors obeys the Gaussian thin lens equation, it is important to remember that the geometric rays actually represent pairs of signal-idler photons, which propagate in different directions.
signal passes through a Young’s double-slit (or single-slit) aperture and then travels about 1 m to be counted by a point-like photon counting detector D ,(0.5 mm in diameter). The idler travels a distance about 1.2 m from BS to the input tip of the optical fiber. In this experiment only the horizontal transverse coordinate, x,, of the fiber input tip is scanned by an encoder driver. Figure 5 reports a typically observed double-slit interference-diffraction pattern. The coincidence counting rate is reported as a function of x,, which is obtained by scanning the detector D, (the fiber tip) in the idler beam, whereas the double-slit is in the signal beam. The Young’s double-slit has a slit width a = 0.15 mm and slit distance d = 0.47 mm. The interference period is measured to be 2.7 ? 0.2 mm and the half-width of the envelope is estimated to be about 8 mm. By curve fittings, we conclude that the observation is a standard Young’s interference pattern, that is, a sinusoidal function oscillation with a sinc function envelope:
R,
0~
sinc 2 ( ~ 2 ~ a l A z 2 ) c o s 2 ( x , ~ d l A z , )
(9)
The remarkable feature here is that z2 is the distance from the slits plane, which is in the signal beam, back through BS to the BBO crystal and then along the idler beam to the scanning fiber tip of detector D, (see Fig. 7). The calculated interference period and half-width of the sinc function from Eq. (9) are 2.67 mm and 8.4 mm, respectively. Even though the interference-diffraction pattern is observed in coincidences, the single detector counting rates are both observed to be constant when scanning detector D ,or D,.Of course it seems reasonable not to have any interference modulation in the single counting rate of D,,which is located in the
10
Yunhua Shih
FIG. 4. Schematic set-up of the two-photon “ghost” interference-diffractionexperiment.
“empty” idler beam. Of interest, however, is that the absence of the interferencediffraction structure in the single counting rate of D which is behind the doubleslit, is mainly due to the divergence of the SPDC beam (>> hld). In other words, the “blurring out” of the first-order interferencefringes is due to the considerably large momentum uncertainty of the signal photon. Furthermore, if D ,is moved to an unsymmetrical point, which results in unequal distances to the two slits, the interference-diffraction pattern is observed to be simply shifted from the current symmetrical position to one side of x 2 . This is quite mind boggling: Imagine that there was a first-order interference pattern behind the double-slit and D ,was moved to a completely destructive interference point (i.e., zero intensity at that point) and fixed there. Then we still can observe the same interference pattern in the coincidences (same period, shape, and counting rate), except for a phase shift! Figure 6 reports a typical two-photon single-slit diffraction pattern. The slit
11
TWO-PHOTON ENTANGLEMENT AND QUANTUM REALITY I
-
'
1
~
1
'
1
'
1
'
I
'
I
'
-
300
250 -
v)
C
3
0
200 -
0
a
0
c
a 9
150 -
0
100
-
s
50
-
.-C
0
2
0
4
6
8
10
14
12
16
Detector 2 position (mm) FIG.5 . Typical observed interference-diffraction pattern: the dependence of the coincidence on the position of D,,which counts the idler, while the signal passes through a double-slit. The solid curve is calculated from Eq. (9). considering the finite size of the detectors. If D ,is moved to an unsymmetrical point, which results in unequal distance to slit C and D, the interference-diffraction pattern is observed to be simply shifted from the current symmetrical position to one side, according to Eq.(13).
500
0)
c
-
400 -
C
3
0
0
a
0
300
-
C
a
-0 .0
200 -
s
100
.-c
-
-6
-4
-2
0
2
4
6
Detector 2 position (mm) FIG. 6. Two-photon diffraction: coincidence counts versus the position of D,. The solid curve is calculated from Q. (15).
width is measured a = 0.4 mm. The pattern fits to the standard diffraction sinc function, that is, the "envelope" of Eq. (9), within reasonable experimental error. Here again z 2 is the unusual distance described in the previous paragraphs.
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Yanhua Shih
a)
Pump
C
z2 FIG. 7. Simplifiedexperimental scheme (a) and its “unfolded” version (b).
To explain this unusual phenomenon, we again present a simple quantum model. The quantum entanglement nature of the two-photon state of SPDC has been described in Eq. (3). Even though for each single photon of the pair the momentum (vector) has a considerably large uncertainty, the measurement of the momentum of either photon determines the momentum of its twin with unit probability. This important peculiarity selects the only possible optical paths in Fig. 7, when the signal passes through the double-slit aperture while the other triggers D,. In the near degenerate case we can simply treat the crystal as a “reflecting” mirror, as discussed in the early “ghost” image section. The coincidencecounting rate R , is determined by the probability PI, of detecting the signal-idler pair by detectors D , and D, simultaneously.
where I ‘P) is the two-photon state of SPDC. Let us simplify the mathematics by using the following “two-modes” expression for the state, bearing in mind that the 6 functions of the “phase matching” conditions have been taken into account based on the “straight line” picture of Fig. 7.
I*)
= 10)
+ ~ [ a f a exp(i(p,) ; + b f b ; exp(i(p,)
10)
(1 1)
TWO-PHOTON ENTANGLEMENTAND QUANTUM REALITY
13
where E << 1 is proportional to the pump field (classical) and the nonlinearity of the crystal, pAand p, are the phases of the pump field at A and B, and af (bf) are the photon creation operators for the upper (lower) mode in Fig. 7 ( j = s, i ) . In terms of the Copenhagen interpretation, one may say that the interference is due to the uncertainty in the birthplace (A or B in Fig. 7) of a signal-idler pair. In Eq. (10) the fields at the detectors are given by:
+ b, exp(ik r B I ) E:+) = aiexp(ik rA2)+ b, exp(ik r,,) E $ + )= a, exp(ik r A l )
(12)
where rAi(rBi)are the optical path lengths from region A (B) along the upper (lower) path to the ith detector. Substituting Eqs. (1 1) and (12) into Eq. (lo),
R,
0:
PI, =
E,
lexp(ik rA + ipA) + exp(ik r,
a 1
+ cos[k(rA - r,)]
+ ip,)12 (13)
where we assume pA= p, in the second line of Eq. (13). (Although this is not a necessary condition to see the interference pattern, the transverse coherence of the pump beam at A and B is crucial.) In Eq. (13) we defined the overall optical lengths between the detectors D ,and D, along the upper and lower paths (see Fig. 7): rA = rAl rA2= rc, rc2, r, = rBl + r,, = rD, + rD2,where rci and rDi are the respective path lengths from the slits C and D to the ith detector. If the optical paths from the fixed detector D ,to the two slits are equal, that is, rci - rDi,and if 2, >> d2/A(far field), then r, - r, = rc2 - rD2= X , ~ / Z , , and Eq. (13) can be written as:
+
+
Rc(x2) 0: cos2(x,~d/Az,)
(14)
Equation (14) has the form of standard Young’s double-slit interference pattern. Here again z , is the unusual distance from the slits plane, which is in the signal beam, back through BS to the crystal, and then along the idler beam to the scanning fiber tip of detector D,. If the optical paths from the fixed detector D to the two slits are unequal, that is, rcl # rDl,the interference pattern will be shifted from the symmetricalposition from that of Eq.(14) according to Eq. (13). This interesting phenomenon has been observed and discussed following the discussion of Fig. 5 . There are two conclusions that can be drawn from Eq. (13):
,
(1) A two-photon interferencepattern can be observed by scanningD, in the transverse direction of one beam, even though the Young’s double-slit aperture is in the other beam if the detectors catch the signal-idler twin. (2) The interference pattern is the same as one would observe on a screen in the plane of D,,if D ,is replaced by a point-like light source and the SPDC crystal by a reflecting mirror (note 6).
14
Yanhua Shih
To calculate the “ghost” diffraction effect of a single-slit as shown in Fig. 6, we need an integral of the two-photon amplitudes over the slit width (the superposition of an infinite number of probability amplitudes results in a click-click coincidence detection event):
1L2 a12
Rc(x2)
cc
dx, exp[-ik ~ ( x , ,x2)]
l2
sinc2(x2~a/Az2) (15)
where r(x,, x2) is the distance between points x, and x2. x, belongs to the slit’s plane, and the inequality z2 >> a2/h is assumed (far field approximation). Repeating the previous calculations, the combined interference-diffractioncoincidence counting rate for the double-slit case is given by: Rr(x2)
sinc2(x2~alAz2)cos2(~2~dIA~2)
(16)
which is exactly the same as Eq. (9) obtained from experimental data fittings. If the finite size of the detectors and the divergence of the pump are taken into account by a convolution, the interference visibility will be reduced. These factors have been considered in the theoretical plots of Figs. 5 and 6. These two experiments demonstrate the striking EPR phenomenon from both a geometrical optics and physical optics point of view. Does a signal or idler photon in Eq. (3) have a defined momentum, in the course of its travel, regardless of whether we measure it or not? Quantum mechanics answers: No (uncertain)! However, if the signal is measured with a certain momentum, the idler is determined with certainty and vice versa.
111. Experimental Testing of Bell’s Inequalities The first experimental test of Bell’s inequality using a two-photon source of SPDC was published in 1986 (Alley and Shih, 1986; Shih and Alley, 1987). Since then, SPDC has been a major tool for this fundamental research. The history is interesting. It was around 1982 that we learned about Klyshko’s method for measuring “absolute quantum efficiency” of single-photon detectors (Klyshko, 1980) by using SPDC. We realized that the state of the signal-idler photon pair is the same as that of atomic cascade decay by means of the phase matching conditions, that is, o, + wi = wp, and k, + k i = k,, which are called energy and momentum conservations in atomic cascade decay. There is, however, no atom recoil involved in SPDC, so the momentum conservation is a true two-particle momentum conservation. It follows that there would be no “collection efficiency loophole” by using the signal-idler pair as the two-photon source. We first experimented with type-II SPDC (note 5). It is natural to use an orthogonal polarized photon pair to realize the EPR-Bohm type entangled states, see Eq. (4), based on the polarization state of photon, for example,
TWO-PHOTON ENTANGLEMENT AND QUANTUM REALITY
15
where IX,) ( IRj)) and IT) (IL,))are the orthogonal linear (circular) polarization bases, i = 1, 2, corresponding to the ith detector (Alley and Shih, 1986; Shih and Alley, 1987). Type-I1 SPDC did not reveal the EPR correlation in a difficult one and a half year, day-and-night effort. Avoiding any mistakes, we decided to try type-I SPDC with the help of a half-wave plate to rotate one of the linear polarization state orthogonals to form a X-Y base or using two quarter-wave plates to rotate both to form a R-L base (Alley and Shih, 1986; Shih and Alley, 1987). The EPR correlation was observed immediately. It took us ten years to finally understand the reason behind the failure of the first attempted type-I1 SPDC experiment (Shih et al., 1994; Rubin et al., 1994). If we had a better understanding of the “two-photon wavepacket,” or “biphoton,” in that time, we would have established the “double entanglement” concept from the beginning. Even though the entangled state of Eq. (17) is based on spin, the space-time part of the state or wavefunction must be taken into account also. The “two-photon wavepacket” (a concept associated with the space-time property of the two-photon state) of the superposed quantum amplitudes must be “indistinguishable,” or “overlapping.” We will discuss in detail the two-photon wavepacket, or biphoton, as well as the peculiarity of that of type-I1 SPDC in the next section, because it is extremely important. To test Bell’s inequality, one can use either entangled states in the form of the original EPR gendankenexperiment based on space-time observable, or in the form of EPR-Bohm based on spin variables. In type-I1SPDC we have both. We will review two types of experiment in this section (Kwiat et al., 1995; Strekalov et al., 1996). These Bell’s inequality measurements took advantageof the “double entanglement” of type-I1 SPDC. As a matter of fact, the experimental set-ups for these two experiments were almost the same, except the measurements were based on different type of observables. A. DOUBLE ENTANGLEMENT OF TYPE-I1 SPDC The two-photon state of SPDC has been briefly discussed in the introduction section. The state of the signal-idler pair is entangled in space-time by means of two S functions, see Eq.(3). The S functions are the results of two mathematical integrals, by considering a SPDC crystal of an infinite size and an infinite interaction time of perturbation. If finite crystal size and finite interaction time are taken into account, the S function will be replaced by a sinc-like function (Rubin et al., 1994). In most of our experiments, a single-mode CW laser beam is used to
16
Yanhua Shih
pump a relatively thin type-I1 SPDC crystal (in the order of mm), so that we may treat the time integral as infinite and still consider the crystal size finite. Furthermore, in the Bell-type experiments, we could assume defining pinholes for signalidler beams to be small enough that the transverse components of the k vectors are ignored. In this case the vector notation is no longer necessary and the twophoton state is thus:
where A, is a normalization constant, and k,(w,) and k,(w,) are the wavenumber (frequency) of the ordinary-ray and the extraordinary ray of the SPDC crystal, respectively. @ ( A k L )is a sinc-like function:
where Ak = kp - k, - k, = 0, and L is the length of the SPDC crystal. The twophoton state, Eq. (19), is very important for understanding the peculiar entanglement nature of type-I1 SPDC as well as for the calculation of the “effective twophoton wavefunction,” or “biphoton,” in next section. The most interesting situation of type-I1 SPDC is for “noncollinear phase matching” (note 7). The signal-idler pair are emitted into two cones, one ordinary polarized, the other extraordinary polarized, as in Fig. 8. Along the intersection lines, where the cones overlap, two pinholes numbered 1 and 2 are used for defining the direction of the k vectors of the signal-idler pair. The state is not only entangled in space-time, but also entangled in spin:
--+ -10
FIG.8. Type-II noncollinear phase matching: a crossection view of the degenerate 702.2 nm cones. The 351.1 nm pump beam is in the center. The numbers along the axes are in degrees. A photograph of the cones can be found in (Kwiat et al., 1995).
TWO-PHOTON ENTANGLEMENT AND QUANTUM REALITY
17
where o, and e , are ordinary and extraordinary polarization, respectively. Equation (21) indicates two “two-photon” amplitudes, which may result in a “clickclick” coincidence detection event; either signal (o polarized) triggers detector 1 and idler ( e polarized) triggers detector 2, or idler (e polarized) triggers detector 1 and signal (o polarized) triggers detector 2. To simplify the expression, we have used eia to indicate the relative phase between the two amplitudes. Note, however, that the relationship between the two amplitudes is much more complicated than that in Eq. (21). We will return to this point in the following section. In order to have interference, or EPR correlation, the two “wavepackets” corresponding to the two amplitudes must be completely overlapped. In other words, the two amplitudes have to be indistinguishable. However, in type-I1 SPDC, the longitudinal “walk-off’ causes a problem. BBO is a negative single-axis crystal, and the extraordinary ray propagates faster than the ordinary ray inside BBO. If the o-e pair is generated in the middle of the crystal, the e-polarization will trigger the detector earlier than the o-polarization by a time T = (no - ne)L/2c. This implies that D, would be fired first in the / o l e 2 )term; but D, would be fired first in le,o,) term. If r is greater than the coherence width of the signal-idler field, one would be able to distinguish which amplitude gives rise to the “click-click” coincidence event. One may compensate the “walk-off’ by introducing an additional piece of birefringent material to delay the e-ray relative to the o-ray by the same amount of time T . However, as SPDC is a coherent process, the signal-idler pair may be created anywhere along the crystal (indistinguishable). So, how to determine the delay time r ? We will defer this question until after the discussion of “two-photon wavepacket.” In fact, we will learn that T = (no - ne)L/2cis the correct solution for the “compensation.” Let us keep the “walk-off’ terminology in this section. After the compensation, a double-entangled EPR state is ready for the testing of Bell’s inequalities based on either spin or space-time observable. B. EXPERIMENT ONE: BELL’SINEQUALITY FOR SPIN VARIABLES The schematic diagram of the experiment is shown in Fig. 9. The 35 1.1 nm pump beam of a single-mode argon ion laser, followed by a dispersion prism to remove the unwanted fluorescence, is directed to a 3-mm-long BBO crystal, which is cut at phase matching angle O,, = 49.2” for collinear degenerate SPDC. The crystal is tilted by 0.72” so that the effective value of ePmis increased to 49.63’ (inside the crystal) for noncollinear phase matching of degenerate 702.2 nm wavelength SPDC. The two cone-overlap directions, selected by irises before the detectors, are consequently separated by 6.0”.Two avalanche photodiodes D , and D,
18
Yanhua Shih
-
\
Detector 1
+
+
! Detector2 FIG. 9. Schematic of one experimental setup for Bell's inequality testing based on polarization of photon.
operated in Geiger mode are used for the signal-idler coincidence detection. Each detector has a narrow-band spectral filter with 5 nm FWHM centered at 702.2 nm, which determines the measured coherence width of the signal-idler fields. Polarization analyzers A , and A, are located in front of detectors D , and D,, respectively. We record coincidence rate R ( 6 , , 6,) as a function of the polarization analyzer angle settings and 6,. Two pieces of additional 1.5 mm BBO crystal C , and C, are inserted in each of the paths, 1 and 2, which play the role of the compensator. Instead of rotating the o-e axes by 90" relative to that of the SPDC BBO for compensation, we use a half-waveplate HWP to exchange the roles of the oray and the e-ray polarizations. It is interesting to see that one can easily produce any of the four EPR-BohmBell states (Bell state in short), =
1 --(lX,Y,)
fi
k
IYJ2))
(22)
where we have defined an X-Y base, in replacing the original o-e base. The k sign (actually the value of a in Eq. (21)) can be realized by rotating C , or C , or using an additional birefringent phase shifter, FS, to slightly change the total optical path (for example the total path of X, and Y2) difference between the two amplitudes, which will be discussed again in the next section. Similarly, a halfwaveplate in one path can be used to change X polarization to I: and vice versa, for realizing states ICP t ).
TWO-PHOTON ENTANGLEMENTAND QUANTUM REALITY
19
TABLE 1 MEASUREMENTS OF PARAMETER S FOR THE FOUR EPR-BOHM-BELL STATES AND THE ASSOCIATED COINCIDENCE RATE FUNCTIONS.
EPR-Bell State IT+) IT-)
I@+) I@-)
~ ( 0 ,e,).
S
sin2 (0, + e,) sin2 (0, - 0,) COSZ (0, + 8,)
-2.6489 2 0.0064 -2.6900 t 0.0066 2.557 ? 0.014 2.529 5 0.013
(e, - e,)
COS~
Note: Measurements for I @) were improved later. The repeated measurements of S for both I*) and 1 @) yield higher accuracy, indicating violations of more than 150 standard deviations.
We observed the expected correlations (see Table 1) for each of the four Bell states. A typical measured fringe visibility is about (98.0 ? l.O)%,indicating a high degree of entanglement of the two-photon state (Kwiat et al., 1995). As is well known, the high visibility sinusoidal coincidence fringes in this kind of experiment imply a violation of a specific Bell's inequality. In particular, the inequality of Clauser, Home, Shimony, and Holt (CHSH, 1969) shows that IS I2 for any local realistic theory, where
I
s = E(e,, 8,) + q e ; , 8,) + ,ye,, e;)
-
~(q 8;),
(24)
and E ( 8 , , 0,) is given by
c(o,,0,) + c(e:,8;) - cw,, 6); - c(et,0,) cv,, e,) + c(e;,6 ; ) + c(e,,0;) + cw:, 0,)
(25)
The measured values of S are reported in Table 1. Fcr each of the four Bell states we took extensive data for the settings: 8, = -22.5", 8: = -67.5'; 8; = 22.5', 0;. = 112.5";and 8, = -45', 8; = 45"; 8; = O", 8;. = 90'. The CHSH inequality is found to be strongly violated in all cases. For one of the measurements, a maximum violation of 102 standard deviations was observed (Kwiat et al., 1995). Our recent unpublished data have shown violations of CHSH inequality with more than 150 standard deviations (Strekalov, 1997). c . EXPERIMENT TWO: BELL'SINEQUALITYFOR SPACE-TIME OBSERVABLE In the second type of experiments (Strekalov et al., 1996). one would be surprised to see how easy it is to turn the polarization-based EPR-Bohm state to a spacetime observable-basedEPR state by taking advantage of the double entanglement
20
Yunhun Shih
-
Detector2
FIG. 10. With polarizing beamsplitters, this scheme implements a Franson interferometer; however, there is no need of a short coincidence time window to “cut o f f the unwanted long-short and short-long amplitudes. These unwanted amplitudes are simply not there, so this experiment is considered “postselection-free.”
nature of t y p e 4 SPDC. This experiment may also be considered an implementation of the Franson interferometer for the study of energy-time entanglement (Franson, 1989), see Fig. 10. The experimental setup shown in Fig. 11 is very similar to that in Fig. 9. We use the same design of SPDC as previously described. The signal-idler beams, propagating at 6” relative to the pump, pass through a 12.8 mm long quartz rod to compensate the longitudinal “walk-off’ of the 3 mm BBO SPDC. The Franson interferometeris implemented by 2 X 20-mm-long quartz rods and a Pockels cell placed in each channel 1 and 2, respectively. The quartz rods delay the slow polarization component relative to the fast one due to their birefringent refraction indexes. This delay corresponds to an optical path difference, AL = AnquartzX L 2 360 pm, of the interferometer, which is greater than the coherence length of the signal-idler field (160 pm);this is basically determined by the 3 nm FWHM bandwidth of the spectral filter for detectors D, and D,. The Pockels cell is for “fine-tuning” of the optical path difference, AL, of the interferometerby applying an adjustable DC voltage. The fast-slow axes of the quartz rods as well as that of the Pockels cell are both oriented at 45” relative to the e-o axes of the SPDC BBO. Polarization analyzers, A , and A,, are installed in each channel following the Pockels cells. The axes of the analyzers are oriented at 45” relative to that of the polarization interferometer, that is, the quartz rod and the Pockels cell. Twophoton coincidence rates are recorded as a function of the optical path difference AL through coincidence circuits with 1.8 ns time window.
TWO-PHOTON ENTANGLEMENT AND QUANTUM REALITY
21
FIG. 1 1 . Schematic of one experimental setup for Bell’s inequality testing based on space-time observable.
Before delving into the more rigorous two-photon wavepacket picture, we first use the simplified SchriSdinger-typepicture, in which the polarization kets evolve along the optical tracks, to analyse this experiment. Let us start from Eq. (21). The polarization kets of the state 19)are projected onto the fast and slow axes of the quartz rod and the Pockels cell: 10) ; = 1s); sin 45” + I f ) i cos 45”,le)i = Is)i sin 45” - If)i cos 45”,i = 1, 2, where s andfare the slow and fast polarizathen becomes tion components, respectively. The state I 9) 19) = Is,sz) -
ei(pI+pZ)
Ifif,)
(26)
where q l and q2 are the phase shifts between the slow and the fast components, introduced by the quartz rods and the Pockels cells in channel 1 and channel 2, respectively. Equation (26) is equivalent to the state proposed by Franson (1989),
19)= (L,L,)
+
ei(pIfp2)
ISIS,)
(27)
where Li and S;correspond to the long and the short paths of the interferometer, respectively. However, there is no need for a short coincidence time window to “cut off’ the I L S, ) and the IS,L, ) amplitudes associated with the original Franson interferometer, so this experiment has been considered “postselectionfree.” Equations (26) and (27) imply that the signal and idler could pass either the long path or the short one of an interferometer with equal probability. However, if one passes the long (short) path of one interferometer the other one must pass the long (short) path of the other spatially separated interferometer.
Yunhua Shih
22 20000 u)
U K
o
16000
0
(I) u)
0
. 0
cu
12000
u)
g C
8000
(I)
0 0
.-I=
4000
-
8 0
i
A
-0.1
0.1
-0.3
I
LLL
A
7
Delay in signal wavelengths FIG. 12. Experimental data and the best fitting curve. The x axis indicates relative delay between fast and slow kets.
,
The coincidence counting rate between D and D, is then calculated by:
+
1 - COS((P, pz) = 1 - COS(O, + wi)7 (28) R, where we assume equal optical delays in the two spatially seperated interferometers, p1 p2= (w, oi)7,where T = AL/c. If the measurement yields a 100% interference visibility, then (0, w i )must be a constant. Although both the signal and the idler can take a wide range of energy, if one is measured to have a certain value the other one is determined with certainty. A typical measured interference pattern is reported in Fig. 12. Part of our early published experimental data reported (95.0 ? 1.4)% visibility (Strekalov et al., 1996). Our recent measurements have shown much higher visibility (=loo%) with much smaller experimental error. It is well known that in order to experimentally infer a violation of Bell’s inequality, the interference visibility has to be greater than l/~‘? = 71%. Thus our early published data (95.0 2 1.4)% exceeds the limit by 17 standard deviations.
+
+
+
IV. Why Two-Photon But Not Two Photons? We always state “two-photon.” Why “two-photon” but not “two photons”? What is the difference? Is it important? If “two-photon” is not “two photons,” then what is it? Do we have single particle reality in an entangled two-particle system? What information is available for “a photon” in two-photon measurements? We will find the answer in this section.
23
TWO-PHOTONENTANGLEMENTAND QUANTUM REALITY
A. Is TWO-PHOTON INTERFERENCE THE INTERFERENCE OF Two PHOTONS? To see the difference between two-photon and two photons, let us review a typical two-photon interferometer (Alley and Shih, 1986; Shih and Alley, 1987; Hong et af., 1987) illustrated in Fig. 13. The entangled signal-idler photon pair generated in SPDC is mixed by a 50-50 beamsplitter BS and detected by two detectors D ,and D , for coincidences. Balancing the signal and idler optical paths by positioning the beamsplitter, one can observe a “null” in coincidences, which indicates destructive interference. When the optical path differences are increased from zero to unbalanced values, a coincidence curve of “dip” is observed. The width of the “dip” equals the coherence length of the signal and idler wavepackets (Hong et al., 1987). Various aspects of this “dip” have been extensively studied (Kwiat et al., 1992; Franson, 1992; Steinberg et al., 1992a,b, 1994; Shih and Sergienko, 1994). Loosely speaking, indistinguishability leads to interference, and it is quite tempting to rely on a picture that somehow envisions the interference as arising from two individual photons of a given signal-idler pair. One sees that when the condition for total destructive interference is held, the two optical paths of the interferometer are of exactly the same length and it appears impossible to distinguish which photon caused which single detector detection event. This can be clearly seen from a conceptual Feynman diagram in Fig. 14. The “two photons interference” picture is further reinforced by the fact that changing the position of the beamsplitter from its balanced position, which begins to make these paths distinguishable, will bring about a degradation of two-photon interference. The coincidence counting rate seems to depend on how much overlap of the signal and idler wavepackets is achieved. Thus the shape of the “dip” is determined by the temporal convolution of the signal and the idler wavepackets, and therefore provides information about them. If this picture is correct, then signal and idler photons do interfere. In his fanbms book, The Principles of Quantum Mechanics,
PD’
1
Rc
0
X
FIG. 13. Schematic of a typical two-photon interferometer. The signal and idler of SPDC are “superposed” at BS and detected by detectors D ,and D,.
24
Yanhua Shih
FIG. 14. Conceptual Feynman diagrams. The beamsplitter is represented by the thin vertical lines. It appears impossible to distinguish which photon fired which detector.
Dirac stated that “. . . photon . . . only interferes with itself. Interferencebetween two different photons never occurs.” One should not be astonished by the comment: “Dirac made a mistake. . . .” As a matter of fact, Dirac was correct. It is not the interference between “two photons.” Although it may lead to correct predictions for some experiments, this mental picture is not generally true. For instance, let us consider anew experiment illustrated in Fig. 15. The experimental setup is similar to that in Fig. 13, except we have two paths for the signal beam. When the beamsplitter BS position is x = 0, the idler arm’s length is Lo, and the signal channel has two paths: one pathlength is L, (short path), the other is L, (longer path), such that L, - Lo = Lo - L, = AL >> I,,, where Zcoh is the coherence length of the signal and idler beams. Because of this condition there is no interference of the signal photon itself. The single detector counting rates remain fairly constant. Based on the idea of “two photons,” the interference arising from the indistinguishability of the signal and idler wavepackets, “dips” are expected to appear for two positions of the beamsplitter only, that is, x = 2 A L/2. In these two cases the idler photon has a 50% chance to overlap with the signal photon. This partial distinguishability results in the contrast of these two dips being at most 50%.However, when x = 0 there is no overlap of the signal and idler photon wavepackets. Moreover, the detectors fire at random: In 50% of the joint detections D, fires ahead of D, by T = AL/c; in the other 50% the opposite happens. Thus no interference is expected in this case according to this single photon interferencepicture. Figure 16 shows the experimental result, which tells a quite different story. We do observe a high contrast interference “dip” in the middle (x = 0). In addition, the “dip” can turn to a “peak” if the experimental conditions are slightly changed. Transition from “dip” to “peak” depends on 4 = 4.rrAL/A, where A is the central signal wavelength. Fixing x = 0 and varying 4, we observe a sinusoidal fringe, which is shown in Fig. 16b, corresponding to a transition from “dip” to “peak” in the center part of the curve shown in Fig. 16a.
TWO-PHOTON ENTANGLEMENT AND QUANTUM REALITY
25
Di
D2 FIG. 15. Schematic of a new experiment. In contrast with Fig. 13, there are two optical paths, L, and L,, for the signal. The idler path is Lo.
0 ~ -0.4
” ” -0.3
” ” -0.2
”
”
-0.1
”
0.0
0.1
0.2
0.3
4
Beamsplitter Position
Phase Difference
FIG. 16. Experimental data. (a) A high contrast “dip” is observed. In addition, the destructive “dip” can turn to a constructive “peak” when L, - L, is slightly changed (X-axis unit: mm). (b) The dip-peak transition is shown as function of 4.
26
Yunhuu Shih
The mental idea of “destructiveinterferencebetween signal and idler photons” has failed to give a correct prediction. Thus the “dip” or “peak” may not be conidered as the interference between signal and idler photons. What is it?
B. ENTANGLED STATE AND TWO-PHOTON WAVEPACKET The most important fact is that we are dealing with a two-photon source, SPDC, a pair of signal-idler photon, and an entangled state, which has been already discussed in the beginning of this chapter,
lq)=
2 &us +
0;
s, i
- w,)&k,
+ ki
- k , ) a f ( w ( k , ) ) a t ( w ( k , ) )10)
It would be helpful to know the wavefunction(s) of the signal-idler pair. Even though there is no wavefunction for photon, the two-photon wavepacket can be evaluated in the following way. According to quantum field theory, the coincidence counting rate, Rc, of detectors D , and D,, on the time interval T is given by Glauber formula (1963a,b):
Rc
0~
=
IT
I ’ d T , dT, T o o
’ 1‘ ’I 1 T o o T
=
dT, dT,
T
T o o
dT, dT,
I W,? t,)12
where are positive and negative frequency components of the field at detectors D , and D,, respectively, 1”) is the entangled SPDC state of Eq. (3), and ti = Ti - Lj/c,i = 1,2, where Ti is the detection time and L; the optical pathlength respective to the ith detector. It is easy to see that the two-dimensional function “ ( t , , t , ) we have defined in Eq. (29), q ( t l , t,)
= (01 E$+)E;+)p)
(30)
plays the role of “wavefunction.” We may name it effective two-photon wavefunction. It is nothing but the probability amplitude for resulting in a “click-click’’ event of detectors D , and D,. Actually, this concept has been seen in the “ghost” interference section. Let us consider a simple experiment, in which we have only one amplitude: signal triggers D and idler triggers D,. The fields at D and D , are given by
,
TWO-PHOTON ENTANGLEMENTAND QUANTUM REALITY
27
where the aj(o)’s are the annihilation operators for the signal and idler, and the fi(w)’s are spectrum distribution functions. It is straightforward to calculate q ( f t 2 ) (Shih and Sergienko, 1994; Sergienko ef al., 1995; Rubin et al., 1994; and Appendix B): q ( t l , t 2 ) = Aoe-“:(fl
+ t z ) 2 e - ~ 2 ( ~ ~ - f ~ ) 2 e - i n , f , , - i n i t z (31)
for type-I SPDC (note 5 ) , where Q j ,j = s, i, is the central frequency for signal or idler, l/a, are coherence times that will be discussed later, y d ti = Ti- Li /2c, i = 1, 2, Tiis the detection time of detector i and Li the optical pathlength of the signal or idler from SPDC to the ith detector. q ( f ,t 2, ) is a two-dimensional wavepacket in configuration space; we may call it the biphoton. For type-I1 SPDC (note 5), one substitutes Eq. (19) into Eq. (30), then the wavepacket q ( t , , t2). or biphoton, is calculated with an unsymmetrical (in respect to “zero” o f t , - t z ) rectangular shape, q ( t , , t 2 ) = A,e-“:(fi+fz)’n(t, - f2)e-insfie-inifz
(32)
where 1 0
if 0 5 t , - t , if otherwise
IDL
and D = l/u, - l h , , u, and u, are recognized as the group velocities of the ordinary and extraordinary rays of the SPDC crystal, and L is the length of the crystal. Figure 17 is a schematic diagram of q ( t I , t 2 )for type-I and type-I1 SPDC, respectively. The unsymmetrical rectangular shape of type-I1 wavepacket is essential for the understanding of the “compensation” of the ‘‘walk-off’ problem, which has been discussed in the last section in a simplified manner.
tl-t2
tl42
FIG. 17. Two-photon wavepacket envelopes for type-I (a) and t y p e 4 (b) SPDC.Note: type-II wavepacket has a rectangular shape in t l - t, and is unsymmetric to t, - t , = 0.
28
Yunhuu Shih
It is clear from Eq. (31) and Eq. (32): The two-photon wavepacket Yr(t,, t , ) is not a product of the wavepackets of signal and idler photons. This again illustrates the entanglement nature of the two-photon state of SPDC. The two-photon wavepacket Y r ( t , , t,), or biphoton, plays important roles in two-photon experiments. We will see this clearly through the discussions that follow. C. EXPERIMENT ONE:TWO-PHOTON INTERFERENCE
This experiment has been briefly described in the beginning of this section. The schematic of the experiment is illustrated in Fig. 15. We have four probability amplitudes that result in a click-click detecting event of D ,and D,. There are two distinct click-click events that can happen. Either detector D ,fires ahead of detector D, by time r = ALIc, where AL = L, - Lo = Lo - L, >> lcoh, or D, fires ahead of D ,by the same time r . The first event happens either when the retarded part of the signal is transmitted to D,, with the idler transmitted to D,, or when the advanced part of the signal amplitude is reflected to D ,, with the idler reflected to D,. Similarly, the second event happens either when the retarded part of the signal amplitude is reflected to D, , and the idler is reflected to D,, or when the advanced part of the signal amplitude is transmitted to D,, and the idler is transmitted to D ] . Each of the four two-photon amplitudes is conveniently represented by a conceptual Feynman diagram in Fig. 18. Y r ( t , , t , ) is a superposition of these four amplitudes, Yr(tl, t,) = A ( t f 0 ,
ti!) + A ( t f s , tie) + A ( t f / , t2) + A ( t F , t!p)
(33)
As we usually do for the single-photon interferometer, we consider here four biphoton wavepackets corresponding to each term in Eq. (33). However, the wavepackets are two-dimensional. This has the form of Eq. (3 1) for type-I SPDC or of Q. (32) for type-I1 SPDC. For further convenience, we will introduce variables t , = t l + t , = T+ - L , lc and t - = t, - t , = T- - L-lc, where T5 = T I 5 T, and L5 = L , ? L,. Thus Eq. (29) becomes
and each of the wavepackets in Eq. (33) has the form A ( t - , t + ) = Aoe-u2f2e-u:':e-irrCt+/Ap
(35)
where we have assumed degenerate central wavelength SPDC, that is, A, = Ai, for simplicity. Note that there are two coherence times, l/a-and lla, , that can be said to localize the two-photon wavepacket or biphoton in t - and t , directions, respectively. This is the essence of the two-photon wavepacket concept. In our
TWO-PHOTON ENTANGLEMENT AND QUANTUM REALITY
29
FIG. 18. Conceptual Feynman diagrams indicating four probability amplitudes resulting in a coincidence detection. In (a) and (b) D ,fires ahead of D,,in (c) and (d) D,fires ahead of D,.
experiment u- = c/21,,, where lcoh is the coherence length of the signal and idler (note 8). It is a short coherence time: llu- < ALIc. On the contrary, the other coherence time is long, u+ = and 1/u+>> ALlc where l{',h is linked to the coherence length of the single-mode CW laser pump beam. It is straightforward to rewrite Eq. (33) in the following form, in the case of AL = L, - Lo = Lo - 1, T C .
It is not hard to see that the first two wavepackets and the last two wavepackets in Eq. (36) are overlapping, respectively, in the t- direction. Because of llu, = 2 f i 1 $ h / c >> ALlc, these wavepackets are also considered as overlapping in the t+ direction, respectively. Interference is expected. Do we have any knowledge about single-photon wavepackets in the previous analysis? No, we do not. What we do know is that the signal and the idler photons are distinguishablefrom the conceptual Feynman diagram in Fig. 18. Two-photon interference is definitely not the interference of two-photons. When we substitute Eq. (36) into Eq. (34) and integrate over d T - , the result breaks up into three disjoint intervals, which is found to be in complete agreement with our experiment:
30
Yunhuu Shih D1
D2 FIG. 19. Schematic of the experimental setup.
where 4 = 4rrAL/h,, i . Setting 4 to be subsequently equal to rr, 0, and 1~12, and varying the relative delay x we observe respectively a peak, a dip, or a flat coincidence rate R , distribution in the center ( x = 0). These three cases are shown in Fig. 16. Separation between the dips is equal to AL, and the widths of all dips (or peak) are equal to lcoh. It is interesting to note that the side-dips do not depend on 4. Instead, they correspond to the third and fourth terms of Eq. (37). The real experimental setup is shown in Fig. 19. A single-mode argon ion laser of 351.1 nm wavelength is used to pump a 3-mm-long BBO for type-I SPDC. The central wavelengths of signal and idler, A, = Ai = 702.2 nm are equal to twice the pump wavelength A,,. Both signal and idler are polarized in the horizontal direction and propagated at about 3.7" from the pump beam. A rod of birefringent material (crystal quartz) oriented at 45" with respect to the signal polarization is inserted in the signal channel. Its function is to provide L, and L, for the signal. Variation of the phase 4 is achieved by a Pockels cell, which is aligned with the quartz rod. A polarizer behind the Pockels cell recovers the initial polarization (note 9). The large-scale optical delay in the longer arm relative to the shorter one is equal to L, - L, = 2AL = AnL = 360 pm, where An is the birefringence and L the length of the quartz rod. The coherence length I,, of both the signal and idler is determined by the 3 nm bandwidth of the spectral filters placed in front of the detectors. For 3 nm FWHM filters, I,, = 160pm is shorter than the delay 2AL. The detectors are photon-counting avalanche photodiodes. The output pulses are brought to a coincidence circuit with a 10 ns acceptance window. D. TWO-PHOTON WAVEPACKET IN BELL'SINEQUALITY MEASUREMENT It is the two-photon wavepacket, or biphoton, that plays the role in two-photon experiments. We have emphasized in the Bell's inequality section that one needs
TWO-PHOTON ENTANGLEMENT AND QUANTUM REALITY
31
to consider the two-photon wavepacket (space-time) even for the EPR-Bohm spin entanglement. The two-photon wavepackets of the superposed amplitudes have to be overlapping. This language is similar to the language one usually uses for single-photon interferometer, except here the statement is for two-photon wavepacket, or biphoton. As a matter of fact, the EPR-Bohm-Bell measurement is a two-photon interference measurement, even though there is no “interferometer” involved. The type-I1 SPDC “compensator” is a good example. Consider a noncollinear type-I1 SPDC. The signal-idler pair is emitted into two cones, one ordinary polarized, the other extraordinary polarized, (o-e is defined by the SPDC crystal) as in Fig. 8. Along the intersection lines where the cones overlap, two detectors D , , and D , are used for two-photon coincidence detection. There are two quantum mechanical amplitudes contributing to a “click-click” event: (1) signal (ordinary) fires D , and idler (extraordinary) fires D,, ( 2 ) the idler (extraordinary) fires D , and signal (ordinary) fires D , . Equation (21) is a simple quantum mechanical description for this statement. We do have a quantum mechanical superposition. Do we have interference, or EPR-Bohm correlation (for spin)? No, if there is no “compensator.” Why? Because the welcher weg information has not been erased yet (Scully and Driihl, 1982). This point can be clearly seen by looking at the picture of two-photon wavepackets. The two-photon wavepacket for amplitude ( 1 ) is
A(q, tl)
= Aoe-u$(fp+fs)*JJ(rO - t ; ) e - i 4 f p + f z ) I
whereas the two-photon wavepacket for amplitude ( 2 ) is different A ( t ; , t z ) = A o e - U : ( f ~ + r 5 ) 2 n (t ; tq)e-i4ft+fq)
These two dimensional wavepackets do not overlap, because of the unsymmetrical rectangular function of rI(t - t,), see Fig. 20. In order to make these two wavepackets overlap, we can either ( 1 ) move both wavepackets a distance DL12, or ( 2 ) move one of the wavepacket a distance DL. The “compensator” described in the last section belongs to case ( 1 ) . We have also proved case (2) experimentally (Kwiat er al., 1995). There is no need to determine the “birthplace of the pair” and we can design a “compensator” correctly. DL12 = r = (no - n,)LMc is the correct choice. From the point of view of “quantum eraser,” the “compensator” can be considered as an “eraser” (Scully and Driihl, 1982). The welcher weg information is erased by the “compensator,” because of the two-photon wavepacket overlapping. However, remember that the welcher weg information here is for two-photon measurement, and the use of the “eraser” makes the “click-click” probability amplitudes indistinguishable. Now we can also understand why it is so easy to realize the four Bell states of Eq. (22) and Eq. (23). The “slight rotation” of the compensators C , , C,, or a
32
Yanhua Shih
-DL
0
DL
FIG.20. Without “compensater,” the two dimensional wavepackets of amplitude ( 1 ) and (2) do not overlap in r , - r2 axis.
phase shifter in the spin variable Bell’s inequality measurement, see Fig. 9, is nothing but tiny shifts between the two biphoton wavepackets. A shift of halfwavelength results in a phase change of T between the two amplitudes. E. EXPERIMENT Two: SINGLE-PHOTON MEASUREMENT OF A TWO-PHOTON STATE All the preceding measurements are coincidence measurements or so-called “click-click’’ events. What happens if we measure a “click” only? The following experiment measures the spectrum of the signal while ignoring the idler of SPDC. The purpose of the measurement is to determine the shape and width of the “single-photonwavepacket” from a two-photon source (note 4). It is a typical Fourier spectroscopy measurement. The schematic setup of the experiment is shown in Fig. 21. The signal-idler two-photon state is generated in a collinear degenerate type-I1 SPDC. After the cleanup of the 702.2 nm wavelength, that is, the signal-idler twin beams, the idler (extraordinary-ray of BBO) is removed by a polarizing beamspliter BS. The signal (ordinary-ray of BBO) is then sent to a Michelson interferometer. A photon-counting detector (avalanche photodiode operated in Geiger mode) is coupled to the output port of the interferometer. A 702 nm spectral filter with Gaussian transmittance function of 83 nm FWHM bandwidth proceeds with the detector. The counting rate of the detector is recorded as a function of the optical arm length difference,AL, of the Michelson interferometer. Note that for the Michelson interferometer the actual optical path difference is 2 X AL. What do we expect from the measurement? A Gaussian spectrum with 83 nm FWHM? The SPDC generate a wide bandwidth of spectrum that is much greater than 83 nm so that the measured spectrum should be determined by the spectral filter. No! Instead we observed an “unexpected” result, which is reported in Fig. 22. The wavepacket of the signal photon is not Gaussian. The envelope of the sinusoidal modulations (in segments) is fitted very well by two “notch” functions
33
TWO-PHOTON ENTANGLEMENTAND QUANTUM REALITY
Prism
Ar Laser
to counter FIG.21. Experimental setup for single-photon measurement of a two-photon source. The Fourier spectroscopy-type measurement determines the shape and width of a “single-photon’’ wavepacket. 70000 60000 50000 40000
30000 20000 10000 0 -1
. . . . ..
. ., .
I
. . .. . .
.
I
..
..
, i
.
. .. .
,.
. . . ..
.. . . .. . .
..
...
-
I
FIG. 22. Experimental data indicated a “double notch” envelope. Each of the dotted single vertical lines contains many cycles of sinusoidal modulation. The width of the triangular base is 21 1 p m roughly corresponding to a spectral bandwidth of 2 nm. Note: The sharp line in the center of the “double notch” has a Gaussian-shape envelope corresponding to 83 nm bandwidth. The interference modulation is close to 100% inside that sharp line envelope.
34
Yunhuu Shih
(upper and lower part of the envelope). The width of the triangular’s base is about 21 1 p m , which roughly corresponds to a spectral bandwidth of 2 nm. We have two questions immediately: (1) Why not Gaussian? (2) Why 2 nm instead of 83 nm? Before finding the answer, let us first ask: (1) Why Gaussian? (2) why 83 nm? Are we sure the spectrum of the signal has nothing to do with the idler? No, we are not sure! We are dealing with an entangled two-photon state. The physics concerned here arise from the two-photon state of SPDC. We have to calculate from the two-photon state. It is interesting to find that even though the two-photon state is a pure state, that is, $2
$=
= $,
IWTI
(38)
where $ is the density matrix operator of the SPDC two-photon state, the corresponding single photon state of the signal and idler
(39) IT) (TI, Bi = tr, IT) (TI are not. To calculate the signal (idler) state from the two-photon state, we have to take a partial trace in Eq. (39), as usual, summing over the idler (signal) modes. It is very clear that any change of the mode structure of idler (signal) would modify the state of signal (idler). In our experiment, the orthogonally polarized signal and idler are degenerate in frequency around w = wJ2, where wp is the pump frequency. Equation (19) can be further simplified to an integral over a frequency-detuningparameter u:
6,
= tri
where @(DLu)is a sinc-like function: 1@(DLu) =
e-iDLv
iDL u
which is a function of the crystal length L, and the difference of inverse group velocities of the signal (ordinary) and the idler (extraordinary), D = I/u, - l/ue. The constant A, is calculated from the normalization tr @ = (*IT)= 1 (dimensionless). Substitute Eq. (40) into Eq. (39), that is, summing over the idler modes, the density matrix of the signal is given by
fiS = A ;
du I@(u)12 a:(@
+ u ) 10) (01 a,(w + u )
where
I@WI2=
DL
DLu sinc2 2
(41)
TWO-PHOTONENTANGLEMENT AND QUANTUM REALITY
35
First, we find immediately that fi: # fi,, so that the signal photon state is a mixed state (as is the idler state). Second, it is very interesting to find that the spectrum of the signal photon depends on the group velocity of the idler photon, which is not measured at all in our experiment. However, this should not come as a surprise, because the state of the signal photon is calculated from the two-photon state by integrating over the idler modes. By now, the experimental results can be well understood. For a spectrum of sinc-square function we do expect a double “notch” envelope in the measurement and the base of the triangle should be DL (we have considered the optical path difference 2 X AL in the Michelson interferometer), which is calculated to be 225 pm, corresponding to a 2.2 nm bandwidth. It is straightforwardto evaluate numerically the Von Neuman entropy, S = -tr
(fi log fi)
of the signal or idler based on the “double notch” fitting function, and find it greater than zero. The numerical evaluation yields S, = 6.4. This is an expected result due to the statistical mixture nature of the subsystem.However, the entropy of the signal-idler two-photon system is zero (pure state). Does it mean that negative entropy is present somewhere in the entangled two-photon system? According to classical “information theory” (see, for example, Shannon and Weaver, 1995, and Bennett, 1995), for the entangled two-photon system, S, S,li = 0, where SSliis the conditional entropy. This conditional entropy must be negative, which means that given the result of a measurement over one particle, the result of measurement over the other must yield negative information. This paradoxical statement is similar and in fact closely related to the EPR “paradox.” It is not only the classical local realism, but also classical information concepts that contradict quantum mechanics.
+
V. Conclusion By now, we may draw at least one conclusion: Two-photon is not two photons. In an entangled two-particle system, 2 f 1 1. In addition, the “physical reality” of a two-particle system is very different from that of a system with two particles. There is no independent single-particle physical reality in a two-particle system.
+
VI. Acknowledgments The author would like to acknowledge years of research collaboration with C. 0. Alley, M. H. Rubin, D. N. Klyshko, A. V. Sergienko, T. B. Pittman, and D. V. Strekalov. This research was supported by the U.S. Office of Naval Research.
36
Yanhua Shih
VII. Notes 1. Klyshko, D. N. Photon and nonlinear optics. New York: Gordon and Breach Science; Yariv, A. Quantum electronics. (1989). New York: John Wiley and Sons. “Spontaneous parametric down conversion” was called “spontaneous fluorescence” and “spontaneous scattering” by the pioneer workers, for example, Harries, S. E., Oshman, M. K., and Beyer, R. L. (1967). Observation of tunable optical parametric fluorescence. Phys. Rev. Lett., 18,732; which are closer to the physics. 2. See, for a review, Clauser, I. F.,and Shimony, A. (1978). Bell’s theorem: experimental tests and implications. Rep. Prog. Phys. 41, 1883. Aspect, A. et al. (1981). Experimental tests of realistic local theory via Bell’s theorem. Phys. Rev. Lett. 4 7 , 4 6 0 (1982). Exprimental realization of EinsteinPodolsky-Rosen gedankenexperiment: A new violation of Bell’s inequalities. 49.91 (1982); Experimental test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett. 49, 1804. 3. Strekalov, D. V., Pittman, T. B., and Shih, Y. H. (1998). What we can learn about single photons in a two-photon interference experiment. Phys. Rev. A 57,567. Another experiment also demonstrated “two-photon is not two photons”: Pittman, T. B., Strekalov, D. V., Migdall, A., Rubin, M. H., Sergienko, A. V. and Shih, Y. H. (1996). Can two-photon interference be considered the interference of two photons? Phys. Rev. Lett., 77, 1917. 4. Strekalov, D. V.,and Shih, Y. H. (1998). Negative entropy in an entangled state, submitted to Phys. Rev. This experiment has been published in two conference proceedings (1997); also see Strekalov, D. V. (1997). 5. In type-I SPDC, signal and idler are both ordinary (or extraordinary) rays of the crystal; however, in type-I1 SPDC they are orthogonally polarized, that is, one is ordinary and the other is extraordinary. 6. For related theory see, Klyshko, D. N. (1988). Combined EPR and two-slit experiments: interference of advanced waves. Phys. Lett. A 132, 299; Klyshko, D. N. (1988). A simple method of preparing pure states of an optical field, of implementing the Einstein-Podolsky-Rosen experiment, and of demonstrating the complementarity principle. Sov. Phys. Usp., 31,74; Belinskii, A. V.and Klyshko, D. N. (1994). Two-photon optics: Diffraction, holography, and transformation of two-dimensional signals. J E W , 78, 259. 7. The noncollinear type-I1 SPDC brought attention to several research groups around 1984 during the Conference on fundamental problems in quantum theory, held at UMBC. Kwiat et al. (1995) and Strekalov et al. (1996) are the results of a research collaboration with Zeilinger’s group. 8. In this experiment, both single-photon coherence length I,, and (+- are determined by interference filters. This is not a general rule; (+- can be completely unrelated to lcoh.In t y p e 4 SPDC (Sergienko et al., 1995) not only width but also shape of the two-photon wavepacket in t--direction ( t , - r2) is quite different: it is rectangular. 9. This device is described in more detail in Strekalov et al. (1996).
VIII. References Alley, C. 0. and Shih, Y. H. (1986). Proceedings of the Second International Symposium on Foundations of Quantum Mechanics in the Light of New Technology. M. Namiki (Ed.) Bell, J. S. (1964). On the Einstein-Podolsky-Rosen paradox. Physics 1, 195. Bell, J. S. (1987). Speakable and unspeakable in quantum mechanics. New York: Cambridge University Press. Bennett, C. H. (1995). Physics today 48 (lo), 24. Clauser, J. F., Home, M. A,, Shimony, A., and Holt, R. A. (1969). Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23,880.
TWO-PHOTON ENTANGLEMENT AND QUANTUM REALITY
37
Bohm, D. (195 I). Quantum theory. New York: Prentice Hall. Bohm, D. (1952a). A suggested interpretation of the quantum theory in terms of ‘hidden variables,’ I, fhys. Rev. 85, 166. Bohm, D. (1952b). A suggeted interpretation of the quantum theory in terms of ‘hidden variables,’ 11. fhys. Rev. 85, 180. Bohm, D. ( I 957). Causality and chance in modern physics. New York Harper. Einstein, A., Podolsky, B., and Rosen, N. (1935). “Can quantum mechanical description of physical reality be considered complete?” fhys. Rev. 47,777. Franson, J. D. (1989). Bell inequality for position and time. fhys. Rev. Lett., 62,2205. Franson, J. D. (1992). Nonlocal cancellation of dispersion. fhys. Rev. A, 45, 3 126. Glauber, R. J. (1963a). The quantum theory of optical coherence. fhys. Rev. 130,2529. Glauber, R. J. (1963b). Coherent and incoherent states of the radiation field. Phys. Rev. 131,2766. Greenberger, D. M., Home, M. A., Shimony, A. and Zeilinger, A. (1990). Bell’s theorem without inequalities. Am. J. Phys. 58, 1131. Hong, C. K., Ou. Z. Y., and Mandel, L. (1987). Measurement of subpicosecond time intervals between two photons by interference. fhys. Rev. Lett., 59,2044. Keller, T. E., Rubin, M. H., Shih, Y. H., and Wu, L. A. (1998). Theory of the three-photon entangled state. fhys. Rev. A 57,2076. Klyshko, D. N. (1980). Using two-photon light for absolute calibration of photoelectric detectors. Sov. J. Quanf.Elec., 10, 1112. Kwiat, P. G., Mattle, K., Weinfurter, H., Zeilinger, A., Sergienco, A. V., and Shih, Y. H. (1995). New high-intensity source of polarization-entangled photon pairs. fhys. Rev. Leff.,75,4337. , The science and the life of Albert Einsfein. Oxford: Oxford Pais, A. (1982). Subtle is the Lord University Press. Pittman, T. B., Shih, Y. H., Strekalov, D. V.,and Sergienko, A. V. (1995). Optical imaging by means of two-photon quantum entanglement, fhys. Rev. A 52, R3429. Rubin, M. H., Klyshko, D. N., and Shih, Y. H. (1994). Theory of two-photon entanglement in type-I1 optical parametric down-conversion. fhys. Rev. A 50,5 122. Schrodinger, E. (1935). Nufunvissenschafren23, 807, 823, 844; translations appear in J. A. Wheeler and W. H. Zurek (Eds.). (1983). Quantum theory and measurement. New York Princeton University Press. Scully, M. 0. and Driihl, K. (1982). Quantum eraser: A proposed photon correlation experiment concerning observations and delayed choice in quantum mechanics. fhys. Rev. A 25,2208. Sergienko, A. V., Shih, Y.H., and Rubin, M. H. (1995). Experimental evaluation of a two-photon wave packet in type-I1 parametric downconversion. JOSAB. 12,859. Shannon, C . E. and Weaver, W. (1949). The mathematical theory of communication. University of Illinois Press. Shih, Y. H. and Alley, C. 0. (1987). New type of Einstein-Podolsky-Rosen experiment using pairs of light quanta produced by optical parametric down conversion. fhys. Rev. Letf.,61,2921. Shih, Y. H. and Sergienko, A. V. (1994). Two-photon anti-correlation in a Hanbury-Brown-Twiss type experiment. fhys. Lett. A, 186.29. Shih, Y. H. and Sergienko, A. V. (1994). Observation of quantum beating in a simple beam-splitting experiment. fhys. Rev. A, 50,2564. Shih, Y. H., Sergienko, A. V., Rubin, M. H., Kiess, T. E., and Alley, C. 0. (1994). Two-photon entanglement in type41 parametric down-conversion. fhys. Rev. A 50, 23. Steinberg, A. M., Kwiat, P. G., and Chiao, R. Y.(1992a). Dispersion cancellation in a measurement of the single-photon propagation velocity in glass. fhys. Rev. Lett., 68,242 I. Steinberg, A. M., Kwiat, P. G., and Chiao, R. Y. (1992b). Dispersion cancellation and high-resolution time measurement in a fourth-order optical interometer. fhys. Rev. A, 45,6659. Steinberg, A. M., Kwiat, P. G., and Chiao, R. Y.(1994). Measurement of the single-photon tunneling time. fhys. Rev. Left., 71,708.
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Yanhua Shih
Strekalov, D. V. (1997). Biphoton optics. Ph.D Dissertation, Graduate School of The University of Maryland at Baltimore. Strekalov, D. V., Pittman, T.B., Sergienko, A. V., Shih, Y. H.,and Kwiat, P.G. (1996). Postselectionfree energy-timeentanglement. Phys. Rev. A 54,R1. Strekalov, D. V., Sergienko, A. V., Klyshko, D. N., and Shih, Y. H. (1995). Observation of two-photon ‘ghost’ interference and diffraction. Phys. Rev. Len., 74,3600. Wheeler, J. A. (1983). Niels Bohrin today’s words. In J. A. Wheeler and W. H.Zurek (Eds.), Quanrurn theory and measurement. New York: Princeton University Press.
Appendix A: The Two-PhotonState We consider calculating the output state of SPDC to first order in perturbation theory:
where X, is the interaction Hamiltonian. In the following, we assume type-I1 phase matching. In type-I1 SPDC, the annihilation of the pump results in the creation of an extraordinary polarized signal (e-ray), and an ordinary polarized idler (0-ray). The standard form of the interaction Hamiltonian is therefore:
where x is an electric susceptibility tensor that describes the crystal’s nonlinearity and H.C. is the Hermitian conjugate. V is the interaction volume covered by the strong laser pump beam, E Y ) , where we chose to be in a simplest form of a plane wave: E Y ) = EP e i ( k p Z - w p ‘ )
(A-3)
The signal and idler operator are given by
(A-4) where j = 0, e and uij is the creation operator for the j-polarized mode of wave vector k j . If we break up the volume integration into transverse and longitudinal parts, the interaction Hamiltonian can be written as
(A-5)
TWO-PHOTON ENTANGLEMENT AND QUANTUM REALITY
39
where L is the length of the crystal and all constant factors have been lumped into the constant A , . If we make a reasonable assumption that the pump beam diameter is very large (>> A), and take the limits of the area integration to infinity, thus giving a delta-function for the transverse components of the k-vectors of the SPDC fields,
We could have a similar integral for the longitudinal part, S(ke, + k,, - k,), leading to a delta-function of the k vectors: 6(ke + k, - k,)
(A-7)
However, if the finite length of the crystal has to be taken into account, the deltafunction of the longitudinal part is replaced by a sinc-like function:
where A k = k, - k , - k,. With these results, substituting the interaction Hamiltonian into Eq. (A-l),
where A, is a new constant, we have used the delta-function for the k vectors. For a reasonable steady-state assumption the two-photon state is thus: r
r
19) = A,
)
d’k,
)
d3k, 6(we + w,
Appendix B: The Biphoton Wavefunction The two-photon effective wavefunction or biphoton wavepacket is defined by
w,,
12)
= (01 E (, + ) E1( + IW )
03-1)
where f j = T, - rj/c, j = 1, 2, ’;. is the time at which detectorj fires and rj is the distance from the SPDC output to the jth detector, and Iq )is the two-photon
40
YaflhuaShih
state. To simplify the calculation, we consider the longitudinal part only. We now write the two-photon state in terms of the integral of k, and k,:
I*)
=
Ah
/
dk,
I
dk, 6(w,
+ w,
- wp)(D(AkL)aLea;, 10) (B-2)
where a type-I1phase-matching crystal with finite length of L is assumed. We also assume a simple coincidence measurement, in which the e-ray triggers detector 1 and the o-ray triggers detector 2. The field operators for D ,and D, are given by
E:f’ =
I
03-31
dw’ f 2 ( w ’ )a,(o’)e-i”’‘4
where a j ( o ’ ) , j= e, o is the photon destruction operator of mode d , f k ( W ’ ) , k = 1, 2, is the spectral transmission function of an assumed filter placed in front of the kth detector, and again, t ; = T, - rI/c, and t5 = T2 - r 2 k . To simplify the calculation, we consider a Gaussian shape function: f k = f,
e-[(o’-n,)*1/20:
03-41
where is the center frequency of the kth spectral filter. It is convenient and actually realistic to treat the filter functions identically, so that the bandwidth parameter crI = cr, = cr. Substitute Eq. (B-3) and Eq. (B-2) into Eq. (B-1): * ( t , , t 2 ) = A,
/ / dk,
dk, S(w,
+ w, - w,)(D(A,L)f(w,)f(w,)e-i(o,t;+ootS) (B-5)
We define we = Cn, + u and w, = Cn, - v, where v is a small returning frequency, so that we + w, = Cn, still holds. Consequently, we can expand k, and k, around K , ( f l , ) and K,(Cn,) to first order in u:
where u, and u, are recognized as the group velocities of the e-ray and o-ray at frequencies Cn, and Cn,, respectively. Using Eq. (B-6) we see that:
hk=kp-k,-ko=u
:i
---
(u:
=vD
TWO-PHOTON ENTANGLEMENT AND QUANTUM REALITY
41
Another useful parameter we can define is a difference frequency Rd =
;(a,- R,) so that in the degenerative case:
R
R e = a-P+ + & 2
"
z L - 0 2
In this way, we will be able to evaluate the biphoton wavepacket:
which can be broken up into terms involving t ; - t; and t f
+ r;:
To further evaluate Y r ( t , , t , ) we perform the frequency integration:
03-91 where t , , = t f - f;. We further break it into two parts, I , yield nearly identical results:
=
I , - I,, which will
(B-10)
To integrate I, we first complete the square in the exponent, and make a change of variables x = (v/a) i(ut,,/2);
+
We now recognize that Eq. (B-1 1) will be solved in terms of the error functions, 1 - erf(-iz)
=
erf c(-iz)
=
ieZZ
with z = (--iut,*/2).We see that I, reduces to
1[1-
I , = DL
erf(y)]
(B-12)
42
Yanhua Shih
It is clear that I, can be solved in the same way. Therefore, the integration yields
[
I , = - erf ( a:2)] DL
-
erf("12
2 ")]
n(t,,)
(B-13)
Substitute this result into Eq. (B-8), the biphoton wavepacket of type-I1 SPDC is thus:
q(f,, t,)
= A,n(tf -
t~)e-ind(l;-'De-i("p'2)(I;+'S)
(B-14)
or in the form: * ( t i , t 2 ) = A,II(r, -
t2)e-in~rle-in212
(B- 15)
of Gaussian where we have dropped the e, o indices. If a finite bandwidth (a,,) spectrum pump laser beam is considered, we have to include an integral of the pump frequency, which yields a two-dimensional wavepacket: q ( t l , t 2 ) = A,e
-g,?&i
n(t,- t,)e - i n ~ he
+1d2
-in92
(B-16)
The shape of n(t,- t 2 )is determined by the bandwidth of the spectral filters and the parameter DL of the SPDC crystal. If the filters are removed or have large enough bandwidth, we thus have a rectangular pulse function l l ( t , - t,).
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 41
QUANTUM CHAOS WITH COLD ATOMS MARK G. RAIZEN Department of Physics, The Universiry of Texas at Austin, Austin. Texas
I. Introduction ....................................................
LI. Two-Level Atoms in a Standing-Wave Potential .......................
43 45
49
..........
...................................
VI. The Modulated Standing Wave ..... ............... A. Introduction .................................. ......... B. Classical Analysis ...... ................................. C. Experiment . . . . . . . . . . . . . . . . VII. Conclusion and Future Directions ................................... Vm.Acknowledgments ............... M.References ......................................
54 59
59 61 64 66 68 70 72 72 73 76 78 79 79
I. Introduction The interface between nonlinear dynamics and quantum mechanics has become an active area of research in recent years. This emerging field, known as quantum chaos, has focused on the quantum behavior of systems that are chaotic in the classical limit (Haake, 1991; Reichl, 1992). One of the key predictions for Hamiltonian systems with a discrete spectrum is a quantum suppression of chaos. A classical ensemble of particles in a chaotic phase space should execute a random walk, leading to diffusive growth in momentum and position (Tabor, 1989; Lichtenberg and Lieberman, 1991). A system of quantum particles, in contrast, was predicted by Casati et al. (1979) to diffuse in phase space only for a limited time, following the classical prediction. The diffusion is predicted to cease after the “quantum break time” due to quantum interference, and then settle into an exponential distribution. This striking effect, known as dynamical localization, has stimulated a great deal of interest and discussion since it was first predicted. Dynamical localization was predicted to occur in a wide range of systems, and was 43
Copyright 0 1999 by Academic Press All rights of reproductionin any form reserved. ISBN 0-12-003841-2/ISSN 1049-250X199$30.00
44
Mark G.Raizen
also shown by Fishman et al. (1982) to be closely related to Anderson localization, a suppression of electronic conduction in a disordered metal at low temperature (Anderson, 1958; Lee and Ramakrishnan, 1985). Experimental observation of dynamical localizationrequires a globally chaotic (classical) phase space, because diffusion can also be restricted by residual stable islands, and by classical boundaries according to the Kolmagorov, Amol’d, Moser (KAM) theorem (Tabor, 1989; Lichtenberg and Lieberman, 1991). The duration of the experiment must exceed the quantum break time, so that quantum effects can be manifested. Finally, the system must be sufficiently isolated from the environment that quantum interference effects can persist. Experimental investigations of quantum chaos started with the study of microwave ionization of hydrogen by Bayfield and Koch (1974). Since those first pioneering experiments, atomic physics has become an important experimental and theoretical testing ground for quantum chaos, with the emphasis on strongly driven or strongly perturbed systems such as Rydberg atoms in strong microwave, or magnetic fields (Delande and Buchleitner, 1994). In particular, suppression of ionization in the microwave experiments was attributed to dynamical localization (Galvez et al., 1988; Bayfield et al., 1989; Blumel et al., 1989). The advantage of these systems is that evolution is nonlinear and Hamiltonian.The one-dimensional model is reasonably accurate for the chosen Rydberg states and linear polarization of the microwave field. One complication is the presence of stable structures in phase space. It also has not been possible to measure the time evolution in phase space in order to observe the initial diffusion followed by dynamical localization after the quantum break time. These limitations created strong motivation to find new experimental systems that can be used to investigate dynamical localization as well as other problems in quantum chaos. This chapter is a review of our work on the motion of atoms in time-dependent potentials. In particular, we study momentum distributions of ultra-cold atoms that are exposed to time-dependent one-dimensional dipole forces. As we will show, the typical potentials are highly nonlinear, so that the classical equations of motion can become chaotic. Because dissipation can be made negligibly small in this system, quantum effects can become important. This work was originally motivated by a theoretical proposal of Graham et al. (1992), and has evolved over the last few years into a series of experiments on dynamical localization and quantum chaos (Collins, 1995). The organization of this chapter is as follows. In Section I1 we give a theoretical background on atomic motion in a far-detuned dipole potential, and provide a classical analysis of the potential in terms of nonlinear resonances. In Section 111 we describe the general experimental approach. In Section IV we discuss the cross-over from classical stability to chaos via the mechanism of resonance overlap, and describe our experimental tests of this phenomena. In Section V we introduce the S-kicked rotor, a paradigm for classical and quantum chaos, and describe our experimentalrealization,leading to the observationof the quantum break time,
45
QUANTUM CHAOS WITH COLD ATOMS
dynamical localization, and quantum resonances. In Section VI we describe our experiments with a modulated standing wave that also exhibits dynamical localization, illustrating the universal nature of this phenomena. Finally, in Section VII we outline some directions for future work in this emerging field.
11. Two-Level Atoms in a Standing-WavePotential Because this work deals with momentum transfer from light to atoms, it is important to review some basic concepts. The relevant unit of momentum is one-photon recoil (fik,). This is the momentum change that an atom experiences when it scatters a single photon, and leads to a velocity change of 3 cmls for the case of sodium atoms. How does an atom scatter light? The most familiar process is absorption, followed by spontaneous emission. The absorption is from the laser beam, however, the emission is in three dimensions. This process is very important in laser cooling and trapping, but is not desirable for coherent evolution. The probability of spontaneous scattering is proportional to the laser intensity, and inversely proportional to the square of the detuning of the laser from atomic resonance (Cohen-Tannoudji, 1992). This scaling law is valid when the detuning is much larger than the natural linewidth of the atomic transition, and when the intensity is low enough (or the interaction time is sufficiently short). The desired process for atom optics (Adams et al., 1994) is a stimulated scattering, where the atom remains in the ground state, and coherently scatters the photon in the direction of the laser beam. In a single beam (traveling wave) the atom scatters in the forward direction, and there is no net momentum transfer. However, in a standing wave of light created by the superposition of two counterpropagatingbeams, the atom can also back-scatter. This process leads to a momentum change of two photon recoils. Because the effective dipole potential that the atom experiences only scales inversely with detuning, it is possible to make the probability of spontaneous scattering negligible, while still having a substantial dipole potential. We begin a more detailed analysis by considering a two-level atom of transition frequency w, interacting with a standing wave of near-resonant light. If the standing wave is composed of two counterpropagating beams, each with field amplitude E, and wavenumbq k, = 2dAL = wL/c, then the atom is exposed to an electric field of the form E (x, t) = j [ E , cos(k,x)e - i w ~ r c.c.] and its Hamiltonian in the rotating-wave approximation is given by
+
H(x, p , t)
P2 2M
= -
+ h o l e ) (el + [ p E , cos(k,x)e-iwL'le)
(81
+ H.c.]
(1)
Here 1 g ) and I e ) are the ground and excited internal states of the atom, x and p are its center of mass position and momentum, M is its mass, and p is the dipole moment coupling the internal states.
46
Mark G. Raizen
Using standard techniques, we obtain two coupled Schrodinger equations for the ground, $Jx, t), and excited, $Jx, t), state amplitudes
where a / 2 = pE,/n is the Rabi frequency of an atom interacting with just one of the light beams. Note that spontaneous emission from the excited state is neglected; this approximation is valid for the case of large detunings SL = w, - wL from the atomic resonance. The large detuning also permits an adiabatic elimination of the excited state amplitude, resulting in a single equation for the ground state amplitude
The wavefunction of the now "structureless" atom obeys a Schrodinger equation with a one-dimensional Hamiltonian P 2 - V, cos 2kLx H(x, p, t) = 2M The potential has a period of one-half the optical wavelength and an amplitude V, that is proportional to the intensity of the standing wave and inversely proportional to its detuning:
- 2 fi(r/2)2 -
3
SL
I Is,,
Here r is the linewidth of the transition and p is its dipole matrix element. I is the intensity of each of the beams comprising the standing wave and Is,, = diwOr/3A2is the saturation intensity for the transition (I,,, = 6 mW/cm2 for the case of sodium atoms). Equation (6) was derived for a standing wave composed of two counterpropagating beams of equal intensities. If the two beams are not perfectly matched the potential amplitude is still given by this equation, with I taken as the geometric mean of the two intensities. The classical analysis of Eq. ( 5 ) is the same as for a pendulum or rotor, except that the conjugate variables are position and momentum rather than angle and angular momentum. Such a potential is also known as a nonlinear resonance, and is a fundamental building block of Hamiltonian nonlinear dynamics. It is important
QUANTUM CHAOS WITH COLD ATOMS
47
10
5
a
0
-5
- 10
-1
- 0.5
0
0.5
1
s/n FIG. 1. Poincad surface of section for a single resonance. Momentum (vertical axis) is in units of two recoils, and position is in units of one period of the standing-wave potential.
to stress at this point that we are interested in the full nonlinear behavior, and will not limit our analysis to the bottom of the wells, where a harmonic approximation is valid. A convenient representation of phase space is obtained by evolving the classical equations of motion in time, with some period Z This results in a graphical representation known as a Poincark surface of section, which is shown for the standing wave in Fig. 1. The position coordinate is shown for one period of the standing wave, and the momentum is in units of the two-photon recoil. There is a stable fixed point at the center corresponding to the bottom of the potential well, and an unstable fixed point corresponding to the top. The closed orbits represent oscillatory motion for particles with energy less than the total well depth 2V0, whereas the continuing paths describe the unconfined motion of higher-energy
Mark G. Raizen
48
particles. Classically, motion is restricted along the lines shown in Fig. 1 according to the KAM theorem (Tabor, 1989; Lichtenberg and Lieberman, 1991). Note that for large momenta the lines become almost straight, indicating free-particle motion. At smaller momenta, the lines bulge out near the center of the well, corresponding to the particle speeding up as it approaches the bottom of the well. This chapter emphasizes quantum-classical correspondence, which is especially appropriate for deep wells. It is worth mentioning that for weak wells, this system is naturally described in terms of Bloch bands, a concept that is most familiar from condensed matter physics. Indeed, quantum transport in optical lattices has become an active area of research with recent experiments on Bloch oscillations, Wannier-Stark ladders, and tunneling. Unlike the periodic potentials in condensed matter systems, the atom optics system is effectively free from dissipation mechanisms such as phonon scattering and imperfections in the lattice periodicity. Experimental work on this subject was recently reviewed by Raizen et al. (1997a), and is continuing in our laboratory as well as in other groups. In yet another direction, time-dependent dipole potentials have also found applications in atomic interferometry (Szriftgiser et al., 1996; Cahn et al., 1997), and manipulation of atomic wavepackets is a rapidly growing area. To address the problem of quantum chaos, we must go beyond the pendulum or stationary standing wave. The connection between atom optics and quantum chaos was first recognized by Graham et al. (1992), who proposed that dynamical localization could be observed in the momentum transfer of ultra-cold atoms in a phase-modulated standing wave of light. More generally, as shown next, quantum chaos can be studied by adding to the one-dimensional Hamiltonian an explicit time dependence. This can be accomplished, for example with a time-dependent amplitude or phase ozthe standing wave. The electric field of the standing wave then takes the form E(x, t ) = j [ E , F , ( t ) cos{k,[x - F 2 ( t ) ] } e - i W + ~ *c.c.]. The time scales for these controls ranged between -25 ns (the response time of our optical modulators) and milliseconds (the duration of the experiments). The amplitude and phase modulations were therefore slow compared to the parameters w, and 8, relevant to the derivation of Eq. ( 5 ) , so they change that equation by simply modifying the amplitude and phase of the sinusoidal potential. The generic time-dependent potential is thus Mx, p,
0'
P2
+
v,F,,(t)
cosPk,x -FpJt)l
(7)
For simulations and theoretical analyses it is helpful to write Eq. (7) in dimensionless units. We take x u = 1/2k, to be the basic unit of distance, so the dimensionless variable t#~ = x/x, = 2k,x is a measure of the atom's position along the standing-wave axis. Depending on the time dependence of the interaction, an appropriate time scale t, is chosen as the unit of time; the variable 7 = tlt, is then a measure of time in this unit. The atomic momentum is scaled accordingly into
QUANTUM CHAOS WITH COLD ATOMS
49
the dimensionless variable p = pt,/Mx, = p2kLt,/M. This transformation preserves the form of Hamilton’s equations with a new (dimensionless)Harniltonian X(+, p, r ) = H(x, p, t) . tfIMx2 = H . 8wrt;/fi. With this scaling, Eq. (7) can be written in the dimensionlessform
w49 p9
7) =
y + kf,,(.r)cOsr+
P2
- f,,(7)1
(8)
The scaled potential amplitude is k = V, . 8wrt2/fi. In these transformed variables, the Schrodinger equation in the position representation becomes
Here the dimensionless parameter k depends on the temporal scaling used in the transformation
k = awrt,
(10)
In the transformation outlined here, the commutation relation between momentum and position becomes [$,p] = i k . Thus k is a measure of the quantum resolution in the transformed phase-space. Another general note on this transformation concerns the measure of the atomic momentum. Because an atom interacts with a near-resonant standing wave, its momentum can be changed by stimulated scattering of photons in the two counterpropagating beams. If a photon is scattered from one of these beams back into the same beam, the result is no net change in the atom’s momentum. However, if the atom scatters a photon from one of the beams into the other, the net change in its momentum is two photon recoils. The atom can thus exchange momentum with the standing wave only in units of 2fik,. In the transformed, dimensionless units, this quantity is
For a sample of atoms initially confined to a momentum distribution narrower than one recoil, the discreteness of the momentum transfer would result in a ladder of equally spaced momentum states. In our experiments the initial momentum distributions were significantly wider than two recoils, so the observed final momenta had smooth distributions rather than discrete structures.
III. Experimental Method The experimental study of momentum transfer in time-dependent interactions consists of three main components: initial conditions, interaction potential, and
50
Mark G. Raizen
For MOTlMolasses Beams (split 6 ways)
FIG. 2. Illustration of optical table setup. An argon ion laser pumps two dye lasers. One dye laser (Coherent 899-21) is locked in saturated absorption to a sodium cell. The main power from this laser is aligned through a phase modulator that operates at 1.7 GHz (EOMI). The intensity of that beam is controlled with an acousto-optical modulator (AOMl) and then aligned into a single-mode fiber. The beam configurationfor the MOTlMolasses is not shown in this figure.
measurement of atomic momentum. The initial distribution should ideally be narrow in position and momentum, and should be sufficiently dilute so that atomatom interactions can be neglected. The time-dependent potential should be onedimensional (for simplicity), with full control over the amplitude and phase. In addition, noise and coupling to the environment must be minimized to enable the study of quantum effects. Finally, the measurement of final momenta after the interaction should have high sensitivity and accuracy. Using techniques of laser cooling and trapping it is possible to realize all these conditions. A schematic of the experimental setup is shown in Fig. 2. Our initial conditions are a sample of ultra-cold sodium atoms, which are trapped and laser-cooled in a magneto-optic trap (MOT) (Chu, 1991; Cohen-Tannoudji, 1992). The atoms are contained in an ultra-high vacuum glass cell at room temperature. The cell is attached to a larger stainless steel chamber, which includes a 20 l/s ion pump. The source of atoms is a small sodium ampoule contained in a copper tube that is attached to the chamber. The ampoule was crushed to expose the sodium to the rest
QUANTUM CHAOS WITH COLD ATOMS
51
of the chamber. Although the partial pressure of sodium at room temperature is below ton-, there are enough atoms in the low-velocity tail of the velocity distribution that can be trapped. The trap is formed using three pairs of counterpropagating, circularly polarized laser beams (2.0 cm beam diameter), which intersect in the middle of the glass cell, together with a magnetic field gradient that is provided by current-carrying wires arranged in an anti-Helmholz configuration. This configuration is now fairly standard and is used in many laboratories. These beams originate from a dye laser that is locked 20 MHz to the low frequency (red) side of the (3SI1,. F = 2) + (3P,,,, F = 3) sodium transition at 589 nm. Approximately lo5 atoms are trapped in a cloud that has an RMS size of 0.15 mm, with an RMS momentum spread of 4.6 hk,. This distribution would be represented in the Poincar6 surface of section of Fig. 1 as a band that is narrow in momentum, but uniform in position on the scale of a standing wave. The potential is provided by a second dye laser that is tuned typically 5 GHz from resonance (both red and blue detunings were used with no difference in the experimental results). The output of this laser is aligned through a fast acousto-optic modulator (25 ns rise time), which is driven by a pulse generator. This device controls the laser intensity in time. The beam is then spatially filtered to ensure a Gaussian intensity profile, and is centered on the atoms, with a lle field waist of w, 1.9 mm. For the single-pulse and kicked rotor experiments (Sections IV and V, respectively) the beam was retro-reflected from a mirror outside the vacuum chamber to create a standing wave, as shown in Fig. 3(a). For the modulated standing wave experiments (Section VI) a more complicated setup was used as shown in Fig. 3(b). To what extent is Eq. (7) a good representation of a sodium atom exposed to an optical standing wave in the laboratory? The two-level atom and rotating-wave approximations are well justified for this optical-frequency transition. The adiabatic elimination of the excited-state amplitude is appropriate for the values of detuning and intensity that were used in the experiments. The detuning was also large compared to the linewidth r and to the recoil shift frequency or. For the sodium D, transition, the values for these quantities are
-
-r -- 10 MHz 2%-
and
The atoms were prepared in a particular hyperfine ground state (in some experiments they were prepared in the F = 2 state, whereas in others they were optically
52
Mark G.Raizen
FIG. 3. Schematic of experimental setup. The vacuum chamber is shown schematically, with MOT/Molasses beams. (a) Configuration of the far-detuned laser that was used in the single-pulse and kicked rotor experiments. The standing wave was created by retro-reflecting a beam from a detuned laser. The beam intensity was controlled with an acousto-optic modulator (AOM2), and was spatially filtered to ensure good beam profile. The optical power was calibrated and monitored on a photodiode (PDl). The phase stability was studied by inserting a beam splitter and a mirror (M3) to form a Michelson interferometer, and the interference fringes were detected on a photodiode (PD2). (b) Configuration of the far-detuned laser that was used in the modulated standing wave experiments. The overall intensity was controlled with AOM3. The beam was split in two paths. Each arm was controlled with separate acousto-optical modulators (AOM4 and AOM5) that were driven at nominally the same frequency. The phase of one arm was modulated with an electro-optic modulator (EOM2). Both beams were spatially filtered. The modulation index was measured and calibrated with a MachZehnder interferometer and the interference fringes were detected on a photodiode (PD2).
pumped to the F = 1 state); however, they were not optically pumped into a particular Zeeman sublevel. This was not a problem because with linearly polarized light and the large detuning, all the m F sublevels experienced the same potential.
QUANTUM CHAOS WITH COLD ATOMS
53
FIG. 4. Two-dimensionalatomic distributions after free expansion. (a) Initial thermal distribution with no interaction. (b) Localized distribution after interaction with the potential.
The one-dimensional nature of Eq. (7) comes from the assumption that the laser beams have spatially uniform transverse profiles. In these experiments the width of the atomic cloud during the illumination by the standing wave was small compared to the width of the laser profile. The detection of momentum is accomplished by allowing the atoms to drift in the dark for a controlled duration, after the interaction with the standing wave. Their motion is frozen by turning on the optical trapping beams in zero magnetic field to form “optical molasses” (Chu, 1991; Cohen-Tannoudji, 1992). The motion of the atoms is overdamped, and for short times (tens of ms) their motion is negligible. The position of the atoms is then recorded via their fluorescence signal on a charged coupled device (CCD) and the time of flight is used to convert position into momentum. The entire sequence of the experiment is computer controlled. In Fig. 4, typical two-dimensional images of atomic fluorescenceare shown. In Fig. 4(a) the initial MOT was released, and the motion was frozen after a 2-ms free-drift time. This enables a measurement of the initial momentum distribution. The distribution of momentum in Fig. 4(a) is Gaussian in both the horizontal and vertical directions. The vertical direction is integrated to give a one-dimensional distribution as shown in Fig. 5(a). In Fig. 4(b), the atoms were exposed to a particular time-dependent potential. The vertical distribution remains Gaussian, but the horizontal distribution becomes exponentially localized due to the interaction potential, as shown in Fig. 5(b). The significance of the lineshape and other characteristics are analyzed next.
Mark G.Raizen
54
1
"
-60
'
~
-40
~
"
~
-20
"
'
~
0
~
"
20
"
"
"
40
'
'
60
P (2%) FIG. 5 . One-dimensional atomic momentum distributions. They were obtained by integrating along the vertical axes of the two-dimensional distributions in the previous figure. The horizontal axes are in units of two recoils, and the vertical axes show fluorescence intensity on a logarithmic scale. (a) Initial thermal distribution with no interaction. (b) Localized distribution after interaction with the potential. The characteristic exponential lineshape is discussed in the text.
IV. Single Pulse Interaction The simplest time-dependent potential that we can impose is the turning on and off of the standing-wave intensity. In the context of atom optics, this type of time-dependent interaction occurs, for example, whenever an atomic beam passes
QUANTUM CHAOS WITH COLD ATOMS
55
through a standing wave of light. Diffraction from a standing wave was first studied by Martin et al. (1987) where the emphasis was on the two regimes of RamanNath and Bragg scattering. The time dependence in those cases was determined by the atoms traversing the Gaussian profile of the standing wave. Initial theoretical models assumed a sudden turn onloff of the standing wave, and it was believed that the details of the temporal profile merely led to an overall correction term. We now reexamine this simple process from the standpoint of classical nonlinear dynamics and find a very different answer. As a first approach to this problem, one expects that for slow turn onloff the evolution is adiabatic. The conditions for adiabaticity are very clear for linear potentials such as the harmonic oscillator. The difficulty with nonlinear potentials is that there are many time scales, so the conditions for adiabaticity must be examined much more carefully. We show that in this case the temporal profile can have important dynamical consequences and find that the intermediate regime between the sudden and adiabatic can lead to mixed phase space and chaos. To analyze this problem in more detail, we assume a generic time dependent potential V(x, t )
=
V,F(t) cos 2k,x
(14)
For the case of atomic beam diffraction (Martin et al., 1987), F(t) = exp -(t/7)2. We consider here the case F(t) = sin2 d T , , which is turned on for a single period T,. This Hamiltonian can be expanded as
H = p 2 / 2 M - V, sin2 d T , cos 2kLx =
p 2 / 2 M - (Vo/2) [COS2kLx
- (COS2 k , ( ~ -
VJ)
(15)
+ cos 2kL(x + vmt))12]
where v,,, = AL/2T,.The effective interaction is that of a stationary wave with two counterpropagating waves moving at +v,. Classically, there are now three resonance zones each of width proportional to fland separation in momentum proportional to T;' . The Poincark surface of section for this Hamiltonian is shown in Fig. 6. Keeping V, constant and increasing T, leads to the overlap of these isolated resonances and a subsequent destruction of the KAM surfaces. This mechanism for cross-over from stability to chaos was formulated by Walker and Ford (1969) and by Chirikov (1979). In this case particle motion is no longer restricted to move along the lines of each isolated resonance. The resulting phase space is generally mixed, with islands of stability surrounded by regions of chaos. This leads to diffusion in certain regions of phase space, and confinement in others. An example of a surface of section in that case is shown in Fig. 7 for parameters that are accessible experimentally. Relative to the atomic diffraction experiments of
56
Mark G.Raizen
FIG. 6. Poincark surface of section for the sin2 potential. In this case there are three isolated resonances.
Martin et al. (1987), this regime requires a combination of deep wells with significant atomic motion (on the scale of the standing-wave period), and is clearly outside the limiting cases of Raman-Nath or Bragg. To experimentally determine the threshold T,, for overlap, we must distinguish the momentum growth associated with spreading within the primary resonance from diffusion that can occur after resonance overlap. This is accomplished by contrasting the momentum transfer from the potential due to a standing wave of fixed amplitude
V’(X) = (Vo/2)cos(2k,x)
(16)
57
QUANTUM CHAOS WITH COLD ATOMS
for duration Tswith V(X,t)
=
(V0/2) [COS2kLx - (COS2k,(x - v m t ) + cos 2k,(x
+ vmt))/2]
(17)
resulting from the sin amplitude modulated standing wave. The experimental setup is shown in Fig. 3(a). The key to the interpretation of the experimental results is the realization that prior to resonance overlap v ’ ( ~ and ) V(x, t ) should give the same result. After overlap of the resonances, V(x, t ) will result in significantly larger momentum transfer than V’(x).The experimental results in Fig. 8(b) show the RMS momentum for both cases as a function of pulse duration (rise and fall times of 25 ns are included in the square pulse duration). These agree well
FIG. 7. Poincark surface of section for the sinZpotential after resonance overlap has occurred. There is a bounded region of global chaos.
58
Mark G. Raizen
-
3 -
/
I
I
I
I
0.5
1
1.5
2
c
0
2.5
T , W FIG.8. (a) RMS momentum computed from a classical simulation for sin2 (solid line) and square (dashed line) pulses. (b) RMS momentum from experimentally measured distributions for sin2 (solid) and square (open) pulses for the same conditions as (a). (c) RMS momentum computed from a quantum simulation for sin2 (solid line) and square (dashed line) pulses for the same conditions as (a). The threshold estimated from resonance overlap is indicated by the arrow.A clear deviation occurs at a pulse duration close to the predicted value (Robinson et aL, 1996).
with classical numerical simulations shown in Fig. 8(a) as well as the estimated resonance overlap threshold (Robinson et al., 1996). The predicted quantum behavior is shown in Fig. 8(c). For the case of V’(x), we find close agreement with the classical simulations and with the experiment. This is an interesting result in its own right, because the coherent oscillations that occur for short times are seen in the experiment with a large ensemble of indepen-
QUANTUM CHAOS WITH COLD ATOMS
59
dent atoms and in the quantum simulation, which uses a single wavepacket approach. For the case of V(x, t) there is also good agreement between the three cases over the entire range of pulse times; however, the quantum widths are slightly lower than the corresponding classical values near the large peak in the RMS width. Although this difference is too small to be of quantitative significance,it is nevertheless the precursor for differences in quantum and classical behavior that can occur when the classical dynamics are globally chaotic. These differences, which form the basis for the study of quantum chaos, are the focus of the next experiments we discuss.
V, Kicked Rotor A.
INTRODUCTION
The classical 6-kicked rotor, or the equivalent standard mapping, is a textbook paradigm for Hamiltonian chaos (Lichtenberg and Lieberman, 1991). A mechanical realization would be an arm rotating about a pivot point. The rotation is free, except for sudden impulses that are applied periodically. The Hamiltonian for the problem is given by 0 .r
X
=
P2
- f K cos 4 2
S(T - n) n = --m
The evolution consists of resonant-kicks that are equally spaced in time, with free motion in between. The quantity K is called the stochasticity parameter, and is the standard control parameter for this system. As K is increased, the size of each resonant-kick grows. Beyond a threshold value of K 2: 4 it has been shown that phase space is globally chaotic (Reichl, 1992). The chaos is due to the fact that the magnitude of each kick depends on the angle of the rotor at that moment and the nonlinearity of the potential. Note that, in contrast, a kicked harmonic oscillator cannot be chaotic because it is a linear system. It is intuitively clear that for a given kick strength, motion can become chaotic if the duration between kicks becomes long enough. This is because after one kick the particle has time to evolve to a completely different point in phase space before the next kick occurs. The quantum version of this problem has played an equally important role for the field of quantum chaos since the pioneering work of Casati et al. (1979) and Chirikov (1979). In particular, dynamical localization was predicted to occur for the kicked rotor and detailed scaling laws were derived. Although this model may seem unique, many physical systems can be mapped locally onto the kicked rotor, so that it is actually a universal paradigm system. To observe dynamical localization in an experimental realization of the kicked rotor, we turned the standing wave on and off in a series of N short pulses with period T This system differs from the ideal kicked rotor in two ways. The first
60
Mark G. Raizen
difference is that the conjugate variables here are position and momentum instead of angle and angular momentum, so that strictly speaking our system consists of kickedparticles. The second is that the pulses have finite duration instead of being 8-kicks. The effect of finite pulse duration was also considered by Blumel et al. (1986) in the context of molecular rotation excitation. The first distinction might seem problematic, because there is a natural quantization of angular momentum, in contrast to a continuum of momentum states for a free particle. In our system, however, the quantization of momentum is imposed by the periodicity of the wells, so that the momentum kicks must occur in units of two recoils. The initial distribution, on the other hand, can be continuously distributed over different momentum states, providing averaging of diffusion and localization. The effects of finite pulse duration are analyzed next, but we note here that if the atoms do not move significantly compared to the spatial period during a pulse, this system is an excellent approximation of the 8-kicked rotor. Atomic motion in this case can be described by the Hamiltonian of Eq. (7) with Famp= Xy=,F(t - n T ) and Fph = 0,
H
P2 2M
= -
N
+ V, c0~(2k,x) 2 n=
F(t - n T )
1
Here the function F(t) is a narrow pulse in time centered at t = 0 that modulates the intensity of the standing wave. The sum in this equation represents the periodic pulsing of the standing-wave amplitude by multiplying V, with a value in the range 0 5 F(t) 5 1. The optical arrangement for the experiment was described in Section I11 and illustrated in Fig. 3(a). The fast acousto-optic modulator (AOM2) provided the amplitude modulation of the standing wave to form the pulse train C.F(f).This modulator had a 10-90% rise and fall time of 25 ns. The number of pulses and pulse period were computer controlled with a arbitrary waveform generator. A sample trace of the pulse profiles recorded on photodiode PD 1 is shown in Fig. 9. With the scaling introduced in Section I1 and the unit of time taken to be ?: the period of the pulse train, the Hamiltonian for this system becomes
X
P'
=-
2
N
+K
cos
4 C, f
( ~ n)
n=l
The train of 8-functions in Eq. (18) has been replaced here by a series of normalized pulsesf(7) = F(TT)/ J-Ym F(TT) dT. Note that the scaled variable T = tlT measures time in units of the pulse period. As described earlier, 4 = 2k,x is a measure of an atom's displacement along the standing wave axis and p is its momentum in units of 2hk,lk. Aside from the temporal profile of the pulses, all the experimental parameters that determine the classical evolution of this system are combined into one quantity, the stochasticityparameter K.As we will see, the quantum evolution depends
61
QUANTUM CHAOS WITH COLD ATOMS
I '
0
I
2
3
5
4
7
6
time @s) FIG. 9. Digitized temporal profile of the pulse train measured on a fast photo-diode. The vertical axis represents the total power in both beams of the standing wave.f(r) and n a a r e derived from this scan (Moore et al., 1995).
additionally on the parameter k . These two dimensionless quantities thus characterize the dynamics of Eq. (20). In terms of the physical parameters of Eq. (19), they are K
5
k
(21)
8VoaTtpwrlfi
(22)
= 8wrT
Here t p is the FWHM duration of each pulse, and a = JEW F(r) dt/tp is a shape factor that characterizes the integrated power for a particular pulse profile: it is the ratio of the energy in a single pulse to the energy of a square pulse with the same amplitude and duration. For a train of square pulses, a = l ; for Gaussian pulses, a = (d4In 2)'12 = 1.06. For the roughly square pulses used in our experiments, a was within a few percent of unity.
B.
CLASSICAL ANALYSIS
Atoms with low velocities do not move significantly during the pulse, so their classical motion can be described by a map. By integrating Hamilton's equations of motion over one period, we obtain the change in an atom's displacement and momentum:
A4=]
n+112
1
dtp=p
n - 112
(23)
n+ 112
Ap =
n - 112
dt K sin
f(7
- n) =
K sin
4
62
Mark G. Raizen
The discretization of these relations is the classical map, =
dn+l
dn
+
Pn+l
(24)
P n + l = Pn + Ksin d n that is known as the “standard” map (Reichl, 1992). For small values of K, the phase space of this system shows bounded motion with regions of local chaos. Global stochasticity occurs for values of K greater than 1, and widespread chaos appears at K > 4, leading to unbounded motion in phase space. Correlations between kicks in the spatial variable d can be ignored for large values of K, so this map can be iterated to estimate the diffusion constant. After N kicks, the expected growth in the square of the momentum is
-
N-
((pN -
po)2)
=
~2
1
2
(sin 4;)
n=O
+ ~2
c (sin 9, sin d n s )
n*nl
(25)
K2
= -N L
The diffusion in momentum is thus ( p 2 ) = ON,
K2 with D = 2
(26)
Note that this description, which follows from the discretization into the standard map, requires the duration of the pulses to be short. To understand the effects of a finite pulse-width, consider the case where the pulse profilef(r) is Gaussian with an RMS width r 0 .In the limit of a large number of kicks N, the potential in Eq. (20) can be expanded into a Fourier series: =P 2 + K cos
2
4
c m
eim2me-(rn~mo)2/2
m = --to m
- P2 +
c
(28)
Km cos(d - m27rr)
with (29) K, 3 K exp[-(m2~r,)~/2] The nonlinear resonances are located (according to the stationary phase condition) at p = d&dr = rn27r. This expansion is similar to the resonance structure of the 6-kicked rotor, in which the Kmare constant for all values of m. In Eq. (28), however, the widths of successive resonances fall off because of the exponential term in the effective stochasticity parameter K,. This fall-off is governed by the pulse profile; the result of Eq. (28) was derived for the case of a Gaussian pulse shape, but in general K , is given by the Fourier coefficients of the periodic pulse train. The nonzero pulse widths thus lead to a finite number of significant resonances in the classical dynamics, which in turn limits the diffusion that results from over-
QUANTUM CHAOS WITH COLD ATOMS
63
lapping resonances to a band in momentum. The width of this band can be made arbitrarily large by decreasing the pulse duration and increasing the well depth, thereby approaching the 8-function pulse result. This can be seen in the result just derived. In the limiting case of T,, + 0 with K fixed (infinitesimal pulse width and large well depth), we recover the resonance structure expected for the 8-function limit in Eq. (18): K , = K. In the experiment, the pulse width only needs to be small enough that the band of diffusion is significantly wider than the range of final momenta and that the effective diffusion constant K,,,is approximately uniform over this range. An example of the bounded region of chaos that arises from the finite pulse duration is illustrated by the classical phase portrait shown in Fig. 10, for typical
FIG. 10. PoincarC surface of section for the pulsed system using a train of Gaussians to represent the experimental sequence. The integrated area under a single pulse is taken to be the same as in the experiment. The standing wave has a spatial RMS value of n,/2~ = 75.6 MHz. T = 1.58 ps, and a = 0.027, leading to K = 11.6. Note that a small intensity variation due to spatial overlap of atoms and laser profile results in a somewhat smaller K than that at peak field (Moore et al., 1995).
64
Mark G.Raizen
experimental parameters. The central region of momentum in this phase portrait is in very close correspondencewith the S-kicked rotor model with K = 11.6. This stochasticity parameter is well beyond the threshold for global chaos. The boundary in momentum can also be understood using the concept of an impulse. If the atomic motion is negligible while the pulse is on, the momentum transfer occurs as an impulse, changing the momentum of the atom without significantly affecting its position. Atoms with a sufficiently large velocity, however, can move over several periods of the potential while the pulse is on. The impulse for these fast atoms is thus averaged to zero, and acceleration to larger velocities is inhibited. The result is a momentum boundary that can be pushed out by making each pulse shorter. Classically, then, the atoms are expected to diffuse in momentum until they reach the momentum boundary that results from the finite pulse width. Equation (27) indicates that the energy of the system (4(~/2hk,)~)thus grows linearly in time. In terms of the number of pulses N, this energy is
c.
QUANTUM ANALYSIS
This system can be expected to exhibit quantum behaviors that are very different from those predicted classically. A qualitative atom-optics picture of the kicked rotor is that of an atom passing through a series of N diffraction gratings and then forming an interference pattern. The entire device can be seen as a multistage atomic interferometer, and is an extension of the three grating interferometerproposed by Chebotayev et d.(1985). Each diffraction grating represents a kick, followed by free evolution between the gratings. From this picture it is clear that this is a manifestly quantum system and the final pattern is determined by complicated interference of amplitudes. From that standpoint, it is perhaps surprising that for a small number of gratings before the “break time,” the resulting interferencepattern appears “classical.” We now discuss two phenomena that are predicted to occur in the kicked rotor, namely dynamical localization and quantum resonances. Dynamical localization is the quantum suppression of chaotic diffusion, which is thought to occur in many physical systems but is most cleanly studied here. Quantum resonances are a quantum feature particular to the S-kicked rotor. A quantum analysis of this system starts with the Schrodingerequation, Eq. (9). For the pulsed modulation of Eq. (20), this becomes
The periodic time dependence of the potential implies that the orthogonal solutions to this equation are time-dependent Floquet states. This system has been
65
QUANTUM CHAOS WITH COLD ATOMS
studied extensively in the ideal case off(7) = 5 ( ~with ) an infinite train of kicks (n = 0, 2 1, 2 2 , . . .) (Casati et al., 1979). An analysis of this system by Chirikov et al. (1981) shows that this system diffuses classically only for short times during which the discrete nature of the Floquet states is not resolved. As shown by Fishman et al. (1982), Eq. (3 1) can be transformed into the form of a tight-binding model of condensed-matter physics. An analysis of that system indicates that the Floquet states of Eq. (31) are discrete and exponentially localized in momentum. Because these states form a complete basis for the system, the initial condition of an atom in the experiments can be expanded in a basis of Floquet states. Subsequent diffusion is limited to values of momentum covered by those states that overlap with the initial conditions of the experiment. If the initial conditions are significantly narrow in momentum, the energy of the system should grow linearly with the number of kicks N, in agreement with the classical prediction in Eq. (30), until a “quantum break time” N*. After this time, the momentum distribution approaches that of the Floquet states that constituted the initial conditions, and the linear growth of energy is curtailed. This phenomenon is known as dynamical localization. The Floquet states are characterized by a “localization length” 6 with l q ( p / k ) 1 2 exp(-lp/k I/,$). The momentum distribution then has a lle halfwidth given by p*/2fikL = p * / k = & where is the average localization length of the Floquet states (Reichl, 1992). The number of Floquet states that overlap the initial condition (and therefore the number of Floquet states in the final state) is roughly ,$, so the average energy spacing between states is Aw l/c. The quantum break time is the point after which the evolution reflects the discreteness of the energy spectrum, hence N*Aw 1, or N* = By combining these estimates with Eq. (30), we see that f is proportional to K2/2k 2. The constant of proportionality has been determined numerically to be $ (Shepelyansky, 1986), and the localization length is thus
-
c
-
-
c.
In our experiments we derive the RMS momentum from the measured lineshapes, because its definition applies as well to the prelocalized Gaussian distributions as to the exponentially localized ones. For an exponential distribution, this quantity is larger than the localization length by a factor of fi:
Because f is also a measure of the number of kicks before diffusion is limited by dynamical localization, we have for the quantum break time
66
Mark G. Raizen
An inherent assumption in the derivation of Eqs. (32-33) is the lack of structure in the phase space of the system. Small residual islands of stability, however, do persist even for values of K > 4. This structure introduces in the dynamics a dependence on the location of the initial conditions in phase space. Nonetheless, this analysis provides a useful estimate of the localization length and the quantum break time.
D. EXPERIMENTAL PARAMETERS It is important to consider these last two relations in choosing experimental parameters. In order for a localized distribution to be observable,p * must be significantly smaller than the region enclosed by the classical boundary. Thus there is a constraint between the duration of the kicks (parameterizedby its FWHM value t,,) and the localization length. As previously described, the simplest estimate for this condition requires that the distance traveled by a particle during a pulse be much less than a period of the standing wave: p*REsatpIM<< A,/2. A better estimate comes from Eq. (29),which indicates the effective stochasticity parameter for a particle with momentum p/2hkL= m 2 d k in a train of Gaussian pulses with FWHM duration t,, = q mT ~ From ~ thisTequation we see that the effective stochasticityparameter drops below 4, resulting in islands of stability for atoms with momenta greater than pmax12hkL= (In 2 In $) */2/2wt . The ‘.p resonance overlap criterion can provide a more accurate expression, but this estimate is sufficient for determining the range of operating parameters. The experimental conditions should be chosen so that the localized momentum width^*^, is much smaller than this limit. On the other hand, the localized momentum distribution needs to be several times wider than the initial distribution so that it can be distinguished from the initial conditions. The atoms in our MOT started with an RMS momentum of upo= -4.6 hk,, imposing a lower limit on the localization length of ~ * ~hk,~> /up,/2 2 hk, = 2.3. Combining these two bounds gives
Another constraint on the localization length comes from its relation to N*, the number of kicks required for the localization to manifest. This time must be short enough to be observable in the experiment. Indeed, the experiment should continue for a time significantly greater than N* so that it is clear that the early period of diffusive growth has ended. An upper limit on the duration of the experiment,
QUANTUM CHAOS WITH COLD ATOMS
67
and therefore on the localization length, comes from the increased probability of spontaneous emission events with longer exposures to the standing wave. Spontaneous emission can randomize the phase of an atomic wavefunction, thereby destroying the coherence necessary for the quantum phenomena under observation. The probability of a spontaneous event during N* kicks of duration t , is 1 - e--YwoJ"*t~. To preserve the coherent evolution of the atomic sample, we require this probability to be small:
Here ysponr = (V,S,/fi) (I'/2)[62 + (r/2)2]-1 is the probability per unit time for an atom to undergo a spontaneous event, and r/2r(= 10MHz) is the linewidth of the sodium D,transition. In addition to these constraints relating to the localization length, there are several other restrictions on the experimental parameters. To ensure that the atoms are all subject to the same well depth, the light field cannot vary greatly over the sample of atoms, and thus a lower limit to the beam waist is given by the spatial width of the atomic sample. In our experiments the interaction times were short enough and the initial temperatures cold enough that the sample of atoms did not spread significantly from its initial MOT width of ax, 0.15 mm (RMS), so it was sufficient for the beam waists to be large in comparison to this initial value,
-
In order to observe dynamical localization, the classical phase space must be characterized by extended regions of chaos evident in the classical phase portraits for K > 4. This requirement set a constraint on the well depth V,, the pulse period I: and the pulse duration r,. The most important constraint is the maximum power available in the beams that make up the standing wave. Large laser powers help satisfy Eq. (37), because the beams can then be made wide while maintaining the desired intensity at the center of the beam profile. In practice, however, the laser power in each beam ( P ) is of course limited and the other experimental control parameters of beam waist (w,), detuning (aL),pulse period (T), and pulse duration (t,) must all be chosen to satisfy the criteria enumerated here. The fact that Eq. (36) can be satisfied is an especially valuable aspect of this experiment. Spontaneous emission is the only significant avenue of energy dissipation from the dilute sample of atoms. By making this dissipation negligible, our system is effectively Hamiltonian. It is interesting to note the features of the system that make this possible. To keep the probability of spontaneousemission small, we take advantage of the different dependencies of the well depth V, and the spontaneous emission rate ysponr on the detuning. The well depth is proportional to the intensity of the standing wave and inversely proportional to the detuning,
68
Mark G. Raizen
Although the spontaneous emission rate is also proportional to the intensity, it varies as the inverse square of the detuning,
rt2
Within the limits of available laser power, a large detuning can therefore provide negligible spontaneous emission during the experiment without too much loss in the well depth. In our experiments, each counterpropagating beam typically had a power of P = 0.2-0.4 W; the waists were in the range of w, = 1.2-2.2 mm, and the detunings from resonance SL/27rwere between 5 and 10 GHz. These operating conditions led to well depths in the range of Vo/h= 5-15 MHz, and to spontaneous emission probabilities of about 1 % per kick. The pulse periods and durations were in the ranges 1-5 ps and 0.05-0.15 ps, respectively.
E.
EXPERIMENTAL RESULTS
We subjected the cooled and trapped atoms to a periodically pulsed standing wave and recorded the resulting momentum distributions as described in Section 111. To study the temporal evolution of the atomic sample under the influence of the periodic kicks, these experiments were repeated with increasing numbers of kicks (N) with the well depth, pulse period, and pulse duration fixed. These successive measurements provided the momentum distributions at different times in the atomic sample’s evolution. Such a series of measurements is shown in Fig. 1 1 . Here the pulse had a period of T = 1.58 ps, and a FWHM duration of tp = 100 ns. For these conditions, k has a value of 2.0. The largest uncertainty in the experimental conditions is in the well depth, V,, which depends on the measurement of the absolute power of the laser beams that make up the standing wave and their spatial profile over the sample of atoms. To within lo%, the well depth for these data had spatial RMS value of V,/h = 9.45 MHz. The pulse profile was nearly square, leading to a stochasticity parameter of K = 11.6, the same value as for the phase portrait in Fig. 10. The distributions clearly evolve from an initial Gaussian at N = 0 to an expo-
QUANTUM CHAOS WITH COLD ATOMS
69
FIG. 1 1 . Experimental time evolution of the lineshape from the initial Gaussian until the exponentially localized lineshape. The quantum break time is approximately 8 kicks. Fringes in the freezing molasses lead to small asymmetries in some of the measured momentum lineshapes as seen here and in the inset of Fig. 12. The vertical scale is measured in arbitrary units and is linear (Moore etal., 1995).
nentially localized distribution after approximately N = 8 kicks. We have measured distributions out until N = 50 and find no further significant change. The small peak on the right side of this graph is due to nonuniformitiesin the detection efficiency.As discussed in Section III, the relative numbers of atoms with different momenta is measured by their fluorescence intensity on a CCD camera. Spatial variations in the MOT beams were due to interference fringes from the chamber windows. This was a minor limitation on the resolution of the momentum measurements, and will be corrected in the future with antireflection coatings on all windows. The growth of the mean kinetic energy of the atoms as a function of the number of kicks was calculated from the data and is displayed in Fig. 12. It shows an initial diffusive growth until the quantum break time N* = 8.4 kicks, after which dynamical localization is observed (Moore et al., 1995). The solid line in this figure represents the classical diffusion predicted in Eq. (30). The data follow this prediction until the break time. The dashed line in the same figure is the prediction for the energy of the localized distribution from Eq. (33). Though not shown here, classical and quantum calculations both agree with the data over the diffusive regime. After the quantum break time, the classical growth slows slightly due to the fall-off in K predicted by Eq. (29) for nonstationary atoms. The observed distribution would lead to a reduction of only 15%in the stochasticity parameter. Thus
70
Mark G. Raizen
0
5
10
15
20
25
N
FIG. 12. Energy ((~/2hk,)~)12 as a function of time. The solid dots are the experimental results. The solid line shows the calculated linear growth from the classical dynamics. The dashed line is the saturation value computed from the theoretical localization length 6. The inset shows an experimentally measured exponential lineshape on a logarithmic scale, which is consistent with the theoretical prediction (Moore et al., 1995).
the classically predicted energy would continue to increase diffusively. The measured distributions, however, stop growing as predicted by the quantum analysis.
F.
QUANTUM RESONANCES
Between kicks, the atoms undergo free evolution for a fixed duration. The quantum phase accumulated during the free evolution is e-ip2T/mh. An initial plane wave at p = 0 couples to a ladder of states separated by 2hk,. For particular pulse periods, the quantum phase for each state in the ladder is a multiple of 27r, a con-
QUANTUM CHAOS WITH COLD ATOMS
71
dition known as a “quantum resonance” (Reichl, 1992). More generally, a quantum resonance is predicted when the accumulated phase between kicks is a rational multiple of 27r. We have scanned T from 3.3 ps to 50 ps and find quantum resonances when the quantum phase is an integer multiple of IT. For even multiples, the free evolution factor between kicks is unity; for odd multiples, there is a flipping of sign between each kick. Quantum resonances have been studied theoretically, and it was shown that instead of localization, one expects the energy to grow quadratically with time (Casati et al., 1979; Izrailev and Shepelyansky, 1979). This picture, however, is only true for an initial plane wave. We have done a general analysis of the quantum resonances (to be published) and show that for an initial Gaussian wavepacket, or for narrow distributions not centered a t p = 0, the final momentum distribution is actually smaller than the exponentially localized one, and settles in after a few kicks. Our experimentalresults are shown in Figs. 13 and 14. Ten quantum resonances are found for T ranging between 5 ps (corresponding to a phase shift of IT) and 50 ps ( 1 0 ~in ) steps of 5 ps. The saturated momentum distribution as a function of T is shown in Fig. 13. The narrower, nonexponential profiles are the resonances between which the exponentially localized profiles are recovered. The time evolution of the distribution at a particular resonance is shown in Fig. 14, from which it is clear that the distribution saturates after very few kicks.
FIG. 13. Experimental observation of quantum resonances as a function of the period of the pulses. The surface plot is constructed from 150 momentum distributions measured, for each I: after 25 kicks. This value of N ensures that the momentum distributions are saturated for the entire range of T shown. On resonance, the profiles are nonexponential and narrower than the localized distributions that appear off-resonance.Note that the vertical scale is linear (Moore et al., 1995).
Mark G.Raizen
72
FIG. 14. Experimental observation of the time evolution of a particular resonance for T = 10 ps (Moore et al., 1995).
VI. The Modulated Standing Wave A.
INTRODUCTION
The last experiment described in this chapter was actually the first to be performed in our laboratory and was originally motivated by a proposal of Graham et al. (1992). It is interesting to note that the same interaction Hamiltonian was derived and analyzed in an earlier paper by Graham er al. (199 1) for driven Josephson junctions; however, that proposal has not yet been realized experimentally. In retrospect, the modulated system is more subtle than the kicked rotor or the single pulse. In our experiment, atoms are subjected to a standing wave of near-resonant light, where the displacement of the standing wave nodes is modulated at a frequency omand with an amplitude AL. Once again, the excited state amplitude is adiabatically eliminated. With this form of the modulation the effective Hamiltonian given in Eq.(7) becomes
H
=
P2 + V, cos[2kL(x - AL sin
2M
wmt)]
Although this Hamiltonian may look somewhat different than the &kicked rotor, it also displays the phenomenon of dynamical localization, as discussed next.
13
QUANTUM CHAOS WITH COLD ATOMS
B.
CLASSICAL ANALYSIS
The Hamiltonian of Eq. (40) can be expanded as a sum of nonlinear resonances using a Fourier expansion. By expanding the temporal dependence of the potential, we obtain the resonance structure of the system, "2
H
=
2M
+ V,[J,(A)
cos 2k,x
+ Jl(A)
cos 2kL(x - v m f )+ L I ( A ) cos 2k,(x
+ J,(A)
cos 2 k L ( ~- 2 ~ , , t )+ J-,(A) cos 2kL(x
+ v,t)
+ 2v,t) +
(41) * *
*]
oc
=
V, ,=
./,(A)
cos 2kL(x - nv,t)
--m
where J , are ordinary Bessel functions, v , = w,/2kL is the velocity difference between neighboring resonances, and A = 2kLAL is the modulation index. As in the case of the 8-kicked rotor, the resonances are located at regular intervals in momentum. The amplitudes of these resonances, however, depend on the modulation index A. The dependence on A allows this system to be tuned between regimes where the classical dynamics are integrable (for example, A = 0) to those in which they are chaotic. The classical resonances are evenly separated in momentum with central values of p,
=
nMv,
(42)
and widths of Ap,
=-4
(43)
There are substantial resonances only for n IA, so for momenta greater than MAv, the phase space is characterizedby essentially free evolution.These regions of free evolution confine the motion of atoms with small initial momentum to the portion of phase space spanned by the resonances. For certain ranges of A, these resonances overlap, leading to a band of chaos with boundaries in momentum that are proportional to A. A sample of atoms starting with initial conditions within this band will remain within it, confined to momenta in the range t M A v , . A simple estimate of the atomic momentum after a long time is a uniform distribution within these bounds (Graham et al., 1992); such a distribution would have an RMS momentum of
74
Mark G. Raizen
The classical dynamics can also be understood in terms of resonant kicks that occur twice during each modulation period. Consider an atom subjected to the modulated standing wave of Eq. (40). When the standing wave is moving with respect to the atom, the time-averaged force is zero, because the sign of the force changes as the atom goes over “hill and dale” of the periodic potential. Momentum is transferred to the atom primarily when the standing wave is stationary in the rest frame of the atom. These resonant kicks occur twice in each modulation period, but they are not equally spaced in time. The magnitude and direction of the resonant kick depends on where the atom is located within the standing wave at that time. The calculated variation of the RMS momentum width as a function of A is shown in Fig. 15 for 0,/27r = 1.3 MHz and V,/h = 3.1 MHz. The estimate of Eq. (44) is shown by the straight solid line. For values of A C 3, this estimate agrees roughly with an integration of the classical Hamilton’s equations by Robinson et al. (1995), (shown in the figure) calculated for an interaction time of 20 ps. For larger values of A, the simulation is lower than the estimate, because in only 20 ps the initial distribution (with pmS/2iikL 2.3) does not have time to diffuse up to the limit represented by the solid line. The longer-durationclassical simulation presented in the figure agrees with the estimate over the entire range of A shown, except for values of A close to 7.0 (explained next). The 20-ps classical simulation also shows oscillations in the diffusion rate as a function of A: peaks in the RMS momentum correspond to values of A leading to large diffusion rates, whereas dips indicate slow diffusion. To understand this variation in diffusion rates, we examine the resonances in Eq. (41). The dependence of the diffusion rate on A is due to oscillations in J,(A), the amplitudes of the resonances. The various resonances grow and shrink as the modulation index A is increased. For certain values of A, a resonance can be significantly diminished, or even removed in the case where A is a zero of one of the Bessel functions. As shown in the phase portraits of Fig. 16 (top panel), this variation in the amplitudes of the resonances strongly influences the dynamics of the system. In general, the phase spaces are mixed, with islands of stabilitysurrounded by regions of chaos. Atoms from the initial distribution that are contained within an island remain trapped, whereas those in the chaotic domain can diffuse out to the boundaries. In the case of a diminished resonance, the islands of stability from neighboring resonances might not be destroyed by resonance overlap. This is the case with A = 3.8, for which J , ( A ) has its first zero. The final momentum spread in this case is governed largely by the surviving island due to the resonance at p o = 0, and the system is nearly integrable. The stability of this system causes the reduced diffusion shown by the dip in the classical simulation of Fig. 15 at A = 3.8. Indeed, all of the dips in this simulation occur at values of A that are near zeros of Bessel functions; the dynamics of the corresponding systems are stabilized by the diminished resonances. This stabilization even affects diffusion in the long-time
-
75
QUANTUM CHAOS WITH COLD ATOMS
14 12 10 Prmsg
2rilL 6 4 2 n "
0
1
2
3
4
5
6
7
h FIG.15. RMS momentum width as a function of the modulation amplitude A, for w,,,/27~= 1.3 MHz and V,/h = 3.1 MHz. Experimental data are denoted by diamonds and have a 10% uncertainty associated with them. The empty diamonds are for an interaction time of 10 ps and the solid diamonds are for 20 ps. The straight line denotes the resonant-kick boundary, and the curved line is the prediction of Graham er al. (1992). The three curves indicate numerical simulations. A classical simulation is shown, one for an interaction time of 20 ps (dot-dash line). The observed data lie well below these curves for some values of A. A 20 ps integration of the Schrodinger equation is also presented for comparison with the corresponding experimental data (heavy dashed line). The heavy solid line is a Floquet-state calculation, and represents the long-time quantum prediction (Robinson er al., 1995).
classical simulation: for values of A close to 7.0 (the second zero of J,(A)), the initial conditions are trapped in a large island of stability at p = 0. For these values of A, the diffusion is limited by the width of the island to a value much smaller than that given by the resonant-kick boundary. Note that the oscillations of the Bessel functions are reflected in the exchange of the location of hyperbolic and elliptic fixed points. At A = 0, there is only one resonance in the expansion of Eq. (41) centered at p o = 0 with an amplitude
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FIG. 16. Poincark surfaces of section (upper panel), classical momentum distributions (middle panel), and experimentally measured momentum distributions with Floquet theory (bottom panel, theory marked by lines) for runs with parameters similar to those in Fig. 15. Note that the vertical scales for the distributions are logarithmic and are marked in decades (Robinson et al., 1995).
+
V,Jo(0) = V,. The potential minima for this resonance are located in space at even multiples of 7~/2k,, so the island of stability is centered at x = 0 in the phase portrait. The phase portraits for A = 3 and A = 3.8 also have islands of stability centered in momentum at p o = 0, but the amplitudes for these resonances are negative: V,J0(3) = -0.40V, and V,J0(3.8) = -0.26V,. The reversal of sign exchanges the location of the potential minima and maxima, so the islands in these portraits are centered in position at x = 17-/2k,. Notice also that the overall amplitude of the oscillations decreases as A is increased due to the reduction in the size of each resonant-kick. This effect can be understood from the impulse approximation,because the maximum classicalforce is fixed but the time that the standing-wave potential is stationary in the rest frame of the atom is inversely proportional to A. The classical diffusion rate is therefore reduced by increasing A, although the classical saturation value of p M s increases with A.
c. EXPERIMENT
m.
The experimental realization of the Hamiltonian of (40)required a somewhat more complicated optical setup for the interaction potential that is illustrated in Fig. 3(b). To modulate the phase of the potential, we vary the phase of one of the two laser beams that make the standing wave. The electro-optic modulator EOM2 in Fig. 3(b) provided this control. For a phase shift of 7~ at 589 nm, this modulator required an applied voltage of V, = 271 V. By applying an oscillating drive
QUANTUM CHAOS WITH COLD ATOMS
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VEo sin w,f we modulated the phase of the beam with an amplitude 7cVEo/Vv and gave the phase of the standing wave a time dependence A sin w,r, with A = 2k,AL = f.rrVE0/V,. To provide the high voltage required for the phase shifts in this experiment, the signal was stepped-up in a helical resonator (Vpp = 2VE0 = 2400 V, corresponds to A = 7). This resonator was designed so that when connected to the capacitive EOM it formed a tuned circuit that had an input impedance of 50 fl at a resonant frequency of l .3 MHz. The circuit had a Q of 108 and the output voltage across the EOM was stepped-up by a factor of 77. The modulation index was calibrated by measuring the FM sidebands in optical heterodyne, and identifying the appropriate zeros of the Bessel functions. The main control parameter in the experimental realization was the modulation index, A. The momentum distribution was measured for a range of A for fixed values of the intensity and detuning. The temporal evolution was not mapped out systematically in these experiments, but the duration was chosen to be long enough to saturate the growth of momenta. The experimental data are shown in Fig. 15 for interaction times of 10 and 20 ps. The 20 ps data match the classical simulations well for small values of A and for values of A that are close to zeros of Bessel functions. For other values of A, however, the experimentally measured distributions are much narrower than those predicted classically.This reduction is a manifestation of dynamical localization in this system. The momentum distributions after 20 ps are exponential, as in our kicked rotor experiment. To observe this effect we must ensure that the location of the resonant-kick boundary is much further than the localization length. As this boundary scales linearly with A, we expect to see the appearance of dynamical localization only beyond some value of A. This experimental requirement is similar to the considerations of the classical boundary in the kicked rotor experiments. There, however, the boundary was due to an effective reduction in K by the motion of an atom over several wells during a single pulse. Here the classical boundary arises from the maximum velocity that can be imparted to an atom by resonant kicks. Note that for small values of A the experiment is good agreement with the classical prediction. At A = 0 the system is integrable and momentum is trivially localized. As A is increased the phase space becomes chaotic, but growth is limited by the resonant-kick boundary. Our measured momentum distributions (in Fig. 16, bottom panel) are characteristically “boxlike” in this regime (0 5 A 5 2). This observation is consistent with the picture of a uniform diffusion limited by the boundaries in momentum. As A is increased beyond a critical value, there are oscillations in the observed RMS momentum. For certain ranges of the modulation index A, the observed values deviate substantially from the classical prediction. These ranges correspond to conditions of large diffusion rates-the peaks in the classical prediction. For these values of A, the classical phase space is predominately chaotic. An example
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of the resulting dynamics is shown in Fig. 16 for A = 3.0. The classicallypredicted distribution (middle panel) is roughly uniform, but the experimentally observed distribution is exponentially localized (Moore et al., 1994; Robinson et al., 1995); hence the RMS value is reduced. As A is increased further, the oscillations in the resonance amplitudes lead to phase portraits with large islands of stability, as in the case A = 3.8. For these values of A, the classical phase space becomes nearly integrable and the measured momentum is close to the classical prediction. Quantum analyses under the conditions of the experiment as well as an asymptotic (long-time limit) Floquet analysis are shown along with the classical simulations and experimental data in Fig. 16. The predicted distributions from the Floquet analysis are displayed along with the experimentally observed ones in the lower panel of Fig. 16. It is clear that there is good quantitative agreement between experiment and the effective single-particle analysis (Robinson et al., 1995; Bardroff et al., 1995). Graham et al. (1992) showed that the modulated system can be approximated by the 6-kicked rotor. Although this connection is valid in certain parameter regimes, it is important to stress that dynamical localization is not restricted to that model system, and can occur in any chaotic phase space. Even in the &kicked rotor, which is the paradigm system, the simple scaling laws that relate diffusion rate with localization length are valid only in the limit of asymptotically large stochasticity parameter. For smaller values of K, the residual structures in phase space can modify local behavior. The same is true for the modulated system and is probably a feature of any experimentally accessible system. Our experimental initial conditions average over a band in phase space, yielding average values for diffusion and localization length. A more complete discussion of this point was covered in a recent series exchange of letters (Latka and West, 1995; Raizen et al., 1997a,b; Menenghini et al., 1997; Latka and West, 1997).
VII. Conclusion and Future Directions In this chapter we reviewed our experiments on dynamical localization with ultracold sodium atoms. There are many interesting questions that can now be addressed experimentally with this system. One direction is to study how dynamical localization may be destroyed by noise or dissipation. This problem has been the topic of a great deal of theoretical work (see for example Ott et al., 1984; Dittrich and Graham, 1987; Fishman and Shepelyansky, 1991). Experiments on microwave ionization of Rydberg atoms studied the effects of amplitude noise, and an increased ionization probability as a function of noise amplitude was observed (Blumel et al., 1989). In our present system of the 6-kicked rotor, noise and dissipation could be introduced as amplitude or phase noise. We can also induce spontaneous scattering by illuminating the atoms with a weak resonant beam dur-
QUANTUM CHAOS WITH COLD ATOMS
79
ing the coherent evolution. We should then be able to follow the growth of momentum as a function of time for different types and levels of noise, and hopefully gain a better understanding of decoherence in this system. One of the limitations of the current sodium experiment is the boundary in phase space. This becomes especially problematic for studies of delocalization, where momentum should grow substantially beyond the localization length. To overcome this problem, we are building a new experiment based on cesium atoms. The boundary (measured in recoil units) should be pushed out by more than an order of magnitude relative to the sodium case. This should enable a detailed study of the effects of noise and dissipation. The role of dimensionalityon dynamical localization has been studied theoretically in detail. A transition to power-law localization in two dimensions and delocalization in three dimensions was predicted. This could be studied experimentally by introducing several spatial or temporal periodicities in the potential (Casati et al., 1989). The spatial periodicity of the standing wave, for example, can be increased by making the angle between the two beams less than 180”. Incommensurate spatial periods can be superimposed with several far-detuned standing waves at different angles. The standing waves must also be detuned from each other so that cross-interferenceterms move at a high velocity and are averagedout. The focus of this work has so far been on cases where the classical phase space is globally chaotic. The more generic situation in nature is a mixed phase space, consisting of islands of stability surrounded by regions of chaos. To study this regime, better initial conditions are needed. We have developed a new method that should enable the preparation of a minimum-uncertainty “box” in phase space, and plan to implement this technique in our cesium experiment.This would enable a detailed study of quantum transport in mixed phase space. Some interesting topics to study would be tunneling from islands of stability, chaos assisted tunneling, and quantum scars (Heller and Tomsovic, 1993).
VIII. Acknowledgments I would like to thank Fred Moore, John Robinson, Cyrus Bharucha, Kirk Madison, and Steven Wilkinson for their important contributions to these experiments. I would also like to thank Bala Sundaram and Qian Niu for excellent theoretical support. This work was supported by the U.S. Office of Naval Research, the Robert A. Welch Foundation, and the U.S. National Science Foundation.
M.References Adams, C. S., Sigel, M., and Mlynek, J. (1994). Phys. Rep. 240, 145. Anderson, P.W. (1958). Phys. Rev. 109, 1492.
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Bardroff, P.J., Bialynicki-Birula, I., Kduner, D. S., Kurizki, G., Mayr, E., Stifter, P., and Schleich, W. P. (1995). Phys. Rev. Lett. 74,3959. Bayfield, J. E. and Koch, P. M. (1974). Phys. Rev. Lett. 33,258. Bayfield, J. E.. Casati, G., Guameri, I., and Sokol, D. W. (1989). Phys. Rev. Lett. 63,364. Bliimel, R., Fishman, S., and Smilansky, U.(1986). J. Chem. Phys. 84,2604. Bliimel, R., Graham, R., Sirko, L., Smilansky, U.,Walther, H., and Yamada, K. (1989). Phys. Rev. Lett. 62, 341. Cahn, S. B., Kumarakrishnan, A,, Shim, U.,Sleator, T., Berman, P. R., and Dubetsky, B. (1997). Phys. Rev. Lett. 79,784. Casati, G., Chirikov, B. V., Izrailev, F. M., and Ford, J. (1979). In G. Casati and J. Ford (Eds.). Srochastic behaviour in classical and quantum Hamiltonian systems, vol. 93 of lecture notes in physics (p. 334). Springer-Verlag (Berlin). Casati, G., Guarneri, I., and Shepelynansky, D. L. (1989). Phys. Rev. Lett. 62,345. Chebotayev, V. P., Dubetsky, B., Kasantsev, A. P., and Yakovlev, V. P. (1985). J. Opt. SOC. Am. E 2, 1791. Chirikov, B. V. (1979). Phys. Rep. 52,265. Chirikov, B., Izrailev, F. M., and Shepelyansky, D. L. (1981). Sov. Sci. Rev. Sec. C 2,209. Chu, S. (1991). Science 253,861. Cohen-Tannoudji, C. (1992). In J. Dalibard, J.-M. Raimond, and J. Zinn-Justin (Eds.). Fundamental systems in quantum optics, les Houches, session LIII (p. 1). Elsevier (Amsterdam). Collins, G. P. (1995). Phys. Today48, 18. Delande, D. and Buchleitner, A. (1994). Adv. At. Mol. Phys. 34,85. Dittrich, T. and Graham, R. (1987). Europhys. Left. 4,263. Fishman, S., Grempel, D. R., and Prange, R. E. (1982). Phys. Rev. Lett. 49,509. Fishman, S. and Shepelyansky, D. L. (1991). Europhys. Lett. 16,643. Galvez, E. J., Sauer, B. E., Moorman, L., Koch, P. M., and Richards, D. (1988). Phys. Rev. Lett. 61,2011. Graham,R., Schlautmann, M., and Shepelyansky, D. L. (1991). Phys. Rev. Lett. 67,255. Graham, R., Schlautmann, M., and Zoller, P,(1992). Phys. Rev. A 45, R19. Haake, F. (1991). Quantum signatures of chaos. Springer-Verlag (New York). Heller, E. J. and Tomsovic, S. (1993). Phys. Today 46,38. Izrailev, F. M. and Shepelyansky, D. L. (1979). Sov. Phys. Dokl. 24,996. Latka, M. and West, B. J. (1995). Phys. Rev. Len. 75,4202. Latka, M. and West, B. J. (1997). Phys. Rev. Lett. 78, 1196. Lee, P. A. and Ramakrishnan,T. V. (1985). Rev. Mod. Phys. 57,287. Lichtenberg, A. L. and Lieberman, M. A. (1991). Regular and chaotic dynamics. Springer-Verlag (Berlin). Martin, P. J., Gould, P. L., Oldaker, B. G., Miklich, A. H., and Pritchard, D. E. (1987). Phys. Rev. A 36,2495. Menenghini, S., Bardroff, P. J., Mayr, E., and Schleich, W. P. (1997). Phys. Rev. Lett. 78, 1195. Moore, F. L., Robinson, J. C., Bharucha, C., Williams, P. E., and Raizen, M. G. (1994). Phys. Rev. Lett. 73,2974. Moore, F. L., Robinson, J. C., Bharucha, C. F., Sundaram, B., and Raizen, M. G.(1995). Phys. Rev. Lett. 75,4598. Ott, E., Antonsen, T. M., and Hanson, J. D. (1984). Phys. Rev. Lett. 53,2187. Raizen, M. G., Salomon, C., and Niu, Q. (1997a). Phys. Today 50,30. Raizen, M. G., Sundaram, B., and Niu, Q.(1997b).Phys. Rev. Lett. 78, 1194. Reichl, L. E. (1992). The transition to chaos in conservative classical systems: quantum manifestations. Springer-Verlag (New York). Robinson, J. C., Bharucha, C., Moore, F. L., Jahnke, R., Georgakis, G. A., Niu, Q., Raizen, M. G., and Sundaram, B. (1995). Phys. Rev. Lett. 74,3963.
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Robinson, J. C., Bharucha, C. F., Madison, K. W., Moore, F., L., Sundaram, B., Wilkinson, S. R., and Raizen, M., G . (1996). Phys. Rev. Letf. 76,3304. Shepelyansky, D. L. (1986). Phys. Rev. Lett. 56,677. Shepelyansky, D. L. (1987). Physica D 28, 103. Szriftgiser, P., Guery-Odelin, D., Arndt, M., and Dalibard. J. (1996). Phys. Rev. Lett. 77.4. Tabor, M. (1989). In Chaos and infegrabiliry in nonlinear dynamics. John Wiley & Sons (New York). Walker, G. H. and Ford, J. (1969). Phys. Rev. 188,416.
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ADVANCES IN ATOMIC. MOLECULAR, AND OFTICALPHYSICS, VOL. 4
STUDY OF THE SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS PASCAL SALIkRES, ANNE L'HUILLIER, PHILIPPE ANTOINE, AND MACIEJ LEWENSTEIN CEA/DSM/DRECAM/SPAM, Centre d'Etudes de Saclay, Gif-sur-Yvette, France I. Introduction
.....
B. Spatio-Temporal Characteristics of High Harmonics . . . . . . . . . . . . . . . . . 1. Experiments .................... 2. Theory .................................................. C. The Scope of the Present Review ......................... 11. Theory of Harmonic Generation in M A. Single-AtomTheories ......................................... B. Single-Atom Response in the Strong Field Approximation ............ C. Propagation Theory ........................................... D. Macroscopic Response .............. 111. PhaseMatching ................................................. A. Source of the Harmonic Emission ................................ B. Dynamically Induced Phase of the Atomic Polarization. ............. C. Influence of the Jet Position on the Conversion Efficiency . . . . . . . D. Modified Cutoff Law ................................. IV. Spatial Coherence . . . . . . . . . . . . . . . . A. Definition .................................................. B. Study of the Spatial Coherence: Atomic Jet After the Focus . . . . . . . . . . C. Study of the Spatial Coherence: Atomic Jet Before the Focus ... V. Temporal and Spectral Coherence ................................. A. Influence of the Jet Position ...................... B. Influence of the Ionization ..................................... C. Consequences of the Phase Modulation .................... D. Influence of Nonadiabatic Phenomena .....................
............... B. Interferometry with Harmonics . . . . . . . . . . . ............ C. Attosecond Physics .......................................... VII. Conclusion . . . . . . . . . . . . . . .... ......... VIII. Acknowledgments ..............................................
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Pascal SaliPres, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
I. Introduction A. SHORT HISTORY OF HIGHHARMONIC GENERATION
In recent years high-order harmonic generation (HG) has become one of the major topics of super-intense laser-atom physics. Generally speaking, high harmonics are generated when a short, intense laser pulse interacts with matter. Although HG has been in the course of recent years mainly studied in atomic gases, it has also been investigated in ions, molecules, atom clusters and solids. Apart from its fascinating fundamental aspects, HG has become one of the most promising ways of producing short-pulse coherent radiation in the XUV range. HG has already been a subject of several review articles (L’Huillier et al., 1992a;Miyazaki 1995; Protopapas et al., 1997a). One can point out the following milestones in the short history of this subject: First observations. High harmonic generation is an entirely nonlinear and nonperturbative process. The spectrum of high harmonics is characterizedby a fall-off for the few low-order harmonics, followed by an extendedplateau, and by a rapid cut08 The first experimental observations of the plateau were accomplished by (McPherson et al., 1987) and (Ferray et al., 1988) at the end of the 1980s. Plateau extension. Most of the early work has concentrated on the extension of the plateau, that is, generation of harmonics of higher order and shorter wavelength (Macklin et al., 1993; L’Huillier and Balcou, 1993a). By focusing short-pulse terawatt lasers in rare gas jets, wavelengths as short as 7.4 nm (143rd harmonic of a 1053-nmNd-Glass laser, Perry and Mourou, 1994),6.7 nm (37th harmonic of a 248-nm KrF laser, Preston et al., 1996), and 4.7 nm (169th harmonic of an 800-nm Ti-Sapphire laser, Chang et al., 1997a) have been obtained. Very recently, with ultra-short intense infrared pulses, it has become possible to generate XUV radiation extending to the water window (below the carbon K-edge at 4.4 nm, Spielmann et al., 1997; Chang et al., 1997b). Simple man’s theory. A breakthrough in the theoretical understanding of the HG process in low-frequency laser fields was initiated by Krause et al. (1992), who have shown that the cutoff position in the harmonic spectrum follows the universal law Zp 3Up, where Zp is the ionization potential, whereas Up = e 2 82/4mw2is the ponderomotive potential, that is, the mean kinetic energy acquired by an electron oscillating in the laser field. Here e is the electron charge, m is its mass, and ’% and w are the laser electric field and its frequency, respectively. Soon an explanation of this universal fact in the framework of “simple man’s theory” was found (Kulander et al., 1993; Corkum, 1993). According to this theory, harmonic generation occurs in the following manner: first the electron tunnels out from the nucleus through the
+
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 85
Coulomb energy barrier modified by the presence of the (relatively slowly varying) electric field of the laser. It then undergoes oscillations in the field, during which the influence of the Coulomb force from the nucleus is practically negligible. Finally, if the electron comes back to the vicinity of the nucleus, it may recombine back to the ground state, thus producing a photon of energy I,, plus the kinetic energy acquired during the oscillatory motion. According to classical mechanics, the maximal kinetic energy that the electron can gain is indeed =3Up. A fully quantum mechanical theory, that is based on strong field approximation and that recovers the “simple man’s theory,” was formulated soon after (L’Huillier et al., 1993c; Lewenstein et al., 1994). Ellipticity studies. The “simple man’s theory” leads to the immediate consequence that harmonic generation in elliptically polarized fields should be strongly suppressed, because the electron released from the nucleus in such fields practically never comes back, and thus cannot recombine (Corkum, 1993; Corkum e l al., 1994). Several groups have demonstrated this effect (Budil et al., 1993; Dietrich et al., 1994; Liang et al., 1994), and have since then performed systematic experimental (Burnett et al., 1995; Weihe et al., 1995; Antoine et al., 1997a; Weihe and Bucksbaum, 1996; Schultze et al., 1997) and theoretical (Becker et al., 1994a; Antoine et al., 1996b; Becker et al., 1997) studies of the polarization properties of harmonics generated by elliptically polarized fields. Optimization and control. Progress in experimental techniques and theoretical understanding has stimulated numerous studies of optimization and control of HG depending on various parameters of the laser and the active medium. These studies involved among others: - Optimization of laser parameters. These studies concern, for instance
laser polarization (discussed previously), pulse duration or wavelength dependence (Balcou et al., 1992; Kondo et al., 1993; Christov et al., 1996; Balcou, 1993; Salibres, 1995). Although typically infrared (NdGlass or Ti-Sapphire) lasers are used, generation by shorter wavelength intense KrF lasers is also very efficient ([Preston et al., 19961, for theory see [Sanpera et al., 19951). - Generation by multicolored fields. Harmonic generation in combined laser fields of two frequencies was studied in the context of (i) enhancement of conversion efficiency (for theory see Eichmann et al., 1995; Protopapas et al., 1995; Telnov et al., 1995; Kondo et al., 1996; Perry and Crane, 1993, for experiment compare Watanabe et al., 1994; Paulus et al.. 1995), (ii) access to new frequencies and tunability, if one of the fields is tunable (for theory see Gaarde et al., 1996b, for experiments Eichmann et al., 1994, Gaarde et al., 1996a), and (iii) the control of HG process in general (Ivanov et al., 1995).
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- Optimization of the generating medium. First of all, optimization with respect to atomic gases was studied (Balcou et al., 1992; Balcou and L‘Huillier, 1993). Other active media apart from noble gases have been used to generate harmonics: ions (these involve ionized noble gas atoms [Sarukura et al., 1991; Wahlstrom et al., 1993; Kondo et al., 1994; Preston et al., 19961, and alkaline ions [Akiyama et al., 1992;Wahlstrom et al., 1995]), molecular gases (Chin et al., 1995; Fraser et al., 1995; Lyngi et al., 1996), atomic clusters (Donnelly et al., 1997), and so on. At this point it is worth adding that harmonic generation from solid targets and laser-induced plasmas has also been intensively studied in recent years (for theory see [Gibbon, 1996; Pukhov and Meyer-ter-Vehn, 1996; Lichters et al., 1996; Roso et al., 19981, for experiments compare [Carman et al., 1981; Kohlweyer et al., 1995; von der Linde et al., 1995; Norreys et al., 19961). - Optimization and characteristics of spatial and temporal properties. Those studies concern spatial, temporal, and spectral properties of harmonic radiation, and in particular their coherence properties (Salibres, 1995); they are closely related to the subject of this review and will be discussed separately later. Other examples of such studies involve spatial control of HG using spatially dependent ellipticity (Mercer et al., 1996), control of phase-matching conditions for low (Meyer et al., 1996) and high harmonics (Salibres et al., 1995), role of ionization and defocusing effects (Altucci et al., 1996; Miyazaki, 1995; Ditmire et al., 1995). - Other control schemes. Other control schemes of harmonic generation have been proposed that involve for example the coherent superposition of atomic states (Watson et al., 1996; Sanpera et al., 1996). Applications. High harmonics provide a very promising source of coherent XUV radiation, with numerous applications in various areas of physics. In particular, applications in atomic physics are reviewed by (Balcou et al., 1995; L’Huillier et al., 1995). Harmonics have already been used for solid state spectroscopy (Haight and Peale, 1993) and plasma diagnostics (Theobald et al., 1996). Further applications that directly employ coherence properties of harmonics will be discussed in this review. Attosecond physics. Future applications of high harmonics will presumably involve attosecond physics, that is, the physics of generation, control, detection, and application of subfemtosecondlaser pulses. Two types of proposals on how to reach the subfemtosecond limit have been put forward over the last few years: those that rely on phase-locking between consecutive harmonics (Farkas and Toth, 1992; Harris et al., 1993; Antoine et al., 1996a; Corkum et al., 1994; Ivanov et al., 1995; Wahlstrom et al., 1997), and those that concern single harmonics (Schafer and Kulander, 1997; Salibres, 1995).
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 87
B. SPATIO-TEMPORAL CHARACTERISTICS OF HIGHHARMONICS 1. Experiments From both the fundamental and practical points of view it is very important to know and understand the spatial and temporal coherence properties of high harmonics. Information on the spatial coherence of the beam and on its focusability, its spectrum, and time profile are of direct interest for applications. However, they also help in understanding the physics of the process, as there are many possible causes for distortion of the spatial and temporal profiles, and their interpretation implies a rather refined and deep study of the problem. The spatial distribution of the harmonic emission has been investigated by several groups in various experimental conditions. Peatross and Meyerhofer (1995a) used a 1-,um l-ps Nd-Glass laser loosely focused (f/70) into a very diluted gaseous media (1 torr) in order to get rid of distortions induced by phase matching and propagation in the medium. The far-field distributions of the harmonics (11 to 41) generated in heavy rare gases were found to be quite distorted, with pedestals surrounding a narrow central peak. These wings were attributed to the rapid variation of the harmonic dipole phase with the laser intensity. Tisch et al. (1994) studied high-order harmonics (71 to 111) generated by a similar laser, focused (f/50) in 10 torr of helium. Complex spatial distributions are found for harmonics in the plateau region of the spectrum. However, in the cutoff, the measured angular distributions narrow to approximately that predicted by lowest-order perturbation theory. The broad distributions with numerous substructures observed in the plateau are attributed to the influence of ionization, and in particular of the free electrons, on phase matching. The influence of ionization on spatial profiles has also been investigated experimentally by L'Huillier and Balcou (1993b) for low-order harmonics in xenon, and, more recently, by Wahlstrom et al. (1995) for harmonics generated by raregas-like ions. Generally speaking, ionization induces a significant distortion of the harmonic profiles, thus complicating their interpretation. In a recent letter (SaliBres et al., 1994), we presented results of an experimental study of spatial profiles of harmonics generated by a 140 fs Cr:LiSrAlF6 (Cr:LiSAF) laser system. Thanks to this very short pulse duration, it was possible to expose the medium to high intensities while keeping a weak degree of ionization. Under certain conditions, the resulting harmonic profiles were found to be very smooth, Gaussian to near flat-top, without substructure. In Salitres et al. (1996), we present systematic experimental studies of harmonic angular distributions, investigating the influence of different parameters, such as laser intensity, nonlinear order, nature of the gas, and position of the laser focus relative to the generating medium. We show that when the laser is focused before the atomic medium, harmonics with regular spatial profiles can be generated with reasonable conversion efficiency. Their divergence does not depend
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Pascal SaliPres, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
directly on the nonlinear order, the intensity, or even the nature of the generating gas, but rather on the region of the spectrum the considered harmonic belongs to, which is determined by the combination of the three preceding elements. When the focus is drawn closer to the medium, the distributions get increasingly distorted, becoming annular with a significant divergence for a focus right within or after the jet. A first endeavor to measure the degree of spatial coherence of the harmonic radiation has been recently made by Ditmire et al. (1996) with a Young two-slit experiment. They investigated how the coherence between two points chosen to be located symmetrically relative to the propagation axis depends on the degree of ionization of the medium. Finally, very recently far-field interference pattern created by overlapping in space two beams of the 13th harmonic, generated independently at different places in a xenon gas jet was observed (WahlstriSm et al., 1997; Zerne et al., 1997). The experimental studies of temporal and spectral properties of high-order harmonics have also been carried out in different experimental conditions. Temporal profiles of low harmonics generated by relatively long pulses (several tens of ps) were measured using a VUV streak camera by Faldon et al. (1992) and Starczewski et al. (1994). In order to measure the duration of harmonic pulses in the femtosecond regime, Schins et al. (1996) developed a cross-correlation method in which they ionize helium atoms by combining two pulses: the fundamental (800 nm, 150 fs) from a Ti:Sapphire laser, and its 21st harmonic (38 nm). These pulses generate characteristic electron spectra whose sidebands scale as the cross-correlationfunction, which can be mapped out by varying the delay between the two pulses. Using variants of this method, Bouhal et al. (1997a) and Glover et al. (1996) were able to measure the duration of 21st to 27th harmonics within a subpicosecond accuracy. For instance, in Bouhal et al. (1997a) for a fundamental pulse of 190-fs FWHM, durations of 100 ? 30 fs and 150 ? 30 were found for the FWHM of 2 1st and 27th harmonic, respectively. Concerning the spectral properties of the individual harmonics, the blue shift due to ionization has been reported in Wahlstrom et al. (1993), whereas spectral properties of harmonics generated by chirped pulses have been discussed in Zhou et al. ( 1996). 2. Theory
The theoretical description of spatial distributions and temporal profiles of harmonics requires combining a reliable single-atom theory that describes the nonlinear atomic response to the fundamental field with a propagation code that accounts for phase matching, dispersion, and so on. Peatross er al. have studied the spatial profiles of low-order harmonics in the loosely focused regime (Peatross
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 89
and Meyerhofer, 1995b; Peatross et a[., 1995). Muffet et al. (1994) modeled the results of Tisch et al. (1994), and showed that, depending on the focusing conditions, substructures could be due either to ionization or to resonances in the intensity dependence of the atomic phase. Temporal profiles of low-order harmonics were discussed in Faldon et al. (1992) and Starczewski et al. (1994). Rae et al. (1994) performed calculations outside the slowly varying envelope approximation by solving simultaneously the equations for the atomic dynamics and propagation, using a one-dimensional approximation. Temporal and spectral profiles were studied in the strongly ionizing regime. In the series of papers (L’Huillier et al., 1993c; Lewenstein et al., 1994), we have developed a single-atom theory that is a quantum-mechanicalversion of the two-step model of Kulander et al. (1993) and Corkum (1993). This theory has been combined with the theory of HG by macroscopic media (L’Huillier et al., 1992b) to describe experimental results in a realistic manner. In a recent letter (Salibres et al., 1995) we have stressed the role of the dynamically induced phase of the atomic polarization in phase matching and propagation processes. In particular we have demonstrated the possibility of controlling the spatial and temporal coherence of the harmonics by changing the focusing conditions of the fundamental. We have performed numerical simulations of the angular distributions. The simulated profiles reproduced remarkably well the experimental trends and are thus used to interpret them in Salikres et al. (1996). The role of the intensitydependent phase of atomic dipoles was elaborated in more detail in Lewenstein et al. (1995) (see also Kan et al., 1995). In Antoine et al. (1997b) we present a short review of the various consequences of the intensity-dependentphase.
c. THESCOPE OF THE PRESENT REVIEW As already stated previously, the knowledge of the coherence properties of high harmonics is of major importance both for applications and from the fundamental point of view. Although experimental and theoretical work has been already devoted to this subject, a systematic study of the spatial and temporal coherence properties of harmonics is still lacking. In particular, the implications of the existence of a phase of the harmonic dipole have not been fully explored. The aim of this review is to present a detailed theoretical study of this important subject. The theoretical approach used in this paper is very well established, and has been confronted numerous times with experimental results with great success. Some of the results presented in this review have been published before and compared with experiments. Many of the results, however, are either new, or have been only reported in the PhD thesis of P. Salikres (Salibres, 1995). Nevertheless, we feel that very soon these results will find their experimental confirmation. To some extent this review has a character of a case study, that is, we discuss
90
Pascal Salikres, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
here in great detail quantitative coherence properties of a specific high harmonic in specific conditions among other things. It is important, however, to keep in mind that the presented results are not only qualitatively valid, but also quantitatively valid in more general cases. We hope that these results will turn out useful for anyone interested in applications of harmonics in general, and their extraordinary coherence properties in particular. The plan of the review is the following. In Section 11, we give an overview of theoretical approaches and describe our theoretical method: the single-atom theory based on strong field approximation and the propagation equations. Because our theory has been discussed in detail in other publications, we limit ourselves to presenting the final expressions that we use for calculations of the physical quantities. In Section I11 we discuss the phase-matching problem stressing the role of the dynamically induced phase of the atomic polarization (Salibres et al., 1995; Lewenstein et al., 1995). The combined effects of this phase and the phase of the fundamental beam depend on the atomic jet position relative to the focus. We investigate the influence of the jet position on the conversion efficiency. We show how the phase matching effects modify the cutoff law. Section IV starts with a short section devoted to the general definitions of the degree of coherence and characteristicsof partially coherent beams. We then concentrate on the emission profiles and quality of the wavefronts of harmonics, both in the near-field and in the far-field zones. We also present calculations of the degree of spatial coherence of the harmonics. In Section V we turn to the discussion of the temporal and spectral coherence. We show how the intensity dependence of the phase of the atomic polarization leads to a temporal modulation of the harmonic phase and to a chirp of its frequency. In both Sections IV and V, we relate our theoretical findings to experimental results. In particular, we use the parameters corresponding to experiments of Salihres et al. (1996), and discuss systematically the dependence of the coherence properties of harmonics on the focus position of the fundamental. This parameter, as shown in Salihres et al. (1995), allows us to control the degree of coherence; optimal coherence properties are obtained when the fundamental is focused sufficiently before the atomic jet. In the final part of Section V, we discuss the possibility of optimizing and/or controlling the temporal and spectral properties of harmonics by making use of the dynamically induced chirp: temporal compression with a grating pair and spectral compression by using a chirped fundamental pulse. These ideas are confronted with the recent experiment of Zhou er al. (1996) and to the theoretical proposal of Kulander (Schafer and Kulander, 1997). In Section VI we discuss future applications of harmonics with the special emphasis on their coherence properties. In particular we focus on applications in interferometry and on the short pulse effects. We discuss the possibility of generating and applying attosecond pulses. Finally, we conclude in Section VII.
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 9 1
II. Theory of Harmonic Generation in Macroscopic Media The theory of harmonic generation in macroscopic media must necessarily contain two components: (i) a single-atom theory that describes the response of an atom to the driving fundamental laser field, and (ii) a theory of propagation of the generated harmonics in the medium. In this section we outline various approaches to describe these two components of the theory. A. SINGLE-ATOM THEORIES The single-atom theory should describe the single-atom response to a timevarying field of arbitrary intensity, polarization, and phase. In other words, it should allow us to calculate the induced atomic polarization, or dipole acceleration, which then can be inserted as a source in the propagation (Maxwell) equations. In principle it is sufficient to describe the atomic response in the framework of the single active electron (SAE) approximation (compare Kulander, 1987b; for a discussion of two-electron effects, see for instance Lappas et al., 1996; Erhard and Gross, 1997; Taylor et al., 1997). Also for relatively long laser pulses (of duration down to =50-100 fs for infrared lasers) one can use the adiabatic approximation, that is, calculate the atomic response for the field of constant intensity, and only at the end integrate the results over the “slowly” varying envelope of the laser pulse. The discussion of the validity of the adiabatic approximation is presented in more detail in Section V. There are essentially four methods that have been used to solve the problem of the single-atom response : Numerical methods. These methods allow one to solve the time-dependent Schrodinger equation (TDSE) describing an atom in the laser field. Because (at least in the adiabatic case) the field oscillates periodically, one of the possible approaches is to use the Floquet analysis (Potvliege and Shakeshaft, 1989; for a recent review see Joachain, 1997), but the direct integration of the TDSE is used far more often (for a review see Kulander et al., 1992; Protopapas et al., 1997a). In one dimension such integration can be performed using either the finite element (Crank-Nicholson), or split operator techniques; in the context of harmonic generation it has been used first by the Rochester group (Eberly et al., 1989), but then employed by many others as a test method. In three dimensions the numerical method has been initiated by Kulander (1987a, 1987b), who used a two-dimensional finite element (“grid”) method. Soon it was realized that basis expansion methods that employ the symmetry of the problem (i.e., the spherical symmetry of a bare atom, or the cylindrical symmetry of an atom in the linearly polarized field) work much
92
Pascal Sali&es, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
better (DeVries, 1990; LaGutta, 1990). Modem codes typically use expansions in angular momentum basis, and solve the coupled set of equations for the radial wave function using finite grid methods (see Krause et al., 1992), Sturmian expansions (Antoine et al., 1995; Antoine et al., 1997c)or B-spline expansions (Cormier and Lambropoulos, 1996; Cormier and Lambropoulos, 1997). Most of those codes are quite powerful and allow one to calculate the atomic response directly without adiabatic approximation (see Schafer and Kulander, 1997). Unfortunately, they are also quite time consuming, and it is therefore very hard to combine the results obtained from the numerical solutions of TDSE with the propagation codes. The reason is that the singleatom response in the physically interesting regime is typically a rapidly varying function of the laser intensity and other laser parameters. The propagation codes thus require very detailed data from single-atom codes. This problem becomes even more serious in the absence of cylindrical symmetry; real three-dimensional numerical codes (such as the ones describing generation by elliptically polarized fields) have been developed only recently (see Antoine et al., 1995; Antoine et al., 1997c; Protopapas et al., 1997b), and obviously are even more time and memory consuming. Nevertheless, many seminal results concerning harmonic generation have been obtained using direct numerical methods: from the first observation of the Ip + 3Up law (Krause et al., 1992), to the recent proposal of attosecond pulse generation (Schafer and Kulander, 1997). Particularly interesting are the contributions of the Oxford-Imperial College group (for a review see Protopapas et al., 1997a)that concern among others HG by short wavelength lasers (Preston et al., 1996; Sanpera et al., 1995),pulse shape and blue shifting effects (Watson et al., 1995), role of strong ionization in HG (Rae and Burnett, 1993b), temporal aspects of harmonic emission (Rae et al., 1994; Watson et al., 1997), and the generation from the coherent superposition of atomic states (Watson et al., 1996; Sanpera et al., 1996). The TDSE method has also been applied to molecules aligned in the laser field (Zuo etal., 1993; Krause et al., 1991; Plummer and McCann, 1995). Classical phase space averaging method. A lot of useful information about high harmonic generation processes can be gained from a purely classical analysis of the electron driven by the laser field. In order to mimic quantum dynamics, classical Newton equations are solved here for an ensemble of trajectories generated from an initial electron distribution in the phase space. This distribution is supposed to mimic the true quantum initial state of the system, so that averages over this distribution are analogs of quantum averages. Such an approach has been developed in the context of HG by Maquet and his collaborators (Bandarage et al., 1992; VBniard et al., 1993) (see also Balcou, 1993). Strongfield approximation. As already mentioned, the seminal paper on the
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 93
+ 3Vp law (Krause et al., 1992) stimulated the formulation of the “simple man’s theory” (Kulander et al., 1993; Corkum, 1993). Originally,this theory has been formulated as a mixture of quantum and classical elements: First, the tunneling of the electron out from the nucleus was described using the standard ADK (Ammosov et al., 1986; Delone and Krdinov, 1991; Kraihov and RistiC, 1992) theory of tunneling ionization. The subsequent oscillations of the electron in the laser field were described using classical mechanics. Finally, electron recombination back to the ground state was calculated using the classical cross section for the collision and the quantum mechanical recombination probability. A fully quantum mechanical theory that recovers the “simple man’s theory” in the semiclassical limit was formulated soon after (L’Huillier et al., 1993; Lewenstein et al., 1994). This theory is based on the strong field approximation (SFA) to the TDSE. It is a generalization of the KeldyshFaisal-Reiss approximation (Keldysh, 1965; Faisal, 1973; Reiss, 1980), applied to the problem of harmonic generation. It was for the first time formulated in the context of harmonics by Ehlotzky (1992); it is also strongly related to the so-called Becker model of an atom with a zero range pseudopotential interacting with the laser field (see as follows). In our formulation, the theory is based on the following assumptions: (i) it neglects all bound states of the electron in an atom with exception of the ground state; (ii) all the states in the ionization continuum are taken into account, but in their dynamics only the part of the Hamiltonian responsible for the oscillations of the free electron in the laser field is kept. Technically, we disregard all off-diagonal continuum-continuum transitions that change electronic velocity. With these two assumptions, the TDSE becomes exactly soluble, and the resulting solutions are valid provided Up1 I,. In most of applications we treat the electronic states in the continuum as Volkov plane waves, which additionally limits the validity of our approximation to electronic states with high kinetic energy, and thus to the description of the generation of high harmonics (with photon energy 2 Z p ) . It is worth stressing, however, that a heuristic scheme of accounting for Coulomb potential effects in the continuum within the framework of SFA was proposed by Ivanov et al. (1996). Originally our method has been formulated for linearly polarized laser fields in the adiabatic (slowly varying intensity) approximation (L’Huillier et al., 1993; Lewenstein et al., 1994). We have since then generalized it to elliptically polarized fields (Antoine et al., 1996b), two-color fields (Gaarde et al., 1996b), and the fields with periodically time-dependent polarization (Antoine et al., 1997d), where all of those results were obtained in the adiabatic approximation. Finally, the method has been generalized to the fields with arbitrary time dependence without adiabatic approximation (see Zp
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Pascal Salikres, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
Section V). We have also performed intensive studies of the semiclassical approximation applied to our method in order to understand the harmonic emission in terms of semiclassical electron trajectories and Feynman path integrals (Lewenstein etal., 1995; Antoine et al., 1997b). These studies were essential for understanding the role of the intensity-dependent phase of the nonlinear atomic polarization (Salihres et al., 1995). They also allowed understanding the mechanism of electron trajectory selection in propagation, responsible for the generation of attosecond pulse trains (Antoine et al., 1996a;Balcou et al., 1997). The advantage of our method, apart from its very transparent physical sense, is that it gives partially analytic results, allowing rapid calculation of the very precise data required for propagation codes. Last but not least, our method combined with propagation codes gives results in very good agreement with experiment; it has become the standardtheoretical method of analysis of experimental data in the Saclay and Lund groups; it is also used by other groups (Kondo et al., 1996; Dorr et al., 1997). Pseudo-potential model. Many important results in the theory of harmonic generation have been obtained by Becker and his collaborators who have solved exactly (and to a great extent analytically) the zero-range pseudopotential model (Becker et al., 1990). In this model the electron is bound to the nucleus via the potential
where m is the electron mass. This potential supports a single bound state with the energy -Ip = - ~ ~ / 2 m . This model, originally formulated in the case of a linearly polarized field, was also extended to one-color (Becker et al., 1994a), and two-color (Long et al., 1995) fields with arbitrary polarization. It was also used to study the polarization properties of harmonics generated by elliptically polarized fields (Lohr et al., 1996). As our SFA theory (Lewenstein et al., 1994; Antoine et al., 1996b), Becker’s model may rigorously account for the ground-state depletion (Becker et al., 1994b). Structures in the harmonic spectra were associated in this model to the above-threshold ionization channel closings (Becker et al., 1992), rather than with quantum interferences between the contributions of different electronic trajectories (Lewenstein et al., 1995). Nevertheless, Becker’s model leads practically to the same final formulas for the induced atomic dipole moment as our SFA theory, and to very similar results (the small discrepancies are caused by additional approximationsused for numerical elaboration of final expressions; for detailed comparison of the two models see Becker et al., 1997). Becker’s model has also been used by several groups to analyze experimentaldata (Macklin et al., 1993; Eichmann et al., 1995; Paulus et al., 1995).
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 95
B. SINGLE-ATOM RESPONSE IN THE STRONG FIELDAPPROXIMATION In this section we present explicit formulas describing the response of a single atom to the laser pulse in the strong field approximation. Because the details of our version of the SFA can be found in the series of references (L'Huillier et al., 1993; Lewenstein et al., 1994; Salibres et al., 1995; Lewenstein et al., 1995; Antoine et al., 1996b), we limit ourselves to present the relevant expressions and to discuss their physical sense. Within our approach we obtain an approximate solution of the time-dependent Schrodinger equation that describes an atom in the strong electric field of a laser of frequency w in the single active electron approximation.The knowledge of the time-dependent wave function I q ( t ) ) allows us to calculate the time-dependent dipole moment i'(t) = (q(t)li'lT(t)) in the form of a generalized LandauDyhne formula (Delone and Kraihov, 1985; Landau, 1964) (we use atomic units)
+
where v is a positive regul$zation cornstant, A ( t ) denotes the vector potential of the electromagnetic field, % (t) = -8A (t)& = (Elxcos(wt), E sin(wt), 0) is the +'y electric field (polarized elliptically, in general), whereas S( p , t, r ) is the quasiclassical action, describing the motion of an electron moving in the laser field with a constant canonical momentum p',
with I, denoting the ionization potential, In expression (2) we have already performed (using the saddle-point method) the integral over all possible values of the momentum p' with which the electron is born in the continuum. For this reason the integral in Eq.(2) extends only over the possible return times of the electron, that is, the times it spends in the continuum between the moments of tunneling from the ground state to the continuum and recombination back to the ground state. The saddle-point value of the momentum (which is at the same time a stationary point of the quasi-classical action) is
+
Note the characteristic prefactor (Y i ~ / 2 ) - ~in / ' (2) coming from the effect of quantum diffusion. It cuts off very efficiently the contributions from large r's and allows us to extend the integration range from 0 to infinity. The field-free dipole transition element from the ground state to the continuum
96
Pascal Salic?res, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
state characterized by the momentum p’ can be approximated by (Lewenstein ef al., 1994; Bethe and Salpeter, 1957)
with a = 2Zp, for the case of hydrogen-like atoms and transitions from s-states. Finally, r is the ionization rate from the ground state. In the framework of our theory it can be represented as twice the real part of the time-averaged complex decay rate
Note that both expressions (2) and (6) have the characteristic form of semiclassical expressions that can be analyzed in the spirit of Feynman path integral: they contain (from right to left) transition elements from the ground state to the continuum at t - r , propagator in the continuum proportional to the exponential of i times the quasi-classical action, and the final transition elements from the continuum to the ground state. Applying the saddle-point technique to calculate the integral over r (and t if one calculates the corresponding Fourier components or time averages), one can transform both expressions into the sums of contributions corresponding to quasi-classical electron trajectories, characterized by the moment when the electron is born in the continuum t, - r,, its canonical momentum g(ts,7,) (see Eq. (4)), and the moment when it recombines t, (Lewenstein et al., 1994; Lewenstein et al., 1995). Note, however, that due to the fact that we deal here with the tunneling process (i.e., passing through the classically forbidden region), these trajectories will in general be complex. Typically, only the trajectories with the shortest return times Re(7) contribute significantly to the expression (2); there are two such relevant trajectories with return times shorter than one period, that is, 0 C Re(r,) < Re(r,) < 2 d w . Note also that the dipole moment (2) can be written in the form ?(t) =
C
ji’qe-iqmt--Tf
+ c.c.
(7)
4 odd
where xq denote Fourier components. They can be calculated either directly from Eq. (2) using a fast Fourier transform, or analytically as discussed in Antoine et al. (1996b). It is important to remember that expressions (2) and (6) both result from the single active electron approximation. Before inserting these expressions into the propagation equations, one has to account for the contributions of all active electrons, and replace Eqs. (2) and (6) by the total dipole moment and the total ioniza-
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 97
tion rate that are given by the sums of the corresponding (independent)contributions of all active electrons. In the case of helium (two s electrons in the ground state), this amounts to multiplying both expressions by the factor two. In the case of other noble gases (six p electrons in the ground state, two in each of the rn = - 1, 0, 1 states) the procedure is more complex. Both expressions should be replaced by two times the sums of contributions of the given magnetic quantum number rn = - 1, 0, 1; each of those contributions should be calculated replacing Eq. (5) by an appropriate field-free dipole matrix element describing the transition from the 1 = 1, rn = - 1, 0, 1 states to the continuum. Fortunately, the dependence of the dipole moment and ionization rate on the details of the ground-state wave function is rather weak, and typically reduces to an overall prefactor (Lewenstein et al., 1994; Antoine et al., 1996b), that determines the strength of the dipole, but not the form of its intensity dependence. For these reasons, in most of the calculations for noble gases other than helium, we still use the s-wave function to describe the ground state ( 5 ) , but multiply the results by an effective number of active electrons, n,,, that is in the range 2 < n,, = 4 < 6 for other noble gas atoms. Total ionization rates of helium and neon calculated with n,, = 2, = 4, respectively, agree very well with the ADK ionization rates (Ammosov et al., 1986; Delone and Krahov, 1991; Kraihov and RistiC, 1992).
C . PROPAGATION THEORY In order to calculate the macroscopic response of the system, one has to solve the Maxwell equations for the fundamental and harmonic fields. This can be done using the slowly varying envelope and paraxial approximations. The fundamentals of such an approach have been formulated by L’Huillier et al. (1992b). Several groups have used similar approaches to study the effects of phase matching, and to perform direct comparison of the theory with experiments (Muffet et al., 1994; Peatross and Meyerhofer, 1995b; Peatross et al., 1995; Rae and Burnett, 1993a; Rae et al., 1994). To our knowledge, the most systematic studies of this sort have been so far realized by the Saclay-Livermore-Lundcollaboration. In a series of papers we have studied propagation and phase matching effects in the context of the following problems: (i) phase matching enhancement in nonperturbative regime (L’Huillier et al., 1991); (ii) phase-matching effects in tight focusing conditions (L’Huillier et al., 1992c; Balcou and L‘Huillier, 1993); (iii) shift of the observed cutoff position (L’Huillier et al., 1993); (iv) density dependence of the harmonic generation efficiency (Altucci et al., 1996); (v) harmonic generation by elliptically polarized fields (Antoine et al., 1996b); (vi) harmonic generation by two-colored fields (Gaarde et al., 1996b);(vii) coherence control of harmonics by adapting the focusing conditions (Salikres et al.,1995); (viii) influence of the experimental parameters on the harmonic emission profiles
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Pascal Salikres, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
(Salitres et al., 1996); (ix) generation of attosecond pulse trains (Antoine et al., 1996a); (x) generation of attosecond pulses by laser fields with time-dependent polarization (Antoine et al., 1997d), (xi) interference of two overlapping harmonic beams (Wahlstrom et al., 1997); Zerne et al., 1997), and others. We have also formulated generalized phase-matching conditions that take into account the intensity-dependentphase of the induced atomic dipoles (Balcou et al., 1997).
D. MACROSCOPIC RESPONSE In this section we present the Maxwell equations for the fundamental and harmonic fields used in our previously mentioned studies. Using the slowly varying envelope and paraxial approximations, the propagation equations can be reduced to the form (here we use SI units)
z,(z
t), and Zq(2 t) denote the slowly varying (complex)envelopes of the where fundamental and harmonic fields respectively,kt = qw/c, whereas the rest of the symbols are explained next. The slow time dependence in the previous equations accounts for the temporal profile o+f the fundamental field that enters b.(8) through the boundary condition for E l . The solutions of the propagation equations for given t therefore have to be integrated over r. The terms containing Ak;( t) describe dispersion effects due to the linear polarisability of atoms, and in fact can be neglected in the regime of parameters considered (low density). The terms proportional to Skq( t) = -e2Ne( t)/ 2mqcw, with e denoting the electron charge, m its mass, and Ne( t) the electronic density, describe the corrections to the index of refraction due to ionization; here the ionic part of those corrections is neglected. The electronic density is equal to the number of ionized atoms, that is,
z
z
z
where Na(z) is the initial density of the atomic jet and r(1 ZI(Z t’)l) is the total ionization rate, which takes into account the contributions of all active electrons calculated from Q. (6) using r = 2Re[Jiy(t) dtlT] with T= , 2 d w for an instantaneous and local value of the elzctric field envelope E l ( : t’). Note that because I‘ depends functionally on El t f ) , Q. (8) is a nonlinear integro-
(z
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 99
differential equation; it has to be solved first, and its solution is used then to solve Eq. (9). Finally, the Fourier components of the atomic polarization are given by
Z t) denote the harmonic components of the total atomic dipole mowhere i'q( ment, which includes the contributions of all active electrons calculated from Eq. (2) for a field ( lElxl cos(wt), lElyl sin(wth0). The factor of 2 arises from different conventions used in the definitions of f":L and Tq.Finally, dl(Z t) represents the phase of the laser field envelope E,( t), obtained by solving the propagation equation for the fundamental.
<
III. Phase Matching In this section, we study how the harmonic field builds up in the medium, and is influenced by the dynamically induced phase of the atomic polarization. In all the calculations presented in this work, we try to mimic the experimental conditions of (Salihres ef al., 1996; Salihres et al., 1995).The 825-nm wavelength laser is assumed to be Gaussian in space and time, with a 150-fs full width at halfmaximum (FWHM). The laser confocal parameter b is equal to 5 mm and the focus position is located at z = 0. The generating gas is neon and the atomic density profile is a Lorentzian function centered at z with a 0.8-mm FWHM, truncated at z 2 0.8 mm. A. SOURCE OF THE HARMONIC EMISSION The generic intensity dependence of the dipole strength (i.e., absolute value squared) and phase for the 45th harmonic generated by a single neon atom is shown in Fig. 1. At low intensity, when the harmonic is in the cutoff region, the dipole strength (dashed line) increases rather steeply with laser intensity and the phase (solid line) decreases linearly. When the harmonic enters the plateau region, the strength saturates and exhibits many interferences, while the phase decreases twice as fast as in the cutoff region, predominantly linearly but with superimposed oscillations. The regular behavior of the dipole in the cutoff region is due to the existence of only one main electron trajectory leading to the emission of the considered harmonic. On the contrary, in the plateau there are at least two relevant trajectories whose contributions interfere, resulting in a perturbed dipole. In this region the dominant contribution is typically from the trajectory with the longer return time r2 (see Section 1I.B); it is the action along this trajectory that determines the mean slope of the intensity dependence of the phase.
100 Pascal Salikres, Anne L'Huillier, Philippe Antoine, and Maciej Lewenstein
0 -
-e U
-50
-
-w .-%-loo
-
v)
c Q
, , ,
0
I
! I
,
-150 -
!I I
0
2
4
.
,
6
a
Laser intensity (x 1 014 w/crn2)
FIG. 1. 45th single neon atom harmonic intensity (dashed line) and phase (solid line) as a function of the laser intensity.
One should mention, however, that the mean slope of the intensity dependence is not always a good measure of the phase variations in the macroscopic medium. The reason is that propagation and phase matching may lead to a single trajectory selection that depends on the relative position of the atomic jet with respect to the focus (Antoine et al., 1996a). For the case when the jet is before, or close to the focus, phase matching usually selects the dominant trajectory with the return time r2, and the intensity dependence of the phase after the selection follows the mean slope of the total phase. When the jet is after the focus, however, and when intensity in the jet is high enough, the trajectory with the return time is selected, and the slope of the intensity dependence is much weaker than the mean slope of the total phase. The latter case will not be considered in the following; we shall discuss only the situations when the mean slope of the phase can be used to characterize the intensity dependence of the phase in the macroscopic medium. This intensity dependence of the dipole is quite universal for sufficiently highorder harmonics, and depends weakly on harmonic number, laser wavelength, generating gas. These parameters only determine the position of the plateau-cutoff transition. In the following, we will thus concentrate on the study of the 45th harmonic generated in neon, keeping in mind that its behavior is very general.
B. DYNAMICALLY INDUCED PHASE OF THE ATOMICPOLARIZATION Harmonic generation is optimized when phase-matching is achieved, that is, when the difference of phase between the generated field and the driving polarization
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 101
A@ = @, - QPol is minimized over the medium length, allowing an efficient energy transfer. In the case of plane waves and low-order harmonics, this results in the well-known phase matching condition on the wave vectors: Ak = k, - q k , = 0. However, for the very high orders discussed here, the variation of the phase of the polarization is much more important than that of the induced field, so that matching phases amounts to minimizing the phase variation of the driving polarization. One of the main causes of variation of the polarization in the medium is due to the rapid variation of the dipole phase with intensity. It induces a spatially and temporally dependent phase term in the medium, which influences the generation of the macroscopic field. Let us consider this variation on the propagation axis at the maximum of the pulse temporal envelope for the 45th harmonic. It is induced by the variation of the intensity Z(z) = Zo/(l 4z2/b2),and is shown by the short-dashed line in Fig. 2 for a peak intensity, that is, at best focus and at the maximum of the pulse envelope, of 6 X 1014W/cm2. The variation of the dipole phase is less rapid outside the interval [-3 mm, + 3 mm], when the intensity on axis corresponds to the cutoff region (Z(z) 5 2.4 X IOl4 W/cm*, see Fig. 1). The other important contribution to the polarization phase is a propagation term induced by the phase shift of the Gaussian fundamental field in the focus region, equal to -q tanP1(2z/b), q denoting the process order. This function is shown in the long-dashed line in Fig. 2. There are other possible causes of variation of the polarization phase, such as atomic or electronic dispersion, but they are negligible in the conditions
+
6o
T
-120 1
z position
(mm)
FIG. 2. Phase of the polarization on the propagation axis (solid line). The long-dashed line indicates the term due to the propagation of the fundamental, and the short-dashed line the dipole phase for a peak intensity of 6 X 10l4W/cm2. The laser propagates from the left to the right.
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Pascal Salidres, Anne L'Huillier, Philippe Antoine, and Maciej Lewenstein
considered here. The total phase of the nonlinear polarization is represented in Fig. 2 by the solid line. In the region z < 0, the variations of both phases add, leading to a rapid decrease of the total phase. In the region z > 0, they have opposite signs, and almost compensate when the intensity on axis corresponds to the cutoff region. Consequently, phase matching strongly depends on the position of the medium relative to the laser focus. The best phase-matching conditions on axis are those for which the phase variation of the polarization over the medium length (- 1 mm) is minimal, that is, when the laser is focused approximately 3 mm before the generating medium. Note that at the minimum of the curve close to the focus, the superimposed oscillations are detrimental to a good phase matching. So far, we have only considered phase matching on axis, which corresponds to centered harmonic profiles. However, good phase matching oflaxis can be realized in certain conditions. This is illustrated in Fig. 3 with the longitudinal variation of the polarization phase for different radial positions, from r = 0 to 15 p m (relative to the propagation axis) for a peak intensity of 6 X 1014W/cm2. Here, for clarity, the curves have been smoothed so that the superimposed oscillations in the plateau region do not appear. For a gas jet centered in z = - 1 mm, it is possible to minimize the longitudinal phase variation by moving from one curve to the other, that is, by going off the propagation axis. Along these favored directions, the quick variation of the laser intensity on axis (cause of the rapid decrease of the polarization phase) is avoided by going off axis. In these conditions, the harmonic field can build up efficiently in the plateau region. Note that a method
-160'
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(mm)
FIG. 3. Phase of the polarization for different radial positions relative to the propagation axis, for a peak intensity of 6 X 10I4W/cm2: r = 0 (short-dashed), r = 5 prn (solid), r = 10 p m (long-dashed) and r = 15 p m (dot-dashed). The dotted lines indicate the edges of a gas jet placed in z = - 1 mm, and the horizontal solid line, a trajectory r(z) that keeps the phase constant.
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 103
.-c 0
c
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-60
a 0
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z position (rnrn)
FIG. 4. Phase of the polarization on the propagation axis. From the top to the bottom, the laser intensity I = 2, 3.4,5 , 6 X 1014W/cm2.
allowing the systematic study of these generalized phase-matching conditions has been proposed in Balcou et al. (1997). If we now consider another peak intensity, the shape of the total phase is modified. This is illustrated on axis in Fig. 4 for several peak intensities, from 2 to 6 X I O l 4 W/cm2.As the intensity increases, the induced phase becomes more and more important in determining the total phase variation near the focus, which departs more and more from the arctangent term. The optimal phase-matching position on axis is observed at different z depending on the peak intensity, because it always corresponds to the plateaucutoff transition of the dipole (2.4 X 1014 W/cm2). Thus, for a given geometry, there will not be a static phase matching during the laser temporal envelope, but a continuous distortion of the build-up pattern in the medium. This dynamic phase matching complicates the interpretation of the processes.
c. INFLUENCE OF THE JET POSITION O N THE CONVERSION EFFICIENCY Using the numerical methods described in Section 11, we perform the propagation of the generated harmonic fields in the medium, considering for the moment square laser temporal envelopes, that is, static phase matching. In Fig. 5, we study the variation of the conversion efficiency for the 45th harmonic generation as a function of the position z of the center of the atomic medium (relative to the laser focus placed in z = 0), for peak intensities ranging from 3 to 6 X I O l 4 W/cm2. The peak atomic density is 15 torr.
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Pascal Saliires, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
At low intensity, the curve presents only one maximum located in z = 1 mm. Increasing the intensity, this maximum splits into two lobes that become more and more separated. This is the confirmation of the two optimal phase-matching positions previously described. The positions of the maxima z > 0 correspond precisely in Fig. 4 to the best phase-matching positions on axis, that is, to the plateau-cutoff transition of the dipole (2.4 X 1014 W/cm2). The maxima occurring for negative z correspond to optimized phase-matching positions offaxis, as shown in Fig. 3. Note that all the curves are asymmetric compared to the focus position (z = 0), and that the conversion efficiency in z = 1 mm is larger at the lowest intensity. However, this effect disappears when we consider harmonic generation by a Gaussian laser pulse, as shown in Fig. 6 for the same peak intensities as before. During the major part of the laser pulse, the intensity is in the cutoff region where the polarization amplitude drops, resulting in a lower conversion efficiency than for a square pulse. Except for this modification, the general behavior is the same as for square pulses, with enlarged peaks but similar positions and efficiencies. Note that if we take into account the ionization of the medium at 6 X l O I 4 W/cm2 (dots in Fig. 6), we find a marginal influence on the number of photons except very close to the focus position. In the following, we will thus neglect ionization in a first step, and include it afterwards when its effects are not negligible. Wahlstrom et al. (1993) have measured about lo5 generated photons for the 45th harmonic in neon at 6 X 1014W/cm2. The difference of one order of magnitude
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Jet/focus position (mrn)
FIG. 5. Conversion efficiency for the 45th harmonic as a function of the position of the center of the jet relative to the focus, for peak intensitiesranging from 3 to 6 X 10l4W/cm2.The laser temporal envelope is square with a 150-fs width.
I
2
I
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Jet/focus position (mm)
FIG. 6. Conversion efficiency for the 45th harmonic as a function of the position of the center of the jet relative to the focus, for peak intensities ranging from 3 to 6 X I O l 4 W/cm2. The dots show the influence of ionization at 6 X lOI4 W/cmZ.The laser temporal envelope is Gaussian with a 150-f~FWHM.
with the results of our simulations can be explained, at least partly, by the more optimized conditions used in their experiment (higher pressure and longer confocal parameter).
D. MODIFIED CUTOFFLAW The influence of the position of the focus relative to the gas jet on the harmonic conversion efficiency has an interesting consequence on the dependence of the harmonic yield as a function of intensity, for the different geometries. Fig. 7 presents the intensity dependence of the conversion efficiency for the 45th harmonic for a focusing at the center of the jet (solid line, z = 0). The comparison with the strength of the dipole moment (short-dashed line) indicates two main consequences of the propagation: the rapid variations in the plateau region are smoothed out, and the change of slope indicating the plateau-cutoff transition is shifted to a higher intensity. This shift is the same for higher order harmonics, and thus implies that propagation decreases the extent of the plateau of the harmonic spectrum compared to the single-atom response, from a photon energy of I, + 3.2UP to about I, + 2Up.This was observed experimentally by L’Huillier et al. (1993), and explained in terms of the variation of phase matching with intensity. If the laser was not focused in the jet, one would expect that the lower intensity experienced by the nonlinear medium would result in an even larger shift of the
106
Pascal Salikres, Anne L'Huillier, Philippe Antoine, and Maciej Lewenstein 1o6 1o5
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FIG. 7. Intensity dependence of the conversion efficiency for the 45th harmonic for a jet position in z = 0 (solid line) and z = 1 mm (dot-dashed line). Ionization is taken into account. The shortdashed line indicates the strength of the dipole moment (arb. units).
plateau-cutoff transition (recall that we are plotting the curves as a function of the peak intensity, that is, at thefocus and at the maximum of the pulse envelope). In Fig. 7, the intensity dependence for the case when the jet is located in z = 1 mm is shown in dot-dashed line. Amazingly, the shift of the plateau-cutoff transition to higher intensities is less important than for z = 0, corresponding to a cutoff law of about I, + 2.3UP.This is a direct consequence of the optimization of phase matching at low intensity for this particular position, as shown in Fig. 6. This effect is independent of the nonlinear order considered, and would indicate that the maximal extent of the plateau is not obtained for a focusing right into the jet, but rather slightly before it.
IV. Spatial Coherence A. DEFINITION
We recall here some notions of the theory of partial coherence (see, for example, Born and Wolf, 1964). The coherence of a beam is related to the correlation of the temporal fluctuations of the electromagnetic fields inside this beam. It is thus characterized by its mutual coherence function, defined for any two points inside the beam by r12(7) = ( E l ( t ~ ) E z ( t ) where ), E,, E2 are the complex amplitudes of the electric field in these two points, and the angular brackets denote an appropriate time average (here, over the harmonic pulse). The normalized form of the mutual coherence function is the complex degree of coherence:
+
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 107
whose modulus is known as the degree of coherence. The temporal coherence is described by y, (T),whereas the spatial coherence is described by y,2(0). Note that the latter, despite its name, is related to the correlation in time of the fields emitted in two points. These quantities determine the ability of the fields to interfere, and can be measured in interferometry experiments. In Young’s two-slit experiment, with both slits uniformly illuminated, IyI2(O)1 at the slit positions is simply given by the fringe visibility, defined as V = (I,,,,, - Zmin)/(Zma + Zmi,,) (where I,,,,, and Zmin are the maximum and minimum intensities of the fringe pattern). The spatial coherence length of a beam at a given distance from its focus is defined as the length over which the degree of spatial coherence is larger than some prescribed value (between 0.5 and 0.9, depending on the authors, and on the coherence of their own source). Another important aspect related to the coherence of a beam is the quality of its wavefront, an aspect that is often confused with the preceding description. A beam is said to be “diffraction limited” if the product of its spot size (at focus) and of its far-field spread (divergence) is of the order of the wavelength. This is realized both when the focal spot presents a reasonably regular amplitude variation, and when the phase front at focus is very well behaved (typically plane). In particular, any distortion of the phasefront will result in a larger (e.g., N times) angular spread, and the beam will be called “ N times diffraction limited” (Siegman, 1986). In the following, we shall concentrate on the two focusing positions corresponding to well-defined phase-matching conditions at 6 X 1014 W/cm2, namely z = 3 mm (on axis) and z = -1 mm (off axis). For these extremal positions, on either side of the conversion efficiency curve (see Fig. 5), phase matching is mostly efficient close to the maximum of the laser temporal envelope, thus simplifying the study. Note that the main dependences of the harmonic emission profiles (laser intensity, nonlinear order, jet/focus position) have been intensively studied in Salibres et al. (1996), and compared successfully with experimental data. We focus here on the coherence properties.
B. STUDYOF THE SPATIAL COHERENCE: ATOMIC JET AFTERTHE FOCUS First we study the characteristics of the harmonic beam in the near-field, that is, at the exit of the medium in z = 3.8 mm (the half-width of the jet is 0.8 mm) and at the maximum of the laser pulse. Fig. 8 presents in solid lines the harmonic profiles corresponding to different intensities, from 4 to 6 X l o t 4W/cm2 (square pulses). The first two profiles are Gaussian, with 12 and 14 p m radius in l/e2 respectively, while the third is super Gaussian with a 20 p m radius. They are
108
Pascal Sulidres, Anne L’Huillier, Philippe Antoine, and Muciej Lewenstein
I
0
I
I
5
10
Radial coordinate (prn)
15
-
20
25
30
Divergence (x0.26 rnrad)
FIG. 8. Normalized spatial profiles for the 45th harmonic at the maximum of the pulse for a jet position in z = 3 mm, for intensities from 4 to 6 X loJ4W/cm2. The solid lines show the profiles at the exit of the medium as a function of the radial coordinate, and the dashed lines, the far-field profiles (divergence).
narrower than the fundamental (46 pm), but larger than the 7 p m predicted by lowest order perturbation theory. These regular profiles result from a good phase matching on axis (see Section 111) together with a regular intensity dependence of the amplitude of the dipole in the plateau-cutoff transition region. Increasing the intensity, this region is moved to the high-density zone at the center of the jet, leading to a broadening and distortion of the profiles. In Fig. 9 we present the radial phases corresponding to these profiles. They all present a regular parabolic behavior, whose dependence is between 0.045r2 and 0.048r2rad, r being the radial coordinate in pm. To understand the origin of these curved phase fronts, let us consider the phase of the polarization at the exit of the medium. Given the low density, the harmonic field is obviously not mainly generated there, but this gives an estimate of what happens in the medium and can be directly compared to the phase of the generated harmonic field. There are two main contributions to the polarization phase. The first one is the Gaussian fundamental field phase multiplied by the order
where w(z) = w o d l + 4z2/b2and wo is the beam waist, related to the confocal parameter by b = 2.rrwi/A. The second contribution is the dipole phase, which depends on the intensity. The harmonics are here generated in the plateau-cutoff
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 109
transition region, where the dipole phase varies linearly with intensity, with a negative slope, -7.This contribution can then be written as
We can here assume r << w(z),because the harmonic profiles are much narrower than the fundamental. In Section 111, we have considered the first terms of both contributions, which correspond to variations on axis. The second terms of these contributions, related to radial variations, both present a quadratic behavior. The term due to the focusing of the fundamental (Eq. (13)) is in fact the phase corresponding to a Gaussian harmonic field with the same confocal parameter. Its radial dependence is equal to 0.032r2 rad and is thus significantly slower than the ones observed for the harmonic fields. The reason for the bigger curvature of the phase fronts is thus the additional radial variation introduced by the dipole phase (second term in Eq. (14)), which lies between 0.016r2and 0.023r2 rad, depending on the considered intensity. These curved phase fronts have important consequences, as we shall see next. The harmonic fields emitted at 4 X 1014 and 5 X 1014 W/cm2 present Gaussian
"
0
5
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Radial coordinate (pm)
FIG. 9. Radial variation of the phases of the harmonic fields corresponding to the near-field profiles shown in Fig. 8: 4 X lOI4 (dashed), 5 X lOI4 (solid), and 6 X l o t 4W/cm2 (dot-dashed).
110
Pascal SaliPres, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
profiles and quadratic phases at the exit of the medium, and thus correspond to lowest order Gaussian modes, whose characteristics (beam waist position and confocal parameter) can be easily calculated. We find, in both cases, a virtual focus located at the laser focus (like in the perturbative case) but with an extremely small size: about 1.6 p m (even smaller than the perturbative 3.7pm). The corresponding confocal parameters are also very small (about 1 mm), so that the harmonic fields at the exit of the medium are already as if they were in the far-field: spherical phase front centered at the focus and linear increase of the beam size with the distance from the focus. Note that the radial variation of the phase of a spherical wave at a distance of z = 3.8 mm is 0.046r2 rad, close to the observed variations. If we calculate the angular spread of these beams far from the jet, we find similar profiles, with half-angle in l/e2 between 3 and 5 mrad, which can be superimposed on the “near-field” profiles if their spatial dimension is calculated at a distance of z = 3.8 mm. This is shown in dashed lines for the three intensities in Fig. 8, where the horizontal scale is interpreted as the divergence (half-angle) and is simply related to the radial coordinate by: divergence(mrad) = 0.26 X r(pm) (or r = 3.8 X divergence). Note that the profile corresponding to 6 X l O I 4 W/cm2 is more distorted by the propagation, due to its multimode structure, but its divergence is still governed by its curved phase front at the exit of the medium. Finally, we can check the preceding conclusions concerning the virtual source size, and extend them to the case 6 X l O I 4 W/cm2, by “back-propagating’’ these beams to their virtual focus. The solution of the propagation equations in free space for the complex conjugated field is formally equivalent to a time reversal, hence a “back-propagation.” Fig. 10 presents the profiles and phases of the harmonic beams corresponding to the preceding intensities, calculated at the laser focus position ( z = 0). The phases stay about constant over the extent of the profiles, indicating quasi-plane phase fronts. This position is thus close to the best focus for these beams. The spot sizes are very small, close to the above estimates, with a minimum at 6 X 1014W/cm2with a 1.4 p m radius at l/e2. The very small size of the virtual source of the harmonic beams is a valuable information as it gives an indication on the size they can be refocused to (assuming ideal refocusing without aberrations introduced by the optics, and no reduction factor). From this, we can estimate the achievable hamzonic intensities, in view of applications. If we take an harmonic pulse duration of 66 fs (see Section V) and an optimized number of photons of lo5 (obtained by optimizing the pressure, Altucci et al., 1996), the intensity at 18.3 nm (45th harmonic) reaches about 5 X lo8 W/cm2. Note that if argon is used as generating medium instead of neon, a much higher conversion efficiency is obtained for lower order harmonics: for example, lo9 photons for the 19th harmonic (Wahlstrom et al., 1993). Because the behavior of the dipole moment is very general, whatever the (sufficientlyhigh)
SPATIAL AND TEMPORAL COHERENCEOF HIGH-ORDER HARMONICS 111 1 .o
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FIG. 10. Radial variations of the intensity and phase of the harmonic field generated at the peak intensities: 4 X loi4 (dashed), 5 X l o i 4 (solid), and 6 X IOl4 W/cm2 (dot-dashed), and “backpropagated” to the focus position ( 3 mm before the jet).
order or the generating gas, we can extend our results to the 19th harmonic in argon, and we thus find an harmonic intensity of 2 X 10I2W/cm2 at 43.4 nm. If a reduction factor is introduced by the refocusing optics, the very nice harmonic phase front should allow one to reach an even smaller focus size, and thus a higher intensity, provided that the surface figure of the optics is good enough. These intensities have never before been reached at these short wavelengths, and would open the way to nonlinear optics in the XUV region. So far, we have studied the beam spatial characteristics at a given time during the laser pulse. Let us now consider the correlation in time of the fields inside the harmonic beam, by calculating the corresponding degree of spatial coherence, given by Eq. (12). It is shown as the solid line in Fig. 1l(a), calculated at the exit of the medium (z = 3.8 mm) between the central point (on the propagation axis) and the outer points, for a peak intensity of 6 X lOI4 W/cm2 and a peak pressure of 3 torr. The coherence degree is very high, and stays above 0.9 over the whole extent of the time-integrated spatial profile, which is shown as a dashed line. At this low pressure, the influence of the free electrons generated by the (low) ionization of the medium is negligible. The small decrease of the coherence degree can be understood by the slow variation over the pulse of both the phasefront curvature and the spatial profile, as shown for three different intensities in Figs. 8 and 9. At a pressure of 15 ton-, the coherence degree is a little reduced, but stays very high (above 0.8). However, when the pressure is increased to 150 torr, as shown in Fig. 1l(b), the spatial profile is broadened and somewhat
112
Pascal Salieres, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein 1 .o o)
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FIG. 11. Coherence degree (solid line) of the harmonic beam generated at z = 3 mm and at a peak intensity of 6 X lot4W/cm2, calculated at the exit of the medium between the central point and the outer points, for a peak pressure of (a) 3 torr, and (b) 150 ton. The corresponding time-integrated spatial profiles are shown by the dashed lines.
distorted, while the coherence degree drops abruptly between 15 and 25 p m to a value of 0.3. Indeed, the free-electron dispersion is not any more negligible and results in a phase shift that is spatially dependent, due to the spatial distribution of the freeelectron density. Close to the propagation axis, the laser intensity, and thus the free-electron density, varies slowly radially, preserving the coherence. Further away, when the laser intensity drops, the phase shift imparted to the harmonic field is much smaller, resulting in a decorrelation with the center of the beam, and therefore, a smaller coherence degree. Ionization of the generating medium is thus an important cause of degradation of the coherence. Here, we have calculated the coherence degree between the central point (on the axis) and the outer points. Note that for two points taken symmetrically into the beam, the coherence degree is equal to 1, due to the revolution symmetry imposed to the problem. In experiments, this symmetry may be broken by different factors (gas jet and laser spot inhomogeneities, and so on), which results in a
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 113
smaller coherence degree, as recently measured by Ditmire et al. (1996). However, our study shows that in order to truly characterize the coherence of the harmonic beam, it is necessary to measure the coherence degree for nonsymmetrical positions. This point makes the situation very different from that of a beam originating from a completely incoherent source. In the latter case, the coherence degree only depends on the relative position of the two points considered, that is, their distance, not on their absolute positions into the beam, as is the case here. In conclusion for this focus position, at low pressure the very high coherence degree at the exit of the medium indicates that the harmonic beam is extremely coherent, in comparison to soft x-ray lasers (Trebes et al., 1992; Celliers ec al., 1995; Svanberg and Wahlstrom, 1997).The coherence of soft x-ray lasers is often degraded by plasma fluctuations,resulting in a coherence degree equivalent to that of a spatially incoherent disk source with diameter of a few 100 p m (Amendt et al., 1996).The associated transverse coherence length at the output of the x-ray laser is a few pm, whereas it reaches several tens of p m for the harmonics.
c. STUDY OF THE SPATIAL COHERENCE: ATOMICJETBEFORETHE FOCUS The 45th harmonic spatial profile at the exit of the medium for a peak intensity of 6 X 1014 W/cm2 and a focusing 1 mm after the jet is shown in solid line in Fig. 12. It is rather distorted and exhibits an annular structure with three major rings. Note that very little energy is emitted on axis. As shown in Section 111, this 1 .o
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FIG. 12. Normalized spatial profile for the 45th harmonic at the maximum of the pulse for a jet position in z = - 1 mm and an intensity of 6 X lOI4 W/cmZ. The solid line shows the profile at the exit of the medium as a function of the radial coordinate, and the dashed line shows the far-field profile (divergence). The fundamental near-field profile is presented by the dotted line.
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Pascal SaliLres, Anne L'Huillier, Philippe Antoine, and Maciej Lewenstein
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FIG. 13. Radial variation of the phases of the harmonic fields generated for a jet position in z = - 1 mm and intensities of respectively: 4 X lOI4 (dashed), 5 X 1014(solid), and 6 X loL4W/cm2 (dot-dashed).The radial variation of the polarization phase at the exit of the medium at an intensity of 6 X l o i 4W/cm2 is shown by the dotted line.
is the result of the rapid variation of the polarization phase on axis, which favors the off-axis phase matching. The external radius is 27 pm, larger than the 25 p m in l/e2 of the fundamental beam (dotted line). The corresponding phase is presented by the dot-dashed line in Fig. 13 together with phases of beams generated at 4 X 1014 (dashed) and 5 X 1014 W/cm2 (solid). Compared to the case z = 3 mm, these phases are more irregular and vary much more quickly. Surprisingly they correspond to diverging phase fronts, even though the harmonics are generated by a converging beam. Consider the two contributions to the transverse phase of the polarization. The phase induced by the fundamental field, written in Eq. (13), is indeed negative but varies very slowly close to the focus (quasi-plane phase front). On the contrary, the contribution of the atomic phase, shown in J3q. (14),is associated with a very diverging phase front: not only is the radial variation of the intensity very rapid close to the focus, but also the harmonic is generated here in the plateau region where the average slope of the intensity dependence of the phase is twice as large as that in the cutoff region. Note that the quadratic approximation does not hold here, because we consider r w(z).The total polarization phase at the exit of the medium and for an intensity of 6 X 1014 W/cm2 is shown by the dotted line in Fig. 13. Its behavior is very similar to that of the phases of the harmonic beams, except for a slightly concave curvature. These very curved phase fronts dominate the propagation, and the calculation
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS I 15
of the far-field distribution of these beams gives annular profiles, as shown by the dashed line in Fig. 12, where the horizontal scale is interpreted as the divergence and is related to the radial coordinate by r = 1.6 X divergence. The external halfangle is 15 mrad, larger than the 10 mrad at lle2of the fundamental.The harmonic is thus more divergent than the fundamental, which almost leads to a spatial separation of the two beams. As before, we can back-propagate this beam and find the position of the virtual focus. The highest harmonic intensity is obtained in z = - 1.8 mm, which happens to be the entrance of the medium. The profile (solid line) and phase (dashed line) at this virtual focus are presented in Fig. 14. They look like the diffraction pattern of a ring with a spherical phase front. The phase shows a n- shift (i.e., a change of sign) for each profile oscillation, whereas the profile exhibits a narrow central peak (0.5 pm) surrounded by weak rings. A part of the energy is “lost” in these rings, so that the harmonic intensity reached at the central peak is barely larger than that obtained for a focusing 3 mm before the jet. Let us now consider the coherence degree of this beam calculated at the exit of the medium for a peak pressure of 3 ton; shown by the solid line in Fig. 15. Because there is very little energy emitted on axis (see the time-integrated profile, shown by the dashed line), we take as a reference the point located at the center of the ring ( r = 22 pm). The coherence degree drops dramatically on both sides to about 0.3, and then oscillates around 0.2. The coherence of this beam is thus very much degraded, not by the free-electron dispersion (low pressure), but by the rapid phase-front fluctuations induced by the variation of the intensity in the laser 1 .o
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FIG. 14. Radial variations of the intensity (solid line) and phase (dashed line) of the harmonic field generated at z = - 1 mm and 6 X I O l 4 W/cm2 and “back-propagated’’to the entrance of the medium ( z = - 1.8 mm).
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Pascal Salitres, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein 1 .o
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FIG. 15. Coherence degree (solid line) of the harmonic beam generated at z = - 1 m m and a peak intensity of 6 X loL4W/cm2, calculated at the exit of the medium between the point located close to the center of the ring ( r = 22 p m ) and the other points. The time-integrated spatial profile is shown by the dashed line. The peak pressure is 3 tom.
pulse (see Fig. 13). The dynamically induced phase can thus be responsible for a dramatic degradation of the spatial coherence of the harmonic beam. Note that in traditional, that is, perturbative, low-order harmonic generation, where the harmonic dipole moment does not exhibit an intrinsic phase and depends regularly on the laser intensity, the coherence of the laser beam is simply transmitted to the harmonic beam. In conclusion, the dynamically induced phase plays a central role in determining the spatial characteristics of the generated harmonic beams, and especially their coherence properties, which are very sensitive to the focusing conditions.
V. Temporal and Spectral Coherence A.
INFLUENCE OF THE JET POSITION
We have shown in the preceding section that the dynamically induced phase results in a dynamic phase matching due to the intensity distribution in the laser pulse. This will affect the temporal as well as the spectral properties of the generated harmonics. Let us first consider the harmonic temporal profile. Because we calculate in the slowly varying envelope approximation, the temporal profile is simply obtained from the response to elemental square laser pulses, with a Gaussian distribution of the intensities. Fig. 5 shows clearly that the temporal behavior
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 117
will be very much dependent on the position of the jet. On both sides of the conversion efficiency curve, for example in z = - 1 mm and in z = 3 mm, the efficiency increases quickly with intensity. The harmonic yield will thus be high mainly for intensities close to the maximum of the laser temporal envelope, leading to narrow and regular harmonic temporal profiles. On the contrary, for intermediate z positions, phase matching can be more efficient for low intensities than for the maximum of the laser envelope, resulting in large and distorted harmonic profiles. These predictions are confirmed in Fig. 16, which presents the 45th harmonic temporal profiles obtained at 6 X l o t 4W/cm2 for different z positions (ionization is not yet taken into account). When the jet is moved from z = 4 mm to z = 2 mm, the harmonic profile gets larger (from 36- to 125-fs FWHM) but remains regular. On the other side, in z = - 1 mm, the profile is also smooth, narrow, and very similar to the one obtained in z = 3 mm, with the same FWHM (67 fs). On the contrary, in z = 1 mm and z = 0, the profiles are quite distorted, with fast fluctuations. In z = 1 111111, phase matching is much more efficient at 3 X 1014 W/cm2
-100
-50
0 Time (fs)
50
100
FIG. 16. Temporal profiles of the 45th harmonic generated at 6 X IOl4 W/cm*, for different z positions indicated in mm in the captions. The laser profile is presented in dotted lines, the stars indicating the position of the inflection points.
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6 0.8 c W c .G
0.6
-0 N W
5 0.4
E
z" 0.2 0
-2
0 Spectral width
(A)
2
FIG.17. Spectral profiles of the 45th harmonic generated at 6 X lok4W/cm2, for different z positions indicated in mm in the captions.
than at higher intensities (see Fig. 5 ) , which leads to narrow peaks in the harmonic profile at half-maximum of the laser envelope (shown in dotted lines). Rapid fluctuations also appear in the profile corresponding to the position z = 0. They are probably due to the oscillations of the slope in the intensity dependence of the dipole phase in the plateau region, which induce rapid changes of the dynamic phase matching. The corresponding spectral profiles are presented in Fig. 17. For a focusing sufficiently before the medium, the spectra are quite narrow (e.g., 0.4 A FWHM at z = 3 mm). When the jet is located at z = 1 mm, the base of the profile gets broadened with superimposed oscillations. In z = 0, the profile is very broad with a FWHM of 4 A. Finally, in z = - 1 mm, the profile is again regular with a width of 2.2 A. These large variations in spectral width cannot be explained by variations in the intensity temporal profiles. Note for example that in z = - 1 and z = 3 mm, the temporal profiles are very similar with the same FWHM, whereas there is a factor of 5 between the corresponding spectral widths. Consequently, these broad spectra are not due to variations of the amplitude of the harmonic fields, but
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 119
to variations of their phase. Calculations performed without taking into account their phase, that is, assuming that the fields are Fourier transform limited, give indeed much narrower spectra, with widths smaller than 0.15 A. The origin of this phase modulation is the temporal variation of the dipole phase during the laser pulse. Let us model the intensity dependence of the dipole phase by a linear decrease, whose slope -r] depends on the region of the spectrum: in the cutoff 77 = 13.7 X lopL4 rad/(W/cm2) and in the plateau 7 = 24.8 X l o p Lrad/(W/cm2). 4 The induced modulation of the emitted harmonic field reads:
AQ(t)
=
-vZ(t) = -qI, exp(-4 In 2 ( r / ~ ) ~ )
(15)
where T is the FWHM of the Gaussian laser envelope. This results in a modulation of the instantaneous frequency of
a(A@) Aw(t) = -at -
al(t)
vat
with a corresponding broadening of the spectrum. This phenomenon is thus very similar to self-phase modulation of an intense laser pulse in a Kerr medium with a negative index n2 (Shen, 1984). However, the phase modulation is here induced on the harmonic, whose temporal profile is different from that of the laser, and we shall see the consequences next. The rising edge of the harmonic pulse is thus shifted to the blue, and the falling edge, to the red. The extremal instantaneous frequencies are symmetricalon either side of the central frequency, and correspond to the inflection points of the laser temporal envelope: ti = 2 T / v ' ~ .One then finds:
AAex, =
-A2 Awexi = 2 n-c
A2
d
2 -2n-c r
m
exp(- 1/2)v10
(17)
and in our conditions: AAexi = 2 1.6 X IO-*vZ, (A). The maximal broadening of the spectrum should thus be observed at focus, which experiences the largest intensity variation (I, = 6 X 1014W/cm2) and where the intensity at the inflection points is still in the plateau region, ensuring the larger slope q. This results in A h , = k2.4 A, close to the broadening observed for z = 0. However, this spectrum does not exhibit the characteristic grooved shape of the self-phasemodulation spectra. Let us recall the main features of self-phase modulation (Shen, 1984). Close to the inflection points, the instantaneous frequency varies slowly, unlike in the vicinity of the pulse maximum. The spectral density is thus maximum on the sides (in AA,,) and minimal at the central frequency. On the A@(t) curve, there exist two points of the same slope, located on either side of the inflection point. Thus they correspond to the same instantaneous frequency and interfere constructively
120
Pascal Sali?res, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
0.8$
2
0.65 (D
n
.-c
0.4z
E 5 0
0.2*
0
0
(D
u1
I
0 1 .o
0
-0.5 -1.0
-80
- 40
0
40
80
Time (fs)
FIG.18. (a) Temporal dependence of the intensity (dot-dashed) and phase (solid line) of the harmonic field exiting the medium on axis ( r = 0) for a focusing 3 mm before the jet. The variation of the polarization phase at that point is shown as a dashed line. (b) Corresponding frequency chirps for the field (solid) and the polarization (dashed).
or destructively depending on their relative phase. This results in a grooved spectrum, with clear peaks and valleys. In our case, the phase modulation is induced on the harmonic, which is generated efficiently for large enough intensities, located in general above the inflection points (see Fig. 16). Consequently there will be no interferences in the spectra. Thus they are more regular, except the one in z = 1 mm, which corresponds to the only temporal profile sufficiently large to go past the inflection points. In the following, we study in more detail what happens in z = 3 and z = - 1 mm. In the case of z = 3 mm, the integrated spatial profile at the exit of the medium is very regular, with a maximum on axis. The phase temporal variation of the harmonic field in r = 0 thus has a large incidence on the full spectrum, and is characteristic of the phenomenon. Fig. 18(a) presents the temporal variation of the intensity (dot-dashed line) and phase (solid line) of the harmonic field emitted on axis at the exit of the medium. The intensity profile is quasi-square, with a FWHM of 90 fs. The phase presents a very regular behavior. Let us compare it
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 121
with the polarization phase (shown as a dashed line) calculated at this point from Eq. (15). The maximum intensity there is 1, = 1.8 X 1014W/cm2 (the exit of the medium is 3.8 mm away from the focus), which corresponds to the cutoff region (small value of 7).The two curves are very similar over the width of the intensity profile, confirming the origin of the phase modulation. The induced frequency chirps are presented in A in Fig. 18(b). The spectral width corresponding to efficient harmonic emission is about 0.5 A, close to that of the full spectrum. Note that the chirp is almost linear during the harmonic pulse. In z = -1 mm, the near-field spatial profile is annular due to an efficient phase matching off axis (see Section IV). Let us consider the harmonic field in r = 20 p m at the exit of the medium, where the emission is efficient. The intensity profile, shown in dot-dashed line in Fig. 19(a), exhibits rapid oscillations superimposed on a 90-fs-wide curve. These fluctuations do not appear in the spatially integrated temporal profile of Fig. 16, which is very regular. This implies that
1 .o
25 h
U
e
v
20
0.8g
15
0.65
i
W
I
0
Q (D
Q
.g 10 E
0.4s ID
0) 3
0.22
b 5 I 0
-80
0
- 40
0
40
80
Time (fs)
FIG. 19. (a) Temporal dependence of the intensity (dot-dashed line) and phase (solid line) of the harmonic field exiting the medium in r = 20 p n for a focusing 1 mm after the jet. The variation of the polarization phase at that point is shown as a short-dashed line, and in a point of maximum intensity of 3.4 X loi4W/cm2, as a long-dashed line. (b) Corresponding frequency chirps for the field (solid) and the polarization (long- and short-dashed).
122
Pascal Sali2res, Anne L'Huillier, Philippe Antoine, and Maciej Lewenstein
they are compensated by opposed oscillations of the emission in neighboring points. This can be understood by the variation of the phase-front curvature in time, which leads to fluctuations of the angle of the emission cone. In a given position, one then sees the radiation going back and forth, hence these oscillations. The phase of the harmonic field, shown as a solid line, varies much more rapidly than in z = 3 mm. Because the position considered is far from the axis, the maximum intensity corresponds to the cutoff region: I, = 1.7 X l O I 4 W/cm2, and the polarization phase at that point (short-dashed line) varies too slowly to explain the harmonic phase behavior. A good agreement is obtained if we take an intensity of I , = 3.4 X l O I 4 W/cm2,with the plateau slope (long-dashed line). This implies that the harmonic radiation that exits the medium at that point has mainly been generated in a deeper region, close to the axis. The phase-front curvature makes it diverge and exit at that position. The frequency chirp, presented in Fig. 19(b), shows the phase sudden changes of slope, reflecting the intensity fluctuations. The corresponding spectral width is about 2.2 8,close to that of the full spectrum. The origin of the difference by a factor 5 of the spectral widths in z = 3 and z = -1 mm appears thus clearly: in z = 3 mm, the harmonic emission corresponds to the cutoff region, hence a small slope 11 and a low intensity. In z = - 1 mm, it is mainly in the plateau, with a slope twice as large, and for intensities at least two times larger. The spatial and spectral coherence properties are thus closely linked, and are both governed by the variations of the atomic phase. B. INFLUENCE OF THE IONIZATION When the jet is placed several mm away from the focus, the intensity in the medium is sufficiently low to neglect the effects of ionization. In z = 3 mm for example, the profiles are only slightly modified. We will thus concentrate on the case z = - 1 mm, and try to separate the different effects: depletion of the atomic population, and defocusing of the laser and phase mismatch, both due to the generated free electrons. Fig. 20(a) presents the distortions of the temporal profile induced by the different effects. The short-dashed line shows the profile corresponding to a peak pressure of 3 torr. This low pressure ensures that the free-electron density is low enough to neglect its effects. This profile thus characterizes the incidence of the depletion of the medium on harmonic generation (the susceptibility of the generated ions is supposed to be negligible compared to the atoms). At the center of the jet, ionization reaches 67% at the end of the laser pulse. The resulting profile is clearly asymmetric, with a rising edge hardly modified and a decrease of the efficiency on the falling edge. The FWHM is slightly decreased, from 67 to 63 fs. At a pressure of 15 torr, the effects of the free electrons are not any more negligible. We have first made the calculations by neglecting the defocusing of the
SPATIAL AND TEMPORALCOHERENCEOF HIGH-ORDER HARMONICS 123 1 .o
g 0.8 c
c
5 0.6 u
0.4
E.
s 0.2 0 -80
-40
0
80
40
Time (fs) 1.0
G 0.8 c
2
0.6
U
0.4
E
z"
0.2 0 -2
-1
0
Spectral width
1
(A)
2
FIG. 20. Temporal (a) and spectral (b) profiles for a focusing 1 mm after the jet at 6 X 1014W/cmZ: without ionization (solid line), with ionization at a pressure of 3 tom (short-dashed line) and 15 tom (long-dashed line) of neon. The dot-dashed line shows the profile corresponding to 15 tom when the defocusing of the laser is neglected.
laser: we suppose a Gaussian propagation and we only solve the propagation equation for the harmonic field in the ionized medium. The temporal profile (dotdashed line) is even more distorted on the falling edge: the width decreases to 58 fs. This is due to the phase mismatch induced by the free electron dispersion, which reaches Ak = 22 mm-' at the center of the jet and at the maximum of the pulse. The study of the propagation of the laser in the ionizing medium shows clearly an effect of defocusing: at the exit of the medium (0.2 mm from the focus), the maximum intensity is reduced by 17%,to 5 X 10l4W/cm2. The reduced intensity experienced by the medium results in a smaller ionization degree: 54% at the center of the jet and at the end of the pulse (vs. 67%). A consequence of the defocusing is thus to reduce the amplitude of the preceding effects: less depletion and smaller free-electron density. However, the corresponding temporal profile (long-dashed line) is the most distorted one. The efficiency drops abruptly after
124
Pascal SaliPres, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
the maximum of the pulse, leading to a width of 46 fs. Two elements can explain this decrease in efficiency. First, the reduced intensity induces a decrease of the polarization amplitude. Second, the defocusing changes the intensity distribution in the medium compared to the Gaussian. The atomic phase distribution is thus modified, which may result in a less efficient phase matching. In Fig. 20(b) we show the incidence of these phenomena on the spectrum. A striking feature is that, whatever the effect considered, the distortions of the spectrum simply follow the distortions of the temporal profile. The blue side of the spectrum is weakly affected, whereas its red side is progressively cut off. The FWHM decreases first from 2.2 to 1.9 A due to the depletion (3 torr), then to 1.6 due to the phase mismatch induced by the free electrons at 15 torr, and finally to 1.5 A when the defocusing of the laser is taken into account. This is a clear illustration of the phase modulation phenomenon. Because the rising edge of the harmonic pulse is weakly affected by ionization, it is also the case for the blue side of the spectrum to which it is associated. On the contrary, the falling edge, and consequently the red side of the spectrum, are strongly distorted. This results in a blue shift of the central frequency of the spectrum. Note that this phenomenon is different from the blue shift induced by the temporal variation of the free-electron density, and thus of the refractive index (Wahlstrom et al., 1993; Rae et al., 1994). In our conditions, the latter effect shifts the fundamental spectrum by 2.4 A, which results for the 45th harmonic in a shift of only 0.05 A.
c. CONSEQUENCES OF THE PHASE MODULATION The phase modulation studied above can be used to control the temporal and spectral properties of the harmonic radiation. In the time domain, the regular close-toquadratic phase, and the corresponding linear chirp, lead to the possibility of compressing the harmonic pulse with a pair of gratings, as is done in CPA lasers. Schafer and Kulander (1997) studied the single-atom response to very short laser pulses (27 fs), and showed that it should be possible to compress the harmonic pulse to durations below the fundamental period (2.7 fs). We will discuss the short duration problem in the next section, and we concentrate here on the study of the compression of the macroscopic harmonic beam. To simulate the compression, we subtracted a mean quadratic phase from the phase of the spectrum emitted at each point at the exit of the medium. The inverse Fourier transform followed by the spatial averaging give the compressed macroscopic temporal profile. The result of the compression in z = 3 mm for a peak pressure of 3 torr is presented as the solid line in Fig. 21(a). The profile is very narrow, with a FWHM of 10 fs (vs. 66 fs before compression, dashed line). This is close to the Fourier transform limit. The short duration is maintained when the pressure is increased to 15 torr (dot-dashed line). In z = - 1 mm, the com-
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 125 1 .o
2 0.8 In c
(u
+
.E 0.6 -0
a, N
5
0.4
E g 0.2 0
1 .o 0.8 v)
a, c c
.C 0.6 -0
( N u
5 0.4
sE 0.2 0 -80
-40
0 Time (fs)
40
80
FIG. 21. Compressed temporal profiles obtained at 6 X 1014W/cm* and at a pressure of 3 tom (solid line) and 15 torr (dot-dashed line) of neon: (a) in z = 3 mm, (b) in z = - 1 mm. Dashed lines show the corresponding emission profiles before compression when ionization is neglected.
pressed profile for a pressure of 3 torr is even narrower, as shown by the solid line in Fig. 21(b): 6 fs versus 67 fs before compression. Note that on the basis of the spectral width, which is more than three times larger than in z = 3 mm, we would have expected a smaller duration. However, this optimal compression is not reached due to temporal fluctuations in the phase (see Fig. 19) and to spatial inhomogeneities of the phase modulation. Thus phase matching has to be considered carefully in order to get the shortest harmonic pulses. At a pressure of 15 torr, not only is the spectral width reduced (due to defocusing of the laser, see the preceding section), but also the free-electron dispersion introduces a temporally varying phase shift on the harmonic emission. This results in a broader temporal profile of 13-fs FWHM, shown as the dotdashed line. In conclusion, we have shown that it is possible to compress the macroscopic harmonic beam generated by a 150-fs laser to durations below 10 fs. In the frequency domain, it should be possible to compensate for (or increase)
126
Pascal Salieres, Anne L'Huillier, Philippe Antoine, and Maciej Lewenstein 1 .o
ln c al
0.8
.G 0.6 -0 N al
5 0.4
E
22
0.2 0
1 .o
F 0.8 v) 0)
c c
.G 0.6 N al
0.4
E L
P
0.2 0
-6
-4
-2 0 2 Spectral width (A)
4
6
FIG. 22. Spectral profiles of the 45th harmonic generated with a fundamental without chirp (solid line), chirped positively (dashed line) or negatively (dot-dashed line): (a) in z = 3 mm, (b) in z = - 1 nun. Ionization is not taken into account.
the dynamically induced harmonic chirp by introducing an appropriatelydesigned frequency chirp on the fundamental beam. We have calculated the harmonic spectrum generated by a laser pulse of same duration (150 fs) but presenting a quadratic phase in time, such that the fundamental spectrum is broadened to 32 nm, corresponding to a Fourier transform limit of 25 fs. Fig. 22(a) presents the spectra obtained for positive and negative chirps in the case of a focusing 3 mm before the jet. In the preceding section, we have seen that for this position, the dynamically induced harmonic chirp, which is negative, is limited (see Fig. 18), leading to a quite narrow spectrum (0.4 A). It is much smaller than the chirp induced by the fundamental on the harmonic (recall that the fundamental phase is multiplied by the nonlinear order in the polarization). In these conditions, we thus get a broadening of the harmonic spectrum whatever the sign of the fundamental chirp. However, when the fundamental chirp is negative (blue before the red), it adds to the negative dynamic chirp, leading to a 3.4 A width (dot-dashed line). On the other hand, the positive fundamental chirp gets subtracted from the dynamic
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 127
c h q , giving a 2.5 A width (dashed line). Note that the fundamental chirp could be adjusted to reduce the width of the spectrum for this particular position. In z = - 1 mm, the dynamic harmonic chirp is larger resulting in a 2.2-A-wide spectrum (solid line in Fig. 22(b)). It is in fact of the same order as the one induced by the fundamental chirp. When both add (negative fundamental chirp), this results in an extremely wide spectrum (5.6 A, dot-dashed line). Conversely, when they get subtracted (positive fundamental chirp), they compensate each other, leading to a very narrow spectrum of 0.4 A width (dashed line). So far, we have not yet taken ionization into account, which may modify the results in z = - 1 mm at a sufficiently high pressure. At a pressure of 3 tom, only depletion plays a role, and its influence is quite limited (see Fig. 20). The consequences of ionization for a pressure of 15 torr are shown in Fig. 23. The wide spectrum (negative chirp) is reduced to 4.0 A (dotted line) by the same effect that was described in the preceding section: loss of the red part of the spectrum due to a reduced efficiency on the falling edge of the pulse. On the contrary, the “compensated” spectrum (positive chirp) is broadened to 1.1 A (short-dashed). This is certainly caused by phase fluctuations induced by the free-electron dispersion. In conclusion, the regular behavior of the dynamically induced harmonic chirp can be used to control the temporal as well as the spectral properties of the harmonic emission. Concerning the latter, we find a qualitative agreement with the experimental results of Zhou et al. (1996): for negative fundamental chirps, they observe a very pronounced broadening whereas for positive chirps, the peaks I
-6
I
I
-4
-2 0 Spectral width
2
(A)
4
6
FIG. 23. Spectral profiles of the 45th harmonic generated in 15 tom of neon at 6 X 1014 W/cm2, in z = - 1 mm and with a fundamental chirped positively with (and respectively without) ionization: short-dashed (solid line), or negatively with (and respectively without) ionization: dotted (dotdashed line).
128
Pascal Salih-es. Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
remain relatively narrow. However, the experiments were performed in argon at a relatively high intensity, thus with a higher degree of ionization than considered here. Harmonic generation occurs then only on the rising edge of the laser pulse, resulting in significant red shifting or blue shifting (depending on the fundamental chirp) of the harmonic radiation.
D. INFLUENCE OF NONADIABATIC PHENOMENA So far, we have considered harmonic generation by laser pulses of 150-fsduration. The recent advent of lasers of duration 25 fs and less, sufficiently intense to generate harmonics, motivates the investigation of shorter pulse effects. Except for the influence of ionization that depends directly on time, our results are simply scalable to shorter durations as long as the approximations made still apply: for example, going from 150- to 25-fs laser pulses (while keeping the same peak intensity) would result in a shortening of a factor 6 of the harmonic temporal profile and a corresponding broadening of the same factor of the spectrum. What about our approximations? The slowly varying envelope approximation for the propagation of the harmonic field is valid for very short laser pulses as the harmonic period is much less than 1 fs. However, the very broad spectra of the generated harmonics may overlap and interfere, making the propagation calculations more difficult. The main problem, however, is to take into account the nonadiabatic response of the nonlinear polarization to the fast driving field. In our calculations, we consider that the harmonic dipole moment follows adiabatically (or reacts instantaneously to) the intensity distribution of the laser pulse. In particular, we consider that the laser field amplitude does not vary significantly over the optical period. For the very short pulses just mentioned, this may not be true and may result in a distorted atomic response. Such a nonadiabatic effect was invoked by Schafer and Kulander (1997) to explain the phase characteristics of harmonics generated by a 27-fs laser pulse. They calculated the single argon atom response by integrating numerically the time-dependent Schrodingerequation in the single active electron approximation. The plateau region of the spectrum is highly structured, the expected odd harmonics being hard to distinguish. However, harmonics at the end of the plateau are found very broad but distinct, exhibiting a nice quadratic phase. They interprete this phenomenon in terms of the change in laser intensity during the laser period that alters the harmonic generation process (Watson et al., 1995). An electron that enters the continuum while the laser intensity is increasing experiences an additional accelerationbefore returning. This produces a blue shift on the rising edge of the laser envelope. Conversely, electrons ionized after the peak of the pulse are decelerated, returning later, leading to a red shift of the spectrum. In order to investigatethese effects, we calculated the nonadiabaticsingle-atom
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 129
0.75 v
I
N
Y
-10
-5
0
5
10
Time (fs)
FIG. 24. Temporal envelope (lower curves) and phase (upper curves) of the 49th harmonic generated in argon by a 27-fs laser pulse at 3 X 10I4W/cm2: adiabatic (short-dashed lines) and nonadiabatic (solid lines) calculations. The temporal profiles after compression are shown in dot-dashed (adiabatic)and long-dashed (nonadiabatic) lines.
response to the fast driving field in the strong field approximation and compared it with the adiabatic result. We use the same conditions as in Schafer and Kulander (1997), that is, an argon atom interacting with a 27-fs, 810-nm pulse with a peak intensity of 3 X 1014 W/cm2. In these calculations, we neglect ionization as the considered intensity is (slightly) below saturation for this small pulse duration. The 49th harmonic temporal envelopes and phases are presented in Fig. 24. The adiabatic profile (short-dashed line) is very regular with a 7.6-fs FWHM, whereas the corresponding phase (upper short-dashed line) exhibits a close-toquadratic temporal dependence. Indeed, at the considered intensity, the 49th harmonic is in the cutoff region, where the dipole varies regularly with intensity. In particular, the phase depends linearly with intensity, which leads to a temporal quadratic phase close to the top of the Gaussian laser pulse. The nonadiabatic profile (solid) exhibits the same width but is delayed compared to the adiabatic by roughly 1.3 fs, whereas the phase is close to the adiabatic calculation. This shift in time can be related to the physics of the process. In the two-step model, there is one main electron trajectory for the harmonic emission in the cutoff region. The corresponding return time, that is, the time between tunneling and recombination, is about half a period. Because the amplitude of the emission is mainly determined by the ionization probability, this explains the delayed response of the atomic dipole. The spectral envelopes and phases are presented in Fig. 25. The adiabatic
130
Pascal Salikres, Anne L'Huillier, Philippe Antoine, and Maciej Lewenstein
3.0
t
-0 - -5 -
.a
-10%
e
FIG. 25. Spectral envelope (lower curves) and phase (upper curves) of the 49th harmonic generated in argon by a 27-fs laser pulse at 3 X 1014W/cm2: adiabatic (short-dashed lines) and nonadiabatic (solid lines) calculations. The adiabatic phase minus its quadratic component is shown by the dotdashed line.
profile is very broad (1.3 eV FWHM), even more than the nonadiabatic (1.2 eV). The latter is shifted to the red, which is a consequence of the shift in time of the corresponding temporal profile. Indeed, the blue spectral components are associated with the decreasing part of the temporal quadratic phase, and thus are affected by the delay of the rising edge of the temporal envelope. Furthermore, this positive temporal quadratic phase results in a negative spectral quadratic phase, as shown in Fig. 25 (upper lines). The adiabatic phase minus its quadratic component, shown by the dot-dashed line, is almost constant. This operation can be performed experimentally with a pair of gratings, leading to the compression of the harmonic temporal profile. They are presented in Fig. 24. The adiabatic (dot-dashed) and nonadiabatic (long-dashed) profiles are similar with a 1.7-fsFWHM. The compression by more than a factor of 4 results in a corresponding increase in peak intensity. All these profiles are very similar to the ones reported in Schafer and Kulander (1997). In conclusion, the main features in the cutoff region such as broad spectrum and quadratic phase are very well reproduced by an adiabatic calculation. Thus they are the consequence of the intensity dependence of the phase of the adiabatic dipole, and not of a nonadiabatic effect due to the change in intensity during the laser period. In fact, harmonics in the cutoff are created close to the peak of the laser pulse, where the intensity varies slowly. Nonadiabatic phenomena may play a role in the plateau region, where the harmonics are also generated on the edges of the laser envelope. In the cutoff, the main nonadiabatic effect thus seems to be
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 13 1
the shift in time of the atomic response. Note that if we take a sine laser field instead of a cosine, the harmonic temporal profile is slightly modified, exhibiting a double peak structure for the same width. This is an indication that we are close to the limit of validity of the adiabatic approximation. Another important element is the width of the spectrum. It spans one laser photon energy (1.5 eV) on either side of the harmonic frequency. This means that for shorter pulse durations, the spectra of adjacent harmonics will overlap and interfere, leading to a perturbed behavior also in the cutoff region. Note that a nonadiabatic effect has been invoked by Christov et al. to explain the delayed ionization with short laser pulses (25 fs) and the corresponding increase in harmonic plateau extension (Christov et d.,1996).
VI. Future Applications High-order harmonics are currently used in a number of applications such as atomic and molecular spectroscopy, and solid state physics (for a review, see L'Huillier et al., 1995). We focus here on potential applications involving their coherence properties. Indeed, the main message of the studies presented in Sections III-V is that harmonics can be generated in the form of short, but spatially and temporally coherent pulses. Moreover, the properties of such pulses can be to a great extent and on many aspects controlled. Therefore, one can foresee three very interesting classes of possible applications of harmonic pulses: applications in nonlinear optics in the XUV that employ the high refocused harmonic intensities, applications in interferometry that employ spatial coherence properties, and applications in attosecond physics that employ the phase modulation and other phase properties to reduce the duration of harmonic pulses. In this section we present a somewhat speculative discussion of these three classes of applications, as none of them has been realized so far. A. NONLINEAR OPTICSIN THE XUV In Section IV, we showed that the good wavefront quality of the harmonic beam could allow us to refocus it quite easily to very small spots of a few p m in diameter simply using a spherical mirror. Recently, the Saclay group has confirmed this prediction by measuring the size of the focal spot and its dependence on the medium characteristics (Le Dkroff et aZ., 1998). Even better focusing could be achieved by using a Schwarzschildobjective (Kinoshita et al., 1989). If the short pulse character of the emission is preserved, which implies the use of multilayer mirrors to select and refocus the harmonic beam, unprecedented high intensities could be reached in the XUV region. For instance, if argon is used as generating medium, an optimized number of 1 O ' O photons could be generated
132
Pascal Salitres, Anne L’Huillier, Philippe Antoine, and Maciej Lewenstein
around 30 eV. Schafer and Kulander (1997) found that the conversion efficiency at the single-atom level increases roughly as quickly as the laser pulse duration decreases, resulting in the emission of about the same number of harmonic photons. Using the intense 25-fs infrared lasers currently available, an harmonic pulse duration of less than 10 fs could be obtained (note that the generation of subfemtosecond pulses is under consideration, see as follows). The refocused harmonic intensity would thus reach I O l 4 W/cm2,enough to induce multiphoton processes. Some attempts have already been made to observe nonresonant two-photon ionization of rare gases (Bouhal et al., 1997b; Van Woerkom et al., 1997; Kobayashi et al., 1998). They are up until now unsuccessful, except for moderate-order harmonics (Kobayashi et al., 1998), but the rapid progress in short-pulse laser technology (repetition rate, pulse duration and energy) will improve the performances of the harmonic sources and should make possible the observation of such processes in the near future. B. INTERFEROMETRY WITH HARMONICS Interferometry with harmonics is particularly interesting because of their high frequency and short duration. Coherent harmonics of high frequencies would pass many interesting media without significant absorption. Of particular interest are the dense laser-induced plasmas. Short wavelengths are much less sensitive than visible ones to refraction by the large-density gradients involved, and correspond moreover to higher critical densities. Furthermore, due to their short duration, the harmonics may probe dynamics of many systems in a quasi-instantaneous way. Interferometry with harmonics thus appear as a powerful diagnosis for different media, and in particular plasmas. The broad harmonic bandwith (correlated to the short pulse duration) makes it rather difficult to use an amplitude division interferometer (Michelson’s type) at the harmonic frequency, but the good spatial coherence is well suited for wavefront division interferometers (Young’s type). Two kinds of possible interferometry experiments are discussed by experimentalists: in one kind, two correlated harmonic beams are used (like in Zeme et al. (1997)), out of which one may pass and the other does not pass through the medium to be diagnosed; both beams interfere thereafter. However, realization of such experiment may pose difficulties in achieving sufficient spatial and temporal overlap of the two harmonic pulses. The other kind of experiment involves only one harmonic beam that would pass through a region of spatially nonuniform index of refraction, and produce a self-interferencepattern thereafter. Numerous applications of this last scheme in solid state and plasma diagnostics are currently being considered by experimentalists in Saclay. Note finally that the harmonic spatial coherence could be used in holographic or phase-contrast imaging. In particular, the harmonic generation spectrum has
SPATIAL AND TEMPORAL COHERENCE OF HIGH-ORDER HARMONICS 133
now reached the water window (between the K-edges of carbon at 4.4 nm and oxygen at 2.3 nm, Spielmann et al., 1997; Chang et al., 1997b). This region is of particular interest for biological applications as it provides the best contrast for hydrated carbonated structures such as living cells.
C. ATTOSECOND PHYSICS The shortest pulses achieved today (5 fs in the infrared, BaltuSka et al., 1997) are limited by the long period of the radiation. Shorter wavelengths are required for further pulse shortening. High-order harmonics are the most promising way to generate subfemtosecond,that is, attosecond (as), pulses. Different schemes have been proposed to reach these extremely short durations. The first one deals with the relative phase of the harmonics generated in the plateau region (Farkas and Toth, 1992; Harris et al., 1993). If they were emitted in phase, the corresponding temporal profile would consist of a train of pulses separated by half the laser period and of duration in the attosecond range. There is a clear analogy here with mode-locked lasers. However, early calculations of the single-atomresponse showed that the high harmonics were, in general, not in phase, due to the interference of various energetically allowed electronic trajectories leading to the harmonic emission. Recently, Antoine et al. revived this proposal by showing that the propagation in the atomic medium could select one of these trajectories, resulting in the macroscopic emission of a train of ~ 2 0 attosecond 0 pulses (Antoine et al., 1996a). In principle, the proposals based on phase locking require filtering of the phaselocked harmonics from the entire spectrum; in particular they require suppression of the low harmonics from the fall-off region of the spectrum. It is worth mentioning that trains of attosecond pulses could also be generated in laser-induced plasmas from the surface of a solid target. Harmonic generation from solid targets has been intensively studied in recent years both experimentally (Carman et al., 1981; Kohlweyer et al., 1995; von der Linde et al., 1995; Norreys et al., 1996), and theoretically (Gibbon, 1996; Pukhov and Meyer-ter-Vehn, 1996; Lichters et al., 1996) but only very recently have the temporal aspects of such a generation process been studied (Roso et al., 1998). Using a simple model of the oscillating plasma surface, as well as a one-dimensional “Particle-In-Cell” code, these authors have shown that the reflected signal obtained in the case of normal incidence of the driving laser has the form of a train of ultrashort pulses of duration 220 times shorter than the optical period (i.e., -100 attoseconds for the case of TiSapphire laser). This is quite surprising because the harmonic spectrum is monotonically decreasing and does not exhibit any plateau, and moreover, the production of those pulses does not require any filtering. Furthermore, Corkum et al. (Corkum et al., 1994; Ivanov et al., 1995) proposed a way of generating a single pulse by using the high sensitivity of harmonic
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generation in gases to the laser polarization (Budil et al., 1993; Dietrich et al., 1994). By creating a laser pulse whose polarization is linear only during a short time, close to a laser period, the emission could be limited to this interval. Propagation calculations validated this idea (Antoine et al., 1997d). In their original proposal, Corkum et al. suggested using two cross-polarized fundamental beams of slightly different frequencies in order to obtain the desired form of the timedependent polarization. Very recently, the Lund group has realized this idea in a somewhat different way (Wahlstrom et al., 1997),by using a flat birefringent crystal and a short input pulse that is chirped in frequency and polarized at 45 degrees relative to the optical axis of the crystal. The input chirped pulse in such a system is split into two orthogonally polarized chirped pulses delayed in time. When the phase difference between the two polarizations is zero, linear polarization is produced. This enables a single ultrashort time window to be selected for efficient harmonic generation. Moreover, two or more ultrashort time windows with controllable delay may be selected. Another scheme of ultrashort pulse generation has been described in the preceding section, and concerns a single harmonic. The compression of the dynamically induced frequency chirp of a harmonic located at the end of the plateau and generated by a 27-fs laser pulse can result in a pulse of 1.7-fs duration. Moreover, Schafer and Kulander proposed compressing three adjacent harmonics to generate =400 as pulses (Schafer and Kulander, 1997). This is made possible by the fact that harmonics located close to the cutoff exhibit approximately the same frequency chirp. Finally, the last scheme uses very short duration laser pulses to shorten even more the harmonic emission, thanks to the rapid ionization of the medium (less than one cycle). As already mentioned, with 5-fs infrared pulses it has become possible recently to generate soft XUV radiation extending to the K-edge of carbon at 4.4 nm (water window) (Spielmann et al., 1997). Simulations show that the generated pulses could be of sub-500 attosecond duration (Christov et al., 1997). Although the problem of generating attosecond pulses is already difficult, it is even more challenging to consider their detection and applications. So far, the literature on those subjects hardly exists, but theoretical studies have begun. Corkum and his collaborators (Corkum et al., 1997) discuss possible applications of subfemtosecond pulses in physical chemistry. These authors discuss, for instance, Coulomb explosion measurements for molecules in strong laser fields; such measurements allow measuring time-dependent molecular structures. The use of attosecond pulses for such measurements would allow “freezing” the nuclear motion for even highly charged molecular ions by their inertia for the duration of the pulse, and thus achieving much better temporal resolution than is possible so far. An experimental detection of the train of attosecond pulses is equivalent to the detection of phase locking. The latter can be studied for instance in the process of
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two-photon ionization of helium atoms by a filtered harmonic signal consisting of 5-10 harmonics. The ionization yield in such a process should depend on the relative phases between the harmonics (Agostini, 1996). In a series of papers Maquet and his collaborators have investigated the possibility of determining the relative phases between the harmonics and the fundamental field from the above threshold ionization (ATI) spectra. In VCniard et al. (1995) and TaYeb et al. (1996), AT1 in a two-color field (fundamental plus one high harmonic frequency) was considered. The spectra in that case may be strongly affected by quantum interferences between different ionization and electron redistribution channels, and depend on the relative phase between the two fields. In another publication (VCniard et al., 1996) the authors considered AT1 in a mixed multicolor field consisting of the fundamental and several high harmonic components. The detailed study of photoionization spectra provides in this case more or less direct information on the phase differences between successive harmonics. Comparison with experimental results (Agostini, 1996) suggests that indeed harmonics generated in rare gases are phase locked. Another possibility of detection and application of attosecond pulses would be to develop the time-resolved attosecond spectroscopy(TRAS). This could be done in analogy with femtosecond spectroscopy, using a pump and probe attosecond pulses, for example to excite and then ionize a coherent sub-Rydberg wavepacket in an atom. The result of such a pump-probe process should depend on the time delay between the pump and the probe. In the review (Antoine et al., 1997b) we suggested that TRAS could also be realized by mixing an attosecond pulse train (the probe) with the strong fundamental laser beam (the pump). In particular, we have considered harmonic generation by such multicolored fields, expecting that the resulting harmonic signal would depend on the time delay (modulo-one fundamental period) between the train and the pump. Unfortunately, as we already pointed out in (Antoine et al., 1997b), the estimates presented in that paper, based on SFA, are too optimistic. We have checked, using TDSE method, that the dependence of the harmonic signal on the time delay in such a process is much weaker in the experimentally accessible regime of parameters. Nevertheless,the scheme allows at least studying temporal interference between the train of pulses focused in the medium (probe), and the train of pulses generated in the medium by the fundamental laser (pump). This interference can be particularly efficient due to phase-matchingeffects, such as trajectory selection via propagation (Antoine et al., 1996a). Processes other than harmonic generation can be considered in order to make TRAS with attosecond pulse trains or separated attosecond pulses possible. In (Taieb et al., 1997) the authors consider AT1 process by a fundamentallaser pulse that leads to the resonant excitation of dynamically Stark-shifted Rydberg states. Short harmonic pulse (of duration much shorter than the fundamental one) is applied during that process, and its delay with respect to the front of the fundamental
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pulse is controlled. As a result, the highly structured AT1 spectra depend on the delay of the probe, and thus provide the first example of time-resolved spectroscopy of dynamically induced resonances. The quest for other forms of TRAS remains, in our opinion, one of the most interesting challenges of super-intense laser atom physics.
VII. Conclusion We have used a theory of high-order harmonic generation by low-frequency laser fields in the strong field approximation to study the spatial and temporal coherence properties of the harmonic radiation. We show that the intensity dependence of the atomic dipole phase predicted by this model plays a central role in the way phase matching is achieved in the medium. In particular, it introduces an asymmetry relative to the position of the jet compared to the laser focus. When the laser is focused sufficiently before the jet, phase matching is optimized on axis, leading to very regular Gaussian emission profiles. When it is focused in or after the medium, harmonic generation is prevented on axis but can be efficient off axis resulting in annular profiles. Moreover, the dipole phase induces a big curvature on the generated harmonic phase front, leading to quite diverging harmonic beams. Consequently the virtual source of the emission presents a very small size. Temporal fluctuations of the harmonic phase front can degrade the spatial coherence of the radiation, in particular for a focusing after the jet. However, before the jet, the spatial coherence can be very high, indicating that high-order harmonics could be a useful coherent source in the XUV. Induced by the intensity distribution in the laser pulse, the dynamic phase matching results in very regular, narrow temporal profiles for focus positions sufficiently before or after the jet. At intermediate positions, the harmonic profiles are distorted and almost as large as the fundamental. The corresponding spectra are very broad and far from the Fourier transform limit. This is due to the intensity dependence of the dipole phase, which results in a phase modulation of the emitted harmonic field. This phase modulation is all the more important as the peak intensity is high and the harmonic emission is in the plateau region, where the slope of the intensity dependence is the larger. The harmonic beam thus presents a negative chirp, with a rising edge shifted to the blue and a falling edge shifted to the red. In our conditions, ionization is quite limited and affects mainly the falling edge, and thus the red part of the spectrum. The regular phase modulation can be used to control the temporal and spectral properties of the harmonic radiation. Temporally, the chirped pulse could be recompressed by a pair of gratings to very small durations. Spectrally, an appropriately designed chirp on the fundamental can compensate (or increase, depending on its sign) the harmonic dynamic chirp, resulting in a narrow (or extremely
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broad) spectrum. Finally, we show that cutoff harmonics generated by pulses as short as 27 fs can be described by an adiabatic calculation, the nonadiabatic phenomena being negligible. In conclusion, the spatial and temporal coherence properties of high order harmonics are closely related, and depend strongly on the atomic dipole phase. However, the regular behavior of the latter makes it possible to control them by using relatively simple means. These unprecedented coherence properties in the XUV region open the way to a new class of potential applications that are attracting more and more interest. The good wavefront quality, which allows focusing to very small focal spots, together with the short pulse duration could result in the high focused intensities necessary to perform nonlinear optics in the XUV. The high spatial coherence makes it possible to realize XUV interferometry experiments, such as ultrashort plasmas diagnostics. Finally, the harmonic phase characteristics in time could be used to generate attosecond pulses, the shortest ever, opening the way to “attosecond physics.”
VIII. Acknowledgments We acknowledge fruitful discussions with P. Agostini, Th. Auguste, Ph. Balcou, K. Burnett, B. CarrC, P. B. Corkum, T. Ditmire, M. Dorr, M. Gaarde, M. Y. Ivanov, Ch. Joachain, H. C. Kapteyn, K. C. Kulander, A. Maquet, P. Monot, M. M. Murnane, M. D. Perry, A. Sanpera, K. J. Schafer, R. Tdieb, V. VCniard, C.-G. Wahlstrom, and J. Watson.
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ADVANCES IN ATOMIC. MOLECULAR, AND OFTICAL PHYSICS, VOL. 41
ATOM OPTICS IN QUANTIZED LIGHT FIELDS M. FREYBERGER, A. M. HERKOMMER, D. S. K d H M E R , E. MAYR, AND M! P. SCHLEICH Abteilung f i r Qwntenphysik Universitat U r n , Ulm, Germany
I. Introduction ................................................... II. Ante ........................... B. TheHamiltonian
..
.................
A. Photon Statistics from Atomic Deflection . . . . . . . . . . . . . . . . .................. B. Joint Measurements . . . . . . . . . . . . . . . . . C. Influence of Spontaneous Emission . . . , .................. D. Reconstruction of a Quantum Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................ E. Measuring the Phase Operator IV. Atom Optics in Nonresonant Fields . . . . . . . . . . . . . . . . . . . . B. Measurement-InducedLocalization of Atoms
...............
osung
VIII. References
. . .. . . .
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143 145 145 146 148 149 149 151 154
157 160 162 163 164 166 170 170 172 173 174 175
176 176
I. Introduction The interaction of a massive particle with an electromagnetic field has been studied for quite a long time. Kapitza and Dirac (1933) considered the deflection of electrons by a classical standing light field. Following their ideas, Altshuler et al. (1966) and Ashkin (1970) suggested using atoms instead of electrons. A first detailed theoretical description of the momentum exchange between atoms and a classical standing light field was presented by Cook and Bernhardt (1978). In the following years, a variety of experiments (Arimondo et al., 1979; Moskowitz et al., 1985; Martin et al., 1987, 1988) have been performed by different groups 143
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144 M.Freyberger, A. M.Herkommer, D. S. Krahmer, E. Mayr, and W P. Schleich (for a review see for example Kazantsev et al., 1990). In most of these experiments it suffices to describe the electromagnetic field classically, even when the field is contained within a cavity. This is a consequence of the high average photon number in the resonator and the fact that the field tends to be in a coherent statethe state closest to a classical state. The recent progress in cavity QED (Haroche and Kleppner, 1989; Haroche and Raimond, 1994; Raithel et al., 1994; Kimble, 1994; Hinds, 1994) and in particular in creating experimentally nonclassical states of the radiation field (Krause et al., 1987; Rempe et al., 1990; Rempe et al., 1991; Brune et al., 1996a; Davidovich et al., 1996; Raimond et al., 1997) allows us to address new fundamental questions. The observation of the quantization of the electromagnetic energy in a cavity constitutes a direct demonstration of the concept of the photon (Holland et al., 1991; Lamb, 1995; Brune et al., 1996b). Cavity QED is also perfectly suited for studying experimentally the decoherence of a mesoscopic superposition of quantum states often referred to as a Schriidinger cat. The intriguing results provide deep insight into fundamental processes of a quantum measurement (Zurek, 1991; Brune et al., 1996a; Davidovich et al., 1996; Raimond et al., 1997). Besides being a rich and particularly clean experimental tool for investigating the foundations of quantum mechanics, cavity QED also provides deep examples for Gedanken-experiments. In particular the relationship between complementarity and the uncertainty principle has been discussed recently (Scully et al., 1991; Englert et al., 1995; Storey et al., 1994b, 1995; Wiseman et al., 1997). The new subject of atom optics (Kazantsev et al., 1990; Mlynek et al., 1992; Stenholm, 1992; Adams ef al., 1994; Pillet, 1994; Arimondo and Bachor, 1996) combined with the physics of quantized fields opens a new area to study fundamental questions in detail. In particular, we can investigate the interaction of a single atom with a single mode of a quantized cavity field. One important feature of this problem is the atomic deflection pattern containing information about the field state in the cavity (Meystre et al., 1989; Akulin et al., 1991; Herkommer e t al., 1992; Freyberger and Herkommer, 1994; Treussart et al., 1994). An additional measurement on the field aiming to localize the atom in the resonator is another important tool for extracting information, either about the atomic motion (Storey et al., 1992, 1993, 1994a; Rempe, 1995) or about the field (Herkommer et al., 1992, 1994). To resolve the discreteness of a quantized light field via a quantum lens (Averbukh et aZ., 1994; Mayr et al., 1994; Fam Le Kien et al., 1997a) or quantum prism (Domokos et al., 1996) constitutes another interesting possibility offered by the marriage of atom optics and cavity quantum electrodynamics. Because the interaction of atoms with a standing light field may also be used to create an atomic beam splitter (Wright and Meystre, 1990; Sleator et al., 1992a, 1992b; Pfau etal., 1993; Deutschmann et al., 1993; Kunze et al., 1996; Choi et al., 1997), this topic is also relevant for the subject of atomic interferometry (Chebotayev
ATOM OPTICS IN QUANTIZED LIGHT FIELDS
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et al., 1985; Riehle et al., 1991; Kasevitch et al., 1991; Carnal and Mlynek, 1991; Keith et al., 1991; Miniatura et al., 1992; Sterr et al., 1992;Rase1 et al., 1995;Giltner et al., 1995a, 1995b). This presumes that the atom is deflected in a few directions only, which can be achieved in the so-called Bragg regime (Marte and Stenholm, 1992). Here one can also observe oscillations between different diffraction orders, known as the quantum Pendellosung (Meystre et al., 1991), which is, for a quantized field, strongly correlated to the phenomenon of revivals (Eberly et al., 1980). In the following article we will present and highlight some of these ideas. For a more qualitative discussion we refer to Herkommer and Schleich (1997).
11. Ante In this section we lay the groundwork for our study of atom optics in quantized fields discussed in great detail in the following sections. First we introduce our model and the notations and briefly review the essential concepts. Then we present the corresponding Hamiltonian and focus on two cases: (i) The cavity field is resonant with the atomic transition, and (ii) the cavity field is far detuned. We conclude by discussing the Raman-Nath approximation,which is central to the field of atom optics. A. THEMODEL In Fig. 1 we schematically illustrate our model. A two-level atom propagating with velocity v, in the z direction enters a cavity of length L with a single mode of a quantized electromagnetic field. The field is inhomogeneous in the x direction, but for the sake of simplicity we assume it to be uniform in the y direction. We take the longitudinal kinetic energy M u z / 2 to be large compared to the atom-light interaction energy, and hence the interaction with the light field does not change considerably the velocity v, of the atom. Therefore, the z coordinate plays the role of time t = (z + L ) / v ,in the evolution of the state vector describing the transverse center of mass motion of the atom and the field. However, we note that new phenomena occur when the longitudinal kinetic energy becomes comparable to the coupling strength between the atom and the quantized light field. In that case the atomic z motion has to be quantized and the atoms can be reflected (Englert et al., 1991; Battocletti and Englert, 1994) or even trapped (Haroche et al., 1991; Scully et al., 1996)by the cavity field. For a detailed study of this model, see Meyer er al. (1997), Loeffler et al. (1997) and Schroeder et al. (1997). At time t = 0 when the atom enters the cavity at z = -L, the state vector IYr(r = 0)) of the combined system is the direct product of the field state and the
146 M . Freyberger, A. M.Herkommer, D. S. Krahmer, E. Mayr, and W P. Schleich
FIG. 1. Sketch of the experimental setup for atom optics in quantized light fields. An atomic wavepacket passes through an optical resonator and creates a deflection pattern in the far field. The left-hand figures show a magnification of the interaction zones near a node, illustrating the set of optical potentials for the resonant and off-resonant case. The different photon number states of the cavity field reflect themselves in potentials of different steepness. On the right-hand side we show a homodyne scheme to localize the atom in the cavity, as discussed in Section 1V.B. (From Herkommer and Schleich, 1997.)
atomic state. When we expand the field state in photon number states In) and use the position representation to describe the transverse atomic motion, we find Iq(t = 0)) = [field) 8 [atom)
where w, is the probability amplitude of finding n photons in the light field and f ( x ) is the wave function describing the transverse motion of the atom as it enters the resonator. The initial internal atomic state is, in general, a normalized superposition of the atomic ground state 18) and the excited state le) with amplitudes cg and c,, respectively.
B. THEHAMILTONIAN We use the Jaynes-Cummings model (Jaynes and Cummings, 1963; Paul, 1963; Tavis and Cummings, 1968) to describe the dipole interaction between the two-
ATOM OPTICS IN QUANTIZEDLIGHT FIELDS
147
level atom and the single mode of the quantized electromagnetic field. The total Hamiltonian of the atom-field system reads
where$4/(2M) denotes the kinetic energy of the transverse motion of the atom, ri and d t are the annihilation and creation operators of the field mode, and &+, &- , and ezare the Pauli matrices for the two-level atom. The coupling of the atom to the field mode is determined by the spatial mode function sin(kx) of the standing light field of wavenumber k = 2 d A and by the single photon Rabi-frequency g = p%olh,where p is the dipole moment of the atomic transition and So = d h W ( 2 ~ , V )is the so-called electric field per photon. The latter depends on the frequency R of the field mode, the dielectric constant e0, and on the volume of the cavity, which makes it possible to change the value of goby changing the size of the cavity. The longitudinal velocity u, of the incoming atom, combined with the longitudinal profile of the light field, defines the function S(t), determining how the interaction of the atom with the field is switched on and off. For our study we use the case of a rectangular beam profile, that is, 8(t) =
1 0
for01tST else
where the interaction time 7 = Llu, is the time the atom needs to cross the resonator of length L. In an interaction picture defined by the unitary transformation
and in the rotating wave approximation (Walls and Milburn, 1995),the interaction Hamiltonian for 0 5 t 5 7,becomes p 2
1
fi = 2M + -hAi?, + 2
hg sin(kx)(&+d + &-fit)
(3)
where A zs w - R is the detuning between the mode frequency and the atomic transition frequency w . In this article we concentrate on the two limiting cases with either exact resonance, that is, A = 0, or large detuning. For the first case we find from Eq. (3)
148 M. Freyberger, A. M. Herkommer, D.S. Krdhmer, E. Mayr, and U! P. Schleich whereas in the second case we can replace the Hamiltonian, Eq. (3), by the effective Hamiltonian (Brune et al., 1990) =
a,” + -1t i A a z + 2M 2
where 3 indicates the mean photon number in the field. Note that the Hamiltonian Eq. (5) changes neither the photon number in the field mode nor the internal state of the atom. In other words, there is no exchange of energy between the field and the atom, and we may therefore call it a quantum-nondemolition(QND) Hamiltonian (Braginsky e t al., 1977; Unruh, 1978; Caves et al., 1980; Braginsky and Khalili, 1996). However, the phases of the two subsystems do change (Holland e t ul., 1991). C. THERAMAN-NATH REGIME In general the Schrodinger equations corresponding to the full Hamiltonian, Eq. (4)or (3, cannot be solved in terms of simple analytical expressions.For this reason we restrict ourselves in Sections 111 and IV to the Raman-Nath approximation (Raman and Nath, 1933; Born and Wolf, 1970). In this regime we can neglect the transverse kinetic energy@,2/(2M),compared to the interactionenergy h g f i for the resonant case, or h g 2 i i / A for the off-resonant case, respectively. We estimate the momentump,, which is transferred to the atom during the interaction time T by the number of photon exchange processes times the photon momentum, that is, for the resonant case by p , g f i T ( h k ) and for the off-resonant case by p , = g2AT(hk)/A.Hence we find the upper limit for the interaction time
below which our calculations are valid (see also Cook and Bernhardt, 1978). Here w, = hk2/(2M)is the recoil frequency of an atom with mass M emitting or absorbing a photon with momentum tik. We will use the Raman-Nath approximationwhen we consider in Section I11 the deflection of atoms by a resonant quantized light field. A discussion beyond this approximation is given in Section V, where we show that there still exist some limiting cases with analytical solutions. For a numerical treatment of this problem for a classical light field without the Raman-Nath approximation, we refer to Schumacher et al. (1992).
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111. Atomic Deflection by a Resonant Quantum Field In this section we investigate the deflection of an atomic wave by a resonant quantized electromagnetic field. We derive an analytical expression for the momentum distribution of the atoms and show that it is strongly correlated to the photon statistics of the field. We show that a joint measurement of the atomic momentum and an appropriate field variable allows us to extract this photon statistics. Furthermore, we demonstrate that the momentum distribution of atoms scattered at a node of the standing wave follows precisely the photon statistics of the field. A. PHOTONSTATISTICS FROM ATOMIC DEFLECTION According to Eq. (4) we describe the resonant interaction of the atom with the field in the Raman-Nath regime by the Jaynes-Cummings Hamiltonian
A
= fig sin(kx)(C+ ci
+ ~-ci+)
(7)
In order to gain insight into the dynamics of the system it is most useful to introduce the eigenstates
(8) of this Hamiltonian, that is, the so-called dressed states. Note that in the basis of these dressed states the Hamiltonian, Eq. (7), is equivalent to an effective resonant potential via the relation
A I+,
n ) l x > = + f i g G sin(kx)
I+,
n)lx)
(9)
In Fig. 2 we illustrate this resonant potential for two dressed states with different photon numbers. The height of the potential scales with the square root of the photon number and the sign is different for the plus or minus dressed state. We express the initial state, Eq. (l), in terms of these dressed states
where we have assumed that the atom enters the resonator in the ground state Ig). Using Eq. (9) we obtain the quantum state after the interaction
150 M.Freyberger, A. M.Herkommer, D. S. Krahmer, E. Mayr, and W P. Schleich which contains the complete information about the atom and field system. Here we have introduced the resonant interaction parameter K = g7. Note that the interaction of the atom with the field has created a highly entangled state. As a consequence, a measurement on one subsystem provides information about the other subsystem. In order to demonstrate this interplay between atomic and field dynamics we first consider a rectangular transverse distributionf(x) of the atom, which is constant over many wavelengths of the standing light field, that is, a plane atomic wave. When we project the momentum eigenstates Ip) onto Eq.(1 1) and perform the trace over the field and the internal states of the atom, we find for the momentum distribution W( p) of the atom (Herkommer et al., 1992)
x m - c Iw,l’Jmm m
W(P)=
e=
m
(12)
n=O
--m
Here p = p/(hk)is the momentum scaled in units of photon momenta and Jd3are Bessel functions. Note that the spatial periodicity of the standing wave leads to discrete atomic momenta with a spacing of hk. We emphasize that similar expressions appear in other contexts, for example, in classical optics for the scattering of light by ultrasonic standing waves, as discussed by Berry (1966) and Born and Wolf (1970), or in the free electron laser (Bambini and Stenholm, 1979). Our situation is slightly different from the classical Kapitza-Dirac effect, as here the field is quantized and each photon number contributes separately to the diffraction pattern. We may therefore call it the quantum Kapitza-Dirac effect.
-‘IT
0
‘IT
kx +
FIG. 2. Resonant potentials in the dressed state picture, Eq. (9), for two different photon numbers.
ATOM OPTICS IN QUANTIZEDLIGHT FIELDS
151
FIG. 3. Photon statistics of cavity field from deflection pattern. The photon statistics of a squeezed state with squeezing parameter s = 50 and displacement parameter a = 10 (lower curve) and its readout via the momentum distribution of deflected atoms with interaction parameter K = 110. The curve p, rp = 0).defined by Eq. (13, corresponds to a joint measurement of the momentum of the atom and the phase rp = 0 of the field, whereas the distribution W ( p ) , given by Eq. (12), ignores the field phase. The top curve, Wmask(p),Eq. (17). gives the momentum distribution of atoms filtered by a slit of width Ax = All0 placed at a node of the standing wave. The joint measurement strategy gives an adequate readout, whereas ignoring the field phase results in a less effective readout as well as additional rapid oscillations. The momentum distribution obtained by the mask is an exact image of the photon statistics.
w(
We note that due to the sharp maxima of the Bessel functions at @ = KG there is a mapping of the photon statistics onto the momentum distribution of the atoms (Akulin et al., 1991).Therefore, we gain information about the photon statistics of the field inside the cavity by observing the momentum distribution of the atom. As an example we show this mapping in Fig. 3 for a squeezed state, which is defined by the Fock coefficients (Schleich et aL, 1987, 1989)
with squeezing parameters, displacement a, and the Hennite polynomialsH,. We note that indeed the momentum distribution contains some remnants of the photon statistics. However, it also has washed out details and has added more structure.
B. JOINT MEASUREMENTS So far we have ignored all the information stored in the field by tracing over the field as well as the internal atomic variables. We can considerably improve the readout of the photon statistics by making use of the correlation between atomic
152 M. Freyberger. A. M . Herkommer, D. S.Krahmer. E, Mayr. and W P. Schleich and field variables. This can be done, for instance, by replacing the simple measurement of the atomic momentum by a joint measurement strategy (Storey et al., 1992)on atom and field. To get insight into this we calculate the conditional probability amplitude of finding the atom with momentum p provided the field is in a given reference state. We use the London phase state (London, 1926, 1927; Schleich and Barnett, 1993)
as a reference state and project onto the state vector I*(T)), Eq. (11). Then the joint probability for finding the momentum p and the phase p of the field reads (Herkommer et al., 1992) m
-
w ( p , Sp)
=
1 21T Y = - - m
a(@ - 4 )
W,f2-inp./p(Kfi)
/i
(15)
Note that in contrast to Eq. (12) the probability W(p , cp), Eq. ( 1 3 , contains quantum mechanical interference terms w,w:, instead of the probabilities I w, 12. For real coefficients w, the London phase state I cp = 0) provides the best readout of the photon statistics. We illustrate this in Fig. 3, which shows the momentum distributions w ( p ,cp = 0), Eq. (15), and W ( p ) ,Eq. (12), for the highly squeezed and displaced state. We note that follows precisely the photon statistics I w, 12, shown in the lower curve of Fig. 3, whereas only a few features of I w, survive in W This is a consequence of the fact that the kernel J @ ( ~ f i of ) the sum Eq. (15) acts more like a delta function than the kernelJi(Kfi) in Eq. (12), which extends over a broader region. The physical meaning of the conditional probability p, q = 0) is that from all scattering events we select only those atoms that have not altered the phase of the initial field state. These are the atoms that have passed through the nodes of the field. We illustrate this by calculating the joint position distribution w ( x , q) to find the atom at position x, irrespective of its internal state, and the phase cp of the field. From Eq. (1 1) we obtain
w
l2
w(
Figure 4 shows this probability for the same parameters as given in Fig. 3 and proves that the conditional position distribution w(x, cp = 0) is indeed sharply peaked at the nodes of the light field. Note that this localization scheme is very similar to the off-resonant setup of Storey er al. (1992), which we will discuss in more detail in Section N.B.
ATOM OPTICS IN QUANTIZED LIGHT FIELDS
153
. I
I
0
-7r
kx +
FIG. 4. The conditional position distributionw ( x , (p = 0), Eq. (16), for the same parameters as in Fig. 3 is sharply peaked at the nodes of the light field.
We have seen that the joint measurement procedure plays the role of a spatial filter. A mechanical slit or mask placed in front of a node of the electromagnetic field works in a similar way and therefore also gives an adequate readout of the photon statistics. Indeed, when we take the width of the atomic wavepacket to be much smaller than the wavelength A and center it at a node of the field, that is, xo = 0, we can replace the sinusoidal potential in Eq. (1 1) by its linear expansion sin(kx) = kx. In this case we find for the momentum distribution of the scattered atom Wmask(@) =
m
5c 1
n=O
IW,I*{WO(P
-
K G )
+ wo(p +
K
V
a
(17)
where Wo(p) is the momentum distribution of the atom as it enters the light field. The top curve of Fig. 3 shows this momentum distribution for atoms sent through a slit of width Ax <
154 M.Freyberger, A. M. Herkommer, D. S.Krahmer, E. Mayr, and W P.Schleich Hamiltonian but could also be applied to the off-resonance case discussed in Section IV. C. INFLUENCE OF SPONTANEOUS EMISSION
So far we have only considered the coherent evolution of an atom interacting with a quantized field and have neglected spontaneous emission of the atom. For the case of a resonant interaction between atom and field, this approximation is not always applicable, as experimentally demonstrated by Gould et al. (1991). In this section we take spontaneous emission into account and present an analytical expression for the momentum distribution of an atom scattered at a node of the standing field. Tanguy et al. (1984) have discussed the motion of a spontaneously emitting atom in a classical field. We follow their lines and write the equation of motion
for the density matrix fi of the combined system of atom and quantized field mode where the coherent part is governed by the Hamiltonian Eq. (7). The Liouville operator
L,,,fi
= y(2&-fi&+
-
e+e-fi - fie+&-)
(19)
describes the irreversiblespontaneous relaxation of the atom, where y is the spontaneous emission rate of the atom into modes other than the cavity mode (Sargent et al., 1974; Walls and Milburn, 1995; Scully and Zubairy, 1997). Let us take the case of an atomic wavepacket of width much smaller than the wavelength A, that is, kAx << 1, passing through the node of the light field. This assumption allows us to replace the sinusoidal potential in Eqs. (7) by its linear expansion sin(kx) kx. There exists a regime (Herkommer et al., 1994) where one can monitor the coherent and relaxation dynamics in the momentum distribution of the scattered atom. This regime is characterized by two assumptions: (i) The momentum acquired by the atom during the coherence time is larger than the spread of the initial wavepacket, and (ii) interaction time T is not too large so that the typical number of spontaneously emitted photons remains smaller than the average number of photons in the field. Under these conditions we find (Herkommer et al., 1994) for the momentum distribution
ATOM OPTICS IN QUANTIZED LIGHT FIELDS
155
Here the scattering part of the kernel reads
and the spreading part is
with
Here loand I , denote modified Bessel functions and Wo(p ) is the initial momentum distribution of the atomic wavepacket. The term %p) corresponds to atoms that have not lost their coherence with the field and are therefore deflected according to the gradient of the optical potential inside the resonator. This gradient leads to displaced images of the initial momentum distribution, which are weighted with the photon statistics I w, l2 and are cenThis allows us to read off the photon statistics from the tered at I p I = KG. momentum distribution of the atom and thereby to monitor the coherent dynamics. For small interaction times y r = 1, this diffractive part dominates and in the limit y + 0 we obtain the result, Eq. (17), of the previous section. The term %!fP) represents atoms that have lost their coherence with the field due to spontaneousemission. As a result the force changes sign randomly and the atoms perform a random walk in momentum space. These atoms are therefore deflected by an amount Sp = KGZ, where z is the difference of the relative times spent in the symmetric and antisymmetric dressed state and hence I z I < 1. Note that in the limit y r >> 1, the distribution 9(z) given by Eq. (23) reduces to a Gaussian 9 ( z ; y r ) = exp{-yrz2/4}. Therefore, the probability of finding the change of the momentum Sp is given by the Gaussian distribution eXp{- S p 2 y r / ( 4 n ~ 2 )of} Brownian motion. Figure 5 shows the evolution of the momentum distribution for an initially Gaussian wavepacket centered at a node of the light field. The quantum field is again prepared in the squeezed state Eq.(13). As the interaction time increases we note a transition from the diffractive to the diffusive regime, which has also been investigated numerically by Tan and Walls (1991). The figure also shows that the photon statistics are reproduced at the edge of the diffusive background for sufficiently long interaction times. Note that this image of the photon statistics is obtained for times much longer than the spontaneous decay time. This is due to the
~‘m
156 M . Freyberger, A. M. Herkommer, D. S. Krahmer, E. Mayr, and W! P. Schleich
W(P) x 10-2
1
n-
-300
-150
0
150
300
300
600
Y,
-600
-300
0 P
I
600 FIG. 5. Momentum distribution of a spontaneously emitting atom passing through a node of a squeezed displaced light field for different interaction times. The distribution consists of a diffusive part corresponding to the relaxation dynamics and a diffractive part, which reflects the coherent dynamics and images the field photon statistics. On the bottom we magnify the right edge of the momentum distribution shown in the middle and compare it to the photon statistics of the squeezed light field. Note that even the discrete structure of the photon statistics can be resolved. Here we have chosen the interaction parameter Kl(y7) = 15, the squeezing parameter s = 20 and the displacement parameter (Y = 5. The initial wavepacket has a width A @ = 4. (From Herkommer et al., 1994.)
ATOM OPTICS IN QUANTIZED LIGHT FIELDS
157
fact that even though the area under the diffractive part is decaying exponentially, one can still trace it at the edge of the smooth diffusive background, because the diffusive part is by itself exponentially small there. We conclude by noting that the analytical expression for the momentum distribution of the deflected atom allows us to monitor the transition of the dynamics of the system from the coherent evolution to the relaxating regime.
D. RECONSTRUCTION OF A QUANTUM FIELD In Section II1.A we have shown that it is possible to measure the photon statistics of a quantum field by scattering atoms at a node of the field. Can we extract even more information about the quantum state of light (Freyberger et al., 1997; Schleich and Raymer, 1997; Leonhardt, 1997) in a cavity? Nonlinear atomic homodyne detection (Wilkens and Meystre, 1991) is one answer to this question. This concept shows how to uncover the Wigner characteristic function via the ionization probability of a two-level atom. We give a different answer in this section. We present a measurement scheme (Freyberger and Herkommer, 1994) that is based on the deflection of atoms that are initially prepared in a superposition of their ground and excited state. The superposition of the two atomic levels invokes correlations between two neighboring Fock states, which manifest themselves in the momentum distribution of the deflected atom. This momentum distribution in connection with a suited measurement strategy enables us to reconstruct each individual coefficient w, of the field quantum state I $) = E n w,ln). For a different atom optics scheme we refer to Schneider et al. (1 997). In our model a de-Broglie wave of two-level atoms enters the quantized standing field in a superposition of ground state and excited state as illustrated in Fig. 6. Hence the initial state of the combined system of atom and field reads
where the normalized position distribution lf(x)I2 results from a narrow slit of width A x << A, centered around x = 0. A classical field before the resonator prepares the internal structure of the atom (Sargent et al., 1974). We emphasize that this coherent superposition of the ground state 18) and the excited state l e ) with an adjustable relative phase 40 is crucial for the envisioned state reconstruction. The initial state evolves in the time interval 0 5 t 5 T according to the resonant interaction Z?, Eq. (4). Here we use again the Raman-Nath approximation and neglect the kinetic energy during this time interval. For times t > T the atom is a free particle. That is, the spatial distribution of the atoms on a screen far away from the resonator reflects the momentum distribution of the atoms at t = 7,that is, right after their passage through the field.
158 M.Freyberger, A. M.Herkommer, D. S. Krahmer, E. Mayr, and W P. Schleich
FIG. 6. Quantum state reconstruction using atomic deflection: data collection. A classical field prepares a two-level atom in a superposition Ig) + ei*l e) of the ground state Ig) and the excited state Ie). The atom thus-prepared passes a narrow slit confining it to a region Ax << A centered around the node of the standing light field at x = 0. After the resonant interaction with the field, the deflected atom is detected on a screen in the far field. We reprepare the field and the atom and repeat the experiment many times. The atomic position distribution on the screen in the far field then reflects the momentum distribution of the atoms right after the interaction zone. (From Freyberger and Herkommer, 1994.)
Following the reasoning of Section 1II.Awe can calculate the probability W( p ) for finding an atom with scaled momentum p = p/(hk) and arrive at (Freyberger and Herkommer, 1994)
The initial momentum distribution Wo(p ) describes the atoms before they enter the field. The expression, Eq. (25),for the momentum distribution W( p )clearly demonstrates that the interferingprobability amplitudesw, and w, contribute to W( @). But how should we analyze W( p ) to bring out the important interference terms w X - w,, which conserve the relative phase between w, and w, - To answer this question we consider a spatial distributionf(x), which is a normalized Gaussian with width Ax much smaller than the wavelength 27dk of the field, that is, Ax << 27dk. Its Fourier transform yields the initial momentum distribution
-,
ATOM OPTICS IN QUANTIZED LIGHT FIELDS
159
We insert this expression into Eq. (25) and find the momentum distribution W(p ) after the interaction with the quantized field. In Fig. 7 we compare W(p ) for different values of the interaction parameter K to the bare superposition terms I w, ?eipw,-, 12. The initial state of the field in this example is a coherent state with w, = e-l'12'2an/fl and amplitude (Y = 1.5 exp(id6) and we have chosen a phase 4p = 0. This comparison clearly indicates that for a large value of K we can distinguish the different peaks at = ? GK. Their heights give us directly the value of I w, ?ei+w,- 12, because the contributions of the peaks in the neighborhood are exponentially small, as apparent from Eqs. (25) and (26). Consequently, for large values of K the factor Wo( ? KG) acts almost like a delta function. For @ > 0 and @ < 0 it selects the two sequences of peaks
for n
2
1. The height of the central peak
kAx 0) = --Iw0l2 2v5 reflects the vacuum overlap of I +). Now when we take two corresponding peaks from both sides for the same n value and subtract them, we directly find the interference term
W(@
=
FIG. 7. Quantum state reconstruction using atomic deflection: data analysis. The momentum distribution W(p) in arbitrary units for three different values of the interaction parameter K = g7 compared and contrasted to the probabilities Iw, 2 e % - , l2 shown at the points @ / K = ?fi. We compare the left side of W(p),for p < 0, to the probabilities I w, - ei*w,-, l2 and the right side, for p 2 0 , to the probabilities I w, ei*w,-, 12. For large K values W(p ) follows the peak structure of these probabilities, whereas for a decreasing K the discrete character gets lost. The initial field is a coherent state with a = 1.5ei"I6.The atoms enter the light field through a slit with Ax = All0 and we have chosen the phase cp = 0.
+
160 M . Freyberger, A. M . Herkommer, D. S. Krahmer, E. Mayr, and W P. Schleich
:-
The real part of w I w, emerges for the phase (o = 0 of the classical field, whereas for (o = d 2 this difference measurement is proportional to the imaginary part. Consequently, we reconstruct the whole sequence of products w w, via a simple evaluation of the momentum distribution, which is in fact the position distribution on a screen far away from the resonator. Note, however, that the relevant distributions follow from two ensembles of atoms: The phase (o = 0 of the classical field defines one ensemble and (o = 7 ~ / 2the other. Each atom in these ensembles interacts with a quantized field, which is in a state identical to the original quantum state I $). Starting now from the height of the central peak, Eq. (28), we immediately find I w oI, that is, the vacuum overlap without its phase. Therefore, the hierarchy w 8 w I , wTw2,. . . allows us to derive all the coefficients w,exp(-i@,), which are the original coefficients up to the phase of w o = I w o 1 exp(ia0). Via this measurement strategy we have reconstructed the entire state I $) up to a common phase factor. When we simulate this procedure with the help of Eqs. (28) and (29), we can visualize the reconstructed state by a Wigner function (Hillery et al., 1984)
:-
We show such a reconstruction for the example of a coherent state in Fig. 8. Note that even for K = 10 (Fig. 8c), where the peaks of W ( p )are already washed out (Fig. 7), the reconstructed Wigner function reveals almost the exact Gaussian (Fig. 8a). E. MEASURING THE PHASE OPERATOR Even in the case of a diffuse peak structure in W( p),we may extract information about the field from the momentum data. This strategy results in the detection of an important property of the quantized electromagnetic field: The expectation value
h
+
of the phase operator ei& = C.;=,ln)(n 1 I (Carruthers and Nieto, 1968; Pegg and Barnett, 1988; Schleich and Barnett, 1993), which offers the possibility of deducing the phase uncertainty (LBvy-Leblond, 1976) of the quantized field. Our
ATOM OPTICS IN QUANTIZED LIGHT FIELDS
161
2
0
X
0
2
40
2
4
FIG. 8. Reconstruction of the Wigner function W(&, ,y) of a coherent state la)with the amplitude a = 1.5e'"". We compare the contour lines of the exact Wigner function (a) to three reconstructed Wigner functions that we have calculated via Eqs. (28) and (29). For a small interaction parameter, K = 5 , the reconstructed contour lines are strongly distorted (b), but already for an intermediate value K = 10 the original Gaussian is almost recovered (c). Finally, for K = 40 the reconstructed function (d) is indistinguishable from the exact function. (From Freyberger and Herkommer, 1994.)
measurement proposal consists of detecting all the atoms with positive momenta p and all the atoms with negative momenta separately. That is, we count the number N , of atoms with $ ,I 2 0 that reach the screen in the far field in the region x 2 0 and we count the number N - of atoms with p < 0 in the region x < 0. The relative difference of the two parts with respect to the total number of atoms N,, reads
/
m
\
Here we have introduced the complement of the error function (Gradshteyn and Ryzhik, 1965)
162 M. Freyberger, A. M. Herkommer, D. S. Krahmer, E. Mayr, and W.I? Schleich erfc(6,) =
5 1;
exp[-x21&
with the argument 6, = G K k A x . In the limit of becomes exponentially small and we obtain
(33)
6, >> 1 the function erfc(5,)
Hence we perform one experiment with a classical field that prepares the atomic meter system in a superposition with 4p =0 - and the difference in atomic counts in the far field yields the real part of (I)I e i4 I $). Then we repeat the experiment with a phase 4p = 7r/2 and find the imaginary part. Note that this analysis of the momentum data also works for a strongly washed-out peak structure in W(@). We conclude this section by noting that the presented measurement scheme is completely different from quantum state measurements based on photon counting. These approaches reconstruct quasi-probability functions like Wigner functions or Q functions via optical homodyne techniques. Because Wigner or Q functions contain the complete information about the density matrix of the system (Hillery et al., 1984), such measurements indeed reveal the complete state of the system (Bertrand and Bertrand, 1987; Vogel and Risken, 1989; Royer, 1989; Schleich and Raymer, 1997). Indeed, recently optical homodyne tomography has been applied successfully (Smithey et al., 1993; Schiller et al., 1996; Breitenbach et al., 1997) to reconstruct experimentally quasi-probability functions of several nonclassical states of the quantized light field. For a more detailed review of state reconstruction of light and matter fields, we refer to the articles in the special issue entitled Quantum State Preparation and Measurement edited by Schleich and Raymer (1997).
IV. Atom Optics in Nonresonant Fields Quantum nondemolition (QND) schemes (Braginsky et al., 1977; Braginsky and Khalili, 1996; Unruh, 1978; Caves et al., 1980) rely on a nonresonant interaction between the atom and the field that does not provoke an energy exchange between the two subsystems. Such interactions offer the possibility of measuring the photon number using a phase-sensitive detection of the atom (Brune et al., 1990 and 1992) or by atomic deflection (Aspect et al., 1988, Holland et al., 1991). Because the interaction does not change the internal atomic state, an atom initially in the ground state will remain there, and therefore spontaneous emission is not a problem.
ATOM OPTICS IN QUANTIZED LIGHT FIELDS
163
A. MOMENTUM DISTRIBUTION OF DEFLECTED ATOMS In this section we consider the deflection of a two-level atom by an off-resonant standing field. For large detuning A we describe the interaction of the field and the atom by the effective Hamiltonian, Eq. (5) 1 fi = -fiAeZ + 2
in the Raman-Nath regime. Note that the states lg, n ) and I e, n - 1 ) are eigenstates of this Hamiltonian because
Thus, as in the resonant case, we associate an effective potential with this Hamiltonian. These effective potentials are illustrated in Fig. 9 for two different photon numbers. The period of the sinusoidal potentials for the ground and excited states of the atom is twice the period of the resonant case. Therefore, we expect similar features to appear in the momentum distribution of a deflected atom as in the resonant case. Using Eq. (36) we can easily solve the corresponding Schrodinger equation for the initial state, Eq. (1) with cg = 1 and c , = 0. We find for the quantum state after the interaction
FIG. 9. Off-resonant potentials, Fq. (36), for the atomic ground and excited states, for two different photon numbers.
164 M.Freyberger, A. M . Herkornrner, D.S. Krahrner, E. Mayr, and U! P. Schleich
where we have introduced the interaction parameter T = g27/(2A). We again consider a rectangular transverse distributionf(x) of the atom, which is constant over many wavelengths of the standing light field. By projecting a momentum eigenstate Ip) onto Eq. (37) and taking the trace over the field states we obtain the momentum distribution m
W(@) =
c
e=
-m
x m
&@ - 2l)
lwA2 J2,,,(rln)
(38)
n=O
where we have again scaled the momentum p = p/(hk) in units of photon momenta. Note that in contrast to Eq. (12) the atomic momentum is now discretized in steps of 2hk. This is due to the virtual two-photon processes during the offresonant interaction of the atom with the field or, in other words, due to the doubled period of the effective potential. As discussed in Section 1II.A in the resonant case, we can also read out the photon statistics of the field mode in the off-resonant case when we take the width of the atomic wavepacket to be much smaller than the wavelength h of the light field. Indeed, when we center this wavepacket around the position where the gradient of the potential assumes a maximum, that is, at xo = A/8, we again can approximate the sinusoidal potential in Eq. (37) by its linear expansion cos(2kx) = -2kx for x = xo. In this case we find for the momentum distribution of the scattered atom
where W o (p)is the momentum distribution of the atom as it enters the light field. This momentum distribution reproduces precisely the photon statistics similar to expression (17) obtained for the resonant case.
B. MEASUREMENT-INDUCED LOCALIZATION OF ATOMS How do we localize the atom within a region smaller than the optical wavelength of the light in the resonator? One possibility consists of channeling the atoms in a strong standing light field, as suggested by Salomon et al. (1987) and Balykin el al. (1989). Another possibility is to measure the atomic position via resonance imaging of Raman-transitions,as suggested by Gardner et al. (1993). Storey et al. (1992) investigated a joint measurement performed on atom and field aimed to localize the atom. Marte and Zoller (1992) used a similar scheme, based on an inter-
ATOM OPTICS IN QUANTIZED LIGHT FIELDS
165
action with several resonators, to measure the atomic position in a nondemolishing way. Furthermore, the position of slow atoms passing through a cavity field can be detected by observing the intensity or phase of the light leaking out of the cavity (Quadt et al., 1995; Rempe, 1995; Herkommer et al., 1996 Mabuchi et al., 1996) or by observing internal states via Ramsey interferometry (Kunze et al., 1994; Fam Le Kien et al., 1997; Kunze et al., 1997). Moreover, Autler-Townes microscopy (Herkommer et al., 1997) provides information about the position of an atom. For a comprehensive review on precision position measurement of moving atoms we refer to Thomas and Wang (1995). We now illustrate a measurement-induced localization procedure in a simplified way by applying it to our model. We calculate the conditional probability amplitude of finding the atom with momentum &I provided the field is in a given reference state. For this reference state we use the London phase states, Eq. (14), instead of the quadrature field states used by Storey et al. (1992, 1993). After projecting the states Ix)l p) onto the state vector I ~ ( T ) ) , Eq. (37), we obtain the joint probability distribution
to find the atom at position x and the phase p of the field. For real coefficients w, we choose p = 7,which gives the best localization, and we arrive at
which is a sharply peaked function at the nodes of cos(2kx). We note that a different choice for the value of p leads to a localization at a different position as shown in Storey et al. (1994a). We now compare this localization scheme to the resonant scheme presented in Section 1II.A. From Eq. (16) we obtain, for real coefficients w, and phase cp = 0, the following very similar expression
To be more specific, we now concentrate on a coherent state with mean photon number a 2 >> 1. When we replace the sums in Eqs. (41) and (42) by integrals and substitute for the Poissonian photon statistics its asymptotic expansion (Schleich et al., 1988)
166 M.Freyberger, A. M . Herkommer, D.S. Krahmer, E. Mayr, and W F! Schleich we can perform the integration and find for the off-resonant case -
Woff-res(x,Q = 77)
cc
If(x)I2 e ~ p [ - 2 ~ ~ a ~ c o s 2 ( 2 k x ) l
(4)
and for the resonant case after expanding fiat n = a -
w,,(x, cp
=
0)
cc I f ( X ) 1 2 e - ~ ~ s i n ~ ( k x ) / 2
(45)
These functions are sharply peaked at the zeros of the oscillating functions cos2(2kx) and sin2(kx). We compare the widths S(kx) of the maxima for these two cases and find
and 1 S(kx),, = K
(47)
implies 477a << K, the Because the off-resonance condition IA I >> 2ag, Eq. (3, resolution S(kx),, obtained by the resonant interaction is better than the resolution S ( ~ X ) , ~ -achieved ~~, by the off-resonant interaction. We conclude by noting that the localization of the atoms at certain positions constitutes a virtual grid for the atoms and therefore leads to interference effects in the far-field and in the momentum distribution of the atoms. This has been studied in detail by Storey et al. (1993). Moreover, a similar scheme based on the interaction with a running light wave can be used to create a virtual grid for the atoms in momentum space, as shown by Sleator and Wilkens (1993).
c. FOCUSING OF ATOMIC WAVES In Section 1V.A we have shown that the individual Fock states building up the field state deflect an atomic wavepacket into different directions. In the present section we show that they also focus the wavepacket, which allows us to resolve the individual photon states (Averbukh et al., 1994;Rohwedder and Orszag, 1996; Domokos etal., 1996).This effect is shown schematicallyin Fig. 10. However,before we start our discussion of this quantum lens, we note that focusing of atomic matter waves also manifests itself in the case of classical light fields (Sleator et al., 1992a, 1992b). Moreover, such classical atom lenses have been applied successfully to create nanostructures (Timp et al., 1992; Mc Clelland et al., 1993; Kaenders et al., 1995; Nowak et al., 1996). We again describe the interaction of an atom in its ground state I g ) with the field by the effective Hamiltonian
ATOM OPTICS IN QUANTIZED LIGHT FIELDS
167
FIG. 10. Quantum lens: sketch of experimental setup. A beam of off-resonant atoms propagates along the z-direction and interacts with the light field in the region - L 5 z 5 0. Different Fock states of the cavity field deflect atoms in different directions and focus them at different points. (From Averbukh et al., 1994.)
given by Eq. ( 3 3 , where g(x) is a position-dependent coupling strength. In the cavity we neglect the kinetic energy fifree = @,2/(2M)of the transverse motion of the atom compared to H e f i . Because we assume a constant velocity u, in the z direction, the connection between time t and the z-coordinate is simply given by z = u,(t - T ) , where the interaction time T = L/u, is defined by the length L of the resonator. Therefore, the state vector of the system at the exit of the cavity at z = 0 reads
Outside the cavity the dynamics of the atom in the x direction is governed by and hence the state vector at a position z is given by
afree,
From this expression we obtain the probability oc
m
W(x, z) =
c
n=O
I(x, nl*(z))(* =
c
(wf112Rfl(x,z)
fl=O
of finding the particle at the point with the coordinatesx and z.
(51)
168 M. Freyberger, A. M. Herkommer, D. S. Krdhrner, E. Mayr, and W P. Schleich We evaluate the intensities R, of the atomic partial wave corresponding to the nth Fock state for a Gaussian profile of the atomic beam of width Ax << A centered at x = 0 and find (Averbukh et al., 1994)
(Lr
Here we have used the definitions D,(z)
and
=
[
AxMv,
+ Ax2 (1
-
Ez)’]”’
(53)
withgj = (ajg/axj)lx=oforj= 1,2. Hence the nth partial wave with intensity R,(x, z) is a Gaussian beam propagating in the x-z plane under an angle
en = -arctan(n/~)
(55) with respect to the z-axis. Positive g, result in a positive value for N and thus in negative deflection angles 8,. Negative g result in positive deflection angles. Note that in the framework of our approximation H , << He, << MvZl2, we always have N >> n and hence 10, I << 1. For positive g, the width D, of the partial beam reaches its minimum for positive z and thus the nth partial wave focuses at the point
,
x, = 8, tan
en =
-g,ig,
Hence all foci lie along a straight line parallel to the direction of incident atoms. The net probability W(x, z), Eq. (51). for an arbitrary field state is a superposition of partial Gaussian beams weighted with the photon statistics I w, 12. This results in a multifocal structure for W(x, z) as shown in Fig. 11. In order to resolve the contributions from different Fock states, the widths of the foci have to be narrower than their spacings. A velocity spread of the atoms translates itself into an additional spread of the foci. Therefore, we need an atomic beam of good monochromaticity. Furthermore, in order to guarantee the applicability of our approach based on the nondemolition Hamiltonian, Q.(48). we have to fulfill further conditions concerning transitions to the upper level and the influence of spontaneousemission. For a detailed discussion of all these points we refer
ATOM OPTICS IN QUANTIZED LIGHT FIELDS
169
FIG.1 I. Quantum lens: Contour plot (a) for probability W(x, z) of finding the atom at the point with coordinates x and z. The Gaussian atomic beam centered at x = 0 leaves the cavity at z = 0. The cavity field is in a coherent state with amplitude a = 1. The undeflected and unfocused partial wave R , associated with the cavity vacuum state represents the profile of the incident beam. The deflected partial waves R , , R,, R,, and R , associated with the photon states n = 1, 2, 3, and 4 in the coherent state of the field focus along the line x = -g,/g,. The intensity of atoms along this line is shown in ,) 3 d 2 . (b). Parameters used are g , / ( g , A x ) = 2; ( g , A x ~ / f i ) ( L / u=
to Averbukh et al. (1994) and note that the experimental facilities necessary for this approach are presently available. We also note that the required atomic wavepacket of width Ax << A could be produced by a localization scheme similar to that of the previous section and discussed by Storey et al. (1994a). We conclude this section by noting that a discussion of this quantum lens in terms of the Wigner function and beyond the Raman-Nath approximation can be found in Mayr et al. (1994).
170 M.Freyberger, A. M.Herkommer, D. S. Krahmer, E. Mayr, and U! P. Schleich
V. The Bragg Regime In this section we investigate a more general situation than discussed in the previous sections. First, we allow for nonorthogonal incidence of the atom with respect to the resonator axis, that is, the atom has a non-vanishing initial momentum p x oin the x-direction; second, we do not use the Raman-Nath approximation,that is, we will keep the p,2/(2M) term in the Hamiltonian, Eq. (3). Note that in the Raman-Nath regime only momentum conservation is taken into account. Now the coupling between transverse momenta is limited to those states where both energy and momentum are conserved. A. EQUATIONS OF MOTION
The mechanism of momentum exchange between atom and quantized standing wave comes out most clearly in an expansion of the state vector in momentum eigenstates (Stenholm, 1992; Marte and Stenholm, 1992). Starting from the initial momentump,,, the interaction with the standing wave leads only to a coupling to the componentsp,, + @nk, where @ is an integer. This suggests the ansatz
for the state vector of the system. Here the prefactor is a Galilean transformation to the frame moving with the initial atomic momentum p x oalong the x-axis. The coefficients [ g : Iz and I e; l2 give the probabilities of finding a ground-state and an excited-state atom with momentum p,, + @hk while the field is in the nth and (n - 1)th Fock state, respectively. When we insert this state into the Schrodinger equation corresponding to the Hamiltonian, Eq. (3), we find
Here we have introduced the initial Doppler shift A, = kp,,/M of the atomic motion and the recoil frequency w, = fik2/(2M). Equations (59) contain the whole physics of scattering atoms from a quantized standing light field. In Fig. 12 we illustrate the coupling between different states in the form of an energy diagram. Here we have plotted the bare atomic energies
171
ATOM OPTICS IN QUANTIZED LIGHT FIELDS
I
-6
I
I
I
I
I
-4
-2
0
2
4
P-)
6
FIG. 12. Bare atomic energies E Y , Eq. (60),for different momentum states, but for one fixed Fock state In). For the sake of simplicity we have suppressed the subscript n in the probability amplitudes e f and g g. The arrows indicate the coupling between the initial state and other diffraction orders.
of the various states (without the atom-field interaction), where ej is - 1/2 for the ground state and 1/2 for the excited state. Hence the energies lie on two parabolas, separated by the atomic transition energy hw. Their curvature is determined by the recoil frequency orand their shift in p is proportional to the Doppler shift A,. Only when the energy difference between the initial state and the final state is equal to the photon energy ha is the total energy conserved for this transition and we find a strong interaction, that is, a resonance. With the help of the diagram, Fig. 12, we can now discuss the influence of the different parameters in Eq. (59): One important parameter is the atom field detuning A = w - a, which can be used to control the occurrence of one, two, or higher photon processes over others. For example, on resonance, A = 0, transitions between lower and upper atomic state are enhanced and single photon processes, corresponding to the exchange of momenta in units of hk, dominate. In contrast, far off-resonance only virtual transitions between the lower and upper electronic states take place, and the scattering orders correspond to transverse momenta, which are multiples of 2hk indicating a two-photon process. In general, if a @,,-process is dominant, the important scattering components are multiples of p&k. The other important parameter is the kinetic energy, corresponding to an effective detuning AeH= A, p + w , p *. If it is large compared to the coupling strength g 6 , there will be very little coupling between the incoming beam, p = 0, and other momentum states, which simply means there will be no deflection. This
172 M . Freyberger, A. M.Herkornrner, D. S. Krahrner, E. May& and W P. Schleich implies that interesting situations arise when the detuning AeRis either negligible compared to the coupling, as in the Raman-Nath regime, or when it effectively vanishes for a certain transition between the initial and final state. The last case is usually referred to as Bragg or Doppleron resonances. Note that in the Raman-Nath regime all states are in resonance, whereas in the Bragg regime only a few diffraction orders are resonant. This implies that the Bragg regime is more suited to realizing an atomic beam splitter (Sleator et al., 1992a, 1992b; Pfau et al., 1993; Deutschmann et al., 1993; Giltner et al., 1995a, 1995b; Kunze et al., 1996). A general discussion of this situation for the case of a classical field was given by Arimondo et al. (1981a, 1981b) and Bernhardt and Shore (198 1). A correspondingexperiment was suggested by Pritchard and Gould (1985) and performed by Martin et al. (1988). Later, Glasgow et al. (1991) and Wilkens et al. (1991) discussed the problem using a band theoretical approach. Following these ideas we now consider some limiting cases of the previous equations. In Section B, we investigate a situation similar to the Raman-Nath regime, where we still neglect wr, but consider grazing incidence. In Section C we then discuss the problem in the so-called Bragg regime, where the influence of kinetic energy @ 20, can no longer be neglected and certain types of resonances occur. We will illustrate this regime in Section D for a special case. B. THERAMAN-NATH SOLUTION FOR GRAZING INCIDENCE The analogous case of pure diffractive scattering considered in Section II1.A emerges when the recoil frequency is small compared to the Rabi frequency coupling of the atom-field system, that is, when w, @ << A. @ < gfi. In this case for exact atom-field resonance, A = 0,we obtain from Eq. (59) the following equations of motion d i-gf dt
= @Aogf
gvn + i-(ef+' 2
- ef-l)
As the initial condition we take an atom in the ground state with transverse momentum pxoand a field described by the pure state I r/l) = XnwnIn), that is, g f ( t = 0) = wnSaJ,o
and
ef(t =
0) = 0
(62)
Note that for normal incidence, that is, for A. = 0, these equations are completely equivalent to the ones discussed in Section III. In contrast, for A,, # 0 we prove by direct insertion that the amplitudes
ATOM OPTICS IN QUANTIZED LIGHT FDELDS - i @ A o t / 2 JP
gPn =
(y
for p odd
173
(63)
and
eg
=
{
for p even
0 --wne-ipAot/2
(64)
solve Eqs. (61). From Eq. (58) we therefore obtain the probability m
W(p)=
e = -=
m
S(p - 4)
lwn12J L (=sin(AOt/2) A0
n=O
+
of finding the atomic momentum p = p x o ghk. We note that in contrast to the Raman-Nath expression, Eq. (12), for the momentum distribution under normal incidence the argument of the Bessel functions is now oscillatory. Therefore, the width of the momentum distribution is no longer a linear function of time, as for normal incidence, but is limited at any time by p < 2gfi/A,. This implies that the Doppler shift A, should be smaller than the effective Rabi coupling g f i so as to observe considerable diffraction. The physical interpretation is clear: If the atom has a large transverse velocity p,,/M it is Doppler shifted far from resonance. Therefore, it cannot exchange photons with the field and is not deflected.
C. BRAGGRESONANCES We now turn to the regime opposite to the one of the previous section, that is, we consider the case w , p >> gfi. Again we assume A = 0. Now in contrast to the situation in the Raman-Nath regime, energy conservation plays an important role. From the structure of Eq. (59) we see that the term A, p + w, p * takes on the role of an effective detuning. When this term is large there is no coupling between different diffraction orders, and no diffraction can be observed. However, the coupling term X g G becomes important when the effective detuning A, p + w, 63 vanishes. Then only diffraction orders p that fulfill the resonance condition
A,@
+ o,p2 = 0
are populated. The trivial solution p whereas the second solution reads
=
(66)
0 corresponds to the incoming beam,
174 M. Freyberger, A. M.Herkommer, D. S.Krahmer, E. Mayr, and W! P. Schleich Because g is an integer, this condition can only be fulfilled if pxo is an integer multiple of hk/2. If we start with the initial momentum p , = pxo= - g?ik/2 we only get a coupling to the final momentum p r f = p , g h k = g h k / 2 , which means Ipx,iI = I p , This is nothing else but conservation of kinetic energy. Note that the resonance condition, Eq. (67), corresponds to the Bragg condition in x-ray scattering from crystals.
+
fl.
D. QUANTUM PENDELLOSUNG To gain deeper insight into resonant Bragg scattering we now consider the case pxo = hk/2. For an atom initially in the ground state, only the coefficient g:(O) = w, is nonzero. According to Fig. 12, the amplitude g: is coupled to many other amplitudes. However, for most of these transitions the required energy is larger than the available photon energy. Hence for a qualitative analysis of the dynamics it is sufficient to take into account only the coupling between gg and e ; ' . Using this crude approximation we find from Eqs. (59) the two coupled equations
which have the solution gjj(t) = w, c o s ( q ) e;'(t)
=
w, sin
(Y) -
Hence the amplitudes for the two diffraction orders oscillate with the effective Rabi frequency g f i / 2 . The probability of finding an atom that has not been diffracted is m
w(g
=
0, t ) =
IWJ
n=O
(Tt)
cos2 -
whereas the probability of finding an atom in the first diffraction order reads
This solution illustrated in Fig. 13, is called the quantum Pendellosung and has been derived by Meystre et al. (1991) in the limit of large atom-field detuning.
ATOM OPTICS IN QUANTIZED LIGHT FIELDS
175
1
0.5
0
20
40
60
gf
FIG. 13. Quantum Pendellosung. The population W(@ = 0) of the diffraction order @ = 0, Eq. (69),as a function of the interaction parameter gf.The field was initially prepared in a coherent state with amplitude a = 4, which leads to the characteristic collapse-and-revival phenomenon of the population.
Note that the above expressions also show the typical collapse and revival structure, as it is well-known from the Jaynes-Cummings model and wavepacket dynamics. For a detailed discussion of the properties of these sums, see Eberly et al. (1980), Averbukh and Perel’man (1989), Fleischhauer and Schleich (1993), and Leichtle et al. (1996a, 1996b). We conclude by noting that a more accurate description could be achieved by an adiabatic elimination of higher diffraction orders (Marte and Stenholm, 1992). Moreover, the coherent splitting of an atomic beam into two different diffraction orders can be used to create an atomic interferometer, as discussed theoretically by Wright and Meystre (1990) and realized experimentally by Rase1 et al. (1995) and Giltner er al. (1995a, 1995b).
VI. Conclusion In this paper we reviewed the new field of atom optics in quantized light fields. Here we treat not only the internal degrees of freedom of the atom and its centerof-mass motion quantum mechanically, but have also quantized the cavity field. A simple but experimentally relevant model has allowed us to study various effects. In particular, the Raman-Nath approximation makes it possible to investigate analytically the deflection of atoms from a quantized electromagnetic field. We discussed two important cases: (i) exact resonance of atom and field and (ii) the quantum nondemolition case with large detuning. In both cases the deflection
176 M. Freyberger, A. M.Herkommer, D. S. Krahmer, E. Mayr, and W P. Schleich pattern created by the scattered atoms contains information about the quantum state of the light field. Moreover, we have shown that a joint measurement on the atom and on the field improves the scheme and allows us to read out the photon statistics of the field. A further improvement consists of sending the atoms through the nodes of the field by using a mechanical mask. We have also seen that spontaneous emission of the atom will not immediately destroy the information contained in the momentum distribution of the deflected atoms. Furthermore, it turned out that a measurement on the field results in a reduction of the state vector and thereby in a localization of the atom inside the resonator. In the last section we discussed the evolution of the system beyond the Raman-Nath approximation and found the occurrence of resonances, as well as quantum revivals. So far many experiments on atom optics in classical light fields have been performed. However, to the best of our knowledge no experiment on quantum light fields exists. However, Ton van Leeuwen in Eindhoven ( K r i e r et al., 1994) has built an impressive setup to test these ideas for the first time. He expects to obtain the first experimental results in the year 1998. These experiments will open a new era in this marriage of atom optics and cavity QED.
VII. Acknowledgments We thank V. M. Akulin, I. Sh. Averbukh, V. Balykin, P. J. Bardroff, M. V. Berry, V. B. Braginsky,H. Carmichael,Fam Le Ken, S. Haroche, H. J. Kimble, P. Knight, Ch. Kurtsiefer, K. V. A. van Leeuwen, M. A. M. Marte, B. Mecking, J. Mlynek, P. Meystre, D. O’Dell, T. Pfau, D. Pritchard, M. Raizen, E. Rasel, G. Rempe, S. Schneider, M. 0. Scully, S. Stenholm, D. F. Walls, H. Walther, M. Wilkens, V. P. Yakovlev, A. Zeilinger, and P. Zoller for many fruitful and enlightening discussions during the course of this work. We also express our gratitude to H. Walther for inviting us to write this article and for patiently awaiting the manuscript. This work was partially supported by the Deutsche Forschungsgemeinschaft.
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Mlynek, J., Balykin, V., and Meystre, P. (Eds.). (1992). Optics and interferometry with atoms (Special issue). Appl. Phys. B 54, 319. Moskowitz, P. E., Could, P. L., and Pritchard, D. E. (1985). J. Opt. Soc. Am. B 2, 1784. Nowak, S., Pfau, T., and Mlynek, J. (1 996). Appl. Phys. B 63,203. Paul, H. (1963). Ann. Phys. (Leipzig) 11,411. Pegg, D. T. and Barnett, S. M. (1988). Europhys. Lett. 6,483. Pfau, T., Kurtsiefer, C., Adams, C. S., Sigel, M., and Mlynek, J. (1993). Phys. Rev. Lett. 71, 3427. Pillet, P. (Ed.). (1994). Optics and interferometry with atoms. (Special issue). J. Phys. 114, 1877. Pritchard, D. E. and Could, P. L. (1985). J. Opt. SOC.Am. B 2, 1799. Quadt, R., Collett, M., and Walls, D. F. (1995). Phys. Rev. Lett. 74, 35 1. Raimond, J. M., Brune, M., and Haroche, S. (1997). Phys. Rev. Lett. 79, 1964. Raithel, G., Wagner, C., Walther, H., Narducci, L. M., and Scully, M. 0. (1994). In P. Berman (Ed.), Cuviry quantum electrodynamics. Academic Press (New York). Raman, C. W. and Nath, N. S. (1933). Proc. Ind. Acad. Sci. 2,406. Rasel, E. M., Oberthaler, M. K., Batelaan, H., Schmiedmayer, J., and Zeilinger, A. (1995). Phys. Rev. Lett. 75, 2633. Rempe, G. (1995). Appl. Phys. B 60,233. Rempe, G., Schmidt-Kaler, F., and Walther, H. (1990). Phys. Rev. Lett. 64,2783. Rempe, G., Thompson, R. J., Brecha, R. J., Lee, W. D., and Kimble, H. J. (1991). Phys. Rev. Lett. 67, 1727. Riehle, T., Kisters, T., Witte, A,, Helrncke, J., and Borde, C. J. (1991). Phys. Rev. Lett. 67, 177. Rohwedder, B. and Orszag, M. (1996). Phys. Rev. A 54,5076. Royer, A. (1989). Found. Phys. 19,3. Salomon, C., Dalibard, J., Aspect, A., Metcalf, H., and Cohen-Tannoudji, C. (1987). Phys. Rev. Lett. 59, 1659. Sargent, M., Scully, M. O., and Lamb, W. E. (1974). Laserphysics. Addison-Wesley (Reading, MA). Schiller, S., Breitenbach, G., Pereira, S. F., Muller, T., andMlynek, J. (1996). Phys. Rev. Lett. 77,2933. Schleich, W. P. and Barnett, S. M. (Eds.). (1993). Quantumphase andphase dependent measurements. (Special issue). Physica Scripta T48. Schleich, W. P. and Raymer, M. (Eds.). (1997). Quantum state preparation andmeasurement. (Special issue). J. Mod. Opt. 44, ( I I , 12). Schleich, W. P., Walls, D. F.,and Wheeler, J. A. (1989). Phys. Rev. A 38, 1177. Schleich, W., Walther, H., and Wheeler, J. A. (1988).Found. Phys. 18, 953. Schleich, W. P. and Wheeler, J. A. (1987).Nature 326, 574. Schneider, S., Herkommer, A. M., Leonhardt, U., and Schleich, W. P. (1997). J. Mod. Opt. 44,2333. Schroeder, M., Vogel, K., Schleich, W. P., Scully, M. O., and Walther, H. (1997). Phys. Rev. A 56,4164. Schumacher, E., Wilkens, M., Meystre, P., and Glasgow, S. (1992).Appl. Phys. B 54,45 1. Scully, M. 0..Englert, B.-G., and Walther, H. (1991).Nature 351, 111. Scully, M. O., Meyer, G. M., and Walther, H. (1996).Phys. Rev. Lett. 76,4144. Scully, M. 0. and Zubairy, M. S. ( I 997). Quantum optics. Cambridge University Press (Cambridge). Sleator, T., Pfau, T., Balykin, V., and Mlynek, J. (1992a).Appl. Phys. B 54, 375. Sleator, T., Pfau, T., Balykin, V., Carnal, O., and Mlynek, J. (1992b).Phys. Rev. Lett. 68, 1996. Sleator, T. and Wilkens, M. (1993). Phys. Rev. A 48,3286. Smithey, D. T., Beck, M., Raymer, M. G., and Faridani, A. (1993). Phys. Rev. Lett. 70, 1244. Stenholm, S. ( I 992). Proceedings of the International School of Physics “Enrico Fermi, ” Course CXVffI.E. Arimondo, W. D. Phillips, and F. Strumia (Eds.). (Varenna). Sterr, U., Sengstock, K., Mueller, J. H., Bettermann, D., and Ertmer, W. (1992).Appl. Phys. E54,341. Storey, P., Collett, M., and Walls, D. (1992).Phys. Rev. Lett. 68,472. Storey, P., Collett, M., and Walls, D. (1993). Phys. Rev. A 47,405. Storey, P., Sleator, T., Collett, M., and Walls, D. (1994a). Phys. Rev. A 49,2322.
180 M. Freyberger, A. M. Herkommer, D. S. Krahmer, E. Mayr, and W P Schleich Storey, E. P., Tan, S. M., Collett, M. J., and Walls, D. F,(1994b). Nature 367, 626. Storey, E. P., Tan, S. M., Collett, M. J., and Walls, D. F. (1995). Nature 375, 368. Tan, S. M. and Walls, D. F.(1991). Phys. Rev. A 44, R2779. Tanguy, C., Reynaud, S., and Cohen-Tannoudji, C. (1984). J. Phys. B 17,4623. Tavis, M. and Cummings, F.W.(1968). Phys. Rev. A 170,379. Thomas, J. E. and Wang, L. J. (1995). Phys. Rep. 262,31 I . Timp. G., Behringer, R. E.,Tennant, D. M., Cunningham, J. E., Prentiss, M., andBerggren, K. (1992). Phys. Rev. Lert. 69, 1636. Treussart, F., Hare, J., Collot, L., Lefevre, V., Weiss, D. S., Sandoghdar, V.,Raimond, J. M., and Haroche, S. (1994). Opt. Lett. 19, 1651. Unruh, W. G. (1978). Phys. Rev. D 18, 1764. Vogel, K. and Risken, H. (1989). Phys. Rev. A 40,2847. Walls, D. F.and Milbum, G. J., (1995). Quantum optics. Springer (New York). Wilkens, M. and Meystre, P. (1991). Phys. Rev. A 43,3832. Wilkens, M., Schumacher, E., and Meystre, P. (1991). Phys. Rev. A 44,3130. Wiseman, H. M., Harrison, F. E., ColIett, M. J., Tan, S. M., Walls, D. F.,and Killip, R. B. (1997). Phys. Rev. A 56,55. Wright, E. M. and Meystre, P.(1990). Opt. Commun. 75,388. Zurek, W. H. (1991). Phys. Today, 44, (10). 36.
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 4
ATOM WAVEGUIDES VICTOR I. BALYKIN Institute of Laser Science, University of ElectroCommunications, Tokyo, Japan Institute of Spectroscopy, Russian Academy of Sciences, Moscow. Russia
I. Introduction ................................................... 11. Guiding of Atoms with Static Electrical and Magnetic Fields
............
A. Electrical Field ... B. Magnetic Field . . . III. Evanescent Light Wave ................. ............................... A. Forces on Atoms in 1. Near-Resonant Light Force on Atoms .........................
...............
..................
B. Evanescent-Wave
.............................
...........................
.......................
182 184 184 186 187 187 187 190 190 191 191 193 193 194 195 196 198 202
202
............ ............................. A. ElectromagneticField in Optical Hollow Fiber .................... B. Cylindrical Hollow Fiber as Atom Waveguide ..................... I. Quantum Mechanics of a Cylindrical 2. Losses in Atom Waveguide ................................. a. SpontaneousEmission .......... b. Tunneling to Dielectric Surface. ........................... c. Nonadiabatic Transitions ................................. C. Horn Shape Hollow Fiber .................................. D. Planar Waveguides .......................... A. Gaussian Laser Beam ........................................ B. Laser Light Inside of Hollow Fiber ..................
181
204 207 207 208 210 213 213 219 219 22 1 222 223 223 224 226 234 236 236 237
Copyright 8 1999 by Academic Press All rights of reproductionin any form reserved.
ISBN 0- 12-003841-2/1SSN 1049-250X/99 $30.00
182
Victor I. Balykin C. Dark Spot Laser Beams ....................................... 1. Mode Conversion Method .................................. 2. Computer-Generated Hologram Method 3. Micro-Collimation Technique ............................... ................ D. Atom Guiding with a Standing Light Wave 1. Atom Potential in a Standing Wave ........................... 2. Guiding Time in a Single Potential Well ....................... 3. Experiments with a Standing Wave . . . . . . . . . . . . ..... VI. Experiments with Atom Guiding .................................. A. Atom Guiding with Grazing Incident Mode ....................... C. Atom Guiding with a Donut Mode .............................. VII. Acknowledgments VIII. References ....................................................
238 239 24 1 242 243 244 245 246 250 250 252 255 257 251
I. Introduction The established area of matter-wave optics-electron and neutron optics-is enriched by a new type of optics: atom optics (Balykin and Letokhov, 1989b; Mlynek et al., 1992; Adams et al., 1994a; Adams et al., 1994b; Pillet, 1994; Rempe and Schleich, 1995; Balykin and Letokhov, 1995; Arimondo and Bachor, 1996; Baldwin, 1996). Atom optics, in analogy to neutron and electron optics, deals with the realization of traditional elements, such as lenses, mirrors, beam splitters, and atom interferometers,as well as new “dissipative” elements such as a slower and a cooler, which have no analogy in another types of optics. The important subfield of atom optics is atom guiding. Atom guiding can occur in an analogous fashion to the fiberoptics for light. Optical fibers were first envisioned as optical elements in the early 1960s. Later Kao and Hockham (1966) suggested the possibility that low-loss optical fiber could be competitive with coaxial cable for telecommunication applications. In 1970 Corning Glass Works announced a low-loss optical fiber and today we see a tremendous variety of commercial and laboratory applications of optical fibers. It is predicted that atom guiding also has a lot of promise: Atoms can be guided over a long distance without losses with high spatial accuracy, which opens a novel form of atom deposition in lithography. The spatial resolution of traditional lithography is limited by the diffraction of light waves. The guiding of atoms by means of near field configurationspermits us to overcome these limitations. The state and species selectivity of atom guiding permits us to extract and deliver the chosen atoms from one vacuum chamber to another. When the de Broglie wavelength of an atom becomes comparablewith the transverse dimension of a waveguide, the atom propagation through the waveguide is similar to a single-mode light propagation in conventional optical fibers. If single-mode atom propagation will maintain coherence, then application of atom waveguide to a large area of atom interferometryis very promising.
ATOM WAVEGUIDES
183
Various approaches were proposed to guide free atoms. The simplest way to guide atoms is to employ the iris waveguides; this is commonly used to produce collimated atomic beams in a vacuum. The aperture waveguides have found practical commercial and laboratory applications. However, the tremendous geometrical transmission loss makes this commonly used method far from ideal. Moreover, the atomic beam path can never be bent in this method. The advent of lasers not only stimulated the development of an optical fiber but also gave a creative impulse for a guiding of free atoms. Probably the first proposed scheme to guide atoms was the scheme published by Letokhov in 1968. Later several groups realized this scheme of atom guiding, which is presently known as the channeling of atoms in a standing wave (Prentiss and Ezekiel, 1986; Salomon et al., 1987; Balykin et al., 1989a). The Gaussian laser beam was the first propagating laser configuration that attracted a great deal of attention for focusing and guiding of atoms. In 1978 Bjorkholm et al. demonstrated the focusing of an atomic beam that was propagating coaxially with the Gaussian laser beam. Actually, this pioneering work in atom optics could be considered largely as the first guiding experiments with laser light. In 1992, Ol’shanii et al. proposed combining two experimental techniques: the guiding of the radiation itself in a conventional fiber and the guiding of the atomic beam in an optical fiber. Savage et al. (1993) and Marksteiner el al. (1994) developed the extended theory of the quantum motion of atoms in a hollow fiber. The first successful experiment of the guiding of atoms in a hollow fiber was demonstrated by a Colorado group (Renn et al., 1995). The more promising concept of atom guiding is based on the use of an evanescent wave atom mirror (Cook and Hill, 1982). Since the time of the first demonstration of the atom reflection (Balykin et al., 1987), the atom mirror was extensively studied (see the review paper by Dowling and Gea-Banacloche, 1996). The same Colorado group (Renn et al., 1996) and the Japanese-Korean group (Ito et al., 1996) have successfully demonstrated the guidance of atoms by using optical near fields. Guidance of atoms by optical near fields is complicated by two processes: the diffusive scattering of laser light on the dielectric surface (Henkel et al., 1997) and the attractive van der Waals force between the dielectric wall and the guided atom (Landragin et al., 1996). Guidance of atoms with a propagating “dark spot laser beam” (for instance, a donut mode) is free from these limitations (Kuppens et al., 1996; Yin et al., 1997); at the same time a spatial “rigidity” of the laser beam limits applicability of such kinds of atom waveguide. In the past a series of electrical and magnetic fields were used to focus and to deflect atomic beams. The present status of atom guiding by static electrical and magnetic fields is also discussed in this review.
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II. Guiding of Atoms with Static Electrical and Magnetic Fields A. ELECTRICAL FIELD
A static electrical field was successfully used for focusing of the polar molecules and the molecules with induced electrical dipole moment (Ramsey, 1956). Could the focusing technique be used for guiding of ultra-cold atoms? To answer this question let us first recall the basic physics of the spatial confinement of neutral particles. To keep a neutral particle in a static equilibrium two conditions must be fulfilled. First, the applied force must vanish at a certain point r,
F(r,) = 0 (1) Second, the force field should tend to restore the particle to equilibrium point r,,. The second condition can be met if the partial derivatives aFx/ax, aF,/ay, aF$z are negative ones. Then the necessary requirement upon force F(r) is V*F
W
=
1 2
-- C X ~ E ~ ~
(3)
where a is called the polarizability of the atom, and E is the magnitude of the external electrical field. In a free-source region the Maxwell equation V . E = 0, and the electrical field E is expressed in terms of the scalar potential 4 alone by the relation
E = -V$ (4) The force on the atom is the gradient of the potential energy W (Eq.(3)). The components of the force become
and the sum of the partial derivatives in Eq. (5) (Jackson, 1975) equals
The right side of Eq. (6) consists of a sum of squares of the field gradients and hence only the sign of atom polarizability determines the sign of the divergence of the force Eq. (5). For polar molecules the requirement of Eq. (2) can easily be fulfilled for prop-
185
ATOM WAVEGUIDES
erly chosen rotational quantum states. The practical arrangement for focusing such molecules is four (six)-pole electrostatic field. The field strength in the quadmpole case is given by
(El
=
(UoIrg)r
(7)
where Uo is the voltage applied across adjacent wires, ro is the distance between a surface at potential U, and the axis of the multipole, and r is the distance from the axis of the multipole. The magnitude of the electrical field in this focuser increases with increasing distance from the axis and molecules are directed toward the axis. For neutral atoms the sign of atom polarizability is positive one and the necessary requirement V * F < 0 for the force to be restorative could not be met. For the particle with the positive polarizability the dynamic focusing methods were developed. The basic element of the dynamically focusing (actually guiding) system is a two-wire charge element with a static electrical potential applied between the wires. Atoms move along the axis midway between the wires. The transverse motion can be separated in two planes. In a plane of the two wires the effect of the applied potential is a diverging force that tends to drive atoms away from the axis. In the perpendicular plane the force is a restoring force. The force constant for atoms is given by k, = -k
Y
=
2E0a
d2 '
k,
=
0
where d is the distance between the wires, and E, is the value of the electrical field at the z-axis. The second two-wire pair is placed behind the first one and turned 90" with respect to the first pair around the symmetry axis. An atom that feels a divergent force in the first field arrangement is influenced by a converging force in the second field and vice versa. The whole guiding arrangement consists of an array of truncated line charges alternately arranged in mutually perpendicular planes along the z-axis. The atomic trajectories in the xz and yz planes are analogous to the path of a ray of light that passes through a sequence of lenses with equal focal length, but is alternative in sign. The alternative approach to guiding neutral atoms is to use two pairs of wires in the quadrupole configuration and applied ac voltage and d 2 phase difference between the two pairs. Such a configuration was successfully used to focus metastable neon atoms released from a magneto-optical trap (Shimizu et al., 1992). Another guiding scheme is based on the interaction of a neutral atom with the electrical field of a single charge wire. Consider a neutral atom placed in an electrical field of a charge wire. The interaction potential with the electrical field is then given by
186
Victor I. Balykin
where q is the charge per unit length of the wire. The interaction potential of Eq. (9) is attractive and has the form l/r*. A stable confinement of an atom in such a form potential is not possible: the classical trajectory of atom moving around the wire will end on the surface of the wire or in an infinity (Schmiedmayer, 1995). To stabilize the motion of a neutral atom around the charge wire, Hau et al. (1992) proposed applying an oscillating charge on the wire. B. MAGNETIC FIELD The interaction potential of a neutral atom with a static magnetic field is
W,
= -pB
(10)
where ,u is the magnetic moment of the atom. In a nonuniform magnetic field the potential of Eq. (10) was used to focus or guide (Friedburg and Paul, 1950; Benewitz and Paul, 1954) and to trap neutral atoms (Migdall er al., 1985). In all these experiments, the atoms were focused to a local minimum and trapped in a local minimum of the magnetic field. In both cases the magnetic moment is antiparallel to the magnetic field: the atoms are in an excited state of the atom-field system. A spin-flip transition to the ground state will change a sign of the potential of atom-field interaction. The atom will leave the focusing (guiding) configuration. A focusing or trapping of atoms around a local maximum of the magnetic field will avoid the spin-flip problem; however, the Earnshaw theorem forbids the creation of a local maximum of the magnetic field in a free space. In 1961, Vladimirski considered a scheme in which a maximum of the magnetic field can be found in the region of a current-carrying wire. The scheme of a currentcarrying wire was later used by Schmiedmayer (1995) to guide neutral sodium atoms in a circular orbit along the wire. The interaction potential of atom with a current-carryingwire is
where I is a current flowing through the wire, m F is the projection of the total angular momentum of the atom on the magnetic field, y = g F p Bis the gyro magnetic ratio of the atom, c is the velocity of light, and r is the distance from the center of the wire. To provide a stable guiding, the atom has to be kept away from the wire. Because the potential of Eq. (1 1) has a form llr, it can be compensated by the centrifugal potential (Schmiedmayer, 1995). The atoms move in Kepplerlike orbits around the wire and angular momentum prevents them from an absorption on the wire surface. The experiment with a guiding by a current-carrying wire was successfully performed with sodium atoms (Schmiedmayer, 1995). A well-collimated thermal atomic beam was directed along the wire l-m long and
ATOM WAVEGUIDES
187
150-pm in diameter. The length of the wire limits the guiding time. The typical magnetic field generated by a current 2 A at radius 200 p m was 20 G. The experimental arrangement permits a binding of the atoms with typical transverse energy of lop7eV or transverse velocities on the order of 1 m/s.
III. Evanescent Light Wave A. FORCES ON ATOMSIN A LASER FIELD 1. Near-Resonant Light Force on Atoms
There are a number of ways to describe the radiation force on atoms. The simplest one is the semiclassical approach, which is based on the assumption that the electromagnetic field can be described classically (Cook, 1979; Letokhov and Minogin, 1981; Cohen-Tannoudji, 1991). We consider so-called two-level atoms, with a ground state I g ) and an excited state I e ) separated an energy interval E, - E, = h,,where w,, is the frequency of atomic transition. The atomic medium is supposed to be dilute, so that one can ignore atom-atom interaction. The Hamiltonian of atoms in the laser field can be written as
H
=
HA
+ H , + H,
(12)
where HA, H,, and H,are the Hamiltonians of the free atom, the vacuum field, and the interaction of the atom with the laser and vacuum fields, respectively. The Hamiltonian for a two-level atom P2
H A = - + hw,le)(el 2m
(13)
describes the atomic center mass motion and the internal degrees of freedom of the atom. In Eq. (13), P is the atomic momentum, and wo is the transition frequency between the ground I g ) and excited I e ) states. The second term in Eq. (12)
describes the energy of the quantum vacuum radiation field where the summation takes place over the various modes and a: and aj are the creation and annihilation operators of a photon of energy hwj. The last term in Eq. (12) describes the interaction of the atom with the laser and the vacuum fields H,
=
V,
+ V, = -d*E,(R,
t ) - d*E,(R)
(15)
where d is the atomic dipole moment operator, and E,(R, t) and E,(R) are the electrical laser field operator and the electrical field operator corresponding to the vacuum field.
188
Victor I. Balykin
Because our main interest is the dynamics of the center of atom mass, the most general approach is to use the Heisenberg equations for the position operator R and the atomic impulse operator P
R f'
=
MR
=
1 P -[R, HI = -
ih
1
= - [P, HI =
ih
M
SH - - = -VV'(R)
SR
-
VV,(R)
(16b)
Then the force operator is determined from the second Heisenberg equation 16(b)
F = P
(17)
In the following we assume that the atomic wavepacket is well localized in position and in momentum space so that the quantum discretion of atomic motion is close to the classical one. In such a semiclassical approach the atom is described by a wave function $(r) centered on r = (R)and p = (P). The motion of atoms in a laser field can be described as semiclassical if the following conditions are fulfilled. First, the spread A r of the atomic wavepacket must be smaller than the wavelength of atomic transition
Ar << A
(18)
In this case all parts of the wavepacket experience the same radiation force. Second, the impulse width of the wavepacket must be such that the corresponding Doppler broadening is smaller than the natural absorption linewidth of the atom
kAv << y
(19)
As long as both conditions of Eq. (18-19) are not fulfilled, the atomic wavepacket is so strongly deformed that there is no physical meaning in definition of the concept of the radiation force on the atom. In semiclassical approximation the force on an atom equals the expectation value of the force operator in Eq. (17) f =
(F((R)))
(20)
In the dipole approximation the expectation value of the force, Eq. (20), equals
f = (V(d * E))
(21)
where the gradient operator is taken with respect to the classical variable r = (R). The atomic dipole moment operator does not depend on position center mass motion r, as it is only a function of the atomic electron relative coordinate operator; therefore
189
ATOM WAVEGUIDES
where C is a unit polarization vector. Because in our semiclassical approximation the atomic wavepacket is small in comparison with the characteristic length over which the electrical field E varies, Eq. (22) may be written as f = ((d . C))VE(r, t)
(23) where E(r) is the classical electrical field of laser light at the position of the atomic center mass. We assume in the following field E(r, t ) that the is monochromatic and can be written as
E(r, t ) = QE,(r)cos(wt
+ 4p(r))
(24) where C, Eo(r),and q(r) are, respectively, the polarization vector, the amplitude, and the phase of the laser field. Then the equation for the radiation force takes the form
-
f = ((d C))V[E,(r)cos(wt
+ &-))I
(25)
The dipole moment d in Eq.(25) has the expectation value (d) =Tr(ud) (26) where u is the atomic density operator. The dynamic of internal degrees of freedom of the atom is described by the Liouville equation, which in the interaction representation can be written as in& = [H’, u] (27) where the Hamiltonian H’describes the internal motion of the atom and the atomfield interaction
H‘ = hw,le)(el - (d &)E,(r)cos(wt + p@)) The introduction of the time-independent matrix elements ege, eeg, eee
ege=
g,,e-i~t,
egg-
ugg,
8 ,
= u
(28)
eiWt
eg
eee= gee
and the so-called Bloch vector components u, v, w 1 2
u = -(Cge
+ eeg),
v =
1
-(ege 2i
ifeg),
w =
bring Eq. (27) to the Bloch equations U = -YU
+ AV
it =
-Au - Y V - R,w
w =
0,
-
2y(w
+
);
1
-(eee - egg) (29b) 2
190
Victor I. Balykin
where flR is the Rabi frequency, 2 y is the natural linewidth, and A = w - w,. The Bloch equations permit us to find the mean value of the atomic dipole moment
+ eeg)= 2d,,(u
(d-e) = d,,(B,,
cos wt - u sin wt)
(31)
where d,, = ( g Id I e ) . Inserting the mean value of the atomic dipole moment from Eq. ( 3 1 ) into Eq. (25) gives for the radiation force
f
=
d,,(uVE,
+ uE,Vcp)
(32)
The first term in this equation is the dipole or gradient force, which acts in the direction of the gradient of the field intensity. The second term is proportional to the gradient of the phase of the electrical field. Depending on the spatial and temporal structure of the light field, its strength, and wavelength, the radiation force may be a very complicated function of atom position and velocity. Because all the known studies on the application of the radiation force have been carried out using four types of light fields, namely, a plane traveling wave, the Gaussian laser beam, a standing light wave, and an evanescent wave, or their combinations, we consider here only these types of fields. 2. Traveling Plane Wave Consider the case of the interaction of an atom with a plane traveling wave E(r, t )
=
E,(r)cos(wt - kr)
(33)
The phase factor in this particular case is cp = -kr and the amplitude E, is constant. Equation (32)then yields for the radiation force
f
= -d,,kv
=
(
hky A2
fly2
+ y2 + n i l 2
(34)
and it is usually called the spontaneous light pressure force.
3. Standing Plane Wave The electrical field of a standing plane wave can be written as E(r, r ) = E(r)cos wt
(35)
where E(r) is a solution of the wave equation
+ (w/c)E2 = 0
(36)
and the phase factor cp = 0. The general Eq. (32) yields for this particular laser field case and zero atomic velocity the expression for the radiation force f =
--
A2
+ y2 +
(37)
ATOM WAVEGUIDES
191
This force is called the gradient force as it is proportional to the gradient of laser intensity. In the case of nonzero small atomic velocity (h<< y ) and a small intensity of the laser field ( G << l), the radiation force of the standing wave equals the sum of the light pressure forces (Eq. (34)) of each individual traveling wave force. For arbitrary laser intensity and atom velocity the nonlinear interaction between atom and counterpropagating waves makes the situation rather complicated and the analytical treatment no longer becomes feasible (Minogin and Letokhov, 1987).
4. The Gaussian Laser Beam In the case of a laser beam propagating in the z direction and with the Gaussian intensity profile E(x, y ) = E , exp[-(x2
+ y2)/w$]cos(kz - wt)
(38)
both the electrical field amplitude and the phase are different from zero. The general Eq. (32) for the radiation force gives for the force in the Gaussian laser beam
The force of Eq. (39) depends on the atomic velocity through the Doppler shift and the force is directed inward the laser beam for the negative frequency detuning (A - kv < 0) and outward for the positive detuning (A - kv > 0).
5. Dressed-Atom Approach to Dipole Force The treatment of atomic motion in an intense laser field based on the “dressedatom” approach gives in many cases a clear physical picture of a considered effect (Dalibard and Cohen-Tannoudji, 1985). In such a treatment a laser field is an excitation of a single mode with a frequency w,. The Hamiltonian H, of the laser mode (see Eq. (14)) is
where ii and li are the creation and annihilation operators of laser photons. The . eigenstates IN) of the Hamiltonian H, satisfy the equation +
and describe the state of the field with N photons in the mode. The laser field is supposed to be a coherent state with a mean value of photons (N) >> 1. In the
192
Victor I. Balykin
+
absence of coupling, the combined “atom laser photons” system is characterized by two quantum numbers of e or g for atoms and N for the number of laser photons. When a laser frequency is close to an atomic transition frequency, these states form the two-dimensional manifolds EN
= {Jg, N
+
I),
le, N ) ) , EN-^ =
{la N ) ,
le, N - I)),
...
(42)
+
with the distance between levels I g, N 1) and Ie, N) equal to hA and between manifolds and E N - 1 equal to no,. The interaction VNbetween the laser field and the atom couples the states
where and p(r) are the Rabi frequency and the phase of the laser mode. This coupling gives rise to two new “dressed” states I 1(N)) and I2(N)), which are a linear combination of the unperturbed states I g, N + 1 ) and I e, N). The frequency difference between these “dressed” states is
W(r) = i’li(r)
+ A2
(44)
Spontaneous transitions couple these “dressed” states. There are four allowed transitions between the dressed states with the following frequencies wo fl, oo, wo - i’l. The energy of “dressed” states equals
+
They can be considered as the potential energy of an atom in a laser field and, in the absence of spontaneous emission, the gradient of the dressed-state energy of Eq. (45) gives the gradient forces
The spontaneous emission produces transitions between dressed states. This changes in a random way the sign of the instantaneous dressed-stateforce and the. mean gradient force is then given by
(f)= nlfl + nzfz where IIiis the probability of a population of the ith dressed state.
(48)
193
ATOM WAVEGUIDES
6. Impulse Diffusion The physical origin of the radiation force on an atom is the quantized process of stimulated absorption of laser photons followed by stimulated or spontaneous emission. The variation in atomic momentum depends on the recoils coming from both induced and spontaneous transitions. The quantitative characteristics of the diffusion of atomic momentum in a laser field is the momentum diffusion coefficient, which is defined as
where S p i is the deviation of momentum projection p i from the average value (pi).In the simplest case of an atom in the traveling plane wave (Eq. (33)), the diffusion coefficient has the form (Letokhov and Minogin, 1981)
D = -(hk)2y 2 (A2
Oil2
+ y 2 + flit2
and reaches its maximum value D,, = ( h l ~ )at~an y exact resonance of the atomic transition with the laser frequency. The order of magnitude of the momentum diffusion coefficient due to the fluctuation of dipole force can be estimated by considering the resonance case (A = 0) of atom-laser interaction (Dalibard and Cohen-Tannoudji, 1985). In that case, both dressed states Il(N)) and I2(N)) are equally populated and because from Eqs. (46-47)A = -&, then the average dipole force in Eq. (48) (f) = 0. The diffusion coefficient is determined by the correlation function of the instantaneous dipole force
D,, =
I
dt (f(t)f(f +
(51)
7))
The correlation time of the dipole forcef(t) is of the order of the mean time between two successive changes of the dressed states due to a spontaneous transition, and it is of the order of the spontaneous decay time y Equation (51) gives the order of magnitude of the momentum diffusion
-'.
B. EVANESCENT-WAVE ATOMMIRROR There are several configurations of laser field that can be used for the reflection of atoms. The first scheme for atom mirror was suggested by Cook and Hill (1982). In their scheme a plane traveling light wave is totally reflected internally at the
194
Victor I. Balykin
surface of a dielectric in a vacuum: A thin evanescent light wave is generated on the surface. It is this surface wave that can be served as an atomic mirror for atoms running into it. Several authors have proposed alternative schemes for atom mirror. They mainly are based on laser beams focused into “sheets” of light (Wilkens et al., 1993; Davidson et al., 1995). Let us first consider the Cook and Hill atom mirror based on the evanescent wave.
1. Simple Evanescent Wave Suppose we have a laser light incident on a smooth interface of a dielectric in a vacuum and its electrical field has a form Ei
EoieKWf-kir)
2
(53)
The transmitted field through the interface is E, =
Eotei(Wf-k,r)
(54)
For incident angles greater than the so-called critical angle @,(sin 8,= n - l , n is a refractive index of the dielectric), there is a total internal reflection of the incident wave. Using Shell’s law we can write for the perpendicular k, and parallel k, components of wave vector of the transmitted wave
k,x
k n
= -Isin
8;
Then the electrical field of the transmitted field has the form E ev
=
E, = E01e-BYe;(W‘-klxsinei/fl)= Eo,e-PY cos(wt - qx) q = k, sin 8 ; / n
(57) (58)
and it is called the surface or evanescent wave. Its amplitude decays at a distance
p-’
=
w ( n 2 sin2 Oi - l)-1/2/c= A/21r
(59)
where A is the wavelength of the incident wave, and c is the speed of light in vacuum. The evanescent wave propagates parallel to the dielectric vacuum interface with the wave vector of the evanescent wave q = k, sin O i / nand it is substantial for a few wavelengths from the interface. The amplitude of the evanescent wave at the dielectric interface is greater than the electrical field in the incident
ATOM WAVEGUIDES
195
laser field E,a, (outside of the dielectric) and the relationship between them is given by the following equation (Jackson, 1975; Kaiser et al., 1994)
where rn = 0, 1 for TE or TM polarizations, respectively. 2. Su$ace Plasmon-Enhanced Evanescent Wave Surface plasmons are electromagnetic charge-density waves propagating along a metal-dielectric interface (Raether, 1988). A metal-dielectric interface is prepared usually by evaporating a thin metal layer on the surface of a prism. For a surface plasmon the wave vector component k , along the interface is
where E , = E ; + icy is the relative permittivity of the metal of the surface layer. Surface plasmons can be excited by the attenuated total reflection methods. In this method a laser beam, polarized parallel to the plane of incidence, undergoes total internal reflection on the interface with the metal layer. If the wave vector k , (Eq. (56)) of the evanescent wave propagating along the interface matches the wave vector of the plasmon wave (Eq. (61)) then there is a resonant coupling into the plasmon wave. The resonance condition defines an optimum angle of incidence of the laser beam (Seifert et al., 1994a)
where e2 = n 2 is a relative permittivity of the dielectric. An attractive feature of a plasmon wave in atom optics is the large field enhancement of the initial laser beam intensity. The field enhancement factor 7,defined as the ratio between the maximum intensity of the evanescent wave with and without plasmon excitation, is: =
(2) -€;(E2
2E24
- 1) - E 2
1-
E;
The maximum evanescent wave enhancement factor 7 depends strongly on the values 6 ;and E;. A quartz prism (n = = 1.45) with a silver layer ( E ; = -30) and at optimum angle of incident (Oi,op, = 44,6")has the maximum enhancement factor between 15 and 100 (Johnson and Christy, 1972). There are a number of limitations in the use of the plasmon wave as an atom mirror. First, the field enhancement is limited by the surface imperfections such
196
Victor I. Balykin
as metal surface corrugations, which increase the radiative damping. Second, the resonance condition of plasmon excitation depends on the thickness of the metal layer, which results in a thickness-dependence resonance of plasmon excitation. This could cause a strong fluctuation of the field intensity. Third, if the film layer on the prism surface is rather rough (the mean square height of corrugation is larger than 5 nm), plasmon waves are strongly scattered. This scattering leads to accumulation of electromagnetic field density, which is locally higher than that of an extended surface plasmon wave on a smooth surface. The theoretical consideration (Raether, 1988) gives for the roughness in several hundred angstroms an enhancement factor of 20. The surface plasmon technique has the advantage in its simplicity over other techniques of the enhancement of the evanescent wave (see the next section).
3. Enhancement with Dielectric Waveguide Another approach to enhancing an evanescent field is to use a multilayer dielectric structure (Kaiser et al., 1994; Seifert et al., 1994b). Figure 1 illustrates the principle of enhancement by the multilayer structure: The layer with a refractive index n is a thin-film waveguide. The dielectric surface is separated from the waveguide by a small gap of low refractive index n2. The incident laser light is totally reflected at the dielectric interface and the laser light in the prism and a mode in the waveguide are coupled through the evanescent field in the gap. This is a wellknown principle of a distributed coupling of a laser light through the evanescent field to the modes of thin-film waveguides (Tien and Ulrich, 1970; Ulrich, 1970). When a mode of the waveguide is excited, even a moderate incident laser power gives a large light intensity in the waveguide. The enhancement in the waveguide depends upon two main factors: (1) the coupling of a waveguide mode with the incident laser field, and (2) the losses in the waveguide. The maximum enhancement is reached when the coupling equals the losses in the waveguide.
,
FIG. I. Evanescent wave atom mirror with a dielectric waveguide. The layer with index of refraction n , is a thin waveguide. The dielectric with index of refraction n, is separated from the waveguide by a small gap of a low refractive index nz.The incident laser beam is totally reflected at the dielectric surface and the laser field in the dielectric and a mode in the waveguide are coupled through the evanescent field in the gap.
ATOM WAVEGUIDES
197
The enhancement factor exhibits a sharp resonance, which corresponds to the resonance condition for the modes of the waveguide w
2 d , - n , cos 8,= C
x, + x2 + 27rm
(64)
where d , is the waveguide thickness; 8,is the angle of refraction in the waveguide; x, and ,y2 are the phase changes at total internal reflection on the waveguide-vacuum boundary and the waveguide-gap boundary, respectively; and m = 0, 1, 2, . . . , . From Q. (64)it follows that the incident angle for optimal coupling is determined by the thickness of the waveguide. This condition also determines the modes of the waveguide. The enhancement of the evanescent wave with a thin dielectric waveguide was described in detail by Kaiser et al. (1994) and Seifert et al. (1994b). Their waveguides were realized on a surface of a glass prism (a refractive index n = 1.9), which was coated with 350-nm thick layer of Si,O with refractive index n2 = 1.46 (the gap) and covered by a layer of TiO, with n3 = 2.37 (the waveguide). The realized enhancement of an evanescent wave, which was defined as the ratio of the evanescent wave intensity at the interface with and without the waveguide, was about 130. Results obtained in this paper show that a larger enhancement factor can be achieved. The theoretically predicted enhancement factor is about several thousands (Tien and Ulrich, 1970; Ulrich, 1970). In the latest publication, Labeyrie et al. (1996) reported a considerable step forward in the waveguide design: With an improved waveguide structure the enhancement factor of 1690 was achieved. The enhancement with waveguides has several major advantages among other techniques. First, according to Eq. (59) a short decay length of evanescent wave can be chosen. This makes the atom-field interaction time during an atom reflection by the evanescent wave very short, and hence it minimizes a spontaneous emission loss. It is particularly important if the reflection of atoms is to be without loss of the coherence of the atomic de Broglie wave. Second, the enhancement parameter is limited by losses in the layer. The modern evaporative technology can build the dielectric layers with the negligible losses and the enhancement of several thousands can be possible. Third, the dielectric layer is chemically stable and withdraws high laser intensity. In concluding this section we notice once more the main advantages and disadvantages of different types of evanescent waves as an atom mirror. A simple evanescent wave permits us to create a homogeneous field along a dielectric surface without a limitation on the field intensity in the evanescent wave. A spot size of an atomic mirror based on this type of evanescent wave is usually limited by available laser power. The finite, rather limited spot size of the evanescent wave give rise to a gradient of laser intensity along the surface and to violation of “the mirror-like” character of atom reflection. The main motivation for using the plasmon wave and the evanescent wave of dielectric waveguides as an atomic
Victor I. Balykin
198
mirror is to achieve a high field intensity with a lower laser power and shorter decay length. 4. Atom-Surface Interaction
We now consider an atom located in the vicinity of a vacuum-dielectricinterface. It is well known that a surface interface will modify the spontaneousradiation rate and the energy of atomic levels (the latest reviews on this subject are Haroche and Kleppner, 1989; Hinds, 1991; Haroche, 1991). The interface-induced atom level shift can manifest itself in atomic motion in the evanescent wave mirror through a modification of the force experienced by the atom. For atom-interface distance smaller than the wavelength of the atomic transition divided by 27r (nonretardedregime of the interaction), the interactionbetween an atom and a dielectric can be viewed as a result of coupling between the instantaneous atomic dipole and its electrostatic image in the dielectric. In this classical electrostatic image model (Jackson, 1975) the effect of arbitrary flat surface on an atom-dipole d, located in a vacuum at distance z from the surface, can be modeled by assuming the presence of a dipole image located symmetrically at position -z. Then the dipole interaction energy is given by 1 E - 1 d ; + 2dZ v,, = - 4m0 E + 1 16z3 where E is the dielectric constant, and d p and d , are the components of the electric dipole moment parallel and perpendicular to the interface. The corresponding quantum mechanical interaction operator is given by Eq. (65),where d p and d , are now the atom-electric dipole operators. Then the energy shift of the atomic level (known as the van der Waals energy shift) is given by (Fichet et aL, 1995)
where d? = ( a 1 d p In) is the transition dipole moment and the summation in Eq. (66) over all atomic transitions. Equation (66) is valid when the electric dipole transitions of the atom lie outside the absorption range of the dielectric. Otherwise the dielectric response is characterizedby the frequency-dependentpermittivity e(w)and the van der Waals energy shift will be
where r(w,,) is the so-called frequency-dependent dielectric reflection coefficient
199
ATOM WAVEGUIDES
P in Eq. (68) denotes the principal value of the integral. In the case of long wavelength (W + 0) the dielectric reflection coefficient becomes I(W
+ 0) = .I
=
E(0) - 1 E(0) + 1
where ~ ( 0is) the static dielectric constant and the expression for the van der Waals energy shifts takes the form of Eq. (65). If the atom is in the excited state there is an additional energy shift that oscillates with an amplitude that is for a small distance to interface (z << A / ~ T of ) the order AE,, = fiy (Hinds, 1991). The excited atom-surface interaction will be enhanced when some atom transition frequency is resonant with the dielectric absorption band (Fichet et al., 1995). It can be seen from Eqs. (66)-(67) that the van der Waals energy shift is determined, on the one hand, by the dielectric substance (the permittivity E ( W ) of the dielectric, its energy bands); and, on the other hand, by the internal structure of the atom (through the dependence of the atomic matrix elements from the atomic internal variables). It is possible to describe some general characteristics of the interface-induced energy level shift from the symmetry properties of the atomic system under rotations of the internal atomic variables (Courtois et al., 1996). First, the system atom-flat interface has invariance under rotation around the perpendicular to the interface. As a result, the component of the total angular momentum M on this axis is a good quantum number. Second, the invariance under time reversal determines the dependence of energy shift only on the magnitude of the component of the total angular momentum M. Hence the total Hamiltonian V, associated with the energy shift, splits into a scalar part VS) and a quadrupolepart V(Q!The scalar part V ( s )produces a global shift energy level but does not lift Zeeman degeneracy of atomic sublevels and does not mix levels with different angular momentum J. It modifies the atomic transition frequency and makes it dependent on the atomsurface separation. The quadrupole part V ( Q )does not shift the net average of the Zeeman shift but produces a Zeeman splitting (Courtois et al., 1996)
SE,,
cc
3M2 - J ( J
+
1)
(70)
Moreover, as can be seen from Eq. (70), in the case of alkali atoms in ground state with J = 1/2 (the case of a great practical interest in the field of laser manipulation of atoms), the quadrupole part V ( Q does ) affect the ground state of atoms. Details of the calculations of the dependence of van der Waals energy shift on the internal atomic variables can be found in the paper of Courtois ef al. (1996). When the atomic-interfacedistance is not small compared to the wavelength of the atomic transitions, the r3 law of the energy shift associated with instantaneous electrostatic interaction is no longer valid. The retardation effect becomes important and the van der Waals atom energy shift changes to the Casimir-Polder
200
Victor I. Balykin
energy shift (Casimir and Polder, 1948; Spruch, 1986), which for the atom in the ground state takes the form SE,,
=
1 3hc 4m, 8,rrz4
- --ff
Sf
where a,,is the static electric polarizability. All the previously considered interface-induced atom levels shifts can alter the atom-evanescent mirror potential. In addition, they modify the detuning of the evanescent wave, because they change the atomic transition frequency when the atom approaches the interface (Desbiolles et al., 1996). Raskin and Kusch (1969) reported the first experimental results on the van der Waals measurement interaction between atom and surface. In their experiment the deflection of a Cs atomic beam in the ground state from the metal surface was studied. Close to the surface the van der Waals force attracts Cs atoms to the surface with the following deflection of an initial Cs beam. With the assumption of the van der Wads interaction dependence on the separation z between atom and , and Kusch were able to calculate the value K , which was surface as K / z ~ Raskin in quantitative agreement with theory. The improved version of the deflection experiment (Shih and Parsigian, 1975) gave the results, which were in agreement with the K / Z ~law of the surface-atom interaction. In the first experiment with atoms in ground states the observed effects were so weak that no precise comparison between the theory and the experiment data was possible. The studies of Rydberg atoms in a parallel-plate metallic cavity gave more precise data on the van der Waals interaction (Sandoghdar et al., 1992; Sukenik et al., 1993). A series of experiments on selective-reflection spectroscopy at an interface between a dielectric surface and an atomic vapor has given information on the interaction between atoms in an excited state and dielectric media (Chevrollier et al., 1992). The first measurement of interaction between atoms and a surface by using an evanescent wave mirror was done by a Stanford group (Kasevich et al., 1991). In their experiment the long range van der Waals potential was modified by adding a repulsive potential from a totally reflected laser beam in a prism. The combined potential is the repulsive dipole potential and the attractive van der Waals potential. Atoms with sufficiently low incident energy will be reflected from the repulsive part of the potential. If the initial kinetic energy is higher than the height of the combined potential, the atom will be in most cases stuck to the surface. The height of the combined potential can be calculated as a function of the evanescent wave parameters (the laser intensity and the detuning). The measurement of the number of reflected Cs atoms with a given atomic kinetic energy was compared with the theory. The experiment and the theory were in good agreement. Kasevich et al. (1991) have also observed the reflection of atoms at the kinetic
ATOM WAVEGUIDES
F
3
20 1
1
0
FIG. 2. Interaction potentials seen by rubidium atom near dielectric surface with an evanescent wave. Udipis the dipole potential due to the evanescent wave; UVdw is the van der Wads potential.The sum of the dipole potential and the van der Wads potential is shown by solid line. The dotted line shows the total resulting potential taking into account the QED expression of the van der Waals potential (from Landragin et a!., 1996, Fig. 1, Phys. Rev. Lett. 77, N. 8, 1464, reprinted with permission).
energy larger than the maximum potential barrier, the evanescent wave plus van der Waals interaction. The appearance of nonzero reflection can be considered as evidence for the quantum reflection of atoms. The comprehensive measurement of van der Waals force between atoms in a ground state and a dielectric surface was done by Landregin et al. (1996). In their experiment the laser-cooled Rb atoms with a well-defined kinetic energy were released on an evanescent wave atomic mirror. Figure 2 shows the calculation in this work of the repulsive potential Udipdue to the evanescent wave, the van der Waals potential UYm,and the sum of the van der Waals and the dipole potentials (solid line) as a function of the distance between an atom and a surface in units of A/27~.The evanescent wave potential and the total potential were calculated for the light shift at the surface A = 16.8* 2y, which corresponds to a barrier height equal to the kinetic energy of atom E,, = 7hy. It can be seen that the van der Waals interaction reduces the barrier height of the evanescent wave potential by a factor of 3. The position of the maximum of combined potential is at the atomsurface distance 47 nm. The results of the measurement of the number of reflected atoms as a function of the light shift (at the surface) give the barrier height of the combined potential. Figure 3 (Landregin et al., 1996) shows a plot of the number of reflected atoms as a function of the light shift for various laser power and detuning. The arrows show the predicted thresholds calculated with the electrostatic
202
Victor I. Balykin
n
?
1,o-
m
A
v
* 0
0
-0
0
A
4.4w 2.75W 2w
I
100
10
A,/T FIG. 3. The test of the van der Waals theory with an evanescent wave atom mirror. The figure shows the number of reflected atoms from an evanescent wave mirror as a function of the light shift for various laser power and detuning. The arrows show the predicted thresholds for atom reflection: (1) ignoring the van der Waals interaction (AfP); (2) using the electrostatic model for the van der Waals force (AY); (3) using the QED model (A?’’) (from Landragin et al., 1996, Fig. 3, Phys. Rev. Lett. 77, N. 8, 1466, reprinted with permission).
model van der Waals potential and QED prediction. The experimental data are in good agreement with the theory.
c. REFLECTIONOF ATOMSBY EVANESCENT WAVE 1. Specular Rejection If the evanescent wave is created by a total internal reflection of the Gaussian laser beam at a dielectric vacuum interface, than the electrical field in the vacuum above the surface takes the form
where w, and w yare the beam waists along the surface, x and z the coordinates parallel and perpendicular to the surface. The corresponding Rabi frequency
For two-level atoms the radiative force has two components: a normal to the dielectric surface and a component parallel to the surface. The normal component
ATOM WAVEGUIDES
203
is associated with a gradient of the field amplitude (72) perpendicular to the surface (Cook and Hill, 1982; Seifert et d.,1994a) and it is equal
and A = w - wo - k,v. The where R, is the Rabi frequency, R = d m , parallel component of the radiative force consists of the dipole force associated with a gradient of the field amplitude along the surface and the spontaneous light pressure force
F,=
-n
(75)
Consideration of the geometry of the atomic initial and final velocity vectors shows that the law of atom reflection of atoms from the evanescent wave (Cook and Hill, 1982) is tan rpr
=
tan pi - 2R
(76)
where rp, and pi are the angles of reflection and incidence and R is the ratio of the parallel to the normal radiation force R = F,/F, - normal. When the parameter R is negligibly small, Eq. (76) reduces to the equation of the specular reflection law: rpr = rpi. It occurs when the spontaneous force and the parallel component of the dipole force (Eq. (75)) are considerably smaller then the normal component of the dipole force (Eq. (74)). Because the spontaneous force is negligible for a large laser intensity (R, >> y ) and for a large laser detuning (A >> y ) , it is evident that a specular reflection of atoms can be expected only for a large laser detuning and a large diameter of laser beam. Fluctuations in the sign of the dipole force induced by spontaneous emission lead also to the nonspecular character of atomic reflection. The effect of the Gaussian profile of a laser beam on the reflection of an atom is a defocusing of the reflected beam. The parallel components of dipole force arise due to the finite size of the laser beam and their effect is equivalent to a divergent lens. The focal length can be estimated from the effective radius of curvature of the evanescent wave. For the case of the normal incident of atoms on the surface wave it is easy to show that the focal length of such a concave mirror is
Because w , , ~>> 1/p, the defocusing effect is usually relatively small. For a glancing incidence of atoms on the surface wave the effective focal length is larger and astigmatic aberrations become important. One additional reason for nonspecular reflection of atoms should be recog-
204
Victor I. Balykin
nized. In a derivation of the radiation pressure force, it is usually assumed that the amplitude of the light field varies adiabatically slow in comparison with the spontaneous relaxation time. However, in the reflection by a thin evanescent wave, the interaction time can be comparable with the relaxation time. This could cause a modification of the gradient force and, as a result, a violation of the law of specular reflection (Ol’Shanii et al., 1992). In a typical atom reflection experiment, however, a characteristic time of the interaction is considerably larger than the spontaneous lifetime of the excited state of the transition used and this rather fundamental effect can be negligible. Reflections of atoms by the evanescent waves were reported in many papers (Balykin et al., 1987; Kasevich et al., 1990; Seifert et al., 1994a; Kaiser et al., 1994; Esslinger et al., 1993). Probably the most accurate measurement of atom reflection was done by Seifert et al. (1994a). In their experiment a very wellcollimated beam of metastable argon atoms was reflected by both simple and plasmon-enhanced evanescent waves. The angular divergence of the initial atomic beam was around rad and the angular resolution of the detection system was also lop4 rad. Figure 4 shows the atomic intensity distribution produced by the reflection of a metastable argon beam from a simple evanescent wave (Section 1II.B). The peak at q i= 0 mrad corresponds to the initial atomic beam. The peaks to the right correspond to the atoms that were reflected by the evanescent wave. The position of the reflected peaks is equal to the position expected for specular reflection indicated by the dashed line. In the experiment the near specular reflection was expected, because the Rabi frequency was much larger than the spontaneous decay rate and the laser beam waist on the surface was much larger the evanescent wave decay length. The broadening of the reflected atomic beam was caused mainly by the defocusing effect and by fluctuation in the dipole force. The significance of the dipole force fluctuation can be seen from Fig. 5, where the atomic intensity distribution produced by the reflection of the beam of metastable argon from a plasmon-enchanced evanescent wave is shown (Seifert et al., 1994a). The solid curve is a Monte-Carlo simulation of the reflection. From the computer simulations the relative importance of the different broadening mechanisms can be estimated. The broadening is dominated mainly by the fluctuation in the dipole force. The narrow peak corresponds to atoms, which never experience fluctuation of the dipole force (in the dressed state model of the reflection process these atoms are always in one particular repulsive dressed state). This peak was smeared out in the experiment due to experiment imperfection.
2. Coherence of Matter Wave
To a large extent the concept of the coherence of the atomic matter wave can be introduced in direct analogy with conventional optics (Born and Wolf, 1984), the electromagnetic wave being replaced by the atomic de Broglie waves. If the
ATOM WAVEGUIDES
I 1 ,It,
50 0 50
-
5
b
X G 3
.0
E
2
0 50
-
205 cpi = 0.1 mrad
' \
0.3 mrad \
\
-
1.6 mrad
\
-
0
0 50
1.9 mrad
\
50-
\
n
2.3 mrad
'\
-
\
deflection angle (pd (mrad)
FIG. 4. The atomic intensity distribution produced by the reflection of a metastable argon atomic beam from an evanescent wave. The peak at (pi = 0 corresponds to the position of the initial atomic beam. The peaks to the right correspond to the atoms that were reflected by the evanescent wave. The position of the reflected peaks was equal to the position expected for specular reflection, indicated by the dashed line (from Seifert et al., 1994a, Fig. 3, Phys. Rev. Lett. 71, N. 5 , 3818, reprinted with permission).
+(?,r ) is a wave function of the atomic matter waves then the coherence of the atomic matter waves is described by the autocorrelation function
rG,, r , ;
-)
r 2 , t 2 ) = <(/I*(?,, r , ) ( / ~ ( ? ~ , t 2 ) )
(78)
Victor I. Balykin
206
10
5
0
6
7
a
9
10
11
deflection angle qd (mrad) FIG. 5. Figure illustrates the significance of the dipole force fluctuation on the atom beam reflection by an evanescent light wave. The points show atomic intensity distribution produced by the reflection of the atomic beam from a plasmon-enhanced evanescent wave. The solid curve is a Monte-Carlocomputer simulation of the reflection process. The narrow peak corresponds to the atoms that never experience the fluctuation of the dipole force (from Seifert et nl., 1994a. Fig. 3, Phys. Rev. Lett. 71, N. 5 , 3818, reprinted with permission).
where the average denotes the ensemble average. As in the case of conventional + + optics, the autocorrelation function for equal times r(r , , t; r2, t) describes the + + spatial coherence and for equal positions r(r , t , ; r , t 2 ) describes the temporal coherence of the atomic matter wave. It should also be noted that there are a number of fundamental differences in the concept of the coherence of the atomic matter wave and the electromagnetic waves. First, atoms have internal structure; second, the spontaneous transitions influence the coherence of the atomic matter wave; third, there is an interaction between atoms; and, finally, in contrast to conventional optics, the free-space dispersion relations are different for conventional optics and atom optics.
207
ATOM WAVEGUIDES
The rigorous treatment of a coherence of the atomic matter wave requires the use of second-quantized fields (Mandel, 1983). A normalized N-particle state I @)N may be written as (Taylor et al., 1994)
I@)N
1 =
dl
. . . dN f,(l,
2, . . . , N ) $ + ( N )
. . . 1$+(1)
10)
(79)
where the N-body wave functionfN( 1,2, . . . ,N) is totally symmetric in its arguments for bosons and totally antisymmetric in its arguments for fermions, $ +(1) is a creative operator, and 10) is a vacuum state. For the model of atom detection similar to Glauber’s theory of photodetection, the normally ordered and timeordered correlation function of an atomic beam can be defined as r(r, t) = (
C
$:(r, t)$,.(r, t ) )
(80)
,.PI
3. Coherence of Atom Mirror In the following we will briefly discuss the main processes that can destroy the atomic wave coherence during the reflection of atoms from an evanescent wave. In atom optics there are mainly two types of sources of free atoms: ( 1 ) thermal atomic beams produced from thermal sources by a suitable set of collimating slits and (2) beams of slow atoms released from a magneto-optical trap. Both types of atomic beams are non-coherent sources of atoms and the spatial and temporal coherence of atomic beam from these sources is rather limited. The recent spectacular experiments with Bose-Einstein condensate (Andrews et al., 1997) and the proposed laser-like source of atoms (Ol’shanii et al., 1995; Guzman et al., 1996) could give the opportunity to work with coherent atomic beams. To which extent the atom mirror can preserve the initial coherence of atomic beam? As we have mentioned already there are mainly two threats to the coherence of an atomic wave in the process of their reflection: ( 1 ) spontaneousemission and ( 2 ) diffusive scattering of the evanescent wave. a. Spontaneous Emission. Spontaneous emission partially destroys the mutual coherence between the incident and the reflected de Broglie waves. So far there was no direct measurement of the coherence loss of atomic beam during its reflection from an atomic mirror and even the theoretical treatment of this problem does not exist. Experimental researches are mainly concerned with establishing the conditions at which the minimum amount of spontaneous emission occurs during the reflection process. Considering the atomic center of mass classically,the number of spontaneously emitted photons by atoms during the time interval dt and its motion through the evanescent field is given by
dn,
=
2yp2, dt
208
Victor I. Balykin
where 2 y is the natural linewidth, and p22is the population of the excited state. In the case when the laser detuning is much larger than the natural linewidth the number of spontaneously emitted photons by atom (Kasevich et al., 1990) is
which depends only on the normal component of the atomic velocity ul,the laser detuning, and the decay length of the evanescent wave. From Eq. (82) it follows that the number of spontaneouslyemitted photons does not depend on the intensity of the evanescent wave. This is due to the fact that the repulsive potential depends exponentially on the atom-surface interaction and an increase in the intensity only shifts the turning points of the atomic trajectory away from the surface but has no influence on the field strength seen by the atom. To measure the amount of spontaneous emission during the reflection process Seifert et al. (1994a) have made an experiment on the reflection of metastable argon atoms using “open” and “closed” transitions. Transition Is,-lp, in metastable argon is a “closed” transition. Transition ls,-2p, is a partially “open” one: The 2p, state decays either to Is, or Is, and the state Is, decays further to the electronic ground state. On the average for each spontaneous emission, roughly 71% atoms end up in the electronic ground state. Because the reflected atoms were detected by a channeltron (which is sensitive only to the metastable atoms), the detector did not see the reflected atoms in ground state. By comparing and “closed” transition (N,) the number of reflected atoms for the “open” (No) (and assuming that the loss of atoms is caused only by spontaneous emissions) it is possible to estimate the maximum amount of spontaneous emission, which was equal to 0.28. The open transition atom mirror gives an opportunityto improve the coherence of reflection in comparison with a normal atom mirror by selecting atoms that have not undergone spontaneous emission. In the experiment of Seifert et al. (1994a)the “open” transition atom mirror improved the coherence of the reflected beam (in comparison with a two-level atom case) by selecting atoms that have not undergone spontaneousemission. In this case, the average number of spontaneous emissionsfor reflected atom was less than 0.1 1. As follows from Eq. (82). another way to deduce the probability of spontaneous emission during the reflection is to use the evanescentwave with a short decay length p-’. Seifert et al. (1994b) reported the excitation of the mode in the waveguide layer with a decay length of the evanescent wave of 97 nm. b. Difisive Atom Rejection. A detailed treatment of the influence of surface roughness on the specular character of atomic reflection and on the coherence of a reflected beam was done by Henkel et al. (1996) and Landragin et al. (1996). The optical potential of the evanescent wave prevents the atoms from the direct
ATOM WAVEGUIDES
209
interaction with the surface supporting the evanescent wave. The interaction of atoms with the evanescent wave minimizes the diffusive scattering of atoms by the surface and van der Wads sticking. However, there are several mechanics that are responsible for an indirect interaction of atoms with the surface. These include (1) the interaction with a light scattered from the interface and (2) the modification of the van der Waals-Casimier interaction due to the surface roughness. The most important fact here is that the laser light, scattered at the rough dielectric surface, interferes with the unscattered evanescent wave and, as a result, creates a random spatial variation of the optical near-field intensity above the surface. As a result, the repulsive optical potential hence acquires some “roughness” that scatters the atoms. In the absence of surface roughness the evanescent electrical field (Eq.(57)) is E(0) =
E 0 ,(iqr-pY)
(83)
where q and r are the wave vector and the coordinate vector components parallel to the surface. The electrical field above the rough surface also contains a scattering part. If the surface roughness CT is small compared to the optical wavelength, the Rayleigh theory of scattering can be applied. To the first order in the surface roughness, the scattered electrical field amplitude E(’)is proportional to the transmitted zeroth order field amplitude E(O) and proportionality factor is of the order CT/A << 1. It is useful to make the Fourier expansion of the scattered light (Henkel et al., 1997)
The Fourier coefficients of the scattering light in the Rayleigh approximation are proportional to the Fourier components S(q’ - q) of the surface profile s(r) E(l)(q’) cc S(q’ - q) where the Fourier component S(q’ - q) =
I
d2r s(r)exp[-i(q’ - q)rl
(85)
(86)
In other words a specific Fourier component of the surface roughness excites scatter field modes from the incident wave, and the scattering modes can be evanescent modes with a real decay constant p’ or plane waves that propagate above the surface with imaginary p. The tot$ electrical field above the rough surface consists of two parts: unscattered E m ( r ) and scattered E(I)(?) parts
E(?) and E ( l ) ( ? ) << E(O)(?).
= E(O)(?) + E(l)(?)
(87)
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Victor I. Balykin
The optical potential corresponding to the electrical field of Eq.(87) contains three parts. Zero-order potential V(O)(r)corresponds to the flat mirror; the firstorder potential V(l)(r)corresponds to the interference term between the scattering field E(l)(?) and zeroth order field E(O)(?). The roughness of the evanescent wave mirror is described by a potential in which the intensity is the interference term E(O)(?) E(’)(?). As a next step Henkel et al. (1997) used the potentialscattering theory in the Born approximationto characterize the diffusivereflection of atoms. The calculation of the total probability w of the diffusive reflection gives the following inequality
+
In Eq. (88) y o is a classical turning point where the magnitude of the optical potential is equal to the incident kinetic energy of the atom, and u is the rms surface roughness. As can be seen from Eq. (88), the total probability of diffusive reflection w depends on the surface parameters and the parameters of the incident atomic beam. The specular reflection from the evanescent wave mirror can be expected only if the roughness of the dielectric surface u is smaller than the de Broglie wavelength of the incident atoms. The increase of optical potential (by increasing the power of the laser beam forming the evanescent wave) will shift the turning points further from the surface and we can expect the decrease of the influence of the surface on the reflection. Equation (88), however, gives us the opposite result: The reflection probability w increases by the factor ePY? Physical explanation of “unexpected” behavior of the atom mirror is the following: There are the scattered light modes that decay more slowly than the zeroth order evanescent wave and also there are propagating scattering modes. Both types of scattering modes contribute to the diffusive reflection. The diffusive character of atom reflection influences the spatial coherence of the reflected atomic wave. In the case of quasi-specular region of reflection, the spatial coherence on the reflected beam decreases as (Henkel et al., 1997) I‘(rl, r 2 ) = r 0 ( l - w
+ . . .)
(89)
where To is the spatial coherence of an incident atom wave. 4. Dissipative Refection
Since the first experimental realizations of the atom evanescent mirror (Balykin et aL, 1987; Kasevich et al., 1990), it has been used in a wide variety of experiments in atom optics (see the review paper by Dowling and Gea-Banacloche, 1996). A main concern in the use of the evanescent wave mirror is to ensure that a reflection at the mirror is specular and coherent. It is possible to choose the
ATOM WAVEGUIDES
21 1
parameters of the evanescent wave that in one reflection event the reflection is a specular and coherent one. Fluctuation of dipole force in the evanescent wave and spontaneous impulse diffusion cause the heating of the reflected atom and it violates the specular character of the reflection. It is rather attractiveto find mechanics that can compensate the heating of atoms and even cool them down. Helmerson et al. (1993) proposed such cooling mechanics. Later Soding et al. (1996) and Nha and Jhe (1997) considered its applications in the gravitational atomic traps. The basic idea of cooling of atoms in an evanescent wave is the following. The inhomogeneous electromagnetic field of the surface wave (see the Section II1.B)
E
= eO@exp(-pz)
(90) creates the dipole potential for the atom that depends on the ground state of the atom. For the laser detuning larger than the natural line width (A >> 2y), the potential has the form
where fl,(O) is the value of the Rabi frequency on the surface. During the reflection the atom can experience spontaneous emission of photons. The probability for a spontaneous emission during the time interval dt is dn, = 2y-
4A2
exp(-2pz) dt
The total number of spontaneous decays per single reflection (Kasevich et al., 1990) is
and it depends only on the atomic mass m and the its velocity uo and does not depend on the electrical field strength. Let us now consider three level atoms with two stable ground states and one excited state. For definiteness, we assume a sodium atom with two hyperfine ground states (IF = l ) , IF = 2)), used in the experimental demonstration of the cooling mechanism in an evanescent wave (see Fig. 6). The main simplification of the real scheme of levels is that the hyperfine structure in the excited state is ignored, which is justified for the laser detuning larger than the splitting of the excited state. The laser frequency w of the surface wave is blue detuned (A = w - w , ) from the frequency w 1 of the transition IF = 1) + I e). The atom interaction potentials for the two ground states IF = 1), I F = 2) are
212
Victor I. Balykin
FIG.6. Energy level of the Na atom and position of the laser frequency w relative to the atomic transitions in the evanescent wave cooling experiment.
The potential U2(z) due to larger detuning ( A + A H F S ) is weaker than the potential Ul(z). If the spontaneous emission occurs during the reflection, the atom may then fall back in either one of the two ground states. The spontaneous transition to the sublevel IF = 1 ) makes only a small perturbation on the atomic motion through the impulse kick in the absorption-spontaneousemission event. The transition to another sublevel IF = 2 ) changes the atomic potential from (94a) to (94b) and, as a result, the kinetic energy of the atom after reflection is smaller than the incident one. The average transverse energy lost ( A E f r )in one reflection (Soding et al., 1995) is
where q is the branching ratio to the lower hypertine ground state. The dielectric surface makes the dynamics of the atomic motion more complicated (Desbiolles et al., 1996). As the atom approachesthe surface, it starts to feel the attractive van der Waals potential (see Section 1II.B). The total resulting potential is then (96) ' 2 = 'ev + u v ~ and it could be quite different from the evanescent wave potential Uev.As a result, the atom feels the repulsive potential in a few wavelengths distance from the sur-
ATOM WAVEGUIDES
213
face, which becomes attractive only at the shorter distance. The combined potential (96) reaches its maximum values for two sublevels IF = 1) and IF = 2) at the different distances from the surface. Consequently, there is a certain probability for a sticking of the atom to the surface after the cooling event, because the atom can go to the attractive part of the combined potential. And finally, the van der Waals interaction affects the cooling process in the evanescent wave as the spontaneous emission rate of an atom near a dielectric surface is enhanced. The experimental realization of the cooling of atoms in the evanescent surface wave was performed with both a thermal atomic beam (Ovchinnikov et al., 1995) and with laser-cooled atoms (Desbiolles et al., 1996). To observe cooling of sodium atoms in the evanescent wave, Ovchinnikov et ul. (1995) measured the change of transverse velocity of atoms during the reflection processes. In one single reflection event the initial transverse momentum of 42 hk decreased to 28 hk, which corresponds to an almost 50% loss of the transverse kinetic energy of the atom.
IV. Guiding Atoms with Evanescent Wave A. ELECTROMAGNETIC FIELDIN OPTICAL HOLLOWFIBER We first consider a simple cylindrical-core hollow dielectric waveguide as shown in Fig. 7. The refractive index of the cylindrical core and cladding are denoted by n and n 2 . The inner diameter and the thickness of the waveguide are 2p and Ap, respectively. Shortly we will review some general properties of a simple cylindrical dielectric waveguide (Jackson, 1975; Okoshi, 1982).The electromagneticfield propagating in the z direction along the waveguide with angular frequency w and the phase constant p can be expressed as
,
E(x, y , z ) = E(x, y)e[i(wr-@)l
(974
H(x, y , z) = H(x, y)eIi(ot-@)l
(97b)
From Maxwell equations we have the following relationship between the transverse components E x , E,, H,,and H , and the z-components of the fields
E = - -P:
(aaE; P--
wP%j
214
Victor I. Balykin
/
/
/
/
/
/
hollow region
r
no.
FIG. 7. Diagram of a cylindrical-core hollow fiber. The refractive index of the cylindrical core and cladding are denoted by n , and n2, respectively. The inner diameter and the thickness of the fiber are 2p and Ap, respectively.
The parameter P, is called the transverse phase constant (or propagation constant)
Pf = k2n2
-
P 2
(99)
In Eqs. (98-99) n is the refractive index of the core or cladding, k2 = w ~ EE , e0, and p, po denote the permittivities and permeabilities of the mediums and vacuum, respectively.The longitudinalcomponentsof the field obey the Helmholz equation
The similar equation holds for the magnetic field component H,. Solutions of Eq. (100) are well known. The solution of the wave equation is separable in polar coordinates. If we express the longitudinal component of the electrical field as a product of the radial part R,(r) and the azimuth one a,(@
E,(r) = R,(z)@.,(@
(101)
then for the radial and the azimuthal components there are two independent equations
~
~
~
ATOM WAVEGUIDES
215 ( 102a)
d 2@,
-
ae*
+ n2@, = 0
(102b)
The general solutions of these equations are
R,(r)
=
AJ,,(P,r) + A’N,(P,r) CK,(lP,l,> + C’Z, @,(@
=
cos(n6 + rp) sin(n6 + rp)
(P, - real) (P, - imaginary) (rp = const)
( 103a)
(103b)
where n is the azimuthal mode number; A, A’, C, C’ are constants; J,, N , denote the nth-order Bessel functions of the first and second kind, respectively; and I,, K , are the nth-order modified Bessel functions of the first and second kind, respectively. The transverse components of the electrical field are derived from the longitudinal component (Eqs. (103)) and from Eq. (98). The solutions of Eqs. (103) hold true in each region of the optical fiber. These solutions have to be matched at the interfaces so as to satisfy the boundary conditions. Let us recall the mode classification in optical fiber. The existence of different kinds of field configuration follows from the solution of a wave equation that satisfies the boundary conditions. In a hollow metallic cylindrical waveguide for perfectly conductive walls, the boundary condition demands that on the surface for electrical field E,I,
=
0
( 104a)
and for the magnetic component on the surface
-dB Isan
= 0
The two different boundary conditions specify two different eigenvalue problems of the solution of the wave equation and as a result two different categories of electromagnetic modes: the transverse magnetic ( T M ) modes ( H , = 0; boundary condition (104a)) and the transverse electrical (TE) modes (E, = 0; boundary condition (104b)). In optical dielectric fiber the boundary conditions are other than in Eq. (104) and the modes picture is richer. The TE and TM modes can still exist, but only for the azimuthal number n = 0 (see Eq. (103)). When n # 0, the boundary condition at the core-cladding can be satisfied only when we choose different linear combinations of TE and TM modes in the core and the cladding. Such combinations of TE and TM modes are called “hybrid modes” (EHand HE). Usually the fiber modes are first classified according to azimuth mode number n,
216
Victor I. Balykin
then by the TE, TM, EH, and HE mode designations, and, finally, by radial mode number 1 (1 shows the number of the radial variation of the longitudinal component of the field). According such classifications the lowest order mode in a dielectric waveguide is the HE, mode. To find the explicit expression for the field (103) it is necessary to find the propagation constant pt, Eq. (99). The propagation constant can be found from the boundary conditions at the core-cladding interface. The explicit dispersion equation for the dielectric hollow waveguide can be found in It0 ef al. (1995). In a general case the dispersion equation is rather complicated but in many practical cases the so-called “weakly guiding approximation” (WGA) holds true. WGA can be applied to the cases where the refractive index difference An = (n: n3/2n: between the core and the cladding is smaller than unity. This approximation greatly simplifies the dispersion equation and, as a result, gives convenient formulas for the electromagnetic fields and the atom-field potential. In the WGA case it is useful to introduce a new parameter m (Okoshi, 1982) instead of the azimuth mode number n
[:
m= n
+1
(for TE, TM modes) (forEHmodes) (for EH modes)
(105)
Then the dispersion equation can be written in the unified, very simple, and compact form
uJ,
,
(u)lJ, (u) = w K , - (w)/K,(w) (106) where u = &a and w = I p, la are the normalized propagation constants for the core and cladding, respectively. In WGA all modes with a common set of m and 1 (the radial mode number) satisfy a common dispersion Eq. (106).These degenerate modes are called “LP” modes (Gloge, 1971). The abbreviation “LP” means “Linear Polarized.” The field distributions of the transverse componentEx, E,, are identical for those modes that belong to the same LP mode and the rather confusing name “linear polarized” is related to this fact. Let us return to the problem of a hollow dielectric waveguide. The solution of the wave equation (100) can be in this case written for the hole, the core, and cladding in polar coordinates ( p , 8, z ) as (Marksteineret al., 1994) -
1 n
E, =
XP)eine
[A,J,(kp) + A,Y,,(kp)]eine ! A,K,,(ap)eine I(
( 107)
The constantsA A,, A,, A , can be defined from the conditions of a continuity of the field at the interfaces between the core, the cladding, and the central vacuum region of the fiber. The transverse propagation constant k and the transverse attenuation constants a and y, obey the relationship
ATOM WAVEGUIDES
217
10 9 8 7 6
x 1 5 Q 4
3 2 1
0 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2
Aplh FIG. 8. The map of the mode structure regime of the dielectric waveguide in the p-Ap plane, where p is the radius of the hollow region and Ap is the thickness of the core. In the case of a large radius of the hole the fiber can be considered as a planar dielectric waveguide. At a small thickness of the cladding it is still possible to excite only a fundamental mode, which in this case corresponds to the TE,, and TM,,modes of a slab waveguide. For a larger hollow region and a cladding thickness there is only a multimode regime. When the radius of the hole is on the order of the optical wavelength, the hollow fiber is a single mode fiber (from Marksteiner et a/., 1994, Fig. 2, Phys. Rev. A 50, N. 3, 2680, reprinted with permission).
x 2 = p2
k2 =
- k2
- ( p 2 - n:k2)
cy2 = p2
- n$k2
(108a) ( 108b)
(108c)
The detailed treatment of the electromagnetic field in a general case of multimode fiber can be found in Savage et al. (1993), Marksteiner et al. (1994), It0 et al. (1995a) and Ito et al. (1995b). The mode picture of a hollow fiber depends on the geometrical parameters of the waveguide. Decreasing the radius of the hole and the thickness of the core leads to disappearance of high-order modes and under certain conditions all modes except the fundamental mode HE,, are cut off. Figure 8 shows the map of mode structure regime of a dielectric waveguide in the p-Ap plane where p is the radius and A p is the thickness of the core. The refractive index of the core and the cladding are chosen as n = 1.5 and n2 = 1.497, respectively. In the case of large radius of the hole the fiber can be considered as a planar dielectric waveguide. At a small thickness of the cladding it is still possible to excite only a fundamental
218
Victor I. Balykin
10
I
9
Hole
I I
0.5
1
0
I
Core
I I
1.5
2
Cladding
2.5
3
r/P FIG.9. Electrical field amplitude for the lowest order HE,,mode in the annular core of a hollow fiber. The hollow radius and the core thickness equal 2.889 A (from Marksteiner er al., 1994, Fig. 3, Phys. Rev. A 50, N. 3,2680, reprinted with permission).
mode, which in this case corresponds to the TE,, and TM,, modes of a slab waveguide. For a larger hollow region and a cladding thickness there is only a multimode regime. When the radius of the hole is on the order of the optical wavelength, the hollow fiber is a single-mode fiber. The electrical field strength in the single-mode fiber has the form (Marksteineret al., 1994)
E(r, 0, z) = E(r)u(r, O)eivz-or)
+ C.C.
(109)
where E(r) is the electrical field strength and u(r, 0) is a complex unit vector. The electrical field (109) is the field of HE,, mode with azimuth mode number n = 1. The dependence of the electrical strength on radius for this mode is shown in Fig. 9. From the general expression (103) it follows that the electrical field (109) has angular dependence; however, for a circular polarized light field the time average field is circularly symmetric on the slow time scale of the atomic motion. The most interesting case for the guiding of atoms in a hollow fiber corresponds to the fundamental mode excitation in the fiber: The single-mode fiber guarantees that the electrical field inside the fiber has no state zero anywhere on the wall. In the multimode case the electromagnetic field inside the fiber is a superposition of all allowed eigenmodes. The amplitude and the phase of the resulting field depend strongly on the way the fiber is coupled to the external laser source.
ATOM WAVEGUIDES
219
B. CYLINDRICAL HOLLOW FIBERAS ATOMWAVEGUIDE The use of a hollow optical fiber (HOF) for a guiding of atoms was first considered by Savage, Marksteiner and Zoller (1993). In their first publication (Savage et al., 1993) they showed the feasibility of HOF as a new element of atom optics. In the next publication (Marksteiner et al., 1994) they developed the theory of quantum motion of atoms in a hollow fiber and analyzed in detail the different loss mechanisms in HOE Very soon it was also considered a possible application of HOF: atomic gravitational cavities from optical fibers (Harris and Savage, 1995). The most severe limitation on the HOF for the guiding of atoms is spontaneous emission, as it limits the atom confinement lifetime. One of the ways to overcome this rather fundamental drawback of all laser light guiding schemes is to use the confining force with a reduced spontaneous emission rate. The mechanical potential due to Raman transition could be the solution of this problem (Hope and Savage, 1996). 1. Quantum Mechanics of a Cylindrical Atom Waveguide We consider the motion of atoms in the hollow cylindrical fiber with the following assumptions. A laser light is coupled to the fiber and the fundamental mode HE,, (Eq. 109) is excited. The evanescent field of the HE,, mode in the hollow region produces the repulsive potential (Eq. (9 1)) over the surface on the inner wall. Near the surface atoms are attracted to the dielectric by the long range van der Waals force (Section III.B.4). The resulting potential remains repulsive in a few wavelength distances from the surface and becomes attractive close to the surface. The van der Waals force lowers the effective potential barrier seen by the atoms. Spontaneous emission during the guiding of atoms leads to a recoil heating of atoms and the fluctuation of the dipole force of the optical potential (Section II1.A). Spontaneous emission is considered here only as one of the loss mechanisms in the atom guiding. Other loss mechanisms are the tunneling of atoms to the surface and the nonadiabatic transitions. We are interested in the atom wave model structure and the atom energy eigenvalues. The stationary Schrodinger equation for an atom in the waveguide is
where V(r) is the combined evanescent wave-van der Waals potential in the waveguide. Because the potential V(r) is axially symmetric and depends only on the distance r of the atom from the surface, the cylindrical coordinates ( p , 8, z) are best adapted to the problem. Then the Laplacian in cylindrical coordinates (Jackson, 1975) is
Victor I. Balykin
220
A = -a+2 - - +1 -a- + - 1 ap2
pap
p2
a2
d2
ae2
az2
and the wavefunction q(r) is a function of the variables ( p , 8, z). The separation of variables leads to the Schrodinger equation in the form (Dowling and GeaBanacloche, 1996)
with the wavefunction as p(p, 8, Z) = R(p)exp(~il8)exp(~ik,z)
(1 13)
where k, = 21r/A,, and A, is the atom de Broglie wavelength. The solution of the Schrodinger equation (1 12) with the combined potential V(r) is a rather complicated problem. The comprehensive theory of atom guiding with only the evanescent wave optical potential can be found in Savage et al. (1993) and Marksteiner et al. (1994). However, main features of the atom guiding can be earned by considering a more simple case of an infinite step potential on the wall of the waveguide
For the lowest order atom wave modes the transverse de Broglie wavelength is much larger than the penetration length of the evanescent wave and the simplified step potential model fits the guiding process adequately. Such a model leads to considerable simplification of the picture of the guiding of atoms. From Eqs. (112) and (113) with the potential V ( p ) in the form of 4.(114), the radial equation for the wave function takes the form (Dowling and GeaBanacloche, 1996)
with the boundary condition R(a) = 0 and E = (2m/h2)E. The last equation can be transformed into the Bessel equation by the changing of variable u = p v q
=p a
The solution of Eq. (1 16) is the Bessel function of the order 1 R ( p ) = const J,(up)
(1 17)
ATOM WAVEGUIDES
22 1
Finally, the atom wavefunction becomes
where 1 is the number of azimuth nodes and q is the number of radial modes in the atom wavefunction. The boundary condition on the wall of the fiber gives the quantization conditions on the transverse motion (119a)
clqa = jlq
which can be rewritten for total atom energy as (119b) where j , is the qth root of the Bessel function (Eq.(1 17)). The transverse quantization condition of Eq. (119) couples the longitudinal velocity of the atoms ulq = fik,q/mwith the transverse velocity u$ = fia,/m. If we assume that the waveguide is loaded from the external atom source then only atoms with the transverse velocity u t will be guided in the fiber and the quantization condition (1 19) will determine the corresponding longitudinal atom velocity uIq.For the basic atom wave mode qolthe transverse velocity atom in the fiber is of the order fig/,
vt =m
--
1.2 A ~a
-vr
and considerably smaller than recoil velocity u,. For the hollow fiber parameters of Marksteiner et al. (1994) the transverse atom velocity of the basic mode equals uk, = 0.12 u,. Such small atom velocity can be reached, for instance, by releasing atoms from a magneto-optical trap and by a following spatial selection of atoms with small transverse velocities. The scheme of laser cooling of atoms to subrecoil temperatures (Aspect et al., 1988; Reichel et al., 1995) can also be used for a loading of the basic mode of atom waveguides. 2. Losses in Atom Waveguide
As we already pointed out there are mainly three loss mechanisms for atoms guided in the hollow fiber (Balykin and Letokhov, 1989; Marksteiner et al., 1994): (1) spontaneous emission, (2) tunneling of atoms to the dielectric surface, and (3) nonadiabatic transitions between states corresponding to the attractive and repulsive potential of the surface wave. Each of these processes limits the lifetime of atoms in the basic mode and, finally, the length of the atom waveguide. The comprehensive theory of the different loss mechanisms can be found in Marksteiner et al. (1994). Here we consider only the physics of the loss mechanisms
222
Victor I. Balykin
and will make simple estimations on the atom lifetime and the length of the atom waveguide. It is assumed that the losses are small enough that they can be considered independently. a. Spontaneous Emission. An atom in a waveguide moves in two different regions: in the surface light wave and in the light free space near the center of the waveguide. In the case of a single-mode atom waveguide with a small hollow region such a picture is oversimplified. A more appropriate physical picture will be that the atom moves in the adiabatic confining state, which converges to the atom ground state at the limit of a zero guiding electrical field (Kazantsev er al., 1990). The adiabatic confined state has admixtures of the excited internal state and there is nonzero probability for emission of a spontaneousphoton for the atom in this state. When an atom moves in the guiding field, it may remit a spontaneous photon. There are two different physical processes, which accompany each individual spontaneous emission event. (1) After the re-emission of the spontaneous photon the atom remains in the same adiabatic state. Then the only result on the atom motion is the change of its momentum on the recoil value. As we saw in the previous section, for a single-mode atom waveguide the transverse velocity of the guided atom is smaller than its recoil velocity. That means that a single spontaneous emission event transfers the atom from the basic modes to the higher order modes. (2) If, as result of the re-emission of the photon, the atom changes its confined state to the unconfined one the atom moves to the wall and sticks to it. In both cases a single spontaneous emitted photon destroys the single-mode guiding regime. We conclude that in both cases the first re-emitted atom photon will drive it out of the basic mode and the atom lifetime in the fundamental mode will be determined by the photon re-emission rate. The probability for the atom to undergo spontaneousemission during the time interval dt is given by dP = 2Y V ( P ) dt
(121)
where ~ ( pis) the population of the excited state of the atom (which has to be considerably smaller than unity). The population of the excited state depends on the surface light intensity (through the Rabi frequency f l , ( p ) and the light detuning A) and for the case A >> fl, it can be estimated by
The total probability of atom undergoing spontaneous emission in a round trip between the wall of the fiber is
ATOM WAVEGUIDES
223
The ratio aR/A in Eq. (123) is not a free parameter of the problem. With a given laser power the detuning is determined by the maximum height of the surface potential, which has to be higher than the transverse kinetic energy of atom Urn,, = haa/2A 2 r n ( ~ & ) ~ There / 2 . is also a limitation on the value of the Rabi frequency. It is not wise to put too much laser light in the evanescent wave: The scattering light on the dielectric surface of the fiber limits maximum intensity of the evanescent wave (see Section III.C.3). Scattering light in the fiber will determine the final achievable parameters of the atom waveguide. b. Tunneling to Dielectric Surface. When the transverse kinetic energy of an atom in the waveguide is close to the potential barrier of the combined evanescent wave plus the van der Waals potential, the atom may undergo tunneling over the resulting potential. In that case, the atom reaches the dielectric surface and it is either scattered in a diffuse manner (with a considerable gain of its transverse kinetic energy) or absorbed on the surface. In both cases, the atom is lost from the waveguide. The tunneling limits the guiding length ltUnof the fiber to the value ‘tun
-
- u
‘tun
( 124)
where rtunis the tunneling-limited atom lifetime in the waveguide and u is atom velocity along the fiber. The lifetime 7tuncan be estimated quasi-classically by noting that an atom in its motion in fundamental mode bounces in transverse direction at frequency v = uL/2a, where v Lis the transverse velocity of atom and a is the radius of the hollow region of the fiber. Then the lifetime r,,, = ( a / u L ) T where T is the transmissivity of the resulting potential barrier. In the case that the transverse velocity is less than the maximum reflected velocity u L < u,,,, the transmissivity equals
where Q 2 ( p ) = 2rn[U(p) - E ] / h 2 ,p , , p 2 are the classical turning points, and U ( p ) is the atomic potential energy. For the fundamental atom wave mode pol, the maximum guiding length through the tunneling becomes
Assuming uL/urn,, = 0.8 and characteristic penetration depth of the evanescent wave p = h/2, the transmissivity is of the order T = lop6. Then for atoms with a longitudinal velocity u = 10-100 cm/s, the guiding length is very long lguid = lo4 -+ lo5cm. c. Nonadiabatic Transitions. This loss mechanism is due to the transition from the confined state to the nonconfined states. The nonadiabatic transitions appear
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Victor 1. Balykin
if the atom cannot adiabatically follow the local changes of the confining field intensity. For the typically used parameters of atom waveguides the nonadiabatic transition loss rate is negligible in comparison with the spontaneousemission loss rate (see Kazantsev et af., 1990; Wilkins et al., 1993; Marksteiner et al., 1994). 3. Loading of Atom Waveguide
The scheme of exposing the fiber’s input hole to an atom vapor is probably the most simple scheme of a loading of atom waveguides. In this scheme the input fiber face is in one vapor cell (which is considered a source of atoms) and the flux of atoms is directed into another empty cell (where the atoms can be detected). One of the main parameters of such a scheme is the total flux of atoms through the fiber (Savage et al., 1993; Renn et al., 1997). Suppose that in the first cell we have the Maxwell velocity distribution of atoms and the input fiber facet and the laser beam, coupled into the fiber, does not influence the velocity and the spatial distribution of atoms. In the case of a single-mode guiding regime the fiber selects from the Maxwell distribution the atoms with velocities that fulfilled the transverse quantization conditions (Eq. (1 19)).In a multimode regime of atom guiding the fiber is coupled to the atoms having transverse velocity less than the maximum reflected by the surface wave velocity u,,,,. The atoms at the input facet of the fiber have the uniform spatial distribution and the Maxwell distribution
n(v) = n(u,, u,, u z ) = no (2fkTy
-mv212kT
(127)
where no is the density of atoms, T is their temperature, and k , is the Bolzmann constant. The total flux of atoms Q coupled to the fiber equals to the number of atoms striking the entrance surface of the fiber per second and having the transverse velocity less than u,, Q =S
1;
du,
rm::1 dux
du, n(v) = SnouL,/fi ( u )
(128)
-vm
where ( u ) = (2k,T/m) is the average velocity of atoms in the source cell and S is the area of the entrance surface of the fiber. In the case of a single-mode atom waveguide the transverse velocity (Eq.(120))is uo, = u,h/u and the corresponding flux Q, of atoms through the fiber is Q, = noupA2/ ( 6 )
(129)
For the room temperature atom cell, Eq. (128) gives Q, = lO-l7nO atoms/sec. Actually the flux will be even less if we take into account the transverse quantization condition (Eq. ( 1 19)).From this estimation it is clear that for a single-mode
225
ATOM WAVEGUIDES
fiber the considered scheme of the loading of atoms from a vapor cell is not practical. If we consider a MOT as the initial source for a loading of a single-mode fiber then the longitudinal velocity of atoms is considerably smaller than in the previous case but the maximum achievable atom density is also smaller than in a vapor cell and, once more, the flux is discouragingly small. In the next section we will consider the hornfiber, which permits achieving high coupling efficiency of the loading of atom waveguides. In the case of a multimode fiber the loading is a considerably easier problem. For instance, the flux of atoms coupled to the fiber with a radius of a hollow region 50 ,urn and the maximum transverse velocity urn, = 100 cm/sec equals Q=3* no. At the density of atom no = 1014cmv3the guiding flux (see Eq. (120)) will be about lo3 atomdsec. These estimations hold true for the case of a straight fiber. For a curved fiber there are additional losses: Atoms moving through a bent section of the fiber experience a centrifugal force, which presses them into the wall of the fiber. When the guiding potential exerts insufficient force to overcome the centrifugal force the atoms are lost (Renn et al., 1997). Savage et al. (1993) considered the possibility of making a loop of the fiber between the source and the detection chambers. In such a loop the relatively large axial velocity is converted into transverse velocity due to the bend in the fiber. From the condition that the centrifugal force is equal to the maximum guiding force we can easily estimate the minimum radius of the loop rmin= ~ ( ( u ) / u ~ ) ~ / / ?
( 1 30)
where p - I is a decay length of evanescent wave in the fiber and u; is the transverse velocity of the atom. For atoms with a thermal longitudinal velocity the minimum radius is very large: rmin= lo3 meters for Cs and 100 meters for Nu atoms. For cooled Cs atoms the minimum radius is only rmin= 10 cm and the idea of the atomic fiber loop can be feasible. A bending of a fiber decreases atom flux transmitted through the fiber. Using the expression (128) we can estimate the flux of atoms through the bent fiber. In this case the limit of the integration in Eq. (128) over u, is uzm,, where uzmax is determined from Eq. (130). Finally, the flux of atoms through the bent fiber becomes Qbend
= S no fa
u:rnax/fi
(v>~>
(131)
For a multimode fiber with the following parameters urn, = 100 cm/sec, uzm, = lo3 cm/sec, and ( u ) = 500 m/sec, the atom flux Qbed is several hundreds of atoms per second and it can be easily detected. The main interest in the bent fibers comes from exciting opportunities of building the atom interferometers similar to the conventional optical interferometersbased on the fibers.
Victor I. Balykin
226
/ X
FIG. 10. Hornfiber: Atoms are injected continuously from a magneto-optical trap into a hollow waveguide with an evanescent light wave formed on its inside surface. The atoms channel over the waveguide while undergoing reflections from the evanescent wave.
C. HORNSHAPEHOLLOW FIBER As we saw in the previous section there are severe problems in an efficient loading of a single-mode atom waveguide. To overcome the loading problem of the atom waveguide, Balykin el al. (1996) proposed using a special type of atom waveguide: a horn-shape hollow fiber. Their waveguide has a shape of a horn with a large entrance diameter (see Fig. 10). The entrance diameter is about one millimeter to fit the typical size of a magneto-opticaltrap. At the end of the horn fiber its diameter is decreased to the diameter of a single-mode atom waveguide. As in
ATOM WAVEGUIDES
227
the case of a cylindrical hollow fiber the internal wall of the horn fiber is supposed to be covered by an evanescent light wave. A magneto-optical trap is considered an external source to load the horn fiber. A simple geometrical consideration of a motion of an atom from a MOT source into the horn fiber shows that after several specular reflections from the inner fiber wall the atom will be directed out of the fiber. To overcome this particular loading problem, it was proposed to use at the initial loading-guiding stage the dissipative reflection of atoms from the evanescent wave (see Section III.C.4). The role of dissipative reflection is not only to load atoms into the horn fiber. The dissipative reflection also implements (a) the cooling of the transverse velocities of atoms in the fiber (and also the longitudinal and the azimuth velocity components through the coupling all degree of freedom); (b) the accumulation of atoms in the narrow end of the fiber; and (c) a population of the basic mode of the final cylindrical part of the fiber. Let us first consider the behavior of an atom in a plane hollow waveguide formed by two parallel dielectric plates with an evanescent wave on their inner surfaces (Subbotin et al., 1997).For definiteness, let us consider the sodium atom. The evanescent wave frequency is assumed to be blue-detuned to all transitions between the ground state and the excited state. Assume also that the waveguide is filled with another light whose frequency is tuned to resonance with the transition IF = 2)-1 P) ( “repumping” light). The evanescent waves provide reflection of the atom in the course of its being channeled over the waveguide. This reflection may be either elastic (the atom remains in the state IF = 1)) or inelastic (the atom moves to the state IF = 2)). An atom in the state IF = 2 ) residing outside the evanescent waves is moved back to the sublevel IF = 1) by the repumping light. Let the initial atomic velocity projection onto the waveguide axis be positive, and the transverse atomic velocity be other than zero. In that case, the atom will start to be channeled over the waveguide, while undergoing reflections from the evanescent waves. The average reduction of the transverse energy of the atom in a single reflection event is given by Eq. (95). The average rate at which the atomic energy decreases in the waveguide is governed by its change in a single reflection event and the time between two consecutive reflection events, A t = d h L , where d is the diameter of the fiber and u I is the transverse velocity of the atom. Dividing both sides of Eq. (95) by At, we get the following expression for the average atomic energy reduction rate
From Eq. (132) follows a differential equation for the rate of extraction from the transverse kinetic energy component E , of the atom in the course of its being channeled over the waveguide:
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Victor I. Balykin
A simple estimate gives the dependence of the transverse kinetic energy of the atom on the longitudinal coordinate z
where E,o is the initial transverse kinetic energy of the atom, u, is the longitudinal velocity, and C is a constant that depends on the system levels of the atom and the parameters of the laser light. The expression (134) holds when the transverse kinetic energy of the atom is much higher than the recoil energy. To determine the function E , at a small transverse kinetic energy of the atom and for a rather complicated shape of the horn waveguide, the Monte-Carlo method was used (Balykin et al., 1996; Subbotin et al., 1997). The equation of motion of the atom in the waveguide is (135) mi: = -VUl,2,3(r) + f(t) where U1,2,3(r)is the potential energy of the atom in one of the states I F = 1 ), I F = 2 ) , or I P ) ( 136a)
( 136c)
The term f(t) takes into account the change in the momentum of the atom accompanying the change in the state. When atoms being channelled along the waveguide, some of them may be lost as a result of, first, spontaneous emissions and second, subbarrier tunnelling. Both these mechanisms were taken into account in the model by excluding from further consideration those atoms that reached the surface of the dielectric in the course of reflection. The atoms channel along the waveguide while experiencing numerous reflections from the evanescent wave. Some of them (around 30%) are of inelastic character and lead to a reduction of the radial velocity component; its azimuthal counterpart remains unchanged. Figure 11 shows the results of the computed average transverse velocity of atoms propagating along the cylindrical fiber as a function of the longitudinal coordinate. The inside diameter of the waveguide was chosen as d = 10 p m and
ATOM WAVEGUIDES
229
60
50
40 u)
2
30
0
‘k
>
20
10
n 00
02
04
06
08
1.o
z; cm FIG. 1 1. Figure shows the transverse velocity component of atoms propagating over a cylindrical waveguide as a function of the coordinate z, along the waveguide at the presence of a mechanism by which the atom in the waveguide loses some of its kinetic energy in inelastic reflection from the evanescent wave (Reprinted from Opt. Comm., Vol. 139, Subbotin e?al., 1997, p. 107, with kind permission of Elsevier Science-NL,Sara Burgerhartstraat25, 1055 KV Amsterdam, The Netherlands).
its length as L = 0.8 cm. The solid line is the result of parametric adjustment. The minimum temperature to which the atomic ensemble can be cooled in such a cyK. The corresponding transverse velindrical waveguide is about T = 8 locity is ulm = 7 cm/sec, which is comparable with the recoil velocity. With the waveguide being 0.8-cm long, an atom undergoes about 30 reflections from the evanescent wave. There was also observed insignificant narrowing of the mimuthal velocity distribution due to the coupling between the radial and azimuthal degree of freedom in the gravity field. The extent of this coupling is determined by the parameter a = mgd/Elmax,which in the considered case was 10 -*. The narrowing of the velocity and spatial distributions of atoms in an ensemble in the course of their channeling over a cylindrical waveguide points to the possibility of using this effect to increase the atomic space phase density. Now consider a two-dimensional curved horn shape waveguide formed by two dielectric surfaces with evanescent wave on their inner surfaces (Balykin e l al., 1996; Subbotin et al., 1997). The waveguide has an entrance size of 500 p m and an exit size of 10 pm. Figure 12 shows various types of atomic trajectories in such a waveguide. Atoms at the initial instant of time were placed near the entrance in the waveguide and were imparted some initial velocity directed inside the waveguide. The absolute value of the velocity obeyed a thermal distribution with a mean value of 50 cm/s. The atoms channeled over the waveguide while undergoing numerous reflections from the evanescent wave. a
230
Victor I. Balykin
FIG. 12. Typical trajectories of atoms in a homfiber: (1) damped oscillations; (2) trajectory of an atom that has tunneled through the potential barrier to the wall and has been lost; (3) strongly damped oscillations (Reprinted from Opt. Cornrn.,Vol. 139, Subbotin et al., 1997, p. 108, with kind permission of Elsevier Science-NL, Sara Burgerhartstraat25, 1055 KV Amsterdam, The Netherlands).
The presence of a mechanism by which an atom in the waveguide loses some of its kinetic energy in inelastic reflections from the evanescent light wave causes the equilibrium position of the atom to move gradually downward. Depending on the relationship between the dissipation rate and the natural oscillation frequency of the atom, the system exhibits one of the following two modes of behavior typical of linear oscillatory systems: (1) oscillations dying over many periods of oscillation (see Fig. 12), and (2) an exponential decay of the initial energy without oscillations. Figure 12 also presents the trajectory of an atom that has tunneled through the potential barrier formed by the evanescent light wave and “settled” on the waveguide wall. This fact is marked in Fig. 12 by a tombstone. Consider now the behavior of an atomic ensemble in a hollow threedimensional tapering curved waveguide shown schematically in Fig. 10, along with a cloud of atoms confined in a magneto-optical trap. Assume that atoms are being continuously injected from a magneto-optical trap into the waveguide cavity and their velocity distribution corresponds to a Maxwell one with a temperature of T = W / k , that can easily be attained in the trap. The entrance inside diameter of the waveguide corresponds to the characteristic size of the cloud of atoms in the trap and amounts to 500 pm. The average transverse atomic velocity in the ensemble at the exit from the waveguide amounts to some 10 cm/s, whereas the mean absolute velocity is around 20 cm/s. However, at the expected high densities of atoms in such a waveguide, there will take place the equalization of their kinetic energy distributions among all their degrees of freedom because of colli-
ATOM WAVEGUIDES
23 1
sions and a long channeling time. Therefore, one might expect that the average atomic velocity will be around 10 cm/s, which corresponds to an ensemble temperature of T = K. One can also see from Fig. 12 that the density of the atomic trajectories increases as the waveguide narrows down, which means that the atomic space phase density grows higher. Figure 13 shows the space phase density normalized to its initial value pph(z)/pphO as a function of the coordinate z along the fiber. One can see that while the atoms channel in the waveguide over a distance of L = 1 cm, their space phase density is increased by five orders of magnitude. The achievement of an extremely cold and dense sample of weakly interacting bosons gives an opportunity to use the horn fiber for investigating the quantum statistics phenomena and wave propagation of matter.
z; cm FIG. 13. Space phase density of an atomic ensemble in a hornfiber as a function of the longitudinal coordinate (Reprinted from Opt. Comm., Vol. 139, Subbotin et al., 1997, p. 109, with kind permission of Elsevier Science-NL, Sara Burgerhartstraat25, 1055 KV Amsterdam, The Netherlands).
232
Victor I. Balykin
If the space phase density of noninteracting bosons in an external potential exceeds a certain value governed by the form of the potential, a perceptible proportion of the particles in the ensemble will reside at the lower energy level. This phenomenon, known as the Bose-Einstein condensation, gives rise to some interesting physical properties of the ensemble, associated with the high degree of coherence of the wave functions of individual atoms. A Bose-Einstein condensate is formed when the average distance between the particles in the ensemble becomes comparable with the de Broglie wavelength. Let us reveal the population pattern of the atomic waveguide modes. We use the results of Marksteiner et al. (1994). An approximate equation for the radialrotational energy levels En,, of atoms propagating in a quantum fashion over a hollow cylindrical waveguide with an evanescent wave in the case where the atomic de Broglie wavelength is shorter than the optical wavelength is
Here n, m = 0, 1, . . . , are the radial and rotational quantum numbers, respectively; V,, is the barrier height of the evanescent wave; U is a dimensionless parameter characterizing the penetration depth of the evanescent wave into the vacuum; and eR = 2h2U2/Md2is an energy of the same order of magnitude as the recoil energy. Equation (137) can be numerically solved for E,,,,. In the system at hand, there is no longitudinal coordinate dependence, and so, for the sake of simplicity, let us examine a cylindrical waveguide section of fixed length L, assuming that the atomic wave functions go to zero at its ends (“blank” walls). Obviously the choice of the value of L is arbitrary and is only required to satisfy the condition L >> Am. The total kinetic energy of an atom in the waveguide is the sum of the energies of the radial-rotational and translational atomic motions
where 1 = 1,2, . . . ,is the translational quantum number characterizingthe motion of the atom along the waveguide. Let 10%of the total number of atoms confined in the magneto-optical trap, N,,, = 10 lo atoms, enter the waveguide cavity every second, this corresponding to an atomic flow of jin= lo9 atoms/s. Inasmuch as the flow j remains approximately constant (the loss of atoms is low), the density of the atomic ensemble in the cylindrical part of the waveguide is p =j/vS = 4 j r /
233
ATOM WAVEGUIDES P (cm-3)
1.0 x 1015
1.5 x 1015
2.0 x 1015
2.5 x 1015
3.0x
loi5
4
3
0
FIG. 14. Relative population of the fundamental mode of a cylindrical waveguide with a diameter of d = 4 p m in which the hornfiber terminates, as a function of the number of atoms confined in a magneto-optical trap, of which 10% are being injected every second into the hornfiber (Reprinted from Opt. Comm., Vol. 139, Subbotin et al., 1997, p. 109, with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands).
?rLd = 10 l4 cm-3, where v is the velocity of the collective motion of the atoms along the waveguide. The distribution of the atoms among the waveguide energy levels is defined by the Bose function 1
(139)
where p is the chemical potential of the system depending on the number of particles, N = p(7rd2/4)L, and the temperature Z which can be calculated numerically from the normalization equation
Figure 14 presents the relative population G = N,/N of the minimum-energy mode of the waveguide as a function of the number of particles in the magneto-
234
Victor I. Balykin
optical trap, NMoT,where N o is the number of atoms in the fundamental waveguide mode. One can see that at a certain number density of the particles, reached with realistic NMoTvalues, there occurs a sharp increase in the proportion of atoms in the fundamental waveguide mode. The relative population of the fundamental mode amounts to a few percent. We should note, however, the existence of numerous factors capable of preventing the attainment of sufficiently high particle densities and the formation of a Bose-Einstein condensate. These include the already noted three-particle collisions giving rise to the Nu, molecules, light-induced collisions in the evanescent wave, and excitation by diffuse light. Because a perceptible proportion of atoms in the output section of the waveguide are in one and the same quantum state, such a waveguide can be viewed as a coherent source of de Broglie waves. Let us estimate the main parameters of the source. The output flow of atoms with an average velocity of ( u ) = 10 cm/s is j,,, = jinG = lo7 atoms/s. The corresponding de Broglie wavelength is A,, = 0.2 pm. In that case, the divergence of the atomic beam issuing from the waveguide is cp = AdB/d = 5 X lo-, rad, and its brightness, b = jout/2q= lo8 atoms/s sr.
-
D. PLANAR WAVEGUIDES A two-dimensional atom waveguide permits us to prepare an atomic sample as two-dimensional gas whose behavior at small temperature and sufficient atom density could be quite different from three-dimensional atomic ensembles (Kosterlitz and Thouless, 1973; Svistunov et al., 1991; Stoof and Bijlsma, 1993). Several schemes were proposed for a two-dimensional waveguide. In one of them (Ovchinnikov et al., 1991) (considered as the atom trap) the waveguide is formed with two different evanescent waves at one dielectric interface. The evanescent waves are different in a sense of frequency detuning with respect to atomic transition and the depth of penetration in the vacuum. One evanescent wave is obtained from the total internal reflection of a red-detuned laser beam at a very small angle of incidence only slightly greater than the critical value. The decay length may in this case be as great as a few wavelengths of light. The second evanescent wave is produced by a blue-detuned laser beam incident upon the interface at a much larger angle of incidence. The decay length for such a wave may be only of the order of one tenth of the laser wavelength. If the atom is placed close enough to the interface, the first wave will tend to push the atom away from the surface, while the second one will tend to pull it toward the interface. In the case of weak atomic transition saturation, G << (1 + A2/y2). the resulting force on the atom may be represented as a sum of the forces acting on the atom in each of the two evanescent waves. If the additional condition A >> y is satisfied, the guiding potential may be represented in the form (Gordon and Ashkin, 1980)
ATOM WAVEGUIDES
U
f i y ( y G 1 / 2 A ,+ yG2/2A2)
235
(141) where A , and A2 and are the frequency detuning of the two evanescent waves and G, and G, are their local saturation parameters. The resulting potential is a Morse potential. The corresponding potential for the laser parameters G , = G I = 4 * lo7, A , = 5 . 105y, A2 = -106y, and the angle of incidence 8,= 47", and 8,= 45.7" (n = 1.4) has the potential minimum at a distance of xmin= h from the dielectric surface and the depth of the well is AU = 7hy. The width of the well at the A U / 2 level is of the same order of magnitude as the wavelength of light. The guiding time of atoms is determined mainly by their heating as a result of impulse diffusion: the atoms spend a lot of time in the high intensity red-detuned laser light. For the previously chosen laser parameters and for the sodium atom, the guiding lifetime is about one second (Ovchinnikov et al., 1991). Several possible schemes were considered to inject atoms in a double evanescent wave trap-waveguide. The promising scheme of Desbiolles and Dalibard (1996) is based on the Sisyphus effect and it is suitable for atoms with two internal ground states (alkali atoms). In their scheme the laser frequency has been chosen such that atoms in the upper ground internal state see only a strong repulsive potential. On the contrary, atoms in a lower internal ground state see the potential of both laser fields (the Morse potential) and can be confined in a direction transverse to the interface. The injection procedure works as follows (Desbiolles and Dalibard, 1996). Atoms are prepared in the upper internal ground state. They are introduced into the region of two evanescent waves where the repulsive potential keeps them away from the dielectric surface. A spontaneous Raman transition from upper to lower ground internal state can occur in the region of two evanescent waves. This procedure transfers the atom to the one of the states of the Morse potential. The calculation, perfumed by Desbiolles and Dalibard (1996), shows that a noticeable fraction of atoms can be loaded into the ground state of the Morse potential. Another scheme for two-dimensional atom guiding is based on the use of the standing wave with a large period (>> A) located in the vicinity of the dielectric surface. Originally this scheme was used for atom focusing (Sleator et al., 1992) and in the demonstration of an optical Stern-Gerlach effect (Sleator et al., 1992). To load such a two-dimensional waveguide the Konstanz group (Gauck et al., 1996) realized a magneto-optical trap in contact with a dielectric surface. Gauck et al. (1996) consider as a next step the continuous transfer of atoms from the surface MOT into the waveguide. An atom cavity with evanescent waves as mirrors (Balykin and Letokhov, 1989), with the mirrors separated in the micrometer region (Wilkens et al., 1993) and with a sufficiently large transverse mirror size, is another example of a twodimensional waveguide (Dowling and Gea-Banacloche, 1996). The basic physics of such waveguides is similar to that of a cylindrical evanescent wave waveguide. =
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Victor I. Balykin
V. Atom Waveguide with Propagating Laser Fields In this section we consider the different configurations of propagating laser fields for the guiding of atoms. The Gaussian laser beam was the first propagating laser beam that attracted a great deal of attention for the focusing and guiding of atoms. The beam can create the transverse confinement of atoms and does not restrict the motion of atoms along its axis. In 1978, Bjorkholm et al. demonstrated the focusing of an atomic beam that was propagating coaxially with the Gaussian laser beam. Actually, this pioneering atom optics experiment could be considered to a great extent as a first guiding experiment with laser light. In their experiment the light from a continuous single-frequency laser was superimposed upon a beam of sodium atoms. They used in the experiment negative detuning, and when the laser frequency is smaller than atomic absorption frequency, the force on the atoms is directed toward higher intensity and the atoms are attracted into the center of the laser beam. The atoms did not experience periodic focusing and defocusing (which actually means a guiding) only due to an optical pumping of sodium atoms: The duration of resonant interaction of atoms with a single frequency mode laser was less than the time of flight atoms through the superimposed laser beam. Let us consider briefly the main features of atom guiding in propagating laser beams. A. GAUSSIAN LASERBEAM The Gaussian laser beam has the following z-dependent beam radius (z along the laser beam) w(z)
=
wOd1
+ (Z/ZR)*
(142)
where wo is the minimum spot size of the laser beam and zR = (rw:/A) is the Rayleigh range. (The Rayleigh range equals the distance along the laser beam at which the laser beam diameter will be increased by a factor fi). If we consider the Gaussian laser beam as an atom waveguide, then the Rayleigh range determines the effective length of such a waveguide. For a single-mode guiding the internal radius of an atom waveguide has to be in diameter about several laser light wavelengths. At the minimum spot size wo = 2A, the Rayleigh range is very short and equals only zR = 4rA. It means that the Gaussian laser beam can be considered only as a multimode atom waveguide. Another limitation of the Gaussian laser beam as an atom waveguide comes from the character of the radial intensity profile of the beam: The maximum field intensity is on the beam axis. In the guiding of atoms by the Gaussian laser beam they are localized predominantly in the region of high intensity where the impulse diffusion is the most severe. The limitation imposed by the impulse diffusion was
ATOM WAVEGUIDES
237
observed even in the focusing experiment (Bjorkholm et al., 1978) where the time of atom-light interaction was only half of the period of transverse oscillation of the atom. Another limitation of the scheme based on the Gaussian laser beam (and all another schemes with the propagating laser beams) comes from a physical impossibility of “bending” the Gaussian beam as waveguide in a vacuum, which closes many potential applications of such an atom waveguide. All these limitations of a beautiful Gaussian laser beam as an atom waveguide push the numerous research groups to explore other laser field configurations. B. LASERLIGHTINSIDE OF HOLLOW FIBER Laser light itself can be easily and efficiently guided by the different types of optical fibers. The confinement of laser light inside a hollow fiber immediately solves two problems of atom guiding by the Gaussian laser beam: (1) the laser light can propagate without transverse spreading over a long distance; (2) inside an optical fiber the laser field can be “bent” almost without limitation. The first scheme for guiding of atoms with laser light inside a hollow fiber is proposed by Ol’shanii et al. (1993). In their scheme the light propagating along a hollow optical fiber is the lowest order propagating mode EH, I (see Section 1V.A). The transverse electrical field profiles of the mode E H , is a zero-order Bessel function
,
E(p, z, t)
=
eEoJO(Xp)ei(pz-ot)
(143)
and the intensity transverse profile is
0 )= 4J;(XP)
(144)
with a maximum intensity along the fiber axis. The mode EH,I can be excited in the fiber by a laser beam that propagates along the axis of the fiber (Renn et al., 1993). With a red-detuned laser frequency, atoms are attracted to the highintensity region along the fiber axis, as in the case of the Gaussian laser beam. This means that the scheme still has the same impulse diffusion limitations as in the case of the Gaussian laser beam: Atoms are localized predominantly in the region of high laser intensity where the impulse diffusion leads to their heating and, finally, to sticking of atoms to the inner fiber wall. The impulse diffusion limits atom guiding time and effective guiding length of this type of atom waveguide (Ol’shanii et al., 1993). The atom guiding time is just equal to the time it takes for the atom to enlarge its transverse kinetic energy to the value of the maximum potential depth V,,, of the guiding laser field r = mVmaxlD
(145)
where D is the diffusion coefficient. For the large laser detuning (A >> y ) the diffusion coefficient can expressed as (Ol’shanii et aZ., 1993)
238
Victor I. Balykin
D = -f i 2 k 2 yS 20 where s is the saturation parameter of the atomic transition s = 2sZi(p)/(y2 + 4A2). For guiding of thermal atoms with an average velocity ( v , ) = 5 . lo4 cm/ sec the corresponding atom guiding time, according to Eq. (145), is r = + 10-l sec. The corresponding guiding length is L = ( u z ) r= 5 . (lo2 t lo3) cm. For the guidance of ultra-cold atoms the heating mechanism limited the guiding length to a fraction of a centimeter, and in turn it limits the applicability of such a type of waveguide. Another serious drawback of the hollow fiber atom waveguide with a grazing incidence mode is that these modes decay exponentially themselves as they propagate along the fiber. The propagation constant of the grazing modes (Eq. 143)p = p’ + ip” has an imaginary part p”,which in an approximation of multimode fiber ka >> 1 has the form (Renn et aL, 1995) cc a3
(147)
Because the attenuation length decreases as the cube of the hole radius a, even for a relatively large (40 pm) fiber diameter, used in the Renn et al. (1995) experiment, the corresponding propagation attenuation length is only Up” = 6.2 cm. The attenuation of a propagating fiber mode limits the useful length of a fiber for atom guiding. For the curved fiber the imaginary part of the propagation constant has an additional term, which depends on the radius of curvature of the fiber and in turn it makes the guiding potential of an asymmetric elliptical shape.
c. DARKSPOT LASERBEAMS A simple circumvention of the serious diffusion problem in the atom guiding can be the use of a dark spot laser beam (DSLB): a laser beam with a minimum intensity in the beam center. The best known example for this type of laser beam is a TEM& beam (known also as a donut mode). A copropagating TEM;, beam was first considered as an atom focusing lens (Balykin and Letokhov, 1987; Gallatin and Gould, 1991; McClelland and Scheinfein, 1991). To realize a focusing and a guiding with a donut mode, the positive laser detuning has to be used, so the dipole force is directed toward the hollow center of the laser beam. In this type of atom lens and waveguides, atoms move predominantly through a relatively lowintensity region, where the rate of spontaneousemission is a minimum. Nowadays a rich variety of methods exist to create the dark spot laser beams, such as the transverse mode selection method (Wang and Littman, 1993), geometrical optical method (Herman and Wiggins, 1991), optical holographic methods (Lee et aZ., 1994),computer generated hologram method (Paterson and Smith, 1996; Kuppens et al., 1996). the method of Gaussian mode conversion (Beijers-
239
ATOM WAVEGUIDES
bergen et al., 1992) and the method based on the use of hollow optical fibers (Jhe et al., 1997).We shall only mention some of them and will consider in some detail those that found an application in atom optics and, particular, in atom guiding. In atom guiding application of the dark spot light beam there are three most important parameters of the beam: (1) the smallest achievable dark-spot size; (2) the degree of diffraction efficiency of the first bright ring; and, finally, (3) the divergence of the beam. 1. Mode Conversion Method
The simplest method of generating a dark spot laser beam is to place inside a laser resonator a small absorbing dot close to the optical axis of the resonator. Then the transverse intensity distribution of the laser output recalls the distribution close to the TEM;, mode. Unfortunately, the quality of the laser beam is rather pure. If, instead of the absorbing dot, a thin wire is placed inside the resonator, then the laser can be forced to generate high-quality Hermite-Gaussian modes. These modes have nodal lines at the position of the wires. The next step is a conversion of the Hermite-Gaussianmode to a dark spot laser beam. Let us consider this conversion procedure in some detail. Quite generally the modal structure of a laser resonator can be described by two group modes: (1) the Hermite-Gaussian (HG) and (2) the Languerre-Gaussian (LG) modes. The characteristic features of HG modes are that the spatial transverse distribution consists of series of stripes or dots. The spatial transverse distribution of LG modes consists of series-concentric circular rings. The amplitude of the HG and LG modes can be written as E:~(X,
Ef;,G(x,y, z)
eiE(x, y , z)exp[-i(n - m)$l
=
y, z)exp[-i(n
+m+
E,","(x, y, z) =
1)$1
(148) (149)
where 6:: and 6;: are rather complicated functions describing the spatial dependence of mode amplitudes (the exact expressions of 6:: and ~f;,"are not essential for our mode conversion consideration), $(z) = arctan(z/z,), zR is the Rayleigh range, and n and m are the indices of the modes. One can show (Beijersbergen et al., 1992) that a LG mode can be decomposed into HG modes of the same order as N
ikb(n, m, k )
Ef;,G(&y, Z) =
EG?k,k(X,
y, Z)
( 150)
k=O
+
where b(n, m, k ) are the coefficients of the decomposition, N = n m is called the order of the mode, and the factor i k in Eq. (150) corresponds to a 7 ~ / 2relative phase difference successive components. The physical significanceof Eq. (150) is the following: If we already have high-quality HG modes, then we can construct
240
Victor I. Balykin
a high-quality LG mode. The problem is that the laser can generate only one particular HG mode, but according to Eq. (150) it is necessary to have a complete set of HG modes. There is a particular property of HG modes that can help to overcome this problem: A HG mode whose principal axis makes an angle of 45" with the (x, y) axes can be decomposed into exactly the same constituent set
with the same coefficient b(n, m, k ) as in the expansion (150). What we need further is to find the way to rephase the term in the decomposition (15 1). This can be done by exploiting the dependence of the phase Gouy i,b in Eqs. (148-149) on the z-coordinate. For an isotropic Gaussian beam the Gouy phase appears in Eqs. (148-149) as (n m + l)$, where +(z) = arctan(z/z,) and the Gouy phase does not depend on the x, y coordinates. For an astigmatic HG beam the Rayleigh ranges into planes (x, z) and (y, z) are different. The amplitude of the mode has to be considered separately in these planes. The Gouy phase for the astigmatic case has two contribution for each transverse direction (Siegman, 1986)
+
with $,(z) and i,b,(z) dependent on the position of the beam waists and the Rayleigh ranges in the (x, z) and (y, z) planes, respectively. The mode converter can be constructed by using two cylindrical lenses. An initial isotropic HG mode beam passes through a first cylindrical lens whose axis makes an angle 45" with the mode principal axis (its nodal lines). After the cylindrical lens the mode became astigmatic and it accumulated two different Gouy phases for each transverse direction. At the position where the accumulated phase difference became 7r/2, the second cylindrical lens makes the mode once more isotropic and, finally, a LG type mode. Figure 15 shows the experimental results obtained with such a type of mode converter (Beijersbergenet al., 1992). The top row shows the input HG modes and the bottom shows the output LG modes. As can be seen from Fig. 15, the dark spot hollow light beam can be generated from the four different HG modes. A laser beam in the form of TEM,, mode can be easily transformed into a donut mode TEM;,, because a donut mode is a linear superposition of two HG modes, TEM,, and TEM,,. Kuppers et al. (1996) used the just described mode conversion method to focus and guide metastable Ne atoms. First, they generated a TEM,, laser mode by inserting a 20-pm diameter wire into the ring dye laser. Then, by using a mode converter consisting of two cylindrical lens, the TEM,, mode was transformed into a donut mode without a significant loss of power.
ATOM WAVEGUIDES
24 1
FIG. 15. Transverse intensity distributions of the HG,, mode (the top row) and LG,, mode (the bottom). The index of the mode n,m is indicated above the mode distributions (Reprinted from Opr. Cornrn., Vol. 96, N. 1, 2, 3, Beijersbergen et al., 1992, p. 128, with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands).
2. Computer-Generated Hologram Method A hologram is, in essence, an interference pattern arising from a coherent reference wave and a second wave (usually scattering from the object). Commonly, the interference pattern is recorded by a photographic emulsion (and very often it is called a hologram). During a reconstruction step, the reference field illuminates the hologram and the hologram regenerates the original light field. In the rare cases when a mathematical form of the second wave is known (in our particular case it is the TEMG, mode) it is possible to calculate the interference pattern (the hologram) and to omit the recording process. The interference pattern of a hologram contains the sinusoidal variations in the field intensity. In a standard technique of a photographic recording of a hologram, the sinusoidal variation in the field will lead to sinusoidal variation in optical density on the photo plate. By using a computer graphic technique it is possible to print out the calculated hologram pattern. However, in practice, it is much easier to print a binary hologram with a “square wave” transmission function, which incorporates the sinusoidal variation in the field intensity. The next step is a photo reduction of the binary hologram onto a photographic film. The result is the amplitude hologram, which always has rather low efficiency. To obtain a hologram
242
Victor 1. Balykin
with a high efficiency the amplitude hologram has to be transformed into the phase hologram. This can be done by using the bleaching technique: The bleached hologram is produced by making a contact print of the photo reduced binary hologram onto holographic plate and then bleaching the developed plate. The amplitude of phase modulation of the phase hologram can be adjusted by varying the exposition time. The efficiency of the phase hologram is limited only by the light scattering and the residual light absorption and it can be very high. Heckenberg et al. (1992) demonstrated a computer-generated hologram method to create the TEMA, mode. The Gaussian mode TEM, is used as a reference field. The reconstructed light beam is not strictly TEM;, mode donut mode but in the far field the spatial intensity dependence equivalent to the donut mode. By using such a dark spot laser beam He et al. (1995) demonstrated the trapping of reflective and absorptive microscopic-size particles in the dark central spot of the focused light beam. Kuppens et al. (1996) made extensive studies of two methods of generating a dark spot-size light beam: (1) the conversion method, and (2) the computer-generated hologram method. The quality of the donut mode obtained in the computer-generatedhologram method was higher than the quality of the mode generated by the conversion method. In the second method they used an external cavity diode laser beam, which was filtered by Fabry-Perot cavity. The obtained high-quality TEM, mode beam was sent through a computer-generated phase hologram. Kuppens et al. (1996) also noted some intrinsic limitation of the computer-generated hologram method: Light beam leaves the hologram in not a pure TEM& mode (the binary hologram is only an approximation of the exact interference pattern) and it can be decomposed into a series of the LaguerreGaussian eigenmode with different Gouy shift. The Gouy phases of the various modes evolve differently upon their propagation. This leads to different intensity distribution of the superposition of these modes at a different position after the hologram and the resulting field can even have an intensity maximum on the beam axis. Kuppens et al. (1996) solved this problem by placing behind the hologram a Fabry-Perot cavity to select the pure TEM& mode. The total conversionefficiency of the TEM, mode into the TEM;, mode was 44%.
3. Micro-Collimation Technique Recently, Yin et al. (1997) proposed and demonstrated a relatively simple and very efficient method to generate a dark hollow laser beam with a radial intensity distribution similar to one of a donut mode laser beam. Their method is based on the use of the modes of a hollow optical fiber outside of the fiber. As we saw in the Section (1V.B) the lower order modes of a hollow fiber are LPol, LP, 1, LP2,. All these modes are already the dark spot modes but inside the fiber. Is it possible to extract and preserve these modes outside of the fiber? The experiment of Yin et al. (1997) answers this question very positively. To demonstrate their idea the
ATOM WAVEGUIDES
243
authors used a He-Ne laser or diode lasers as a laser source. A focusing lens coupled the laser beam into a hollow fiber. The inner and outer diameters of the fiber were 7 pm and 14.6 p m , respectively. At the output of the hollow fiber a microscope objective was used to collimate the output beam from the hollow fiber. In the experiment the dark spot light beams corresponding to the LPol,LP, I , LP2, modes of the fiber were observed. The intensity distribution of the dark spot light beams in the central region were similar to the distribution of the TEMG, mode. The dark spot size of the output beam collimated by M-20 X microscope objective was about 50 p m at distance 100 mm from the output end facet and about 100 p m at distance 500 mm. The beam divergence at the near field was equal to 6.5 * l o p 5rad, whereas at the far field it was 4.0 * rad. The main limitations on the output parameters of the dark spot beam come from the divergence on the core of the fiber. Yin et al. (1997) expect to decrease further the diameter of the dark spot by using a large core. Guiding of atoms with the donut mode configurations overcomes the diffusion problem; however, it suffers still from inevitable divergence of a free propagating laser beam and its “rigidity” in a vacuum.
D. ATOMGUIDING WITH A STANDING LIGHT WAVE In this section we consider a standing wave of laser light as an atom guiding configuration. Intensity variation within a standing wave creates a series of parallel dipole-force potential wells, spaced by half of the wavelength. A single potential well of a standing wave is an almost ideal atom waveguide (at least theoretically). Indeed, for blue-detuning of laser frequency with respect to the atomic transition, the potential minimum lies along the standing wave node. It means that during their guiding the atoms experience a minimum of impulse diffusion. The shape of the potential well is perfectly defined by a mirror setup (used to form the standing wave) and the laser light phase. Both parameters can be controlled in the guiding experiments with very high precision. A spatial imperfection and a spatial variation of the laser beam profile have an insignificant influence on the shape of the individual well. The bottom of the individual well can lie along a straight line (with a flat mirror setup) or along a curved line (with a curved mirror setup). The shape of the potential well can be an even more complicated form with a use of computer-generated optical elements. There are no spatial spreading and variations of the potential well formed by a standing wave. A transverse size of the potential well is suited to a single-mode regime of atom guiding. There is one drawback (rather obvious and more technical than physical) of a standing wave as an atom waveguide: It is almost impossible to localize all laser power in one or several individual potential wells. As a consequence of that fact, the standing wave atom waveguide needs either a high laser power or its length is rather limited.
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Victor I. Balykin
The idea of atomic guiding in a standing wave was set first by Letokhov (1968) when the author suggested using the effect of guiding an atom in a standing wave to eliminate the Doppler broadening of an atomic spectral line by Dike narrowing. The effect was indeed observed many years ago by the group of NIST, Gaitherberg in the fluorescence spectrum of sodium in a three-dimensionalstanding wave (Westbrooket al., 1990). 1. Atom Potential in a Standing Wave An appropriate description of atom motion in far frequency detuned from the
atomic resonance standing wave is a “dressed” state approach (Dalibard and Cohen-Tannoudji, 1985). Let us consider a two-level atom with a ground state 1 g ) and an excited state I e) in a laser standing wave E(z) = 2E, cos kz cos wLt,where E , is the maximum field strength of one traveling wave component and k is the wave vector that is along the z-axis. The uncoupled states of “atom + photons of laser field” can be written as I g, n 1) and Ie, n ) (see Section 1II.A). The first state corresponds to the atom in the initial state lg) in the presence of n 1 photons and the second state corresponds to the atom in the excited state I e) and the presence of n photons. These states are bunched in a manifold separated by energy hA. When the coupling is taken into account, the two unperturbed states transform into perturbed states I 1, n ) and 12, n 1). Bunches of these states are called “dressed” states. Each dressed state is a linear superposition of the unperturbed state I g, n + 1 ) and e, n ) (Dalibard and Cohen-Tannoudji, 1985)
+
+
+
I
+ e-i(T/4)sin 8 lg, n + 1 ) 12, n ) = -&(“I4) sin 8 le, n ) + e-i(T/4)cos 8 Ig, n + 1 )
11, n ) = ei(T/4)cos 8 le, n )
(153a) (153b)
The eigen energies of the atom in the dressed states are
E , = hw,(n
+
+ hR(z)/2
( 154a)
E,
+ 1) - hA/2 - hfl(z)/2
(154b)
=
hw,(n
1) - hA/2
where fl(z) is off-resonance Rabi frequency, cos 2 8 = -A/fl, sin 2 8 = = fl(z) = [fli(z) + A2I1l2, and fl,(z) = sin kz. The last terms in Eqs. (154) are related to an interaction of the atom with the laser field and can be interpreted as a potential energy of the atom in the laser field. The potential energy function Ul(z) and U2(z),corresponding to the states 11, n ) and 12, n ) , respectively, can be derived from the total energy E l and E, (Eq. 154) (Chen et al., 1993)
a,
fl, sin kzlfl, fl
U,
=
n
- [(a;sin2 kz 2
+
- [A[]
(155a)
245
ATOM WAVEGUIDES
U,
=
h -[(a,,, - (0; sin2 kz + 2
(155b)
a, +
The depth of the potential wells is U,,, = (h/2)(Rm- I A I), where = [a: + A2] ‘ I 2 . If we take into account the coupling of the dressed states with a vacuum field, then through spontaneous emission both states 11, n ) and 12, n 1 ) can decay to the state I 1, n - 1) and 12, n ) ; that is, the atom changes its energy from E l to E,. Spontaneous emission (or “nonaddiabaticity” of atomic motion in an inhomogeneousfield) transfers the “atom + laser field system” from one dressed state to another one. When the spontaneous emission rate is sufficiently high that the atom does not move a significant fraction of a wavelength between spontaneous events, the atom can be described as moving in a “mean” potential
u = UIPI
+ U2P2
(156)
where p , and p2 are the relative populations of the dressed states. This expression for the potential energy can be written in the more familiar form (see Section II1.A)
U
=
1 -hA In(] 2
+ s)
(157)
where s is a saturation parameter. From Eq. ( I 57) it is clear that the “mean” potential is less than the potentials of the atom in certain dressed states.
2. Guiding Time in a Single Potential Well The dressed-state picture is quite appropriate for an estimation of the atomguiding time in a single potential well of a standing wave. For a blue-detuned standing wave, atoms being in the state I 1, n ) move near the nodes if their maximum transverse energy is less than the potential depth U,,,. The transverse atomic motion is periodic with interruption caused by spontaneous emission. For sufficiently large detuning, the rate of spontaneous emission is very small compared to the atomic oscillation frequency. When spontaneous emission occurs the atom from state I 1, n ) can go to state 12, n - 1) where the atom is not localized. The rate of spontaneous transition from state I 1, n ) to state 12, n - 1 ) averaged over an oscillation period equals (Dalibard and Cohen-Tannoudji, 1985; Chen et al., 1993)
r,
=
For sufficiently large detuning (A >> of the guided atom is
2y(cOs4
e)
(158)
a,)cos 0 = ( QR)/Aand the mean lifetime
246
Victor I. BaIykin
(a,)
where is the Rabi frequency average over an oscillation period. From Eq. (159) we can conclude that the guiding time can easily be made sufficiently long by increasing the detuning. However, at the same time the potential depth is also decreased. It is rather interesting to estimate the guiding time for a singlemode guiding regime as it requires the minimum potential well depth. For the quantum mechanical ground state with spatial spot size zo = h / 2 ~the , transverse energy spot is about the recoil energy R = (hk)*/2m.Then at the potential depth comparable with recoil energy U,,, = R (we neglect for a moment the tunneling between the neighbor potential wells), the guiding time becomes
where w , = R/h is the recoil frequency. At the Rabi frequency f l R = 108wr,the guiding time can be of the order of 1 second. For deeper localization of atoms near the node of the standing wave (to avoid tunneling atoms between the neighboring potential wells), it is necessary to increase the Rabi frequency with an appropriate increasing of the detuning.
3. Experiments with a Standing Wave
Three different methods were used to investigate the guiding of atoms in a standing wave. When an atomic beam crosses a standing wave and the atoms have a low enough transversal kinetic energy, they are guided into the channel where they move along the channel and oscillate in the transverse direction. Prentiss and Ezekiel (1986) detected an increase of atomic concentration in the vicinity of nodes of the wave by measuring the fluorescence line shape of a beam of sodium atoms that crossed a plane standing wave at a right angle. The detected asymmetry in the fluorescence line shape was attributed to the action on the atoms of the gradient force that caused the concentration of atoms near the nodes of the standing wave. Salomon et al. (1987) used absorption of the additional weakly resonant wave to measure the atomic density distribution in a standing wave. The atoms inside the standing wave were chosen as probes of their position. Because of spatial varying of the laser field the light shift depends on the position of atoms in the standing wave: atoms at a node have no light shift; elsewhere the absorption line is shifted. The calculation showed that the absorption spectrum of a uniformly distributed atoms in a standing wave is quite different from the spectrum of atoms with a periodic spatial distribution of atoms near the nodes. The density of atoms was found to increase near the nodes or loops of the standing wave, depending on whether the light frequency detuning was positive or negative with respect to the atomic transition frequency. The experiment has been performed with a Cs atomic beam. The atoms were prepared as two-level atoms by a method of optical pumping. The intensive standing wave (150 mW in each traveling wave, and 2.3-mm
ATOM WAVEGUIDES
247
FIG. 16. Figure shows the potential energy of the localized (1) and nonlocalized (2) atoms in a standing spherical light wave, which reflects their trajectories in the laser field.
diameter) has irradiated the atomic beam at right angles. The weak probe beam was also orthogonal to the atomic beam and traveled through the central part of the standing wave. The maximum height of the potential hill was 2 mK, which corresponds to the maximum trapping velocity equal to 50 cm/s. From the experimental absorption, spectra have been deduced corresponding to spatial distributions of atoms that have shown the concentration of atoms near the nodes of the standing wave. A clear demonstration of guiding of atoms was obtained using a curved standing wave formed by a spherical laser wavefront, Fig. 16 (Balykin et al., 1988a;
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Victor I. Balykin
1989a). The atomic beam transverses the spherical standing wave at a point far from the beam waist. In the polar system of coordinates related to the spherical wave the effective potential has the form
U(z) = U, cos2 kz
+ f .r
(161)
where the first term is the atomic potential in the laser field and the second one is an inertial potential.* The inertial field gives rise to the force averaged over the standing wave period. This force accelerates the nonlocalized atoms and hence causes their spatial separation from the localized atoms. This makes it possible, first, to measure the atomic localization effect itself by observing the spatial separation of the atoms, and second, to isolate cold (localized) atoms from nonlocalized ones. In the experiment the spherical standing wave was produced by focusing a laser beam on the center of curvature of a spherical mirror and by reflecting the beam back by this mirror. The standing wave diameter at the point of interaction with the atomic beam was 0.6 mm and the wavefront radius was 40 mm. The atomic beam profile was measured by means of a probe laser field tuned to resonance with the atomic transition. For this purpose, the narrow probe beam transversed the atomic beam at a small angle and scanned it in space at the certain point from the spherical wave. Figure 17 presents the experimental results of the guiding of sodium atoms in the spherical standing wave. The curves are the spatial profiles of the atomic beam in the measured region. It can be seen from Fig. 17 that after interacting with the standing wave, the atomic beam gets split into two beams. The left peak corresponds to localized atoms and the right to nonlocalized atoms. This conclusion follows from comparison with the calculated curve. The distance between the peaks in experiment and theory agrees accurately enough. The wave-front curvature and the size of the laser beam in the atom interaction region determine this distance. The effects of guiding in a plane standing wave on spatial profile are not so clear and require a detailed analysis of the resulting transverse spatial atomic beam profile in terms of trajectories in the standing wave (Chen et al., 1993; Li et al., 1994). At the end of this section we briefly mention a series of very successful experiments with a standing wave as microlenses: Each of the potential wells of a standing wave can produce focusing of atoms in the near field, that is, within the standing wave. (Actually atom guiding is a succession of focusing-defocusingof atoms inside of the potential well). Focusing was performed by placing the deposition surface parallel to the laser standing wave field and inside of the laser beam *f-the
centrifugal force, r-the
wavefront radius at the point of atom-wave interaction.
ATOM WAVEGUIDES
10
249
c
~ 6 1 m/s 1
s
1 0 4 5
z 1-1
FIG. 17. Experimental results of a guiding of sodium atoms in the spherical standing wave. After interacting with the standing wave, the atomic beam gets split into two beams. The left peak corresponds to localized atoms and the right to nonlocalized ones. The curves are the spatial profiles of the atomic beam after its interaction with the standing wave. The atomic velocity values are u = 61 1,800, and 1035 mkec.
where the flux of the incident atomic beam is focused (Timp et al., 1992; Berggren et al., 1994; McCleland et al., 1993; McCleland, 1995). By this method atoms were focused and deposited onto a silicon substrate. The resulting nanostructure consists of a series of narrow lines of around 65-nm width and with the spacing equal to the standing wave period. Recently a similar experiment was performed with a two-dimensional standing wave where a dot-like pattern of chromium atoms was created (McCleland et al., 1996). In Sleator et al. (1992) an experiment demonstrated atom focusing by a single
250
Victor I. Balykin
potential well of standing wave with a large period. This was performed using a large-period standing wave field generated by reflection of a laser beam at a small angle from a glass surface. In the region near the surface, the incoming and outgoing laser beams interfere, producing a standing wave parallel to the glass surface and with a very large period (45 pm). In this experiment the helium metastable atoms were focused to the spot 4 p m in size.
VI. Experiments with Atom Guiding In this section we discuss the main experimental achievement in atom guiding by laser light. At the time of writing this review, three atom-guiding schemes were realized: ( I ) atom guiding with the grazing-incidence light mode (Renn et al., 1995; Renn et al., 1997); ( 2 ) evanescent wave atom guiding (Renn et al., 1996; Ito et al., 1996; Ito et al., 1997; Ito and Ohtsu, 1997); and (3) atom guiding in a donut mode (Kuppers et al., 1996). In Sections IV and V we discussed and compared the potential applicability of all these guiding schemes; here we present the main experimental achievement in atom guiding. A. ATOMGUIDING WITH GRAZING INCIDENT MODE
In the JILA experiment (Renn et al., 1995; Renn et al., 1997) to generate an atom guiding laser mode E H , ,the laser beam was launched into the hollow region of a glass capillary. The laser light is coupled into various modes, and propagates along the fiber by grazing incidence reflection from the glass wall. In the experiment the capillary fiber has an outer diameter of 144 p m and hollow core diameter of 40 pm. The propagation attenuation length, for a chosen diameter fiber and at the wavelength of rubidium transition 780 nm, is only 6.2 cm. The attenuation length limits the guiding distance and the fiber length used was from 3 to 15 cm. The coupling efficiency into the lowest order E H , , mode for the chosen fiber diameter was around 50% and it was achieved when the laser beam waist at the entrance of the fiber is approximately the size of E H , mode, and the axes of the laser beam and the fiber coincide. The E H , mode diameter (defined as a diameter at which the intensity falls to l/e of the maximum value) is substantially smaller than the diameter of the core. It means that the guided atoms are localized in a transverse direction to a size considerably smaller than the internal diameter of the fiber and, as consequence, the effect of van der Waals interaction and quantum tunneling can be ignored in this type of atom waveguide. The fiber connects two vacuum chambers: The first one, the source chamber, contains rubidium vapor with a partial pressure of tom Atoms with small transverse velocities and appropriate trajectories pass into the fiber and are guided through the fiber into a second detection chamber. At the exit of the fiber the atoms are ionized and the ions detected with a channeltron electron multiplier.
,
,
-
,
ATOM WAVEGUIDES
2
0.5
25 1
i;' Intensity (MW/m2)
FIG. 18. Atom guiding with grazing incidence mode. Figure shows the intensity dependence for a guided atom flux for a red laser detuning of A = -8 GHz from resonance (from Renn et al., 1997, Fig. 3, Phys. Rev. A 55, N. 2,3686, reprinted with permission).
For the chosen parameters of the experiment the spontaneous scattering rate is sufficiently high and the atoms are guided by the effective dipole force (Eq. (48)). The main experimental data are described satisfactorily by the model based on the effective dipole potential: Atoms were guided when the laser frequency was reddetuned to the atom transition and the effective potential is an attractive one. For a blue-detuning of the laser frequency there was no guiding: The effective potential is a repulsive one. In the measurement of a guided atom flux as a function of the laser detuning, the flux was increased to its maximum value at the detuning of several GHz (correspondingto the maximum of the dipole potential) and then falls off for larger detuning as l/A. With a large laser intensity the maximum value of the guided atom flux is shifted to a larger laser detuning. At intermediate laser detuning the heating of atoms by the velocity dependent component of the dipole force was observed: Atoms moving in the high-intensity laser field experience the dissipative force (see Section 11I.A). which is directed along the radial atom velocity. The heating of atoms by this force was observed as the substantial loss of guided atoms at the intermediate laser detuning. For large frequency detuning the heating force falls off as l/A5 while the guiding dipole force depends on the detuning only as l/A. This difference in the frequency dependence of the forces allows atoms to be guided at the larger detuning. Figure 18 shows the intensity dependence for an atom flux for a red detuning of the laser of A = -8 GHz from atom resonance. At low intensity, the flux increases linearly with laser intensity, as expected for guiding atoms in the dipole potential.
252
Victor I. Balykin
However, at high laser intensity the guided flux is decreased as a result of the action of the dissipative force and the impulse diffusion. Renn et al. (1995) showed that atoms may be guided in curved fiber. Atom guiding in curved fiber is complicated by several additional effects: (1) Bending of the fiber alters the optical field distribution from the lowest order grazing incidence E N , mode; (2) there is an additional centrifugal force acting on atoms to push atoms into the wall; (3) for atom guiding in the bent fiber it is necessary to launch a larger laser intensity and as a result the optical pumping to other hyperfine sublevels starts to play a significantrole. In the Renn et al. (1995) experiment the atom guiding was detected at the minimum bend radius R = 5 cm. The authors concluded that “the tightest bend through which atom guiding can be achieved is limited by a critical radius for effective optical guiding and not by a radius that depends on atomic properties.”
B. EVANESCENT WAVEATOMGUIDING Experimental demonstration of atom guiding with an evanescent wave were performed in (1) a hollow fiber with relatively large hollow diameter -20 p m (Renn et al., 1996). and ( 2 ) in a micron-sized hollow fiber (It0 et al., 1996; It0 et al., 1997; Ito and Ohtsu, 1997). In all experiments Rb was used as a guided atom. Atom guidance with a fiber of large hollow diameter shows a number of limitations of the principal character. During launching light in a core of the fiber, there is inevitable excitation of the grazing-incidence modes in the hollow region besides the main guiding mode in the core: Some fraction of the laser light scatters in the fiber and couples to grazing incidence modes, in particular, the E H , mode. The propagation attenuation length of a grazing mode depends on the inner diameter as a and with a case of a relatively large fiber diameter the grazing modes accompany the evanescent mode. However, because they are now blue-detuned, they push atoms to the wall through the weaker evanescent wave. The attenuation length of the E H , mode for the Renn et al. (1996) fiber is around 4 mm. Renn et al. (1996) found that an effective guiding is possible when the light intensity of the evanescent wave exceeds the intensity of the basic grazingincidence mode by a factor of 10. To achieve this ratio the scattering of the laser light on the fiber wall must be less than 0.05%for use in their experiment fiber. It is rather hard to fulfill this lower level scattering condition in a real experiment. Another limitation of a large-size fiber comes from inevitable multimode excitation in the glass core region of the fiber. The interference between these modes give rise to an optical speckle pattern on the inner glass wall and, as a result, the modulation of the intensity of the evanescent wave. In the “dark” region on the wall of the fiber atoms are attracted by van der Waals force and may be lost from guided atomic flux. To circumvent this problem of a large size fiber, Renn et al. (1996) used an
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additional red-detuned laser beam. The second laser coupled mainly into the lowest order grazing incidence E H , , mode, which has the maximum intensity on the fiber axis and the potential of this mode, attracts atoms to an axial region of the fiber. Both the grazing incidence E H , , mode and the modes that appeared as a result of scattering of the main laser beam in the core have an intensity attenuation length around several millimeters. The attractive potential of the additional mode compensates the repulsive potential of the scattering modes at the initial stage of atom guiding until the atoms will not come to the region of a pure evanescent wave potential. The experimental setup for evanescent wave guiding in the Renn et al. (1996) experiment is similar to one in their previous experiment on the grazing-incidence atom guiding. The main results were reported for a 6-cm-long fiber with a 20 p m core diameter. To create the evanescent field a laser beam of 500-mW power was focused into the annual region of the fiber facet. It was coupled mainly to the evanescent field but other modes were also unfortunately excited. The additional laser beam was focused into the hollow region of the fiber where it was mainly coupled to the EH,, grazing-incidence mode. It was sufficient only at 10-mW power of the second laser to escort atoms at the initial launching stage of their guiding through the fiber until the evanescent potential begins to dominate. When only the red-detuned “escort” laser was used, the guided flux through the fiber was around 200 s - l , which is by a factor 500 less than the initially launched flux. Addition of the evanescent field in the fiber enhances the flux by a factor 3 at the optimal detuning of both lasers. The measurements of guided atom flux as a function of evanescent wave detuning show the dispersive character as expected from the conservative component of dipole force and the flux was increased to its maximum value at the positive detuning of -2 GHz, which corresponds to the maximum of the dipole potential of the evanescent wave. The character of evanescent wave atom guiding is qualitatively different than for the grazing incidence mode. In grazing incidence guiding the atoms are concentrated near the axis of the fiber and the influence of the van der Waals force is not significant. In evanescent wave mode guiding the van der Waals force plays an essential role especially when the evanescent wave intensity is relatively low (see Section III.B.4 and III.C.4). Renn et al. (1996) observed the threshold intensity behavior for guiding atoms: For the evanescent wave intensity below 6 MW/ m2 there was no optical guiding of atoms; above this threshold the ejected flux of guided atoms increases linearly with laser intensity. At the threshold intensity the dipole force from the evanescent wave exceeds the van der Waals force. The detailed treatment of the influence of the van der Waals force and the cavity QED effect on the atom guiding was done by the Japanese-Korean group (It0 et aL, 1997; Ito and Ohtsu, 1997) with a micron-size diameter fiber. A great advantage of the use of a micron-size diameter fiber for atom guiding is due to the intrinsic ability of that kind of fiber to support only a desirable guiding mode and
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strongly suppress all another parasitic modes. An additional attractive feature of a small fiber is that even a small coupled laser power can produce a sufficiently high potential barrier for atom guiding. Ito ef al. (1996) successfully demonstrated the guiding of Rb atoms by a cylindrical-core hollow fiber with 7- and 2-pm hollow diameter. They manage to reach a high coupling efficiency: About 40% of laser power was coupled into the core of the fiber. For this fiber’s parameters and at the wavelength of 780 nm, three modes can be excited: LPo,, LP,, , and LP2, (see Section 1V.A). The measured light pattern at the exit facet of the 3-cm-long fiber showed that an effectively dark-spot mode could be excited. In the case of using a smaller fiber diameter (2 pm), the guiding mode beam is a single LPo, mode. In the Ito et al. (1996) experiment a well-collimated atomic beam was used as the atom source for the fiber. A straight section of an optical fiber of 3-cm length was aligned with respect to the atomic beam. The atoms that did not enter the fiber were blocked by a fiber holder. The collimated atomic beam and its alignment with respect to the fiber provided a transverse velocity of atoms up to 0.3 m/sec. With the used laser power (several hundred mW) the maximum transverse atom velocity that can be reflected is around 2 t 4 m/sec. Therefore, most atoms impinging on the entrance facet of the fiber are expected to be guided. The two-step photoionization method was used to detect atoms and this detection technique also permitted an isotope selective detection of guided atoms. The hollow fiber (7-pm diameter) was coupled with a laser beam of 130-mW power. Figure 19 shows the transmitted 85Rbflux in the sublevel F = 3 as a function of the frequency detuning of the guide laser. In the red-detuned region the atomic flux is decreased even below the background level (the curve b), which testifies to the action of the attractive dipole force with the result of absorption of atoms on the fiber wall. The maximum transmitted flux was 3 * lo4 s-’ and a comparison with the background transmitted flux gives a rather high enhancement factor of 20. It also reached a very high total guidance efficiency (43%) and the pure optical guidance efficiency (37%). The same group also demonstrated a first application of atom guidance: It performed an on-line isotope separation for two stable 85Rband 87Rbisotopes. The quantum state-selectivecharacter of the atom mirror reflection was demonstrated before in a single reflection of sodium atoms (Balykin et al., 1988). That the atomic reflection is quantum-stateand isotope selective follows from the character of the relationship between dipole force (Eq. (39)) and a laser detuning with respect to atomic transition. When the detuning is positive, the gradient force repels the atom from the fiber surface and thus the atom guiding is effected. With a negative detuning the force attracts the atoms to the surface where the atoms are absorbed and lost from the guided flux. Ito et al. (1996) select a specific isotope by adjusting the guide laser frequency. Figure 20 shows a demonstration of two isotopes 85Rband 87Rbseparation by the 7-pm hollow fiber. In the experiment the atomic transmission flux was recorded as a function of the guided laser frequency.
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FIG. 19. Evanescent wave atom guiding with a micron-size optical hollow fiber. Figure shows the transmitted @Rb flux in the sublevel F = 3 as a function of the frequency detuning of the guide laser (It0 et nl., 1996, Fig. 2, Phys. Rev. Left. 76, N. 24,4502, reprinted with permission).
Figure 20(a) corresponds to a large blue-detuning for both Rb isotopes and the transmission flux contains both isotopes. When the laser frequency was bluedetuned for 87Rbatoms but red-detuned for 85Rbatoms, the transmitted flux of the 85Rbisotopes is strongly suppressed (the lower trace on Fig. 20).
c. ATOMGUIDINGWITH A DONUTMODE In Section 1V.D we discussed the different methods of generating the dark spot laser beam (DSLB). The Bonn group (Kuppens et al., 1996) successfully demonstrated the guiding and focusing of metastable neon atoms with DSLB. The DSLB was created by two methods. In the first method, the lowest order HermiteGaussian TEM,, mode was derived from a ring dye laser by inserting a 20-pm diameter wire into the laser cavity. A mode converter,consisting of two cylindrical lens, transforms the TEM,, mode into a donut mode. The TEMG, mode obtained in this way contains about l-W laser power. The donut mode intensity profile has a slightly asymmetrical ring-shape form. An ultra-cold beam of metastable neon
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500 MHz
85Rb, F=3
1
"Rb, F=2 4
+
LD Frequency
FIG. 20. Separation of two isotopes 85Rband 87Rbthrough a guiding of atoms by the 7-pm hollow fiber: (a) the laser frequency is blue detuned for both isotopes; (b) the laser frequency is blue-detuned for s7Rb and red-detuned for ssRb atoms (It0 et aL, 1996, Fig. 4, Phys. Rev. Lett. 76, N. 24, 4503, reprinted with permission).
atoms was injected coaxially into the donut mode beam. The radius of the donut mode laser beam at the injection plane was 400 pm; at the distance of 20 cm after the injection plane the donut mode waist was 100 pm. The slow atomic beam was prepared by Zeeman slowing of a thermal beam with a further compression and deflection by a two-dimensional magneto-optical molasses (Nellessen et al., 1990; Riis ef al., 1990). The atomic beam prepared in this way had a sub-Dopplertransverse temperature and its longitudinal velocity was 25 mls with a 3 m/s velocity spread. The spatial distribution of the guided atoms was detected at the waist of the donut mode. Without the guide laser beam the width of the atomic spatial distribution of the atomic beam was a 750 pm. The guided laser beam decreased the spatial size of the neon beam to the value of 17 pm. The peak intensity of guided atoms was increased by two orders of magnitude. The total flux of guided atoms contains 10% of the initial value. The Kuppens et al. (1996) experiment pointed out several effects that can be responsible for a rather high loss of atoms during the guiding. There is one inevitable loss mechanism due to the fact that neon atom
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transition is J = 2 + J = 2: The atoms at one of the Zeeman sublevels do not experience a light shift and they cannot be guided. Another loss mechanism is due to spontaneous scattering of photons, which leads to the heating of atoms. In the case of metastable neon atoms there is an additional loss of the guided atoms due to deexcitation of atoms to the ground state. The authors suppose that the imperfection of their donut mode could also cause the losses of the atoms.
VII. Acknowledgments This article was written largely during my stay at the University of ElectroCommunications,Institute for Laser Science, Tokyo. I would like to acknowledge the Institute of Laser Science for their support and especially thank K. Shimizu and F. Shimizu. I would also like to thank all those who kindly provided reprints of their work. I am grateful to D. Lapshin, M. Subbotin, and V. Letokhov for their comments and reading of the manuscript.
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ADVANCES IN ATOMIC. MOLECULAR, AND OPTICAL PHYSICS, VOL. 4
ATOMIC MATTER WAVEAMPLIFICATION BY OPTICAL PUMPING ULF JANICKE Meersburg, Germany
MARTIN WILKENS Institut fur Physik, Universitat Potsdam, Potsdam, Germany
I. Introduction ................................................... 11. Model of an Atom Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Principle B. Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Resonator . . . 2. Pump .................................................. 3. Loss ................................ .......... C. Kinetic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Stationary State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Atom-Laser Versus Bose-Einstein Condensation . . . . . . . . . . . . . . . . . 111. Master Equation . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Atom-Laser Master Equation . . .. . . . .. B. Resonant Dipole Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Derivation of the Pump Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Atom-Laser Rate Equations IV. Photon Reabsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. An Exact Two-Atom Problem . . . . . . . . . . . . . . . . . ........... 1. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Final Mode Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Technicalities . . . . . . . . . . . . . . . . . . . . . . B. Results.. . . . . . . . . . . . . . . . C. Modified Kinetic Equations V. Summary ..................................................... VI. Acknowledgments.. . . VII. Appendix A: N-Atom M 1. Fundamental Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Elimination of the Fluorescence Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . a. Exact Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . h. Markoff Approx c. Born Series . . . 3. Atomic Master Equ a. Empty-Bath Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . h. Single-Line Approximation .... c. Coupling in the Markoff Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1
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Copyright 0 1999 by Academic Press All rights of reproduction in any form reserved.
ISBN 0-12-003841-2/ISSN 1049-250x199S30.M)
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Uv Janicke and Martin Wilkens 4. Self-Energy and Pair-Interaction Energy
.........................
a. Pair-Interaction-Dispersive Part . . . . . . . . . . . . b. Dissipative Interaction ..................................... 5 . Rotating-Wave Approximation ........... 6. Two-Level Approximation .................................... 7. Second Quantization VIII. References . . . . . . . . . . .
300 300 30 I 30 1
302 302 303
I. Introduction When the concept of the atom-laser was conceived in the early 1990s, its mother-the Bose-Einstein condensation of atomic gases-was not yet even born, leaving plenty of space for theorists to speculate about the prospects and properties of a laser-like source of atoms. Meanwhile, experimental ingenuity has not only demonstrated the celebrated Bose-Einstein condensation for a variety of alkaline atoms (Anderson et al., 1995; Davis et al., 1995; Bradley et al., 1995; Rempe et al., 1997), but has also delivered a device that may be called the first prototype of a “pulsed atom laser” (Mewes et al., 1997). In close analogy to the ordinary laser, an atom-laser is a source of particles that is characterized by four properties: (i) the source is monochromatic,(ii) the source is intense, (iii) the intensity is well defined, and (iv) the source is coherent. In mathematical terms, these features translate into
No >> N,, No >> gi(t)g,,(O) = N , e - K f ,
V Z O
1
K
l/No
(4)
where No is the mean occupation of the “atom-lasing” mode (denoted by the subscript Y = 0), and 2; is the bosonic creation operator of a particle in that mode. Leaving questions of beam formation aside, the conditions (1)-(4) are most prominently met by the Bose-Einstein condensate of a trapped Bose gas. This point is illustrated in Fig. 1, which depicts the counting statistics of particles in the ground state of a one-dimensional ideal Bose gas in a harmonic oscillator trap for temperatures above and below the condensation temperature. Two aspects of this simple illustration are worth being emphasized. First, in contrast to the ordinary laser, a macroscopic population of the trap ground state with a narrow peaked counting statistics is a property of the thermal equilibrium, and does not require any kind of nonequilibrium inversion, say. Second, in contrast to common convictions, the occurrence of a macroscopic occupation of the ground state is universal, that is, it occurs in virtually all trapping potentials in arbitrary spatial dimensions.
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10
0.25
5
0.50
/
+.......J ..._..
0 0
0.25
0.50
0.75
1.00
nlN FIG. 1 . Probability Po(n) of finding n particles in the trap ground state for a total number of particles N = lo00 cqnfined by a one-dimensional harmonic oscillator trapping potential. Temperatures are TIT, = 1.5 (a), 0.8 (b), 0.2 (c). Inset: Probability P , ( n l N ) of finding n particles in the first excited state for the same parameter values. The dotted curves depict the predicitions of the textbook grandcanonical statistics for TIT, = 0.8. Below the condensation temperature, this prediction must be rejected as unphysical; for details see Wilkens and Weiss (1997) and Weiss and Wilkens (1997).
The system underlying Fig. 1, for example, does not undergo a Bose-Einstein condensation in the orthodox sense, but the counting statistics clearly displays a well-defined, macroscopic population of the ground state below the condensation temperature. Thus any trapped Bose gas provides a natural source for the atom laser if only the temperature is sufficiently low to allow for a Bose-Einstein condensate to form. Yet the atom-laser was initially perceived somewhat differently. In particular it should operate in a continuous manner, resembling more the ordinary laser in cw operational mode than the cavity dump of a Q-switched device. The first such scheme was proposed in 1994 by Holland et al. (1996). Very much like in the ordinary Bose-Einstein condensation, the scheme is based on evaporative cooling of a thermally driven atom trap where high energy atoms are quickly removed from the trap, and the remaining atoms rethermalize due to atom-atom collisions. A macroscopic population of the trap ground state builds up only if the driving is sufficiently strong, and the loss rate for hot atoms is larger than the out-coupling rate of the trap ground state. The model was further elaborated on by Wiseman et al. (1996) who demonstrated that although it leads to a macroscopic population of the atom-laser mode (the trap ground state), it will not have a well-defined phase, which is mostly due to the peculiarities of the thermal driving and the assumed scalar nature of the atoms. However, atoms are characterized by a rich internal structure, that is, magnetic and electronic degrees of freedom, which not only allows atoms to be trapped in
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magnetic traps, but also allows them to be cooled or otherwise manipulated using resonant or near-resonant laser light. Indeed, one of the early hopes was to achieve Bose-Einstein condensation solely by means of laser-cooling in a magneto-optical trap. This hope turned out to be void, however, because of the many fluorescence photons that are released by an atom in course of its cooling process (Walker et al., 1990; Ellinger et a/., 1994). These photons may get reabsorbed-a process that generally leads to heating and thereby suppresses the onset of Bose-Einstein condensation. This result does not rule out other schemes where atom-light interaction plays a major role in the formation of a Bose-degenerate state. To date, models of the optically driven atom-laser have been proposed that involve dark-state cooling (Wiseman and Collett, 1995), laser-induced dipole-dipole interactions (Guzm6n et al., 1996), Raman-transitions (Moy et al., 1997), and optical pumping (Olshanii et al., 1995; Spreeuw et al., 1995). A covariant formulation of the latter class of models has been developed by Bordi (1999, and the relation to the problem of superradiance has been discussed by Wallis (1997). Closely related is an all-optical scheme for the creation of a Bose-degenerate state, which was proposed by Cirac et al. (1996). Given that the problem of reabsorption can be overcome, the advantage of the all-optical methods would lie in their great variability. Indeed, the tunability of lasers with its ensuing control of the strength of the atom-laser interaction would allow using any atomic species for the atom-laser, and not only the paramagnetic alkalines, say, which are used in present-day experiments on BoseEinstein condensation. In this review we will concentrate on a scheme of an atom-laser that is based on optical pumping. In Section I1 the model is presented in terms of kinetic equations, and its relation to the ordinary laser and the Bose-Einstein condensation is discussed. In Section III we derive a master equation for the quantum statistical dynamics of the atom-laser. Neglecting inelastic reabsorption processes, the master equation is solved and the counting statistics is derived. In Section IV, the effects of the inelastic reabsorption processes are investigated for the particular case of two atoms. It is shown that the onset of atom-lasing is suppressed in large resonators, but may be achieved in small and/or low dimensional resonators.
II. Model of an Atom Laser A. PRINCIPLE The proposed atom-laser operates for atoms with three electronic levels in a A configuration-see Fig. 2. The three levels are denoted a (“auxiliary”), e (“excited’’), and g (“ground-state”). The atoms in the electronic state a constitute
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FIG. 2. The atom laser model. Atoms that are precooled in state a are pumped with rate r into the electronically excited state e. Starting from e they decay into the electronic ground state g under the emission of a photon. In state g they occupy one of the modes v of the matter wave resonator. Here S, is the Franck-Condon factor for the transition into mode v, and K,, is the loss-rate of that mode.
a particle reservoir from which the “resonator” of the atom-laser is replenished. The resonator is an atom trap that is only “seen” by atoms in the ground electronic state, and the transfer of a-state atoms into the resonator-a process that is necessarily irreversible-takes place via a two-step sequence of internal transitions a + e + g. Here the first step is induced by the absorption of a laser photon, whereas the second step occurs via radiative decay e + g, which is accompanied by the spontaneous emission of an optical photon. Including motional degrees of freedom, the “pumping” of the atom laser is described by
+
where p labels the motional state of the a-state atoms, p hk,,, is the momentum after the atom has absorbed the laser photon, and v labels a bound state of the trapping potential, that is, a mode of the atom-laser resonator. The key observation is that the e + g transition into the resonator mode v, say, is modified by the g-state atoms already present in the resonator. The basic mechanism for this is Bose-enhancement-a mechanism that also governs the operation of the ordinary laser, as in Fig. 3(a). In the ordinary laser, atoms in an electronically excited state e decay into the electronic ground state g, thereby emitting a photon into any mode of an optical resonator, say, the kth mode. If prior to the emission there are already Nk photons present in that mode, and all the other modes are empty, the rate of emission into that mode is enhanced by a factor of Nk + 1 (light amplification by stimulated emission of radiation). From a technical point of view, the enhancement factor has its origin in the symmetry properties of an ensemble of identical, bosonic particles, which in the case of the ordinary laser are just the photons in mode k. For the atom-laser, the roles of photons and ground-state atoms are reversed,
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Ulf Janicke and Martin Wilkens
FIG. 3. (a) Light amplification by stimulated emission of radiation. (b) Matter wave amplification based on optical pumping.
see Fig. 3(b). Again, an atom in the electronically excited state decays into the electronic ground state, which is concomitant with the emission of a spontaneous photon. However, now it is the ground-state atoms that are confined by a resonator, whereas the photon can escape from the system. If prior to the decay there are already N,,atoms in the vth resonator mode, the rate of transition into that mode is enhanced by a factor of N,,+ 1. Thus, if the resonator fundamental mode v = 0, say, displays the largest population of all modes, a stimulated emission of bosonic matter waves into that particular mode seems feasible.
B. INGREDIENTS In this subsection we outline a specific implementation of the atom-laser. The reader who is not interested in the details of the implementation may skip this section and resume with Section 1I.C. 1. Resonator
The atom-laser resonator is made of light. A standing-wave laser field that drives the e-g transition effectively amounts to a periodically varying potential, each minimum of which may be viewed as a small resonator for atoms. For bluedetuned laser light, w >> w, (w, is the Bohr transition frequency of the e-g transition), minima coincide with the nodes of the standing-wave laser field, and the g-state atom is effectively trapped in the dark. In leading order, atomic motion in such a trap is characterized by harmonic oscillations with oscillation frequencies Oi in the three Cartesian directions i = x, y, z. For cubic lattice arrangements of the trapping laser the oscillation frequency is given by ll = ER,(o,/A)~’*, where w,, = h k 3 2 M is the recoil frequency with Plank’s constant h = 2 ~ 6M, is the mass of the atom, and A = w - wo is the laser-atom detuning. The harmonic oscillator eigenstates lv) define the mode functions ( x l v ) = &,(x) of the
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
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resonator, each mode being labeled by a collective index v = {vx,vy,vz}, where vi = 0, 1, 2, . . . is the number of vibronic quanta in the i Cartesian direction. We also assume that the detuning of the trapping laser with respect to the a-e transition is much larger than the detuning for the e-g transition, which implies a reduced coupling of the reservoir of a-state atoms to the trapping laser. In addition we may assume that the branching ratio of the spontaneous decay favors the channel e + g, which leads to an additional suppression of the coupling of the trapping laser to the a-e transitions. Under any of these conditions the atoms in state a are not affected by the trapping laser and may be considered as moving effectively freely.
2. Pump Loading of the resonator proceeds by optical pumping. A precooled atom in level a is excited into level e from where it decays under the spontaneousemission of a photon into state g, thereby occupying one of the modes of the resonator. Spontaneous decay back into level a is also possible, but for a given intensity of the pump-laser, this process merely affects the effective rate with which the optical driving produces e-atoms, which will decay via the channel e + g. Denoting ruethe single-atom rate for the induced transition a + e, the effective pump rate is given by reR= ruey/( y + y,), where y, is the spontaneous decay rate in channel e + a, and y is the spontaneous decay rate in channel e + g. Working with the effective pump rate allows us to bypass the the process e + a, that is, from now on every atom that is found in e may be considered to exclusively decay into level g. The spontaneous photon that is released in this decay may be reabsorbed and re-emitted several times before it eventually escapes the system. Ignoring the inelastic effects of this process, the probability that as a result of the e + g transition the atom ends up in mode v, say, is given by P, M S,(N, + l), where S, is the single-atom transition probability, and N, is the number of atoms in mode v prior to the arrival of the atom. The factor N, 1 follows from Fermi’s golden rule, which states that the probability for a transition from state IN,,) with N, atoms in mode v to state IN, 1) is proportional to the matrix element I ( N , + 1I g: IN,) I = N, + 1, where g: is the bosonic creation operator for g-state atom in mode v. The Franck-Condon factor S,, is given by the recoil-corrected overlap between the thermal wavepacket of an atom in state a with the mode function of mode v, S, = [ [ ( ( V I P - hk)12];],, where [. . .Ip is the thermal average over momenta p of the atom in state e and [. . .] is the average over the dipole radiation pattern of the spontaneous emission in direction 2. For a thermal gas of a-state atoms, where the temperature is larger than the recoil temperature, T >> ( f ~ k ) ~ / 2 M kthe , , recoil-correction of S,, may be neglected. Assuming a harmonic oscillator trapping potential, and neglecting the
+
+
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Ulf Janicke and Martin Wilkens
effects of the photon recoil, the Franck-Condon factor in one spatial dimension evaluates to
Here A,/2 is the spatial period of the standing wave, A , = [ h * / ( 2 ~ M k , T ) ] ” ~ is the thermal de-Broglie wavelength, H Y i ( [ )are Hermite polynomials, 5 = p / ( M ~ 5 C l ~ )and ” ~ ,(. . .) indicates a Gaussian average with (5)= 0 and ( t 2=) (2 hCl,/k,T)-’. The Franck-Condon factors for a three-dimensionalresonator are simply the product of the corresponding Franck-Condon factors of Eq. (6),
+
SP
=
s,
SY, SY,.
Here and in what follows we assume a quantization volume (A,/2)3,that is, S,, defines the single-particle probability for an atom in that volume to end up in mode v,The Franck-Condon factors become larger for energetically lower modes, for example SOBl= 1 + hCli/2k,Z This fact leads to a natural preference of transitions into the fundamental mode v = 0. For temperatures k,T 5 haj,a typical value in three dimensions is So = 3. Loss
Very much like in the optical case, the resonator of the atom-laser is not perfect. For a single atom in mode v, the imperfection is described by an effective loss rate K,
= K ~ c a+ Ktun Y
Y
+ Kout Y
(7)
where the various loss mechanisms, which we assume to be independent, are characterized by corresponding loss rates K ; ‘ .. Within our model of the atom-laser resonator, the scattering of a photon from the trapping laser, for example, is an important loss mechanism. The corresponding loss rate is easily estimated to be given by
Another loss mechanism is tunneling through the light-induced potential barriers. The corresponding loss rates K? are typically of the same order as the loss rates for photon scattering. Both loss rates increase for higher lying modes. The outcoupling of the atoms from the resonator may be described by yet another loss rate K O ~ ‘ . Switching off the trapping laser would be the simplest outcoupling mechanism (“Q-switched” atom laser). More desirable is a continuous output, which could be achieved by a fast modulation of the trap depth. Such a modulation changes the effective tunneling rate without disturbing the internal dynamics. Detailed studies about the mechanisms of out-coupling, in particular for the case of magnetically trapped alkalines, and the ensuing properties of the
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICALPUMPING
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out-coupled atoms may be found in Mewes et al. (1997), Ballagh et al. (1997), Naraschewski et al. (1997), and Steck et al. (1998).
C . KINETICEQUATIONS A simple description of the atom laser is given by the so-called kinetic equations, which describe the temporal evolution of the mean values of the mode populations. Assuming that at any given instance, at most one atom is found in the excited state e, the kinetic equations are given by
where Re and N,,are the average numbers of e-state atoms and g-state atoms in mode v, respectively. The rate r is the effective pump rate per unit volume with which atoms are pumped from level a to level e. The term (d/dt)N,Ireab accounts for the inelastic part of the photon reabsorption. The elastic part is contained in the gain terms as will be shown next.
I . Stationary State Neglecting the inelastic processes in Eqs. (9) and (lo), it is easy to find the mean mode populations in the stationary regime. For illustrative purposes we only take the two lowest lying resonator modes into account, and treat the collection of all other modes as an effective mode with a large loss rate (no Bose-enhancement) and Frank-Condon factor 1 - So - S, . Within this approximation,the mean mode populations are given by
N o =2so L { 'L0 -
1
+
[(;-
1)*+4so~]"*}
with ro = K ~ / Sand , r, = K ~ / S ~ . In Fig. 4 the mean occupation of the resonator fundamental mode v = 0 and first excited mode v = 1 are displayed as a function of the normalized pump rate r/ro for parameter values So = 0.01, S, = 0.95S0, K~ = (5/3)~,, K~ = 1 s - l . The population of the fundamental mode-the atom- "lasing" mode-clearly displays threshold behavior at r = ro, whereas the population of the excited resonator mode remains virtually unchanged even for strong driving r >> ro. For weak driving, that is, far below threshold, the slope efficiency of the atom-lasing mode,
Ulf Janicke and Martin Wilkens
270 150
100
50
0
FIG. 4. Mean populations of the resonator fundamental and first excited mode, in the stationary , population of the lowest regime, as a function of the normalized pump rate. At r = ro = K ~ / S the mode reaches threshold. The parameters are So = 0.01,S, = 0.95S0, K , = (5/3)~,, K~ = 1 s-I.
(d/dr)No,is given by S J K ~ which , is proportional to the single-particletransition probability into that mode. Far above threshold, the slope efficiency is given by I / K ~Physically . this means that above threshold not a fraction So, but all atoms that are pumped into the resonator end up in the atom-lasing mode. Although in its single-atom characteristics the fundamental mode is only slightly preferred as compared to all the other modes (largest Frank-Condon factor, smallest loss rate), this tiny imbalance is sufficient to deplete all the other modes above threshold, and let the fundamental mode win the mode competition. We note that if only this imbalance is preserved, the qualitative behavior remains unchanged if more modes are modeled explicitly. 2. Atom-Laser Versus Bose-Einstein Condensation The threshold behavior and mode competition bears strong resemblance to the Bose-Einstein condensation of a trapped Bose gas (Olshanii et al., 1995). From Eqs. (9) and (lo), the atom-laser stationary mean population are given by the implicit equation
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING -
N,
1 =
For simplicity we assume S, sible resonator modes. Then
K,(1
f
2, S,N,)/(rS,)
= S, where
27 1 (13)
- 1
1/Scorresponds to the number of acces-
with
For a trapped ideal Bose gas, on the other hand, the mean occupation of a given single-particle state v is given by
where E , is the single-particle energy of that state, and z = exp[p/k,T] is the fugacity with p the chemical potential. Comparing Eq. (14) with Eq. (16) one observes that the atom-laser stationary state may be viewed as a thermodynamic equilibrium state of a trapped ideal Bose gas, with single-particle energies ~ , / ( k T, ) = ln(K,/K,) and chemical potential p / ( k , T ) = In z, where z is defined in Eq. (15). A change of the chemical potential p corresponds to a change of the inverse pump rate of the atom-laser.Figures 5 and 6 show the "fugacity" Eq. (15) and the fraction of atoms in the atom-laser mode as a function of the inverse pump rate.' For small resonators (large values of S), the threshold behavior is much less pronounced, as it is for large resonators (small values of s).In the thermodynamic limit S + 0 one observes a step-like threshold behavior, which is also observed for an ideal Bose gas with a logarithmic energy spectrum (Weiss, 1997). The results indicate a certain analogy between the behavior of an atom-laser and Bose-Einstein condensation of noninteracting particles in a trap. However, there are distinct differences between the two systems. First, the atom-laser is an open system with a stationary state far from thermodynamic equilibrium. Temperature only enters indirectly via the thermal Frank-Condon factors. Second, a macroscopic population of the atom-lasing mode is achieved by atom-photon interactions and not, like in evaporative cooling, by thermalizing atom-atom collisions. Finally, the single-particle "energy spectrum" E , = ln(K,/K,) can be 'We here assume that photon scattering is the dominant loss mechanism. Therefore, K, = K,,(I + 2n13) with n = Z:=, vi.The numbers vi label the single-particle states in an isotropic, threedimensional, parabolic potential with degeneracy factors g, = f(n I)(n 2).
+
+
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Ulf Janicke and Martin Wilkens
1
0.8
or
0.6
0.5
1
1.5
2
rolr FIG. 5. The “fugacity” as a function of the inverse pump rate. Small Franck-Condon factors S correspond to large traps and vice versa.
changed independently of the form of the resonator, which is not possible in ordinary Bose-Einstein condensation.
111. Master Equation The results of the previous section indicate that the atom-laser conditions (i) and (ii) are indeed obeyed by our scheme, yet so far nothing has been said about the definiteness properties of the intensity, the phase coherence, the impact of the inelastic processes of photon reabsorption, and-last but not least-the impact of atom-atom collisions. The general framework for a study of these issues is provided by a master equation, which governs the temporal evolution of the quantum statistical density operator of the resonator state. Alternatively, one may invoke a description in terms of a macroscopic stochastic wave function, which would be more in the spirit of semiclassical laser theory (Wallis, 1997; b e e r et al., 1998). Such a description has its merits, which mostly lie in its convenience, but a few remarks seem in order before we proceed. First, working with macroscopic wave functions necessarily invokes symmetry breaking-a concept that does not follow from orthodox theory but rather must be postulated. Although this postulate has been
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proven to be quite successful for the properties of first-order coherence, virtually nothing is known on its meaning and possibly necessary modifications if questions of higher order coherence are addressed (Javanainen and Wilkens, 1997). Second, in contrast to the ordinary laser, where “semiclassical” has a clearly defined meaning, this is not the case for the matter fields; it appears that the better a macroscopic wave function describes reality, the less classical the system in fact behaves. These remarks do not rule out the possibility of a theory based on the macroscopic wave function. To the contrary, such theory may well exist, and it would be very interesting to compare its predictions with the results of the more orthodox analysis presented here. A. ATOM-LASER MASTER EQUATION Denoting pg the quantum statistical state of the g-state atoms in the resonator, the temporal evolution of this state is governed by a master equation of the form
where the first part accounts for the loading of the resonator and the second part accounts for resonator losses.
1
0.6
0.8
O6
I \
1s
0.4
0.2
0 0.5
1
1.5
2
rO/r FIG. 6. The fraction of atoms in the resonator fundamental mode as a function of the inverse pump rate. Small Franck-Condon factors S correspond to large traps and vice versa.
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Ulf Janicke and Martin Wilkens
Loading results from optical pumping a + e + g, which we model as a Poissonian process for the creation of e-state atoms without paying attention to the details of the laser excitation. We assume that spontaneous emission is the fastest of all processes, y >> r, K ~ which , means that at any instance at most one atom is found in the electronically exited state and that resonator losses may be neglected during the e + g transition. Under these assumptions, the update of the resonator state, pg +p g’, which is due to the arrival of a new atom, is given by the mapping where p g r describes the state of the resonator with one atom more than in state p g . Technically, the pump-map 9 is given by a certain Greens function of a more general master equation, which describes the temporal evolution of a system of two-level atoms, say, and their interaction with the quantized electromagnetic field. The derivation of this master equation and the construction of 9 is the subject of the next section. Here we assume that 9 is known. Because loading is a Poissonian process, the temporal evolution of the resonator state on a time increment At >> y-I is given by pg(t
+ Ar)Ipump= rAt9pg(t) + (1 - rAt)pg(t)
Dividing by At, taking the limit A t + 0, and adding resonator losses, one obtains the atom-laser master equation
We note that by means of a suitable interpretation of the pump map 9, virtually all models of the optically driven atom-laser may be described by a master equation of this form.
B. RESONANT DIPOLEINTERACTION For the derivation of pump map we must study the temporal evolution of a gas of atoms, where initially one atom is in the electronically excited state e. This is a difficult problem because the spontaneous photon, which is released on the e-g transition, may be reabsorbed by the resonator g-state atoms before it eventually leaves the system. The reabsorption causes an effective interaction between the atoms, which is the resonant dipole interaction (Lewenstein & You, 1996). For a gas of g-state atoms with one e-state atom, the resonant dipole interaction is a transient phenomenon, which terminates when the photon is eventually gone. The decay is described by a master equation that is derived by eliminating the electromagnetic field degrees of freedom from a quantum electrodynamics Liouville-von Neumann equation, which governs the temporal evolution of the combined system atoms + field. Details of the derivation are given in Appendix A. Denoting p the statistical density operator for the atomic gas, the result reads
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
275
where
He,
= HA
+ HD, - i i h r
(22)
Here HA is the Hamiltonian of the noninteracting atoms, HDD describes the dispersive effects of the resonant dipole-dipole interaction (exchange of virtual photons), whereas r and the associated superoperator 9describes the dissipative effects like spontaneous emission and Dicke superradiance. Utilizing a two-level approximation for the atoms and using the language of second quantization, the resonant dipole interaction is given by2
where the Bose operators obey the commutation relation
rg, 'El = &'
P, .^:I
=
a,,
(26)
and SvqPp= SuqPp(k0) all other commutators being zero. The coefficients DvqPp are given by
where rvYPp (k)=
1
d 2 ~ @ ( ; ) ( qI eik,;I p ) ( v I e-ik'ilp)
(28)
Here 2 = k/k, @(2) is the dipole radiation pattern with J d 2 K @ ( 2 )= 1, k , is the wavenumber of spontaneous emission, and P denotes the principal value. Other types of interaction, which may be included in HA, are the Van der Waals/Casimir Polder interaction of g-state atoms, which result from an exchange of virtual photons, and the overlap of atomic orbitals, which dominates at very short distances. The latter two types of interaction give rise to the atom-atom collision potential that governs the evaporative cooling. The exchange of a resonant photon, in contrast, is characteristic for the optically pumped atom-laser, ZInthis section the atoms are described as effective two-level systems with levels e and g. In a more detailed description, the excited state e consists in the simplest case (P-state) of three sublevels. This implies .? + { .?, , ey , e, }, where for example iX is the annihilation operator for an atom in the electronicallyexcited state with the electronic alignment in the x-direction.
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UlfJanicke and Martin Wilkens
and we shall concentrate on the effects of this resonant dipole interaction in the following.
C. DERIVATION OF THE PUMPMAP The pump map is defined in terms of a solution of the Eq.(21), assuming that initially there is one atom in the electronically excited state, and an unspecified number of atoms being distributed over the resonator modes in the electronic ground state. The initial condition reads
(29)
p = pe @ pg
with
where IN) = IN,, N , , . . .) is a multimode Fock state of the resonator g-state atoms, I l p )is a Fock state with one e-state atom in the momentum eigenstate p, and Z is a normalization. Because for the particular initial condition (31) only one jump e + g may occur, the pump map 9 is easily found
where Tr, {. . .} is the trace over the states of the electronically excited atom, and i % Y = --(H,,Y h
- YHif‘),
i
Xe,Y = --[H*,Y ] h
(33)
We note that the pump map 9 may be viewed as the T-matrix of spontaneous emission of an atomic Bose gas. It reveals the mechanical impact of the photon, which is released on the e + g transition, on the motional state of the gas.
D. ATOM-LASER RATEEQUATIONS In order to recover the kinetic theory outlined in Section 1I.C we employ the socalled secular approximation, Dvqpp
-
1 TiSwqpp
+
[Dwp -
$is~pl~py~pq
(34)
in which inelastic effects of the dipole-dipole interaction are neglected. Later we shall study in detail the inelastic effects using the exact pump map of a two-atom system. From Eq. (27) we read off the representation for the matrix element
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
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(35) SYP = [ b l P - w 1 2 1 ; where 1 k I = k o , and [. . .I ;denotes an average over the directions of spontaneous emission, see Eq. (28). In the secular approximation, the diagonal elements of the density matrix, are not coupled to the off-diagonal elements. The temporal evolution pP pf‘’,” P of the diagonal elements is given by the rate equation [see Eq. (20)],
d
-dtP ~ N = r ( Y p g ) N - r~gN+ C KY[(NV + l)pgN+lY- N Y p h l Y
(36)
+
where N I, is a configuration with one more atom in mode v than in configuration N. Instead of using the compact expression (32), the pump map is here derived by solving the master equation (21) in the secular approximation (34). The probability of finding the g-state atoms in configuration N and one e-state atom with momentum p , P ~ , ~ ,changes ,, due to the decay of the excited atom according to
from which it follows that
The probability of finding no atom in the electronically excited state changes according to
from which it follows that PN,O(r)
=Y
2 VP
I,’
s ~ p N ~
dr’pN-l,,lp(T’)
(40)
Inserting Eq. (38) into Eq. (40), carrying out the time integration, and taking the limit r -+ w, one obtains
where [. . .Ip denotes a thermal average with the distribution given in Eq. (31). With the approximation [AIB] = [A],/[B] the pump map may be read off from
with& = [[[(vlp - fik)(2];]pasbefore.
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UlfJanicke and Martin Wilkens
The rate equation Eq. (36) together with the pump map Eq. (42) allow a detailed study of the population statistics for the different resonator modes. The kinetic equations Eq. (10) are recovered by taking the first moments of the rate equation (36). Following our discussion of the kinetic equations, we model only the two lowest modes explicitly with K~ (5/3)~,, K , = 1s-*, S, = 0.95S0,So = 0.01 and treat all other resonator modes as an effective continuum. Figure 7(a) shows the counting statistics for the lowest mode in the stationary regime as a function of the scaled pump rate r/ro,r, = K,/S,. Figure 7(b) shows the corresponding mean No = ( N ) , and the variance c-ri = ( N 2 ) , - ( N ) ; . Figure 8 depicts three cuts from the distribution for pump rates below, at, and above the threshold pump rate r, . The distribution is exponentially decreasing below threshold and is Poisson-like with a single peak above threshold. The atom-counting statistics may be compared with the photon number distribution in an optical laser. In fact, with our approximations the counting statistics that results from Eqs. (36) and (42) is identical to the photon number statistics of a three-level laser,3
P,=,(N) = x
( r k l1
(l/So + N - l)(l/So + N - 2 ) . . . (US,
+ l)(l/So)
(43)
where X is a normalization constant. The temporal evolution of the atom-counting statistics in the fundamental mode and first excited mode is depicted in Fig. 9. At the onset of amplification, the atom-lasing mode v = 0 displays enhanced fluctuations, and the occupation of the first excited mode approaches its maximum value, see Fig. 10. As time goes by, the atom-counting statistics of the atom-lasing mode develops into a single peaked distribution, whereas the other modes become depleted, as in Fig. 9. In this figure the thin lines depict the results of a calculation in which only the fundamental mode was modeled explicitly. Comparing the two-mode results with the simplified single-mode result, we find that the modes v > 0 have only little influence on the atom counting statistics of the atom-lasingmode.
IV. Photon Reabsorption The pump map Eq. (42) is based on the secular approximation (34) where the inelastic effects of the dipole-dipole interaction, which may change the mode populations, are neglected. However, these population changes may substantially affect the threshold behavior and need to be studied carefully. %ee for example Chapter 7 in Loudon (1983).
200
150
I ilSn- 1
100
50
0 0
0.5
1
1.5
2
rlro FIG. 7. (a) The atom-counting statistics for the resonator fundamental mode. Dark regions correspond to large probabilities. (b) The corresponding mean occupation R0 = and the variance u; = ( N * ) o - (N)$
UlfJanicke and Martin Wilkens
280
0
20
60
40
80
100
N FIG. 8. The atom-counting statistics for three values of the pump rate below, at, and above the threshold value r,,.
The dipole-dipole interaction results from the reabsorption of photons, which are emitted by the optically pumped atoms. For two atoms that are located in a large resonator of size L >> A (A is the photon wavelength of the e-g transition),
FIG.9. The time evolution of the atom-counting statistics for the resonator fundamental mode (v = 0) and first excited mode (v = 1).
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
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250
200
150
1lSo - 1
100
-
Nn
50 -
-
0 0
0.1
0.2
0.3
0.4
t FIG. 10. The variances and mean occupations of the two lowest modes as a function of time (time ) ) . thin lines denote the results if only the lowest mode is modeled is given in units of I / ( K ~ S ~ The explicitly (three-level model).
the effects of reabsorption are easily estimated (Olshanii et al., 1995).We consider the situation in which one g-state atom occupies the fundamental mode of the resonator when a second, electronically excited atom enters the system. Let P2 denote the probability that both atoms occupy the fundamental mode after the photon has eventually left the system (“gain”). Let Po denote the probability that no atom occupies the fundamental mode in the end (“loss”). The necessary condition for the atom laser to reach threshold reads4 p2
’Po
(44)
At least for large resonators, a simple estimate for the probabilities P2 and Po is easily derived. For an electronically excited atom that is cooled to about the recoil temperature, we find from the definition of the thermal Franck-Condon factors 41n general, the condition P2 > Po is not sufficient, because the pump efficiency for the fundamental mode must be large enough to overcome possible resonator losses, which are neglected in this discussion.
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UlfJanicke and Martin Wilkens
Eq. (6) that P2 cc where D is the spatial effective dimension of the resonator. In order to estimate Po we evoke a ballistic model for the photon exchange between the two atoms: The photon is emitted by the e-state atom in an arbitrary direction and is reabsorbed by the g-state atom, which is situated at a distance O ( L )>> A. The absorption cross-section is of the order of A2 giving a probability Because the momentum kick that comes along for reabsorption preab with the reabsorption almost certainly implies loss, we have Po = preab,and therefore
-
The estimate implies that in a large resonator with L >> A, the atom-laser cannot reach threshold because of inelastic photon reabsorption processes. However, in quasi-two-dimensional or quasi-one-dimensional configurations, the relative probability that the photon leaves the system without being reabsorbed is enlarged and threshold may be reached. The simple estimate (45) assumes that the atom-atom interaction results from an exchange of ballistic photons. However, this assumption is only justified if the mean distance of the atoms in the resonator is much larger than the photon wavelength (far-field limit). For small resonators, where the two atoms are situated in their respective induction zone or even near-field zone, the estimate is not valid and the problem of photon reabsorption must be considered more carefully. A. AN EXACT TWO-ATOM PROBLEM In the following we study the effects of photon reabsorption processes on the threshold behavior of the atom laser. For a detailed exposition see Janicke and Wilkens (1996). In order to keep the discussion transparent, we use a simplified model of the atom laser where one g-state atom occupies the fundamental mode of an ideal resonator when a second, electronically excited atom is pumped into the system. We neglect resonator losses, and we assume perfect mode matching, that is, the center-of-massdegrees of freedom of the newly arriving atom are prepared in the resonator fundamental mode. The dominant interaction between the two atoms is the resonant dipole-dipole interaction, which leads to a redistribution of the atoms over the resonator modes. The theory of the interaction between photons and ultra-cold atoms is an active field of research (Lewenstein et al., 1994; Cirac et al., 1994; You er al., 1995). The rapid progress in laser cooling and atom trapping has led to an increasing interest in the mechanical effects of the dipole-dipole interaction of trapped atoms (Goldstein et al., 1995; Vogt et al., 1996; Goldstein et al., 1996; Cirac et al., 1996; Naraschewski et al., 1997). In close connection with the subject of this
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
283
section is a recent work by Lewenstein and Cirac (1996), who studied the stability of a Bose-Einstein condensate against the mechanical perturbations caused by spontaneously emitted photons. Using perturbation theory, these authors demonstrate that a Bose-Einstein condensate with a sufficiently large number of atoms is stable against photon emission. In contrast to this investigation, the present study is nonperturbative, and we show that it is possible to create a Bose-Einstein condensate despite the mechanical effects of spontaneousemission, provided that initially the spatial extent of the system is sufficiently small. I. Model
The temporal evolution of the two-atom state p is governed by the master equation (21). The Hamilton operator of the noninteracting atoms is given by
H A
=fi21 + A 2 k1 + -Mf12(f: 2M
2M
+ f;)
2
where we assume that the resonator is described by an isotropic harmonic oscillator potential that is not sensitive to the atomic electronic state. This assumption is well justified if the single-atom rate of spontaneous emission is much larger than the trap frequency, y >> a. In the opposite limit, y << fI, the e-state atom will undergo many oscillations before spontaneous emission is complete. Such a situation may well be encountered in situations where the loading of the resonator proceeds by means of Raman transitions, see Cirac et al. (1996) for details. Like in the preceding sections, we model the atoms as two-level atoms with electronic states e and g. Translating Eqs. (23)-(25) into the language of “first” quantization, the interaction part of the master equation (21) is described by HDD
r
= fiyD(P1, f 2 ) [ a I a 2 =
222)[a;a2
+ aia11
+
(47)
+ y [ a ; a , + (+;(+21
(48)
e-ik.Pz
(50)
with
A
= a,e-ik.f~
+ a
2
Here fa is the position operator of the a t h atom, a = 1,2, and aa = I g)( e I is the electronic transition operator of that atom. The functions D(i,, i2)and S(f,, i2)are nothing but the corresponding matrix elements (27) in the position basis. These functions depend on the distance of the atoms, r = I r, - r2I, and also on the angle of the distance vector rl - r2 relative to the alignment of the transition dipole elements This latter dependency
3.
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Ulf Janicke and Martin Wilkens
makes the evaluation of the time evaluation quite awkward, as it also requires keeping track of the angular variables. In order to simplify matters, here we use the replacement D + D(r),S + S(r) with the isotropic functions D(r),S(r) given by [see Eq. (A50) in the Appendix] = DLW
(51)
The description of the dipole-dipole interaction by a scalar and isotropic model deserves some comments. From a fundamental point of view, this interaction must be described by a second rank tensor as it depends on the relative orientation between the two dipole moments. In a local frame, where the quantization axis for the electronic degrees of freedom is oriented along the distance vector between the two atoms, r = r1 - r,, the tensors become diagonal with elements given by D,,DIIS,,SII, see the Appendix Eqs. (A50)-(A53). These functions describe the interaction between two dipoles at fixed positions, which are aligned perpendicular or parallel with respect to their distance vector. In the radiation zone k,r >> 1, the perpendicular components M llr dominate. In the near field k,r < 1, S, and SIIapproach unity, whereas ID, I and I diverge cc l/r3(see Fig. 11). In order to reduce the complexity of the tensor interaction one is tempted to (i) either resort to a local diagonalization, or (ii) invoke a spatial average. Both these approximations, however, would miss essential points. A local diagonalization of the tensors D, and S,, which is frequently used in physical chemistry and in studies of atom-atom collisions (Kurizki and Ben-Reuven, 1987), is not useful here because the motional degrees of freedom must be treated quantum dynamically and a local diagonalizationdoes not commute with the kinetic operator of the center-of-mass motion of the atoms. The spatial average, on the other hand, would miss the potentially hazardous lh3-dependence which dominates the dipole-dipole interaction at small distances. With the substitution (51)-(52), the dipole-dipole interaction is modeled faithfully for all distances ranging from the near field ( D ( r )c~ l h 3 , S(r) = 1) to the radiation zone (D(r), S(r) lh). We note that the function D, describes the dispersive interaction between the atoms if the system is driven by a bluedetuned light field. Because a near field llr3-repulsion is assumed for all directions, the scalar model (51)-(52) provides an upper limit of the heating effects of the dipole-dipole interaction. We also note that the Hamiltonian HA in Eq. (46) can be separated by introducing two-atom center-of-mass coordinates R = (rl r2)/*, and relative coordinates, r = (rl - r,)/*, that is, HA = HR H,.This separability is of crucial importance for an analytical treatment of the dipole-dipole interaction, which in
+
+
0
5
10
15
20
15
20
kx (b)
0
5
10
kx FIG. 11. The basic tensor components of the dipole-interaction (a) D and (b) S.
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UlfJanicke and Martin Wilkens
the approximation (51)-(52) is only a function of the distance r. For anisotropic trapping potentials, such a simple separation is not possible and an analytical treatment is substantially limited. We shall use for dimensionless units r' = G k o r , R' = f l k o R , R ' = R/4wrec, y' = yl4w,,, t' = t X 4w,, with w, = fiki/2M where the primes are omitted in the following. In these units, both HR and Hr are given by the Hamilton operator of a harmonic oscillator of unit mass with frequency R = 1 and n = 1. 2. Final Mode Populations The state of the two-atom system is described by the density operator p ( t ) . For indistinguishable,bosonic atoms, p(t) must be symmetric under the exchange of the atom labels. Because by assumption both atoms are prepared in the same motional state initially, the initial state is given by P(0) = W(0)@;(leg>
+ Ige)>((egl + (gel)
(53)
-++ +-+
where W(0)= I OO)( 00 I with Id) the resonator fundamental mode. After the photon has left the system, both atoms are in the electronic ground state and occupy certain resonator modes. We want to calculate the probability P, of finding n atoms in the resonator fundamental mode, where n = 0, 1 , 2 . Because the probabilities are exclusive, it is sufficient to calculate PI and P2, say, from which it follows Po = 1 - P2 - P , . The probability P2 is given by the matrix element [see Eq. (32)]
P2 = lim m I'
with the abbreviation (0) = one finds
I'
dt' ( O ~ e ~ ~ ( f - f ' ) & ~ f ' ~ O )
0
(54)
ldd) €3 Igg). After (quite) some boring calculation m
m
P2 = 1 2 8 f i I 0
0
drdr' v(r', r)p(r', r)
(55)
where
{ :I
p(r', r) = Re y
with
d$*(r'; t)&r; t )
(57)
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
287
Here the function V ( T , r ’) accounts for the photon recoil on the relative and centerof-mass motion whereas the function ,u(T, r ’ ) accounts for the relative motion before the photon emission is complete. In order to calculate the probability PI we define the projection operator
n = l ~ > ( ~ €3l l1 2 6 3 Igg>(ggI
(60)
which leaves the state of particle 2 unchanged and projects the state of particle 1 onto the resonator fundamental mode. With this operator, the probability P, reads
P, = 2Tr{IIp(t+m)} - 2P2
(61)
After (quite) some more boring calculation one finds
where
+ sinh ( $ f l r r ’ ) [ l +K ( r ) + K ( r ’ ) ] and p(r, r ’ ) as in Eq. (57). The somewhat baroque appearance of the last few formulas notwithstanding, the evaluation of the desired probabilities Po and P, is actually quite straightforward if only the temporal evolution of the atomic state vector prior to the emission, Eq. (58),is known. 3. Technicalities
The temporal evolution of the relative motion between the two atoms in Eq. (58) was solved numerically using wavepacket simulations. Because of the isotropy of both the Hamiltonian and the initial state, only the radial part of the wavefunction &r; t ) has to be calculated. With the substitution u(r) = r&r), the time evolution of the radial part can be written in the form of the Schrodinger equation for a onedimensional, harmonic oscillator. The dipole-dipole interaction enters as a nonHermitian perturbation potential of the form y [ D ( r )- i i ( l + S ( r ) ) ] . The function D ( r ) diverges for r + 0, which causes a problem in a numerical calculation. A regularization was achieved by modeling the excitation process,
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Ulf Janicke and Martin Wilkens
which prepares the e-state atom. During the excitation into the electronically excited state, the resonant dipole interaction causes a spatially dependent detuning between the electronic ground and excited state. At small distances between the two atoms, the detuning is so large that the pumping laser is out of resonance with the electronic transition and the excited state is effectively not populated in this region. We assumed a pump laser with Rabi frequency wp that is in resonance with the perturbed electronic transition at the center-of-mass point r = F of the wavepacket, which describes the relative motion of the two atoms. The spatially dependent detuning is then given by 6(r) = y[D(F) - D(r)].For a pump cycle that corresponds to half a Rabi oscillation, the perturbed wavepacket has the form
u’(r) = [l
+ S*(r)/w;]-1/2u(r)
(64)
where u(r) is the unperturbed wavepacket at time t = 0. The Rabi frequency was chosen such that the total pump efficiency for the excited state was at least 98%. The resulting state &r; t = 0 ) = u’(r)/rwas normalized and used as the initial state. The time evolution of the wavepacket u(r; t) was solved numerically observing the constraint u(r = 0; t) = 0. We extended the coordinate system to negative values of r with an initial u antisymmetric and the Hamilton operator symmetric with respect to I: With these settings, the boundary condition u(r = 0; t ) = 0 was maintained for all times.
B. RESULTS Figures 12-14 depict the probabilities P2, P,,Po as a function of the so-called Lamb-Dicke parameter 7 = 1/(4!?,)’/2 (in physical units 7 = (w,/n) * I 2 ) for various values of the Einstein-A coefficient y. For large traps 7 >> 1, the probability P, is close to unity. In this regime, the dipole-dipole interaction is negligible; the electronicallyexcited atom decays into the electronic ground state and populates certain modes of the resonator that are accessible within one recoil energy, whereas the other atom remains unperturbed in the lowest resonator mode. The probability of the initially excited atom ending up in the lowest mode is very small for large traps (high mode densities) and hence PI = 1. In the regime v2 >> lly, the results do not depend on the specific value of y. This is the so-called Raman-Nath regime, where the natural lifetime of the excited state is much smaller than the oscillation period in the resonator potential so that the motion of the atoms during the interaction may be neglected. Technically this means that we may neglect the kinetic operator in the Hamilton operator HR. With this simplification, the time evolution in Eq. ( 5 8 ) can be carried out analytically and one obtains
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
289
1
0.8
0.6
2 0.4
0.2
0
FIG. 12. The probability P2 as a function of the Lamb-Dicke parameter 7 = ( w , , / ~ ) ~ ’The ~. bold line depicts the Raman-Nath approximation.
with f[2 + S(r) + S(r’)] f(r,r’) =
[ D ( r )- D(r’)I2+ 1[2 + S(r) + S(r’)I2
(66)
The probabilities that follow from Eq. (65) (thick lines in Figs. 12-14) show perfect agreement with the exact calculations for q 2 >> lly. Figure 15 depicts the fraction P,IP, as a function of the Lamb-Dicke parameter q. For small values of q (small resonators), the gain is completely suppressed by the near-field dipole-dipole interaction. For large values of q (large resonators, high mode densities), gain is suppressed by the effects of photon recoil whereas the effects of the dipole-dipole interaction are negligible. For medium-size resonators, however, there exists a region around q = 1, where gain exceeds loss, that is, P2 > P I .In the Raman-Nath regime, this region is given by 0.3 < q < 1.7. This result clearly indicates that for resonators of linear dimension L = A (77 = l),
U gJanicke and Martin Wilkens
290
1
0.8
0.6
0.4
0.2
0 1
3
2
rl FIG. 13. The probability P, as a function of the Lamb-Dicke parameter bold line depicts the Raman-Nath approximation.
=
(umc/n)’’*. The
matter wave amplification should be possible in a three-dimensionalconfiguration despite photon reabsorption processes.
C. MODIFIED KINETIC EQUATIONS In order to demonstrate the effects of photon reabsorption on the threshold behavior of the atom-laser, we model the reabsorption terms in the kinetic equations (9) and (10)
Here
where pabs is the probability that a spontaneous photon is reabsorbed by a resonator atom. In leading order we may identifypabs= Po and 2S0/(l + S o ) = P2.
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
291
The mean mode population of the atom-lasing mode in the stationary regime is plotted in Fig. 16 as a function of the pump rate r for various values of the ratio P,IP,. The smaller this ratio is, the less pronounced is the onset of atom-lasing. Choosing PJP, = 2, which is the optimal gain according to the results shown in Fig. 15, threshold is reached at about r = 2r0 with a 50% reduced slope efficiency above threshold.
V. Summary We have introduced a particular scheme for the atom-laser that is based on optical pumping. Neglecting inelastic processes that result from photon scattering, the system displays mode competition, threshold behavior, and single-peaked Poissonian counting statistics above threshold. For the study of the inelastic processes, we have developed a two-atom model of the resonant dipole-dipole interaction. The results indicate that matter wave amplification is possible despite photon reabsorption, provided the size of the
0.8
0.6
e 0.4
0.2 0 2
1
3
T FIG. 14. The probability Po as a function of the Lamb-Dicke parameter 7 = ( ~ , / f i ) ” ~The . bold line depicts the Raman-Nath approximation.
Ulf Janicke and Martin Wilkens
292 15 14 13 12 11
10
c g c 7 \
a
6
5 4 3 2 1
0
FIG. 15. The fraction Pz/Poas a function of the Lamb-Dicke parameter r ) = ( ~ ~ ~ J i 2The )''~. threshold condition for matter wave amplification is P,lP, > I . The bold line depicts the Raman-Nath approximation.
atom-laser resonator is of the order of the photon wavelength. For large resonators, threshold cannot be reached because of the photon recoil associated with the reabsorption. Experiments with small resonators were carried out with metastable argon (Mueller er al., 1997). Argon atoms (a + Is,, e + 2p,, g + Is,) were precooled in a magneto-optical trap to temperatures of about 10 mK and pumped into a strongly detuned, dark, three-dimensional optical lattice. Each of the lattice points corresponds to a micro-resonator for atomic matter waves. Typical parameters are y / 2 = ~ 5 MHz, S/277 = 2 THz, and ai/27r = 30 kHz. Both the quantized motion of the atoms in the resonator modes and mode selection due to different loss rates could be observed. The loss rates for the lowest mode were typically Ksca Ktun 0.4s-l. The achievable pump rate, however, was of the order of Hz, which is by several orders of magnitude smaller than the pump rate at threshold, which is about 10 Hz. The use of micro-resonators is problematic for two reasons. First, because of the small pump volume, it is difficult to achieve high pump rates. Second, even with only a few atoms in the resonator volume, the atom density becomes so high %
%
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
500
s=,
0.01, Kv= (3/2+ V)Ko 5 explicit levels
r/
400
293
/
P2= 1
,s
300
’* B
200
100
0 1
2
3
4
5
6
rho FIG. 16. The results of the modified kinetic equations for different values of the fraction P,IP,.
that collisions between the atoms may become important. In this case, atom-atom interactions are difficult to control and chemistry takes over. However, even with a large, isotropic resonator an atom-laser could be realized if only the pump rate is sufficiently large. In the initial stages, the system is pumped until threshold is reached in a micro-resonator. Then the resonator volume is increased adiabatically while pumping continues. In the case of a resonator created by an optical standing wave, the change of resonator volume could simply be achieved by a continuous change of the standing wave period. According to the results of Cirac and Lewenstein (1996),* the system should remain above threshold despite photon reabsorption processes, provided the number of atoms in the amplified mode is sufficiently large in the beginning. This would allow the creation of a Bose-Einstein condensate with a large number of atoms using all-optical methods. A possible alternative is the use of resonators with an macroscopic extension in only one or two spatial dimensions. Such configurations could support sufficiently high pump rates and at the same time suppress the hazardous effects of photon reabsorption (Pfau and Mlynek, 1997). *In a recent letter by Castin et al., Phys. Rev. Lett, 80, 5305 (1998). it is argued that photonreabsorption must not necessarily be detrimental for the atom laser proposed by Cirac et al. (1996).
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UlfJanicke and Martin Wilkens
VI. Acknowledgments This work was supported by the Forschergruppe Quantengase of the Deutsche Forschungsgemeinschaft. Both authors gratefully acknowledge the inspiring discussions with Jurgen Mlynek, Tilman Pfau, Maciej Lewenstein, Pierre Meystre, Ewa Rodriguez, and Anna Ciszewska.
VII. Appendix A: N-Atom Master Equation 1. FUNDAMENTAL HAMILTONIAN
We begin with a brief review of the Power-Zienau formulation of nonrelativistic quantum electrodynamics. For a detailed account see Craig and Thirunamachandran, Molecular Quantum Electrodynamics, Academic ( 1984). Neglecting magnetic interactions, the Hamiltonian of the system atoms field can be expressed as a sum of four terms
+
Htot
= H A + HF + HAF+ HPZ
('41)
which refer, respectively, to the noninteracting atoms, the free electromagnetic field, the atom-field coupling, and a term that stems from the Power-Zienau transformation. The Hamiltonian of the atoms is given by
HA=
P2 c a+ 2M
N
,=I
H,
where H , accounts for the dynamics of the electronic degrees of freedom of the ath atom and pzl(2M) is the atomic center-of-mass kinetic energy. Here and in what follows, p, denotes the canonical momentum operator and r, denotes the conjugate position operator of atom a = 1,2, . . . ,N.The electronic Hamiltonian Ha need not be explicitly given here; it involves the kinetic energies of electrons and nucleus in the center-of-mass frame of the atom, the Coulomb interaction between the charged particles, possibly the fine structure and hyperfine structure interactions, and so on. The Hamiltonian of the electromagneticfield is given by
LA,
where a d k , are photon creation and annihilation operators, k is a wave vector, A = 1 , 2 is a polarization index, and w k = ck where k = Ik I is a wave number. The interaction of atoms with the radiation field is described by
I
-+
HAF= - d 3 x 9 ( x ) * E(x)
(A41
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
295
where E(x) is the electric component of the radiation field and $(x) is the atomic polarization field.5 The electric component of the radiation field may be decomposed
E(x) = E + ( x ) + E-(x)
645)
where E (x) denotes the positive frequency part. Utilizing a plane wave expansion we have +
where %k = v f i W k / ( 2 E o v ) is the electric field strength per photon with v being the quantization volume, and ekA,A = 1, 2, are polarization unit vectors with ekl X ek2 = klk. The atomic polarization field is additive over the atoms
a=l
+
The single-atom contribution 9,(x) is a certain functional of the charge carriers of the a t h atom. This functional is usually evaluated in terms of a multipole expansion, which is truncated at an appropriate level; in the most common electric dipole approximation, for example, one has +
Pa@)= d,S[x - ra]
(A8)
where d, is the electric dipole operator of the atom, which is located at position r, . We shall not perform the electric dipole approximation until the final stage of our derivation. That has the advantage of notational clarity and also provides deeper insight into the structure of the theory. The term Hpz in Eq. (Al) is characteristic for the Power-Zienau formulation of electrodynamics;it may be decomposed into two parts
where the second sum extends over all pairs of atoms. Here HE: is part of the single-atom self-energy, pz
/ /
1 =d3X
2E0
d3x’S,:.(x - x‘)9,,;(x)9a,j(x’)
SFormally, in the Power-Zienau scheme, E(x) is the electric displacement D, but we shall continue to use the more familiar notation E instead of D.
296
Ulf Janicke and Martin Wilkens
and HE; is an atom-atom contact interaction
Here and in what follows, i, j = x, y, z are Cartesian indices, 9 a , i ( x ) is the ith Cartesian component of the polarization field of the ath atom, and we adopt the Einstein summation convention of summing over repeated Cartesian indices. Furthermore, 8, is the Kronecker delta, S(r) is the usual delta function, and 6; (r) is the transverse delta function
where r is a spatial vector with Cartesian components ri, i = x, y, z, and r = I r I. In the Power-Zienau formulation all interaction between the atoms (except for the contact interaction) is mediated by the radiation field. The elimination of the radiation field, which is described next, then yields a description that entails in leading order-besides the spontaneous emission and the Lamb-shift-the atom-atom interaction such as the dipole-dipole interaction, and in yet higher orders the Van der Wads and Casimir-Polderinteraction. 2.
ELIMINATION OF THE FLUORESCENCE FIELD
a. Exact Relations
To prepare for the elimination of the radiation degrees of freedom, we consider the Liouville-von Neumann equation d -W= dt
i - - [ H , W] =%W h
which describes the dynamics of the density operator W of the composed system atoms + radiation field. The formal solution of that equation reads W(t) = e2' W(0)
(A14)
The density operator of the atoms, p, is obtained by tracing over the degrees of freedom of the radiation field p([)
TrFIW(t)l = TrF[e2rpp(0)lp(0)
(-415)
where we assume that the radiation field and the atoms are uncorrelated initially, that is, W(0) = pF(0)C3 p(0). Within our formalism, this assumption is not really necessary, but it makes life a little easier. The dynamical equation for p is now obtained by taking the time derivative
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
d -p dt
=e
297
wp
where
is the exact Liouville operator for the atomic system.
b. Markoff Approximation In the Markoff approximationEq. (A16) is replaced by the equation
d - p = ep dt where
The physics described by the exact equation (A16) and its approximation (A18) is the same on the time scale on which t ( t ) becomes stationary. For the frequently studied case of a single atom this is the time scale of fluctuations of the radiation field T ~this~time~scale ; is usually very short, even shorter than an optical period. In the present case of many atoms the situation is slightly different. Here stationarity is reached on a longer time scale rwa,, which is given by the time it takes the light to traverse the coherence volume of the atomic sample. Thus Eq. (A18) is valid on time scales t >> rwav. c. Born Series
We shall calculate the atomic Liouvillian (A19) perturbatively in a Born expansion where the noninteracting Hamiltonian
Ho = HA
+ HF
(A201
is treated exactly, whereas the remainder Htot- H, is treated as a perturbation. In our perturbation scheme, the atomic Liouvillian is represented as
c = e ( o ) + e ( l )+ e(*)+ . . .
(A2 1)
where t(") is nth order in the atomic polarization field. Obviously the zeroth-order contribution is given by i eco) = - fi [HA ' . ]
(A23
298
Ulf Janicke and Martin Wilkens
is proportional to (E(x)) where the expectation The first-order contribution value refers to the initial distribution of photons p F ( 0 ) .Except for special circumstances this distribution does not carry a phase, that is, (E(x)) = 0, and hence there is no contribution in first order,
ecu = 0
(A23)
The second-order contribution is given by
e m = - -hi [H,,,
XI+
~ ( 2 )
(A241
with
where we have defined Y o = - ( i / h ) [ H o , -1, Y l = - ( i / h ) [ H A F , . ] , and we have assumed stationarity of the radiation state with respect to the free evolution, [HF’PF(o)I = O.
3. ATOMIC MASTER EQUATION Collecting (A22)-(A25), the atomic master equation may be cast in the form
d i -dt~ ( t )= - [HeffP- ~ H i f f+ I $P
(‘426)
where
(A271 He, = HA + Hpz + ‘He, is a non-Hermitian effective Hamiltonian, and $ is a super-operator, which assures that Eq. (A26) is a proper master equation. In second order of the atom-radiation interaction
dt’(Ej(x, t)E’(x’,t’))q.(x)q(x’, t‘ - t ) (A28)
$X =
d/
d 3 x / d3x’lim [dt’(Ei(x, t)Ej(x’, t’))q(x’,t’ - t)X%(x) + H.c. 1-m
0
(A29) where the time dependency of 9(t)and E ( t ) is governed by the free Hamiltonians HA and HF, respectively:
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
299
a. Empty-Bath Assumption
To proceed we must specify the state of the fluorescence field, pF,and choose an appropriate model for the polarization field. We here assume that the fluorescence field is in the vacuum state, pF = I { O } ) ( { 0) I. With this assumption the two-point correlation function evaluates to
where 8 ; ( k ) = S i j - k i k j / k 2is the transverse delta function in Fourier space. b. Single-Line Approximation
In the single-line approximation, the polarization field is modeled +
9 ( x , t ) = e-imor$+(x)
+ + eimo'g-(x)
(A33)
where w, is the Bohr transition frequency, and $ ( x ) = $ + ( x ) + $ - ( x ) is the operator of the polarization field in the Schrodinger picture. The conjugate polarization field is defined +
-3
2 E @ j - - $+)
(A341
In passing, we note that the positive frequeflcy part @ ( x ) transfers atoms from the excited state into the ground state, and 8 - ( x ) acts in reverse.6 +
c. Coupling in the Markoff Limit
Substituting Eqs. (A32) and (A33) into Eq. (A28), one realizes that SH,, is bilinear in the polarization fields with the coupling given by the half-sided Fourier transform (A33 for k = ? k o , k , = w,/c being the wavenumber of the atomic line. In passing we note that Ciiobeys a Kramers-Kronig relation 6Do not confuse the single-line approximation with a two-level approximation. In the single-line approximation, only the principal quantum numbers are selected but nothing is said about additional quantum numbers such as magnitude or direction of angular momentum.
300
Ulf Janicke and Martin Wilkens
Im C,(x, x‘; k ) = -P
k - k‘
IT
A useful representation of the real part is provided by
hk3 Re Cij(x,x‘; k) = -6(k)Tij(x,x’; k) 6~e,
with
I
~ ~ ~x’;( k x) , d2KQij(g ) e i k ’ ( x - x,‘ )
Qij( K +)
=3
IT
(aij - !$)
(A38)
where 2 = k/k is the unit wave-vector, and d2K is the associated solid angle. 4. SELF-ENERGY AND PAIR-INTERACTION ENERGY
We decompose the interaction effective Hamiltonian into a Hermitian and an antiHermitian part,
Hpz
+ SH,,
h
= Hi,, - i - T
(A39)
2
The interaction-Hamiltonian is further decomposed into a self-energy part and a pair-interaction part, Hi,, = Z, Ha, + Z,,p, H a p . For the self-energy part, the Power-Zienau contribution is exactly cancelled by a corresponding term of SH,,. The remaining self-interaction
I
Ha, = - di dj Im(C; - Cij 4h
92,,jl
(A401
is still infinite and will therefore be ignored. Here we have introduced the abbreviations C,? = C,(x, x’; +k,), and J di = Z?,, J d3x. a. Pair-Interaction-Dispersive Part
The pair-interaction energy is given by
where the first term is the Power-Zienau interaction energy: combined with the second term, which also contains a delta-function contribution, the resulting ex-
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
301
pression is easily identified with the Greens-function tensor of the mathematical dipole. The second integral is finite, and vanishes in the static limit k , + 0. It will be discarded in the rotating-wave approximation-see following. b. Dissipative Interaction
The decay operator r and jump super-operator$ are given by
’I I 5I I
r =7 fi $p =
d 3 x d3x‘ReCij(x,x’; k , ) 9 ; ( x ) 9 f (x’)
(A421
d 3 x d3x’ReCjj(x,x’; k , ) 9 : ( x ’ ) p 9 ; (x)
(A43)
where terms rn 9 9+,9 - 9-, which are all finite, have been ignored. +
5. ROTATING-WAVE APPROXIMATION
In the rotating-wave approximation all terms rn 9 9+,9- 9 - are discarded. In this approximation, the atomic master equation +
d dt
-p =
-i
[HA
+ HDD
9
p1 -
1
2{
r9
p 1 + $p
(AW
is given in terms of the following quantities:
where we have introduced the dimensionless polarization field9 = 9/63,with 63 the dipole reduced matrix element of the atomic line, and y = p 2 k i / ( 3 ~ e , his) the Einstein-A coefficient. The tensors D, and S , are defined by
37T Djj(x,x‘) = - S i j S ( x ki
- x’)
+ -7TP
k4 T ~ ~ (x’; x , k) dkki k z - k2 ’
10
(A47)
S 1J. .(x, x’) = T~~(x, x’; k , )
where T~~is defined in Eq. (A38). In the position representation, these tensors evaluate to 7T
D i j ( x , x ’ )= --S..S(x - x’) k i lJ
+
D,(k,r)
r.r. + ‘--J.Dll(k,r) r2
(A48)
302
UlfJanicke and Martin Wilkens
S,(k,r)
rj5 + -S,I(k,r) r2
w h e r e r j = x j - x ] , r = Irl,and
D11(5)=
--(y+F) 3 sin5
2
5
S,,(() = -3 (c;t--):;s
6. TWO-LEVEL APPROXIMATION In the two-level approximation a particular alignment (linear, circular) of the atomic dipole moment is assumed. Denoting the alignment GI@,one postulates --f
s , ; ( x ) = $C+;S(X - ra>
(A54)
where u = Ig)(el is the atomic lowering operator. The expressions for HDD,r, and are obtained by inserting Eq. (A54) into Eqs. (A45)-(A46), HDD
= hy
2 D(ra, rp)(aLup + u;ua)
(a@
9 p = y 2 J d2K@(K')e-'ko"r~u apu;e'ko;.ra 0.B
(A551
(A571
where S, D, and @ are given by the contraction of the tensors S,, D,, and CPij, which are defined in Eqs. (A47) and (A38), for example CP = @Faij Qj/Q2.
7. SECOND QUANTIZATION For a gas of identical two-level atoms a formulation in terms of second quantized atomic fields is convenient. Introducing a suitable set of single-atom states {I gv), I ep)}, and associated creation and annihilation operators, {g:, ZL,g,, Z p } , the second quantized atomic-matterfield reads
@(x) =
c 4,(x)8, + c 4,(x>@, Y
(XI
P
(-458)
where +,(x) = v) and 4p= (xlp) specifies the motional state of an atom in the electronic ground and excited state, respectively.
ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING
303
In the language of second quantization the dipole pair-interaction assumes the form ir,D = f
C
i ~ DvqpptL2itp2p Y W P
and the decay operator and jump super-operator are given by
Here the quantities DYqpp, SYqpp are just matrix elements of the operators D and S, which have been introduced in Eqs. (A55)-(A57). By inspection of Eq.(A47),
they are all given in terms of Twqpp(k) =
I
d2K~(~)(qle;k.rlCL)(vle-'k.'Ip)
('462)
We note that the expressions (A59)-(A61) hold for both Fermi and Bose statistics.
References Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E., and Comell, E. A. (1995). Observation of Bose-Einstein condensation in a dilute atomic vapor. Science 269, 198-201. Ballagh, R. J., Bumett, K., and Scott, T. F. (1997). Theory of an output coupler for Bose-Einstein condensed atoms. Phys. Rev. Lett. 78,1607-161 1. BordC, C. (1995). Amplification of atomic fields by stimulated emission of atoms. Phys. Lett. A 204, 217-222. Bradley, C. C., Sackett, C. A., Tollett, J. J., and Hulet, R. G. (1995). Evidence of Bose-Einstein condensation in an atomic gas with attractive interaction. Phys. Rev. Lett. 75, 1687-1690. Cirac, J. I. and Lewenstein, M. (1996). Pumping atoms into a Bose-Einstein condensate in the bosonaccumulation regime. Phys. Rev. A 53,2466-2476. Cirac, J. I., Lewenstein, M., and Zoller, P. (1994). Quantum statistics of a laser cooled ideal gas. Phys. Rev. Lett. 72,2977-2980. Cirac, J. I., Lewenstein, M., and Zoller, P. (1996). Collective laser cooling of trapped atoms. Europhys. Lett. 35, 647-651. Davis, K. B., Mewes, M.-O., Andrews, M. R., van Druten, N. J., Durfee, D. S., Kurn, D. M., and Ketterle, W. (1995). Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, 3969-3973. Ellinger, K., Cooper, J., and Zoller, P. (1994). Light-pressure force in N-atom systems. Phys. Rev. A 49,3909-3933. Goldstein, E., Pax, P., Schernthanner, K. J., Taylor, B., and Meystre, P. (1995). Influence of the dipoledipole interaction on velocity selective coherent population trapping. Appl. Phys. B 60,161-167. Goldstein, E., Pax, P., and Meystre, P. (1996). Dipole-dipole interaction in a three-dimensional optical lattice. Ph.ys. Rev. A 53,2604-2615. Guzmh, A. M., Moore, M., and Meystre, P. (1996). Theory of a coherent atomic-beam generator. Phys. Rev. A 53,977-984. Holland, M., Burnett, K., Gardiner, C., Cirac, J. I., and Zoller, P. (1996). Theory of an atom laser. Phys. Rev. A 54, 1757-1760.
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Janicke, U., and Wilkens, M. (1996). Prospects of matter wave amplificationin the presence of a single photon. Europhys. Lerr. 35,561-566. Javanainen,J., and Wilkens, M. (1997). Phase and phase diffusion of a split Bose-Einsteincondensate. Phys. Rev. Lett. 78,4675-4678. Kneer, B., Wong, T., Vogel, K., Schleich, W. P., and Walls, D. F. (1998). Generic model o f a n atom laser. eprint cond-mat19806287. Kurizki, G., and Ben-Reuven, A. (1987). Theory of cooperative fluorescence from products of reactions or collisions: Identical neutral atomic fragments. Phys. Rev. A 36.90-104. Lewenstein, M., and You, L. (1996). In B. Bederson and H. Walther (Eds.), Advances of atomic, molecular and opticalphysics 36,221. Academic Press (New York). Lewenstein,M., You, L., Cooper, J., and Burnett, K. (1994). Quantum field theory of atoms interacting with photons: Foundations. Phys. Rev. A 50,2207-223 1. Loudon, R. (1983). The quantum rheory oflight. Oxford University Press (Oxford). Mewes, M.-O., Andrews, M. R., Kurn, D. M.,Durfee, D. S., Townsend,C. G., and Ketterle, W. (1997). Output coupler for Bose-Einsteincondensed atoms. Phys. Rev. Left. 78,582-585. Moy, G. M., Hope, J. J., and Savage, C. M. (1997). Atom laser based on Raman transitions. Phys. Rev. A 55,3631-3638. MUller-Seydlitz, T., Hartl, M., Brezger, B., Hhsel, H., Keller, C.. Schnetz, A., Spreeuw, R., Pfau, T., and Mlynek, J. (1997). Atoms in the lowest motional band of a three-dimensionaloptical lattice. Phys. Rev. Lett. 78, 1038-1041. Naraschewski,M., Schenzle, A., and Wallis, H. (1997). Phase diffusion and the output properties of a cw atom-laser. Phys. Rev. A 56,603-606. Olshanii, M., Castin, Y.,and Dalibard, J. (1995). In A. Sasso, M. Inguscio, M. Allegrini, (EMS.), Proceedings ofrhe XI1 Conference on Laser Spectroscopy. World Scientific (New York). Pfau, T., and Mlynek, J. (1997). A 2D quantum gas of laser cooled atoms. OSA Trends in Optics and Photonics Series on Bose-Einstein Condensation, 7,33-37. Rempe, G. et al. (1997). BEC in a gas of Rubidium atoms. Private communication. Spreeuw, R. J. C., Pfau, T., Janicke, U.,and Wilkens, M. (1995). Laser-like scheme for atomic-matter waves. Europhys. Lett. 32,469-474. Steck, H., Naraschewski, M., and Wallis, H. (1998). Output of a pulsed atom laser. Phys. Rev. Lett. 80, 1. Vogt, A. W., Cirac, J. I., and Zoller, P.(1996). Collective laser cooling of two trapped ions. Phys. Rev. A 53,950-968. Walker, T.,Sesko, D., and Wieman, C. (1990). Collectivebehavior of optically trapped neutral atoms. Phys. Rev. Lert. 64,408-41 1. Wallis, H. (forthcoming). Matter wave amplification and collective internal excitations in a trapped Bose gas. MPQ Munich. Weiss, C. (1997). Diploma thesis, University of Konstanz (unpublished). Weiss, C., and Wilkens, M. (1997). Particle number counting statistics in ideal Bose gasses. Optics Express 1,272-283. Wilkens, M., and Weiss, C. (1997). Particle number fluctuationsin an ideal Bose gas. J. Mod. Opt. 44. 1801-1814. Wiseman, H. M. and Collett, M. J. (1995). An atom laser based on dark-state cooling. Phys. Lett. A 202,246-252. Wiseman, H. M., Martins, A., and Walls, D. (1996). An atom laser based on evaporative cooling. Quantum Semiclass. Opr. 8, 737-753. You, L., Lewenstein, M., and Cooper, J. (1995). Quantum field theory of atoms interacting with photons, part It. Phys. Rev. A 51,4712-4726.
Index Above threshold ionization (ATI), 135136 Acousto-optic modulator (AOM), 60 Adiabatic approximation,91,93-94 nonadiabatic phenomena, influence of, 128-13 1,232-224 Anderson localization, 44 Anti-Helmholz configuration,5 1 Atom guiding applications, 182-183 channeling in a standing wave, 183 donut mode, 238,255-257 electrical field guiding, 184-186 experiments, 250 -257 grazing incident mode, 250-252 iris waveguides, 183 lasers, role of, 183 magnetic field guiding, 186-187 nonadiabatic transitions, 223-224 optical near fields, 183 spontaneous emission, 207-208,222223 tunneling to dielectric surface, 223 Atom guiding, evanescent waves and cylindrical hollow optical fiber for, 219-225 electromagnetic field in hollow optical fiber, 213-218 experiment, 252-255 horn shape hollow optical fiber for, 226 -234 planar waveguides, 234-235 Atom guiding, propagation of laser fields for dark spot laser beams, 238-243 Gaussian laser beam, 183, 191,236237 in hollow optical fiber, 237-238 standing light waves, 243-250 Atomic deflection joint measurements, 151-154 phase operator measurements, 160162 photon statistics from, 149-151, 163
reconstruction of a quantum field, 157160 spontaneous emission, influence of, 154-157 Atom-laser background information, 262-264 components, 266-269 kinetic equations and, 269-272,29029 1 loss rate, 268-269 master equation, 272-278,294-303 optical pumping, 267-268 photon reabsorption, 278-291 principle of, 264-266 pump map derivation, 276 rate equations, 276-278 resonant dipole interaction, 274-276 resonator, 266 -267 two-atom problem, 282-288 versus Bose-Einstein condensation, 270-272 Atom mirror schemes, 193 atom-surface interaction, 198-202 coherence of, 207-210 dielectric waveguide, 196-198 simple evanescent wave, 194-195 surface plasmons, 195-196 Atom optics Bragg regime, 170-175 Bragg resonances, 172, 173-174 focusing of atomic waves, 166-169 grazing incidence, 172-173 Hamiltonian, 146-148,149,163, 166167 localization of atoms, 164-166 model, 145-146 momentum distribution of deflected atoms, 163-164 motion equations, 170-172 in nonresonant fields, 162-169 quantum Pendellosung, 174-175 Raman-Nath approximation, 148, 157, 169,172-173 research on, 144-145 305
306
INDEX
Atoms See also Quantum chaos, cold atoms and coherence of matter wave, 204-207 methods of light scattering, 45 radiation forces on, in a laser field, 187193 reflection of, using atom mirrors, 193202 reflection of, using evanescent waves, 202-2 13 surface interaction, 198-202 two-level atoms in standing-wave potential, 45-49 Attosecond physics, 86, 133-136 Autler-Townes microscopy, 165 Autocorrelation function, 205-206 Becker model, 93,94 Bell’ s inequality, 4, 14 double entanglement of type-I1 SPDC, 15-17 for space-time observable, 19-22 for spin variables, 17-19 two-photon wavepacket in, 30-32 Bessel functions, 73,74,75-76,77, 150, 151,215,220 Biphoton. See Two-photon wavepacket Bloch bands, 48 Bloch equations, 189-190 Bloch vector components, 189 Born series, 297-298 Bose-Einstein condensation, 232,262-264 atom-laser versus, 270-272 Bose function, 232-233 Bragg regime, 170-175 Bragg resonances, 172, 173-174 Bragg scattering, 55,56 Casimir-Polder energy shift, 199-200 Classical phase space averaging, 92 Click-click coincidence detection event, 14, 17,26,28, 31, 32 Coherence See also Spatial coherence; Spectral coherence; Temporal coherence of atom mirror, 207-2 10 of matter wave, 204 -207 Collection efficiency loophole, 4, 14 Conceptual Feynman diagram, use of, 23 24, 28,29
Copenhagen interpretation, 2, 13 Cylindrical hollow optical fiber, atom guiding and loading of atom waveguide, 224-225 losses in atom waveguide, 221-224 quantum mechanics of, 219-221 Dark spot laser beams (DSLBs) computer-generated hologram method, 241-242 donut mode, 238,255-257 micro-collimation technique, 242-243 mode conversion method, 239-240 de Broglie wave, 197,204,220 Dielectric waveguide, 196-198 tunneling to, 223 Diffusive reflection, 208 -2 10 Dissipative reflection, 210-213 Doppleron resonances, 172 Doppler shift, 170, 171, 173 Dressed-atom approach to dipole force, 191-192 Dynamical localization, 43-44,48 future for, 78-79 kicked rotor, 59,64-65,67 modulated standing wave, 77 Earnshaw theorem, 186 Einstein-Podolsky-Rosen (EPR), 1,2-5 Electrical field guiding of atoms, 184186 Electro-optic modulator (EOM), 76-77 Evanescent waves See also Atom guiding, evanescent waves and atom mirror schemes, 193-202 atom-surface interaction, 198-202 coherence of atom mirror, 207-210 coherence of matter wave, 204-207 dielectric waveguide, 196-198 diffusive reflection, 208-210 dissipative reflection, 210-213 dressed-atom approach to dipole force, 191-192 forces on atoms in a laser field, 187-193 Gaussian laser beam, 191 impulse diffusion, 193 near-resonant light forces, 187-190 simple, 194-195 specular reflection, 202-204
INDEX standing plane wave, 190-191 surface plasmons, 195-196 traveling plane wave, 190 used in reflection of atoms, 202-213 Fermi’s golden rule, 267 Floquet analysis, 91 Floquet states, 64-65 Fock coefficients, 151, 166, 168 Fourier series, 62,73,96,209 Franck-Condon factor, 267,268, 269, 270, 281-282 Franson interferometer,20-21 Frequency-dependentdielectric reflection coefficient, 198-199 Gaussian laser beam, 183, 191,236-237 Gaussian thin lense equation, 8 Gedankenexperiment, I , 2, 144 Ghost image, 5.6-8 Ghost interference-diffraction,5-6,914,26 Glauber formula, 26 Gouy phase, 240 Gradient force, 191 Grazing incident mode, 250-252 Greenberger-Home-Zeilinger(GHZ), 5 Hamiltonian atom optics, 146-148, 149, 163, 166167 of atoms in a laser field, 187 chaos, 59 fundamental, 294-296 Jaynes-Cummings, 146-148,149 one-dimensional,46,48,49 quantum-nondemolition(QND), 148, 162 two-level atoms in standing-wave potential, 45-49 Harmonic beam, macroscopic, 124-125 Harmonic chirp, 126-128 Harmonic generation (HG), high-order applications, 86, 131- 136 cutoff position, 84 ellipticity studies, 85, 94 history of, 84-86 optimization and control studies, 85-86 plateau extension, 84 spatio-temporalcharacteristics of, 87-89
307
Harmonic generation, phase matching and atomic polarization, 100-103 at the focus, 106 cutoff law, modified, 105-106 dynamic, 103 harmonic emission sources, 99-100 jet position and conversion efficiency, 103-105 off axis, 102, 104 on axis, 104 static, 103 Harmonic generation, spatial coherence and atomic jet after the focus, 107-1 13 atomic jet before the focus, 113-1 16 defined, 106-107 Harmonic generation, temporal and spectral coherence ionization, influence of, 122-124 jet position, role of, 116-122 nonadiabatic phenomena, influence of, 128-131 phase modulation, consequences of, 124-1 28 Harmonic generation theories macroscopic response, 98-99 propagation theory, 97-98 single-atom response in strong field approximation, 95 -97 single-atom theories, 9 1-94 Heisenberg equations, 188 Helmholz equation, 214 Hermite-Gaussian mode, 239-240 Hidden variable theory, 4 Hollow optical fiber (HOF), atom guiding and cylindrical, 2 19-225 electromagnetic field in, 213-218 horn shape, 226-234 laser light inside, 237-238 Holograms, computer-generated,24 1-242 Horn shape hollow optical fiber, atom guiding and, 26-234 Impulse diffusion, 193 Interferometry Franson, 20-21 harmonics and, 132-133 Michelson, 32, 132 Ramsey, 165 Ionization, influence of, 122-124
308
INDEX
Jaynes-Cummings Hamiltonian, 146-148, 149 Jet position conversion efficiency and, 103-105 influence of, 116-122 Josephson junctions, 72 Keldysh-Faisal-Reiss approximation, 93 Kerr medium, 119 Kicked particles, 60 Kicked rotor background information, 59-61 classical analysis, 61-64 dynamical localization, 59,64-65,67 experimental parameters, 66-68 experimental results, 68-70 quantum analysis, 64-66 quantum resonances, 64,70-71 Kinetic equations, 269-272,290-291 Kolmagorov-Amol’d-Moser (KAM) theorem, 44,48,55 Kramers-Kronig relation, 299-300 Lambe-Dicke paramter, 288,289 Landau-Dyhne formula, 95 Languerre-Gaussian mode, 239-240 Laser field propagation. See Atom guiding, propagation of laser fields for Liouville operator, 154 Liouville-von Neumann equation, 274, 296 -297 Magnetic field guiding of atoms, 186-1 87 Magnetic-optic trap (MOT), 50-54,235 Markoff approximation, 297,299-300 Master equation, 272-278,294-303 Maxwell distribution, 224 Maxwell equations, 97.98-99,213-214 Michelson interferometer, 32, 132 Micro-collimation technique, 242 -243 Momentum diffusion coefficient, 193 Momentum distribution of deflected atoms, 163-164 Motion equations, 170-172 Nonadiabatic phenomena, influence of, 128-131,223-224 Nonlinear atomic homodyne detection, 157, 162
Pair-interaction energy, 300-301 Phase matching atomic polarization, 100-103 at the focus, 106 conditions, 8 cutoff law, modified, 105-106 dynamic, 103 harmonic emission sources, 99-100 jet position and conversion efficiency, 103-105 off axis, 102, 104 onaxis, 104 static, 103 Phase modulation, consequences of, 124-128 Phase operator measurements, 160-162 Photon reabsorption, 278-291 Photon statistics, 149-151, 163 Planar waveguides, 234-235 Pockels cell, 20-21, 30 Poincar6 surface of section, 47,5 I, 55 Polarizability of atoms, 184 Power-Zienau formulation, 294 -296 Propagation theory, 97-98 Pseudo-potential model, 94 Quantum break time, 43,65-66,69 Quantum chaos, cold atoms and dynamical localization, 4 3 4 4 4 8 future for, 78-79 kicked rotor, 59-71 modulated standing wave, 72-78 momentum transfer, experimental methods used, 49-54 research on, 43-44 single pulse interaction, 54-59 two-level atoms in standing-wave potential, 45-49 Quantum electrodynamics(QEDs), cavity, 144 Quantum Kapitza-Dirac effect, 150 Quantum-nondemolition (QND), 148, 162 Quantum Pendellosung, 174-175 Quantum resonances, 64,70-71 Rabi frequency, 46, 147, 172, 174, 190, 202,203 Raman-Nath approximation atom optics and, 148, 157, 169 grazing incidence, 172-173
INDEX Raman-Nath regime, 288 Raman-Nath scattering, 55,56 Ramsey interferometry, 165 Rayleigh approximation, 209 Rayleigh range, 236 Resonant dipole interaction, 274-276 Resonant kicks, 74 Resonant quantum field, atomic deflection and joint measurements, 151-154 phase operator measurements, 160-162 photon statistics from, 149-151 reconstruction of a quantum field, 157160 spontaneous emission, influence of, 154-1 57 Rotating-wave approximation, 30 1-302 Rydberg atoms, 44 Saddle-point value of momentum, 95 -96 Schrodinger cat, 144 entangled state, 2, 21 equations, 46,49,64, 170,219-220 time-dependent, 91,95 Second quantization, 302-303 Shell’s law, 194 Simple man’s theory, 84-85,93 Sinc function envelope, 9, 16,34 Single active electron (SAJZ) approximation, 91 Single-atom response in strong field approximation, 95-97 Single-atom theories classical phase space averaging, 92 numerical methods, 9 1-92 pseudo-potential model, 94 strong field approximation, 92-94 Single-line approximation, 299 Single-photon measurement, of a twophoton state, 32-35 Single pulse interaction, 54-59 Snell’s law, 8 Space-time observable, Bell’s inequality for, 19-22 Spatial coherence atomic jet after the focus, 107-1 13 atomic jet before the focus, 113-116 defined, 106-1 07
309
Spatio-temporal characteristics of high harmonics, 87-89 Spectral coherence ionization, influence of, 122-124 jet position, role of, 116-122 nonadiabatic phenomena, influence of, 128-13 1 phase modulation, consequences of, 124-128 Spin variables, Bell’s inequality for, 17-19 Spontaneous emission, influence of, 154157,207-208,222-223 Spontaneous light pressure force, 190 Spontaneous parametric down conversion (SPDC),1-2,3 Bell’ s inequality, 4, 14-22 double entanglement of type-11, 15-17 ghost image, 5,6-8 ghost interference-diffraction, 5 -6, 9-14 type-11, 15,-17,31 Standing light waves, atom guiding with, 243 atom potential in, 244-245 experiments with, 246-250 single potential well, 245-246 Standing plane wave, 190-191 Standing wave, modulated background information, 72 classical analysis, 73-76 experiment, 76 -78 Standing-wave intensity, turning on and off of, 54-59 Standing-wave potential, two-level atoms in, 45-49 Stern-Gerlach effect, 235 Strong field approximation (SFA), 92-94 single-atom response in, 95 -97 Surface plasmons, 195-196 Temporal coherence ionization, influence of, 122-124 jet position, role of, 116-1 22 nonadiabatic phenomena, influence of, 128-13 1 phase modulation, consequences of, 124-I 28 Three-photon entanglement, 5 Time-dependent Floquet states, 64-65
310
INDEX
Time-dependent potential, single pulse interaction and, 54-59 Time-dependent Schrodinger equation (TDSE), 91,95, 135 Time-resolved attosecond spectroscopy (TRAS), 135-136 Transverse phase constant, 214 Traveling plane wave, 190 Tunneling to dielectric surface, 223 Two-level approximation, 302 Two-level atoms in standing-wave potential, 45-49 Two-photon entanglement Bell’ s inequality, 4, 14-22 double entanglement of type-I1 SPDC, 15-17 Einstein-Podolsky-Rosen (EPR), 1,2-5 entangled state and two-photon wavepacket, 26-28 gedankenexperiment, 1 , 2 ghost image, 5,6-8 ghost interference-diffraction, 5-6, 9-14 hidden variable theory, 4 Schrodinger entangled state, 2 single-photon measurement of a twophoton state, 32-35 singlet state of two spin +particles,3-4 space-time observable and Bell’s inequality, 19-22 spin variables and Bell’s inequality, 17-19
Two-photon interference experiment, 28-30 versus interference of two photons, 23 -26 Two-photon state, 38-39 Two-photon wavepacket in Bell’s inequality measurement, 30-32 defined, 39-42 entangled state and, 26-28 T y p e 4 SPDC, 15,-17,31 Unsymmetrical rectangular shape, 27,31 van der Waals-Cashier interaction, 209 van der Waals-Casimir-Polder interaction, 275,296 van der Waals energy shift, 198, 199-202 Von Neuman entropy, 35 Walk-off problem, 17,27 Weakly guiding approximation (WGA), 216 Welcher weg information, 3 I Wigner function, 160, 162, 169 XUV radiation, harmonics generation and, 86, 131-132 Young’s double-slit aperture, 9, 13, 88, 107 Zeeman splitting, 199
Contents of Volumes in This Serial Volume 1
Volume 3
Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A. I: Amos Electron Affinities of Atoms and Molecules, B. L Moiseiwitsch Atomic Rearrangement Collisions, B. H. Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J. P. Toennies High-Intensity and High-Energy Molecular Beams, J. B. Anderson, R. P. Andres, and J. B. Fen
The Quanta1Calculation of Photoionization Cross Sections, A. L. Stewart Radiofrequency Spectroscopy of Stored Ions I: Storage. H. G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C. Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J. Wood
Volume 4 Volume 2 The Calculation of van der Waals Interactions, A. Dalgamo and U! D. Davison Thermal Diffusion in Gases, E. A. Mason, R. J. M u m , and Francis J. Smith Spectroscopy in the Vacuum Ultraviolet, W R. S. Carton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J . de Heer Mass Spectrometry of Free Radicals, S. N. Foner
H. S. W. Massey-A Sixtieth Birthday Tribute, E. H. S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D. R. Bates and R. H.G. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal Positrons and Positronium in Gases, P. A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I. C. Percival Born Expansions, A. R. Holf and B. L. Moiselwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionizations, C. B. 0. Mohr Recent Measurements on Charge Transfer, J. B. Hasted Measurements of Electron Excitation Functions, D. W 0. Heddle and R. G. U! Keesing 31 1
312
CONTENTS OF VOLUMES IN THIS SERIAL
Some New Experimental Methods in Collision Physics, R. E Stebbings Atomic Collision Processes in Gaseous Nebulae, M. J. Seaton Collisions in the Ionosphere, A. Dalgarno The Direct Study of Ionization in Space, R. L E Boyd
Volume 5 Flowing Afterglow Measurementsof Ion-Neutral Reactions, E. E. Ferguson, E C. Fehsenfeld, and A. L Schmeltekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy,H. G. Dehmelt The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuven The Calculation of Atomic Transition Probabilities, R. J. S. Crossley Tables of One- and Two-Particle Coefficientsof Fractional Parentage for Configurationss,s’, pq,C, D. H. Chisholm, A. Dalgarno, and E R. Innes Relativistic2-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle
Volume 6 Dissociative Recombination,J. N. Bardsley and M. A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A. S. Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagiand Yukikazu Itikawa The Diffusion of Atoms and Molecules, E. A. Mason and 1: R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A. E. Kingston
Volume 7 Physics of the Hydrogen Master, C. Audoin, J. P. Schermann, and P Grivet Molecular Wave Functions: Calculations and Use in Atomic and Molecular Processes, J. C. Browne Localized Molecular Orbitals,Hare1 Weinstein, Ruben Pauncz, and Maurice Cohen General Theory of Spin-CoupledWave Functions for Atoms and Molecules, J. Gerrati Diabatic States of Molecules- QuasiStationary Electronic States, Thomas E O’Malley Selection Rules within Atomic Shells, B. R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H. S. Taylor, and Robert Yaris A Review of Pseudo-Potentialswith Emphasis on Their Application to Liquid Metals, Nathan Wiser and A. J. GreenJield
Volume 8 Interstellar Molecules: Their Formation and Destruction, D. McNally Monte Car10 Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck NonrelativisticOff-Shell Two-Body Coulomb Amplitudes,Joseph C. 1 Chen and Augustine C. Chen Photoionization with Molecular Beams, R. B. Cairns, Halsread Harrison, and R. I. Schoen The Auger Effect, E. H. S. Burhop and U? N.Asaad
Volume 9 Correlation in Excited States of Atoms, A. U? Weiss The Calculation of Electron-Atom Excitation Cross Sections,M. R. H. Rudge Collision-InducedTransitions between Rotational Levels, Takeshi Oka
CONTENTS OF VOLUMES IN THIS SERIAL The Differential Cross Section of Low-Energy Electron-Atom Collisions, D. Andrick Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy
313
Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M. Broyer, G. Goudedard, J. C. Lehmann, and J. ViguC Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C. Reid
Volume 10 Relativistic Effects in the Many-Electron Atom, Lloyd Annstrong, Jr. and Serge Feneuille The First Born Approximation, K. L Bell and A. E. Kingston Photoelectron Spectroscopy, W C. Price Dye Lasers in Atomic Spectroscopy, W Lange, J. Luther and A. Steudel Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcett A Review of Jovian Ionospheric Chemistry, Wesley 7: Huntress, Jr.
Volume 11 The Theory of Collisions between Charged Particles and Highly Excited Atoms, I. C. Percival and D. Richards Electron Impact Excitation of Positive Ions, M. J. Seaton The R-Matrix Theory of Atomic Process, P. G. Burke and W D. Robb Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R. B. Bernstein and R. D.Levine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M. E Golde and B. A. Thrush
Volume 12 Nonadiabatic Transitions between Ionic and Covalent States, R. K. Janev Recent Progress in the Theory of Atomic Isotope Shift, J. Bauche and R . 4 Champeau
Volume 13 Atomic and Molecular Polarizabilities-A Review of Recent Advances, Thomas M. Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, Paul R. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, I. R Hertel and W Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J. Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K.Nesbet Microwave Transitions of Interstellar Atoms and Molecules, W B. Somerville
Volume 14 Resonances in Electron Atom and Molecule Scattering, D. E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J. Jamieson, and Ronald E. Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transitions in One- and TwoElectron Atoms, Richard Marrus and Peter J. Mohr Semiclassical Effects in Heavy-Particle Collisions, M. S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in IonAtom Collisions, S. V Bobashev Rydberg Atoms, S. A. Edelstein and 1: F. Gallagher
314
CONTENTS OF VOLUMES IN THIS SERIAL
UV and X-Ray Spectroscopy in Astrophysics, A. K. Dupree
RelativisticEffects in Atomic Collisions Theory, B. L.Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experment, E. N. Forrson and L. Wilets
Volume 15 Negative Ions, H. S. U! Massey Atomic Physics from Atmospheric and Astrophysical Studies,A. Dalgarno Collisions of Highly Excited Atoms. R. E Stebbings Theoretical Aspects of Positron Collisions in Gases, J. W Humberston ExperimentalAspects of Positron Collisions in Gases, I: C. Grifith Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion-Atom Charge Transfer Collisions at Low Energies, J. B. Hasted Aspects of Recombination,D. R. Bares The Theory of Fast Heavy Particle Collisions, B. H. Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H. B. Gilbody Inner-Shell Ionization, E. H. S. Burhop Excitation of Atoms by Electron Impact, D. W 0.Heddle Coherence and Correlation in Atomic Collisions, H. Kleinpoppen Theory of Low Energy Electron-MoleculeCollisions, l? G. Burke
Volume 16 Atomic Hartree-Fock Theory, M. Cohen and R. P. McEachran Experiments and Model Calculationsto Determine Interatomic Potentials, R. Duren Sources of Polarized Electrons, R. J. Celorta and D. I: Pierce Theory of Atomic Processes in Strong Resonant ElectromagneticFields, S. Swain Spectroscopyof Laser-ProducedPlasmas, M. H. Key and R. J. Hutcheon
Volume 17 CollectiveEffects in Photoionization of Atoms, M. Ya. Amusia Nonadiabatic Charge Transfer, D. S. F. Crothers Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot Supefluorescence, M. F. H. Schuurmans, Q. H. F. Vrehen, D. Polder, and H. M.Gibbs Applications of Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M. G. Payne, C. H. Chen, G. S. Hurst, and G. W Folb Inner-Shell vacancy Production in Ion-Atom Collisions, C. D. Lin and Patrick Richard Atomic Processes in the Sun, P. L Dufton and A. E. Kingston
Volume 18 Theory of Electron-Atom Scattering in a Radiation Field, Leonard Rosenberg Positron-Gas ScatteringExperiments, Talbert S. Stein and Walter E. Kauppila Nonresonant Multiphoton Ionization of Atoms, J. Morellec, D. Normand, and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions,A. S. Dickinson and D. Richards Recent Computational Developmentsin the Use of Complex Scaling in Resonance Phenomena, B. R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-ElectronSystems, N. Anderson and S. E. Nielsen Model Potentials in Atomic Structure, A. Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D. W Norcross and L. A. Collins
CONTENTS OF VOLUMES IN THIS SERIAL Quantum Electrodynamic Effects in FewElectron Atomic Systems, G. W E Drake Volume 19 Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B. H. Bransden and R. K. Janev Interactions of Simple Ion-Atom Systems, J. Z Park High-Resolution Spectroscopy of Stored Ions, D. J. Wineland, Wayne M.Itano, and R. S. Van Dyck, JK Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K. Blum and H. Kleinpoppen The Reduced Potential Curve Method for Diatonic Molecules and Its Applications, E JenE The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization of Atomic and Molecular Photoelectrons, N. A. Cherepkov Volume 20 Ion-Ion Recombination in an Ambient Gas, D. R. Bates Atomic Charges within Molecules, G. G. Hall Experimental Studies on Cluster Ions, T D. Mark and A. W Castleman, Jr. Nuclear Reaction Effects on Atomic Inner-Shell Ionization, W E. Meyerhof and J.-E Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstrong On the Problem of Extreme UV and X-Ray Lasers, I. L Sobel’man and A. 1.! Vinogradov Radiative Properties of Rydberg State, in Resonant Cavities, S. Haroche and J. M. Ralmond Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction-Rydberg Molecules, J. A. C. Gallas, G. Leuchs, H.Walther and H. Figger
315
Volume 21 Subnatural Linewidths in Atomic Spectroscopy, Dennis I? O’Brien, Pierre Meystre, and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen Theory of Dielectronic Recombination, Yukap Hahn Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu Scattering in Strong Magnetic Fields, M. R. C. McDowell and M. Zarcone Pressure Ionization, Resonances, and the Continuity of Bound and Free States, R. M.More Volume 22 Positronium-Its Formation and Interaction with Simple Systems, J. U! Humberston Experimental Aspects of Positron and Positronium Physics, I: C. Grifirh Doubly Excited States, Including New Classification Schemes, C. D. Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H. B. Gilbody Electron-Ion and Ion-Ion Collisions with Intersecting Beams, K. Dolder and B. Pearl Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion-Atom Collisions, R. Anholt and Harvey Gould Continued-Fraction Methods in Atomic Physics, S. Swain Volume 23 Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C. R. Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M. Quiney Point-Charge Models for Molecules Derived from Last-Squares Fitting of the Electric Potential, D. E. Williams and Ji-Min Yun
316
CONTENTS OF VOLUMES IN THIS SERIAL
Transition Arrays in the Spectra of Ionized Atoms, J. Bauche, C. Bauche-Amoult, and M. Klapisch Photoionizationand Collisional Ionization of Excited Atoms Using Synchroton and Laser Radiation, E. J. Wuilleumier,D. L. Ederer, and J. L. PicquP
Volume 24 The Selected Ion Flow Tube (SIDT): Studies of Ion-Neutral Reactions, D. Smith and N. G. Adams Near-ThresholdElectron-Molecule Scattering, Michael A. Morrison Angular Correlation in Multiphoton Ionization of Atoms, S.J. Smith and G. Leuchs Optical Pumping and Spin Exchange in Gas Cells, R. J. Knize, Z Wu, and W Happer Correlations in Electron-Atom Scattering,A. Crowe
Volume 25 Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgamo: Contributions to Aeronomy, Michael B. McElroy Alexander Dalgarno: Contributions to Astrophysics, David A. Williams Dipole Polarizability Measurements, Thomas M. Miller and Benjamin Bederson Flow Tube Studies of Ion-Molecule Reactions, Eldon Ferguson Differential Scatteringin He-He and He+-He Collisions at KeV Energies, R. E Stebbings Atomic Excitation in Dense Plasmas,Jon C. Weisheit Pressure Broadening and Laser-InducedSpectral Line Shapes, Kenneth M. Sando and ShihI Chu Model-PotentialMethods, G. Laughlin and G. A. Victor Z-Expansion Methods, M. Cohen
Schwinger VariationalMethods, Deborah Kay Watson Fine-Structure Transitionsin Proton-Ion Collisions, R. H. G. Reid Electron Impact Excitation, R. J. W Henry and A. E. Kingston Recent Advances in the Numerical Calculation of lonization Amplitudes, Christopher Bottcher The Numerical Solution of the Equations of Molecular Scattering,A. C. Allison High Energy Charge Transfer, B. H. Bransden and D. P. Dewangan RelativisticRandom-Phase Approximation, W R. Johnson Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G. W E Drake and S. P. Goldman Dissociation Dynamics of Polyatomic Molecules, 'c: Uzer PhotodissociationProcesses in Diatoniic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine F. v a n Dishoeck The Abundances and Excitation of Interstellar Molecules, John H. Black
Volume 26 Comparisons of Positrons and Electron Scattering by Gases, Walter E. Kauppila and Talbert S.Stein Electron Capture at RelativisticEnergies, B. .Z. Moiseiwitsch The Low-Energy, Heavy Particle Collisions-A Close-CouplingTreatment, Mineo Kimura and Neal E Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V Sidis AssociativeIonization: Experiments, Potentials, and Dynamics, John Weiner, Francoise Masnou-Sweeuws, and Annick Giusti-Suzor On the p Decay of '*'Re: An Interface of Atomic and Nuclear Physics and Cosmochronology,Zonghau Chen, Leonard Rosenberg, and Larry Spruch Progress in Low Pressure Mercury-RareGas
CONTENTS OF VOLUMES IN THIS SERIAL Discharge Research, J. Maya and R. Lagushenko
Volume 27 Negative Ions: Structure and Spectra, David R. Bates Electron Polarization Phenomena in ElectronAtom Collisions,Joachim Kessler Electron-Atom Scattering,I. E. McCarfhy and E. Weigold Electron-Atom Ionization, I. E. McCarfhy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, V I. Lengyel and M. I. Haysak Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule
Volume 28 The Theory of Fast Ion-Atom Collisions, J. S. Briggs and J. H. Macek Some Recent Developments in the Fundamental Theory of Light, Peter W Milonni and Surendra Singh Squeezed States of the Radiation Field, Khalid Zaheer and M. Suhail Zubairy Cavity Quantun, Electrodynamics,E. A. Hinds
Volume 29 Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin and L. U? Anderson Cross Sections for Direct Multiphoton Ionionization of Atoms, M. K Ammosov, N. B. Delone, M. Yu. Ivanov, I. I. Bondar, and A. V; Masalov Collision-InducedCoherences in Optical Physics, G. S. Aganval Muon-Catalyzed Fusion, Johann Rafelski and Helga E. Rafelski
317
Cooperative Effects in Atomic Physics, J. P. Connerade Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, J. H.McGuire
Volume 30 Differential Cross Sections for Excitation of Helium Atoms and Helium-Like Ions by Electron Impact, Shinobu Nakazaki Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Trajmar and J. C. Nickel The Dissociative Ionization of Simple, Molecules by Fast Ions, Colin J. Latimer Theory of Collisions between Laser Cooled Atoms, P. S . Julienne, A. M. Smith, and K. Burnen Light-Induced Drift, E. R. Eliel Continuum Distorted Wave Methods in IonAtom Collisions, Derrick S. F. Crorhers and Louis J. Dub6
Volume 31 Energies and Asymptotic Analysis for Helium Rydberg States, G. W F. Drake Spectroscopy of Trapped Ions, R. C. Thompson Phase Transitions of Stored Laser-Cooled Ions, H. Walther Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, RudolfDiiren, and Jacques Robert Atomic Physics and Non-MaxwellianPlasmas, MichLle Lamoureux
Volume 32 Photoionization of Atomic Oxygen and Atomic Nitrogen, K. L. Bell and A. E. Kingston Positronium Formation by Positron Impact on Atoms at Intermediate Energies, B. H. Bransden and C. J. Noble Electron-Atom ScatteringTheory and Calculations, P. G. Burke
318
CONTENTS OF VOLUMES IN THIS SERIAL
Terrestrial and Extraterrestrial H,+, Alexander Dalgarno Indirect Ionization of Positive Atomic Ions, K. Dolder Quantum Defect Theory and Analysis of HighPrecision Helium Term Energies, G. W R Drake Electron-Ion and Ion-Ion RecombinationProcesses, M. R. Flannery Studies of State-SelectiveElectron Capture in Atomic Hydrogen by Translational Energy Spectroscopy,H. B. Gilbody RelativisticElectronic Structure of Atoms and Molecules, I. l? Grant The Chemistry of Stellar Environments,D. A. Howe, J. M. C. Rawlings, and D. A. Williams Positron and Positronium Scatteringat Low Energies, J. W Humberston How Perfect are Complete Atomic Collision Experiments?, H. Kleinpoppen and H. Handy Adiabatic Expansions and Nonadiabatic Effects, R. McCarroll and D. S. F. Crothers Electron Capture to the Continuum,B. L, Moiseiwirsch How Opaque Is a Star? M. J. Searon Studies of Electron Attachment at Thermal Energies Using the Flowing Afterglow-Langmuir Technique,David Smith and Patrik span81 Exact and ApproximateRate Equations in Atom-Field Interactions, S. Swain Atoms in Cavities and Traps, H. Walrher Some Recent Advances in Electron-Impact Excitation of n = 3 States of Atomic Hydrogen and Helium, J. F. Williams and J. B. Wang
Volume 33 Principles and Methods for Measurementof Electron Impact Excitation Cross Sections for Atoms and Molecules by Optical Techniques, A. R. Filippelli, Chun C. Lin, L. W Andersen, and J. W McConkey Benchmark Measurementsof Cross Sections for Electron Collisions: Analysis of Scattered Electrons, S. Trajmar and J. W McConkey
Benchmark Measurements of Cross Sections for Electron Collisions: Electron Swarm Methods, R. W Crompfon Some Benchmark Measurements of Cross Sections for Collisions of Simple Heavy Particles, H. B. Gilbody The Role of Theory in the Evaluation and Interpretation of Cross-Section Data, Barry I. Schneider Analytic Representation of Cross-Section Data, Mirio Inokuti, Mineo Kimura, M. A. Dillon, Ism Shimamura Electron Collisions with NI.O2and 0 What We Do and Do Not Know, Yukikazu Itikawa Need for Cross Sections in Fusion Plasma Research, Hugh P. Summers Need for Cross Sections in Plasma Chemistry, M. Capitelli, R. Celiberto, and M.Cacciatore Guide for Users of Data Resources, Jean W Gallagher Guide to Bibliographies, Books, Reviews, and Compendia of Data on Atomic Collisions, E. W McDaniel and E. J. Mansky
Volume 34 Atom Interferometry, C. S.Adams, 0.Carnal, and J. Mlynek Optical Tests of Quantum Mechanics, R. Y Chiao, l? G. Kwiar, and A. M. Steinberg Classical and Quantum Chaos in Atomic Systems, Dominique Delande and Andreas Buchleirner Measurementsof Collisions between Laser-Cooled Atoms, Thad Walker and Paul Feng The Measurement and Analysis of Electric Fields in Glow Discharge Plasmas, J. E. Lawler and D. A. Doughty Polarization and Orientation Phenomena in Photoionization of Molecules, N. A. Cherepkov Role of Two-Center Electron-Electron Interaction in Projectile Electron Excitation and Loss, E. C. Montenegro, W E. Meyerhof and J. H. McGuire Indirect Processes in Electron Impact Ionization of Positive Ions, D. L. Moores and K. J. Reed
CONTENTS OF VOLUMES IN THIS SERIAL Dissociative Recombination: Crossing and Tunneling Modes, David R. Bates
Volume 35 Laser Manipulation of Atoms, K. Sengstock and W Ertmer Advances in Ultracold Collisions: Experiment and Theory, J. Weiner Ionization Dynamics in Strong Laser Fields, L. E DiMauro and F? Agostini Infrared Spectroscopy of Size Selected Molecular Clusters, U. Buck Femtosecond Spectroscopy of Molecules and Clusters, T. Baumer and G. Gerber Calculation of Electron Scattering on Hydrogenic Targets, I. Bray and A. Z Stelbovics Relativistic Calculations of Transition Amplitudes in the Helium Isoelectronic Sequence, W R. Johnson, D. R. Plante, and J. Sapirsrein Rotational Energy Transfer in Small Polyatomic Molecules, H. 0. Everitt and E C. De h c i a
Volume 36 Complete Experiments in Electron-Atom Collisions, Nils Overgaard Andersen and Klaus Bartschat Stimulated Rayleigh Resonances and RecoilInduced Effects, J.-I! Courtois and G. Grynberg Precision Laser Spectroscopy Using AcoustoOptic Modulators, W A. van Wijngaarden Highly Parallel Computational Techniques for Electron-Molecule Collisions, Carl Winstead and Vincent McKoy Quantum Field Theory of Atoms and Photons, Maciej Lewenstein and Li You
Volume 37 Evanescent Light-Wave Atom Mirrors, Resonators, Waveguides, and Traps, Jonathan F? Dowling and Julio Gea-Banacloche Optical Lattices, P. S. Jessen and I. H. Deutsch Channeling Heavy Ions through Crystalline Lattices, Herbert F. Krause and Sheldon Datz
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Evaporative Cooling of Trapped Atoms, Wolfgang Ketterle and N. J. van Druten Nonclassical States of Motion in Ion Traps, J. I. Cirac, A. S. Parkins, R. Blurt, and F? Zoller The Physics of Highly-Charged Heavy Ions Revealed by StoragelCooler Rings, P. H. Mokler and Th. Stohlker
Volume 38 Electronic Wavepackets, Robert R. Jones and L. D. Noordam Chiral Effects in Electron Scattering by Molecules, K. Blum and D. G. Thompson Optical and Magneto-Optical Spectroscopy of Point Defects in Condensed Helium, Serguei I. Kanorsky and Antoine Weis Rydberg Ionization: From Field to Photon, G. M. Lankhuijzen and L. D. Noordam Studies of Negative Ions in Storage Rings, L. H. Andersen, T Andersen, and P. Hvelplund Single-Molecule Spectroscopy and Quantum Optics in Solids, W E. Moerner, R. M. Dickson, and D. J. Norris
Volume 39 Author and Subject Cumulative Index Volumes 1-38 Author Index Subject Index Appendix: Tables of Contents of Volumes 1-38 and Supplements
Volume 40 Electric Dipole Moments of Leptons, Eugene D. Commins High-Precision Calculations for the Ground and Excited States of the Lithium Atom, Frederick W King Storage Ring Laser Spectroscopy, Thomas U. Kiihl Laser Cooling of Solids, Carl E. Mungan and Timothy R. Gosnell Optical Pattern Formation, L. A. Lugiato, M. Brambilla. and A. Gatti
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CONTENTS OF VOLUMES IN THIS SERIAL
Volume 41 Two-Photon Entanglement and Quantum Reality, Yanhua Shih Quantum Chaos with Cold Atoms, Mark G. Raizen Study of the spatial and Tempo& Coherence of High-OrderHarmonics, Pascal Sali.?res, Ann L’Huiller, Philippe Antoine, and Maciej Lewenstein
Atom Optics in Quantized Light Fields, Matthias Freyburger, Alois M. Herkornrner, Daniel S.Kriihrner, Erwin Mayr, and Wolfgang P. Schleich Atom Waveguides, Victor I. Balykin Atomic Matter Wave Amplification by Optical Pumping, Ulf Janicke and Martin Wilkens
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