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
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Contents CONTRIBUTORS
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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 . . . . . . . . . . . . . . . . . . . . . . .
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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
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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
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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
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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
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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
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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 .
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C. Interpretation within the R e s c a t t e r i n g M o d e l
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D. S-Matrix C a l c u l a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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E. T i m e - d e p e n d e n t Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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
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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
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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
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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
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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
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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.
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[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
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[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
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[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)
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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
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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
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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
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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,
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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.
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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
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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).
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ATI: CLASSICAL TO QUANTUM
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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
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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.
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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
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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 ~< ~ <~ 1)
F_,(t) = ~
coA
(~ sin cot - ~S' cos cot)
(5)
A A(t) = ~ (~ cos cot + ~S' sin cot) V/1 +~2
(6)
V/1 +~2
with vector potential
and ponderomotive energy Up = (eA)e/4m. The drift energy is restricted to the interval 2~ 2 1+
~2 Up ~
2 Edrift <~ 1 + ~2 Up.
Edrif t --
(eA(to))2/2m
(7)
For linear polarization, it can acquire any value between 0 and 2Up, while for circular polarization it is restricted to the value Up. Quantum mechanics considerably softens these classical bounds. However, these bounds are useful as benchmarks in the analysis of experimental spectra (Bucksbaum et al., 1986), in particular for high intensity (Mohideen et al., 1993; Reiss, 1996) or low frequency (Gallagher and Scholz, 1989). In general, it is important to recall that the ionization probability depends on the electric field, while the drift momentum p = eA(t0) is proportional to the vector potential, both at the time to of ionization. The probability of a certain drift momentum is weighted with the ionization rate at time to. The electron is preferably ionized when the absolute value of the electric field is near its maximum. Then, for linear polarization, the vector potential and, hence, the drift momentum are near zero. In order to reach the maximal drift energy of 2Up, the electron must be ionized when the electric field is zero and, hence, the ionization rate is very low. This explains the pronounced drop of the ATI electron spectrum for increasing energy, see Fig. 1. Sometimes, this interplay between the instantaneous ionization rate and the drift momentum has surprising consequences, notably for fields where the connection between the two is less straightforward than for a linearly polarized sinusoidal field, e.g. for a two-color
II]
ATI: CLASSICAL TO QUANTUM 100
' ' --~ 16.1eV
'
~
k~
I
o ._~
=>"
f
\
' =104~. . . . . lin.'pollt E= 103[ ~ ]
8
1
o,o,
101
\ -~1ooI . . . . . . . . ,~
f
_O9 o
43
p
\
0
10 20 30 40 50
v]
_..m (D
"O (D
.N_ E
tO r
'
-olso
'
o.oo
'
o.5o
'
.oo
ellipticity FIG. 3. Dependence of the photoelectron yield as a function of the ellipticity ~ of the elliptically polarized laser field (5) for electrons with an energy of 16.1 eV. Only electrons emitted parallel to the major axis of the polarization ellipse are recorded. The ATI spectrum corresponding to linear polarization (~ = 0) is shown in the inset. The laser intensity was 0.8x 1014 W/cm 2 at a wavelength of 630 nm. The figure illustrates the dodging effect mentioned in Sect. II.A.I: ionization primarily takes place when the electric field is near an extremum. For elliptical polarization, the electric field then points in the direction of the major axis of the polarization ellipse, and the vector potential in the direction of the minor axis. Hence, the electron's drift momentum p = eA(t0) is in the direction of the minor axis. It is the larger, the larger the ellipticity ~ is. Consequently, emission in the direction of the large component of the field decreases with increasing ellipticity: the electron dodges the strong component of the field. The effect vanishes when circular polarization is approached and the distinction between the major and minor axes disappears. From Paulus et al. (1998).
field (Paulus et al., 1995; Chelkowski and Bandrauk, 2000; Ehlotzky, 2001). Another illustration is the dodging phenomenon for the direct ATI electrons in an elliptically polarized laser field (Paulus et al., 1998; Goreslavskii and Popruzhenko, 1996; Mur et al., 2001), see Fig. 3. We have tacitly assumed that pulses are short enough to pass over the electron before it has a chance to experience the transverse spatial gradient of the focused pulse. In this event, the spatial dependence of the vector potential A(t) can truly be neglected. Hence the drift momentum p is conserved and is indeed the momentum recorded at the detector outside the field (Kibble, 1966; Becker et al., 1987). The wiggling energy Up is lost or, in a self-consistent description, returned to the field when the electron is left behind by the trailing edge of the pulse. In the opposite case, where the electron escapes from the pulse perpendicularly to its direction of propagation, the wiggling energy is converted into drift energy (Kruit et al., 1983, Muller et al., 1983). This limit is also realized in ATI by microwave ionization of Rydberg states (Gallagher and
44
[II
W. Becker et al.
Scholz, 1989). The effects of a space-dependent ponderomotive potential Up were observed in the "surfing" experiments of Bucksbaum et al. (1987). B. QUANTUM-MECHANICAL DESCRIPTION OF DIRECT ELECTRONS
There is an enormous body of work on the quantum-mechanical description of laser-induced ionization. For reviews, we refer to Delone and Krainov (1994, 1998). Here we want to concentrate on the analytical approach dating back to Keldysh (1964) and Perelomov, Popov and Terent'ev (Perelomov et al. 1966a,b; Perelomov and Popov, 1967). The goal is to find a suitable approximation to the probability amplitude for detecting an ATI electron with drift momentum p that originates from laser irradiation of an atom that was in its ground state ]~P0) before the laser pulse arrived: Mp =
t ~
e~,
lim
--(X3
t t ~
(lpp(t)
Iu(t, t')l ~p0(t')).
(8)
Here, U(t, t') is the time-evolution operator of the Hamiltonian (h = 1)
1
~r2
H ( t ) = -2--mm
- er . s
+ V(r),
(9)
which includes the atomic binding potential V(r) and the interaction - e r . g ( t ) with the laser field. Furthermore, we introduce the Hamiltonians for the atom without the field and for a free electron in the laser field without the atom, Ha - Hatom --
1 i~72 + V(r), 2m
1 V2
H f ( t ) - Hfield(t) = -~mm
- e r . g(t).
(10) (1 1)
The corresponding time-evolution operators are denoted by Ua and Uf, respectively. In Eq. (8), [~pp) and [~P0) are a scattering state with asymptotic momentum p and the ground state, respectively, of the atomic Hamiltonian Ha. The eigenstates of the time-dependent Schr6dinger equation with the Hamiltonian H f ( t ) are known as the Volkov states and are of compact analytical form. In the length gauge, one has Iw~vv)(t)) - IP - eA(t)) e-isp(t),
(12)
with ]p - eA(t)) a plane-wave state [(rip - eA(t)) = (2:r) -3/2 exp i[(p - eA(t)) 9r] and Sp(t) = ~ dr [p - eA(r)] 2. (13) The lower limit of the integral is immaterial. It introduces a phase that does not contribute to any observable.
II]
ATI: CLASSICAL TO QUANTUM
45
The time-evolution operator U(t, t') satisfies integral equations (Dyson equations), which are convenient if one wants to generate perturbation expansions with respect to either the interaction H t ( t ) = - e r . g(t) with the laser field, t
U(t, t')
= Ua(t, t') -
i / dr U(t, r) Hi(r)
U a ( r , t'),
(14)
tt
or the binding potential V(r), t
U(t, t') = Uf(t, t') - i /
dr Us(t,r) V U(r, t').
(1 5)
t /
tt
Equation (14) also holds if U and Ua in the second term on the right-hand side are interchanged. The equivalent is true of Eq. (15). With the help of the integral equation (14), using the orthogonality of the eigenstates of Ha, we rewrite Eq. (8) in the form Mp = - i
lim
dr (~pp(t)IU(t, r)H~(r)l ~0(r)),
t ----+ (N~
(16)
0<3
which is still exact. A crucial simplification occurs if we now introduce the strong-field approximation. That is, we make the substitutions I~pp) ---. I~ptpvv)) and U ~ Uy, with the result
Alp = - i
dto (~(pVV)(t0)IHt(t0)l ~0(t0)) 9
(17)
oo
The physical content of this substitution is that, after the electron has been promoted into the continuum at time to due to the interaction Ht(to) = - e r . F-.(to) with the laser field, it no longer feels the atomic potential. This satisfies the above definition of a "direct electron." Amplitudes of the type (17) are called KeldyshFaisal-Reiss (KFR) amplitudes (Keldysh, 1964; Perelomov et al., 1966a,b; Faisal, 1973; Reiss, 1980); for a comparison of the various forms that exist, see Reiss (1992). In the amplitude (17), one may write - e r . e(t0) =
Y Y H/(tol-Ha + V(r) = - i ~ 0 - i ~ 0 + V(r).
(is)
Via integration by parts, the amplitude (17) can then be rewritten as
Mp
:
dto (w~vv)(t0)I V(r)l
-i
Wo(to)).
(19)
oo
This form is particularly useful for a short-range or zero-range potential, since these restrict the range of the spatial integration in the matrix element.
W. Becker et al.
46
[II
Further evaluation of the amplitudes (17) or (19) leads to expansions in terms of Bessel functions. For sufficiently high intensity (small Keldysh parameter ~,), the saddle-point method (method of steepest descent) can be invoked (Dykhne, 1960). This consists in expanding the phase of the integrand about the points where the phase is stationary. Given the form of the Volkov wave functions (12) and the time dependence of the ground-state wave function, ]~P0(t)) = exp(iEipt)]~po), this amounts to determining the solutions of d
dt
[Eipt + Sp(t)] = EIp + ~ [ p - eA(t)] 2 = 0.
(20)
Let us consider a periodic (not necessarily monochromatic) vector potential with period T = 23:/co. In terms of the solutions ts of Eq. (20), the amplitude (19) can then be written as (Gribakin and Kuchiev, 1997; Paulus et al., 1998) /lip oc
Z 6
(p2
)
-~-mm+ Eip + mp - n oJ
n
• ~s
(21)
2~i S~'(ts)
ei[Ewts+Sp(ts)](p-eA(ts)lV[lP~
where S~~ denotes the second derivative of the action (13) with respect to time. The sum over s extends over those solutions of Eq. (20) within one period of the field (e.g. such that 0 ~< Re t, < T) that have a positive imaginary part. Obviously, the saddle points are complex unless Eip = 0. For Eip - 0, we retrieve the classical drift momentum (2) provided p is such that p = eA(t) at some time t. The imaginary part of to can be related to a tunneling time (Hauge and Stovneng, 1989). In Eq. (21), the ionization amplitude is represented as the coherent sum over all saddle points within one period of the field. The fact that the spectrum consists of the discrete energies
p2 Ep - 2m - n m - U p - Eip
(22)
can be attributed to interference of the contributions from different periods. This interference is destructive, unless the energy corresponds to one of the discrete peaks (22). Depending on the shape of the vector potential A(t) and its symmetries, there will be several solutions ts (for a sinusoidal field, there are two in the upper half plane and two in the lower, which are complex conjugate to the former) within one period of the field. Their interference creates a beat pattern in the calculated spectrum. This is, however, difficult to observe due to its sensitive dependence on the laser intensity, which is not very well controlled in an experiment so that the interference effects are usually washed out.
II]
ATI: C L A S S I C A L TO Q U A N T U M
47
Interferences also exist for an elliptically polarized laser field for fixed electron momentum as a function of the ellipticity. Since, in experiments, the ellipticity is better defined than the intensity, these interferences have been observed (Paulus et al., 1998); see next subsection. The amplitude (19) admits a vector potential A(t) of arbitrary shape; it is by no means restricted to a monochromatic field of infinite extent. For a pulse of finite extent, the saddle points are still determined by Eq. (20). They have, however, no longer any periodicity. Hence, the discreteness of the spectrum is lost. Interference from different parts o f the pulse may lead to unexpected effects (Raczyfiski and Zaremba, 1997). While for an infinitely long monochromatic pulse the spectrum is symmetric with respect to p ~ - p , this forward-backward symmetry no longer holds for a finite pulse. Analysis of the spatial asymmetry of the spectrum may aid in determination o f the pulse length or the absolute carrier phase (Dietrich et al., 2000; Hansen et al., 2001; Paulus et al., 2001b); see Sec. VII.B. C. INTERFERENCESOF DIRECT ELECTRONS For linear polarization and a drift m o m e n t u m p = p~ with [p] <~ eA, there are two possible ionization times cot01 = Jr/2 + 6 and O)to2 = 3 3 r / 2 - 6. The corresponding classical orbits are illustrated in Fig. 4. As discussed above, while A(t01) = A(t02), the field satisfies s = -•(t02). Hence, electrons ionized at t01 and t02 depart in opposite directions right after the instant of ionization. As illustrated in Fig. 4, the electric field changes sign soon after t01. Hence, the electron ionized at this time turns around at a later time and acquires the same drift momentum as the electron ionized at time t02, which keeps its original direction. We expect quantum-mechanical interference of the contribution of these two ionization channels. to~ =
Jr/2 + 6
0
2
4
rot~Jr
6
8
0
2
4
~ot # r
6
8
FIG. 4. Classical trajectories (dashed lines) of electrons having the same drift momentum. The solid line is the effective potential V(x)- exg(t) at times t01 (left) and t02 (right). The electron ionized at to] is turned around by the field shortly after ionization. In contrast, the electron ionized at t02 maintains its original direction. This is a strongly simplified picture of the physics underlying the interferences of direct electrons.
48
[II
W. B e c k e r et al.
3.0
a) E l ~=0.78
.,qo ~2.0
0.0 0.2 0.4 0.6 0.8 1.0
~=o.7~..-.--T---....~=o.7
E
,,....=
1.0 rt/2
3rd2
1"1;
,
,
0.25
0.50
0.75
Re[o~ts] F1G. 5. (a) Positions of the saddle points Ogts in the upper half of the complex ~ot plane in the interval 89 ~
cos
Ots =
v/1
+
I-v7
+
v / c ~ 2 - ~ip(1 - ~2) - ~2~] ,
(23)
where ~ = ( 1 - ~2)/(1 + ~2), ~ = E / 2 U p , and eip = E i p / 2 U p . Obviously, the solutions ts come in complex conjugate pairs. Those in the upper half-plane enter the amplitude (21). The solutions are plotted in Fig. 5. F o r - ~ 0 < ~ ~< ~0 [with ~0 given by the zero of the square root in Eq. (23)], the second square root on the right-hand side o f Eq. (23) is imaginary, and the solutions are symmetric with respect to Re tOts = Jr. For I~l ~> ~0, this square root is real, and all solutions have Re cot~ - Jr. This has important consequences for the saddle-point amplitude (21). In the first case, both solutions contribute to the amplitude and an interference pattern results. This corresponds to the case o f linear polarization discussed above. In the second case, inspection o f the integration contour in the complex plane shows that it has to be routed only through the one solution that is closest to the real axis (Leubner, 1981). Hence,
II]
ATI: CLASSICAL TO QUANTUM 10 0
,
,
f - o - 8.23eV ~ 10.03eV 11.38eV t 48.88eV
49
,
~ ,,d %, ~ ,o~2~ 4~a ~%,
"~"~
~104
o=o103
'-
-~102
i"
1~
10 20 30 40 50 60 70
c--
ln-2
"-1 .00
,
-0.50
,
0.00
,
0.50
1.00
ellipticity
FIG. 6. ATI spectra in xenon for an intensity of 1.2x 1014 W/cm 2 for various energies in the direction of the large component of The inset shows the energy spectrum display the interference phenomenon highest energy belongs to a plateau Paulus et al. (1998).
the elliptically polarized field as a function of the ellipticity. for linear polarization. The three traces for the lower energies of the direct electrons discussed in Sect. II.C; the one for the electron. The interference dips are related to Fig. 5c. From
there is no interference. In the first case, the amplitude can be written in the form Mp ~ exp(Re ~)cos(Im 9 + ~p), in the second case the cosine is absent. The two arguments are also plotted in Fig. 5. The corresponding interferences have been observed by Paulus et al. (1998); see Fig. 6. They are responsible for the undulating pattern in the ellipticity distribution, which moves to smaller ellipticity for increasing energy. The same tendency can be observed in the numerical evaluation of the amplitude (21); see Paulus et al. (1998) for an example. For elliptical polarization, the KFR amplitude (19) must be applied with due caution: it predicts fourfold symmetry of the angular distribution, while the experimental distributions only display inversion symmetry (Bashkansky et al., 1988). Mending this deficiency requires improved treatment of the binding potential (Krsti6 and Mittleman, 1991). More discussion of this point has been provided elsewhere by Becker et al. (1998), who also give further references. The spatial dependence introduced by Coulomb-Volkov solutions in place of the usual Volkov solutions (12) already suffices to destroy the fourfold symmetry, and angular distributions have been calculated with their help by Jarofi et al. (1999). However, even for a zero-range potential the fourfold symmetry is
50
[III
W. B e c k e r et al.
broken provided the effects of the finite binding energy are treated beyond the KFR approximation (Borca et al., 2001). Very similar interferences have been seen by Bryant et al. (1987) in the photodetachment of H- in a constant electric field. Here the electron, once detached, has the choice of starting its subsequent travel either against or with the direction of the electric field, by close analogy with the opposite directions of initial travel for the ionization times t01 and/02 in the present case; see Fig. 4. A spatial resolution of the same effect is observed by the photodetachment microscope of Blondel et al. (1999). The theoretical description reproduces the observed patterns. Additional bottle-neck structures develop when a magnetic field is applied parallel to the electric field (Kramer et al., 2001).
III. Rescattering: The Classical Theory Thus far, we have dealt with "direct" electrons, which after the first step of ionization leave the laser focus without any additional interaction with the ion. In this and the next section, we will consider the consequences of one such additional encounter. The classical model becomes much richer if rescattering effects are taken into account. To this end, we integrate the electron's velocity (2) to obtain its trajectory x(t) = -m
(t - to) A(t0) -
to
dr A(r)
.
(24)
The condition that the electron return to the ion at some time tl > to is x(tl) - 0. For linear polarization in the x-direction, this implies x ( t l ) = 0, and y ( t ) = z ( t ) - O. This yields tl as a function of to. We defer discussion of elliptical polarization to Sect. IV.H. When the electron returns, one of the following can happen (Corkum, 1993): (1) The electron may recombine with the ion, emitting its energy plus the ionization energy in the form of one photon. This process is responsible for the plateau of high-order harmonic generation. (2) The electron may scatter inelastically off the ion. In particular, it may dislodge a second electron (or more) from the ionic ground state. This process is now believed to constitute the dominant contribution to nonsequential double ionization. (3) The electron may scatter elastically. In this process, it can acquire drift energies much higher than otherwise. In the following, we will concentrate on this high-order above-threshold ionization (HATI). We will, however, also briefly discuss high-order harmonic generation in Sect. VI.
III]
ATI" CLASSICAL TO QUANTUM
51
From Eq. (2), the kinetic energy of the electron at the time of its return is Eret -
e2 ~m [A(tl) - A(t0)] 2 .
(25)
Maximizing this energy with respect to to under the condition that X(tl ) = 0 yields 3.17Up for oJt0 = 108 ~ and ootl = 342 ~ (Corkum, 1993; Kulander et al., 1993). It is easy to see that after rescattering the electron can attain a much higher energy: Suppose that at t = tm the electron backscatters by 180 ~ so that m v ( t l -- O) = e[A(to) - A ( t l )] j u s t before and mv(tl + O) = - e [ A ( t o ) - A(tl )] just after the event of backscattering. Then, for t > 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<3
which now describes both the direct and the rescattered electrons. The physical content of the amplitude (29) corresponds to the recollision scenario: The electron is promoted into the continuum at some time to; it propagates in the continuum subject to the laser field until at the later time tl it returns to within the range of the binding potential, whereupon it scatters into its final Volkov state. Exact numerical evaluation of the amplitude (29) for a finite-range binding potential is very cumbersome. For a zero-range potential, however, the spatial integrations in the matrix element become trivial, and the computation is rather straightforward. If the field dependence of the Volkov wave function and the Volkov time-evolution operator is expanded in terms of Bessel functions, one of the temporal integrations in the amplitude (29) can be carried out analytically and yields the same 6 function as in Eq. (21), specifying the peak energies. The remaining quadrature with respect to the travel time tl - t o has to be carried out numerically; see Lohr et al. (1997) and Milo~evi6 and Ehlotzky (1998a), where explicit formulas can be found; for elliptical polarization see Becker et al. (1995) and Kopold (2001). Alternatively, the integral over the travel time may be done first, and the integral over the return time tl is then evaluated by Fourier transformation (Milo~evi6 and Ehlotzky, 1998b). The relevance of the rescattering mechanism to ATI and multiple ionization was suggested early by Kuchiev (1987) and by Beigman and Chichkov (1987). Improvements of the customary KFR theory by including further interactions with the binding potential were already discussed by Reiss (1980). The first explicit calculations of angular-resolved energy spectra were carried out by Becker et al. (1994a, 1995) and by Bao et al. (1996). Closely related rescattering models were presented by Smirnov and Krainov (1998) and by Goreslavskii and Popruzhenko (1999a,b, 2000). The physics of high-order ATI is related to electron scattering at atoms in the presence of a strong laser field. For high-order ATI, the initial state of the electron is a wave packet created by tunneling, while for electron-atom scattering it is a plane-wave state. This latter problem was studied theoretically by Bunkin and Fedorov (1966) and by Kroll and Watson (1973). Corresponding experiments
IV]
ATI: CLASSICAL TO QUANTUM
55
were done by Weingartshofer et al. (1977, 1983). Some quantum features of electron scattering in intense laser fields are remarkably similar to HATI; see Kull et al. (2000) and G6rlinger et al. (2000). A.
SADDLE-POINT METHODS
For sufficiently high intensity, the temporal integrations in the amplitude (29) can be carried out by the saddle-point method, as in the case of the direct amplitude (19). This procedure provides much more physical insight than Bessel-function expansions, and establishes the connection with Feynman's path integral, to be discussed below. In this context, rather than taking advantage of the explicit form of the Volkov time-evolution operator, we expand it in terms of the Volkov states (12),
S,'k
I,
(30.
so that the amplitude Mp is represented by the five-dimensional integral
Mp ~
f o~
/'f
dtl
dto
with the function
d3k exp[iSp(tl , to, k)] mp(tl, to, k)
mp(tl, to, k) = (p - eA(t,)[ VI k - eA(tl)) (k - eA(to) lV[ ~Po).
(31 )
(32)
For ATI, the action
Sp(tl,to, k) -
1 2m
2m
ftl ~
d r [p - e A ( r ) ] 2
(33) dr [k - eA(T)] 2
+
dr EIp
in the exponent consists of three parts, according to the three stages discussed above. As above in Eq. (20), we approximate the amplitude (31) by expanding the phase (33) of the integrand about its stationary points. In this process, we assume that the function mp(tl, to, k) depends only weakly on its arguments. Indeed, for a zero-range potential, it is a constant. We now have to determine the stationary points with respect to the five variables tl, to and k. They are given by the solutions of the three conditions (Lewenstein et al., 1995a) [k - eA(t0)] 2 = -2mEip,
(34)
dr eA(r),
(35)
[k - eA(tl)]2 = [p _ eA(tl)]2.
(36)
( t l - t0) k =
The first condition (34) attempts to enforce energy conservation at the time of tunneling. The second condition (35) ensures that the electron returns to its
56
W. Becker et al.
[IV
parent ion, and the third one (36) expresses that, on this occasion, it rescatters elastically into its final state. In general, the saddle-point equations have several solutions (tls, t0s, ks), (s = 1,2,...), of which only those are relevant for which Re tls > Re t0s, such that the recollision is later than ionization, cf. the limits of the integral in Eq. (31). The matrix element can be written as
k15)
1/2
det( O2So/Oq~S)Oq~S))j,
eiSp(t's't~
tos, ks),
(37)
where ql s) (i = 1 , . . . , 5 ) runs over the five variables tls, tos and ks. As we noted already for the direct electrons in the context of Eq. (21), the sum has to be extended only over a subset of the solutions of the saddle-point equations (34)-(36). However, in the present case, determining this subset may be tricky (Kopold et al., 2000a). For a periodic field, the sum over the periods in Eq. (37) can be carried out by Poisson's formula. This leaves a sum over the saddle points within one period and produces a 6 function as in Eq. (21). The computation of ATI now consists of two separate tasks. First, the solutions of the saddle-point equations (34)-(36) have to be determined and, second, the appropriate subset has to be inserted into expression (37). Note that we apply the saddle-point approximation to the probability amplitude for given final momentum p, and not to the complete wave function of the final state. This is the reason why only few solutions contribute, while a semiclassical computation of the wave function, which contains all possible outcomes, requires consideration of a very large number of trajectories (van de Sand and Rost, 2000). Since EIp > 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 <~ t <~ Re tls), (t
I
tls)P -ftts dr eA(r)
(t ~> Re tls).
(38)
We regard the orbit as a function of the real time t. The conditions x(t0) = 0 and X(tl) = 0, however, are satisfied for the complex times to and tl. As a consequence, the quantum orbit (38) as a function of real time does not depart from the origin but, rather, from the "exit of the tunnel." This is clearly visible in Figs. 14, 15, 17 and 20 below. In contrast to the start time to, the return time tl is real to a good approximation, see Fig. 10 (below). Accordingly, the orbits return almost exactly to the origin. B. CONNECTION WITH FEYNMAN'S PATH INTEGRAL Any quantum-mechanical transition amplitude, such as the ionization amplitude (8), can also be represented in terms of Feynman's path integral. To this end, we recall the path-integral representation of the complete time-evolution operator of the system atom + field,
U(rt, r't') = f(rt) ~ ~r't')
7)[r(r)]e is(''t'),
(39)
where S(t, t') = ft t, dr L[r(r)], r] is the action calculated along a system path, and the integral measure D[r(r)] mandates summation over all paths that connect (rt) and (r' t') (see, e.g., Schulman, 1977). The path integral (39) sums over the functional set of all continuous paths. In the quasi-classical limit, this can be reduced to a sum over all classical paths, which are those for which the action S(t,t') is stationary. For quadratic Hamiltonians, this WKB approximation is exact. In our case, motivated by the success of the classical three-step model of Sect. III, we have reduced the exact transition amplitude to the form (31). In implementing the strong-field approximation, we have approximated the exact action of the system appropriately at the various stages of the process: before the initial ionization, in between ionization and rescattering, and after rescattering, as in the decomposition (33) of the action. This still left us with a five-dimensional variety of paths. Out of those, finally, the saddle-point approximation (37) selects the handful of "relevant paths" (Antoine et al., 1997; Kopold et al., 2000a; Sali~}res et al., 2001). These are essentially the orbits of the classical model, yet quantum mechanics is fully present: Their coherent superposition as expressed in the form (37) allows for interference of the contributions of different orbits, and the fact that they are complex accounts for their origin via tunneling.
W. Becker et al.
58
[IV
C. CONNECTION WITH CLOSED-ORBIT THEORY
There appears to be a close similarity to the concepts of periodic-closed-orbit theory, see, e.g., Du and Delos (1988), Gutzwiller (1990), and Delande and Buchleitner (1994). The photoabsorption cross section o(E) of an atom in the state ]~pi) with energy Ei can be expressed in the form (Du and Delos, 1988)
o(E) = 4~ hcc Re
dt e iEt (lpi DU(t, 0)D I 1])i )
]
,
(40)
where D = r . e is the dipole operator responsible for photoabsorption of the field with polarization e and E ,.~ Ei + ]'tO). The quantity U(t, 0) is the timeevolution operator in the presence of the binding potential as well as additional static external electric and magnetic fields that may be present. In effect, the timeevolution operator propagates wave packets at constant energy that emanate from the atom and are reflected by the caustics of the potential back to the atom where they interfere with each other and with the starting wave packets. This leads to oscillations in the photoabsorption spectrum. In a semiclassical approximation, the time-evolution operator can be expanded in terms of classical closed orbits that start from and return to the vicinity of the atom, defined by the spatial range of the wave function ]~Pi). Since the classical problem is chaotic, there are more and more such orbits when the energy nears zero. Fourier transformation of the photoabsorption spectrum reveals the recurrence times of the classical orbits. There are several differences to the quantum orbits we are considering here. In our case, the role of the binding potential is, in effect, reduced to acting as a coherent source of electrons and to causing rescattering, while in closed-orbit theory the interplay of its spatial shape with the external static fields generates the rich structure of the closed orbits. In our case, closed orbits are entirely due to the time dependence of the laser field. The most important difference is that closed-orbit theory is concerned with total photoabsorption rates as a function of frequency, while we consider differential electron spectra for a laser field with fixed frequency. In other words, our orbits depend on the final state of the electron. From Eq. (19), in view of the completeness of the Volkov states, the total ionization probability due to direct electrons is
d 3p IM, i2 = 2e 2
[ f~ L a --OO
ft,d,o d --OO
(~or
F-,(t,)u(Vv)(t,,to)r
s
9
~'o(to)).1
(41)
This differs from the photoabsorption cross section (40) only by the presence of the Volkov time-evolution operator u(Vv)(tl,t0), which reflects the strongfield approximation, instead of the exact time-evolution operator U(t, 0) of the
time-independent problem [for which the time-evolution operator U(t, t ~) only
IV]
ATI: CLASSICAL TO QUANTUM
59
depends on the time difference t - t~]. In the total ionization probability (41), via the same partial integration (18) as above, the electron-field interaction r. s can be replaced by V(r). The result then looks like the differential HAIl amplitude (29) except that it is sandwiched by the ground state. This correspondence is a manifestation of the optical theorem. D. THE ROLE OF THE BINDING POTENTIAL
The improved Keldysh approximation (29) has been written down for an arbitrary binding potential V(r). The expansion in terms of the binding potential, introduced via the Dyson equation (15), is a strong-field approximation (SFA), which is valid when the electron's quiver amplitude is so large that most of its orbit is outside the range of the binding potential. This is trivially guaranteed for the three-dimensional binding potential of zero range, V(r) : mtr 6(r)Or"
(42)
This potential supports a single (s-wave) bound state at the energy -tc2/2m and a continuum that is undistorted from the free continuum except for the s wave, as required by completeness (Demkov and Ostrovskii, 1989). Without the regularization operator (O/Or)r, which acts on the subsequent state, the potential does not admit any bound state. There are several possibilities to adjust the one parameter tr to an individual atom or ion. In most cases one will determine it so as to reproduce the ionization potential; see, however, Sect. IV.G. The zero-range potential (42) underlies many of the explicit results exhibited in this chapter. However, we emphasize that the amplitude (29), as well as its saddle-point approximation (37), hold for a much wider class of potentials. Regardless of the potential, the saddle-point equations (34)-(36) have the electron start from and return to the center of the binding potential, which is the origin, and do not depend on its shape. The potential only enters via the form factors in Eq. (32). For the SFA to be applicable, they must depend on time only weakly. The procedure corresponds to the Born approximation. It will be the better justified, the shorter the range of the potential is, so that the form factor depends only weakly on the momenta. Excited bound states do not enter the amplitude (29) regardless of the potential used. For a comparison of a high-order ATI spectrum calculated for the zero-range potential (42) with the same spectrum extracted from a solution of the threedimensional TDSE for hydrogen, see Cormier and Lambropoulos (1997) for the latter and Kopold and Becker (1999) for the former. There is good qualitative agreement within the ATI plateau; in particular, the positions of the dips in the spectrum that are due to destructive interference agree within a few percent. The comparison confirms that the detailed shape of the potential has only minor significance for the HATI spectrum. This holds for hydrogen and the rare gases,
W. Becker et al.
60
[IV
but not for the alkali-metal atoms (Gaarde et al., 2000). The height of the rescattering plateau with respect to the direct electrons does depend on the atomic species; for the theoretical modeling, see Goreslavskii and Popruzhenko (1999a, 2000). Clearly, however, the real physical systems best described by a zerorange potential are negative ions with a s-wave ground state. Angular-resolved photoelectron spectra in H- have been recorded recently by Reichle et al. (2001). Attempts have been made at including the effects of the binding potential of the residual ion beyond the first-order Born approximation. Kamifiski et al. (1996) and Milogevi6 and Ehlotzky (1998c) have employed the so-called Coulomb-Volkov (CV) states, which are obtained from the ordinary Volkov states (12) by replacing the plane wave [ p - eA(t)) by a Coulomb outgoing-wave scattering state with momentum p (ordinary CV state) or p - eA(t) (improved CV state). These are then substituted in the transition amplitudes (17) or (29) for the ordinary Volkov states. A systematic assessment of the merits of this approach appears not to have been made; recently, however, it has been compared with the solution of the TDSE and was shown to work well for ionization by ultrashort pulses having a duration shorter than the orbital period of the initial bound state (Duchateau et al., 2001). E. A HOMOGENEOUS INTEGRAL EQUATION An alternative route to the standard KFR matrix element (19) and its improved version (29) starts from the homogeneous integral equation
IvP(t)) = - i
/'
dr Uf(t, r) VIW(r)),
(43)
OO
which holds for the state that develops out of the unperturbed ground state due to its interaction with the laser field. This integral equation can be derived immediately from the Dyson equation (15) if one applies both sides of the latter to the atomic ground state [~P0(t~)) in the limit where t ~ ---, -oo. By inspection, one may convince oneself that the term Uf(t,t')[~Po(t')) makes no contribution for t - t ~ --, cx~ so that Eq. (43) is left. This equation was first introduced in the context of the quasi-energy formalism by Berson (1975) and by Manakov and Rapoport (1975) for circular polarization and Manakov and Fainshtein (1980) for arbitrary polarization. Inserting on the right-hand side of the integral equation (43) the expansion (30) of Uf in terms of Volkov states and replacing [W(r)) by the unperturbed atomic ground state [~P0(~')), one can read off the matrix element (19) for direct ionization. Iterating Eq. (43) one gets [W(t)) = -
dr' Uf(t, r)
dr s
VUf(T, .gt) V klJ(Tjt)),
(44)
(X)
which yields the improved KFR amplitude (29) in the same fashion. The integral equation (43) is particularly useful for the zero-range potential (42), since in this case it allows one to calculate the wave function in all space
IV]
ATI: CLASSICAL TO QUANTUM
61
provided it is known at the origin. For the latter, to a first approximation, one may employ the unperturbed wave function. Better approximations are obtained by using more accurate expressions. These incorporate the possibility that the ionized electron revisits the core, as illustrated by Eq. (44). For the zero-range potential and a monochromatic plane wave with circular polarization, it can be shown that the wave function near the origin exactly obeys qJ(r,t) o( ( 1 / r - t c ) e x p ( - i E t ) for all times. The complex quasi-energy E has to be determined as the eigenvalue of a nonlinear integral equation (Berson, 1975; Manakov and Rapoport, 1975). For any polarization other than circular, the time dependence at the origin is given by a Floquet expansion (Manakov and Fainshtein, 1980; Manakov et al., 2000). The interaction with a laser field for a finite period of time was considered along similar lines by Faisal et al. (1990) and Filipowicz et al. (1991); see also Gottlieb et al. (1991) and Robustelli et al. (1997). The integral equation (43) was also used for two-center potentials in order to model molecular ions (Krstid et al., 1991; Kopold et al., 1998). F. QUANTUM ORBITS FOR LINEAR POLARIZATION For linear polarization, Fig. 9 presents a calculated ATI spectrum that is typical of a high laser intensity, cf. the data of Walker et al. (1996). The solid circles that make up the topmost curve of the upper panel were calculated from the amplitude (29) by means of a zero-range potential, while the other curves give the results of including an increasing number of quantum orbits in the saddle-point approximation (37). The spectra that result from just the sth pair (which comprises the orbits 2 s - 1 and 2s) are displayed in the lower panel. Quantitatively, the first pair dominates the entire spectrum, but the contribution of the second pair comes close, in particular near its cutoff around 7 Up. The contribution of the third pair is already weaker by almost one order of magnitude, and the subsequent pairs hardly play a role anymore. Indeed, in the upper panel, already the third curve from bottom virtually agrees with the result of the exact calculation. The dependence of the parameters tls, t l s - tos (the travel time), and ks on the electron energy E v is illustrated in Fig. 10 for the two orbits (s = 1,2) having the shortest travel times. These parameters uniquely specify the quantum orbits in space and time. Their behavior is very different for energies below and above the classical cutoff at 10Up. Below the cutoff, the imaginary parts of the parameters are only weakly dependent on the energy. Both orbits have to be included in the sum (37), and their interference leads to the beat pattern, which is visible in the spectrum of Fig. 9. Notice that the imaginary parts of both the return times tls and the momenta kx, are small. In contrast, the imaginary part of the travel times tls - t0s, which are related via t0s to the tunneling rate, is substantial; see Fig. 5, where the ionization time ts is plotted for the direct electrons. Hence, after rescattering, the orbits are real for all practical purposes: the electron has forgotten its origin via tunneling. The parameter values of the two orbits (s = 1,2) approach each other closely near the cutoff. At some point, one of the two orbits
W. B e c k e r et al.
62
[IV
-8 "1""I 1'[ E~~ '/,, integration . . . . . . "E - 1 2
-o
-16
10
!
~-~ -22
~ -24 -26
-28 0
|
!
J
1
2
"N,
P,
2
3
4
g t5 "7 8 9 electron energy [Up]
10 11 12
FIG. 9. Upper panel: ATI spectrum in the direction of the laser field for linear polarization for 1015 W/cm 2, hto = 0.0584 a.u., and a binding energy of Eip = 0.9 a.u. The electron energy is given in multiples of Up. The curve at the top (solid circles) is the exact result from Eq. (29). The other curves were calculated from the saddle-point approximation (37). From bottom to top, more and more quantum orbits are taken into account; the results are displaced with respect to each other for visual convenience. The curve at the bottom incorporates just the pair of orbits with the shortest travel times, the next one up includes in addition the pair with the next-to-shortest travel times, and so on. The occasional small spikes are artifacts of the saddle-point approximation, cf. Goreslavskii and Popruzhenko (2000) and Kopold et al. (2000a). Lower panel: The envelopes of the contributions of the individual pairs are shown all on the same scale so that the quantitative relevance of the various pairs is put in perspective. The cutoffs of the various orbits agree with those displayed in Fig. 8. From Kopold et al. (2000a).
IV]
ATI: C L A S S I C A L TO Q U A N T U M Re c o ( t - t ' )
Re cot 2 i
0.5
3
'
i
'
3
i
,
_
4
1
,
5
i
[!11.51
,
-
0 -0.5
_a o
2.5
~,, tl
-0.5
-o-
9
- o-
i
i
~ ,
i
J i
I
1,15
-2 , _
I
'
- 11.5 _ ~
_.,. ~-.z-.:_ ..
,,
E -0.1 -
~ -
i
-0.8
,
i
-0.6
, "
~
2.54
~ 11.5 9 , v
-0.4
--
t11.5 ,
,
s E
-1.5
I
m
< 0.1 i1) 0
-1
- -iBt~ 11.5
e91 1 5 9 i
0.2
4
11.5~
_
63
I
-0.2
Re kx/leAI
,
t
'
'
I
'
'
~ ~ 2 5 0
1
/,o-~
_
'
'
-
""
11.5
i
< m
-
-0.2 x " E m
...2.,s i
-0.1
i
-0.1
1
i
i
0
Reky/leAI
FIG. 10. Saddle points (ts, t~,ks) for the orbits (s = 1,2) having the two shortest travel times. In this figure, ts is the return time (elsewhere denoted by t]s), and ts~ the start time (elsewhere denoted by tos). The figure shows a comparison of elliptical polarization (~ --0.5, solid circles) and linear polarization (open squares). The values of the other parameters are those of Fig. 9 (eA = 2.04 a.u.). The symbols identify electron energies of 11.5, 10.4, 8.92, 6.01, and 2.49, all in multiples of Up. The dashed orbits have to be dropped from the sum (37) after the cutoff. With the scaling of k given on the ordinate, the saddle points depend only on the Keldysh parameter y = v/IEoI/2Up. (drawn dashed in the figure) has to be dropped from the sum (37). This causes the artifact of the small spikes visible in Fig. 9. For energies above the cutoff, just one orbit contributes and, as a consequence, the spectrum smoothly decreases without any trace o f interferences. The real part o f the parameters stays approximately constant, while the imaginary part increases strongly with increasing energy. This is responsible for the steep drop o f the spectrum after the cutoff. Similar behavior, as a function o f ellipticity, occurs in Fig. 5. The procedure o f dropping one o f the orbits o f each pair after its cutoff can be replaced by a more rigorous method. In the vicinity o f the cutoffs, an approximation in terms of Airy functions was used by Goreslavskii and Popruzhenko (2000). A uniform approximation was described in a different context by Schomerus and Sieber (1997). It reproduces the spectra o f Fig. 9 without the spikes (Figueira de Morisson Faria et al., 2002b). G. ENHANCEMENTS IN ATI SPECTRA In several experiments, pronounced e n h a n c e m e n t s o f groups o f ATI peaks in
64
[IV
W. B e c k e r et al.
105
==
103 . . . . . . . . . .
8 lo 2 101 I
0
10 20 30 40 electron energy [eV]
FIc. 11. ATI spectra in argon at 800 nm recorded in the direction of the linearly polarized field for various intensities rising by increments of 0.110 from 0.5 I0 (bottom curve) to 1.010 (top curve). The horizontal lines mark the maxima of the ATI plateaus for each intensity. For intensities I > 0.8 I0 a group of ATI peaks between 15 eV and 25 eV rears up quickly. (The spectra shown here represent only a fraction of those actually measured.) From Paulus et al. (2001a). the plateau region (by up to an order of magnitude) have been observed upon a change of the laser intensity by just a few percent (Hertlein et al., 1997; Hansch et al., 1997; Nandor et al., 1999). This behavior suggests a resonant process. Near the resonances, the contrast of the spectra is remarkably reduced (Cormier et al., 2001). For the experiments reported so far, the effect is most pronounced for argon. This holds not only for a laser wavelength of 800nm but also for 630nm (Paulus et al., 1994c). The enhancements are so strong that in experiments implying significant focal averaging the observed spectral intensity may well be dominated by these enhancements, regardless of the actual peak intensity. In this sense, ATI in toto has been called a resonant process (Muller, 1999b). A big step towards understanding the physical origin of the enhancements was made in theoretical studies that reproduced the enhancements in the single-active-electron approximation by numerical solution of the oneparticle time-dependent Schr6dinger equation in three dimensions (Muller and Kooiman, 1998; Muller, 1999a,b; Nandor et al., 1999), thereby ruling out any mechanism that invokes electron-electron correlation. In Fig. 11 we show results of a measurement of the same effect, but for a shorter pulse length of 50 fs (Paulus et al., 2001a). Spectra in an intensity interval of 0.5 to 1.0 • I0 in steps of 0.1 • I0 are displayed. The maximum intensity I0 was calibrated by using the cutoff energy of 10Up. This leads to I0 ~ 8• 1013 W/cm 2. There is a striking difference between the spectra for I ~< 0.810 and those for higher intensity: within a small intensity interval a group
IV]
ATI: CLASSICAL TO QUANTUM
65
of ATI peaks corresponding to energies between about 15 eV and 25 eV grows very quickly. In the figure this is emphasized by horizontal lines drawn at the maximal heights of the plateaus. Increasing the intensity above 0.910 leads to a smaller growth rate of these peaks. The plateau, however, preserves its shape. For an interpretation, it should be kept in mind that a measured ATI spectrum is made up of contributions from all intensities I ~< I0 that are contained within the spatio-temporal pulse profile. This means that a spectrum for a fixed intensity would show the enhanced group of ATI peaks only at that intensity where it first appears in our measurement, namely at I ~ 0.8510 = 7• 1013 W/cm 2. In other words, the enhancement happens at a well-defined intensity or at least within a very narrow intensity interval. Analyzing the wave function of the atom in the laser field, Muller (1999a) suggested that the enhancements are related to multiphoton resonances with ponderomotively upshifted Rydberg states. In some cases, in particular for electrons with rather low energy, one particular Rydberg state could be definitely identified as responsible. In others, notably for the strong enhancement that for appropriate intensities dominates the middle of the plateau, this was not possible (Muller, 2001 a). A closer look at the data of the measurement shown in Fig. 11 reveals that under the conditions of this experiment (i.e. a pulse duration of 50 fs as compared with more than 100fs in the other measurements mentioned) resonantly enhanced multiphoton ionization does not play an essential role. This can be deduced from the different intensity dependence of the enhanced ATI peaks in the plateau and of the low-energy ATI peaks, see Fig. 12. The latter are known to originate from atomic resonances (Freeman et al., 1987; Agostini et al., 1989). In Muller's numerical simulations, the existence of excited bound states appears to be instrumental for the enhancements. Yet, the modified KFR matrix element (29), which does not incorporate any excited states, produces much the same enhancements (Paulus et al., 2001a; Kopold et al., 2002). An example is shown in Fig. 13. In these calculations, the enhancements occur for intensities for which an ATI channel closes. This is the case when EIp + Up = khco.
(45)
For an intensity slightly higher than specified by this condition, k + 1 is the minimum number of photons required for ionization in place of k. Such channel closings are distinctly visible in the multiphoton-detachment yields of negative ions (Tang et al., 1991) and have been shown to produce a separate comb of peaks in the low-energy ATI spectrum (Faisal and Scanzano, 1992). Comparison of the channel-closing condition (45) with the ATI energy spectrum (22) shows that at a channel closing electrons may be produced with zero drift momentum p. In this event, the energy of the k photons is entirely used to overcome the binding potential raised by the ponderomotive energy, and no energy is left for a drift
66
[IV
I'I4. B e c k e r et al.
I
o d
0.00
iI
. _
co
" -0.05 O o I1) N
E
I
-0.10
0
-0.15
tI
-
plateau
a
7.3 eV
•
6.4eV
I
0.4
=
t
I
0.6
\
I
I/I o
0.8
~
J
1.0
FIG. 12. Comparison of the intensity dependence of ATI electrons with different energies. For visual convenience, the overall increase in yield with increasing intensity has been subtracted. The electrons at 6.4eV and 7.3 eV are due to the strongest Freeman resonances, i.e. resonance with atomic states. Those labeled "plateau" are electrons in the plateau region of the spectra. As a consequence of the subtraction of the overall increase, the resonance-like behavior corresponds to those intensities where the respective curves start rising. It is evident that for the plateau electrons this does not happen at those intensities where the atomic states shift into resonance. Quite to the contrary, the intensity at which the yield of the plateau electrons starts its rise is reflected in the yield of the low-energy electrons by a brief halt in their rise. This is indicated by the dashed circles. motion. An electron having a drift m o m e n t u m near zero has many recurring opportunities to rescatter. Indeed, the quantum-orbit analysis of the spectra o f Fig. 13 shows that at the channel closings, and only there, an exceptionally large number o f orbits are required to reproduce the exact result. All of these orbits conspire to interfere constructively to produce the observed enhancements. In the tunneling regime, this can be proved analytically (Popruzhenko et al., 2002). In Muller's numerical simulations, inspection of the temporal evolution reveals that at the intensities that produce the enhancements electrons linger about the ion for m a n y cycles of the field before the final act of rescattering. A detailed comparison between Muller's numerical simulations and results based on Eq. (29) has been made by Kopold et al. (2002). This paper also includes an assessment of the consequences of focal averaging. It is noteworthy that both approaches predict ATI enhancements also for helium deeply in the tunneling regime, in spite o f the obvious multiphoton character of the channel-closing condition (45). Unfortunately, the helium data of Walker et al. (1996) and Sheehy et al. (1998) do not allow one to draw conclusions about the presence or absence of enhancements. The interference interpretation just given requires the existence of a sufficient number of orbits to contribute to the energy considered. The lower panel o f
IV]
ATI: C L A S S I C A L TO Q U A N T U M I 12
!
4~
/ -13 ~
~
-14
1"1= 2.626 ........
,'- "-.
"'-'~- '" '
-15
'" ''• ,,,
,'
': ,,'
, , I
10
',;~
I
20
~
67
I
30
,
~
"" -
-
2
-
6
I
J
i
t,rajec] ~
-
40
40 I
50
ffl
~ 9 -12
& -13
9
_
"o
~, -14 tO i_ tO
-15 ~",
o 0
10 - 12 j
I ,
' ,
20
30
40
50
q = 2.326
oo
-13
e ~
-14 ,....,, -15""
,, ','
10
20
30
---
6
9
trajectories
04
40
50
electron energy [eV]
Fic. 13. ATI spectra for Eip = 14.7eV, oJ = 1.55 a.u., and three intensities: at a channel closing (71 = Up/oJ = 2.526, middle panel), below the channel closing (7/= 2.326, lower panel), and above
(7/= 2.626, upper panel). In each panel, the exact result calculated from Eq. (29) is shown (solid symbols) and approximations involving the first 2 (dashed line), 6 (dot-dashed line), and 40 (solid line) quantum orbits in Eq. (37). From Kopold et al. (2002). Fig. 9 shows that too few orbits contribute for energies above about 8Up. Indeed, the enhancements observed experimentally are restricted to the lower two-thirds of the plateau. The interpretation also implies that the enhancements should disappear for ultrashort pulses, where late returns do not occur. This has been observed in experiments by Paulus et al. (2002). In numerical simulations o f H H G based on the three-dimensional TDSE, the same effect has been noticed by de Bohan et al. (1998). W h e n the modified K F R matrix element (29) is used to describe data for real atoms, the ionization potential EIp has to be replaced by an effective (lower) value that corresponds to the d e f a c t o onset o f the continuum (Paulus et al.,
68
W. B e c k e r et al.
[IV
2001 a; Kopold et aL, 2002). It is a fact that, for a Coulomb potential, the actual onset of the continuum is hard to see and may better be replaced by an effective value. This is illustrated, for example, by the photoabsorption spectra of Garton and Tomkins (1967). Numerical simulations predict very similar enhancements in high-order harmonic spectra (Becker et al., 1992; Toma et al., 1999; Kuchiev and Ostrovsky, 2001; Kopold et al., 2002) and in nonsequential double ionization (NSDI) of helium (Muller, 2001c). Experimentally, in argon irradiated by a flat-top pulse from a Ti:Sa laser, resonant-like enhancement of the 13th harmonic was observed by Toma et al. (1999). In HHG in one dimension, the dependence of the enhancements on the shape of the potential and the presence or absence of excited bound states has been investigated (Figueira de Morisson Faria et al., 2002a). The results are largely compatible with the quantum-orbit picture. In a semiclassical framework, the binding potential can be incorporated into the orbits. This leads to Coulomb refocusing (Ivanov et al., 1996; Yudin and Ivanov, 2001a): orbits that would miss the ion in the absence of the binding potential are refocused to the ion in its presence. This emphasizes the importance of late returns and leads to a substantial increase of rescattering effects without, however, resonant behavior. If late returns are cut off due to an ultrashort laser pulse, the rate of NSDI should decrease. Indeed, this has been experimentally confirmed by comparison of 12-fs and 50-fs pulses (Bhardwaj et al., 2001). H.
QUANTUM ORBITS FOR ELLIPTICAL POLARIZATION
Formulation of a classical model of the simple-man variety to describe rescattering for an elliptically polarized laser field meets with difficulties. The problem is that an electron that starts with zero velocity almost never returns exactly to its starting point if the laser field has elliptical polarization. Formally, this shows in the saddle-point equations (34)-(36) as follows. For EIp = 0, Eq. (34) yields k = eA(t0) if real solutions are sought. For linear polarization, this leaves two equations to be solved for to and tl: Eq. (36) and the x-projection of Eq. (35). Real solutions are obtained, provided the final momentum p is classically accessible. In contrast, for elliptical polarization, three equations are left since now both the x-projection and the y-projection of Eq. (35) have to be considered. Hence, there is no simple-man model for elliptical polarization, even when EIp = 0. The same situation occurs for HHG. This does not mean that there is no HATI or HHG for elliptical polarization: quantum-mechanical wave-function spreading assures overlapping of the wave packet of the returning electron with the ion (Dietrich et al., 1994; Gottlieb et al., 1996). The complete absence of HHG for a circularly polarized laser field is sometimes taken as confirmation of the rescattering mechanism. This conclusion is not rigorous since there is still sufficient overlapping. Rather, the absence of HHG is due to angularmomentum selection rules or, equivalently, destructive interference.
IV]
ATI: C L A S S I C A L TO Q U A N T U M
~
5Up
edlrec~ons~.
-18
-~o
69
20 a.:u. electric field ellipse
' +~~'~~~" ~
"=
7Up
~ y ~
-
-20
.m r
.
.
.
.
.
.
.
.
.
.
.
.
8.5u
.
~"o. -22
"
2 -24 o
"
"
"
o
..... ~ -26 . . . . . . . o ....... I , -28 3
"
exact
"
-""~"
'
'\
integration
11+212 13+412 15+612 Isuml 2
O
-
2
I
4
~
j
"
2
o~ ,,, "~ ,, o ',,, % '\ -',, '~ ', ',,, '\ ',,,
_
~
I
5
~
I
~
I
~
~,1
6 7 8 electron energy [ Up]
,
]'. . . .
9
10
11
FIG. 14. ATI spectrum in the direction of the large component of the elliptically polarized driving laser field (5) for ff = 0.5 (see the field ellipse in the upper right corner of the figure) and electron energies between 2.5 and 10.5Up. The other parameters are o9 = 1.59eV, EIp = 24.5eV, and I = 5• 1014 W/cm 2. The open circles give the yields of the individual ATI peaks calculated from the integral (29) for the zero-range potential. The other curves represent the contributions to the quantum-path approximation (37) of the shortest trajectories 1 and 2 (dot-dashed line), 3 and 4 (long-dashed line), and 5 and 6 (short-dashed line), as well as the sum of all six (solid line). Note that some of these curves overlap partly or entirely. The orbits responsible for each part of the spectrum, viz. 1 and 2, 3 and 4, and 5 and 6, are presented near the margins of the figures. The position of the ion is marked by a cross; notice that the orbits do not depart from there, but rather from a point several atomic units away from it. This is the point where the electron tunnels into the continuum. The electron travels the orbits in the direction of the arrows. Experimental data for a similar situation are shown in Fig. 15. From Kopold et al. (2000b).
One m i g h t try to f o r m u l a t e a s i m p l e - m a n m o d e l for elliptical p o l a r i z a t i o n by relaxing the r e q u i r e m e n t that the e l e c t r o n return exactly to the posi t i on o f the ion or by a d m i t t i n g a n o n z e r o initial velocity, but in d o i n g so a large a m o u n t o f arbitrariness is unavoidable. Instead, we will just solve the s a d d l e - p o i n t e q u a t i o n s ( 3 4 ) - ( 3 6 ) and accept and interpret the c o m p l e x solutions. The results o f such a calculation are p r e s e n t e d in Fig. 14. W h a t u s e d to be the rescattering p l a t e a u for linear p o l a r i z a t i o n has t u r n e d into a staircase for elliptical polarization. E a c h step can be attributed to one p a r t i c u l a r pair o f orbits, and for each step the real parts o f such orbits are d i s p l a y e d in the figure. T h e orbits are closely related to their analogs in the case o f linear polarization, e x h i b i t e d in Fig. 9. In particular, their cutoffs oscillate with i n c r e a s i n g travel time as
70
W. B e c k e r et al.
[IV
FIG. 15. ATI spectrum in xenon for an elliptically polarized laser field with ellipticity ~ = 0.36 and intensity 0.77• 1014W/cm2 for emission at an angle with respect to the polarization axis as indicated in the upper right. The spectrum has a staircase-like appearance. The respective steps are shaded differently. For each step, the real parts of the responsible quantum orbits are displayed. The dots with the crosses mark the position of the atom, and the length scale is given in the upper left of the figure. From Sali~res et al. (2001).
illustrated in Figs. 8 and 9. The main difference is that for elliptical polarization the orbits are two-dimensional and encircle the ion. The pair of orbits with the shortest travel times generates the part of the spectrum preceding the final (highest-energy) cutoff. However, this part is very weak in relation to the yields at lower energies. The latter are generated by orbits with longer travel times, whose contributions for linear polarization are marginal, see Fig. 9. Intuitively, this staircase structure can be understood as follows. Return o f the electron to the ion is possible if the electron has a nonzero initial velocity. This velocity is largely in the direction o f the small component o f the elliptically polarized field. The larger this velocity is, the smaller is the contribution that the associated orbit makes to the spectrum. [This can be compared with the distribution of transverse momenta in a Gaussian wave packet (Dietrich et al., 1994; Gottlieb et al., 1996).] For the shortest orbit, while the large component of the field changes sign so that the electron is driven back to the core in this direction, the small component has the same sign for the entire duration o f the orbit. Hence, a particularly large initial velocity in this direction is required in order to compensate the drift induced by the small field component. For the longer orbits, the small component changes direction, too, during the travel time and, consequently, a smaller initial transverse velocity suffices to allow the electron to return to the ion. Support for these qualitative statements can be found in the orbits depicted in Fig. 14. The parameters of the two shortest quantum orbits can be read from Fig. 10
IV]
ATI: CLASSICAL TO QUANTUM
71
and compared with the case of linear polarization. For elliptical polarization, the momentum ks has two nonzero components, kxs and kys. Both have substantial imaginary parts, in particular ky~. This is a consequence of the lack of a classical simple-man model for elliptical polarization, as discussed above. For the orbits ( s - 3, 4) (not shown), the imaginary parts are much smaller, in keeping with the fact that they make a larger contribution to the spectrum. Figure 15 presents a corresponding measurement of an ATI spectrum and displays the staircase structure predicted by the theory. The first step (the one corresponding to energies below 10 eV) is due to direct electrons and does not concern us here. The other ones correspond to the steps of Fig. 14. The real parts of representative orbits, calculated from Eqs. (34)-(36), are shown in the figure. In order to reach a maximum contrast for the steps, the spectrum was recorded at 30 ~ to the major axis of the polarization ellipse.
I.
INTERFERENCE BETWEEN DIRECT AND RESCATTERED ELECTRONS
In the lower part of the plateau, the electron can reach a given energy either directly or after rescattering so that one expects interference of these two paths. However, Fig. 9 shows that, for linear polarization and high intensity, the transition region where both paths make a contribution of comparable magnitude is very narrow. The situation is more favorable for elliptical polarization: since the plateau turns into a staircase (Fig.14), the yields of the two paths remain comparable over a larger energy region. This has permitted experimental observation of this interference effect in the energy-resolved angular distribution (EAD) (Paulus et al., 2000). Figure 16 shows a comparison of the EAD's for linear and for elliptical polarization at the same intensity. For linear polarization, the standard plateau in the direction of the laser polarization is very noticeable. The side lobes corresponding to rainbow scattering, mentioned in Sect. III, are also visible. For elliptical polarization, the plateau has split into two, one to the left of the direction of the major axis of the field and another weaker one to its right. The lower panel of Fig. 16 exhibits (on the right) EAD's of a sequence of ATI peaks where the interference is best developed and (on the left) compares them with theoretical calculations from the amplitude (29). The parameters underlying the calculation do not exactly match the experiment. This is mostly attributable to the insufficient description of the direct electrons for elliptical polarization. The theoretical results, however, show the same interference pattern. In order to make sure that this pattern is really due to interference between direct and rescattered electrons, the two contributions have been displayed separately for one of the peaks (s = 17): neither one shows a pronounced dip, only their coherent superposition does. For more details of the theory we refer to Kopold (2001).
72
W. B e c k e r et al.
[IV
FIG. 16. Upper panels: density plots of measurements of the energy-resolved angular distributions for Xe at an intensity of 7.7 • 1013 W/cm 2 and a wavelength of 800 nm for (a) linear polarization and (b) elliptical polarization with ellipticity ~ = 0.36. The direction of the major axis of the polarization ellipse is at 0 ~ Dark means high electron yield. Yields can only be compared horizontally, not vertically, since the data were normalized separately for each ATI peak. For linear polarization, the cutoff is at I OUp = 46eV. Lower panels: (a) theoretical calculation from Eq. (29) of the angular distribution for the ATI peaks s = 11,... ,21 for elliptical polarization (~ = 0.48). The other parameters are Eip = 0.436 a.u. Oust below the binding energy of xenon in order to stay away from a channel closing) and I = 5.7 • 1013 W/cm 2. For the ATI peak s = 17, the contributions of the direct electrons (dashed line) and the rescattered electrons (dotted line) are displayed separately. The slight variation in the former is unrelated to the interference pattern of the total yield (solid line), which results from the coherent sum of the two contributions. (b) Experimental angular distribution extracted from the upper panel (b) of the ATI peaks s - 1 5 , . . . , 25. From Paulus et al. (2000).
V]
ATI: C L A S S I C A L TO Q U A N T U M
73
V. ATI in the Relativistic Regime A sufficiently intense laser field accelerates an electron from rest to relativistic velocities Ivl ~ c within one cycle. Such intensities are characterized by the ponderomotive energy Up becoming comparable with or exceeding the electron's rest energy m c 2. We will briefly summarize the kinematics o f an otherwise free electron in the presence of such a field. In other words, we will generalize the simple-man model of Sect. II.A. 1 to the case o f "relativistic intensity." The changes are surprisingly few.
A. BASIC RELATWISTIC KINEMATICS
For a four-vector potential A~' = (A0,A), the electron's four-vector velocity is (Jackson, 1999) m~t ~ = pU - eA ~, (46) where flu = y(c, v) with v = v(t) the ordinary velocity d x / d t , and the usual relativistic factor ~, = [1 - ( v / c ) 2 ] - 1 / 2 (not to be confused with the Keldysh parameter). The four-velocity satisfies ~2 ~ ~ . ~ ~ ~/t~/~ -- C2 SO that the four-vector mfi is on the mass shell. Equation (46) is the analog o f the nonrelativistic Eq. (2). We will consider a plane-wave field o f arbitrary polarization, 2
A" = Z ai(k. x)e;
(47)
i=1
with the four-dimensional wave vector k u = (co~c, k) so that k 2 - 0 and k . ei = O. The field (47) differs from the field (6) by the fact that the wave fronts are now given by k - x = const, in place o f t = const., that is, we do no longer make the dipole approximation. We will assume that the laser field propagates in the z-direction so that k = Iklez. The four-vector p" = ( E / c , p ) is the canonical momentum. For a vector potential whose cycle average vanishes, its spatial components p have the physical meaning o f the drift m o m e n t u m as in the nonrelativistic case. Since the electron-field interaction depends only on u =_ k . x / c o = t - z / c ,
(48)
the canonical m o m e n t u m Pv -= ( p x , P y , 0) transverse to the propagation direction as well as p . k = co(p0-pz)/C are constants o f the motion inside the field (47). If we assume that the laser field (47) is turned on and off as a function o f u = t - z/c, then Pv a n d p . k are also conserved when the electron enters and leaves the field.
74
W. B e c k e r
[V
e t al.
As we did for the nonrelativistic simple-man model in Sect. II.A. l, we assume that the electron is initially, at some space-time instant u0, at rest. Then Pr = eA(u0) From the condition that as a function of u,
~, -
and
~2 = C 2,
E
mc 2
po-pz
for all times.
= mc
(49)
using the conditions (49), we obtain the energy
_ 1+
e2
2m 2c 2
( A ( u ) - A(uo))2
(50)
This yields the cycle-averaged kinetic energy (Ekin) = ( g ) -
mc 2 = ~m + Up.
(51)
This is exactly the same decomposition into drift energy and ponderomotive energy as in the nonrelativistic case, Eq. (3). The ponderomotive energy is still defined by Eq. (4), and the classical bounds of the spectrum discussed in Sect. II.A.1 are unchanged. However, velocity and canonical momentum are connected by the relativistic expresion myv = P r - eA, and the cycle average was performed with respect to u rather than the time t. Since it can be shown that u is proportional to the electron's proper time, this was, actually, the proper thing to do (Kibble, 1966). The fact that p . k is a conserved quantity implies that the electron's velocity in the propagation direction of the laser field is given by pz = myvz
= mc(y-
1) = Ekin/C.
(52)
The presence of this momentum reflects the fact that a laser photon has a momentum in the direction of its propagation or, alternatively, that the magnetic field via the Lorentz force causes a drift in the propagation direction or, alternatively, that the laser field exerts radiation pressure. All three statements are essentially equivalent. As a consequence, electrons born with zero velocity in a relativistic laser field are no longer emitted in the direction of its polarization, but acquire a component in the propagation direction of the laser so that, for circular polarization, they are emitted in a cone given by the angle 1
tan 0 -
IvT(t)l Iv~(t)l
2m _ . /
2
Ipr/ Vyo~-1
(53)
with respect to the propagation direction. The subscript oc characterizes quantities outside of the laser field. In the derivation, Eqs. (49)-(52) were used. The angle 0 has been observed by Moore e t al. (1995, 1999) for intensities
V]
ATI: CLASSICAL TO QUANTUM
75
of several 1018 W/cm 2 and ~ = 1.053~tm and was used to draw conclusions regarding the actual (nonzero) value of the initial velocity (McNaught et al., 1997), which can be introduced as discussed in the nonrelativistic case in Sect. II.A. 1. There are, however, still some unresolved issues in the interpretation of these experiments (Taieb et al., 2001). The cycle average of Eq. (50) can be written in the covariant form p Z = m Z c 2 - ((eA)2)_ m ,2c 2
>
m2c 2,
(54)
where p2 and (cA) 2 < 0 are invariant four-dimensional scalar products. This relation is often used to introduce the so-called "relativistic effective mass" m,. It occurs very naturally in the context of the Klein-Gordon equation [(iO~ - eA~) 2 - m2c 2] q / = 0,
(55)
which explicitly displays the effective mass. However, one has to keep in mind that this apparently increased mass is just due to the transverse wiggling motion of the electron, viz. the ponderomotive energy, and that there is nothing especially relativistic about it. All the same, envisioning the ponderomotive energy as a mass increase makes sense since, like the rest mass, it is an energy reservoir that is not easily tapped. The classical kinematics just discussed are embedded in quantum-mechanical calculations, which can be carried out along the lines of the strong-field approximation (17), taking the relativistic instead of the nonrelativistic Volkov wave function (Reiss, 1990; Faisal and Rado2ycki, 1993; Crawford and Reiss, 1997). In particular, the stationary-phase approximation is well justified, leading to a form similar to Eq. (21) (Krainov and Shokri, 1995; Popov et al., 1997; Mur et al., 1998; Krainov, 1999; Ortner and Rylyuk, 2000).
B. RESCATTERING IN THE RELATIVISTIC REGIME With increasing laser intensity, the first relativistic effect to become significantbefore the ponderomotive potential becomes comparable with the electronic rest mass - is the drift momentum (52) in the direction of propagation of the laser field, which can be traced to the Lorentz force. This has virtually no effect on the initial process of ionization where the electron's velocity is low, but since it is always positive it prevents the electron from returning to the ion. Therefore, with increasing intensity it gradually eliminates the significance of rescattering processes. This effect can be estimated by calculating the distance by which the electron misses the ion in the z-direction when it returns to the ion in the
76
[VI
W. B e c k e r et al.
x - y plane (approximately at the time/ret
~
T/2). From Eqs. (52) and (51) (where,
for simplicity, we only kept Up), we obtain
(Vz) T/2 ~
Up 2mc 2
)~
(56)
with/l the wavelength of the laser field. Obviously, this distance can exceed the width of the wave packet of the returning electron to the point where it does not overlap anymore with the ion, even when Up/mc 2 << 1. In HHG the consequences have been investigated in a number of recent theoretical works (see Sect. VI.D) and were found to cause a dramatic drop of the plateau. The same should be expected for high-order ATI, but to our knowledge, this has not been explored in detail. However, in the analysis of multiplenonsequential-ionization experiments of neon at 2 • 10 ~s W/cm 2 a conspicuous suppression of the highest charge state has been attributed to the magnetic-fieldinduced drift (Dammasch et aL, 2001).
VI. Quantum Orbits in High-order Harmonic Generation According to the rescattering model, the physics of high-order ATI and highorder harmonic generation differ only in the third step: elastic scattering versus recombination. Correspondingly, the description in terms of quantum orbits can be applied to HHG as well; in fact, quantum orbits were introduced for the first time in the analysis of HHG by Lewenstein et al. (1994). It is from the practical point of view that the two processes differ greatly: HHG by one single atom has never been observed, only HHG by an ensemble of atoms. This introduces phase matching as an additional consideration, equal in significance to the single-atom behavior (Sali~res et al., 1999; Brabec and Krausz, 2000). Below we will consider examples of a quantum-orbit analysis of HHG for several nonstandard situations. The first example is an elliptically polarized laser field. A bichromatic elliptically polarized laser field was considered by Milogevi6 et al. (2000), and in Sect. VI.C we concentrate on a special case of such a field: a two-color bicircular field. Finally, in Sect. VI.D the quantum-orbits formalism is extended into the relativistic regime. A bichromatic linearly polarized laser field was investigated by Figueira de Morisson Faria et al. (2000), and a simplified version of the quantum-orbits formalism was used to deal with problems in the presence of a laser field and an additional static electric field (Milogevi6 and Starace, 1998, 1999c) or a laser field and an additional magnetic field (Milo~evi6 and Starace, 1999a,b, 2000).
VI]
ATI: CLASSICAL TO QUANTUM
77
A. THE LEWENSTEINMODEL OF HIGH-ORDERHARMONIC GENERATION The matrix element for emission of a photon with frequency fl and polarization e in the HHG process in the context of the strong-field approximation (Lewenstein et al., 1994),
Me(f2) "
dtl
dto
(57)
d3k exp [iSn(tl, to, k)] me(tl, to, k),
OO
has the same structure as the corresponding expression (31) for ATI. The function
me(tl,to, k) : (~0o l e r - e I k - eA(h)) ( k - eA(to)ler. E(to)l ~Po)
(58)
is the product of two matrix elements: one that describes the ionization at time to due to interaction with the laser field, and another one at time tl that corresponds to recombination into the ground state followed by emission of the high-order harmonic photon having the polarization e. The difference to ATI is mostly in the first term of the action:
Sn(tl, to, k) =
dr
(EIp - ~ ) - ~mm
dr [k - eA(r)] 2 +
dr EIp,
(59)
Oo
which now refers to the emitted photon. The corresponding saddle-point approximation of Eq. (57) is like the HATI approximation (37), except that the summation is now over saddle points that are solutions of the system of equations (34), (35) and (Lewenstein et aL, 1995b, Kopold et al., 2000b) [k - eA(tl)]2 = 2m(~ - EIp).
(60)
The last equation corresponds to the condition of energy conservation at the time of recombination and replaces the condition (36) of elastic rescattering in HATI. For a linearly polarized monochromatic field, quantum orbits were employed from the very beginning for the evaluation of HHG in the Lewenstein model (Lewenstein et al., 1994, 1995b) and routinely applied in the theoretical analysis and interpretation (Sali6res et aL, 1999). Conversely, numerical solutions of the TDSE were analyzed in terms of the short (rl) and the long (r2) quantum orbit, and the dominance of these two orbits was corroborated (Gaarde et al., 1999; Kim et al., 2001). The contributions of the long and the short orbit could be spatially resolved in an experiment by Bellini et al. (1998). Spectral resolution was achieved by exploiting the dependence of phase matching on the position of the atomic jet with respect to the laser focus by Lee et al. (2001) and by Salibres et al. (2001).
78
[VI
W. Becker et al.
At the end of Sect. II.B we remarked that the quantum-orbit formalism is not restricted to periodic fields, but can equally well be applied to finite pulses. For a periodic field, interference of contributions from different cycles generates a discrete spectrum. For a finite pulse, it enhances or suppresses particular frequency intervals. This was dubbed "intra-atomic phase matching" by Christov et al. (2001) and has been calculated in terms of quantum orbits; in the context of the TDSE, see Watson et al. (1997). This mechanism underlies the engineering of a HHG spectrum by tailoring the pulse shape in a feedbackcontrolled experiment (Bartels et al., 2000, 2001). Individual HHG peaks could be enhanced by up to an order of magnitude. A description of HHG that is practically equivalent to the Lewenstein model is based on the integral equation (43) and the zero-range potential (42) (Becker et al., 1990, 1994b). The equivalence implies that the contribution of "continuum-continuum terms" is insignificant (Becker et al., 1997). The threestep nature of H H G - direct ATI followed by continuum propagation followed by laser-assisted recombination- is particularly emphasized in the approach of Kuchiev and Ostrovsky (1999, 2001), where the integration over the intermediate momentum k is replaced by a discrete summation over ATI channels. The latter is carried out by a variant of the saddle-point approach, which is reminiscent of Regge poles and leads to a complex effective channel number. B. ELLIPTICALLY POLARIZED FIELDS High-order harmonic generation by an elliptically polarized field is of great interest for applications such as the generation of sub-femtosecond pulses (Corkum et al., 1994). For theoretical calculations in the context of the SFA, see Becker et al. (1994b, 1997) and Antoine et al. (1996); for a fairly comprehensive list of references, see Milogevid (2000). Fields having polarization other than linear generate particularly appealing quantum orbits since they allow them to unfold in a plane. As an example, Fig. 17 shows a HHG spectrum for the elliptically polarized laser field (5) (Kopold et al., 2000b; Milogevid, 2000). The figure confirms that the "exact results" are well approximated by the contributions of only the six shortest orbits. This figure is the analog of Fig. 14 for HATI. The spectrum exhibits the same staircase structure, and everything said there applies here as well. C. H H G BY A TWO-COLOR BICIRCULAR FIELD
The bichromatic m - 2 m laser field s
1 (gle+e-iCot + r = gi
-2it~
+ c.c. ,
(61)
whose two components are circularly polarized and counter-rotating in the same plane (e+ = (i 4- i~,)/v/2), is known to generate high harmonics very efficiently;
VI]
ATI: CLASSICAL TO QUANTUM 20 a.u. <
24
I
~
electric field ellipse
>
(4)
..-
I
/
///
~ -2
co
.0
~- -28 o
Eo -30
_
//
.'.~.7.~.'.~.....~.."~
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-'-.
-
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,
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-
-
//
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79
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. . . .
a
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,
,
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~
60
,"".,
4
""121
,612""...
50
'
1112~__t ,
,
~
70
,
,
,~tl " '
harmonic order FiG. 17. High-order harmonic spectrum for an elliptically polarized laser field with the same parameters as in Fig. 14 and harmonic orders between 25 and 77. The open circles are calculated from the integral (57), and the curves labeled 1 through 6 represent the individual contributions to the quantum-orbit approximation of the six shortest quantum orbits, numbered as in Fig. 14. The contributions from quantum orbits 2, 4 and 6 have to be dropped above their intersections with curves 1, 3 and 6, respectively. The coherent sum of all six orbits is represented by the solid line. Typical orbits responsible for each part of the spectrum are depicted as in Fig. 14. From Kopold et aL (2000b).
see Eichmann et al. (1995) for experimental results and Long et al. ( 1 9 9 5 ) for a theoretical description. We will call this field "bicircular." This high efficiency was surprising because, for a monochromatic field, the harmonic emission rate decreases with increasing ellipticity (cf. the preceding subsection) and a circularly polarized laser field does not produce any harmonics at all. A more detailed analysis, based on the quantum-orbits formalism, gives an explanation of this effect (Milo~evi6 et al., 2000, 2001 a,b). The harmonics produced this way can be of a practical importance because of their high intensity (Milo~evi6 and Sandner, 2000) and temporal characteristics (attosecond pulse trains; Milo~evi6 and Becker, 2000). The more general case of an roo-s~o (with r and s integers) bicircular field was considered by Milo~evi6 et al. (2001 a). For the laser field (61), selection rules only permit emission of circularly polarized harmonics with frequencies s = (3n + l)6o and helicities +1. Similar selection rules govern harmonic generation by a ring-shaped molecule (Ceccherini and Bauer, 2001) or a carbon nanotube (Alon et al., 2000).
80
[VI
W. Becker et al.
~
.m t--
,
i
.
.
.
.
.
.
.
.
.
r
.
.
.
.
.
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.
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.
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.
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.
.
.
.
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.
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i
. . . .
-13 ID I,._ r
o ~
-15
E ~O r
o E
~- -17' 4 O
o -19
...........................................................
10
20
30 40 harmonic order
50
60
Flo. 18. Harmonic-emission rate as a function of the harmonic order for the bicircular laser field (61) with 09 = 1.6eV and intensities I l = 12 = 4x1014 W / c m 2. The ionization potential is Eip = 15.76 eV (argon). The inset shows the laser electric-field vector in the x - y plane for times - 89T ~< t ~< 89T, with T = 23r/o9 being the period of the field (61). The arrows indicate the time evolution of the field. The ionization time to and the recombination time t I of the three harmonics Q = 19~o, 31o9 and 43~o are marked by asterisks and solid circles, respectively. These times and harmonics correspond to the dominant saddle-point solution 2 in Fig. 19. In between the ionization time (asterisks) and the recombination time (solid circles) the x-component of the electric field changes from its negative maximum to its positive maximum, whereas its y-component remains small and does not change sign. From Milo~evi6 et al. (2000).
Figure 18 presents an example of the harmonic spectrum for the bicircular field (61). The results are obtained from Eq. (57) by numerical integration. Compared with the spectrum of a monochromatic linearly polarized field (see, for example, the nonrelativistic curve in Fig. 22), the spectrum is smooth. Furthermore, the cutoff is less pronounced and there are small oscillations after the cutoff. These features can be explained in terms of the quantum orbits. Figure 19a shows the first eleven solutions (those having the shortest travel times) of the system of the saddle-point equations (34), (35) and (60), while Fig. 19b shows the individual contributions to the harmonic emission rate of the first eight of these solutions (Milo~evi6 et al., 2000). Obviously, in the plateau region the contribution of a single orbit, corresponding to solution 2, is dominant by one order of magnitude, while in the cutoff region more solutions are relevant (in particular solution 5). This is just the opposite of the standard situation of the monochromatic linearly polarized field (Lewenstein et al., 1995b) where essentially two orbits contribute in the plateau and just one in the cutoff region. Figure 19a shows which solutions are dominant. The probability of harmonic
VI]
ATI: CLASSICAL
0.05
.
.
.
2
TO QUANTUM
.
4
.
.
5
.
7
81
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--
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52
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42
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413
21 ,,.~'~"'~
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1
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15
Re (tl-t0)/T
2
"~ -11
-13
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--
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-15
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"
11
.0 m tO
~
"~___
9
;,"
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7
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',,
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,,,s
o
o
-19
10
20
30 40 harmonic order
50
60
Fie. 19. Saddle-point analysis of the results of Fig. 18. (a) The imaginary part of the recombination time tl as a function of the real part of the travel time tl - t o , obtained from the solutions of the saddle-point equations (34), (35) and (60). Each point on the curves corresponds to a specific value of the harmonic frequency f2, which is treated as a continuous variable. For the interval of Re(t1 - to) covered in the figure, eleven solutions were found, which are labeled with the corresponding numbers at the top and bottom of the graph. Values of the harmonic order that approximately determine the cutoffs for each particular solution are marked by stars with the corresponding harmonic numbers next to them. Those values of the harmonic order for which I Im tl] is minimal are identified as well. (b) The partial contributions to the harmonic-emission rate of each of the first eight solutions of the saddle-point equations. From Milo~evi6 et al. (2000).
82
[VI
W. Becker et al.
/ I
20
i"
/ "7'.
-',
0
/
'1
"X
,
/ \
ii
"\jl
/
/
,,
i
\
\
X
\\ x
'\
/..---,~..
x XX5
i
>,,
/ -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 << q). The strength of the sidebands can be changed by temporally delaying the fundamental with respect to the harmonic by a time r. Optimal overlapping of the pulses (r = 0) leads to a maximum in the strength of the sidebands, whereas complete separation entirely eliminates them. The strength of the sidebands as a function of r can be used to determine the duration of the harmonic pulse. For theoretical modeling, the simple ansatz of Becker et al. (1986) can be used, which assumes that an electron is born in the presence of the laser field with a positive initial energy Ei, which will be identified with Eq. For Up << hoo, which is well satisfied for the weak field we will consider, the differential ionization rate in the field direction is given by (in atomic units) 02F
OEOff2
oc Ipl- m =-~']2
~f V/2(Eio)2+ moo) O(E - moo - Ei).
(68)
Here, ,5'f is the amplitude of the electric field of the fundamental, m is the order of the sideband, p is the momentum of the photoelectron (]p[ = V/2(Ei + moo)), and tim is the Bessel function of the first kind. The intensities of the side bands are not, in general, symmetric. However, for sufficiently weak fields, both fields can be treated by lowest-order perturbation theory. It follows that a sideband of ,r r m ] order m is proportional to "~h'~y , where ,5'h is the field strength of the harmonic radiation. In this case, the cross correlation for a sideband of order m can be (X3 calculated as Cm(T ) =
f g~(t)" E~lml(t - r)dt.
(69)
--OO
Figure 23 shows the result of a corresponding calculation, which is compared with results from a numerical solution of the appropriate one-dimensional Schr6dinger equation. The agreement is nearly perfect. Hence, if the NIR pulse is precisely known, the pulse duration of the harmonic (and even its shape) can be determined by deconvolution of the cross-correlation
88
[VII
W. Becker et al.
+ x
m=+l rn=-I i i•
=
s-10-s
0
i
i
w
-
+ m=+2 x m=-2
w
s
lo
w
+ x
m=+3 m=-3 i i
=
1
=
w
-10-5
o
s To---
0-5
0
5
10 - ' - 15 "
delay [number of optical cycles] Fie. 23. Cross-correlation of near-infrared and soft-X-ray pulses. A harmonic of order q creates photoelectrons at the kinetic energy qho9- Eip. Sidebands are created by simultaneous irradiation with the fundamental of frequency o9. Plotted are the heights of the sidebands for various side-band orders m versus the delay r between the fundamental and the harmonic. The solid line represents the analytical approximation (69), whereas the points were calculated by numerically solving the appropriate (one-dimensional) Schr6dinger equation. In each case, the analytical approximation was normalized to the maximum of the numerical result.
functions. Numerical and experimental investigations of this problem were made by V6niard et al. (1995) and Schins et al. (1996), respectively. A. 1. Measurement o f attosecond pulses
Clearly, an experiment as discussed above will not be able to determine harmonic-pulse durations significantly shorter than that of the fundamental in the NIR spectral region. In 1996 already, V~niard et al. pointed out that the cross correlation of harmonic and NIR radiation provides access to the relative phase of neighboring harmonics. This is an extremely important insight because the phase dependence of the harmonics as a function of their order (or frequency) determines whether they are mode-locked and whether the corresponding pulses - which would constitute attosecond pulses in the softX-ray region if bandwidth-limited- are chirped. In fact, Paul et al. (2001) used this scheme for the first observation of a train of attosecond pulses. In order to achieve phase measurement of adjacent harmonics, the conditions have to be chosen such that only sidebands of order m = +1 are generated with appreciable amplitude. This calls for intensities of the NIR beam below 1012 W / c m 2. Along with the fact that the NIR field generates only oddorder harmonics this ensures that only two adjacent harmonics contribute to each sideband. An electron with energy Eq - q h o o - EIp, with q an even integer,
VII]
ATI: CLASSICAL TO QUANTUM i
89
1.35 fs,
10
2~.50 as
5
0
0
,
!
9
~1 \
,
\
/
,,,~
,,
I
t i m e [fs]
3
',
4
\
'
\
I /
"
5
'
FIG. 24. Reconstruction of a train of attosecond pulses synthesized from the five harmonics q = 11 . . . . . 19. The attosecond pulses are separated by 1.35 fs, which is half the cycle time of the driving laser. The latter is represented by the dashed cosine function. Reprinted with permission from Paul et al. (2001), Science 292, 1689, fig. 4. 9 2001 American Association for the Advancement of Science.
can be generated by absorption of the lower harmonic plus one NIR photon (Eq -- ( q - 1)ho) + boo) or by absorption of the upper harmonic and emission of one NIR photon (Eq = (q + 1)hco- ha)). Each of these two channels receives contributions from two different quantum paths, which are related to the temporal order of the interaction with the harmonic and the NIR field. (In contrast to the quantum orbits we considered elsewhere in this chapter, the quantum paths here are defined in state space rather than position space.) The photoelectron yield at energy Eq is proportional to the square of the (coherent) sum of the amplitudes of all four quantum paths. Due to the fact that two paths represent absorption from the NIR field whereas the other two represent emission into it, the interference term between these two contributions is essentially proportional to cos(~q_l --~q+l 4- 2htor). By varying the delay r between the harmonic and the NIR radiation, the difference ~q-1 --~q+l of the phases of the two harmonics can be recorded. The result of the corresponding experiment (Paul et al., 2001) is that the phase of the harmonics depends almost linearly on their frequency. Hence, the harmonics considered in the experiment (q = 11 to 19) are modelocked and make up a train of attosecond pulses of 250 as FWHM duration, see Fig. 24. A.2. Isolated attosecond pulses
With respect to applications, isolated attosecond pulses appear more useful than a train of pulses separated by half the period of the fundamental. Isolated attosecond pulses could be generated by sufficiently short fundamental pulses, i.e. pulses of about 5 fs, which consist of less than two optical cycles (few-cycle
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regime). Then, however, the spectral width of the harmonics will be so broad that it is no longer possible to identify individual sidebands as necessary for the method of Paul et al. (2001). Nevertheless, Drescher et al. (2001) and Hentschel et al. (2001) succeeded in performing measurements of the harmonic-pulse length with a resolution of 1.8 fs and 150 as, respectively. The experimental setup, in principle, resembles that of Paul et al. with the difference that higher intensities of the NIR radiation are used for the photoionization cross correlation. In addition, only photoelectrons ejected perpendicularly to the laser polarization are detected. The motivation for choosing these conditions can be deduced from a classical analysis of trajectories of electrons that were injected into the electric field of the few-cycle NIR pulse by absorption of a harmonic photon. If the duration of the X-ray pulse is shorter than the optical period T in the NIR, then the final kinetic energy of the photoelectrons depends on the phase tot0 when the injection took place, i. e. it exhibits a modulation with a period of T/2. By delaying the fundamental with respect to the harmonic, the modulation can be recorded. This was done in the experiment of Drescher et al. (2001). Hentschel et al. (2001) realized that the width of the photoelectrons' kinetic energy distribution also exhibits such a modulation, and is measureable with much higher precision than the center of mass of the distribution. For the two approaches, it is not the envelope of the fundamental that enters the correlation function, but rather the optical period. The restriction to photoelectrons emitted perpendicularly to the laser polarization suppresses the influence of effects related to the emission and absorption of photons from the laser field, i.e. the sidebands which were crucial for the experiment of Paul et al. (2001). B. THE "ABSOLUTE PHASE" OF FEW-CYCLE LASER PULSES
The need for highest intensities and extremely broad bandwidths in several areas of the natural sciences is driving the development to shorter and shorter laser pulses. At a FWHM duration shorter than a few optical cycles the time variation of the pulse's electric field depends on the phase q~ of the carrier frequency with respect to the center of the envelope, the so-called "absolute phase." The electric field should be written as E(t) = C0(t)ex cos(rot + ~),
(70)
where the function g0(t) is maximal at t = 0. Clearly, for a long pulse the phase q~ can be practically eliminated by resetting the clock. For a short pulse, however, the shape of the field (70) strongly depends on this phase, which, therefore, will influence various effects of the laser-atom interaction. This is one reason for the significance of this new parameter of laser pulses. The precise knowledge and control of the absolute phase will pave the way to new regimes in coherent X-ray
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FIG. 25. Evidence of absolute-phase effects from few-cycle laser pulses. In this contingency map, every laser shot is recorded according to the number of photoelectrons measured in the left and the right arm of the "stereo" ATI spectrometer. The number of laser shots with electron numbers according to the coordinates of the pixel is coded in grey shades. For visual convenience the darkest shades were chosen for medium numbers of laser shots. (The most frequent result of the laser pulses was about 5 electrons in each of both arms.) The signature of the absolute phase is an anticorrelation in the number of electrons recorded with the left and the right detector. In the contingency map they form a structure perpendicular to the diagonal. Shown here is a measurement with krypton atoms for circular laser polarization, a pulse duration of 6 fs, and an intensity of 5 x 1013 W/cm 2. From Paulus et al. (2001b).
generation and attosecond generation; for an overview see Krausz (2001). In addition, such extremely well-defined laser pulses are likely to have applications for the coherent control of chemical reactions and other processes. Another reason is that phase control of femtosecond laser pulses has already had a huge impact on frequency metrology. This is because phase-stabilized femtosecond lasers can be viewed as ultra-broadband frequency combs that can be used to measure optical frequencies with atomic-clock precision; see, e.g., Jones et al. (2000). With current laser technology, only femtosecond laser oscillators can be phasestabilized (Reichert et al., 1999; Apolonski et al., 2000), which is sufficient for frequency metrology. Strong-field effects require amplified laser pulses. Nisoli et al. (1997) demonstrated that it is possible to generate powerful (>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.
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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.
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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
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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
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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.
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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.
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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
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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.
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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).
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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.
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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
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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
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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
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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
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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
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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
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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.
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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).
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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.
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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
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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
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!~~;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
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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
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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
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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.
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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
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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
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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
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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
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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.
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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,
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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.
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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,
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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
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DARK OPTICAL TRAPS FOR COLD ATOMS
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/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.
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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
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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.
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[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
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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
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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).
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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.
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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
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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
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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-
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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.
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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).
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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
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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
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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
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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.
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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
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[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 << 61,62), the dipole force, the spontaneous force, and their respective diffusion tensors can be obtained as the following simple expressions [ 13]: h VG Fd = - ~ (ql ~1 + q262) --~-a '
(6)
Fr =
(7)
hi{l-"Co' 1
G
Dii = -i-~h2k2Fc0 ,
(8)
gd =
(9)
8
hF 2 (ql ~1 -t- q262) e l '
where Co = ql/fl + qz/f2 + 3G/2 + 4Cl/l -'2, Cl = qlOZ/fl + q262/f2, the optical potential is obtained by Ua = - f Fa. dr, and qj = ))/(Yl + 7'2) defines the relative spontaneous-emission rates for two photon-emission channels with j = 1,2. The above general relations can now be applied directly to our problem of cold atoms guided in a hollow laser beam. For an HLB, the dipole gradient force in Eq. (6) pushes atoms to the central dark hollow region at blue laser detunings. We assume that HLB is linearly polarized and described by the intensity distribution
I=I(p,z)
-4Ppz315w 2 W2 exp ( ) -2p2 --~- ,
w = wo v / l + ~ ( z / z R ) 2,
zR=jrw2/~,
(10) where P is the laser power, w = w(z) and w0 are the beam waists at distance z and at z = O, respectively, and zR is the Rayleigh length. The parameters w0 and can be obtained by comparing the measured intensity distribution with Eq. (10). Note that at a given coordinate z, the radial intensity distribution has a maximum
II]
M A N I P U L A T I O N OF C O L D ATOMS IN HLBs
157
Table I The mean relative transition strengths, J), j = 1,2, o f several alkali atoms. Line
ft.
D2-1ine
3q
or a + or a -
D 1-line
Jr or a + or o-
23Na(87Rb, 39K)
85Rb
133Cs
j~
2 j 2j
2 j 2j
2 j 2 j
fl
j
J
J
3~
1
1
31
1
1
j
1
J
Table II Mean branch ratio qi, i = 1,2 from the excited state 13) decay to the ground state li) for the excitation o f the ground state lJ) by the H L B , j = 1,2.
Line
Initial State IFj)
qi
D2-1ine
~ = FI
qI
23Na(87Rb, 39K)
8SRb
133Cs
13 T8 5 l--g
20 ~ 7 ~
3
q2 Fj=F2
ql
Fj = F l
1
/~ = F2
7
ql
$ 5 ~ ~4
~ 22 2--7 ~13
3-6 29 3-6 I
q2
5 ~
14 ~-7
1
q2 Dl-line
5
l 2[
1
ql
~ 2 j
q2
10
~ 17 2--7
7
1--8 11 1-8
value of Im = 2 P / ~ e w 2 at the distance Pm = w / v / ~ . For Eq. (10), the reduced saturation parameter becomes I p2 (2p2) G = ~ = Go ~-5 exp - - - ~
4P ,
Go - ~ w 2 i S .
(11)
For numerical calculation of atomic dynamics, the values o f f for several alkali atoms are listed in table I. We find that in the case of the traveling-wave laser field, the values o f f are independent of the polarization o f the laser field. In the case of D2-1ine transitions, for example, J] =j~ = 2 for all alkali atoms as listed in table I. In addition, table II presents the relative spontaneous emission rates, qj (j = 1,2), from le) to [ g j ) [ 1 4 , 1 6 ] . B.
DRESSED-ATOM MODEL
In this subsection we analyze the atomic interaction in the dressed-atom representation by using the secular approximation, 6j >> Qj >> F, ( j = 1,2),
158
[II
H.-R. Noh et al.
where F is the spontaneously decay rate of the excited state [e) [1 4]. Furthermore, under the electric-dipole and rotating-wave approximations, the dressed-atom Hamiltonian at point r within the manifold En can be written as [ 17,1 8]
2 H,(r) = Z
j=l
{-h(coL - 6j) bjbf - [dj- EL(r)bfaL + dj . t~(r)bja~]} + hcoLa+taL,
(12) where the first term describes the atomic energies, the second term is the atomic dipole interaction Hamiltonian, and the last term represents the laser field. Here bj = Igj) (el is the lowering operator, bf = ]e) (gj] is the raising operator, dj is the electric-dipole-moment operator, eL(r) is the laser field, and a~ (aL) is the creation (annihilation) operator of a photon. In this model, the laser field is also chosen as a spatially inhomogeneous monochromatic laser wave as given in Eq. (1). In a manifold E~, each eigenstate is given by a linear superposition of three basis states, ]i(n)) = aTl ]gl,n + 1) +a72 ]g2,n + 1) + a73 ]e,n),
(i = 1,2,3),
(13)
where ~-~=, lai~ 12 = 1 according to the time-independent Schr6dinger equation, H~ [i(n))- E,i li(n)). Therefore, the matrix form of H~ becomes H,, = h
n~OL + 62 Q2/2 ff22/2 n~OL
0
Ql/2
,
(1 4)
with s = - 2 v / n + ldj. tc(r)/h. The coefficients in Eq. (13) are then a~. =
b~.
~
v/l + ~--~2=l (bg.) 2' where
with i
ai3--
1
V/1 + ~-~2= 1(bg.) 2 ,
ff2j/2 bij = n~oc + 6j - E n i / h '
=
l, 2, 3 and j
=
(15)
(16)
1,2. Consequently, the eigen-energies of Hn,
E~i = nhO)L + Ei, (i = 1,2, 3), are given by E ~ ~ nh~oL + h6~ + ~
En3 ~
nhOJL-
j=l
A~." "T I_Ij
(i = 1,2),
(17) (18)
For the intensity distribution of the laser beam described by Eq. (1 0), the energy levels of the dressed atoms are shown in Fig. 1 for the case of 6j > 0, which shows the dressed-level energies, as a function of the position p of the atom, belonging to the manifolds E, and En-1 separated by the laser energy hCOL. At
II]
MANIPULATION OF COLD ATOMS IN HLBs
2(n))
/ ~ [gl'n+l)/ ~
En n~
'
le, n)
159
l(n)) 3(n))
'
V-
03 L
En_l
_/2 ,
(n-l)
[3(n-l))
le, n - 1 )
cot
P
Pm
o
Pm
P
FIG. 1. Dressed states for a three-level A-atom interacting with a HLB for positive detunings
(aj, 62 > 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
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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
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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
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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
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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 << 1, the ring diameter is given by d = 2 0 ( n - 1)D ,,~ 800 ~tm, where n = 1.51 is the refractive index of the glass substrate and D = 85 mm is the distance between the axicon and the focal plane. The diameter of the ring in the focal plane can be changed easily by varying D, i.e. by simply moving the axicon along the optical axis. While the diameter of the incoming laser beam does not affect the resulting ring diameter, it does determine the 1/e-width of the ring because of diffraction [48]. Manek et al. produced a one-to-one image of the ring-shaped beam profile with a second achromatic doublet ( f ' = 150 mm). In order to clean the dark inner region, the dark spot (an opaque disk of 725 ~tm diameter) was also introduced. The beam profile of the hollow beam in the image plane was recorded by a CCD camera. They observed the conical propagation of the hollow beam with a divergence of ~50 mrad, which was comparable to the propagation of a usual Gaussian laser beam with the corresponding waist in the focal plane. With
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FIG. 4. Generation of HLB by a double-cone prism. the dark spot inserted, the intensity at the center was ~0.1% of the maximum intensity of the ring. Song et al. [11] used a series of axicons (each with a 3 ~ base angle) and a simple lens. A hollow beam generated thereby has different focal points for the inner and outer walls. Therefore, two additional axicons were required to control the core diameter and wall thickness. An important byproduct of this arrangement was that most of the diffraction originating from the tip of the first axicon was located outside the core. Since the LG beams are not generated by axicons, this HLB generally does not propagate indefinitely, nor is the dark core preserved when the beam is focused by a simple lens. Nevertheless, a HLB that is usable for tens of centimeters can be produced routinely, which is useful for guiding atoms. A double-cone prism was also used for the generation of HLB by Ito et al. [29]. Figure 4 explains the conversion mechanism by a double-cone prism. A Gaussian beam is divided into two parts by the first refraction at the apex of the prism, and then a nondivergent doughnut-shaped hollow beam appears from the other side after the second refraction. When a prism with a length of 4.3 mm and a full apex angle of 90 ~ is used, the inner and outer diameters of the doughnut-shaped light beam are 0.6 and 1.4 mm at e -1 intensity, respectively. Although the characteristic feature is not better than can be produced with multiple axicons, this method is very simple and convenient and is useful, for example, for generating evanescent waves at the conical hollow prism. An axicon was also employed for the transformation of an LG beam to a high-order Bessel beam by Arlt and Dholakia [49]. If an LG mode with azimuthal mode index l is used to illuminate an axicon placed at its beam waist, an approximation to a Bessel beam of order I is generated (Fig. 5). First, they obtained LG modes by the computer-generated hologram method [42]. The LG beam had a waist of w0 = 2.5 mm and the axicon was positioned at its beam waist. They generated Bessel beams with orders l = 1 to 4. The radius of the inner ring of the generated first-order Bessel beam was only rm = 21.2 ~tm, and it propagated about Zma• = 29 cm without any spread. This should be compared with an LG beam with I = 1 and the same ring size at its waist, which would have a Rayleigh range of only about 4 mm. The conversion efficiency was almost 100%, limited only by the CGHs used to produce the LG beams.
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FIG. 5. Generation of a high-order Bessel beam within the shaded region by illuminating an axicon with an LG mode (from Arlt et al., 2000, Opt. Commun. 177, 297, Fig. 1, reprinted with permission).
E. MICRO-IMAGING METHOD FOR HOLLOW FIBER MODES
A hollow-core optical fiber (HOF) has many interesting applications in sensors [50], harmonic generation [51 ], and in particular, atom optics [4,6,52]. Yin et al. [30] obtained an HLB by using a micro-collimation technique for the output beam of a micron-sized hollow optical fiber. The principle of this method is very simple: for a fiber waveguide consisting of a hollow cylindrical core, some low-order modes can be guided in the hollow core, such as the LP01, LPll, LP21 and LP31 modes [53]. Therefore, when one uses a microscope objective with a short focal length to image the output intensity distribution at the facet of a hollow fiber, a simple HLB can be obtained. The inner and outer diameters of the hollow core of the fiber were 7 ~tm and 14.6 ~tm, respectively, and the outer diameter of the cladding of the fiber was 123.4~tm. The relative refractive index difference, An = (n 2 - n Z ) / Z n 2, was 0.0018 and n2 - 1.45, where nl and n2 are the refractive index of the core and the cladding, respectively. The numerical aperture is about 0.124. Figure 6 shows the relationship between the dark spot size (DSS) and the propagation distance Z of the dark HLB. It can be observed that (i) the DSS of the dark hollow beam collimated by a M-20• objective is about 50 ~tm at Z = 100 mm and about 100 ~tm at Z = 500 mm, and (ii) the relative divergent angle in the near field of HOF is about 6.5 • 10-5, whereas the divergent angle in the far field is 4.0 x 10-4. If one uses an HOF having a slightly larger hollow core, an HLB with a smaller DSS and better propagation invariance may be obtained. F. NEAR-fiELD DIFFRACTION METHOD FOR HOLLOW FIBER MODES Instead of using an imaging lens, an HLB can be obtained by superposing two orthogonal LPll guided modes, which have a node line. Let us first describe the characteristics of near fields diffracted by an HOF and the resulting generation of an HLB [54,55]. In the weakly guiding approximation, the excited mode is well
III]
MANIPULATION OF COLD ATOMS IN HLBs eoo
(a)
7o0
E= L S O O
M - 2 0 X Objective
~4oo
0') (/) soo t'~ 2o0 100 0
~,,.
~ 0
t .... 2oo
....
t
4oo
i, 000
J
Z(mm)
~.,
1 ....
Boo
I , , 1000
(b)
700
~eoo
E=1
167
r~o~
M - 4 0 X Objective
CO IZI z~0 lO0
0
100
2o0
moo
400
500
Z(mm)
FIG. 6. The relationship between the DSS and the propagation distance Z of the HLB measured with (a) M-20• and (b) M-40• objective lens. described by the dominant transverse component, the so-called linear-polarized LPlm mode, where l (m) is the azimuthal (radial) mode number [56]. For angular frequency r wave number k, and propagation constant fi, a guided mode in a HOF with hollow diameter 2a and core thickness d = b - a is given in cylindrical coordinates by its transverse component [52,53],
I Al/(ur) sin[(10 Et(r, O) =
(r < a),
+ 0)]
(BJ/(ur) + CN/(ur))sin[(10 + 0)]
(a ~< r ~< b),
DKl(wr) sin[(/0 + 0)]
(r > b),
(27)
where u = V/fi 2 - k 2, u = v/k2n, 2 - fi2, and w = v/fi 2 - kZn22. Here Jl and Nl (It and Kl) are the (modified) Bessel functions of the first and the second kind of order l, respectively, nl (n2) is the refractive index of the core (cladding), and q) is a phase constant. All the coefficients and fl can be determined by the boundary conditions at a and b, and the solutions are then given as the LPtm modes according to the number of radial nodes. Since one knows the electric fields (Eq. 27) on the facet of the HOF (z = 0), one can calculate the diffraction pattern at (x,y,z) using the Huygens-Fresnel integral [57]:
z E(x,y,z) = - ~
~
~ E~
1 P - ik
exp(ikp) clxo dyo, p2
(28)
168
(~tm)
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H.-R. Noh et al.
(~trn)~//
J~/
O, ,-11,
,
0
10
r (~tm)
(a) LPol
10
0
r (~tm)
(b) LPll
FIG. 7. Development of the radial intensity distributions due to the diffraction of (a) LP01 mode and (b) LP11 mode near the facet of HOF (z = 0 at the facet). with (x0,Y0) the coordinate of a source point and p = v/z 2 + (x--X0) 2 + ( y - y 0 ) 2. For a given LPlm mode represented by El~ at z = 0, one can obtain
Elm(r, O,z) -- 2~r(-i) t sin(10 + q~0) x
U~~
- X2~z]Jt(2~r)~d~,
(29)
where (r, 0) are the cylindrical coordinates of (x, y). Here U)~ that is, the Fourier transform of Et~ can be obtained analytically as described in Ref. [54]. Figure 7a explains how the LP01 mode diffracts in free space near the HOF: the two peaks at z - 0, representing a cross section of ring-shaped mode, diminish away while an additional central peak grows. In Fig. 7b one can find that the peaks of LPll also diminish whereas another pair of peaks grow. In this case, however, there still does exist a dark column along the central axis. A doughnutshaped HLB can then be produced as follows. For a given propagation constant, there are four degenerate LPll modes, whose respective polarizations and angular variations are described by ~ sin(0 + q~),)3 cos(0 + q~),~ cos(0 + r and)3 sin(0 + q~) [58]. One can easily find that superposition of the first two or the last two modes can produce an azimuthally symmetric mode. The output beam then forms a doughnut-shaped HLB since it is just a superposition of the output fields of two orthogonal LPll modes. The resulting combined beam in front of the HOF may look like a single linear-polarized beam with its plane of polarization rotated by 45 ~ with respect to the horizontal plane. One should note, however, that each beam can be adjusted separately, which is important in exciting modes that differ from one another. The first two images in Fig. 8 represent the independent mode patterns of two perpendicular modes at z = 0 before they are merged, and the last image
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169
Fie. 8. Superposition of two orthogonal LPll modes to obtain HLB. Transverse intensity profiles at z = 0 are shown: (a) and (b) present the images before superposition, (c) shows the superposed image. Polarization and angular variation of the corresponding electric field can be described by, for example, (a) ~ sin(0 + r and (b) ~ cos(0 + r
Fl~. 9. Characteristic dimensions of the diffraction-limited dark hollow spot, measured in terms of the dark-spot radius Rmax and the halfwidth w, which is in good agreement with the numerical simulation.
shows their combined pattern, which is similar to that of the LP01 mode. At a distance z = 250 ~m, the peak-to-peak distance is about 17 ~tm and the dark spot size is about 8.2 ~tm. Its azimuthal isotropy was checked by measuring the beam profiles along eight different radial axes and they showed good uniformity within the maximum uncertainty of about 7%. In a later experiment, the same group [59] also obtained the diffraction-limited dark spot near the HOF facet. Figure 9 shows the development of the radial intensity and the size of the dark hollow region, which is equivalent to the calculations presented in Fig. 7b. The minimum size of the dark spot, Rmax, is about 2 txm, the halfwidth of the dark spot, w, is under 1 ~tm, and the inner peaks are diverging with a 40-mrad diffraction angle. The experimental results were in good agreement with the numerical calculations obtained by the RayleighSommerfeld theory and the weak guiding approximation. The small dark spot may be applicable as an atomic lens which focuses atoms to a small spot or as an optical dipole-force microtrap by combining several beams.
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IV. Cold Atom Manipulation in Hollow Laser Beams A. ATOMIC GUIDANCE IN HOLLOW LASER BEAMS
A.1. HLB guidance of Rb Xu et al. [12] performed optical guiding of trapped cold atoms by a hollow laser beam (HLB) produced by micro-collimation and micro-imaging technique as discussed in Sect. III.E. The atomic guiding direction was downward along the gravity (+z direction), whereas the HLB propagated along the - z direction (counterpropagating scheme) or along the +z direction (copropagating scheme). A Ti:sapphire laser was used as the guiding laser source with a maximum output power of 1.8 W. It was coupled to the core of the HOF with a coupling efficiency of about 30%. The typical HLB power used for guiding atoms was 250 mW. They used a micron-sized HOF with a hollow diameter of 4 Ftm, core thickness 2 Ixm, and length 25 cm. In both guiding schemes they obtained identical values for the radius of the maximum-intensity ring pm(z) that varies linearly with the distance z [pm(Z) -- pro(0)- aZ, where pm(0) = 1.4 mm is the value at the trap center (z - 0) and a = 1.27(4)x 10-3]. They used a standard vapor-cell magneto-optical trap (MOT) of 85Rb atoms. The number of trapped atoms was typically 2x 107, and the trap diameter was about l mm so that the loading efficiency of the atoms trapped into the HLB was 98%. By time-of-flight measurement, the temperature of atoms in the MOT was found to be about 140FtK, which was further cooled down to 16 txK by polarization-gradient cooling. After sub-Doppler cooling, the cooling and repumping lasers were blocked by mechanical shutters, and the HLB was simultaneously introduced to the atoms to guide their gravitational falling. The number and the temperature of guided atoms were detected by observing the probe-induced fluorescence with a photomultiplier. The probe laser beam was placed horizontally at 105 mm below the trap center. Figure 10 shows time-of-flight signals of guided cold atoms in both guiding
.~ 2.5] (a) 2.0 t
16GHz--j
~m~ 2.5
0OHz
0.5 0
0 ,~
"E00
ree "~ .
~
-
0;05 0;10 0;15 -time of Flight (s)
-
"
>
"
-
.
-"~.
(b)
2.0
-~.-
9. " ; - " ~
---=-
~ 0.5 0"8 0 0.~12 0.~14 0.~160.~180.20 0.22. 0J24 Time of Flight (s) .
.
-.~-,e-----.
--
FIG. 10. Typical TOF signals of atoms guided by a single HLB. (a) In the copropagating scheme, the laser detuning 62 is 1, 2, 6, 10 and 16 GHz, respectively. (b) In the counterpropagating scheme, 62 is 6, 10 and 16 GHz, respectively.
IV]
MANIPULATION OF COLD ATOMS IN HLBs 0-
"~
40-
~..~
171
9 Copropagating
Counterpropagating
30,
20, .-~ 10. (-9
O
-4-~_ o 2 ~, 6 8 io i2 i4 i6 Detuning~i2(GHz)
FIc. 11. Guiding efficiency as a function of the detuning 62 in the (a) copropagating and (b) counterpropagating schemes. The solid curves represent numerical simulation results.
schemes at various laser detunings with respect to the 5S1/2,F - 2 ~ 5P3/2 transition line. For comparison, the detected signal without the HLB for the freely falling atoms is also shown. In particular, it is observed that the number of atoms guided by the copropagating HLB is about 20-fold enhanced with respect to that without the HLB at 2 GHz detuning. In this case, the guided atoms also become accelerated along the +z direction due to the increased radiation pressure at small detunings (Fig. 10a). In the counterpropagating case, on the other hand, the guided atoms are decelerated as the detuning is decreased (Fig. 10b). Figure 11 presents experimental and numerical guiding efficiencies versus detuning in the copropagating (a) as well as in the counterpropagating (b) scheme. Note that in numerical simulation the HLB was assumed, to a good approximation, as the lowest Laguerre-Gaussian (LG~) mode given by Eq. (10) in Sect. II.A. It can be observed that at small detuning, atoms are most efficiently guided in the copropagating scheme (for example, the maximum guiding efficiency is about 50% at a detuning of 2GHz). Note that when atoms are released from the molasses without HLB, only 2.5% of the initial trapped atoms are detected. On the other hand, the counterpropagating guiding is generally less efficient as seen in Fig. 11. However, for large detunings, both schemes provide similar guiding efficiencies and the maximum efficiency of 23% is obtained at 10 GHz detuning in the counterpropagating scheme. Note that the efficiency of the copropagating guiding can be further enhanced with a higher laser power at a given atomic temperature (e.g., 60% with 500 mW at 16 9K) or with a lower atomic temperature at a given power (e.g., 75% with 1 ~tK at 250mW). Moreover, if the quality of the HLB is improved by more careful fiber treatment and optical alignment to excite the LG~ mode, the guiding efficiency is also expected to be much enhanced. Xu et al. find that the maximum guiding efficiency can be 80% (50%) in the copropagating (counterpropagating) scheme. In particular, the guiding efficiency at large detunings in both schemes can be increased twice. They also find that if the coherence of guided atoms such as Bose condensate is to be preserved, both schemes may be exploited
H.-R. Noh et al.
172
[IV
with higher power, larger detuning, and colder atoms. For example, simulation shows that when the HLB has 500 mW power and 220 GHz detuning, more than 30% of the atoms at 2 ~tK temperature can be guided over a distance of 30 cm without any spontaneous emission and with an average photon scattering rate much less than 1 s-1. A.2. HLB guidance o f Ne* and Rb condensate Schiffer et al. [ 10,60] have demonstrated guiding and focusing of atoms in the dark region of a holographically generated HLB. They performed experiments with metastable neon atoms. A laser-decelerated and compressed atomic beam, having a high brightness of 5• 1012/sr/s [61], is injected into the dark region of a blue-detuned doughnut mode. The longitudinal velocity is 28 rn/s with a rms width of 4 m/s. The atoms are injected into the doughnut mode through a 30-mm-radius hole in a dielectric mirror at 45 ~ Behind the mirror, the flux is 1.4 • 10 6 atoms per second with a transverse Gaussian velocity distribution, having the spread a. = 7.8 cm/s=2.5Vrec, where Ure c = hk/m is the photon recoil velocity. They used a computer-designed blazed phase hologram that was produced by a direct laser-writing technique [62]. Figure 12 shows the normalized atomic intensity in the focal plane as a function of the doughnut-mode well depth for a waist w0 of 50 ~tm and a power of 300mW. For a shallow potential, only a minor part of the atomic beam was captured by the guiding potential when entering the doughnut mode. By increasing potential height, however, a growing number of atoms was trapped and guided. The experimental curve in Fig. 12 provides slightly lower guiding 1/
0.00 |
[1/OHz]
0.02
0.04
i
!
0.06
0.08
,,
I-,
2000
I
1500 o N
I000
Q
500
z
0
:-AO
......
51
.........
i .....
110
50
I00
i,.
Urnax[ 103Eree] FIG. 12. The dependence of the atomic intensity enhancement on the maximum light shift for atom guiding in a blue-detuned doughnut-mode HLB (open circles) and in a red-detuned Gaussian mode (solid circles). The dashed and solid curves present the results of the numerical simulation. The plot for a wider range is shown in the inset (from Schiffer et al., 1998, Appl. Phys. B 67, 705, Fig. 2, reprinted with permission).
IV]
MANIPULATION OF COLD ATOMS IN HLBs
173
Fie. 13. Atoms loaded into a HLB waveguide from a Rb BEC with different evolution times inside the waveguide (from Bongs et al., 2001, Phys. Rev. A 63(3), 031602, Fig. 4, reprinted with permission). efficiency compared to the simulation result. For Umo~ - 8500Eree, the atomic intensity is enhanced by a factor of 1600. In this case, (60+ 10)% of all atoms injected into the doughnut mode are captured and guided. The inset of Fig. 12 shows that for high Umo~ the enhancement factor is expected to reach values well above 3000. For Uma~ = 2000Erec, the flux is 8 • 105 atoms/s, the density is 2 • 108 cm -3, and the intensity is 6 • 1011 (cm -2 s -1). In particular, they achieved a factor of 10 improvement in atomic flux by using the doughnut mode, and a factor of 80 with a TEM~)5 mode. Recently, the transfer of Bose-Einstein condensate (BEC) into a quasi-lD waveguide created by a blue-detuned HLB was also demonstrated by the same group [63]. The combined optical dipole and magnetic (DM) trap consists of a waveguide added to the 3D potential of a Ioffe-type magnetic trap. It allows for a natural connection between the magnetic trap and a pure 1D waveguide geometry created by an LG (TEMPI) laser beam. With a power of P = 1 W at 532 nm and a beam waist of r0 ~ 10 ~tm, a dipole potential at the focal plane with a maximum value of ~120~tK and a transverse oscillation frequency of ~6 kHz (corresponding to 570nK) for 87Rb atoms can be realized. They investigated the transfer process of BEC into a blue-detuned dipole waveguide and studied the subsequent evolution of the ensemble in a quasi-lD waveguide. Figure 13 shows guiding for evolution times up to 500 ms. On time scales above 40 ms, the conversion of mean-field energy into kinetic energy is nearly complete and the ensemble is expected to expand with constant velocity, keeping its parabolic density distribution. In these experiments they demonstrated that a fully coherent transfer is possible, and they observed a mean-field-dominated expansion of the ensemble for adiabatic loading conditions. A.3. HLB guidance o f Cs
Song et al. [11] demonstrated guiding of ~108 Cs atoms through an 18-cmlong, 1-mm-diameter core HLB. The generation of HLB used in their guiding is
174
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H.-R. Noh et al. 2.0
I
~
'
'
'
I
'
'
1.5
o x v
1.0
0.5
o.o . . . . . 0
I 50
~
UKE (=,)
c~| 100
150
Fl6. 14. Number of atoms in the HLB tunnel as a function of time, measured from longitudinal images (squares) with the repumper, those without the repumper (diamonds), and those in the presence of the repumper but with the kicking beam placed 10 cm below the MOT (triangles) (from Song et al., 1999, Opt. Lett. 24(24), 1805, Fig. 4, reprinted with permission).
described in Sect. III.D. The source of cold atoms was a vapor-loaded Cs MOT containing - 1 0 9 atoms. The guide beam and the major axis of the MOT were aligned and oriented in the vertical direction. The atoms are transferred to the tunnel by turning the MOT beams off. They then monitored the evolution of the cloud of atoms (size, location, and density) by the absorption shadow the cloud of atoms casts on a CCD camera. They showed that the center of mass of the cloud in the transverse direction has an acceleration o f - 1 5 m / s 2, appreciably larger than the gravitation g ( ~ 9.8 m/s2). They also found that when the guide beam is directed against gravity and detuned by less than 1 GHz, it is possible to levitate the center of mass of the cloud. In the one-dimensional case, in which the atoms are restricted to radial motion, they find that the average acceleration /asc)ensemble ~ 1/V/A to first order, in qualitative agreement with the observations (A is the laser detuning). They also measured the number of atoms in the tunnel for times well beyond 35 ms by taking images in the longitudinal direction (Fig. 14). With the r e p u m p e r , - 1 0 s atoms make it to the bottom of the chamber, whereas without the repumper, they detect no atoms at the bottom. As the triangles in Fig. 14 show, nearly all the atoms are kicked out of the beam before hitting the window. Although only 3% of the atoms go through the tunnel as shown in Fig. 14 (A ~ 1.5 GHz), the efficiency was about 10% when they reduced the detuning to increase the atomic speed.
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M A N I P U L A T I O N OF C O L D ATOMS IN HLBs
175
FiG. 15. (a) CCD image of the low-velocity rubidium beam split and guided by the HLB. (c) Spatial profile of the rubidium beam taken at the place indicated by the arrow. (b) and (d) show the corresponding image and the spatial profile without the HLB (from Yan et al., 2000, J. Opt. Soc. Am. B 17(11), 1817, Fig. 4, reprinted with permission).
A. 4. H L B s p l i t t i n g o f Rb
Yan et al. [64] guided a continuous low-velocity atomic beam and achieved incoherent splitting of the atomic beam with the HLB. A low-velocity intense source of atoms (LVIS) was generated by the method demonstrated by Lu et al. [65]. An HLB was generated by an axicon setup that converts a Gaussian beam from a Ti:sapphire laser into the HLB as explained in Sect. III.D [28]. A convergent HLB with a full convergence angle o f 5 ~ was generated from a collimated HLB with a dark center diameter o f ~1 cm. With the intercepting angle a = 0, they investigated the effect o f the HLB for the guiding. The result was that, owing to the HLB guiding and collimating or focusing, the rubidium flux was increased by ~20% and the spatial width was reduced from 0.9 mm to 0.7 mm. Figure 15a shows the CCD image of the LVIS split and guided by an HLB with a convergence angle o f ~ 5 ~ and intercepting the LVIS at a ~ 8 ~ The top of the image was about 4 m m below the center o f the MOT. Spatial separation of the LVIS is observed: one atomic beam is propagating downward and a second atomic beam is produced by the atoms guided by the HLB along the propagating direction of the HLB. This effectively realizes an incoherent atomic beam splitter. For comparison, Fig. 15b shows the CCD image of the LVIS without the HLB. Figures 15c and 15d plot the cross-sectional spatial profiles of the rubidium atomic beams taken at a distance o f 9 m m below the MOT. The maximum atomic flux guided by the HLB is 50% o f the flux intensity of the free-traveling rubidium beam without the HLB.
176
H.-R. Noh et al.
[IV
B. ATOMIC FOUNTAIN WITH HOLLOW LASER BEAMS The development of an atomic fountain based on laser-cooled atoms [66, 67] has created prospects for an improved accuracy and stability of frequency standards. In such a clock, one approach to solve the line shift due to cold collisions is to use laser light for guiding the upward-launched atoms [68]. This is because the guiding can enhance the number of atoms which come back into the microcavity without increasing the atomic densities. Optical guiding of an atomic fountain by using a cylindrical HLB was recently demonstrated by Kim et al. [69]. They generated an HLB by using the microimaging method described in Sect. III.E. It was collimated by the objective lens and propagated downwards toward the center of the Rb MOT. The power of the guiding laser was 250 mW and the beam waist was 3 mm. The HLB was nearly collimated in order to remove the dipole force of the guiding direction, which can cause broadening of the spatial distribution of guided atoms. With an intensity of 3 mW/cm 2 in each beam, the typical diameter of an atomic cloud in the MOT was about 1 mm, and the number of trapped atoms was typically 2 • 107. Cold atoms were then launched upwards in a rather simple way by rapidly varying the vertical magnetic field, resulting in the atomic Zeeman shift. After 1-ms acceleration, the detuning of the laser beams was changed from -2.5F to -70F, lowering the atomic temperature to 33.7 ~tK in the frame moving upwards. A typical launching velocity of ascending atoms was 1.4 m/s and the atoms were launched up to 10 cm. The number of guided atoms was detected by observing with a photomultiplier tube the fluorescence induced by a horizontally placed probe laser at 10.5 cm below the center of the MOT. They observed that 0.5% of the launched atoms were detected without the HLB. On the other hand, a tenfold enhancement of the HLB-guided atomic fountain was clearly obtained without appreciable heating. In Fig. 16, curve (a) is the time-of-flight (TOF) signal of atoms launched without the HLB, while curve (b) is the TOF signal with the HLB at a detuning of 19 GHz. From this TOF signal, one can deduce the guiding efficiency of atoms and the temperature. Without the HLB, the temperature was about 33.7 (-+-2.1) ~tK; with the HLB it was about 34.4 (• ~K. To characterize the enhancement due to the guiding HLB, they introduced the enhancement factor, defined as the ratio of the number of atoms guided with the HLB to that without HLB. In Fig. 17, the curve with solid squares shows the relationship between the enhancement factor and the detuning measured with respect to the 5S1/2,F = 2 ~ 5/93/2 transition line. The inset shows the enhancement of the guiding efficiency for larger detuning. The curve with open circles shows how the number of scatterings, or the heating, is changed with the detuning. They observe that for small detuning the enhancement factor is more than 35, but there is serious heating. As the detuning increases, the enhancement factor as well as the heating decrease. Note that the number
IV]
MANIPULATION OF COLD ATOMS IN HLBs
177
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of scatterings decreases more rapidly (,~(~-2) than the enhancement factor. At a detuning of 19GHz, the enhancement factor is over 10 and an atom experiences spontaneous emissions about 40 times during the launching and falling processes. According to the calculation, however, they found that the heating due to spontaneous emissions was not so serious. In order to reduce the loss of atomic coherence, an HLB with a large detuning may be used. For example, if a 15-W Ar-ion laser is used for a tenfold enhancement of guiding efficiency, the average rate of spontaneously
178
H.-R. Nob et al.
[IV
scattered photons is calculated to be l0 -3 Hz. While the number of atoms being guided in the fountain is increased, the HLB introduces inhomogeneous energy shifts of the ground-state hyperfine levels. In a trap based on a sheet of blue-detuned light supporting against gravity, a Stark shift of 270mHz is obtained for 4 s trapping time, which is larger than the line shift due to coldatom collisions [70]. One possibility for reducing the light shift in the HLB is to use a Bessel beam of much higher order for the HLB, or to use the evanescent waves of a hollow optical fiber. Since the evolution of atoms in an HLB depends on the shape of the HLB, it is suggested that the ensemble-averaged heating and the light shift will be changed with the shape of the HLB [71]. In principle, if the potential of the HLB is square, then atoms in the HLB may not feel any scattering or light shift. C. ATOMIC TRAPS WITH HOLLOW LASER BEAMS The first optical dipole trap consisted of a strongly focused Gaussian laser beam, detuned to the red side of the atomic resonance line [72]. Since then, several schemes have been realized to reduce the scattering rate that limits the trap lifetime [73], such as a far-off-resonance trap (FORT)[74] and far-detuned traps operating with Nd:YAG [75] or CO2 [76] lasers. Since the large scattering rate or the ac-Stark energy shift is a serious obstacle to precision spectroscopy, blue-detuned optical dipole traps have been designed to provide lower scattering rate and less energy shift by using the fact that atoms mostly remain in the region of low laser intensity. Before the advent of HLB, Davidson et al. [70] at Stanford demonstrated the first blue-detuned dipole trap by using two sheets of Ar + laser beams. The beams intersect at 90 ~ forming a V-shaped cross section, and provide a strong confinement perpendicular to the beam propagation axis. Confinement along the laser propagation axis is provided by the divergence of the focused light sheets. In this trap, a 1/e coherence decay time of 4.4 s was obtained. A linewidth of 0.125 Hz and a Ramsey fringe contrast of 43% were also obtained. The coherence time was 300 times longer than that achieved in a red-detuned Nd:YAG laser dipole trap with a comparable trap depth. In a later experiment, they used two pairs of light-sheets, forming an inverted pyramid [77]. After Raman cooling, 4.5• 105 atoms were loaded at a temperature of 1.0~tK to a final density of 4 • 10 ll cm -3 with 1/e lifetime 7.0 s. In this subsection, we discuss several types of atom traps that use blue-detuned hollow laser beams.
C. 1. Single-HLB trap of Rb Kuga et al. [8] have demonstrated a novel optical dipole trap using an LG laser beam. Precooled Rb atoms were trapped in the dark core of the doughnut beam (2D trap). Because there was no restoring force along the axial direction, they
IV]
MANIPULATION OF COLD ATOMS IN HLBs ,
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added two "plugging" laser beams to make a three-dimensional optical trap (3D trap). To obtain the plugging beams, they recycled the doughnut-shaped HLB that was divided into two beams and redirected to the trap with a separation of 2 mm. The HLB was 1.5 mm in diameter and the plugging beam had a diameter of 0.7 mm. An LG-mode HLB was generated from a Hermite-Gaussian (HG) beam by the cylindrical-lens mode conversion method (Sect. III.A), which was produced by a Ti:sapphire (TS) laser pumped by an all-line Ar-ion laser. A tungsten wire of 20ram diameter was inserted into the TS laser cavity and its position was adjusted to generate the HG03 laser beam. It was converted to the LG30 mode by an astigmatic mode converter composed of a pair of cylindrical lenses (focal l e n g t h f = 25 ram) separated a distance d = x/2j. Figure 18 shows the decay of the number of trapped atoms as a function of the trapping time. The lifetime of the 3D trap was determined to be 150 ms. The decay of the 2D trap was faster than that of the 3D trap and did not fit a simple exponential function. By extrapolating the decay curve to zero trapping time, they could estimate that one-third of the atoms in the MOT was initially loaded in the dipole trap. The temperature of the trapped atoms was approximately 18 p~K, almost independent of the trapping time. These results also suggested that the lifetime of the 2D trap was determined mainly by the radiation pressure. From the decay curve of the 2D trap, they estimated that the photon scattering rate was -~100 S-1 . When they used an LG~ beam for trapping, the lifetime was only a few tens of milliseconds, shorter than that with the LG 3 beam. The reason is that a lower-/ doughnut beam has a smaller dark spot. Therefore, they expected that the lifetime of the novel trap would be considerably extended by using a doughnut beam that is not only intense at a large detuning but also high in I. Such a beam can be efficiently produced from a Gaussian laser beam by a blazed phase hologram [Sect. III.B]. In a later experiment, they applied polarization-gradient cooling (PGC) to extend the lifetime of the trap [78]. They estimated the number of atoms initially
H.-R. Noh et al.
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loaded in the dipole trap to be 2x 108 (loading efficiency from the MOT to the dipole trap was 30%). The time constant of r = 1.5 s was consistent with losses due to collisions with background gas. This indicates that the trap loss due to heating could be efficiently suppressed by pulsed PGC. The temperature of the trapped atoms was 13 gK, close to the temperature of 10 ~tK achieved by PGC just before the dipole trap was turned on.
C.2. Single-HLB trap of Cs Ovchinnikov et al. [9] have presented a gravito-optical surface trap (GOST) in which they used evanescent-wave cooling to store Cs atoms just above a flat dielectric surface. Horizontal confinement was provided by the conservative optical dipole potential due to a cylindrical HLB, far blue-detuned from the atomic resonance. The evanescent-wave (EW) cooling mechanism in the GOST is based on the splitting of the 2S1/2 ground state of Cs into two hyperfine sublevels with F = 3, 4. As an inelastic reflection takes place when the atom enters the EW in the F = 3 lower ground state, it is pumped into the less repulsive F = 4 upper ground state by scattering an EW photon during the reflection process. The EW that forms the bottom of the GOST was produced on the flat horizontal surface of a fused-silica prism by total internal reflection of the 60-mW beam delivered by a laser diode. The upward directed HLB used for horizontal confinement in the GOST was produced by imaging the 300-mW output of a TS laser with a spherical lens (achromatic doublet, focal length 100mm) in combination with an axicon (Sect. III.D]. The GOST was loaded from a standard MOT placed right at the center of the HLB about 800 gm above the prism surface. They measured the lifetime of atoms in the GOST by recapturing them in the MOT after a variable time. Figure 19 shows the results they obtained for two different values of the rest-gas pressure in the apparatus (p = 4.2x 10-l~ mbar e
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FIG. 19. Number of atoms in the GOST versus storage time at a rest gas pressure of 4.2x 10 - l ~ mbar (solid circles) and 7.6x 10 - l ~ mbar (open circles). The solid lines are exponential fits (from Ovchinnikov et al., 1997, Phys. Rev. Lett. 79(12), 2225, Fig. 3, reprinted with permission).
IV]
MANIPULATION OF COLD ATOMS IN HLBs
181
and p = 7.6 • 10-l~ mbar). In about 1 s, the decay is found to be exponential with 1/e lifetimes of (6.0-+-0.1)s and (3.2• respectively. These results suggest that, besides small transfer losses of ~30% observed in the first second, losses from the GOST are essentially due to rest-gas collisions. From the TOF method, they obtained the vertical and horizontal temperatures To = (3.0+0.1) ~tK and Th = (3.1-+-0.3) ~tK after 4 s of storage and cooling in the GOST. Both have equal values within experimental uncertainty. In order to study the cooling dynamics, they also performed a series of measurements on the temperatures To and Th as a function of the storage time in the GOST. They obtained the EW cooling rate to be 1/fi = (380-+-20)ms, which is in good agreement with the predicted value of 1/fl = (400• ms [16]. Ovchinnikov et al. also demonstrated a gravito-optical dipole trap, which used an intense blue-detuned conical HLB together with gravity to confine an ensemble of cesium atoms in a dark spatial region, producing a conical atom trap (CAT) [79]. The conical trapping beam with an opening angle of 0 ~ 150 mrad is generated by using two identical axicons: in a telescope-like arrangement, a Gaussian laser beam is transformed into a collimated HLB. This tubular beam is then focused with a spherical lens to generate the conical HLB with its apex located in the focal plane. In the focal plane, the beats profile is roughly Gaussian with a diameter of about 100 ~tm. Within a few millimeters, the beam profile evolves into a ring shape in resemblance to a higher-order LG mode [8]. Along the symmetry axis, the intensity distribution is approximately Gaussian and decreases to 1/e of its peak value (maximum optical potential of ~50 mK at 3 GHz detuning) within an estimated distance of ~700 ~tm. They have demonstrated the storage of Cs atoms in the CAT with a lifetime of several seconds, limited only by collisions with the residual gas in the vacuum chamber. After a variable storage time in the CAT, atoms were retrapped into the MOT and a fluorescence image was taken with a CCD camera. The corresponding experimental results are shown in Fig. 20 for two different values of the residual gas pressure in the vacuum chamber (p - 2.5 x 10-1~ mbar and 6.9• 10-1~ mbar). An exponential decay is observed with lifetimes of 7.8s and 2.8 s, respectively. This indicates that collision with the residual gas is also the predominant loss mechanism of atoms in the CAT. The measurement also demonstrates a very high transfer efficiency from the MOT into the CAT of about 80%. This corresponds to an absolute number of ~8 • 105 atoms initially stored in the CAT. The atoms in the CAT are cigar shaped, with the diameter of about 100 ~tm and the length of about 1 mm. The peak density is on the order of 1011 cm -3 with about 10 6 atoms in the CAT. Webster et al. [80] demonstrated a dipole trap by using a blue-detuned LG-mode HLB based on the geometry proposed by Morsch and Meacher [81 ]. An LG-mode doughnut-shaped beam was produced by a mode converter (Sect. III.A) from an HG beam emitted by an argon-ion laser. The doughnut-
H.-R. Noh et al.
182
[IV
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t i m e (s) FIG. 20. Measurement of the storage time in the CAT as a function of storage time. The measured lifetime is 7.8 s at a background pressure of 2.5 • 10-1~ mbar (solid circles). At 6.9x 10-l~ mbar (open circles), a lifetime of 2.8 s is observed (from Ovchinnikov et al., 1998, Europhys. Lett. 43(5), 510, Fig. 2, reprinted with permission).
shaped beam was expanded by a telescope to a beam waist of 3.5 m m and then focused by a lens of focal length 25 mm, located one focal length from the centre of the trap. To make the temperature o f the atomic cloud less than the height o f the potential barrier, and to make the cloud small enough to fit inside the cone, the cesium atoms were evaporatively cooled in a magnetic trap [82] to give a temperature of 1 ~tK and a cloud diameter in the horizontal plane of 150 ~tm. The size of the trapped cloud was roughly 100 ~tm radially and 200 ~tm axially, elongated in the direction of the beam axis. To increase the trap depth, they noted that a better method for achieving a higher dipole potential would be to use a holographic technique for producing the HLB, since holography enables one to reproduce an arbitrary waveform from a Gaussian laser beam with an efficiency greater than 90% [60]. In particular, they demonstrated application to a rubidium BEC. A condensate was formed in a magnetic trap positioned above the apex of a focused doughnut-mode HLB and then released so that, falling downwards, it would be funnelled towards the focused region, which thus realizes a condensate propagating through a small orifice. This may permit investigation of the superfluid flow in Bose condensates produced by evaporative cooling in the magnetic trap. C.3. C r o s s e d - H L B trap o f Rb
Xu et al. [83] constructed a blue-detuned optical dipole trap by intersecting two horizontal, cylindrical HLBs at a right angle in the center o f a Rb MOT. The polarizations of the beams were chosen to be orthogonal in the crossed region in order to suppress standing-wave effects. The detuning the HLBs was 20 GHz from the 5 8 1 / 2 , F = 3 ~ 5P3/2,F t = 4 transition line of 85Rb, and the trap depths were about 10 ~tK in the x-direction and 90 ~tK in the z-direction, respectively.
IV]
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They estimated that 60% of the atoms in the MOT were initially loaded in the HLB trap. The number of atoms trapped, and their temperature, could be deduced by TOF measurements. They found that the temperature of the trapped atoms was about 7 gK and this value was close to the minimum height of potential barrier of 10 gK, where about 105 atoms stayed inside the trap for 100 ms and the lifetime of trapped atoms was about 20ms as shown in Fig. 21. Figure 21 also shows the trapping efficiency as a function of the trapping time. For comparison, the simulation results based on Eqs. 6 - 9 in Sect. II.A are also shown as the dotted curve in Fig. 21. Since the detuning was much larger than the splitting between the hyperfine-structure levels of the excited state, the three-level interaction mode was quite good for the simulation.
C.4. Single-beam hollow optical traps There has been interesting progress in the generation of light beams having a null intensity region surrounded by light walls in all three directions with simple geometries. In this subsection, we discuss theoretical and experimental results on the application of such HLBs to the blue-detuned dipole traps reported thus far. Ozeri et al. [84] trapped atoms in a blue-detuned dipole trap that uses a single, three-dimensional HLB detuned by 0 . 1 - 3 0 n m above the 5S~/2 ~ 5P3/2 transition in 85Rb. To produce a light distribution that is completely surrounded by light, a collimated beam is passed through a circular phase plate that imposes a phase difference of exactly Jr radians between the central and the outer part of the beam, and then focused by a lens. Destructive interference between these two parts yields a dark region around the focus, which is surrounded by light in all directions. To form the circular Jr-phase plates, they evaporated a thin dielectric layer (e.g., MgF2, with a refractive index of n = 1.38) through a 4-mm diameter radial mask on a glass substrate by using a commercial optical coating machine. To generate a phase difference of exactly Jr radians, the thickness of the dielectric
184
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FIG. 22. Lifetime versus detuning. The inset shows a typical lifetime measurement for 6 = 5 nm (dots). The exponential fit (solid line) gives the lifetime of 340ms (from Ozeri e t a l . , 1999, P h y s . Reu. A 59(3), R1750, Fig. 3, reprinted with permission).
coating should be d = ~,/[2(n- 1)] = 676.3 nm for ~, = 514.5 nm. The darkness factor, defined as the ratio between the light intensity at the center of the trap and the light intensity at the first surrounding ring, was measured to be 1/750. A TEM00 Gaussian laser beam was magnified into a collimated Gaussian beam with w0 = 6 mm, passed through the :r-phase plate, and was focused with a f - 250-mm lens into the vacuum chamber. Figure 22 shows the measured lifetime for various trapping beam wavelengths. For detunings 6 > 0.5 nm the 1/e trap lifetime is nearly independent of 65 at around 300ms and is inversely proportional to the background pressure in the vacuum chamber. This indicates that the trap lifetime is governed by collisions with background atoms. For 6 > 0.5 nm, the trap lifetime is approximately proportional to 6. This is consistent with a heating-induced lifetime, since the heating rate is proportional to 0 -2, whereas the trap depth is approximately proportional to 6 -1. The peak density was measured to be 6.8• l0 II atoms/cm 3. In a later experiment [85], they made a similar dipole trap with a much larger volume and a more symmetric shape than before. They achieved this by simultaneously exploiting two diffraction orders of a properly designed binary phase element (BPE)[86], which was composed of concentric phase rings with a phase difference of ;r between subsequent rings, thus creating a radial grating with uniform spacing. Figure 23 depicts the 1st a n d - l s t diffraction orders from the radial grating, each having a diffraction efficiency of 40.5%, which together form an outgoing cone of light. When focused, the cone appears in the focal plane as a narrow ring. The ring closes upon itself in both sides of the focal plane owing to the beam divergence to form a dark region completely surrounded by light. Zem~nek and Foot [87] proposed a blue-detuned dipole trap by using two
IV]
MANIPULATION OF COLD ATOMS IN HLBs
185
FIG.23. Generation of a dark, hollowoptical trap by a binary phase element (BPE). The 1st (grey) and-lst (dashed) diffracted beams are focused and generate a dark region around the focal plane (from Ozeri et al., 2000, J. Opt. Soc. Am. B 17(7), 1113, Fig. 1, reprinted with permission).
counterpropagating Gaussian waves with different beam waists. At the nodes, the field intensity of the standing wave is not completely cancelled at all radial positions across the beam. This creates an intensity dip in both the axial and radial directions that can be used as an atomic trap with blue-detuned laser light. In particular, the intensity dips in the standing wave correspond to an array of small traps. They found that the bigger the difference in beam waists is, the bigger is the trap width in proportion to the width of the smallest beam. In particular, the radial width changes more rapidly with axial position. They also found that the bigger the difference in beam waists, the wider and deeper the trap becomes. Arlt and Padgett [88] demonstrated that a CGH can be used to form an optical beam with a well-localized intensity null at its focus. The beam is a superposition of two LG modes that are phased in such a way that they interfere destructively to give a beam focus that is surrounded in all directions by regions of higher intensity. When two LG modes are appropriately adjusted to give the same on-axis intensity, the relative phases of these two modes can be controlled so that they interfere destructively at their common focus to give zero on-axis intensity. To demonstrate the generation of such an optical bottle beam, they chose to produce the required superposition of LG modes by employing a CGH. They observed good agreement between the calculated and observed beam cross sections and profiles. They concluded that there were no holes due to unwanted interference effects and the optical bottle was complete. The residual on-axis intensity was measured to be less than 1% of the surrounding radial maximum. Cacciapuoti et al. [89] proposed a single-beam blue-detuned optical trap by using an axicon in combination with two spherical lenses. A collimated laser beam passes through a converging lens and then enters the axicon. If the distance d between the first lens and the axicon is greater than the lens focal length f , the optical rays produce a ring-shaped intensity distribution in the back focal plane of the first lens. Since this radiation field is imaged by the second lens in the image plane, it is possible to observe a dark region surrounded by light. The generated optical potential is cylindrically symmetric and bounded in all directions. They experimentally obtained an optical bottle beam for a refractive index of the axicon n = 1.51, base angle a ___ 100 mrad, f = 12.5 cm, d = 14 cm,
186
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H.-R. Nob et al.
focal length of the second lens f l = 10.0 cm, and distance between the axicon and the second lens d I = 93 cm. As a result, they observed a ring radius of 43 gm in the image plane, which is in qualitative agreement with the theoretical prediction of 90 gm.
C.5. Rotating-beam hollow optical trap Friedman et al. [90] demonstrated a single-beam blue-detuned hollow optical trap, based on a rapidly rotating laser beam that traps atoms in a dark volume completely surrounded by light. The rotating beam optical trap (ROBOT) is composed of a linearly polarized, tightly focused Gaussian laser beam ( 1 / e 2 radius of w0 = 16 gm), which is made to rotate rapidly (up to 400 kHz) by using two perpendicular acousto-optic scanners. If the rotation frequency is high enough, the optical dipole potential can be approximated as a time-averaged quasistatic potential. With a laser power of 200mW, a detuning of 6 = 1.5 nm above the D2 line of 85Rb, and the radius of rotation r = 100~tm, the minimal potential height is estimated to be 22 gK, which scales approximately a s r - 2 . The ROBOT was loaded from a compressed 85Rb MOT [91] having ~3 x 108 atoms, with temperature 9 gK and peak density 1.5 x 101~ cm -3. The measured 1/e lifetimes are shown in Fig. 24 for two rotation radii. For r = 67 gm, the trap becomes stable for for > 20kHz, where the lifetime is -600ms. For r = 32 gm, on the other hand, stable trapping is achieved only forfrot > 60 kHz. To increase the density of the atomic sample adiabatically, the trap was slowly compressed from rinitial = 100~tm to rfinal = 27~tm in 150ms. Furthermore, the density was increased by optically cooling the atomic sample. The lowest final temperatures were T: = 28 ~tK and Tx = 41 ~tK. From the measured temperatures and oscillation frequencies, the peak atomic density is calculated to be 2 • cm -3. As an application, they also measured the weak transition rate in the dark ROBOT [92]. They measured the 5S1/2 -~ 5D5/2 transition rate of 750"
@
9
e' ~ 5O0
o
o
o
9
o
.z 250
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|~
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o
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i
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9 o
r=67p m r=32~m
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9
i
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,
,
,
i
40 60 80 100 120 Rotation frequency [kHz]
9
i
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160
FIG. 24. Atomic lifetime in ROBOT as a function of the frequency of the trapping beam (from Friedman et al., 2000, Phys. Rec. A 61(3), 031403, Fig. 2, reprinted with permission).
IV]
MANIPULATION OF COLD ATOMS IN HLBs
187
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100
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20 30 detuning (GHz)
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50
Ff6. 25. Number of atoms captured by RODiO as a function of trap radius (from Rudy et al., 2001, Opt. Exp. 8(2), 159, Fig. 3, reprinted with permission).
85Rb atoms using an extremely weak (25 gW) probe laser beam. The measured transition rate was as small as 0.09 s-~. Note that the striking feature was the long spin-relaxation times combined with tight confinement of the atoms in a dark, hollow optical dipole trap. Rudy et al. [93,94] demonstrated a Rotating Off-resonant Dipole Optical (RODiO) trap, a similar all-optical dynamical dark trap. A circular scan of radius 1-5 mm was created by using two perpendicular linear scans phase-shifted by 90 ~ In this way, they created an average trap potential with a spatially averaged height of ~< 240 gK. After the RODiO trap was loaded, it was left on for a variable time duration, At. Figure 25 shows the atomic confinement at different scan radii taken after 30 ms at A - 30 GHz. For small radii, the averaged potential well was deep, but the trap beam cut into the initial trapped atoms, expelling many atoms. As shown in the inset, the decay time is faster in this regime because the interaction time with the light is significant. When the radius is increased, since the interaction time as well as the time-averaged scattering force decrease, the number of trapped atoms rises sharply. They also examined the dependence of confinement time rRoDio on the scan rate by implementing a two-dimensional RODiO at 2, 5 and 11 kHz scan rates. For the 11-kHz case, they found rRODiO ~ 40ms at the optimum conditions of A = 30GHz. At 5kHz, rRODiO ~ 13 ms and retention was observed beyond 25 ms. At 2 kHz, rRODiO ~ 6 ms and finite retention was found well beyond 10 ms. In particular, they discussed the merits of RODiO over other blue-
188
H.-R. Noh et al.
[VI
detuned dipole traps, such as the flexibility with respect to the number of laser beams, the realization of an arbitrarily sized dark region, and the completely dark interior.
V. A c k n o w l e d g m e n t The authors acknowledge the support of the Creative Research Initiatives of the Korean Ministry of Science and Technology.
VI. R e f e r e n c e s 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
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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
CONTINUOUS STERN-GERLACH EFFECT ON ATOMIC IONS G(J/NTHER WERTH l, H A R T M U T Hf4"FFNER l and WOLFGANG Q U I N T 2 l Johannes Gutenberg University, Department of Physics, 55099 Mainz, Germany," 2Gesellschaft fiir Schwerionenforschung, 64291 Darmstadt, Germany I. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. A Single Ion in a P e n n i n g Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. C o n t i n u o u s S t e r n - G e r l a c h Effect
.................................
IV. D o u b l e - T r a p Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. C o r r e c t i o n s and S y s t e m a t i c Line Shifts
.............................
VI. C o n c l u s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
206 209 212 213
VII. O u t l o o k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. A c k n o w l e d g e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. R e f e r e n c e s
191 195
................................................
214 216 216
I. I n t r o d u c t i o n In 1922 Stern and Gerlach succeeded in spatially separating a beam of silver atoms into two beams, utilizing the force exerted in an inhomogeneous magnetic field on the magnetic moment of the unpaired electron in silver. This so-called Stern-Gerlach effect was the first observation of the directional quantization of the quantum-mechanical angular momentum, and represents a cornerstone of quantum physics (Stern and Gerlach, 1922). Apart from its historical role the effect has been used in numerous atomic beam experiments to determine the magnetic moments of electrons bound in atomic systems: An atomic beam enters a first inhomogeneous magnetic field where one spin direction is separated from the other. The polarized atoms then enter a region with a homogeneous magnetic field where they are subjected to a radio-frequency (rf) field which changes the spin direction. Finally, a second inhomogeneous magnetic field region analyzes the spin direction. Variation of the frequency w of the rf field and recording the spin-flip probability as a function of w yields a resonance curve. Together with a measurement of the field strength in the homogeneous magnetic field, the magnetic moment can be derived. Usually the size of the magnetic moment ~ is given in units of the Bohr magneton ktB = eh/(2m) and expressed by the g-factor: = gskts, 191
(1) Copyright 9 2002 Elsevier Science (USA) All rights reserved ISBN 0-12-003848-X/ISSN 1049-250X/02 $35.00
192
[I
G. Werth et al.
where s is the spin quantum number. Accurate g-factor measurements represent a critical test of atomic physics calculations for complex systems (Veseth, 1980, 1983; Lindroth and Ynnerman, 1993). There has been an extensive discussion that the Stern-Gerlach effect could be employed for neutral atoms only. For charged particles the magnetic force is overshadowed by the Lorentz force acting on a moving charged particle in a magnetic field. Proposals to use the acceleration of electrons moving along the field lines of an inhomogeneous B-field to separate the spin directions (Brillouin, 1928) have been discarded by Bohr (1928) and Pauli (1958) on the basis of Heisenberg's uncertainty principle. Nevertheless attempts have been made, however unsuccessful, to separate the two spin states in an electron beam by using the different signs of the force on the spin exerted by a longitudinal inhomogeneous magnetic field which results in a deceleleration or acceleration of the electrons (Bloch, 1953). Recently, new proposals have come up to perform Stern-Gerlach experiments on electron beams which w o u l d - in contrast to the analysis by Bohr and Pauli - result in a high degree of spin separation under carefully chosen initial conditions (Batelaan et al., 1997; Garraway and Stenholm, 1999). Dehmelt has pointed out that confining a charged particle by electromagnetic fields provides a way to circumvent Bohr's and Pauli's reasoning (Dehmelt, 1988): the force of an inhomogeneous magnetic field on the spin of a particle which oscillates in a parabolic potential well leads to a small but measurable difference of its oscillation frequency for different orientations of the spin. Thus a precise measurement of the oscillation frequency gives information on the spin direction. Dehmelt et al. used this effect for monitoring induced changes of the spin direction of an electron by observing the corresponding changes in the electron's oscillation frequency. Since the sensitive electronics monitors the trapped particle's spin direction continuously, Dehmelt coined the term "continuous Stern-Gerlach effect" for this technique. In a series of experiments he and his coworkers have applied this method to measure the magnetic moment of the electron and the positron, which culminated in the most precise values of a property of any elementary particle (Van Dyck et al., 1987). The experimental value gexp = 2.002319304 3766(87) (2) agrees to 10 significant digits with the result of calculations based on the theory of quantum electrodynamics (QED) for free particles (Hughes and Kinoshita, 1999), gth = 2.002 319 304432 0(687) (3) and provides one of the most stringent tests of QED. The theoretical result for the free-electron g-factor can be expressed in a perturbation series as gfree =
2 Ao +A1
+A2
~
+A3
+A4
~
+-"
,
(4)
I]
CONTINUOUS STERN-GERLACH EFFECT ON ATOMIC IONS
193
--~,1017 10TM
,,,,"."'.".'".'".""'.". . . . . . . . . . . . . . . . . . . .
W"1012101310141015
1011 / 10lo 109
10
20
30 40 50 60 70 80 90 nuclearchargeZ
Fie. 1. Calculated expectation values for the electric field strength for hydrogen-like ions of different nuclear charge Z. (Courtesy Thomas Beier.)
where a ,-~ ~-~ i is the fine-structure constant. The coefficients An in Eq. (4) have been calculated by evaluating the Feynman diagrams of different orders using a plane-wave basis set; they are of the order of unity. In contrast to a free electron, an electron bound to an atomic nucleus experiences an extremely strong electric field. The expectation value of the field strength ranges from 109V/cm in the helium ion (Z = 2) to 1015 V/cm in hydrogen-like uranium (Z = 92) (Fig. 1), and gives rise to a variety of new effects. The largest change of the bound electron's g-factor was analytically derived by Breit (1928) from the Dirac equation: gBreit =
(5)
The conditions of extreme electric fields also necessitate changes to be made in the methods of calculations for the QED contributions to the electron's magnetic moment. In a perturbative treatment a series expansion in (Za) is made in addition to that in a. The expansion parameter Z a, however, i s - at least for large Z - no longer small compared to 1. In addition, the expansion coefficients can be large. For small values of the nuclear charge Z the perturbation expansion may give reliable results, and calculations were performed which include terms up to order (Za) 2 (Grotch, 1970a; Close and Osborn, 1971; Karshenboim et al., 2001). For more accurate theoretical predictions, non-perturbative methods have been developed where the solutions of the Dirac equation for the hydrogen-like ion rather than those for the free case are used as a basis set (Beier et al., 2000). The most recent summary of the status of QED for bound systems has
194
[I
G. Werth et al. 20
10 0
40
60
80
100 100 10 -1
10 -1 I. O o '
10-2
jag
10 .3
9;'~
104
:
~o o .~ 10 -s 0
o tJ
/.
.
9
9
~"
_
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9
)r
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)r
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+
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+
+
+
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+
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E
AgBsQED 1. order in ot 9
10 .2
free
Y
lO-S 10 .9
+~
+
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10 ~ 10 .9
+ +
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,
0
I
20
,
I
,
40
I
60
,
I
80
100
nuclear charge Z FIG. 2. Contributions to the g-factor of a bound electron in hydrogen-like ions for different nuclear charges Z. (Courtesy Thomas Beier.)
Table I Theoretical contributions to the g-factor in 12C5+ Contribution Dirac Theory
Size 1.998 721 3544
Reference Breit (1928)
QED, free (all orders)
+0.002 319 3044
Hughes and Kinoshita (1999)
QED, bound, order (a/K)
+0.000 000 844 6( 1)
Yerokhin et al. (2002)
QED, bound, order ( a / ~ ) 2, (Za) 2 term
-0.000 000 0012(3)
Yerokhin et al. (2002)
Recoil (in Z a expansion)
+0.000000087 6(1)
Yelkovsky (2001 )
Finite size correction
+0.0000000004
Beier et al. (2000)
been published by Beier (2000). A graphical representation of the bound-state contributions to the electron g-factor is shown in Fig. 2. In this contribution we describe an experiment which for the first time applies the "continuous Stern-Gerlach effect" to an atomic ion (Hermanspahn et aL, 2000). We measured the magnetic moment of the electron bound to a nucleus with zero nuclear magnetic moment, hydrogen-like carbon (12C5+). For the bound-state contributions of order a(Za)" the existing calculations (Grotch, 1970a) deviate already for Z = 6 by as much as 10% from the non-perturbative calculations to all orders in (Za). The numerical results of the calculations for C 5+ are summarized in Table I. The leading term comes from the solution
II]
CONTINUOUS STERN-GERLACH EFFECT ON ATOMIC IONS
195
of the Dirac equation and deviates from the value g = 2 for the free electron (Breit, 1928). The next-largest part is the well-known QED contribution for the free electron (Hughes and Kinoshita, 1999). The bound-state contributions of order a (calculated to all orders in Z a) are given with error bars which represent the numerical uncertainty of the calculations. The quoted uncertainty of the aZ(za) 2 term is an estimate of the contribution from non-calculated higherorder terms. Finally, nuclear recoil contributions have been calculated to lowest order in Z a by Grotch (1970b), Faustov (1970) and Close and Osborn (1971). Recently, Shabaev (2001) presented formulas for a non-perturbative calculation (in Za), and Yelkovsky (2001) presented further numerical results. The nuclearshape correction was considered numerically by Beier et al. (2000), and recently (for low Z) also analytically by Glazov and Shabaev (2001). The sum of the different contributions leads to a theoretical value for the g-factor in hydrogenlike carbon of gtheor(12C5+) = 2.001 041 590 1 (3). (6)
II. A Single Ion in a Penning Trap The experiment is carried out on a single C 5+ ion confined in two Penning traps. In a Penning trap a charged particle is stored in a combination of a homogeneous magnetic field B0 and an electrostatic quadrupole potential. The magnetic field confines the particle in the plane perpendicular to the magnetic field lines, and the electrostatic potential confines it in the direction parallel to the magnetic field lines. In our experiment we use two nearly identical traps placed 2.7 cm apart in the magnetic field direction. They consist of a stack of 13 cylindrical electrodes of 2r0 = 7 mm inner diameter. The difference between the traps is that in one trap the center electrode is made of ferromagnetic nickel while all others are machined from OFHC copper. Figure 3 shows a sketch of this setup. The nickel ring distorts the homogeneity of the superimposed magnetic field in the corresponding trap while the field remains homogeneous in the other trap (see Fig. 3). As will become evident below, the inhomogeneity of the field is the key element to analyze the direction of the electron spin through the continuous Stern-Gerlach effect. Therefore we call the corresponding potential minimum "analysis trap" while we call the one in the homogeneous magnetic field "precision trap." Each trap uses five of these electrodes to create a potential well, which serves for axial confinement. We apply a negative voltage U0 to the center electrode while we hold the two endcap electrodes at a distance z0 from the center at
196
[II
G. Werth et al.
FIc. 3. Sketch of the electrode structure and potential distribution of the double trap.
ground potential. The potential 9 inside this configuration can be described in cylindrical coordinates r , z , 0 by an expansion in Legendre polynomials Pi:
Uo ~
(~)( r , I.~q) : -~- Z Ci i=0
(~); -~
Pi(COS L~),
(7)
where d 2 - (z 2 + r 2 / 2 ) / 2 is a characteristic dimension of the trap (Gabrielse et al., 1989). Two correction electrodes are placed between the center ring and the endcaps. The coefficient (74 which is the dominant contribution to the trap anharmonicity can be made small by proper tuning of the voltages applied to the correction electrodes. Essentially then the potential depends on the square of the coordinates, and is a harmonic quadrupole potential ,I, -
Uo z 2 -
2
r2/2 d2
(8)
We optimize the trap by changing the voltages on the correction electrodes until the ion oscillation frequency is independent of the ion's oscillation amplitude as characteristic for a harmonic oscillator. With this method we can reduce the
II]
CONTINUOUS STERN-GERLACH EFFECT ON ATOMIC IONS
197
axial motion magnetron m
TB
cyclotron motion FIG. 4. Ion oscillation in a Penning trap. dominant high-order term C4, the octupole contribution, to less than 10-5. The ion's frequency in the harmonic approximation is then given by
O)z =
qUo Md 2 .
(9)
Radial confinement is achieved by the homogeneous magnetic field directed along the trap axis. This results in three independent oscillations (axial, cyclotron, and magnetron oscillation) for the ion motion, as depicted in Fig. 4. The fast radial oscillation frequency of the ion in the Penning trap is a perturbed cyclotron frequency ~o~. It differs from the cyclotron frequency o)~ = q B M
(10)
of a free particle with charge q and mass M because of the presence of the electric trapping field, and is given by , toe v/~O2 coc = - ~ + 4
co2 2
It can be expressed also as ~o~ = ayc - OOm,
(11)
(12)
where (-/)mis the magnetron frequency, a slow drift of the cyclotron orbit around the trap center, given by
Om-2
V7--5-"
(13)
198
G. Werth et al.
[II
For calibration of the magnetic field we use the cyclotron frequency of the trapped ion. It can be derived either from Eq. (12) or more reliably from the relation
since this equation is independent of trap misalignments to first order (Brown and Gabrielse, 1986). In this case the measurement of the magnetron frequency (_gm is required in addition to a measurement of to~ and COz. The traps are enclosed in a vacuum chamber placed at the bottom of a helium cryostat at a temperature of 4 K and located at the center of a superconducting NMR solenoid. The helium cryostat provides efficient cryopumping. As an upper limit we estimate the vacuum in the container to be below 10 -16 mbar. The estimation was derived from the measurements on a cloud of highly charged ions, whose storage time would be limited by charge exchange in collisions with neutral background particles. We observed no ion loss in a cloud of 30 hydrogenlike carbon ions stored for 4 weeks. Together with the known cross section for charge exchange with helium as the most likely background gas at 4 K we obtain an upper limit of 10 -16 mbar for the background gas pressure. The magnetic field of the superconducting magnet is chosen to be 3.8 T. At this field strength the precession frequency of the electron spin is 104 GHz. Microwave sources of sufficient power and spectral purity are commercially available at this frequency. We load the trap by bombarding a carbon-covered surface with electrons. This process releases ions and neutrals of the element under investigation as well as of other elements present on the surface. Higher charge states are obtained by consecutive ionisation by the electron beam. We detect the ions by picking up the current induced by the ion motions in the trap electrodes. For this purpose superconducting resonant circuits and amplifiers are attached to the electrodes. Upon sweeping the voltage of the trap, and thus the ions' axial frequencies, the ions get in resonance with the circuit and their signal is detected. Figure 5 shows such a spectrum, where we identified different elements and charge states. We eliminate unwanted ion species by exciting their axial oscillation amplitude with an rf field until the ions are driven out of the trap. Ions of the same species have different perturbed cyclotron frequencies in the slightly inhomogeneous magnetic field of the precision trap, because they have different orbits. Therefore, for small ion numbers, single ions can be distinguished by their different cyclotron frequencies. Figure 6 shows a Fourier transform of the induced current from 6 stored ions. We excite the ions' cyclotron motion individually and thus eliminate them from the trap until a single ion is left. Typical cyclotron energies for signals as shown in Fig. 6 are of the order of several eV. In order to reduce the ion's kinetic energy we use the method of "resistive cooling" which was first applied by Dehmelt and collaborators (Wineland and Dehmelt, 1975; Dehmelt 1986). The ion's oscillation is brought into resonance
II]
CONTINUOUS S T E R N - G E R L A C H EFFECT ON ATOMIC IONS
3,4
mass-to-charge ratio m/q 3,0 2,8 2,6 2,4 2,2
3,2
|
!
,
|
l
|
199
2,0
,
,
0,4 a) with impurity ions 1606+
= 0,3 "~ 0,2
0,0
l
:
I
,
I
,
I
,
I
,
I
,
I
,
I
,
b) without impurity ions =" 0,3
"~ 0,2 r
,w
~ o,1
0,0_ 14 trapping potential [Volt] FIG. 5. Mass spectrum of trapped ions after electron bombardment of a carbon surface showing different charge states of carbon ions as well as impurity ions (a) before and (b) after removal of unwanted species. with the circuits attached to the electrodes. The induced current through the impedance of the circuit leads to heating of the resonance circuit, and the ion's kinetic energy is dissipated to the surrounding liquid helium bath (Fig. 7). This leads to an exponentially decreasing energy with a time constant r given by q2 r -1 -
Md 2
R,
(15)
where R is the resonance impedance of the circuit. For the axial motion we use superconducting high-quality circuits (Q = 1000 at 1 MHz in the precision trap and Q = 2500 at 365 kHz in the analysis trap). With the resonance impedances of R = 23 M ~ (analysis trap) and 10 Mr2 (precision trap) the cooling time constants are 8 0 m s and 235 ms, respectively. For cooling the cyclotron motion we employ a normal-conducting circuit at 24 MHz (Q = 400) with a resonance impedance of 80 kf~. Here we reach cooling time constants of a few minutes. Figure 8 shows the exponential decrease of the induced currents from the ion oscillations as the result of axial cooling.
200
[II
G. Werth et al. .~
30
"f~
25
~
6 C 5+ions I
o 76000
,
i
J
76500
,
I
,
77000
I
77500
~
I
78000
cyclotron frequency [Hz] - 24 MHz FIG. 6. Fourier transform of the voltage induced in one of the trap electrodes from the cyclotron motion of 6 stored 12C5+ ions. The inhomogeneous magnetic field of the trap causes ions at different positions to have slightly different cyclotron frequencies. |
Ion trap
I
l
Superc0nductang
Cryogenic GaAsamphfier
FIG. 7. Principle of resistive cooling.
We do not cool the magnetron motion in a similar way because it is metastable: in the radial plane the ion experiences an electrostatic force towards the negatively biased center electrode. Ion loss is prevented by the presence of the magnetic field. Thus the potential energy is an inverted parabola. Therefore reduction of the ion's magnetron energy results in an increase in the magnetron radius. It is, however, essential to reduce the magnetron radius because of the magnetic field inhomogeneities. This is achieved in a well-defined way by coupling the magnetron motion to the axial motion by a radio-frequency field at the sum frequency of both oscillations (Brown and Gabrielse, 1986; Cornell et al., 1990). In the quantum-mechanical picture for the ion motion, the absorption of a photon from this field increases the quantum number of the axial oscillation by 1 while that of the magnetron oscillation is decreased by 1. An analysis of the absorption probabilities in the framework of a harmonic oscillator
II]
CONTINUOUS STERN-GERLACH EFFECT ON ATOMIC IONS
~
201
t--
ms
t',-
T =4 K
~~,..,__....~
_
cooling time [s] FIG. 8. Exponential energy loss of the axial motion of trapped C 5+ ions by resistive cooling.
yields that the quantum numbers tend to equalize. This leads to the expectation value (Em) for the magnetron energy,
I<Em>I=
1 boom ((Am) + 1) __ ho)m ((kz) + .~) _ 1 (-Om o)m h ooz (
(16)
r
The axial oscillation is continuously kept in equilibrium with the cooling circuit and we thus reduce the magnetron orbit to about 10 ~tm. The mean kinetic energy of a single ion is often expressed in terms of temperature. This is justified by the statistical equilibrium of the ion and the resonant circuit. The statistical motion of the electrons in the resonance circuit causes Johnson noise in the trap-electrode voltages, which in turn leads to varying energies of the ion as a function of time (Fig. 9). Extracting a histogram of the cyclotron energies results in a Boltzmann distribution (Fig. 10) with a temperature of 4.9 K close to the temperature of the environment. Calculating the temporal autocorrelation function of the energy gives, as expected, an exponential (Fig. 11) with a time constant well in agreement with the measured cooling time constant. In order to calibrate the magnetic field at the ion's position with high precision from Eq. (14) the three oscillation frequencies have to be measured. Because of their different orders of magnitude (~Oc~/2;r = 24 MHz, Ogz/2:r = 1 MHz, O)m/2~ 18 kHz) the required precision is different, o~ is determined from the Fourier transform of the current induced in a split electrode. Figure 12 shows that the relative linewidth of the resonance, well described by a Lorentzian, is of the order of 10-9 and the center frequency can be determined with an accuracy of 10-1~ In order to obtain sufficient signal strength the energy of the cyclotron oscillation has to be raised to about 1 eV. Due to the inhomogeneity of the =
202
G. Werth et al.
[II
FIG. 9. Noise power of the induced voltage in a trap electrode from the cyclotron oscillation of a single trapped ion while its frequency is continuously kept in resonance with an attached tank circuit.
FIG. 10. Histogram of the probabilities for cyclotron energies. The curve can be well fitted to a Boltzmann distribution, giving a temperature of 4.9(1) K.
II]
CONTINUOUS STERN-GERLACH EFFECT ON ATOMIC IONS %,
%,
%,
203
C(t) = C o exp(-th:) %,
%, %,
".% %,
-c = 5.40 + 0.07 min %
"~,
-% % ,~, %
4
%.
".,, %. e.
I
'
0
I
'
2
[
4
'
'
15 Time [min]
I
'
8
1~0
FIG. 11. The time-correlation function of the noise in fig. 10 shows an exponential decrease. The time constant o f 5 . 4 0 ( 7 ) m i n corresponds to the time constant for resistive cooling of the cyclotron motion.
~ .,...
30
~
25
.,,..
~
2O
lO 5
0 --
-0.6
'
i
7 -
-0.4
- - 1 -
-0.2
, - - -
~
. . . . . .
0.0
-w---
- T - - -
"T~
0.2
......
1
0.4
i
1
0.6
reduced cyclotron frequency [Hz] - 24 075 552.802 6 Hz FIG. 12. High-resolution Fourier transform of the induced noise at the perturbed cyclotron frequency. A Lorentzian fit gives a fractional width o f 1.4>< 10 -9.
magnetic field this changes the mean field strength along the cyclotron orbit. This has to be considered in the final evaluation of the measurements. The axial frequency OJz is determined while the ion is in thermal equilibrium with the resonance circuit. At a given temperature the thermal noise voltage in the impedance Z(e)) of the axial circuit, given by Unoise =
v/4kTRe[Z(oo)] 6v,
(17)
excites the ion motion within the frequency range 6v of the ion's axial resonance. This motion in turn induces a voltage in the endcap electrodes, however at a phase difference of 180 ~ as can been shown by modeling the system as a driven harmonic oscillator. Consequently the sum of the thermal noise voltage and
204
G. Werth et al.
[II
FIG. 13. Axial resonance of a single trapped C5+ ion. The noise voltage across a tank circuit shows a minimum at the ion's oscillation frequency.
FIG. 14. High-resolution Fourier transform of the axial noise near the center of the resonance frequency of the axial detection circuit. the induced voltage leads to a reduced total power around the axial frequency of the ion. This appears as a m i n i m u m in the Fourier transform of the axial noise as shown in Fig. 13. A spectrum with a resolution of 1 0 m H z (Fig. 14) shows that the center frequency can be determined to about 24 mHz. A different approach to explain the appearance of a m i n i m u m in the axial noise spectrum was taken by Wineland and Dehmelt (1975) considering the equivalent electric circuit of an oscillating ion in the trap. The magnetron frequency tOm is measured by sideband coupling to the axial motion. If the ion is excited at the difference between the axial and magnetron frequencies, the ion's axial
II]
CONTINUOUS STERN-GERLACH EFFECT ON ATOMIC IONS
205
energy increases, leading to an increased current at the ion's axial frequency. We detect this as a peak in the detection circuit signal. The uncertainty in this frequency determination is below 100 mHz. Imperfections in the trap geometry may change the motional frequencies. These changes have been calculated by different authors (Brown and Gabrielse, 1982; Kretzschmar, 1990; Gerz et al., 1990; Bollen et al., 1990). Considering only an octupole contribution to the trap potential, characterized by a coefficient Ca in the potential expansion (5), these shifts amount to
tOz
~
Ez-3
(18)
Ec +6Em,
A6OctC4~ 1 (~cc)2[-3Ez + -~3 (~cc)2Ec - 6Era] tO,c
(19)
~Om
(20)
qUo
E~ +6Em .
Ez, Ee and Em are the energies in the axial, cyclotron and magnetron degrees of freedom. The tuning of the trap potential results in a coefficient C4 as small as 10-5 . For the energies of the motions in thermal equilibrium the corresponding frequency shifts are below 10-l~ and need not be considered here. The coefficient C6 of the dodekapole contribution to the trapping potential has been calculated to an accuracy of 10-3 for our trap geometry. The frequency shifts arising from this perturbation scale with (E/qUo) 2 and are negligible here. The residual inhomogeneity of the magnetic field in the precision trap arising from the nickel ring electrode in the analysis trap 2.7 cm away changes the value of the oscillation frequencies of an ion with finite kinetic energy as compared to the ion at rest. A series expansion of the B-field in axial direction, Bz = Bo + Blz
+
B2z2 + " "
gives frequency shifts Am~_
1 B2 1
mtO2z Bo hoOc ACOz ~Oz Ao)m
mm2z Bo hoe
(Dm
mm 2 Bo hmc
1 B2 1
B2
1 1
-
[Ee
~
(21)
j
Ec+Ez+2E~
-Em],
[2Ec - Ez - 2Era].
(22) (23) (24)
The size of the inhomogeneity term B2 is measured by application of a bias voltage between the endcap electrodes. This shifts the ion's position in the axial
206
[III
G. Werth et al.
direction by a calculable amount, and the cyclotron frequency is measured at each position. We obtain B2 = 8.2(9)~tT/mm 2. The shift in the perturbed cyclotron frequency which is of most interest here is dominated by the axial energy Ez, and amounts to A~o~/o9~ = 7x 10 -9 for an axial temperature of 100 K.
III. Continuous Stern-Gerlach Effect The g-factor of the bound electron as defined by Eq. (1) can be determined by a measurement of the energy difference between the two spin directions in a magnetic field B: A E = hVL = g ~ s B , (25) where VL is the Larmor precession frequency. We induce spin flips by applying magnetic dipole radiation which is blown into the trap structure by a microwave horn. For the detection of an induced spin flip we follow a route developed in the determination of the g-factor of the free electron (Dehmelt, 1986; Van Dyck et al., 1986): the quadrupole potential of the Penning trap depends on the square of the coordinates (6) leading to a linear force acting upon the charge of the stored ion. Considering the force upon the magnetic moment of the bound electron by the inhomogeneous field in the analysis trap we get F - -27(g 9B).
(26)
The nickel ring in the analysis trap creates a bottle-like magnetic field distortion which can be described in first approximation by B=B0+2B2
2
b-z
.
(27)
The odd terms vanish in the expansion because of mirror symmetry of the field. The corresponding force on the magnetic moment in axial direction is (28)
F= = - 2 1 z z B 2 z ,
which is linear in the axial coordinate. It adds to the electric force from the quadrupole trapping field acting on the particle's charge. Since both forces are linear in the axial coordinate the ion motion is still described by a harmonic oscillator (Fig. 15). The axial frequency, however, depends on the direction of the magnetic moment/t with respect to the magnetic field: ~ 6COz = % 0 +
% = %0 + ~
ltzB2
M (OzO
9
(29)
The value of B2 in our set-up was calculated using the known geometry and magnetic susceptibility of the nickel ring electrode. We also determined it
III]
C O N T I N U O U S S T E R N - G E R L A C H E F F E C T ON ATOMIC IONS
207
z-a~s FIo. 15. Axial parabola potential for an ion in a quadrupole trap including the magnetic potential for the spin-magnetic moment in a bottle-like magnetic field. The strength of the potential depends on the spin direction. Upper curve: spin down; lower curve: spin up.
spin up ,~ ~
J
,
---- ~---0.7Hz
,
r
,
,
i i _.
-10
,
-5 ~
,
0 I
'
5 r
,
1~0
axial frequency- 364 423.07 [Hz] FIG. 16. Axial frequencies of a single C5+ ion for different spin directions. The averaging time for each resonance line was 1 min. experimentally by applying a bias voltage between the endcap electrodes o f the analysis trap and measuring the cyclotron frequency of the ion at different axial positions. The calculated and experimental values for B2 in our experiment agree within their uncertainties of 10% and yield B2 - 1T/cm 2. For hydrogen-like carbon the frequency difference &Oz/2:r between the two spin states amounts to 0.7 Hz at a total frequency of tOzO/2:r- 365 kHz. As evident from Fig. 16, the axial frequency can be determined to better than 100mHz. Fig. 17 demonstrates that after 1 min. averaging the expected frequency difference between the two spin states becomes obvious. Driving the spin-flip transition, we can distinguish the two possible axial frequencies, 0.7 Hz apart as calculated from the trap parameters. Varying the frequency o f the microwave field and counting the number of induced spin flips per unit time yields a resonance curve as shown in Fig. 18. The shape of this resonance is asymmetric due to the inhomogeneity of the magnetic field. The general shape of the Larmor resonance in an inhomogeneous magnetic field has been derived
208
[III
G. Werth et al.
)"
1,0 0,5 0,0
quantum jumps
~,.1 I 111 I l i
u~
down
oo
~ -1,o~ '
go
'
time [min]
Ibo
FI6. 17. Center of the axial frequency for a single C 5+ ion when irradiated continuously with microwaves at the Larmor precession frequency showing two distinct values which correspond to the two spin directions of the bound electron.
3O
E E
20
2~
~
L_
lO
E m m
m m
I
-5
m m m m
I
0
I
5
I
10
I
15
Larmor precession frequency- 103 958 [MHz] FIG. 18. Number of observed spin flips per unit time vs. the frequency of the inducing field. The solid line is a fit according to Eq. (29).
by Brown (1985) as a complex function of the trap parameters, the ion's energy and the field inhomogeneity. However, assuming that the ion's amplitude z(t) is
IV]
CONTINUOUS STERN-GERLACH EFFECT ON ATOMIC IONS
209
constant during the time 1/AwL, the line profile is given by a 6-function averaged over the Boltzmann distribution of the energy:
Z ( ~~ ) =
io ( dE 6
~oL - O)Lo
-- O(O)L { A--O09LO) ) L exp -
( '"))".-"" 1 + m---~z
(30)
(-OL--O')LO}Ao)L"
Here O(~oL - coL0) is the step function, which is 0 for o)L < coL0 and 1 for col > 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
<3
1,2 0,8 0,4 o,o
4He +
;
t
12C5+
,
t, 8
1607+
,t
9
,'2
'
9
,'6
9
,
9
9
2'0
9
n
uu
'
2'4
'
9
2'8
(q * m)'/2 FIG. 23. Difference in axial frequency for two spin directions in a bottle-like magnetic field for various hydrogen-like ions. The parameters B0 = 3.8T, B2 = I T/cm2, and tOz/2ar = 365kHz are those of our experiment. The frequency difference scales linear with the magnetic field inhomogeneity B2. frequency difference for the two spin states significantly. This would also have the advantage that it reduces the amount of ferromagnetic material placed in the analysis trap, and so helps to improve the homogeneity in the precision trap. To further improve the homogeneity in the precision trap the distance between the two traps can be increased. Finally, shim coils may be used to make the field in the precision trap more homogeneous. We believe that we can maintain the presently achieved precision with other ions as well, and hope to even increase it when we apply some of the measures for improvement. This would result in a more significant test of higher-order bound-state QED contributions since they increase quadratically with the nuclear charge (see Fig. 2). The method can also be applied to more complicated systems: a measurement of the electronic g-factor in lithium-like ions would test not only the QED corrections in these systems but also correlation effects with the remaining electrons which change the g-factor significantly. When applied to hydrogen-like ions with non-zero nuclear spin the transition frequencies between spin states depend on the nuclear magnetic moment. Measuring the different transition frequencies yields the magnetic moment of the nucleus. This would be of special interest, because all nuclear magnetic moments so far have been determined using neutral atoms or singly ionized ions. The effective magnetic field seen by the nucleus in these systems differs from the applied magnetic field by shielding effects of the electron cloud. In a measurement on hydrogen-like ions this shielding is strongly
216
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[IX
reduced, and comparison with data obtained on neutral systems would, for the first time, test atomic-physics calculations on electron shielding.
VIII. Acknowledgements The measurements described above are performed in close collaboration to GSI/Darmstadt. We gratefully acknowledge financial support from its Atomic Physics group (Prof. H.-J. Kluge). Several doctoral and diploma students were and are actively involved in the experiments: Stefan Stahl, Nikolaus Hermanspahn, Jose Verdfi, Tristan Valenzuela, Slobodan Djekic, Michael Diederich, Markus Immel, and Manfred T6nges. We appreciated stimulating discussions with our colleagues: Thomas Beier, Andrzej Czarnecki, Ingvar Lindgren, Savely Karshenboim, Vasant Natarajan, Hans Persson, Sten Salomonson, Vladimir Shabaev, Gerhard Soft, Alexander Yelkovsky, and Vladimir Yerokhin. The experiments are part of the TMR network ERB FMRX CT 97-0144 "EUROTRAPS" of the European Community.
IX. R e f e r e n c e s Batelaan, H., Gay, T.J., and Schwendiman, J.J. (1997). Phys. Rec. Lett. 79, 4517. Beier, Th. (2000). The g j factor of a bound electron and the hyperfine structure splitting in hydrogenlike ions. Phys. Rep. 339, 79-213. Beier, Th., et al. (2000). The g-factor of an electron bound in a hydrogenlike ion. Phys. Rev. A 62, 032510, pp. 1-31. Beier, Th., et al. (2001). New determination of the electron's mass. Phys. Rev. Lett. 88, 011603-1-4. Bloch, E (1953). Experiments on the g-factor of the electron. Physica 19, 821-831. Bohr, N. (1928). The magnetic electron. Collected Works of Niels Bohr (J. Kalckar, Ed.), Vol. 6. North-Holland, Amsterdam, p. 333. Bollen, G., et al. (1990). The accuracy of heavy-ion mass measurements using time of flight-ion cyclotron resonance in a Penning trap. J Appl. Phys. 68, 4355-4374. Breit, G. (1928). The magnetic moment of the electron, Nature 122, 649. Brillouin, L. (1928). Proc. Natl. Acad. Sci. U.S.A. 14, 755. Brown, L.S. (1985). Geonium lineshape. Ann. Phys. 159, 62-98. Brown, L.S., and Gabrielse, G. (1982). Precision spectroscopy of a charged particle in an imperfect Penning trap. Phys. Rec. A 25, 2423-2425. Brown, L.S., and Gabrielse, G. (1986). Geonium Theory: Physics of a single electron or ion in a Penning trap. Rev. Mod. Phys. 58, 233. Close, EE, and Osborn, H. (1971). Relativistic extension of the electromagnetic current for composite systems. Phys. Lett. B 34, 400-404. Cornell, E.A., et al. (1990). Mode coupling in a Penning trap: :r pulses and a classical avoided crossing. Phys. Rev. A 41, 312-315. Dehmelt, H. (1986). Continuous Stern-Gerlach effect: Principle and idealized apparatus. Proc. Natl. Acad. Sci. US.A. 53, 2291. Dehmelt, H. (1988). New continuous Stern-Gerlach effect and the hint of "the" elementary particle. Z. Phys. D 10, 127-133.
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CONTINUOUS
STERN-GERLACH
EFFECT
ON ATOMIC IONS
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Farnham, D.L., Van Dyck, R.S., and Schwinberg, P.B. (1995). Determination of the electron's atomic mass and the proton/electron mass ratio via Penning trap mass spectrometry. Phys. Rev. Lett. 75, 3598-3601. Faustov, O. (1970). The magnetic moment of the hydrogen atom, Phys. Lett. B 33, 422-424. Gabrielse, G., Haarsma, L., and Rolston, S.L. (1989). Open endcap Penning traps for high-precision experiments. Int. J. Mass Spectrosc. Ion Proc. 88, 319-332. Garraway, B.M., and Stenholm, S. (1999). Observing the spin of a free electron. Phys. Rev. A 60, 63-79. Gerz, Ch., Wilsdorf, D., and Werth, G. (1990). A high precision Penning trap mass spectrometer. Nucl. Instrum. Methods B 47, 453-461. Glazov, D.A., and Shabaev, V.M. (2001). Finite nuclear size correction to the bound-state g factor in a hydrogenlike atom, Phys. Lett. A 297, 408-411. Grotch, H. (1970a). Electron g factor in hydrogenic atoms. Phys. Rev. Lett. 24, 39-45. Grotch, H. (1970b). Nuclear mass correction to the electronic g factor. Phys. Rev. A 2, 1605-1607. H/iffner, H., et al. (2000). High-accuracy measurement of the magnetic moment anomaly of the electron bound in hydrogenlike carbon. Phys. Rev. Lett. 85, 5308-5311. Hermanspahn, N., et al. (2000). Observation of the continuous Stern-Gerlach effect on an electron bound in an atomic ion. Phys. Rev. Lett. 84, 427-430. Hughes, V.W., and Kinoshita, T. (1999). Anomalous g values of the electron and muon. Rev. Mod. Phys. 71, 133-139. Karshenboim, S., Ivanov, V.G., and Shabaev, V.M. (2001). Can. J. Phys. 79, 81-86. Kretzschmar, M. (1990). A theory of anharmonic perturbations in a Penning trap. Z. Naturf 45a, 965-978. Lindroth, E., and Ynnerman, A. (1993). Ab initio calculations of g j factors for Li, Be +, and Ba +. Phys. Rev. A 47, 961-970. Mohr, P.J., and Taylor, B.N. (1999). CODATA recommended values of the fundamental physical constants: 1998, J. Phys. Chem. Ref Data 28, 1713-1852. Natarajan, V., et al. (1993). Precision Penning trap comparison of nondoublets: atomic masses of H, D, and the neutron. Phys. Rev. Lett. 71, 1998-2001. Pauli, W. (1958). Prinzipien der Quantentheorie. In "Handbuch der Physik" (S. Fliigge, Ed.), Vol. 5. Springer, Berlin, p. 167. Shabaev, V.M. (2001). QED theory of the nuclear recoil effect on the atomic g factor. Phys. Rev. A 64, 052104-1-14. Stern, O., and Gerlach, W. (1922). Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld, Z Phys. 9, 349-352. Van Dyck, R.S., Schwinberg, P.B., and Dehmelt, H.G. (1986). Electron magnetic moment from Geonium spectra: Early experiments and background concepts. Phys. Rev. D 34, 722-736. Van Dyck, R.S., Schwinberg, P.B., and Dehmelt, H.G. (1987). New high-precision comparison of electron and positron g factors. Phys. Rev. Lett. 59, 26-29. Van Dyck, R.S., Farnham, D.L., and Schwinberg, P.B. (1993). Tritium-helium-3 mass difference using the Penning trap mass spectroscopy. Phys. Rev. Lett. 70, 2888-2891. Veseth, L. (1980). Spin-extended Hartree-Fock calculations of atomic g j factors. Phys. Rev. A 11, 421-426. Veseth, L. (1983). Many-body calculations of atomic properties: I. g j factors. J. Phys. B 16, 2891-2912. Wineland, D.J., and Dehmelt, H.G. (1975). Principles of the stored ion calorimeter. J. Appl. Phys. 46, 919-930. Yelkovsky, A. (2001). Recoil correction to the magnetic moment of a bound electron. E-print archive, hep-ph/O108091 (http://xxx.lanl.gov). Yerokhin, V.A., Indelicato, P., and Shabaev, V.M. (2002). Self-energy correction to the bound-electron g factor in H-like ions. E-print archive, physics~0205245 (http://xxx.lanl.gov).
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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
THE CHIRALITY OF BIOMOLECULES R O B E R T N. C O M P T O N 1,2 a n d R I C H A R D M. P A G N I 1
IDepartment of Chemistry and 2Department of Physics, University of Tennessee, Knoxville, Tennessee 37996 I. II. III. IV. V. VI. VII. VIII. IX. X.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Nature of Chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . True and False Chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Galaxies, Plants, and Pharmaceuticals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plausible Origins of Homochirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A s y m m e t r y in Beta Radiolysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Amplification and Degradation M e c h a n i s m s . . . . . . . . . . . . . . . . . . . . . . . Possible Effects of the Parity-Violating Energy Difference (PVED) in Extended Molecular Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
219 219 230 233 236 243 247 252 257 257 258
I. Introduction It is often said that the feature that distinguishes the physical sciences from the biological sciences is the precision with which the terms and concepts in each field are defined and measured. Terms in the physical sciences are precisely defined while those in the biological sciences are often much less so. Even the definition of life, which is the sine qua non of biology, is difficult to pin down in all cases. Is a virus or a prion living, for example? Nonetheless, there are features that all living things have in common. Most of the molecules that serve structural and functional roles in all living systems exist in only one of two seemingly identical forms. This property is called homochirality, and the molecules that possess it are said to be optically active. This chapter will investigate the present state of understanding of how homochirality in living systems may have come to be on the earth. We refer to the fact that overwhelmingly only one of these identical forms is found in life. Before doing this, however, it will be necessary to find out what homochirality means and what are the properties of the molecules that potentially posses it.
II. Fundamental Nature of Chirality Chirality is a term familiar to physicists, mathematicians and chemists. What follows below is a condensed description of chirality from a chemist's perspective. 219
Copyright 9 2002 Elsevier Science (USA) All rights reserved ISBN 0-12-003848-X/ISSN 1049-250X/02 $35.00
220
R.N. Compton and R.M. Pagni
[II
FIG. 1. Left and right hands. A more detailed discussion may be found in the excellent book on the subject by Eliel et al. (1994). The English word chirality is derived from the Greek word Zetp (kheir) meaning hand, and refers to an interesting geometrical property of left and right hands (and feet) and other three-dimensional objects. Neither the left nor the right hand has any internal symmetry but both are related to each other via a mirror plane. If you take your left hand and reflect it into a mirror, what you see in the mirror is an image of your right hand (Fig. 1). Likewise, reflection of your right hand into the mirror also yields an image of your left hand. All objects (except vampires) have a mirror image. In most cases, such as with spheres (balls) and planes (pieces of papers), the mirror image is superposable or congruent with the original object. This means that the mirror image object is identical in every respect (constituents and geometry) to the original. What makes the relationship of the left and right hands different is the fact that they cannot be superposed one on top of the other even though the connectivities of the fingers to the palms are the same in both cases. No matter how hard one tries to attain the superposition, there will always be mismatches. The left and right h a n d s - and any other objects with non-superposable mirror i m a g e s - are said to be chiral. There are myriad objects in the physical world that are chiral. Coils, screws and springs exist in chiral forms (right- and left-handed helices), for example. Perhaps the most well-known of these is the alpha (right-handed) helix of DNA.
II]
THE CHIRALITY OF BIOMOLECULES H
CH4
221
H H\
HmC--H
.... H
H / C "~H
I
H formula
bond connectivities; no three-dimensional structure implied
tetrahedron representation; three-dimensional structure implied
shorthand tetrahedral representation
FIG. 2. Various representations of the molecule CH 4.
On a larger scale, vines coil up (twine) trees and shrubs in one of the two mirror image forms whose direction of coiling (helicity) is species dependent. When only one of the two mirror image forms exists, the system is said to be homochiral. Homochirality at all size scales is a hallmark of biology. Many seashells also coil in only one direction, with the direction of coiling being species dependent. Spiral staircases are chiral as well. The massless photon with a unit spin and moving in a straight line at the speed of light carries a spin angular momentum (+h). Thus, a single photon can be thought of as "right-" or "left-" handed. Linearly polarized light is an equal mixture (racemic) mixture of these photons. Electrons also have one-half integer spin, but at rest the electron and its mirror image are superposable. When electrons are in motion, their mirror images are not superposable, thus yielding spin-polarized particles if the z component of the spin angular momentum can be along or against the direction of propagation of the electron. Thus, for elementary particles such as photons and electrons, it is the combination of spin and translation that is responsible for their chirality (physicists speak of helicity) and circular polarization. Many molecules, including biological ones, are chiral. To see how this is possible, consider first the structure and properties of the simplest organic molecule, methane (CH4) (Fig. 2). This molecule consists of a central carbon atom (C) connected or bonded to four hydrogen atoms (H). Each bond connecting a carbon atom to a hydrogen atom is represented by a straight line. When a carbon atom is bonded to four other atoms, it exists in a tetrahedral environment in which the carbon resides in the center of an imaginary Platonic tetrahedron and the four hydrogen atoms reside at apices of the tetrahedron. This yields a three-dimensional structure with four identical carbon-hydrogen bond lengths and six identical hydrogen to carbon to hydrogen bond angles, in agreement with experiment. Rather than draw a cumbersome tetrahedron every time a structure contains a tetrahedral carbon atom, it is convenient to draw it in a different, simplified manner: In this picture of methane two straight lines from C to H represent bonds in the plane of the paper, while a solid triangle represents a carbon-hydrogen bond projecting from the paper, where the carbon resides, to the hydrogen in front of the paper and a dashed triangle represents a C - H bond projecting behind the paper. Methane is thus a highly symmetric molecule with Td symmetry. It should
R.N. Compton and R.M. Pagni
222
H~
....,\\
H
Ho, t,,.
[II
/H
H~'~C ~ H
i _ i / c "~H
FIG. 3. Methane and its mirror image.
H
I
HmC ~H
I
H
H ~
I
H~G ~F
/
H
H
I
H--CmF
I
H ~
CI
I
Br--CmF
I
C1 CHIRAL
FIG. 4. Methane and several of its derivatives.
be noted, however, that the picture of methane with Ta symmetry represents the time-averaged or equilibrium structure. As the bonds in molecules are constantly vibrating, an instantaneous picture of methane would be of lower symmetry. Now consider methane as was done above for left and right hands, i.e., reflect the image of methane into a mirror (Fig. 3). It is not difficult to see, either in one's mind or with models, that the mirror image object is superposed exactly on the original methane molecule. Methane thus lacks chirality and is said to be achiral. What happens when one of the hydrogen atoms in methane is replaced with a different atom (or a group of atoms)? Consider fluoromethane, CH3F, where the replacement is a fluorine atom (Fig. 4). This molecule also contains a tetrahedron carbon atom at its center but with C3~ symmetry has lower symmetry than methane. Nonetheless, CH3F and its mirror image are superposable on one another and the molecule is achiral. Likewise, when a second atom is replaced with a still different atom (or a group of atoms), an achiral molecule results. For example, chlorofluoromethane (CHzC1F), where the second replacement is with a chlorine atom (C1), is also achiral. When a third replacement is made with a still different substituent, a chiral molecule is formed. Bromochlorofluoromethane, CHBrC1F, where the third substituent is a bromine atom (Br), is a fascinating example of such a molecule (Fig. 5). Here the mirror image molecule cannot be superposed on the original. It is always possible to match up the carbon and two of its attached atoms (and their bonds) in the two structures, but the other two attached atoms (and their bonds) will always be out of alignment. Thus CHBrC1F is chiral.
II]
THE CHIRALITY OF BIOMOLECULES
H\ J C -,.~ICI Br~
223
F/,..... / H CIw~'" C ~ B r
FIG. 5. Bromochlorofluoromethane and its mirror image.
It is worth noting that the achiral CH4, CH3F and CH2FC1 all have one or more planes of symmetry while the chiral CHBrC1F does not. It can be shown that all achiral molecules have at least one improper axis of symmetry Sn, defined as rotation by 2sr/n radians followed by reflection in a plane perpendicular to the rotation axis. A plane of symmetry is thus equivalent to S I and a center of symmetry is equivalent to $2. The two forms of CHBrC1F are stereoisomers of one another, i.e., they have the same connectivities of atoms and yet the very same atoms are oriented differently in space. The carbon atoms in the stereoisomers of CHBrC1F are called stereogenic centers. Stereoisomers such as the two forms of CHBrC1F that are also mirror images of one another are called enantiomers. Interestingly, enantiomers have identical chemical properties except when they react with other molecules which are also enantiomeric; reaction or interaction with chiral forces may also yield a difference in behavior. Enantiomers also have identical physical properties except in the way they interact with plane-polarized or circularly polarized light or other chiral objects. From this observation it would appear to be difficult to separate, i.e., resolve, mixtures of enantiomers into their individual forms. Fortunately, as described below, there are methods to accomplish this task. If one has samples of each pure enantiomer, one enantiomer would rotate the plane of monochromatic plane-polarized light a given number of degrees in the clockwise direction while the other enantiomer would rotate the light the same number of degrees but in the counterclockwise direction. The device in which these measurements are made is called a polarimeter. Molecules that rotate plane-polarized light are said to be optically active, and the wavelength dependence of the optical activity is called optical rotary dispersion (ORD). A collection of achiral molecules does not rotate light and is thus said to be optically inactive. An individual achiral molecule can also rotate the plane of polarization, but the rotation averaged over many non-oriented molecules goes to zero. A 1:1 mixture of two enantiomers, which is called a racemic mixture, also is optically inactive because the rotation of one enantiomer is canceled by
224
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the opposite rotation of the other enantiomer. The enatiomeric excess is defined as the difference in amounts of the two enantiomers in a sample divided by their sum times 100. A pure enantiomer will have an enatiomeric excess of 100% and is thus homochiral; a racemic mixture will have an enantiomeric excess of 0%. Enantiomeric excess is often abbreviated as ee. CHBrC1F is prepared in racemic form by the reaction of achiral CHBrzC1 with achiral HgF2 (Hine et al., 1956): 2CHBr2C1 + HgF2 --~ 2CHBrC1F + HgBr2. Generally speaking when achiral or racemic compounds react with one another, the resulting chiral products will be racemic. Only when a r e a g e n t - or c a t a l y s t - is optically active will a chiral product be optically active as well. Chiral forces such as circularly polarized light may also induce optical activity (enantiomeric excess) in the product as well. Chiral reagents, chiral catalysts or chiral forces are required to bring about enantiomeric excess in reaction products. Thus, at first glance it is difficult to see how optically active biological compounds such as amino acids and carbohydrates came to be on the earth when they resulted in the first place from the chemical combination of achiral molecules. CHBrC1F can be prepared in optically active or resolved form in two ways: (1) by the reaction of a precursor that is already optically active (Doyle and Vogl, 1989) and (2) by the reaction of a chiral but optically inactive precursor in a chiral, optically active environment (Wilen et al., 1985). It always takes an optically active molecule or chiral force to produce a product that is optically active. There is nothing unique about CHBrC1E Any molecule containing a tetrahedral carbon or other central atom such as silicon (Si) bonded to four different substituents, regardless of their nature, will exist in enatiomeric forms. Thus the property of chirality, which is manifested in chemical compounds, is in reality geometric in nature. Many other geometric orientations of molecules in space also yield enantiomeric forms. CHBrC1F (and like molecules) has permanent chirality because its stereogenic, tetrahedral carbon atom is attached to four different substituents. The innate chirality of CHBrC1F does not depend on the lengths of its four bonds or the fact that the lengths are constantly changing due to molecular vibrations. CH4, on the other hand, behaves differently. It is achiral because its averaged equilibrium structure of Td symmetry possesses six planes of symmetry and six $4 axes of symmetry as well. Because it is vibrating, any instantaneous structure of CH4 lacks Td symmetry and may be chiral if all four bond lengths are different. In another moment this chiral structure will disappear and be replaced by another, perhaps with a mirror-image relationship to the first. This constantly shifting chirality will disappear when averaged through an ensemble of methane
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THE CHIRALITY OF BIOMOLECULES
225
molecules, thus yielding a net optical rotation of zero degrees. Even crystals and liquids made up of achiral molecules may display instantaneous chirality which averages to zero. Monochromatic, linearly polarized light (LPL) may be viewed as a superposition of left-handed and right-handed circularly polarized light (CPL) (Michl and Thulstrup, 1986). Alternately, one could describe LPL as a racemic mixture of photons with spin angular momentum oriented along and against the direction of propagation. When LPL passes through a sample of enantiomers, one circular component of the light passes through the medium faster than the other component because the real parts of the refractive indices, nL and ne, of the left-handed and right-handed CPL through the molecule are different (Atkins, 1978). This is the origin of optical activity. The angle of rotation is equal to ( n L - nR)Jrl/~,, where l is the length of the sample and Jl is the wavelength of the light. If the rotation of the plane of polarization is clockwise as seen by an observer looking at the source of light, this is designated in two ways: (1) with a + sign or (2) with a lower case "d" which stands for dextrorotatory. Counterclockwise rotation is designated with a - sign or with a lower case 'T' which stands for levorotatory. One enantiomer of CHBrC1F will thus be (+)-CHBrC1F and the other (-)-CHBrC1F or d-CHBrC1F and 1-CHBrC1E Unfortunately, there is no simple method to relate the direction of the optical rotation that the ensemble of enantiomers generates with the absolute configuration of the enantiomer. Absolute configuration is the property of the enantiomer that distinguishes the orientation of its substituents in space from that of its mirror image partner. Amino acids are the building blocks of peptides, proteins and enzymes which serve an extraordinary number of structural and catalytic roles in cells. Enzymes, for example, are the biological agents that catalyze virtually all cellular reactions, most often with amazing speed and selectivity. Structurally, an amino acid that is used biologically consists of an amino group NH2, consisting of a nitrogen atom N bonded to two hydrogen atoms, and a carboxylic acid group COOH, consisting of a carbon atom b o n d e d - actually double-bonded- to one oxygen atom O and a hydroxyl group OH, and to a central CHR group, where R varies in structure. As there are 20 amino acids commonly used in cellular chemistry, there are 20 different R groups. All of the amino acids except glycine with R = H contain a stereogenic center at the CHR carbon atom. Two of the amino acids, isoleucine and threonine, contain a second stereogenic center in the R group itself.
H
I
H2N~CHR~COOH
H2N~ l ~COOH R AMINO ACID
226
[II
R.N. Compton and R.M. Pagni
..... //COOH
H l,
CH3W~"C ~
NH2
one enantiomer of alanine
HOOC
....,,\H
H2N/C "~CH3 other enantiomer of alanine
I rotate
rotate structure
structure
COOH
COOH
_
H2ND-- C ~ H
H~ ' - C ~ N H 2 _
CH3
_
CH3
I project into paper
I project into paper
COOH
H2N~
I C ~H I CH3
COOH
I I CH3
H~ C ~NH2
L-alanine
D-alanine
FIG. 6. The enantiomers of alanine. To see the consequences of the amino acids having a stereogenic center at the CHR group, consider alanine with R = CH3 (methyl group, where a carbon atom is bonded to three hydrogen atoms) (Fig. 6). With one stereogenic center alanine exists in two enantiomeric forms, as shown in Fig. 6. It is common to draw the enantiomers of the amino acids in the so-called Fischer projection, named after Emil Fischer, the great German chemist. Here the COOH and CH3 groups of one of the enantiomers are aligned vertically with the COOH group at the top. This forces the hydrogen and NH2 group attached to the stereogenic center to occupy horizontal positions. If one imagines the stereogenic carbon to be in the plane of the paper, the two vertical groups lie behind the paper and the two horizontal groups lie in front. At this point the vertical and horizontal groups and their bonds are projected onto the plane of the paper, resulting in the Fischer projection of one of the enantiomers of alanine. The enantiomer which places the NH2 group to the left of the vertical axis in the Fischer projection is assigned the L absolute configuration. The other enantiomer, with the NH2 group to the right of the vertical axis, is assigned the D absolute configuration. All 19 amino acids with a stereogenic center at the CHR carbon thus have L and D enantiomers. It
II]
THE CHIRALITY OF BIOMOLECULES
227
is important to note that the upper case D and L refer to absolute configuration while the lower case "d" and "1" refer to the clockwise/counterclockwise rotation of plane polarized light through the enantiomers; knowing one quantity does not tell you anything about the other. A strange fact of living material is the almost exclusive use of L-amino acids in cellular chemistry. The origin of this homochirality is difficult to understand if one assumes that the amino acids were synthesized on the primitive earth from achiral and racemic compounds without any apparent intervening chiral force. Chiral products generated from achiral or racemic compounds are themselves racemic. Homochirality is an accepted tenant in any theory involving the origins of life. Science continues to ponder the seemingly mystical question of the origins of specific homochirality: Did it occur by chance or some fundamental bias? Carbohydrates or sugars are a second class of important biological molecules. Most people are familiar at least with the names of the carbohydrates glucose, fructose, lactose, and sucrose (table sugar). Although they generally serve as energy sources in cells, they have other functions as well. Glucose, for example, is the building block for both cellulose, the fibrous material of plants, and starch. Other carbohydrates can be found in cell walls. Ribose and deoxyribose, two other carbohydrates, are building blocks of the nucleic acids which are responsible for the genetic code and ultimately for all of the chemistry that takes place in cells. The name carbohydrate is a misnomer as it implies a hydrated form of carbon, i.e., a compound in which carbon is bonded to water. This arises because sugars have formulas of the type Cn(HzO)m, where H20 is the formula for water, and n and m are integers. Instead, most carbohydrates are polyhydroxyaldehydes; here hydroxy implies an OH group and aldehyde a CHO group, which can only occur at the terminus of a molecule. Thus carbohydrates are aldehydes containing two or more, thus poly, OH groups, one at each of the non-aldehydic carbons: HOCHz(CHOH)nCHO. Because most carbohydrates contain more than one stereogenic center, it is important to see what stereochemical consequences there are for such molecules. Consider the two stereoisomers of the carbohydrate 2,3,4-trihydroxybutanal (HOCHzCHOHCHOHCHO) in Fischer projection: D-erythrose and L-erythrose, each with stereogenic centers at carbons 2 and 3 (Fig. 7). These mirror-image isomers are enantiomeric because they are non-superimposable. The enantiomer at the left has the D absolute configuration because its OH group at carbon 3 (the one attached to the stereogenic center farthest removed from the aldehyde group) is to the right of the vertical axis of carbon atoms, while its partner to the right has the L absolute configuration because its carbon-3 OH group is to the left of the vertical axis. As D- erythrose and L-erythrose are enantiomeric, their properties and relationships are identical to those described above for enantiomers with one stereogenic center.
R.N. Compton and R.M. Pagni
228
CHO
I
OHC
I
H-- C2 --OH
HO-- C2--H
H--C3--OH
HO-- ~3--H
I I
CH2OH D-erythrose
[II
I
HOH2C
~L-erythrose
FIG. 7. The enantiomers of the carbohydrate erythrose.
CHO
[
HO-- C2--H
I
H--~3--OH CH2OH D-threose
OHC
I
H-- C2--OH
I
HO-- ~3--H HOH2C L-threose
FIG. 8. The enantiomers of the carbohydrate threose.
There are two other stereoisomers that have the same bonding pattern as D-erythrose and L-erythrose (Fig. 8). One can imagine forming them from the erythrose enantiomers by interchanging the H and OH groups at carbon 2. The resulting structures, which are called D-threose and L-threose, are enantiomeric with one another because, even in Fischer projection, it is easy to see that the molecules are mirror images of one another and yet are not superimposable. As with D-erythrose, D-threose has the D absolute configuration because its OH group at carbon 3 is to the right of the vertical axis of carbon atoms, while its mirror image partner has the L absolute configuration because its carbon-3 OH group is to the left of the axis. What is new for molecules that have two stereogenic centers is the relationship of the erythrose enantiomers to the threose enantiomers. They are clearly stereoisomers of one another because the bond connectivities in the erythrose isomers are the same as those in the threose isomers but the atoms are oriented differently in space. Furthermore, neither erythrose isomer is superposable on or has a mirror-image relationship with the threose enantiomers and vice versa. Stereoisomers that do not have a mirror-image relationship are called
II]
THE CHIRALITY OF BIOMOLECULES D-threose
COOH
I
HOmCmH
I
H~CmOH
I
COOH 1-tartaric acid
L-threose
D-erythrose
HOOC
I
HmC~OH
I
HO~C~H
I
HOOC d-tartaric acid
L-erythrose
COOH
HOOC
I
H--C ~OH
.......
f
229
I
HOmC--H .......
.
.
.
.
.
.
.
~ .......
.
HmCmOH
I
COOH
HO--CmH
I
HOOC
meso-tartaric acid (dashed line represents the plane of symmetry $1)
FIG. 9. Conversion of threose and erythrose into tartaric acid.
diastereomers. Thus, the erythrose enantiomers are diastereomeric with the threose enantiomers. Unlike enantiomers whose physical and chemical properties are identical except under very special circumstances, diastereomers have different chemical and physical properties. One can take advantage of this fact to easily separate diastereomers from each other, a process that is not feasible for enantiomers. Methods that are available to separate the threose diastereomeric pair from the erythrose diastereomeric pair, for example, will not separate the threose enantiomers from each other or the erythrose enantiomers from each other. In the examples just described, molecules with two stereogenic centers afforded 4 stereoisomers consisting of 2 pairs of enantiomers. In general, a structure with n stereogenic centers will yield a maximum number of stereoisomers equal to 2 n, consisting of 2n/2 or 2 "-~ pairs of enantiomers. Three centers thus will yield a maximum of 4 pairs of enantiomers, 4 centers yield 8 pairs etc. The maximum number will not be obtained if the molecule of interest possesses an internal improper axis of symmetry such as S] or $2. To visualize this, consider a known classical reaction of carbohydrates that converts both termini of D- and L-erythrose and D- and L-threose into carboxylic acid groups (COOH) (Fig. 9). The threose isomers yield a pair of enantiomeric molecules called d- and 1-tartartic acid, d because one enantiomer rotates plane-polarized light to the right and 1 because the other enantiomer rotates the light to the left. Neither of these enantiomers possesses an improper axis of symmetry in any of its various conformations. It is thus not possible to superpose d-tartaric acid onto 1-tartaric acid. The situation is different for D- and L-erythrose. When the ends of both molecules are converted into COOH, a single molecule called meso-tartaric acid is produced, i.e., both enantiomers yield the same product. It is easy to see from the two Fischer projections of meso-tartaric acid in Fig. 9 that each possesses a place of symmetry (S1), which renders the two structures superposable and thus equivalent. Tartaric acid with two stereogenic centers
230
R.N. Compton and R.M. Pagni
[III
represents then a set of three stereoisomers consisting of a pair of enantiomers, d,l-tartaric acid, and meso-tartaric acid which is diastereomeric with the first two. meso-Tartaric acid has the unusual property of having two stereogenic centers and yet being achiral. As with the amino acids (except glycine which lacks a stereogenic center), carbohydrates such as glucose, fructose, ribose and many others, regardless of how many stereogenic centers each possesses, occur as pairs of enantiomers with D and L absolute configurations. Emil Fischer, who elucidated the structure of glucose and many other sugars, never knew whether D-glucose or L-glucose was naturally occurring. Determination of absolute configurations came later with the development of X-ray diffraction methods and, more recently, circular dichroism (CD). Circular dichroism is the differential absorption of LCPL and RCPL of an enantiomer undergoing an electronic or vibrational transition. Today it is known that all naturally occurring carbohydrates have the D absolute configuration. What mechanism did nature use to achieve homochirality in the carbohydrates? Before moving on to a discussion of true and false chirality, there is one additional issue of nomenclature that must be discussed because it will appear on occasion in the remainder of the chapter. D and L, as already seen, refer to the absolute configuration of an entire molecule, be it with one stereogenic center such as in alanine or with many centers such as in glucose which has 4. It is actually preferable to assign the absolute configuration of every stereogenic center. The unambiguous methodology for doing this, which will not be presented here, was developed by Cahn, Ingold and Prelog (1966). Each stereogenic center is assigned either the (upper case) R (rectus _= right) absolute configuration or the S (sinister--left) absolute configuration.
III. True and False Chirality Thus far we have defined a static three-dimensional object to be chiral if its coordinates are non-superposable on that of its mirror image. In the language of group theory a stationary object is chiral if it has no mirror plane, center of inversion, or improper axis of rotation. However, as we discussed earlier for the case of an electron, an object in motion requires further consideration since the chirality is now defined by both space- and time-dependent quantities. Barron has provided a broader definition of molecular chirality through the concept of "true and false" chirality (Barron, 1981, 1986a,b). "True chirality is shown by systems existing in two distinct enantiomeric states that are interconverted by space inversion, but not by time reversal combined with any proper spatial rotation." This can be illustrated through considerations of the operations involved. The parity operator P represents the spatial inversion, SI, of the positions of all particles in an object through a fixed origin (mirror
III]
THE CHIRALITY OF BIOMOLECULES
231
reflection involves spatial inversion) plus a rotation through Jr, Rn, about an axis perpendicular to the mirror plane. This is illustrated by the S-(-) and R-(+) enantiomers of bromochlorofluromethane" F
Mirror Reflection .,,,~ Br C "'~H
F Brm,....
SI + R~ . . . .
H""
C
/
\
Cl
Classical time reversal, T, represents the reversal of the motions of all the particles in the system. Barron presented a vivid picture of the distinction between "true" and "false" chirality by considering two rotating cones, one in a state of linear translation and the other rotating but translationally at rest (Barron, 1986a,b). A stationary rotating cone exists as two mirror image states which are interconverted by SI. Since P plus R:r also interconverts these two cones, the object exhibits "false chirality." However, a spinning cone translating along the axis of the cone, although interconverted by SI, is not interconverted by T and R and thus is said to possess "true chirality." This is further illustrated by replacing the spinning cones with rotating ammonia molecules: FALSE CHIRALITY
,,,
H
H
TRUE CHIRALITY
,...... "N" HH~'(~ H
S..4-...[.-~I
H~.
H ~
.N.
,......"I~ H~'~~ H ~
T
H
Thus the instantaneous snapshot (i.e., time short compared to the tunneling time) of a translating and rotating ammonia molecule represents a truly chiral system. However, the fact that ammonia is rapidly converting (tunneling) from one form to the other, requires that the two states would have definite chirality but not definite parity. The two true eigenstates of definite parity are described by the symmetric or antisymmetric combination of the two mirror-image forms of the ammonia molecule. If the molecule is in one state, after a time r it will exist in the other state. Thus, if observed over a time long compared to r, the probability of being in either state is equal (racemization). This also illustrates the so-called Hund's paradox (Hund, 1927), where a quantum-mechanical chiral molecule is
232
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R.N. Compton and R.M. Pagni
therefore thermodynamically unstable (i.e., able to exist in either energetically degenerate form). Hund evoked the long tunneling time between enantiomers to obtain stability. Right- and left-circularly polarized light (RCPL and LCPL) consists of massless particles of unit spin (I = 1) angular momentum along (+h, RCPL) or against (-h, LCPL) the direction of propagation. Circulary polarized photons exhibit true chirality since P interconverts RCPL and LCPL, but T does not. Likewise, translating particles with half-integer spin such as electrons and neutrons are also fundamentally chiral. The possible effects of spin-polarized electrons originating from beta decay will be discussed at length later. Barron's criteria can be applied to other combined sources. As a final example consider the combination of a static magnetic field, B, and static electric field, E: E ~
D
SI ,
~
E ~
D
.
E
~ -
T .,,q_..............~
E
-"
B
Y
B
B
where E represents a time-even polar vector and B a time-odd axial vector. Therefore, E changes sign under spatial inversion, SI, and B does not, whereas the reverse is true for time reversal operation, T. As illustrated above, false chirality is seen for this system because R~ plus T has the same effect as does SI. Avertisov, Goldanskii and Kuz'min (1991) have used the prescriptions of Barron for true and false chirality to provide a convenient summary of most of the physical situations in nature which are capable of producing an enantiomeric excess in chemical reactions. They divide the "advantage factor" for mirror symmetry breaking into a "local advantage" for those which might have existed in a particular region on the Earth and "global advantage" as those due directly or indirectly to the electroweak interaction. They define an advantage factor, g, as the relative difference in the rate constants, k L and k o, for the mirror-conjugated reactions, i.e. k L _ kD g =
kL + k D
9
Neither a static magnetic field (SMF), static electric field (SEF) or their combinations are chiral. Gravitational fields (GF) are also achiral. However, certain combinations of these fields together with rotations are chiral, as seen in Table I. All of the advantage factors shown are dependent upon a molecular factor, Z, determined by the structure of the molecule under consideration. The chiral influence of longitudinally polarized [3 particles also depends upon the electron helicity h, and the relative difference in the spin-polarized electron scattering cross section ((yL _ ( j D ) / / ( o L _+_ (yD). A report of asymmetric induction/resolution occurring in static magnetic fields (Zadel et al., 1994) has been retracted (Breitmaier, 1994) and was shown
IV]
THE CHIRALITY OF BIOMOLECULES
233
Table I Chirality of field combination with rotation
Advantage factors
Local Circularly polarized light
10-4 to I 0 -1
Rotation + SMF + SEF
<10 -4
Rotation + SMF + GF
<10 -4
Linearly polarized light + SMF
<10 -4
Global Weak neutral currents Beta particles
IO-20ZrZ5kBT~ 10-13 to 10-17 Xqh3A(oL-oD)/(OL +oD)~ 10-9
to 10-11
to be theoretically impossible (Barron, 1994), in agreement with experiment. Recently, Rikken and coworkers have shown, however, that enantioselective photochemistry can occur using plane-polarized light in a parallel magnetic field which exhibits true chirality (Rikken and Raupach, 2000; Raupach et al., 2000). This research was recently "highlighted" by van Wfillen (2001). The enantiomeric excesses were quite small even in the presence of large magnetic fields. It is generally believed that the Earth's magnetic field is too small to have produced an enantiomeric advantage factor large enough to have influenced the development of homochirality of biological molecules.
IV. Galaxies, Plants, and Pharmaceuticals Chirality exists in our universe at many levels of size. On the grand scale, spiral galaxies are chiral. When observed from the earth onto the galactic plane, they have an "S" or "Z" shape. Of course, when observed side-on, these shapes are not apparent. A non-receding, stationary, S-shaped galaxy would be achiral because its mirror image, which is Z-shaped, is superposable on the original, even if the galaxy were rotating about the galactic center. Likewise, a stationary Z-shaped galaxy would be achiral. Because of the Big Bang origin of the universe, all galaxies are receding from the earth. Thus, the receding and rotating S- and Z-shaped galaxies are chiral due to the combined linear and rotational motion. Interestingly, a study of more than 500 spiral galaxies reveals an equal number of S- and Z-shaped galaxies (Kondepudi and Durand, 2001; Iye and Sugai, 1991; Sugai and Iye, 1995), although there is a significant asymmetry in certain subclasses of the galaxies (Sugai and Iye, 1995). There is also no apparent segregation of S- and Z-shaped galaxies based on their recessional velocities.
234
R.N. Compton and R.M. Pagni
FIG. 10.
[IV
Twisted eucalyptus tree.
Many plants also exhibit handedness. Charles Darwin, the father of the theory of evolution, made detailed observations of the "twining" of certain climbing plants, that is, their propensity to grow in a spiral or corkscrew manner (Darwin, 1906). In nature, twining plants are homochiral; honeysuckle twines to the left and morning glory twines exclusively to the right. Most other plants grow in a right-handed spiral and most seashells also exhibit a right-handed helicity. The human umbilical cord, on the other hand, twists in either direction. The left tusks of the Narwhal, a species of toothed whale, twist in a counterclockwise direction. Figure 10 shows the right-hand twist exhibited by a large Blue Gum tree (genus Eucalyptus) located in the Botanical Gardens of Christchurch, New Zealand.
IV]
THE CHIRALITY OF BIOMOLECULES CH 3
CH3
CH3
235 CH3
o
i
i
_
~cg3
3
(R)-carvone
~"CH3
(S)-carvone
H3
(R)-limonene
(S)-limonene
OH
O
,I
H
II
OCH2CHCH2NHCH(CH3)2
,I
H2NCCH2m C mCOOH
I
NH2 asparagine
propranolol O
O
OH
I,
HOCH2CCH2CI
I
H 1-chloro-2,3-propanediol
(R)-thalidomide
(S)-thalidomide
FIG. 11.
Many other examples of homochirality may be found in the book by Gardner (1990). In addition to amino acids and sugars, many other life-supporting molecules are homochiral as are many pharmaceuticals. The chemistry of the human body is vitally dependent upon the subtle stereochemical differences of enantiomers. Different sensations, for example, are often exhibited by naturally occurring enantiomers. To about 90% of people (R)-carvone (see Fig. 11; the asterisk locates the stereogenic center) tastes and smells like spearmint (from spearmint oil) while the S enantiomer smells and tastes like caraway (from caraway seeds). The R enantiomer of limonene, a terpene, tastes like orange while the other enantiomer tastes like lemon. The S enantiomer of asparagine, a chemical (alkaloid) found in asparagus, tastes bitter while its R mirror twin is sweet to the taste. These differences are not always observed, however. The
236
R.N. Compton and R.M. Pagni
[V
naturally occurring D-glucose and the unnatural L-glucose are equally sweet. If an inexpensive synthesis of it were available, L-glucose would be an ideal nonnutritive sweetener. Enantiomerically pure drugs are becoming increasingly important even though they are generally more expensive to prepare than their racemic counterparts (Deutsch, 1991; Stinson, 1995, 1998, 2000, 2001). The world-wide market for single-enantiomers drugs already exceeds $120 billion per year. Often one enantiomer is therapeutic while the other is benign or even harmful. Separate enantiomers of barbituric acid derivatives can function as a narcotic or anticonvulsant. (-)-Propranolol acts as a beta blocker for heart disease while its enantiomer acts as a contraceptive. The S form of 1-chloro-2,3-propanediol is a pharmaceutical while the R form is poisonous. One enantiomer of the steroid estrone is a sex hormone while the other is inactive. Thalidomide, which possesses one stereogenic center (at the starred position), is undoubtedly the most notorious compound where one enantiomer is reported to be therapeutic while the other is harmful. The R enantiomer of this compound has been used to alleviate the effects of morning sickness of pregnant women while the S enantiomer gives rise to teratoidism, i.e., birth defects, in developing fetuses. It has been estimated that 12 000 children have been born with thalidomide-induced birth defects in the 1950s, primarily in Europe and South America. This molecule was not approved for use in the United States except for research purposes. It had been presumed that if enantiomerically pure (R)-thalidomide were given to pregnant women the problem of birth defects would be avoided. However, it is now known that thalidomide racemizes rapidly in blood plasma so that providing an enantiomerically pure form of thalidomide would not alleviate the problem of teratoidism. Thalidomide may still prove useful in certain circumstances. The compound suppresses the growth of blood vessels (angiogenesis) which may prove useful in the treatment of certain cancers and AIDS-related conditions. Research is now underway on the development of resolved derivatives of thalidomide which are therapeutic but do not racemize easily. Approximately 530 synthetic chiral drugs are now marketed worldwide. Because of the technical difficulties and corresponding expense of making enantiomerically pure drugs, only about 60 of these compounds are marketed in their enantiomerically pure form. Increasing pressure from the scientific community along with tighter restrictions imposed by the U.S. Food and Drug Administration means that more enantiomerically pure pharmaceuticals will become available in the near future.
V. Plausible Origins of Homochirality Louis Pasteur, the 19th-century chemist and microbiologist, was the first person to realize that many of the molecules of life are homochiral. We know today
V]
THE CHIRALITY OF BIOMOLECULES
237
that carbohydrates, which are found in starch, cellulose, and the nucleic acids, exist in nature exclusively with the D absolute configuration. Most amino acids, which are the building blocks of proteins and enzymes, have the L absolute configuration. This is true regardless of the type or complexity of the organism, be they found in a bacterium, betel (palm or pepper), beetle, beagle, or the Beatles. Thus specific homochirality of biomolecules is a fundamental characteristic of life. Any plausible theory of the origins of life on earth must take this fact into account. Two general types of theories have been invoked to explain the origin of homochirality on the earth: (1) those involving a chance or random selection of the homochiral molecules that were incorporated into the first living organisms; and (2) those that require an internal or external chiral force to bring about the asymmetric synthesis of the homochiral molecules. As will be seen in due course, many theories in the second grouping also have a random quality about them because the chiral influence itself may appear randomly. Leaving aside the accidental "chance scenario," theories have concentrated on how chiral forces may influence the prebiotic asymmetric synthesis of biological molecules in nature. Ideas that specific homochirality has resulted from some prebiotic asymmetric synthesis have become more credible in recent years following the analysis of organic compounds brought to the earth in meteors. Cronin and Pizzarello (1997) for example, found a 7 to 9% excess of L amino acids in the Murchison meteorite found in Australia. It is believed that this enantiomeric excess did not arise by terrestrial contamination because some of the amino acids found in the meteorite do not exist in the biosphere. The Murchison meteorite, a carbonaceous chondrite, was created about 4.5 billion years ago in the asteroid belt, strongly suggesting an asymmetric prebiotic chemical synthesis of the carbon-containing molecules in the universe. The "normal" laboratory synthesis of amino acids, of course, yields a racemic mixture. For example, an electric discharge (a mimic of lightning) in a mixture of methane, nitrogen, ammonia, and water produced racemic mixtures of the amino acids, alanine (1.7% yield), asparagine (0.024%), and glutamine (0.051%); achiral glycine was found in 2.1% yield (Miller, 1959). Assuming an extraterrestrial asymmetric synthesis of the amino acids found in the Murchison meteorite, some chiral influence has been at work on the meteorite when it was first synthesized or during the intervening 4.5 billion years. It is plausible that the source of homochiral molecules necessary for the formation of life on the earth has an extraterrestrial origin. Even if this be true, the question remains as to how the asymmetric synthesis occurred. Several possible explanations follow. Circularly polarized light is a plausible chiral influence. It is well known that the photochemistry of racemic mixtures of molecules initiated with circularly polarized light can lead to an excess of one enantiomer over the other, either in product or recovered reactant (Feringa and van Delden, 1999; Bonner,
238
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R.N. Compton and R.M. Pagni
1996; Rau, 1983; Balavoine et al., 1974). This photochemistry can occur by photoequilibration of a racemic mixture of molecules without any loss of reactant or by the selective destruction of one enantiomer over the other. Both processes take advantage of the fact that enantiomers have different absorptivities (cross sections) when exposed to LCPL and RCPL. Both methods, however, have serious drawbacks. In the first instance enantiomeric excesses tend to be low because differences in absorptivities usually are very small while the absorptivities are generally large. In the second case, very high enantiomeric excesses can be obtained but only at the expense of destroying most of the reactant. There are several notable photoreactions in the literature initiated by CPL. Shimizu and Kawanishi have studied the two-photon photochemistry of D,L, i.e., racemic tartaric acid with 351-nm CPL from a XeF excimer laser (Shimizu and Kawanishi, 1996a,b; Shimizu, 1997). Irradiation with RCPL led to a selective destruction of the L enantiomer while the concentration of the D enantiomers changed little; this led to a maximum enantiomeric excess of recovered tartaric acid of 7.5%. Irradiation with LCPL was unfortunately not carried out. COOH
I
COOH
~HOH
351-nm RCPL from XeF excimer laser
HO
r
CHOH
I
H
COOH
H OH
+
CO2, CO, H 2
COOH
racemic D,Ltartaric acid
D-tartaric acid
Inoue and coworkers have also investigated a photoreaction using a different intense light source, CPL with 190 nm synchrotron radiation from a storage ring (Inoue et al., 1996). They studied the photointerconversion of racemic transcyclooctene, a chiral molecule with no stereogenic centers, and the achiral ciscyclooctene:
CPL
R enantiomer
trans-cyclooctene
S enantiomer
cis-cyclooctene
Although enantiomeric enrichment was found, the effect was small. Photolysis of the racemic amino acid, D,L-leucine, with RCPL and LCPL in aqueous solution (in water) at 212.8 nm also afforded small, but meaningful enantiomeric excesses
V]
THE CHIRALITY OF BIOMOLECULES
239
in recovered reactant, a result with implications for the origins of homochirality on the earth (Flores et al., 1977). Thus circularly polarized light in nature could provide a mechanism for the prebiotic synthesis of optically active amino acids, carbohydrates, and other relevant molecules. A number of sources of circularly polarized light on earth and in the universe have been identified. Sunlight passing through the atmosphere generates slightly elliptically polarized light (not quite circular), LCPL before sunrise and RCPL after sunset (Angel et al., 1992). The polarization can reach and +0.2%, respectively. At sunrise and sunset the polarization is zero. This is clearly a small effect that averages to zero over the course of a day. However, the opposite sides of a mountain ranging from north to south would experience more or less of a given elliptical polarization. Supernovae are also known to emit CPL at right angles to the normal linear Bremsstrahlung generated from electrons rotating in a circle. CPL is emitted perpendicular to the plane of Bremsstrahlung: RCPL in one direction and LCPL in the other direction. Thus, in the sunrise/sunset effect of our sun and in supernovae explosions, both right- and left-CPL are produced so that any advantage in the photosynthesis of biomolecules would have to occur through a chance encounter with one form of the light (Bonner, 1997). There has also been observation of strong circular polarization of infrared radiation in regions such as Orion OMC-1, where star formation is occurring (Bailey et al., 1998). The circular polarization was attributed to Mie scattering of linearly polarized light from either spherical grains of dust or nonspherical grains that had been aligned by a magnetic field. The percentage of circular polarization varied from -5% to +17% depending upon the region observed. Although only polarized radiation at 2.2 ~m was detected, calculations show that the circular polarization would persist at shorter wavelengths. Bailey and coworkers noted that this type of light may be responsible for the synthesis of homochiral biomolecules because enantioselective synthesis of amino acids using CPL of much shorter wavelength (<200 nm) has been reported (Norden, 1997). Recent calculations also suggest that photochemistry initiated with elliptically polarized infrared radiation may also be possible (Fujima et al., 1999). Unfortunately, infrared photochemistry is usually a multi-photon process because a single infrared photon does not possess enough energy to break chemical bonds. We are investigating the use of intense, short-pulsed, high-power circularly polarized infrared light from a free-electron laser to initiate enantioselective photochemistry. All of the chiral influences discussed thus far are local. When averaged over space and time, these asymmetries average to approximately zero. One source of CPL that is chiral over space and time comes from the beta decay of radioactive atoms. This effect will be described shortly. Many review articles describe other physical processes which could have been responsible for the mirror symmetry-
240
R.N. Compton and R.M. Pagni
[V
breaking in the biological world (Avertisov et al., 1991; Ponnamperuma and MacDermott, 1994; Mason, 1988, 2000; Quack, 1989; Frank et al., 2000; Cline, 1996; Bonner, 1988; Avalos et al., 1998, 2000a; Buschmann et al., 2000). Of particular note are the possible effects of parity-nonconserving electroweak interaction because the phenomenon is innately homochiral. All matter interacts through four known forces (in order of strength): nuclear, electromagnetic, weak, and gravitational. Of these forces only the weak interaction is fundamentally chiral because it can distinguish between left and right. The weak interaction is responsible for radioactive beta decay, the slow but spontaneous emission of electrons from metastable nuclei. Electrons possess an intrinsic spin of l h (h = h/27c) and spin-angular momentum (z component) of 1v~h . An electron in motion can have its spin-angular momentum vector aligned along or against its direction of propagation, thus yielding a circularly polarized particle. Thus an electron in motion is chiral and its spin describes a right- or left-handed helix as it moves through space. In the 1950s it was predicted (Lee and Yang, 1956) and observed (Wu et al., 1957) that the weak nuclear force does not conserve parity. The consequence of parity nonconservation in beta decay is that leptons emitted from the nucleus are primarily spin polarized in one helicity. Beta rays are primarily left-handed. Positron decay, on the other hand, gives rise to right-handed anti-electrons. Thus, it is often said that we live in a left-handed universe since it is made of matter rather than antimatter (Barrow and Silk, 1983). Shortly after the discovery of parity violation, the predicted effects of the chiral electroweak interaction in atomic physics were tested. Since the nucleus of an atom contains a chiral force, all atoms are thereby chiral and should exhibit optical activity and circular dichroism (Khirplovich, 1991). This force will also mix states of opposite parity, giving rise to the observation of "parity-forbidden transitions" in atoms (Bouchait and Bouchait, 1997). In combination with theory, these experimental measurements have provided a sensitive test of the standard model of nuclear physics at low energy (Masterson and Wieman, 1995). Similar predictions can be made concerning the effects of the electroweak interaction in molecular physics, i.e., optically active diatomic molecules and the mixing of parity-forbidden states; however, such effects have yet to be reported. The observation of the parity-violating effects in atoms poses a broad question in the context of the subject of this article: can the weak interaction play a role in chemistry in general and the origins of homochirality in biomolecules in particular? There are at least two ways that the weak interaction could influence molecules and their chemical reactivity: (1) by altering the energies and energy levels of molecules (parity-violating energy difference, PVED); and (2) by the interaction of left-handed electrons with matter following beta decay. We examine these effects separately. The parity-violating (P-odd) weak interaction is the only known chiral force in non-decaying atoms and molecules. Weak interaction effects in molecules of the
V]
THE CHIRALITY OF BIOMOLECULES
241
type observed for atoms have not been reported. Instead, research in P-odd effect in molecules has concentrated on the prediction and detection of PVED between R and S enantiomers. Rein (1974) and Letokhov (1975) first suggested that the electroweak interaction would lift the degeneracy between enantiomers predicted by normal quantum electrodynamics. A number of calculations of the PVED for small molecules have been published (Hegstrom et al., 1980; Zel'dovich et al., 1977; Mason, 1984; Mason and Tranter, 1984; MacDermott et al., 1992; Bakusov et al., 1998; Lazzeretti and Zanasi, 1997; Zanasi et al., 1999; Laerdahl and Schwerdtfeger, 1999; Laerdahl et al., 2002; Berger et al., 2001). The Hamiltonian operator describing the PVED for a molecule containing i electrons and n nuclei is given by
Hpv= H~v+ H2v =(GF -~) ZOw'5pn(ri)+-~)
Z
I11(I~ + 1)
di " [nPn(ri).
The Fermi electroweak coupling constant, GF (2.22255• 10-14 a.u.-1.16637• 10-l~ eV), sets the scale for the small magnitude of this effect. 111 is the nuclear spin vector operator, K11 and tr are nuclear parameters for nucleus n, y5 _- i y I y Z y 3 y 4 represents a pseudoscalar chirality operator formed from a product of Dirac g-matrices, p11(ri) is the normalized nucleon density, and Qw is the weak charge given by Qw = -N11 + Z11(1 - sin20w), where N11 and Z11 are the numbers of neutrons and protons, respectively, and 0w is the Weinberg mixing angle. An excellent review of the state of affairs of the theoretical calculations of the parity-violating energy difference for small molecules has been published by Buschmann, Thede and Heller (2000). In the 1980s, ab initio calculations gave PVED values on the order of 10-14 J/mol and, interestingly, concluded that the L-amino acids and D-sugars were lower in energy. Until recently, this stabilization energy was believed to be too small to be of significance to prebiotic asymmetric synthesis. Although autocatalytic processes such as those evoked by Kondepudi and Nelson (1983, 1984a,b, 1985) could play a role in providing enantiomeric excesses over long times, the small energy differences expected did not encourage further speculation. However, more recent calculations, particularly those by Quack and co-workers (see Berger and Quack, 2000 and others contained therein) and Schwerdtfeger et al. (see Laerdahl, Wesendrup and Schwerdtfeger, 2000) are about an order of magnitude larger and give new interest to possible PVED effects in prebiotic asymmetric chemistry. However, both of these groups have shown that the magnitude and indeed the sign of the PVED for alanine depend upon its exact molecular conformation (shape that may change by rotating atoms or group of atoms
242
R.N. Compton and R.M. Pagni
[V
around single bonds) and therefore may not lead to a preferential stabilization of one enantiomer over the other. There is an obvious critical need for an experimental measurement of the PVED. Unfortunately, a measurement on the order of 10- 1 4 J/mol is beyond any present experimental technique. Fortunately, the PVED increases with the magnitude of the atomic number. Specifically, the PVED scales with Z 6 (Laerdahl and Schwerdtfeger, 1999). Thus, molecules with atoms having large atomic numbers will show the largest PVED and may provide an experimental test for PVED. There have been a number of experimental attempts to measure PVED. Most notably, high-resolution infrared absorption spectroscopy has been used on two occasions. Arimondo, Glorieux and Oka (1977) observed inverted Lamb-dip ro-vibrational transitions for the two enantiomers of camphor and found them to agree to within 10-8. Recently, Daussy et al. (1999) demonstrated that the C - F stretching mode of the two CHFBrC1 enantiomers agreed to within 13 Hz. This places an upper limit on AEpv/E of <4x 10 -13. Although these measurements are five orders of magnitude more sensitive than the results for camphor, they are still not sensitive enough. Theoretical studies on CHFBrC1 give an energy difference of only 2 mHz, (-8 x 10-18 eV) for the C - F frequency (Laerdahl et al., 2002). Keszthelyi (1994) used M6ssbauer spectroscopy to establish an upper limit of 4x 10 -9 for the energy difference of 1- and d-tris(1,2-ethanediamine)iridium complexes. A Z 6 scaling was used to estimate the energy difference of 1.1 x 10-12 eV between the two enantiomers. Thus the measured upper limit is three orders of magnitude higher than the expected Z-scaling result. This report contains very few details on the experimental procedures used, and no M6ssbauer spectra were shown. Although all of the L amino acids are predicted to lie lower in energy than the corresponding D amino acids, the differences are so small that it is hard to envision how this could lead to a dominance of the L amino acids in biology. A large amplification mechanism would be required. However, the PVED scales with Z 6, as noted earlier. Thus, for larger molecules containing atoms with larger atomic numbers, the energy differences between enantiomers should be sufficiently large to be observable with currently available spectroscopic techniques such as nuclear magnetic resonance, M6ssbauer and perhaps infrared spectroscopies. Lahamer et al. (2000), for instance, reported an energy difference between enantiomers of an iron complex using M6ssbauer spectroscopy. Due to the fact that these experiments were performed on crystalline material, the energy difference may be due to crystalline effects, although the difference in energy (-10 -l~ eV) is close to that predicted by scaling the PVED with atomic number. In 1991 Abdus Salam (1991), who along with Glashow and Weinberg united the electromagnetic and weak force theory into one electroweak theory, speculated that the Z ~ interactions might provide an explanation of the
VI]
THE CHIRALITY OF BIOMOLECULES
243
domination of L amino acids in living systems in a manner which has not previously been considered. His explanation was mainly heuristic and involved quantum-mechanical cooperative and condensation effects similar to those in Cooper pairing and Bose-Einstein condensation. He postulated that these effects could give rise to second-order phase transitions below a critical temperature Tc. Realizing that phase transitions as a general rule occur at a low temperature he speculated as to the location for the production of L amino acids and suggested possible laboratory tests of these ideas. In a recent paper, Wang et al. (2000) presented three experimental tests of Salam's ideas using single crystals of L- and D-alanine (and valine). In one measurement they report a X phase transition at 270+ 1 K for both sets of crystals using differential scanning calorimetry. The L-enantiomer was reported to be lower in energy. They also reported temperature-dependent mass susceptibilities which are different for L- and D-alanine. These differences are attributed to a PVED-dependent difference in the intramolecular geometry of chiral density. Finally, Raman spectroscopy measurements show that the second-order Ca-H deformation modes at 2606 cm -l and 2724 cm -I for D-alanine vanish at 270 K but reappear at 100 K whereas L-alanine shows no such effect. All of these experiments are presented as experimental evidence of the possible relevance of the Z ~ interaction as presented by Salam at a phase transitions of Tc ~ 270 K. One notes the proximity of this 270 K temperature to that of the freezing point of water, the solvent used to recrystallize the samples. Verifications of these results are crucial and are ongoing in our laboratories.
VI. Asymmetry in Beta Radiolysis Because the energy difference between a pair of enantiomers is extremely small regardless of the conformations of the molecules, a significant amplification or deracemization mechanism would have had to operate in the prebiotic world for an extended period of time for the homochirality of biological molecules to have evolved. This is indeed a difficult task. A perhaps more viable way in which PVED could have brought about homochirality is through beta decay. Many experiments have been performed to test this hypothesis, often with ambiguous or contradictory results. Vester et al. (1959) were the first to suggest a connection between the asymmetry of beta ([3) decay and the evolution of homochirality in the biological world. Their mechanism involves the energetic spin-polarized [3-rays and spinpolarized secondary electrons producing a net left-circularly polarized )'-rays (Bremmsstrahlung) which could be a source of asymmetric photochemical reactions. In an extensive set of experiments using radioactive phosphorus (32p), strontium (9~ and europium (152Eu), all beta-particle emitters, to initiate asymmetric reactions of organic compounds, the same researchers found no
244
[VI
R.N. Compton and R.M. Pagni
H
,I
HzN~C~COOH
I
R
O (+)-camphor
R
tyrosine
H O ~ H 2 ~ CH 2
] CH3CH2~COOH
J
tryptophan H
]~~/~
leucine
(CH3)2CHCH 2
2-phenylbutyric acid aspartic a c i d
HOOCCH2
FIG. 12.
positive results (Ulbricht and Vester, 1962). Garay (1968), on the other hand, found D-tyrosine, an amino acid (Fig. 12), to degrade more rapidly in water containing radioactive strontium chloride (9~ than the naturally occurring L-tyrosine enantiomer. Unfortunately, the method of analysis that Garay u s e d - UV spectroscopy- is not definitive. Darge et al. (1976), likewise, found positive results. In these experiments a frozen aqueous solution of racemic D,L-tryptophan, another amino acid, and radioactive phosphate (32po43) underwent self-radiolysis for 12 months; an astounding 19% enantiomeric excess of the D enantiomer was reported. As with the Garay experiments, the methods of analysis- UV spectroscopy and polarimetry- are not definitive. When the Garay and Darge experiments were repeated (with considerable modification) on D,L-tyrosine, D,L-tryptophan, and other amino acids by Bonner (Bonner, 1974; Bonner and Flores, 1975, Bonner et al., 1979) using gas chromatography, where the enantiomers are separated and quantitated, as the analytical tool, no asymmetric degradation of any amino acid was found even though extensive overall degradation of the amino acids had occurred. Calvin and colleagues (Bernstein et al., 1972) and then Bonner (Bonner et al., 1978) also examined the internal beta radiolysis of several 14C-labeled amino acids (14C is a betaparticle emitter), Calvin by polarimetry and Bonner by gas chromatography; no positive results were obtained. Bonner did have success, however, with polarized electrons not from beta decay but using longitudinally polarized electrons (120 KeV) from a linear accelerator (Bonner et al., 1975, 1976/77).
VI]
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The target was a solid sample of racemic leucine, a still different amino acid. Antiparallel electrons afforded a slight excess of L-leucine (~1% enantiomeric excess) after about 50% degradation of the sample, while parallel electrons afforded a slight excess of the D enantiomer (~0.9% enantiomeric excess) after about 75% degradation. More recently, Akaboshi et al. (1999) looked for the asymmetric decomposition of the racemic amino acid, D,L-aspartic acid, in tritiated water (3H20 , 3H--tritium, a beta emitter). When the recovered amino acid was examined by HPLC, another technique that separates and quantitates the enantiomers, no enantiomeric excess was detected. Except for Bonner's results using polarized electrons generated in a linear accelerator, none of the other experiments involving amino acids, all of which involved beta radiolysis, led to asymmetric degradation. Several related experiments did give positive results. Beta radiolysis of polycrystalline amino acids yields free radicals (species with one unpaired electron) which are detected by electron-spin resonance spectroscopy, a very sensitive technique. Irradiation of D- and L-alanine with internal or external beta radiation affords an excess of radicals from the D enantiomer (Akaboshi et al., 1978, 1981, 1982; Conte et al., 1986). Beta particles passing through the liquid enantiomers, R- and S-2-phenylbutyric acid scatter differently as evidenced by emitted Cerenkov radiation (Garay and Ahlgren-Beckendorf, 1990). The scattering of polarized, low-energy electrons from enantiomers such as D- and L-camphor also occurs asymmetrically (Campbell and Farago, 1985; Blum and Thompson, 1998). To date there is no unambiguous evidence that the beta radiolysis of amino acids leads to the faster degradation of one enantiomer as compared to the other, i.e. deracemization. There is evidence, on the other hand, for the racemization of enantiomers of amino acids by beta radiation. The self beta radiolysis of optically pure 14C-labeled amino acids over the course of decades led to the compounds' partial racemization. (Bonner, 1996). Radiolysis of L-alanine, D-2-aminobutyric acid, L-norvaline, L-norleucine, and D- and L-leucine, all amino acids, with y-rays from a cobalt source (6~ results in partial racemization of the amino acids in all cases (Bonner and Lemmon, 1978a,b). Thus any mechanism involving radiolysis that builds up enantiomeric excess in recovered amino acid would also provide a mechanism for its loss. There is a way in which beta radiolysis has led to significant amplification of enantiomeric excess. This involves the behavior of sodium chlorate (NaC103) in water, for which relevant background material will now be presented. NaC103 is an achiral molecule consisting of a spherical sodium cation (Na +) and a chlorate anion (C103) of C3v symmetry. Solutions of this compound in water are likewise achiral and thus optically inactive. The colorless, transparent crystals of this molecule, on the other hand, are chiral because they belong to the chiral P213 space group. Crystals of NaC103 thus exist in enantiomeric forms which are easily distinguished by polarimetry (Pagni and Compton, 2002).
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R.N. Compton and R.M. Pagni
[VI
When an aqueous solution of NaC103 is evaporated in a Petri dish, wellformed crystals form; a distribution of (+) and (-) crystals, and thus an enantiomeric excess, can be determined by counting crystals. Kondepudi has shown that in the absence of a perturbation, a random distribution of (+) and (-) crystals is produced, as expected (Kondepudi et al., 1993). This is a consequence of the fact that the seed crystals are generated in one enantiomeric form or the other randomly. Stirring, surprisingly, has a dramatic effect on the distribution of crystals (Kondepudi et al., 1990). Under these circumstances a Petri dish will generally contain exclusively all (+) or all (-) crystals. The direction of stirring has no bearing on which enantiomers are produced in a dish. Although the mechanism by which this bifurcation occurs has been debated (Kondepudi and Sabanayagam, 1994; Martin et al., 1996), it is clear that, when stirring is involved, the initially formed crystals provide the seed(s) for the subsequent production of all crystals that arise in the dish. Even though bifurcation has occurred with stirring, a 50:50 distribution of (+) and (-) crystals is still obtained when one counts all the crystals from a large number of dishes. [3 radiolysis of an evaporating aqueous solution of NaCIO3 has a profound influence on the distribution of the resulting (+) and (-) crystals (Mahurin et al., 2001). In two sets of experiments (54 Petri dishes) carried out in front of an intense 9~176 source, enantiomeric excesses (based on numbers of crystals) of +32.1% and +42.6% favoring the (+) crystals were obtained. The enantiomeric excess distribution per dish versus frequency was now bifurcated asymmetrically, with dishes having enantiomeric excesses of +100% being four times more prevalent than those with enantiomeric excesses o f - 1 0 0 % . There was still a smattering of enantiomeric excesses between these two extreme values. Because the [3 particles are approximately 80% left-handed, one can rationalize these results in the following manner: The more common left-handed electrons induce the preferential formation of (+) crystals which then seed the solution to produce more of the same (+) crystals. By the same token, the less common right-handed electrons initiate the formation of (-) seed crystals. The details of how this remarkable result occurs are not currently known. Random crystallization still competes with the particle-induced crystallization. This explains the smattering of enantiomeric excesses occurring between +100% a n d - 1 0 0 % . Similar, but less extensive experiments were carried out in front of the positron source (22Na). The energetic, positively charged electrons are predominantly right-handed. In this instance an overall enantiomeric excess o f - 5 5 . 2 % favoring the (-) crystals was obtained. Petri dishes with 100% of the (-) crystals were about four times more prevalent than dishes with 100% of the (+) crystals. The results with positrons were in essence the mirror image of those obtained with beta particles. The chirality of one lepton thus strongly influenced the chirality of the crystals. Before proceeding to the next section, it is worth describing a recent result
VI]
THE CHIRALITY OF BIOMOLECULES
247
that presents an interesting juxtaposition to the results of Kondepudi described above. Stirring of the aqueous NaC103 solutions had a profound effect on the distribution of + and - crystals, but in a random fashion. Rib6 and coworkers (2001) have now found a clear correlation. The researchers found that certain achiral porphyrins form chiral and optically active aggregates in solution. When the aggregations are carried out without stirring, a 50:50 distribution of + and aggregates are generated. When the aggregate-forming solutions are stirred in one direction, 88% of the experiments yield + aggregates whereas, when stirred in the other direction, 85% of the experiments y i e l d - aggregates. There is thus a distinct correlation between the direction of stirring and the chirality of the aggregates that result. Stirring of the solutions creates vortices in the fluids which possess true chirality according to the definition of Barron. The mechanism by which the chirality of a vortex is transferred to that of the aggregate is not clear. It is also unclear how this very interesting observation can be applied to the origin of optically active molecules in the prebiotic world because the optically active vortices in the primitive aqueous environment would have been generated randomly.
A. AMPLIFICATION AND DEGRADATION MECHANISMS If the physics and chemistry described above generally yield enantiomers with very small enantiomeric excesses, a large amplification mechanism is required to attain the homochirality of the biological world. This is particularly true if PVED was the initial source of asymmetry because its effect is exceedingly small. Where crystallization and aggregation phenomena are concerned, however, the initial enantiomeric excesses may be substantial and only a modest amplification may be required to attain homochirality. The crystallization effect is best exemplified with sodium chlorate. Seeding of an evaporating aqueous sodium chlorate solution with (+) and (-) crystals yields (+) or (-) crystals, exclusively. Stirring has the same effect but the direction of the induced homochirality is random. Circularly polarized electrons and positrons generated from radioactive nuclei, as noted above, have a similar effect on the distribution of (+) and (-) sodium chlorate crystals. Although it is difficult to see how sodium chlorate crystals could have played a role in the origin of biological homochirality because they are synthetic, naturally occurring quartz, which exists in enantiomeric forms, could have played such a role. Unfortunately, there is essentially no excess of one enantiomeric form of quartz over the other on the earth although local excesses may exist (Frondel, 1978; McBride, 2001). Interestingly, asymmetric adsorption of the alanine enantiomers onto (+) and (-) quartz has been demonstrated by Bonner (Bonner et al., 1974). Until the recent work of Soai, which will be described below, quartz had not served as a medium for an asymmetric chemical reaction.
248
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R.N. Compton and R.M. Pagni
Calcite, a form of calcium carbonate (CaCO3), also occurs naturally. Unlike quartz, calcite is achiral. Its adjacent crystal faces turn out to be optically active and of opposite handedness, however. Recent experiments have shown that one face of calcite selectively adsorbs L enantiomers of a few amino acids from aqueous solution while the other face selectively adsorbs the D enantiomers (Hazen et al., 2001). On a related note, naturally occurring clay has been suggested as the environment in which life evolved (Cairns-Smith, 1982). Even if this were so, clay could not have been the progenitor of homochirality because it is achiral. The photochemistry of homochiral crystals, which are made up of achiral molecules, usually affords photoproducts with significant enantiomeric excesses. The following example illustrates this fact. The so-called dibenzobarrelene exists in two crystalline space groups: Pbca, which is achiral, and P212121, which is chiral. Photolysis in the achiral crystalline form generates a dibenzosemibullvalene containing four stereogenic centers in racemic form, identical to what is obtained in solution (Evans et al., 1986):
CO2CH(CH3)2 (CH3)2CHO2C~
(CH3)2CHO2C
CO2CH(CH3)2
hv
dibenzobarrelene
dibenzosemibullvalene
Photolysis of homochiral crystals of the dibenzobarrelene in the P212121 dimorph, on the other hand, yields the photoproduct with 100% enantiomeric excess. It is difficult, however, to see how results of this type could be responsible for the origin of biological homochirality. As with quartz, one expects to get a 50:50 mixture of crystals on crystallization, with photolysis of (+) crystals yielding one enantiomer of the photoproduct and photolysis of (-) crystals yielding the other enantiomer. Frank (1953) was the first person to propose a kinetic model for the spontaneous synthesis of optically active products from achiral precursors. In this model both enantiomers of the products are produced initially. Each enantiomer in turn acts as a catalyst for its own production and an inhibitor for the production of the other enantiomer. Under some circumstances the system becomes unstable and the subsequent course of the reaction is subject to small perturbations. The net result is to generate an excess of one enantiomer, with the one in excess being produced randomly. Kondepudi and Nelson (1983, 1984a,b, 1985) have proposed a related model in which two reagents A and B react reversibly to
VI]
THE CHIRALITY OF BIOMOLECULES
249
give enantiomeric products R and S, each of which catalyzes its own formation reversibly and with each other irreversibly to produce a product P: A + B ~,-~--R,
A+B~----S,
A + B + R ~,-~-2R,
A + B + R ~,-~--2S,
R + S--,P.
This system is open because A and B are continually added while P is continually removed. As with Frank's model, this one becomes unstable under certain conditions, thus yielding one enantiomeric product in excess. If this system is biased, however, with an initial excess of R or S, even ever so slightly due to PVED, one pathway becomes dominant. If R is in excess initially, the pathway leading to production of R will be favored, and vice versa if S is in excess initially. Calculations show that in periods as small as tens of thousands of years, the initial very small enantiomeric excess due to PVED can be amplified to enantiomeric excesses close to 100% favoring the L-amino acid. It is unfortunate that there is no possibility of testing this mechanism. Soai and coworkers have developed in recent years a remarkable autocatalytic reaction which also results in significant amplification of chirality. (Shibata et al., 1996, 1998; Soai et al., 1999, 2000; Soai and Shibata, 1999; Sato et al., 2000a, 2000b.) In this chemistry the chiral product also functions as a catalyst for its own production. Although this chemistry could not have occurred in the prebiotic world, prebiotic chemistry based on the model of this chemistry could have played a role in the origin of homochirality. Let us begin by examining the reaction under consideration. Treatment of achiral pyrimidine-5-carboxaldehyde which contains a carbonoxygen double bond (carbonyl group) with achiral diisopropylzinc [isopropyl = -CH(CH3)2] affords a zinc alkoxide product. Treatment of the zinc alkoxide in turn with water yields an alcohol product. The initial reaction involves the transfer of an isopropyl group from the zinc reagent to the carbonyl carbon of the pyrimidine reagent. Both the initially formed zinc alkoxide and the isolated alcohol contain a single stereogenic center. Because there are no chiral forces or optically active reagent involved in the chemistry, both products are racemic.
~ ~
0
H + N pyrimidine-5carboxaldehyde
"...-
[(CH3)2CH]2Z n diisopropylzinc
/
OH ] ~ H CH(CH3)2 ~ N
alcohol
OZnCH(CH3)2 ~ Jc"CH(CH3)2 -~- ~ "N zinc alkoxide
R.N. Compton and R.M. Pagni
250
[VI
The zinc alkoxide still has a reactive isopropyl group which also adds to the carbonyl group of the pyrimide-5-carboxaldehyde to yield a zinc dialkoxide with a second stereogenic center. Recall that a product with two stereogenic centers yields a maximum of 4 stereoisomers. In this instance a pair of enantiomers (RR and SS) in racemic form, and a meso compound (RS=SR) are generated in unequal amounts. On treatment with water this mixture of zinc dialkoxide stereoisomers yields the same alcohol as shown above in racemic form. CH(CH3)2
CH(CH3) 2
,I
,I
I
I
Ar--- C ~ O ~ Z n ~ O ~ C H
~Ar
Ar = pyrimidine ring
H zinc dialkoxide RS = SR (meso)
RR and SS (enantiomeric pair)
The system becomes very interesting when a small amount of optically active alcohol is added to diisopropylzinc before addition of the carboxaldehyde. The added alcohol reacts with the zinc reagent to yield the optically active (S)-zinc alkoxide. If the added alcohol has the S absolute configuration, so does the zinc alkoxide. Before addition of the carboxaldehyde, the system contains mostly achiral diisopropylzinc and a little (S)-zinc alkoxide. When the carboxaldehyde is then added, it reacts preferentially with the optically active zinc alkoxide. This can happen in two ways, creating either an S or an R stereogenic center in the zinc dialkoxide. For steric reasons, the second addition yields the S absolute configuration at the new center. By a disproportionation reaction the (S)-zinc dialkoxide reacts with diisopropylzinc to generate more (S)-zinc alkoxide which is then used to add to still more pyrimidine-5-carboxaldehyde. Overall then, the addition of a little (S)-alcohol to the systems yields lots of (S)-alcohol. (CH3)2CHx Ar,, .)C -- OH H (S)-alcohol
+
[(CH3)2CH]2Zn
(CH3)2CHx, Ar,"~C --OZnCH(CH3)2 O Ar H
~
(CH3)2CHx Ar,,')C--OZnCH(CH3)2 + Propane H (S)-zinc alkoxide
...._ (CH3)2CHN /QCH3)2CH "~C_O_ZnOC~,,H~....x [ (S,S)-zinc dialkoxide original ~ S center J J ~ [(CH3)2CH]2Zn
2 (S)-zinc alkoxide
I new S center
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THE CHIRALITY OF BIOMOLECULES
251
Table II Amplification of chirality in the Soai reactions Initiator
% Enantiomeric excess
% Enantiomeric excess of alcohol after one reaction
Leucine
2
(L)
21
(R)
Leucine
2
(D)
20
(S)
Valine
1
(L)
51
(R)
Valine
1
(D)
47
(S)
2-Butanol
0.1
(S)
73
(S)
2-Butanol
0.1
(R)
76
(R)
Quartz
100
(+)
89
(S)
Quartz
100
(-)
85
(R)
NaC103
100
(+)
98
(S)
NaCIO3
100
(-)
98
(R)
Even more amazing results are obtained when the added alcohol is only slightly enriched in one enantiomer (2% enantiomeric excess). After one reaction, the alcohol has enantiomeric excess = 10%. If this alcohol in turn is used to initiate a second reaction, the resultant alcohol has enantiomeric excess = 57%. After 4 such cycles, the original 2% enantiomeric excess has become 88%, a 44-fold amplification of chirality. Similar but even more spectacular results are also obtained on the reaction of pyrimidine carboxaldehydes using the amino acids leucine (2% enantiomeric excess favoring D or L) and valine (1% D or L), alcohols (0.1% enantiomeric excess R or S), (+) and (-) quartz and (+) and (-)-NaC103 as initiators (Table II). Amplifications of close to a thousand have been obtained after one reaction cycle in some cases. The results discussed above provide clear theoretical and experimental evidence that autocatalysts can lead to enantioamplification. Autocata|ysis is, however, not the only mechanism by which enantiomer enrichment may have occurred in the prebiotic world. Yamagata has proposed that amplification may have occurred by an accumulation of small enrichments during each step of the synthesis of a polymer such as a protein or nucleic acid (Yamagata, 1966; Yamagata et al., 1980). The starting material in this scheme is a racemic mixture of a monomer to be incorporated into the polymer; the monomer will be called R-M and L-M. Now allow R-M to react with itself to make a dimer (R-M)2. A similar reaction converts L-M into (L-M)2, the enantiomer of (R-M)2. Because of PVED, the two reactions occur at minutely different rates. Repeat the reactions again to form (R-M)3 and (L-M)3. These also occur at slightly different rates.
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Thus, if homopolymers (R-M)n and (L-M)n are ultimately formed, the PVED kinetic effect will have been amplified n - 1 times. Although this amplification would occur several thousand times in forming a protein, it would occur tens of millions of times to form DNA. In forming the protein the PVED kinetic effect occurs within the D- and L-amino acids, 20 of which are commonly used in life, whereas for DNA, the differences occur in cyclic forms of the carbohydrates Dand L-deoxyribose. The simple scheme involving the build-up of a polymer from a single pair of enantiomers is too simple because R-M and L-M can also react with one another although at rates different from the R-M + R-M and L-M + L-M reactions. This produces a pair of dimer enantiomers, (R-M)(L-M) and (L-M)(R-M). The PVED kinetic effect will also operate when R-M and L-M react with one another. If this "random" polymerization continues, one will obtain in theory 2 n pairs of enantiomers for a polymer with n monomeric units. Each pair of enantiomers will be subject to the cumulative PVED effect; each pair of enantiomers will be formed in unequal amounts. If the starting pool of reactants consists of 20 different racemic molecules, as would be the case for the amino acids, the chemistry becomes incredibly complex because each of the 40 reactants can react to form 1600 dimeric products consisting of 800 pairs of enantiomers. The dimers would then yield 64000 trimeric products.
VII. Possible Effects of the Parity-Violating Energy Difference (PVED) in Extended Molecular Systems As discussed above, the PVED for individual molecules made up of elements with small atomic numbers Z is expected to be exceedingly small. However, since the initial work of Yamagata (1966) a number of researchers have considered the possible effects of "amplification" of molecular PVED in extended molecular systems such as crystals and polymers. MacDermott (1995), for example, has attempted to relate the PVED expected for chiral quartz crystals to an enantiomeric excess of quartz crystals on the Earth. This comparison suffers for a lack of sufficient data on quartz crystals and possible local enantiomeric excesses. Others have considered the kinetic effects of PVED on reaction stereochemistry (Szabo-Nagy and Keszthelyi, 1999a; Avalos et al., 2000a,b). From equilibrium thermodynamics, the reaction rate for either R-R or S-S molecules is proportional to e -PVED/kT. Thus, the ratio of reaction rates for R-R or S-S molecules is roughly proportional to PVED/kT, or epsilon for brevity. Accordingly, a chemical reaction of achiral starting materials would yield an enantiomeric excess, (R-S)/(R+S), of chiral products on the order of e. The influence of this small energy difference for polymers or crystals is expected to increase with the number of monomers or molecules in the bulk material.
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This is expected to yield an enantiomeric excess favoring the lower-energy enantiomer roughly in proportion to n times c. Avalos et al. (2000b) have recently provided a complete review of this subject. All of these studies involve a statistical analysis of chiral crystal growth. Optical rotation or circular dichroism is used to characterize the enantiomeric excess before and after crystallization. Crystallization from low-Z materials such as (+)- and (-)-sodium ammonium tartrate show no appreciable enantiomeric selection (Szabo-Nagy and Keszthelyi, 1999b). That is, crystallization of a racemic mixture of sodium ammonium tartrate gave a Gaussian distribution of (+) and (-) crystals centered about an enantiomeric excess of zero. However, in the cases of crystallization of tris(1,2-ethanediamine)cobalt and tris(1,2-ethanediamine)iridium(III), both containing atoms with large Z, the authors obtained a Gaussian distribution of crystals which was broadened and shifted from zero relative to that of triply distilled water or the starting racemic material. Using a statistical analysis, these authors calculated an e of 8.3 • 10-j4 for the cobalt complex and 4.5 • 10-11 for the iridium complex. The estimated value for the sodium ammonium complex was ~<10-17. These data were presented as evidence for the expected increase with atomic number Z discussed above. The interested reader should refer to the review article by Avalos et al. (2000b) to find an in-depth analysis of some of the controversy surrounding these interpretations. The Z-dependence of asymmetric crystallization of chiral crystals has received other attention. Crystallization of potassium dodecatungstosilicate in aqueous solution is found to give a small excess of (+) crystals (Mason and Tranter, 1984). The actinide compound sodium uranyl acetate also crystallizes into a chiral crystal (Suh et al., 1997). Interestingly, in 1954, before the discovery of parity violation, Havinga (1954) found that the crystallization of allyl ethyl methylanilinium iodide gives mainly dextrorotatory crystals. Our own studies involving beta radiolysis of sodium chlorate crystals discussed earlier also revealed a rather strange crystallization behavior of sodium bromate which crystallizes in the same chiral space group as sodium chlorate. Careful examination of numerous crystallizations yielding thousands of sodium bromate crystals in a wide variety of locations (Wake Forrest, NC, USA; Christchurch, New Zealand; Statesville, GA, USA; Knoxville and Oak Ridge, TN, USA) under extreme conditions of cleanliness (e.g., autoclaving of solutions, filtering, glove box conditions in rare gas or pure nitrogen atmosphere) revealed a consistent ee of >95% favoring (+) crystals. Clearly a bias exists toward the growth of dextrorotatory crystals. This was not seen in the case of crystallization of the sodium chlorate solutions. In view of the results described herein, one could logically evoke the PVED as an explanation of this result. Currently, we are extending these measurements to sodium iodate, which contains iodine with Z = 53. Unfortunately, this is a difficult task because the chiral crystals are extremely small.
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[VII
A large number of interesting, but complicated, studies, too numerous to discuss in detail here, have been carried out on the polymerization of amino acids and their derivatives and other molecules in the light of the origins of life and homochirality. The interested reader is referred to the reviews of Feringa (Feringa and van Delden, 1999) and Bonner (1988) for details. A few pertinent points are in order about this work. (1) Peptide synthesis, which involves the polymerization of amino acids and their derivatives or the linking of smaller peptide units, may be self-replicating (Lee et al., 1996; Bonner, 1972) provided the growing chain adopts an a-helical structure. In a recent paper, Saghatelian et al. (2001) have reported such a chiroselective amplification in which a 32-residue peptide replicator demonstrated efficient amplification of homochiral products from a racemic mixture of peptide products through a chiroselective autocatalytic cycle. This work showed that even within a simple helical molecular architecture, a peptide replicator can contain information capable of self-replication, homochirality and resistance toward accumulation of errors (stereochemical mutations). Although this article does not address the question of specific homochirality, the chiral forces in nature as discussed above (CPL, beta rays, electroweak force, etc.) may act as the subtle influence which provides the initial bias toward a specific handedness. (2) Polymerization of racemic amino-acid derivatives does not lead to any measurable enantioamplification in the polymer or unreacted monomer. However, when one of the enantiomers of the amino acid is in excess, enantioamplification may occur if the growing polymer adopts an a-helical structure (Matsura et al., 1965). These results also suggest an autocatalytic mechanism is operating in which the c~ helix preferentially assists the incorporation of the correct amino-acid enantiomers. (3) Degradation of the polymer via reaction with water (hydrolysis) may compete with the polymerization itself. When the polymer being hydrolyzed is enantiomerically impure, amplification of ee is observed in the reaction products (Blair et al., 1981). The non-linear effect is another way in which an initially small ee can be amplified (Girard and Kagan, 1998). This is generally observed in the case where an enantiomerically impure catalyst yields products with a larger ee. Although there are many ways in which this phenomenon can occur, let us look at one example. Consider the case in which a reaction is catalyzed by a catalyst consisting of a metal M and a ligand L existing in enantiomeric forms LR and Ls. The reaction of interest creates a product with a single stereogenic center. The reaction of LR and Ls with M yields two complexes, MLR and MLs. One of the complexes produces the R enantiomer of the product, PR, and the other produces Ps. In this case the enantiomeric excess of the product (eeproduct) is linearly proportional to the enantiomeric excess of the ligand (eeligand) and thus
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THE CHIRALITY OF BIOMOLECULES
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does not result in amplification. Thus, eeproduct= (eeo)(eeligand), where eeo is the maximum enantiomeric excess that is attainable for the reaction of interest.
kR
M + LR ~ MLR---+ PR,
M + Ls ~ MLs
ke
' Ps.
On the other hand, if the metal contains two ligands, three catalysts, MLRLR, MLsLs and MLsLR, are possible, each of which catalyzes the asymmetric reaction to yield, respectively, PR, Ps and racemic product, each at its own distinctive rate. The expression relating eeproduct to eeligand is now more complex in that it contains/3, which is the ratio of the amount of meso complex, MLRLs, to the sum of the homochiral complexes, MLRLR and MLsLs, and g, which is the ratio of the rate constant for the reaction of the meso complex, kRs, to that of the homochiral complex MLRLR, kRR (kRR = kss): M + LR + Ls ~ MLRLR + MLsLs + MLRLs,
f
I
+ kaa
+ kss
PR
Ps
I
+ kRs
+
eeproduct = (eeo)(eeligand) where
Racemic product
l+/q 1 +g/~'
MLRLs /~ = MLRLR + MLsLs'
g-
kRs kRR
When /3 = 0, i.e., the meso complex is not present, or g = l, i.e., the meso and homochiral complexes have the same rate of reactivity, the new equation for eeproduct reverts to the original one where a linear response is anticipated. When g < 1, a positive non-linear effect (amplification) is expected which reaches its maximum effect at g = 0, where the meso complex MLRLs is unreactive. Values of g > 1 result in a negative non-linear or deamplification effect. In order to calculate the exact positive, zero or negative non-linear effect it is necessary to know K = (MLRLs)2/(MLRLR)(MLsLs), the equilibrium distribution of the complexes. If the complexes are formed randomly, K = 4, for example. It should be noted here that the chemistry of Soai described in detail above is almost certainly an example of a positive non-linear amplification of chirality involving a diastereomeric zinc complex of unknown structure. Diastereomeric catalysts are required for this behavior to occur; enantiomeric complexes are excluded. As seen above, there is significant theoretical speculation and experimental evidence for enantioamplification. What is not always appreciated is that racemization chemistry will always compete. Degradation chemistry will also remove the substrate from the racemic pool being amplified. As seen earlier in
R.N. Compton and R.M. Pagni
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[VII
the article, beta radiation racemizes amino acids. Amino acids also racemize in water with a half-life of about 6000 years, the exact value depending on the structure of the amino acid (Bada, 1972). COOH I H2N - C --H I R L amino acid
-...
COOH I H-- C --NH 2
water
1;1/2- 6000 years
/
R D amino acid
It is possible to date bone and teeth in archaeological samples using this fact. Dating is difficult, however, because the half-life of racemization is environment dependent. The optically active amino acids in the Murchison meteorite, for example, may have remained that way for 109 years. Carbohydrates, as noted near the beginning of the article, are polyhydroxyaldehydes and ketones. As is the case with all aldehydes and ketones with similar structure, they readily form enols and enolates reversibly in the presence of acid and base, respectively. In so doing the stereogenic center next to the carbonyl group (C=O) is lost initially and reformed in two ways: with the original absolute configuration or with the other absolute configuration. By such facile pathways, D-glyceraldehyde is converted into L-glyceraldehyde, thus racemizing the molecule. The enol and enolate also react to give an isomeric ketone, 2,3-dihydroxy-2-propanone, with no stereogenic centers. The racemization of amino acids in waters also occurs through intermediate enols and enolates: H\
CHO I
aci ~
H-- C --OH I CH2OH D-glyceraldehyde ~
enol, enolate
/OH C II
HO/ C \ CH2OH ~ . ~ d
CHO
enol
I
HO-- C --H I CH2OH L-glyceraldehyde
~ H\ /OC II /C\ HO CH2OH enolate
HOCH2\
/c=o
HOCH2
If the carbohydrate contains more than one stereogenic center, racemization does not occur since the absolute configuration of only the center next to the carbonyl group is altered. Instead a diastereomer is formed if the enol/enolate
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THE CHIRALITY OF BIOMOLECULES
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regenerates the center, and an isomeric compound if the center is not regenerated. D-glucose, for example, yields the diastereomer D-mannose and the isomer D-fructose when treated with base in water. CHO I H - - ~ --OH HO-- C --H
I
CHO I HO-- ~ - - H base
~
H--C --OH
I
HO-- C --H
I
H--C --OH
I
HOCH2~ //O
+
HO-- C --H
I
H--C --OH
J
H - - C --OH
H - - C --OH
H - - C --OH
CH2OH
CH2OH
CH2OH
D-glucose
D-mannose
I
I
I
D-fructose
Thus most carbohydrates will not racemize but will quite readily be converted into diastereomeric and isomeric compounds. How such chemistry would affect the amplification of chirality is unclear but important. Interestingly, the carbohydrates, ribose and deoxyribose, that exist in nucleic acids are not susceptible to the above chemistry because they exist in cyclic forms that mask the carbonyl group. Under such circumstances enols and enolates cannot form. In conclusion it is fair to say that any mechanism that amplifies a small enantiomeric excess into a larger one must do so at rates faster than mechanisms that racemize or degrade the compound of interest.
VIII.
Conclusions
Considerable progress has been made in the last few decades in understanding the origin and evolution of the molecules of life (Brack, 1998; Mason, 1991). This cannot be said about the origins of specific homochirality. As this article attests, the field is active and many interesting results have been obtained, but there are still a very large number of plausible explanations for specific homochirality. Hopefully future research will reject some explanations and refine others to the point where a likely cause is found. This may never occur, however, because the time scales of some theories are vast and thereby difficult to study experimentally. The origin may have arisen from a random event and is thus not explainable by any theory or testable by experiment, because of our inability to go back and see it transpire in the first place. Thus, the origins of specific homochirality may remain one of the many mysteries of life.
IX.
Acknowledgment
The authors thank the National Science Foundation for support of this work, and graduate student Rodney Sullivan for his assistance.
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ADVANCES IN ATOMIC, 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
MICROSCOPIC ATOM OPTICS." FROM WIRES TO A N A TOM CHIP RON FOLMAN, PETER KROGER and JORG SCHMIEDMAYER Physikalisches Institut, Universit?it Heidelberg, 69120 Heidelberg, Germany
JOHANNES DENSCHLA G Institut fiir Experimentalphysik, Universita't Innsbruck, 6020 Innsbruck, Austria
CARSTEN HENKEL Institut fiir Physik, Universitdt Potsdam, 14469 Potsdam, Germany I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Designing Microscopic Atom Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Magnetic Interaction B. Electric Interaction
....................................... ........................................
C. Traps and Guides formed by C o m b i n i n g the Interactions . . . . . . . . . . . . . . . D. Miniaturization and Technological Considerations . . . . . . . . . . . . . . . . . . . . III. Experiments with Free-Standing Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Magnetic Interaction
.......................................
B. Charged Wire Experiments
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IV. Surface-Mounted Structures: The Atom Chip
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263 265 265 283 286 289 292 292 300 303
A. Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
304
B. Loading the Chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Atom Chip Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
307 314
V. Loss, Heating and Decoherence A. Loss Mechanisms B. Heating
..................................
.........................................
................................................
C. Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
324 324 330 335
VI. Vision and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Integrating the Atom Chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
342 343
B. Mesoscopic Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
346
C. Q u a n t u m Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Conclusion
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viii. Acknowledgement IX. References
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348 351 351 352
I. I n t r o d u c t i o n Scientific and technological progress in the last decades has proven that miniaturization and integration are important steps towards the robust application of fundamental physics, be it electronics and semiconductor physics in integrated circuits, or optics in micro-optical devices and sensors. The experimental effort described in this work aims at achieving the same for matter wave optics. 263
Copyright 9 2002 Elsevier Science (USA) All rights reserved ISBN 0-12-003848-X/ISSN 1049-250X/02 $35.00
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Matter wave optics beautifully illustrates quantum behavior. Realizations using neutral atoms are attractive because of the well established techniques of coherently manipulating internal and external degrees of freedom, and their weak coupling to the environment. Miniaturizing electric and magnetic potentials is essential to building versatile traps and guides for atoms at a scale < 1 ~tm which will enable controlled quantum manipulation and entanglement. Integration with other quantum optics, micro-optics and photonics techniques will allow for robust creation, manipulation and measurement of atomic quantum states in these microtraps. In our vision we see a monolithic integrated matter wave device which will allow us to establish a new experimental toolbox and enable new insights into fundamental quantum physics, for example in issues such as decoherence, entanglement and nonlinearity, low-dimensional mesoscopic systems, and degenerate quantum gases (Bosons and Fermions) beyond mean-field theory. A successful implementation may lead to widespread applications from highly sensitive sensors (time and acceleration) to quantum information technology. The goal of this review is to sum up the 10 year long exciting journey into the miniaturization and integration of matter wave optics resulting in devices mounted on surfaces, so called atom chips. It brought together the best of two worlds: the vast knowledge of quantum optics and matter wave optics and the mature techniques of microfabrication. The first experiments started in the early 1990s with the guiding of atoms along free-standing wires and investigating the trapping potentials in simple geometries. This later led to the microfabrication of atom-optical elements down to 1 ~m size on atom chips. Very recently the simple creation of Bose-Einstein condensates in miniaturized surface traps was demonstrated, and the first attempts to integrate light optics on the atom chip are in progress. Even though there are many open questions, we firmly believe that we are only at the beginning of a new era of robust quantum manipulation of atomic systems with many applications. The review is organized as follows. We begin in Sect. II by describing microscopic atom-optical elements using current-carrying and charged structures that act as sources for electric and magnetic fields which interact with the atom. In the following sections we describe first the experiments with free-standing structures - the so called atom wires (Sect. III), investigating the basic principles of microscopic atom optics, and then the miniaturization on the atom chip (Sect. IV). In Sect. V we discuss one of the central open questions: what happens with cold atoms close to a warm surface, how fast will they heat up, and how fast will they lose their coherence? The role of technical noise, the fundamental noise limits and the influence of atom-atom and atom-surface interactions are discussed. We conclude with an outlook of what we believe the future directions to be, and what can be hoped for (Sect. VI). The scientific progress regarding manipulation of atoms close to surfaces has been enormous within the last decade. Besides the atom wire and atom
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MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP 265
chip described here, it covers a whole spectrum: from reflection experiments on atom mirrors to studying Van der Waals interactions and quantum reflection; from using micromagnets to trap atoms to employing evanescent light field traps. Many of these have been reviewed recently and will not be included here. We will almost exclusively concentrate on manipulation of atoms with static microscopic electric and magnetic fields created by charged and/or currentcarrying (microscopic) structures. For related experiments and proposals, which are not discussed in this review, we refer the reader to the excellent reviews referenced throughout the text, e.g. Dowling and Gea-Banacloche (1996), Grimm et al. (2000), Hinds and Hughes (1999).
II. Designing Microscopic Atom Optics Neutral atoms can be manipulated by means of their interaction with magnetic, electric, and optical fields. In this review the emphasis is put on the magnetic and the electric interaction. The designing of traps and guides using charged and current-carrying structures and the combination of different types of interaction to form devices for guided matter wave optics are discussed. It is shown how miniaturization of the structures leads to great versatility where a variety of potentials can be tailored at will. We start with some general statements and then focus on the concepts that are important for surface-mounted structures and address issues of miniaturization and its technological implications. A. MAGNETIC INTERACTION
A particle with total spin F and magnetic moment/u = gFttBF experiences the potential
Vmag = - / ~ " B = --gFktBmFB,
(1)
where /re is the Bohr magneton, gF the Land6 factor of the atomic hyperfine state, and mF the magnetic quantum number. In general, the vector coupling/u. B results in a complicated motion of the atom. However, if the Larmor precession (mL = ItBB/h) of the magnetic moment is much faster than the apparent change of direction of the magnetic field in the rest frame of the moving atom, an adiabatic approximation can be applied. The magnetic moment then follows the direction of the field adiabatically, mF is a constant of motion, and the atom is moving in a potential proportional to the modulus of the magnetic field B = ]B]. Depending on the orientation of/u relative to the direction of a static magnetic field, one distinguishes two cases: (1) If the magnetic moment is pointing in the same direction as the magnetic field (Vmag < 0), an atom is drawn towards increasing fields, therefore it is in a strong-field seeking state. This state is the lowest energy state of the system.
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Minima o f the potential energy are found at maxima of the field. Maxima of the magnetic field in free space are, however, forbidden by the Earnshaw theorem 1. This means that for trapping atoms in the strong-field-seeking state, a source of the magnetic field, such as a current-carrying material object or an electron beam, has to be located inside the trapping region. (2) If the magnetic moment of an atom is pointing in the direction opposite to the magnetic field (Vmag > 0), the atom is repelled from regions with high magnetic fields; it is then in the metastable weak-field seeking state. In this case, minima o f the modulus of the field correspond to potential minima. Because a minimum of the modulus of the magnetic field in free space is not forbidden by the Earnshaw theorem, traps of this type are most c o m m o n for neutral atom trapping. Losses from the traps are a potential problem (see Sect. V), especially when non-adiabatic transitions to the energetically lower high-field-seeking states become likely in regions of low or even vanishing fields.
A. 1. Kepler guide A possible realization of a trap for an atom in the strong-field-seeking state is a current-carrying wire with the atom orbiting around it (Vladimirskii, 1961; Schmiedmayer, 1992, 1995a,b; Schmiedmayer and Scrinzi, 1996a,b; Denschlag, 1998; Denschlag et al., 1999b). The interaction potential is given by 2
Vmag = - / I / - B = -
~
1 Iw-%'la,r
(2)
where Iw is the current through the wire, % is the azimuthal unit vector in cylindrical coordinates, and /z0 = 4 ; r m m G / A is the vacuum permeability. This potential has the 1/r form of a Coulomb potential, but the coupling / u - B is vectorial. Using the adiabatic approximation, Vmag corresponds to a 2-dimensional scalar ( l / r ) potential, in which atoms move in Kepler orbits 3. In the quantum regime, the system looks like a 2-dimensional hydrogen atom in a (nearly circular) Rydberg state. The wire resembles the "nucleus" and
l The Earnshaw theorem can be generalized to any combination of electric, magnetic and gravitational fields (Wing, 1984; Ketterle and Pritchard, 1992). 2 This and all other expressions for magnetic and electric fields in this section are given in the limit of an infinitely thin wire, unless stated otherwise. 3 From corrections to the adiabatic approximation to the next order, we obtain an effective Hamiltonian for the orbital motion of the atom where the Coulomb-like binding potential is corrected by a small repulsive 1/r2 interaction (Shapere and Wilczek, 1989; Aharonov and Stern, 1992; Stern, 1992; Littlejohn and Weigert, 1993; Schmiedmayer and Scrinzi, 1996a,b). As a result, the Keplerlike orbits show an additional precession around the wire. A very similar potential can be realized for small polar molecules with a permanent dipole moment interacting with the electric field of a charged wire (Sekatskii and Schmiedmayer, 1996).
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MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP 267
FIG. 1. Guiding neutral atoms using a current carrying wire. (a) Guiding the atoms in their strong .field seeking state as they circle around the wire. (b) Atoms in the weakfield seeking state can be held in a 2-dimensional magnetic quadrupole field which is created by adding a constant bias field to the wire field. Typical trajectories of atoms are shown on the right-hand side of the figure.
the atom takes the place of the "electron ". Considerable theoretical work has been published on the quantum mechanical treatment of this system showing a hydrogen-like energy spectrum (Pron'kov and Stroganov, 1977; Blfimel and Dietrich, 1989, 1991; Voronin, 1991 ; Hau et al., 1995; Burke et al., 1996; BergSorensen et al., 1996) with a characteristic quantum defect (Schmiedmayer and Scrinzi, 1996a,b). The magnetic field, the potential, and typical classical trajectories are presented in Fig. la.
A.2. Side guide Originally, Frisch and Segr6 (1933) presented the idea that a straight currentcarrying wire (Iw) and a homogeneous bias field (Bb) pointing in a direction orthogonal to the wire form a quadrupole field with a well-defined 2-dimensional field minimum (Fig. lb). The bias field cancels the circular magnetic field of the wire along a line parallel to the wire at a distance r0 =
~-~)
{ ~to
s Bb
(3)
Around this line the modulus of the magnetic field increases in all directions and forms a tube with a magnetic field minimum at its center. Atoms in the weakfield seeking state can be trapped in this 2-dimensional quadrupole field and can
268
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Fie. 2. Upper left: potential for a side guide generated by one wire and an external bias field perpendicular to the wire direction. The external bias field can be replaced by two extra wires (lower left). Upper right: field configuration for a two-wire guide with an external bias field perpendicular to the plane containing the wires. This external bias field may also be replaced by surface mounted wires (lower right).
be guided along the side of the wire, i.e. in a side guide. At the center of the trap the magnetic field gradient is dr ro
Iw
(4)
ro
As long as the bias field is orthogonal to the wire, the two fields cancel exactly, and trapped atoms can be lost due to Majorana transitions between trapped and untrapped spin states (see Sect. V.A). This problem can be circumvented by adding a small B-field component Bip along the wire direction which lifts the energetic degeneracy between the trapped and untrapped states. This potential is conventionally called a Ioffe-Pritchard trap (Gott et al., 1962; Pritchard, 1983; Bagnato et al., 1987). At the same time, the potential form of the guide near the minimum changes from linear to harmonic. The guide is then characterized by the curvature in the transverse directions dT 2 r~ =
-~-
gipI 2
T2 g i p .
(5)
In the harmonic oscillator approximation, the trap frequency is given by
2Jr
2:r V
M
-ff~r2 J (3( --r0
m-Oip '
(6)
where M is the mass of the atom. When mounting the wire onto a surface, the bias field has to have a component parallel to the surface in order to achieve a side guide above the surface. The bias field can be formed by two additional wires on each side of the guiding wire. The direction of the current flow in these wires has to be opposite to the current in the guiding wire (Fig. 2). This is especially interesting because the wires can be mounted on the same surface (chip), and a self-sufficient guide can be obtained.
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MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP 269
A.3. Two-wire guides A.3.1. Counter-propagating currents. A different way to create a guide is by using two parallel wires with counter-propagating equal currents Iw with a bias field which has a component Bb orthogonal to the plane containing the two wires (Fig. 2) (Thywissen et al., 1999a). The important advantage of this configuration is that the two wires and therefore the atom guide can be bent in an arbitrary way in the plane perpendicular to the bias field, whereas the single-wire guide direction is restricted to angles close to the line perpendicular to the bias field. If there is an additional bias field Bip applied along the wires, a Ioffe-Pritchard guide is obtained. Again, two added wires can replace the external bias field (see Sect. IV for an example of an experimental implementation). The field generated by the wires compensates the bias field Bb at a distance
ro=
-Y-
1,
(7)
where d is the distance between the two wires. When Bb > 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)
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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).
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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
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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.
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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.
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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
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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
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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).
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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. =
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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
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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.
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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
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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
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(a)
/
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(b)
2so 200 150 100 50
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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.
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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
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12 10
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.
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0.1
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5
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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.
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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 (<
Vguid(r) = AKZ(Br)- (
1 ) 22~
4:re0
r2
,
(15)
where A and B are constants that depend on specifics of the optical fiber as well as on light power, wavelength and atomic properties (Batelaan et al., 1994). K0 is the modified Bessel function of the second kind. Figure 13 shows a typical example of such a potential. Cold atoms are bound in radial direction by the effective potential but free along the z-direction, the direction of the charged optical fiber. 2
~
T
1
g o ~
-2
-
-4.
~
\ ~
0.5
/ ~
............. Van der Waals ...... evanescent wave electric interaction
1.0 radius
1.5
-
2.0
[l~m]
FIG. 13. Typical radial potential for a neutral lithium atom trapped around a charged (5 V) optical quartz fiber (diameter 0.5 ~tm) with 1-mW light and a detuning of A/F = 3 • 105. The attractive potential (1/r 2) is created by the interaction of the induced dipole moment with the electric field of the charged fiber. The repulsion is due to the evanescent wave from blue-detuned light propagating in the fiber. Close to the wire surface the Van der Waals interaction becomes important.
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C. TRAPS AND GUIDES FORMED BY COMBINING THE INTERACTIONS
C. 1. Charged wire on a mirror As we have seen above, a static charged wire alone cannot form the basis for stable trapping. Cylindrical solutions such as the charged light fiber have the disadvantage that they cannot be m o u n t e d on a surface. An alternative solution would be to mount a charged wire onto the surface o f an atom mirror. The combination o f the attractive 1/r 2 potential with the repulsive potential o f the atom mirror s Vm(z) gives:
Vguid(r)=Vm(Z)--(
1 ) 22~
4;re0
r2
,
(16)
where z is the height above the mirror and r the distance from the wire. This creates a potential tube for the atoms as shown in Fig. 14 which can be viewed as a wave guide for neutral atoms. Typical parameters for guides f o r m e d by a magnetic mirror and a charged wire are given by Schmiedmayer (1998). They can be very similar to the magnetic guides discussed in Sect. II.A. Using typical mirror parameters (Roach et al., 1995; Sidorov et al., 1996), one can easily achieve deep and narrow guides with transverse level spacings in the kHz range for both light (Li) and heavy (Rb) atoms. In a similar fashion microscopic traps can be created by mounting a charged tip (point) at or close beneath the atom mirror surface. A point charge on the surface o f an atom mirror creates an attractive 1/r 4 interaction potential:
Vpol(r) = -
1 ) 2aq 2 8yt60 r4
(17)
where q is the tip charge. Together with the atomic mirror it forms a microscopic cell for the atoms. It can be viewed as the atom-optical analog to a quantum dot (Schmiedmayer, 1998; Sekatskii et al., 2001). This approach o f combining a charged structure with an atom mirror is compatible with well-developed nanofabrication techniques. This opens up
8 There are two main types of atom mirrors: The first type utilizes evanescent waves (e.g. above the surface of a reflecting prism) of blue-detuned light which repels the atoms. Here the potential takes the form Vm(z) = VOexp(-tCmZ) where trm is of the order of the light wave number and z is the distance from the mirror (Cook and Hill, 1982). The second type is based on a surface with alternating magnetic fields. Here, ~ = 2;r/lcm is the periodicity of the alternating magnetic field. The approaching atom experiences an exponentially increasing field, and consequently the weakfield seekers are repelled (Opat et al., 1992; Roach et al., 1995; Sidorov et al., 1996; Hinds and Hughes, 1999).
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MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP 287
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Flo. 14. Typical potential for a neutral atom guide. The attractive potential (1/r 2) is created by the interaction of the induced dipole moment in the electric field of the charged wire mounted directly on the surface of an atomic mirror. The action of the atomic mirror (evanescent wave or magnetic mirror) prevents the atom from reaching the surface and creates a potential tube close to the surface illustrated by the contour graph. The two adjacent plots give the potential in a direction orthogonal to the charged wire and orthogonal to the mirror surface. Distances are given from the location of the charged wire and the surface of the atom mirror.
a wide variety of possibilities ranging from curved and split guides to interferometers or even complex networks.
C.2. Combined electric-magnetic state-dependent traps The magnetic guides and traps (Sect. II.A) can be modified by combining them with the electric interaction, thereby creating tailored potentials depending on internal (e.g. spin) states. For example, supplementary electrodes located between independent magnetic traps can be used to lower the magnetic barrier between them by the attractive electric potential the electrodes create. Since the magnetic barrier height depends on the magnetic substate of the atom, whereas the electric potential does not, this allows state-selective operation. This is especially interesting since it can lead to implementing quantum information processing with neutral atoms in microscopic trapping potentials where the logical states are identified with atomic internal levels (see Sect. VI). A simple example, showing such a controllable state dependence, is a magnetic wire guide approached by a set of electrodes (Fig. 15a). Applying a high voltage to the electrodes introduces an electrostatic potential which provides confinement along the direction parallel to the magnetic side guide, and also shifts the trapping minimum towards the surface, possibly breaking the magnetic
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FIG. 15. State-dependent potential: (a) top view of an actual chip design; the wire in the center is used as a side guide wire, the additional electrodes create a spatially oscillating electric field providing confinement also along the wire. The contour plot shows a typical potential configuration for 7Li a t o m s in the IF = 2, m F = 2) magnetic substate using experimentally accessible parameters. Dark areas correspond to attractive potentials, the trap minima are located 50 gm above the surface. (b,c) The side views show that only one state (IF = 2, m F = 2)) is trapped in the combined potential (b), while the other (IF = 1, m F = - 1 ) ) is not, because the weaker magnetic barrier to the surface is compensated by the attractive electric potential (c). The parameters used in a simulation with the electromagnetic field solver MAFIA were lw = 500 mA, B b = 20 G for the side guide and a voltage of 600 V on the electrodes. potential barrier in the direction p e r p e n d i c u l a r to the surface itself. T h e charge can be adjusted in such a way that d e p e n d i n g on the strength o f the m a g n e t i c barrier created by the wire current, the a t o m s either i m p a c t onto the surface or are trapped above it. Since the strength o f the m a g n e t i c barrier d e p e n d s on the m a g n e t i c substate o f the a t o m or, m o r e precisely, d e p e n d s linearly on the q u a n t u m n u m b e r mF, this can be e x p l o i t e d to f o r m a state-selective m a g n e t i c trap (Fig. 15b,c). C.3. The e l e c t r i c m o t o r
In general, electric fields are always p r e s e n t in m a g n e t i c wire traps since an electric potential difference is n e e d e d to drive a c u r r e n t t h r o u g h a wire w i t h finite resistance. For large wires, the voltages in q u e s t i o n are low and if the distances o f the a t o m s f r o m the wire are large e n o u g h , the attractive electric interaction
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FIG. 16. Two-wire guide configuration with currents of 1 A running in opposite directions with a vertical bias field of 150 G. The combined magnetic and electric potential is shown in contour plots perpendicular to the wire and along the wire at the minimum height (inset). The parabolic potential shape offers the possibility to drive the atoms (87Rb) along the wire. In the example, the voltages applied to the wires are chosen to be 0 V with respect to ground in the wire center.
can be neglected. However, for micron-sized wires, one finds that if the currentcarrying wire becomes long, at some point the voltage is strong enough to create a significant driving force for the atoms or even to destroy the traps. On the other hand, one can actually exploit this effect and turn it into an 'electric motor' by using the electric potential gradient inside the magnetic minimum to accelerate and decelerate the atoms at will. Figure 16 illustrates the mechanism used for the motor for the example of a two-wire guide with a vertical bias field. The wires carry counter-propagating currents, and the electric interaction is zero in the middle of the guide (see inset) where both wires have the same voltage. By adding a homogeneous electric potential relative to ground, the zero electric field point may be moved at will to achieve any acceleration or deceleration rate. A constant acceleration is obtained when the zero electric field point is maintained at a constant distance from the position of the atoms. D. MINIATURIZATION AND TECHNOLOGICAL CONSIDERATIONS To achieve very robust and highly controlled atom manipulation one would like to localize atoms in steep traps or guides which can be fabricated with high precision. The large technological advances in precise nanofabrication, with the
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achievable size limit on chips smaller than 100 nm, makes the adaptation of these processes for mounting the wires onto surfaces very attractive. D. 1. Miniaturization The main motivations behind miniaturization and surface fabrication are: 9 Large trap level spacings help to suppress heating rates. To achieve the necessary large trapping gradients and curvatures with reasonable power consumption, miniaturization is unavoidable (see Sect. II.D.4). 9 The tailoring resolution of the potentials used for atom manipulation is given by the resolution of the fabrication of the structures used. It is, for example, important for the realization of atom-atom entanglement by controlled collisions as suggested by Calarco et al. (2000) (see Sect. VI) to reduce the distances between individual trapping sites to the micron regime or below. This would be virtually impossible with (large) free-standing structures. 9 Nanofabrication is a mature field which allows one to place wires on a surface with great accuracy (< 100 nm). Surface-mounted structures are very robust and the substrate serves as a heat sink allowing larger current densities (see Sect. II.D.4). In addition, nanofabrication allows parallelism in production of manipulating elements (scalability). 9 Nanofabrication also allows us to contemplate the integration of other techniques on the chip (see Sect. VI for details).
D.2. Finite size effects The formulae presented in Sects. II.A to II.C are exact only for infinitely small wire cross sections. In the case of a physical wire with a finite cross section, they are a good approximation only as long as the height above the wire is greater than the width of the wire. For experiments requiring a trap height smaller than the width of the wire, finite size effects have to be taken into account. In Fig. 17, we present examples of calculations showing how the trap gradient is limited by finite size wires. One clearly sees that at trap heights of the order of the width of the wire the resulting gradient starts to deviate from the expected value. The effect is small for wires with a square cross section, while it becomes considerably more important when rectangular wires with high ratios of width to thickness are used. D.3. Van der Waals interaction The Van der Waals interaction becomes important at distances of the order of a few 100nm from the surface. The interaction can be strong enough to
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Distance from wire surface [in units of d] FIG. l 7. Deviations from the field of an infinitely thin wire become important as the surface of a physical wire is approached. The plot shows the trap gradient for a side guide (see Sect. II.A.2) when differently shaped wires are used. The solid line corresponds to a circular cross section as a reference since the field outside the wire equals that of an infinitely thin wire at the wire center. A wire with a square cross section (dotted line) shows very small deviations, while broad and thin wires (dashed lines) deviate more and more as the thickness/width ratio decreases. Here, all wires are chosen to have the same cross section d 2. Therefore, the widths of the rectangular wires are 2d and x/]--dd = 3.2d for the ratios 1:4 and 1:10, respectively.
significantly alter the trapping potentials (Grimm et closer than 100 nm from the surface will be very hard der Waals potential attracts the atoms to the surface (in the non-retarded regime where the distance d is wavelength).
al., 2000). Traps much to achieve since the Van and increases with 1/d 3 smaller than the optical
D. 4. Current densities A limiting factor in creating steep traps and guides is the maximally tolerable current density of a current-carrying structure. Considering a side guide potential created by a wire with finite width d and a constant thickness, the highest possible gradient is achieved at a distance from the wire comparable with d. The bias field needed for such a trap is given by the ratio of the m a x i m u m current that can be pushed through the wire and d; therefore the bias field is proportional to the m a x i m u m current density j. This leads to the conclusion that the highest possible gradient is given by j / d which favors smaller wires. If a square wire cross section d 2 is assumed, the m a x i m u m gradient is proportional to j . Even in this case, smaller d will allow for larger gradients b e c a u s e j has been observed to increase with smaller wire cross sections. The drive for smaller width is stopped
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at a distance of about 100 nm where surface decoherence effects (see Sect. V) and Van der Waals forces may be too strong to endure.
D.5. Multi-layer chips Last, one should also note that as more complex operations are demanded from the atom chip (see Sect. VI), it will have to move on from a 2-dimensional structure into a 3-dimensional structure in which not only current- and chargecarrying wires are embedded, but also light elements and wave guides. These highly complex devices will force upon the fabrication a whole range of material and geometrical constraints.
III. Experiments with Free-Standing Structures The basic principles of microscopic atom optics have been demonstrated using free-standing structures: current-carrying and charged wires. The interaction potentials are in general shallow, typically only a few mK deep. Hence experiments use cold atoms from a MOT or a well collimated atom beam (even the moderate collimation of 1 mm over 1 m results in a typical transverse temperature of <1 mK). Free-standing wire structures can be installed close to a standard six beam MOT without significantly disturbing its operation (as long as the wire is thin enough), and offer large optical access which has advantages when probing the dynamics of the atoms and their spatial distribution within the wire potentials. They have the disadvantages that they are not very sturdy, they deform easily due to external forces, and they cannot be cooled efficiently to dissipate energy from ohmic heating. This limits the achievable confinement and the potential complexity of wire networks. Nevertheless there are some special potentials which can only be realized with free-standing wires.
A. MAGNETIC INTERACTION
As discussed in Sect. II.A there are two possibilities for magnetically trapping a particle with a magnetic moment: traps for strong field seekers and traps for weak field seekers. In the following we describe experiments with magnetic microtraps which are based on small, free-standing wires or other magnetic structures. Typical wire sizes range from 10 ~tm to a few mm and the wires carry electrical currents of up to 20 A. All experiments but the first example start with a conventional MOT of alkali atoms (lithium or rubidium) which is initially situated a few mm away from the magnetic field producing structures. This distance prevents the atoms in the MOT from coming into contact with the
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structure surface where they would be adsorbed. It also provides the necessary optical access for the MOT laser beams. To load the magnetic wire traps and guides, the MOT laser light is simply switched off and the magnetic trap fields are turned on. The loading rate into the miniature magnetic traps has been enhanced in some experiments (a) by optically pumping the unpolarized MOT atoms to the right trapping state (Key et al., 2000); (b) by first loading the MOT atoms in a size-matched magnetic trap which is then further adiabatically compressed (Vuletic et al., 1996, 1998; Key et al., 2000; Fortagh et al., 1998; Haase et al., 2001); (c) by moving the MOT closer to the trapping region shortly before the light is turned off, which can be done with an additional magnetic bias field (Denschlag et al., 1999b). In this way the efficiency of transferring the atoms into the miniature magnetic traps reached between 1 and 40 %. In general the spatial distribution of the trapped atoms was imaged with a CCD camera by shining a resonant laser beam onto the atoms and detecting its absorption or the atomic fluorescence.
A. 1. Magnetic strong-field-seeking traps." the Kepler guide A magnetic strong-field-seeker trap for cold neutral atoms was demonstrated in two experiments: in 1991 by guiding an effusive beam of thermal sodium atoms (mean v e l o c i t y - 6 0 0 m / s ) along a 1-m long current-carrying wire (Schmiedmayer, 1992, 1995a,b) and in 1998 with cold lithium atoms loaded from a MOT (Denschlag et al., 1999b). The setup of the beam experiment is given in Fig. 18. The atom beam is emitted from a 1-mm diameter nozzle in a 100~ oven and is collimated to 1 mrad. Introducing a small bend in the wire (-1 mrad), one can guide some of the atoms along the wire around the beam stop. The atomic flux was measured with a hot wire detector. The guiding wire was 150 ~tm thick and carried 2 A of electrical current. In the second experiment, lithium atoms were cooled in a MOT (1.6mm diameter FWHM) to about 200 ~tK (which corresponds to a velocity of about 0.5 m/s). By shifting the MOT onto a 50 ~tm thick wire and releasing the atoms from the MOT, about 10% of the unpolarized atomic gas could be trapped magnetically in orbits of about 1 mm diameter around the wire that carried about 1 A of current. Monte Carlo calculations indicate that by optically pumping the atoms and optimizing the trap size and current through the wire, it should be possible to guide over 40% of the atoms from a thermal cloud with the Kepler guide. The loading efficiency is limited to this amount, because atoms in highly eccentric orbits hit the wire and are lost. The bound atoms are guided along the wire corresponding to their initial velocity component in this direction. Consequently, a cylindrical atomic cloud forms that expands along the wire. After 40 ms of guiding, the atoms typically had propagated over a 2cm distance along the wire (see left-hand panel of
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FIG. 18. (a) Experimental setup: The schematics at the bottom show in detail the relative geometric arrangement between the apertures, the movable beam shutter used to bend the wire, and how the wire is mounted. (b) Guiding of Na atoms along the 1-m long, 150-~tm diameter tungsten wire (at detector position 0 indicated by the vertical line). Experimental count rates, n(I)- n(O) (left), and Monte Carlo simulations (right), are shown for 0.0, 0.50 and 1.00mrad bends in the wire. The different symbols represent currents of 0.5 A (circles), 1.0 A (diamonds), 1.5 A (crosses) and 2.0 A (triangles) through the wire. The thick line shows the fraction of atoms of the direct beam that gets to the detector when no current is on (right-hand vertical axis). Its form corresponds to the shadow of the bender that is cast onto the detector. Fig. 19). For long guiding times the bound atoms leave the field of view, and the fluorescence signal of the atoms decreases. The top left view images of Fig. 19 show a round atom cloud that is centered on the wire suggesting that atoms circle around it. By studying the ballistic expansion of the bound atoms after switching off the guiding potentials, the m o m e n t u m distribution of the guided atoms can be extracted. The center panel of Fig. 19 shows a picture sequence demonstrating how the atomic cloud expands as a function of time. Starting from a welllocalized cylindrical cloud of guided atoms at t = 0 the spatial atomic distribution transforms into a doughnut-like shape. This shows that there are no zero-velocity atoms in the Kepler guide. In order to be trapped in stable orbits around the wire the atoms need sufficient angular m o m e n t u m and therefore velocity. Atoms with too little angular m o m e n t u m hit the wire and are lost. Guiding in the Kepler guide is very sensitive to the presence of uncompensated bias fields. Such additional magnetic bias fields, even if homogeneous, destroy the rotational symmetry of the Kepler potential and angular m o m e n t u m is not conserved anymore. Over the course of time, the Kepler orbits become increasingly eccentric and thus finally hit the current-carrying wire leading to loss, which was confirmed by Monte Carlo calculations. The right-hand panel of Fig. 19 shows the results of an experiment investigating the dependence of the
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FIG. 19. Left: guiding of atoms along a current-carrying wire in their strong field seeking state (Kepler guide). Pictures of the atomic clouds are shown, taken in axial and transverse directions with respect to the wire. For times shorter than 15 ms the expanding cloud of untrapped atoms is also visible. The location of the wire is indicated by a line (dot). The pictures show a 2-cm long section of the wire that is illuminated by the laser beams. Center: Atomic distribution after free expansion of 0 to 9 ms for atoms that have been guided in Kepler orbits around the wire. The expanded cloud is doughnut-shaped due to the orbital motion of the atoms around the wire. Right: Experimentally measured stability of the Kepler guide as a function of the magnitude of bias fields. The signal is proportional to the number of atoms trapped in the guide after an interaction time of 20 ms. magnetic trap stability on the magnetic bias field. The remaining atom number in the Kepler guide was measured after 20 ms interaction time. It clearly decreases with increasing bias field strength: the larger the bias field, the faster the atoms get lost (Denschlag, 1998). In case of a weak disturbance the orbits can be stabilized by an additional 1/r 2 potential which leads to a precession of the orbits.
A.2. Magnetic weak-field-seeking traps and guides The development of miniature weak-field-seeker traps, as discussed in Sects. II.A.2 and II.A.3, lays the foundations of miniaturized atom optics on chips. Here and in the following sections we restrict our discussion explicitly to experiments with free-standing structures. Surface-mounted guides and traps are discussed in Sect. IV. In the following experiments the circular symmetric magnetic field of a straight current-carrying wire is combined with a magnetic bias field as described in Sect. II.A.2. The two fields cancel each other along a line that is parallel to the wire creating a magnetic field minimum (side guide). In the simplest case, the bias field can be created by an additional wire (Fig. 20a) (Fortagh et al., 1998) or by an homogeneous external field (Fig. 20b) (Denschlag, 1998; Denschlag et al., 1999b). Four wires also create a 2-dimensional quadrupole field (Fig. 20c) (Key et al., 2000). The experiments of the group of C. Z i m m e r m a n n (Fortagh et al., 1998) used additional endcap ('pinch') coils (see Fig. 20a) to confine the atoms also in the direction along the wire. They succeeded in adiabatically transferring and
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FIG. 20. Three realizations of magnetic quadrupole traps with straight wires. (a) Trap realized by Fortagh et al. (1998) with a thin wire (50~tm) glued onto a thick wire (1 mm). The current through both wires flows in opposite directions. (b) A homogeneous bias field is combined with a single straight wire (Denschlag, 1998; Denschlag et al., 1999b). (c) Four wires with alternating current direction produce a quadrupole field minimum in the center. In the experiment the four wires were embedded in a silica fiber (Key et al., 2000). (d) Images of atoms in guide (b).
compressing the magnetic t r a p - reaching a relatively high transfer efficiency of 14% from the MOT into a microtrap without losing phase space density. In experiments in Innsbruck (Denschlag, 1998; Denschlag et al., 1999b) and Sussex (Key et al., 2000) (Figs. 20b and c, respectively) cold atoms released from a MOT were guided along the wires at a distance of one to two centimeters (Fig. 20d). In addition, the vertical Sussex experiment used one bottom pinch coil to confine the falling atoms from exiting the guide. The atoms bounced back and were imaged at the top exit. By choosing appropriate bias field strengths and wire currents, a wide range of traps with different gradients have been realized, and the scaling properties (see Sect. II.A.4) were studied. With a fixed trap depth (given by the magnitude of the bias field Bb) the trap size and its distance from the wire can be controlled by the current in the wire. The trap gets smaller and steeper (gradient ~ B 2 / I ) for decreasing the current in the wire, which was confirmed experimentally (Denschlag, 1998; Denschlag et al., 1999b). For example, a trap with a gradient of 1000 G/cm can be achieved with a moderate current of 0.5 A and an offset field of 10 G. The trap is then be located 100 ~tm away from the wire center. A different weak-field-seeker trap has been experimentally realized by placing a current-carrying wire right through the minimum of a magnetic quadrupole field (Denschlag, 1998; Denschlag et al., 1999a). If the wire is aligned along the direction of the symmetry axis of the quadrupole field, a ring-shaped potential is obtained with a non-vanishing minimum field strength (see Sect. II.A.4 and Fig. 5a).
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A.3. Beam splitters Although free-standing wire experiments are certainly limited in their architectural complexity because of mechanical stability, some variations of the straight wire geometry have been explored. By combining two free-standing wires one can form a "Y" or fork, which can be used as an atomic beam splitter (see Fig. 21) (Cassettari et al., 1998; Denschlag et al., 1999a). Choosing an arm of the fork through which an electrical current is conducted, the atomic flow can be switched from one arm to the other. If current is sent through both arms, the atom beam is split in two.
FIG. 21. Atomic beam switch for guided atoms using a "Y"-shaped current-carrying wire. By controlling the current through the arms, one can send cold lithium atoms along either arm or split the beam in two. The images here show the switch operated in the Kepler guide mode and the "weak-field-seeker" mode.
A.4. Free-standing bent wire traps Experiments with free-standing wires that are bent in shape of a "U" or "Z" have been reported by Denschlag et al. (1999a), Haase (2000) and Haase et al. (2001). Bending the wire has the effect of putting potential endcaps on the wire guide, which turns it into a 3-dimensional weak-field-seeker trap (see Sect. II.A.4.2). A simple Z-wire trap achieves trapping parameters similar to the ones currently used in BEC production, here, however, with moderate currents and very low power consumption (see Sect. IV.C.4). In their experiment, Haase et al. used a 1-mm thick copper wire, with the central bar being about 6 mm long. The wire can carry 25 A without any sign of heating. Figure 22c shows the scaling
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FIG. 22. (a) Schematic description of the experiment. Camera 1 is looking along the central bar of the magnetic trap and camera 2 along the leads. In addition to the two laser beams shown in the figure, there is the third MOT beam parallel to the central bar. (b) The Z-wire held by two Macor blocks is mounted on a flange. (c) The cloud of trapped atoms monitored by camera 1. By changing the bias field Bb from 5 to 52 G, the trap size and position change. Also, the trap frequency increases from 30 to 1600Hz. The experiment confirms the predicted scaling laws concerning trap distance, frequency and bias field.
properties of the Z-trap. The atomic cloud can be compressed by raising the bias field or by lowering the wire current.
A.5. The tip trap Vuletic et al. (1996, 1998) have demonstrated a miniature magnetic quadrupole trap (the tip trap) by mounting small coils on a combination of permanent magnets and ferromagnetic pole pieces (see Fig. 23). In this way they exploited the fact that for a given magnetic field Bo the maximum possible field gradient scales like Bo/R where R is the geometric size of the smallest relevant element. The central element of the tip trap is a 0 . 6 5 m m steel pin of which one tip is sharpened to a radius of curvature of 10gm. Thus with R = 1 0 g m and Bo -- 1000G the magnetic field gradient exceeded 105 G/cm. Working with lithium atoms, this gradient implies a ground-state size of the atomic wavefunction smaller than the wavelength of the optical transition at 671 nm. The microtrap is loaded by adiabatic transport and compression: The atoms of the lithium MOT are transferred to a volume-matched, but still relatively shallow magnetic potential after turning off the MOT light. By adiabatically changing the currents through the miniature coils the magnetic trap compresses its size by a
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MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP 299
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factor of 6.5 within 100 ms. A total of 3% of the MOT atoms could be transferred to the microtrap at moderate currents of 3 A through the tip trap coils. A.6. Scattering experiments with a current-carrying wire In 1995 the Melbourne group (Rowlands et al., 1995, 1996a,b) performed an experiment where a beam of laser cooled cesium atoms, after being released from a MOT, is scattered off a current-carrying wire. As the atoms pass through the static inhomogeneous magnetic field of the wire they are deflected by a force 27(/tB) dependent on the magnetic substate of the atom (see Fig. 24). 30
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With currents of up to 45 A through the wire, the positions of the atoms in the individual magnetic substates are resolved and deflection angles as large as 25 ~ are observed. State preparation of the atoms using optical pumping increases the number of atoms deflected through essentially the same angle. A. 7. A storage ring f o r neutral atoms
Very recently Sauer et al. (2001) have demonstrated a storage ring for neutral atoms using a two-wire guide (Sect. II.A.3). A pair of wires (separation-840 gm) which forms a ring of 2cm diameter, produces a 2-dimensional quadrupole magnetic field (see Fig. 25). The wires carry currents of 8 A in the same direction which produces a field minimum between the two wires with a field gradient of 1800 G/cm and a trap depth of 2.5 mK for the F = 1, mF = -1 ground state of 87Rb (weak-field seeker). The ring is loaded from a MOT via a similar second two-wire waveguide. The MOT is turned off and the second waveguide is ramped up in 5 ms. Approximately 106 laser-cooled rubidium atoms (longitudinal temperature 3 ~tK) fall 4 cm under gravity along the guide after which they enter the storage ring with a velocity of about 1 m/s. To transfer the atoms to the ring, the current in the guide is ramped off while simultaneously increasing the current in the ring. Using fluorescence imaging the position and the number of the atom cloud can be probed. Up to seven revolutions of the atoms in the ring have been observed.
FIG. 25. (a) Schematic of the storage ring. (b) Cross section of the overlap region. The trap minimum is shifted from between the guide wires to the ring wires by adjusting the current. (c) Contour plot of a two-wire potential. The contours are drawn every 0.5 mK for the wire distance d = 0.84mm and I = 8 A. (d) Successive revolutions in the storage ring. The points represent experimental data, the curve is a theoretical model. Courtesy M. Chapman.
B.
CHARGED WIRE EXPERIMENTS
Two types of experiments have used the 1/r 2 potential (Eq. 12) of a charged
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wire. One investigated the effect of a charged wire in atom interferometry. The other investigated atomic motion in the singularity of the 1/r 2 potential. Here, laser cooled atoms fall into the attractive singularity and are lost as they hit the charged wire. B.1. A charged wire a n d interferometry Shimizu et al. (1992) used a straight charged wire to shift (deflect) the interference patterns of a matter wave interferometer in a Young's double slit configuration. In a recent experiment of the same group (Fujita et al., 2000) (Fig. 26, at right), this work is expanded by combining a binary matter wave
FIG. 26. Experimental set-ups and data for interferometry and holography experiments with charged wires. Left: charged-wire interferometerfor metastable helium. Different voltages applied to the electrodes: the data sets are plotted with a vertical offset. The dotted horizontal lines indicate the zero level for the respective measurements. Courtesy J. Mlynek. Right: Selective atom holography: switching between atomic images "r and "~". For the upper figure the wire array is uncharged, whereas for the lower figure it is electrically charged. The squares in the lower part of each figure are nondiffracted atom patterns. Courtesy E Shimizu. hologram with an array of straight charged wires. By changing the electric potential applied to the electrodes on the hologram the holographic image patterns can be shifted or erased, and it is even possible to switch between two arbitrary holographic image patterns 9. These experiments were performed using laser-cooled metastable neon in the l s3 state. After releasing them from the MOT, the atoms fell under gravity onto a double slit or a binary hologram. A few centimeters further down the atoms formed an interference pattern which was detected by a multi-channel plate (MCP). The binary hologram pattern held an array of 513 regularly spaced parallel wires of platinum on its surface. Each electrode was either grounded or connected to
9 Similarly it was suggested by Ekstrom et al. (1992) that charged patches on a grating can be used to modify the diffraction properties.
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FIG. 27. (a) Two classical trajectories: An atom falls into the 1/r2 singularity of an electrically charged wire if the atomic angular momentum Lz < Lcrit. If Lz > Lcrit it scatters and escapes from the singularity. (b) When moved onto the wire the atom trap decays exponentially, as can be seen by monitoring the atomic fluorescence signal. Charging the wire (100 V ~ 6.4 pC/cm) creates an attractive 1/r2 potential and enhances the decay rate. Inset: experimental steps. Loading of the trap, shifting it onto the wire and observing its decay. (c) Dependence of the trap decay rates on the wire charge for different wire thicknesses. The decay rate for uncharged wires is proportional to their actual diameters. For increasing charges the absorption rate becomes a linear function of the charge, a characteristic of the 1/r2 singularity. The slope is independent of the wire diameter. a terminal. The width and the spacing of each wire was 0.5 ~tm and the holes for the binary hologram in between the wires were 0.5 ~tm • 0.5 ~tm in size. The electric field E generated between two wires shifted the energy of the neon atom b y - o t E 2 / 2 . When two adjacent electrodes had the same potential, the atoms in the gap were unaffected. If they had different potentials, the atoms accumulated an additional phase while passing through the hole. In an experiment in Konstanz, Nowak et al. (1998) sent a collimated thermal beam of metastable helium atoms onto a charged wire (tungsten, 4 ~tm diameter) where it was diffracted (see Fig. 26, left). 1.3 m further downstream they observed an interferometric fringe pattern which depended on the wire charge and on the de Broglie wavelength. The data agreed well with the theoretical predictions for scattering polarizable particles off a 1/r 2 potential. B.2. A charged wire in gas o f cold atoms: studying a singular p o t e n t i a l
The motion in a 1/r 2 singularity can be studied by placing a cloud of cold atoms in the potential of a charged wire. In this experiment the n u m b e r of cold lithium atoms of a M O T is monitored while the atoms move in the 1/r 2 potential of the wire (Denschlag et al., 1998). At extremely low light levels the M O T acts as a box holding a gas o f atoms. Atoms falling into the attractive 1/r 2 singularity are lost as they hit the wire. This loss m e c h a n i s m leads to an exponential decay of the trapped atom n u m b e r (see Fig. 27b). The corresponding loss rate is characteristic for the 1/r 2 singularity and its
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MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP 303
strength. Atoms with angular momentum Lz < Lcrit (see Eq. (14) in Sect. II.B) fall into the singularity. The loss rate is a linear function of q because Lcrit is proportional to the line charge q and the atoms are uniformly distributed over angular momentum states (see Fig. 27c). This is actually true only for high charges, since for lower q, the finite thickness of the charged wire becomes apparent. The MOT decay rate for an uncharged wire is proportional to its actual diameter. The radii of the wires in the experiments ranged between 0.7 ~tm and 5 ~tm. A detailed analysis of the absorption data reveals that Van der Waals forces also contribute to the atomic absorption rate (Denschlag, 1998). This effect was found to be important for thin wires with diameters of less than 1 ~tm. Hence this system should allow for detailed future studies of Van der Waals interaction and retardation in nontrivial boundary conditions. The 1/r 2 potential would be especially interesting to study in the quantum regime where the de Broglie wavelength of the atoms is much larger than the diameter of the charged wire; the quantization of angular momentum then begins to play a role (Denschlag and Schmiedmayer, 1997). This can be used for example in order to build an angular momentum filter for atoms (Schmiedmayer, 1995a).
IV. Surface-Mounted Structures: The Atom Chip Free-standing structures, as those described in the previous section, are extremely delicate, and one arrives quickly at their structural limit, when miniaturizing traps and guides. Wires mounted on a surface are more robust, can be made much smaller, and heat is dissipated more easily which allows significantly more current density to be sent through the wires. This together with strong bias fields allows for tighter confinement of atoms in the traps. Consequently, ground-state sizes <10nm become feasible. Existing accurate nanofabrication technology provides rich and well established production procedures, not only for conducting structures, but also for micromagnets. Optical elements such as micro-optics, photonic crystals and microcavities can also be included to arrive at a highly integrated device. The small ground-state size of such microtraps implies that we know the exact location of the atom relative to other structures on the surface to the precision of the fabrication process (typically <100 nm), allowing extremely close sites to be addressed individually for manipulation and measurement. We have named nanofabricated surfaces for cold atom manipulation 'atom chips' in reminder of the similarity of these atom-optical circuits to electronic integrated circuits (Folman et al., 2000). In designing atom chips one attempts to bring together the best of two worlds: the well-developed techniques of quantum manipulation of atoms, and the mature world of nanofabrication in electronics and optics, to build complex experiments utilizing the above techniques.
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In the following we describe the atom chip and its present experimental status. Future goals will be addressed in Sect. VI. A. FABRICATION There are many different techniques of atom manipulation which can be integrated into an atom chip. Present atom chip experiments follow a simple scheme based on wires that carry currents or charges. These allow to miniaturize the free-standing devices discussed in Sect. III. We will focus here on these simple integrated structures, leaving issues of further integration to the outlook in Sect. VI. To build an atom chip one has to solve the following problems: first of all, the microstructures have to withstand high current densities and high electric fields. This requires structures with low electric resistance. The material of choice for the wires is gold, though other materials such as copper are also used. For the substrate one wants good heat conductivity with high electric insulation withstanding large electric fields (created at sharp (r ~ l gm) corners even by small voltages), and ease of fabrication. Typical materials are silicon, gallium arsenide, aluminum nitride, aluminum oxide and sapphire (A1203), though glass has also been used. Another requirement lies in the fact that cold atoms have to be collected and then transferred towards the small traps on the chip. If one wants to avoid transferring the atoms from a distant MOT, the chip has to be either transparent or reflecting, to allow lasers to address the atoms from all directions near the surface. Nevertheless, experiments exist in which the atoms have been brought from a distance to a chip (Ott et al., 2001; Gustavson et al., 2002). Presently, atom chips are built mainly using two technologies: thin film hybrid technology, or plain nanofabrication which is the first step of the two-stage hybrid technology.
A.1. Thin film hybrid technology In this approach one starts from an insulating substrate (e.g. sapphire) and patterns, using lithographic techniques, a layout of the desired structure onto a thin (<100nm) metallic layer. In the second stage, the wires are grown by electroplating: Metal ions from a solution are deposited onto the exposed metallic layer, which is now charged. With this process one obtains wires with quite large cross sections (typical structure widths are 3 to 100~tm) that support high currents. However, miniaturization will be limited to a few micron wire width. Furthermore, surface roughness is quite large, which makes such surfaces less suitable for the reflection MOT and atom detection. These drawbacks and the expected shadows from large etchings between wires, have been dealt with successfully by covering the chip with an insulating layer and then with a metallic
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FIG. 28. Electroplating. Left: cross section of the Munich-group chip. The metallic layer on top of the wires gives the chip enhanced surface quality in order to form a mirror MOT. Right: the layout of the chip. The magnetic 'conveyor belt' explained in Sect. II is visible. The wires are connected to the chip pads from the outside by means of wire bonding. Recently, this chip was used to achieve Bose-Einstein condensation. Courtesy J. Reichel. reflection layer (e.g. the Munich chip as shown in Fig. 28) (Reichel et al., 2001). This, however, carries the price of not enabling atoms to be closer than some 20 gm from the wires themselves. A technical advantage of electroplating is that it wastes less gold or copper because one avoids evaporation of large amounts of metal, which mostly cover the evaporation chamber and not the chip. Atom chips fabricated using using this technique have been used sucessfully by the groups at Harvard (M. Prentiss), Munich (J. Reichel and T.W. HS.nsch), JILA, Boulder (D. Anderson and E. Cornell) and Ttibingen (C. Zimmermann).
A.2. Nanofabrication Atom chip structures can also be fabricated into an evaporated conductive layer with state-of-the-art processes used for electronic chips. To the best of our knowledge, this approach is only used by the Heidelberg (formerly Innsbruck) group. In these atom chips a 1-5 gm gold layer is evaporated onto a 0.6-mm thick semiconductor substrate (GaAs or Si). As GaAs or Si tend to leak currents, especially in the presence of light, a thin isolating layer of SiO2 is put between the substrate and the gold layer. The chip wires are defined by 2-10 gm wide grooves from which the conductive gold has been removed. This leaves the chip as a gold mirror that can be used to reflect MOT laser beams (the 10-gm grooves impede the MOT operation only in a very slight way). The mirror surface quality is very high, achieving an extremely low amount of scattered light. The chips were produced at the microfabrication centers of the Technische UniversitS, t Wien and of the Weizmann Institute of Science, Rehovot, see Fig. 29. Atom chips fabricated with this method have the advantage that the structure size is only limited by nanofabrication (<100nm). The drawback is that the conductive layer cannot be too thick. This is due mainly to restrictions on the available thickness of the photoresist used in the process. The thin wires support only smaller currents, and therefore only smaller traps closer to the surface can
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FIG. 29. Nanofabricated atom chips (Heidelberg). Left: a mounted chip, ready to be put into the vacuum chamber. The mechanical clamp contacts to the pads are visible. The mounting also includes cooling in order to remove heat produced by the currents. Center (from top to bottom): details of fabrication and assembly: (i) a chip in the middle of the fabrication process, after some gold has been evaporated and before the photoresist has been removed. The visible wires have a cross section of 1• 1 ~tm2; (ii) an electron-microscope view of the surface: a 'T' junction of a 10-~tm wide wire is visible as well as the 10 ~tm etchings which define it; (iii) typical design of the U- and Z-shaped wires placed underneath the chip to help in the initial loading process; the wires can support >50 A of current in DC operation without degrading a p < 10-11 mbar vacuum. Right: a typical design of an atom chip. On both sides contact pads are visible. The center of the chip is used for loading the atoms, which are then released into the physics areas: on top, a magnetic guide with arrays of electric leads, on the bottom, a spiral formed by two parallel wires enables atom guiding in all directions on the chip.
be built. This d i s a d v a n t a g e can be c o r r e c t e d by a d d i n g larger wires b e l o w the chip surface, as p r e s e n t e d in Fig. 29. At this stage it is hard to j u d g e w h a t is the best fabrication process. T h e r e are still m a n y o p e n questions. For e x a m p l e , is there a sizable difference in the resistivity b e t w e e n e v a p o r a t e d gold and e l e c t r o p l a t e d g o l d ? For a direct c o m p a r i s o n , one w o u l d have to unify all other p a r a m e t e r s such as substrates a n d i n t e r m e d i a t e layers. A n o t h e r q u e s t i o n c o n c e r n s the final fabrication r e s o l u t i o n one w i s h e s to realize. A s s u m i n g one aspires to achieve the s m a l l e s t possible trap height above the chip surface for sake o f low p o w e r c o n s u m p t i o n and h i g h potential tailoring resolution, the limit will be at a h e i g h t b e l o w w h i c h surface i n d u c e d d e c o h e r e n c e b e c o m e s too strong (see Sect. V). This height, t o g e t h e r with the finite size effects d e s c r i b e d in Sect. II, will d e t e r m i n e the fabrication resolution needed. S m o o t h n e s s o f r e s o l u t i o n will also be r e q u i r e d as fluctuating wire widths will c h a n g e the current density and therefore the trap f r e q u e n c y in a way that m a y h i n d e r the transport o f B E C due to potential hills. Finally, as m u l t i layer chips u s i n g m o r e elaborate 3 - d i m e n s i o n a l designs are i n t r o d u c e d e.g. for
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wire crossings and more complicated structures including photonic elements, it may be that conductor layers thicker than a few microns will have to be abandoned. In order to fully exploit the potential of the atom chip in the future, the technology used will have to be such that all elements could be made with a suitable process into a monolithic device. Finally, we note that although usual current densities used in the experiments range between 106 and 107 A/cm 2 (higher with smaller cross sections and depending on pulse time, work cycle and heat conductivity of substrate), densities of up to 108 A/cm 2 have been reported for cooled substrates (Drndi6 et al., 1998). Gold wires have been found to be the best, achieving superior performance even when compared to superconductors.
B. LOADING THE CHIP
In general there are two different approaches to loading cold atoms into the chip traps: (i) Collect and cool the atoms at a different location and transport the cold ensemble to the surface traps. This may be achieved using direct injection from a cold atomic beam coming from a low-velocity intense source (LVIS) (Mfiller et al., 1999, 2000, 2001) or a released MOT whereby the atoms are pulled by gravity (Dekker et al., 2000). Transferring the atoms with magnetic traps has been achieved by (Ott et al., 2001), and a Bose-Einstein condensate (BEC) has been loaded using optical tweezers (Gustavson et al., 2002). (ii) Cool and trap atoms close to the surface in a surface MOT, and transfer the atoms from there to the microtraps on the chip (Reichel et al., 1999; Folman et al., 2000). For this method the atom chip has to be either transparent or reflecting. In the following, we describe experiments performed at Heidelberg (resp. Innsbruck), Sussex and Munich using the second approach. Further on, several experiments using the first approach will also be discussed (see Figs. 37 and 38).
B. 1. Mirror M O T
The first problem to solve is how to obtain a MOT configuration close to a surface. This problem has an easy solution if we recall that a circularly polarized light beam changes helicity upon reflection from a mirror. To the best of our knowledge, the idea of a reflection MOT was first put into practice with a pyramid of mirrors and one beam (Lee et al., 1996), as presented in Fig. 30a. Almost in parallel, a single planar surface with four beams impinging at 45 ~ degrees onto the surface was used, thus realizing an eight beam MOT (Pfau
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FI6. 30. (A) a 'pyramid MOT' is obtained when one single laser beam is retro-reflected by a four-sided pyramid in the center of a magnetic quadrupole. The reflections ensure the correct helicities of the laser beams when the quadrupole field (field lines) has the same symmetry as the pyramid. (B) The mirror MOT is generated from the pyramid by leaving out 3 of the 4 reflecting walls. Two MOT beams (I and II) impinge from opposite directions on the reflecting surface of the atom chip. The correct MOT configuration is ensured together with the magnetic quadrupole field rotated 45 ~ to the atom chip surface as illustrated by the field lines. The magnetic field can be obtained either by a set of external quadrupole coils, or by a U-shaped wire on the chip. Top: the Sussex mirror MOT chip setup with the external quadrupole coils on the mounting, inside the vacuum; two parallel wires embedded in a fiber are positioned on the surface of the mirror, forming a two-wire guide and a time-dependent interferometer (see Sect. II). Two small 'pinch' coils visible at the edges of the mirror provide longitudinal confinement. Courtesy E. Hinds.
et al., 1997; S c h n e b l e et al., 1999; G a u c k et al., 1998; S c h n e b l e et al., 2 0 0 1 ) 1 0 . The surface MOT
most common
today derives f r o m the p y r a m i d MOT. The
M O T b e a m c o n f i g u r a t i o n is g e n e r a t e d f r o m f o u r b e a m s b y r e f l e c t i n g o n l y t w o
10 In another version of this experiment, an evanescent field just above the extremely thin metal surface, formed by light beams impinging on the back of the surface, was used as an atom mirror. This allowed to produce a MOT with reasonable surface induced losses even at the extreme proximity of 100nm from the surface.
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beams off the chip surface (see Fig. 30b) (Reichel et al., 1999; Folman et al., 2000). The magnetic quadrupole field for the MOT can be obtained either by a set of external quadrupole coils, or by superimposing a homogeneous bias field with the field generated from a U-shaped wire on or below the chip ('U-MOT'). External quadrupole coils generate the correct magnetic field configuration if one of the reflected light beam axes coincides with the coil axis. If the U-MOT is used, the reflected light beams must lie in the symmetry plane of the U. Trapping in the U-MOT has the advantage that the MOT is well aligned with respect to the chip and its microtraps. If the mirror MOT is sufficiently far from the surface (a few times the MOT radius), its loading rate and final atom number are very similar to a regular free space MOT under the same conditions (laser power, vacuum, supply of cold atoms, etc.). In agreement with earlier observations using wires, the shadows (diffraction patterns) from the 10~tm etchings in the gold surface of the nanofabricated atom chip do not disturb the MOT significantly (Denschlag, 1998; Denschlag et al., 1999b). Such atom chip mirror MOTs have been loaded from an atomic beam in Innsbruck/Heidelberg (Folman et al., 2000), from the background vapor in Munich (Reichel et al., 1999), and in a double MOT system in Innsbruck/Heidelberg (>108 atoms at lifetime <100 s), using either external coils or the U-wire for the quadrupole field. In addition, at Sussex and Harvard surface MOTs were realized using permanent and semi-permanent (magnetizable cores) magnetic structures. As an example we describe the Innsbruck/Heidelberg lithium setup. Figure 31 (overleaf) shows a top view of the mirror MOT just above the chip with some of its electric connections. For the transfer into the U-MOT, the large external quadrupole coils are switched off while the current in the U-shaped wire underneath the chip is switched on (up to 25 A), together with an external bias field (8 G). This forms a nearly identical, but spatially smaller quadrupole field as compared to the fields of the large coils. By changing the bias field, the U-MOT can be compressed and shifted close to the chip surface (typically 1-2 mm). The laser power and detuning are changed to further cool the atoms, giving a sample with a temperature of about 200 ~tK.
B.2. Transferring atoms to the chip surface
After the U-MOT phase, atoms are cooled using optical molasses, optically pumped and transferred into a matched magnetic trap, typically produced by a thick Z-shaped wire plus bias field. From there atoms are transferred closer and closer to the chip and loaded sequentially into smaller and smaller traps. In general, it is favorable to lower the trap towards the surface by increasing the magnetic bias field. This way the trap depth increases and less atoms are lost due to adiabatic heating during compression. Unfortunately this is not feasible
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FIG. 31. Loading of cold atoms close to the surface of an atom chip. Top left: Picture of the mirror MOT, taken from above; the cloud is visible at the center while the electrical contacts can be seen at the edges. Top right: schematic of the MOT beams and quadrupole coils. Center row: Atoms trapped in the U-MOT created by a current in the large U-shaped wire underneath the chip and a homogeneous bias field. Bottom row: Atoms in a magnetic trap generated by the U-wire field; from left to right, the columns show the top, front and side (direction of bias field) views respectively, the far right column shows the schematics of the wire configuration; current-carrying wires are marked in black. The front and side views show two images: the upper is the actual atom cloud and the lower is the reflection on the gold surface of the chip. The distance between both images is an indication of the distance of the atoms from the chip surface. The pictures of the magnetically trapped atomic cloud are obtained by fluorescence imaging using a short laser pulse (typically < 1 ms).
all the way: Finite size effects limit small traps to thin wires, at the price o f not being able to p u s h high currents. The basic transfer principle from a large wire to a small wire is to first switch on the current for the s m a l l e r trap, and then to r a m p d o w n the c u r r e n t in the b i g g e r trap m a i n t a i n e d by a thicker wire (Fig. 32). F u r t h e r c o m p r e s s i o n is a c h i e v e d by using s m a l l e r and s m a l l e r currents. Care has to be taken that the transfer is adiabatic, especially with r e s p e c t to the m o t i o n o f the p o t e n t i a l m i n i m u m . B y an appropriate c h a n g e o f the bias field, the c o m p r e s s i o n o f the a t o m s in the shrinking trap can be p e r f o r m e d v e r y smoothly. T r a n s f e r r i n g into
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Fl6. 32. Principle of compressing and loading wire guides. The position of the surface-mounted wires and equipotential lines of the trapping potential are shown. Top row: the transfer from two large 200 ~tm wires to one small 10 ~tm wire. In (a)-(c) the current in the small wire is constant at 300 mA and the bias field is constant at 10 G. The current in the two large wires is decreased from 2 A in each wire to zero. This transfers the atoms to the small wire. (d) By increasing the bias field the trap can be compressed further. Bottom row: the transfer from one large 200 ~tm wire to one small 10 ~tm wire. In (a)-(d) the current in the large wire drops from 2 A to zero. The thick line shows how the trap center moves during transfer. A much weaker confinement during transfer is obtained in this configuration.
m o r e c o m p l i c a t e d potential configurations one has to avoid the o p e n i n g o f escape routes for the trapped atoms. For an adiabatic transfer o f relatively hot atoms, the m a i n loss is due to heating: w h e n c o m p r e s s i n g by lowering the current, high-lying levels m a y eventually spill over the potential barrier. Significant loss occurs if the trap depth is m u c h smaller than 10 times the t e m p e r a t u r e o f the atomic ensemble. Other loss m e c h a n i s m s are described in Sect. V. For t h e r m a l clouds, typical achieved transfer efficiencies from the M O T to the m a g n e t i c chip trap can be as high as 60%. As a detailed e x a m p l e we describe the loading o f the first o f the I n n s b r u c k experiments ( F o l m a n et al., 2000). After a c c u m u l a t i n g atoms in a m i r r o r M O T and transferring t h e m to the U-MOT, the laser b e a m s are switched off and the quadrupole field generated by the U - s h a p e d wire below the chip surface
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FIG. 33. Compressing a cloud of cold atoms on an atom chip: Top row: view from the top; center row: front view; bottom row: side view. The first three columns show atoms trapped on the chip with the two U-shaped wires. The compression of the trap is accomplished by increasing the bias field. The last row displays images from a Z-trap created by 300 mA current through the 10 gm wire in the center of the chip. The pictures of the magnetically trapped atomic cloud are obtained by fluorescence imaging using a short (<1 ms) molasses laser pulse. serves as a magnetic trap for weak-field seekers (Fig. 3 l: U-trap). The magnetic trap is lowered further towards the surface of the chip by increasing the bias field. Atoms are now close enough to be trapped by the chip fields. Next, a current of 2 A is sent through two 2 0 0 g m U-shaped wires on the chip, and the current in the U-shaped wire located underneath the chip is ramped down to zero. This procedure brings the atoms closer to the chip, compresses the trap considerably, and transfers the atoms to a magnetic trap formed by the currents on the chip surface. This trap is further compressed and lowered towards the surface (typically <100~tm) by increasing the bias field (Fig. 33). From there the atoms are transferred to a microtrap created by a 10 g m Z-shaped wire. In the lowest height and most compressed trap achieved to date, a 1 x 1 ~tm2 Z-shaped wire is used with a current of 100 mA (Heidelberg). With a bias field of 30 G the atoms are trapped at a height of about 7 ~tm above the surface and at an angular oscillation frequency co ~ 2Jr x 200 kHz (magnetic field gradients of 50 kG/cm) for several tens of ms (see Eq. 6 for typical trap frequencies). At such a small trap height several problems come into play: First, with the present
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2 mm distance between the bends of the Z-shaped wire, the Ioffe-Pritchard configuration is nearly lost and one is left with a single-wire quadrupole field where atoms can suffer Majorana flips. Two easy remedies would involve smaller Z lengths or a slight tilt of the bias field direction. Second, the trap is so tight that the number of atoms that survive the transfer and compression is small. This limitation should not be applicable in the case of a BEC as there are no high-lying states where atoms run the risk of spilling over the finite trap barrier. A third problem has to do with the observation of the atoms: even with negligible stray light from surface scattering or blurring by atomic motion, it is found that direct observation of extremely tight traps close to the surface (<20 ~tm) is very hard. The signal suppression is probably due to large Zeeman shifts in the cloud, which together with optical pumping processes dramatically reduce the scattered light. In such a case, one can observe the atoms after trapping by 'pulling' them up, away from the surface into a less compressed trap. This may be done simply by increasing the wire current or decreasing the bias field. B.3. Observing atoms on the chip
A simple way to observe the trapped atoms is by fluorescence imaging. For this, one illuminates the cloud with near-resonant molasses laser beams for a short time (typically much less than l ms). The scattered light is imaged by CCD cameras as shown in Fig. 31 and Fig. 33. One should use short enough exposure times to avoid blurring of the image due to atomic motion. One also has to select the camera positions wisely to avoid stray light caused by scattering off the grooves in the atom chip surface. Furthermore, it is important that the metal surface itself shows minimal light scattering. Here, the excellent surface quality of evaporation on semiconductor surfaces is advantageous. A different possibility is to use absorption imaging. If the probe beam is directed parallel to the chip surface, the surface quality is not as critical, and one does not have to take care of diffraction peaks from the grooves. Such absorption imaging is used by the Munich, Tfibingen, Heidelberg, and MIT groups. Profiting from an excellent surface mirror quality, the Heidelberg group has also implemented absorption imaging with laser beams reflected from the chip surface, which allows direct distance measurements. More sophisticated methods such as phase contrast imaging will be important for more complicated atom-optical devices on atom chips, where non-destructive observation very close to the chip surface becomes essential. For an overview of these methods, we refer the reader to the many BEC review papers (see, for example, Ketterle, 1999). Finally, future light optical elements incorporated on the chip, such as microspheres or cavities, will allow for much better detection sensitivity, possibly at the single-atom level (see Sect. VI.A.3). Such work has been started in several laboratories.
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C. ATOM CHIP EXPERIMENTS Since the first attempts two years ago, the atom chip has now become a 'tool box in development' in numerous labs around the world. To the best of our knowledge these include (in alphabetical order) the groups at Boulder/JILA (D. Anderson and E. Cornell), CalTech (H. Mabuchi), Harvard (M. Prentiss), Heidelberg (J. Schmiedmayer), MIT (W. Ketterle), Munich (J. Reichel and T.W. H~insch), Orsay (C. Westbrook and A. Aspect), Sussex (E. Hinds), and Tfibingen (C. Zimmermann). Unfortunately, we will not be able to present in detail all the extensive work done, nor will we be able to touch upon other surface-related projects, such as the atom mirror. C.1. Traps
The simplest traps (i.e 3-dimensional confinement) are usually based on a straight wire guide with some form of longitudinal confinement, which is produced either by external coils or by wires on the chip (Sect. II.A.4). Additional wires for onboard bias fields may also be added. As an example, we start with the simple microtraps realized in Innsbruck/Heidelberg with lithium (Folman et al., 2000) and Munich with rubidium atoms (Reichel et al., 1999; Reichel et al., 2001). Here, the traps are based on wires of 1 to 30 ~tm width with which surface-trap distances below 10 ~tm were achieved. The wires used are either U-, Z- or H-shaped. In these experiments, the compression of traps and guides was also investigated (Folman et al., 2000; Reichel et al., 2001). This is done by ramping up the bias magnetic field. In this process one typically achieves gradients of >25 kG/cm. With lithium atoms, trap parameters with a transverse ground-state size below 100nm and angular frequencies of 2:r x 200kHz were achieved (Folman et al., 2000), thus reaching the parameter regime required by quantum computation proposals (Calarco et al., 2000; Briegel et al., 2000). In addition, an on-board bias field for the thin wire trap was also created by sending currents through two U-shaped wires in the opposite direction with respect to the thin wire current, substituting the external bias field (see Sect. II.A.2). Hence, trapping of atoms on a self-contained chip was demonstrated (Folman et al., 2000). An example of a different configuration was realized in Munich with rubidium atoms. Three-dimensional trapping was achieved by crossing two straight wires and choosing an appropriate bias field direction, as discussed in Sect. II.A.4 (Reichel et al., 2001) (Fig. 34). Here the additional wire actually provides the endcaps that were previously provided in the Z- and U-shaped traps by the same wire. This type of trap will be useful for the realization of arrays of nearby traps. In Tfibingen and Sussex longitudinal confinement has been achieved by additional coils.
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FIG. 34. Ioffe-Pritchard trap created by two intersecting wires. The left-hand column corresponds to I l > 12 and ]B0,y] > [B0,x[; in the right-hand column both relations are reversed. Top row: conductor pattern; the thickness of the arrows corresponds to the magnitude of the current; dashed arrows indicate the bias field direction. Middle row: calculated contours of the magnetic field modulus ]B(x,y)] indicating how the long trap axis turns; the left potential continuously transforms into the right one when the parameters are changed smoothly. Bottom row: absorption images corresponding to the two situations. Courtesy J. Reichel.
Finally, the splitting o f a s i n g l e trap into t w o has b e e n d e m o n s t r a t e d
in
H e i d e l b e r g , M u n i c h a n d Sussex. S u c h a t i m e - d e p e n d e n t p o t e n t i a l is p r e s e n t e d in Fig. 35; as e x p l a i n e d in Sect. I|, it m a y f o r m the basis o f an i n t e r f e r o m e t e r . It is also the first step in c r e a t i n g m u l t i - w e l l traps or a r r a y s o f traps.
FIG. 35. Top view of a thermal 200 ~tK cloud of lithium atoms in a double-well potential 40 ~tm above the chip surface. The minima are separated by 350 ~tm. The imaging flash light pulse is 100 ~ts long. The splitting may be done as slow as needed in order to achieve adiabaticity.
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More sophisticated designs have been suggested by Weinstein and Libbrecht (1995) (see Sect. II.A.5) and fabricated (e.g. Harvard: Drndid et al., 1998). C.2. Guiding and transport To achieve mesoscopic atom optics on a chip, it is essential to have reliable means of transporting atoms. One such device is an atomic guide using a single wire with a bias field. Such an experiment is shown in Fig. 36a. The Z-trap is transformed into an L-shaped guide by re-routing the current from one of the Z leads. The atoms expand along the guide due to their thermal velocity (Folman et al., 2000). Similarly, it was demonstrated that one can directly load the guide from a larger magnetic trap on the chip and skip the small surface trap. It is also possible to achieve a continuously loaded magnetic guide using a leaky microtrap (see Fig. 36b). This is achieved by lowering the barrier between the trap and the guide, the barrier being simply the trap end cap, whose height may be controlled by changing the current in the microtrap (Brugger et al., 2000). However, there are limitations to such a simple guide. Using a homogeneous external bias field, the guide has to be straight (linear), since the bias field must be perpendicular to the wire as discussed in Sect. II.A.2. This considerably limits the potential use of the whole chip surface. A possible solution is to create the bias field using on-chip wires (3-wire configuration shown in Fig. 2) or the two-wire guide configuration discussed in Sect. II.A.3, in which the currents are counter-propagating and the bias field is perpendicular to the chip surface. A first experiment was conducted by M. Prentiss' group at Harvard (Dekker et al., 2000). Here, cesium atoms were dropped from a MOT onto a vertically positioned chip, on which a two-wire guide managed to deflect the atoms from their free fall (see Fig. 37). Furthermore, a four-wire guide was realized whereby the two extra wires served as the source for the bias field (see also Fig. 2).
Fie. 36. (a) Cold atoms in a microtrap (left) and released all at once into a linear guide (right). (b) Continuous loading of an atom guide by leaking atoms from a reservoir created by a U-trap into the guide by ramping down the current in the U. Propagation is due to thermal velocity.
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FIG. 37. Vertical bias field: this Harvard experiment realized two-wire vertical guides, enabling the guiding of atoms in a variety of directions. Left: setup. Right: absorption of probe beam versus position along x at 3.5 mm below the output of the guide. The left and right peaks are attributed to unaffected atoms and atoms deflected by the outside of the guide potential, respectively. The open triangles are the data collected while the guide is turned off. Courtesy M. Prentiss.
G u i d i n g a l o n g a c u r v e d two-wire guide has been achieved in H e i d e l b e r g , and e x p e r i m e n t s to guide a t o m s along a spiral are in progress (for the chip design see Fig. 29). Several e x p e r i m e n t s have achieved g u i d i n g w i t h o u t any bias field by trapping the weak-field seekers in the m i n i m u m existing exactly in b e t w e e n two parallel wires with c o - p r o p a g a t i n g currents (see Sect. II.A.3). In Fig. 38, we p r e s e n t such
FIG. 38. The JILA setup in which a 'low-velocity intense source' (LVIS) was used to directly load the two-wire guide. The data show the need for strong potentials with which the magnetic guide can overcome the kinetic energy in order to deflect the atoms, thereby bypassing the beam block. Courtesy E. Cornell.
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Fic. 39. Moving atoms using a magnetic conveyor belt. (a) Potential for various phases of the movement. The numbers indicate the position of the atoms as shown in the absorption images in column (b). (c) Linear collider experiment: left, time evolution of the centers of mass of the two clouds; right, absorption images of the colliding atoms. Courtesy J. Reichel.
a setup (Mfiller et al., 1999). Another similar use of this principle (in this case, not surface-mounted), in which a storage ring has been realized is presented in Sect. III.A.7. Although advantageous for the lack of bias fields, this concept may be hard to implement on miniaturized atom chips as the atoms would be extremely close to the surface for 1-2-~tm thick wires. Guiding with semi-permanent magnets has also been achieved (Rooijakkers et al., 2001; Vengalattore et al., 2001). These materials enhance the magnetic fields coming from current-carrying wires (see Sect. II.A.10 for a description). Completely permanent magnets are also being contemplated to avoid current noise. However, to the best of our knowledge, only atom mirrors have thus far been realized this way (Hinds and Hughes, 1999). A further limitation of the guides described above is that they rely on thermal velocity. Much more control can be achieved by transporting atoms using moving potentials, as described in Sect. II.A.7. Such a transport device was implemented in an experiment in Munich. Using the movable 3-dimensional potentials of their 'motor', atoms can be extracted from a reservoir and moved or stopped at will (Hfinsel et al., 2001b) (Fig. 39). This considerably improves the possibilities of the chip, as demonstrated by the 'linear collider' shown in Fig. 39c, in which the motor was used to split a cloud in two and then to collide the two halves (Reichel et al., 2001).
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C.3. Beam splitters As discussed in Sect. II.A.8, one may combine the wire guides as described in the previous section to build more complicated atom-optical elements. One such element is a beam splitter. A simple configuration is a Y-shaped wire (Fig. 40a) which creates a beam splitter with one input guide for the atoms, the central wire of the Y, and two outputs guides, the right and left arms. The atoms are split by
FIG. 40. Beam splitter on a chip. (a) Chip outline; (b) fluorescence images of guided atoms. Two large U-shaped 200 ~tm wires are used to load atoms into the input guide of a 10-~tm Y-beam splitter. In the first two pictures in (b), a current of 0.8 A is driven only through one side of the Y, therefore guiding atoms either to the left or to the right; in the next two pictures, taken at two different bias fields (12 G and 8 G, respectively), the current is divided in equal parts and the guided atoms split into both sides. (c) Switching atoms between left and right is achieved by changing the current ratio in the two outputs and keeping the total current constant as before. The points are measured values while the lines are obtained from Monte Carlo simulations with a 3-G field along the input guide. The kinks in the lines are due to Monte Carlo statistics.
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means of symmetric scattering off the potential hill, which they encounter at the splitting point. Such a beam splitter on an atom chip was realized by Cassettari et al. (2000) in Innsbruck. Atoms were released from a chip microtrap and guided into the beam splitter. Depending on how the current in the input wire is sent through the Y, atoms can be switched from one output arm of the Y to the other, or directed to the two outputs with any desired ratio (Fig. 40). Similar beam splitters have been widely used for the splitting of guided electron waves in solidstate quantum electronics devices. For example, two Y splitters were put back to back to form an Aharonov-Bohm type interferometer (Buks et al., 1998). A four-port beam splitter has been realized at JILA by the group of E. Cornell and D. Anderson by making a near X-shape out of two wires which avoid a full crossing (Mtiller et al., 2000). In this experiment, two input guides formed by two current-carrying wires, merge at the point of closest approach of the wires so that the two minima merge into one, and then again split into two independent minima. C. 4. B E C on a chip
A degenerate quantum gas in a microtrap is an ideal reservoir from which atoms can be extracted for the experiments on the chip. For example a BEC will take a similar role as source of bosonic matter waves as the Fermi sea has in quantum electronics. A clear advantage of a BEC is the higher efficiency of the transfer to the smallest compressed surface traps, which involves high compression, leading to large losses for thermal atoms if the trap depth is not appropriate. The condensate occupies the trap ground state and should follow any adiabatic compression of the trap. Second, a BEC in a microtrap also provides the initial atomic state needed to initiate delicate quantum processes such as interference or even a well-defined entanglement between atoms in two nearby traps. In the last year three groups in Tfibingen, Munich and Heidelberg succeeded in making and holding a Bose-Einstein condensate in a surface trap (Ott et al., 2001; HS.nsel et al., 2001a; Reichel, 2002), and the MIT group managed to transfer a BEC to a surface trap, and load it into it (Leanhardt et al., 2002). These experiments showed that making the BEC in a surface trap can be much simpler. For example in very tight microtraps the BEC is formed in much shorter time as the tightness of the traps allows for fast thermalization and consequently fast evaporative cooling which relaxes the vacuum requirements, permitting the use of a very simple one-MOT setup to collect the atoms (H~insel et al., 2001 a). In the Tfibingen experiment (Ott et al., 2001), a relatively large condensate of 4 • 105 87Rb atoms has been formed at a height of some 200 ~tm above the surface. The experiment made use of a pulsed dispenser as an atom source, allowing ultra high vacuum (2x 10-11 mbar) while the dispenser was off. This enabled the use of a simple single-MOT setup. In the experiment, atoms were transferred magnetically from a distant six-beam MOT to the chip using two
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FIG. 41. Left: the Ttibingen setup. The first pair of coils (right) produced the MOT and then the atoms were conveyed to the trap formed by the second pair of coils. The chip mounting is visible within the second pair of coils. Right: absorption images of the compression and final cooling stage; (a) compression into the microtrap; (b) RF cooling in the microtrap; (c) release of the condensate after 5, 10, and 15 ms time of flight. Courtesy C. Zimmermann.
adjacent pairs of coils (Fig. 41). In the chip trap, condensation was reached after 10 to 30 s of forced RF evaporative cooling. Aside from being the first surface BEC, the chip used in Tfibingen with its 25-mm long wires provides a highly anisotropic BEC (aspect ratio 105), approaching a quasi one-dimensional regime. In recent work, the BEC was taken to a height of only 20~tm without observing substantial heating (Fortagh et al., 2002). The smallest structure holding the BEC was a 3 x 2 . 5 / t m 2 cross section copper wire with a current of 0.4 A. The BEC had a lifetime of 100 ms in the compressed trap (limited by 3-body collisions) and a 1 s lifetime once it was expanded into a larger trap. The second experiment producing a BEC in a microtrap was performed in Munich (H~insel et al., 2001a) (Fig. 42). Here an attractively simple setup with a continuous dispenser discharge was used. Consequently, the vacuum background pressure was high (10 -9 mbar) and evaporative cooling had to be achieved quickly. RF cooling times were as short as 700 ms thanks to the strong compression in the microtrap which results in a high rate of elastic collisions. The final BEC included some 6000 atoms at a height o f 70 ~tm. The trapping wire was 1 . 9 5 m m long and had a cross section of 50• the current density approached 106 A / c m 2. Strong heating of the cloud was observed in this experiment but the source remains elusive (possibly, current noise). A beautiful feature of this experiment is the use of the magnetic 'conveyor belt' described before (Sects. II.A.7 and IV.C.2, Fig. 39), in order to transport the BEC during a time of lOOms over a distance o f 1.6 m m without destroying it (see Fig. 42). Furthermore, the ability of the 'motor' to split clouds was used to show that a
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FI6. 42. Munich atom chip BEC experiment. Left: schematics of the simple vapor cell apparatus. Center: time-of-flight images showing the formation of a BEC. Right: (a) the BEC is transported in a movable 3-dimensional potential minimum; (b) at the end it is released and is observed falling and expanding. Courtesy J. Reichel. BEC survives such a splitting. Two such halves were then released into free fall exhibiting interference fringes as they overlapped. In the third experiment, performed in Heidelberg, typically 3 • 105 87Rb atoms were condensed in a Z-wire Ioffe-Pritchard trap, created by a structure underneath the chip, and subsequently transferred to a Z-trap on the chip (Fig. 43). First, more than 3• 108 atoms are loaded into a mirror MOT ( < 1 0 -li torr) created by external quadruople coils using a double MOT configuration with a continuous push beam. The atoms are then transferred into a U-MOT, where they are compressed and after molasses-cooling loaded into a Z-wire trap. The BEC is formed by forced RF evaporation in typically 20 seconds. Creating the BEC using a wire structure underneath the chip allows to place other surfaces close to the BEC while maintaining the high precision of a microtrap for manipulating the cold atoms. This will open up the possibility to study surfaces with the cold atoms and to transfer the BEC to surface traps based on dipole forces in light fields created by micro-optic elements and evanescent fields. The MIT group transported a BEC of the order of 10 6 Na atoms into an auxiliary chamber and loaded it into a magnetic trap formed by a Z-shaped wire (Gustavson et al., 2002) (Fig. 44). This was accomplished by trapping the condensate in the focus of an infrared laser and translating the location of the laser focus with controlled acceleration. This transport technique avoids the optical and mechanical access constraints of conventional condensate experiments. The BEC was consequently loaded into a microstructure (Leanhardt et al., 2002). Finally, we would like to note that currently other groups are also working
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FIG. 43. Heidelberg atom chip BEC experiment. (a) Schematics of the double MOT setup. Atoms from a lower vapor cell MOT are transferred to a UHV mirror MOT using a continuous push beam. (b) Photograph of the upper (UHV) chamber. (c) The mounted chip and the U- and Z-shaped wire structure underneath the chip (inset). (d) Thermal cloud, BEC with thermal background, and pure BEC released and expanded for 15 ms.
FIG. 44. Transfer of a BEC to a microtrap. Left: schematics of the setup with the science chamber housing the Z-trap on the far left and the BEC production chamber on the right. Right: condensates in the science chamber (a) optical trap and (b) Z-trap. The condensate was (c) released from an optical trap and imaged after 10 ms time of flight and (d) released from a wire trap and imaged after 23 ms time of flight. (e) Schematic of the Z-trap. Courtesy W. Ketterle. towards BEC experiments
in s u r f a c e t r a p s , a n d w e e x p e c t to s e e m a n y d i f f e r e n t s u c c e s s f u l shortly.
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V. Loss, Heating and Decoherence For atom chips to work, three main destructive elements have to be put under control: (i) Trap loss: It is crucial that we are able to keep the atoms inside the trap as long as needed. (ii) Heating: Transfer of energy to our quantum system may result in excitations of motional degrees of freedom (e.g. trap vibrational levels), and consequently in multimode propagation which would render the evolution of the system ill-defined. (iii) Decoherence or dephasing as it is sometimes referred to also originates from coupling to the environment. While heating requires the transfer of energy, decoherence is more delicate in nature (Stern et al., 1990). Nevertheless, the effect is just as harmful because superpositions with a definite phase relation between different quantum states are destroyed. This has to be avoided, e.g. for interferometers or quantum information processing on the atom chip. In discussing these three points, we focus on the particularities of atoms in strongly confined traps close to the surface of an atom chip. The small separation between the cold atom cloud and the 'hot' macroscopic environment raises the intriguing question of how strong the energy exchange will be, and which limit of atom confinement and height above the surface can ultimately be reached. We review theoretical results showing that fluctuations in the magnetic trapping potential give a fairly large contribution to both atom loss and heating. In addition, thermally excited near fields are also responsible for loss and may impose limits on coherent atom manipulation in very small (~tm-sized) traps on the atom chip. Estimates for the relevant rates are given, and we outline strategies to reduce them as much as possible. Experimental data are not yet reliable enough to allow for a detailed test of the theory, but there are indications that field fluctuations indeed influence the lifetime of chip traps (Hansel et al., 2001a; Fortagh et al., 2002).
A. Loss MECHANISMS A.1. Spilling over a finite potential barrier
Compression of a thermal atom cloud can lead to losses when the cloud temperature rises above the trap depth. The smallest losses occur if the compression is adiabatic. Atoms then stay in their respective energy levels as the level energy increases. They can nevertheless be lost during trap compression because of the finite trap depth. It should be noted that this loss occurs for the highest energies in the trap and can also be used to evaporatively cool the cloud (see Luiten et al., 1996 for a review).
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A.2. Majorana flips If the atomic magnetic moment is not able to follow the change in the direction of the magnetic field, the spin flips, and a weak-field-seeking atom can be turned into a strong-field seeker which is not trapped (Majorana, 1932; Gov et al., 2000). This occurs when the adiabatic limit (Larmor frequency col much larger than trap frequency co) does not hold. Majorana flips thus happen at or near zeros of the magnetic field. For this reason, additional bias fields are employed to 'plug the hole' in the center of a quadrupole field. For a magnetic field configuration with a zero, loss can be reduced if the atoms circle around it. The loss rate is then inversely proportional to the angular momentum because the latter determines the overlap with the minimum region (Bergeman et al., 1989; Hinds and Eberlein, 2000). In Ioffe-Pritchard traps with nonzero field minimum Bip, there is a finite residual loss rate that has been calculated by Sukumar and Brink (1997). For a model atom with spin 1/2 in the vibrational ground state, one gets ~CO
y = -~
exp(-It[IBip/hco)
__ 6•
s_l
co/2 ~ ((itll/itB)(Bip/1G)) 100 kHz exp - 14 co/2;r 100 kHz
(18) '
where /~11 is the component of the magnetic moment parallel to the trapping field. Note the exponential suppression for a sufficiently large plugging field Bip, typical of nonadiabatic (Landau-Zener) transitions. Choosing a Larmor frequency coc = ItljBip/h > 10co, one gets a lifetime larger than '~104 trap oscillation periods. A ratio coL/co > 20 already pushes this limit to ~ 108.
A.3. No&e-induced flips Fluctuations in the magn~/tic trap fields can also induce spin flips into untrapped states, and lead to losses. These fluctuations are produced by thermally excited currents in the metallic substrate or simply by technical noise in the wire currents. Fluctuations of electric fields and of the Van der Waals atom-surface interaction have been shown to be less relevant for typical atom traps (Henkel and Wilkens, 1999; Henkel et al., 1999). The trapped spin is perturbed via the magnetic dipole interaction and flips at a rate given by Fermi's Golden Rule: 1
Y - 2h 2
Z
(il/tklf)(fl/t'li)S~'(COL),
(19)
k,l = x,y,z
where S~t(COL) is the noise spectrum of the magnetic fields, taken at the Larmor frequency COL.We use the following convention for the noise spectrum:
SO(co) = 2
dr e i~~ (Bi(t + r)Bj(t)) , oo
(20)
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Table II Trace of the geometric tensor Y/j that determines the loss due to the thermally fluctuating magnetic near field, according to the rate (22) a
Geometry
Half-space
Layer
Wire
Tr Yij
~/h
Jrd/h 2
jr2 a2/(2h 3)
a The metallic layer has a thickness d, assumed much smaller than the distance h to the trap center. The wire has a radius a << h, and h << 6 is assumed where 6 is the skin depth of the metal. Taken from Henkel and P6tting (2001). A more accurate calculation of Tr Y/j corrects the results of table 2 by a factor of 1/2 for the half-space and the layer (Henkel and Scheel, 2002).
where (...) is a time average (experiment) or an ensemble average (theory). The rms noise is thus given by an integral over positive frequencies
(Bi(t)Bj(t))=
Z
~dco
O
~-~SB(o)).
(21)
For example, the rms magnetic field in a given bandwidth Af for a white noise spectrum $8 is given by Brms = v/AfSB. The spectrum SB thus has units G2/Hz.
A.3.1. Thermally excited currents. An explicit calculation of the magnetic noise due to substrate currents ('near field noise') yields the following estimate for the loss rate (Henkel et al., 1999; Henkel and P6tting, 2001): g "-~ 75 S-1 ( t 2 / ~ B ) 2 ( T s / 3 0 0 K) (TRY/)- x 1 gm),
(e/ecu)
(22)
where 1/e is the substrate conductivity (for copper, ~)Cu = 1.7x 10-6f~cm) and Ts is the substrate temperature. Note that the Larmor frequency COLactually does not enter the loss rate. The 'geometric tensor' YO has dimension (1/length) and is inversely proportional to the height h of the trap center above the surface (Table II). The loss rate (22) is quite large for a trap microns above a bulk metal surface. One can reduce the loss by two orders of magnitude when bulk metal in the vicinity of the trap is replaced by microstructures. For a thin metallic layer of thickness d, the loss rate (22) is proportional to d/h 2, and for a thin wire (radius a), a faster decrease o( a2/h 3 takes over (Table II). The estimates of Table II apply only in an intermediate distance regime, d,a << h << 6((OL): on the one hand, when the trap distance h is smaller than the size of the metallic structures, one recovers a 1/h behaviour characteristic for
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FIG. 45. Loss rates in a magnetic trap above a copper surface. Results for two different Larmor frequencies tOL/2Jr = 1 MHz (curve a) and 100 MHz (curve b) are shown. The arrows mark the corresponding skin depths 6(~OL). Equation (22) applies to the region h << 6(~OL). See Henkel et al. (1999) for details. Parameters: spin S = 89 magnetic bias field aligned parallel to the surface. The loss rate due to the black-body field is about 10-13 s-I at 100MHz (not shown). Reproduced from Henkel et al. (1999), Appl. Phys. B 69 (1999) 379, Fig. 3, with permission. 9 Springer-Verlag.
a metallic half-space; on the other hand, steeper power laws take over at large distances, when h gets comparable to or larger than the skin depth
(~( coL ) =
0 = 160 gm v/0/~)Cu [JOcoL V/coL /2Yr 1 MHz"
(23)
Recall that the skin depth characterizes the penetration of high-frequency radiation into a metal. This crossover can be seen in Fig. 45 where the flip rate (22) is plotted vs. the trap height h for a metallic half-space. For details, we refer to Henkel et al. (1999) and Henkel and P6tting (2001). Note that an increase of the Larmor frequency only helps to reduce the substrate-induced flips in the regime where h >> 6(coc). Equation (23) shows that this requires, for h _~ 1 gm, quite large Larmor frequencies coL/2~ >> 10 GHz, meaning large magnetic (bias) fields. Additional loss processes may be related to fluctuations in the currents used in the experiment, for example in the chip wires and in the coils producing the bias and compensation fields. Let us focus on the wire current, and denote by St(co) its noise spectrum. Neglecting the finite wire size, the magnetic field Bw = ~ I w / 2 ~ h is given by Eq. (3), and we find the following upper limit for the noise-induced flip rate:
A.3.2. Technical noise.
~2
]1 "~ - ~
( ~lO -~/
)2
SI((DL) ~ 1.3 S-1
(~/~B)2
SI((DL) (h/1 ~tm) 2 SSN '
(24)
where the reference value Ssy = 3.2 x 10 - 1 9 A2/Hz corresponds to shot noise at a wire current of 1 A (Ssy = 2elw). This estimate is pessimistic and assumes equal
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noise in both field components parallel and perpendicular to the static trapping field. Nevertheless, it highlights the need to use 'quiet' current drivers for atom chip traps. In future chip traps with strong confinement, it may turn out necessary to reduce current noise below the shot-noise level. This can be achieved with superconducting wires or permanent magnets, as discussed in Sect. II.A. 10 and reviewed by Hinds and Hughes (1999). See also Varpula and Poutanen (1984). A. 4. Collisional losses A.4.1. Background collisions. Here, collisions between background gas atoms and trapped atoms endow the latter with sufficient energy to escape the trap. In order to estimate the loss rate per atom ~', let us assume that the background gas is dominated by hydrogen molecules and at room temperature. We then get: ]r = nbg Ubg O ---
3.8 • 10 -3 s-l
P bg
O"
10 -l~ mbar 1 nm 2'
(25)
w h e r e Pbg is the background pressure. Typical collision cross sections o are in the l nm 2 = 100 ~2 range (Bali et al., 1999). As a general rule, one gets a trap lifetime of a few seconds in a vacuum of 10 -9 mbar. It is clear that vacuum requirements will become more stringent as longer interaction times are required.
A.4.2. Collisions o f trapped atoms. For traps in UHV conditions, and especially for highly compressed traps and high-density samples, the dominant collisional loss mechanisms involve collisions between trapped atoms. The scattering of two atoms leads to a loss rate per atom scaling with the density, while 3-body collision rates scale with the density squared. Spin exchange. This process corresponds to inelastic two-body collisions where the hyperfine spin projections mF are conserved, but not the spins F themselves. In the alkali atoms 7Li, 23Na, and 87Rb, for example, a collision between two weak-field-seeking states IF = 1, mF = -1) can lead to the emergence of two strong-field seekers ]2,-1 ) that are not trapped. This transition requires an excess energy of the order of the hyperfine splitting to occur, which is typically not available in cold atom collisions. Exothermic collisions between the weak-field seekers 11,-1), 12, +1), and 12, +2) are not suppressed, however. The corresponding rate constant is proportional to n ( a s - aT) 2 where as (at) are the scattering lengths in the singlet (triplet) diatomic potential (C6t6 et al., 1994). For 87Rb, these scattering lengths accidentally differ very little, leading to a very small spin flip rate (Moerdijk and Verhaar, 1996; Burke et al., 1997). As a consequence, 87Rb is practically immune to spin exchange and can form stable condensates, even of two hyperfine species (Myatt et al., 1997; Julienne et al., 1997; Kokkelmans et al., 1997). Spin-polarized samples consisting only of 12, +2) cannot undergo spin exchange because of mF conservation, the other available states having smaller F. For more details, we refer to the review by Weiner et al. (1999) and references therein.
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Spin relaxation. This process also results from inelastic two-body collisions, but does not conserve mF. Spin relaxation is caused by a flip of the nuclear spin and occurs at a lower rate because of the smaller nuclear magnetic moment. In 87Rb for example, the trapped weak-field seeker I1,-1) may be changed into the untrapped strong-field seeker I1, + 1). More details can be found in the theoretical work by Burke et al. (1997), Julienne et al. (1997), Timmermans and C6t6 (1998), the experimental work of Gerton et al. (1999) and S6ding et al. (1998), and the review paper by Weiner et al. (1999). Three-body recombination. In this process, two atoms combine to form a molecule. Although the molecules may have a definite magnetic moment and still be trapped, the reaction releases the molecular binding energy that is shared as excess kinetic energy between the molecule and the third atom. The binding energy being typically quite large (larger than 100 ~teV), both partners escape the trap (Fedichev et al., 1996; Moerdijk and Verhaar, 1996; Moerdijk et al., 1996; Esry et al., 1999). For references to experimental work, see Burt et al. (1997), S6ding et al. (1999) and Weiner et al. (1999). We expect three-body processes to be the dominant collisional decay channel in strongly compressed traps because the collision rate per atom increases with the square of the atomic density. A.5. Tunneling Traps very close to the surface might also show loss due to tunneling of atoms out of the local minimum of the trap towards the surface. The rate can be estimated by 1 ?'~ e) exp\2m[U(z)- E]/dz, (26)
(,/
rrier width ~
where U ( z ) - E is the height of the barrier above the energy of the trapped particle. Tunneling will therefore only be important for states close to the top of the potential barrier. Low-lying states in traps where the magnetic field magnitude rises for long distances will have very little tunneling. Even for atom waveguide potentials as close as 1 ~tm from the surface, tunneling lifetimes of more than 1000s have been estimated (Pfau and Mlynek, 1996; Schmiedmayer, 1998).
A.6. Stray light scattering Residual light can flip the atomic spin via optical pumping. For resonant light, this happens at a rate of the order of ['(Istray/Isat) where F is the linewidth of the first strong electric dipole transition (typically, F/2Jr _~ 5 MHz) and/sat the saturation intensity (typically a few mW/cm2). It is highly desirable to perform atom chip experiments 'in the dark': a shielding from any stray light at the level 10-6/sat is required for manipulations on a scale of seconds. For more detailed estimates, we refer to the review by Grimm et al. (2000) on optical traps.
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Table III Loss mechanisms for the atom chip (overview) Mechanism
Scaling a
Magnitude a
Remedy
Background collisionsb
Pbg
0.01 s-1
ultra-high vacuum
Majorana flips c
m e -C~176
~ 1 s-1
avoid B = 0
deep trap potential
Spilling over
Near-field noise d
Ts/~h a
10 s-1
little metal
Current noise e
Si(tOL)/h 2
_~ 1 s-1
low-noise drivers
2-body spin exchange f
n
10-4 s- l
spin polarize
2-body spin relaxation g
n
10-2-10 -4 s- 1
3-body collisions h
n2
10-9-10 -7 s-1
Tunneling Stray light
10-3 s- 1 /stray
deep potential, wide barriers keep in the dark
a The columns 'Scaling' and 'Magnitude' refer to loss rates per atom at typical atom chip traps: density n = 101~ -3, height h = 10~tm, trap frequency to/2Jr = 100kHz, Larmor frequency tOL/2Jr = 5 MHz. b Eq. (25). c Flip rate (18) from trap ground state. d Eq. (22). The exponent ct = 1,2, 3 for metal half-space, layer, and wire (see table 2). The estimate 10s -l is for a half-space. e Eq. (24). SI(m)/SsN = 100. f Experimental result for 87Rb (Myatt et al., 1997). g Experimental result for Cs and 7Li, respectively (S6ding et al., 1998; Gerton et al., 1999). h Experimental results for 87Rb and 7Li, respectively (Burt et al., 1997; S6ding et al., 1999; Gerton et al., 1999). To s u m m a r i z e , an o v e r v i e w o f the p r e v i o u s loss m e c h a n i s m s is g i v e n in Table III. We e x p e c t that on the route t o w a r d s ~tm-sized traps with high c o m p r e s sion, inelastic collisions and m a g n e t i c field n o i s e will d o m i n a t e the trap loss.
B. HEATING In the t r e a t m e n t o f loss m e c h a n i s m s in the p r e v i o u s section, h e a t i n g w a s m e n t i o n e d in relation to adiabatic c o m p r e s s i o n w h e r e s o m e a t o m s gain e n e r g i e s larger than the trap depth. H e r e w e discuss a different f o r m o f h e a t i n g , in w h i c h the a t o m e x c h a n g e s e n e r g y w i t h the e n v i r o n m e n t . S u c h h e a t i n g d o e s not n e c e s s a r i l y cause the a t o m to be lost, but it is still v e r y h a r m f u l as e x c i t a t i o n s o f vibrational d e g r e e s o f f r e e d o m lead to an ill-defined q u a n t u m state o f the s y s t e m . In the case o f the a t o m s y s t e m and the chip e n v i r o n m e n t , the e n v i r o n m e n t is always hot c o m p a r e d to the system. E n e r g y e x c h a n g e thus i n c r e a s e s b o t h the system's m e a n e n e r g y and its e n e r g y spread. In the f o l l o w i n g , w e first d e s c r i b e
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the influence of position and frequency noise using the harmonic oscillator model, then turn to substrate and technical noise, and finally touch upon the issue of heating due to light fields. B.1. Harmonic oscillator model
Let us consider the trap potential to be a one-dimensional harmonic potential with angular frequency 6o and with a ground-state size of a0 = ( h / ( 2 M o ) ) ) 1/2, where M is the mass of the vibrating atom. Assume for simplicity that the atom is initially prepared in the oscillator ground state 10). Heating can occur as a result of a fluctuating trap either in frequency or position (see, for example, Gehm et al., 1998; Turchette et al., 2000). These processes may be described by transition rates to higher excited states of the oscillator. For example, fluctuations in the trap position (amplitude noise) are equivalent to a force acting on the atom. They drive the transition 0 ~ 1 between the ground and first excited vibrational states, with an excitation rate given by (Gehm et al., 1998; Henkel et al., 1999)
['0---* 1 --
-•2
SF(CO) SF(O0) -- 4ho)M
(27)
that is determined by the noise spectrum of the force at the oscillator frequency SF(~O). The rate of energy transfer to the atom ('heating rate') is simply F0~lho) or SF(O))/4M. Note that this estimate remains valid for an arbitrary initial state. We may make contact with the work of Gehm et al. (1998) by noting that fluctuations Ax of the trap center are equivalent to a force F = Me)2Ax.
(28)
In terms of the fluctuation spectrum of the trap center Sx(~O), the excitation rate (27) is thus given by Me) 3 F0___, 1 =
4h
09 2
Sx((D ) = -x-Sx/ao((_D), ~5
(29)
which is equivalent to the heating rate (12) of Gehm et al. (1998), given our definition (20) of the noise spectrum. Fluctuations of the trap frequency are described by the Hamiltonian MxZo)Aco and heat the atom by exciting the 0 ~ 2 transition. The corresponding transition rate is (Gehm et al., 1998) ro_~2 = 1S~o(2o))
(30)
and involves the frequency noise spectrum at twice the trap frequency. Using the rates given by Gehm et al. (1998), one can show that the heating rate due
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to frequency fluctuations is equal to F0~2(4(E) + hog), where the mean energy (E) = 89 in the ground state. In the following, we differentiate between thermal fluctuations and technical ones. To get the total heating rate, one simply adds the force fluctuation spectra SF(~) of all the relevant sources (e.g. electromagnetic noise from radio stations).
B.2. Thermal fluctuations Magnetic fields generated by thermally excited currents in the metallic substrate correspond to a force given by the gradient of the Zeeman interaction - p . B. An explicit calculation of the magnetic gradient noise gives the following force spectrum (Henkel and Wilkens, 1999; Henkel et al., 1999) SF(O)) --
3-~
h3
,
(31)
where p is the magnetic moment and /tll its component parallel to the static trapping field. The expression (31) applies to a planar metallic substrate (halfspace) and an oscillation perpendicular to its surface. Again, the noise spectrum is actually frequency independent as long as h << 6(~o) where 6(09) is the skin depth (23). The average magnetic moment is taken in the trapped spin state [see Henkel et al. (1999) for details]. We thus obtain the following estimate for the excitation rate (27): F0~l '~ 0.7s -1
(~/ptB)Z(Ts/300 K) (M/amu)(oo/2er 100 kHz)(e/eCu)(h/1 ~tm) 3"
(32)
Fig. 46 shows a plot of heating rates for varying substrates and trap heights according to Eq. (32).
FIG. 46. Heating rate for a trapped spin above copper and glass substrates. Parameters: trap frequency to/2Jr = 100 kHz, M = 40 amu, magnetic moment/t =/t B = 1 Bohr magneton, spin S - 1. The heating rate due to the magnetic black-body field (not shown) is about 10-39 s-1 . For the glass substrate, a dielectric constant with Re e = 5 and a resistivity e = 1011 ~2cm are taken. Reproduced from Henkel et al. (1999), Appl. Phys. B 69 (1999) 379, Fig. 5, with permission. 9 Springer-Verlag.
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For lithium atoms, a typical trap frequency of 100kHz and a height of h = 101am, we estimate a heating rate of 10 -4 s-1. For time scales typical of atom chip experiments (1-100ms), thermal fluctuations thus lead to tolerable heating only for traps with h > 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).
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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).
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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
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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.
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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)
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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).
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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 << lc which decohere more slowly because the interferometer arms see essentially the same noise potential. Note, however, that the spin lifetime will always be the upper limit to the coherence time of the cloud. The previous theory allows to recover the decoherence model of Eq. (42) at long times t > 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.
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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
(51)
where the last two terms contain the correlation between the noise in both arms. They may therefore be expressed through the normalized correlation function CRC =-- C(s) with s the separation between the left and right arms. The reasonable assumption that both arms 'see' the same white noise spectrum, say SE(W), yields
= [I - C(s)] SE(o) ~ 0) 6(t - t'), 1 - C(s)
Ydec = ] I d e c ( S )
=
2h 2
SE(o) ~ 0),
(52) (53)
where we recover the decoherence rate (46) obtained for the quasi-free longitudinal motion. We also recover the trivial result that the contrast stays constant if both interferometer arms are subject to the same noise amplitude (perfect correlation C ( s ) - 1).
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The previous argument shows that transverse and longitudinal coherence are affected in a similar way by magnetic noise. Again, near field noise is a serious threat due to its short correlation length. Since the decoherence rate is so small that Ydec(CX3)t << 1 for interaction times not longer than a few hundred ms, the phase noise remains small even for widely separated arms subject to decorrelated noise (separation larger than the guide height). This is a worstcase estimate: a more careful approach would take into account the form of the interferometer, where the arm separation is not constant. Current noise should neither be underestimated. It is certainly possible to reduce dephasing by feeding the same current through both left and right wire guides, as shown by Eq. (53). But this does not seem to help at the shot-noise level because each electron randomly follows one or the other wire. The wire current fluctuations are thus uncorrelated, leading to a transverse decoherence rate comparable to the longitudinal decoherence rate. Both rates are thus of the order of the flip rate (24), typically a few s-1 . Let us estimate as another example the dephasing due to technical noise in a magnetic field gradient. This may be introduced by an imperfect Helmholtz configuration or coil misalignment. For small gradients b, we have AE(t) =(ktll) s. b(t)
(54)
where s is the spatial separation between the interferometer arms. To be precise, b(t) = VBx(t) gives the gradient of the bias field component along the direction of the (static) trapping field. Ignoring a possible anisotropy in the gradient noise, we find the estimate (/tl I) 2 S 2 )tdec(S) ~'~ 4h 2 Sb(o9 ~ 0), (55) where Sb(og) is related to the power spectrum of the current difference in the Helmholtz coils. We may take as the worst case completely uncorrelated Helmholtz currents, and a magnetic gradient b ~_ B b / R where R is the size of the Helmholtz coils. The dephasing rate is then of the order of Ydec(S) ~
10-6 S-1 Q/b)2 $2 (Bb/G)2 Sl(fO --+ O) tA2 R 2 (Ib/m) 2 SSN
(56)
where Ib and Bb are the Helmholtz current and the bias field. The experimentally reasonable parameters Ib = 1 A, s = 100 ~tm, R = 10 cm, Bb = 10 G yield the small value I/dec(S) "~ 10-10 S -1 X SI(O) --+ 0 ) / / S s N . We note that the residual gradient of imperfect Helmholtz coils is usually less than 0.1 G/cm which is an order of magnitude below the estimate Bb/R - 1 G/cm taken here. Finally, let us estimate the phase noise due to fluctuations in the spring constant of the guide potential. Even in the adiabatic limit where the transitions between transverse quantum states are suppressed (no heating), these fluctuations shift
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Table V Decoherence mechanisms for atom chip interferometers (overview) Mechanism
Scaling
Magnitude a
Remedy
s << h
Tss2/eh a+2
<< 10s -1
little metal,
s >> h
Ts/eh a
10 s -1
small splitting correlate currents
Substrate fields b
Current noise c
w 3 S I / B 2 ~ ogSi/h
1 s -1
Bias fluctuationsd
s 2B b2S I / R 2 I~2 ~ s2SI/R 4
10 -8 s -1
Trap frequency noise e
~ 2 3 i / I 2 ~ Si/h 4
10 -5 s -1
a 'Magnitude' refers to the decoherence rate Ydec(S) for a typical guided interferometer: lithium atoms, height h = 10 gm, separation s - 10 ~tm, transverse guide frequency o9/2:r = 100 kHz. Along the waveguide axis, the atomic motion is free. b Exponent a = 1,2, 3 for metal half-space, layer, and wire (Eq. 22 and table 2). c Eq. (34). d Eq. (56). The bias field scales as B b ~ Ib/R where R is the size of the bias coils. e Eq. (36).
the energy of the guided state. In the harmonic approximation, we have for the ground state of the guide AE = l hA~o where Ao9 is the relative shift of the vibration frequency. This gives a dephasing rate Ydec : ~S,,,(Oo --~ 0).
(57)
We have neglected noise correlations between the interferometer arms that would reduce decoherence because of correlated phase shifts in both arms. The rate (57) is of the same order as the heating rate (30, 36) due to frequency noise (~_ 10-5 s-~). It thus appears that fluctuations of the trap frequency have a larger impact than bias field gradients, but still they lead to negligible dephasing. In Table V we give an overview of the different decoherence mechanisms discussed in this subsection. For interferometers with large path differences (compared to the waveguide height), we expect current shot noise and thermal near field fluctuations to be the dominant sources of decoherence. They give quite 'rough' potentials (small correlation length) and perturb both the quasifree motion along the waveguide axis and the relative phase between spatially separated wavepackets in an interferometer. An increase in the trap frequency does not help, rather the amount of metallic material in the vicinity of the guide should be kept to a minimum.
VI. Vision and Outlook Much has been achieved in the field of micro-optics with matter waves in the last 10 years. We have seen a steady development from free-standing wires to micron-size traps and guides, from trapping thermal atoms to the creation of
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BEC on an atom chip. Where to go from here? What can we expect from future integrated matter wave devices? There are still many open questions before we can assess the full promise of integrated microscopic atom optics. In the following we try to pinpoint the relevant future developments and directions. Some of them, like the study of the influence of the warm thermal surface and the fundamental noise limits on lifetime, heating and coherence of atoms, are already under way. Hopefully in a few years we will know how far micromanipulation of atoms on chips can be pushed. A. INTEGRATING THE ATOM CHIP
A. 1. Chip fabrication technology We will see continued development of atom chip fabrication techniques. Depending on how close to the surface one is able to place atoms before significant decoherence occurs, the commonly used technology will be either state-of-the-art nanofabrication with scale limits below 100 nm or thicker and larger wires built by a combination of less demanding techniques. Another limitation would be smoothness of fabrication: as fluctuations in wire widths would cause changing current densities and consequently changing trap frequencies, potential 'hills' may appear which may be large enough to hinder the transport of a BEC or control its phase evolution. In the near future many advances in fabrication techniques are expected. One of the first steps will be to build multilayer structures that will enable for example crossing wires in order to realize more elaborate potentials and give more freedom for atom manipulation. Thin film magnetic materials should allow to build permanent magnetic microscopic devices, which can be switched on and off for loading and manipulation of atoms. Such structures would have the advantage that the magnetic fields are much more stable, and consequently one can expect much longer coherence times, when compared with current generated fields.
A.2. Integration with other techniques With cold atoms trapped close to a surface, integration with many other techniques of atom manipulation onto the atom chip is possible. One of the first tasks will be to integrate present day atom chips with existing micro-optics (see for example Birkl et al., 2001) and solid-state optics (photonics), for atom manipulation and detection. We envision for example microfabricated wave guides and/or microfabricated lenses on the atom chip for bringing to and collecting light from atoms in the atom-optical circuits. Light can also be used for trapping (Grimm et al., 2000). Having cold atoms close to a surface will allow efficient transfer and precise loading of atoms into light surface traps, which would be otherwise difficult because of their small volume and inaccessible location. For example, an atom chip with integrated
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micro-optics will allow to load atoms into evanescent-wave guides and traps, as proposed by Barnett et al. (2000). Such traps and guides would be a way to circumvent the decoherence caused by Johnson noise in a warm conducting surface (Sect. V). With the standing wave created by reflecting light off the chip surface one will be able to generate 2-dimensional traps with strong confinement in one direction, resembling quantum wells, as demonstrated by Gauck et al. (1998). Adding additional laser beams or additional electrodes on the surface restricts the atomic motion further, yielding 2-dimensional devices as in quantum electronics (Imry, 1987). Similarly one can build and load optical lattices close to the surface where each site can be individually addressed by placing electrodes on the chip next to each site. In principle, many other quantum optical components can be integrated on the atom chip. For example, high-Q cavities combined with microtraps will allow atoms to be held inside the cavity to much better than the wavelength of light providing a strong coupling between light and atoms. For recent experimental work concerning the manipulation and detection of atoms in cavities, we refer the reader to Berman (1994), Pinkse et al. (2000), Hood et al. (2000), Osnaghi et al. (2001) and Guth6hrlein et al. (2001). Regarding cavities one can think of examining a wide variety of technologies ranging from standard high-Q cavities consisting of macroscopic mirrors to optical fiber cavities (with Bragg reflectors or with mirrors on the ends); from photonic band gap structures to microcavities like microspheres and microdiscs fabricated from a suitable transparent material. One proposed implementation is presented in Fig. 48 (Mabuchi et al., 2001).
FIG. 48. A proposed implementationof an integrated nanofabricated high-Q cavity from CalTech. The cavity is made of a 2D photonic crystal utilizing holes with diameters of order 100nm. A Weinstein-Libbrecht-type Ioffe magnetic trap will hold the atom in the cavity. CourtesyH. Mabuchi.
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A.3. Atom detection For future applications, it would be advantageous to have a state-selective singleatom detector integrated on the atom chip. Such detectors could be based on different methods. The most direct method would be to detect the fluorescent light of the atom using surface-mounted micro-optics. More accurate nondestructive methods could be based on measuring an optical phase shift induced by an atom in a high-Q cavity.
A.3.1. Single-atom detection using near field radiation. To detect light scattered from single atoms near a chip surface, the main challenge will be to minimize the stray light scattered from the surface. One possible solution may be to collect a large fraction of the light scattered by the atom using near field apertures and/or confocal microscope techniques. An atom could also be used to couple light between two wave guides, as used in some micro-optic detectors for molecules and directional couplers in telecommunication. A.3.2. Detecting single atoms by selective ionization. This may be achieved using a multistep process up to a Rydberg state. The electron and the location from where it came can then be detected with a simple electron microscope. Using a dipole blockade mechanism as discussed by Lukin et al. (2001) one should be able to implement an amplification mechanism, which will allow 100% detection efficiency (Schmiedmayer et al., 2002). A.3.3. Transmission of resonant light through a small cavity. Such a scheme may be used to detect single atoms even for moderate Q values of the cavity. The cavity could be created by two fibers with high reflectivity coatings at the exit facets, or even by a DBR fiber cavity with a small gap for the cold atoms. Fiber ends molded in a lens shape could considerably reduce the light losses due to the gap. Having atoms localized in steep traps should allow a small gap that would reduce the losses even further. A.3.4. Transmission of light through a high-Q cavity. Here, the transmission is modified by the presence of single atoms, and the light may be quite far from atomic resonance and the atoms are still detected with high probability. The basic mechanism of this detector is that atoms inside the cavity change the dispersion for the light. The high Q value makes it possible to detect very small modifications of the dispersion. In addition the cavities can be incorporated into integrated optics interferometers to measure the phase shift introduced by the presence of the atoms. Off resonant detection would allow for nondestructive atom detection (see for example Domokos et al., 2000).
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B. MESOSCOPIC PHYSICS The potentials created on an atom chip are very similar in scale and confinement to the potentials confining electrons in mesoscopic quantum electronics (Imry, 1987). There electrons move inside semiconductor structures, in our case atoms move above surfaces in atom-optical circuits. In both cases they can be manipulated using potentials in which at least one dimension is comparable to the de Broglie wavelength of the guided, trapped particle. To find similarities and differences between mesoscopic quantum electronics and mesoscopic atom optics will probably become a very rich and fascinating research field. Electrons in semiconductors interact strongly with the surrounding lattice. It is therefore hard to maintain their phase coherence over long times and distances. An atomic system on the contrary is well isolated. Furthermore, atoms (especially in a BEC) can be prepared in such a way that the temperature is extremely low with respect to the energy level spacing. The consequence is that phase coherence is maintained over much longer times and distances. This might enable us to explore new domains in mesoscopic physics, which are hard to reach with electrons. B.1. Matter wave optics in versatile potentials
A degenerate quantum gas in the atom chip will allow us to study matter wave optics in confined systems with non-trivial geometries, such as splitters, loops, interferometers, etc. One can think of building rings, quantum dots connected by tunnel junctions or quantum point contacts (Thywissen et al., 1999b), or even nearly arbitrary combinations thereof in matter wave quantum networks. For many atomic situations the electronic counterparts can easily be identified. Atom chips will allow to probe a wide parameter range of transverse ground state widths, confinement and very large aspect ratios of 105 and more. Atomic flow can be monitored by observing the expansion from an on-board reservoir along the conduit. Further perturbations and corrugations can be added at any stage to the potential by applying additional electric, magnetic or light fields to modify the quantum wire or quantum well. In this manner we can also explore how disorder in the guides may change the atomic behavior. In the following, we give details regarding three exemplary matter wave potentials on the atom chip. B.2. Interferometers
In the near future it will be essential to develop and implement interferometers, and to study through them the decoherence of internal states and external motional states. Atom chip interferometers have been discussed in detail in Sect. II.A.9. They can be built either in the spatial (Andersson et al., 2002) or in the temporal domain (Hinds et al., 2001; H~insel et al., 2001 c). Integrated
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on an atom chip, they are very sensitive devices that may be used to measure inertial forces or even to perform computation (Andersson and Barnett, 2000). Coherence properties in more complicated networks can be studied by observing interference and speckle patterns. Interferometers can also serve as probes for the understanding of surfaceatom interactions, allowing for a quantitative test of the limits imposed on the atom chip by the warm surface for both internal state and external (motional) state coherence. Since many of the important parameters scale with the spin flip life time in a trap (see Sect. V), a first important step would be to measure the (BEC) lifetime in a microtrap as a function of distance to the surface. Aside from heating and spin flips, the surface also induces 'phase noise'. Interferometers will be able to measure this subtle effect as a function of surface material type and temperature as well as atom-surface distance and spatial spread of the atomic superposition, through a reduction in the fringe visibility. Finally, by coupling microtraps (atomic quantum dots) to one of the interferometer arms, similar to the mesoscopic electron experiments (Buks et al., 1998), subtle interaction terms may be investigated, e.g., 1/r second-order dipole interactions as discussed by O'Dell et al. (2000). Internal state superpositions of atoms close to surfaces can be studied using internal state interferometers. Using Raman transitions or microwave transitions we can create superpositions, observe their lifetime and put theoretical estimates to the test. B.3. Low-dimensional systems
Much is known about the behavior of fermions in low-dimensional strongly confining systems (one- and two-dimensional systems) from mesoscopic quantum electronic experiments. By designing low-dimensional experiments using atoms (weakly interacting bosons or fermions) we expect to obtain further insight also about electronic phenomena. The role of interactions inside an atomic matter wave can range from minimal in a very dilute system to dominating in a very dense system. Low-dimensional systems are especially interesting in this context, since it is expected that the interactions between the atoms will change for different potentials. The study of the dependence of the interactions (scattering length) on the dimensionality and the degree of confinement of the system (Olshanii, 1998; G6rlitz et al., 2001; Petrov et al., 2000) will benefit from the variety of potentials available on the atom chip. B. 4. Non-linear phenomena
Another example of an interesting regime for the study of atom-atom interaction or non-linearity are multi-well potentials. Again, as mentioned in the context
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of interferometers, the splitting of a cloud of atoms into these multi-sites can be either temporal or spatial. Here calculations beyond mean field theory are relevant, and new insight may be acquired. For example, one expects a crossover from coherent splitting to number splitting in different potential configurations, depending on the height of the potential barrier, the density, and the scattering length (Menotti et al., 2001; Vardi and Anglin, 2001; Orzel et al., 2001). This phase transition has already been observed experimentally by Greiner et al. (2002). B.5. Boundary between macroscopic and microscopic description
Let us end this subsection concerning mesoscopics by noting that the ability to change the number of atoms in a system, or alternatively to address specific atoms in an interacting ensemble, will allow us to probe the boundary between the macroscopic and microscopic description. Starting from a large system, we will try to gain more and more control over the system parameters, imprinting quantum behavior onto the system. On the other hand we can try and build larger and larger systems from single quantum objects (called qubits in modern lingo), and keep individual control over the parameters. Success in such an undertaking would bring us much closer to implementing quantum information transfer and quantum information processing as discussed below.
C. QUANTUMINFORMATION The implementation of quantum information processing requires (DiVincenzo, 2000): (i) storage of the quantum information in a set of two-level systems (qubits), (ii) the processing of this information using quantum gates, and (iii) reading out the results. For a review of quantum computation we refer the reader to Bouwmeester et al. (2000). We believe that quantum optical schemes where the qubit is encoded in neutral atoms can be implemented using atom optics on integrated atom chips (Schmiedmayer et al., 2002). These promise to combine the outstanding features of quantum optical proposals, in particular quantum control and long decoherence times, with the technological capabilities of engineering microstructures implying scalability, a feature usually associated with solid-state proposals. Let us review some of the requirements: - The qubit. Using neutral atoms, the qubit can be encoded in two internal, longlived states (e.g. two different hyperfine electronic ground states). Single-qubit operations are induced as transitions between the hyperfine states of the atoms. These are introduced by external fields, using RF pulses like in NMR or in Ramsey-Bord6 interferometers, Raman transitions or adiabatic passage.
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One method to realize a qubit is to write the qubits into single atoms, which requires selective cooling and filling of atoms into the qubit sites. However, recently it was proposed that single qubits can also be written into an ensemble of atoms using 'dipole blockade' (Lukin et al., 2001). This may be simpler as it avoids the need for single-atom loading of traps. As will be pointed out below, the dipole blockade mechanism can also be used to manipulate the qubit. - E n t a n g l i n g qubits. The fundamental two-qubit quantum gate requires stateselective interaction between two qubits, which is more delicate to implement. A two-qubit quantum gate is a state-dependent operation such as a control NOT gate:
Io) 1o) Io)1 ) 11>1o>
Io) 1o), FO)ll), I1>11>,
I1)[1) ~
I1)1o).
(58)
A good way to implement such a quantum gate is by state-selective interactions, which can be switched on and off at will. This interaction can be between the qubits themselves, or mediated by a 'bus'. Neutral atoms naturally interact with each other. To achieve different phase shifts for different qubit states, either the interaction between the qubits has to be state selective, or it has to be turned on conditioned on the qubit state. There are different ways to implement quantum gates in atom optics: depending on the interaction, we distinguish between (a) the generic interactions between the atoms, like the Van der Waals interaction (Jaksch et al., 1999; Calarco et al., 2000; Briegel et al., 2000) and (b) interactions which can be switched on and off, like induced electric dipole-dipole interactions (Brennen et al., 1999; Brennen and Deutsch, 2000), including highly excited Rydberg states (Jaksch et al., 2000; Lukin et al., 2001). - Dipole-blockade quantum gates between m e s o s c o p i c atom ensembles. Lukin et al. (2001) devised a technique for the coherent manipulation of quantum information stored in collective excitations of many-atom mesoscopic ensembles by optically exciting the ensemble into states with a strong atomatom interaction. Under certain conditions the level shifts associated with these interactions can be used to block the transitions into states with more than a single excitation. The resulting dipole-blockade phenomenon closely resembles similar mesoscopic effects in nanoscale solid-state devices. It can take place in an ensemble with a size that can exceed many optical wavelengths and can be used to perform quantum gate operations between distant ensembles, each acting as a single qubit. - Cavity QED. The 2-qubit processing operation may be realized through a direct interaction (entanglement) between two atoms or through an inter-
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FIG. 49. A possible implementation of a neutral atom qubit processor on an atom chip which includes a reservoir of cold atoms in a well-defined state (for example a BEC or a degenerate Fermi gas). From there the atoms can be transported using guides or moving potentials to the processing sites. Either single atoms or small ensembles of atoms are then loaded into the qubit traps. Each qubit can be addressed either by bringing light to each individual site separately, or by illuminating the whole processor and addressing the single qubits by shifting them in and out of resonance using local electric or magnetic fields created by the nanostructures on the atom chip. We think that electric fields are preferable, since magnetic fields might produce qubit-state dependent phase shifts, which have to be corrected. A different method would also be to address the single sites using field gradients like in NMR.
mediate 'bus'. A light mode of a high-Q cavity can serve as such a 'bus' acting on an array of atoms trapped inside the cavity (Pellizzari et al., 1995). Atoms in high-Q cavities which in turn are connected with fibers, can also act as a converting device between 'flying' qubits (photons) which transverse distances, and storage qubits (atoms). The same principle can be used for entangling atoms in different cavities for a 'distributed' computation process (van Enk et al., 1998, 1999). In all of the above, the atom chip promises to enhance the feasibility of accurate a t o m - c a v i t y systems. - Input~Output. Even without high-Q cavities, an integrated atom chip, with atoms trapped in well-controlled microtraps and with individual site light elements, can probably provide input/output processes by making use o f techniques such as light scattering from trapped atomic ensembles (Duan et al., 2001), slow light (Hau et al., 1999; Vitali et al., 2000), stopped light (Phillips et al., 2001; Liu et al., 2001; Fleischhauer and Lukin, 2002) or macroscopic spin states (Duan et al., 2000; Julsgaard et al., 2001). Let us summarize the road map for quantum computation with the atom chip: one would need to implement (a) versatile traps to accurately control atoms up to the stage of entanglement;
VIII] MICROSCOPIC ATOM OPTICS: FROM WIRES TO ATOM CHIP 351 (b) controlled loading of single qubits (atoms or excitations) into these traps in well-defined internal and external states; (c) manipulation and detection of individual qubits; (d) control over decoherence; and (e) scalability to be able to achieve controlled quantum manipulation of a large number of qubits. A schematic view of a possible realization is shown in Fig. 49.
VII. Conclusion Neutral-atom manipulation using integrated microdevices is a new and extremely promising experimental approach. It combines the best of two worlds: the ability to use cold a t o m s - a well-controllable quantum system, and the immense technological capabilities of nanofabrication, micro-optics and microelectronics to manipulate and detect the atoms. In the future, a final integrated atom chip will have a reliable source of cold atoms with an efficient loading mechanism, single-mode guides for coherent transportation of atoms, nanoscale traps, movable potentials allowing controlled collisions for the creation of entanglement between atoms, highresolution light fields for the manipulation of individual atoms, and internal state-sensitive detection of atoms. All of these, including the bias fields and possibly even the light sources and the read-out electronics, could be on-board a self-contained chip. Such a robust and easy to use device would make possible advances in many different fields of quantum physics: from applications such as clocks, sensors and implementations of quantum information processing and communication, to new experimental insight into fundamental questions relating to decoherence, disorder, non-linearity, entanglement, and atom scattering in lowdimensional physics.
VIII. Acknowledgement Foremost we would like to thank all the members of the Innsbruck, now Heidelberg, atom chip group for their enthusiasm and the enormous effort they put into the experiments. We would like to thank our long-time theoretical collaborators Peter Zoller, Tommaso Calarco and Robin C6t6. The atom chips for the Innsbruck-Heidelberg experiments were fabricated by Thomas Maier at the Institut ft~r Festk6rperelektronik, Technische Universit~t Wien, Austria, and by Israel Bar-Joseph at the Sub-micron center, Weizmann Institute of Science, Israel. We would also like to extend a warm thanks to the entire atom chip community for responding so positively to our requests for information and figures. Our work was supported by many sources, most notably the Austrian
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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 , V O L . 48
ME THOD S OF MEA S URING ELE C TR ON-A TOM COLLISION CROSS SECTIONS WITH A N ATOM TRAP R.S. SCHAPPE Department of Physics, Lake Forest College, Lake Forest, Illinois 60045
M.L. KEELER Department of Physics, Wesleyan University, Middletown, Connecticut 06459
TODD A. Z I M M E R M A N and M. L A R S E N Department of Physics, University of Wisconsin, Madison, Wisconsin 53706
PAUL FENG Department of Physics, University of St. Thomas, St. Paul, Minnesota 55105
RENEE C. NESNIDAL New Focus, Inc., Middleton, Wisconsin 53562
J O H N B. BOFFARD, THAD G. WALKER, L. W A N D E R S O N and C H U N C. LIN Department of Physics, University of Wisconsin, Madison, Wisconsin 53 706 I. Introduction
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II. General E x p e r i m e n t O v e r v i e w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. T r a p p e d A t o m s as a Target B. A p p a r a t u s
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III. M e t h o d s for M e a s u r i n g Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Loss Rate M e a s u r e m e n t s
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B. E x p e r i m e n t s B a s e d on Direct D e t e c t i o n o f Final State IV. C o n c l u s i o n s
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V. A c k n o w l e d g m e n t s
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VI. 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 VII. R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I. I n t r o d u c t i o n Measurements of electron-atom collision cross sections are a subject of great interest. Until recently, a typical experiment consisted of an electron beam incident on target atoms in the form of either an atomic beam or a static gas 357
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target. In this chapter we describe a new type of atomic target: ultra-cold trapped atoms (Schappe et al., 1995, 1996; Keeler et al., 2000). An atom trap has three features that make it desirable for use as a target in scattering experiments. First, the atoms in a trap can be confined to a very small volume (diameter < 1 mm). Hence, an atom-trap target can be treated practically as a point target of atoms. This property greatly simplifies the calculation of the overlap of the target with the electron beam; absolute calibration can be done without undertaking the difficult task of determining the profiles of both the electron beam and atomic beam and then finding their convolution as must be done in a crossed beam experiment (Mark and Dunn, 1985). Second, atoms in a typical optical trap are cold, with temperatures on the order of hundreds of gK or atomic speeds on the order of ten cm/s. Thus the atoms in a trap are for all practical purposes at rest. Consequently, an atom that has undergone even a weak electron-atom collision can pick up a sufficient atomic recoil velocity to differentiate it from atoms that have not undergone a collision. Furthermore, for the investigation of excited states, the linewidth of an atomic transition for the trapped atoms is determined by the natural linewidth of the transition, and not a Doppler profile. This vastly reduces the laser intensity needed to saturate a transition. Third, atoms in the trap are constantly scattering photons from the trapping laser beams and thus the trap fluorescence is directly proportional to the number of atoms in the target. Because this signal is large ( - g W ) it can be measured easily with excellent time resolution using even a simple photodiode. The high rate of photon scattering, Doppler-free transition width, and small target size can be combined to yield a target with a large laser-excited state fraction using only low-power lasers. For example, to study electron-impact excitation out of the Na(3P) level with a conventional crossed beam target, Stumpf and Gallagher (1985) needed -100 mW of laser power to create an excited state fraction of~20%, while an atom-trap target can yield ~40% excited state fraction with only ~10 mW of laser power (Keeler et al., 2000). Collisions between cold atoms in a trap have been extensively studied since the start of atom trapping (Walker and Feng, 1994; Weiner et al., 1999). To our knowledge, however, the first application where all of the advantages of using a trap as a separate target for measuring cross sections was the work of Dinneen et al. (1992) on photoionization of Rb(5P). In their work, trapped atoms were ionized (and thus lost from the trap) by a Kr-ion laser. Atom traps have subsequently been used to study photoionization of the Rb(5D) level (Duncan et al., 2001), and of the Li(2P) (Wippel et al., 2001), Na(3P) (Wippel et al., 2001), and Cs(6P) (Marag6 et al., 1998) levels. Atom traps have also been used in the study of heavy ion collisions (Flechard et al., 2001; Turkstra et al., 2001; van der Poel et al., 2001). In this chapter we describe the techniques for using trapped atoms for
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measuring a wide variety of electron-atom cross sections including total, excitation, and ionization cross sections, from both the ground state and a laser excited state of trapped atoms. In Sect. II we briefly describe the technique of atom trapping in the context of electron-atom collision studies. In Sect. III we discuss two major ways to use atom traps to measure cross sections, the trap-loss technique and the direct detection technique.
II. General Experiment Overview A. TRAPPED ATOMS AS A TARGET
A.1. Basics of magneto-optical trapping In our experiments we use a standard Magneto Optical Trap (MOT) (Chu et al., 1985) as the target for our collision experiments. The general principles of atom trapping are described elsewhere (Foot, 1991; Metcalf and van der Straten, 1999; Wieman et al., 1995); in this section we briefly touch upon those subjects that are relevant to the use of atom traps as a target in electron-impact studies. An atom trap requires two forces: a velocity-dependent viscous force to cool atoms and a spatially-dependent restoring force to confine atoms. In an optical trap both forces are derived from modulating the photon scattering rate, and thus the resultant momentum transfer rate, between an atom and an opposing laser beam. A standard MOT consists of (a) three pairs of mutually perpendicular, circularly-polarized, counter-propagating laser beams tuned to a frequency slightly below resonance ("red-detuned") and (b) a quadrupole magnetic field (formed using a pair of anti-Helmholtz coils) with zero (minimum) field located at the intersection of the laser beams. An atom moving opposite the direction of a red-detuned laser beam is Doppler-shifted closer into resonance, increasing the photon scattering rate, which produces the velocity-dependent force that cools the atoms (i.e., an optical molasses). For an atom situated away from the magnetic field zero of the trap, the Zeeman effect shifts one of the mj sub-levels of the atom closer into resonance with the corresponding circularly polarized laser beam. This beam pushes the atom back towards the center of the trap, resulting in the restoring force that confines the atoms. In order to make our discussion concrete we analyze the particular trap we use that functions on the 5281/2 --+ 52P3/2 (780nm) transition in 85Rb (see Fig. 1). The 5281/2 level of 85Rb has two hyperfine levels with F = 2 and F = 3. The trapping laser is red-detuned from 5281/2 F = 3 --~ 52p3/2 F ~ = 4 recycling transition. On the order of 1 in 102 to 103 absorption cycles, however, will be into the 52p3/2 F ~ = 3 level which can decay into the 5281/2F - 2 level. Atoms in this F = 2 "dark" level are non-resonant with the primary trapping laser beam and are free to drift out of the trapping region. To prevent this, a second laser beam
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FIG. 1. Hyperfine structure of the 52S1/2 and 52P3/2 levels of 85Rb. [the "hyperfine (re)pump"] is needed to excite atoms to either the 52p3/2 F t = 2 or F ~ - 3 levels which can both decay back to the 52S1/2 F - 3 trapped level. A.2. Using a M O T as a target
There are a number of issues involved in using a MOT as a target for collision studies. Some of these issues are challenges; for example, a MOT relies on a quadrupole magnetic field that interferes with the propagation of an electron beam. On the other hand, the very complexity of atom trapping allows the experimentalist many options for achieving a given task. For instance, the trap (i.e., the confining and cooling forces) can be turned on/off in many ways: (a) by turning on/off the magnetic field, (b) by modulation of the trapping laser, or (c) by modulation of the hyperfine repump laser. Each method has its own advantages and disadvantages in terms of complexity, cost, and time response. In fact, there are a number of different natural time scales involved in the trapping of atoms as listed in Table I. For example, if the trapping forces on a cloud of trapped atoms are turned off for 5 ms and then restored, few if any atoms will be lost from the trap. But, if the trapping forces are left off for 30 ms, all of the atoms will be lost from the trap due to ballistic expansion. This sets a natural upper bound on the time the trap can be left off. Alternatively, the decay time of the magnetic field sets a lower bound on the time the trap must be turned off to propagate an electron beam. Finally, we note that trap parameters and controls are richly interconnected. For instance, to increase the fraction of trapped atoms in the 5P excited level one might increase the photon scattering rate, which depends on both the intensity and frequency detuning of the trapping laser. This will, however, also affect the forces on the trapped atoms, which causes changes in
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Table I Approximate times for processes in a typical Rb MOT Process
Time
Lifetime o f 52P3/2 excited level
27 ns
Transfer time by Raman scattering into 52S1/2F = 2 dark state Decay time of magnetic field (L/R)
~25 gs ~0.5 ms
Time to escape the trap region after the trap is turned off
~20 ms
Time to load trap (trap lifetime)
seconds
the loading rate of atoms into the trap, the trap depth, and even the total number of atoms in the trap.
A.3. Polarization o f atoms in the trap One problem of interest in electron-scattering experiments is whether or not the target atoms have some preferred polarization relative to the electron-beam axis. When the trap is on, the three sets of counter-propagating laser beams assure that average light polarization taken over the entire trap is zero (Metcalf and van der Straten, 1999). At any particular location in the trap, however, the atoms can be polarized along the local magnetic field direction due to the optical pumping of the atoms; only the spatial average is zero 1. We also consider what happens when the magnetic field and hyperfine repumping lasers are both turned off, allowing atoms to fall into the F = 2 dark state. When the current through the anti-Helmholtz coils is turned off, the magnetic field gradually dies out, and only the local earth's magnetic field is left. Since the magnetic field decay time (~500 gs) is much longer than the precession time of an atom, the polarization of the atoms will adiabatically follow the local magnetic field direction so that the atom cloud may have a small residual polarization along the earth's field 2. To estimate the magnitude of these effects, we have developed a simple rate equation model for a one-dimensional atom-trap model as described in Appendix 1. For a typical trap, the model yields a polarization at the edge of the atom cloud of about 0.16, and a value of 0.1 averaged over the entire volume of trapped atoms. This value is an overestimate since it ignores the other four laser beams that can depolarize the atoms via absorption followed by spontaneous
1 Intensityimbalances in the counter-propagatinglaser beams could also lead to polarization effects. 2 This potential problem could be greatly reduced by leaving the hyperfine repump on until the trapping magnetic field has completely decayed away.
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emission. We conclude that the polarization of the cloud of trapped atoms (both before and after the trap's anti-Helmholtz coils are turned off) is small (P < 0.1). B. APPARATUS
In the following four sub-sections we describe the components of our atom trap common to all our measurements: the laser system, the vacuum system, the magnetic field coils, and the electron gun. Further details on the components unique to each type of cross-section measurement are described in Sect. III. B. 1. Laser system
The optics associated with the frequency stabilization of the trapping laser are shown in Fig. 2, while Fig. 3 illustrates the optics associated with the trap. We use a commercially available 30 or 70mW, 780nm diode laser in an external cavity arrangement (MacAdam et al., 1992; Arnold et al., 1998) using a Littrowmounted 1200 groove/mm holographic grating for feedback. Tuning of the laser frequency is achieved by varying the laser diode current and temperature, and coarse adjustment of the grating angle. A piezo-electric stack is installed on the horizontal grating adjustment to provide fine-control of the laser wavelength by changing the effective cavity length. The linearly-polarized, elliptical output beam from the diode laser is converted into a circular profile with an anamorphic prism pair, and passed through a 40 dB optical isolator to prevent disruptive optical feedback from retro-reflected laser light. A Doppler-free saturated absorption spectrometer is used to determine the laser wavelength relative to the Rb (52S1/2 F = 3 + 52p3/2 F t = 4) trapping transition. Since we want to lock the laser to a frequency A less than this trapping transition, we use an acoustic-optical modulator (AOM) to downshift the frequency of the beam entering the saturated absorption spectrometer relative to the trapping beam. Feedback from the saturated absorption spectrometer (Wieman et al., 1995) is used to stabilize the laser frequency at the peak of the crossover feature between the F = 3 ---+ F ' = 2 and F - 3 --+ F ' = 4 transitions, located 92 MHz below 52S1/2 F = 3 ---+ 52p3/2 F ' = 4 transition. A single mode optical fiber is used to decouple the laser alignment from the trap optics alignment and to provide spatial filtering. A beam expanding telescope is used to convert the output of the optical fiber into a collimated beam approximately 1 cm in diameter. The linearly polarized laser beam is split into three beams using half-wave plates and polarizing beamsplitter cubes. Since the magnetic field gradient is twice as large along the axis of the anti-Helmholtz coils (the z-axis) as along the x and y axes, the power in the beams are set in the ratio 2:2:1 so as to roughly equalize the x , y , z restoring forces. The linearly polarized beams are converted into circularly polarized beams with zero-order quarter-wave plates. Since the direction of the
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Fie. 2. Trapping laser. 780 nm light is generated by an external cavity, grating-stabilized 70 mW diode laser. The portion of the beam directed into the Doppler-free saturated absorption spectrometer is downshifted by -80 MHz by an AOM to allow locking to the large 2-4 crossover peak (see insert for sample spectrum). The purpose of the LCVR is described in Sect. III.B.2. magnetic field gradient is opposite for the z-axis relative to the x&y axes, the beam incident on the trap along the z-axis is left circularly polarized, while the x&y beams are right circularly polarized. The hyperfine repump beam is obtained by coupling a 2 . 9 1 5 G H z microwave modulation signal into the primary trapping diode laser drive current (Feng and Walker, 1995). This modulation creates a sideband approximately 1% the intensity of the primary laser transition at the frequency of the 5281/2F = 2 ~ 5 2 p 3 / 2 F t -- 3 transition (see Fig. 1). Since only -1 in 200 photons result in a Raman scattering event that populates the dark state, this weak sideband intensity is enough to keep the dark state essentially empty. If, however, the hyperfine repump is turned off (by shifting the microwave
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FIG. 3. Top view of the atom trap chamber (and optics) used in electron-impactexcitation studies (Sect. III.B.4), but contains all the elements used in our earlier trap-loss measurements(Sect. Ill.A). Photodiode and CCD camera are used to monitor the trap. For clarity, the z laser beam and associated optics are not shown. modulation frequency -`200 MHz off-resonance), it takes only --400 photon scattering cycles (< 50 ~ts) to transfer all of the atoms in the trap into the dark state. This is the primary way we switch the optical forces on the atom on/off and convert the atoms in the trap from a mixture of atoms in the 52S1/2 F = 3 and 52p3/2 F ~ = 4 levels when the hyperfine repumping is on, to only atoms in the 52S1/2 F = 2 level when the hyperfine repumping is off. Thus it is straightforward to produce a target of all ground level atoms or a mixed target of ground and excited level atoms. Typically we have 30 mW of power directly out of the external-cavity diode laser. The total laser power (before being split into the three trapping beams) is 10 mW. Most of the losses occur in coupling the light into the optical fiber. The
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laser frequency is very stable (< 1 MHz), typically remaining locked >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
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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
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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
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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.
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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
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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 '
!
'
!
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i
'
no e l e c t r o n b e a m
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.= ..,--,,..:~. " b ~ - _-~r"or'-=~.,L lb-l
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371
]
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.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
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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
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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
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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
.
.
.
.
.
!
.
.
.
.
!
.
.
.
.
!
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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.
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[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.
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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 <~ 5% of the atoms are in the F = 2 dark state at any one time. This has a negligible effect on our f~ measurements. If the intensity of the sidebands is reduced to 0.5%, the fraction in the dark state soars to 22%. To verify that there are essentially no atoms in the dark state we increase the size of the microwave-modulation sideband intensity until the trap fluorescence is essentially independent of the repump intensity.
B.3. Ionization cross section out o f the 5P level In this experiment trapped atoms ionized via electron-atom collisions are directly detected using a channel electron multiplier (CEM) detector (Keeler et al., 2000). A pulsed electric field is used to extract the ions, allowing time of flight (TOF) analysis to separate out the individual Rb +, Rb++, ... channels. The ionization cross section of the 52p level is determined relative to the known 52S ionization cross section (Schappe et al., 1996) using three different measurements: (1) the ion signal recorded when no trap is present, Coff; (2) the ion signal from a trap target with the hyperfine repump off at the time of the electron beam pulse (only 52S atoms), Css; and (3) the signal with the hyperfine repump on (a mixed target of 5 2 S and 52P atoms), C m i x . The Coff signal is the sum of dark counts (D) and ions created by ionizing the background Rb gas in the vacuum chamber (BG). The Css count is equal to the Coff signal plus ions created by electron-atom collisions with the 52S atoms in the trap (Sss). In addition to the background
380
[III
S c h a p p e et al. 4.0 ms
Magnetic Fiel'~']'
I trapoZ. I
Hypermfi;e~' 5"~on'~Ysigna 1 ~ 5S&5P mixed signal
i_ .I [0.6 ms-]
Electron Beam Ion Extractor Ion Detection Gate
]
FIG. 11. Timing diagram for measuring ionization out of the Rb(5P) level. The target is controlled by turning the hyperfine pumping laser on and off: no trapped atoms are present when it is always off, a 5S target is achieved by turning it off only during the electron beam pulse, and a mixed 5S&5P target is obtained by leaving it on during the electron beam pulse.
sources, the Cmix signal includes a contribution from the fraction of atoms in the excited state, fe Ssp, and a contribution from the fraction of atoms in the ground state, (1 - f e ) Sss. Hence, Cmix = D + B G + feS5p + ( 1 - f~) $5s.
(10)
In general, the electron-impact signal rates (BG, S5s, S5p) depend upon a vast collection of experimental parameters including: the electron beam current, the number of target atoms (in the trap and in the background vapor), and the 52S and 52p ionization cross sections. Assuming a constant number of atoms in the trap for both the 52S and mixed 52S & 52p targets, and a constant overlap of the electron beam with the trapped atoms, it is possible to find the ratio of the 52p to the 52S ionization cross sections from the expression o+(5P)
a+(ss)
(Cmix - Coff) - (1 - f e ) ( C 5 s
L(Css -Co~)
-
Coff)
(11)
Note that only measurements of the three signal rates and of the excited state fraction are needed. Quantities such as the electron beam current, number of atoms in the trap, and ion collection efficiency are essentially the same for the ground state and mixed targets and thus divide out. A timing diagram of the three phases of the experiment is shown in Fig. 11. After a delay to allow the magnetic field to decay, the trap is hit with electrons for 0.6 ms. After a 100 ~ts delay to make sure there are no longer any electrons in the collision region, an ion extractor electrode (used to pull ions into the
III]
ELECTRON-ATOM COLLISION CROSS SECTIONS t
16
i
|
Rb++
.mr
12
~
8
r._~
4
o
0
m
'
2
3
|
!
!
6
7
381
Rb +
4
5
Time
(gs)
8
Fie. 12. Time of flight spectra for detection of Rb ions at 150 eV. The dotted line is background only (no trap), the solid line is with an atom trap target (53 only).
CEM detector assembly) receives a high voltage pulse. After the ion signal is recorded, the magnetic field is turned back on. To replace atoms lost in electron-atom collisions, additional atoms are added to the trap for 100 ms. We obtain reasonable signal rates by averaging counting rates for 300 pulses, which corresponds to 30 s. To obtain a signal from only trapped atoms in the 5231/2 F = 2 dark state, the hyperfine repump is turned off for the duration of the electron-beam pulse. A mixed 523 & 5 2 p target is obtained by leaving the hyperfine repump on during the electron beam pulse. To maintain a constant number of atoms in the trap for the two cycles the hyperfine pumping is turned off for 0.6 ms shortly before the electron beam pulse to maintain a constant repump duty cycle. A background run (no trap) is obtained by leaving the hyperfine pump (or magnetic field) off. When switching from the trap off data run to one of the trap signal runs, we first reload the trap for 10 seconds before acquiring data. A time of flight spectra at an electron energy of 150 eV is shown in Fig. 12. Singly ionized Rb atoms from the trap arrive at the detector at (4.5+0.5)~ts. As seen in Fig. 12, ions formed by ionization of background gas atoms arrive at the detector with a much broader range of arrival times, approximately (3.5+2)~ts. To maximize the SNR ratio, the ion signal is recorded with a box-car integrator with a signal gate centered 4.5 ~ts after the start of the ion-extractor pulse with a total width of 1 ~ts. Based upon the arrival time of the Rb + ions, the Rb ++ ions are expected to arrive at the detector at 3.2 pts, versus an observed peak arrival time of 2.9 pts. This illustrates another advantage of the trapped atom target in that the Rb + and Rb ++ yields can be determined separately by means of a simple time-of-flight detection system. The largest single source of ions in this experiment is from ionization of background gas atoms. The number of ions produced by the electron beam is proportional to f n(F)J(F)dY, where n is the number density of atoms, and J is the electron beam current density. For ionization of trapped atoms, this is equal to NJo, where N is the number of atoms in the trap, and J0 is the current density
382
Schappe et al.
[Ill
at the trap location. For ionization of atoms in the Rb background vapor, ions are created along the entire length of the electron beam from the electron gun to the Faraday cup. The number of background ions created is proportional to nscleL, where nBc is the number density of background atoms, Ie is the total electron beam current, and L is the effective length of the electron beam from which ions are extracted into the detector. Using typical values (N ~ 106, J0 ~ 2 mA/cm 2, nsa ~ 109 cm -3, Ie ,~ 100gA, L ~ 1 cm), the ion yield from the background vapor is a factor of fifty larger than the ion yield from the trap. Two factors help minimize the contribution from background gas ionization: (1) the ion-extractor is designed to minimize the effective range L from which ions are collected (since ions from the trap are well localized at the trap center), and (2) TOF selection is used to separate the 'cold' trap ions from the 'hot' ions created by ionization of room temperature background gas atoms. Details about the design and operation of the ion extractor are given in Keeler et al. (2000). Computer modeling of ion trajectories indicate that the transmission of the ions to the detector is essentially constant for ions within +0.25 cm of the extraction axis. In principle, the extractor could be designed to have a smaller extraction volume (i.e., the size of the ball of trapped atoms) to further eliminate the inclusion of background gas ions, but this would require the placement and alignment of the extractor relative to the cloud of trapped atoms to tolerances beyond our control. To maximize the alignment of the trap to the fixed location of the ion-extractor, shim coils are used to move the zero of the trapping magnetic field. The contribution from ionization of background gas atoms can also be improved by switching from a vapor loaded trap to one fed by a collimated, slowed atomic beam. The estimated uncertainty in our o+(5P)/o+(5S) measurements is on the order of 28%. The 52p ionization cross section is obtained by multiplying the measured ratio in Eq. (11) by the ionization cross section out of the 52S ground state. Note that the total ionization cross section measured in Sect. III.A.3 using the lossrate technique (Schappe et al., 1996) is the sum of the Rb +, Rb++, ... ionization cross sections, i.e., Eq. (7). Thus the contribution from higher ionization levels must be removed before these values can be used to place the 52p results on an absolute scale. Combining the uncertainties in the ratio measurement, the ground state total ionization cross section, and the correction of Ocount to o +, the total uncertainty in our 52p cross sections (Keeler et al., 2000) is estimated to be +33%. It is interesting to compare the 52p ionization cross-section results obtained by direct detection of the ions with those using the trap-loss technique as described in Sect. III.A.4. The loss-rate measurements must be interpreted with some trepidation since the method forces us to assume that loss rate from atoms in the trap during a brief electron beam pulse is the same if the hyperfine repumping is on or off. That is, the recapture rate of atoms is the same for 52S atoms with no optical molasses and for a mixture of 52S and 52p with an optical molasses.
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ELECTRON-ATOM COLLISION CROSS SECTIONS
383
Nonetheless, at 50 eV, the loss rate measurement (Keeler et al., 2000) yields a total ionization cross section of 2.2x 10-15 cm 2 in remarkably good agreement with the o+(5P) direct detection value of (2.5+0.7)x 10-15 cm 2. B.4. Excitation cross sections out o f the 5P leoel Measurement of electron-impact excitation cross sections into discrete levels of Rb from atoms starting in the 52p level can be accomplished in much the same way as the ionization cross sections described in the last subsection except that one detects photons from the decay of excited atoms rather than ions themselves. The general methods for measuring electron-impact excitation cross sections by optical detection with a static or atomic-beam target have been discussed extensively in the literature (Filippelli et al., 1994). Generally the determination of excitation cross sections requires difficult absolute measurements of both the radiation from the excited atoms and the target-atom number density. Similar to what was done to measure the 52p ionization cross section, we bypass these difficult steps by using the trap to measure the ratio of the 52p --+ n2L to 52S ~ nZL excitation cross sections and then using previously measured 52S ground state excitation results to place the 52p results on an absolute scale. Excitation cross sections from the Rb(52S) ground level into the 52p, 72S, 82S, 52D, and 62D levels are available in the literature (Chen and Gallagher, 1978; Wei et al., 1993). The apparatus for the electron-impact excitation work is shown in Fig. 3. The electron gun and Faraday cup are almost identical to those used in our earlier loss-rate measurements described in Sect. II.B.4. The component unique to this experiment is the optical system to detect the fluorescence from atoms excited by the electron beam. An f~ 1.33 lens located within the trapping chamber collimates light emitted from the electron-impact excitation of trapped atoms. To cut down on the large amount of scattered light at the wavelength of the trapping lasers, we use two holographic notch filters each having an effective optical density of 4 at 780 nm. A narrowband interference filter (1 nm FWHM) is used to provide spectral isolation of an individual transition line. Scattered laser light is further blocked from reaching the PMT by using a 2 mm field stop that acts as the limiting aperture of the optical system. Photons are detected with a Hamamatsu R943-02 (GaAs photocathode) PMT operating in photon-counting mode. The large background signal that comes from the intense trapping laser beams represents one of the greatest challenges to this experiment. To illustrate the difficulties due to this background, consider an attempt to measure the electronimpact excitation cross section into the 52p1/2 level which decays to the ground state with transition wavelength of 795 nm. To simplify our illustration, we consider light from only three sources: electron-impact excitation signal at 795 nm, trap fluorescence at 780nm, and scattered laser light at 780 nm. There are approximately 106 atoms in the trap (assumed to have a radius of 0.5 mm).
384
S c h a p p e et al.
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Accounting for the solid angle of the optical system, and assuming a typical cross section value of 10- 1 6 cm 2, and an effective electron beam current of 10 gA passing through the trap volume, the electron-impact excitation signal is expected to be approximately 3 x 10-14 W. If we are measuring the excitation cross section out of the laser-excited level, the atoms in the trap will be fluorescing at the same time we are looking for electron-impact excitation signal. The 780 nm trap fluorescence radiated into the solid angle of the optical system is approximately 2 x 10-6 W. Note that as this light is co-located in space and time with the electron excitation signal, the only way to remove it is by wavelength selection. For a 1 nm FWHM filter centered at 795 nm, the transmission at 780 nm is ~ 10--4. With only 10 -4 attenuation, the trap fluorescence is four orders of magnitude larger than the expected excitation signal, hence the need for the holographic notch filters which have a high transmission (>80%) at all wavelengths except the 780nm laser wavelength. Another source of scattered laser light arises from a beam reflecting off of surfaces near the trap region. For example, to keep the space charge spreading of the electron beam to a minimum, the electron gun is located within 12 mm of the collision region. For a laser beam with a 1/e 2 diameter of 1 cm, we might expect ~1% of the beam to be reflected off of the electron gun. For an average beam power of 3 mW, this corresponds to 3 x 10-s W of scattered 780nm light. Two additional measures beyond the wavelength-selection filters are used to minimize this light source. First, surfaces close to the viewing region are coated with colloidal graphite which has a measured reflection coefficient of <1% at 780 nm. Second, since this light does not originate at the exact location of the trap, most of the scattered light can be removed by limiting the viewing region to the trap volume. Data is taken in three phases similar to those used in the 5P ionization experiment (see Sect. III.B.3). However, in comparison to the ionization experiment where there were four major sources of ion counts (two signal sources and two background sources), there are seven major sources of photon counts (two signal sources and five background sources). A detailed description of these sources and typical count rates from each source are listed in Table II. The signal is extracted from these numerous background sources using the timing cycles shown in Fig. 13. In the first phase two counting gates are used. Within each of these gates the electron gun is on half the time and the hyperfine repump is on half the time. In the A-gate the timing of electron beam and laser pulses are synchronous, while in the B-gate the two pulses are asynchronous. When the pulses are synchronous, the resulting signal is from a mixed 52p & 52S target, while the asynchronous gating produces excitation from only the 52S level. This leads to a very simple determination of the relative difference of the 52p and 52S signal rates. Using the definitions provided in Table 2: A1 = 2 S L + 2L + B G + B F + T F + (1 - f e ) E s + f e E p ,
(12a)
B 1 = 2 S L + 2L + B G + B F + T F + E s ,
(12b)
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ELECTRON-ATOM COLLISION CROSS SECTIONS
385
Table II Light sources in an atom-trap experiment measuring the optical emission cross section for the i ~ j transition Name
BG
Process
/l
Size a
Excitation from the laser excited 52p level into level i
/l~j
100
Excitation from the 52S ground state into level i
/lij
130
Excitation of background gas (52S) atoms into level i
/1#-
6
TF
Trap fluorescence
~rap
1
BF
Fluorescence from background gas
/1trap
1
L
Scattered laser light
/1trap
2
SL
Other stray light sources (i.e., cathode blackbody emission)
all/1
20
a Relative signal sizes are based upon measurements of the 52P1/2 ~ 52S1/2 optical emission cross section, after taking all measures to reduce scattered laser light at/1trap = 780 nm.
L
Magnetic Field Hyperfine Pump
[
Electron Beam
1.35 ms
.L
T
i
i
i
!
i
ilj
37.8 ms
J
i
i
i i
!ll
I
!
r
Photon Counting Gate ~0.50 ms'],, 9
Magnetic Field
I
i
.ypo ne um
I
iil
!
il [
Electron Beam
l
Photon Counting Gate
~-~ .~,
! ,~.-
0.25 ms Magnetic Field
I
j!
Hyperfine Pump
!
Electron Beam
[
jJI
i
|
Photon Counting Gate FIG. 13. Timing diagram for 5P electron-impact excitation experiment. Phase-1 measures excitation from the mixed target. Phase-2 measures excitation from the 5S target, and Phase-3 measures the background. We accumulate counts in each phase for 2000 cycles (~100 s).
386
[IV
Schappe et al.
SO
A1 - B1
: fe(Ep
(~3)
- Es).
In phase two, we record the electron-impact excitation signal from the trap, but with the hyperfine repumping off during the entire counting gate; hence signal arises only from excitation of the 52S level. Phase three is the same as phase two, except the hyperfine repump is always off, so we never have a trap. The difference in these two counts gives the signal rate for excitation of trapped atom in the 52S ground state, A2 - A3 : (SL + L + BG + Es) - (SL + L + BG) : Es.
(14)
The signal rates Es and Ep are proportional to the cross sections for the respective 52S ~ n2L and 52P ---. nZL electron-impact excitation processes. In addition, they are also proportional to the total number of atoms in the trap, the electron beam current density at the location of the trap, the optical system detection efficiency, and the branching ratio of the particular nL ~ n~L ' transition observed. All of these other factors, however, are the same for excitation from both the initial levels, so the ratio of cross sections can be found from the ratio o(SeP--~neL)_ 1 (A1-B1) o(52S~n2L) fe A 2 - A 3
+1.
(15)
Thus the unknown o(52P -~ n2L) excited state excitation cross section can be found from the four photon counter readings, the measured excited state fraction (see Sect. III.B.2) and the known o(52S -~ n2L) ground state excitation cross section. Due to the large amount of background light, the signal to noise ratio of these measurements is very low. Long counting times (-2 days per energy) are needed to reduce the statistical uncertainty in our measurements to less than 10%. We estimate the total uncertainty in our measurements including that from the f~ measurement and from the uncertainty in the ground state excitation cross section to be 35%. Preliminary measurements for excitation into the 72S indicate that the 52p ~ 72S cross section is significantly larger than the 52S -+ 72S cross section as predicted by earlier theoretical calculations (Krishnan and Stumpf, 1992). A further discussion of the results of this experiment is outside the scope of this chapter and will be published separately.
IV. Conclusions The unique characteristics of trapped-atom targets makes them a powerful tool for measuring collision cross sections. The small size of the target minimizes
VI]
ELECTRON-ATOM COLLISION CROSS SECTIONS
387
the need for beam profile and overlap calculations. The low temperature (low velocity) of the trapped atoms makes them very sensitive to collisions so that even small angle scattering (low-recoil) events can be detected. Accurate measurements of the relative trap population are made by simple measurements of the trap fluorescence and suitable experimental design can eliminate the need to determine the absolute target density. These distinctive features allow trap-loss rate measurements which produce very accurate total scattering cross sections (uncertainty <10%) that include scattering events down to very shallow angles. Trap-loss measurements also produce ionization cross sections out of the ground state with uncertainties of < 14%. Also, it is possible to create a trap target with a significant and well-known fraction of atoms in an excited state. By directly detecting the collisionally excited atoms (or ions), it is also possible to measure ionization and excitation cross sections out of the 5 2 p laser-excited level. Our present discussion has mainly been limited to measuring Rb cross sections with an atom trap. However, a wide variety atomic species can be trapped, including all of the alkali atoms, alkaline-earth, and the noble gases (in metastable levels) (Metcalf and van der Straten, 1999). Work is also ongoing to cool and trap molecules (Bethlem et al., 2000). In comparison to the low polarization of a typical trap (see Sect. II.A.3), it is also possible to create spinpolarized (P ~ 0.7) atom traps (Walker et al., 1992), which could be used as targets for a new class of electron collision experiments.
V. Acknowledgments This work was supported by the National Science Foundation and the U.S. Air Force Office of Scientific Research.
VI. Appendix. Numerical Model for Residual Polarization In this appendix we derive a numerical value for the residual polarization of atoms in a simplified model atom trap using an idealized alkali atom with no nuclear spin. We further simplify the system by considering only a onedimensional trap with two oppositely directed incident laser beams as shown in Fig. 14. The polarization in the target is proportional to the difference in 1 sublevels in the 2S~/2 level. the populations of the m j = +~1 and m j - For a stationary atom located at the center of the trap where the magnetic field is zero, the atom is equally likely to absorb photons from either the o + beam approaching from the left, or the a - beam incident from the right, resulting in no net optical pumping, and thus no polarization. If the atom is displaced to the right, however, the Zeeman splitting caused by the non-zero magnetic field in this region shifts the A m j = --1 transitions closer into resonance, causing a
Schappe et al.
388 (a)
[VI
(3.+
(y-
I
(b)
m=
f -3/2
- 1/2
1/2
3/2
3
4
5
6
1
2
FIG. 14. (a) A 1D model atom trap is located at zero magnetic field. The spins of the photons entering the trap are anti-parallel to the local direction of the magnetic field. (b) We denote the 2S1/2 m=- 89and+l states by the labels l&2, and the 2P3/2 m - ~,- 89 and +3 states by the labels 3 through 6. slight excess in scattering from the o - beam approaching from the right. Besides pushing the atom back towards the center of the trap, this will also tend to optically pump atoms into the mj -_ - ~ 1 sublevel The local force on the atoms in the trap due to the trapping lasers depends on the local polarization. The force is increased by optical pumping which pumps the atoms into states with a high differential scattering rate between o + and o - light. We denote the states in the 2S1/2 ground level with m -- - ~i a n d m = + l by the 3 labels 1 and 2, respectively, and the states in the 2p3/2 excited level with m - 2, l ~, I and ~3 by the labels 3 , 4, 5 , and 6, respectively (see Fig. 14). At a given ~., location in space the rate of change of the population NI of state 1 is given by
dN1 _ 2 Rz4N2 - 2RlsN1 dt 3 3
dN2 , dt
(16)
where RI5 and R24 are the absorption rates for the levels 1 and 2 into states 5 and 4, respectively. The absorption of laser light between levels 2 and 6 and between levels 1 and 3 do not affect the ground level populations since the 2 excited levels all decay back to the same initial state. The factors 5 are the branching ratios out of the excited states 3 and 4 to return to the opposite spin ground state. The population of state 2 is given by N2 = 1 - N1. The population of state 2 in the steady state is given by N2 = R15/(R15 + R24). The local polarization of the atoms at a particular location in space is given by P = N2 - N 1 = (R15 - R z 4 ) / ( R 1 5 + R24). The rate of absorption of light between state i and state j is given by 16~ij 2
Rij = [09- o90 + (gumu -gtml)ff2] 2 + F2/4 '
(17)
where f2 = ~ B is the Larmor precession frequency of the atoms in the magnetic field, gu = 4 and gt = 2 are the Land6 g-values of the 2p3/2 and 2S1/2 levels,
VII]
E L E C T R O N - A T O M C O L L I S I O N CROSS SECTIONS
389
F is the natural linewidth of the atomic transition, o00 is the natural frequency o f the transition, and o) is the frequency o f the laser light with a Rabi frequency e~. If we define the detuning of the laser as A = o) - o)0, in the approximation where A >> ~ , the denominator can be expanded, resulting in
Rij ~ A2 + F2/4
I ]
1 - A2 + F2/4 (gumu - g t m t )
9
(18)
Inserting the definition o f the oscillator strength o f the transition, f u , we obtain Rij = R o f u
2A~ ) 1 - A2 + F2/4 (gumu - gzml) 9
(19)
From this we can obtain the individual absorption rates of the m sublevels, R26
=
R15
:
R0(1 - b),
(20a)
R0(1
(20b)
R13 = R0(1 + b), l R24 = ~R0(1 + 5 b),
(20c) (20d)
where b = 2Aff2/(A 2 + F2/4). The polarization of the atoms at a particular location in space is R15-R24
_
P = Rl5 + R24
5b"
(21)
3
Typical values for an atom trap used in our work are a detuning A/2Jr - 12 MHz, a magnetic field gradient of 9G/cm, and an atom-trap size of approximately 0.5 mm diameter. Thus the magnetic field at the edge of the atom cloud is approximately 0.2 G, which yields a precession frequency o f f2 =/z0B - 6.3 x 105 Hz. Inserting the F/2sr = 5.9 MHz natural linewidth of the Rb resonance transition along with these values into Eq. (21) yields a polarization value at the edge o f the atom cloud of 0.16. We have extended this model to the F = 3 case of interest here; numerical solution gives P = - 1 . 2 9 b , implying a polarization at the edge of the cloud of 0.21.
VII.
References
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Dedman, C.J., Baldwin, K.G.H., and Colla, M. (2001). Rev. Sci. Inst. 72, 4055. Dinneen, T.E, Wallace, C.D., Tan, K.Y.N., and Gould, EL. (1992). Opt. Lett. 17, 1706. Duncan, B.C., Sanchez-Villicana, V., Gould, EL., and ~adeghpour, H.R. (2001). Phys. Rev. A 63, 043411. Feng, E, and Walker, T. (1995). Am. J. Phys. 63, 905. Filippelli, A.R., Lin, C.C., Anderson, L.W., and McConkey, J.W. (1994). Adv. At. Mol. Opt. Phys. 33, 1. Flechard, X., Nguyen, H., Wells, E., Ben-Itzhak, I., and DePaola, B.D. (2001). Phys. Rev. Lett. 87, 123203. Foot, C.J. (1991). Contemp. Phys. 32, 369. Hansen, E.W., and Law, E-L. (1985). J. Opt. Soc. Am. A 2, 510. Hoffmann, D., Bali, S., and Walker, T. (1996). Phys. Rev. A 54, R1030. Javanainen, J. (1993). J. Opt. Soc. Am. B 10, 572. Keeler, M.L., Anderson, L.W., and Lin, C.C. (2000). Phys. Rev. Lett. 85, 3353. Krishnan, U., and Stumpf, B. (1992). At. Data Nucl. Data Tables 51, 151. MacAdam, K.B., Steinbach, A., and Wieman, C. (1992). Am. J. Phys. 60, 1098. Marag6, O., Ciampini, D., Fuso, E, Arimondo, E., Gabbanini, C., and Manson, S.T. (1998). Phys. Rev. A 57, R4110. Mark, T.D., and Dunn, G.H., eds. (1985). "Electron Impact Ionization." Springer, New York. McDaniel, E.W. (1989). "Atomic Collisions: Electron and Photon Projectiles." Wiley, New York. Metcalf, H.J., and van der Straten, P. (1999). "Laser Cooling and Trapping." Springer, New York. Phelps, J.O., Solomon, J.E., Korff, D.E, Lin, C.C., and Lee, E.T.P. (1979). Phys. Rev. A 20, 1418. Schappe, R.S., Feng, P., Anderson, L.W., Lin, C.C., and Walker, T. (1995). Europhys. Lett. 29, 439. Schappe, R.S., Walker, T., Anderson, L.W., and Lin, C.C. (1996). Phys. Rev. Lett. 76, 4328. Stumpf, B., and Gallagher, A. (1985). Phys. Rev. A 32, 3344. Turkstra, J.W., Hoekstra, R., Knoop, S., Meyer, D., Morgenstern, R., and Olson, R.E. (2001). Phys. Rev. Lett. 87, 123202. van der Poel, M., Nielsen, C.V., Gearba, M.-A., and Andersen, N. (2001). Phys. Rev. Lett. 87, 123201. Walker, T., and Feng, P. (1994). Adv. At. Mol. Opt. Phys. 34, 125. Walker, T., Feng, P., Hoffmann, D., and Williamson, R.S. (1992). Phys. Rev. Lett. 69, 2168. Wei, Z., Flynn, C., Redd, A., and Stumpf, B. (1993). Phys. Rev. A 47, 1918. Weiner, J., Bagnato, V.S., Zilio, S., and Julienne, P.S. (1999). Rev. Mod. Phys. 71, 1. Wieman, C., Flowers, G., and Gilbert, S. (1995). Am. J. Phys. 63, 317. Wippel, V., Binder, C., Huber, W., Windholz, L., Allegrini, M., Fuso, E, and Arimondo, E. (2001). Eur. Phys. J. D 17, 285.
Index A Absolute configuration of enantiomers D and L notation, 227 definition, 225 determination, 230 R and S notation, 230 Absolute phase of few-cycle laser pulses, 90 ADK (Ammosov-Delone-Krainov) tunneling rate in strong fields, 40 Amino acids chirality, 225 Murchison meteorite, 237 homochirality, origin of, 227 parity-violating energy difference (PVED), 241 polymerization, 254 racemization, 256 stereogenic centers, 225 ATI (above-threshold ionization), 7, 3592 angle-resolved electron spectra, 38 application of COLTRIMS, 38 applications, 86 few-cycle laser pulses, 90 measurement of pulse durations, 88 boundary conditions in a focused pulse, 43 classical model of direct ionization, 40 classical orbits, 41, 51 cutoff, 52 dependence of spectrum on quantum orbits, 61, 68 direct cutoff, 42 direct ionization, 40 discrete energies due to interference, 46 electric field versus vector potential, 42 elliptical polarization described by saddle-point equations, 68 experimental methods, 38
generation of attosecond pulses, 89 HATI (high-order ATI), 50, 51, 59 history, 36 intensity-dependent enhancements in plateau, 63-65 interference between direct and rescattered electrons, 71 between direct electrons, 47 KFR amplitudes for elliptical polarization, 49 lobes in energy-resolved angular distributions, 52 need for femtosecond laser, 38 origin of staircase structure for elliptical polarization, 70 plateau, 39, 52, 59 plateau enhancement caused by channel closing, 65 dependence on late returns, 67 resonance versus channel closing, 66 quantum-mechanical description of direct electrons, 44 of rescattering, 53 quantum orbits two-dimensional for elliptical polarization, 70 quasi-energy formalism, 61 relativistic regime, 73-76 rescattering for zero-range potential, 54 in the relativistic regime, 75 into arbitrary angle, 51 mechanism, 50-71 in MPI, 27 model, 37 relation to Feynman's path integral, 57 relation with closed-orbit theory, 58 role of the binding potential, 59 391
392
INDEX
ATI (above-threshold ionization) (cont'd) saddle-point method for calculating ionization amplitudes, 46 saddle-point solution for elliptical polarization, 48 for rescattering at high intensity, 55 simple-man model compared to QM solution, 56 in relativistic regime, 73 not applicable to elliptical polarization, 68 strong-field approximation, 45 strong-field approximation to Feynman's path integral, 57 structure in energy-resolved angular distributions, 71 TDSE solutions, 39 theoretical methods, 39 time-of-flight detection methods, 38 zero-range potential, 59 Atom chips, 264, 303-323 arrays of traps, 275 Bose-Einstein condensate (BEC), 273 cold atoms on a warm surface, 264 decoherence (dephasing) problems, 324 decoherence problems, 335-342 definition, 335 fringe contrast, 335 longitudinal decoherence, 337 loss of coherence between internal states of a trapped atom, 335 motional decoherence, 337 interferometers, 337-342 transverse decoherence, 339 design problems, 324-342 experiments, 314-323 beam splitters, 319 Bose-Einstein condensate (BEC), 320-323 guiding and transport, 316-318 microtraps, 314 fabrication, 304-307 nanofabrication, 305-307 thin-film hybrid production technology, 304 future developments, 342-351 guiding and transport, 316-318
heat loss problems, 330-335 current fluctuations in chip wires, 333 fluctuations in trap potential, 331 light heating, 334 Lamb-Dicke parameter, 334 overview, 334 thermal fluctuations in trap potential, 332 heating problems, 324 interferometers, 278 loading, 304, 307-313 into sequentially smaller traps, 309 magneto-optical trap, 307-309 observing the atoms, 313 transfer from bigger to smaller wires, 310 trap loss problems, 324, 330 mechanisms, 324-330 collisions between trapped atoms, 328 current fluctuations in chip wires, 327 dependence on Larmor frequency, 327 dependence on skin depth, 327 Majorana flips, 325 noise-induced spin flips, 325 stray light, 329 thermally induced noise, 326 tunneling, 329 reduction utilizing permanent magnets, 328 wire guides, 271 Atom optics applications of miniaturization, 264 atom chips, 264 atom wire, 264 beam splitters, 276 guidance of atoms, 264 arrays of traps, 275 atom chips, 303-323 combining magnetic and electric fields, 286-289 electric interactions, 283-285 free-standing structures, 292-303 Ioffe-Pritchard trap, 268 Kepler guide, 266, 293
INDEX magnetic interactions, 265-283 magnetic traps, 273 moving potentials, 275 simple traps, 272 strong-field seekers, 265 two-wire guides, 269 U- and Z-trap, 273 weak-field seekers, 266 Weinstein-Libbrecht traps, 274 HOF for atomic lens, 169 importance for transfer of trapped cold atoms, 154 interferometers made of wire guides, 278 manipulation of cold atoms in hollow laser beams (HLBs), 153-188 microscopic, 263-351 miniaturization, 289-292 permanent magnets, 281 Van der Waals interaction, 290 Atom-optics billiards, 141 Atom traps advantages for scattering experiments, 358 coherence time of dark optical traps, 107 dark optical traps for cold atoms, 99147 depth of potential in single-beam trap, 119 difference between red- and bluedetuned dipole traps, 103 general experimental principles, 359 measurement of scattering rates, 117 storage capacity of dark optical traps, 108, 114 storage time of dark optical traps, 108 use as target in collision experiments, 358-389 use of HLBs, 178-188 use of light sheets, 178 volume of single-beam dark optical traps, 117 Atom wires, 264 loss rate depends on angular momentum, 302 Mach-Zehnder interferometer, 278
393
Atomic beam splitter using HLB, 175 Atomic fountain, efficiency improved with HLB, 176 Atomic Hamiltonian dressed-atom model, 158 kinetic theory of cold atoms in laser light, 154 Axicon use in dark optical traps, 113 use in hollow laser beam (HLB), 106 B Beam splitters, 276 Ioffe-Pritchard configuration, 277 X-beam splitters, 278 Y-beam splitters, 276 BEC (Bose-Einstein condensate), 273 compression of trap to improve cooling rate, 132 created in CO2 laser trap, 103 created in optical trap, 131 investigation of movement in billiard, 146 on atom chips, 320-323 preserving coherence in manipulation, 171 produced by evaporative cooling, 131 propagating through orifice made by HLB, 182 quantum information processing, 320 Bessel laser field mode high-order beams are HLB, 162 Beta decay, asymmetry in, 243-252 Blue-detuned traps, s e e Dark optical traps Born approximation in ATI rescattering mechanism, 54 Bose-Einstein condensate, s e e BEC Bright optical traps, s e e Red-detuned traps B-splines used in TDSE calculations of ATI, 39
C Carbohydrates chirality, 227 enantiomers, 227
394
INDEX
Carbohydrates (cont 'd) homochirality in naturally occurring carbohydrates, 230 CAT (conical atom trap), 181 CGH (computer-generated hologram) construction, 161 use in dark optical traps, 119 CH4 (methane), symmetry properties, 221 Chaotic motion dependence on wall softness, 145 in intensity-modulated HLB, 146 of cold atoms in 'gravitational wedge' billiard, 143 Characterization of high-order harmonics utilizing ATI, 87 Charged-particle impact ionization mechanisms, 24 two-step picture of double ionization, 24 CHBrC1F (bromochlorofluoromethane), 222 chirality, 222 symmetry properties, 222 CH3F (fluoromethane) symmetry properties, 222 Chirality amino acids, 225 Murchison meteorite, 237 amplification processes, 247-252 autocatalysis, 248-251 kinetic PVED effect, 251 optically active alcohol, 250 photochemistry, 248 seeding, 247 archaeological dating, 256 Barron's definition, 230 beta decay, 243-252 carbohydrates, 227, 230 CHBrC1F, 222 crystal growth, 253 definition, 220 degradation, 255 diastereomers, 229 effects of stirring, 247 effects on drugs, 236 electroweak interaction, 240 enantiomers, 223
group theory, 230 homochirality, 221 of molecules, 219 homochirality of amino acids in cellular chemistry, 227 influence on taste, 235 mirror symmetry breaking, advantage factor, 232 non-linear production, 254 of biomolecules, 219-257 of electrons and neutrons, 232 of fields, 233 of galaxies, 233 of pharmaceuticals, 235 of plants, 234 of right and left circularly polarized light, 232 optical activity, 225 measurement of, 223 wavelength dependence, 223 optical activity of chiral molecules, 219 parity-violating energy difference (PVED), 241,252-257 permanent versus instantaneous, 224 polymerization of amino acids, 254 production of chiral molecules, 224 racemization, 256 relation to parity, 231 relation to time reversal, 230 stereogenic centers, 223,225 two stereogenic centers, 228 stereoisomers, 223 thalidomide, 236 theories of origin, 236-243 beta decay, 240 circularly polarized light, 237 extraterrestrial origin, 237 true and false, 230-233 Hund's paradox, 231 of nonstationary objects, 230 Circularly polarized light influence on chirality, 237 natural sources, 239 Classical orbits, see ATI Cold atom manipulation using HLB, 170188
INDEX COLTRIMS (cold-target recoil-ionmomentum spectroscopy), 3-6 applied to ATI, 38 experimental setup for MPI, 3 target density for MPI experiments, 4 Continuous Stern-Gerlach effect on atomic ions, 191-216 applied to C 5+, 194 detection of spin flip, 206-209 Cooling in magnetic versus optical traps, 128 Correlated electron momenta in MPI, 2030 dependence on photon polarization, 2023 influence of electron repulsion, 22 parallel and transverse momentum in double ionization of Ar, 22 relation between electron and ion recoil momenta, 20 S-matrix calculations, 27, 28 time-dependent calculations, 28 CR-BPE (concentric-rings binary phase element), use in dark optical traps, 119, 184 Cross-correlation ATI applied to measurement of pulse duration, 87 ATI applied to phase measurement, 88 Cross sections measurements with an atom trap, 357389
D Dark optical traps efficiency of loading single-beam traps, 119 application of HLB, 108 applications, 127-147 atom-optics billiards, 141 based on single laser beams, 113-124 coherence time, 107 comparing blue-detuned with reddetuned traps, 126 comparing different designs, 124 comparing dynamics in different billiards, 142
395
compression of scanning wave trap, 132 cooling mechanisms, 127 dark optical lattices, 111 definition of "darkness factor", 125 density-dependent heating, 133, 134 effect of cooling on phase-space density, 128 effects of gravity, 127 evanescent-wave traps, 109 evaporative cooling, 111, 131 generated by multiple laser beams, 106113 heating and loss mechanisms, 128, 133136 loading atoms into, 103 loss mechanisms, dependence on density, 134 losses due to light-assisted collisions, 135 measurement of two-photon transition in Rb, 140 measurements of weak optical transitions, 140 multiple light beams added incoherently, 107 normalized properties, 107 Jr-phase plate trap, 116 polarization-gradient cooling, 128 precision measurements, 136 of ground-state hyperfine splitting in Na, 136 Raman cooling, 129 reflection (Sisyphus) cooling, 130 scanning single-beam trap, 122 scattering rate dependence on detuning, 133 for CR-BPE trap, 120 for LG HLB trap, 126 for scanning beam trap, 124 single beams combined with diffractive optical elements, 115 refractive optical elements, 113 storage capacity of axicon trap, 114 of CR-BPE trap, 120 of evanescent-wave trap, 110 of scanning beam trap, 123, 124
396
INDEX
Dark optical traps (cont'd) storage time in dark optical lattices, 111 of axicon trap, 115 of evanescent-wave trap, 110 of Jr-phase plate trap, 117 of scanning beam trap, 123, 124 trap designs using a single laser beam, 113-124 use of atomic trampoline, 109 use of axicons, 113, 119 volume of combined CR-BPE/'axicon telescope' trap, 122 CR-BPE trap, 120 Jr-phase plate trap, 116 scanning beam trap, 123 single-beam traps, 117,. 119 Dark state, 359 Density matrix, 335 time evolution of an atom in a laser field, 154 Diastereomers definition, 229 properties, 229 Dipole interaction between atom and laser field, 154 dipole force in dressed atom model, 160 dipole potential for two-level atom, 102 expression for gradient force, 156 gradient force in HLB, 156 in dressed-atom model, 158 momentum transfer, 101 multi-level atom, 103 optical potential, 156 repulsive force used in dark optical traps, 101 rotating-wave approximation, 155 Directional quantization observed as Stern-Gerlach effect, 191 Double ionization by charged particle impact, 23, 24 double peak momentum structure, 14 electron energy distribution in MPI, 19
measured for He, Ar, Ne and Xe in MPI, 19 in MPI, 2 mechanisms, 9 rescattering mechanism, 10 in MPI, 25 S-matrix calculations, 14 sequential ionization, 9 shake-off mechanism, 9 single-photon absorption, 23 TS 1 (two-step-one) mechanism, 10 TS2 (two-step-two) mechanism, 9 Double peak momentum structure in MPI approximate calculations, 15 consistent with rescattering mechanism in MPI, 18 dependence on polarization, 18 in CTMC calculations, 18 in nonsequential ionization, 11 S-matrix calculations, 14, 15 Wannier theory, 16 Dressed-atom model density matrix for HLB, 159 energy levels in HLB, 158 for cold atoms in laser field, 157 Dyson equation, use in ATI, 45, 60
E EAD (energy-resolved angular distribution), 71 EDM (electric dipole moment) advantages of measurements in dark optical traps, 139 error analysis for Cs atoms in a dark optical lattice, 138 of experiment in dark optical traps, 138 precision measurements in dark optical traps, 137 using dark optical lattices, 113 ee, see Enantiomeric excess Electron mass: continuous Stern-Gerlach experiment, 214 Electron scattering in intense laser fields relation to HATI, 55
INDEX Electron-atom collisions cross sections for Rb, 357-389 measurement of cross sections with an atom trap, 357-389 differential scattering, 373 excitation, 383 ionization, 374, 379 total scattering, 370 Electroweak interaction in atoms, 240 in molecules, 240 Z ~ interactions, 242 Enantiomeric excess (ee), 224, 237 Enantiomers absolute configuration, 227 definition, 225 carbohydrates, 227 definition, 223 diastereomers, 229 Fischer projection, 226 properties, 223 racemic mixture, 223 stereogenic centers number of, 229 two stereogenic centers, 228 Evaporative cooling in dark optical traps, 128, 131 Excited state fraction, 377
397
F FEL (Free Electron Laser) experiments, two-photon double ionization of He, 30 Feynman's path integral, in rescattering mechanism, 57 Fischer projection, 226-228 Fokker-Planck equation for cold atoms in laser light, 155 Frequency metrology, impact of phase control of femtosecond laser, 91 Frequency standard, use of atomic fountain, 176
extension of measurements to heavier ions, 214 for hydrogen-like carbon calculated, 194 measured, 213 measured using Stern-Gerlach effect, 191 measurement in Penning trap, 206 of free electron, 192 Gouy phase shift, 119, 161 Guidance of atoms, 264 arrays of traps, 275 atom chips, 303-323 charged optical fibers, 285 charged-wire experiments, 300 cold gas, 302 interferometry, 301 combining magnetic and electric fields, 286-289 current-carrying wire, 299 electric AC traps, 284 electric interactions, 283-285 "electric motor" for ac(de)celeration of atoms, 288 free-standing structures, 292-303 Ioffe-Pritchard trap, 268 Kepler guide, 266, 293 magnetic, 265 magnetic interactions, 265-283 magnetic traps, 273 moving potentials, 275 relation between traps and guides, 272 state-selective combined electric and magnetic traps, 287 storage ring, 300 strong-field seekers, 265 tip trap, 298 two-wire guides, 269 U- and Z-trap, 273 weak-field seekers, 266 in magnetic structures, 295, 296 Weinstein-Libbrecht traps, 274
G g-factor, 192 electron bound in neutral atom, 192 electron bound to ion, 193
It H- photodetachment in constant electric field, 50
398
INDEX
HATI (high-order ATI), s e e u n d e r ATI Helicity of elementary particles, 221 HHG (high-order harmonic generation) cutoff, 36 in two-color bicircular light, 80 experimental facts, 36 in elliptically polarized light, 78 in the relativistic regime, 84 in two-color bicircular light, 78 Lewenstein model, 77 with linearly polarized light, 77 multiplateau structure, 86 origin via rescattering, 50 plateau, 36 in two-color bicircular light, 80 quantum orbits, 76 relativistic effects on plateau, 76 relativistic versus non-relativistic results, 85 saddle-point approximation, 77 semiclassical model, 37 sequence of attosecond pulses, 84 HLB (hollow laser beam) axicon used to form three dimensional trap, 185 CGH used to form three dimensional trap, 185 cold atom manipulation, 153, 170-188 definition, 104, 160 definition of detuning, 154 definitions of laser beam parameters, 156 dependence of guiding efficiency on detuning, 171 dougnut shape using HOF, 168 effect of dipole gradient force, 156 efficiency of guiding cold atoms, 171 efficiency of loading HLB trap, 183 generated by axicon, 106, 164 generated using double-cone prism, 165 generation by phase-plate method, 163 generation of Bessel beams, 162 generation using geometric optics, 163165 generation using hologram, 161 guidance of BEC, 105, 173
of cold atoms, 104 of Cs, 173 of metastable Ne, 105, 172 of Rb, 170 guiding efficiency for cold atoms, 176 horizontal confinement in gravito-optical surface trap, 180 LG laser beam as dipole trap, 178 LG modes, 105 methods of generation, 160-169 minimizing loss of atomic coherence, 177 optical dipole trap made by two HLBs, 182 Rabi frequencies in three level model, 155 single HLB used as three-dimensional trap, 183 splitting of an atomic beam, 175 storage time in three-dimensional dipole trap, 184 storage time of optical dipole trap, 179 storage times in CAT limited by rest gas collisions, 181 theoretical models for cold atoms, 154160 three-dimensional trap constructed from rotating laser beam, 186 three-dimensional trap made by two Gaussian beams, 185 three-dimensional trap RODiO, 187 three-level atom model, 154 transfer efficiency from MOT to CAT, 181 use as atomic trap, 178-188 used as two-dimensional trap for cold atoms, 104 used for creating an atomic fountain, 176 used for levitation, 174 used to produce CAT, 181 waist size, 156 HOF (hollow-core optical fiber) applications to constructing hollow laser beam (HLB), 106, 166-169 cold atom manipulation, 153 diffraction pattern, 167
INDEX electric field structure, 167 near-field diffraction, 166 Homochirality, s e e Chirality I Interferometers, 278, 346 decoherence problems, 342 in the spatial domain, 278 effect of excitation, 280 in the time domain, 280 Ioffe-Pritchard trap, 273 field potential, 268 relation to two-wire guides, 269 Ionization in strong fields, 40 K Keldysh parameter, 41 Kepler guide comparison with 2-dimensional hydrogen, 266 experimental setup, 293 experiments, 293-295 strong-field seeking state, 293 use of strong-field seekers, 266 KFR (Keldysh-Faisal-Reiss) ionization amplitudes in strong-field approximation, 45 Klein-Gordon equation in relativistic ATI, 75 quantum orbits in relativistic HHG, 84
L Larmor frequency, 327 Laser field elliptically polarized, 42 Gouy phase shift for Gaussian beams, 161 HG (Hermite-Gaussian) mode, 160 in HLB, 154 intensity distribution in HLB, 156 LG (Laguerre-Gaussian) mode, 160 propagation of HG and LG modes, 161 LG (Laguerre-Gaussian) laser field modes angular momentum of, 161, 163 constituting a HLB, 161 generated from CGH, 105
399
generated from HG modes, 105, 161 generation by hologram, 162 generation by phase-plate method, 163 intensity distribution, 105 waist size, 105 LOPT (lowest-order perturbation theory), applied to MPI, 36 Low-dimensional systems, 347 M Mach-Zehnder interferometer, 278 Magnetic moment determination utilizing Stern-Gerlach effect, 191 Magnetic traps 3-dimensional wire, 273 magnetic quadrupole trap, 298 Magneto-optical trap, s e e MOT Mechanisms of double ionization in MPI, 9 Methods for measuring cross sections in MOT, 367-389 Microscopic trapping potentials, 287 Miniaturization, 290-292 application in atom "motors" or "conveyor belts", 275 finite size effects, 290 limiting factors, 291 nanofabrication, 289 permanent magnets, 281 Van der Waals interaction, 290 Momentum distribution for multiphoton absorption, 6, 8 for single photon absorption, 6, 8 Momentum transfer in interaction between light and atom, 101 s e e a l s o Dipole interaction MOT (magneto-optical trap), 292, 298, 301 atom-chip loading, 307-309 basic principles, 359 comparison with crossed-beam experiments, 369 cooling and confining in, 359 dark state, 359 differential cross sections, 373 ESD (electron stimulated desorption), 374
400
INDEX
MOT (magneto-optical trap) (cont'd) excitation cross sections for excited levels, 383 experimental setup, 362-367 electron beam, 366 laser system, 362 magnetic field, 365 avoiding interaction with electron beam, 366 vacuum chamber, 365 final-state measurements of cross sections, 376-386 hyperfine repump, 360 ionization cross section, 374 for excited states, 379 ionization of background gas atoms, 381 load rates versus loss rates, 371 loading efficiency of dark optical traps, 114, 115, 120 loading microscopic guidance structures, 293 loss rate measurements, 368-376 measurement of excited state fraction, 377 measurements on mixed ground and excited state population, 379 measuring total cross sections, 370 minimization of contribution from background ions, 382 natural time scales, 360 optical molasses, 359 photoionization experiments, 358 polarization of atoms in trap, 361,387 pyramid MOT, 307 rate equations involving cross sections, 368 rate loss measurement for excited states, 376 red-detuning, 359 reflection MOT, 307 signal-to-noise ratio in excitation experiments, 386 source of cold atoms, 103, 170, 174, 176, 180, 186 sources of photons in fluorescence experiments, 384, 385
target for collision experiments, 360 timing cycles for excited-state ionization measurements, 380 for fluorescence measurements, 384 for ground-state ionization measurements, 374 for measuring total cross sections, 370 TOF measurements of ions for excited level cross sections, 379 transfer efficiency into CAT, 181 MPI (multiphoton ionization) comparison to single-photon and charged particle double ionization, 23 laser described as classical electromagnetic field, 2 role of electron correlation, 2 two-step picture, 8 use of COLTRIMS, 2 Multi-well potentials, 347 Multilevel atoms: dipole interaction with light beam, 103 Multiple ionization in MPI, 2 Murchison meteorite, 237, 256 N Nanofabrication, 289 atom chips, 303, 305-307 Nonlinear optics applications of ATI, 86 Nonsequential ionization, 2 correlated electron momenta, 20 double peak momentum structure, 11 mechanisms for double peak momentum structure, 13 NSDI (nonsequential double ionization), origin via rescattering, 37, 50 O Optical activity, 219 designation of rotation, d and 1 notation, 225 measurement of, using polarimeter, 223 origin of, 225 Optical billiards effect of soft-wall potential, 145
INDEX for cold atoms, 123 loading scheme for dark optical traps, 145 survival probability for cold atoms as measure of dynamics, 144 Optical Bloch equations, 102 Optical bottle beam HLB, 185 use as atomic trap, 118 Optical molasses, 359 Optical rotary dispersion (ORD), wavelength dependence of optical activity, 223 O R D , s e e Optical rotary dispersion
P Parity-violating energy difference (PVED), 241, 251-257 chirality, 241 experiments, 242 for molecules, 240 Z 6 dependence, 242 Penning trap calibration of magnetic field, 201 detection of spin flip, 206 double-trap technique, 209-212 error analysis, 212, 213 for a single ion, 195 for continuous Stern-Gerlach experiment, 195 influence of magnetic field inhomogeneities on motional frequencies, 205 influence of trap imperfections on motional frequencies, 205 ion motion, 197 ion temperature, 201 loading the trap, 198 measurement of oscillation frequencies, 201 Permanent magnets, 281 PGC (polarization-gradient cooling) in dark optical traps, 127, 129 Phase-space density, dependence on adiabatic change in potential shape, 132
401
PNC (parity nonconservation) measurement for Fr in dark optical trap, 140 tests in dark optical traps, 137 Polarimeter for measurement of optical activity, 223 Polarization-gradient cooling, s e e PGC Ponderomotive energy, 41 Precision measurements ground-state hyperfine splitting in Na measured in dark optical trap, 136 limiting accuracy in dark optical traps, 136 weak optical transitions measurable in dark optical traps, 140 Precision spectroscopy using dark optical traps, 127 PVED, s e e Parity-violating energy difference
Q QED (quantum electron dynamics) corrections for C 5+, 195 expansion in Za for bound electrons, 193 g-factor of bound electron, 193 g-factor of free particle, 192 test of, using magnetic moments, 214 Quantum information processing, 287, 348 using BEC, 320 using dark optical lattices, 113 Quantum orbits applied to finite pulses, 78 in elliptically polarized light, 78 in HHG versus ATI, 76 in two-color bicircular light, 79, 82 interference, 61 quantum-mechanical description of HHG and ATI, 37 specifying parameters, 61 Qubit manipulation, 348 R Racemic mixture, definition, 223 s e e a l s o Chirality Rainbow scattering in ATI, 52, 71
402
INDEX
Raman cooling applied to dark optical lattices, 130 in dark optical traps, 128, 129 Raman scattering, used to determine interaction between atoms and trapping light in optical trap, 117 Ramsey spectroscopy for EDM measurement, 139 for precision measurement of groundstate hyperfine splitting in Na, 136 Recoil ion momentum in MPI, 11-19 measured for He, Ne and Ar, 11 origin of double peak structure, 13-19 parallel and perpendicular distributions in He and Ne, 17 saddle potential, 17 S-matrix calculations, 14 integration of one-dimensional Schr6dinger equation, 18 Red-detuning, 359 Reflection cooling in dark optical traps, 128 in evanescent-wave trap, 110 cooling rate, 130 unique to blue-detuned optical traps, 130 Relativistic effects in ATI, 73 ponderomotive energy, 73-75 radiation pressure, 74 in HHG, 84 magnetic-field-induced drift, 76 "relativistic effective mass", 75 suppression of rescattering effects, 76 v xB drift in HHG, 86 Rescattering mechanism classical theory in ATI, 50 connection with closed-orbit theory, 58 in HHG versus ATI, 76 in MPI, 10, 25-27 dependence on polarization, 10 double ionization, 25, 26 double peak structure, 13 electron energy distribution, 19 influence of photon polarization, 26
low energy electrons in ATI of Ar, 27 Wannier theory, 16 linear polarization, 50 quantum-mechanical description, 53 saddle-point methods, 55 use of Born-approximation, 54 use of zero-range potential, 54 Rescattering model of HHG and ATI, 37 "Resistive cooling" in Penning traps, 198 ROBOT (rotating beam optical trap), 186 RODiO (rotating off-resonant dipole optical trap), 187 storage time, 187 Rotating wave approximation (RWA), 102 S S-matrix calculations are time-independent, 17 double peak momentum structure, 14 of correlated electron momenta in MPI, 27, 28 SAE (single-active-electron) approximation applied to ATI, 39 in MPI, 39 Scattering length: importance for spinexchange collisions in atom traps, 328 Scattering rate: dependence on detuning, 133 Sequential ionization, 2, 9 correlated electron momenta, 20 single peak momentum structure, 12 Side guide experiments, 295 Side guide, weak-field-seeking state, 267 Simple-man model of HHG and ATI, 37 Simple traps bent guide wire, 273 crossed wires, 273 straight guide wire and bias field, 272 U- and Z-trap, 273 Single ionization in MPI, 2 momentum distribution in single- and multiphoton absorption, 6-9
INDEX Single-photon absorption influence of electron repulsion, 23 Standard Model, tests by tabletop experiments, 138 Stereogenic centers, 227 absolute configuration, 230 amino acids, 225 definition, 223 number of, 229 two stereogenic centers, 228 Stereoisomers, 228 chiral molecules, 223 diastereomers, 229 Stern-Gerlach effect, s e e Continuous Stern-Gerlach effect Strong-field approximation in HHG, 77 Strong-field-seeking state, 265, 292, 293, 328 Sub-Doppler cooling in MOT, 170 T TDSE (time-dependent Schr6dinger equation) HHG analyzed in terms of quantum orbits, 77 numerical solution in one or more dimensions for ATI, 39 solution by split-operator method, 40 Thalidomide, chirality, 236 Tilted Bunimovich stadium billiard, 142 chaotic motion found in dark optical trap, 142 Time-dependent calculations CTMC (Classical Trajectory Monte Carlo) approach, 18 for three-body systems, 18
403
of correlated electron momenta in MPI, 28-30 recoil ion momentum distribution, 18 Time-independent calculations; solution of dressed-atom model for cold atoms in laser field, 158 TOF (time-of-flight) analysis for ATI electrons, 38 Two-step picture of MPI, 8 single peak momentum structure, 12 Two-wire guides, 269 U U-trap (quadrupole trap), 273 V Van der Waals interaction, 290 Volkov methods, alternative to TDSE for ATI, 40 Volkov states for free electron in laser field, 44 W Weak-field-seeking state, 266, 267, 292, 295, 328 Weinstein-Libbrecht traps, 274
Z Z-trap (Ioffe-Pritchard trap), 273 Zero-range potential bound states, 59 calculation of wave function, 60 in ATI rescattering, 54 in HHG, 78
This Page Intentionally Left Blank
Contents of Volumes in This Serial Radiofrequency Spectroscopy of Stored Ions I: Storage, H.G. Dehmelt
Volume 1
Molecular Orbital Theory of the Spin Properties of Conjugated Molecules,
Optical Pumping Methods in Atomic Spectroscopy, B. Budick
G.G. Hall and A.T. Amos
Energy Transfer in Organic Molecular Crystals: A Survey of Experiments,
Electron Affinities of Atoms and Molecules, B.L. Moiseiwitsch
H. C. Wolf
Atomic Rearrangement Collisions,
Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney
B.H. Bransden
The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi
Quantum, Mechanics in Gas CrystalSurface van der Waals Scattering, E. Chanoch Beder
The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J.P Toennies
Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J. Wood
High-Intensity and High-Energy Molecular Beams, J.B. Anderson, R.P Andres and J.B. Fen
Volume 4
Volume 2
The Calculation of van der Waals Interactions, A. Dalgarno and W.D. Davison Thermal Diffusion in Gases, E.A. Mason, R.J Munn and Francis J. Smith
H.S.W M a s s e y - A Sixtieth Birthday Tribute, E.H.S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D.R. Bates and R.H. G. Reid
Spectroscopy in the Vacuum Ultraviolet,
Applications of Quantum Theory to the Viscosity of Dilute Gases,
W.R.S. Garton
The Measurement of the Photoionization Cross Sections of the Atomic Gases,
R.A. Buckingham and E. Gal
Positrons and Positronium in Gases,
James A.R. Samson
PA. Fraser
The Theory of Electron-Atom Collisions,
Classical Theory of Atomic Scattering,
R. Peterkop and V. Veldre
A. Burgess and I.C. Percival
Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, EJ. de Heer
Born Expansions, A.R. Holt and B.L. Moiseiwitsch
Resonances in Electron Scattering by Atoms and Molecules, P G. Burke
Mass Spectrometry of Free Radicals, S.N. Foner
Relativistic Inner Shell Ionizations,
Volume 3
C.B. 0. Mohr
The Quantal Calculation of Photoionization Cross Sections, A.L. Stewart
Recent Measurements on Charge Transfer, J.B. Hasted
405
406
C O N T E N T S OF V O L U M E S IN THIS SERIAL
Measurements of Electron Excitation Functions, D. W.O. Heddle and R. G. W Keesing Some New Experimental Methods in Collision Physics, R.F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M.J. Seaton Collisions in the Ionosphere, A. Dalgarno The Direct Study of Ionization in Space, R.L.E Boyd Volume 5
Flowing Afterglow Measurements of Ion-Neutral Reactions, E.E. Ferguson, EC. Fehsenfeld and A.L. Schmeltekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions II: Spectroscopy, H.G. Dehmelt The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuven
The Diffusion of Atoms and Molecules, E.A. Mason and T.R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical mechanics in the Treatment of Collisions between Massive Systems, D.R. Bates and A.E. Kingston Volume 7
Physics of the Hydrogen Maser, C. Audoin, JP. Schermann and P. Grivet Molecular Wave Functions: Calculations and Use in Atomic and Molecular Processes, J. C. Browne Localized Molecular Orbitals, Harel Weinstein, Ruben Pauncz and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J. Gerratt Diabatic States of Molecules Quasi-Stationary Electronic States, Thomas EO'Malley
The Calculation of Atomic Transition Probabilities, R.JS. Crossley
Selection Rules within Atomic Shells, B.R. Judd Green's Function Technique in Atomic and Molecular Physics, Gy. Csanak, H.S. Taylor and Robert Yaris
Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations sZs'Upq, C.D.H. Chisholm, A. Dalgarno and ER. Innes
A Review of Pseudo- Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A.J. Greenfield
Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle
Volume 8
Volume 6
Interstellar Molecules: Their Formation and Destruction, D. McNally Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck
Dissociative Recombination, JN. Bardsley and M.A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A.S. Kaufman
Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C.Y. Chen and Augustine C. Chen
The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa
Photoionization with Molecular Beams, R.B. Cairns, Halstead Harrison and R.I. Schoen
CONTENTS OF VOLUMES IN THIS SERIAL The Auger Effect, E.H.S. Burhop and W.N. Asaad Volume 9
Correlation in Excited States of Atoms, A. W. Weiss
The Calculation of Electron-Atom Excitation Cross Sections, M.R.H. Rudge Collision-Induced Transitions between Rotational Levels, Takeshi Oka The Differential Cross Section of Low-Energy Electron-Atom Collisions, D. Andrick
Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English
Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy Volume 10
Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong Jr. and Serge Feneuille
The First Born Approximation, K.L. Bell and A.E. Kingston
Photoelectron Spectroscopy, W.C. Price Dye Lasers in Atomic Spectroscopy, W. Lange, J. Luther and A. Steudel
Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C Fawcett
A Review of Jovian Ionospheric Chemistry, Wesley T. Huntress Jr.
407
Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R.B. Bernstein and R.D. Levine
Inner Shell Ionization by incident Nuclei, Johannes M. Hansteen
Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M.E Golde and B.A. Thrush Volume 12 Nonadiabatic Transitions between Ionic and Covalent States, R.K. Janev
Recent Progress in the Theory of Atomic Isotope Shift, J. Bauche and R.-J. Champeau
Topics on Multiphoton Processes in Atoms, P Lambropoulos Optical Pumping of Molecules, M. Broyer, G. Goudedard, J.C. Lehmann and J Vigu~
Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith
Ion Chemistry in the D Region, George C. Reid Volume 13 Atomic and Molecular PolarizabilitiesReview of Recent Advances,
Thomas M. Miller and Benjamin Bederson
Study of Collisions by Laser Spectroscopy, Paul R. Berman
Collision Experiments with Laser-Excited Atoms in Crossed Beams, I. V. Hertel and W. Stoll
Volume 11
The Theory of Collisions between Charged Particles and Highly Excited Atoms, I.C. Percival and D. Richards Electron Impact Excitation of Positive Ions, M.J Seaton The R-Matrix Theory of Atomic Process, P G. Burke and W.D. Robb
Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J. Peter Toennies
Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R.K Nesbet Microwave Transitions of Interstellar Atoms and Molecules, WB. Somerville
408
CONTENTS OF VOLUMES IN THIS SERIAL
Volume 14
Resonances in Electron Atom and Molecule Scattering, D.E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J. Jamieson and Ronald F. Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy
Forbidden Transitions in One- and Two-Electron Atoms, Richard Marrus and Peter J. Mohr
Semiclassical Effects in Heavy-Particle Collisions, M.S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin
Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H.B. Gilbody
Inner-Shell Ionization, E.H.S. Burhop Excitation of Atoms by Electron Impact, D. W.O. Heddle
Coherence and Correlation in Atomic Collisions, H. Kleinpoppen Theory of Low Energy Electron-Molecule Collisions, PG. Burke Volume 16
Atomic Hartree-Fock Theory, M. Cohen and R.P. McEachran
Experiments and Model Calculations to Determine Interatomic Potentials, R. Diiren
Quasi-Molecular Interference Effects in Ion-Atom Collisions, S. V. Bobashev
Sources of Polarized Electrons,
Rydberg Atoms, S.A. Edelstein and
Theory of Atomic Processes in Strong Resonant Electromagnetic Fields,
T.E Gallagher
R.J. Celotta and D. T. Pierce
S. Swain
UV and X-Ray Spectroscopy in Astrophysics, A.K. Dupree
Spectroscopy of Laser-Produced Plasmas,
Volume 15
Relativistic Effects in Atomic Collisions Theory, B.L. Moiseiwitsch
Negative Ions, H.S.W. Massey Atomic Physics from Atmospheric and Astrophysical Studies, A. Dalgarno Collisions of Highly Excited Atoms, R.E Stebbings
Theoretical Aspects of Positron Collisions in Gases, J. W. Humberston Experimental Aspects of Positron Collisions in Gases, T.C. Griffith Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein
Ion-Atom Charge Transfer Collisions at Low Energies, J.B. Hasted Aspects of Recombination, D.R. Bates The Theory of Fast Heavy Particle Collisions, B.H. Bransden
M.H. Key and R.J. Hutcheon
Parity Nonconservation in Atoms: Status of Theory and Experiment, E.N. Fortson and L. Wilets Volume 17
Collective Effects in Photoionization of Atoms, M. Ya. Amusia Nonadiabatic Charge Transfer, D.S.E Crothers
Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot
Superfluorescence, M.F.H. Schuurmans, Q H . E Vrehen, D. Polder and H.M. Gibbs
Applications of Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M.G. Payne, C.H. Chen, G.S. Hurst and G. W. Foltz
CONTENTS OF VOLUMES IN THIS SERIAL Inner-Shell Vacancy Production in Ion-Atom Collisions, C.D. Lin and Patrick Richard Atomic Processes in the Sun, PL. Dufion and A.E. Kingston
Volume 18 Theory of Electron-Atom Scattering in a Radiation Field, Leonard Rosenberg
Positron-Gas Scattering Experiments, Talbert S. Stein and Walter E. Kauppila Nonresonant Multiphoton Ionization of Atoms, J.. Morellec, D. Normand and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A.S. Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B.R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N. Andersen and S.E. Nielsen Model Potentials in Atomic Structure, A. Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D.W. Norcross and L.A. Collins Quantum Electrodynamic Effects in FewElectron Atomic Systems, G. W.E Drake
Volume 19
Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B.H. Bransden and R.K. Janev Interactions of Simple Ion Atom Systems, JT. Park High-Resolution Spectroscopy of Stored Ions, D.J. Wineland, Wayne M. Itano and R.S Van Dyck Jr. Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K. Blum and H. Kleinpoppen
409
The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, E Jen6 The Vibrational Excitation of Molecules by Electron Impact, D.G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization of Atomic and Molecular Photoelectrons, N.A. Cherepkov Volume 20
Ion-Ion Recombination in an Ambient Gas, D.R. Bates Atomic Charges within Molecules, G. G. Hall Experimental Studies on Cluster Ions, T D. Mark and A. W. Castleman Jr. Nuclear Reaction Effects on Atomic Inner-Shell Ionization, WE. Meyerhof and J.-F Chemin Numerical Calculations on ElectronImpact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstrong On the Problem of Extreme UV and X-Ray Lasers, I.I. Sobel'man and A. V. Vinogradov Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J.M. Raimond Rydberg Atoms: High-Resolution Spectroscopy and Radiation InteractionRydberg Molecules, J.A. C. Gallas, G. Leuchs, H. Walther, and H. Figger Volume 21
Subnatural Linewidths in Atomic Spectroscopy, Dennis P O'Brien, Pierre Meystre and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch.Jungen
410
CONTENTS OF VOLUMES IN THIS SERIAL
Theory of Dielectronic Recombination, Yukap Hahn
Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu Scattering in Strong Magnetic Fields, M.R.C. McDowell and M. Zarcone
Pressure Ionization, Resonances and the Continuity of Bound and Free States, R.M. More Volume 22
Positronium- Its Formation and Interaction with Simple Systems, 2 W. Humberston
Experimental Aspects of Positron and Positronium Physics, T. C. Griffith Doubly Excited States, Including New Classification Schemes, C.D. Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H.B. Gilbody Electron Ion and Ion-Ion Collisions with Intersecting Beams, K. Dolder and B. Peart
Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn
Relativistic Heavy-Ion-Atom Collisions, R. Anholt and Harvey Gould
Continued-Fraction Methods in Atomic Physics, S. Swain Volume 23
Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C.R. Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M. Quiney
Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D.E. Williams and Ji-Min Yan
Transition Arrays in the Spectra of Ionized Atoms, J. Bauche, C. Bauche-Arnoult and M. Klapisch
Photoionization and Collisional Ionization of Excited Atoms Using Synchroton and Laser Radiation, F.J. Wuilleumier, D.L. Ederer and J.L. Picqu~ Volume 24
The Selected Ion Flow Tube (SIDT): Studies of Ion-Neutral Reactions, D. Smith and N.G. Adams
Near-Threshold Electron-Molecule Scattering, Michael A. Morrison Angular Correlation in Multiphoton Ionization of Atoms, S.J. Smith and G. Leuchs
Optical Pumping and Spin Exchange in Gas Cells, R.J. Knize, Z. Wu and W Happer
Correlations in Electron-Atom Scattering, A. Crowe Volume 25
Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor
Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane
Alexander Dalgarno: Contributions to Aeronomy, Michael B. McElroy Alexander Dalgarno: Contributions to Astrophysics, David A. Williams Dipole Polarizability Measurements, Thomas M. Miller and Benjamin Bederson
Flow Tube Studies of Ion-Molecule Reactions, Eldon Ferguson Differential Scattering in He-He and He+-He Collisions at keV Energies, R.E Stebbings
Atomic Excitation in Dense Plasmas, Jon C. Weisheit
Pressure Broadening and Laser-induced Spectral Line Shapes, Kenneth M. Sando and Shih-I Chu
Model-Potential Methods, C. Laughlin and G.A. Victor
CONTENTS OF VOLUMES IN THIS SERIAL Z-Expansion Methods, M. Cohen Schwinger Variational Methods, Deborah Kay Watson Fine-Structure Transitions in Proton-Ion Collisions, R.H. G. Reid Electron Impact Excitation, R.J.W. Henry and A.E. Kingston Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher The Numerical Solution of the Equations of Molecular Scattering, A.C. Allison High Energy Charge Transfer, B.H. Bransden and D.P. Dewangan Relativistic Random-Phase Approximation, WR. Johnson Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G. W.E Drake and S.P. Goldman Dissociation Dynamics of Polyatomic Molecules, T. Uzer
411
On the [3 Decay of 187Re: An Interface of Atomic and Nuclear Physics and Cosmochronology, Zonghau Chen, Leonard Rosenberg and Larry Spruch Progress in Low Pressure Mercury-Rare Gas Discharge Research, J. Maya and R. Lagushenko Volume 27
Negative Ions: Structure and Spectra, David R. Bates Electron Polarization Phenomena in Electron-Atom Collisions, Joachim Kessler Electron-Atom Scattering, I.E. McCarthy and E. Weigold Electron-Atom Ionization, I.E. McCarthy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, V.I. Lengyel and M.I. Haysak Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule
Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine E van Dishoeck The Abundances and Excitation of Interstellar Molecules, John H. Black
The Theory of Fast Ion-Atom Collisions, J.S. Briggs and J.H. Macek
Volume 26
Some Recent Developments in the Fundamental Theory of Light, Peter W. Milonni and Surendra Singh
Comparisons of Positrons and Electron Scattering by Gases, Waiter E. Kauppila and Talbert S. Stein Electron Capture at Relativistic Energies, B.L. Moiseiwitsch The Low-Energy, Heavy Particle Collisions- A Close-Coupling Treatment, Mineo Kimura and Neal E Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V.Sidis Associative Ionization: Experiments, Potentials and Dynamics, John Weiner, Franfoise Masnou-Seeuws and Annick Giusti-Suzor
Volume 28
Squeezed States of the Radiation Field, Khalid Zaheer and M. Suhail Zubairy Cavity Quantum Electrodynamics, E.A. Hinds Volume 29
Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin and L. W Anderson Cross Sections for Direct Multiphoton Ionionization of Atoms, M. V. Ammosov, N.B. Delone, M. Yu. Ivanov, I.L Bondar and A. V. Masalov
412
CONTENTS OF VOLUMES IN THIS SERIAL
Collision-Induced Coherences in Optical Physics, G.S. Agarwal Muon-Catalyzed Fusion, Johann Rafelski and Helga E. Rafelski Cooperative Effects in Atomic Physics, J.P Connerade Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, J.H. McGuire
Positronium Formation by Positron Impact on Atoms at Intermediate Energies, B.H. Bransden and C.J. Noble Electron-Atom Scattering Theory and Calculations, PG. Burke Terrestrial and Extraterrestrial H~-, Alexander Dalgarno Indirect Ionization of Positive Atomic Ions, K. Dolder
Differential Cross Sections for Excitation of Helium Atoms and Helium-Like Ions by Electron Impact, Shinobu Nakazaki
Quantum Defect Theory and Analysis of High-Precision Helium Term Energies, G. W.E Drake Electron-Ion and Ion-Ion Recombination Processes, M.R. Flannery
Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Trajmar and J C. Nickel
Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy, H.B. Gilbody
The Dissociative Ionization of Simple Molecules by Fast Ions, Colin J Latimer Theory of Collisions between Laser Cooled Atoms, PS. Julienne, A.M. Smith, and K. Burnett
Relativistic Electronic Structure of Atoms and Molecules, I.P Grant
Volume 30
Light-Induced Drift, E.R. Eliel
The Chemistry of Stellar Environments, D.A. Howe, J.M.C. Rawlings and D.A. Williams Positron and Positronium Scattering at Low Energies, J. W. Humberston
Continuum Distorted Wave Methods in Ion-Atom Collisions, Derrick S.E Crothers and Louis J. Dub~
How Perfect are Complete Atomic Collision Experiments?, H. Kleinpoppen and H. Handy
Volume 31
Energies and Asymptotic Analysis for Helium Rydberg States, G.W.E Drake Spectroscopy of Trapped Ions, R. C. Thompson Phase Transitions of Stored Laser-Cooled Ions, H. Walther Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, Rudolf Diiren and Jacques Robert Atomic Physics and Non-Maxwellian Plasmas, MichOle Lamoureux Volume 32
Photoionization of Atomic Oxygen and Atomic Nitrogen, K.L. Bell and A.E. Kingston
Adiabatic Expansions and Nonadiabatic Effects, R. McCarroll and D.S.F Crothers Electron Capture to the Continuum, B.L. Moiseiwitsch How Opaque Is a Star?, M.J. Seaton Studies of Electron Attachment at Thermal Energies Using the Flowing Afterglow-Langmuir Technique, David Smith and Patrik Span~l Exact and Approximate Rate Equations in Atom-Field Interactions, S. Swain Atoms in Cavities and Traps, H. Walther Some Recent Advances in ElectronImpact Excitation of n = 3 States of Atomic Hydrogen and Helium, J.E Williams and J.B. Wang
C O N T E N T S OF V O L U M E S IN THIS SERIAL Volume 33 Principles and Methods for Measurement of Electron Impact Excitation Cross Sections for Atoms and Molecules by Optical Techniques, A.R. FilippellL Chun C. Lin, L.W. Andersen and J. W. McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Analysis of Scattered Electrons, S. Trajmar and J. W. McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Electron Swarm Methods, R.W. Crompton Some Benchmark Measurements of Cross Sections for Collisions of Simple Heavy Particles, H.B. Gilbody The Role of Theory in the Evaluation and Interpretation of Cross-Section Data, Barry I. Schneider Analytic Representation of Cross-Section Data, Mitio Inokuti, Mineo Kimura, M.A. Dillon, Isao Shimamura Electron Collisions with N2, O2 and O: What We Do and Do Not Know, Yukikazu Itikawa Need for Cross Sections in Fusion Plasma Research, Hugh P Summers Need for Cross Sections in Plasma Chemistry, M. Capitelli, R. Celiberto and M. Cacciatore Guide for Users of Data Resources, Jean W. Gallagher Guide to Bibliographies, Books, Reviews and Compendia of Data on Atomic Collisions, E. W McDaniel and E.J. Mansky
413
Classical and Quantum Chaos in Atomic Systems, Dominique Delande and Andreas Buchleitner Measurements of Collisions between Laser-Cooled Atoms, Thad Walker and Paul Feng The Measurement and Analysis of Electric Fields in Glow Discharge Plasmas, J.E. Lawler and D.A. Doughty Polarization and Orientation Phenomena in Photoionization of Molecules, N.A. Cherepkov Role of Two-Center Electron-Electron Interaction in Projectile Electron Excitation and Loss, E.C. Montenegro, WE. Meyerhof and J.H. McGuire Indirect Processes in Electron Impact Ionization of Positive Ions, D.L. Moores and K.J. Reed Dissociative Recombination: Crossing and Tunneling Modes, David R. Bates
Volume 35 Laser Manipulation of Atoms, K. Sengstock and W. Ertmer Advances in Ultracold Collisions: Experiment and Theory, J. Weiner Ionization Dynamics in Strong Laser Fields, L.E DiMauro and P Agostini Infrared Spectroscopy of Size Selected Molecular Clusters, U Buck Fermosecond Spectroscopy of Molecules and Clusters, T. Baumer and G. Gerber
Volume 34 Atom Interferometry, C.S. Adams, O. Carnal and J. Mlynek
Calculation of Electron Scattering on Hydrogenic Targets, I. Bray and A. T. Stelbovics Relativistic Calculations of Transition Amplitudes in the Helium Isoelectronic Sequence, W.R. Johnson, D.R. Plante and J Sapirstein
Optical Tests of Quantum Mechanics, R.Y. Chiao, P G. Kwiat and A.M. Steinberg
Rotational Energy Transfer in Small Polyatomic Molecules, H.O. Everitt and F.C. De Lucia
414
CONTENTS OF VOLUMES IN THIS SERIAL
Volume 36
Complete Experiments in Electron-Atom Collisions, Nils Overgaard Andersen and Klaus Bartschat
Stimulated Rayleigh Resonances and Recoil-induced Effects, J.-Y Courtois and G. Grynberg
Precision Laser Spectroscopy Using Acousto-Optic Modulators, W.A. van Wo'ngaarden
Highly Parallel Computational Techniques for Electron-Molecule Collisions, Carl Winstead and Vincent McKoy
Quantum Field Theory of Atoms and Photons, Maciej Lewenstein and Li You Volume 37
Evanescent Light-Wave Atom Mirrors, Resonators, Waveguides, and Traps, Jonathan P. Dowling and Julio Gea-Banacloche Optical Lattices, P.S. Jessen and I.H. Deutsch
Channeling Heavy Ions through Crystalline Lattices, Herbert E Krause and Sheldon Datz
Evaporative Cooling of Trapped Atoms, Wolfgang Ketterle and N.J. van Druten
Nonclassical States of Motion in Ion Traps, J.L Cirac, A.S. Parkins, R. Blatt and P. Zoller
The Physics of Highly-Charged Heavy Ions Revealed by Storage/Cooler Rings, P.H. Mokler and Th. St6hlker
Rydberg Ionization: From Field to Photon, G.M. Lankhuijzen and L.D. Noordam
Studies of Negative Ions in Storage Rings, L.H. Andersen, T. Andersen and P. Hvelplund
Single-Molecule Spectroscopy and Quantum Optics in Solids, WE. Moerner, R.M. Dickson and D.J. Norris Volume 39
Author and Subject Cumulative Index Volumes 1-38 Author Index Subject Index Appendix: Tables of Contents of Volumes 1-38 and Supplements Volume 40
Electric Dipole Moments of Leptons, Eugene D. Commins
High-Precision Calculations for the Ground and Excited States of the Lithium Atom, Frederick W. King Storage Ring Laser Spectroscopy, Thomas U. Kiihl
Laser Cooling of Solids, Carl E. Mungan and Timothy R. Gosnell
Optical Pattern Formation, L.A. Lugiato, M. Brambilla and A. Gatti Volume 41
Two-Photon Entanglement and Quantum Reality, Yanhua Shih Quantum Chaos with Cold Atoms,
Volume 38
Electronic Wavepackets, Robert R. Jones and L.D. Noordam
Chiral Effects in Electron Scattering by Molecules, K. Blum and D. G. Thompson
Optical and Magneto-Optical Spectroscopy of Point Defects in Condensed Helium, Serguei I. Kanorsky and Antoine Weis
Mark G. Raizen
Study of the Spatial and Temporal Coherence of High-Order Harmonics, Pascal Sali~res, Ann L'Huillier, Philippe Antoine and Maciej Lewenstein
Atom Optics in Quantized Light Fields, Matthias Freyburger, Alois M. Herkommer, Daniel S. Kriihmer, Erwin Mayr and Wolfgang P. Schleich
Atom Waveguides, Victor I. Balykin
CONTENTS OF VOLUMES IN THIS SERIAL Atomic Matter Wave Amplification by Optical Pumping, Ulf Janicke and Martin Wilkens Volume 42
Fundamental Tests of Quantum Mechanics, Edward S. Fry and Thomas Walther
415
Electron Impact Ionization of Organic Silicon Compounds, Ralf Basner, Kurt Becker, Hans Deutsch and Martin Schmidt Kinetic Energy Dependence of IonMolecule Reactions Related to Plasma Chemistry, PB. Armentrout
High-intensity Laser-Atom Physics, C.J. Joachain, M. Dorr and N.J Kylstra
Physicochemical Aspects of Atomic and Molecular Processes in Reactive Plasmas, Yoshihiko Hatano Ion-Molecule Reactions, Werner Lindinger, Armin Hansel and Zdenek Herman Uses of High-Sensitivity White-Light Absorption Spectroscopy in Chemical Vapor Deposition and Plasma Processing, L. W. Anderson, A.N. Goyette and J.E. Lawler Fundamental Processes of Plasma-Surface Interactions, Rainer Hippler
Coherent Control of Atomic, Molecular and Electronic Processes, Moshe Shapiro and Paul Brumer
Recent Applications of Gaseous Discharges: Dusty Plasmas and UpwardDirected Lightning, Ara Chutjian
Resonant Nonlinear Optics in Phase Coherent Media, M.D. Lukin, P Hemmer and M. O. Scully
Opportunities and Challenges for Atomic, Molecular and Optical Physics in Plasma Chemistry, Kurl Becker, Hans Deutsch and Mitio Inokuti
Wave-Particle Duality in an Atom Interferometer, Stephan Diirr and Gerhard Rempe Atom Holography, Fujio Shimizu Optical Dipole Traps for Neutral Atoms, Rudolf Grimm, Matthias Weidemiiller and Yurii B. Ovchinnikov Formation of Cold (T ~< 1 K) Molecules, J.T. Bahns, PL. Gould and W.C. Stwalley
The Characterization of Liquid and Solid Surfaces with Metastable Helium Atoms, H. Morgner Quantum Communication with Entangled Photons, Harald Weinfurter Volume 43
Plasma Processing of Materials and Atomic, Molecular, and Optical Physics: An Introduction, Hiroshi Tanaka and Mitio Inokuti The Boltzmann Equation and Transport Coefficients of Electrons in Weakly Ionized Plasmas, R. Winkler Electron Collision Data for Plasma Chemistry Modeling, W.L. Morgan Electron-Molecule Collisions in LowTemperature Plasmas: The role of Theory, Carl Winstead and Vincent McKoy
Volume 44
Mechanisms of Electron Transport in Electrical Discharges and Electron Collision Cross Sections, Hiroshi Tanaka and Osamu Sueoka Theoretical Consideration of PlasmaProcessing Processes, Mineo Kimura Electron Collision Data for Plasma-Processing Gases, Loucas G. Christophorou and James K. Olthoff Radical Measurements in Plasma Processing, Toshio Goto Radio-Frequency Plasma Modeling for Low-Temperature Processing, Toshiaki Makabe
416
CONTENTS OF VOLUMES IN THIS SERIAL
Electron Interactions with Excited Atoms and Molecules, Loucas G. Christophorou and James K. Olthoff Volume 45
Comparing the Antiproton and Proton, and Opening the Way to Cold Antihydrogen, G. Gabrielse Medical Imaging with Laser-Polarized Noble Gases, Timothy Chupp and Scott Swanson Polarization and Coherence Analysis of the Optical Two-Photon Radiation from the Metastable 22S1/2 State of Atomic Hydrogen, Alan J Duncan, Hans Kleinpoppen and Marian O. Scully Laser Spectroscopy of Small Molecules, W. DemtrSder, M. Keil and H. Wenz Coulomb Explosion Imaging of Molecules, Z Vager Volume 46
Femtosecond Quantum Control, T Brixner, N.H. Damrauer and G. Gerber Coherent Manipulation of Atoms and Molecules by Sequential Laser Pulses, N.V. Vitanov, M. Fleischhauer, B. W. Shore and K. Bergmann Slow, Ultraslow, Stored, and Frozen Light, Andrey B. Matsko, Olga Kocharovskaya, Yuri Rostovtsev, George R. Welch, Alexander S. Zibrov and Marlan O. Scully Longitudinal Interferometry with Atomic Beams, S. Gupta, D.A. Kokorowski, R.A. Rubenstein, and W.W. Smith Volume 47
Nonlinear Optics of de Broglie Waves, P Meystre Formation of Ultracold Molecules (T ~< 200 ~tK) via Photoassociation in a Gas of Laser-Cooled Atoms, Frangoise Masnou-Seeuws and Pierre Pillet
Molecular Emissions from the Atmospheres of Giant Planets and Comets: Needs for Spectroscopic and Collision Data, Yukikazu Itikawa, Sang Joon Kim, Yong Ha Kim, and Y C. Minh Studies of Electron-Excited Targets Using Recoil Momentum Spectroscopy with Laser Probing of the Excited State, Andrew James Murray and Peter Hammond Quantum Noise of Small Lasers, J.P Woerdman, N.J van Druten and M.P. van Exter Volume 48
Multiple Ionization in Strong Laser Fields, R. DSrner, Th. Weber, M. Weckenbrock, A. Staudte, M. Hattass, R. Moshammer, J. Ullrich, H. Schmidt-BScking Above-Threshold Ionization: From Classical Features to Quantum Effects, W. Becker, F. Grasbon, R. Kopold, D.B. Milo~eviO, G.G. Paulus and H. Walther Dark Optical Traps for Cold Atoms, Nir Friedman, Ariel Kaplan and Nir Davidson Manipulation of Cold Atoms in Hollow Laser Beams, Heung-Ryoul Noh, Xinye Xu and Wonho Jhe Continuous Stern-Gerlach Effect on Atomic Ions, Giinther Werth, Hartmut Hdffner and Wolfgang Quint The Chirality of Biomolecules, Robert N. Compton and Richard M. Pagni Microscopic Atom Optics: From Wires to an Atom Chip, Ron Folman, Peter Kriiger, J6rg Schmiedmayer, Johannes Denschlag and Carsten Henkel 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
