Advances in Atomic, Molecular, and Optical Physics: Volume 48

Advances in

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 48

Editors BENJAMIN BEDERSON

New York University New York, New York HERBERT WALTHER

Max-Plank-Institut j~ir Quantenoptik Garching bei Miinchen Germany

Editorial Board ER. BERMAN

University of Michigan Ann Arbor, Michigan M. GAVRILA

EO.M. Instituut voor Atoom- en Molecuulfysica Amsterdam, The Netherlands M. INOKUTI

Argonne National Laboratory Argonne, Illinois CHUN C. LIN

University of Wisconsin Madison, Wisconsin

Founding Editor SIR DAVID 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.

A D VANCES IN

ATOMIC MOLECOLAR AND OPTICAL PHYSICS Edited by

Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK

Herbert Walther UNIVERSITY OF MUNICH AND MAX-PLANK INSTITUT FOR QUANTENOPTIK MUNICH, GERMANY

Volume 4 8

ACADEMIC PRESS An imprint of Elsevier Science A m s t e r d a m . Boston- London. New York- Oxford. Paris San D i e g o . San Francisco. Singapore. S y d n e y - T o k y o

This book is printed on acid-free paper. 9 2002 Elsevier Science (USA) 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 the code at the bottom of the first page of a chapter in this book indicates the 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, Massachusetts 01923), 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 pre-2002 chapters are as shown on the title pages. If no fee code appears on the title page, the copy fee is the same as for current chapters. 1049-250X/02 $35.00 Explicit permission from Academic Press is not required to reproduce a maximum of two figures or tables from an Academic Press chapter in another scientific or research publication provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given. A c a d e m i c Press An Elsevier Science Imprint 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com International Standard Book Number: 0-12-003848-X Printed and bound in Great Britain by MPG Books Ltd, Cornwall, UK 02

03

04

05

06

07

MP

9

8

7

6

5

4

3

2

1

Contents CONTRIBUTORS

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

ix

Multiple Ionization in Strong Laser Fields

R. D6rner, Th. Weber, M. Weckenbrock, A. Staudte, M. Hattass, R. Moshammer, J. Ullrich, H. Schmidt-B6cking 1

Io I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. III. IV. V. VI. VII. VIII. IX. X.

C O L T R I M S - A C l o u d C h a m b e r for A t o m i c P h y s i c s . . . . . . . Single I o n i z a t i o n and the T w o - s t e p M o d e l . . . . . . . . . . . . . . . Mechanisms of Double Ionization ..................... Recoil Ion M o m e n t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron Energies ................................ Correlated Electron Momenta ........................ Outlook ...................................... Acknowledgments ............................... References ....................................

3 6 9 11 19 20 30 30 31

Above-Threshold Ionization" From Classical Features to Quantum Effects

W. Becker, E Grasbon, R. Kopold, D.B. MilodeviO, G.G. Paulus and H. Walther I~ I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. III. IV. V. VI. VII. VIII. IX.

Direct I o n i z a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R e s c a t t e r i n g : T h e Classical T h e o r y . . . . . . . . . . . . . . . . . . . . Rescattering: Quantum-mechanical Description . . . . . . . . . . . . ATI in the Relativistic R e g i m e . . . . . . . . . . . . . . . . . . . . . . . Q u a n t u m Orbits in H i g h - o r d e r H a r m o n i c G e n e r a t i o n . . . . . . . . A p p l i c a t i o n s o f ATI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments ............................... References ....................................

36 40 50 53 73 76 86 92 92

Dark Optical Traps for Cold Atoms

Nir Friedman, Ariel Kaplan and Nir Davidson I. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. B a c k g r o u n d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. M u l t i p l e - L a s e r - B e a m s D a r k O p t i c a l Traps

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

99 101 106

Contents

vi IV. V. VI. VII.

S i n g l e - B e a m D a r k Optical Traps . . . . . . . . . . . . . . . . . . . . . Applications ................................... Conclusions .................................... References ....................................

113

127 147 148

Manipulation of Cold Atoms in Hollow Laser Beams

Heung-Ryoul Noh, Xinye Xu and Wonho Jhe I. II. III. IV. V. VI.

Introduction ..................................... T h e o r e t i c a l M o d e l s for C o l d A t o m s in H o l l o w L a s e r B e a m s G e n e r a t i o n M e t h o d s for H o l l o w L a s e r B e a m s . . . . . . . . . . . . . . C o l d A t o m M a n i p u l a t i o n in H o l l o w L a s e r B e a m s . . . . . . . . . . . Acknowledgment ................................. References ......................................

....

153 154 160

170 188 188

Continuous Stern-Gerlach Effect on Atomic Ions

Giinther Werth, Hartmut Hdffner and Wolfgang Quint I. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. IIl. IV. V. VI. VII. VIII. IX.

A Single Ion in a P e n n i n g Trap . . . . . . . . . . . . . . . . . . . . . . C o n t i n u o u s S t e r n - G e r l a c h Effect . . . . . . . . . . . . . . . . . . . . . Double-Trap Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . C o r r e c t i o n s and S y s t e m a t i c Line Shifts . . . . . . . . . . . . . . . . . Conclusions .................................... Outlook ...................................... Acknowledgements ............................... References ....................................

191 195 206 209 212 213 214 216 216

The Chirality of Biomolecules

Robert N. Compton and Richard M. Pagni I. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F u n d a m e n t a l N a t u r e o f Chirality . . . . . . . . . . . . . . . . . . . . . . True and False Chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . G a l a x i e s , Plants, and P h a r m a c e u t i c a l s . . . . . . . . . . . . . . . . . . Plausible O r i g i n s o f H o m o c h i r a l i t y . . . . . . . . . . . . . . . . . . . . A s y m m e t r y in B e t a R a d i o l y s i s . . . . . . . . . . . . . . . . . . . . . . . Possible Effects o f the P a r i t y - V i o l a t i n g E n e r g y D i f f e r e n c e ( P V E D ) in E x t e n d e d M o l e c u l a r S y s t e m s . . . . . . . . . . . . . . . . . . . . . . VIII. C o n c l u s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. A c k n o w l e d g m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. III. IV. V. VI. VII.

219 219 230 233 236 243 252 257 257 258

Contents

vii

Microscopic Atom Optics" From Wires to an Atom Chip

Ron Folman, Peter Kriiger, J6rg Schmiedmayer, Johannes Denschlag and Carsten Henkel I. II. III. IV. V. VI. VII. VIII. IX.

Introduction .................................... Designing Microscopic Atom Optics ................... E x p e r i m e n t s with F r e e - S t a n d i n g S t r u c t u r e s . . . . . . . . . . . . . . . S u r f a c e - M o u n t e d Structures: T h e A t o m Chip . . . . . . . . . . . . . Loss, H e a t i n g and D e c o h e r e n c e . . . . . . . . . . . . . . . . . . . . . . Vision and O u t l o o k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion .................................... Acknowledgement ............................... References ....................................

263 265 292 303 324 342 351 351 352

Methods of Measuring Electron-Atom Collision Cross Sections with an Atom Trap

R.S. Schappe, M.L. Keeler, T.A. Zimmerman, M. Larsen, P. Feng, R.C. Nesnidal, J.B. Boffard, T.G. Walker, L. W. Anderson and C.C. Lin I. II. III. IV. V. VI. VII.

Introduction .................................... General Experiment Overview ........................ M e t h o d s for M e a s u r i n g C r o s s S e c t i o n s . . . . . . . . . . . . . . . . . . Conclusions .................................... Acknowledgments ................................ A p p e n d i x . N u m e r i c a l M o d e l for R e s i d u a l Polarization References .....................................

.......

357 359 367 386 387 387 389

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

391

CONTENTS OF VOLUMES IN THIS SERIAL . . . . . . . . . . . . . . . . . . . . . . .

405

This Page Intentionally Left Blank

Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin. L.W. ANDERSON(357), Department of Physics, University of Wisconsin, Madison, Wisconsin 53706 W. BECI~R (35), Max-Born-Institut, Max-Born-Str. 2A, 12489 Berlin, Germany J.B. BOFFARD(357), Department of Physics, University of Wisconsin, Madison, Wisconsin 53706 ROBERT N. COMPTON (219), Department of Chemistry, and Department of Physics, University of Tennessee, Knoxville, Tennessee 37996 NIR DAVIDSON(99), Weizmann Institute of Science, Department of Physics of Complex Systems, Rehovot, Israel JOHANNES DENSCHLAG(263), Institut f'tir Experimentalphysik, Universitfit Innsbruck, 6020 Innsbruck, Austria R. DORNER (1), Institut f'tir Kernphysik, August Euler Str. 6, 60486 Frankfurt, Germany P. FEN6 (357), Department of Physics, University of St. Thomas, St. Paul, Minnesota 55105 RON FOEMAN (263), Physikalisches Institut, Universitfit Heidelberg, 69120 Heidelberg, Germany NIR FRIEDMAN(99), Weizmann Institute of Science, Department of Physics of Complex Systems, Rehovot, Israel E G~SBON (35), Max-Planck-Institut f'tir Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany HARTMUT H~FFNER (191), Johannes Gutenberg University, Department of Physics, 55099 Mainz, Germany M. HATTASS(1), Institut f'tir Kernphysik, August Euler Str. 6, 60486 Frankfurt, Germany CARSTENHENKEL(263), Institut ffir Physik, Universit~it Potsdam, 14469 Potsdam, Germany WONHO JIqE (153), School of Physics and Center for Near-field Atom-photon Technology, Seoul National University, Seoul 151-742, South Korea

x

Contributors

ARIEL KAPLAN (99), Weizmann Institute of Science, Department of Physics of Complex Systems, Rehovot, Israel M.L. KELLER(357), Department of Physics, University of Wisconsin, Superior, Wisconsin 54880 R. KOPOLD(35), Max-Born-Institut, Max-Born-Str. 2A, 12489 Berlin, Germany PETER KRUGER (263), Physikalisches Institut, Universitfit Heidelberg, 69120 Heidelberg, Germany M. LARSEN (357), Department of Physics, University of Wisconsin, Madison, Wisconsin 53706 C.C. LIN (357), Department of Physics, University of Wisconsin, Madison, Wisconsin 53706 D.B. MILO~EVI~(35), Faculty of Science, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Hercegovina R. MOSHAMMER(1), Max-Planck-Institut ftir Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany R.C. NESNIDAL(357), New Focus, Inc., Middleton, Wisconsin 53562 HEUNG-Ru NOH (153), School of Physics and Center for Near-field Atomphoton Technology, Seoul National University, Seoul 151-742, South Korea RICHARD M. PAGNI (219), Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996 G.G. PAULUS(35), Max-Planck-Institut ftir Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany WOLFGANG QUINT (191), Gesellschaft f'tir Schwerionenforschung, 64291 Darmstadt, Germany R.S. SCHAPPE(357), Department of Physics, Lake Forest College, Lake Forest, Illinois 60045 H. SCHMIDT-BOCKING(1), Institut ffir Kernphysik, August Euler Str. 6, 60486 Frankfurt, Germany J6RG SCHMIEDMAYER(263), Physikalisches Institut, Universitfit Heidelberg, 69120 Heidelberg, Germany A. STAUDTE(1), Institut for Kernphysik, August Euler Str. 6, 60486 Frankfurt, Germany J. ULLRICH(1), Max-Planck-Institut for Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

Contributors

xi

T.G. WALKER(357), Department of Physics, University of Wisconsin, Madison, Wisconsin 53706 H. WALXHER(35), Ludwig-Maximilians-Universit~it Mfinchen, Germany TH. WEBER (1), Institut for Kernphysik, August Euler Str. 6, 60486 Frankfurt, Germany M. WECKENBROCK(1), Institut fiir Kernphysik, August Euler Str. 6, 60486 Frankfurt, Germany GONTHERWERTH(191), Johannes Gutenberg University, Department of Physics, 55099 Mainz, Germany XINYE Xu (153), School of Physics and Center for Near-field Atom-photon Technology, Seoul National University, Seoul 151-742, South Korea T.A. ZIMMERMAN (357), Department of Physics, University of Wisconsin, Madison, Wisconsin 53706

This Page Intentionally Left Blank

A D V A N C E S IN A T O M I C , M O L E C U L A R , A N D O P T I C A L P H Y S I C S , VOL. 48

M UL TIPL E IONIZATION IN S TR ONG LASER FIELD S R. DORNER*, Th. WEBER, M. WECKENBROCK, A. STAUDTE, M. HATTASS and H. SCHMIDT-BOCKING Institut fiir Kernphysik, August Euler Str. 6, 60486 Frankfurt, Germany

R. MOSHAMMER and J. ULLRICH Max-Planck-Institut fiir Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany 1

I. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. C O L T R I M S - A C l o u d C h a m b e r for A t o m i c Physics

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

III. Single I o n i z a t i o n and the Two-step M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. M e c h a n i s m s o f D o u b l e Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Recoil Ion M o m e n t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. F r o m N o n s e q u e n t i a l to Sequential D o u b l e I o n i z a t i o n

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

B. The Origin o f the Double Peak Structure . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Electron E n e r g i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. C o r r e l a t e d Electron M o m e n t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. E x p e r i m e n t a l Findings

3 6 9

11 11 13

19 20

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

20

B. C o m p a r i s o n to S i n g l e - P h o t o n and C h a r g e d Particle I m p a c t Double Ionization .

23

C. Interpretation within the R e s c a t t e r i n g M o d e l

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

25

D. S-Matrix C a l c u l a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

E. T i m e - d e p e n d e n t Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

VIII. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. A c k n o w l e d g m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. R e f e r e n c e s

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

30 30 31

I. I n t r o d u c t i o n 70 years ago Maria G6ppert-Mayer [ 1] showed that the energy of many photons can be combined to achieve ionization in cases where the energy of one photon is not sufficient to overcome the binding. Modern short-pulse Ti:Sa lasers (800 nm, 1.5 eV) routinely provide intensities of more than 1016 W/cm 2 and pulses shorter than 100 femtoseconds. Under these conditions the ionization probability of most atoms is close to unity. 1016 W/cm 2 corresponds to about 10 l~ coherent photons in a box of the size of the wavelength (800nm). This extreme photon density

* E-mail: d o e r n e r @ h s b . u n i - f r a n k f u r t . d e

Copyright 9 2002 Elsevier Science (USA) All rights reserved ISBN 0-12-003848-X/ISSN 1049-250X/02 $35.00

2

R. D6rner et al.

[I

allows highly nonlinear multiphoton processes such as multiple ionization, where typically more than 50 photons can be absorbed from the laser field. Such densities of coherent photons in the laser pulse also suggests a change from the "photon perspective" to the "field perspective": The laser field can be described as a classical electromagnetic field, neglecting the quantum nature of the photons. From this point of view the relevant quantities are the field strength and its frequency. 1016 W/cm 2 at 800 nm corresponds to a field of 3 x 1011 V/m, comparable to the field experienced by the electron in a Bohr orbit in atomic hydrogen (5 • 1011 V/m). Single ionization in such strong fields has been intensively studied for many years now. The experimental observables are the ionization rates as function of the laser intensity and wavelength, the electron energy and angular distribution as well as the emission of higher harmonic light. We refer the reader to several review articles covering this broad field[2-4]. Also the generation of femtosecond laser pulses has been described in a number of detailed reviews [5-8]. The present article focuses on some recent advances in unveiling the mechanism of double and multiple ionization in strong fields. Since more particles are involved, the number of observables and the challenge to the experimental as well as to the theoretical techniques increases. Early studies measured the rate of multiply charged ions as a function of laser intensity. The work reviewed here employs mainly COLTRIMS (Cold Target Recoil Ion Momentum Spectroscopy)[9] to detect not only the charge state but also the momentum vector of the ion and of one of the electrons in coincidence. Today such highly differential measurements are standard in the fields of ion-atom, electron-atom and high-energy single-photon-atom collision studies. The main question discussed in the context of strong fields as well as in the above-mentioned areas of current research is the role of electron correlation in the multiple ionization process. Do the electrons escape from the atom "sequentially" or "nonsequentially," i.e. does each electron absorb the photons independently, or does one electron absorb the energy from the field and then share it with the second electron via electron-electron correlation? Despite its long history the underlying question of the dynamics of electron correlation is still one of the fundamental puzzles in quantum physics. Its importance lies not only in the intellectual challenge of the few-body problem, but also in its wide-ranging impact to many fields of science and technology. It is the correlated motion of electrons that is responsible for the structure and the evolution of large parts of our macroscopic world. It drives chemical reactions, it is the ultimate reason for superconductivity and many other effects in the condensed phase. In atomic processes few-body correlation effects can be studied in a particularly clear manner. This, for example, was the motivation for studying theoretically and experimentally the question of double ionization

II]

MULTIPLE IONIZATION IN STRONG LASER FIELDS

3

by charged-particle (see ref. [10] for a review) or single-photon [1 l, 12] impact in great detail. As soon as lasers became strong enough to eject two or more electrons from an atom, electron correlation in strong light fields became subject of increased attention, too. As we will show below, in comparison with some of the latest results on double ionization by ion and single-photon impact, the laser field generates new correlation mechanisms, thereby raising more exciting new questions than settling old ones.

II. C O L T R I M S - A Cloud Chamber for Atomic Physics For a long time the experimental study of electron correlation in ionization processes of atoms, molecules and solids has suffered from the technical challenge to observe more than one electron emerging from a multiple ionization event. The main problem lies in performing coincidence studies employing conventional electron spectrometers, which usually cover only a small part of the total solid angle. COLTRIMS (Cold Target Recoil Ion Momentum Spectroscopy) is an imaging technique that solves this fundamental problem in atomic and molecular coincidence experiments. Like the cloud chamber and its modern successors in nuclear and high-energy physics, it delivers complete images of the momentum vectors of all charged fragments from an atomic or molecular fragmentation process. The key feature of this technique is to provide a 4:r collection solid angle for low-energy electrons (up to a few hundred eV) in combination with 4:r solid angle and high resolution for the coincident imaging of the ion momenta. As we will show below, the ion momenta in most atomic reactions with photons or charged particles are of the same order of magnitude as the electron momenta. Due to their mass, however, this corresponds to ion energies in the range of ~teV to meV. These energies are below thermal motion at room temperature. Thus, the atoms have to be cooled substantially before the reaction. In the experiments discussed here this is achieved by using a supersonic gas jet as a target. More recently, atoms in magneto-optical traps have been used to further increase the resolution [ 13-16]. A typical setup as used for the experiments discussed here is shown in Fig. 1. The laser pulse is focused by a lens of 5 cm focal length or a parabolic mirror into a supersonic gas jet providing target atoms with very small initial momentum spread of under 0.1 au (atomic units are used throughout this chapter) in the direction of the laser polarization (along the z-axis in Fig. 1). For experiments in ion-atom collisions or with synchrotron radiation the ionization probability is very small: That is why one aims at a target density in the range of up to 10-4 mbar local pressure in the gas jet. Accordingly, a background pressure in the chamber in the range of 10-8 mbar is sufficient. In contrast, for multiple ionization by femtosecond laser pulses the single ionization probability

4

R. D 6 r n e r et al.

[II

FIG. 1. Experimental setup. Electrons and ions are created in the supersonic gas-jet target. The thin copper rings create a homogeneous electric field and the large Helmholtz coils an additional magnetic field. These fields guide the charged particles onto fast time- and position-sensitive channel plate detectors (Roentdek, www.roentdek.com). The time-of-flight (TOF) and the position of impact of each electron-ion pair is recorded in list mode. From this the three-dimensional momentum vector of each particle can be calculated. easily reaches unity. Thus, within the reaction volume defined by the laser focus of typically (10~m) 2 • 100~tm all atoms are ionized. Since for coincidence experiments it is essential that much less than one atom is ionized per laser shot, a background pressure of less than 10 -1~ mbar is required. The gas jet has to be adjusted accordingly to reach single-collision conditions at the desired laser peak power. With standard supersonic gas jets this can only be achieved by tightly skimming the atomic beam, since a lower driving pressure for the expansion would result in an increase of the internal temperature of the jet along its direction of propagation. Single ionization (see Sect. III) allows for an efficient monitoring of the resolution as well as on-line control of single-collision conditions. The ions created in the laser focus are guided by a weak electric field towards a position-sensitive channel plate detector. From the position of impact and the time-of-flight (TOF) of the ion all three components of the momentum vector and the charge state are obtained. A typical ion TOF spectrum from the experiment reported in ref. [17] is shown in Fig. 2. The electric field also guides the electrons towards a second position-sensitive channel plate detector. To collect electrons with large energies transverse to the electric field a homogeneous magnetic field is superimposed parallel to the electric field. This guides the electrons on cyclotron trajectories towards the detector. Depending on their time-of-flight the electrons perform several

II]

MULTIPLE IONIZATION IN STRONG LASER FIELDS

5

FIG. 2. Time-of-flight distribution of ions produced by a 6.6 x 1014 W/cm 2 laser pulse. The gas target was 3He; the residual gas pressure in the chamber was about 2 x 10-l~ mbar. The double peak structure in the 3He2+ peak can be seen. The total count rate was about 0.1 ion per laser shot.

FIG. 3. Horizontal axis: Electron time-of-flight. Vertical axis: radial distance from a central trajectory with zero transverse momentum on electron detector, see text. full turns on their w a y to the detector. F i g u r e 3 s h o w s the e l e c t r o n T O F versus the radial distance o f the p o s i t i o n f r o m a central t r a j e c t o r y w i t h z e r o transverse m o m e n t u m o f the electron. W h e n the T O F is an i n t e g e r m u l t i p l e o f the c y c l o t r o n f r e q u e n c y the electrons hit the d e t e c t o r at this position, i n d e p e n d e n t l y o f their m o m e n t u m t r a n s v e r s e to the field. T h e s e T O F s r e p r e s e n t p o i n t s in p h a s e space w h e r e the s p e c t r o m e t e r has no r e s o l u t i o n in the t r a n s v e r s e direction.

6

R. D6rner et al.

[III

For all other TOFs the initial momentum can be uniquely calculated from the measured positions of impact and the TOE Using a magnetic field of 10 Gauss, 4:r solid angle collection is achieved for electrons up to about 30 eV. The typical detection probability of an electron is in the range o f 3 0 - 4 0 % . Thus, even for double ionization in most cases only one electron is detected. The positions o f impact and the times-of-flight are stored for each event in list mode. Thus the whole experiment can be replayed in the off-line analysis. A detailed description of the integrated multi-electron-ion m o m e n t u m spectrometer can be found in ref. [ 18].

III. Single Ionization and the Two-step Model The m o m e n t u m distribution o f singly charged helium ions produced by absorption of one 85-eV photon (synchrotron radiation) and by multiphoton absorption at 800 nm and 1.5 • 1015 W/cm 2 is shown in Fig. 4. In both cases the m o m e n t u m of the photon is negligible compared to the electron momentum. Therefore, electron and He ~+ ion are essentially emitted back-to-back compensating each others momentum (The exact kinematics including the photon m o m e n t u m can be found in section 2.3.1 of ref. [9]). Hence, for single ionization the spectroscopy of the ion momentum is equivalent to electron spectroscopy. This can be directly confirmed by looking at the coincidence between the ions and electrons in Fig. 5. All true coincidence events are located on the diagonal with equal momenta Pz in the TOF direction. The width of this diagonal gives the combined resolution

FIG. 4. Momentum distribution of He l+ ions. Left: For 85 eV single-photon absorption. Right: 1.5 eV (800nm), 220 fs, 1.5• 1015W/cm2. The polarization vector of the light is horizontal. The photon momentum is perpendicular to the (ky,kz) plane. In the left-hand panel the momentum component in the third dimension out of the plane of the figure is restricted to • au. The right-hand panel is integrated over the momenta in the direction out of the plane of the figure.

III]

MULTIPLE IONIZATION IN S T R O N G L A S E R FIELDS

7

FIG. 5. Single ionization of argon by 3.8 • 1014W/cm2. The horizontal axis shows the momentum component of the recoil ion parallel to the polarization. The vertical axis represents the momentum of the coincident electron in the same direction. By momentum conservation all true coincidences are located on the diagonal. Along the diagonal ATI peaks can be seen. The z-axis is plotted in linear scale.

of the electron and ion m o m e n t u m measurement for the pz component (in this case 0.25 au full width at half maximum). All events off the diagonal result from false coincidences in which the electron and ion were created in the same pulse but did not emerge from the same atom. This allows a continuous monitoring o f the fraction o f false coincidences during the experiment. Knowing this number the false coincidences can also be subtracted for double-ionization events. For single-photon absorption the electron energy is uniquely determined by the photon energy Ev and the binding energy plus a possible internal excitation energy of the ion. The resulting narrow lines in the photoelectron energy spectrum correspond to spheres in m o m e n t u m space. The left-hand panel of Fig. 4 shows a slice through this m o m e n t u m sphere. The outer ring corresponds to He 1+ ions in the ground state, the inner rings to the excited states. The photons are linearly polarized with the polarization direction horizontal in the figure. The angular distribution of the outer ring shows an almost pure dipole distribution according to the absorption of one single photon. On the contrary, in the laser field any number of photons can be absorbed, leading to an almost continuous energy distribution o f the electrons (right-hand panel in Fig. 4). Structure o f individual ATI (above threshold ionization) peaks spaced by the

8

R. D6rner et al.

[III

photon energy (1.5 eV) is not seen here. This is in agreement with electron spectra, where at comparable laser intensities ATI structure is not observed either. The electrons and ions are emitted in narrow jets along the polarization axis. Such high-angular-momentum states, needed to produce this kind of distribution, are accessible due to the large number of photons absorbed. How do the ions and electrons get their momenta? For the case of singlephoton absorption the light field is so weak that there is no acceleration. Also, the photon carries no significant momentum into the reaction. The photon cuts the tie between nucleus and electron by providing the energy. The momenta observed in the final state thus have to be present already in the initial-state Compton profile of the atom. Single-photon absorption is therefore linked to a particular fraction of the initial-state wave function, which in momentum representation coincides with the final-state momentum. The scaling of the photo ionization cross section at high energies follows, besides a phase space factor, the initial-state momentum space Compton profile, i.e. the probability to find an electron-ion pair with the appropriate momentum in the initial state. In the strong-field case the situation changes completely. The field is strong enough to accelerate the ions and electrons substantially after the electron is set free. The momentum balance, however, is still the same as in the single-photon limit: The laser field accelerates electron and ion to the opposite directions resulting again in their back-to-back emission (see Fig. 5). This changes only if the laser pulse is long enough that the electron can escape from the focus during the pulse. In that case, which we do not consider here, the momenta are balanced by a huge amount of elastically scattered photons. In the regime of wavelength and binding energies under consideration here, a simple two-step picture has been proven useful. In the first step the electron is set free by tunneling through the potential barrier created by the superposition of the Coulomb potential of the atom and the electric field of the laser. This process promotes electrons and ions with zero momentum to the continuum. Then they are accelerated in the laser field and perform a quiver motion. In this model the net momentum in the polarization direction, which is observed after a pulse with an envelope of the electric field strength E(t) being long compared to the laser frequency, is purely a function of the phase of the field at the instant of tunneling (tunneling time to): PzHeI+(too) =

ft0 t~

E(t) sin ~ot dt.

(1)

Tunneling at the field maximum thus leads to electrons and ions with zero momentum. The maximum momentum corresponding to the zero crossing of the laser field is x/~Up, where Up = I/4~o 2 is the ponderomotive potential at intensity I and photon frequency ~o (Up = 39.4eV at 6.6x1014W/cm2). Within this simple model the ion and electron momentum detection provides a measurement of the phase of the field at the instant of tunneling. We will generalize this idea below for the case of double ionization.

IV]

MULTIPLE IONIZATION IN STRONG LASER FIELDS

9

Single ionization is shown here mainly for illustration. Much more detailed experiments have been reported using conventional TOF spectrometers (see ref. [4] and references therein) and photoelectron imaging [19,20].

IV. M e c h a n i s m s of Double Ionization What are the "mechanisms" leading to double ionization? This seemingly clearcut question does not necessarily have a quantum-mechanical answer. The word "mechanism" mostly refers to an intuitive mechanistical picture. It is not always clear how this intuition can be translated into theory, and even if one finds such a translation the contributions from different mechanisms have to be added coherently to obtain the measurable final state of the reaction [21,22]. Thus, only in some cases mechanisms are experimentally accessible. This is only the case if different mechanisms occur at different strengths of the perturbation (such as laser power or projectile charge) or if they predominantly populate different regions of the final-state phase space. In these cases situations can be found where one mechanism dominates such that interference becomes negligible. With these words of caution in mind, we list the most discussed mechanisms leading to double ionization: (1) TS2 or Sequential Ionization: Here the two electrons are emitted sequentially by two independent interactions of the laser field with the atom. From a photon perspective one could say that each of the electrons absorbs photons independently. From the field perspective one would say that each electron tunnels independently at different times during the laser pulse. This is equivalent to the TS2 (two-step-two) mechanism in ion-atom and electron-atom collisions. In this approximation the probability of the double ejection can be estimated in an independent-particle model. Most simply one calculates double ionization as two independent steps of single ionization. A somewhat more refined approach uses an independent-event model, which takes into account the different binding energies for the ejection of the first and the second electron (see, e.g., ref. [23] for ion impact, ref. [24] for laser impact). (2) Shake-Off: If one electron is removed rapidly (sudden approximation) from an atom or a molecule, the wave function of the remaining electron has to relax to the new eigenstates of the altered potential. Parts of these states are in the continuum, so that a second electron can be "shaken off" in this relaxation process. This is known for example from beta decay, where the nuclear charge is changed. Shake-off is also known to be one of the mechanisms for double ionization by absorption or Compton scattering of a single photon (see the discussion in ref. [25] and references therein). However, only for very high photon energies (in the keV range) it is the dominating mechanism. For helium it leads to a ratio of double to

10

R. D 6 r n e r et al.

[IV

single ionization of 1.66% [26,27] for photoabsorption (emission of the first electron from close to the nucleus) and 0.86% for Compton scattering (averaged over the initial-state Compton profile) [28]. (3) Two-Step-One (TS1): For single-photon absorption at lower photon energies (threshold to several 100eV [22]) TS1 is known to dominate by far over the shake-off contribution. A simplified picture of TS1 is that one electron absorbs the photon and knocks out the second one via an electron-electron collision on its way through the atom [29]. A close connection between the electron impact ionization cross section and the ratio of double to single ionization by single-photon absorption as function of the energy is seen experimentally [29] and theoretically [22], supporting this simple picture. For the TS 1 mechanism the electron correlation is on a very short time scale (a few attoseconds) and confined to a small region of space (the size of the electron cloud). (4) Rescattering: Rescattering is a version of the TS1 mechanism which is induced only by the laser field. The mechanism was proposed originally by Kuchiev [30] under the name "antenna model." He suggested that one of the electrons is driven in the laser field acting as an antenna absorbing the energy which it then shares with the other electron via correlation. Corkum [31] and Schafer [32] extended this basic idea and interpreted the process in the two-step model: First one electron is set free by tunneling. Then it is accelerated by the laser field and is driven back to its parent ion with about 50% probability. Upon recollision with the ion the electron can recombine and emit higher harmonic radiation. Besides that it could be elastically scattered and further accelerated or it could be inelastically scattered with simultaneous excitation or ionization of the ion. In contrast to TS1 in this case there is a femtosecond time delay between the first and the second step. Also the wave function of the rescattered electron explores a larger region of space than in the case of TS1 [33-35]. Strong experimental evidence favoring the rescattering process to be dominantly responsible for double ionization by strong laser fields was later provided by the observation that double ejection is strongly suppressed in ionization with circularly polarized light [36,37] (see also Fig. 19 of ref. [3]). The rescattering mechanism is inhibited by the circular polarization since the rotating electric field does not drive the electrons back to their origin. The other mechanisms, in contrast, are expected to be polarization independent. Further insight in the double ionization process clearly necessitates differential measurements beyond the ion yield. Two types of such experiments have been reported recently: Electron time-of-flight measurements in coincidence with the ion charge state [38,39] and those using COLTRIMS, where at first only the ion momenta [40-42] and later the ion momenta in coincidence with one electron [ 17,43-45] have been measured.

V]

MULTIPLE IONIZATION IN S T R O N G L A S E R FIELDS

11

V. Recoil Ion M o m e n t a A. FROM NONSEQUENTIAL TO SEQUENTIAL DOUBLE IONIZATION Recoil ion m o m e n t u m distributions have been measured for helium (He 1+, He2+)[40], neon (Ne 1+, Ne 2+, Ne3+)[41] and argon (Ar l+, Ar2+)[45,46]. Figure 6 summarizes some of the results for neon. The m o m e n t u m distribution of the singly charged ion is strongly peaked at the origin as in the case o f helium (Fig. 4), reflecting the fact that tunnel ionization is most likely at the m a x i m u m of the field (see Eq. 1). The structure o f the m o m e n t u m distribution of the doubly charged ions changes strongly with the peak intensity. In the region where the rates suggest the dominance of nonsequential ionization the ion momenta show a distinct double peak structure (Fig. 6(2)). At higher intensities, where rates can be described by assuming sequential ionization, the momenta o f the Ne 2+ ions are peaked at the origin as for single ionization. The studies for helium show a similar double peak structure at 6.6 x 1014 W/cm 2 (see Fig. 9). The evolution of the ion momentum distributions with laser peak power has been studied in detail for argon [46], too, confirming the fact that at the transition to the nonsequential regime an increase in laser power results in colder ions. The argon data, however, show no distinct double peak structure (see Fig. 7), where the sequential ionization already sets in at about 6.6x 1014 W/cm 2. The reason might be that the sequential contribution fills "the valley" in the m o m e n t u m

FIG. 6. Neon double ionization by 800 nm, 25 fs laser pulses. Left-hand panel: Rate of single and double ionization as a function of the laser power (from ref. [47]). The solid line shows the rate calculated in an independent event model. Right-hand panel: Recoil-ion momentum distributions at intensities marked in the left-hand panel. A projection of the double-peaked distribution (2) is shown in Fig. 10. Horizontal axis: Momentum component parallel to the electric field. Vertical axis: One momentum component perpendicular to the field (data partially from ref. [41]).

12

[V

R. D 6 r n e r et al.

(a)

~3

0

9;, 9 .......

/1: //"

,

........

-

. ,

\\

" \\

o,~

.........

0

kl'/7,,.~,,~ .... t ; , -4

-3

-2

.......

J .........

-1

~. . . . . . . . . J ....... La, . . . . 0

1

2

5a.aP~i 3

4

Prz (a.u.) FIG. 7. Momentum distribution of Ar 2+ ions created in the focus of a 220 fs, 800 nm laser pulse at peak intensities of (a) 3.75• l0 TMW/cm 2 and (b) 12• 1014 W/cm 2 in the direction of the polarization. The distributions are integrated over the directions perpendicular to the polarization. Solid circles: distribution of Ar 2+ ions; dotted line: distribution of Ar 1§ ions; dashed line: results of the independent electron model of convoluting the Ar l+ distribution with itself; solid line: results of the independent-electron ADK model (see text); open circles in (a): distribution of He 2+ ions at 3.8• l014 W/cm 2 (figure from ref. [46], helium data from ref. [40]).

distribution at the origin before a double peak structure has developed. In ref. [45] it has been argued based on classical kinematics that excitation of a second electron during recollision followed by tunneling ionization of the excited electron might be responsible for "filling the valley." Due to the open 3d shell in Ar, excitation cross sections are much larger than in Ne. At the highest intensity the single peak distribution can be at least qualitatively understood in an independent two-step picture (see Fig. 7). The dash-dotted lines in Fig. 7 show the measured momentum distributions of Ar 1+ ions, the dashed line is this distribution convoluted with itself. Such a convolution models two sequential and totally uncorrelated steps of single ionization spaced in time by a random number of optical cycles. Figure 7 shows that for argon at 12• W/cm 2 (which,

V]

MULTIPLE IONIZATION IN STRONG LASER FIELDS

13

judging from the rates, is in the sequential regime), this very simple approach describes the ion momentum distributions in double ionization rather well. One obvious oversimplification of this convolution procedure is that it implicitly assumes that the momentum distributions do not change with binding energy. A more refined independent-event approach would use different binding energies for both steps. As an alternative simple model, the momentum distribution for removal of the first electron and the second electron have been calculated in the ADK (Ammosov-Delone-Krainov) model (see, e.g., ref. [48], Eq. 10) using the correct binding energies for both steps. The result of convoluting these two calculated distributions is shown by the solid lines in Fig. 7. Clearly such modeling fails in the regime where sequential ionization dominates (Fig. 7a). B. THE ORIGIN OF THE DOUBLE PEAK STRUCTURE

The recoil ion is an important messenger carrying detailed information on the time evolution of the ionization process. It allows not only to distinguish between sequential and nonsequential ionization but also to rule out some of the nonsequential mechanisms as we will show now. Analogous to the situation for single ionization discussed above one can estimate the net momentum accumulated by the doubly charged ion from the laser pulse as He2+ ~t, 2 Pz (t~) = E ( t ) sin tot dt + 2

~tlt~

E ( t ) sin tot dt.

(2)

2

The first electron is removed at time tl and the ion switches its charge from 1+ to 2+ at time tl 2. It is assumed that there is no momentum transfer to the ion from the first emitted electron during double ionization. Thus, as in the case of single ionization the phase of the field at the instant of the emission of the first and of the second electron is encoded in the ion momentum. Shake-off and TS2 will both lead to a momentum distribution peaked at zero, similar to single ionization. In both cases the emission of the second electron follows the first with a time delay, which is orders of magnitude shorter than the laser period. Hence tl 2 = t~ in Eq. (2), and since the first electron is emitted He2+ most likely at the field maximum Pz would also peak at zero for shake-off and TS1. Consequently, the observed double peak structure for He and Ne directly rules out these mechanisms. For the rescattering there is a significant time delay between the emission of the first electron and the return to its parent ion. Estimating tl 2 for a rescattering trajectory which has sufficient energy to ionize leads to ion momenta close to the measured peak positions [40,41,49]. The high momenta of the doubly and triply charged ions are direct proof of the time delay introduced by the rescattering trajectory. It is this time delay with respect to the field maximum

14

[V

R. D 6 r n e r et al. time

kb

ka

{.

2

1

FIG. 8. Feynman diagram describing the rescattering and TS1 mechanism (from ref. [50]). See text.

that is responsible for multiple ionization and allows an effective net momentum transfer to the ion by accelerating the parent ion. Within the classical rescattering model the final momentum of the doubly charged ion will be the momentum received from the field (as given by Eq. 2) plus the momentum transfer from the recolliding electron to the ion. Soon after the measurement of the first ion momentum distributions Becker and Faisal succeeded in the first theoretical prediction of this quantity. They calculated double ionization of helium using (time-independent) S-matrix theory. They evaluated the Feynman diagram shown in Fig. 8. Time progresses from bottom to top. Starting with 2 electrons in the helium ground state at time ti, the laser field couples once at tl to electron 1 (VATI). Electron 1 is then propagated in a Volkov state (k) in the presence of the laser field, while electron 2 is in the unperturbed He l+ ground state (j). Physically the Volkov electron does not have a fixed energy but can pick up energy from the field. This describes e.g. an acceleration of the electron in the field and its return to the ion. At time t2 one interaction of the two electrons via the full Coulomb interaction is included. This allows for an energy transfer from the Volkov electron to the bound electron. Finally, both electrons are propagated independently in Volkov states, describing their quiver motion in the field. By evaluating this diagram Becker and Faisal obtained excellent agreement with the observed ion yields (see ref. [51,52] for helium and ref. [53] for an approximated rate calculation on other rare gases). The ion momentum distribution calculated as the sum momentum of the two electrons predicted by this diagram is shown in Fig. 9b. The calculation correctly predicts the double peak structure and the position of the maxima. The minimum at momentum zero is more pronounced in the calculation than in the data. The major approximations which might be responsible for this are: Only one step of electron-electron energy transfer is taken into account (see ref. [57] for a discussion of the importance of multiple steps); no intermediate excited states are considered; and the laser field is neglected for all bound states as in turn the Coulomb field is neglected in the continuum states. To unveil the physical mechanism producing the double hump structure Becker and Faisal

V]

MULTIPLE IONIZATION IN STRONG LASER FIELDS

15

I = 6 . 6 . 1 0 '4 W / c m 2 1.0

'

I

'

I

'

I

'

I

'

0.8 - a]

I

'

I

'

I

'

i

o4

,,

;

o,~ _.,.' ~"

0.8-

-~

0.6

!

9

'

I

'

I

'

I

'

I

'

I

'

i

'

;

i

9

:. .

|

-

,

-

,

9

|

.

9

,

.

-

b]

9

|

.

9

|

9

|

9

|

9

|

9

|

9

!

9

_

,

I

,

~z 0,4 "-

I

J

........

9

'

.. d]

i

0.2

_ , ~ . '

0,8-

I

!

, . , . , .~,m I

'

I

'

i

|

i

,

I

'

. i

'

c]

9

|

'

|

'

i

9

|

9

!

9

!

9

|

9

.. f]

..

0,6-

;%

;I

0,40 , 2 -/ 0,0 l

-8

, a ,

-6 -4 -2

I

0

2

4

6

,

L" ,

I

-6

___

,

-4

I

-2

,

I

,

0

I

2

,

L .

I

4

6

,

i

8

Prz [a.u.]

FIG. 9. Momentum distribution of He 2+ ions at an intensity of 6 . 6 •

1014 W / c m 2 for all panels.

Prz is the component parallel to the laser polarization. (a) Experiment (from ref. [40]); (b) results of the S-matrix calculation (from Becker and Faisal [ 5 0 ] ) ; (c) S-matrix with additional saddle-point approximation (from Goreslavskii and P o p r u z h e n k o [ 5 4 ] ) ; (d) solution of the one-dimensional Schr6dinger equation (from Lein et al. [34]); (e) Classical Trajectory Monte Carlo calculations (from Chen e t al. [55]); (f) Wannier-type calculation (from Sacha and Eckhardt [56]).

have evaluated the diagram also by replacing the final Volkov states by plane waves. Physically this corresponds to switching off the laser field after both electrons are in the continuum. In the calculation this led to a collapse of the double peak structure to a single peak similar to single ionization. This confirms our interpretation given above, that it is the acceleration of the ion in the field after the rescattering (starting at tl 2 in Eq. 2) that leads to the high momenta. The S-matrix theory also yielded good agreement with the observed narrow momentum distribution in the direction perpendicular to the laser field. Later, different approximations in the evaluation of the diagram (Fig. 8) have been introduced. First, Kopold and coworkers [58] replaced the electron-electron interaction by a contact potential and additionally used a zero-range potential for the initial state. This simplified the computation considerably while still yielding the observed double peak structure, not only for helium but also for neon

16

[V

R. D 6 r n e r et al. 1.0 0,8

.%,

~

,,X

0.6

..-9

0.4

~9~

..

9

9

-

13.10~4 W/cm2

0.2

O.8 -o

0.6

8 - 1 0 ~4 W / c ~

0,4 0,2

0,8

c)

0.6

13" 10~4W/crn:

0,4 0,2

0.0

-8 -6 -4 -2

0

2

4

6

8

Prz (a.u.)

FIc. 10. Momentum distribution ofNe2+: (a) projection of data in Fig. 6 (2) at 13• 1014 W/cm 2 (from Moshammer et al. [41]); (b) S-matrix calculation evaluating the diagram in Fig. 8 with contact potentials at 8• W/cm 2 (from Kopold et al. [58]); (c) Wannier-type calculation at 13• 1014 W/cm 2 (from Sacha and Eckhardt [56]).

(Fig. 10b) and other rare gases. They found that the inclusion of intermediate excited states of the singly charged ion yields a filling of the minimum at zero momentum. Goreslavskii and Popruzhenko [54,59] used the saddle-point approximation for the intermediate step. This additional approximation did not change the calculated ion momenta strongly (see Fig. 9c) but simplified the computation, allowing to investigate also the correlated electron emission discussed in the next section. A conceptionally very different approach was used by Sacha and Eckhardt[56]. They argued that the rescattering will produce a highly excited intermediate complex, which will then decay in the presence of the field. This decay process will not have any memory of how it was created. They assumed a certain excitation energy as free parameter in the calculations and then propagated both electrons in the classical laser field semiclassically in reduced dimensions. Therefore they analyzed this decay by a Wannier-type analysis. Wannier theory is known to reproduce the electron angular dependence as well as the recoil ion momenta for the case of single-photon double ionization [6062,25]. In this case the Wannier configuration would be the emission of both

V]

MULTIPLE IONIZATION IN STRONG LASER FIELDS

17

electrons back-to-back, leaving the recoil ion at rest on the saddle of the electronelectron potential. For single-photon absorption from an S state this configuration is forbidden by selection rules; it would allow, however, the absorption of an even number of photons. In the multiphoton case the external field has to be included in addition to the Coulomb potential among the particles. This leads to a saddle in the potential, which is not at rest at the center between the electrons but at momenta which correspond to the observed peaks. Sacha and Eckhardt analyzed classical trajectories in the saddle potential created by the field and the Coulomb potentials. At a given laser field the decay of the excited complex in the field is characterized only by two parameters: The time when the complex is created and the total energy. Interestingly the recoil ion momentum obtained this way exhibits a double peak structure, which does not depend strongly on the creation time but on the energy. They find parallel and perpendicular momentum distributions, which for helium (Fig. 9f) and for neon (Fig. 10c) are in reasonable agreement with the experiment. This argument of a time-independent intermediate complex seems to contradict the claim that the high recoil momenta and the double peak result from the time delay due to the rescattering. One has to keep in mind, however, that within the rescattering model the recollision energy and hence the total energy of the complex analyzed by Sacha and Eckhardt is uniquely determined by the recollision time. In a recent work they extended this model to examine the decay of highly excited three-electron atoms [63]. The S-matrix approaches discussed above are based on the time-independent Schr6dinger equation. One of the advantages of such approaches is that they allow a precise definition of a mechanism (see, e.g., ref. [ 10]). Each particular diagram represents one mechanism. The price that has to be paid is the loss of information on the time evolution of the system. The diagram contains the time order of interactions, but not the real time between them. Starting from the time-dependent Schr6dinger equation in contrast gives the full information on the time evolution of the many-body wave function in momentum or coordinate space. In these coordinate space density distributions it is, however, often difficult to clearly define what one means with a mechanism. Lein and coworkers found a very elegant way to solve this problem [34,35]. Instead of plotting the density in coordinate space they calculated the Wigner transform of the wave function, depending on momentum and position. Integrated over the momentum coordinate it is the density in coordinate space and integrated over the position it is the distribution in momentum space. The Wigner transform can be read as a density in phase space. Lein and coworkers plotted for example the phase space evolution of the recoil ion in the polarization direction. This presentation of a quantum-mechanical wave function is very close to the presentation of the classical phase space trajectories. The rescattering mechanism can be seen very clearly in this presentation.

18

R. D 6 r n e r et al.

[V

Computation of the time-dependent Schr6dinger equation for three particles in three dimensions is extremely challenging. Even though great progress has been made in this field (see e.g. refs. [64-69]), there are no predictions of recoil ion momenta or other differential information based on the solution of the timedependent Schr6dinger equation in three dimensions for the "long" wavelength regime of presently available high-intensity lasers. To allow for a practical calculation of the time evolution of the three-body system two rather different approximations have been made: (a) Reducing the dimensions from three for each particle to only one along the laser polarization and (b) keeping the full dimensionality but using classical mechanics instead of the Schr6dinger equation [70]. Lein et al. [34] reported the first results on recoil ion momenta based on an integration of the one-dimensional Schr6dinger equation (see Fig. 9d). The momentum distribution of the He 2+ ions at 6.6 x 1014 W/cm 2 in Fig. 9d) peaks at zero momentum in contrast to all other results. It will become clear in Sect. VII that there is evidence in Lein et al. 's calculation for a correlated emission of both electrons into the same hemisphere. A well-known problem of one-dimensional calculations is that the effect of electron repulsion is overemphasized. This might be partially responsible for "filling the valley" in these calculations. For further discussion see Sect. VII. Chen et al. [55] have performed a Classical Trajectory Monte Carlo calculation (CTMC) in which they solved the classical Hamilton equations of motion for all three particles in the field. Instead of a full classical simulation of the process (see, e.g., ref. [71] for CTMC calculation for single ionization and ref. [70] for refined classical calculation of double ionization rates) they have initialized one electron by tunneling and then propagated all particles classically. This also yields the observed double peak structure. Such CTMC calculations have proven to be extremely successful in predicting the highly differential cross sections from ion impact single and multiple ionization (see refs. [72-79] for some examples). One of the virtues of this approach is that the output comprises the momenta for each of the particles for each individual ionizing event, exactly like in a COLTRIMS experiment. In addition, however, each particle can be followed in time, shedding light on the mechanism. Such detailed studies would be highly desirable for the strong field case, too. All theoretical analyses of the observed double peak structure in the recoil ion distribution confirm the first conclusion from both experimental teams reporting these structures: (a) It is an indication of the nonsequential process and (b) it is consistent with the rescattering mechanism, which is included in one or the other way in the various theoretical models. In the direction perpendicular to the polarization the observed and all calculated distributions are very narrow and peak at zero. Since there is no acceleration by the laser field in this direction the transverse momentum of the

VI]

MULTIPLE IONIZATION IN STRONG LASER FIELDS

19

ion is purely either from the ground state or from the momentum transfer in the recollision process. All theories which are not confined to one dimension agree roughly with the experimental width of the distribution. This direction should be most sensitive to the details of the recollision process since that is where the parallel momentum acquired from the field is scattered to the transverse direction. Hence, a closer inspection of the transverse momentum transfer is of great interest for future experimental and theoretical studies.

VI.

Electron

Energies

Electron energy distributions for double ionization have been reported for helium [39], argon [80], neon [81] and xenon [38]. All these experiments find in the sequential regime that the electron energies from double ionization are much higher than those generated in single ionization. This is in full agreement with the recoil ion momenta discussed above, since the mechanism being predominantly responsible for producing high-energy electrons is exactly the same: It is a fact that due to the rescattering the electrons from double ionization are not promoted to the continuum at the field maximum but at a later time. Depending on the actual time delay an energy of up to 2Up (see Eq. 1) can be acquired. The work for helium (Fig. 11) and neon shows that the electron spectra extend well above this value. Energies beyond 2Up are only obtainable if the recolliding electron is backscattered during the (e,2e) collision. In this case the momentum they have after the recollision adds to the momentum acquired in the field. A large amount of elastically backward-scattered electrons has been observed for single ionization where a plateau in the energy distribution is found extending to energies of up to l OUp. Electron energy (Up)

Electron energy (Up)

1 > 1000

N C

100

oO

10

9

!

2 9

,

9

3

4

5

,

,

,

1 ,

9

2 .

,

3 ,

,

4 .

,

5 .

,

~N

E O

z

0.1 0.01

(a)

k._

(b) ,

,

50

,

,

100 150 200 250

Electron energy (eV)

2's 5'o 7; loo Electron energy (eV)

FIG. 11. Electron energy spectra from single ionization (solid line) and double ionization (dots) of helium at (a) 8• W/cm 2 and (b) 4x 1014 W/cm 2 (from ref. [39]).

20

R. D 6 r n e r et al.

[VII

VII. Correlated Electron M o m e n t a More information can be obtained from the momentum correlation between the two electrons. In an experiment one possible choice would be to observe the momenta of both electrons in coincidence. In this case the recoil ion momentum could be calculated employing momentum conservation. From an experimental point of view however, it is easier to detect the ion and one of the electrons, in which case the momentum of the second electron can be inferred from momentum conservation. It is experimentally simpler since the additional knowledge of the ion charge state allows for an effective suppression of random coincidences. Moreover, electron and ion are detected by opposite detectors circumventing possible problems of multihit detection. Many successful studies for single-photon double ionization have been performed this way [25,62,82, 83]. Up to present, however, no fully differential experiment has been reported for multiphoton double ionization. Weber et al. [ 17] and Feuerstein et al. [45] reported measurements observing only the momentum component parallel to the field of electron and ion integrating over all other momentum components. Weckenbrock et al. [43] and Moshammer[41] have detected the transverse momentum of one of the electrons in addition to the parallel momenta. In these experiments, however, the transverse momentum of the ion could not be measured with sufficient resolution, mainly due to the internal temperature of the gas jet for argon and neon targets. Experiments on helium have not yet been reported but are in preparation in several laboratories. A. EXPERIMENTALFINDINGS

The correlation between the momentum components parallel to the polarization is shown in Fig. 12. The electron momenta are integrated over all momentum components perpendicular to the field direction. Events in the first and third quadrants are those where both electrons are emitted to the same hemisphere, the second and fourth quadrants correspond to emission to opposite half spheres. The upper panel shows the electron momenta at an intensity of 3.6• 2, which is in the regime where nonsequential ionization is expected. The distribution shows a strong correlation between the two electrons, they are most likely emitted to the same hemisphere with a similar momentum of about 1 au. At higher intensity, where double ionization proceeds sequentially, this correlation is lost (lower panel in Fig. 12). To interpret the correlation pattern it is helpful to consider the relationship between the electron and the recoil ion momenta. We define the Jacobi momentum coordinates kz+ and kz: k+z = kezl + ke~2,

(3)

k z = kez~ -- kez2,

(4)

VII]

M U L T I P L E I O N I Z A T I O N IN S T R O N G L A S E R F I E L D S

21

FIG. 12. Momentum correlation between the two electrons emitted when an Ar 2+ ion is produced in the focus of a 220 fs, 800nm laser pulse at peak intensities of 3.8x 1014 W/cm 2 and 15x 1014 W/cm2. The horizontal axis shows the momentum component of one electron along the polarization of the laser field; the vertical axis represents the same momentum component of the corresponding second electron. Same sign of the momenta for both electrons represents an emission to the same half sphere. The data are integrated over the momentum components in the direction perpendicular to the polarization direction. The gray shading shows the differential rate in arbitrary units on a linear scale (adapted from ref. [ 17]). Also compare this figure to Fig. 17.

with kzion = -kz+. T h e s e coordinates are a l o n g the d i a g o n a l s o f Fig. 12. H e n c e the recoil ion m o m e n t u m distribution is s i m p l y a p r o j e c t i o n o f Fig. 12 onto the diagonal kz+. T h e coordinates k] and k z are helpful to illustrate the relative i m p o r t a n c e o f the two c o u n t e r a c t i n g effects o f e l e c t r o n - e l e c t r o n r e p u l s i o n and acceleration o f particles by the optical field. B o t h influence the final-state m o m e n t a in different ways. E l e c t r o n r e p u l s i o n (and t w o - b o d y e l e c t r o n - e l e c t r o n scattering) does not c h a n g e kz+ but contributes to the m o m e n t u m kz. O n the other hand, once both electrons are set free, the m o m e n t u m transfer r e c e i v e d from the field is identical for both. T h e r e f o r e , this part o f the a c c e l e r a t i o n does not change kz b u t adds to kz+. The o b s e r v e d wide kz+ and n a r r o w k z distributions

22

R. D 6 r n e r et al.

[VII

FiG. 13. Momentum correlation between the two electrons emitted when an Ar 2+ ion is produced in the focus of a 150 fs, 780nm laser pulse at peak intensities of 4.7• 1014 W/cm 2. Axis as in Fig. 12. Each panel panel represents a part of the final state for a fixed transverse momentum (p• of one of the electrons. (a) One of the electrons has a transverse momentum of p • < 0.1 au; (b) 0.1 < p • < 0.2au; (c) 0.2 < p • < 0.3 au; (d) 0.3 < p • < 0.4au. The gray scale shows the differential rate in arbitrary units and linear scale (from ref. [43]).

thus indicate that the joint acceleration of the electrons in the laser field clearly dominates over the influence of electron repulsion. For argon double ionization Weckenbrock et al. [43] and Moshammer et al. [84] measured in addition to the momentum parallel to the field also the transverse momentum of the detected electron. Both find that the correlation pattern strongly depends on this transverse momentum (see Fig. 13). If one electron is emitted with any transverse momentum larger than 0.1 au (i.e. at some angle to the polarization axis) one mostly finds both electrons with a similar momentum component in the field direction. It is this configuration that dominates the integrated spectrum in Fig. 12. If, however, one electron is emitted parallel to the polarization with a very small transverse momentum window of p• < 0.1 au one finds that the parallel momentum distribution does no longer peak on the diagonal. In this case most likely one electron is fast and the other slow. This might be due to the fact that the 1/rl 2 potential forces the electrons into different regions in the three-dimensional phase space. Consequently, for electrons to have equal parallel momentum some angle between them is required.

VII]

MULTIPLE IONIZATION IN STRONG LASER FIELDS

23

Accordingly, the peak at Pezl = P e z 2 = 1 au is found to be most pronounced if at least one of the electrons has considerable transverse momentum. In tendency, this feature can be explained by (e,2e) kinematics as discussed in ref. [84]: Unequal momentum sharing is known to be most likely in field-free (e,2e) reactions. The rescattered electron is only little deflected, losing only a little of its longitudinal momentum during recollision. At the same time, the ionized electron is low-energetic, resulting in very different start momenta of both electrons at recollision time tl 2. At intensities not too close to the threshold this scenario leads to asymmetric longitudinal energy sharing as calculated in refs. [54,85].

B. COMPARISON TO SINGLE-PHOTON AND CHARGED PARTICLE IMPACT DOUBLE IONIZATION

One might expect that the pure effect of electron repulsion could be studied in double ionization by single-photon absorption with synchrotron radiation. In this case there is no external field in the final state that could accelerate the electrons. Many studies have shown however, that the measured momentum distribution is not only governed by the Coulomb forces in the final state, but also by selection rules resulting from the absorption of one unit of angular momentum and the accompanying change in parity. For helium for example the two-electron continuum wave function has to have ~p0 character. Since these symmetry restrictions on the final state are severe it is misleading to compare distributions of kezl versus kez2 a s in Fig. 12 directly to those from single-photon absorption (this distribution can be found in ref. [86]). The effect of electron repulsion can be more clearly displayed in a slightly different geometry as shown in Fig. 14. Here one electron is emitted along the positive x-direction and the momentum distribution of the second electron is shown. The data are integrated over all directions of this internal plane of the three-body system relative to the laboratory. Clearly electron repulsion dominates the formation of this final state distribution: there is almost no intensity for emission to the same half sphere. There is also a node for emission of both electrons back-to-back. This is a result of the odd symmetry of the final state. In the multiphoton case this node is expected for those events where an odd number of photons is absorbed from the field (see e.g. [88]). Another instructive comparison is the process of double ionization by charged particle impact. Experiments have been reported for electron impact [89-91] and fast highly charged ion impact [79,92]. The latter is of particular interest from the strong field perspective since the potential "shock" induced at a target atom by a fast highly charged projectile is in many aspects comparable to a half cycle laser pulse. The time scale however is much shorter than that accessible with lasers today. For their experiment colliding 1 GeV/u U 92+ projectiles on helium for example Moshammer and coworkers [93] estimated a power density of > 1019 W/cm 2 and a time of sub attoseconds. Under such conditions ion-

24

R. D 6 r n e r et al.

[VII

Fie. 14. Single-photon double ionization of He at l eV and 20eV above threshold by linearly polarized light (synchrotron radiation). Shown is the momentum distribution of electron 2 for fixed direction of electron 1 as indicated. The plane of the figure is the internal momentum plane of the two particles. The data are integrated over all orientations of the polarization axis with respect to this plane. The figure thus samples the full cross section and all angular and energy distributions of the fragments. The outer circle corresponds to the maximum possible electron momentum; the inner one represents the case of equal energy sharing (from ref. [87]; compare also ref. [83]). atom collisions can be successfully described by the Weizs/icker-Williams formalism [94,95,93,96], which replaces the ion by a flash of virtual photons (for a detailed discussion on the validity and limitations of this method see ref. [97]). Since such an extremely short "photon field" also has contributions from very high frequencies, i.e. virtual photon energies, the ionization is dominated by the absorption of one photon per electron. This is contrary to the femtosecond laser case discussed here. Multiple ionization in fast ion-atom collisions is dominated by either the TS2 or the TS 1 process (with only a small amount of shake-off) depending on the strength of the perturbation, i.e. the intensity of the virtual photon field. The ratio of the projectile charge to the projectile velocity is usually taken as a measure of the perturbation. Figure 15 shows the electron m o m e n t u m correlation of double ionization of helium by 100 MeV/u C 6+ impact parallel to the direction of the projectile. The dominant double ionization mechanism at these small perturbations is TS1 [98], or, in a virtual photon picture, one photon is absorbed during a collision by either one of the electrons and the second is taken to the continuum due to electron-electron correlation. Under these conditions the electron repulsion in the final state drives the electrons to opposite half spheres, whereas the projectile itself passes so fast that during this short time essentially no momentum is transferred to the system. Similar studies have been performed with slower and more highly charged projectiles [79]. In this case the dominant double ionization mechanism is TS2. The experiments show a joint forward emission of both electrons. This effect has been interpreted in a two-step picture: First the initial-state momentum distribution is lifted to the continuum by absorption of two virtual photons, then in a second step the strong potential of the

VII]

MULTIPLE IONIZATION IN STRONG LASER FIELDS

25

FIG. 15. Double ionization of helium by 100 MeV/u C 6+ impact. The horizontal and vertical axes (Pill and P211) show the momentum components of electrons 1 and 2 parallel to the direction of the projectile. The dashed curves demarcate the region of the two-electron momentum space which is not accessed by the spectrometer. The gray scale is linear (adapted from ref. [92]).

projectile accelerates both electrons into the forward direction (see also ref. [96] for a theoretical interpretation of the double ionization process; see ref. [99] for another experiment showing directed multiple electron emission; see ref. [78] for an analysis of the acceleration of an electron in the field of the projectile). C. INTERPRETATION WITHIN THE RESCATTERING MODEL

The data shown in Fig. 12 can be qualitatively understood by estimating the momentum transfer in the rescattering model. From this one obtains kinematical boundaries of the momenta for different scenarios. For simplicity we restrict ourselves to a single return of the electron. If the electron recollides with an energy above the ionization threshold clearly double ionization is possible. The electron will lose the energy (and hence the momentum) necessary to overcome the binding of the second electron. The remaining excess energy can be freely distributed among the two electrons in the continuum. From electron impact ionization studies it is known that the electron energy distribution is asymmetric, i.e. one fast, one slow electron is most likely, for excess energies above 10 to 20 eV. The momentum vector of the electrons can point in all directions, but forward scattering of one electron is most likely (see ref. [100] for a review of electron impact ionization). After recollision the electrons are further accelerated in the field yielding a net momentum transfer at the end of the pulse, which is equal for both electrons and given by Eq. (1) (replace to by the rescattering time tl 2). For each recollision energy this leads to a classically allowed region of phase space, which is a circle centered on the diagonal in Fig. 12. An example is shown in Fig. 16.

26

[VII

R. D 6 r n e r et al. 4

3 2

5" 1 d

-2 -3 -4

-3

-2

-1 0 1 P z, e l ( a . u . ]

2

3

4

FIG. 16. Classically allowed region of phase space within the rescattering model for double ionization of argon by 4.7 x 1014W/cm2, 800 nm light. Each circle corresponds to a fixed recollision energy. Axis as in Fig. 12 (adapted from ref. [44]).

If the recollision energy is below the field-flee ionization threshold it is still possible that double ionization occurs. Any detailed scenario for this case without an explicit calculation is rather speculative since one deals with an excitation process in a very strong field environment for which no experiment exist so far. Already the levels of excited states are strongly modified compared to the field-free case. The same will certainly be true for the cross sections. We can however distinguish two extreme cases: An excited intermediate complex is formed, which either is quenched immediately by the field or may survive at least half a cycle of the field and will be quenched close to the next field maximum. The probability of such survival will depend on the field at the time of the return. For 3.8 or 4.7x1014 W/cm 2 the field at times corresponding to a return energy sufficient to reach the first excited states of an Ar l+ ion (at about 16-17 eV field flee) is so high that such a state would be above the barrier and hence would not be bound. A scenario which leads to the observed momentum of a 0.9-1 au for 3.8 and 4.7x 1014 W/cm 2 (data of figures 12, 13) is the following [17,43]: Electron 1 has a return energy of about 17 eV, which corresponds to the first exited states of the Ar l+ ion. Electron 1 is stopped, electron 2 is excited and immediately field ionized. Both electrons thus start with momentum zero at the time of the recollision. They are accelerated in the field and, hence, end up with the same m o m e n t u m of about 0.9-1 au after the pulse. This is in good agreement with the experimental observations for electrons emitted in the same hemisphere. It does not explain, however, a considerable number of events ejected into opposite hemispheres along the laser polarization. Feuerstein et al. [45] performed the same experiment in argon at a lower laser intensity of 2.5 x 1014 W/cm 2 (see Fig. 17). In this case sufficiently high return energies for excitation correspond to a return time close to the zero crossing

VII]

MULTIPLE IONIZATION IN STRONG LASER FIELDS I

I

-

I

_

Q_

-I

I

I 9

~

I

. , - m I I

I --O

. . . . I I

o

I I

I

I

I

I I

27

II n l i ~ l l l I l i - .

9= , m ,

m l i m i I

---

ii

/

emInm,

~

i

I I

I I I I

/

I I I l m ,

I

nImn

I

I I I

I

I W

I

I m I

I . . . . . .

_

I

I I I I

I. . . . . . I

-3

1

-2

I

-1

I . . . . .

I

I

0 1 pl I [a.u.]

I

2

I

3

FIG. 17. Correlated electron momentum spectrum of two electrons emitted from argon atoms ii

at 0.25x 1015 W/cm 2. plI is the electron momentum component along the light polarization axis of electron 1. Dashed line: kinematical constraints for recollision with excitation, assuming the excited state is not immediately quenched. Solid line: kinematical constraints for recollision with (e,2e) ionization (from ref. [45]).

of the field. Therefore one can expect that the excited state survives at least until the next field maximum. Feuerstein et al. estimated an expected region in phase space for excitation as shown in Fig. 17. For recollision events where the second electron is lifted into the continuum the allowed region of phase space is somewhat smaller than in Fig. 16 and confined to the two circles on the diagonal. Feuerstein et al. used this argument to separate events in which the recollision leads to an excited state and those which involve electron impact ionization. Supporting this notion of an intermediate excited complex Peterson and Bucksbaum [80] reported an enhanced production of low-energy electrons in the ATI electron spectrum of argon previously unobserved which can be interpreted in terms of inelastic excitation of Ar + or of multiple returns of the first electron. Electrons from excited states field ionized at the field maximum will be detected with very little momentum as they receive almost no drift velocity in the laser field. D. S-MATRIX CALCULATIONS The full diagram shown in Fig. 8 has not yet been evaluated to obtain the correlated electron momentum distribution. Goreslavskii and Popruzhenko succeeded, however, in calculating those distributions by making use of the saddlepoint approximation in the integration (see Fig. 18). The calculations shown in Fig. 18 are restricted to zero transverse momentum; similar distributions for

28

R. D 6 r n e r et al.

[VII

FIG. 18. Two-electron momentum distributions for double ionization of argon (similar to Fig. 12), calculated by evaluating diagram 8 in the saddle-point approximation at an intensity of 3.8 x 10TMW/cm2. Contrary to the experimentthe calculations are not integrated over all momentum components transverse to the field but restricted to electrons with no transverse momentum. The right-hand panel presents the same distribution with the classically forbidden region of phase space shown in white (compare Fig. 16) (adapted from ref. [54]). neon and argon integrated over all transverse momenta can be found in ref. [85]. These calculations do not include intermediate excited states but only the direct (e,2e) process. The calculations do not show a maximum on the diagonal as seen in the experiments. To the contrary, they favor the situation where one electron is slow and the other is fast. The authors of ref. [54] point out that this is a direct consequence of the sharing of the excess energy in the (e,2e) collision; the long-range Coulomb potential favors small momentum transfer in the collision. By replacing the Coulomb potential with a contact potential Goreslavskii and coworkers find a distribution which peaks on the diagonal, much like the experimental results. The main reason is that a contact potential does not emphasize small momentum transfers. It has to remain open at present how well justified such a modification of the interaction potential is. These calculations have been restricted to electrons with zero transverse momentum. The trend seen in these calculations is in agreement with the observation by Weckenbrock et al. [43] shown in Fig. 13a, where one electron was confined to small transverse momenta. The calculations do not include intermediate excited states but only direct electron impact ionization during rescattering. Therefore the theoretical results are not too surprising since electron impact ionization favors unequal energy sharing at the return energies dominating here. E. TIME-DEPENDENT CALCULATIONS Calculations by the Taylor group solving the time-dependent Schr6dinger equation in three dimensions predicted the emission of both electrons to the same side prior to the experimental observation [64]. Similar conclusions have been drawn from one-dimensional calculations [101]. In the low-field, shortwavelength regime the full calculations have proven to be able to predict

VII]

MULTIPLE IONIZATION IN STRONG LASER FIELDS

29

FIG. 19. Two-electron momentum distributions for double ionization of helium (similar to Fig. 12) calculated by solving the one-dimensional time-dependent Schr6dinger equation at the following intensities: (a) 1 x 1014 W/cm 2, (b) 3x 1014 W/cm 2, (c) 6.6x 1014 W/cm 2, (d) 10x 1014 W/cm 2, (e) 13x 1014 W/cm 2, (f) 20x1014 W/cm 2 (adapted from [34]).

electron-electron angular distributions and the energy sharing among the electrons as well as the total double ionization cross section [69,102]. In the strong field case at 800 nm, however, the calculations are extremely demanding. No electron-electron momentum space distributions have been reported up to now. The total double ionization rates however are in good agreement with the observations at 380 nm [66]. Several one-dimensional calculations have been performed at 800nm. All calculations show the majority of electrons emitted to the same side [34,103, 104]. From the calculated electron densities in coordinate space Lein and coworkers have obtained momentum distributions (Fig. 19). The enhanced emission probability in the first and third quadrants at intermediate in panels (d) and (e) is clearly visible. Different from the experiment, however, a strongly reduced probability is observed along the diagonal, which is most likely an artifact of the one-dimensional model. While in three dimensions electron

30

R. D 6 r n e r et al.

[IX

repulsion can lead to an opening angle between the electrons having the same momentum component in the polarization direction, this is impossible in one dimension. Here the electron repulsion necessarily leads to a node on the diagonal for electrons emitted at the same instant in the field. For 400nm radiation these calculations have also shown clear rings corresponding to ATI peaks in the sum energy of both electrons [ 105]. Analogous to ATI peaks in single ionization they are spaced by the photon energy. Similar rings have been seen also in three-dimensional calculations at shorter wavelength [65].

VIII. Outlook The application of COLTRIMS yielded the first differential data for double ionization in strong laser fields. Compared to the experimental situation in double ionization by single-photon absorption, however, the experiments are still in their infancy. So far correlated electron momenta have been measured only for argon and neon. Clearly experiments on helium are highly desirable since this is where theory is most tractable. Also, mainly the momentum component in the polarization direction has been investigated so far, resulting in a big step forward in the understanding of multiple ionization in strong laser fields. None of the experiments up to now has provided fully differential data since not all six momentum components of the two electrons were analyzed. Therefore, no coincident angular distributions as for single-photon absorption are available at this point (see ref. [88] for a theoretical prediction of these distributions). Most important for such future studies is a high resolution of the sum energy of the two electrons, which would allow to count the number of photons absorbed. From single-photon absorption it is known that angular distributions are prominently governed by selection rules resulting from angular momentum and parity, hence, from the even or oddness of the number of absorbed photons. Another important future direction is a study of the wavelength dependence of double ionization. The two cases of single and multiphoton absorption discussed here are only the two extremes. The region of two- and few-photon double ionization is experimentally completely unexplored. Experiments for two-photon double ionization of helium will become feasible in the near future at the VUV FEL facilities such as the TESLA Test facility in Hamburg.

IX. Acknowledgments The Frankfurt coauthors would like to thank H. Giessen, G. Urbasch, H. Roskos, T. L6ttter and M. Thomson for collaboration on some of the experiments described here and C. Freudenberger for preparation of many of the figures. The Heidelberg coauthors are indebted to the Max-Born-Institute in Berlin,

X]

MULTIPLE IONIZATION IN STRONG LASER FIELDS

31

providing the laser facilities for the experiments. Moreover, H. Rottke, C. Trump, M. Wittmann, G. Korn and W. Sandner made decisive contributions to the experimental setup, helping in the realization of the experiments during beamtimes and contributed strongly in the evaluation and interpretation of the data. We thank A. Becker, E Faisal and W. Becker for many helpful discussions and for educating us on S-matrix theory. We have also profited tremendously from discussions with K. Taylor, D. Dundas, M. Lein, V. Engel, J. Feagin, L. DiMauro and P. Corkum. This work is supported by DFG, BMBF, GSI. R.D. acknowledges supported by the Heisenberg-Programm of the DFG. R.M., B.E and J.U. acknowledge support by the Leibniz-Programm of the DFG. T.W. is grateful for financial support of the Graduiertenf'6rderung des Landes Hessen.

X. R e f e r e n c e s 1. G6ppert-Mayer, M. (1931). Ann. d. Phys. 9, 273. 2. Burnett, K., Reed, V., and Knight, R (1993). J. Phys. B 26, 561. 3. DiMauro, L., and Agostini, R (1995). "Advances in Atomic and Molecular Physics." Academic Press, New York. 4. Protopapas, M., Keitel, C., and Knight, R (1997). Phys. Rep. 60, 389. 5. Backus, S., Durfee, C., Murnane, M., and Kapteyn, H. (1998). Rev. Sci. Instrum. 69, 1207. 6. Brabec, T., and Krausz, E (2000). Rev. Mod. Phys. 72, 545. 7. Diels, J.-C., and Rudolph, W. (1995). "Ultrashort Laser Pulse Phenomena." Academic Press, New York. 8. Rulliere, C. (1998). "Femtosecond Laser Pulses. Principles and Experiments." Springer Verlag, New York. 9. D6rner, R., Mergel, V., Jagutzki, O., Spielberger, L., Ullrich, J., Moshammer, R., and SchmidtB6cking, H. (2000). Phys. Rep. 330, 96. 10. McGuire, J. (1997). "Electron Correlation Dynamics in Atomic Collisions." Cambridge University Press, Cambridge. 11. McGuire, J., Berrah, N., Bartlett, R., Samson, J., Tanis, J., Cocke, C., and Schlachter, A. (1995). J. Phys. B 28, 913. 12. Briggs, J., and Schmidt, V. (2000). J. Phys. 33, R1. 13. Wolf, S., and Helm, H. (1997). Phys. Rev. A 56, R4385. 14. van der Poel, M., Nielsen, C.V., Gearba, M.-A., and Andersen, N. (2001). Phys. Rev. Lett. 87, 123201. 15. Turkstra, J.W., Hoekstra, R., Knoop, S., Meyer, D., Morgenstern, R., and Olson, R.E. (2001). Phys. Rev. Lett. 87, 123202. 16. Flechard, X., Nguyen, H., Wells, E., Ben-Itzhak, I., and DePaola, B.D. (2001). Phys. Rev. Lett. 87, 123203. 17. Weber, T., Giessen, H., Weckenbrock, M., Staudte, A., Spielberger, L., Jagutzki, O., Mergel, V., Urbasch, G., Vollmer, M., and D6rner, R. (2000a). Nature 404, 608. 18. Moshammer, R., Unverzagt, M., Schmitt, W., Ullrich, J., and Schmidt-B6cking, H. (1996a). Nucl. Instr. Meth. B 108, 425. 19. Schyja, V., Lang, T., and Helm, H. (1998). Phys. Rev. 57, 3692. 20. Helm, H., Bjerre, N., Dyer, M., Huestis, D., and Saeed, M. (1993). Phys. Rev. Lett 70, 3221. 21. Hino, K., Ishihara, T., Shimizu, E, Toshima, N., and McGuire, J.H. (1993). Phys. Rev. A 48, 1271.

32

R. D 6 r n e r et al.

[X

22. Kheifets, A., Bray, I., Soejima, K., Danjo, A., Okuno, K., and Yagishita, A. (2001). J. Phys. B 34, L247. 23. Shingal, R., and Lin, C. (1991). J. Phys. B 24, 251. 24. Lambropoulos, P., Maragakis, E, and Zhang, J. (1998). Phys. Rep. 305, 203. 25. D6rner, R., Br~iuning, H., Feagin, J., Mergel, V., Jagutzki, O., Spielberger, L., Vogt, T., Khemliche, H., Prior, M., Ullrich, J., et al. (1998a). Phys. Rev. A 57, 1074. 26. Byron, E, and Joachain, C. (1967). Phys. Rev. 164, 1. 27. Spielberger, L., Jagutzki, O., D6rner, R., Ullrich, J., Meyer, U., Mergel, V., Unverzagt, M., Damrau, M., Vogt, T., Ali, I., et al. (1995). Phys. Rev. Lett. 74, 4615. 28. Spielberger, L., Br~iuning, H., Muthig, A., Tang, J., Wang, J., Qui, Y., D6rner, R., Jagutzki, O., Tschentscher, T., Honkim~iki, V., et al. (1999). Phys. Rev. 59, 371. 29. Samson, J. (1990). Phys. Rev. Lett. 65, 2863. 30. Kuchiev, M.Y. (1987). Soy. Phys.-JETP Lett. 45, 404. 31. Corkum, P. (1993). Phys. Rev. Lett. 71, 1994. 32. Schafer, K., Yang, B., DiMauro, L., and Kulander, K. (1993). Phys. Rev. Lett. 70, 1599. 33. Watson, J., Sanpera, A., Lappas, D., Knight, P., and Burnett, K. (1997). Phys. Rev. Lett. 78, 1884. 34. Lein, M., Gross, E., and Engel, V. (2000a). Phys. Rev. Lett. 85, 4707. 35. Lein, M., Gross, E., and Engel, V. (2001a). Opt. Express 8, 441. On-line: http:// www.opticsexpress.org/oearchive/source/30744.htm. 36. Fittinghoff, D., Bolton, P., Chang, B., and Kulander, K. (1994). Phys. Rev. A 49, 2174. 37. Dietrich, P., Burnett, N.H., Ivanov, M., and Corkum, P.B. (1994). Phys. Rev. A 50, R3585. 38. Witzel, B., Papadogiannis, N.A., and Charalambidis, D. (2000). Phys. Rev. Lett. 85, 2268. 39. Lafon, R., Chaloupka, J.L., Sheehy, B., Paul, P.M., Agostini, P., Kulander, K.C., and DiMauro, L.E (2001). Phys. Rev. Lett. 86, 2762. 40. Weber, T., Weckenbrock, M., Staudte, A., Spielberger, L., Jagutzki, O., Mergel, V., Urbasch, G., Vollmer, M., Giessen, H., and D6rner, R. (2000b). Phys. Rev. Lett. 84, 443. 41. Moshammer, R., Feuerstein, B., Schmitt, W., Dorn, A., Schr6ter, C., Ullrich, J., Rottke, H., Trump, C., Wittmann, M., Korn, G., et al. (2000). Phys. Rev. Lett. 84, 447. 42. Weber, T., Jagutzki, O., Hattass, M., Staudte, A., Nauert, A., Schmidt, L., Prior, M., Landers, A., Br/iuning-Demian, A., Br~iuning, H., et al. (2001a). J. Phys. B 34, 3669. 43. Weckenbrock, M., Hattass, M., Czasch, A., Jagutzki, O., Schmidt, L., Weber, T., Roskos, H., L6ffler, T., Thomson, M., and D6rner, R. (2001). J. Phys. B 34, L449. 44. Weckenbrock, M. (2001). Diploma Thesis. J.W. Goethe University, Frankfurt/Main. On-line: http ://hsbpc 1.ikf. physik, uni- frankfurt, de/publications/Diplom_Doktor.html. 45. Feuerstein, B., Moshammer, R., Fischer, D., Dorn, A., Schr6ter, C.D., Deipenwisch, J., LopezUrrutia, J., H6hr, C., Neumayer, P., Ullrich, J., et al. (2001). Phys. Rev. Lett. 87, 043003. 46. Weber, T., Weckenbrock, M., Staudte, A., Spielberger, L., Jagutzki, O., Mergel, V., Urbasch, G., Vollmer, M., Giessen, H., and D6rner, R. (2000). J. Phys. B 33, L127. 47. Larochelle, S., Talebpour, A., and Chin, S.L. (1998). J. Phys. B 31, 1201. 48. Delone, N., and Krainov, V. (1998). Phys. Usp. 41,469. 49. Feuerstein, B., Moshammer, R., and Ullrich, J. (2000). J. Phys. B 33, L823. 50. Becker, A., and Faisal, E (2000). Phys. Rev. Lett. 84, 3546. 51. Becker, A., and Faisal, E (1996). J. Phys. B 29, L 197. 52. Becker, A., and Faisal, E (1999a). Phys. Rev. A 59, R1742. 53. Becker, A., and Faisal, E (1999b). J. Phys. B 32, L335. 54. Goreslavskii, S., and Popruzhenko, S. (2001). Opt. Express 8, 395. On-line: http:// www.opticsexpress.org/oearchive/source/30694.htm. 55. Chen, J., Liu, J., Fu, L., and Zheng, W. (2000). Phys. Rev. 63, 011404R. 56. Sacha, K., and Eckhardt, B. (2001a). Phys. Rev. 63, 043414.

X]

M U L T I P L E I O N I Z A T I O N IN S T R O N G L A S E R F I E L D S

33

57. Bhardwaj, V.R., Aseyev, S.A., Mehendale, M., Yudin, G.L., Villeneuve, D.M., Rayner, D.M., Ivanov, M.Y., and Corkum, P.B. (2001). Phys. Rev. Lett. 86, 3522. 58. Kopold, R., Becker, W., Rottke, H., and Sandner, W. (2000). Phys. Rev. Lett. 85, 3781. 59. Goreslavskii, S., and Popruzhenko, S. (2001). J. Phys. B 34, L239. 60. Feagin, J. (1995). J. Phys. B 28, 1495. 61. Feagin, J. (1996). J. Phys. B 29, L551. 62. D6rner, R., Feagin, J., Cocke, C., Br/iuning, H., Jagutzki, O., Jung, M., Kanter, E., Khemliche, H., Kravis, S., Mergel, V., et al. (1996). Phys. Rev. Lett. 77, 1024. Erratum: (1997). Phys. Rev. Lett. 78, 2031. 63. Sacha, K., and Eckhardt, B. (2001b). Phys. Rev. 64, 053401. 64. Taylor, K., Parker, J., Dundas, D., Smyth, E., and Vitirito, S. (1999). Laser Physics 9, 98. 65. Parker, J.S., Moore, L.R., Meharg, K.J., Dundas, D., and Taylor, K.T. (2001). J. Phys. B 34, L69. 66. Parker, J.S., Moore, L.R., Dundas, D., and Taylor, K.T. (2000). J. Phys. B 33, L691. 67. Parker, J., Taylor, K., Clark, C., and Blodgett-Ford, S. (1996). J. Phys. B 29, L33. 68. Dundas, D., Taylor, K., Parker, J., and Smyth, E. (1999). J. Phys. B 32, L231. 69. Colgan, J., Pindzola, M.S., and Robicheaux, E (2001). J. Phys. 34, L457. 70. LaGattuta, K., and Cohen, J. (1998). J. Phys. B 31, 5281. 71. Feeler, C., and Olson, R. (2000). J. Phys. 33, 1997. 72. Cassimi, A., Duponchel, S., Flechard, X., Jardin, P., Sortais, P., Hennecart, D., and Olson, R. (1996). Phys. Rev. Lett. 76, 3679. 73. D/Srner, R., Ullrich, J., Schmidt-B6cking, H., and Olson, R. (1989). Phys. Rev. Lett. 63, 147. 74. D6rner, R., Mergel, V., Ali, R., Buck, U., Cocke, C., Froschauer, K., Jagutzki, O., Lencinas, S., Meyerhof, W., Nfittgens, S., et al. (1994). Phys. Rev. Lett. 72, 3166. 75. D6rner, R., Khemliche, H., Prior, M., Cocke, C., Gary, J., Olson, R., Mergel, V., Ullrich, J., and Schmidt-B6cking, H. (1996b). Phys. Rev. Lett. 77, 4520. 76. Frohne, V., Cheng, S., Ali, R., Raphaelian, M., Cocke, C., and Olson, R. (1993). Phys. Rev. Lett. 71,696. 77. Mergel, V., D6rner, R., Achier, M., Khayyat, K., Lencinas, S., Euler, J., Jagutzki, O., Nfittgens, S., Unverzagt, M., Spielberger, L., et al. (1997). Phys. Rev. Lett. 79, 387. 78. Moshammer, R., Ullrich, J., Unverzagt, M., Schmidt, W., Jardin, P., Olson, R., Mann, R., D6rner, R., Mergel, V., Buck, U., et al. (1994). Phys. Rev. Lett. 73, 3371. 79. Moshammer, R., Ullrich, J., Kollmus, H., Schmitt, W., Unverzagt, M., Jagutzki, O., Mergel, V., Schmidt-B6cking, H., Mann, R., Woods, C., et al. (1996b). Phys. Rev. Lett. 77, 1242. 80. Peterson, E., and Bucksbaum, P. (2001). Phys. Rev. 64, 053405. 81. Moshammer, R., Feuerstein, B., Fischer, D., Dorn, A., Schr6ter, C., Deipenwisch, J., LopezUrrutia, J., H6hr, C., Neumayer, P., Ullrich, J., et al. (2001). Opt. Express 8, 358. On-line: http://www.opticsexpress.org/oearchive/source/30944.htm. 82. Mergel, V., Achier, M., D6rner, R., Khayyat, K., Kambara, T., Awaya, Y., Zoran, V., Nystr6m, B., Spielberger, L., McGuire, J., et al. (1998). Phys. Rev. Lett. 80, 5301. 83. Achler, M., Mergel, V, Spielberger, L., R. D6rner, Y.A., and Schmidt-B6cking, H. (2001). J. Phys. B 34, L965. 84. Moshammer, R., Feuerstein, B., Urrutin, I.C.L., Dorn, A., Fischer, D., Schr6ter, C., Schmitt, W., Ullrich, J., Rottke, H., Trump, C., et al. (2002). Phys. Rev. A 65, 035401. 85. Goreslavskii, S., Popruzhenko, S., Kopold, R., and Becker, W. (2001). Phys. Rev. 64, 053402. 86. Weber, T., Weckenbrock, M., Staudte, A., Hattass, M., Spielberger, L., Jagutzki, O., Mergel, V., H. Schmidt-B6cking, G.U., Giessen, H., Br/iuning, H., et al. (2001b). Opt. Express 7(9), 368. On-line: http://www.opticsexpress.org/oearchive/source/30623.htm. 87. D6rner, R., Mergel, V., Br/iuning, H., Achier, M., Weber, T., Khayyat, K., Jagutzki, O., Spielberger, L., Ullrich, J., Moshammer, R., et al. (1998). In "Atomic processes in Plasmas" (E. Oks, M. Pindzola, Eds.). AlP Conf. Proc. 443, 137.

34

R. D 6 r n e r et al.

[X

88. Becker, A., and Faisal, E (1994). Phys. Rev. A 50, 3256. 89. Dorn, A., Moshammer, R., Schr6ter, C., Zouros, T., Schmitt, W., Kollmus, H., Mann, R., and Ullrich, J. (1999). Phys. Rev. Lett 82, 2496. 90. Dorn, A., Kheifets, A., Schr6ter, C.D., Najjari, B., H6hr, C., Moshammer, R., and Ullrich, J. (2001). Phys. Rev. Lett 86, 3755. 91. Taouil, I., Lahmam-Bennani, A., Duguet, A., Duguet, A., and Avaldi, L. (1998). Phys. Rev. Lett 81, 4600. 92. Bapat, B., Keller, S., Moshammer, R., Mann, R., and Ullrich, J. (2000). J. Phys. B 33, 1437. 93. Moshammer, R., Ullrich, J., Schmitt, W., Kollmus, H., Cassimi, A., D6rner, R., Dreizler, R., Jagutzki, O., Keller, S., Lfidde, H.-J., et al. (1997). Phys. Rev. Lett. 79, 3621. 94. von Weiz/icker, C. (1934). Z. Phys. 88, 612. 95. Williams, E. (1934). Phys. Rev. 45, 729. 96. Keller, S., Lfidde, H., and Dreizler, R. (1997). Phys. Rev. A 55, 4215. 97. Voitkiv, A., and Ullrich, J. (2001). J. Phys 43, 1673. 98. Keller, S. (2000). J. Phys. B 33, L513. 99. Unverzagt, M., Moshammer, R., Schmitt, W., Olson, R., Jardin, R, Mergel, V., Ullrich, J., and Schmidt-B6cking, H. (1996). Phys. Rec. Lett. 76, 1043. 100. Coplan, M.A., et al. (1994). Rev. Mod. Phys. 66, 985, and references therein. 101. Lein, M., Gross, E., and Engel, V. (2000b). J. Phys. B 33, 433. 102. Pindzola, M., and Robicheaux, E (1998). Phys. Rec. A 57, 318. 103. Haan, S.L., Hoekema, N., Poniatowski, S., Liu, W.-C., and Eberly, J.H. (2000). Opt. Express 7, 29. On-line: http://www.opticsexpress.org/oearchive/21863.htm. 104. Muller, H. (2001). Opt. Express 8, 417. On-line: http'//www.opticsexpress.org/oearchive/source/ 30932.htm. 105. Lein, M., Gross, E., and Engel, V. (2001b). Phys. Rev. A 64, 023406.

A D V A N C E S IN ATOMIC, M O L E C U L A R , A N D O P T I C A L PHYSICS, VOL. 48

A B 0 VE- THRESHOLD IONIZATION: FROM CLA SSICA L FEATURES TO QUA N T UM EFFE C TS W. BECKER l, E GRASBON 2, R. KOPOLD 1, D.B. MILOSEVIC 3, G. G. PAUL US 2 and H. WAL THER 2,4 1Max-Born-Institut, Max-Born-Str. 2A, 12489 Berlin, Germany," 2Max-Planck-Institut fffr Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany; 3Faculty of Science, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Hercegovina; 4Ludwig-Maximilians-Universitdt Miinchen, Germany I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Theoretical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Direct Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Classical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Quantum-mechanical Description of Direct Electrons . . . . . . . . . . . . . . . . . C. Interferences of Direct Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Rescattering: The Classical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Rescattering: Quantum-mechanical Description . . . . . . . . . . . . . . . . . . . . . . . . A. Saddle-point methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Connection with Feynman's path integral . . . . . . . . . . . . . . . . . . . . . . . . . .

V.

VI.

VII.

VIII. IX.

C. Connection with closed-orbit theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. The role of the binding potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. A homogeneous integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E Quantum orbits for linear polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Enhancements in ATI spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Quantum orbits for elliptical polarization . . . . . . . . . . . . . . . . . . . . . . . . . I. Interference between direct and rescattered electrons . . . . . . . . . . . . . . . . . . ATI in the Relativistic Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Basic Relativistic Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Rescattering in the Relativistic Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Orbits in High-order Harmonic Generation . . . . . . . . . . . . . . . . . . . . A. The Lewenstein Model of High-order Harmonic Generation . . . . . . . . . . . . . B. Elliptically Polarized Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. H H G by a Two-color Bicircular Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. H H G in the Relativistic Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of ATI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Characterization of High Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The "Absolute Phase" of Few-cycle Laser Pulses . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

36 38 39 40 40 44 47 5O 53 55 57 58 59 60 61 63 68 71 73 73 75 76 77 78 78 84 86 87 90 92 92

Copyright 9 2002 Elsevier Science (USA) All rights reserved ISBN 0-12-003848-X/ISSN 1049-250X/02 $35.00

36

W. Becker et al.

[I

I. I n t r o d u c t i o n With the discovery of above-threshold ionization (ATI) by Agostini et al. (1979) intense-laser atom physics entered the nonperturbative regime. These experiments recorded the photoelectron kinetic-energy spectra generated by laser irradiation of atoms. Earlier experiments had measured total ionization rates by way of counting ions, and the data were well described by lowest-order perturbation theory (LOPT) with respect to the electron-field interaction. This LOPT regime was already highly nonlinear (see, e.g., Mainfray and Manus, 1991), the lowest order being the minimal number N of photons necessary for ionization. An ATI spectrum consists of a series of peaks separated by the photon energy, see Fig. 1. They reveal that an atom may absorb many more photons than the minimum number N, which corresponds to LOPT. In the 1980s, the photon spectra emitted by laser-irradiated gaseous media were investigated at comparable laser intensities and were found to exhibit peaks at odd harmonics of the laser frequency (McPherson et al., 1987; Wildenauer, 1987). The spectra of this high-order harmonic generation (HHG) display a plateau (Ferray et al., 1988), i.e., the initial decrease of the harmonic yield with increasing harmonic order is followed by a flat region where the harmonic intensity is more or less independent of its order. This plateau region terminates at some well-defined order, the so-called cutoff.

FIG. 1. Photoelectron spectrumin the above-threshold-ionization(ATI) intensityregime. The series of peaks corresponds to the absorption of photons in excess of the minimumrequired for ionization. The figure shows the result of a numerical solution of the SchrSdinger equation (Paulus, 1996).

I]

ATI: CLASSICAL TO QUANTUM

37

A simple semiclassical model of HHG was furnished by Kulander et al. (1993) and by Corkum (1993): At some time, an electron enters the continuum due to ionization. Thereafter, the laser's linearly polarized electric field accelerates the electron away from the atom. However, when the field changes direction, then, depending on the initial time of ionization, it may drive the electron back to its parent ion, where it may recombine into the ground state, emitting its entire energy - the sum of the kinetic energy that it acquired along its orbit plus the binding energy- in the form of one single photon. This simple model beautifully explains the cutoff energy of the plateau, as well as the fact that the yield of HHG strongly decreases when the laser field is elliptically polarized. In this event, the electron misses the ion. This model is often referred to as the simple-man model. The model suggests (Corkum, 1993) that the electron, when it recollides with the ion, may very well scatter off it, either elastically or inelastically. Elastic scattering should contribute to ATI. Indeed, the corresponding characteristic features in the angular distributions were observed by Yang et al. (1993), and an extended plateau in the energy spectra due to this mechanism, much like the plateau of HHG, was identified by Paulus et al. (1994c). Under the same conditions, a surprisingly large yield of doubly charged ions was recorded (l'Huillier et al., 1983; Fittinghoff et al., 1992) that was incompatible with a sequential ionization process. A potential mechanism causing this nonsequential ionization (NSDI) is inelastic scattering. It was only recently, however, that this inelastic-scattering scenario emerged as the dominant mechanism of NSDI, through analysis of measurements of the momentum distribution of the doubly charged ions (Weber et al., 2000a,b; Moshammer et al., 2000). The semiclassical rescattering model sketched above has proved invaluable in providing intuitive understanding and predictive power. It was embedded in fully quantum-mechanical descriptions of HHG (Lewenstein et al., 1994; Becker et al., 1994b) and ATI (Becker et al. 1994a; Lewenstein et al., 1995a). This work has led to the concept of "quantum orbits," a fully quantum-mechanical generalization of the classical orbits of the simple-man model that retains the intuitive appeal of the former, but allows for interference and incorporates quantum-mechanical tunneling. The quantum orbits arise naturally in the context of Feynman's path integral (Sali6res et al., 2001). This review will concentrate on ATI and the various formulations of the rescattering model, from the simplest classical model to the quantum orbits for elliptical polarization. Alongside with theory, we will provide a review of the experimental status of ATI. We also give a brief survey of recent applications of ATI. High-order harmonic generation is considered only insofar as it provides further illustrations of the concept and application of quantum orbits. We do not deal with the important collective aspects of HHG, and no attempt is made to represent the vast literature on HHG. For this purpose, we refer to the recent reviews by Sali~res et al. (1999) and Brabec and Krausz (2000). Earlier

38

W. Becker et al.

[I

reviews pertinent to ATI have been given by Eberly et al. (1991), Mainfray and Manus (1991), DiMauro and Agostini (1995), and Protopapas et al. (1997a). The entire field of laser-atom physics has been succinctly surveyed by Kulander and Lewenstein (1996) and, recently, by Joachain et al. (2000). Both of these reviews concentrate on the theory. Nonsequential double ionization is well covered in a recent focus issue of Optics Express, Vol. 8.

A. EXPERIMENTAL METHODS

ATI is observed in the intensity regime 1012W/cm 2 to 1016W/cm 2. At such intensities, atoms may ionize so quickly that complete ionization has taken place before the laser pulse has reached its maximum. This calls, on the one hand, for atoms with high ionization potential (i.e. the rare gases) and, on the other, for ultrashort laser pulses. Owing to the rapid progress in femtosecond laser technology, in particular since the invention of titaniumsapphire (Ti:Sa) femtosecond lasers (Spence et al., 1991), generation of laser fields with strengths comparable to inner atomic fields has become routine. The prerequisite of detailed investigations of ATI, however, has been the development of femtosecond laser systems with high repetition rate. Owing to the latter, the detection of faint but qualitatively important features of ATI spectra with low statistical noise has become possible. This holds, in particular, if multiply differential ATI spectra are to be studied, such as angle-resolved energy spectra, or spectra that are very weak, such as for elliptical polarization or outside the classically allowed regions. State-of-the-art pulses are as short as 5 fs (Nisoli et al., 1997) and repetition rates reach 100 kHz (Lindner et al., 2001). The most widespread method of analyzing ATI electrons is time-of-flight spectroscopy. When the laser pulse creates a photoelectron, it simultaneously triggers a high-resolution clock. The electrons drift in a field-free flight tube of known length towards an electron detector, which then gives the respective stop pulses to the clock. Now, their kinetic energy can easily be calculated from their time of flight. This approach has by far the highest energy resolution and is comparatively simple. However, the higher the laser repetition rate, the more demanding becomes the data aquisition. Other approaches include photoelectron imaging spectroscopy (Bordas et al., 1996), which is able to record angle-resolved ATI spectra, and so-called coldtarget recoil-ion-momentum spectroscopy (COLTRIMS) technology (D6rner et al., 2000), which is capable of providing complete kinematic determination of the fragments of photoionization, i.e. the electrons and ions. It requires, however, conditions such that no more than one atom is ionized per laser shot. Therefore, it can take particular advantage of high laser repetition rates. The disadvantage of COLTRIMS is the poor energy resolution for the electrons and the extracting technology.

I]

ATI: CLASSICAL TO QUANTUM

~

~ ~LA

39

b) Calculation, 66. I

.~.

.L

,,,,l,,,,l,,,,iiiiii-,,,,l,,,,

I0

i,,,,I,,,,I,,,,

20 KE (eV) :30

40

FIG. 2. (a) Measured and (b) calculated photoelectron spectrum in argon for 800 nm, 120 fs pulses at the intensities given in TW/cm 2 in the figure (10Up = 39eV). From Nandor et al. (1999).

B. THEORETICAL METHODS

The single-active-electron approximation (SAE) replaces the atom in the laser field by a single electron that interacts with the laser field and is bound by an effective potential so optimized as to reproduce the ground state and singly excited states. Up to now, in single ionization no qualitative effect has been identified that would reveal electron-electron correlation. The SAE has found its most impressive support in the comparison of experimental ATI spectra in argon with spectra calculated by numerical solution of the three-dimensional time-dependent Schr6dinger equation (TDSE) (Nandor et al., 1999); see Fig. 2. The agreement between theory and experiment is equally remarkable as it has been achieved for low-order ATI in hydrogen; cf. D6rr et al. (1990) for the Sturmian-Floquet calculation and Rottke et al. (1990) for the experiment. For helium, a comparison of total ionization rates with and without the SAE in the above-barrier regime has lent further support to the SAE (Scrinzi et al., 1999). Numerical solution of the one-particle TDSE in one dimension was instrumental for the understanding of ATI in its early days; for a review, see Eberly et al. (1992). For the various methods of solving the TDSE in more than one dimension we refer to Joachain et al. (2000). Comparatively few papers have dealt with high-order ATI in three (that is, in effect, two) dimensions. This is particularly challenging since the emission of plateau electrons is caused by very small changes in the wave function, and the large excursion amplitudes of free-electron motion in high-intensity low-frequency fields necessitate a large spatial grid. This is exacerbated for energies above the cutoff and for elliptical polarization. Expansion of the radial wave function in terms of a set of B-spline functions was used by Paulus (1996), by Cormier and Lambropoulos (1997), and by Lambropoulos et al. (1998). Matrix-iterative methods were employed by

40

[II

W. Becker et al.

Nurhuda and Faisal (1999). The most detailed calculations have been carried out by Nandor et al. (1999) and by Muller (1999a,b, 2001a,b). The techniques are detailed by Muller (1999c). To our knowledge, no results for high-order ATI for elliptical polarization based on numerical solution of the TDSE have been published to this day. Recently, numerical solution of the TDSE for a two-dimensional model atom by means of the split-operator method has been widely used in order to investigate various problems such as elliptical polarization (Protopapas et al., 1997b), stabilization (Patel et al., 1998; Kylstra et al., 2000), magnetic-drift effects (V~izquez de Aldana and Roso, 1999; V~izquez de Aldana et al., 2001) and various low-order relativistic effects (Hu and Keitel, 2001). Efforts to deal with the two-electron TDSE and, in particular, to compute double-electron ATI spectra are under way (Smyth et al., 1998; Parker et al., 2001; Muller, 2001c). In one dimension for each electron, such spectra have been obtained by Lein et al. (2001). An approach that is almost complementary to the solution of the TDSE starts from the analytic solution for a free electron in a plane-wave laser field, the socalled Volkov solution (Volkov, 1935), which is available for the Schr6dinger equation as well as for relativistic wave equations, and considers the binding potential as a perturbation. The stronger the laser field, the lower its frequency, and the longer the pulse becomes, the more demanding is the solution of the TDSE, and the more the Volkov-based methods play out their strengths. This review concentrates on methods of the latter variety.

II. Direct Ionization A. THE CLASSICAL MODEL

The classical model of strong-field effects divides the ionization process into several steps (van Linden van den Heuvell and Muller, 1988; Kulander et al., 1993; Corkum, 1993; Paulus et al., 1994a, 1995). In a first step, an electron enters the continuum at some time to. If this is caused by tunneling (Chin et al., 1985; Yergeau et al., 1987; Walsh et al., 1994), the corresponding rate is a highly nonlinear function of the laser electric field ,~(t0). For example, the quasistatic Ammosov-Delone-Krainov (ADK) tunneling rate (Perelomov et al., 1966a,b; Ammosov et al., 1986) is given by (in atomic units) 2n*-Iml-I r'(t) = A E w

[~7(t)l

exp ' 3[E(t) I

(1)

where ,~(t) is the instantaneous electric field, EIp > 0 is the ionization potential of the atom, n* = Z~ 2x/2E~p is the effective principal quantum number, Z is the charge

II]

ATI: CLASSICAL TO QUANTUM

41

of the nucleus, and m is the projection of the angular momentum on the direction of the laser polarization. The constant A depends on the actual and the effective quantum numbers. The rate F(t) was derived on the assumption that the laser frequency is low, excited states play no role, and the Keldysh parameter y = v/Exp/2Up is small compared with unity [Up is the ponderomotive potential of the laser field; see Eq. (3) below]. Instantaneous rates that hold for arbitrary values of the Keldysh parameter have been presented by Yudin and Ivanov (2001b). For the discussion below, the important feature of the instantaneous ionization rate F(t) is that it develops a sharp maximum at times when the field ,~(t) reaches a maximum. The classical model considers the orbits of electrons that are released into the laser-field environment at some time to. The contribution of such an orbit will be weighted according to the value of the rate F(t0). Classically, an electron born by tunneling will start its orbit with a velocity of zero at the classical "exit of the tunnel" at r ~ Eip/[eF~ l, which, for strong fields, is a few atomic units away from the position of the ion. We will, usually, ignore this small offset and have the electronic orbit start at x(t = to) = 0 (the position of the ion) with x(t = to) = 0. If, after the ionization process, the interaction of the electron with the ion is negligible, we speak of a "direct" electron, in contrast to the case, to be considered below in Sects. III and II.B, where the electron is driven back to the ion and rescatters. An unambiguous distinction between direct and rescattered electrons, in particular for low energy, is possible only in theoretical models. A.1. Basic kinematics

The second step of the classical model is the evolution of the electron trajectory in the strong laser field. During this step, the influence of the atomic potential is neglected. For an intense laser field, the electron's oscillation amplitude is much larger than the atomic diameter, and so this is well justified. For a vector potential A(t) that is chosen so that its cycle average (A(t))r is zero, the electron's velocity is mv(t) = e(A(t0) - A(t)) -- p - eA(t),

(2)

where e = - l e l is the electron's charge. The velocity consists of a constant term p - eA(t0), which is the drift momentum measured at the detector, and a term that oscillates in phase with the vector potential A(t). The kinetic energy of this electron, averaged over a cycle T of the laser field, is p2 e2 E = - 2 m (v(t)Z)v = ~mm + ~mm (A(t)2)T -- Edrifl + Up.

(3)

The ponderomotive energy e2 Up = z-- (A(t)z)v, zm-

(4)

W. Becker et al.

42

[II

viz. the cycle-averaged kinetic energy of the electron's wiggling motion, is frequently employed to characterize the laser intensity. A useful formula is

Up [eV] = 0.09337I[W/cm 2]/~2[m] for a laser with intensity I and wavelength/l. If the electron is to have a nonzero velocity v0 at time to, one has to replace eA(t0) by eA(t0)+ mv0 = p in the velocity (2). Most of the time, we will be concerned with the monochromatic elliptically polarized laser field (-1 ~< ~ t l , the electron's velocity is again given by Eq. (2), but with Px = e [ 2 A ( t m ) - A(t0)] so that Eret, m a x =

e2

Ebs - ~m [2A(tl) - A(t0)] 2.

(26)

Maximizing Ebs under the same condition as above yields Ebs, max = 10.007Up (Paulus et al., 1994a) for tot0 = 105 ~ and totl = 352 ~ These values are very close to those that afford the maximal return energy. It is important to keep in mind that for maximal return energy or backscattering energy, the electron has to start its orbit shortly after a maximum of the electric field strength. As a consequence, it returns or rescatters near a zero of the field, see Fig. 7. This also provides an intuitive explanation of the energy gain through backscattering: if the electron returns near a zero of the field and backscatters by 180 ~ then it will be accelerated by another half-cycle of the field. In general, the equation X(tl) = 0 for fixed to may have any number of solutions. This becomes evident from the graphical solution presented in Fig. 7. If the electron starts at a time to just past an extremum of the field, it returns to the ion many times. These solutions having long "travel times" t l - to are very important for the intensity-dependent quantum-mechanical enhancements of the ATI plateau to be discussed in Sect. IV.G. Here we will be satisfied with mentioning another property of the classical orbits" obviously, the return energy will have extrema, e.g. the maximum of Ebs, max -- 10.007Up mentioned above, which is assumed for a certain time t0,max (t0,max = 108 ~ in the example). If we are interested in a fixed energy Ebs < Ebs, max, there are two start times that will lead to this energy: one earlier than t0, max, the other one later. From the graphical construction of Fig. 7 it is easy to see that the former has a longer travel time than the latter. In the closely related case of HHG, these correspond to the "long" and the "short" orbit (Lewenstein et al., 1995b). The cutoffs of the solutions with longer and longer travel times are depicted in Fig. 8. If we consider rescattering into an arbitrary angle 0 with respect to the direction of the linearly polarized laser field, we expect a lower maximal energy since part of the maximal energy 3.17Up of the returning electron will go into the

52

[III

W. Becker et al. [

'

I

'

[

'

[

--

....-....... 9

.

9

~149 ~

t~ t',

.

"" "'............. .

.

3:/4

......'"

.

. 3:/2

"'" . 33:/4

23:

FIG. 7. Graphical solution of the return time t 1 for given start time to; cf. Paulus et al. (1995): The return condition X(tl) = 0 can be written in the form F ( t l ) - F(to) + (tl - to)Fl(to), where the function F(t) = f t dr A ( r ) ~ sin ~ot (solid curve) is an integral of the vector potential A ( r ) ~ cos w r (dotted curve) 9 The thick solid straight line, which is the tangent to F ( r ) at r = to, intersects F ( r ) for the first time at r = tl. The start (ionization) time to was chosen such that the kinetic energy Ere t (Eq. 25) at the return time tl is maximal and equal to Eret,ma x = 3.17Up. The two adjacent straight lines both yield the same kinetic energy Ere t < Eret,ma x. The figure shows that one starts earlier and returns later while the other one starts later and returns earlier. Obviously, there can be many more intersections with larger values of ti provided the start times are near the extrema of F ( r ) . They correspond to the orbits with longer travel times.

transverse motion. This implies that, for fixed energy E b s , there is a cutoff in the angular distribution; in other words, rescattering events will only be recorded for angles such that 0 ~< 0 ~< 0max(Ebs). This is a manifestation of rainbow scattering (Lewenstein et al., 1995a). All of this kinematics is contained in the following equations (Paulus et al., 1994a): Ebs = ~1 [A(t0)2 + 2A(tl) [A(tl) - A(t0)] (1 + cos 00)], cot 0 = cot 00 -

A(tl) sin 00 IA(t0) - A(h)["

(27) (28)

Here 00 is the scattering angle at the instant of rescattering, which may have any value between 0 and Jr, as opposed to the observed scattering angle 0 at the detector (outside the field). In Eq. (27), the upper (lower) sign holds for A(to) > A(h) (A(to) < A(tl)). Pronounced lobes in the angular distributions off the polarization direction were first observed by Yang et al. (1993), while the rescattering plateau in the energy spectrum with its cutoff at 10Up was identified by Paulus et al. (1994b,c).

IV]

ATI: C L A S S I C A L TO Q U A N T U M

53

Fie. 8. Maximum drift energy after rescattering (ATI plateau cutoff) upon the mth return to the ion core during the ionization process. Electrons with the shortest orbits (m = 1) can acquire the highest energy, whereas electrons that pass the ion core once before rescattering at the second return (m = 2) have a rather low energy. Each return corresponds to two quantum orbits: the mth return corresponds to the quantum orbits 2m + 1 and 2m + 2. These spectra prominently display the classical cutoffs at 0max and Ebs,max. The classical features become the better developed the higher the intensity is. Hence, they are particularly conspicuous in the strong-field tunneling limit. This has been shown theoretically by comparison with numerical solutions of the Schr6dinger equation (Paulus et al., 1995) and experimentally for He at intensities around 1015 W/cm 2. Indeed, the latter spectra show an extended plateau for energies between 2Up and 10Up (Walker et al., 1996; Sheehy et al., 1998). For comparatively low intensities, angular distributions have been recorded in xenon with very high precision by Nandor et al. (1998). They also show the effects just discussed, but with much additional structure that appears to be attributable to quantum-mechanical interference and to multiphoton resonance with ponderomotively upshifted Rydberg states (Freeman resonances; Freeman et al., 1987).

IV. Rescattering: Quantum-mechanical Description In order to incorporate the possibility of rescattering into the quantummechanical description, we have to allow the freed electron once again to interact with the ion (Lohr et al., 1997). To this end, we return to the exact equation (16) and insert the Dyson integral equation (15). This yields two terms. Next, as we did in Sect. II.B, we replace the exact scattering state ]~pp) by a Volkov state and

54

W. Becker et al.

[IV

the exact time-evolution operator U by the Volkov-time evolution operator UU. In other words, we disregard the interaction with the binding potential V(r), except for the one single interaction that is explicit in the Dyson equation. This procedure corresponds to adopting the Born approximation for the rescattering process. Of the two terms, the first is identical with the "direct" amplitude (17) or (19). The second describes rescattering. Via integration by parts similar to that explained in Eq. (18) the two terms can be combined into one, Mp = - i

dtl

dto (lp;Vv)(tl)IVUf(tl,to)V I ~P0(t0)),

(29)

O 0, the condition (34) of "energy conservation" at the time of ionization cannot be satisfied for any real time to. As a consequence, all solutions (tls, tos, ks) become complex. If the ionization potential EIp is zero, then, for a linearly polarized field, the first saddle-point equation (34) implies that the electron starts on its orbit with a speed of zero. Provided the final momentum p is classically accessible, the resulting solutions are entirely real. They correspond to the so-called "simple-man model" (van Linden van den Heuvell and Muller, 1988; Kulander et al., 1993; Corkum, 1993). For EIp r 0, so long as the Keldysh parameter 72= Eip/(2Up) is small compared with unity, the imaginary parts of the solutions of Eqs. (34)-(36) are still not too large, and the real parts are still close to these simple-man solutions. In this case, approximate analytical solutions to the saddle-point equations can be written down, which yield an analytical approximation to the amplitude (31) (Goreslavskii and Popruzhenko, 2000). On the other hand, for elliptical polarization, the solutions are always complex, even when EIp = 0. This reflects the fact that, for any polarization other than linear, an electron set free at any time during the optical cycle with velocity zero will never return to the point where it was released. Equation (34) then only implies that k - eA(t0) is a complex null vector.

IV]

ATI: CLASSICAL TO QUANTUM

57

With the solutions (tls, to,, ks) (s = 1,2 .... ) of Eqs. (34)-(36), the sth quantum orbit has the form rex(t) =

(t - tos)ks - fttos dr eA(r) (Re tos ,,

/ -20

/

/

-40

,/

/

i "

-~"-.-\--~\ ~ "

'\ 3

~

]2.-,,,

""4

1

,/

,/

t"

..................................... -20 0 20 x [a.u.]

40

FIG. 20. Real parts of the quantum orbits for the same parameters as in Fig. 18 and for the harmonic fl = 43r Five orbits are shown that correspond to the saddle-point solutions 2, 3, 4, 5 and 8 in Fig. 19. The direction of the electron's travel is given by the arrows. In each case, the electron is "born" a few atomic units away from the position of the ion (at the origin), where its orbit almost exactly terminates. The dominant contribution to the 43rd harmonic intensity comes from the shortest orbit number 2, whose shape closely resembles the orbit in the case of a linearly polarized monochromatic field. From Milo~evi~ et al. (2000).

emission decreases with increasing absolute value of the imaginary part of the recombination time tl. The possible cutoff of the harmonic spectrum can be defined as the value of the harmonic order after which I Im tll becomes larger than (say) 0.01T. The probability of HHG is maximal when I Im tll is minimal. For each solution in Fig. 19a, these points are marked by asterisks and by the corresponding harmonic order. As a consequence of wave-function spreading, the emission rate decreases with increasing travel time ti - t o . This gives an additional reason why the contribution of solution 2 is dominant in the plateau region. Let us now consider the quantum orbits. In Fig. 20 for the fixed harmonic = 43~o, we present the five orbits that correspond to saddle-point solutions 2, 3, 4, 5 and 8 in Fig. 19. The dominant contribution comes from the shortest orbit 2 (thick line). It starts at the point (4.06, 0.66) by setting off in the negative y-direction, but soon turns until it travels at an angle of 68 ~ to the negative y-axis. Thereafter, it is essentially linear, as would be the case for a linearly polarized field. This behavior can be understood by inspection of the driving bicircular field depicted in the inset of Fig. 18, where the start time and the recombination time of the orbit are marked. During the entire length of the orbit, the field exerts a force in the positive y-direction. The effect of this force is canceled by the electron's initial velocity in the negative y-direction. The force in the x-direction

VI]

ATI: C L A S S I C A L TO Q U A N T U M

83

0 I

,

0

,

0'.2

i

0.4

,

,

0'.6

time [optical cycle]

i

0.8

,

1

FIG. 21. Parametric polar plot of the electric-field vector of a group of harmonics during one cycle of the bicircular field (61) on an arbitrary isotropic scale. The position of the origin is indicated in the upper and the left margin. The parameters are 11 = 12 = 9.36• 1014 W/cm 2, h~o = 1.6eV, and E[p = 24.6eV. The plot displays two traces: The circular trace is generated by the ten harmonics = (3n + 1)co with n = 10. . . . . 19, all having positive helicity. The starlike trace is generated by all harmonics ~ = (3n + 1)co between the orders 31 and 59, regardless of their helicity. The curve at the bottom represents the x-component of the field of the latter group over one cycle, the time scale being given on the horizontal axis. It shows that the field is strongly chirped. The black blob at the center is due to the fact that the field is near zero throughout most of the cycle, cf. the trace of the x-component. From Milo~evi6 and Becker (2000).

is m u c h like that in the case o f a linearly p o l a r i z e d driving field. Since H H G by a linearly p o l a r i z e d field is m o s t efficient, this m a k e s plausible the high efficiency o f H H G by the bicircular field. The orbit that c o r r e s p o n d s to solution 3 has a shape similar to that o f orbit 2, but is m u c h longer. The c o r r e s p o n d i n g travel time is longer, too, and, consequently, the c o n t r i b u t i o n o f s o l u t i o n 3 to the e m i s s i o n rate o f the 43rd h a r m o n i c is smaller. The other orbits are still l o n g e r and m o r e c o m p l i c a t e d so that their c o n t r i b u t i o n is negligible. The electric field o f a g r o u p o f p l a t e a u h a r m o n i c s is displayed in Fig. 21. It shows interesting behavior, w h i c h again reflects the t h r e e f o l d s y m m e t r y o f the field (61), see the inset o f Fig. 18. If the g r o u p o f h a r m o n i c s includes h a r m o n i c s o f either parity, then the field consists o f a s e q u e n c e o f essentially linearly polarized, strongly c h i r p e d a t t o s e c o n d pulses, each rotated by 120 ~ with r e s p e c t to the previous one. If, on the other hand, one were able to select h a r m o n i c s o f definite helicity, i. e. either ~2 = (3n + 1)co or ~ = ( 3 n - 1)co, then one w o u l d

84

[VI

W. B e c k e r et al.

obtain a sequence of attosecond pulses with approximately circular polarization. Both cases are illustrated in Fig. 21.

D. HHG IN THE RELATIVISTIC REGIME Quantum orbits can also be employed in the relativistic regime starting from the Klein-Gordon equation (55). Milo~evi6 et al. (2001c, 2002) found that the relativistic harmonic-emission matrix element has a form similar to that in Eq. (57), but with the relativistic action (h = c = 1) Sfa(tl, to, k) =

L Cx~du (EIp -

m - f2) -

L tl du ek(u) + ftOcxDdu (EIp -

where

k + ~A(u) Ek-~.k

ek(u) = Ek + eA(u) 9

m),

(62)

(63)

and Ek = ( k 2 + m2) 1/2, u ( t - z)/co. Solving the classical Hamilton-Jacobi equation for Hamilton's principal function it can be shown that ek(u) is the classical relativistic electron energy in the laser field. In the relativistic case, the function m E ( t l , t o , k ) in Eq. (57) consists of two parts: the dominant part is responsible for the emission of odd harmonics ~ = (2n + 1)~, while the other one originates from the intensity-dependent drift momentum of the electron in the field and allows for emission of even harmonics ~ = 2n~. Similarly to the nonrelativistic case, the integral over the intermediate electron momentum k can be calculated by the saddle-point method. The stationarity condition ftto' du Oek(u)/Ok = 0, with Oek/Ok = d r / d t , implies r(t0) = r(tl), so that the stationary relativistic electron orbit is such that the electron starts from and returns to the nucleus. As above, the start time and, to a lesser degree, the recombination time are complex. In the relativistic case, the stationary momentum k = ks is introduced in the following way. For fixed to and tl, its component ks• perpendicular to the photon's direction of propagation f~ is given by -

-

tl

(tl - to) k s • =

Introducing .A/I 2 = e 2 fttoI d r / A 2 ( u ) / ( t l

k 2 = ks2

L

- to)-

du cA(u).

ks2

(64)

> O, one h a s

(.A/j2 _ ks2_L)2 + 4 ( m 2 + .A/j2) ,

(65)

VI]

ATI: C L A S S I C A L TO Q U A N T U M

85

-20 ~0 r

~

=

nonrelativistic

-40

_ _Z [a'u'] ~-

tO ffl ffl

E o tO

-100

-60

~

relativistic

l

-80

E

c.-

0

750 _

.m

I._

-50

-100

o TO}

0

-120

0

50000

100000 harmonic order

150000

Fie. 22. Harmonic-emission rate as a function of the harmonic order for ultrahigh-order harmonic generation by an Ar8+ ion (Eip = 422eV) in the presence of an 800-nm Ti:Sa laser having the intensity 1.5• 1018 W/cm 2. Both the nonrelativistic and the relativistic results are shown. The corresponding relativistic electron orbit with the shortest travel time that is responsible for the emission of the harmonic ~ -- 100000o9 is shown in the inset. The arrows indicate which way the electron travels the orbit. The laser field is linearly polarized in the x-direction and the v • B electron drift is in the z-direction. From Milo~evi6 et al. (2002). which yields ek~ as a function of to and tl. The two stationarity equations connected with the integrals over to and tl are eks (to) -

m - EIp,

~2 - t~k~(tl) + EIp - m.

(66) (67)

As in the nonrelativistic case, they express energy conservation at the time of tunneling to and at the time of recombination t~, respectively. The final expression for the relativistic harmonic-emission matrix element has the form (37) with (62), where the summation is now over the appropriate subset o f the relativistic saddle points ( 6 s , tos, ks) that are the solutions of the system of equations (64)-(67). In the relativistic case it is very difficult to evaluate the harmonic-emission rates by numerical integration. For very high laser-field intensities and ultra-high harmonic orders, this is practically impossible, so that the saddle-point m e t h o d is the only way to produce reasonable results. Figure 22 presents an example. The nonrelativistic result is obtained from Eq. (37) where the summation is over the solutions of the system o f the nonrelativistic saddle-point equations (34), (35) and (60). It is, o f course, inapplicable for the high intensity o f

86

W. B e c k e r et al.

[VII

1.5x1018 W/cm 2 at 800nm and is only shown to demonstrate the dramatic impact of relativistic kinematics. For the relativistic result, the summation in Eq. (37) is over the relativistic solutions of Eqs. (64)-(67). The relativistic harmonic-emission rate assumes a convex shape, and the difference between the relativistic and nonrelativistic results reaches several hundred orders of magnitude in the upper part of the nonrelativistic plateau. The origin of this dramatic suppression is the magnetic-field-induced v x B drift. The significance of this drift for the rescattering mechanism was emphasized early by Kulyagin et al. (1996). This is illustrated in the inset of Fig. 22, which shows the real part of the dominant shortest orbit for the harmonic f2 = 100000o). In order to counteract this drift so that the electron is able to return to the ion, the electron has to take off with a very substantial initial velocity in the direction opposite to the laser propagation. The probability of such a large initial velocity is low, and this is the reason for the strong suppression. As in the nonrelativistic case, the electron is "born" at a distance of 7.5 a.u. from the nucleus. The nonrelativistic harmonic yield shows a pronounced multiplateau structure. While this is an artifact of the nonrelativistic approximation for the intensity of Fig. 22, it is a real effect for lower laser-field intensities where relativistic effects are still small (Walser et al., 2000; Kylstra et al., 2001; Milogevid et al., 2001 c, 2002). In this case, the three plateaus visible in the nonrelativistic curve of Fig. 22 are related to the three pairs of orbits, whose contribution to the harmonic emission rate is dominant in the particular spectral region (see Figs. 2 and 3 of Milogevid et al., 2001c). These are very similar to the pairs of orbits that we have discussed for the elliptically polarized laser field in Fig. 17. However, for the very high intensity of Fig. 22, the contribution of the shortest of these orbits becomes so dominant that the multiplateau and the interference-related oscillatory structure disappear completely. The reason is that the effect of the v x B drift increases with increasing travel time; see Eqs. (52) and (56) in Sect. V.A. This is in contrast to the nonrelativistic case of elliptical polarization, where longer orbits may be favored because the minor component of the field oscillates and, therefore, for a longer orbit a smaller initial velocity may be sufficient to allow the electron to return.

VII. Applications of ATI Experimental and theoretical advances in understanding A T I - some of which have been treated in this review - permit its application to the investigation of other effects. One obvious idea is to exploit the nonlinear properties of ATI. This is particularly relevant to characterization of high-order harmonics and measurement of attosecond pulses in the soft-X-ray regime. In this spectral region (vacuum UV) virtually all bulk non-linear media are opaque. ATI, in contrast, is usually studied under high- or ultra-high-vacuum conditions.

VII]

ATI: CLASSICAL TO QUANTUM

87

Another advantage over conventional nonlinear optics is that the nonlinear effect of photoelectron emission can be observed from more or less any direction, whereby different properties of the effect can be exploited. A.

CHARACTERIZATION OF H I G H HARMONICS

The most straightforward approach to characterize high-order harmonics is a cross-correlation scheme: An (isolated) harmonic of frequency qoo, where q is an odd integer, produces electrons by single-photon ionization with a kinetic energy Eq = qhoo - EIp. Simultaneous presence of a fraction of the fundamental laser beam in the near infrared (NIR) produces sidebands, i.e. electrons with energies qhoo- Ew + mhoo (m 500~tJ) laser pulses in the few-cycle regime. However, these are not stabilized and, accordingly, the absolute phase changes in a random fashion from pulse to pulse.

92

W. B e c k e r et al.

[IX

In a recent experiment, Paulus et al. (2001b) were able to detect effects due to the absolute phase by performing a shot-to-shot analysis of the number of photoelectrons emitted in opposite directions. To this end, a field-free drift tube is placed symmetrically around the target gas. Each end of the tube is equipped with an electron detector. Because of its characteristic appearance, this was dubbed a stereo-ATI spectrometer. A characteristic feature of few-cycle pulses such as (70) is that, depending on the absolute phase, the peak electric-field strength (and thus also the vector potential) is different in the positive and negative x-directions. Recall from Eq. (2) that the electron's drift momentum depends on the vector potential at its time of birth. Therefore, depending on the value of the absolute phase, such a laser pulse creates more electrons in one direction than in the other. A theoretical analysis of the photoelectrons' angular distribution was given by Dietrich et al. (2000) and Hansen et al. (2001) for the nonperturbative intensity regime. Interestingly, the effect is predicted to be much less pronounced in the perturbative regime (Cormier and Lambropoulos, 1998). Equivalent to a leftright asymmetry of the photoelectron angular distribution is that the number of electrons emitted to the left vs. those emitted to the right is anticorrelated: A laser shot for which many electrons are seen at the right detector is likely to produce only a few that go left, and vice versa. This can be proved by correlation analysis. Each laser shot is sorted into a contingency map according to the number of electrons recorded at both detectors. Anticorrelations can then be seen in structures perpendicular to the diagonal, see Fig. 25.

VIII. Acknowledgments We learned a lot in discussions with S.L. Chin, M. DSrr, C. Faria, S.P. Goreslavskii, C.J. Joachain, M. Kleber, V.P. Krainov, M. Lewenstein, A. Lohr, H.G. Muller, S.V. Popruzhenko, and W. Sandner. This work was supported in part by Deutsche Forschungsgemeinschaft and Volkswagen Stiftung.

IX. References Agostini, P., Fabre, E, Mainfray, G., Petite, G., and Rahman, N.K. (1979). Phys. Rev. Lett. 42, 1127. Agostini, E, Antonetti, A., Breger, E, Crance, M., Migus, A., Muller, H.G., and Petite, G. (1989). J. Phys. B 22, 1971. Alon, O.E., Averbukh, V., and Moiseyev, N. (2000). Phys. Rev. Lett. 85, 5218. Ammosov, M.V., Delone, N.B., and Krainov, V.P. (1986). Zh. Eksp. Teor. Fiz. 91, 2008 [Soy. Phys.JETP 64, 1191]. Antoine, P., l'Huillier, A., Lewenstein, M., Sali6res, P., and CarrY, B. (1996). Phys. Rev. A 53, 1725. Antoine, Ph., Gaarde, M., Sali~res, P., CarrY, B., l'Huillier, A., and Lewenstein, M. (1997). In "Multiphoton Processes 1996" (P. Lambropoulos and H. Walther, Eds.), Institute of Physics Conference Series No. 154. Institute of Physics Publishing, Bristol, p. 142.

IX]

ATI" C L A S S I C A L TO Q U A N T U M

93

Apolonski, A., Poppe, A., Tempea, G., Spielmann, Ch., Udem, Th., Holzwarth, R., H~insch, T.W., and Krausz, E (2000). Phys. Rev. Lett. 85, 740. Bao, D., Chen, S.G., and Liu, J. (1996). Appl. Phys. B 62, 313. Bartels, R., Backus, S., Zeek, E., Misoguti, L., Vdovin, G., Christov, I.P., Murnane, M.M., and Kapteyn, H.C. (2000). Nature (London) 406, 164. Bartels, R., Backus, S., Christov, I., Kapteyn, H., and Murnane, M. (2001). Chem. Phys. 267, 277. Bashkansky, M., Bucksbaum, P.H., and Schumacher, D.W. (1988). Phys. Rev. Lett. 60, 2458. Becker, W., Schlicher, R.R., and Scully, M.O. (1986). s Phys. B 19, L785. Becket, W., Schlicher, R.R., Scully, M.O., and W6dkiewicz, K. (1987). s Opt. Soc. Am. B 4, 743. Becker, W., Long, S., and McIver, J.K. (1990). Phys. Rev. A 42, 4416. Becker, W., Long, S., and McIver, J.K. (1992). Phys. Rev. A 46, R5334. Becker, W., Lohr, A., and Kleber, M. (1994a). s Phys. B 27, L325. Corrigendum: 28, 1931. Becker, W., Long, S., and McIver, J.K. (1994b). Phys. Rev. A 50, 1540. Becker, W., Lohr, A., and Kleber, M. (1995). Quantum Semiclass. Opt. 7, 423. Becket, W., Lohr, A., Kleber, M., and Lewenstein, M. (1997). Phys. Rev. A 56, 645. Becker, W., Kleber, M., Lohr, A., Paulus, G.G., Walther, H., and Zacher, E (1998). Laser Phys. 8, 56. Beigman, I.L., and Chichkov, B.N. (1987). Pis "ma Zh. Eksp. Teor. Fiz. 46, 314 [JETP Lett. 46, 395]. Bellini, M., LyngA, C., Tozzi, A., Gaarde, M.B., H/insch, T.W., l'Huillier, A., and Wahlstr6m, C.-G. (1998). Phys. Rev. Lett. 81, 297. Berson, I.J. (1975). J. Phys. B 8, 3078. Bhardwaj, V.R., Aseyev, S.A., Mehendale, M., Yudin, G.L., Villeneuve, D.M., Rayner, D.M., Ivanov, M.Yu., and Corkum, P.B. (2001). Phys. Rev. Lett. 86, 3522. Blondel, C., Delsart, C., Dulieu, E, and Valli, C. (1999). Eur. Phys. J. D 5, 207. Borca, B., Frolov, M.V., Manakov, N.L., and Starace, A.E (2001). Phys. Rev. Lett. 87, 133001. Bordas, C., Paulig, E, Helm, H., and Huestis, D.L. (1996). Rev. Sci. Instrum. 67, 2257. Brabec, T., and Krausz, F. (2000). Rev. Mod. Phys. 72, 545. Bryant, H.C., Mohagheghi, A., Stewart, J.E., Donahue, J.B., Quick, C.R., Reeder, R.A., Yuan, V., Hummer, C.R., Smith, W.W., Cohen, C., Reinhardt, W.P., and Overman, L. (1987). Phys. Rev. Lett. 58, 2412. Bucksbaum, P.H., Bashkansky, M., Freeman, R.R., Mcllrath, T.J., and DiMauro, L.F. (1986). Phys. Rev. Lett. 56, 2590. Bucksbaum, P.H., Bashkansky, M., and Mcllrath, T.J. (1987). Phys. Rev. Lett. 58, 349. Bunkin, F.V., and Fedorov, M.V. (1966). Zh. Eksp. Teor. Fiz. 49, 1215 [Soy. Phys.-JETP 22, 844]. Ceccherini, E, and Bauer, D. (2001). Phys. Rev. A 64, 033423. Chelkowski, S., and Bandrauk, A.D. (2000). Laser Phys. 10, 216. Chin, S.L., Yergeau, E, and Lavigne, P. (1985). J. Phys. B 18, L213. Christov, I.P., Bartels, R., Kapteyn, H.C., and Murnane, M.M. (2001). Phys. Rev. Lett. 86, 5458. Corkum, P.B. (1993). Phys. Rev. Lett. 71, 1994. Corkum, P.B., Burnett, N.H., and Ivanov, M.Y. (1994). Opt. Lett. 19, 1870. Cormier, E., and Lambropoulos, P. (1997). J. Phys. B 30, 77. Cormier, E., and Lambropoulos, P. (1998). Eur. Phys. J. D 2, 15. Cormier, E., Garzella, D., Breger, P., Agostini, P., Ch6riaux, G., and Leblanc, C. (2001). J. Phys. B 34, L9. Crawford, D.P., and Reiss, H.R. (1997). Opt. Express 2, 289. Dammasch, M., D6rr, M., Eichmann, U., Lenz, E., and Sandner, W. (2001). Phys. Rev. A 64, 061402(R). de Bohan, A., Antoine, P., Milo~evi6, D.B., and Piraux, B. (1998). Phys. Rec. Lett. 81, 1837. Delande, D., and Buchleitner, A. (1994). Adv. At. Mol. Opt. Phys. 34, 85. Delone, N.B., and Krainov, V.P. (1994). "Multiphoton Processes in Atoms." Springer, Berlin. Delone, N.B., and Krainov, V.P. (1998). Usp. Fiz. Nauk 168, 531 [Phys. Usp. 41, 469].

94

W. B e c k e r et al.

[IX

Demkov, Yu., and Ostrovskii, V.N. (1989). "Zero-Range Potentials and their Applications in Atomic Physics." Plenum, New York. Dietrich, E, Burnett, N.H., Ivanov, M., and Corkum, EB. (1994). Phys. Rev. A 50, R3585. Dietrich, E, Krausz, E, and Corkum, EB. (2000). Opt. Lett. 25, 16. DiMauro, L.E, and Agostini, E (1995). Adv. At. Mol. Opt. Phys. 35, 79. D6rner, R., Mergel, V., Jagutzki, O., Spielberger, L., Ullrich, J., Moshammer, R., and SchmidtBScking, H. (2000). Phys. Rep. 330, 95. D6rr, M., Potvliege, R.M., and Shakeshaft, R. (1990). Phys. Rev. A 41,558. Drescher, M., Hentschel, M., Kienberger, R., Tempea, G., Spielmann, Ch., Reider, G.A., Corkum, P.B., and Krausz, F. (2001). Science 291, 1923. Du, M.L., and Delos, J.B. (1988). Phys. Rev. A 38, 1896, 1913. Duchateau, G., Cormier, E., Bachau, H., and Gayet, R. (2001). Phys. Rev. A 63, 053411. Dykhne, A.M. (1960). Zh. Eksp. Teor. Fiz. 38, 570 [Soy. Phys.-JETP 11, 411]. Eberly, J.H., Javanainen, J., and Rz~2ewski, K. (1991). Phys. Rep. 204, 331. Eberly, J.H., Grobe, R., Law, C.K., and Su, Q. (1992). Adv. At. Mol. Opt. Phys. Suppl. 1, 301. Ehlotzky, E (2001). Phys. Rep. 345, 175. Eichmann, H., Egbert, A., Nolte, S., Momma, C., Wellegehausen, B., Becker, W., Long, S., and Mclver, J.K. (1995). Phys. Rev. A 51, R3414. Faisal, EH.M. (1973). J. Phys. B 6, L89. Faisal, F.H.M., and Rado2ycki, T. (1993). Phys. Rev. A 47, 4464. Faisal, EH.M., and Scanzano, P. (1992). Phys. Rev. Lett. 68, 2909. Faisal, EH.M., Filipowicz, P., and Rz~• K. (1990). Phys. Rev. A 41, 6176. Ferray, M., l'Huillier, A., Li, X.E, Lompr6, L.A., Mainfray, G., and Manus, C. (1988). J. Phys. B 21, L31. Figueira de Morisson Faria, C., Milo~evi6, D.B., and Paulus, G.G. (2000). Phys. Rev. A 61, 063415. Figueira de Morisson Faria, C., Kopold, R., Becker, W., and Rost, J.M. (2002a). Phys. Rev. A 65, 023404. Figueira de Morisson Faria, C., Schomerus, H., and Becker, W. (2002b). physics~0206028. Phys. Rev. A, to be published. Filipowicz, P., Faisal, F.H.M., and Rz~.ewski, K. (1991). Phys. Rev. A 44, 2210. Fittinghoff, D.N., Bolton, P.R., Chang, B., and Kulander, K.C. (1992). Phys. Rev. Lett. 69, 2642. Freeman, R.R., Bucksbaum, P.H., Milchberg, H., Darack, S., Schumacher, D., and Geusic, M.E. (1987). Phys. Rev. Lett. 59, 1092. Gaarde, M.B., Salin, E, Constant, E., Balcou, Ph., Schafer, K.J., Kulander, K.C., and l'Huillier, A. (1999). Phys. Rev. A 59, 1367. Gaarde, M.B., Schafer, K.J., Kulander, K.C., Sheehy, B., Kim, D., and DiMauro, L.F. (2000). Phys. Rev. Lett. 84, 2822. Gallagher, T.E, and Scholz, T.J. (1989). Phys. Rec. A 40, 2762. Garton, W.R.S., and Tomkins, F.S. (1967). Astrophys. J. 158, 839. Goreslavskii, S.P., and Popruzhenko, S.V. (1996). Zh. Eksp. Teor. Fiz. 110, 1200 [JETP 83, 661]. Goreslavskii, S.P., and Popruzhenko, S.V. (1999a). Phys. Lett. A 249, 477. Goreslavskii, S.P., and Popruzhenko, S.V. (1999b). J. Phys. B 32, L531. Goreslavskii, S.P., and Popruzhenko, S.V. (2000). Zh. Eksp. Teor Fiz. 117, 895 [JETP 90, 778]. G6rlinger, J., Plagne, L., and Kull, H.-J. (2000). Appl. Phys. B 71, 331. Gottlieb, B., Kleber, M., and Krause, J. (1991). Z. Phys. A 339, 201. Gottlieb, B., Lohr, A., Becker, W., and Kleber, M. (1996). Phys. Rev. A 54, R1022. Gribakin, G.E, and Kuchiev, M.Yu. (1997). Phys. Rev. A 55, 3760. Gutzwiller, M. (1990). "Chaos in Classical and Quantum Mechanics." Springer, Berlin. Hansch, P., Walker, M.A., and Van Woerkom, L.D. (1997). Phys. Rev. A 55, R2535. Hansen, J.P., Lu, J., Madsen, L.B., and Nilsen, H.M. (2001). Phys. Rev. A 64, 033418. Hauge, E.H., and Stovneng, J.A. (1989). Rev. Mod. Phys. 59, 917.

IX]

ATI: C L A S S I C A L TO Q U A N T U M

95

Hentschel, M., Kienberger, R., Spielmann, Ch., Reider, G.A., Milosevic, N., Brabec, T., Corkum, P., Heinzmann, U., Drescher, M., and Krausz, E (2001). Nature (London) 414, 509. Hertlein, M.E, Bucksbaum, P.H., and Muller, H.G. (1997). J. Phys. B 30, L 197. Hu, S.X., and Keitel, C.H. (2001). Phys. Rev. A 63, 053402. Ivanov, M.Yu., Brabec, Th., and Burnett, N. (1996). Phys. Rev. A 54, 742. Jackson, J.D. (1999). "Classical Electrodynamics," 3rd edition. Wiley, New York. Jarofi, A., Kamifiski, J.Z., and Ehlotzky, E (1999). Opt. Commun. 163, 115. Joachain, C.J., DSrr, M., and Kylstra, N. (2000). Adv. At. Mol. Opt. Phys. 42, 225. Jones, D.J., Diddams, S.A., Ranka, J.K., Stentz, A., Windeler, R.S., Hall, J.L., and Cundiff, S.T. (2000). Science 288, 635. Kamifiski, J.Z., Jarofi, A., and Ehlotzky, E (1996). Phys. Rev. A 53, 1756. Keldysh, L.V. (1964). Zh. Eksp. Teor. Fiz. 47, 1945 [Soy. Phys.-JETP 20, 1307]. Kibble, T.W.B. (1966). Phys. Rev. 150, 1060. Kim, J.-H., Lee, D.G., Shin, H.J., and Nam, C.H. (2001). Phys. Rev. A 63, 063403. Kopold, R. (2001). Ph.D. Dissertation. Munich Technical University. In German. Kopold, R., and Becker, W. (1999). J. Phys. B 32, L419. Kopold, R., Becker, W., and Kleber, M. (1998). Phys. Rev. A 58, 4022. Kopold, R., Becker, W., and Kleber, M. (2000a). Opt. Commun. 179, 39. Kopold, R., Milo~evi6, D.B., and Becker, W. (2000b). Phys. Rev. Lett. 84, 3831. Kopold, R., Becker, W., Kleber, M., and Paulus, G.G. (2002). J. Phys. B 35, 217. Krainov, V.P. (1999). J. Phys. B 32, 1607. Krainov, V.P., and Shokri, B. (1995). Laser Phys. 5, 793. Kramer, T., Bracher, C., and Kleber, M. (2001). Europhys. Lett. 56, 471. Krausz, E (2001 ). Phys. World 14, 41. Kroll, N.M., and Watson, K.M. (1973). Phys. Rev. A 8, 804. Krsti6, E, and Mittleman, M.H. (1991). Phys. Rev. A 44, 5938. Krsti6, ES., Milo~evi6, D.B., and Janev, R.K. (1991). Phys. Rev. A 44, 3089. Kruit, P., Kimman, J., Muller, H.G., and van der Wiel, M.J. (1983). Phys. Rev. A 28, 248. Kuchiev, M.Yu. (1987). Pis 'ma Zh. Eksp. Teor. Fiz. 45, 319 [JETP Lett. 45, 404]. Kuchiev, M.Yu., and Ostrovsky, V.N. (1999). J. Phys. B 32, L189. Kuchiev, M.Yu., and Ostrovsky, V.N. (2001). J. Phys. B 34, 405. Kulander, K.C., and Lewenstein, M. (1996). In "Atomic, Molecular, & Optical Physics Handbook" (G.W. Drake, Ed.). American Institute of Physics Press, Woodbury, NY, p. 828. Kulander, K.C., Schafer, K.J., and Krause, J.L. (1993). In "Super-Intense Laser-Atom Physics" (B. Piraux, A. l'Huillier and K. Rz~ewski, Eds.), Vol. 316 of NATO Advanced Studies Institute, Series B: Physics. Plenum, New York, p. 95. Kull, H.-J., G6rlinger, J., and Plagne, L. (2000). Laser Phys. 10, 151. Kulyagin, R.V., Shubin, N.Yu., and Taranukhin, V.D. (1996). Laser Phys. 6, 79. Kylstra, N.J., Worthington, R.A., Patel, A., Knight, EL., Vfizquez de Aldana, J.R., and Roso, L. (2000). Phys. Rev. Lett. 85, 1835. Kylstra, N.J., Potvliege, R.M., and Joachain, C.J. (2001). J. Phys. B 34, L55. Lambropoulos, P., Maragakis, E, and Cormier, E. (1998). Laser Phys. 8, 625. Lee, D.G., Shin, H.J., Cha, Y.H., Hong, K.H., Kim, J.-H., and Nam, C.H. (2001). Phys. Rev. A 63, 021801 (R). Lein, M., Gross, E.K.U., and Engel, V. (2001). Phys. Rev. A 64, 023406. Leubner, C. (1981). Phys. Rev. A 23, 2877. Lewenstein, M., Balcou, Ph., Ivanov, M.Yu., l'Huillier, A., and Corkum, P.B. (1994). Phys. Rev. A 49, 2117. Lewenstein, M., Kulander, K.C., Schafer, K.J., and Bucksbaum, P.H. (1995a). Phys. Rev. A 51, 1495. Lewenstein, M., Sali6res, P., and l'Huillier, A. (1995b). Phys. Rev. A 52, 4747. l'Huillier, A., Lompr6, L.A., Mainfray, G., and Manus, C. (1983). Phys. Rev. A 27, 2503.

96

W. B e c k e r et al.

[IX

Lindner, E, Dreischuh, A., Grasbon, E, Paulus, G.G., and Walther, H. (2001). IEEE J. Quantum Electron., to be published. Lohr, A., Kleber, M., Kopold, R., and Becker, W. (1997). Phys. Rev. A 55, R4003. Long, S., Becker, W., and McIver, J.K. (1995). Phys. Rev. A 52, 2262. Mainfray, G., and Manus, C. (1991). Rep. Prog. Phys. 54, 1333. Manakov, N.L., and Fainshtein, A.G. (1980). Zh. Eksp. Teor. Fiz. 79, 751 [Sot). Phys.-JETP 52, 382]. Manakov, N.L., and Rapoport, L.E (1975). Zh. Eksp. Teor. Fiz. 69, 842 [Soy. Phys.-JETP 42, 430]. Manakov, N.L., Frolov, M.V., Starace, A.E, and Fabrikant, I.I. (2000). J. Phys. B 33, R141. McNaught, S.J., Knauer, J.P., and Meyerhofer, D.D. (1997). Phys. Rev. Lett. 78, 626. McPherson, A., Gibson, G., Jara, H., Johann, U., Luk, T.S., McIntyre, I.A., Boyer, K., and Rhodes, C.K. (1987). J. Opt. Soc. Am. B 4, 595. Milo~evi6, D.B. (2000). J. Phys. B 33, 2479. Milo~evi6, D.B., and Becker, W. (2000). Phys. Rev. A 62, 011403(R). Milo~evi6, D.B., and Ehlotzky, E (1998a). Phys. Rev. A 57, 5002. Milo~evi6, D.B., and Ehlotzky, E (1998b). Phys. Rev. A 58, 3124. Milo~evi6, D.B., and Ehlotzky, E (1998c). J. Phys. B 31, 4149. Milo~evi6, D.B., and Sandner, W. (2000). Opt. Lett. 25, 1532. Milo~evi6, D.B., and Starace, A.E (1998). Phys. Rev. Lett. 81, 5097. Milo~evi6, D.B., and Starace, A.E (1999a). Phys. Rev. Lett. 82, 2653. Milo~evi6, D.B., and Starace, A.E (1999b). Phys. Rev. A 60, 3160. Milo~evi6, D.B., and Starace, A.E (1999c). Phys. Rev. A 60, 3943. Milo~evi6, D.B., and Starace, A.E (2000). Laser Phys. 10, 278. Milo~evi6, D.B., Becker, W., and Kopold, R. (2000). Phys. Rev. A 61, 063403. Milo~evi6, D.B., Becker, W., and Kopold, R. (2001a). In "Atoms, Molecules and Quantum Dots in Laser Fields: Fundamental Processes" (N. Bloembergen, N. Rahman and A. Rizzo, Eds.), Conference Proceedings Vol. 71. Italian Physical Society/Editrice Compositori, Bologna, p. 239. Milo~evi6, D.B., Becker, W., Kopold, R., and Sandner, W. (2001b). Laser Phys. 11, 165. Milo~evi6, D.B., Hu, S., and Becker, W. (2001c). Phys. Rev. A 63, 011403(R). Milo~evi6, D.B., Hu, S.X., and Becker, W. (2002). Laser Phys. 12, 389. Mohideen, U., Sher, M.H., Tom, H.W.K., Aumiller, G.D., Wood II, O.R., Freeman, R.R., Bokor, J., and Bucksbaum, EH. (1993). Phys. Rev. Lett. 71,509. Moore, C.I., Knauer, J.E, and Meyerhofer, D.D. (1995). Phys. Rev. Lett. 74, 2439. Moore, C.L., Ting, A., McNaught, S.J., Qiu, J., Burris, H.R., and Sprangle, E (1999). Phys. Rev. Lett. 82, 1688. Moshammer, R., Feuerstein, B., Schmitt, W., Dorn, A., Schr6ter, C.D., Ullrich, J., Rottke, H., Trump, C., Wittmann, M., Korn, G., Hoffmann, K., and Sandner, W. (2000). Phys. Rev. Lett. 84, 447. Muller, H.G. (1999a). Phys. Rev. A 60, 1341. Muller, H.G. (1999b). Phys. Rev. Lett. 83, 3158. Muller, H.G. (1999c). Laser Phys. 9, 138. Muller, H.G. (2001a). Opt. Express 8, 44. Muller, H.G. (2001b). Opt. Express 8, 86. Muller, H.G. (2001c). Opt. Express 8, 417. Muller, H.G., and Kooiman, EC. (1998). Phys. Rev. Lett. 81, 1207. Muller, H.G., Tip, A., and van der Wiel, M.J. (1983). J. Phys. B 16, L679. Mur, V.D., Karnakov, B.M., and Popov, V.S. (1998). Zh. Eksp. Teor. Fiz. 114, 798 [J. Exp. Theor. Phys. 87, 433]. Mur, V.D., Popruzhenko, S.V., and Popov, V.S. (2001). Zh. Eksp. Teor. Fiz. 119, 893 [J Exp. Theor. Phys. 92, 777]. Nandor, M.J., Walker, M.A., and Van Woerkom, L.D. (1998). J Phys. B 31, 4617. Nandor, M.J., Walker, M.A., Van Woerkom, L.D., and Muller, H.G. (1999). Phys. Rev. A 60, R1771.

IX]

ATI: C L A S S I C A L TO Q U A N T U M

97

Nisoli, M., De Silvestri, S., Svelto, O., Szip6cs, R., Ferencz, K., Spielmann, Ch., Sartania, S., and Krausz, E (1997). Opt. Lett. 22, 522. Nurhuda, M., and Faisal, F.H.M. (1999). Phys. Rev. A 60, 3125. Ortner, J., and Rylyuk, V.M. (2000). Phys. Rev. A 61, 033403. Parker, J.S., Moore, L.R., Meharg, K.J., Dundas, D., and Taylor, K.T. (2001). J. Phys. B 34, L69. Patel, A., Protopapas, M., Lappas, D.G., and Knight, P.L. (1998). Phys. Rec. A 58, R2652. Paul, P.M., Toma, E.S., Breger, P., Mullot, G., Aug6, E, Balcou, Ph., Muller, H.G., and Agostini, P. (2001). Science 292, 1689. Paulus, G.G. (1996). "Multiphotonenionisation mit intensiven, ultrakurzen Laserpulsen." Utz, Mfinchen. Paulus, G.G., Becker, W., Nicklich, W., and Walther, H. (1994a). J. Phys. B 27, L703. Paulus, G.G., Nicklich, W., and Walther, H. (1994b). Europhys. Lett. 27, 267. Paulus, G.G., Nicklich, W., Xu, H., Lambropoulos, P., and Walther, H. (1994c). Phys. Rev. Lett. 72, 2851. Paulus, G.G., Becker, W., and Walther, H. (1995). Phys. Rev. A 52, 4043. Paulus, G.G., Zacher, F., Walther, H., Lohr, A., Becker, W., and Kleber, M. (1998). Phys. Rev. Lett. 80, 484. Paulus, G.G., Grasbon, E, Dreischuh, A., Walther, H., Kopold, R., and Becker, W (2000). Phys. Rev. Lett. 84, 3791. Paulus, G.G., Grasbon, E, Walther, H., Kopold, R., and Becker, W. (2001a). Phys. Rev. A 64, 021401 (R). Paulus, G.G., Grasbon, E, Walther, H., Villoresi, P., Nisoli, M., Stagira, S., Priori, E., and De Silvestri, S. (2001b). Nature (London) 414, 182. Paulus, G.G., Grasbon, E, Walther, H., Nisoli, M., Stagira, S., Sansine, G., and De Silvestri, S. (2002). To be published. Perelomov, A.M., and Popov, V.S. (1967). Zh. Eksp. Teor. Fiz. 52, 514 [Soy. Phys.-JETP 25, 336]. Perelomov, A.M., Popov, V.S., and Terent'ev, M.V. (1966a). Zh. Eksp. Teor. Fiz. 50, 1393 [Soy. Phys.-JETP 23, 924]. Perelomov, A.M., Popov, V.S., and Terent'ev, M.V. (1966b). Zh. Eksp. Teor. Fiz. 51, 309 [Soy. Phys.-JETP 24, 207]. Popov, V.S., Mur, V.D., and Karnakov, B.M. (1997). Pis 'ma Zh. Eksp. Teor. Fiz. 66, 213 [JETP Lett. 66, 229]. Popruzhenko, S.V., Goreslavskii, S.P., Korneev, P.A., and Becker, W. (2002). Phys. Rev. Lett. 89, 023001. Protopapas, M., Keitel, C.H., and Knight, P.L. (1997a). Rep. Progr. Phys. 60, 389. Protopapas, M., Lappas, D.G., and Knight, P.L. (1997b). Phys. Rev. Lett. 79, 4550. Raczyfiski, A., and Zaremba, J. (1997). Phys. Lett. A 232, 428. Reichert, J., Holzwarth, R., Udem, Th., and H~insch, T.W. (1999). Opt. Commun. 172, 59. Reichle, R., Helm, H., and Kiyan, I.Yu. (2001). Phys. Rev. Lett. 87, 243001. Reiss, H.R. (1980). Phys. Rev. A 22, 1786. Reiss, H.R. (1990). J. Opt. Soc. Am. B 7, 574. Reiss, H.R. (1992). Prog. Quantum Electron. 16, 1. Reiss, H.R. (1996). Phys. Rev. A 54, R1765. Robustelli, D., Saladin, D., and Scharf, G. (1997). Helv. Phys. Acta 70, 96. Rottke, H., Wolff, B., Brickwedde, M., Feldmann, D., and Welge, K.H. (1990). Phys. Rev. Lett. 64, 404. Salibres, P., l'Huillier, A., Antoine, Ph., and Lewenstein, M. (1999). Adv. At. Mol. Opt. Phys. 41, 83. Salibres, P., Carr6, B., le D6roff, L., Grasbon, F., Paulus, G.G., Walther, H., Kopold, R., Becker, W., Milo~evi6, D.B., Sanpera, A., and Lewenstein, M. (2001). Science 292, 902. Schins, J.M., Breger, P., Agostini, P., Constantinescu, R.C., Muller, H.G., Bouhal, A., Grillon, G., Antonetti, A., and Mysyrowicz, A. (1996). J. Opt. Soc. Am. B 13, 197.

98

W. B e c k e r et al.

[IX

Schomerus, H., and Sieber, M. (1997). J. Phys. A 30, 4537. Schulman, L. (1977). "Techniques and Applications of Path Integration." Benjamin, New York. Scrinzi, A., Geissler, M., and Brabec, Th. (1999). Phys. Rev. Lett. 83, 706. Sheehy, B., Lafon, R., Widmer, M., Walker, B., DiMauro, L.E, Agostini, P.A., and Kulander, K.C. (1998). Phys. Rev. A 58, 3942. Smirnov, M.B., and Krainov, V.P. (1998). J. Phys. B 31, L519. Smyth, E.S., Parker, J.S., and Taylor, K.T. (1998). Comput. Phys. Commun. 114, 1. Spence, D.E., Kean, EN., and Sibbett, W. (1991). Opt. Lett. 16, 42. Taieb, R., V6niard, V., and Maquet, A. (2001). Phys. Rev. Lett. 87, 053002. Tang, C.Y., Bryant, H.C., Harris, P.G., Mohagheghi, A.H., Reeder, R.A., Sharifian, H., Tootoonchi, H., Quick, C.R., Donahue, J.B., Cohen, S., and Smith, W.W. (1991). Phys. Rev. Lett. 66, 3124. Toma, E.S., Antoine, Ph., de Bohan, A., and Muller, H.G. (1999). J. Phys. B 32, 5843. van de Sand, G., and Rost, J.M. (2000). Phys. Rev. A 62, 053403. van Linden van den Heuvell, H.B., and Muller, H.G. (1988). In "Multiphoton Processes" (S.J. Smith and P.L. Knight, Eds.), Vol. 8 of Cambridge Studies in Modern Optics. Cambridge University Press, Cambridge, p. 25. V~izquez de Aldana, J.R., and Roso, L. (1999). Opt. Express 5, 144. V~zquez de Aldana, J.R., Kylstra, N.J., Roso, L., Knight, EL., Patel, A., and Worthington, R.A. (2001). Phys. Rev. A 64, 013411. V~niard, V., Ta'ieb, R., and Maquet, A. (1995). Phys. Rev. Lett. 74, 4161. V6niard, V., Taieb, R., and Maquet, A. (1996). Phys. Rev. A 54, 721. Volkov, D.M. (1935). Z. Phys. 94, 250. Walker, B., Sheehy, B., Kulander, K.C., and DiMauro, L.E (1996). Phys. Rev. Lett. 77, 5031. Walser, M.W., Keitel, C.H., Scrinzi, A., and Brabec, T. (2000). Phys. Rev. Lett. 85, 5082. Walsh, T.D.G., Ilkov, EA., and Chin, S.L. (1994). J. Phys. B 27, 3767. Watson, J.B., Sanpera, A., Burnett, K., and Knight, P.L. (1997). Phys. Rev. A 55, 1224. Weber, Th., Giessen, H., Weckenbrock, M., Urbasch, G., Staudte, A., Spielberger, L., Jagutzki, O., Mergel, V., Vollmer, M., and D6rner, R. (2000a). Nature (London) 405, 658. Weber, Th., Weckenbrock, M., Staudte, A., Spielberger, L., Jagutzki, O., Mergel, V., Afaneh, E, Urbasch, G., Vollmer, M., Giessen, H., and D6rner, R. (2000b). Phys. Rev. Lett. 84, 443. Weingartshofer, A., Holmes, J.K., Caudle, G., Clarke, E.M., and Krfiger, H. (1977). Phys. Rev. Lett. 39, 269. Weingartshofer, A., Holmes, J.K., Sabbagh, J., and Chin, S.L. (1983). J. Phys. B 16, 1805. Wildenauer, J. (1987). J. Appl. Phys. 62, 41. Yang, B., Schafer, K.J., Walker, B., Kulander, K.C., Agostini, P., and DiMauro, L.E (1993). Phys. Rev. Lett. 71, 3770. Yergeau, E, Chin, S.L., and Lavigne, P. (1987). J. Phys. B 20, 723. Yudin, G.L., and Ivanov, M.Yu. (2001a). Phys. Rev. A 63, 033404. Yudin, G.L., and Ivanov, M.Yu. (200 l b). Phys. Rev. A 64, 013409.

ADVANCES IN ATOMIC, M O L E C U L A R , A N D O P T I C A L PHYSICS, VOL. 48

DARK OPTICAL TRAPS FOR COLD ATOMS NIR FRIEDMAN, ARIEL K A P L A N and NIR D A V I D S O N Weizmann Institute of Science, Department of Physics of Complex Systems, Rehovot, Israel I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Trapping Cold Atoms using Repulsive Dipole Forces . . . . . . . . . . . . . . . . . B. Loading Atoms into Dark Optical Traps . . . . . . . . . . . . . . . . . . . . . . . . . . C. Hollow Laser Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Multiple-Laser-Beams Dark Optical Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Adding Beams Incoherently: Light-sheets and Hollow-beam Traps . . . . . . . . B. Evanescent-wave Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Using Interference: Dark Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . IV. Single-Beam Dark Optical Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Generating Dark Volumes using Refractive Optical Elements: Axicon Traps . . B. Creating Single-beam Dark Traps with Diffractive Optical Elements . . . . . . . C. Scanning-beam Dark Optical Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Comparing Different Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Manipulations in Phase Space: Cooling and Compression . . . . . . . . . . . . . . B. Precision Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dynamics of the Trapped Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 101 101 103 104 106 107 109 111 113 113 115 122 124 127 127 136 141 147 148

I. I n t r o d u c t i o n Trapping of neutral atoms has become possible in the last 15 years, thanks to the advances in laser cooling techniques (Adams and Riis, 1997). Several kinds of traps were developed and investigated (Balykin et al., 2000), the most useful ones being magnetic, magneto-optical and dipole force traps. Optical dipole traps use the interaction between the electric field of the light and an electric dipole, which is induced in the atom by this field. This force is weaker than the magneto-optical and magnetic forces, and typical dipole trap depths are below 1 mK. These traps offer the possibility to trap atoms in all internal states, as well as a possibility to lower the dissipative component of the interaction between the atoms and the trapping light by increasing the detuning of the laser from the atomic resonance (which also unavoidably reduces the depth of the trapping potential). These traps and their properties have been described in a recent review by Grimm et al.

(2000). 99

Copyright 9 2002 Elsevier Science (USA) All rights reserved ISBN 0-12-003848-X/ISSN 1049-250X/02 $35.00

1O0

N. Friedman et al.

[I

Fla. 1. Fluorescence images of atoms in optical dipole traps. (a) Atoms trapped inside a red-detuned trap; imaging is performed when the trapping beam is on. Atoms close to the focus of the trapping beam are not observed, since they experience a large ac Stark shift, which shifts them out of resonance with the probe beam. (b) Same as (a), but imaging is performed a short time after the trapping beam is shut off. All atoms are detected. (c,d) Atoms trapped inside a blue-detuned trap, with the same detuning and laser power as those of the red-detuned trap. All atoms are observed, irrespective of whether the trapping beam is on (c) or off (d), indicating reduced perturbation in this dark trap. To further reduce the interaction between the atoms and the trapping laser, dark optical traps were developed, in which the atoms are trapped by a repulsive dipole interaction with the laser. The repulsive force is achieved by detuning the trapping laser above the atomic resonance, such that there is a phase difference between the field and the induced dipole. In these dark optical dipole traps, cold atoms are trapped inside a dark region which is surrounded by a repulsive dipole potential wall. To visualize the lower perturbations that are observed by atoms inside a dark optical trap, consider Fig. 1, in which fluorescence images of trapped atomic clouds are presented, for both attractive and repulsive optical dipole traps. In the attractive (red-detuned) trap, atoms around the b e a m focus are not observed, since they are shifted out of resonance with the fluorescence beams, due to the large ac Stark shift induced by the trapping laser. W h e n the measurement is repeated just after the trapping b e a m is shut off, these atoms are easily observed. For the repulsive (blue-detuned) dark trap, which had the same laser power, the same detuning and the same dimensions, there is no

II]

DARK OPTICAL TRAPS FOR COLD ATOMS

101

difference between the measurements with and without the trapping beam. This result indicates that perturbations due to the trapping beam are largely suppressed for dark traps. Throughout this review we will present quantitative evidences for this suppression. Experimentally, dark traps (also called "blue-detuned" traps) are harder to realize than attractive (or "red-detuned") dipole traps, where already a single focused beam constitutes a trap. Several configurations have been demonstrated, the first of which used multiple laser beams to form a confinement by light in three dimensions, or used repulsive light structures to support the cold atoms against gravity, which provided the trapping potential in the vertical direction. Later, simpler trap designs were realized, which use only a single laser beam, and traps with optimized properties were designed for various applications. In this review, we describe dark optical traps, the techniques in which they are experimentally realized, and their main applications in atomic physics. We start with a short background section discussing the optical dipole force and the process of loading cold atoms into the trap. We also describe hollow laser beams, methods by which they can be formed, and their application as atom guides, which confine atoms in two dimensions. In Sect. III we discuss dark traps created using multiple laser beams, including gravito-optical traps, traps that use evanescent light fields as the confining walls, and far-detuned dark optical lattices. In Sect. IV, single-beam dark optical traps are considered, with special emphasis on the optical techniques used in their construction. In both sections, the focus is on trap designs that have been experimentally realized, and the measured properties of the trapped atomic ensembles are described. In Sect. V, the main applications of dark traps are discussed, including advanced laser-cooling methods for dense atomic samples, the use of dark traps as a favorable environment for precision spectroscopy, and the study of the dynamics of trapped atoms.

II. Background A. TRAPPING COLD ATOMS USING REPULSIVE DIPOLE FORCES

Light can exert a force upon atoms by momentum exchange due to photon absorption and emission. The force that a laser field applies on an atom is usually separated into two terms which correspond to a scattering force and a dipole force. The scattering force results from absorption followed by spontaneous emission of a photon in a random direction. The average momentum exchanged is therefore one photon momentum, hkL, in the direction of the absorbed photon (kf = 2:r//~, where/~ is the wavelength of the laser). In the case of the dipole force, the photon is emitted in a stimulated way into a mode of the laser field. The momentum transfer in this case is the vector difference between the momenta of the absorbed and emitted photons.

N. Friedman et al.

102

[II

For a two-level atom, the forces can be calculated by using the solutions of the optical Bloch equations, while translational degrees of freedom are taken into account (Cohen-Tannoudji et al., 1992). The resulting expression for the average scattering force is then Fscattering = hkL ~

(~2 + ( ),2/4 ) + (ff22/2)

= hkL ),s,

(1)

where ), is the natural linewidth of the atomic transition, 6 = 00c-000 is the detuning of the laser from the atomic resonance, f2 is the Rabi frequency that characterizes the atom-field interaction, and ),s is the spontaneous photon scattering rate. The average dipole force can be written as

-h6( Fdipole(r) = ~

~7(Q2(r)/2) 62 + (),2/4) + (ff22(r)/2)

) .

(2)

This force depends on the sign of the detuning. If the laser is detuned below the atomic resonance (red-detuned), the force will attract atoms into regions with higher intensity. If the laser is detuned above resonance (blue-detuned) the force will be repulsive, and atoms will be repelled by the light into lower-intensity regions. The dipole force is conservative, and can be written as a gradient of the potential 7

(

f22(r'/2

Udipole(r) = - - In 1 + 62 + (),2/4)

)

h6 ( l(r)/Is ) . = m In 1 + 1 + (462/), 2)

(3,

In the last expression, the Rabi frequency is given in terms of measurable quantities: ~2 = (),2/2)(i/ls), where I is the intensity of the laser, and Is is the saturation intensity of the transition, given by Is = 2:r2h),c/3)t 3. For a large detuning of the laser, 6 >> ),, f2, which is the usual case in optical dipole traps, the potential has a simpler form: h), 2 I(r) _ 3:rrc2 ~I(r). Udipole(r)-

86

Is

2003

(4)

The last term is equal to Eq. (12) of Grimm et al. (2000), which was derived for the classical oscillator model of the atom, using the rotating wave approximation. The same result can also be derived in the dressed-state model, where the combined Hamiltonian for the atom and the laser field is solved. (See, for example, Cohen-Tannoudji et al., 1992). Equation 4 indicates that the dipole potential is proportional to the laser intensity and inversely proportional to its detuning. Comparing the expressions for the dipole force and scattering rate, under the above approximations, yields the relationship Udipole _ ~,

h),s

),

(5)

which means that a trap with a reduced scattering rate can be made by increasing the detuning while maintaining the ratio I/6.

II]

DARK OPTICAL TRAPS FOR COLD ATOMS

103

In the case of multi-level atoms, Eq. (4) should be modified to include the electric dipole interaction between the ground-state and all the excited states, with their respective detunings and transition strengths. In practice, only energy levels which are close to resonance with the laser frequency have to be considered. In the case of far blue-detuned dipole traps, levels above the first electronic excited state might have a considerable influence on the potential, if the laser frequency is close to resonance with the transition. The laser detuning is limited at the blue side by the ionization energy of the atom. These consideration do not apply to red-detuned traps, where the contribution of transitions other than the lowest one are much smaller. As an example, we consider a bluedetuned trap for Rb atoms. The ionization energy of the ground state (5S1/2) corresponds to a photon wavelength of ~300 nm. Note that ionization from the excited state (5Pj) will occur at ~480 nm. If a trap is realized at this wavelength, excited-state ionization will lead to trap loss when laser cooling is performed on the trapped atoms. The lowest transitions from the ground state are the D lines at 795 nm and 780nm. The next line (5S1/2 ~ 6P1/2) is at 421.5 nm, and its transition strength is about 100 times lower than that of the D lines. Hence, this line becomes relevant only for a laser which is detuned about 2 nm with respect to it. For very large detunings, which are comparable to the optical frequency, a multiplicative correction factor of the order of unity is needed in the potential calculation, as a correction to the rotating wave approximation.

B. LOADING ATOMS INTO DARK OPTICAL TRAPS

The usual loading scheme of atoms into optical dipole traps starts with a magneto-optical trap (MOT) (Raab et al., 1987), which traps atoms from a vapor or an atomic beam and cools them to a typical temperature of 100 ~tK. Since most dipole traps are relatively shallow and small as compared with the MOT, it is advantageous to further cool the atoms and increase their density in order to enhance the loading efficiency. The loading of red-detuned dipole traps was thoroughly investigated both for trapping laser detunings of few nm (or -3 • 105 V) (Kuppens et al., 2000), for larger detunings (~ 1 • 107 V) (Han et al., 2001), and also for CO2 laser traps (O'Hara et al., 2001), where the detuning is comparable to the atomic resonance frequency. In the CO2 trap, spatial and phase-space densities much higher than that of a MOT are achieved, a fact that led recently to the first demonstration of a Bose-Einstein condensate (BEC) created in an all-optical way (Barrett et al., 2001). There are some differences in the loading process between red- and bluedetuned dipole traps. For red-detuned traps, the atoms are loaded into a region of high trapping light intensity, which may interfere with the loading process. The trap may reduce the optical cooling efficiency, since it results in a spatially inhomogeneous Stark shift of the cooling line. The level shifts may influence also

104

N. Friedman et al.

[II

the photon reabsorption since both the spontaneous emission and the absorption spectra are altered. Combined with the trap's potential, which is usually steeper than that of the MOT, it can result in a higher atomic density in the trap. For blue-detuned traps, the loading seems to be simpler since the atoms are loaded into a dark region, hence their interaction with the trapping light is much smaller than their interaction with the MOT. This is supported by several experimental observations. First, in many experiments the density and temperature of the atoms loaded into the dark trap are very close to those in the MOT. Second, the number of trapped atoms almost does not change if the trap is present during the whole loading stage, or is turned on just at the end of the MOT operation. These findings suggest that the loading is purely geometrical - those atoms that are inside the dark "box" will stay there, those outside the box will not be trapped, while the trap does not interfere with the operation of the MOT. The different loading mechanism of bright and dark dipole traps is emphasized by an experiment where atoms are first loaded from a MOT into a red-detuned dipole trap, and are then transferred into an overlapping dark trap. In this manner, an increase of about x2.5 in the number of atoms in the dark trap was observed, as compared to direct loading of the dark trap from the MOT (Friedman et al., 200 l a). This indicates that the red-detuned trap can enhance the atomic density, while the dark trap leaves it almost unchanged. In this way, a dark trap may be used as a probe for investigating loading into other traps, since it can sample the atomic density and temperature at a given time. The advantage is that the measurement can be performed at a later time, when the MOT atoms that were not trapped have expanded and have fallen out of the detection region.

C. HOLLOWLASER BEAMS As an introduction to the discussion of three-dimensional dark optical traps, it is useful to first treat hollow laser beams, which can serve as two-dimensional traps, or guides, for cold atoms. Here, we will describe how such hollow beams can be produced, and the main results of atom guiding in such beams. This subject has been discussed thoroughly by Balykin (1999). A hollow beam has a light distribution with a minimum (ideally equal to zero) along its axis. With a laser beam detuned above the atomic resonance, such beams act as linear guides for cold atoms, where the atoms propagate inside the dark "tube" created by the dipole potential of the beam. It is possible to distinguish between two types of hollow beams. The first type contains structurally stable beams (modes), which have a constant intensity cross section that scales in size as they propagate. The second type of hollow beams is not structurally stable but remains hollow along a relatively large propagation distance, which can be made long enough for many applications.

II]

DARK OPTICAL TRAPS FOR COLD ATOMS

105

Laguerre-Gaussian modes: An example for structurally stable beams are Laguerre-Gaussian modes, LG/, which form a complete basis set of solutions of the paraxial wave equation 1. For p = 0, l ;~ 0, the intensity distribution of this beam has the form of an annulus, and is given by

jr[l[!w(z) 2 ~w(z)2

exp W(Z) 2

,

(6)

where P is the total laser power, the waist size is w ( z ) = w0v/1 + (2/ZR) 2, and zR = :rw2/)~ is the Rayleigh range. The phase of the beam changes linearly with the azimuthal angle, 0, and there is a phase singularity on the beam axis. These modes were extensively studied in the last decade, mainly due to their special property of having orbital angular momentum lh per photon. For a review of this subject see (Allen et al., 1999). LG modes were generated experimentally from high-order Hermite-Gaussian modes (which are easy to produce directly from laser resonators), by a mode converter composed of two cylindrical lenses (Beijersbergen et al., 1993). Such a cylindrical-lens mode converter was used to produce a LG g mode that was part of a 3D dark optical trap (Kuga et al., 1997), as will be discussed in Sect. III.A. LG modes were generated also by using a computer generated hologram, with a "fork" in the grating pattern. When illuminated with a plane-wave (or a TEM00 laser beam), a "charge-one" phase singularity will be created in the beam, centered around the fork defect. The resulting 1st diffraction order is a good approximation of the required LG~ field distribution, although it is not a pure mode (Heckenberg et al., 1992). Such computer-generated holograms were used to create hollow beams which served as guides for cold atoms (Kuppens et al., 1998; Schiffer et al., 1998). In these experiments, metastable neon atoms were guided inside a focused LG~ hollow beam, a first demonstration of guiding cold atoms in free space 2. Atoms were guided along a distance of 30cm and focused to a spot size o f - 6 . 5 ~tm. Polarization-gradient cooling (PGC) in the transverse direction was applied in order to further increase the phase-space density (see Sect. V.A.1). In later experiments, a BEC was adiabatically transferred into a hollow LG 1 beam and its propagation inside this optical guide was studied (Bongs et al., 2001). A TEM00 Gaussian beam will be projected with a high efficiency onto a LGt0 beam when it passes through a spiral phase element having a transmission

1 Another example for structurally stable hollow beams is constituted by high-order Bessel beams, which are non-diffractive and were considered as atom guides (Arlt et al., 200 l b). 2 Cold atoms have been guided previously with light propagating inside hollow fibers (Balykin, 1999).

106

N. F r i e d m a n et al.

[III

function of e ilO (Beijersbergen et al., 1994). Alternatively, intra-cavity spiral phase elements can force a pure helical mode output directly from the laser (Oron et al., 2000). Output from a hollow fiber: A conceptually different method to generate a hollow beam is to use the output beam from a hollow optical fiber (Yin et al., 1997). Here, the laser is coupled to a high-order mode which propagates in the fiber's cylindrical core. When the mode exits from the fiber's end it is collimated with a strong lens to form the hollow beam, which has an intensity distribution that is similar to the LG~ mode. Efficient guiding of falling cold atoms in such a beam was demonstrated along a vertical distance of 11 cm (Xu et al., 1999, 2001). Axicons: An axicon is a refractive optical element with a conical surface. A TEM00 laser beam incident on its center will be refracted into a conical beam. If a second axicon with the same base angle is then placed into the beam it will collimate it, resulting in a light "tube." The intensity cross section of this beam is similar to that of a high-order LG mode, but it has a constant phase and it is not structurally stable. However, such tubes can remain hollow over relatively large propagation distances, large enough for guiding cold atoms. This was recently demonstrated (Song et al., 1999), by transporting a cloud of -~108 atoms through an 18 cm long optical guide, with a diameter of 1 mm. In another work, a continuous low-velocity atomic beam was guided inside a hollow laser beam which was produced using axicons (Yan et al., 2000). Axicons have some advantages in producing hollow beams for cold-atom guiding and trapping, as demonstrated by Manek et al. (1998). First, the characteristic width of the guide walls, w, can be very narrow, and the radius of the dark center, r, can be large. In the above experiment, for example, R = r/w ~ 11 was demonstrated, equivalent to a LGt0 mode with a very large l that is much more difficult to realize efficiently 3. Second, the optical setup is very simple: a Gaussian laser beam is used as the input, and simple commercially available optical elements are utilized. When illuminated with a LG mode with l > 0, the axicon-lens system converts it into a hollow beam with a much thinner ring (hence much larger R), which still has the orbital angular momentum of the original beam (Arlt et al., 2001a).

III. Multiple-Laser-Beams Dark Optical Traps During the last decade, several types of dark optical traps were proposed and demonstrated. The first dark traps were formed by incoherently adding several laser beams that served as the trap's walls. These first traps were shallow and

3

For LG modes, R _~ v/1. Hence R - 11 will require l as large as 120.

III]

DARK OPTICAL TRAPS FOR COLD ATOMS

107

had a relatively small volume, hence captured only a modest number of atoms. However, these first experiments demonstrated the main advantages of dark traps, namely low photon scattering rates and long coherence times of the trapped atoms. Later improvements in trap design led to traps with larger volumes, in which large number of atoms could be captured and manipulated. In this chapter, traps based on multiple laser beams are described in a comparative way. The main dark-trap designs are considered, and the performance of experimentally realized traps is discussed. In order to compare traps made for different atomic species on a common basis, trap properties are given in a normalized way: the detuning is normalized by y (the linewidth of the relevant excited state) and the trap depth is normalized by the recoil energy of the atom, Erec = hZk~/2m, where m is the mass of the atom.

A. ADDING BEAMS INCOHERENTLY: LIGHT-SHEETS AND HOLLOW-BEAM TRAPS

In the first dark optical dipole trap for cold atoms, realized in 1995 in Stanford (Davidson et al., 1995), sodium atoms where trapped using light from an Ar-ion laser. This trap consisted of two elliptical light sheets (generated by focusing a Gaussian beam with a cylindrical lens) intersecting at 90 ~ and forming a "V"-shaped cross section. Confinement was provided by gravity in the vertical direction and by the beams' divergence in the longitudinal direction. The two beams had powers of 4 and 6 W, were linearly polarized, and had different wavelengths (488 and 514.5 nm) so that they did not interfere in the overlap region and formed a smooth potential. The large detuning of the trap beams, ~ 107 y, resulted in a relatively low potential of ~10E~ec, and hence a low number of trapped atoms (~3000), but also an extremely low photon scattering rate, calculated as ~ 10-3 s-1 , and a long lifetime of 5 s, limited by the background vacuum. A spectroscopic measurement of the hyperfine splitting of Na, which was performed on the trapped atoms (see Sect. V.B), yielded a coherence time of 7 s. This coherence time was 300 times longer than that achieved in a red-detuned trap having the same potential height and a larger detuning, emphasizing the advantage of a dark trap for precision measurements. The coherence time was limited by inhomogeneous broadening, since different atoms acquire different Stark shifts, depending on the velocity distribution and the dynamics of the trapped atoms. In a later work (Lee et al., 1996), this trap was improved by intersecting two such "V" traps at a right angle, resulting in an inverted-pyramid trap (see Fig. 2). The polarization of the beams in the second pair was rotated by 90 ~ with respect to the first in order to prevent interference in the trapping region, which would lead to trap loss. A much larger number of atoms (4.5 • 105) were confined in this trap, and its shape produced coupling between the motion in all three dimensions. This allowed cooling in three dimensions by applying one-dimensional Raman cooling (see further discussion in Sect. V.A.2). The issue of atom dynamics

108

N. F r i e d m a n et al.

[III

FIG. 2. Schematic illustration of the inverted pyramid dark optical trap, which is composed of four blue-detuned light sheets. (From Lee et al., 1996, Phys. Rev. Lett. 76, 2658, Fig. 1).

FIG. 3. Schematic illustration of the dark hollow beam trap, which is based on a hollow Laguerre-Gaussian beam and two plugging beams. (From Kuga et al., 1997, Phys. Rev. Lett. 78, 4713, Fig. 2). and its possible implication for cooling and spectroscopy will be discussed later, Sect. V.C. A different trap configuration was demonstrated by Kuga et al. (1997): it consisted of a hollow laser beam (LG 3) and two additional "plug" beams that confined the atoms in the propagation direction of the hollow beam (see Fig. 3). The hollow beam had a power of 600 mW and a radius of 600 ~m. A detuning of ~1047 resulted in a potential of ~100Erec, higher than the typical energy of PGC-cooled Rb atoms. The deep potential, combined with a very large volume of 2 • 10-3 cm 3, enabled the loading of a much larger number of atoms (1 • 108), with a very good loading efficiency of about one-third from the MOT. However, the relatively small detuning resulted in a high photon scattering rate of~100 s-1 , which limited the lifetime of the trap to 150 ms due to heating of the atoms above the potential barrier. To reduce this heating effect, pulsed PGC was applied, resulting in a longer lifetime of 1.5 s (Torii et al., 1998). There have been several proposals for traps using a vertical hollow beam combined with a horizontal plug beam that supports the atoms against gravity, so forming a gravito-optical trap (Morsch and Meacher, 1998; Yin et al., 1998). The hollow beam can be produced in either of the ways discussed in the previous section, and can be focused to create a conical shape, forming a funnel that

III]

DARK OPTICAL TRAPS FOR COLD ATOMS

109

potentially increases the loading efficiency from a MOT into the dipole trap. These proposed schemes suggest the use of inelastic reflection of atoms from the trapping light (reflection Sisyphus cooling, see Sect. V.A.3) to reduce the relatively high kinetic energy of atoms that fall from the MOT into the trap, and also to balance the heating due to spontaneous photon scattering. A similar configuration was used by Webster et al. (2000): 103 Cs atoms were trapped above the focus of a vertical LG 1 beam from an Ar-ion laser. No plug beam was used here, but the loss of atoms through the very small hole at the bottom of the trap is negligible due to the relatively high temperature of the atoms. The trap was loaded from a magnetic trap, in which evaporative cooling was performed to lower the kinetic energy of the atoms below the dipole trap potential. Since no reflection cooling occurs at this large detuning, gravity was balanced by a magnetic field gradient, such that atoms were falling very slowly into the dark trap. B. EVANESCENT-WAVE TRAPS

In the early 1990s, normal-incidence reflection of cold atoms from an evanescent wave was demonstrated by Kasevich et al. (1990). The evanescent wave is produced by total internal reflection of a linearly polarized blue-detuned laser beam at the surface of a dielectric (glass) prism, which forms a steep potential wall above the surface, of the form U ( z ) = Uo exp(-2z/A). Here, A =

2~ V/n 2 sin 2 0 - 1

(7)

is the characteristic interaction length scale, where n is the refractive index of the prism, and 0 is the angle of incidence. U0 is the dipole potential on the prism surface, which can be calculated as discussed in Sect. II.A. Atoms with a kinetic energy lower than the potential height are reflected from the evanescent light sheet without hitting the surface. The photon scattering per bounce is given by np = 7 m o • where v• is the vertical component of the atom's velocity before it enters the interaction region. A is typically much smaller than the size of a focused Gaussian beam, resulting in a much steeper potential, which is advantageous for supporting atoms against gravity while minimizing their interaction with the supporting sheet of light. The height of the potential barrier is reduced due to the attractive van der Waals force between the atoms and the surface of the prism, which becomes relevant at distances of ~~./2:r. In another experiment, a gravitational atom cavity was realized, in which atoms bounced several times on an evanescent wave which was formed on a curved glass surface (Aminoff et al., 1993). This "atomic trampoline" can be regarded as a dark optical trap, with a potential depth of 5000Erec in the vertical direction (created by the evanescent wave), and ~30Erec in the radial direction

110

[III

N. F r i e d m a n et al. repumping l l beam ,~::z~,,..~] atoms in MOT V ~ ; [

atoms in GOST " ~

1

1

evan.

T

wave

P

v

D

9.2 GHz ,~

dielectric vacuum EW laser W beam

F=4 F=3

~ hollow beam

(a)

(b)

FIG. 4. (a) Schematic illustration of the experimental setup of the gravito-optical surface trap. The trap is formed by an evanescent light wave that supports atoms against gravity, and a hollow beam that provides confinement in the horizontal direction. (From Ovchinnikov et al., 1997, Phys. Reu. Lett. 79, 2225, Fig. 1). (b) The reflection cooling cycle, with the relevant energy levels for the Cs D 2 line. An atom moves towards the mirror in the lower hyperfine level (F -- 3). Close to the classical turning point, it may undergo a spontaneous Raman transition to the upper level ( F - 4), by scattering a photon from the evanescent wave. It is then reflected from the mirror along the lower potential observed by the upper level. The cycle is closed by spontaneous scattering of a photon from a repumping beam, which takes the atom back to the lower level. (From Engler et al., 1998, Appl. Phys. B 67, 709, Fig. 2).

(induced by the surface curvature). The trap's lifetime was ~lOOms, limited mainly by scattering o f stray light from the surface, which either heats the atoms or optically pumps them into a state for which the potential is weaker, hence the effective area o f the mirror is smaller. In order to form a trap with a longer lifetime and a better confinement, a vertical hollow beam was added to confine the atoms in the horizontal direction, as shown in Fig. 4a (Ovchinnikov et al., 1997). In this trap, the evanescent wave was produced using a 6 0 - m W laser b e a m which was detuned by only ~ 2 0 0 7 from the lower hyperfine level of the atomic ground state, creating a potential barrier of 104Erec for that level (taking into account the van der Waals interaction with the surface). For the upper hyperfine level, the detuning is larger by the hyperfine splitting ( - 1 7 5 0 y in Cs), hence the potential is lower. The different potential for these two levels is used in the reflection Sisyphus cooling to be discussed in Sect. V.A.3. The 1 2 0 - m W hollow b e a m had a radius o f 360[am and a characteristic ring width o f w0 - 18 [am. It was detuned by 0.3 n m from resonance (6 = 2 x 104) ,) and produced a potential barrier o f ~500Erec. The high potential barrier and large volume m a d e it possible to trap a large n u m b e r o f Cs atoms: 2 x 105 in the first experiment, which was later increased up to 2 x 107 by improving the loading scheme o f the M O T ( H a m m e s et al., 2000). From a

III]

DARK OPTICAL TRAPS FOR COLD ATOMS

111

measurement of the heating rate of the trapped atoms, the upper bound for the total photon scattering rate in this trap was calculated to be 50 s -1. The large number of atoms at relatively low temperature and high density make this trap a good candidate for evaporative cooling towards quantum degeneracy in an all-optical way in a dark trap. Evaporative cooling of Cs atoms was indeed demonstrated by lowering the potential height, as will be discussed in Sect. V.A.4. Due to the very different confinement in the vertical and horizontal directions, this trap has some promising applications in the investigation of quantum degeneracy of a 2D gas (Petrov et al., 2000). For this purpose, the atoms can be confined more tightly in the vertical direction by adding a reddetuned evanescent wave that will create a narrower potential well (Engler et al., 1998), or by adding a vertical standing wave above the surface. In this scheme, an atom falls on the evanescent wave and at the turning point is transferred by a spontaneous Raman transition to a different internal state, which is uncoupled from the evanescent wave but is coupled to the standing-wave potential that has a minimum at this point. Such a scheme was demonstrated experimentally with metastable Ar atoms, which were preferentially loaded into a single potential well of a red- or blue-detuned standing wave with an increase of 100 in spatial density (Gauck et al., 1998). This scheme can be extended to alkali atoms by a proper choice of laser polarizations, as discussed by Spreeuw et al. (2000). Finally, the evanescent-wave surface trap can also be used to explore atomsurface interaction, which is of theoretical as well as practical importance for various atom-optics devices such as waveguides and traps close to surfaces.

C. USING INTERFERENCE: DARK OPTICAL LATTICES Cold atoms can be trapped in the dark nodes of interference patterns of bluedetuned light. Only three-dimensional (3D) blue-detuned optical lattices will provide 3D trapping, while even a 1D red-detuned optical lattice is capable of confining atoms in three dimensions. In a first demonstration of a 3D dark optical lattice, lithium atoms were trapped in a non-dissipative way (Anderson et al., 1996). The lattice was formed by the interference of four intersecting laser beams, detuned either to the red (10 nm) or to the blue (1 nm, or - 1 x 105 y) of the atomic resonance. Typical lifetime of the trapped atoms was on the order of 50ms, limited by heating due to photon scattering with a scattering rate of ~100s -1. This scattering rate is relatively high as compared to far-detuned dark optical traps since the lattice potential height was comparable to the kinetic energy of the atoms, so no effective "darkness factor" was obtained 4.

Note that a higher potential is needed for trapping the lighter Li atoms, since the typical momenta of laser-cooled alkalis are comparable, resulting in a higher kinetic energy for the lighter atoms.

4

112

N. Friedman et al.

[Ill

A much lower scattering rate was demonstrated for metastable Ar atoms trapped in the lowest energy band of a 3D dark optical lattice (Muller-Seydlitz et al., 1997). Atoms in higher bands suffer higher loss rates due to heating caused by photon scattering since their broader wavefunctions have a larger overlap with the trapping light. Quantitatively, the scattering rate for atoms in the (l, m, n) band can be estimated within a harmonic approximation of the potential around a node, as ! ysO,

9 (t + m + n + 3 ) .

With the experimental parameters, gs was as low as 6 S-1 for the (0, 0, 0) ground band. Another band-dependent loss mechanism is tunneling of a trapped atom to a neighboring lattice site, and eventually out of the lattice by diffusion. The detuning of the lattice beams was 6 = 2.5x 105,/, and the trap depth was 54Erec, not much higher than the initial kinetic energy of the atoms. Due to the low potential, only the lowest bands with l, m, n ~< 3 are bound and initially populated, with a total number of 104 atoms. The RMS momentum of the atoms remaining in the trap decreased as a function of trapping time, indicating the band-dependent loss described above. The population in each band was resolved by ramping down the potential slowly such that higher lying bands were released first, and atoms in the lower state were released in a later, resolvable time. The decay time for the lowest band was found to be 0.31 s, as compared to 0.13 s for the next-higher band, leading to a preparation of an atomic sample in the ground band after about 0.45 s of storage in the lattice. Since the preparation was done by selection and not by cooling to the ground state, only a very small number of about 50 atoms were trapped in the ground state. Muller-Seydlitz et al. (1997) constructed the lattice by three orthogonal standing waves with mutually orthogonal linear polarizations. Since the atoms are trapped in a J = 0 state, the dipole potential is independent of the local polarization of the light and is proportional to the sum of intensities. For atoms in a J ;~ 0 state, the relative phases of the three orthogonal standing waves have to be stabilized. Alternatively, standing waves with a frequency difference between them can be used such that on average the polarization is linear (DePue et al., 1999; Chin et al., 2001), or an inherently stable configuration with only four beams can be used, as did Anderson et al. (1996). A related proposal describes a way of producing a 1D blue-detuned optical lattice using two counter-propagating Gaussian beams with different waists such that confinement is achieved also in the radial direction (Zemanek and Foot, 1998). Here, as opposed to the 3D lattice, the intensity in the nodes is zero only at z = 0, where the waists of the two waves are located, and the intensities are equal. Out of this plane the two waves have different divergence angles due to the different waist sizes, resulting in a gradual increase in the intensity at the well bottom and a decrease in the darkness factor. Since the confinement in

IV]

DARK OPTICAL TRAPS FOR COLD ATOMS

113

the radial direction results from non-perfect destructive interference, the radial potential depth is much smaller than that in the longitudinal direction. Dark optical lattices are promising candidates for performing precision measurements, such as of the electron's permanent electric dipole moment, as discussed recently by Chin et al. (2001), and to be described in Sect. V.B. Finally, far blue-detuned optical lattices have attracted much attention as a possible system for the realization of quantum information processing (Brennen et aL, 1999). Here, the main advantages are the low interaction of an atom with its environment leading to long coherence time, and the possibility to control the interactions between individual atoms to induce entanglement.

IV. Single-Beam Dark Optical Traps In this section we describe dark optical traps created with a single laser beam. These traps are simpler to align than traps using several beams, hence it is easier to optimize the trap properties. The simplicity of these traps also permits easy manipulation and dynamical control of the trapping potential, its size and its shape. As opposed to the 2D case, where any desired light distribution can be generated using diffractive or refractive optical elements, there is no simple procedure to design an arbitrary 3D light distribution. Actually, there are not enough degrees of freedom in the design of an optical element to achieve a full 3D arbitrary light distribution 5. Nevertheless, with the combined use of refractive and holographic optical elements, it is possible to extend the methods described in Sect. II.C in order to produce light distributions which are suitable for trapping atoms in the dark using a single laser beam. Such light distributions, which comprise of a dark volume completely surrounded by light, were realized using either combinations of axicons and spherical lenses, diffractive optical elements, or rapidly scanning laser beams. The following subsections describe the various trap designs, and analyze the properties of the resulting traps. In the last subsection, traps of different classes are compared quantitatively. A. GENERATING DARK VOLUMES USING REFRACTIVE OPTICAL ELEMENTS: AXICON TRAPS A combination of axicons and spherical lenses is capable of transforming a TEM00 Gaussian beam into a light distribution that is suitable for 3D dark optical trapping. The first example was actually a gravity-assisted trap, where a cone of

5 This issue has been discussed, for example, by Piestun and Shamir (1994), Spektor et al. (1996) and Shabtayet al. (2000), who designed hologramsthat produce some specific 3D light distributions, and discussed the limits of these design procedures.

114

N. F r i e d m a n et al.

[IV

Fic. 5. Schematic illustration of the conical beam optical trap. The conical hollow beam, directed upwards, is generated with two equal axicons (an "axicon telescope") and a spherical lens. (From Ovchinnikov et al., 1998, Europhys. Lett. 43, 510, Fig. l a). light facing upward was realized (Ovchinnikov et al., 1998). The optical setup, illustrated in Fig. 5, consisted of two axicons with identical angles (an "axicon telescope") followed by a spherical lens. As opposed to the funnel trap described previously (Sect. III.A), the apex of the cone in this case is not hollow and no plug beam is needed. In the experiment, a 250-mW laser beam was used to form the conical trap with an opening angle of 150mrad. The detuning of the laser was 3 GHz (6 = 560),) with respect to the lower hyperfine level of Cs (F = 3), producing a potential barrier o f ~ 3 x 10SErec in the focal plane (apex of the cone), decreasing linearly with height. The trap was loaded from a MOT located 5 mm above the apex. A Cs atom that falls 5 mm acquires a kinetic energy o f - 8x 103Erec and hence is easily confined by the trap potential. Six molasses beams were left on during the experiment, to realize a combined polarization gradient cooling and reflection Sisyphus cooling. The measured loading efficiency from the MOT into the trap was very high, ~ 80%, resulting in -8 x 105 atoms in the trap, at a temperature of-10Trec. The trapped atomic cloud had an elongated shape with a length of about 1 mm in the vertical direction, and a radial size of 100 ~tm. Two drawbacks of this trap are the high photon scattering rate from the conical beam, ~3 x 103 s-1 , and the poorly defined potential shape near the apex [probably caused by interference (Webster et al., 2000)]. This can be inferred from the elongated shape of the resulting atomic cloud, which does not compress towards the apex as would be expected from the equipartition of energy. Recently, a related scheme producing a 3D single-beam dark trap has been demonstrated by two groups (Cacciapuoti et al., 2001; Kulin et al., 2001). This scheme uses a single axicon placed between two spherical lenses (see Fig. 6).

IV]

D A R K OPTICAL TRAPS F O R C O L D ATOMS

115

FIG. 6. Schematic illustration of the experimental setup for a single-beam dark optical trap, based on an axicon and two spherical lenses. The axicon and the first lens provide a virtual image of the trap (with radius r) indicated by the dashed lines, which is imaged by the second lens to form the dark trap (with radius R). (From Cacciapuoti et al., 2001, Eur. Phys. J. D 14, 373, Fig. 1). After passing through the first lens, a divergent Gaussian beam hits the axicon. The second lens focuses the resulting divergent conical beam into a ring o f light with a dark center. Since the divergence angle o f the beam is larger than the opening angle o f the cone of light, the trap is closed also along the beam propagation direction. Traps in a large range o f dimensions (40 ~tm < r < 740 ~tm) were experimentally realized, demonstrating the flexibility of this design 6. In the second experiment (Kulin et al., 2001), this optical scheme was used to produce a large trap, with r = 740 ~tm, L = 150 mm, and V = 8 x 10 -2 cm 3. The optical darkness factor (defined as the ratio between the light intensity inside the trap and the intensity in the focal plane ring) was about 1 : 1000, while along the axis there was a residual peak, probably caused by light going through the apex of the axicon. Since the opening angle of the conical beam is not zero, the two cusps on the optical axis are not of the same height, the one closer to the lens being about 10 times higher. The lowest points in the potential barrier are located off-axis, close to the far end o f the trap. This trap was used to capture Xe atoms in a metastable state. The high trap volume resulted in a good loading efficiency o f 50% from the MOT which contained a few million atoms. The lifetime o f the trap was only about 2 0 m s , limited by the gravitational energy of the heavy Xe atoms in the large trap, which is larger than the height of the potential barrier at the far end o f the trap. B. CREATING SINGLE-BEAM DARK TRAPS WITH DIFFRACTIVE OPTICAL ELEMENTS A different approach to produce dark volumes surrounded by light is to create a destructive interference in some region in space between two coherent light fields which have different propagation characteristics. Outside this dark region the intensity will rise in all directions due to the different propagation constants.

6 Actually, the trap dimensions can be changed by moving only the axicon, which does not change the location of the focal plane.

1 16

N. Friedman et al.

[IV

FIG. 7. (a) Schematic illustration of the experimental setup for the single-beam dark optical trap based on a Jr-phase plate element. A destructive interference between the inner part of the beam (shifted by Jr radians) and the outer ring is formed around the focus of the third lens. (b) Contour map of the calculated light intensity distribution around the focus, for the parameters described in the text. The dark minimum is labeled m, and three bright maxima are labeled M. The trap height is given by the two saddle-points at r _~ 0.017 mm, Z _~ 2504-1 mm. (From Ozeri et al., 1999, Phys. Rev. A 59, R1750, Fig. 1).

The :r-phase plate trap: The simplest trap o f this kind is realized by placing a circular phase-plate element into a Gaussian laser beam (Chaloupka et al., 1997; Ozeri et al., 1999), as shown in Fig. 7a. The phase plate imposes a phase difference o f exactly Jr radians between the central and outer parts o f the beam, which have equal intensities. W h e n the b e a m is focused by a lens, destructive interference between the two parts ensures a dark region around the focus, which is surrounded by light from all directions, as required. Let b denote the radius of the inner (phase-shifted) circle, a the outer (clipped) radius o f the Gaussian beam, and w0 its waist. Equal intensities in the two regions are achieved when b = w 0 v / - l n {~1 [1 + exp(-a2/w2)] }.

(9)

The resulting intensity distribution calculated numerically using the Fresnel diffraction integral, for the parameters used by Ozeri et al. (1999), are presented in Fig. 7b. The radius o f the trap is similar to the waist of the focused Gaussian beam in the absence o f the phase element, w0, focus = ) f / Z w o , w h e r e f is the focal length o f the focusing lens. The length o f the trap is L ~ 2ZR = 2Yt'w2,focus//],, which gives a volume o f 7

V

2:rK4wZ, focusZe = 2;rZK4w4,focus/) ~ =

;

( wo)4

~3.

(10)

Note that this is the maximal volume of the trap. The trapped atoms usually fill a smaller volume, depending on the ratio between their kinetic energy and the height of the trapping potential.

7

IV]

DARK OPTICAL TRAPS FOR COLD ATOMS

117

Here, K is a constant on the order of 1-3, which depends on the clipping radius a. The volume of many single-beam dark traps can be expressed in the form V : C . (f/wo~ ~3,

(11)

where the constant C depends on the exact realization of the trap. Note that the volume of a focused Gaussian red-detuned trap is given by V -- 2zR. Jrw2, focus, which can be written in the form of Eq. (11), with C = 2/jr 2. With the experimental parameters w0 = 6mm, a = 5 m m (resulting in K ~ 2.3), f = 250 mm, and/l = 799 nm, the volume of the dark trap is V ~ 9• 10-6 cm 3. The potential height of the trap is determined by the minimal light intensity on the trap surface, which is ~ 10% of the peak intensity of the unaltered Gaussian beam. In the experiment described by Ozeri et al. (1999), 105 85Rb atoms were confined in a trap that was realized using this optical setup. The Jr phase plate was produced by evaporating a thin dielectric layer of an exact thickness on a glass plate (Davidson et al., 1999). An optical darkness ratio of 750 was measured, which depends on the amount of light scattered into the dark region, the degree to which condition 9 is fulfilled, the deviations of the incoming beam from a Gaussian, and the deviations of the phase shift from Jr. These effects were studied in detail by Chaloupka and Meyerhofer (2000). In the experiment of Ozeri et al. (1999), a 1-W laser beam was used, which could be detuned in the range 4 • 104)' < 6 < 1x 106) ' above resonance. In this range the lifetime of the trap was 300 ms, limited by collisions with background atoms. For smaller detunings, the lifetime decreased linearly with 6, which is consistent with heating-induced lifetime, since the heating rate is proportional to 6 -2 whereas the trap depth is proportional to 6 -! . A similar optical configuration was used to optically trap high-energy electrons with an intense single laser beam, using the ponderomotive force (Chaloupka and Meyerhofer, 1999). A laser intensity distribution with a local minimum is essential for trapping electrons with light, since the electrons are always repelled by the laser field towards the intensity minimum. In this experiment, the Jr phase shift was realized with a segmented wave plate which can endure a very high laser intensity. Spontaneous Raman scattering rate: The amount of interaction between the atoms and the trapping light was accurately determined by measuring the spontaneous Raman scattering rate that results from the trapping light (Cline et al., 1994). This is a very useful experimental technique which permits the measurement of even very low scattering rates, hence we will present it in some detail. For this measurement, the trapped atoms were first prepared in the lower hyperfine level of the ground state, F - 2. The number of atoms in F = 3 after a variable time t, N3(t), was measured by detecting the fluorescence after a short pulse of a resonant laser beam. For normalization, the total number of atoms in

118

[IV

N. Friedman et al. 0.51 . . . . . . .

!~~;i '

.

,~ . 9

oo,

. . . . . .

000

Ume (msec)

FIG. 8. A measurement of spontaneous Raman scattering rate for atoms in the z-phase plate dark optical trap. At t = 0 all atoms are pumped to the lower hyperfine level. The fraction of atoms in the upper hyperfine level (F = 3, for 85Rb) is measured as a function of time in the trap. The fit to Eq. (12) gives a Raman scattering time of 164 ms, from which the total scattering rate can be calculated. (From Ozeri et al., 1999, Phys. Rev. A 59, R1750, Fig. 4).

the two sublevels of the ground state, N3(t)+ Nz(t), can be measured by turning on also the repumping beam (which is resonant with F = 2) during the detection pulse. Actually, N3(t) and N2(t) are measured in the same experimental run. Since the detection beam excites a closed transition, it accelerates the atoms in F = 3, and these are rapidly shifted out of resonance. Then the repumping laser is turned on and the number of atoms in F = 2 (which were not accelerated) is measured. This normalized detection scheme is insensitive to shot-to-shot fluctuations in atom number as well as fluctuations in frequency and intensity of the detection laser (Khaykovich et al., 2000). Typical experimental data for the F = 3 fraction as a function of time in the trap, for a detuning of 0.5 nm (4• 104),), are shown in Fig. 8. The data are well fitted by the function

N3(t)+ N2(t) = c 1 - e x p

--r-~R

'

(12)

where c is the equilibrium fraction of atoms in F = 3 at long times, and rSR is the measured spontaneous Raman scattering time, which is 164 ms in this case. The total scattering rate ?'~. is given by 1/(qrSR), where q is the branching ratio for a spontaneous Raman transition which ends in the upper hyperfine level. The branching ratio depends on the detuning of the trapping laser and its polarization. For laser trap detunings larger than the fine-structure splitting of the excited state (e.g. 15 nm in Rb), destructive interference exists between the transition amplitudes for Raman scattering, summed over the intermediate (excited) states (Cline et al., 1994). In this case, most spontaneous scattering events leave the internal state of the atom unchanged. Hence, the probability for a spontaneous Raman transition is strongly suppressed, and the spin-relaxation times are largely increased. The "bottle beam" trap: Another way to create a destructive interference between two different light fields emerging from a single laser beam is

IV]

DARK OPTICAL TRAPS FOR COLD ATOMS

119

FIG. 9. Contour map of the calculated light intensity distribution for the CR-BPE dark trap. O indicates the trap center, A the transverse maximum, B the axial maximum, and C the lowest barrier height. (From Ozeri et al., 2000, J. Opt. Soc. Am. B 17, 1113, Fig. 2). described by Arlt and Padgett (2000). A computer-generated hologram produces a superposition of the LG ~ and LG ~ modes from an incoming Gaussian beam, such that the two modes have equal on-axis intensities in the focal plane and an opposite phase. The relative phase between the modes changes with propagation due to the different Gouy phase around the focal plane, hence the zero-intensity region is surrounded by regions of higher intensity in all directions. The dimensions and shape of this trap as well as the height of the potential barrier are very similar to those of the Jr phase-plate trap. "Diffractive axicon" traps: The main disadvantages of the traps described above are their low volumes and high aspect ratios, which leads to low loading efficiency of atoms from a MOT. It is possible to optimize the trap parameters by combining diffractive optical elements and axicons to form a more symmetric trap with a large volume while still enjoying the advantages of a single-laserbeam setup. A main optical element for this optimization is the concentric-rings binary phase element (CR-BPE), which is an extension of the Jr phase plate (Ozeri et al., 2000). The CR-BPE is composed of concentric phase rings with a Jr phase difference between sequential rings, thus creating a radial grating with a uniform spacing. This radial grating functions similar to a refractive axicon, however, it produces two cones of light corresponding to the :kl diffraction orders, as opposed to only one cone in the axicon. To form a dark trap, the CR-BPE is illuminated with a Gaussian beam of waist w0, which is then focused by a lens. At the focal plane, the interference between the two cones of light results in a ring with a dark center, and due to the different propagation characteristics, a dark volume surrounded by light is formed. A numerical solution of the Fresnel diffraction integral for this case is shown in Fig. 9. This light-intensity map reveals that out of the focal plane the interference pattern of the two diffraction orders is not smooth but rather contains radial fringes, forming channels with reduced dipole potential height. In particular, the lowest

120

[IV

N. F r i e d m a n et al.

point in the potential barrier (point C in Fig. 9) is - 1 0 times lower than the transverse potential height (point A) and 60 times lower than the maximal on-axis intensity (point B). Hence, this trap is relatively shallow, and a very large detuning cannot be used. The radius of this trap can be estimated as r = (Jr/2)Mwo, focus, and its length as L - JrMzR, where M = wold, and d is the width of a phase ring in the radial grating. Since M and w0, focus can be controlled nearly independently, it is possible to design a trap with a smaller aspect ratio and a larger volume than the previous ones. Specifically, the volume of this trap is given by 1 ~ 4 M 3 w 2 0, focusZR = 1-~jrm3 V ~ ~1jrr2L = ]-~

()4 f W0

~3.

(13)

This trap was realized with the following parameters" w0 = 400 [xm, d = 50 ~tm a n d f = 16mm, giving a volume o f - l . 6 • -4 cm 3. The CR-BPE was formed as a binary surface-relief phase element, as described by Ozeri et al. (2000). With a laser power of 120 mW and a detuning of 6 = 1 • 1057 above resonance, -3 • 106 Rb atoms were loaded into the trap, with a loading efficiency of 5% from the MOT. The total photon scattering rate was determined from a measurement of the spontaneous Raman scattering rate to be 10 s-~ . For larger detunings, part of the atoms were trapped only in two dimensions, and escaped from the trap through the lowest point in the potential barrier. An improved trap configuration was recently demonstrated by adding an axicon telescope before the CR-BPE of the above setup (Kaplan et al., 2002a). This configuration maximizes the trap depth for a given laser power and trap dimensions, and greatly reduces the light-induced perturbations to the trapped atoms. These properties are achieved by surrounding a large dark volume with a light envelope with (a) an almost minimal surface area for a given volume, (b) the minimal wall thickness that is allowed by diffraction, and (c) an almost constant wall height over the entire envelope. The stiffness of the trap walls, combined with the large detuning allowed by the efficient intensity distribution, yield a very low calculated spontaneous photon scattering rate for the trapped atoms. The optical configuration for the creation of this trap is illustrated in Fig. 10a. The thin ring of light created by the axicon telescope is diffracted into two cones by the CR-BPE. When these two diffraction orders are imaged, a light distribution is generated that consists of two equal hollow cones attached at their bases and completely surrounding a dark region. The height of the confining potential of this double-conical trap can be approximated by U(z) U ( z - O)

=

L 4 [ ~L _ z]

9

1

V/1 + (Z/ZR) 2 '

(14)

where L is the total trap length. The first term accounts for the linear decrease in the trap radius away from the focal plane (in both directions) and the second term

IV]

DARK OPTICAL TRAPS FOR COLD ATOMS

121

FIG. 10. The double-conical single-beam dark optical trap. (a) Schematic illustration of the optical setup. A CR-BPE is placed after an "axicon telescope." The +1 and -1 diffraction orders are imaged to generate the trap (in black). (b) Contour map of the calculated light intensity distribution for the trap. Note the very thin walls, which provide a very good darkness factor. (c) Measured intensity cross sections at different planes along the trap's axis [parameters of the optical setup are different than in (b)]. The two diffraction orders are observed, and the inner one provides the trap walls. At z = 0 the two orders overlap exactly. accounts for the diffraction o f the f o c u s e d b e a m . U s i n g a s p h e r i c a l - l e n s t e l e s c o p e before the a x i c o n s (see Fig. 10a), the b e a m waist on the p h a s e e l e m e n t is c h o s e n to b a l a n c e these two effects, and to m a i n t a i n a n e a r l y c o n s t a n t intensity a l o n g z: 1)3/2 I =

~

p rw0,

(15)

focus'

w h i c h is o b t a i n e d for L ~ 7ZR 8 (r is the trap radius in the focal plane). T h e

8 Two exceptions are z = 0, where the two diffraction orders overlap, yielding a double potential height, and z "~ L / 2 , where the singularity in Eq. (14) yields an extremely high potential.

122

[IV

N. F r i e d m a n et al.

FIG. 11. (a) Measured cross sections of the time-averaged light intensity in three positions along the scanning-beam optica! trap, together with a schematic drawing of the trap formation. (b) Lifetime of atoms trapped in a scanning-beam trap, as a function of the scanning frequency of the trapping beam, for two trap radii. (From Friedman et al., 2000a, Phys. Reo. A 61, 031403(R), Figs. 1,2).

dimensions of the trap can be changed while maintaining this optimal ratio between L and zR. The trap volume is given by 1 ~ r 2 L ~ 7yg2r2

Z ~ 5

5

w2 o,focus

/l

"

(16)

This optical configuration was demonstrated experimentally, and a trap with w0,focus = 22.6gm, r = 1.47mm and L -- 13mm was generated. The aspect ratio of this trap was 1"4.4, which is very small compared to the other configurations discussed above. Figure 10b presents a calculated cross section of the trap's potential in the x - z plane, and Fig. 10c shows the measured intensity cross sections along the trap axis. The measured intensity is nearly constant over the entire envelope, evidence for the optimal use of the laser power.

C. SCANNING-BEAM DARK OPTICAL TRAPS

Trapping particles with a time-dependent potential was successfully applied to trap ions in oscillating electric fields (Major and Dehmelt, 1968), cold atoms in magnetic traps (Petrich et al., 1995), and larger particles in optical tweezers (Sasaki et al., 1992). It is possible to extend this approach to construct a dynamical optical trap for cold atoms with a rapidly scanning laser beam. If the scan frequency is high enough, the optical dipole potential can be approximated as a time-averaged quasi-static potential. For a blue-detuned laser beam, and a radius of rotation, r, larger than the waist of the focused beam, w0,focus, a dark volume suitable for 3D trapping is obtained, as shown in Fig. 1 la. Let us calculate the properties of the trap for a circular scan of the beam in the focal

IV]

DARK OPTICAL TRAPS FOR COLD ATOMS

123

plane. The time-averaged intensity in the focal plane ring (which corresponds to the radial potential barrier) is given by

/r =

2)1/2 i0 2P ~ 4---R ~ 5~rwo, focus'

(17)

where P is the laser power, I0 is the peak intensity of the static focused beam, and R = r/wo, rocus is the resolution of the scan. The two lowest points in the potential barrier of the trap are located on the optical axis, and the intensity at these points (which corresponds to the trap depth) is Io Iz -- 2eR2

P Jrer 2 .

(18)

Hence, the trap depth depends only on the rotation radius and not on the resolution. The length of the trap is L = 2zRv/2R 2 - 1, which gives a trapping volume of 4 V ~ 89 = 2zR2v/ZR2 - 1 -~3 (19) 3Jr

The scanning is realized either with two perpendicular acousto-optic scanners (AOS) (Friedman et al., 2000a) or with two mechanical scanners (Rudy et al., 2001), by electronically modulating the deflection angle. The possibility of electronically controlling the shape of the trap is an important advantage of the scanning-beam trap. It is possible to create traps of different shapes by changing the modulating signal, as was demonstrated by creating "optical billiards" for cold atoms (Milner et al., 2001; Friedman et al., 200 lb), as will be discussed in Sect. V.C. Moreover, it is possible to dynamically change the shape and size of the trap during an experiment, as demonstrated by compression of a cold atomic cloud to very high densities (see Sect. V.A). A scanning-beam optical trap was demonstrated by Friedman et al. (2000a) using a linearly polarized laser with a power of 200mW, scanned at a rate of 100 kHz. Stable trapping of atoms was obtained for a detuning in the range 1 x 104)' < 6 < l x 1 0 6 ) ' above resonance, and a trap radius of r = 24-105 ~tm (R = 1.5-6.5). For r = 100 ~tm and 6 = 1.2 x 1057' the potential height was "~60Erec as compared to the atomic kinetic energy which was -25Erec. More than 106 8SRb atoms were loaded into this trap from a MOT, and 3.5x 106 atoms were loaded into a deeper trap with 6 = 4 x 104. The lifetime of the trap was 600ms, limited by collisions with background gas. Several measurements were performed in order to prove that the potential can be regarded as a timeaveraged potential. The radial and axial oscillation frequencies of atoms in the trap were measured and found to be in very good agreement with the predicted frequencies for a time-averaged potential. In addition, the trap stability was studied by measuring the trap lifetime as a function of the scanning frequency

124

N. F r i e d m a n et al.

[IV

of the laser beam (see Fig. 11 b). A large trap (r = 67 ~tm) was found to be stable for scan frequencies larger than -20 kHz, while for a smaller trap (r = 32 ~tm) stable trapping was achieved only for higher frequencies (above -60 kHz). As the scan frequency is decreased, stability is reduced in a gradual way. One reason for this gradual behavior is the velocity distribution of the trapped atoms, since slow atoms are easier to trap even with a slowly scanning beam. However, numerical simulations for a monoenergetic ensemble reveal that the stability of the trap depends not only on the velocity of the atoms, but also on the exact position in phase space, and on the shape of the trap. The stability of the scanning-beam trap is an interesting subject, which will probably be further studied. A spin-relaxation time of rSR = 380 ms was measured for the trapped atoms by investigating the spontaneous Raman scattering. Using the branching ratio for the experimental parameters, this corresponds to a total photon scattering rate of 7 s-1 . In another realization of a scanning-beam optical trap (Rudy et al., 2001), a 500-mW laser beam was scanned with mechanical scanners at lower frequencies (2-11kHz). This trap had larger dimensions (w0 = 200~tm, r = 1.5mm), which made it more stable at low scanning frequencies, but demanded lower detuning ( o n l y 1 x 103),) to achieve a sufficient height of the trapping potential. To increase the height of the potential barrier, the scanning beam was re-imaged after passing through the vacuum chamber and crossed the original in an orthogonal direction, such that a nearly spherical trapping volume was achieved, with a potential height of 100Erec. Up to 8x 105 Na atoms were loaded into the trap with a lifetime o f - 5 0 m s , limited by heating due to spontaneous photon scattering from the trapping beam, calculated to be 500 s -1. In the realization of scanning-beam traps, two contradicting requirements exist: a fast scan and a high resolution. Mechanical scanners are usually limited to scan frequencies below 10 kHz. Acousto-optic scanners are capable of much faster scans, up to a few 100 kHz, but with nonlinear scans the resolution is decreased at high frequencies due to the chirp of the acoustic grating over the laser beam. As demonstrated recently by Friedman et al. (2000b), this limitation can be corrected through the use of two counter-propagating acoustic waves, such that the chirp is canceled.

D. COMPARING DIFFERENT TRAPS The appropriate design considerations, and the advantages or disadvantages of the various trap schemes, depend on the specific experimental details (the application, the atomic species, etc.). Hence, no absolute preference of one configuration over the others, nor a general recipe for an optimization procedure can be presented. Nevertheless, the purpose of this section is to shed some light

IV]

D A R K OPTICAL TRAPS F O R C O L D ATOMS

125

on the common design considerations and trade-offs usually met when choosing a certain scheme and optimizing its performance. As a specific example, we assume that the requirement is to trap most o f the atoms from a magneto-optical trap (MOT) into the dipole trap, and minimize the spontaneous photon scattering rate o f the trapped atoms. We assume a laser with a fixed power P -- 1 W and a sample o f alkali atoms, laser-cooled in a MOT to a temperature o f ks T ,.~ 25Erec, and forming a nearly spherical cloud with radius -0.5 mm. We will compare the performance o f three dark traps o f the different classes described previously: a trap based on a Laguerre-Gaussian (LG) beam (Kuga et al., 1997), a trap based on a scanning beam (Friedman et aL, 2000a), and one based on a diffractive axicon element (the double-conical trap, Kaplan et al., 2002a). Adopting a criterion o f > 90% geometrical loading efficiency from the MOT, we choose a radius r - 0.5 mm for all traps. The beam waist is chosen as w0 = 50 ~tm (and therefore R = 10) for the scanning-beam trap, and w0 = 10 ~tm (and therefore R = 50) for the double-conical trap. The length o f the latter is an independent parameter, chosen as L = 3 m m to optimize the power distribution as explained in Sect. IV.B. For the LG trap, a LG 3 mode is assumed, with w0 - 0.5 mm. For comparison, we look also at a red-detuned trap, formed by two focused Gaussian beams, intersecting at a right angle 9 (Adams et al., 1995). We neglect the enhanced loading efficiency of red-detuned traps (Kuppens et al., 2000) (see Sect. II.B), and assume for the crossed red-detuned trap w0 = 0.6 mm, which corresponds to > 90% overlap with the MOT. Following Grimm et al. (2000), we introduce the parameter to, defined as the ratio of the ensemble-averaged potential and kinetic energies of the trapped atoms, tr <Epot>/ (Ek >, and refer to it as the "darkness factor" o f the dark trap. Assuming a trapped atomic gas in thermal equilibrium, and neglecting gravity, the ensemble-averaged potential energy is given by =

fdr(U(r)-Uo)

<Epot)=

e x p [ - U ( r ) - U0]

kbT fdrexp[-U(r)-kSTUo]

,

(20)

with U(r) the three-dimensional dipole potential function, U0 the potential at the bottom o f the trap, and the integration taken over the entire trap volume. 3 The ensemble-averaged kinetic energy is (Ek> = ~ksT. To generalize the darkness factor also for red-detuned traps, we use a modified definition o f

We choose a crossed trap, and not a simpler focused Gaussian beam trap, since with a single focused beam a trap radius of 0.6 mm will result in an extremely large axial size of > 1 m.

9

126

N. F r i e d m a n

[IV

et al.

Table I Required detuning, calculated atomic darkness factor and mean spontaneous photon scattering rate for Rb atoms confined in the traps analyzed in the text Trap Red-detuned trap "Laguerre-Gaussian" trap

Detuning [nm] -0.7 0.23

to'

()'s) [s-1 ]

4.9

166.9

0.6

87.6 21.2

Scanning-beam trap

0.19

0.2

Double-conical "optimized" trap

4.69

0.02

0.09

to' = ] U o / ( E k ) + tr For r e d - d e t u n e d traps U0 < 0, while U0 = 0 for m o s t b l u e - d e t u n e d traps 10.

The e n s e m b l e - a v e r a g e d s p o n t a n e o u s p h o t o n scattering rate is then given in terms o f the darkness factor by 3 kb T . K.t. ( Ys ) = - ~Y -~

(21)

The detuning in the c o m p a r i s o n is c h o s e n such that the trap depth is 3 times larger than the m e a n kinetic energy o f the atoms. Since a fixed laser p o w e r is assumed, less efficient traps w o u l d require s m a l l e r detunings to provide the s a m e trap depth. In Table I, the required d e t u n i n g is p r e s e n t e d for Rb a t o m s in each o f the different traps, with the p a r a m e t e r s discussed above. Also s h o w n are the numerically calculated to', and the m e a n s p o n t a n e o u s p h o t o n scattering rate, (Ys), which d e t e r m i n e s the heating and d e c o h e r e n c e rates o f the trapped atoms. As expected, all b l u e - d e t u n e d traps have a better d a r k n e s s factor than the reddetuned trap. Their scattering rates are smaller as well, even for traps requiring a smaller detuning. The d o u b l e - c o n i c a l trap has a significantly better d a r k n e s s factor (to' = 0.02) than all other schemes. The a d v a n t a g e o f the d o u b l e - c o n i c a l trap is even larger w h e n the scattering rate is considered, since the i m p r o v e d darkness factor is c o m b i n e d with the efficient distribution o f optical p o w e r that enables an increased detuning ll. We p e r f o r m e d a similar calculation also for a

10 For example, for an harmonic trap given by U = Uo + ax 2 + by 2 + cz 2, the ensemble-averaged potential and kinetic energies are equal, and therefore the darkness factor is always tr ~ = 1 for a blue-detuned trap, independent of laser power, detuning or atomic temperature, but tr t cx Uo/kB T for a red-detuned trap with U0 >> kBT. This is the case for the crossed red-detuned trap considered here, for which tr = 1.1, while tr r = 4.9. 11 A scattering rate comparable to that achieved with the double-conical trap could be achieved with the LG trap, with a similar trap resolution R-50. But since for a LG mode R-x/7 an extremely highorder mode would be required, which is experimentally impractical. On the other hand, as a result of the large contribution of light in the out-of-focus regions, the darkness ratio of the scanning-beam trap depends only very moderately on R.

V]

DARK OPTICAL TRAPS FOR COLD ATOMS

127

3D blue-detuned optical lattice, composed of three orthogonal standing waves. Here, a power of gi W was taken for each of the standing waves, and other parameters were w0 = 737 ~tm, 6 = 0.25 nm, such that the lattice potential at the MOT radius (0.5 mm) was 3 times the kinetic energy of the atoms. We then assumed further cooling of the atoms to the ground state of the lattice, and calculated the scattering rate, averaged over the lattice (inside the MOT volume). The result is (Ys) -- 29.1 s-1, similar to that in the rotating-beam trap and considerably higher than in the double-conical trap. In the next section, other considerations for applying dark traps for precision measurements are discussed, such as collision-induced level shifts, which are dramatically reduced inside a dark lattice (Chin et al., 2001). So far, we have neglected the effects of gravity. Assuming gravity is in the radial direction, the relatively large dimensions of our traps result in a large maximal gravitational potential of ~500Erec (larger than the assumed trap depth), which can seriously affect the loading efficiency, and increase the temperature of the atoms. We neglect this excess energy, assuming that it is efficiently dissipated by some cooling mechanism (e.g. PGC) during the loading process. In this case, the only effect of gravity is to modify the distribution of the atoms inside the trap. Including gravity in our calculations results in an increase of 10-60% for the scattering rate for Rb atoms in the above dark traps. Similar calculations were performed also for Na atoms. Since Na atoms have a much smaller mass than Rb atoms, and a similar resonance frequency, their recoil energy is much higher, hence deeper traps are needed. For a fixed power of the trap laser smaller detunings are required, resulting in higher scattering rates. An additional consequence of the smaller mass is that the scattering rate of trapped atoms is less affected by gravity. In our calculations, the inclusion of gravity yields an increase of no more than 5% in the scattering rate for Na atoms (and even less for the lighter Li atoms).

V. Applications Dark optical traps offer a relatively interaction-free environment for the confined cold atoms. This property has opened up a way for several applications, including precision spectroscopic measurements and the preparation and investigation of atomic samples at high spatial and phase-space densities. Recently, the dynamics of atoms inside a dark optical trap has been studied as a versatile experimental realization of the well-known billiard system. All these applications are discussed in this section, with a focus on their experimental demonstrations. A. MANIPULATIONS IN PHASE SPACE: COOLING AND COMPRESSION In this subsection we describe the main cooling mechanisms that have been realized inside dark optical traps. These include polarization-gradient cooling,

128

N. Friedman et al.

[V

Raman cooling and evaporative cooling, and also the reflection cooling mechanism, which is unique for blue-detuned traps. Cooling inside a trap is favorable since density increases as the atoms get colder, hence the gain in phase-space density is larger. Moreover, coupling between the different directions which results from trap anisotropy or collisions between the atoms makes it possible to achieve 3D cooling while performing laser cooling along only one dimension. As opposed to magnetic traps, in an optical dipole trap it is possible to trap and cool atoms independently of their magnetic sublevel, so that a BEC composed of different m-states can be achieved (Barrett et al., 2001), it is possible to investigate a BEC in an arbitrary magnetic field (Inouye et al., 1998) and to study spinor condensates composed of atoms at different m-states (Stenger et al., 1998). Moreover, in some atomic species such as Cs, the lowest ground state cannot be trapped in a magnetic trap, and those states that can be trapped suffer from a very high inelastic loss rate which limits the achievable phase-space density below the BEC transition (S6ding et al., 1998). As a result, experimental effort is directed towards cooling Cs atoms in optical traps. Experimental and theoretical effort has been devoted to find the limitations of the various cooling schemes, and many heating and loss mechanisms have been identified and investigated. [Relevant examples for the present discussion include Bali et al. (1994), Castin et al. (1998), Winoto et al. (1999), Wolf et al. (2000), Kerman et al. (2000), Kuppens et al. (2000)]. A detailed description of these mechanisms is beyond the scope of this review. Here, we will briefly describe the limiting mechanisms which are relevant for cooling inside dark optical traps. At the end of this subsection we will discuss compression of the trapped atomic cloud, as it can lead to better starting conditions for evaporative cooling, and is also interesting for measurements of cold collisions. A.1. Polarization-gradient cooling

Polarization-gradient cooling is among the most widely used sub-Doppler laser cooling techniques for non-trapped atoms. It was applied also to cool atoms inside bright and dark optical dipole traps. Inside a bright trap, the positiondependent light shift induced by the trapping light might be comparable to the shift induced by the cooling laser, hence cooling efficiency is reduced. In dark traps, this effect is suppressed due to the lower interaction with the trapping light. A first investigation of the effect of a dark trap on the PGC process was performed with metastable neon atoms guided inside a focused, blue-detuned hollow beam (Kuppens et al., 1998). A cold atomic beam was adiabatically focused by the hollow guiding laser beam, and hence heated from -5Erec to more than 1000Erec. When 2D PGC was applied just before the focus of the beam, sub-Doppler cooling of the atoms was observed. A residual interaction of the atoms with the guiding beam alters the optical pumping of the cooling light.

V]

DARK OPTICAL TRAPS FOR COLD ATOMS

129

As a result, a higher intensity is required in the cooling beams to overcome the light shift of the guiding beam, and cooling efficiency is reduced with increasing depth of the guiding potential. PGC was also applied to 3D dark traps. In some experiments it was used to suppress heating due to photon scattering of the trapping light which was relatively close to resonance (Torii et al., 1998; Ovchinnikov et al., 1998). Torii et al. (1998) applied pulsed PGC to reduce loss of atoms due to inelastic lightassisted collisions. Here, as well, the achievable temperature was slightly higher than in free-space PGC with the same parameters and atomic densities. In the work of Ovchinnikov et al. (1998) the PGC beams were continuously on, but the cooling had an effective low duty cycle because no repumping laser was used, hence atoms spent most of the time in a dark state and the trapping laser supplied a slow repumping. The main limit to optical cooling inside traps is densitydependent heating and loss, which are discussed at the end of this subsection. A.2. R a m a n cooling

Raman cooling was first used to cool untrapped alkali atoms below the photonrecoil limit in one dimension (Kasevich and Chu, 1992) and then in two and three dimensions (Davidson et al., 1994). This cooling method is based on accumulating cold atoms in a velocity dark state. In alkalis, the scheme is realized by transferring atoms between the two hyperfine levels of the ground state, using pulses of counter-propagating Raman laser beams. The parameters of these pulses are chosen such that atoms in different velocity classes (except a velocity class around zero) are transferred to the upper level. A repumping beam transfers the atoms back to the lower level, via spontaneous emission. This process is continued until most of the atoms accumulate in the lower level, in the dark velocity class near v = 0. This scheme was successfully applied to sodium atoms trapped in the invertedpyramid dark trap described in Sect. III.A (Lee et al., 1996). Since the trap mixes the motion in the three spatial dimensions, cooling in 3D was achieved by applying the Raman beams in only one dimension. In the trap, 4.5 • 105 atoms were cooled to a temperature of 0.4Trec at a final density of 4• 1011 cm -3. The phase-space density was increased by a factor of 320 as compared to the MOT, to a final value of n/laB ,~ 6• l 0 -3 [with AdB the thermal de-Broglie wavelength], which is the highest yet achieved inside a dark optical trap. Since the atoms are trapped, their velocity might change during their interaction with the Raman pulse, resulting in motional sidebands which reduce the velocity selectivity of the Raman pulses, and limit the achievable temperature as compared with untrapped atoms. In a later experiment, a modified scheme was used to simultaneously Raman-cool the atoms and optically pump them into one magnetic sublevel of the lower hyperfine state (Lee and Chu, 1998). The modified cooling scheme resulted in a slightly higher temperature and also some loss of trapped atoms,

130

N. Friedman et al.

[V

such that the final phase-space density was comparable to that of the previous experiment. For very tight traps, the motional sidebands are resolved, enabling the realization of Raman sideband cooling schemes. Raman sideband cooling was applied so far only to atoms trapped inside red-detuned optical lattices (Hamann et al., 1998; Perrin et al., 1998; Vuleti6 et al., 1998; Kerman et al., 2000), but it can be applied also for dark optical lattices to cool atoms to the lowest vibrational level. Cooling inside optical lattices [using either Raman sideband cooling or PGC (Winoto et aL, 1999)] can also be used to produce a cold and dense source of atoms, to improve loading into optical dipole traps (Han et al., 2001). A.3. Reflection Sisyphus cooling

Reflection Sisyphus cooling is a unique cooling scheme for atoms in bluedetuned dipole traps. It exploits the fact that a different potential height is observed by different internal energy levels of the atom. By scattering a photon during a reflection, an atom may lose some of its kinetic energy to the light field. This cooling mechanism was first proposed (Ovchinnikov et al., 1995; S6ding et al., 1995) and realized (Desbiolles et al., 1996) for traps consisting of evanescent waves. The cooling cycle begins with a bouncing atom that enters the potential in an internal level which feels a high potential value. Close to the classical turning point the atom has a large chance to perform a spontaneous Raman transition into another internal state for which the potential is lower. As a result, the atom loses part of its potential energy to the light field, and is reflected with a lower velocity. The cooling cycle is closed by optically pumping the atom back to the initial state when it is out of the potential, without changing its potential energy. In the case of alkali atoms, the two hyperfine levels of the ground electronic state are used as illustrated in Fig. 4b, for a Cs atom. The average energy loss per reflection, for motion perpendicular to the surface, is given by A E • 1 7 7 ~ - ~2 ~6hf qnp, where 6hf is the frequency difference between the two hyperfine levels and q is the branching ratio as defined in Sect. IV.B. As an example, consider reflection cooling performed inside the gravito-optical surface trap (see Sect. III.B). With the experimental parameters of Ovchinnikov et al. (1997), a Cs atom initially loses on average 6% of its kinetic energy per bounce. The mean time between reflections is rr = 2v• (with g the gravitational acceleration), thus the cooling rate in the case of an evanescent wave is independent of v• and given by , ~

AE• q 6hf mgA fi - rrE• - 3 6 h(b + Ohf ) Y"

(22)

For Cs, one has q = 0.25, dihf = 2Jr • 9.2 GHz, and )' = 2Jr • 5.3 MHz. With the experimental values of 6 = 1 GHz and A = 300nm, the cooling rate is

V]

DARK OPTICAL TRAPS FOR COLD ATOMS

131

/3 ~ 5 • 10-77 ~ 2st x 2.5 Hz. This cooling rate is limited by the relatively long time between inelastic collisions, and is much lower than the cooling rates of Doppler cooling and polarization-gradient cooling. The equilibrium temperature can be estimated by equating the average cooling and heating per bounce. With the above experimental parameters, a temperature of T ~ 10Trec ,~ 2~tK is expected, which is similar to that achieved in PGC. This temperature was experimentally obtained for a sample with a small number of atoms. Higher temperatures were obtained in dense samples, due to multiple photon scattering, caused mainly by stray light from the hollow beam. Although the Sisyphus cooling acts only along the vertical direction, the horizontal direction is also cooled to the same temperature. This coupling is probably due to evanescentwave diffusive reflection from the non-perfect surface of the prism. Reflection cooling was also demonstrated with traveling waves, in the Axicon conical trap (Ovchinnikov et al., 1998). Here, cooling was observed only for larger detunings of the conical beam, 6 = 30GHz. The larger detuning is required in order that the condition np < 1 be fulfilled [np is the number of photons scattered per bounce, see Sect. III.B], but it lowers the cooling rate (22). In traveling-wave dark optical traps with a larger spring constant, such as the scanning-beam trap, the lower cooling rate might be partially compensated by the high rate of reflection from the walls, which is of the order of the oscillation frequency in the trap potential.

A.4. Evaporative cooling Despite the progress of optical cooling schemes, the only way by which BEC has been achieved until now is evaporative cooling. For a detailed review on evaporative cooling, see Ketterle and VanDruten (1996). Briefly, cooling is initiated by inducing a loss of the more energetic atoms from a dense sample, which is followed by rethermalization via elastic collisions, to a lower equilibrium temperature. By gradually decreasing the cutting temperature, the phase-space density of the trapped sample increases while the number of trapped atoms and their temperature decrease. The common way to produce a BEC uses evaporative cooling of atoms inside a magnetic trap. Evaporative cooling was demonstrated for atoms inside a red-detuned optical trap (Adams et al., 1995), and very recently 87Rb atoms were cooled below the condensation limit in a quasi-electrostatic crossed trap (two crossed CO2 laser beams), creating for the first time a BEC in an all-optical way (Barrett et al., 2001). Evaporative cooling was studied also inside dark optical traps, with Cs atoms inside the gravito-optical surface trap (see Sect. III.B) (Hammes et al. 2000, 2001). The starting conditions provided by efficient loading and reflection cooling w e r e 10 7 atoms, at a temperature of 52Trec (10~tK), and a density of 6 • 1011 c m -3. The sample was unpolarized in the seven sublevels of the F = 3 lower hyperfine level, hence the corresponding phase-space density was -10 -5.

132

N. Friedman et al.

[V

Evaporation was forced by lowering the dipole potential of the evanescent wave by gradually increasing its detuning. This simultaneously reduced photon scattering and loss due to light-assisted collisions. In the experiment, the laser detuning was changed from ~l.5x103y up to ~5x104y within 4.5 seconds. At the end of the evaporation ramp, the temperature of the atoms dropped to 1.5Trec (300 nK), the number of atoms in the trap was ~3 z 104, and their density was ~6x 10 l~ cm -3. The phase-space density was increased by a factor of 30 to 3 x 10 -4. Since the potential in the vertical direction is linear with z due to gravity, the density increases with decreasing temperature as T -1, only slightly less than the T -3/2 dependence in a harmonic potential. The phase-space density is thus proportional to N T -5/2, as was confirmed experimentally. The regime of runaway evaporation was not reached in the experiment. The evaporation was limited by heating caused by residual on-resonant light from the evanescent-wave laser, which was greatly reduced by passing the laser beam through a heated cell of Cs vapor. Another limiting mechanism was the interaction between the atoms and the imperfect surface of the prism at low potentials. A.5. Compression

When the volume of a trap is reduced, the atomic density and temperature are increased. The increase in density leads to better starting conditions for evaporative cooling since the cooling rate is limited by the elastic collision rate or, in the hydrodynamic limit, also by the trap oscillation frequency (Han et al., 2001; Ketterle and VanDruten, 1996), which are both increased with compression. Compression of magnetic traps is a common procedure in many BEC production schemes. In optical traps, compression might have a larger effect since even higher densities and oscillation frequencies can be achieved. For example, in the first realization of an all-optical BEC formation (Barrett et al., 2001), the evaporative cooling time was only 2 s, as compared to > 10 s usually needed in magnetic traps. Compression was demonstrated in the scanning-beam dark optical trap described in Sect. IV.C. Since the shape of the trap is controlled electronically, it can be changed easily at a desired rate. As an example, when the trap radius was decreased from 100 ~tm to 27 ~m in 150ms, a 350 times adiabatic increase in spatial density was observed. These results were improved by adding a PGC pulse during the compression, which resulted in a cloud of 106 atoms at a density of 2x1013 cm -3, an axial temperature of 75Trec and a radial temperature of 110Tr~. This represented a x 130 increase in spatial density and a x 16 increase in phase-space density over the initial conditions, to a value of 1.2x 10-4. An interesting effect in this context is that even an adiabatic change in the potential shape can lead to a change in the maximal phase-space density if the potential functional dependence is modified, and the elastic collision rate is high

V]

DARK OPTICAL TRAPS FOR COLD ATOMS

133

enough to allow for thermalization. For a potential which is approximated as U ( x , y , z ) = axX nx + ayy ny + azz nz, an adiabatic change in the exponents will lead to a change in phase-space density which can be expressed as exp( ]"final- ~'initial), where ), = nx 1 + ny 1 + nz 1. This effect was first observed for magnetically trapped hydrogen by Pinkse et al. (1997). In the scanning-beam trap experiment, it was demonstrated by decreasing the final radius to 24 ~m, for which the trap is nearly harmonic in all directions (as compared to nx = ny = 28, nz -- 6 in the initial 100 ILtmtrap). In this experiment, 3.5 • 106 atoms were compressed to a final density of 5 • 1013 cm -3, and the phase-space density was increased by a factor of ~4, in agreement with the above calculation. Loss of atoms during the compression was negligible, ensuring that no evaporation process caused the observed gain in phase-space density. Compression can be accomplished also by mechanical movement of a lens, as was recently demonstrated for a red-detuned crossed dipole trap (Han et al., 2001), and can be applied to dark optical traps based on axicons or diffractive optical elements, as well. A.6. H e a t i n g a n d loss

In this section we briefly discuss the main heating and loss mechanisms for dark optical traps. Heating can result either from interaction with the trapping light itself, or due to photon re-absorption when laser cooling is applied to dense atomic samples. Loss may result from interaction of atoms with the environment and with the trapping light, or from inelastic collisions between atoms. Finally, light-assisted collisions, where two colliding atoms interact with the trap light, also contribute to trap loss. Photon scattering: A major source for heating in optical traps is photon scattering from the trapping light. To minimize the scattering rate, traps with large detunings are favorable, and dark traps have an advantage over bright traps (see Sect. IV.D). From a practical viewpoint, it is important to reduce the amount of stray light scattered into the dark region. Very small amounts of residual on-resonance light in the trap laser beam might also lead to heating, and should be filtered in cases where the latter should be kept minimal. Other heating sources: Intensity or pointing instabilities of the trapping laser (Savard et al., 1997), and quantum diffractive collisions with background gas in the vacuum chamber (Bali et al., 1999) cause heating in both bright and dark optical traps. However, in some experiments the measured heating rate exceeds the estimated rate based on these processes (Han et al., 2001), indicating that some other heating mechanisms may exist. Density-dependent heating: In optically dense samples, reabsorption of spontaneously scattered photons during a laser-cooling process causes heating, and limits the attainable equilibrium temperature. A quantitative estimation of this effect was obtained experimentally for PGC performed on Cs atoms in

N. Friedman et al.

134

[V

free space (Boiron et al., 1996). In this experiment, a density-dependent increase in temperature, at a rate of 0.6~tK/10 l~ cm -3, was measured. Density-dependent heating limits also sub-recoil laser-cooling techniques, such as Raman cooling (Perrin et al., 1999), since the dark state is only dark with respect to the cooling laser photons, but not with respect to spontaneously emitted photons, which have a different frequency spectrum. The effect of photon re-absorption can be reduced in elongated traps with a large surface to volume ratio, since the photons have a higher probability to escape from the trap before being rescattered. This was demonstrated for a reddetuned dipole trap (Boiron et al., 1998), where atoms at a density of 1012 cm -3 where cooled to a temperature of 2 ~tK, about 30 times colder than a free-space sample with the same density. Cooling becomes limited again when the optical density of the cloud in the transverse (smallest) direction is larger than unity, as was demonstrated by applying PGC in a compressed dark trap (Friedman et al., 2000a). A cooling pulse was able to cool the atoms to their free-space molasses temperature up to a density which resulted in a radial optical density of the order of one, while for higher densities optical cooling failed. Density-dependent heating was observed also for reflection cooling in the evanescent-wave trap (Hammes et al., 2000), for densities that corresponded to an optical density larger than unity even in the small direction. Recently, suppression of density-dependent heating was observed and investigated in tightly confining red-detuned optical lattices, using either PGC (Winoto et al., 1999) or Raman sideband cooling (Han et al., 2000; Vuleti6 et al., 1998; Kerman et al., 2000). The proposed mechanisms for this suppression should also be valid for a tightly confining dark trap. Loss: The loss of atoms from a trap can be approximated as

dN_dt

aN(t)-fifvn2(r't)d3r-cJvn3(r't)d3r'

(23)

where n is the atomic density, and integration is over the trap volume (Grimm

et al., 2000). The first term corresponds to loss processes which do not depend on the atomic density, mainly collisions of the trapped atoms with hot background atoms in the vacuum chamber. In ultra-high vacuum chambers, the backgroundlimited lifetime can be of the order of tens of seconds. In shallow traps, heating of the atoms is also translated into trap loss. In tightly confining traps, the high density results in two-body loss mechanisms, which are described by the second term in Eq. (23). The third term corresponds to three-body loss which plays a role only at very high densities. The density-dependent loss can be observed and quantified by measuring the decay of the number of trapped atoms as a function of time. As an example, the decay curves for atoms from the compressed scanning-beam dark trap (Friedman et al., 2000a) are presented in Fig. 12a, for atoms in either the lower or the upper

V]

DARK OPTICAL TRAPS FOR COLD ATOMS

!

06

F,=2

135

Ur

F,=3 I I

10~ ~ 104

I

U=(R)

I~

s+p

oro

o'.5

I'.o

Time [sl

(a)

~15

i.o

I I I

(!) S+S

' R

(b)

FIG. 12. (a) Measured decay of the number of trapped atoms from a scanning-beamtrap, for atoms in either of the two hyperfine levels of the ground state. A high density was obtained by compression of the trap, and resulted in two-body collision loss. (From Friedman et al., 2000a, Phys. Rev. A 61, 031403(R), Fig. 4). (b) Diagram illustrating the light-assisted collision loss mechanism, for two atoms colliding in the presence of a blue-detuned optical field. A pair of atoms in the ground state (1) approach each other. At the Condon point (Rc) the laser is in resonance with a repulsive molecular excited state. The pair might be excited by the laser (2), and reach the turning point (Rtp). Then, the atoms are repelled (3) and, if not brought back to the ground state by the laser, they share a gain in kinetic energy which is asymptotically equal to h6. (From Suominen, 1996, J. Phys. B 29, 5981, Fig. 4b).

ground-state hyperfine level. The data are well fitted by the solution to Eq. (23) neglecting the third term (e = 0). The two-body loss coefficients found from the fit are flF=3 = 2.0• 10-ll cm3/s and flF=2 = 1.2• 10 -ll cm3/s. The larger twobody collision loss from the upper hyperfine level is due to hyperfine exchange collisions, since the energy difference between the two ground-state hyperfine levels in this case (85Rb) is "-4 • 105Erec, which is much higher than the trap depth of ~ 103Erec. For this reason, it is important to keep atoms in the lower hyperfine level in high-density traps. The two-body loss from the lower hyperfine level is attributed to light-assisted collisions. When two atoms collide in the presence of a light field, absorption of a photon will transfer them into an excited molecular state. In the case of red-detuned light, excitation is possible into an attractive molecular state, which gives rise to loss processes (Weiner et al., 1999; Suominen, 1996) like radiative escape and photoassociation. Blue-detuned light can excite the colliding atoms into a repulsive molecular state (Bali et al., 1994) (see Fig. 12b). The atoms are then accelerated along the repulsive potential curve and obtain a kinetic energy which is asymptotically equal to the detuning of the exciting laser from the atomic resonance. This energy is usually much larger than the potential barrier of the trap, hence both atoms will be lost. Light-assisted binary collisions in the presence of blue-detuned light were further investigated in the gravito-optical surface trap (Hammes et al., 2000). Another blue-detuned beam was applied

136

N. Friedman et al.

[V

to the trap, and the induced two-body loss coefficients,/3, were measured for different detunings and intensities of this "catalysis" laser. The loss was found to be proportional to//62, in the range 5-80 GHz.

B. PRECISIONMEASUREMENTS The strong suppression of Doppler and time-of-flight broadening due to the ultralow temperatures, and the possibility to obtain very long interaction times, are obvious advantages of using cold atoms for precision optical and rf spectroscopic measurements. To obtain long atomic coherence times, spontaneous scattering of photons and energy-level perturbations caused by the trapping laser should be reduced. This is achieved by increasing the laser detuning from resonance and trapping the atoms in dark traps. These advantages were demonstrated already in the first experimental realization of a dark optical trap for cold atoms (Davidson et aL, 1995), where the ground-state hyperfine splitting of sodium, dihf, was measured with a very long coherence time of 4s, yielding a linewidth of 0.125Hz. The magnetic-field insensitive transition between the IF = 1,mF = 0) and IF = 2, mF = 0) states was excited with a ~l.77-GHz linearly polarized rf wave. A magnetic field parallel to the rf polarization direction separated the required transition from the magnetic-field sensitive transitions. During the experiment, the trap was loaded with atoms which were optically pumped to F = 1. Then, the rf sequence was applied and the number of atoms making the transition was measured by a stateselective fluorescence detection of atoms in the F = 2 state. The rf transition was excited using Ramsey's method of separated oscillatory fields (Ramsey, 1956) by applying two Jr/2 pulses separated by the measurement time T. The resulting central Ramsey fringes are shown in Fig. 13a, together with a sinusoidal fit. The fit yields a fringe contrast of 43%, which was found to decay exponentially with T, with a decay constant of 4.4 s. The uncertainty in the line central frequency was + 1.3 mHz, for 200 data points collected during 900 s. This is higher than the shot-noise-limited frequency sensitivity of the Ramsey method, which is given by A v = (4:r2NtT) -1/2, where N is the number of atoms and t is the integration time. The accuracy resolution of the spectroscopic measurement are limited by the interaction of atoms with the trapping laser field. First, this interaction causes an average shift in the line center, since it shifts the energy levels of the atom (in proportion to I/6). This ac Stark shift is different for the two levels used in the experiment, due to the (very small) difference in the detuning. (For linearly polarized light, the dipole matrix elements are identical for all sub-levels of the ground state.) As a result, the ac Stark shift of the hyperfine transition is lower than the optical Stark shift by a factor of 6/6hf, which is ~ 4.5 • 10 4 in this case. In the above experiment, a linear dependence of the Stark shift on the trapping

V]

DARK OPTICAL TRAPS FOR COLD ATOMS

137

Hyperfine light shift (Hz)

0

350

0.11 I

0.22 I

0.33 I

0.44

1"0-i; .

.-. 300 r

= 250 ~.~ 200

8 " 150 8

O

u_

0.5

100 50

0 -0.3

0.0 -0.2

-0.1

0.0

0.1

f- 1,771,626,130 Hz

(a)

0.2

0.3

"- ~

I

0

5000

.

.... "'"'" ~ 1 7 6 "'" , o . . I,,

~, ~ ,,.

I

10000 15000 Light shift (Hz)

20000

(b)

FIG. 13. (a) Central Ramsey fringes of the IF = 1,mf = 0 ) ~ IF = 2, mf = 0 ) rf transition, measured in the "V"-shaped trap with a 4s measurement time. (b) Calculated ac Stark shift distributions for atoms stored in blue-detuned dipole traps. The dotted line corresponds to the "V"-shaped trap, and the solid line is for a more ergodic inverted pyramid trap (composed of three laser beams). (From Davidson et al., 1995, Phys. Reo. Lett. 74, 1311, Fig. 4).

laser intensity was observed, resulting in a 270-mHz shift of the line center in the displayed data. Second, it is important to note that this shift is not equal for atoms which have different trajectories in the trap, yielding an inhomogeneous distribution of Stark shifts that limits the coherence time of the trap. Hence, coherence time is related to the dynamics of the trapped atoms. A more chaotic trap will increase temporal averaging between the atoms and lead to a narrower distribution and longer coherence times. This averaging effect was calculated using numerical Monte Carlo simulations; the results are presented in Fig. 13b. In our discussion, we neglected the contribution of spontaneous photon scattering to the decoherence rate. This is justified because )'s is smaller than the ac Stark shift by the factor t~hf/Y ~ 170-2000 for most alkalis. This means that by the time it takes for a spontaneous scattering event to occur, the inhomogeneous phase shift of the rf transition due to the ac Stark shift is already hundreds of 2Jr radians. The Stark shift in the dark optical trap is a great limitation for its performance as an atomic clock 12. On the other hand, it seems that a dark optical trap is a very good candidate for precision experiments in atomic physics, such as paritynonconservation and permanent electric dipole moment (EDM) measurements (Bijlsma et al., 1994). Such tabletop experiments are very appealing for tests

12 This limitation can be partially solved by using an additional, very weak, optical field, which is spatially mode-matched to the trap laser and whose detuning is in the middle of the hyperfine splitting. The relative Stark shift introduced by this laser compensates that introduced by the trap (Kaplan et al., 2002b).

138

N. Friedman et al.

[V

of the standard model and extensions of it. In some of these theories, for example, a non-zero EDM value is predicted which is within experimental reach. Measurement of EDM with optically trapped cold atoms was proposed and discussed in two recent papers (Romalis and Fortson, 1999; Chin et al., 2001), and dark optical traps were found to be a promising tool for performing such measurements with much higher sensitivity than is currently available. The use of cold atoms can overcome the two limiting factors of current experiments, namely systematic errors due to the atomic velocity in beam experiments, and leakage currents in cell experiments. The EDM measurement is based on measuring a possible energy shift between two Zeeman sublevels when a static electric field is applied. Therefore, any perturbation of the Zeeman sublevels should be kept minimal. In dark optical dipole traps, the limits on the accuracy of the measurement are due to interactions between the trapping light and the atoms, which cause frequency shifts between the Zeeman sublevels. The three leading terms of these interactions are (Romalis and Fortson, 1999): a vector shift caused by a residual circular polarization of the trapping laser; tensor shifts which result from the interaction of the atoms with the trapping light in the presence of the static electric field; and a thirdorder effect which represents interaction of an induced electric dipole with the laser field through magnetic-dipole or electric-quadrupole interaction. The enhanced cross section for cold atomic collisions may result in frequency shifts (Gibble and Chu, 1993; Bijlsma et al., 1994) and should also be avoided. These limiting factors have been analyzed by Romalis and Fortson (1999) for the case of cesium atoms confined in either a red- or a blue-detuned dipole trap. The vector light shift can be lowered by reducing the residual circular polarization of the trapping beam, and aligning the beam propagation direction perpendicular to the static magnetic field (which defines the quantization axis). When the trapping laser is detuned above resonance, destructive interference between the amplitudes of the vector light shift from two resonance lines lowers the total shift 13. The tensor shift is eliminated at a "magic angle" (54.7 ~ which satisfies 3 cos 2 r 1 = 0) between the direction of the electric field of the light and the quantization axis, or by measuring a IF, mF) ~ I F , - m F ) transition (as suggested by Chin et al., 2001), for which the tensor shift, which depends on m~, vanishes. In another work, Chin et al. (2001) have made a detailed error analysis for a specific experimental realization of an EDM measurement, for Cs atoms in a dark optical lattice. The calculation was made for a lattice which is realized with a 532-nm laser (detuning of 4 x 107}I above the Cs 6S1/2 ---+ 6P3/2 transition). The optimal trap depth is "~130Erec, and the photon scattering rate for atoms in the lattice ground state is -~7x 10-3 s-1. The proposed lattice

13 For Cs, it actually vanishes for two wavelengths: 464 and 474nm, but no sufficiently strong laser lines are available at these wavelengths.

V]

DARK OPTICAL TRAPS FOR COLD ATOMS

139

FIG. 14. (a) Proposed experimental configuration for a measurement of the electron's electric dipole moment using Cs atoms trapped in a dark optical lattice. The lattice is composed of three linearly polarized standing waves, with different frequencies. The polarizations of the beams are perpendicular to the quantization axis defined by the external electric and magnetic fields along the z-axis. (b) Ground-state tunneling rate (solid line) and scattering rate for blue- (dotted) and red-detuned (dashed) 3D lattices. Lattice detuning is assumed constant, and the lattice depth is changed by increasing the laser intensity. The arrow marks the operation point of the proposed scheme, where tunneling and scattering rates are equal for a blue-detuned lattice. (From Chin et al., 2001, Phys. Rev. A 63, 033401, Figs. 1, 2). configuration has 3 linearly polarized standing waves at different directions (see Fig. 14a), which have a frequency difference of few MHz between them. In this way, the polarization is effectively linear in every point in the lattice. The optical lattice is a very flexible trap, and its parameters (beam polarization and propagation directions) are chosen in a way which minimizes systematic errors and decoherence processes. The energy difference between the IF = 3, m F = 2) and IF = 3, m F = --2) states can be measured in the Ramsey method, using pulses of circularly polarized light to induce two-photon Raman transitions inside the F = 3 level of the ground state. The main advantage of using a lattice is the great reduction in collision rate, since atoms are isolated in different lattice sites, and the collision rate is then limited by tunneling between neighboring sites. In Fig. 14b, tunneling and scattering rates for the ground states of a redand a blue-detuned lattice are plotted. When the lattice is made deeper while keeping its detuning constant, the tunneling rate is decreased but the scattering is increased (see Eq. 8). The working point is chosen so as to equalize the two rates. As can be seen from the figure, the tunneling rate at the working point is 20 times lower in the blue-detuned lattice than in the red-detuned one. For a singly occupied lattice with the chosen parameters, and atoms cooled to the ground state (e.g. by Raman sideband cooling, see Sect. V.A.2), the collision rate is lowered by a factor of 106 as compared to free atoms with the same bulk density and kinetic energy. Another advantage of the lattice is that

140

N. Friedman et al.

[V

a different atomic species (Rb, for example) can be trapped in the same lattice, and serve as a "co-magnetometer," to investigate and correct potential systematic effects induced by the external fields. It is estimated that using this system with 108 trapped atoms, a 1 s coherence time, and a measurement time of 8 hours, the sensitivity of the EDM measurement will be 100 times better than current experiments. The main limitations are stringent requirements on the intensity and polarization stability of the trapping laser, and the 3rd-order polarizability effect. Another application for which dark optical traps are considered (Kulin et al., 2001) is parity-nonconservation measurements (PNC). For these experiments, the radioactive alkali atom francium appears as a good candidate, since it is predicted to have a large PNC effect (18 times larger than Cs). In the last few years, Fr was efficiently trapped in a MOT (Simsarian et al., 1996; Lu et al., 1997), and its energy structure was investigated spectroscopically (Sprouse et al., 1998). Precision spectroscopy experiments on Fr atoms in a dark optical trap seem feasible, and might improve experimental tests of the standard model. Dark traps are useful also for spectroscopic measurements of extremely weak optical transitions. While preserving long atomic coherence times, those traps can provide large spring constants and tight confinement of trapped atoms, which ensure good spatial overlap even with a tightly focused excitation laser beam. Therefore, the atoms can be exposed to a much higher intensity of the excitation laser, yet being relatively unperturbed by the trapping light. This yields an increased sensitivity for very weak transitions, and especially for multi-photon transitions. This property was demonstrated by measuring a twophoton transition in cold Rb atoms trapped in a scanning-beam optical trap, with a very weak probe laser (Khaykovich et al., 2000). A spectroscopy scheme which exploits the long spin-relaxation time of the dark trap was used. In this scheme (see Fig. 15a), atoms with two ground-state hyperfine levels (Ig,), Ig2)) are stored in the trap in a level Ig,) that is coupled to the upper (excited) state, [e), by an extremely weak transition which is excited with a laser. An atom that undergoes the weak transition may be shelved, through a spontaneous Raman transition, in ]g2), which is uncoupled to the excited level. After waiting long enough, a significant fraction of the atoms will be shelved in Ig2)- The detection benefits from multiply excited fluorescence of a strong cycling transition from the shelved level ]g2). Thanks to the use of a stable ground state ([g2)) as a "spin shelf," the quantum amplification is limited only by spin-relaxation processes which are strongly suppressed in a dark trap. This scheme was realized on the 5S1/2 ----+ 5D5/2 two-photon transition in cold and trapped 85Rb atoms (see Fig. 15a for the relevant energy levels). The trapped atoms were optically pumped to the lower (F = 2) hyperfine level. The spectroscopy was made with an extremely weak (25 ~tW), frequency-stabilized laser beam, which excited the two-photon transition. The scanning-beam trap was loaded and then compressed to a lower radius (Friedman et aL, 2000a), to best

V]

DARK OPTICAL TRAPS FOR COLD ATOMS

• •

5D,a le)

6P,~__~_~ /

F'--4

6p,~ - ~ a - ~ / 777.9tim

~

5P~ 5P,a

o .g

,..

q

F=3 ig~...,~ F--2 I ~

:

0.30 5Sla

(a)

':

,~

~ 0.40

0.35

780.24mn

F,=3

,.',,

0.45

///

420.:t

141

5

9

F'--2

q'

k }

b)

I'=

~

10

' ~ V'=l

15

20

4 25

Frequency [MHz]

(b)

FIG. 15. (a) Energy levels of 85Rb and the relevant transitions for a two-photon spectroscopy experiment inside a scanning-beam dark trap. Spectroscopy of the Igl) ~ ]e) transition ( 5 S 1 / 2 , F = 2 --~ 5 D 5 / 2 , F t in the case of 85Rb) is performed. Atoms which undergo the transition are shelved in the level ]g2) (5S1/2, F = 3), from which they are detected using a cycling transition (to 5 P 3 / 2 , F = 4). (b) Measured spectrum of the 5 S I / 2 , F = 2 ~ 5 D 5 / 2 , F t = 4,3,2, 1 two-photon transition. Each point corresponds to an experimental cycle, in which atoms are exposed during 500ms to a 25~tW excitation laser. (From Khaykovich et al., 2000, Europhys. Lett. 50, 454, Figs. 1, 4b).

overlap with the excitation laser. After 500 ms, the fraction of atoms transferred to F = 3 was measured. In Fig. 15b, the measured F = 3 fraction is presented as a function of the frequency of the two-photon laser. The hyperfine splitting of the excited state was resolved, and the measured frequency differences and relative line strengths are in a good agreement with theory and previous measurements (performed with much higher laser intensities). A transition rate as small as 0.09 s -I was measured, with a "quantum rate amplification," due to spin shelving, o f ~ 1 0 7 . This optical spectroscopy technique can be applied for other weak (forbidden) transitions such as optical clock transitions (Ruschewitz et al., 1998; Kurosu et aL, 1998) and parity-violating transitions where a much lower mixing with an allowed transition could be used.

C. DYNAMICS OF THE TRAPPED ATOMS

In a dark optical trap, atoms move freely in the dark region, and reflect elastically from the trapping potential. The similarity of this system to the well-studied billiard problem has recently led to the realization of "atom-optics billiards" (Milner et al., 2001; Friedman et aL, 2001b) in which cold atoms move inside dark traps of various shapes. The motion of the atoms is governed by the shape of the trap, and can exhibit different types of dynamics, from regular to chaotic. The billiards were formed as scanning-beam traps, using two perpendicular acousto-

142

N. F r i e d m a n et al.

[V

FIG. 16. Classical numerical simulations of two-dimensional trajectories of Rb atoms in atom-optics billiards of various shapes. A 16 ~tm blue-detuned laser beam scans along the boundary (shown in the figure) at a 100kHz rate. (a) Circular billiard: an example of a nearly periodic trajectory, which requires an increasinglylong time to sample all regions on the boundary. (b) "Tilted" Bunimovich stadium: the motion is chaotic, every trajectory would reach a certain region on the boundary with a comparable time scale. (c) Elliptical billiard: an example of an "internal" trajectory, which is confined by a hyperbolic caustic and thus excluded from a certain part of the boundary. The other type of trajectories ("external," not shown) reaches every region on the boundary. (From Friedman et al., 2001b, Phys. Rev. Lett. 86, 1518, Fig. 1). optic scanners, as described in Sect. IV.C. In some cases, a stationary bluedetuned standing wave was applied along the optical axis, to confine the atomic motion to two-dimensional planes. In this case the atoms are tightly confined near the node planes of the standing wave, forming "pancake"-shaped traps separated b y - 4 0 0 n m (half the wavelength of the standing-wave laser), all of them with a nearly identical billiard potential in the radial direction. The dynamics of the trapped atoms was probed by opening a small hole in the boundary, and measuring the decay of atoms from the billiard through this hole. The hole is opened by switching off one of the AOSs, synchronously with the scan. The decay of atoms through the hole depends on their dynamics, as illustrated by a comparison of the atomic trajectories for a circular billiard (Fig. 16a) and a tilted Bunimovich stadium billiard (two semicircles with different radii, connected with straight lines, Fig. 16b). For the circular billiard, in which the motion is integrable (regular), nearly periodic trajectories exist (see Fig. 16a) that require an increasingly long time to sample all regions on the boundary 14. This yields many time scales for the decay through a small hole on the boundary, and results in an algebraic decay at the long-time limit (Bauer and Bertsch, 1990). For the tilted-stadium billiard, phase space is chaotic, hence each trajectory samples every point on the boundary with an equal probability. This results in a pure exponential decay (Bauer and Bertsch, 1990; Alt et al., 1996), with a

14 Exactly periodic trajectories that are completely stable have only a zero measure, and hence can be neglected.

V]

DARK OPTICAL TRAPS FOR COLD ATOMS

143

FIG. 17. Survival probability of atoms in the gravitational wedge billiard, 300 ms after opening the hole, as a function of the wedge half-angle. Lower curve: experimental data. Upper curve: results of a classical numerical simulation (divided by 2 and shifted up by 0.1). The inset shows the measured intensity cross section at the focal plane of the billiard (with a hole) (From Milner et al., 2001, Phys. Rev. Lett. 86, 1514, Figs. 3, 4). (I/e) decay timescale of rc = ( ~ A ) / ( v L ) , with A the billiard area, v the atomic velocity and L the length of the hole (Bauer and Bertsch, 1990)15. In a recent experiment (Milner et al., 2001), cold Cs atoms were trapped inside a "gravitational wedge" billiard (shown as inset to Fig. 17). The dynamical behavior of this billiard can be tuned from stability to chaos with a single parameter, the vertex half-angle of the wedge, 0. Numerical simulations demonstrate that for 0>45 ~ the system is fully chaotic. For 0 < 45 ~ the system has a mixed phase space, i.e. it has chaotic regions coexisting with stable periodic and quasiperiodic trajectories. For 0 = 90~ (n = 3, 4 , . . . ) , the chaotic part of phase space is minimal, and most of the phase space is regular. In between these angles, the fraction of chaos is larger. The billiard was realized experimentally by loading Cs atoms from a MOT into a scanning-beam optical trap with the required wedge shape. (This trap is actually gravity-assisted, similar to the "V"-shape trap described in Sect. III.A). A hole was opened at the apex of the wedge, and the number of atoms left in the trap after 3 0 0 m s was measured. In Fig. 17, the measured survival probability in the billiard is presented as a function of 0. As expected, maxima are observed in the survival probability for 0 = 22.5 ~ and 30 ~ (90~ and 90~ whereas for intermediate angles and above 45 ~ the motion is less stable. These observations are in very good agreement with theory, and with the results of classical numerical simulations of the system, which are also shown. The factor of 2 discrepancy between the simulation and the experiment is attributed either to collisions between atoms,

15 A "tilted" stadium (and not the more common Bunimovich stadium) is used in the experiment in order to reduce the number of nearly stable trajectories (Vivaldi et al., 1983).

144

N. Friedman et al.

[V

FIG. 18. Survival probability of atoms in billiards of various shapes, as a function of time after opening a hole in the boundary. (a) Elliptical billiard. The solid symbols denote the unperturbed case, in which the surviving fraction for the ellipse with the hole on the long side (solid squares) decays much faster than for the hole on the short side (solid circles). The open symbols show the case in which 10~ts velocity randomizing molasses pulses are applied every 3 ms. (b) Decay of atoms from circular and stadium billiards: The decay from the stadium billiard (solid circles) shows a nearly pure exponential decay. For the circle (solid diamonds) the decay curve flattens, indicating the existence of nearly stable trajectories. The solid lines represent numerical simulations, including all the experimental parameters, and no fitting parameters. The dashed line represents exp(-t/rc), where r c is the escape time calculated for the experimental parameters. The insets show CCD-camera images of the billiards' cross sections at the beam focus. The size of the images is 300x 300 ~tm. (From Friedman et al., 2001b, Phys. Rev. Lett. 86, 1518, Figs. 2, 4).

which are not included in the simulation and may decrease the stability, or to imperfections in the trapping beam. In another set of experiments, billiards of various shapes were investigated (Friedman et al., 2001b). First, "macroscopic" separation in phase space was measured, for the elliptical billiard. Here, phase space is divided into two separate regions (Koiller et al., 1996): "external" trajectories that are confined outside elliptical caustics (smaller than the billiard itself but with the same focal points), and "internal" trajectories confined by hyperbolic caustics, again with the same focal points, as shown in Fig. 16c. Hence, if a hole exists at the short side of the ellipse (upper inset of Fig. 18a), atoms in those trajectories remain confined and never reach the hole. Alternatively, all trajectories, excluding a zero-measure amount, reach the vicinity of a hole on the long side of the ellipse (lower inset of Fig. 18a) and hence the number of confined atoms decays indefinitely. Figure 18a shows the measured survival probability for cold Rb atoms in the elliptical optical billiard with a hole on either the long or the short side. At long times, the survival probability for the hole on the short side is much higher than for the hole on the long side, as expected from the discussion above. Next, a controlled amount of randomization was introduced to the atomic motion, by exposing the confined atoms to a series of short PGC pulses (using the six MOT beams). During each pulse, an atom scatters "--30 photons, and hence its direction of motion is completely randomized, whereas the total velocity distribution remains statistically unchanged. The measured decay curves for this case, with a

V]

DARK OPTICAL TRAPS FOR COLD ATOMS

145

PGC pulse every 3 ms, are also shown in Fig. 18a, for the two hole positions of the ellipse. As seen, for the hole on the long side, the randomizing pulses cause little change. However, for the hole on the short side, a complete destruction of the stability occurs, and the decay curves for the two hole positions coincide approximately. The stability decrease was found to be gradual, and the pulse rate required to significantly reduce the stability is approximately one pulse per rc, the average 1/e decay time. Next, decay curves were measured for the circular and the tilted-stadium billiards. Since the atoms were loaded into the billiard from a cloud in thermal equilibrium, their velocity was distributed around zero, with a measured RMS velocity distribution of 1 It)recoil. To better approximate a mono-energetic ensemble of atoms, and to reduce the relative contribution of gravitational energy (~< 50Erecoil), the loading scheme was modified. After loading from the MOT into the trap, the atoms were illuminated with a short pulse of a strong, on-resonance pushing beam perpendicular to the billiard beam. Following this pushing beam, the center of the velocity distribution was shifted to 20t)recoil, while the RMS width was barely changed. The hole was opened 50ms after the push, to allow for a randomization of the direction of the transverse velocity through collisions with the billiard's boundaries. The measured decay curves for the circular and the tilted-stadium billiards, with equal area and hole size, are shown in Fig. 18b. The decay from the circular billiard is slower, indicating the existence of nearly stable trajectories, whereas that of the stadium is a nearly pure exponential. Also shown in the figure are the results of full numerical simulations that contain no fitting parameters. The simulations include the measured three-dimensional atomic and laser-beam distributions, atomic velocity spread, laser-beam scanning, and gravity. It is seen that there is fairly good agreement between the simulations and the data. As opposed to ideal billiards which have an infinite potential wall, optical billiards inherently have a soft-wall potential, which may affect the dynamics (Rom-Kedar and Turaev, 1999; Gerland, 1999; Sachrajda et al., 1998). For example, a soft wall may introduce stable regions into an otherwise chaotic phase space, and create "islands of stability" immersed in a chaotic sea. This structure greatly affects the transport properties of the system, since trajectories from the chaotic part of phase space are trapped for long times near the boundary between regular and chaotic motion (Zaslavski, 1999). This was demonstrated in a recent experiment by Kaplan et aL (2001), who compared the decay from billiards with hard and soft walls. The softness of the billiards' wall was experimentally changed by varying w0 of the scanning beam. In Fig. 19, experimental results for the decay from a tilted stadium billiard with a harder (w0 = 14.5~m) and a softer (w0 = 24~tm) wall are presented. When the hole is located entirely inside the big semi-circle (Fig. 19a), the soft wall causes an increased stability, and a slowing down in the decay curve. When the

146

N. F r i e d m a n et al.

[V

FIG. 19. The effect of wall softness: measured survival probability of atoms in a tilted-stadium shaped billiard, with two different values of wall softness: w0 = 14.5 ~tm (o), and w0 = 24.5 ~tm (+). (a) The hole is located inside the big semicircle. The smoothening of the potential wall causes a growth in stability, and a slowing down in the decay curve. (b) The hole includes the singular point, no effect for the change in w0 is seen. Also shown are results of numerical simulations, with the experimental parameters and no fitting parameters. The dashed line represents exp(-t/rc), the decay curve for an ideal (hard-wall) billiard. The insets show measured intensity cross sections for the soft-wall billiards, in the beam's focal plane. The size of the images is 300• ~tm. (From Kaplan et al., 2001, Phys. Rev. Lett. 87, 274101, Fig. 1).

hole includes the singular point, where the straight line and semi-circle meet, (Fig. 19b), no effect for the change in w0 is seen. These results can be explained by the formation of a stable island around the trajectory that connects both singular points, and a "sticky" region around it. This explanation is supported by numerical simulations, which predict the formation of a stable island around the singular trajectory when the wall becomes soft. Around this stable island there is a "sticky" area in which chaotic trajectories spend a long time before escaping back to the chaotic part of phase space. Similar decay measurements and simulations for a circular atom-optics billiard showed no dependence on w0 in the range 14.5-24 ~m, and no dependence on the hole position. In another work, it was shown that adding a force field across the billiard can also stabilize specific orbits in otherwise chaotic billiards (Andersen et al., 2002a). Two theoretical works (Liu and Milburn, 1999, 2000) are related to the optical billiard system. In these works, the classical and quantum dynamics of a gas of cold atoms trapped inside a circular hollow laser beam or a hollow fiber was investigated, when the intensity of the trapping light is periodically modulated. In this system, chaotic dynamics exists for certain values of the modulation index, and causes atoms to accumulate in rings corresponding to fixed points of the system. The ability to form billiards of arbitrary shape which can also be varied dynamically, together with the precise control of parameters offered by lasercooling techniques, provide a powerful tool for the study of dynamical quantum effects. These effects are expected to become important at lower temperatures and smaller billiards. Very interesting in this context is the investigation of the properties of a BEC trapped in an integrable or a chaotic billiard. Other

VI]

DARK OPTICAL TRAPS FOR COLD ATOMS

147

problems in nonlinear dynamics can also be addressed using atom-optics billiard, including the influence of many-body interactions, external fields and noise on the dynamics of the atoms. The precise control of the atomic motion in these optical traps can also find useful application in stochastic cooling of atoms (Raizen et al., 1998) and precision spectroscopy. In stochastic cooling, the existence of chaotic motion is an important requirement for mixing the velocities of the atoms after a cooling step. In precision spectroscopy, chaotic motion may reduce inhomogeneous line broadening, as described in the previous section. The performance of an echo scheme, in which a Jr-pulse is added in the middle of the measurement time (between the two Jr/2 pulses in the Ramsey method), should also depend on the dynamics (Andersen et al., 2002b), and may be used to investigate connections between dynamics and decoherence (Jalabert and Pastawski, 2001).

VI. Conclusions In this chapter, we have reviewed the main configurations that are used to form dark optical dipole traps, and their principal applications. The formation of a dark, blue-detuned trap is less obvious than that of a red-detuned one, and, as discussed in Sects. III and IV, some of the effort in the last few years was directed towards the generation of improved schemes, which are also easier to implement experimentally. This effort has led to the development of traps with larger volumes, better loading efficiencies, more efficient use of the laser power and a larger darkness factor. We believe that dark optical traps have matured and will now enter into more applications, in which their advantages will be important. These will include precision spectroscopic measurements, where the reduced interaction with the trapping field is crucial, and investigation of atomic dynamics inside atom-optics billiards, both as a model system for quantum and mesoscopic dynamics and as a tool to further improve the accuracy of spectroscopic measurements. Other applications may benefit from the ability to confine atoms with reduced interactions, including quantum information processing in dark optical lattices, and possibly also quantum-optics experiments which require long relaxation times of the atomic spins, such as slow light (Hau et al., 1999; Kash et al., 1999), stopped light (Liu et al., 2001; Phillips et al., 2001), and entangled atomic samples (Julsgaard et al., 2001). Finally, the special light distributions employed to trap cold atoms can also be used to trap electrons (Chaloupka and Meyerhofer, 1999), Rydberg atoms (Dutta et al., 2000) and molecules (Seideman, 1999) using the ponderomotive force, and to manipulate larger objects when used as dark optical tweezers (Sasaki et al., 1992). In all these cases, the advantages of dark traps are twofold: The ability to trap dark-field seeking objects (e.g. electrons, or metallic beads) and the reduced light intensity to which the trapped object is exposed.

148

N. Friedman et al.

[VII

VII. References Adams, C.S., and Riis, E. (1997). Prog. Quantum Electron. 21, 1. Adams, C.S., Lee, H.J., Davidson, N., Kasevich, M., and Chu, S. (1995). Phys. Rev. Lett. 74, 3577. Allen, L., Padgett, M.J., and Babiker, M. (1999). Prog. Opt. 39, 291. Alt, H., Gr~if, H.D., Harney, H.L., Hofferbert, R., Rehfeld, H., Richter, A., and Schard, P. (1996). Phys. Rev. E 53, 2217. Aminoff, C.G., Steane, A.M., Bouyer, P., Desbiolles, P., Dalibard, J., and Cohen-Tannoudji, C. (1993). Phys. Rev. Lett. 71, 3083. Andersen, M.E, Kaplan, A., Friedman, N., and Davidson, N. (2002a). J. Phys. B: At. Mol. Opt. Phys. 35, 2183. Andersen, M.F., Kaplan, A., and Davidson, N. (2002b). quant-ph/0204082. Anderson, B.P., Gustavson, T.L., and Kasevich, M.A. (1996). Phys. Rev. A 53, R3727. Arlt, J., and Padgett, M.J. (2000). Opt. Lett. 25, 191. Arlt, J., Kuhn, R., and Dholakia, K. (2001 a). J. Mod. Opt. 48, 783. Arlt, J., Dholakia, K., Soneson, J., and Wright, E.M. (200 l b). Phys. Rev. A 63, 063602. Bali, S., Hoffmann, D., and Walker, T. (1994). Europhys. Lett. 27, 273. Bali, S., O'Hara, K.M., Gehm, M.E., Granade, S.R., and Thomas, J.E. (1999). Phys. Rev. A 60, R29. Balykin, V.I. (1999). Adv. Atom Mol. Opt. Phys. 41, 181. Balykin, V.I., Minogin, V.G., and Letokhov, V.S. (2000). Rep. Prog. Phys. 63, 1429. Barrett, M.D., Sauer, J.A., and Chapman, M.S. (2001). Phys. Rev. Lett. 87, 010404. Bauer, W., and Bertsch, G.F. (1990). Phys. Rev. Lett. 65, 2213. Beijersbergen, M.W., Allen, L., van der Veen, H.E.L.O., and Woerdman, J.P. (1993). Opt. Commun. 96, 123. Beijersbergen, M.W., Coerwinkel, R.P.C., Kristensen, M., and Woerdman, J.P. (1994). Opt. Commun. 112, 321. Bijlsma, M., Verhaar, B.J., and Heinzen, D.J. (1994). Phys. Rev. A 49, R4285. Boiron, D., Michaud, A., Lemonde, P., Castin, Y., Salomon, C., Weyers, S., Szymaniec, K., Cognet, L., and Clairon, A. (1996). Phys. Rev. A 53, R3734. Boiron, D., Michaud, A., Fournier, J.M., Simard, L., Sprenger, M., Grynberg, G., and Salomon, C. (1998). Phys. Rev. A 57, R4106. Bongs, K., Burger, S., Dettmer, S., Hellweg, D., Arlt, J., Ertmer, W., and Sengstock, K. (2001). Phys. Rev. A 63, 031602. Brennen, G.K., Caves, C.M., Jessen, P.S., and Deutsch, I.H. (1999). Phys. Rec. Lett. 82, 1060. Cacciapuoti, L., de Angelis, M., Pierattini, G., and Tino, G.M. (2001). Eur. Phys. J D 14, 373. Castin, Y., Cirac, J.I., and Lewenstein, M. (1998). Phys. Rev. Lett. 80, 5305. Chaloupka, J.L., and Meyerhofer, D.D. (1999). Phys. Rev. Lett. 82, 4538. Chaloupka, J.L.,'and Meyerhofer, D.D. (2000). J. Opt. Soc. Am. B 17, 713. Chaloupka, J.L., Fisher, Y., Kessler, T.J., and Meyerhofer, D.D. (1997). Opt. Lett. 22, 1021. Chin, C., Leiber, V., Vuletic, V., Kerman, A.J., and Chu, S. (2001). Phys. Rev. A 63, 033401. Cline, R.A., Miller, J.D., Matthews, M.R., and Heinzen, D.J. (1994). Opt. Lett. 19, 207. Cohen-Tannoudji, C., Dupont-Roc, J., and Grynberg, G. (1992). "Atom-Photon Interactions." Wiley, New York. Davidson, N., Lee, H.J., Kasevich, M., and Chu, S. (1994). Phys. Rev. Lett. 72, 3158. Davidson, N., Lee, H.J., Adams, C.S., Kasevich, M., and Chu, S. (1995). Phys. Rev. Lett. 74, 1311. Davidson, N., Ozeri, R., and Baron, R. (1999). Rev. Sci. Instrum. 70, 1264. DePue, M.T., McCormick, C., Winoto, S.L., Oliver, S., and Weiss, D.S. (1999). Phys. Rev. Lett. 82, 2262. Desbiolles, P., Arndt, M., Szriftgiser, P., and Dalibard, J. (1996). Phys. Rev. A 54, 4292. Dutta, S.K., Guest, J.R., Feldbaum, D., Walz-Flannigan, A., and Raithel, G. (2000). Phys. Rev. Lett. 85, 5551.

VII]

DARK OPTICAL TRAPS FOR COLD ATOMS

149

Engler, H., Manek, I., Moslener, U., Nill, M., Ovchinnikov, Yu.B., Schl6der, U., Schfinemann, U., Zielonkowski, M., Weidemfiller, M., and Grimm, R. (1998). Appl. Phys. B 67, 709. Friedman, N., Khaykovich, L., Ozeri, R., and Davidson, N. (2000a). Phys. Rev. A 61, 031403. Friedman, N., Kaplan, A., and Davidson, N. (2000b). Opt. Lett. 25, 1762. Friedman, N., Kaplan, A., and Davidson, N. (2001a). unpublished. Friedman, N., Kaplan, A., Carasso, D., and Davidson, N. (2001b). Phys. Rev. Lett. 86, 1518. Gauck, H., Hartl, M., Schneble, D., Schnitzler, H., Pfau, T., and Mlynek, J. (1998). Phys. Rev. Lett. 81, 5298. Gerland, U. (1999). Phys. Rev. Lett. 83, 1139. Gibble, K., and Chu, S. (1993). Phys. Rev. Lett. 70, 1771. Grimm, R., Weidemuller, M., and Ovchinnikov, Y.B. (2000). Adv. Atom Mol. Opt. Phys. 42, 95. Hamann, S.E., Haycock, D.L., Klose, G., Pax, P.H., Deutsch, I.H., and Jessen, P.S. (1998). Phys. Rev. Lett. 80, 4149. Hammes, M., Rychtarik, D., Druzhinina, V., Moselner, U., Manek-Honninger, I., and Grimm, R. (2000). J. Mod. Opt. 47, 2755. Hammes, M., Rychtarik, D., and Grimm, R. (2001). C.R. Acad. Sci. IV 2, 625. Han, D.J., Wolf, S., Oliver, S., McCormick, C., DePue, M.T., and Weiss, D.S. (2000). Phys. Rev. Lett. 85, 724. Han, D.J., DePue, M.T., and Weiss, D.S. (2001). Phys. Rev. A 63, 023405. Hau, L.V., Harris, S.E., Dutton, Z., and Behroozi, C.H. (1999). Nature, 397, 594. Heckenberg, N.R., McDuff, R., Smith, C.P., Rubinsztein-Dunlop, H., and Wegener, M.J. (1992). Opt. Quantum Electron. 24, $951. Inouye, S., Andrews, M.R., Stenger, J., Miesner, H.-J., Stamper-Kurn, D.M., and Ketterle, W. (1998). Nature 392, 151. Jalabert, R.A., and Pastawski, H.M. (2001). Phys. Rev. Lett. 86, 2490. Julsgaard, B., Kozhekin, A., and Polzik, E.S. (2001). Nature 413, 400. Kaplan, A., Friedman, N., Andersen, M., and Davidson, N. (2001). Phys. Rev. Lett. 87, 274101. Kaplan, A., Friedman, N., and Davidson, N. (2002a). J. Opt. Soc. Am. B 19, 1233. Kaplan, A., Andersen, M.E, and Davidson, N. (2002b). physics~0204082. Phys. Rev. A, in press. Kasevich, M., and Chu, S. (1992). Phys. Rev. Lett. 69, 1741. Kasevich, M.A., Weiss, D.S., and Chu, S. (1990). Opt. Lett. 15, 607. Kash, M.M., Sautenkov, V.A., Zibrov, A.S., Hollberg, L., Welch, G.R., Lukin, M.D., Rostovtsev, Y., Fry, E.S., and Scully, M.O. (1999). Phys. Rev. Lett. 82, 5229. Kerman, A.J., Vuleti6, V., Chin, C., and Chu, S. (2000). Phys. Rev. Lett. 84, 439. Ketterle, W., and VanDruten, N.J. (1996). Adv. Atom. Mol. Opt. Phys. 37, 181. Khaykovich, L., Friedman, N., Baluschev, S., Fathi, D., and Davidson, N. (2000). Europhys. Lett. 50, 454. Koiller, J., Markarian, R., Oliffson Kamphorst, S., and Pinto de Carvalho, S. (1996). J. Stat. Phys. 83, 127. Kuga, T., Torii, Y., Shiokawa, N., and Hirano, T. (1997). Phys. Rev. Lett. 78, 4713. Kulin, S., Aubin, S., Christe, S., Peker, B., Rolston, S.L., and Orozco, L.A. (2001). J. Opt. B: Quantum Semiclass. Opt. 3, 353.

Kuppens, S., Rauner, M., Schiffer, M., Sengstock, K., Ertmer, W., van Dorsselaer, EE., and Nienhuis, G. (1998). Phys. Rev. A 58, 3068. Kuppens, S.J.M., Corwin, K.L., Miller, K.W., Chupp, T.E., and Wieman, C.E. (2000). Phys. Rev. A 62, 013406. Kurosu, T., Zinner, G., Trebst, T., and Riehle, E (1998). Phys. Rev. A 58, R4275. Lee, H.J., and Chu, S. (1998). Phys. Rev. A 57, 2905. Lee, H.J., Adams, C.S., Kasevich, M., and Chu, S. (1996). Phys. Rev. Lett. 76, 2658. Liu, C., Dutton, Z., Behroozi, C.H., and Hau, L.V. (2001). Nature 409, 490. Liu, X.M., and Milburn, G.J. (1999). Phys. Rev. E 59, 2842.

150

N. F r i e d m a n et al.

[VII

Liu, X.M., and Milburn, G.J. (2000). Phys. Rev. A 61, 053401. Lu, Z.-T., Corwin, K.L., Vogel, K.R., Wieman, C.E., Dinneen, T.P., Maddi, J., and Gould, H. (1997). Phys. Rev. Lett. 79, 994. Major, EG., and Dehmelt, H. (1968). Phys. Rev. 170, 91. Manek, I., Ovchinnikov, Y.B., and Grimm, R. (1998). Opt. Commun. 147, 67. Milner, V., Hanssen, J.L., Campbell, W.C., and Raizen, M.G. (2001). Phys. Rev. Lett. 86, 1514. Morsch, O., and Meacher, D.R. (1998). Opt. Commun. 148, 49. Muller-Seydlitz, T., Hartl, M., Brezger, B., Hansel, H., Keller, C., Schnetz, A., Spreeuw, R.J.C., Pfau, T., and Mlynek, J. (1997). Phys. Rev. Lett. 78, 1038. O'Hara, K.M., Granade, S.R., Gehm, M.E., and Thomas, J.E. (2001). Phys. Rev. A 63, 043403. Oron, R., Davidson, N., Friesem, A.A., and Hasman, E. (2000). Opt. Commun. 182, 205. Ovchinnikov, Y.B., Soding, J., and Grimm, R. (1995). JETP Lett. 61, 21. Ovchinnikov, Y.B., Manek, I., and Grimm, R. (1997). Phys. Rev. Lett. 79, 2225. Ovchinnikov, Y.B., Manek, I., Sidorov, A.I., Wasik, G., and Grimm, R. (1998). Europhys. Lett. 43, 510. Ozeri, R., Khaykovich, L., and Davidson, N. (1999). Phys. Rev. A 59, R1750. Ozeri, R., Khaykovich, L., Friedman, N., and Davidson, N. (2000). J. Opt. Soc. Am. B 17, 1113. Perrin, H., Kuhn, A., Bouchoule, I., and Salomon, C. (1998). Europhys. Lett. 42, 395. Perrin, H., Kuhn, A., Bouchoule, I., Pfau, T., and Salomon, C. (1999). Europhys. Lett. 46, 141. Petrich, W., Anderson, M.H., Ensher, J.R., and Cornell, E.A. (1995). Phys. Rev. Lett. 74, 3352. Petrov, D.S., Holzmann, M., and Shlyapnikov, G.V. (2000). Phys. Rev. Lett. 84, 2551. Phillips, D.E, Fleischhauer, A., Mair, A., and Walsworth, R.L. (2001). Phys. Rev. Lett. 86, 783. Piestun, R., and Shamir, J. (1994). Opt. Lett. 19, 771. Pinkse, P.W.H., Mosk, A., Weidemfiller, M., Reynolds, M.W., Hijmans, T.W., and Walraven, J.T.M. ( ! 997). Phys. Rev. Lett. 78, 990. Raab, E.L., Prentiss, M., Cable, A., Chu, S., and Pritchard, D.E. (1987). Phys. Rev. Lett. 59, 2631. Raizen, M.G., Koga, J., Sundaram, B., Kishimoto, Y., Takuma, H., and Tajima, T. (1998). Phys. Rev. A 58, 4757. Ramsey, N.E (1956). "Molecular beams." Oxford University Press, London. Rom-Kedar, V., and Turaev, D. (1999). Physica D 130, 187. Romalis, M.V., and Fortson, E.N. (1999). Phys. Rev. A 59, 4547. Rudy, P., Ejnisman, R., Rahman, A., Lee, S., and Bigelow, N.P. (2001). Opt. Express 8, 159. Ruschewitz, E, Peng, J.L., Hinderthfir, H., Schaffrath, N., Sengstock, K., and Ertmer, W. (1998). Phys. Rev. Lett. 80, 3173. Sachrajda, A.S., Ketzmerick, R., Gould, C., Feng, Y., Kelly, P.J., Delage, A., and Wasilewski, Z. (1998). Phys. Rev. Lett. 80, 1948. Sasaki, K., Koshioka, M., Misawa, H., Kitamura, H., and Masuhara, H. (1992). Appl. Phys. Lett. 60, 807. Savard, T.A., O'Hara, K.M., and Thomas, J.E. (1997). Phys. Rev. A 56, R1095. Schiffer, M., Rauner, M., Kuppens, S., Zinner, M., Sengstock, K., and Ertmer, W. (1998). Appl. Phys. B 67, 705. Seideman, T. (1999). J. Chem. Phys. 111, 4397. Shabtay, G., Zalevsky, Z., Levy, U., and Mendlovic, D. (2000). Opt. Lett. 25, 363. Simsarian, J.E., Ghosh, A., Gwinner, G., Orozco, L.A., Sprouse, G.D., and Voytas, P.A. (1996). Phys. Rev. Lett. 76, 3522. S6ding, J., Grimm, R., and Ovchinnikov, Y.B. (1995). Opt. Commun. 119, 652. S6ding, J., Gu6ry-Odelin, D., Desbiolles, P., Ferrari, G., and Dalibard, J. (1998). Phys. Rev. Lett. 80, 1869. Song, Y., Milam, D., and Hill, W.T. (1999). Opt. Lett. 24, 1805. Spektor, B., Piestun, R., and Shamir, J. (1996). Opt. Lett. 21,456.

VII]

DARK OPTICAL TRAPS FOR COLD ATOMS

151

Spreeuw, R.J.C., Voigt, D., Wolschrijn, B.T., and van den Heuvel, H.B.V. (2000). Phys. Rev. A 61, 053604. Sprouse, G.D., Orozco, L.A., Simsarian, J.E., and Zhao, W.Z. (1998). Nucl. Phys. A 639, 316c. Stenger, J., Inouye, S., Stamper-Kurn, D.M., Miesner, H.-J., Chikkatur, A.P., and Ketterle, W. (1998). Nature 396, 345. Suominen, K.A. (1996). J. Phys. B 29, 5981. Torii, Y., Shiokawa, N., Hirano, T., Kuga, T., Shimizu, Y., and Sasada, H. (1998). Eur. Phys. J. D 1,239. Vivaldi, E, Casati, G., and Guarneri, I. (1983). Phys. Rev. Lett. 51,727. Vuleti6, V., Chin, C., Kerman, A.J., and Chu, S. (1998). Phys. Rev. Lett. 81, 5768. Webster, S.A., Hechenblaikner, G., Hopkins, S.A., Arlt, J., and Foot, C.J. (2000). J. Phys. B 33, 4149. Weiner, J., Bagnato, V.S., Zilio, S., and Julienne, P.S. (1999). Rev. Mod. Phys. 71, 1. Winoto, S.L., DePue, M.T., Bramall, N.E., and Weiss, D.S. (1999). Phys. Rev. A 59, R19. Wolf, S., Oliver, S.J., and Weiss, D.S. (2000). Phys. Rev. Lett. 85, 4249. Xu, X., Kim, K., Jhe, W., and Kwon, N. (2001). Phys. Rev. A 63, 063401. Xu, X.Y., Minogin, V.G., Lee, K., Wang, Y.Z., and Jhe, W.H. (1999). Phys. Rev. A 60, 4796. Yan, M., Yin, J.E, and Zhu, Y.E (2000). J. Opt. Soc. Am. B 17, 1817. Yin, J., Zhu, Y., and Wang, Y. (1998). Phys. Lett. A 248, 309. Yin, J.P., Noh, H.R., Lee, K.I., Kim, K.H., Wang, Y.Z., and Jhe, W. (1997). Opt. Commun. 138, 287. Zaslavski, G.M. (1999). Phys. Today 52(8), 39. Zemanek, P., and Foot, C.J. (1998). Opt. Commun. 146, 119.

This Page Intentionally Left Blank

A D V A N C E S IN ATOMIC, M O L E C U L A R , A N D O P T I C A L PHYSICS, VOL. 48

MANIPULATION OF COLD A TOMS IN HOLLOW LASER BEAMS HEUNG-RYOUL NOH, XINYE XU* and WONHO JHE** School of Physics and Center for Near-field Atom-photon Technology, Seoul National University, Seoul 151-742, South Korea I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Theoretical Models for Cold Atoms in Hollow Laser B e a m s . . . . . . . . . . . . . . . . A. Strict Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Dressed-atom Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Generation Methods for Hollow Laser Beams . . . . . . . . . . . . . . . . . . . . . . . . . . A. Mode-Conversion Method with Cylindrical Lens . . . . . . . . . . . . . . . . . . . . . B. Computer-Generated Hologram Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Spiral Phase-plate Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Geometric Optics Method with Axicons . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Micro-imaging Method for Hollow Fiber Modes . . . . . . . . . . . . . . . . . . . . . . E Near-field Diffraction Method for Hollow Fiber Modes . . . . . . . . . . . . . . . . . IV. Cold Atom Manipulation in Hollow Laser Beams . . . . . . . . . . . . . . . . . . . . . . . A. Atomic Guidance in Hollow Laser Beams . . . . . . . . . . . . . . . . . . . . . . . . . . B. Atomic Fountain with Hollow Laser Beams . . . . . . . . . . . . . . . . . . . . . . . . . C. Atomic Traps with Hollow Laser Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 154 154 157 160 160 161 163 163 166 166 170 170 176 178 188 188

I. Introduction Atom optics has become an active and attractive research field, and numerous novel atom-optical components that use optical or magnetic methods have been developed [1-3]. Although magnetic atom optics is a promising approach to realize coherent atom optics or miniaturized atom-optical elements, atom optics utilizing optical schemes also provides unique and versatile tools for such studies. In particular, optical atom optics becomes more powerful when combined with microscopic atom-optical elements on the surface or even with magnetic atom-optical techniques. Cold atoms have been manipulated by optical dipole potentials produced by a hollow-core optical fiber (HOF)[4-7] and a hollow laser beam (HLB)[8-12].

* Present address: JILA, University of Colorado, Boulder, CO. ** Corresponding author: [email protected] 153

Copyright 9 2002 Elsevier Science (USA) All rights reserved ISBN 0-12-003848-X/ISSN 1049-250X/02 $35.00

154

H.-R. Noh et al.

[II

Guidance of atoms by HLB, in particular, has advantages over that by the evanescent waves in HOF, since the van der Waals attraction due to the fiber walls can be ignored and collisions with background gas are much less probable. Moreover, an HLB configuration can be controlled more conveniently than one with HOE Therefore, the development of efficient optical manipulation of cold atoms is of much interest and importance in applications related to the transfer of trapped cold atoms to a region of dense atoms or to a lower-dimensional space, which can be used for such experiments as high-resolution spectroscopy, atom lithography, atom microscopy, atomic fountains, and cold atoms in confined space. In this review, we present quantitative experimental and theoretical studies associated with optical manipulation of trapped cold atoms by HLB.

II. Theoretical M o d e l s for Cold A t o m s in H o l l o w L a s e r B e a m s In this section, we present two theoretical models describing the motion of cold atoms in laser light, in particular in HLB [13,14]. The optical dipole force, the radiation-pressure force, and the diffusion coefficients are derived. We assume a three-level A-atom that consists of two hyperfine-structure groundstates (]g,) = I1) and Ig2) = 12)) and one excited state (]e) = ]3)). The atom is assumed to interact with a single-mode traveling laser beam of frequency coL. The detunings of the laser wave with respect to the atomic resonant transitions are 6/ = col-coj (j = 1,2), where col (092) is the atomic transition frequency between the ground state I1) (]2)) and the upper state ]3). The hyperfine-structure splitting between two ground-state levels is denoted by 6hfs = co~ --092. The laser field is chosen as a spatially inhomogeneous monochromatic laser wave given by

E = E0 cos (k- r - col t),

( 1)

where E0 = E0(r) is the coordinate-dependent amplitude of the laser field and k = coL/c is the magnitude of the wave vector. A. STRICT KINETIC THEORY In this interaction scheme, the atomic Hamiltonian can be expressed by ]/2 V 2 H = Ho - ~ - D . E,

where Ho describes describes the dipole complete description the equations for the

(2)

the atomic energy levels E 1 , E 2 , E 3 , and the last term interaction between the atom and the laser field. The of the time evolution of an atom can be then given by atomic density matrix in the Wigner representation. The

II]

MANIPULATION OF COLD ATOMS IN HLBs

155

dipole interaction is considered as usual in the rotating-wave approximation [ 15]. To simplify the derivation of the equations of motion, we first write down the initial microscopic density-matrix equations neglecting the spatial variation of the laser beam amplitude E0:

d /933 =

i/r

( ei(k'r-6't) P(-)13 -- e-i(k'r-'5~t) P(-)'~31)

+iK'2 -dt- P22 = itr

dld [ l l

--iK'l

(e i(k'r-6zt) p~;) --e -i(k'r-62t) p(-)) - 2( Yl + 32 e i(k r eat) P32 ..(+) _ ei(k.r-(ht) /d23 ..(+) + 2 72

(e_i(k.r_(~,t) p~l )-

ei(k.r_61t )

pi3 )) + 2y,

Y2)P33

~(n) p(n) 33 d2n,

/

*(n)

p(n) dan, 33

d = iKl ei(k.r_6~ t ) ( p l l ) - p(+)) -dt/331 33 + itr

e i(k'r-6zt) P~l)--( Y1 + 72)P31,

d

ei(k.r-O,t) p(-) 12 - - ( Yl + Y2)P32,

dt P32 =

iir ei(k .r-,~t)( P~2) _ p(+)) 33 +

d Pi2 = i1r e -i(k" r-r t)p(+) dt 32 _

iK'l

(3)

iic2ei(k.r-r

Here the density matrix elements are defined with respect to the timedependent, stationary atomic eigenfunctions such that pr

= (a IP ( r , p + 89 t)I b),

pI~) = (a p (r, p + n h k , t) I b>,

where (a,b) - (1,2,3) and n is a unit vector that defines the direction of the spontaneous photon emission. The halves of the Rabi frequencies, ~ , and the partial spontaneous decay rates, )9, are defined by (j = 1,2) _ f2j 2-

dj3" Eo 2h

2~. = W J p - 4 ~23 m 3 '

3

hc 3"

(4)

In Eqs. (3), the function ~(n) describes the angular anisotropy of the spontaneously emitted photons. In our simplified model that neglects the atomiclevel degeneracy, the function ~(n) can be chosen to be isotropic such that 9 ( n ) - 1/4sz and f ~(n)dZn - 1. Note that the microscopic equations for the considered model scheme do not include the integral term for the ground-state coherence Pl2. Assuming the interaction time exceeds the spontaneous decay time (tint >> l"sp = 1/Wsp), one can expand the density-matrix elements in powers of photon momentum hk. Moreover, when Tint >> Tsp , the Wigner density-matrix elements become the functionals of the Wigner distribution function, w = w(r, p , t ) = }-~3a . = l Paa As a result, one can derive the Fokker-Planck kinetic equation for

H.-R. Noh et al.

156

[II

the Wigner distribution function w(r, p,t), to the second order in the photon momentum hk,

0W (914' Ot + v Or

_

0 (Fw)+ E Op

02

~(Diiw). Opi

(5)

For a three-level atom, the Rabi frequencies are f2j = r v/[/(2[s)s 1/2 (j = 1,2), where F = 27' = 2(7'1 + 7'2), I is the intensity of the laser beam, Is = ,rrhcF/(3A 3) is the saturation intensity, fj. = @/(3areohc3F)l(eldj .eLlgj)l 2 is the relative transition strength, ~. is the resonant transition frequency from le) to ]gj), and eL is the polarization unit vector of the electric field EL. Note that the saturation parameters can be written as G1 = j i G and G2 = J)G, where the reduced saturation parameter are given by G = I/Is. Consequently, for large positive detunings and slowly moving atoms (]k. v I 0). the center of HLB, ff2j tends to vanish and thus the atomic levels approach the uncoupled-state levels. Taking into account the coupling of the dressed atom with the vacuum-field reservoir, responsible for spontaneous emission between adjacent manifolds, one can write a master equation for the density matrix o of the dressed atom, which describes both the internal free evolution of the dressed atom and the relaxation due to the atom-vacuum coupling. If we denote three reduced populations by ;ri(r), corresponding to three dressed states ]i(n)) defined by ~i(r) = ~-~n (i(n) l~ the evolution of 3-gi is described by ~ = ~-]~i~j(-FijJrj. + Fji:ri) with (i, j) = (1,2,3). Here the rate of transfer Fik 2 2 The resulting steadyfrom ]k(n)) to ]i(n - 1)) is given by Fik = ~ - ] 2 Fjaoak3" state solutions are then calculated as MS~t =

q , f 262 qlf262 + qzf162,

(19)

qzflb2 :r~t = qlf262 + qefl 62,

(20)

;r~t = JiJ~GZF4 qlflb2 + qzfzb2 6462 62 q l j~ 62 + q zj] 62.

(21)

160

[III

H.-R. Noh et al.

Consequently, the radiation force, the momentum diffusion tensor, and the dipole force can be obtained as [ 15,17]

Fr=hk

--~

i'

.=

Dii

1 2 k 2 rp~', : gh

Fa

=

(23)

2

- Z ygstvui,

(24)

i=1

where (dN/dt)i = ~-~2=1 rki"7"gst, D st3 -- ~ - ~ : 1 zcSta2i3' and Ui = fhr2G/(86j) Note that by explicit substitution of Eqs. (15), (16) and (19)-(21) in Eqs. (22)-(24), one can recover Eqs. (6)-(8) obtained by the strict kinetic theory. As a result, both independently derived results can be equally employed in numerical simulation of atomic dynamics in HLB [ 13,14].

III. Generation Methods for Hollow Laser Beams A hollow laser beam (HLB) is a laser beam whose intensity along the central axis vanishes, having a doughnut-shaped intensity distribution. HLBs include a ring-shaped TEM~)l mode, high-order Laguerre-Gaussian (LG) beams, highorder Bessel beams, and vortex solitons. Here we review several methods that have been developed to generate an HLB, such as the vortex grating method [ 19], transverse-mode selection [20], direct production from a laser [21 ], the optical holographic method [22], computer-generated holography [23], mode conversion from Hermite-Gaussian to LG by use of two cylindrical lenses [24,25], spiral phase-plate methods [26,27], geometrical methods with axicons [11,28] or a double-cone prism [29], and use of a hollow-core optical fiber (HOF)[30]. A. MODE-CONVERSION METHOD WITH CYLINDRICAL LENS Both the Hermite-Gaussian (HG) and the Laguerre-Gaussian (LG) modes form complete sets of solutions to the paraxial wave equation [31]. The rectangularly symmetric HG modes are described by the product of two independent Hermite polynomials, describing the field distribution in the x and y directions. They are characterized by integer subscripts m and n representing the order of the two polynomials, that is, the number of nodes in the electromagnetic field. In contrast, the circularly symmetric LG modes are similarly denoted by LGlp, where l is the number of 2Jr cycles in phase around the the circumference and (p + 1) is the number of nodes across the radial field distribution.

III]

MANIPULATION OF COLD ATOMS IN HLBs

161

Since the LG mode possesses an azimuthal phase dependence of exp(i/q~), it has a helical wavefront and null intensity along the propagation axis. Therefore, the LG/ beam has an orbital angular momentum of lh per photon [32] as well as a spin angular momentum +h (-h) for a o+-polarized (o--polarized) light beam. The existence of the orbital angular momentum has led to a number of exciting studies such as transfer of the orbital angular momentum to macroscopic objects[33-35], second-harmonic generation[36], interaction with atomic systems [37,38], high-resolution spectroscopy in a magneto-optical trap [39], and blue-detuned optical dipole traps (see Sect. IV.C, and the chapter by Friedman et al. in this volume). The propagation of laser beams with an HG or an LG mode can be described in the usual language of Gaussian beams. In the vicinity of the beam waist, a Gaussian beam experiences a phase shift compared to that of a plane wave of the same frequency. This phase shift lp is called the Gouy phase shift [31] and is given by ~p(z) = (n + m + 1 ) a r c t a n ( z / z R ) for the HGm,n mode, whereas it is ~p(z) = (2p + l + 1)arctan(z/zR) for the LGtp mode, where z is the distance along the axis from the beam waist in each case and zR is the Rayleigh range. If the Gaussian beam is focused by a cylindrical lens, the situation becomes more complex since the Rayleigh ranges in the x - z and y - z planes are not equal, Zp.x ~ ZRy. Such a beam is called an elliptical Gaussian beam, and the corresponding Gouy phase shift for the HGm,n mode is given by

l(Z)

~p(z) = (m + ~) arctan

ZRx

+(n+ 89

(z) ~

ZRy

.

(25)

Note that it is the Gouy phase shift occurring in the presence of a cylindrical lens that forms the basis of the mode converter. The generation of an LG beam was first demonstrated by Beijersbergen et al. [24,25] by transforming an HG mode of arbitrarily high order to an LG mode. They used a mode converter that consisted of two cylindrical lenses. Unlike other methods discussed in the remaining part of this section, this method can produce pure LG modes. Figure 2 shows how the HG~,0 mode rotated at 45 ~ with respect to the x- or y-axis is equivalent to the sum of the HG~,0 and HG0,~ modes, and how these two modes are related to the LG~ mode. Specifically, the LG~ mode can be formed by a superposition of HG1,0 and HG0,1 modes with a phase difference of Jr/2. B. COMPUTER-GENERATEDHOLOGRAM METHOD One can generate an HLB by using a computer-generated hologram (CGH), which is created by the interference between an electromagnetic field of interest and a reference laser beam. An optical holographic method was carried out by Lee et al. [22] to generate a nondiverging hollow beam, which is similar to a J1

162

[III

H.-R. Noh et al.

FIG. 2. Generation of LG1 mode HLB by mode-conversionmethod (from Padgett et al., 1996, Am. J. Phys. 64(1), 77, Fig. 2, reprinted with permission). Bessel beam. By using the CGH method, Heckenberg et al. generated a TEM~)1 doughnut mode [40], a TEM~0 doughnut mode [41], and later an LG~ mode [42]. In particular, by using the LG beam, they demonstrated trapping of reflective and absorptive microscopic particles, which cannot be trapped by using a Gaussian spot due to the strong repulsive forces. Paterson and Smith [23] produced high-order Bessel beams by using an axicon-type CGH, where an azimuthal phase factor, exp(inq)), is added to the phase of the hologram. The Bessel beam [43] is one of the propagation-invariant waves and has an amplitude proportional to Jn(por)exp(in(~), where J,, is the nth-order Bessel function of the first kind, r is the radial coordinate, q~ is the azimuthal coordinate, and P0 is the radial spatial frequency. The zero-order Bessel beam has a sharp intensity peak at its center, while higher-order Bessel waves have zero-intensity minima at their centers. Paterson and Smith calculated the amplitudes of the waves produced by an axicon-type hologram by using the Kirchhoff integral, and experimentally demonstrated the production of Bessel beams of orders 1 and 10. Clifford et al. [44] generated LG laser modes with an azimuthal mode index l ranging from 1 to 6 (p = 0) by using an external cavity diode laser. The transmittance function is given by T(r, 4))= exp[i6H(r, r in polar coordinates, where 6 is the amplitude of the phase modulation, and the holographic pattern is given by H(r, r = ~

mod

lq~ - --~r cos 0, 2~

,

(26)

with mod (a, b) = a - b int(a/b). As the azimuthal index l increases, the inner

III]

MANIPULATION OF COLD ATOMS IN HLBs

163

dark region of the light becomes larger and the outer ring becomes narrower. The conversion efficiency was as high as 40% and the efficiency was claimed to increase by using a phase hologram and blazing it to maximize the power in a chosen diffracted order. In general, when a hologram is irradiated by a fundamental Gaussian mode, the output becomes a superposition of an infinite number of LG modes with the same l and different p. The fraction of p = 0 mode was 78.5% in the first diffracted order. In an analogous method, they also generated multi-ringed (p > 0) LG modes with azimuthal index l = 1 [45].

C. SPIRAL PHASE-PLATE METHOD

Beijersbergen et al. [26] demonstrated that a spiral phase plate can convert a Gaussian laser beam into an LG mode with a phase singularity on its axis. A spiral phase plate is a transparent plate whose thickness increases in proportion to the azimuthal angle q~ around a point in the middle of the plate. If u(p, q~,z) is the complex amplitude of the incident beam, the amplitude u I directly after the plate is given by u' = u exp(-iA/r where Al is the height of the step in wavelengths given by Al = A n h / X , h is the step height at r = 0, An is the difference of refractive index between the plate and its surrounding, and X is the vacuum wavelength. Beijersbergen et al. chose an acrylic (PMMA, n = 1.49) as a phase plate, where h = 0.72mm or Al = 577 at 633 nm wavelength. To make Al = 1, the plate was immersed in a liquid with nearly the same index of refraction. The effective step size was tuned by controlling the temperature, and they obtained Al = 1 with An = 8.7x 10-4. They used an LG ~ or LG~ beam as an incident laser that passed through a phase plate, and the output beam was imaged by a lens in the focal plane. For each incident beam, the output beam was obtained with various values of Al ranging from -1 to 2.5 with a step of 0.5. For the LG ~ mode, a nearly LG~ beam was obtained ( A / = 1). For the LG 1 mode, which itself is already a helical mode, a mode similar to the input beam was obtained when Al = 2. Turnbull et al. [27] generated free-space LG modes at millimeter-wave frequencies (-100 GHz) by using a spiral phase plate. Due to the large frequency difference of-104 with respect to the optical field, the orbital angular momentum is also -104 times smaller. The phase plate was made of polyethylene, which has a refractive index of 1.52 at millimeter-wave frequencies. They could generate LG 1 and LG 2 modes with phase plates of step height 6.7 mm and 13.4 mm, respectively. D. GEOMETRIC OPTICS METHOD WITH AXICONS Since the time Herman and Wiggins [46] used an axicon [47] to produce a

164

H.-R. Noh et al.

[III

FIG. 3. Generation of HLB by axicons (from Manek et al., 1998, Opt. Commun. 14"/, 67, Fig. l, reprinted with permission).

propagation-invariant zero-order Bessel beam, hollow laser beams have been produced by axicons [11,28] or a double-cone prism [29]. Manek et al. [28] generated an HLB for atom trapping by using an axicon in combination with a spherical lens (Fig. 3). They used this method in a recent demonstration of a gravito-optical surface trap for Cs atoms that was based on evanescent-wave cooling (See Sect. IV.C). The axicon has one flat and one conical surface (base angle 0 ~ 10mrad), and is used in combination with a spherical lens (achromatic doublet, f = 100 mm). For small base angles 0 I=

1 boom ((Am) + 1) __ ho)m ((kz) + .~) _ 1 (-Om o)m h ooz ( cot0, and e is a linewidth parameter so that the Larmor frequency depends as ooL = ~Oo(1 + ez 2) on the axial coordinate. A least-squares fit of this function to the data points of Fig. 18 yields the Larmor frequency with a relative uncertainty of 10-6 (Hermanspahn et al., 2000). This is sufficient to measure the binding correction to the g-factor in C 5+. The bound-state QED corrections for C 5+, however, are 4 • 10-7 and were not observed in this measurement.

IV. Double-Trap Technique The limitation in accuracy of the experiment described above stems from the inhomogeneous magnetic field as required for the analysis of the spin direction via the continuous Stern-Gerlach effect. In fact the inhomogeneity of the field was chosen to be as small as possible, but still large enough to be able to distinguish the two spin directions. We obtained an improvement of three orders of magnitude in the accuracy of the measured magnetic moment by spatially separating the processes of inducing spin flips and analyzing the spin direction (H/iffner et al., 2000). This is achieved by transferring the ion after a determination of the spin direction from the analysis trap to the precision trap. The voltages at the trap electrodes are changed in such a way that the potential minimum in which the ion is kept is moved towards the precision trap. The transport takes place in a time of the order of 1 s, which is slow compared to any oscillation period of the ion and is therefore adiabatic. Once in the precision trap, the ion's motional amplitudes are prepared by coupling the ion to the resonant circuits. We then apply the microwave field to induce spin flips. After the interaction time, typically 80 s, and an additional cooling time, the ion is moved back to the analysis trap. Here the spin direction is analyzed again. In principle one measurement of the axial frequency would be sufficient to determine whether it has changed by 0.7 Hz as compared to the value before transport into the precision trap. If, however, the ion is not brought back with the same radial motional amplitudes to the analysis trap, the axial frequency may have changed by as much as 1 Hz. This is because of the magnetic moment connected with the cyclotron and magnetron motion. To circumvent this problem

210

[IV

G. Werth et al. spin flips in 0,4

1,0

analysis trap

0,2 ~"

0,0 ~

(D o

-0,2

:' ,'

", : ',,

-0,4

~.~ =I 0,2

:' :,

~r ~

,

..~

0,4

%

0,0

-0,6

-

)$

-0,2 -0,8 i

0

i

i

1 2

I

3

i

4

5

I

6

I

i

7

0

m i c r o w a v e excitations

i

l

1

2

i

3

i

4

1

5

1

6

i

7

m i c r o w a v e excitations

FIG. 19. D e t e r m i n a t i o n o f the spin direction in the analysis trap after transport from the precision trap. A change in axial frequency o f about 0.7 Hz indicates that the spin was up (left) or d o w n (right) when the ion left the precision trap.

we induce an additional spin flip in the analysis trap to determine without doubt the spin direction after return to the analysis trap. Figure 19 shows several cycles for a spin analysis. The total time for a complete cycle is about 30 min. While the ion is in the precision trap its cyclotron frequency COc = ( q / M ) B is measured simultaneously with the interaction with the microwaves. This ensures that the magnetic field is calibrated at the same time as the possible spin flip is induced. The field of a superconducting solenoid fluctuates at the level of 10-8-10 -9 on the time scale of several minutes. Figure 20 shows a measurement of the cyclotron frequency of the ion in the precision trap over a time span of several hours. Every 2 min the center frequency of the cyclotron resonance was determined. The change in cyclotron frequency has approximately a Gaussian distribution with a full-width-at-half-maximum of 1.2x 10 -8. This may impose a serious limit on the precision of measurements as in the case of high-precision mass spectrometry using Penning traps (Van Dyck et al., 1993; Natarajan et al., 1993). However, the simultaneous measurement of cyclotron and Larmor frequencies eliminates most of this broadening. Using Eqs. (10) and (25) we obtain the g-factor as the ratio of the two measured frequencies (DL m

g=2~--. mc M

(31)

The mass ratio of the electron to the ion can be taken from the literature. In our case of 12C5+, Van Dyck and coworkers (Farnham et al., 1995) measured it with high accuracy using a Penning trap mass spectrometer. We measure the induced spin flip rate for a given frequency ratio of the microwave field and the simultaneously measured cyclotron frequency. When we

IV]

CONTINUOUS

STERN-GERLACH

EFFECT ON ATOMIC IONS

211

807O 6O e--

-I~ 50 .~. ._~ 4o v/v = 1.2. 10-8I 30 Q.

20 10 0

A.~

I

I

I

I

I

'

I

'

I

.

.

.

.

i.

I

-0,6-0,5-0,4-0,3-0,2-0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 change of reduced cyclotron frequency [Hz]

FIG. 20. Distribution of magnetic field values measured by the cyclotron frequency of a trapped ion in a period of several hours. Data were taken every 2 min. The distribution is fitted by a Gaussian with a full width of 1.2 x 10-8. 35 ,-, 3O ~>' 25

.m -0

.0 o r

20

i,..

.m

.~

15 10 5 ~'

-60

~

~

-40

'

I

-20

'

Wmw/We

1

0 -

'

I

20

'

40

w

,

~

I

60 x l 0

6

4 3 7 6 . 2 1 0 499

Fie. 21. Measured spin-flip probability vs. ratio of Larmor and cyclotron frequencies. The data are least-squares fitted to a Gaussian. plot the spin flip probability, i.e. the n u m b e r o f successful at t empt s to c h a n g e the spin direction d i v i d e d by the total n u m b e r o f attempts, we obtain a r e s o n a n c e line as s h o w n in Fig. 21. The m a x i m u m attainable p r o b a b i l i t y is 50% w h e n the a m p l i t u d e o f the m i c r o w a v e field is high e n o u g h . To avoid those saturation effects we take care to k e e p the a m p l i t u d e o f the m i c r o w a v e field at a level that the m a x i m u m p r o b a b i l i t y for a spin flip at r e s o n a n c e f r e q u e n c y is b e l o w 30%. In addition we can take saturation into a c c o u n t using a s i m p l e r a t e - e q u a t i o n m o d e l .

G. Werth et al.

212

[V

In contrast to the single-trap experiment the lineshape is now much more symmetric. For a constant homogeneous magnetic field in the precision trap the lineshape would be a Lorentzian with a very narrow linewidth determined by the coupling constant ~, to the cooling circuit. However, the observed lineshape can be well described by a Gaussian. The fractional full width is 1.1 • 10 -8. This reflects the variation of the magnetic field during the time the ion spends in the precision trap which is of the same order of magnitude (see Fig. 20). The line center can be determined from a least squares fit to 1 x 10-l~

V. Corrections and Systematic Line Shifts The main systematic shifts of the Larmor and cyclotron resonances arise from the fact that the field in the precision trap is not perfectly homogeneous. As mentioned above, the ferromagnetic nickel ring placed 2.7 cm away in the analysis trap causes a residual inhomogeneity in the precision trap. The expansion coefficient from Eq. (21) gives B2 = 8 ~tT/mm 2, three orders of magnitude smaller than in the analysis trap. Therefore we still have to consider an asymmetry in the line profile. Performing such an analysis gives a maximum deviation as compared to the symmetric Gaussian fit of 2• 10 -1~ In addition, the inhomogeneity of the magnetic field causes a shift of the line with the ion's energies. In order to obtain a sufficiently strong signal of the induced current from the cyclotron motion in the precision trap, the ion's energy has to be raised to about 1 eV. This finite cyclotron energy has a large magnetic moment and thus shifts the axial frequency as compared to vanishing cyclotron energy even in the precision trap by about 1 Hz. To account for this shift we grouped our data of the spin flip probabilities according to the different axial frequency shifts in the precision trap corresponding to different cyclotron energies, and extrapolated the ratios o)L/coc to zero cyclotron energy (Fig. 22). We find a slope of A(~oL/~oc)/Ec = -1.09(5)• 10-9 eV -l . Other systematic shifts are less important:

0',

-10-

0

c-q -20t~

~"

-30-402

4

6

8

10

1

14

16

Cyclotron energy E+ [eV] FIG. 22. Extrapolation of measured frequency ratios to vanishing cyclotron energy.

VI]

CONTINUOUS STERN-GERLACH EFFECT ON ATOMIC IONS

213

Table II Systematic uncertainties (in relative units) in the g-factor determination of 12C5+ Contribution

Relative size

Asymmetry of resonance

2 x 10-10

Electric field imperfections

1 x 10- l 0

Ground loops in apparatus

4 x 10 -11

Interact. with image charges

3 x 10 -11

Calibration of cyclotron energy

2 x 1O-11

Sum

2.3 x 10- l 0

From the residual imperfection of the electric trapping field (C4 - l0 -5) we calculate a shift of the cyclotron frequency of 1 x 10 -10. O f the same order of magnitude are frequency shifts caused by changes of the trapping potential due to ground loops when the computer controls are activated. The interaction of the ion with its image charges changes the frequencies by 3 x 10-1~ but can be calculated with an accuracy of 10%. Relativistic shifts are of the order of 10-l~ at typical ion energies, but do not contribute to the uncertainty at the extrapolation to zero energy. A list of uncertainties of these corrections is given in Table II. The quadrature sum of all systematic uncertainties amounts to 3x 10-~~ The final experimental value for the frequency ratio WL/OOCin 12C5+ is COL O9C

- 4376.2104989(19)(13).

(32)

The first number in parentheses is the statistical uncertainty from the extrapolation to vanishing cyclotron energy, the second is the quadrature sum of the systematical uncertainties. Taking the value for the electron mass in atomic units (M(12C) = 12) from the most recent CODATA compilation (Mohr and Taylor, 1999) we arrive at a g-factor for the bound electron in 12C5+ of gexp( 12 C 5+) =

2.001 041 596 3 (10)(44).

(33)

Here the first number in parentheses is the total uncertainty of our experiment, and the second reflects the uncertainty in the electron mass.

VI. Conclusions A comparison of the experimentally obtained result of Eq. (33) to the theoretical calculations presented in Table I shows that the bound-state QED effects of

214

G. Werth et al.

[VII

order a / ~ in hydrogen-like carbon are verified at the level of 5 x 10-3. Bound QED contributions of order (a/:r) 2 are too small to be observed. The nuclear recoil part has been verified to about 5%. It is believed that uncalculated terms of higher-order QED contributions do not change the theoretical value beyond the presently quoted uncertainties. Taking this for granted we can use experimental and theoretical numbers to determine a more accurate value for the atomic mass of the electron, since this represents by far the largest part in the total error budget (Beier et al., 2001). Using Eqs. (6) and (33) we obtain from Eq. (31) the electron's atomic mass as m = 0.000 548 579909 3(3).

(34)

This is in agreement with the CODATA electron mass (Mohr and Taylor, 1999) based on a direct determination by the comparison of its cyclotron frequency to that of a carbon ion in a Penning trap (Farnham et al., 1995): m : 0.000 548 579 911 0(12).

(35)

VII. Outlook The continuous Stern-Gerlach effect, using the frequency dependence of the axial oscillation on the spin direction of an ion confined in a Penning trap when an inhomogeneous field is superimposed, is a powerful tool to measure magnetic moments of charged particles with great precision. This accurate knowledge of magnetic moments is very important for tests of QED calculations. The g - 2 experiment on free electrons by Dehmelt and coworkers (Van Dyck et al., 1987) was a first example, followed now by the first application to an atomic ion. The method described above is applicable to any ion having a magnetic moment on the order of a Bohr magneton, provided it can be loaded into the trap. For a given axial frequency and magnetic inhomogeneity B2, the frequency splitting depends as 1/x/-qM on the mass M of the ion and its charge state q (Fig. 23). This will impose technical limitations when working with heavier hydrogen-like ions. Currently the stability of the electric trapping field limits the maximal resolution of the axial frequency measurements: a jitter of the trapping voltage by 1 ~tV, typical for state-of-the-art high-precision voltage sources, induces frequency changes of 100 mHz for trap parameters as in our case. However, materials with higher magnetic susceptibilities than nickel, such as Co-Sm alloys, produce a larger magnetic inhomogeneity and therefore a larger frequency splitting, allowing to proceed to heavier ions. In addition, the induced magnetic inhomogeneity scales with the cube of the inverse radius of the ring electrode. Thus a reduction in size of the analysis trap increases the

VII] CONTINUOUS S T E R N - G E R L A C H EFFECT ON ATOMIC IONS

215

2,8 2,4 2,o rn--,I t'q

1,6 N

2~Iw/Jrd, the field from the wires is not capable of compensating the bias field. Two side guides are then obtained, one along each wire in the plane of the wires. In the case Bb < 2ltolw/:rd, the gradient in the confining directions is given by

dB d r ro =

r0

--~

Iw d

(8)

If there is a field component Bip along the wire, the position of the guide is unchanged. However, the shape of the potential near its minimum is parabolic: the curvature in the radial direction is given by

d2B dr 2 r0 =

BipI2 de.

(91

In the special case of r0 = d/2, the gradient and, for the case of a non-vanishing curvature of the potential at the minimum position, are exactly equal to the corresponding magnitudes for the single-wire guide.

Bip, the

A.3.2. Co-propagating currents. The magnetic fields formed by two parallel wires carrying equal co-propagating currents vanishes along the central line between the wires and increases and changes direction like a 2-dimensional quadrupole. The wires form a guide as shown in Fig. 3 allowing atoms to be guided around curves (Mfiller et al., 1999). It is even possible to hold atoms in a storage ring formed by two closed wire loops (Sauer et al., 2001)

270

R. Folman et al.

[II

FIG. 3. Atoms are guided in a two-wire guide that is self-sufficient without external bias fields. Insets (a), (b) and (c) show the magnetic field contour lines for no bias, horizontal bias, and vertical bias fields, respectively. Courtesy E. Cornell.

FIG. 4. Potential for a two-wire guide formed by copropagating currents. The plots show from left to right the equipotential lines for increasing bias fields. As the field is raised, two (quadrupole) minima approach each other in the vertical direction and merge at the characteristic bias field denoted by B = 1 into a harmonic (hexapole) minimum. At higher bias fields this minimum splits into a double (quadrupole) well again; this time the splitting occurs in the horizontal direction.

(Sect. III.A.7). When aiming at miniaturized, surface-mounted structures, the fact that the potential minimum is located between the wires rather than above them, has to be considered. When a bias field parallel to the plane of the wires is added, the potential minimum moves away from the wire plane and a second quadrupole minimum is formed at a distance far above the wire plane where the two wires appear as a single wire carrying twice the current (see side guide in Sect. II.A.2). Depending on the distance d between the wires with respect to the characteristic distance / dsplit =

\

Ito Iw 5-~) Bb

(lO)

one observes three different cases (Fig. 4): (i) If d/2 < dsplit, two minima are created one above the other on the axis between the wires. In the limit of d going to zero, the barrier potential between the two minima goes to infinity and the minimum closer to the wire plane falls onto it; (ii) if d/2 = dsplit, the two minima fuse into one, forming a harmonic guide; (iii) if d/2 > dsplit, t w o minima are created, one above each wire. Splitting and recombination can be achieved by simply increasing and lowering the bias field (Denschlag, 1998; Zokay and Garraway, 2000; Hinds et al., 2001).

II]

MICROSCOPIC

ATOM OPTICS: FROM WIRES TO ATOM CHIP

271

Table I Typical potential parameters for wire guides, based on tested atom chip componentsa Atom

Wire current [mA]

Bias fields

Potential Gradient [kG/cm]

Frequency [kHz]

Size [nm]

Lifetime [ms]

25 5 1

32 400 4000

100 570 3300

120 50 21

> 1000 > 1000 7

25 5 1 1

32 400 4000 20000

41 250 1100 3600

53 21 10 6

>1000 > 1000 > 1000 > 1000

Two-wire guideC (counter-propagating currents) Li 1000 80 2 5.4 25 Li 500 200 10 13 5 Li 100 130 l0 8.7 1.5

32 400 870

100 570 1200

120 50 34

> 1000 > 1000 5

Rb Rb Rb

32 400 870

41 250 490

53 21 15

>1000 > 1000 185

Side guide b Li 1000 Li 500 Li 200 Rb Rb Rb Rb

1000 500 200 1000

1000 500 100

Bb [G]

Bip [G]

Depth [mK]

80 200 400

2 10 30

5.4 13 27

80 200 400 2000

1 4 20 50

5.4 13 27 130

80 200 130

1 4 5

5.4 13 8.7

Distance [~tm]

Ground state

25 5 1.5

a The parameters are given for the two different atoms lithium and rubidium, both assumed to be in the (internal) ground state with the strongest confinement (F = 2, m F = 2). For both types of guide, small bias-field components Bip pointing along the guide were added in order to get a harmonic bottom of the potential and to enhance the trap life time that is limited by Majorana spin flip transitions (see Eq. 18 in Sect. V). It was confirmed in a separate calculation that the trap ground state is always small enough to fully lie in the harmonic region of the Ioffe-Pritchard potential. See also Fig. 2. b Side guide created by a thin current-carrying wire mounted on a surface with an added bias field parallel to the surface but orthogonal to the wire. c Two-wire guide created by two thin current-carrying wires mounted on a surface with an added bias field orthogonal to the plane of the wires. In these examples the two wires are 10 ~tm apart.

Finally we mention a proposal by Richmond

et

al.

(1998) where

a tube

c o n s i s t i n g o f two i d e n t i c a l , i n t e r w o u n d s o l e n o i d s c a r r y i n g e q u a l b u t o p p o s i t e c u r r e n t s c a n be u s e d as a w e a k - f i e l d - s e e k e r guide. T h e m a g n e t i c field is a l m o s t z e r o t h r o u g h o u t the c e n t e r o f the t u b e , b u t it i n c r e a s e s e x p o n e n t i a l l y as o n e a p p r o a c h e s the w a l l s f o r m e d b y the c u r r e n t - c a r r y i n g w i r e s . H e n c e , c o l d l o w f i e l d - s e e k i n g a t o m s p a s s i n g t h r o u g h the t u b e s h o u l d b e r e f l e c t e d b y the h i g h m a g n e t i c fields n e a r the w a l l s , w h i c h f o r m a m a g n e t i c m i r r o r . Examples

of typical guiding parameters

for the a l k a l i a t o m s l i t h i u m a n d

r u b i d i u m t r a p p e d in s i n g l e a n d t w o - w i r e g u i d e s are g i v e n in Table I. T r a p f r e q u e n c i e s o f the o r d e r o f 1 M H z

or a b o v e c a n be a c h i e v e d w i t h m o d e r a t e

272

R. Folman et al.

[II

currents and bias fields. The guided atoms are then located a few ~tm from the wire (above the surface).

A.4. Simple traps An easy way to build traps is to start from the guides discussed above, and close the trapping potential with 'endcaps'. This can be accomplished by taking advantage of the fact that the magnetic field is a vector field, and the interaction potential is scalar (Eq. 1). By varying the angle between the wire and the bias field, one can change the minimum of the potential and close the trap. Simple geometries are either a straight guide and an inhomogeneous bias field, or a homogeneous bias field in combination with a bent wire.

A.4.1.

Straight guide and an inhomogeneous bias field. Traps formed by superposing an inhomogeneous bias field and the field of a straight wire are based on quadrupole fields because the complete change of direction in addition to the inhomogenity is needed to close the trap. An interesting fact is that a currentcarrying wire on the symmetry axis of a quadrupole field can be used to 'plug' the zero of the field. In this configuration a ring shaped trap is formed (Fig. 5a) that has been demonstrated experimentally (Denschlag, 1998; Denschlag et al., 1999a). In the Ttibingen (formerly Munich) group of C. Zimmermann a modified version of this type of trap with the wire displaced from the quadrupole axis

FIG. 5. Creating wire traps: The upper row shows the geometry of various trapping wires, the currents and the bias fields. The lower column shows the corresponding radial and axial trapping potential. (a) A straight wire on the axis of a quadrupole bias field creates a ring-shaped 3-dimensional non-zero trap minimum. (b) A "U"-shaped wire creates a field configuration similar to a 3-dimensional quadrupole field with a zero in the trapping center. (c) For a "Z"-shaped wire a Ioffe-Pritchard type trap is obtained.

II]

MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP 273

(Fortagh et al., 1998, 2000) was used to create a Bose-Einstein condensate on an atom chip (Ott et al., 2001). A.4.2. Bent wire traps: the U- and Z-trap. 3-dimensional magnetic traps can be created by bending the current-carrying wire of the side guide (Cassettari et al., 1999; Reichel et al., 1999; Haase et al., 2001). The magnetic field from the bent leads creates endcaps for the wire guide, confining the atoms along the central part of the wire. The size of the trap along this axis is then given by the distance between the endcaps. Here we describe two different geometries: (1) Bending the wire into a "U"-shape (Fig. 5b) creates a magnetic field that in combination with a homogeneous bias field forms a 3-dimensional quadrupole trap 4. The geometry of the bent leads results in a field configuration where a rotation of the bias field displaces the trap minimum but the field always vanishes completely at this position. (2) A magnetic field zero can be avoided by bending the wire ends to form a "Z" (Fig. 5c). Here, one can find directions of the external bias field where there are no zeros in the trapping potential, for example when the bias field is parallel to the leads. This configuration creates a Ioffe-Pritchard type trap. The potentials for the U- and the Z-trap scale similarly as for the side guide, but the finite length of the central bar and the directions of the leads have to be accounted for. Simple scaling laws only hold as long as the distance of the trap from the central wire is small compared to the length of the central bar (Cassettari et al., 1999; Reichel et al., 1999; Haase et al., 2001). Bending both Z leads once more results in 3 parallel wires. This supplies the bias field for a self-sufficient Z-trap. A.4.3. Crossed wires. Another way to achieve confinement in the direction parallel to the wire in a side guide is to run a current ll < lw through a second wire that crosses the original wire at a right angle (Reichel et al., 2001). Ii creates a magnetic field B~ with a longitudinal component which is maximal at the position of the side guide that is closest to the additional wire. Adding a longitudinal component to the bias field, i.e. rotating Bb, results in an attractive potential confining the atoms in all three dimensions. As a side effect position and shape of the potential minimum are altered by the vertical component of B~. Figure 6 illustrates this type of trap configuration. Experiments of the Munich group have proven this concept to be feasible (see Sect. IV.C.1 and Fig. 34) and it was suggested to use the two-wire cross as a basic module for implementing complex trapping and guiding geometries. 4 The minimum of the U-trap is displaced from the central point of the bar, in a direction opposite to the bent wire leads. A more symmetric quadrupole can be created by using 3 wires in an H configuration. There the side guide is closed by the two parallel wires crossing the central wire orthogonally. The trap is then in between the two wires, along the side guide wire.

274

R. Folman et al.

[II

Flo. 6. Two geometries of crossed-wire traps: different cuts through the potential are displayed without and with a longitudinal bias field component in the left and right column, respectively. The 1-dimensional plots show the potential along the direction of the side guide; in the contour plots the wire configuration is illustrated by light gray bars. Courtesy J. Reichel. A.5. Weinstein-Libbrecht traps Even more elaborate designs for traps than those described previously can be envisioned. For example, Weinstein and Libbrecht (1995) describe planar current geometries for constructing microscopic magnetic traps (multipole traps, IoffePritchard traps and dynamical traps). We focus here on the Ioffe-Pritchard trap proposals. Figure 7 shows four possible geometries: (a) three concentric half loops; (b) two half loops with an external bias field; (c) one half loop, one full loop and a bias field; (d) two full loops with a bias field and external Ioffe

II]

MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP a)

111

b)

275

I1

y 111 cl

111

1! d] |

|

|

|

111 FIG. 7. Four planar (and pseudoplanar) Ioffe trap configurations, as described in the text. Courtesy J. Weinstein/K. Libbrecht. bars. The first of these (a) is essentially a planar analog of the nonplanar IoffePritchard trap with two loops and four bars. Configuration (b) replaces one of the loops with a bias field. Configuration (c) is similar to (b) but provides a steeper trapping potential on-axis and weaker trapping in the perpendicular directions; this makes an overall deeper trap with greater energy-level splitting for given current and size. (d) is a hybrid configuration, which uses external (macroscopic) |offe bars to produce the 2-dimensional quadrupole field, while deriving the onaxis trapping fields from two loops and a bias field. Typical energy splittings in the range of 1 MHz are achievable using experimentally realistic parameters (Drndi6 et al., 1998).

A.6. Arrays of traps The various tools for guiding and trapping discussed above can be combined to form arrays of magnetic microtraps on atom chips. Particularly suitable for this purpose is the technique of the crossed wires which requires, however, a multilayer fabrication of the wires on the surface. Arrays of traps and their applications, especially in quantum information processing, are discussed in Sect. VI.

A. 7. Moving potentials Introducing time-dependent potentials facilitates arbitrary movement of atoms from one location to another. There are different proposals for possible

R. Folman et al.

276

[II

Yl IH2 FIG. 8. Magnetic 'conveyor belt': The wires are configured in a way that allows to transport atoms from one trap to another along a side guide. Together with a homogeneous time-independent bias field, the currents IQ, IH1, and /H2 are used for the confining fields of the source and collecting traps, I0 is the current through the side guide wire. The currents IM1 and IM2 alternate sinusoidally with a phase difference of Jr/2 and provide the moving potential. Courtesy J. Reichel.

implementations of such 'motors' or 'conveyor belts', one of which has already been demonstrated experimentally by Hansel et al. (2001b): Using solely magnetic fields it is based on an approximation of the crossed-wire configuration. Atoms trapped in a side guide potential are confined in the longitudinal direction by two auxiliary meandering wires (Fig. 8). By running an alternating current through both auxiliary wires with a relative phase difference of Jr/2, the potential minimum moves along the guide from one side to the other in a controllable way. In Sect. IV we present experimental results of the above scheme.

A. 8. Beam splitters By combining two of the guides described above, it is possible to design potentials where at some point two different paths are available for the atom. This can be realized using different configurations (examples are shown in Fig. 9) some of which have already been demonstrated experimentally (see Sect. IV).

A.8.1. Y-beam splitters. A side guide potential can be split by forking an incoming wire into two outgoing wires in a Y-shape (Fig. 9a). Similar potentials have been used in photon and electron interferometers 5 (Buks et al., 1998). A Y-shaped beam splitter has one input guide for the atoms, that is the central wire of the Y, and two output guides corresponding to the right and left wires. Depending on how the current Iw in the input wire is sent through the Y, atoms can be directed to the output arms of the Y with any desired ratio. This simple configuration has been investigated by Cassettari et al. (2000) (see Sects. III.A.3 and IV.C.3 for experimental realizations). Its disadvantages are: (1) In a singlewire Y-beam splitter the two outgoing guides are tighter and closer to the surface than the incoming guide. The changed trap frequency and the angle between

5 The Y-configuration has been studied in quantum electronics by Palm and Thyl6n (1992) and Wesstr6m (1999).

II]

M I C R O S C O P I C ATOM OPTICS: F R O M WIRES TO ATOM CHIP

277

FIG. 9. Different wire geometries for a beam splitting potential: The plots show the wire arrangement on the surface of an atom chip, and the directions of current flow and the additional bias field. Each picture also shows a typical equipotential surface to illustrate the shape of the resulting potential. (a) A simple Y-beam splitter consisting of a single wire that is split into two: The output side guides are tighter and closer to the surface than the input guide. Note that a second minimum closer to the chip surface occurs in the region between the wire splitting and the actual split point of the potential; (b) a two-wire guide split into two single-wire guides does not exhibit this 'loss channel'. (c) Here, the output guides have the same characteristics as the input guide, minimizing the backscattered amplitude. The vertical orientation of the bias field ensures exact symmetry of the two output guides. (d) In an X-shaped wire pattern the splitting occurs because of tunneling between two side guides in the region of close approach of the two wires.

incoming and outgoing wires lead to a change o f field strength at the guide minimum and can cause backscattering from the splitting point. (2) In the IoffePritchard configuration (i.e. with an added longitudinal bias field), the splitting is not fully symmetric due to different angles of the outgoing guides relative to the bias field. (3) A fourth guide leads from the splitting point to the wire plane, i.e. to the surface of the chip. The backscattering and the inaccessible fourth guide o f the Y-beam splitter may be overcome, at least partially, by using different beam splitter designs, like those shown in Fig. 9b,c. The configuration in Fig. 9b has two wires which run parallel up to a given point and then separate. If the bias field is chosen so that the height of the incoming guide is equal to the half distance d/2 o f dsplit as defined in Eq. 10 in Sect. II.A.3), the height o f the the wires (d/2 potential m i n i m u m above the chip surface is maintained throughout the device (in the limit of a small opening angle) and no fourth port appears in the splitting region. The remaining problem o f the possible reflections from the potential in the splitting region can be overcome by the design presented in Fig. 9c. Here, a guide is realized with two parallel wires with currents in opposite directions and a bias field perpendicular to the plane of the wires. This type of design creates a truly symmetric beam splitter where input and output guides have fully identical characteristics. =

278

R. Folman et al.

[II

A.8.2. X-beam splitters. A different possible beam splitter geometry relies on the tunneling effect: Two separate wires are arranged to form an X, where both wires are bent at the position of the crossing in such a way that they do not touch (see Fig. 9d). An added horizontal bias field forms two side guides that are separated by a barrier that can be adjusted to be low enough to raise the tunneling probability considerably at the point of closest approach. If the half distance between the wires becomes as small as dsplit (Eq. 10), the barrier vanishes completely, resulting in a configuration that is equivalent to a combination of two Y-beam splitters (Miiller et al., 2000). The choice of the parameters in the wire geometry, the wire current and the bias field governs the tunneling probability and thereby the splitting ratio in this type of beam splitter. The relative phase shift between the two split partial waves in a tunneling beam splitter allows to combine two beam splitters to form a Mach-Zehnder interferometer. Another advantage of the X-beam splitter is that the potential shape in the inputs and outputs stays virtually the same all over the splitting region as opposed to the Y-beam splitter. For a detailed analysis of the tunneling X-beam splitter see Andersson et al. (1999). A.8.3. Quantum behavior o f X- and Y-beam splitters. For an ideal symmetric Y-beam splitter, coherent splitting for all transverse modes should be achieved due to the definite parity of the system (Cassettari et al., 2000). This was confirmed with numerical 2-dimensional wave packet propagation for the lowest 35 modes. The 50/50 splitting independent of the transverse mode is an important advantage over four-way beam splitter designs relying on tunneling such as the X-beam splitter described above. For the X-beam splitter, the splitting ratios for incoming wave packets are very different for different transverse modes, since the tunneling probability depends strongly on the energy of the particle. Even for a single mode, the splitting amplitudes, determined by the barrier width and height, are extremely sensitive to experimental noise. A.9. Interferometers Following the above ideas of position-dependent multiple potentials and timedependent potentials which are able to split minima in two and recombine them, several proposals for chip-based atom interferometers have been put forward (Hinds et al., 2001; H/insel et al., 2001c; Andersson et al., 2002). A.9.1. Interferometers in the spatial domain. To build an interferometer for guided atoms (Andersson et al., 2002) two Y-beam splitters can be joined back to back (Fig. 10a). The first acts as splitter and the second as recombiner. The eigenenergies of the lowest transverse modes along such an interferometer in

II]

MICROSCOPIC

ATOM OPTICS:

FROM

WIRES

TO ATOM CHIP

279

FIG. 10. Basic properties of the guided matter wave interferometer: (a) Two Y-beam splitters are joined together to form the interferometer. (b) Transverse eigenfunctions of the guiding potentials in various places along the first beam splitter. When the two outgoing guides are separated far enough, i.e. no tunnelling between left and right occurs, the symmetric and antisymmetric states become degenerate. (c) Energy eigenvalues for the lowest transverse modes as they evolve along the interferometer. One clearly sees that pairs of transverse eigenstates form disjunct interferometers. (d) The wavefunction of a cold atom cloud starts out in the vibrational ground state of a guide or trap. The wavefunction splits when the guide divides, leaving a part of the wavefunction in each arm of the interferometer. If the phases of the two parts evolve identically on each side, then the original ground state is recovered when the two parts of the wavefunction are recombined. But if a phase difference of Jr accumulates between the two parts (for example due to different gravitational fields acting on them), then recombination generates the first excited state of the guide with a node in the center. Courtesy E. Hinds. (e) 2-dimensional plots of a wave packet propagating through a guided matter wave interferometer for 10) and 11) incoming transverse modes, calculated by solving the time-dependent Schr6dinger equation in two spatial dimensions (x, z, t) for realistic guiding potentials, where z is the longitudinal propagation axis. The probability density of the wave function just before entering, right after exiting the interferometer, and after a rephasing time t are shown for a phase shift of ~ . One clearly sees the separation of the two outgoing packets due to the energy conservation in the guide, e.g. for n = 0 the first excited outgoing state is slower than the ground state.

280

R. Folman et al.

[II

2-dimensional geometry 6 are depicted in Fig. 10c. From the transverse mode structure one can see that there are many disjunct interferometers in Fock space. Each of them has two transverse input modes (]2n) and [2n + 1), n being the energy quantum number of the harmonic oscillator) and two output modes. In between the two Y-beam splitters, the waves propagate in a superposition o f ]n)l and ]n)r in the left and right arm, respectively. With adiabaticity fulfilled, the disjunct interferometers are identical. Considering any one of these interferometers, an incoming transverse state evolves after the interferometer into a superposition o f the same and the neighboring transverse outgoing state (Fig. 10c), depending on the phase difference acquired between [n)l and In)r during the spatial separation o f the wave function 7. While the propagation remains unchanged if the emerging transverse state is the same as the incoming state, a transverse excitation or de-excitation translates into an altered longitudinal propagation velocity (Ao _~ +oo/k where hk is the m o m e n t u m o f a wave packet moving through the interferometer and to/2:r is the transverse trapping frequency), since transverse oscillation energy is transferred to longitudinal kinetic energy, and vice versa. As presented in Fig. 10e, integrating over the transverse coordinate results in a longitudinal interference pattern observable as an atomic density modulation. As all interferometers are identical, an incoherent sum over the interference patterns of all interferometers does not smear out the visibility of the fringes. A.9.2. Interferometers in the time domain. Two different proposals are based on time-dependent potentials (Hinds et al., 2001 ; Hfinsel et al., 2001c). These proposals differ from the interferometer in the spatial domain in several ways: (1) The adiabaticity of the process may be controlled to a better extent due to easier variation of the splitting and recombination time. (2) The interferometers are based on a population of only the ground state. (3) The interference signal amounts to different transverse state populations in the recombined single minimum trap, whereas the above proposal anticipates a spatial interference pattern which may be easier to detect. The first proposal (Hinds et al., 2001) is based on a two parallel wire configuration with co-propagating currents (see Sect. II.A.3). Changing the bias field in this configuration as a function of time produces cases (i), (ii), and (iii) discussed in Sect. II.A.3 depending on the strength of the bias field as compared

In 2-dimensional confinement the out of plane transverse dimension is either subject to a much stronger confinement or can be separated out. For experimental realization see Gauck et al. (1998), Spreeuw et al. (2000), Hinds et al. (2001), Pfau (2001). 7 The relative phase shift Aq~between the two spatial arms of the interferometer can be introduced by a path length difference or by adjusting the potentials to be slightly different in the two arms. In general, Ar is a function of the longitudinal momentum k. 6

II]

MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP 281

to the critical bias field Be = -Y ~o (-3-)" 1~ Starting with Bb < Bc and an atom cloud in the ground state of the upper minimum, a coherent splitting of the corresponding wave function is achieved when Bb is raised to be larger than Bc. As shown in Fig. 10d, the symmetry of the wave function now depends on the relative phase shift introduced between its two spatially separated parts. Thus, when the bias field is lowered again to Bb - Be, a superposition of the symmetric and the antisymmetric state forms in the recombined guide. If the spatial resolution of the detection system is not sufficient to distinguish between the two output states, the following scheme is proposed: The node plane of the excited state is rotated by 90 ~ by turning an additional axial bias field while the guides are combined. If after such an operation the bias field is lowered, atoms in the ground state go to the upper guide whereas the population of the excited state is found in the lower guide. The second proposal (H~insel et al., 2001c) utilizes the crossed-wire concept introduced in Sect. II.A.4. Here, in contrast to the interferometer described above, the splitting of the atomic wavefunction occurs in one dimension whereas the confinement in the other two dimensions is the constant strong confinement of a side guide. Longitudinally, the atoms are trapped by two currents running through wires crossing the side guide wire. The resulting Ioffe-Pritchard potential well is split into a double well and then recombined by a third crossing wire carrying a time-dependent current flowing in the opposite direction. Starting with a wavefunction in the ground state of the combined potential, a relative phase shift introduced between the two parts of the potential after splitting leads to a wavefunction in a (phase-shift dependent) superposition of the ground and first excited states upon recombination. A state-selective detection then displays a phase-shift dependent interference pattern. A detailed analysis of realistic experimental parameters has shown that in this scheme non-adiabatic excitations to higher levels can be sufficiently suppressed. The position and size of the wavefunction are unchanged during the whole process. Therefore, the interferometer is particularly well suited to test local potential variations.

A. 10. Permanent magnets

Although beyond the scope of this chapter, we mention configurations with permanent magnets (Sidorov et al., 1996; Meschede et al., 1997; Saba et al., 1999; Hinds and Hughes, 1999; Davis, 1999). Though less versatile in the sense of not enabling the ramping up and down of fields, permanent magnets might reward us with advantages such as less noise, strong fields, and large-scale periodic structures. As described in Sect. V, technical noise in the currents which induce the magnetic fields may have severe consequences in the form of heating and decoherence. In the framework of extremely low decoherence, such as that

282

[II

R. F o l m a n et al. B, G

(a)

/

~

(b)

2so 200 150 100 50

I B,

G

0.5

1

1.5

2

2.5

3

3.5

25o~

lmm

lOOi \ \

Z,

m

(c) /

50

.... "

~

'

~

'

X

0.5

1

1.5

2

FIc. 11. (a) Two pairs of differently sized magnetic sheets (bottom) are magnetized using current-carrying wires wound around them. The choice of the direction of current flow in these wires establishes the direction of magnetization: the arrows show a possible configuration for which the equipotential lines are plotted (top). (b) The field produced by the sheet pairs measured in the symmetry plane. (c) Scaling of the field due to the combined inner and outer pair of sheets in the plane of symmetry. Courtesy M. Prentiss.

demanded by quantum computation proposals, permanent magnets might be a better choice. An interesting tool is a magnetic atom mirror formed by alternating magnetic dipoles (Opat et al., 1992), creating an exponentially growing field strength as the mirror is approached. This situation can be achieved by running alternating currents in an array of many parallel wires or by writing alternating magnetic domains into a magnetic medium such as a hard disk or a video tape. This has been demonstrated by Saba et al. (1999) and may achieve a periodicity of the order of 100 nm. Current-carrying structures have the disadvantage of large heat dissipation, especially when the structure size is in the submicron region. Another possibility is based on a combination of current-carrying wires and magnetic materials; this was experimentally demonstrated at Harvard in the group of M. Prentiss: Two pairs of ferromagnetic foils that were magnetized by current-carrying wires wound around them were used for magnetic and magnetooptic trapping (Vengalattore et al., 2001). The setup and the potential achieved is illustrated in Fig. 11. The advantages of such a hybrid scheme over a purely current-carrying structure are larger capturing volumes of the traps, less heat dissipation, and enhanced trap depths and gradients because the magnetic field of the wires is greatly amplified by the magnetic material.

II]

MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP

283

The magnets can still be switched by means of time-dependent currents through the wires. B. ELECTRIC INTERACTION

The interaction between a neutral atom and an electric field is determined by the electric polarizability a of the atom. In general, a is a tensor. For the simple atoms we consider, i.e. atoms with only one unpaired electron in an s-state, the electric polarizability is a scalar and the interaction can be written as Vpol(r ) = - ~ l aE 2 (r).

(11 )

B.1. Interaction between a neutral atom and a charged wire We now consider the interaction of a neutral polarizable atom with a charged wire (line charge q) inside a cylindrical ground plate (Hau et al., 1992; Schmiedmayer, 1995a; Denschlag and Schmiedmayer, 1997; Denschlag et al., 1998). The interaction potential (in cylindrical coordinates) given by 1

Vp~

= -

) 2 2aq2

4n'e0

(12)

r2

is attractive. It has exactly the same radial form ( 1/r 2) as the centrifugal potential barrier (VL = LZ/2Mr 2) created by an angular momentum Lz. VL is repulsive. The total Hamiltonian for the radial motion is

H = 2M +2Mr 2 _ p2 -

-

2M -

+

Lz2 _ LcZrit

2Mr 2 '

4Jre0

r2

(13) (14)

where Lcrit = ~ a ]q]/2zrc0 is the critical angular momentum characteristic for the strength of the electric interaction. There are no stable orbits for the atom around the wire. Depending on whether Lz is greater or smaller than Lcrit , the atom either falls into the center and hits the wire (]Lz ] < Lcrit) or escapes from the wire towards infinity ([Lzl > Lcrit). In the quantum regime, only partial waves with hl < Lcrit (l is the quantum number of the angular momentum Lz) fall towards the singularity and thus the absorption cross section of the wire should be quantized (Fig. 12). To build stable traps and guides one has to compensate the strongly attractive singular potential of the charged wire. This can be done either by adding a

284

[II

R. Folman et al.

12 10

"" ...........

8 6 t::~ 4

"

.......

;

.

60

2101

58 -

208

56-

206

54

~

.. ii.: i{ 9

It"

52

t~,i,~,-~

9 " "

"" 50

"

M":"

~'tP

0 0

1i

2,

3i

4i

5

48 ' 25

21 6 2 1 7

'

204 ~

kRw = 0 kR w =

0.1

..........

--

kR w =

1

.....

kR w =

2

.....

kR w =

5

2I8

2t9

202 ,'r 200

30

198 , 100

, 101

, 102

, 103

, 104

105

Line charge q in units of mcrit FIG. 12. Theoretical absorption cross section for a charged wire. The calculations are made for several different relative thicknesses (kRw) of the wire; the charge is given in units of the angular momentum mcrit = Lcrit/h.

repulsive potential, for example from an atom mirror or an evanescent wave (see Sect. II.C.1), or by oscillating electric fields (see Sect. II.B.2). B.2. Stabilizing the motion with an oscillating electric charge: the Kapitza wire

The motion in the attractive electric potential can be stabilized by oscillating the charges. The mechanism is similar to the RF Paul trap (Paul, 1990) where an oscillatory part of the electric fields creates a 3-dimensional confinement for ions. An elementary theoretical discussion of the motion in a sinusoidally varying potential shows that Newton's equations of motion can then be integrated approximately, yielding a solution that consists of a fast oscillatory component superimposed on a slow motion that is governed by an effective potential (Landau and Lifshitz, 1976). An example of a 2-dimensional atom trap based on a charged wire with oscillating charge was proposed by Hau et al. (1992). By sinusoidally varying the charge on a wire, it is possible to add an effective repulsive 1/r 6 potential which stabilizes the motion of an atom around the wire. Sizeable electrical currents appear when the charge of a real wire (with capacitance) is rapidly varied. Magnetic fields are produced which interact with the magnetic moment of an atom. This leads to additional potentials which have not been taken into account in the original calculations. Another AC-electrical trap with several charged wires was proposed by Shimizu and Morinaga (1992). Their setup is reminiscent of a quadrupole mass filter and consists of 4 to 6 charged electrodes that are grouped around the trapping center.

II]

MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP

285

B.3. Guiding atoms with a charged optical fiber Stable orbits for the motion of an atom around a line charge are obtained if the atom is prevented from hitting the wire by a strong repulsive potential near the surface of the wire. Such a strong repulsion can be obtained by the exponential light shift potential of an evanescent wave that is blue-detuned from an atomic resonance. This can be realized by replacing the wire with a charged optical fiber with the cladding removed and the blue-detuned light propagating in the fiber (Batelaan et al., 1994). The fiber itself has to be conducting or coated with a thin ( 100 nm.

B.3. Technical noise Heating due to technical noise may arise from fluctuations in the currents used in the experiment. Noise in the chip wire currents and in the bias and compensation fields, for example, randomly shifts the location of the trap center. Let us focus on fluctuations in the chip wire current Iw. Neglecting finite size effects, the current and the bias field Bb produce the magnetic trap at a height of h = IAOIw/2~Bb (Eq. 3). The conversion from the current noise spectrum SI(O)) to the force spectrum required for the heating rate (27) is simply

SF((,O)=(IA~176 2$I ((o), 2SrBb

(33)

and we end up with an excitation rate F0--,1 = 1.4 s-l (M/amu)(oo/2ar 100 kHz) 3 SI(oo)/SsN (Bb/1 G) 2"

(34)

The reference SSN for the current noise is again the shot-noise level at lw = 1 A. Note that this rate increases with the trap frequency: while a strong confinement suppresses heating from thermal fields (Eq. 32), the inverse is true for trap position fluctuations. This is because in a potential with a large spring constant, position fluctuations translate into large forces (Eq. 28). Typical trap parameters (o)/2:z = 100kHz, Bb = 50G) lead for 7Li atoms to an excitation rate of -~4• 10-3 s-1 • S1(~o)/SsN. This estimate shows that even for very quiet currents technical noise is probably the dominant source of heating on the atom chip. The fluctuations of the trap center (location proportional to Iw/Bb) can be reduced by correlating the currents of the bias field coils and the chip wire so that they have the same fluctuations, up to shot noise. Heating due to fluctuations in the trap frequency may then be relevant, as co is proportional to BZ/Iw (Eq. 6). Let us again calculate an example. For a fixed ratio Iw/Bb (due to correlated currents), we find for the relative frequency fluctuations A(o co

-

AI Iw

(35)

and hence an excitation rate (30) F0-+2 ,-,o 10. 7

S-1

(~o/2.rr 100 kHz) 2 SI(2O)). (Iw / 1 A) 2 SSN

(36)

Typical atom chip parameters (a)/2oz = 100 kHz, Iw = 1 A) lead to F0+2 "~ 10-7 s-1 • Si(2co)/SsN, which is negligible when compared to the rate obtained in Eq. (34).

334

[V

R. Folman et al.

B.4. Light heating

Another source of heating are the external light fields with which the atoms are manipulated and detected. Here the Lamb-Dicke parameter r/ is a convenient tool, where 27ra0 r/~ (37) is the ratio between the ground-state size of the trap a0 and the wavelength of the impinging wave. This becomes clear if we remember that the probability not to be excited P0~0 is simply the well-known Debye-Waller factor exp(-Ak2a 2) ~ exp(-r/2),

(38)

where Ak ~ k is the momentum loss of the impinging photon. Hence, if the atoms are confined below the photon wavelength (the so-called Lamb-Dicke limit t/ < 1), they will not be heated by light scattering. Loss via optically induced spin flips is still relevant, however, as discussed in Sect. V.A.6 and reviewed by Grimm et al. (2000). In Table IV we give an overview of the heating mechanisms discussed above. For microscopic traps, we expect noise from current fluctuations and (to a lesser extent) from the thermal substrate to be the dominant origins of heating. Note the scaling with the trap frequency: trap fluctuations due to technical noise become more important for guides with strong confinement. In this subsection, we have restricted ourselves to heating due to singleatom effects. Collisions with background gas atoms also lead to heating and rate estimates have been given by Bali et al. (1999). Finally, in an onchip Bose condensate, fluctuating forces may be expected to drive collective Table IV Heating mechanisms for the atom chip (overview) Mechanism

Scaling a

Magnitude b

Near-field noise b

Ts/co~)h 3

10-4 s -l

Current noise c

co3S I / B 2 ~ coSl/h 2

1 s- l

Trap frequency noise d

co2S1/I2w ~ S l / h 4

10-5 s-l

Light scattering

1/co,~2

Remedy

correlate currents

reduce stray light

a The columns 'Scaling' and 'Magnitude' refer to transition rates from the ground state of a typical atom chip trap: lithium atoms, height h = 10~tm, trap frequency co/2sr = 100kHz. Harmonic confinement is assumed throughout. b Eq. (32), for a metal half-space. c Eq. (34). Note the scalings c o - B Z / I w and h ~ I w / B b for trap frequency and height. d Eq. (36).

V]

MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP

335

and quasiparticle excitations, leading to a depletion of the condensate ground state (Henkel and Gardiner, 2002). This area deserves further study in the near future.

C. DECOHERENCE

We now turn to the destruction of quantum superpositions or interferences due to the coupling of the atom cloud to the noisy chip environment. This is an important issue when coherent manipulations like interferometry or qubit processing are to succeed on the atom chip. With chip traps being ever closer to the chip substrate, thermal and technical magnetic noise is expected to contribute seriously to decoherence, as it does to loss and heating processes. The theoretical framework for describing decoherence makes use of the density matrix for the trapped atoms. Its diagonal elements give the occupation probabilities, or populations, in some preferred basis, usually the stationary trap states. Their evolution has been discussed in the previous subsections in terms of simple rate equations and constants. Decoherence deals with the decay of off-diagonal elements, or coherences, of the density matrix. Their magnitude can be related to the fringe contrast one obtains in an interference experiment. Magnetic fluctuations typically affect both populations and coherences: field components perpendicular to the trapping fields redistribute the populations, and parallel components suppress the coherences. The latter case illustrates that decoherence can occur even without the exchange of energy, because it suffices that some fluctuations randomize the relative phase in quantum superposition states (Stern et al., 1990). Such fluctuations are sometimes called 'phase noise'. In this subsection we consider first the decoherence of internal atomic states and then describe the impact of fluctuations on the center-of-mass. In the same way as for the heating mechanisms, we leave aside the influence of collisions on decoherence, nor do we consider decoherence in Bose-Einstein condensates. C. 1. Internal states

The spin states of the trapped atom are promising candidates for the implementation of qubits. Their coherence is reduced by transitions between spin states, induced by collisions or noise. The corresponding rates are the same as for the loss processes discussed in Sect. V.A. In addition, pure phase noise occurs in the form of fluctuations in the longitudinal magnetic fields (along the direction of the trapping field). These shift the Larmor frequency in a random fashion and hence the relative phase between spin states. The corresponding off-diagonal density matrix element (or fringe contrast) is proportional to (exp(iA~)) where Aq~ is the phase shift accumulated

336

[V

R. Folman et al.

due to noise during the interaction time t. A 'decoherence rate' Ydec can be defined by '/dec

(A~ 2)

S0)(o)--+ 0)

2t

4

--

'

(39)

where S~(o)) is the spectrum of the frequency fluctuations. Two spin states [mF), Imp-), for example, 'see' a frequency shift (p(t)- g l l B ( m F - m~F)ABil(t)/h, that involves the differential magnetic moment and the component ABII(t ) of the magnetic field noise parallel to the trap field. The spectrum S~(co) is then proportional to the spectrum of the magnetic field fluctuations. Equation (39) is derived in a rotating frame where the phase shift has zero mean and making the assumption that the spectral density Sr is essentially constant in the frequency range o) ~< 1/t. The noise then has a correlation time much shorter than the interaction time t. We consider, as usual in theory, that A~ is a random variable with Gaussian statistics, and get a fringe contrast (40)

(e iaq~) = e -ydect

that decays exponentially at the rate (39). Let us give an estimate for the decoherence rate due to magnetic noise. If AB(r, t) are the magnetic fluctuations at the trap center, the shift of the Larmor frequency is given by AwL(t) -

(il~llli) ~ABil(r,

t).

(41)

Here, the average magnetic moment is taken in the spin state ]i) trapped in the static trap field, thus picking the component ABI[ parallel to the trap field. The noise spectrum of this field component, for thermal near field noise, is of the same order of magnitude as for the perpendicular component (Henkel et al., 1999) and depends only weakly on frequency. We thus get a decoherence rate comparable to the loss rate (22), typically a few l s-1 . The same argument can be put forward for fluctuations in the wire current and the bias field. Assuming a flat current noise spectrum at low frequencies, we recover the estimate (24) for spin flip loss (a few l s-1). Therefore, keeping the atoms in the trap, and maintaining the coherence of the spin states requires the same effort. We finally note that near field magnetic noise also perturbs the coherence between different hyperfine states that have been suggested as qubit carriers. Although these states may have the same magnetic moment (up to a tiny correction due to the nuclear spin), excluding pure phase noise, their coherence is destroyed by transitions between hyperfine states. The corresponding loss rate (relevant, e.g., for optical traps) has been computed by Henkel et al. (1999) and is usually smaller than the spin flip rate.

V]

MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP 337

C.2. Motional decoherence The decoherence of the center-of-mass motion of a quantum particle has been put forward as an explanation for the classical appearance of macroscopic objects since the work of Zeh (1970) and Zurek (1991) (see also the book by Giulini et al., 1996). It has been shown that the density matrix of a free particle subject to a random force field in the high-temperature limit evolves into a diagonal matrix in the position basis (Zurek, 1991)

p(z, zt, t) ~ p(z z,,O)exp I (z-z')ZDt 1 '

h2

9

(42)

Here, the distance z - z t denotes how 'off-diagonal' the element is, and D is the momentum diffusion coefficient. The coherence length thus decreases like h ~:~ - v / ~ .

(43)

At the same time, the momentum spread Ap "-" (2Dt) 1/2 increases, so that the relation Ap~c -~ h is maintained at all times. At long times, ~c will be limited by the thermal de Broglie wavelength at the equilibrium temperature. However, this regime will not be reached on atom chips for typical experimental parameters. For a particle trapped in a potential, the density matrix tends to a diagonal matrix in the potential eigenstate basis if the timescale for decoherence is large compared to the oscillation time 2zr/co. This regime typically applies to the oscillatory motion in atom chip waveguides. The regime in which the two timescales are comparable has been discussed by Zurek et al. (1993) and Paz et al. (1993); it leads to the 'environment-induced selection' of minimum uncertainty states (coherent states for a harmonic oscillator). In the following we discuss different decoherence mechanisms for a typical separated path atom interferometer on the atom chip.

C.3. Longitudinal decoherence We focus first on the quasi-free motion along the waveguide axis (the z-axis), using the free particle model mentioned above. Decoherence arises again from magnetic field fluctuations due to thermal or technical noise. The corresponding random potential is given by (41): V(r, t) = -(il/tll Ii)ABII(r, t),

(44)

where we retain explicitly the position dependence. Henkel and P6tting (2001) have shown that for white noise, the density matrix in the position representation behaves as p(z,z', t) = p(z,z', O) e x p (-Ydec(Z - z t ) t), (45)

R. Folman et al.

338

[V

where the decoherence rate Ydec(s) depends on the spatial separation s = z - z' between the two parts of the atomic wave function being observed: Ydec(S) =

1 -C(s) 2],l 2

Sv(h; co ~ 0).

(46)

Here, C(s) is the normalized spatial correlation function of the potential (equal to unity for s = 0), and the noise spectrum Sv(h; co ---+ 0) characterizes the strength of the magnetic noise at the waveguide center. For an atom chip waveguide perturbed by magnetic near field noise, the decoherence rate is of the order of

0) y =

2h 2

(47)

and hence comparable to the spin flip rate (19, 22). Decoherence should thus typically occur on a timescale of seconds. The correlation function C(s) is well approximated by a Lorentzian, as shown by Henkel et al. (2000), and the decoherence rate (46) can be written as ys 2 Ydec(S)- $2 + 12,

(48)

where lc is the correlation length of the magnetic noise. This length can be taken equal to the height h of the waveguide above the substrate (Henkel et al., 2000). This is because each volume element in the metallic substrate generates a magnetic noise field whose distance-dependence is that of a quasi-static field (a 1/r 2 power law). Points at the same height h above the surface therefore see the same field if their distance s is comparable to h. At distances s >> h, the magnetic noise originates from currents in uncorrelated substrate volume elements, and therefore C(s) ~ O. The corresponding saturation of the decoherence rate (48), Ydec(s >> lc) ---. y, has also been noted, for example, by Cheng and Raymer (1999). Decoherence due to magnetic noise from technical sources will also happen at a rate comparable to the corresponding spin flip rate, as estimated in Eq. (24). The noise correlation length may be comparable to the trap height because the relevant distances are below the photon wavelength at typical electromagnetic noise frequencies, so that the fields produced by wire current fluctuations are quasi-static, and the same argument applies. The noise correlation length of sources like the external magnetic coils will, of course, be much larger because these are far away from the waveguide. These rough estimates for the spatial noise properties of currents merit further investigation, in particular at the shotnoise level (Henkel et al., 2002).

V]

MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP

339

I F(s,t) I 1

0.8 =

0.6 0.4 0.2 1

2

3

4

5

6

s/l c

FIc. 47. Illustration of spatial decoherence in an atomic wave guide. The spatially averaged coherence function F(s, t) = f d z p(z + s,z, t) is plotted vs. the separation s for a few times t. Space is scaled to the field correlation length lc and time to the 'scattering time' 1/y _= 1/Ydec(~ ). A Lorentzian correlation function for the perturbation is assumed. Reproduced from Henkel and P6tting (2001), Appl. Phys. B 72 (2001) 73, Fig. 3, with permission. 9 Springer-Verlag.

Spatial decoherence as a function of time is illustrated in Fig. 47 where the density matrix p(z + s,z, t) averaged over z is plotted. Note that this quantity will be directly proportional to the visibility of interference fringes when two wavepackets with a path difference s interfere. One sees that for large splittings s >> lc, the coherence decays rapidly on the timescale 1/y given in Eq. (47). This is because the parts of the split wavepacket are subject to essentially uncorrelated noise. In a typical waveguide at height h = 10 Ftm, fringe contrast is thus lost after 0.1-1 s (the spin lifetime) for path differences s >> 10 ~tm. Increasing the height to h = 100 ~tm decreases y by at least one order of magnitude as shown by Eq. (22). In addition, the correlation length grows to 100 ~tm, and larger splittings remain coherent. Alternatively, one can choose smaller splittings s 1/y. In this limit, only separations s < lc have not yet decohered, and we can make the expansion s2

~dec(S) ~ ~ [2

(49)

for the decoherence rate (48). From the density matrix (45), we can then read off the momentum diffusion constant D hZy/l 2. =

C.4. Transverse decoherence We finally discuss the decoherence of a spatially split wavepacket in an atom chip interferometer, as described in Sect. II.A.9.

R. Folman et al.

340

[V

C.4.1. Amplitude noise. The excitation of transverse motional states in each arm suppresses the coherence of the superposition at the same rate as the heating processes discussed in Sect. V.B (about 1 s-l). Note that due to the transverse confinement, the relevant noise frequencies are shifted to higher values compared to the longitudinal decoherence discussed before. C.4.2. Phase noise. The coherence between the spatially separated interferometer arms is suppressed in the same way as the longitudinal coherence discussed in Sect. V.C.3. To show this, we use an argument based on phase noise, and focus again on magnetic field fluctuations, either of thermal or technical origin. Magnetic fluctuations affect both the bottom of the trap well and the transverse trap frequency, but are only relevant when they differ in the spatially separated arms. The well bottoms get differentially shifted from an inhomogeneous bias field, e.g., while the trap frequency shifts due to changes in the field curvature. We generalize formula (39) to a phase shift Aq) that is the accumulation of energy-level differences AE(t) along the paths in the two arms. The decoherence (or dephasing) rate is thus given by Ydec =

SAE(O0 ~ O) 4h 2 ,

(50)

where SAE(CO ~ 0) is the spectral density of the energy difference, extrapolated to zero frequency. To make contact with the density matrix formulation of Eq. (45), we write AE(t) = E R ( t ) - EL(t) where ER, L(t) are the energy shifts in the right and left interferometer arms that are 'seen' by an atom travelling through the waveguide. We find 48 hours at a time, with an observed variation in trap fluorescence over 24 hours (due to changes in laser intensity and frequency) of ~< 5%.

B.2. Vacuum chamber The atom trap is located at the center of the vacuum chamber as shown in Fig. 3. This chamber has fourteen ports in the horizontal plane, and two ports in the vertical plane. Four of the horizontal ports and the two vertical ports are for the three pairs of counter-propagating laser beams used for trapping. Three of the horizontal ports are used by components of the electron beam: the electron gun, the Faraday cup, and the translating wire used to measure the electron beam profile. These are described in Sect. II.B.4. Two ports are used to monitor the trap fluorescence: a CCD camera provides a qualitative picture of what is going on, while a photodiode is used to quantitively measure the relative number of atoms in the trap. Also shown in Fig. 3 are the optics used in the electron-impact excitation experiment described in Sect. III.B.4. The ultra-high vacuum chambers we use for trapping are constructed from non-magnetic stainless steel with an electropolish finish. The chambers are initially evacuated with a turbomolecular pump and baked out at 150~ for two to six days. The turbo-pump is then valved off with an all-metal UHV valve and the chamber is pumped by a 201/s ion-pump. Typical base pressure in the chamber is 8 x 10-l~ Torr. A reservoir with a one gram ampule of 99.95% purity Rb is connected to the main chamber by an all-metal valve. The thermal-velocity Rb atoms diffusing off the walls of the vacuum chamber provide the source of Rb for the trap. A higher Rb number density yields more trapped atoms due to an increased trap-loading rate but also results in shorter trap lifetimes due to increased collisions with background atoms. By periodically adding only enough Rb to maintain a good number of trapped atoms (by opening the valve to the reservoir and heating it to ~50~ with a heating tape), long trap lifetimes can be achieved (~4 s) at the price of variations in trap size on the time-scale of days. Alternatively, for a more constant number of trapped atoms but shorter trap lifetimes (~ 1.5 s) the reservoir can be left open at a reduced temperature (~45~

B.3. Magnetic field The quadrupole magnetic field necessary for the trap is generated by two air-cooled coils located above and below the chamber. Each coil consists of 40 turns of 16 gauge high temperature magnet wire. The diameter of each coil is approximately 7.2 cm, with a coil-to-coil distance of 5.3 cm (which is slightly larger than the 3.6 cm spacing for true Helmholtz coils). The coils are wired in series and run in an 'anti-Helmholtz' configuration to provide an approximately

366

Schappe et al.

[II

uniform field gradient. Typically, we run 4.0 A through each coil to generate a measured magnetic field gradient of 9Gauss/cm along the z-axis, although trapping is possible over a wide range of currents (2 to 18 A). Since the trapping magnetic field deflects the electron beam, the magnetic field must be switched off during electron beam pulses. We have used two circuits to perform this function, one using a power MOSFET as a switch, and one using a commercial solid-state relay. We measure the effective decay rate of the magnetic field by monitoring the temporal distortion in the shape of an electron beam current pulse. The minimum measured delay between the time the magnetic field is switched off and the start of the electron beam pulse that does not distort the electron current is 500 ~ts, which is comparable in performance to slightly more advanced switching circuits with eddy-current compensation (Dedman et al., 2001). The center of the trap is determined by the location of the minimum in the magnetic field, while the optimum trap fluorescence and trap-loading rate are located at the intersection of the six laser beams. We use the vacuum chamber to align the two; the coils are attached to the top&bottom viewports, and the laser beams are centered on each viewport. The most uniform trap, however, is achieved by using magnetic field shim coils to steer the magnetic field 'zero' to the intersection of the laser beams. The quality of the trap alignment can be monitored by observing the dispersal of the atom cloud when the magnetic field is turned off. For a well aligned trap the atoms disperse isotropically ("poof"), but for a poorly aligned trap the atoms move off in a directional jet. B.4. Electron beam

The electron gun used in this work consist of five grids and a cathode assembly. The stainless steel grids are 1.78 cm square with alumina spacers between grids. The cathode consists of an indirectly heated BaO cathode. One of the grids can be biased negative relative to the cathode to chop the beam on and off. The electron beam current is measured by a deep Faraday cup (L/D ,~ 5). The back plate of the Faraday cup is conical so that specularly reflected secondary electrons are not reflected back towards the collision region. The back plate is also biased at +18 V to prevent the escape of low energy secondary electrons. The electron gun and Faraday cup are separated by distance of 4.5 cm to allow the trapping laser beams access to the collision region. Space charge expansion of the electron beam over this distance limits the low energy performance of the electron gun. To determine the electron beam current density, J, we translate a thin (0.19 mm diameter) tungsten wire across the electron beam at the location of the atom trap. The current measured on the wire produces a series of line integrals of the beam current density. Assuming the beam is cylindrically symmetric, the measured beam spatial profile is converted into the current density J ( x , y ) using an Abel transform (Hansen and Law, 1985). Due to secondary electron emission from

III]

E L E C T R O N - A T O M C O L L I S I O N CROSS SECTIONS

367

Fl~. 4. Profile of the electron beam (100 eV) at the location of the trap. Measured current values as the wire is traversed along the x-axis are deconvoluted with an Abel transform assuming cylindrical symmetry. The current density is essentially constant over the size of the atom cloud (~< 0.1 cm). the wire, we only use the translating wire measurements to find the shape o f the current density. We put J on an absolute scale using the total electron beam current measured by the Faraday cup. At high energies (>100 eV) the beam is approximately Gaussian, with a F W H M of 2 mm at high energies (see Fig. 4). The value of the peak current density ranges from 0.1 mA/cm 2 at low energies to 3 mA/cm 2 at high energies. Due to the small size of the atom trap, the electron beam is carefully aimed at the atom cloud by using a set o f gimbals (particularly for the trap-loss measurements described in Sect. III.A). The decrease in trap intensity due to electron-atom scattering collisions can be used to aim the electron beam at the trap. Due to residual magnetic fields and the build-up of surface charge on insulators, the center of the electron beam has some dependence on the electron beam energy, requiring that this alignment must be repeated for different energies.

III. Methods for Measuring Cross Sections We have measured electron-atom collision cross sections using trapped atoms with two different classes of experiments. The first class of experiments (Schappe et al., 1995, 1996), which is described in Sect. III.A, monitor the time dependence of the number of atoms in a trap to deduce total-scattering and ionization cross sections. This trap-loss method relies on the advantages of an atom trap listed in Sect. I: the trap fluorescence provides a relative measure o f

Schappe et al.

368

[III

the number of atoms in the trap; the low initial velocity of atoms in the trap differentiates electron-scattered atoms, and the small size of the trap simplifies the absolute calibration. On the other hand, the second class of experiments (Keeler et al., 2000) uses the high excited state fraction in the trapped atom target to measure ionization and excitation cross sections from the laser excited 5P level and is described in Sect. III.B. A. Loss RATE MEASUREMENTS

A.1. Rate equations Before analyzing a particular type of electron-atom collision process, we first derive the general relation between the number of atoms in a trap and a collision cross section. Consider an atom trap that is exposed to a periodically pulsed electron beam. Since the trapping magnetic field interferes with the electron beam propagation, the trap is turned off for a short period of time before the start of the electron beam pulse. The number of atoms in the trap, N, as a function of time is described by the differential equation

dN - A L - r0N - f G N , dt

(1)

where L is the loading rate of atoms into the trap from the background vapor, F0 is the loss rate of atoms out of the trap for all causes other than electron-atom collisions, Fe is the loss rate of atoms out of the trap due to electron collisions, J~ is the fraction of time the trap is on, and f is the fraction of time the electron beam is on. For a vapor loaded trap, the loading rate, L, depends on the Rb partial pressure in the chamber and the laser detuning and intensity. Atoms can only be loaded into the trap during the fraction of time, J~, when both the trapping lasers and magnetic field are 'on'. The loss rate of atoms out of the trap when the electron beam is off, F0, is due to a number of causes. At very high trap densities, atom-atom collisions within the trap are the largest loss mechanism, while at lower densities collisions with hot background gas atoms become important (Walker and Feng, 1994) 3. Additionally, when the trap is turned off, all of the formerly trapped atoms ballistically expand away from the center of the trapping region with a velocity dependent on the temperature of atoms in the trap. If the trap is turned back on after only a very short delay, these atoms are retrapped with almost 100% efficiency. For longer delay times, this ballistic expansion allows atoms to

3 Since the asymptotic number of atoms in the trap is different with and without the electron beam, to eliminate any possibility of density-dependent changes in F0, one should vary fL so that N~ is the same in both cases.

III]

ELECTRON-ATOM COLLISION CROSS SECTIONS

369

escape the physical limits of the trapping laser beams, thus depleting the number of atoms in the trap. The average (rms) speed of Rb atoms in a 150 ~tK trap is 21 cm/s. For a laser beam of radius 0.5 cm, atoms leave the trapping volume in about 24 ms. Since the atoms have a Maxwell-Boltzmann velocity distribution, and the laser beam has a Gaussian profile, this is not a well-defined time limit. Gravity will also eventually pull atoms out of the trapping region. The time it takes to fall 0.5 cm is 32 ms, which is only slightly longer than the ballistic expansion time for Rb. Solving Eq. (1) with the initial condition that there are no atoms in the trap, the number of atoms in the trap as a function of time (a loading curve) is simply N(t)

= N~

(1 - e-(r~

,

(2)

where N ~ is the steady state number of atoms in the trap, and is equal to j~L/(F0 +fFe). If the electron beam is always off, the form of the solution is the same, except Fe = 0. By fitting a rising exponential to two sets of experimental d a t a - one with the electron beam on, and one with the electron beam o f f - F~ can be separated from the background loss rate. The loss rate due to electron collisions, F~, is directly related to the corresponding electron-atom collision cross section, o, by r~ -

oJ

,

(3)

where J is the current density of the electron beam at the location of the trap, and e is the magnitude of the electron charge. Note that since the size of the trapped atom cloud is small compared to the size of the electron beam, we can safely assume J is constant over the volume of the trap. Furthermore, in contrast to the general difficulties of crossed beam experiments, for an atom-trap target the only measurements needed to find absolute cross sections are the electron beam current density (see Sect. II.B.4) and the change in the loss rate with the electron beam on/off. There is no need to measure the absolute number of target atoms. By varying the delay time between the electron beam pulse and the time the trap is turned back on, T, it is possible to measure different types of collision cross sections. To measure total scattering cross sections (Sect. III.A.2), we use a long delay. With a long delay, any atom that has gained any excess recoilmomentum due to an electron-atom collision will have enough time to leave the trapping region. On the other hand, if the trap is turned on immediately after a short electron gun pulse, recoiling atoms will not have had enough time to leave the trapping region, and will be retrapped. Ions formed via electron-impact ionization, though, are not resonant with the trapping lasers, and are lost. Thus only ionizing collisions result in trap loss, allowing us to measure ionization cross sections as is described in Sect. III.A.3.

370

[III

Schappe et al.

Hyperfine~ o f f - - I

Magnetic Field

ton = 30 ms

one, off/

~ ~,.

i

~'"

i"~ -IT= 0- 18 ms a

Electron Beam

off

n

0.8 - 4 ms

FIG. 5. Timing diagram for total scattering experiment. The electron beam pulse is delayed from the the start of the trap-off phase to allow time for the magnetic field to decay (dashed lines).

A.2. Total scattering cross section

Total scattering cross sections (Schappe et al., 1995) are obtained by monitoring the time dependence of the trap fluorescence as the trap is periodically hit with an electron beam pulse. A timing diagram of one electron beam pulse cycle is shown in Fig. 5. Atoms are loaded'into the trap for 30 ms, at which time both the magnetic field and hyperfine repump are turned off. With no repumping, Raman scattering shifts atoms into the F = 2 dark state of Rb, which is non-resonant with the primary trapping laser. Before pulsing the electron beam, however, a delay of 1 ms is needed for the decay of the magnetic field. After a short electron beam pulse (0.8 to 4 ms long), the trap is left off for a variable time of 0 to 18 ms. At the end of this delay, the trap is turned back on (i.e., hyperfine repumping is resumed and the magnetic field is turned on) and the number of atoms in the trap is recorded. After acquiring trap fluorescence data for approximately 12 s, the number of atoms in the trap reaches the asymptotic value, N ~ . To obtain another loading curve, the magnetic field is turned off for 2 s to empty the trap, and the above process is repeated. A loading curve is also obtained with the exact same timing structure, but without the electron beam to obtain the background trap-loss rate F0. Typically, the results from five pairs of loading curves are averaged together to reduce the statistical noise in the data. The relative number of atoms in the trap, for a fixed laser intensity and laser detuning, is proportional to the trap fluorescence. This fluorescence is collected by a lens and detected with a photodiode. The -50 nA signal is amplified by a current to voltage amplifier with a gain of 6 x 107 V/A and recorded by either a digital storage oscilloscope or a data acquisition computer. The computer is also used to control the timing of the electron beam, magnetic field, and hyperfine repumping. In Fig. 6 we show a pair of sample loading curves with and without an electron beam. The trap fluorescence demonstrates a slow exponential rise due to atoms being loaded into the trap via Eq. (2), superimposed on a rapid modulation due

III]

ELECTRON-ATOM COLLISION CROSS SECTIONS '

!

'

!

9

i

'

no e l e c t r o n b e a m

!

'

!

,

.= ..,--,,..:~. " b ~ - _-~r"or'-=~.,L lb-l

9

- ~

l

9 --o

371

]

I

.Q v

3

~

2

e" m e~

e l e c t r o n beam

0.1

0.2

0.3

0.4

0 0

2

4

6

Time

(s)

8

10

12

FIa. 6. Sample loading curves for an electron beam energy of 50 eV with a 2 ms electron beam pulse and 13 ms delay time. The data presented in the main plot has been averaged over one full pulse cycle (46 ms). The raw data for the start of the electron beam off loading curve (shown in the insert) demonstrates the rapid time dependence of trap fluorescence on the modulation of hyperfine repumping. to the modulation of the hyperfine repumping. Since there is no trap fluorescence when the atoms are in the dark state, this portion o f the raw signal is removed from the signal prior to fitting. Since Eqs. (1) and (2) include only the timeaveraged loading rate and electron-impact induced loss rate, they do not model our data for time scales less than the electron-beam pulsing period 4. Thus, before fitting the raw data in Fig. 6 to Eq. (2), the data needs to be time averaged over one timing cycle. Numerical simulations have shown that a direct fit to both the raw data and the time averaged data (over a very wide range of averaging times) produce equivalent fitted loss rates 5 Note that we measure the loss rate of atoms out o f the trap by monitoring the time dependence of atoms being loaded into the trap. It would seem to be more natural, however, to monitor the induced decay rate o f atoms out o f a fully loaded trap. Indeed, this was the approach of Dinneen et al. (1992) to measure photoionization o f trapped Rb atoms. The general difficulty with this later approach is the presence of the loading term in Eq. (1), i.e., the n u m b e r o f atoms in the trap does not decrease as a simple exponential since new atoms are also being loaded into the trap from the background vapor whenever both the trapping lasers and magnetic field are on. Dinneen et al. (1992) overcame this limitation by

4 Technically F0 is not fully time independent, since the background loss rates are different when the trapping lasers are on or off. While some high-frequency residuals in the fit can be removed by including a two parameter background loss rate, only the time averaged value is needed to extract the electron-impact induced loss rate. 5 Only data averaged over integer multiples of a timing cycle will produce fits with no spurious high-order frequency components and valid Z2 values.

372

[III

S c h a p p e et al.

20 eV

g.. E T-v

6

e-

4

co

2

0

0

250 eV

o 0

.

0

i

2

.

i

4

,

i

.

i

,

I

l

I

6 8 10 12 Delay Time toff (ms)

,

I

14

,

i

16

Fie. 7. Variation of loss rate, and thus cross section, with delay time. using a laser-slowed atomic beam that could be switched off during loss rate measurements. Alternatively for a vapor loaded trap, the decay can be measured for a short time interval if the trapping lasers are turned off so that there is no loading term. The presence of the loading term in Eq. (1) also complicates the extraction of loss rates from monitoring only the asymptotic number of atoms in the trap, N ~ . Relative measurements of loss rates (and thus cross sections) can be obtained by monitoring the equilibrium number of atoms in the trap with and without an electron beam. But since N ~ = f L L / ( F o + f F e ) , knowledge of the loading rate is necessary to place these results on an absolute scale. Hence, while being less intuitive, we have found the loading curve method to be easier to implement and analyze experimentally. The measured electron-induced loss rate varies with the delay time as is shown in Fig. 7. For short delay times, only atoms with a very large recoil velocity have sufficient time to leave the retrapping region before the trap is turned back on. Kinematically, the relation between the recoil velocity of the atom, V, and the scattering angle of the electron, 0, is (McDaniel, 1989) 2E0 - AE COS 0 =

M(m+M) V 2

2m

2 ~Eo(Eo - AE -

,

(4)

89M V 2)

where E0 is the energy of the incoming electron, AE is the excitation energy of the collision, and m and M are the masses of the electron and atom, respectively. As small electron scattering angles correspond to low atomic recoil velocities 6,

At 10 eV all inelastically scattered atoms via 5S-5P excitation have recoil velocities in excess of 100 cm/s, and are thus included for delay times in excess of 5 ms. For elastic scattering (at 10 eV), a delay time of 18 ms corresponds to 0mi n ~1.6~

6

III]

ELECTRON-ATOM COLLISION CROSS SECTIONS

373

longer delay times allow more of these slow moving atoms to escape, which is equivalent to probing electron-scattering angles closer to 0 ~ Thus the measured trap loss corresponds to the integral of the relevant differential cross sections 2~

,n(r) d ~

(0) sin 0 dO,

(5)

where T is the delay time, d o / d ~ is the "total" differential cross section, and Omin(T)is found from Eq. (4) assuming V = r/T with r being the effective radius of the retrapping zone. Since the data in Fig. 7 converges to an asymptotic value well before the maximum delay time, we can be assured that our total scattering cross section measurements include the contribution down to all non-negligible scattering angles. In principle, since there is a functional relationship between delay time and the electron scattering angle, the derivative of the trap loss vs. delay time curve can be used to measure differential cross sections, do d---~(T) = -2:r

d0min dT

do (T)~-~(0mi,) sin 0min.

(6)

For electron energies below the first excitation energy (-2eV), this would correspond to the elastic scattering differential cross section. On the other hand, at very high energies the elastic cross section is negligible, leaving only contributions from excitation into all bound levels because the ionization component does not contribute to the variation of trap loss with the delay time as the ions are never retrapped regardless of the delay (see Sect. III.A.3). For alkali atoms the nS ~ nP excitation cross section dominates all the other nS ~ n~L excitation cross sections (Phelps et al., 1979). It may be possible to utilize a detailed measurement of the trap loss as a function of the delay time to obtain information about the 5S ~ 5P differential excitation cross section at very small scattering angles, which is difficult with conventional methods. The effective angular resolution of such a measurement is limited by the length of the electron beam pulse, the number of data points, and the quality of the trap model used to relate the atomic recoil velocity (V) to the delay time (T). As indicated by the simplicity of Eq. (3), there are only two measured quantities, and hence only two major sources of uncertainty, that enter into the determination of the cross section. The measurement of the peak current density which is obtained by taking the Abel transform of the translating thinwire electron beam profile has an estimated uncertainty of 7%. The uncertainty in the extraction of the electron-induced trap-loss rate Fe from fitting the loading curves, and finding the asymptotic delay time value is estimated to be 6%. Results obtained over a wide variety of trap parameters (laser intensity, detuning, laser beam diameters, trap size) and experimental parameters (electron beam pulse length, electron beam spatial width, trap on time) give consistent results.

374

[III

Schappe et al.

1

Trap off

ton=-10-20 ms

J I

Oil

Electron Beam

~ ' -~1 ms

0.167 - 2 ms

off FIG. 8. Timing diagram for ionization measurements. Both the trapping magnetic field and hyperfine repump laser are turned off during the trap-off phase. Thus, in the apparent absence of any secondary effects, we believe the total uncertainty in our measurements is about 9%. One very important secondary effect that can complicate measurements made with the trap-loss technique is electron stimulated desorption (ESD) of Rb atoms from the Faraday cup. The background trap loss is dominated by collisions of trapped atoms with background gas atoms in the trap. Any increase in the background gas pressure synchronous with the electron beam will appear to be due to electron-atom collisions and will be erroneously included in Fe. Due to the low operating pressure of the trap chamber, the small number of atoms/molecules liberated from the Faraday cup by ESD can significantly change the background gas number density. For example, we have observed the pressure in the chamber rise from 10 -9 to 10 -7 Torr when the electron beam first hits the Faraday cup in a chamber that was pumped down after being opened. Two actions minimize the ESD-induced gas load: the chamber is baked out at 250 ~ for 48 hours to remove as much contamination from the chamber as possible, and before data is collected the electron beam is left on at a high current (100 to 400 gA) for an extended period of time. Data is only acquired after the pressure in the chamber is the same with the electron beam gated on or off. A.3. Ionization cross section

As described in Sect. II1.A, if the trap is turned on immediately after the electron gun pulse, most of the trap losses will be due to ionization since the ions are non-resonant with the trapping laser which is tuned to an atomic transition. As seen by the timing diagram in Fig. 8, the ionization experiment is very similar to the previously described total scattering experiment. The trap is periodically turned off by turning off the magnetic field and hyperfine repump. After the delay needed to allow the magnetic field to decay, the electron beam is turned on for a short pulse (0.167 to 2 ms). The trap is turned on immediately following the electron beam pulse and the trap fluorescence is recorded. New atoms are loaded into the trap for 20 ms and the process is repeated. Note that the cross section measured with this technique is related to the number of ions created (or the

III]

ELECTRON-ATOM COLLISION CROSS SECTIONS 1.75

.

.

.

.

.

!

.

.

.

.

!

.

.

.

.

!

1.50

9

,

,

~

375

1-

50 eV

1.25 ,~

1.00

IT"

0.75

co o~ 0

--J G) ._ n'-

2 5 0 eV 0.50 0.25

0.00 0.0

|

|

|

,

l

i

i

i

|

0.5

i

,

1.0

,

1.5

Electron B e a m Pulse W i d t h

2.0 (ms)

FIG. 9. Variation in trap-loss rate versus the width of the electron beam pulse. At high electron energies, collisions are mainly small-angle electron scattering (low atom recoil). At low energies (50eV), however, large-angle (fast atom recoil) scattering is more evident. Measurements at widths less than 0.2 ms are dominated by trap-depth effects.

number of atoms lost from the trap), and does not depend on the charge state of the ion produced, i.e., we measure ~rion "count defined as

ok+,

ount = ~

O-ci o n

(7)

k

where o k+ is the cross section for producing a Rb k+ ion. In addition to the loss of atoms via ionizing collisions, the finite duration of the electron beam pulse allows atoms scattered with a sufficiently high recoil velocity to leave the retrapping zone before the trap is turned on at the end of the electron beam pulse, i.e., Eq. (5) with T approximated by half the electron beam pulse width. For example, with a 1 ms electron beam pulse, an atom receiving an electron-impact recoil velocity in excess of 69 cm/s will escape the retrapping zone. Since the total scattering cross section is dominated by 5 S - 5 P excitation (Schappe et al., 1995), this velocity corresponds to an electron scattering angle of 1.5 ~ for a 50eV electron. By taking measurements with shorter electron beam pulses, this angle can be increased. In principle it appears that taking measurements with ever shorter electron beam pulse widths would completely eliminate this effect. There is, however, an upper limit o n 0mi n which is set by the finite trap depth [estimated to be 3 • 10-5 eV based on measurements of Hoffmann et al. (1996)]. Since the differential cross sections are sharply peaked at low scattering angles, few electrons undergo large angle scattering, so the non-ionizing collisions with scattering angles between 0mi n and 180 ~ do not significantly affect our results. This can be demonstrated by taking loss rate measurements at a variety of electron-beam pulse widths and extrapolating to zero pulse width as shown in Fig. 9.

376

Schappe et al.

[III

Due to the inclusion (and subtraction) of secondary effects in the ionization cross-section measurements versus the total-scattering measurements described in the last section, the estimated uncertainty in our ionization cross sections (Schappe et al., 1996) are slightly higher. At 50 eV, the uncertainty is estimated to be 14%, but due to the diminished large-angle scattering at high energies, the uncertainty drops to 9% at 500 eV.

A.4. Trap-loss measurements f o r collisions involving laser excited atoms When the trap is on (or at least the trapping laser and hyperfine repump), a significant fraction (fe ~ 0.4) of atoms in the trap are in the 52p excited state. Hence, if the hyperfine repump laser is left on for the duration o f the electron beam pulse, it would appear to be possible to repeat the measurements o f Sect. III.A.2 and Sect. III.A.3 to find total-scattering and ionization cross sections from the 52p excited state. Unfortunately, even without the trapping magnetic field, the laser beams alone set up the optical molasses viscous force that slows the escape of electron-scattered atoms. Accounting for this effect may entail nontrivial modifications in the measurements o f the total scattering cross section 7, but only slightly complicates ionization measurements which can be taken in the limit of very short pulses (Schappe et al., 1996; Keeler et al., 2000). Ionization results with this technique are compared to the more straightforward direct detection measurements in Sect. III.B.3. B. EXPERIMENTS BASED ON DIRECT DETECTION OF FINAL STATE

B. 1. Ratio measurements f o r Rb 5P/5S processes In the trap-loss measurements described in the preceding sections, the electron beam pulse occurred when the hyperfine repump was off and the trapped atoms were all in the 52S1/2 F = 2 dark state. If the electron beam and hyperfine repump are on simultaneously, the resulting signal is the weighted sum of cross sections out of both the 52S and 52p levels. If both the excited state fraction o f the target and the cross section from the 52S ground level are known one can derive the corresponding cross section from the 52p level. It is difficult, however, in traploss measurements to fully isolate the 52p signal contribution from changes in the 52S signal due to the lasers being on (Sect. III.A.4). We therefore find it advantageous to directly detect the ions or excited atoms formed by collisions. In Sect. III.B.3 we describe measurements for ionization out o f the 52p level,

7 It may be possible to overcome this limitation on measuring 5P total scattering cross sections, however, by shifting the red-detuned trapping laser beam directly onto resonance. On resonance, the absorption and emission of photons does not provide any net force to the atoms - it will, however, rapidly heat them, leading to increased background trap-loss rates which may pose a new problem.

III]

ELECTRON-ATOM COLLISION CROSS SECTIONS

377

while excitation cross sections out of the 52p level are described in Sect. III.B.4. Both of these measurements, however, depend upon an accurate measurement of the excited state fraction. B.2. Measurement o f excited state fraction

Generally, it has been the practice to calculate the expected excited state fraction based upon the known laser intensity (Walker and Feng, 1994). If we assume the trapped atoms can be treated as a simple two level atom, stimulated emission limits the maximum excited state fraction, fe, to be 0.5 or less in steady state. As a function of the laser intensity, I, the fraction of excited atoms in the trap is (Metcalf and van der Straten, 1999), 1 I/Is fe - 2 1 + I/I, + 4A2/F 2'

(8)

where Is is the saturation intensity of the transition, A is the detuning of the trapping laser from the atomic transition, and F is the natural linewidth of the transition (5.89 MHz for Rb 5 3 - 5P). We generally set A to a value between 7 to 10 MHz. For a two level atom, the saturation intensity can be calculated from /"

;rhc 3/Pr

(9)

where h is Planck's constant, c is the speed of light, A is the wavelength of the trapping laser (780nm), and r is the lifetime of the 52p level (27ns), which yields a value for the saturation intensity of 1.64 mW/cm 2. This value, however, is not directly applicable to atoms in a MOT since it assumes the atoms are in a closed two-state system, i.e., the 5231/2 F = 3, MF = 3 and 5 2 p 3 / 2 F ' = 4, M~ = 4 states. This corresponds to the target being completely polarized; however, as described in Sect. II.A.3, atoms in a MOT are almost completely unpolarized. Assuming equal populations in all 7 MF states (no polarization), we calculate a saturation intensity of 3.6 mW/cm 2. This value is slightly larger than that found in a MOT, since the atoms in a MOT are moving slowly enough that they are optically pumped into an internal state consistent with the local intensity and polarization of the standing wave created by the counter-propagating laser beams. Using a realistic population distribution among the MF states, we calculate a saturation intensity of 3.1 mW/cm 2. Additional complications can arise in calculating the saturation intensity for trapped atoms due to spatial variations of intensity, polarization, and magnetic field across the trap (Javanainen, 1993). The measurement of the laser intensity, I, at the location of the trap also has a degree of uncertainty associated with it. Typically one measures the effective laser intensity at the position of the trap by measuring the laser power that passes

378

Schappe et al.

[III

through a small aperture approximately the size of the trap (d = 1 mm). If the laser beams are well aligned, the total laser intensity is simply the sum of the peak intensities of the six counter propagating laser beams. Any misalignment of the beams, or unaccounted-for losses at windows, etc., would introduce an error into this value. Additionally, for our laser not all of the measured laser power is available for excitation into the 52p3/2F' = 4 level since some of the laser power is in the hyperfine repumping sidebands. Due to these complications, we instead determine the excited state fraction of the trap by fitting a surrogate measure of the excited state fraction to Eq. (8) using Is as a parameter. In steady-state, the 5P ~ 5S trap fluorescence is equal to the number of 5P atoms in the trap times the transition probability for the 5P ---. 5S transition. The number of 5P atoms in the trap is simply feN where N is the total number of atoms in the trap. Thus, for a fixed number of trapped atoms, the trap fluorescence is directly proportional to the excited state fraction. The number of atoms in the trap, however, generally depends upon the laser intensity, detuning, and a vast variety of other parameters. To keep the number of atoms in the trap constant, the trap is always loaded with the same laser intensity, /Load. A fast variable attenuator is then used to rapidly switch to a new laser intensity, Im, and the trap fluorescence is recorded. These relative measurements of the excited state fraction as a function of the easily measured laser power are then fitted to the shape of Eq. (8) with I~ as the only free parameter (i.e., the asymptotic value is forced to 0.5). A liquid crystal variable retarder (LCVR) followed by a linear polarizer is used to rapidly attenuate the trapping laser intensity. The response time of the LCVR we use is on the order of 10ms. Few atoms are lost out of the trap in 10 ms, even with a full attenuation of the laser intensity. However, ballistic expansion of the atom cloud in this time may affect the overlap of the laser beam and atom cloud for very low attenuation powers. This effect should be minimal for Im >~ 0.05ILoad. The trap fluorescence is monitored with a photodiode in the same way as that used in our trap-loss measurements. Since the photodiode also detects scattered laser light (and fluorescence from background atoms), we take the difference of two measurements: one with the trap on, and one with the trap off obtained by turning off the hyperfine repump. A sample plot of excited state fraction versus laser intensity is shown in Fig. 10 for a laser detuning of 9.9 MHz and maximum laser power of 6 mW (total power before being split into separate beams). The fitted value of the saturation intensity, 3.5 mW/cm 2 falls within the range of expected values. Interestingly, the horizontal scaling of Fig. 10 is irrelevant, e.g., we plot fe versus laser power instead of laser intensity as in Eq. (8). If we assume the calibration of the horizontal scale is off by some scale factor (e.g., not properly accounting for window losses), our fitted Is value will be off by the same factor. However, the vertical scale, and thus the extracted f~ value, is unaffected.

III]

ELECTRON-ATOM COLLISION CROSS SECTIONS 0.5

.

,

.

,

.

,

.

,

.

,

.

,

.

,

.

,

.

,

379

.

0.4

.o 0.3

LL

0.2 ._

x

0.1

UJ

0.0

.

.

.

.

.

.

.

.

.

.

L a s e r P o w e r (mW) FIG. 10. T y p i c a l p l o t o f e x c i t e d s t a t e f r a c t i o n v e r s u s One

can obtain

detuning

higher

or focusing

excited

state fractions

the laser beams

(~0.45)

to a smaller

laser power

with a detuning

at a given

laser power

A = 9.9 MHz.

by decreasing

the

diameter.

The one underlying (and possibly problematic) assumption made by using the trap fluorescence as a surrogate forf~N is that the atoms in the trap are assumed to be in either the 52SI/2 F = 3 or 52P3/2 F ~ = 4 levels. However, if the hyperfine repump laser is at a low intensity, a substantial number of atoms in the trap will be in the 5 2 3 1 / 2 F = 2 dark state. Assuming the microwave modulation induced sidebands for hyperfine pumping are 3% of the intensity of the primary trapping laser intensity, we calculate