Advances in Atomic, Molecular, and Optical Physics, Volume 41

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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

ACADEMIC PRESS San Diego New York

London Sydney

Boston Tokyo

Toronto

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.


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


-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.

12

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)


13

where E > 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,


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).


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 ).


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.


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.


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

,


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


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


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


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.


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.

38

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)


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


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


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


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


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.


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


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


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-


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


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-


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


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 /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,


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-


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-


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


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


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

76

Mark G. Raizen

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


77

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-


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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.


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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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).


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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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


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

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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


+ 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

94


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).


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-


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

98


(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


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.

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'

'

-2

-1

I

0

z position

1

2

J

(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.


.-c 0

c

'$-

-60

a 0

-120

-4

-2

0

2

4

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.

104

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

1o4 c 2 c

u)

5

n

c

0

2

f 103 3

z

5 2

.-

in2

-2

-1

0

1

2

3

4

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

-, ’

\

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

g lo4

c

.c

a

z

lo3

L

a

n

5 to2

z

10' 1oo 2

3

4

5

6

7

t

Loser intensity (10" w/crn2)

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:

+


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


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 > 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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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

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A. Electrical Field ... B. Magnetic Field . . . III. Evanescent Light Wave ................. ............................... A. Forces on Atoms in 1. Near-Resonant Light Force on Atoms .........................

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B. Evanescent-Wave

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............ ............................. 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 ..................

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Copyright 8 1999 by Academic Press All rights of reproductionin any form reserved.

ISBN 0- 12-003841-2/1SSN 1049-250X/99 $30.00

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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.

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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> of the guided atom is

2y(cOs4

e)

(158)

a,)cos 0 = ( QR)/Aand the mean lifetime

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(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

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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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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.

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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

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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.

,

,

-

,

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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.

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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.

VIII. References Adams, C. S., Carnal, O., and Mlynek, J. (1994a). Adv. At. Mol. Opt. Phys. 34, 1-33. Adams, C. S., Sigel, M., and Mlynek, J. (1994b). Phys. Rep. 240, 143-210. Arimondo, E., and Bachor, H.-A., eds., (1996). Quant. Sem. Optics. 8 [Spec. Issue], 495-753. Baldwin, K. G. H. (1996). Aust. J. Phys. 49,855-897. Balykin, V. I., Letokhov, V. S., Sidorov, A. I., and Ovchinnikov, Yu. B. (1987a). JETP L e f t (Engl. Transl.)45, 353-356; Pis'ma Zh. Eksp. Teor. Fiz.45,282-285. Balykin. V. I. and Letokhov, V. S. (1987b). Opt. Commun. 64, 151-156. Balykin, V. I., Letokhov, V. S., Ovchinnikov, Yu. B., Sidorov, A. I., and Shulga, S. V. (1988a). Opt. Lett. 13,958-960. Balykin, V. I., Letokhov, V. S . , Ovchinnikov, Yu.B., and Sidorov, A. I. (1988b). Phys. Rev. Lett. 60, 2137-2140; (Errata61,902). Balykin, V. I., Lozovik, Yu. E., Ovchinnikov, Yu. B., Sidorov, A. I., Shulga, S. V., and Letokhov, V. S. (1989). J. Opt. Soc.Am. B6.2178-2187. Balykin, V. I. and Letokhov, V. S. (1989a). Appl. Phys. B48.517-523. Balykin, V. I. and Letokhov, V. S. (1989b). Phys. Today, 4.23-28. Balykin, V.1. and Letokhov, V. S. (1990). Sov. Phys.-llsp. (Engl. Trans/.) 33,79-85. Balykin, V. I. and Letokhov, V. S. (1995). Atom optics with laser light, Laser science and technology, Vol. 18, Harwood Academic Publishers, Australia et al. Balykin, V. I., Laryushin, D. V., Subbotin, M. V., and Letokhov, V. S. (1996). JETP Lett. 63,802-807. and Woerdman, J. P. (1992). Opt. Commun. Beijersbergen, M . W., Allen, L., van der Veen, H. E. L. 0.. 96, 132-132. Benewitz, G . and Paul, W. (1954). B.5 Phys. 139,489-494. Berggren, K . K., Prentiss M., Timp G. L., and Behringer, R. E. (1994). J. Opt. SOC.Am. B 11, 11661176. Born, M. and Wolf, E. (1984). Principles ofoptics (6th ed.). Pergamon (Oxford). Casimir, H.B. G. and Polder, D. (1948). Phys. Rev. 73,360-372.

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Chevrollier, M., Fichet, M., Oria, M., Rahmat, G.,Bloch, D.,and Ducloy, M. (1992). J. Phys. (France) 119 2,631-638. Chen, J., Story, J. G., and Hulet, R. G.(1993). Phys. Rev. A 47,2128-2138. Cohen-Tannoudji, C. (1991). In J. Dalibard, J.- M. Raimond, and J. Zinn-Justin (Eds.). Fundamental system in quantum optics. Les Houche Summer School Session LIII. North-Holland (Amsterdam). Cook, R. J. (1979). Phys. Rev. A 20,224-228. Cook, R. J. and Hill, R. K. (1982). Opf. Commun.43,258-260. Courtois, J.-Y., Courty, J.-M., and Mertz, J. C. (1996). Phys. Rev. A 53, 1862-1878. Davidson, N., Lee, H. J., Adam, C., Kasevich, M., and Chu, S. (1995). Phys. Rev. Lett. 74, 131 11314.

Dalibard, J. and Cohen-Tannoudji, C. (1985). J. Opt. SOC.Am. B 2, 1707-1720. Dalibard, J. and Cohen-Tannoudji, C. (1989). J. Opt. SOC.Am. B 6,2023-2045. Desbiolles, P.,Arndt, M., Szriftgiser, P., and Dalibard, J. (1996). Phys. Rev. A 54,4292-4298. Dowling, J. P.and Gea-Banacloche, J. (1996). Adv. At. Mol. Opt. Phys. 34, 1-94. Esslinger, T., Weidemuller, M., Hemmerich, A., and Hansch, T. W. (1993). Opt. Lett. 18,450-452. Friedburg, H. and Paul, W. (1950). Natuwiss. 37,20-25. Fichet, M., Schuller, F., Bloch, D., and Ducloy, M. (1995). Phys. Rev. A 51, 1553-1564. Gallatin, G.M. and Gould, P. L. (1991). J. Opt. SOC.Am. B 8,502-508. Gauck, H., Schneble, D., Hartl. M., and Pfau, T. (1996).Annual Report of Chiversify Konsfanz. Guzman, A. M., Moore, M., and Meystre, P. (1996). Phys. Rev. A 53,977-984. Gloge, D. (1971).Appl. Opt. 10,2252-2258. Haroche, S . and Kleppner, D. (1989). Physics Today, 42.24-30. Haroche, S. (1991) In J. Dalibard, J.-M. Raimond, and J. Zinn-Justin, (Eds.). Fundamenfal system in quantum oprics. Les Houche Summer School Session LIII. North-Holland (Amsterdam). Harris, D. and Savage, C. (1995). Phys. Rev. A 51,3967-3971. Hau, L., Burns, M., and Golovchenko, J. (1992). Phys. Rev. A 45,6468-6477. He, H., Heckenberg, N. R., and Rubinsztein-Dunlop, H. (1995). J. Mod. Opt. 42,217-223. Heckenberg, N. R., McDuff, R.,Smith, C. P., Rubinsztein-Dunlop, H., and Wegener, M. J. (1992). Opt. and Quant. Electron. 24, S9514962. Helmenson, K., Rolston, S., Goldner, L., and Phillips, W. (1993). Quantum electronics and laser sciences. OSA Technical Digest Series, Vol. 12 (p. 209). Hemmerich, A., Schropp, D., Jr., and H h s c h T.W. (1991). Phys. Rev. A 44, 1910-1921. Henkel, C., Molmer, K., Kaiser, R., Vansteenkiste, N., Westbrook, C. I., and Aspect, A. (1997). Phys. Rev. A 55,1160-1 178. Herman, R. M. and Wiggins, T.A. (1991). J. Opt. SOC.Am. A 8,932-942. Hinds, E. A. (1991). Adv. At. Mol. Opt. Phys. 28,237-289. Hope, J. and Savage, C. (1996). Phys. Rev. A 53, 1697-1701. Ito, H., Sakaki, K., Nakata, T., Jhe, W., and Ohtsu, M. (1995a). Opt. Comrnun. 115,57-64. Ito, H., Sakaki, K., Ohtsu, M., and Jhe, W. (1995b). Appl. Phys. Lett. 70, 1-3. Ito, H., Sakaki, K., Nakata, T., Jhe, W., and Ohtsu, M. (1995~).Ultramicroscopy. 61.91-97. Ito, H., Nakata, T., Sakaki, K., Ohtsu, M., Lee, K. I., and Jhe, W. (1996). Phys. Rev. Lett. 76, 45004503.

Ito, H., Sakaki, K., Jhe, W., and Ohtsu, M. (1997a). Opt. Commun. 141.43-47. Ito, H., Sakaki, K., Ohtsu, M., and Jhe, W. (1997b). Phys. Rev. A 56,712-718. Ito, H. and Ohtsu, M. (1998). Ultramicroscopy, in press. Jackson, J. D. (1975). Classical electrodynamics. Wiley and Sons (New York). Johnson, P.B. and Christy, R. W. (1972). Phys. Rev. B 6,4370-4379. Kaiser, R.. Levy. Y., Vansteenkiste, N., Aspect, A., Seifert, W., Leipold, D., and Mlynek, J. (1994). Opt. Commun. 104,234-240.

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Kasevich, M., Weiss, D., and Chu, S. (1990). Opt. Lett., 15, 607-609. Kasevich, M., Moler, K., Riis, E., Sunderman, E., Weiss, D., and Chu, S. (1991). In Proceedings of the 12th Internutionul Conference on Atomic Physics (pp. 47-56). American Institute of Physics (New York). Kao,K.C.andHockham,G.A. (1966). Proc. IEE, 113, 1151-1155. Kazantsev, A. P., Smirnov, V. S., Surdutovich, G. I., Chudesnikov, D. 0. and Yakovlev V. P. (1985). J. Opt. SOC. Am. B 2, 1731-1742. Kazantsev, A. P., Surdutovich G. I., and Yakovlev, V. P. (1990).Mechanical action of light on atom. World Scientific (Singapore and Teaneck, NJ). Kosterlitz, J. M. andThouless, D. J. (1973).J. Phys. C 6 , 1181-1203. Kuppens, S., Rauner, M., Schiffer, M., Wokurka, G., Slawinski, T., Zinner, M., Sengstock, K., and Ertmer, W. (1996). In K. Burnett (Ed.). OSA TOPS on ultracold atoms and BEC. Vol. 7. 1997 Optical Society of America. Labeyrie, G., Landragin, A., Von Zanthier, J., Kaiser, R., Vansteenkiste, N., Westbrook, C., and Aspect, A. (1996). Quant Semiclass. Opt. 8,603-627. Landragin, A., Labeyrie, G., Henkel, C., Kaiser, R., Vansteenkiste, N., Westbrook, C. I., and Aspect, A. (1996). Opt. Lett. 21, 1591-1593. Landragin, A., Courtois, J.-Y., Labeyrie, G., Vansteenkiste, N., Westbrook, C. I., and Aspect, A. (1996).Phys. Rev. Lett. 77, 1464-1467. Letokhov, V. S. and Minogin, V. G. (1981). Phys. Rep. 73,3-65. Letokhov, V. S . (1968).Pis’ma JETE 7,348-351 [in Russian]. Lee, H. S., Atewart, B. W., Choi, K., and Fenichel, H. (1994). Phys. Rev. A 49,4922-4927. Li, Q., Stenlake, B. W., Littler, I. C. M., Bachor, H.-A,, Baldwin, K. G. H., and McClelland, D. E. (1994). Laser Physics, 4,983-994. McClelland, J. J. and Scheinfein, M. R. (1991). J. Opt. SOC.Am. B 8, 1974-1986. McClelland, J. J., Scholten, R.E., Palm, E. C., and Celotta, R. J. (1993). Science, 262,877-880. McClelland, J. J. (1995). J. Opt. SOC.Am. B 12, 1761-1768. McClelland, J. J., Gupta, R., Jabbour, Z. L., and Cellota, R. J. (1996). Aust. J. Phys. 49,555-565. Mandel, L. (1983). Phys. Rev. A 28,929-943. Marksteiner, S., Savage, C. M., Zoller, P., and Rolston, S. L. (1994). Phys. Rev. A 50,2680-2690. Meschede, D., Jhe, W., and Hinds, E. A. (1990). Phys. Rev. A 41, 1587-1596. Minogin, V. G. and Rozhdestvensky, Yu. V. (1987). Zh. Eksp. Teor Fiz. 93, 1173-1 187 [in Russian]. Minogin, V. G. and Letokhov, V. S. (1987). Laser light pressure on atoms. Gordon & Breach (New York). Mlynek, J., Balykin, V. I., and Meystre, P. (Eds.). (1992). Appl. Phys. B 54 [Spec. Issue], 319-485. Nha, H. and Jhe, W. (1997). Phys. Rev. A 56,729-736. Nellessen, J., Werner, J., and Ertmer, W. (1990). Opt. Commun. 78,300-308. Ol’shanii, M. A., Letokhov, V. S., and Minogin, V. G. (1992). In Mol. cryst. liq. cryst. sci. techno1.Section B: Nonlinear optics (pp. 283-294). Vol. 3. Ol’shanii, M. A., Castin, Y., and Dalibard, J. (1995). In M. Inguscio, M. Allegrini, and A. Sasso (Eds.). Proceedings of the 12th International Conference on Laser Specroscopy (pp. 7-12). World Scientific (Singapore), Gordon and Breach Sciences Publishers. Okoshi, T. (1982). Opticaljbers. Academic Press (New York). Ovchinnikov, Yu. B, Laryushin, D. V., Balykin, V. I., and Letokhov, V. S. (1995). J E W Lett. 62, 113-1 18. Paterson, C. and Smith, R. (1996). Optics Comm. 124, 121-130. Pillet, P. (Ed.). (1994). J. Phys. I14 [Spec. Issue], 1877-2089. Prentiss, M. G. and Ezekel, S. (1986). Phys. Rev. Lett. 56,46 -49. Raether, H. (1988). Surjizce plasmons. Springer (Berlin). Ramsey, N. (1956). Molecular beams. Oxford University Press (New York).

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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.

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a. Pair-Interaction-Dispersive Part . . . . . . . . . . . . b. Dissipative Interaction ..................................... 5 . Rotating-Wave Approximation ........... 6. Two-Level Approximation .................................... 7. Second Quantization VIII. References . . . . . . . . . . .

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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.

ATOMIC MATTER WAVE AMPLIFICATIONBY OPTICAL PUMPING

263

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

264

Ulf Janicke and Martin Wilkens

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

ATOMIC MATTER WAVE AMPLIFICATION BY OPTICAL PUMPING

265

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,

266


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


267

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

+

+

268


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

269

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,


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).

+

+

272


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


273

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.

214


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


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.

276

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


277

(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.

278


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)$


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).


281

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.

282


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


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 > 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.

286


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)


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,

288


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


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.


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.


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 %

%


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).

294


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


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


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


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


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:


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


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-


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


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.


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.

304


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

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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

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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

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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

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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

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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

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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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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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Advances in Atomic, Molecular, and Optical Physics, Volume 41

Advances in Atomic, Molecular, and Optical Physics, Volume 59 (Advances in Atomic, Molecular and Optical Physics)

Advances in Atomic, Molecular, and Optical Physics