Advrinces in
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 38
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Advrinces in
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 38
Editors BENJAMIN BEDERSON New York University New York, New York HERBERT WALTHER Mux-Plunk-Institutfiir Quantenoptik Garching bei Miinchen Germany
Editorial Board P. R. BERMAN University of Michigtrn Ann Arbol; Michigan M. GAVRILA E 0. M. Institure vnor Atoom-en Molecuulfysica Amsterdam, The Netherlands M. INOKUTI Argonne Nutionul Laborciroty Argonne, Illinois W. D. PHILIPS National Institute j b r Standurds and Technology Gaithersburg, Maryland
Founding Editor SIRDAVID BATES
ADVANCES IN
ATOMIC, MOLECULAR, 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 FUR QUANTENOPTIK MUNICH, GERMANY
Volume 38
ACADEMIC PRESS
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This book is printed on acid-free paper. @ Copyright 0 I998 by ACADEMIC PRESS All Rights Reserved. No part of this puhlication may he reproduced or transmitted in any form or by any means. electronic or mechanical, including photocopy. recording, or any information storage and retrieval system. without permission in writing from the publisher. The appearance of the code at the bottom of the first page of a chapter in this hook indicates the Puhlisher’s consent that copies of the chapter may he made for personal or internal use, or lor the personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts 01923). for copying beyond that permitted by Sections 107 or 108 of the U S . Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes. for creating new collective works. or for resale. Copy fees for pre- 1997 chapters are as shown on the chapter title pages; il‘ no lee code appears on the chapter title page. the copy fee is the same as for current chapters. 1049-2SOX/97 $25.00
Academic Press, Inc. 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.apnct.coiii Academic Press Limited 24-28 Oval Road, London NWI 7DX, UK http://www.hhuk.co.uk/ap/ International Standard Serial Number: 1049-2SOX lntemational Standard Book Number: 0- 12-003838-2 PRINTED IN THE UNITED STATES OF AMERICA 97 98 99 00 01 02 IC 9 8 7 6 5 4
3 2
I
Contents CONTKIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vli
Electronic Wavepackets R. R. Jones and L. D.Noordiim I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
11. Rydberg Wavepackets . . . . . . . . . . ......................... 111. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 35 3s 36
IV. Acknowledgments . . . . . . . . . . . . . ......................... V. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chiral Effects in Electron Scattering by Molecules K . Blum und D. C. Thompson 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Chiral Objects and Their Syniietries: True and False Chirality . . . 111. Definitions and Fundamental Symmetries of Spin-Dependent Amp1
1v. V. v1. VII. VIII. IX. X. XI.
Experimental Observahles: Oriented Molecules . . . . . . . . . . . . . . . . . . . . . . . . Experimental Observables: Randonily Oriented Target Systems . . . . . . . . . . . . Experimental Observahles: Attenuation Experiments . . . . . . . . . . . . . . . . . . . . The Physical Cause of Chiral Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical and Computational Details . . . ......................... Results of Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . Experiniental Results . . . . ........................... Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xu. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI11. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40 53 58 62 66
I1
Optical and Magneto-Optical Spectroscopy of Point Defects in Condensed Helium Serguri I. Kunorsky und Antoine Weis 1. lntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Structure of the Point Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Implantation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Optical Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Magnetic Resonance Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88 90 95 97 111
vi
Contents
VI. ConcludingRemark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. References . . . , . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117 117
Rydberg Ionization: From Field to Photon G. M. Lankhuijzen and L. D. Noordum I. Introduction . , . , . . . . . . . 11. 111. Ramped Field Ionization . IV. Microwave Ionization . . . . . . . . . . . . . ..........._... V. THz Ionization . . . . . . . VI. VII. VIII.
..............
IX. X. References . . . , . . . . . . . . . . . . . . .
121 126 131 135 141 143 146 150 150 150
Studies of Negative Ions in Storage Rings L. H. Andersen, T. Andersen, and P: Hvelplund I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11. Lifetime Studies of Negative Ions , . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Electron-Impact Detachment From Negative Ions . . . . . . . . . . . . . . . . . . . . . .
IV. Interactions Between Photons and Negative Ions . . . . , . . . , . . . . , . . . . . . . . . V. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155 158 172 185 188 188
Single-MoleculeSpectroscopy and Quantum Optics in Solids
W E. Moernel; R. M . Dickson, und D. J. Norris I. Introduction . . . . . . . . 11. Physical Principle 111. Methods . . . , . . .
IV. Quantum Optics . . . . . . . . . , . . . . . V. Problems and Promise for Room Temp VI. References . . . . . I
I
............... ................ ............. .................. ................ . . . . . . . . . . . .. . . .
SUBJECTINDbX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS OF VOLUMES IN THIS SERIES . , ., , ., .., . . .. .... .. ., ....
.. .
.
. .. ..
..
193 196 206 22 1 228 232 237 247
Contri butors Nuinhers in parentheses indicate the pages on which the authors’ contributions begin.
L. H. ANDERSEN (155), Institute of Physics and Astronomy, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, DENMARK T. ANDERSEN (155), same address as for L. H. Andersen K. BLUM(401, Institut fur Theoretische Physik 1, Universitat Munster. WilhelmKlemm-Strasse 9,48 149 Munster, GERMANY R. M. DICKSON (193), Department of Chemistry and Biochemistry, University of California, San Diego, 9500 Gilman Drive, Mail Code 0340, La Jolla, CA 92093-0340 P. HVHPI.UND (15.5),Institute of Physics and Astronomy, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, DENMARK ROBEIUR. JONES( 1 ), Department of Physics, Jesse W. Beams Laboratory of Physics, University of Virginia, McCormick Road, Charlottesville, VA 2 9 0 1.
SEKGUE~ I. KANORSKY (87), Max-Planck Institut fur Quantenoptik, HansKopfermann Str. 1, D-8.5748, Garching. GERMANY G. M. LANKHLHJZEN (121), FOM Institute for Atomic and Molecular Physics Kruislaan 407, 1098 SJ Amsterdam, THE NETHERLANDS. W. E. MOERNER (1 93), Department of Chemistry and Biochemistry, University of California, San Diego, 9500 Gilman Drive, Mail Code 0340, La Jolla CA 92093-0340
L. D. NOORDAM (1 ) (121), FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, THE NETHERLANDS
...
Vlll
Contributors
D. J. NORRIS(193), Department of Chemistry and Biochemistry, University of California, San Diego, 9500 Gilman Drive, Mail Code 0340, La Jolla, CA 92093-0340 D. G. THOMPSON (39), Dept. of Applied Mathematics & Theoretical Physics, Queen’s University at Belfast, Belfast BT7 lNN, N. IRELAND
ANTOINE WEIS(87), Institut fur Angewandte Physik, Universitat Bonn, Wegelerstr. 8, D-53115, Bonn, GERMANY
Advrinces in
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 38
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ELECTRONIC WAVEPACKETS
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. RydhcrgWavcpackel\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Monitoring Wavepnckel Dyii;imics . . . . . . . . . . . . . . . . . . . . . . . . . . I . Tiinc-Resolved Photoemissioii and Ahsoi-piion . . . . . . . . . . . . . . . 2. Optical Ramsey Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Impulsive Momen~umK e k i e \ a l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Atomic Streak Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Additional Expcrimcntal Ccmsltlcrations . . . . . . . . . . . . . . . . . . B . Rydbcrg Wavepocket\ i i n d Classii~ulCot-upondence . . . . . . . . . . . . . . I . Radial Wavepackcis . . . . . . . . . . . . . 2. Angular or Oriented WiivqixcLcis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Stark Wavepackis . . . . . . . . . . . . . . . ................. 4 Continutiin Wavcpackcrs 111 ;I Strong Static Field . . . . . . . . . . . . . . . . 5. Tw,o-Electron Wa\ep:rckci\ . . . . . . . . . . . . . . . . . . . . . . . . . . C . Wavepacket\ Cre;itcil by Strong I ~ \ c Fields r ................... I . D w k Wavepackel\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Wavepaclds Excited DuIlng h,Zultll on loni/:\tioii . . . . . . . . . . . . . 3 . Wa\epackets Excilrd with Mitl-IR t Pulsea . . . . . . . . . . . . . . . . 4. Wavepackets kxciietl uitli T H / Hall’-(‘ycle Pulses . . . . . . . . . . . . . . . 5. Wavepackets Excitctl with GHz klalt-Cycle Pul\es . . . . . . . . . . . . . . . D. Wavepackel Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Conclulolls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV Acknowlcdgincnts . . . . . . . . . . V. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I 5 5
6
I 8 0 I0
12 I? 16 Ih 18
I9
77 __ 77 __
21 27 30 33 33 35 35 3h
1. Introduction Recent I y. there has been a great deal of cxpcriinental and theoretical interest in wavepacket studies in atomic, molecular, and condensed matter physics as well as in physical chemistry. Using the most general definition, a “wavepacket” in any quantum system is a “non-stationary” state with time-dependent expectation values for one or more operators. The wavepacket can be described tnathematically as a coherent superposition of non-degenerative, stationary-state wavefunc-
2
R. R. Jones ond L. D.Noordarn
tions, T(7,t ) = 2, u&~&?)e-"'v' a.u. The complex amplitudes, a + b, 6 s d >. Subsequent experiments observed the redistribution (see Fig. 8) as well as the coherent evolution of the dark wavepacket using a strong-field version of ORM (Duncan
I
7
24
R. R. Jones arid L. D. Noordam
FIG. 7. Possible roiitt's for redistrihution of Rydhcrg population during thc creation of a dark wavcpacket in Ba. Stimulated Ranian trnrisitions through real or virtual hound and continuum stiitcs transfer population from a single stationary Rydberg states to other nearby lcvels within the handwidth of tlic short laser pulse (froin Noordam C I d., IC)Y?a).
and Jones, 1995). The 5d7d valence character in these Rydberg states has a continuum coupling significantly hgher than that of pure Rydberg levels. Therefore, the 5Lnd part of these states mediates the production of the observed dark wavepackets in Ba (Hoogenraad et a/., 1994; Vrijeii et al., 1995; Duncan and Jones, 1995). The evolution of dark wavepackets in potassium has also been observed (Jones rt ul., 1993b). In those experiments, efficient redistribution among nf Rydberg levels occurred through resonant coupling of the initial Rydberg state to the 3d states. Although it has been predicted theoretically (Burnett et al., 1993), the creation of a dark Rydberg wavepacket has never been observed when a pure Rydberg state is coupled solely to a flat continuum, even at intensities greater than 10'4Wlcm'. The creation of dark wavepackets is a mechanism through which excited states become less susceptible to ionization, and therefore, is important to studies of atomic systems in strong fields (Burnett et ul., 1993).
2. Wuvepuckets Excited During Multiphoton Ionization The excitation of intermediate resonance plays an important role in the multiphoton ionization of atoms (Freeman et ul., 1987). Typically, only chance multiphoton resonances between bound states in the atom exist at low laser intensities.
ELECTRONIC WAVEPACKETS
25
FIG. 8. Time of Right (TOF) clectron aignal showing the population redistribution from the nominal 5d7d level lo other Rydberg states during the creation of a dark wavepacket in Ba. From hottom to top. the energy of the laser pulse that creates the dark wavepacket is 0.0. 0.05, 0.2. and 0.8 mJ, for an estimated peak laser intensity of 0.0. 0.6, 2.5, and 10 TWlcm'. respectively. At low intensity. only the initial eigenstate is populated. while at higher intensity population is detected in many adjacent Rydherg states. The traces are offset vertically from each other and the true signal level at the left side of the plot is zero for each trace. The uppermost trace wab also taken using a peak intensity of 10 TWlcm' and its amplitude has been inultiplied by a faclor of 8 relative to the other scans (from Duncan and Jones. 1995).
However, AC Stark shifts in the atom due to a pulsed laser field can produce transient resonances as the laser intensity rises and then falls during the pulse as shown in Fig. 9 (Story et al., 1993; Vrijen et al., 1993; Jones, 1995a). These Stark shifts can be greater than the photon energy so that many states may be populated during a single laser pulse. Several recent experiments have explicitly measured the population left in excited states after exposing a ground state atom to an intense laser pulse (de Boer and Muller, 1992; de Boer et ul., 1993; Jones et ul., 1993; Story et al., 1993). The constituents of the resulting wavepacket are not limited to high-lying Rydberg states and can include levels of drastically different principal and angular momentum quantum numbers. The evolution of the bound superposition state that remains after an intense pulse ionizes ground state atoms can be studied theoretically. The combination of large Stark shifts and multiphoton excitation makes it possible to coherently populate a great number of stationary states to produce wavepackets with extremely fast evolutionary time scales. Experimentally observing the dynamics is a real challenge because the evolutionary time scales of the wavepacket can be much
26
H.R. Jones and L. D. Noorclam
Time (ps)
FIG. 9. Plot of the dressed states in potassium as a l'unclion of time in the presence of an intense 590 nni laser pulse. The 4s ground state dreased by two photons shifts through the nd Rydherg serics, producing avoided level crossings between the ground and excited states during the rising and falling edges of the laser pulse. Population transfer to the Rydberg states occurs via Landau-Zener transitions at these avoided level crossings (from Story er ( I / . % 1993).
less than the optical period of the laser field. However, fast wavepackets may ultimately be of practical importance in the amplification of ultra-short laser pulses (Noordam et nl., 1990) and strong-field coherent control of photoprocesses in atoms and molecules. In a recent experiment, a 150 fsec, 777 nm laser pulse was used to four-photon ionize Na atoms (Jones, 1995b). Transient bound state resonances dominate the ionization process and a coherent superposition of several bound states is produced by the pulse. Fig. 10 shows the probability for ionizing the Na atom as a function of delay between two identical pulses. There is only a small amount of ionization during the first pulse, and the dramatic changes in the two-pulse ionization yield are due to ionization of the wavepacket produced by the first pulse as it evolves in time. The two pulses do not overlap in time, so the maximum laser intensity seen by the atom is identical to that with a single pulse. Since the two laser pulses are phase coherent, the oscillation of the wavepacket has a definite phase relationship with the second pulse. This mutual coherence can be exploited to enhance the multiphoton ionization yield. At certain delays between the two laser pulses, the two-pulse ionization yield is nearly a factor of twenty greater than that with a single pulse alone. In addition, Fourier analysis of the two-pulse ionization signal can be used to identify the bound states that are populated. The Fourier transform of the ionization signal is also shown in Fig. 10. The clear resonances correspond to one-, two-, and three-photon transitions between the 3s,
ELECTRONlC WAVEPACKETS
27
F ~ G .10. (A) Temporal interlkrogranl 01’ Na ionization signal vs. the time delay hetween two. IS0 fsec, 0.1 TWlcm’ laser pulses. The oscillation\ iti-c due to Rainsey interference in the excitation of high-lying states during the two pulaca. Below s:ituration. the signal minima and maxima are approximately 4 and I8 times higher than single pulse yield. rebpectively. ( B ) Discrete Fourier translorm 01 the temporal interlcrogr-am ahown i n ( A ) . Notc the three distinct peaks that comespond to one-photon ) , three-photon ( 3 a - 7 ~transitions ) (from Jones, 199Sb). (3s-7p). two-photon ( ~ s - J sand
4s, and 7p levels in Na. Apparently, monitoring non-Rydberg wavepackets can give insight into the evolution of atonlic systems subsequent to their exposure to intense laser fields. 3. W~ivepucketsExcited with Mid-IR FEL Pulses
Rydberg wavepackets excited from a low-lying, stationary, Rydberg state using short-pulse FIR radiation can be quite different from those produced from the ground state using optical frequencies. The size of the ground-state wavefunction limits the area in which population can be transferred to the Rydberg states to about ~ i , as , discussed in Section 1I.B. 1. In the far-infrared, however, the overlap between two Rydberg states extends farther, and one expects that larger areas of the wavefunction should participate in the population transfer. One can estimate the effective radius contributing to the population transfer by inspecting the matrix elements. Assuming that An
and (T = (o,, a2,a3) denote the Pauli matrices. The transformation properties of some relevant vectors are shown in Table 1. Note in particular that time-reversal transforms k, into -k,, and k, into -k,, and that spin vectors are reversed. From the fundamental condition that M must transform as a proper scalar, and from the transformation properties of the basic vectors (see Table 1) we obtain the following symmetry requirements of the g, functions (Johnston et al., 1993). Under spatial inversion we have
g,(-kl, -ko, -ei)
=
g,(k,, k,,, e,) j
=
0, 3
g,(-kl, -k,, -ei) = -g,(k,, k,, e,) j = 1, 2 TABLE 1
TRANSFORMATION PROPERTIES OF S O M E VECTORS. THE Pl.US ( M I N U S ) SIGN DENOTES E V ~ (ODD) N BEHAVIOR OF ’ r H b VECTORS UNDER THE
RELEVANT OPERATION.
Spntial inversion
Time-reversal
+
(3.5) (3.6)
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
51
and for time reversal (k, t)-k,,):
s,(-k,,, -k,, e,) = g,(k,, k,,, ei) j
=
0, 1, 3
g?(-k,,, -k,. ei) = -g2(kl, k,, ei>
(3.7)
(3.8)
Under rotations of the total system. k,,, k,, e,, all g, must remain invariant. We note that M must also remain invariant under a symmetry operation of the molecule alone since all physical properties remain unchanged. (More precisely, a molecular symmetry operation can only change the overall phase of M ,which is physically unimportant and can be put equal to + 1.) As a consequence. the g,-functions remain invariant. If the molecular symmetry operation transform e, into el then we must have
s,(k,.k,, e,) = s,(k,,k,,, el) B.
C H I R A L PROPERTlES OF g, AND
(3.9)
sz
It follows from eqs. 3.5 to 3.8 that go and ‘q3 transform as proper scalar functions (invariant under rotations, inversion, and time-reversal), while g, and gz transform as time-even and time-odd pseudoscalars respectively. g, and g, will therefore be central for the study of chirality. Let us consider this property from a different point of view. Consider a collision between an electron and a non-chird molecule and assume that no screw sense is de$ned hv the geome t q qffrhe collision (for example, a diatomic molecule with its axis in the scattering plane). This means that the image under spatial inversion (-kl, -k,), -ei) of the vectors k,, k,, e, can be brought, by a rotation, into a position (k,, k,,, ei), which is physically indistinguishable from the original one. Under these operations s, and g2 transform in the following way:
g,(k,, k,, ei)
=
-g,(-k,, -k,,, -ei)
(3.10)
-
-R,(kI, k,. ei)
(3.11)
k,,, ei)
(3.12)
= -g,(k,,
for i = 1,2. The first equality (3.10) describes the effect of spatial inversion (eq. 3.6). The second equality (3.11) follows from our assumption that no screw sense has been defined. The last equality (3.12) follows from condition 3.9. Thus g , and g, vanish for non-chiral systems. As an example. consider a diatomic molecule lying in the scattering plane. As discussed in Section 11, no screw sense is defined in this geometry. Assume that the e3 axis is chosen along the internuclear axis. Since the choice of el and e2 is arbitrary we can assume that el is perpendicular to the scattering plane. Under spatial inversion (k,,, k,, e,) and transformed into (-k,, -k,, -ei). A rotation of T about the el axis transforms (-k,,, -k,) into their original positions (k,, k,), and transforms the molecule into a position (-e,, e2, e,) physically indistin-
K . Blum und D. G. Thompson
52
guishable from its original one. A reflection of the molecule alone in the e2 - e3 plane leaves the molecule unchanged but brings (-e,, e2,e,) into (el, e,, e,). In conclusion a necessary condition for g, and g2to be different from zero is that a screw sense is defined, either by the structure of the molecule, or by the geometry of the experiment, or both. These concepts will be further developed in Section 4.2.
C. EXAMPLES In order to illustrate the general results we will give some examples. Since the set gJ must transfer as scalars under rotations of the total collision system, they can only depend on scalar products constructed from $,k, and the three vectors e,.From equations 3.5 to 3.8 it follows that g,, and g, can depend only on proper scalar functions, g , can depend only on time-even pseudoscalars such as ((k, + $).e,)((k+k,).e,), and g2only on time-odd pseudoscalars like @&,).el. For elastic scattering from spinless atoms in their ground state the only scalar combinations available are k; = k: = k2 and kl.kowhich are invariant under spatial inversion and time-reversal. No pseudoscalars can be constructed from k, and k,. From equation (3.6) it follows that g,(k2, kl.ko) = -g,(k2, kl.ko) j
= 1,
2
Hence g, = g2 = 0 because of the invariance of the interaction under spatial inversion. Although this is a sufficient condition, it can be shown that invariance under time-reversal requires that g, = 0 and invariance under a combination of spatial inversion and time-reversal requires that g, = 0. Equation (3.3) then reduces to the well-known expression found in electron-atom scattering:
M
= g,,l
+ g3n3.g
(3.13)
As a second example let us briefly consider interactions violating parity invariance, assuming again isotropic target states. In this case M will not remain invariant under spatial inversion and equations (3.5) and (3.6) will not hold. However, assuming that time-reversal symmetry applies, equations (3.7) and (3.8) will still be valid. The spherical symmetry of the target dictates that the gJ functions depend again on the proper scalars h? and k l . b and from equation (3.8) it follows that g2(k2,k,&) = -g,(k2, k,.ko)
Hence time reversal invariance requires g2to vanish. The most general form of
M that allows for parity violation but is invariant under time reversal is given by
M
= g,l
+ g,n,.a + g,n,.a
(3.14)
if the target possesses spherical symmetry. This case has been discussed in detail by Kessler (1985).
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
53
As a final example consider electron collisions with an oriented diatomic hetero-nuclear molecule (for example. a molecule absorbed at a surface). As discussed in section I1 a screw sense is defined by the geometry of the experiment (that is, by the vectors k,, k,, and the molecular axis n) for certain orientation n. In this simple case it is easy to give a more explicit forni for the scattering matrix M, and for the pseudoscalar functions gI and g2.We write (3.15) R?
=
(n3.n)gi
(3.16)
where now g ; and gi are proper scalars and remain invariant under spatial and time inversion and rotations of the total system. In fact. n,.n - [k,xk,,J.ntransforms as a time-odd pseudoscalar, and n1.n as a time-odd scalar. Hence, equations (3.14) and (3.15) explicitly give the syninmetry properties (3.6) to (3.8) of gI and g_..The matrix M can be written in the form
M
= g,,1
+ g;(n,.n)(n,.n)n,.cr+ gi(n3.n)n2.u+ g3n3.rr
(3.17)
We can read off from equations (3.15) and (3.16) the essential geometrical conditions that must be satisfied in order to observe chiral effects. If n lies in the scattering plane, then n3.n = 0 and g, and g, vanish identically. No screw sense is defined by the geometry of the collision, and the system exhibits no chirality. Furthermore, g , vanishes also if n,.n = 0, that is, if n lies in the n2 - n3 plane (in particular, if n is perpendicular to the scattering plane). This is an example of an oriented system that exhibits time-odd but no time-even chirality (see section IVB). In the present case chirality is defined by the relationship between the vectors k,, k,, and n. The “isomer” is obtained by spatial inversion, or more simply by a reflection, for example in the scattering plane, which transfornis n into its mirror image n’.The functions go, g ; , gi, g, remain invariant under this operation because of their scalar character. Hence, the spin scattering matrix M , describing the “antipodal system”, is simply obtained by substituting n‘ for n in equation (3.12). T h s result illustrates the well-known fact that the dynamics (expressed by g,,, g ; , gi, g,) remain unchanged when a chiral system is transformed into its mirror image.
IV. Experimental Observables: Oriented Molecules A. GENERAL EXPRESSIONS A N D EXAMPLES
Suppose that an electron beam of Larbitrarypolarization P scatters from a medium composed of identically oriented molecules. The resultant differential crosssection I and the polarization P’ of the scattered electrons are given by the expressions (Kessler, 1985):
54
K. Blum and D.G. Thompson
where 1 denotes the identity matrix. Using the identities (A and B denote arbitrary vectors)
(a.A)(o.B) = A.B
+ iu(AxB)
(4.3)
and
rr[(A.a)cr]= 2A
(4.4)
and inserting equation 3.3 for M into equations 4.1 and 4.2 yields, after some algebra, 3 3 3
3
J.k= 1 3
.j= I
1 eukIm(gig;)+ 1{(2~e(g,gr) + I go 1’6, - C ~ ~ ! ~ ~ ~ m (+g ,I g, , g I’S,I} ; ) p.q
IP‘.n, = 2~e(g,g;) -
(4.6)
k= I
where eijkis the totally antisymmetric tensor and ‘Re’ and ‘Im’ denote the real and imaginary part respectively. These equations describing the general dependence of I and P’ on P are complex compared with the atomic case for which the pseudoscalars g , and g2 must equal zero according to the discussion of section IIIC. One important property is taken over from atoms. If I P I = I, it follows that 1 P‘ 1 = 1. It is clear from physical considerations that this must be the case. The change in polarization due to scattering is a function of molecular orientation. If all molecules are identically oriented, each scattering event changes the spin vectors of the electrons in the same way, so that a beam initially in a pure state will remain so after scattering.
B. CLASSIFICATION OF CHIRAL EFFECTS We will now use the results of sections IIIA and B and give a definition and classification of chiral effects with regard to electron scattering. An observable will be called chiml if (and only if) it depends linearly on g,, or g 2 , or both. A further classification of chiral observables can be obtained by considering the transformation properties under time-reversal, equations 3.7 and 3.8. Developing these concepts, and adapting them to our present case of interest, we obtain the following hierarchy of spin-dependent effects.
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
55
(i ) A spiticillv inverted system cun be transjorniecl back into its original position bv rotution (that is, the systeni is non-chirul). In this case we have in general gll
f
0,
g , = 0,
Si
f
0
g? = 0
as has been shown in section IIIB. (ii) A spatially inverted systcvn cut1 be trunsformed back into its originul position bv time-reverscil followed by a rotution. In this case we have in general the relations
so f 0,
s3
# 0,
92 f
0
XI = 0
That g , is necessarily zero follows from eqs. 3.5 to 3.8. Applying successively spatial inversion (eq. 3.6), time-reversal (eq. 3.7), and then a rotation that transforms all vectors back into their original position gives
g,(k,, k,,. ei)= -sl(-kl,
-k,, -eiJ
where in the last step eq. 3.9 has been taken into account, similar to the derivation of eq. 3.12. Hence gI necessarily vanishes. Note that it is sufficient to consider the transformation of b,k, and the axes e, since the g, functions are independent of electron spin. (iii) A time-reversed system can be trtrnsformed back to its original position by orily a rotution. It has been shown by Johnston et al. (1993) that closed-shell systems exhibiting only time-even chirality can exist. Assume that the system is indistinguishable from its image under tiinereversal, followed by a rotation. In this case we have from equation 3.8 gz(kl,k,,, e,) = -g2(-k,,, -kl, ei) =
-gz(ki, k,, e,)
(4.8)
where again eq. 3.9 has been taken into account. Hence g2 vanishes but g, can be different from zero. (iv) Nrither (i), (ii) nor (iii) i s possible. In this case all four g, are different from zero. We will call an observable “titne-add” if it depends linearly on gz, but not g , , and we will speak of a “time-even” observable if it depends linearly on gI and not g 2 .As shown by equations 4.5 and 4.6, it is of course possible for an observable to contain time-odd and time-
56
K. Blirrn and D. G. Thonipson
even components. Time-even and time-odd chirality correspond to “true” and “false” chirality in Barron’s notation, as follows from our discussion in section 11. C. EXAMPLES
In section IVB we have related time-even and time-odd chirality to g , and g2 respectively. A determination of these parameters allows us to obtain detailed information on time-even and time-odd effects, and their relative importance for particular processes. As shown by equations 4.5 and 4.6 the observables (cross-sections and polarization) depend on both g,and gz.Hence collisions with oriented (or partially oriented) molecules allows us to study the relative importance o j time-even arid time-odd e#ects f o r cnllisiorzs. By performing several experiments with different initial polarization, and detecting the electrons with different final polarization, time-even and time-odd amplitudes g , and g? can be extracted separately from the measurements. Some cases have been discussed by Johnston et a/. (1993). Here we will give a few examples of possible experimental interest. Let us consider a beam of initially unpolarized electrons. For collisions with atoms in isotropic states the in-plane components Pi and Pi will be zero. These results are well-known from atomic collision physics (see e.g., Kessler, 1985) and follow from the fact that the total system must be invariant under reflection in the scattering plane. For chiral target systems the collision plane is in general no longer a symmetry plane, and in-plane components P ; and Pi can be produced as shown by eq. 4.6. With regard to the in-plane components it is more common to deal with the longitudinal, PI,,and transversal, P , polarization components with respect to k,. Expressing k, in terms of n, and n2 (Fig. 4) we obtain
k,
=
n1cos 812 + nz sin 812
where 8 is the scattering angle. This yields for initially unpolarized electrons
Now consider an incident beam that is completely longitudinally polarized parallel or antiparallel to k,. The asymmetry is defined by (4.10)
where I , and 1- denote the differential cross-sections for initially parallel or antiparallel longitudinal polarization. Using
k,, = n, cos 812 - nz sin 912
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
57
(Fig. 4 ) we obtain from equation 4.6
A
=
2[Rr(g,,g;) cos 812 -
Re(,y,,g;)sin 012
-
Im(g,g;) sin 0/2
+ Im(g2g;) cos 6/21/I: I g, 1’
(4.11)
Comparing equations 4.9 and 4.1 1 we see that Pi and A differ from each other in the sign of the g,-dependent terms. Pi and A vanish identically if the system exhibits no chirality, that is, if g , and g2 vanish simultaneously. We note that Pliis the sum of a term describing time-even chirality and a term describing time-odd chirality, while A is constructed from the difference of these terms. Hence, the contributions of time-even and time-odd chirality can be obtained by summing and subtracting experimental results of Piland A . It is interesting to express Pi and A in the helicity representation. Let us denote by a(+ -) the differential cross-section when the initial electrons have “spin down” with respect to the initial momentum k,,, and when the scattered electrons have “spin up” with respect to the final momentum k,, and similarly for the other combinations. We obtain
’PI;= [a(++)- a(--)] + [a(+-)- a(-+)]
(4.12)
and the asymmetry A is obtained by reversing the sign in front of the second bracket. It can be shown that the first term of equation 4.12 vanishes if g , = 0, and the second term vanishes if gz = 0. Time-even chirality is therefore responsible for the difference between the non-flip cross-sections, and if the system exhibits time-odd chirality then the helicity flip terms will differ.
tz
FIG. 4. Collision gcometry.
58
K. BIum and D.G. Thompsoii
V. Experimental Observables: Randomly Oriented Target Systems Let us now consider electron collision with an ensemble of isotropically distributed molecules. The expressions for differential cross-section and spin polarization are obtained from equations 4.5 and 4.6 by taking the average over all molecular orientations. Certain terms will vanish as a consequence of the averaging process. Which terms vanish is dictated by symmetry. We will demonstrate this for non-chiral systems, and for time-even and time-odd chiral systems, and at the end of this section we will summarize our main conclusions. A. CHIRAL MOLECULES. ELECTRON CIRCULAR DICHROISM AND OPTICAL ACTIVITY Let us first consider collisions between electrons and (truly) chiral molecules. We will consider the transformation properties of g , and g, under certain symmetry operations that will leave the handedness of the molecules unchanged. Since these functions are independent of the spin we will have only to consider the transformation of k,, k, and the molecular vectors e,. A possible sequence of operations is shown in Fig. 5, where the circle represents the isotropic molecular ensemble. Time-reversal transforms the experimental arrangement 5A into position 5B. Applying the two rotations shown in Fig. 5C and 5D we bring the system 5B back into position 5D,which is indistinguishable from the original situation 5A. The
5a
5b
5c
5d
FIG. 5 . Sequence of symmetry operations: (A) + (9)time-reversal; (9)+ (C) rotation of 7~ Hahoul the y-axis, which brings ( - k , ) into the position of the original k, vector. (C) + (D) rotalion of T about the z-axis, which hrings k; into the position of the original k,.
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
59
molecular system remains unchanged as a whole. Any individual molecule is in general transformed into another orientation, say, the initial vectors e, are brought into a final orientation el. But because of the isotropy of the molecular orientations this does not change the overall appearance of the target system. Any time-odd pseudoscalar or scalar function must vanish under the sequence of symmetry operations, 5A --+ 5D. In particular we obtain for g? g2(k,.k,,, e,) = -g&,
(5.1)
k,,, ei)
when we apply condition 3.8 and use that g, remains invariant under rotations. el denotes the final molecular position in Fig. 5D. However, we also note that under these operations g,&, k,,, e,) = gi(k,,k,,, el)
for
i
f
2
(5.2)
Equations 5.1 and 5.2 can be interpreted in the following way: For any molecular orientation ei we can find a molecule in position ef in such a way that the contributions of these molecules to g2g:, i f 2 cancel each other if the average is taken. Hence, all terms in equations 4.5 m d 4.6 linear in g 2 vanish ifthe averccge
(5.3)
(5.4)
(5.5)
(5.6) Although the symmetry of the target is higher than for a collection of oriented molecules, the symmetry is still low enough to allow effects that would be symmetry-forbidden in the scattering of electrons from isotropic atoms. These effects are characterized by interference terms in the time-even pseudoscalar gI with the proper scalars go and gi. Using the classification of chirality of section IVB we note that this corresponds to time-even chirality. Only time-eve11 chirality is exhibited by ensembles of randomly oriented closed-shell molecules. This result, obtained for electron collisions, is a special case of more general theorems derived by Evans, who applied group theoretical methods for statistical niechanics (Evans, 1988, 1989).
60
K. Blum and D. G. Thompson
Suppose that equations 5.3 to 5.6 give the observables for collisions with a collection of molecules of one enantiomeric form. A reflection in the k, - k, plane leaves the wave vectors invariant, the orientation of the molecules will change in general (say, vectors ei are transformed into orientation e y), and the handedness of the molecules is inverted. Since the reflection is equivalent to spatial inversion followed by a rotation, which returns k, and k,, to their original positions, we can use equations 3.5 and 3.6 to obtain
where the function on the right-hand sides describes the optical isomers of the original molecules. Any bi-linear combination (. . .) occurring in equations 5.3 to 5.6, which is linear in g,. changes its sign when relation 5.7 is inserted. Hence, we deduce that the equations fiw molecules of the opposite hundedness are obtained by reversing the sign ofg, in equations 5.3 to 5.6. For racemic mixtures all terms depending linearly on g , vanish therefore. Let us consider some examples. First, we note that from equations 4.9 and 4.11 we obtain for randomly oriented molecules of one enantiomeric form
Note that (Pi) and ( A ) are equal in this case. It is evident from equation 5.9 that (PI,)and ( A ) change their sign if the handedness of the molecules is reversed, and both parameters vanish for a racemic mixture. Equation 5.9 shows that unpolarized electrons can obtain a longitudinal polarization when interacting with the target system, and left- and right-handed electrons are scattered with different intensity. Both effects are analogous to optical phenomena. When unpolarized light passes through an optically active medium, the emerging beam is generally circularly polarized. In addition, a beam of right-handed circularly polarized light is absorbed differently from a left-handed beam. This phenomena is called “circulardichroism” in optics. The analgous effect for electrons, described by equation 5.9, might be considered as the electronic analogue of circular dichroism. A related phenomena in optics is “optical activity”. Here, the chiral medium causes the polarization axis of linearly polarized light to be rotated in a plane perpendicular to the direction of propagation. The analogous effect in “electron optics” is the rotation of the transverse polarization vector. Assume for example that the initial electrons are polarized in the x-direction of the lab system (see Fig. 4). From equation 5.4 we obtain that the emerging electron beam has obtained a polarization component in the y-direction, corresponding to Pi.Expressing P.n, and P.n, in terms of unit vectors along the x- and z-directions respectively (see Fig. 4). we can rewrite equation 5.4 in the form
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
61
where H is the scattering angle. The tirst term describes the production of a polarization component perpendicular to the scattering plane, which is not a chiral effect, and is well-known from atomic collision physics (Kessler, 1985).The second and third terms can be interpreted as a rotation of the initial polarization vector out of the initial s direction towards they direction perpendicular to the incoming beam. This is truly c k a l effect (since it depends linearly on gI), which vanishes for a racemic mixture. The three effects described by equations 5.9 and 5.10 were first proposed by Farago (1980). It can be deduced from equations 5.5 and 5.6 that a rotation of a polarization vector, initially parallel to the x-axis, in the scattering plane toward the z-direction, is not a chiral effect since the corresponding terms do not depend linearly on g,.In fact, rotation of P in the collision plane has been observed in atomic physics and is described by the so-called U-parameter (Kessler. 1985).
B. RANDOMLY ORIENTED NON-(*HIRAI.MOL~CULES The resulting equations follow immediately from equations 5.7 and 5.8. Since the molecules are non-chiral the k, - k,, plane is now a symmetry plane of the total system. A reflection in this plane now gives the relation
(5.11) instead of equation 5.7, where we have used condition 3.9. Condition 5.1 1 can be interpreted in the following way: The contributions to g , of molecules in positions e, and e‘; cancel each other. Hence. if the average over all molecular orientations is taken, it follows that till terms in eqwtions 5.3 to 5.6 linear in g, iniist vanish. The resulting equations are similar to those obtained for electron scattering from spinless atoms in isotropic states. In-plane and out-of-plane components of P are decoupled. The only remaining evidence of the extended M-matrix required for molecules is in the square moduli terms I g , 1’ and I g 2 1.’ C. MOLECULES WITH TIME-ODD (FALSE) CHIRALITY
Finally, let us consider electron collisions with target systems having time-odd (or “false”) chirality. Let us choose for example the spin-axis polarized molecule discussed i n section 11. The arguments from section VA leading to equation 5.1 do not apply, since time-reversal would change the handedness of the molecule. However, the combined operation of spatial inversion and time-reversal leaves the molecular chirality invariant. y , will change its sign under the coinbined operation, the other three g-functions will not. Hence, by performing a similar sequence of operations as shown in Fig. 5 it can be shown that all terms, lineur
K. Blurn and D. G. Thompson
62
in g,, will vunish in equation 4.6 ifthe average overall orientations is taken. Contributions from g 2 will survive so that time-odd chiral effects can be observed in this case. This may be an interesting new chiral effect.
D. SUMMARY We have obtained the following results for collisions with randomly oriented molecules: (a) For non-chiral molecules (or a racemic mixture of chiral molecules), all terms in equations 4.5 and 4.6 linear in the pseudoscalar functions g ! and g, vanish. (b) The prototype of “chiral” experiments, performed so far with photons or electrons, are collisions from randomly oriented molecules with time-even chirality. In this case, all terms in equations 4.5and 4.6, linear in the timeodd pseudoscalar function g,, vanish and only time-even (or “true”) chiral effects can be observed. The results for molecules with the opposite handedness are obtained by inverting the sign of g , . (c) Collisions with isotropically distributed time-odd chiral systems allow time-odd chiral effects to be experimentally observed, and all terms in equations 4.5 and 4.6 linear in g , will vanish.
VI. Experimental Observables: Attenuation Experiments A.
STRUCTURE OF THE M-MATRIX FOR
FORWARD SCATTERING
We will first develop the general theory of these effects for electron scattering. The situation is more complicated than in the previous sections since now we have to consider forward scattering and thus the interference between scattered and unscattered beams. For non-forward scattering we used the coordinate system defined by the unit vectors 3.1, and expanded M in terms of this set (eq. 3.3). For collisions in the forward direction (k, = k, = k) the set 3.1 is not defined and one has to apply a different approach. It was shown by Blum and Thompson (1989) that the M matrix for forward collisions from molecules with fixed axes can be expressed in the form 3
M
= g,,l
+ 2 g,’(kxei).cr+ gJ[(e,xe,).e,]k.a
(6.1)
r=l
Here el, e2, e3 span the molecular coordinate system as in the preceding sections. The dynamics are contained in the functions g,’ ( i = 1, . . . , 5), which are proper scalars that depend on any scalar functions that can be formed from the set ei and
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
63
k; i.e., all five functions are invariant under rotations, spatial inversion, and timereversal. As shown by Blum and Thonipson (1989) there are only four independent functions. It is convenient, however, to work with the full expression 6.1. We stress the importance of the pseudoscalar [e,xe,].e, in equation 6.1. A term like g(k.a) (where g is a proper scalar) is not invariant under space inversion and could therefore only occur in M if the interaction itself would violate parity conservation. Neglecting parity violating effects, a contribution -k.a can therefore only occur if a pseudoscalar like [e,xe,].e, can be formed from strucrz~rulelements of the target. The importance of pseudoscalars for a description of chiral phenomena has been stressed in previous sections. All molecules of a given isomer have the same value of [e,xe2].e3,independent of the molecular orientation in space. The sign of the pseudoscalar differs for the two optical antipodes. TO THE SCATTERING AMPLITUDES B. RELATION
We have already discussed that, in general, the spin scattering matrix M is characterized by four parameters g,. This number is of course related to the number of possible spin-dependent processes. In the experinients under discussion (with k and the molecular orientation fixed) there are four possible processes, two nonflip ones with t -+ 2,and two spin-flip processes ? + T . The corresponding four sattering amplitudesfim,, wzO) are related to M by equation 3.1:
It is particularly convenient to choose k as the quantization axis and we will do this throughout this section. Choosing the direction of k as the z axis of a laboratory system we have k.cr cr- and the terms kxe, are combinations of u, and CT,.The parameter g4 depends then only on the non-flip processes and g , , g 2 and g 7 are related to the spin-flip processes only. By taking the corresponding matrix elements of A4 we obtain from 6.1 :
-
+
g,, is the spin-independent part of the amplitudesf’( 1/2, + 1/2) andf( - 112, - 112) and g4the spin-dependent part. The flip amplitudes are related to g , , g,, and gl. The explicit relation depends on the direction of k relative to the molecule and will not be derived here. However, assuming that the k is purallel to e, we obtain from 6.1 and 6.2: f(1/2, -1/2)
fC-1/2,
=
-is, - g2
1/2) = ig,
since the vector product kxe, vanishes.
-
g2
(6.5) (6.6)
K. Blurn arid D. G. Thornpsorz
64
For the following it is more convenient to use the amplitude representation and we will write the M matrix in the form: f(1/2, 1/21 ,f(1/2, - 1/21 f( - 1/2, 1/2) f ( - 1/2, - 1/2
(6.7)
C. THEORY OF THE ATTENUATION EXPERIMENT In order to describe the attenuation of electron beams we have to take the interference between scattered and unscattered beams into account. Let us consider an arbitrarily polarized electron beam incident in the i-direction, on a slab of gas with width Az, containing n molecules per unit volume. We introduce the matrix
(6.8)
N=l+iyAzM
where y = 2mdk. The matrix describes how the incident beam is affected by the medium, and takes interference into account (see e.g., Mott and Massey, 1965). We will only sketch the theoretical developments here: for more details see Thompson and Kinnin (1995) and Thompson and Blum (1997). Let I,, and Po be the initial intensity and polarization of the electron beam. The intensity I , and polarization PI after passing through the slab can be obtained from the expressions
+ P,.a)N' I,P, = (I,,/2> tr[N(l + P,,.a))N'a] I , = (I,J2) tr[N(l
(6.9) (6.10)
similar to equations 4.1 and 4.2. The further analysis requires some algebra but is straightforward. Taking the limit Az + clz, and neglecting terms of second order in the amplitudes, we obtain a set of four coupled linear first-order differential equations for I , IP,, IP,,, and IP.. These equations, which are necessary for the study of attenuation in oriented molecules, have been derived by Thompson and Kinnin (1995) and by Fandreyer (1991), and they are formally similar to equations obtained for attenuation of light beams (Baron, 1982, chapter 3 ) . The equations can be simplified for randomly oriented molecules by noting that the spin-flip amplitudes, averaged over all molecular orientations, vanish. The four equations reduce to two sets of two coupled equations, one set for I and 4 the other for IP, and IPS,and solutions can be easily obtained (cf. also Farago, 1981). The equations for I and IP. become dlldz =
-
yl Imof alkali atoms picked up by the helium cluster. The pickedup atoms will eventually recombine and form dimers, trimers, or larger clusters. In this way weakly bound complexes can be synthesized in a controlled way. Using this technique the group at Princeton University has studied the vibrational structure of the 1 t1 transition in the sodium dimer (Stienkemeier et ul., 1995). The observed transitions were found to be only slightly shifted with respect to the free Nu, molecule. The fact that the cluster surface has no effective mechanism that induces spinchanging transitions in the captured atoms enables use of such clusters for the preparation of high spin aggregates. This possibility has been recently realized by Higgins et ul. (1996), who for the first time have observed the formation of quartet spin 3/2 states of alkali trimers on the surface of He clusters, and have studied their excitation and emission spectra. The analysis of the LIF spectra has revealed that upon electronic excitation the quartet trimers undergo intersystem crossing to
'xi
"x:
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
111
the doublet manifold, followed by dissociation of the doublet trimer into an atom and a covalently bound singlet dinier. This work has demonstrated that the alkali aggregates synthesized on the surface of Hr clusters can be used for optical studies of fundamental chemical dynamics processes such as nonadiabatic spin conversion, change of bonding nature, and unimolecular dissociation.
V. Magnetic Resonance Spectroscopy He matrix-isolated particles present, besides their unconventional structure, another outstanding feature-their magnetic properties. In 197 1 it was anticipated that electron bubbles should have an extremely long spin-lattice relaxation time (Huang, 1971). This is a direct consequence of the diamagnetic nature of the matrix atoms: neither the closed electronic S-shell, nor the spinless nucleus of ‘He carry magnetic moments and one expects no first order guest-host spin-spin interaction. Experiments indeed revealed spin-lattice relaxation times in excess of 100 ms for electron bubbles in Hell. In 1993 Amdt et nl. speculated that the same property should hold for paramagnetic atoms immersed in condensed ‘He. The same authors later demonstrated relaxation times in excess of 1 second (Amdt rt ul., 199%). In paramagnetic atoms a high degree of spin polarization can be created by optical pumping. Spin-polarized atoms in condensed He combine the unique features of long storage and hence observation times, long relaxation times due to non-magnetic, isotropic trapping sites, and highly sensitive detection of magnetic resonance by optical means. High-resolution spin physics experiments with such atoms can hence be applied for a number of experiments such as the study of atoniic diffusion by spin-echo techniques, radiation detection of optical pumping, investigation of hyperfine anomalies, or the challenging search for P (parity) and T (time reversal invariance) violating permanent electric dipole moments of atoms (Arndt ef d., 1993; Weis et al., 1995).
A. ELECTRON BUBBLES The ground state of the electron bubble has an S,,>-stateconfiguration and magnetic dipole transitions can be driven in this ideal 2-level quantum system when the degeneracy is lifted by an external uniform magnetic field. Experiments in the 1970s used conventional electron-spin-resonance (ESR) techniques to study the magnetic properties of electron bubbles in liquid ‘He and ’He as well as in ‘He-’He mixtures. Reichert and Dahm (1974) succeeded in the first observation of the ESR transition in a field of 4.8 kG (transition frequency 13.6 GHz). Polarized electrons wcre implanted in superfluid ‘He by field emission from a ferromagnetic tip (iron whisker) Gleich et al., 1971). The observed resonance line-width of 45(10) mG
112
Serguei 1. Kanorsky and Antoine Weis
was later found to be of instrumental origin and did not yield a quantitative determination of spin relaxation times. In subsequent experiments the line-width (HWHM) could be lowered to 5.0(4) mG (Zimmermann et al., 1977) and then to 0.9 mG (Reichert et al., 1979). The electron g-factor was found to coincide with its vacuum value at a level of 10 ppm and the (longitudinal) spin-lattice relaxation time T , was estimated to be greater than 100 ms (Zimmermann et al., 1977). Reichert and coworkers extended the measurements to ’He matrices (Reichert et al., 1979), for which they found a strong increase of the integrated ESR signal intensity with increasing He pressure, while in ‘He matrices the signal decreased with He pressure. This observation was tentatively interpreted as a consequence of the dipolar coupling of the bound electrons to the ‘He nuclear spins. The ESR spectra of positive ions in ‘He, ’He and ‘He-’He mixtures were studied by Reichert and Herold (1984). In contrast to electron bubbles the observed linewidths of the positive ions, mainly He:, in ‘He-’He mixtures, showed a strong temperature dependence, varying from 1 to 200 mG as the temperature was raised from 1 to 4 K. This behavior was explained again in terms of the relaxation induced by the hyperfine coupling to the nuclei of ’He atoms in the snowball’s frozen shell, in which the ’Hel‘He ratio is strongly temperature dependent. B. PARAMAGNETIC ATOMS The conventional ESR spectroscopy of electron bubbles requires many hours of signal averaging in order to yield reasonable signahoise ratios due to the low concentration of the spins and their small (< 1%) degree of polarization. Paramagnetic atoms can be implanted into He matrices at densities comparable to the ones achieved with electron bubbles. Magnetic resonance experiments on these species have the big advantage that optical methods can be used to build up a very large degree of (nuclear and electronic) spin polarization and that the same optical fields can be used to detect the magnetic resonance transitions in double-resonance experiments, for details see review article by Suter and Mlynek (1991). The build-up of polarization is achieved by means of optical pumping, a process in which the angular momentum of circularly polarized light is transferred to the atomic sample by successive absorption-reemission cycles. The polarized atoms are in a non-absorbing (dark) state with respect to the light field, and the magnetic resonance induced by depolarizing radio frequency or microwave radiation is detected via a resonant reappearance of fluorescence. The homogeneous optical absorption linewidth r of He-isolated atoms is approximately 6 orders of magnitude larger than in free atoms, and the peak scattering cross-section is suppressed by the same factor. However, the long electronic spin relaxation times T , , on the order of one second, anticipated for paramagnetic atoms in H e matrices mean that the atoms can be optically pumped with high efficiency, even with moderate cw laser intensities. This can be seen from
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
113
the saturation parameter CI'T,K of the optical pumping process, where R is the optical Rabi frequency. The creation of a substantial degree of spin polarization in foreign atoms isolated in heavier noble gas matrices (Ne, Ar, etc.) is impeded by the strong depolarization due to the coupling of the defect's spins to local crystalline fields present in multiple anisotropic trapping sites. To our knowledge, no optical pumping signals from atoms in such matrices have been reported in the literature so far. Recently the possibility of optically pumping molecules in condensed He has also been discussed (Takami, 1996).
1. Optical Pumping and Magnetic Resonance in Zeemun Multipletts The first optical pumping and magnetic resonance signals from atoms in super$hid He were observed in 1994 by Yabuzaki and coworkers at Kyoto University (Kinoshita et d.,1994; Takahashi rt al., 1995). In that experiment, spin polarizations in excess of 50 percent were created in "Rb, "Rb, and '33Csatoms by optical pumping with circularly polarized light. Magnetic resonance transitions induced by pulsed radio frequency fields in static magnetic fields of 30-50 G were observed. From the resonance magnetic fields and their Breit-Rabi shift when pumping with crf or cr- polarized light, the authors deduced g,-factors of 2.12(2), 2.13(4), and 2.10(8) as well as hyperfine coupling constants of 1.13(6), 3.25(2), and 2.37(11) GHz for X5Rb,X7Rb,and i33Csrespectively. These values agree, within the experimental errors, with the corresponding vacuum values. Because of signal fluctuations due to turbulences induced by the sputtering process (section III), strong r.f. pulses were needed in the experiment, resulting in r.f. power-broadened magnetic resonance line-widths on the order of 500 mG, corresponding to an effective transverse spin relaxation time T2in the psec range. The difficulties related to the short observation times in liquid matrices was overcome by the development of an efficient implantation technique for atoms into solid He. The first observation of optical pumping, optically detected magnetic resonance and magnetic level crossing signals from i33C.yatoms in solid 'He, was reported in 1995 (Arndt et ul., 19951.3; Weis et al., 1995). These experiments yielded resonance line-widths of a few mG, which were limited by magnetic field inhomogeneities. With an improved magnetic field control a resonance line-width (HWHM) of 10Hz, corresponding to 30 pG, could recently be achieved in a driven spin precession experiment with phase-sensitive detection (Kanorsky ef ul., 1996). The signal had an equivalent magnetometric sensitivity of approximately 30 nG with a 1 second integration time. In that experiment it was further shown that even with an optical sample thckness of only lo-', magnetic resonance in He-isolated atoms can also be detected by monitoring the power of the transmitted laser beam.
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2. Relu..xution Times
The optical scattering process used to detect the magnetic resonance affects the lifetime of the sample magnetization and thus contributes to the line-width of the magnetic resonance. The observed line-widths thus do not reflect the intrinsic spin relaxation time of the sample. No measurements of the spin relaxation times of polarized atoms in liquid He have been reported so far. However, both longitudinal T , (Arndt et al., 1995b) and transverse T2 (Kanorksy et id., 1996) electronic spin relaxation times of Cs atoms in the cubic phase of solid He have been measured by the technique of relaxation in the dark (Franzen, 1959). In this threestep process, optical pumping is first used to create a longitudinal/transverse spin polarization, which is then allowed to relax in the absence of light, and whose surviving degree of polarization is finally detected optically. T , values of approximately 1 second were measured (Arndt et al., 1995b) and were found to be independent of the external magnetic holding field for strengths varying from a few pG to a few G. A first measurement of the transverse relaxation time of the same sample yielded T2 = 100 msec (Kanorsky et al., 1996). This value could recently be pushed to 300 msec (Lang et al., 1997), but has still to be considered as a lower bound for the intrinsic T2.The recent observation of a spin echo from Cs atoms in b.c.c. ‘He supports the assumption that part of the dephasing time still originates from magnetic field inhomogeneities. The observed T2 time of 300 msec is equivalent to an average variation of the magnetic field of 2.5 pG over the sample volume of 1 cm’. So far, no quantitative theoretical model for the atomic spin relaxation has been developed and one can only speculate about the exact nature of the depolarization process. Depolarization by a dipolar coupling to the paramagnetic nuclei of the 3He contamination of “He, as well as dipole4ipole interactions between Cs atoms can be estimated to give a negligible contribution to the relaxation rate. The two most likely depolarizing scenarios are spin-spin interactions with paramagnetic Cs clusters, which may be present in large quantities in the sample, or spin couplings to dynamically deformed atomic bubbles. As discussed below, a quadrupolar bubble deformation can couple to the electronic spin by the combination of an electric quadrupole interaction and the hyperfine interaction. A future comparison of the relaxation times of different atomic species, or preferably, of different isotopics of the same species, should shine more light on this open question.
3. Hyperfine Transitions The first experimental determination of the hyperfine coupling constant of Cs in condensed He was performed by the Kyoto team in a microwave-optical double resonance experiment (Takahashi et al., 1995) in superjuid He under saturated
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vapor pressure. The coupling constant was determined to be A,, = 2.3127( 1) GHz, which is 0.63 percent larger than the vacuum value. The first investigation of the hyperfine structure of Cs in a solid (h.c~.c.He 1nntri.v was performed by Lang et crl. (1995) and yielded a blue shift of the transition frequency by 2.1 percent (197 MHz) with a pressure shift rate of 1.4 MHAbar. Unlike the situation encountered in heavier noble gas matrices (Jen er d., 1962; Goldsborough and Koehler, 1964; Coufal et ul., 1984), only a single hypertine resonance line was found in solid helium, thus supporting the bubble model assumption of identical trapping sites for all atoms. The observed blue shift of the hypertine frequency induced by the He matrix can be traced back to the Pauli principle, by which the valence electron wave function Ur(r) is compressed onto the nucleus, thereby increasing the coupling constant proportional to IUr(0)l'. Takahashi er d . ( 1995)calculate an increase of 0.6 percent for lq(0)l' in the liquid phase, which agrees well with the experimental finding. Kanorsky and Weis (1996) obtain a shift of 0.54 percent for Cs in liquid He and a shift of 1.53 percent for Cs in solid He at 27 bar. Both the experiment in the liquid and in the solid phase recorded the IF = 4, M = 24) + IF = 3, M = 43) hyperfine Zeeman component. In the liquid phase a lowest line-width of 10 mG (corresponding to about 25 kHz) was recorded in a free-induction decay experiment (Takahashi et ul., 1995), whereas in the solid matrix a line-width of approximately 100 kHz was found (Lang et ul., 1995). It is interesting to note that the latter result was obtained under the same experimental conditions that yielded a 10 Hz line-width (Kanorsky rr ul., 1996) for the magnetic resonance transitions within the F = 4 niultiplett. It is likely that radial (breathing mode) bubble oscillations and the induced modulation of the hyperfine coupling constant are responsible for this extraordinary broadening of the hyperfine lines. No quantitative calculation of this effect has been performed so far.
4. Mutrix Eflects The control of the matrix pressure and temperature allows implanting the foreign atoms into either the cubic (b.c.c.) or the hexagonal (h.c.p.) phase. So far only spin-polarized Cs atoms have been investigated in both phases, and the spin properties were found to have some very distinct phase dependent features (Weis P t nl., 1996; Lang rt ul., 1996). As a general rule, the efficient build-up of a large and long-lived atomic polarization can only be achieved in the isotropic cubic phase. In the hexagonal phase, local fields depolarize the atomic sample and the longitudinal spin relaxation times are in the msec range with a pronounced magnetic field dependence. As a consequence the magnetic resonance lines are orders of magnitude broader than in b.c.c. The magnetic resonance spectra show further forbidden AM = 2,3 magnetic transitions, which indicate that the cylindrical symmetry imposed by the external magnetic field is broken. These observations can be explained by assuming that in h.c.p. the atoms reside in slightly deformed
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bubbles with random orientations, The following mechanism is likely to explain the coupling of the electron spin to the bubble axis: In lowest order the deformation of the bubbles has a quadrupolar symmetry with a major axis oriented along the c-axis of the local hexagonal matrix environment. The repulsive potential exerted by the bubble wall on the Cs valence electron then leads to a slightly deformed electronic orbit, which can be described quantum mechanically as a small admixture of D-orbital to the ground state S-orbital. The D part of the atomic wavefunction produces an electric field gradient at the nucleus, which couples to the nuclear electric quadrupole moment and hence to the nuclear spin. The latter couples via the hyperfine interaction to the electron spin. The strength of this spin-bubble coupling can be inferred, for example from the magnetic field dependence of the longitudinal spin relaxation time T , . With Cs in a low-pressure h.c.p. matrix a typical decoupling frequency of 10 kHz was observed. The spinbubble coupling also lifts the Zeeman degeneracy of the defect atoms in the absence of a magnetic field, thus enabling the observation of a zero-field magnetic resonance spectrum with a dominating resonance line at 10 kHz. In principle all the observations described above can be described in terms of a single parameter-the static bubble deformation parameter p, defined in Section 11.Band the future quantitative modeling of the effects should enable its experimental determination. Recently we have also obtained some experimental evidence for dynamical bubble deformations when measuring the ratio g,/gl of the nuclear and electronic g-factors. This ratio showed a surprisingly strong magnetic field dependence at low fields, which can be traced back to quadrupolar bubble oscillations. Here again a precise modelling of the effect should enable the measurement of the amplitude of such oscillations. 5. EDMSearch
Experiments searching for permanent electric dipole moments (EDM) of atoms are borderline experiments in which the precision measurement of atomic properties is used to test modern elementary particle theories. The existence of a particle EDM is forbidden by parity conservation (P symmetry) and time reversal invariance (T symmetry). Atomic EDMs can only arise as a consequence of weak interaction processes within the atom and are a sensitive tool for searching for physics beyond the standard model of weak interactions (Ramsey, 1995). The search for an EDM in He-matrix isolated paramagnetic atoms is one of the most challenging applications of spin-polarized atoms in He matrices. It is in fact the outlook toward a possible atomic EDM experiment that has triggered our own interest in these samples some 5 years ago. Sensitive EDM experiments call for samples with extremely long electronic spin relaxation times and very large electrical break-down voltages, both conditions that are met by He-matrix isolated
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atoms. The anticipated sensitivity of EDM experiments in cryogenic samples has been discussed in the literature (Weis, 1995, 1996).
VI. Concluding Remark In the past ten years the traditional field of matrix isolation spectroscopy in solid noble gas matrices has been successfully extended to superfluid and solid He matrices in which isolated atoms and molecules exhibit some quite unique and outstanding features not encountered in any other liquid or solid matrix. So far the properties of the local trapping sites have been investigated in optical experiments that have allowed inference of their gross structure, while magnetic resonance experiments now open the way to study their static and dynamic deformations. As a consequence of the long trapping times and the isotropy of the local trapping sites encountered with many species, ultra-high-resolution magnetic resonance experiments have become feasible. In this review we have discussed the wealth of information gained in the past years by the few, mainly experimental, groups who are developing this novel interdisciplinary branch of spectroscopy. We coiiclude by expressing our hope that in future years the field will attract more attention from both the low-temperature and the traditional matrix isolation community, and that fruitful collaborations will arise from the exchange of cross-fertilizing ideas between these communities.
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ADVANCES IN ATOMIC. MOLECULAR. AND OPTICAL PHYSICS VOL 38
RYDBERG IONIZATION: FROM FIELD TO PHOTON G. M.LANKHUIJZEN AND L. D.NOORDAM FOM Institute for Atomic and Molecular Phy.rics Kruislaan 407, 1098 SJ Amstemhm, the Netherlands tel :020-6081234 email :NOORDAM@AMOLENL (September 27, 1996) 1. Introduction .................................................. A. Properties of Rydberg Atoms. . . . . . . . . . . , . . . , . . . . . . . . . . . . . . . . . . B. Rydberg Ionization: From Field to Photon. . . . . . . . . . . . . . . . . . . . . . . 11. DC Field Ionization . . . . . . . . . . . . . . . . . . . . . . . . . , . . . , . . . . . . . . . . , . , . . A. Stark States. . . . . . . . , B. Wavepacket Decay in an Electric Field . . . . . . . . , . . . . . . , . . . . . . . . . 111. Ramped Field Ionization . . . . . . . . . . . . . . . ............... A. Ionization by Ramped Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Ionization by Half-Cycle Pulses , . . . . . . . . , . , . . IV. Microwave Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Regime I : w < l/n" , B. Regime I1 : o l/n3... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . , , . V. THz Ionization . . . , . . . , . . . . . . . , . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . VI. Far Infrared Ionization . . . . . . . . . . , . . . , . . . . . . . , . . . , , . . . . . . . . . . . . . . A. Far Infrared Dipole Matrix Elements. . . . . . . . . . , . . . . . . . . . . . . . . . . . . B. Multiphoton Ionization of Rydberg Atoms Bypassing a Cooper Minimum. VII. Optical Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . , . . . , . . B. Inner Electron Excitation and Ionization . . . . . . . . . . . . . . . . . . . . . . . . . C. Rydberg States as Population Trap in Multiphoton Processes. . . . , . . . . , . VIII. Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . IX. Acknowledgment . . . . . .............................. X. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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121 122 123 126 127 128 131 13 1 134 135 136 139 141 143
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I. Introduction Atoms where one of the electrons is in a highly excited state, the so-called Rydberg atoms, have proven to be a useful tool for studying the atom-radiation interaction. In the last decade a large variety of mhation sources have been used to study the ionization mechanisms of Rydberg atoms. The physical mechanism of ionization depends on the radiation frequency. For very low frequency thejeld amplitude determines the ionization, while for higher fiquencies the photon energy is most
121
Copyright 0 1997 by Academic Press. Inc. All rights of reproductiw in any form reserved ISBNQ1243lt38-2
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G . M . Lnnkhuijzen and L. D.Noordam
essential. In this article we will review the ionization mechanism of Rydberg atoms exposed to radiation ranging from DC fields to optical frequencies. In the interaction of Rydberg atoms with a radiation field the Rydberg electron can gain enough energy to escape from the ionic potential: The atom ionizes. Important parameters to be considered in the ionization process are: (1) the ratio between the radiation frequency, w , and the binding energy of the Rydberg electron, E,,; (2) the ratio between w and the energy spacing between the Rydberg levels, AE,,; and (3) the coupling interaction between the Rydberg states. By studying the interaction with Rydberg atoms of different principal quantum number n at a given radiation frequency, these ratios vary. Hence to some extent frequency and principle quantum number are interchangeable (see Fig. 1). We consider the ionization of Rydberg atoms in the range of 10 < n < 100. In this section we introduce some properties of Rydberg atoms, and highlight some of the ionization mechanisms when these Rydberg atoms are exposed to various kinds of radiation. In subsequent sections we discuss in more detail the mechanism of Rydberg atom ionization in the different radiation frequency regimes.
A. PROPERTIES OF RYDBERG ATOMS For an extensive overview of Rydberg atoms we refer to the excellent book by T. F. Gallagher (1994). The binding energy of the Rydberg electron in the potential of a singly charged ion is given by
E
= n,‘
1
2(n
-
6,)Z’
where n is the principal quantum number, and 6, the quantum defect of the angular momentum state 1 (atomic units are used unless stated otherwise). This correction term (6,) on the binding energy arises from the presence of the core elec-
FIG. I . The frequency range that will be discussed in this chapter is plotted in the top of the figure. The lower two bars show the principal quantum number n for which the energy spacing between n states (AE,, = 11,i’) arid the binding energy (AE,, = - l/2n2) coincides with the corresponding photon energy of the radiation.
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123
trons. The low angular momentum states will penetrate the (2 - I ) core electrons giving rise to a higher effective potential. This results in a inore deeply bound energy for the low angular momentum states. For instance, the s-state of rubidium has a quantum defect of a,, = 3. I . Theoretically, the hydrogen atom is the easiest to tackle, due to the lack of core electrons and, as a result, a lack of coupling between the Rydberg states. In the experimental study of Rydberg atoms hydrogen has also been used in various experiments. However, alkali metal atoms. which can be considered one-electron atoms, are preferred experimentally because the production of a dilute gas of these atoms is rather simple. State sclective production of Rydberg atoms has become possible with the invention of tunable dye lasers. The Rydberg series of the alkali atoms lies in the operation range of the tunable dye lasers. Furthermore, interesting physics arises from the existence of the core electrons, which introduce coupling between the Rydberg slates. As we will see, this coupling can play a crucial role in the ionization mechanism of the Rydberg atom. especially for the cases of pulsed electric fields and microwave fields. The energy spacing of adjacent Rydberg states in hydrogen is given by
This energy spacing is an important property when resonant transitions between Rydberg states become possible and when the frequency and intensity of the radiation field induce energy shifts of this order. The frequency of the radiation for these transitions to occur lies in the far-infrared regime.
B. RYDBERG IONIZATION: FROMFIELD TO PHOTON We will now briefly mention some highlights in the ionization mechanisms organized in terms of the radiation frequency, covering the range where w > E,l,AE. The mechanisms will be discussed in more detail in the subsequent paragraphs.
I . DC-Field Ionization When an atom is placed in a static electric field, the atomic potential is altered by the presence of the field (see Fig. 2 ) . The tilting of the potential gives rise to a lower ionization threshold. Rydberg states with energy below the threshold can only ionize by tunnelling. We will see that the states that lie above the saddle point energy still have a finite lifetime. This is demonstrated by experiments that use short optical pulses to excite wavepackets above the saddle point energy. The evolution of the short-lived electron can be probed in two ways, either near the
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FIG. 2. The Coulomb potential experienced by a Rydberg electron without (full curve) and with (dotted curve) an external electric field applied. For DC field ionization the potential is lowered on one side, enabling electrons with energies exceeding the saddle point energy to escape from the parent ion. For optical ionization photons are absorbed by the electron leading to ionization.
core by an optical probe or near the saddle point by an atomic streak camera. It is found that the lifetime as measured by an optical techntque is not the same as the time it takes the electron to leave the atom, as measured by the atomic streak camera. 2. Ramped Field Ionizution The ionization mechanism of a Rydberg atom exposed to ramped electric field pulses in the MHz-GHz regime (rise time of the pulses ranging from microseconds to nanoseconds) is already different from the case of DC-field ionization. During the ramp of the electric field the energy of the Rydberg state will change. The change of energy depends strongly on the coupling between the Stark states, giving rise to several ionization threshold fields. Furthermore the lifetime of the “quasi continuum” states that lie above the saddle point energy plays an important role in the ionization dynamics when half-cycle pulses are used. When the lifetime of the continuum state is shorter than the pulse duration, the ionization will be suppressed. 3. Microwave lonizution
Further increasing the radiation frequency brings us into the regime of microwave radiation (GHz). For the case where the GHz frequencies are smaller than the energy spacing between the Rydberg states, Stark states and Landau-Zener tran-
RYDBERG IONIZATION: FROM FIELD TO PHOTON
125
+
sitions are used to describe the ionization. Transitions from n + ti I become possible due to the DC shift of the energy levels by the electric field, bring the levels close together. The amplitude of the field required for ionization is much lower than in the case of DC-tield ionization. For high 17 states the frequency of the radiation field can become comparable to the energy spacing between the Rydberg levels, AE,,, giving rise to multiphoton type transitions.
4. TH: Ioni7ution The production of THz radiation has recently stimulated a lot of experimental and theoretical work on the ionization of Rydberg atoms by THz radiation. Experiments have been performed with essentially unitary half-cycle pulses (HCP) with a pulse duration < 1 ps, giving a large \pectral bandwidth with a central frequency in the THz regime. Because the pulse duration is much shorter than the Kepler period of a Rydberg electron (2.5 ps for 11 = 2 5 ) , the Rydberg electron IS frozen on the time scale of the interaction with the HCP. The ionization is described using a model where the electron experiences a momentum kick from the HCP. The atom ionizes when the energy gained by the electron is larger than its original binding energy. 5. Fur hlfi-urrd Icitiizntion
Increasing the radiation frequency even further. the photon energy of the radiation becomes comparable to the binding energy of Rydberg electrons. Using far infrared radiation, transitions between Rydberg states where An >> 1 have been studied. Starting from a Rydberg state the ionization occurs via a few resonant intermediate states and ionization is very efficient. A surprising exception is the ionization of lithium Rydberg atoms. The two-photon ionization cross-section of a lithium ns ( n - 17) state is very small due to a Cooper minimum in the lithium bound-bound ns -+ n’p transition. Therefore, ionization only proceeds in a complicated manner. The Rydberg electron is first deexcited by one photon to the lower lying np state, and then climbs up the ladder (nd + n’p + continuum) bypassing the s -+ p Cooper minimum in the bound-bound transitions.
6. Optictrl Ioni:ation In the optical regime, ionization can occur by absorbing a single photon ( w >> E,,) (see Fig. 2). Although in lowest order perturbation theory the frequency determines the ionization cross-section, we will see that the field aniplitude can still play an important role at these high frequencies. Several mechanisms of stabilization of the Rydberg electron against ionization will be discussed depending on the laser intensity and laser pulse duration. Furthermore, at specific optical frequen-
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G. M . knkhuijzen and L. D.Noordam
cies the inner electrons can also be excited, giving rise to autoionizing mechanisms. Finally, Rydberg states can act as population traps in the multiphoton ionization process of ground-state atoms.
11. DC Field Ionization When an atom is placed in a static electric field, the potential experienced by a Rydberg electron is given by 1 V = - - -r F z
(3)
where F is the electric field strength in the -z direction (see Fig. 3). In the +z direction the potential is lowered by the external electric field, giving rise to a saddle point in the potential surface at z = l/ /AT) that the states are quasi stable even though their energy exceeds the saddle point energy in the potential. For non-hydrogenic atoms the Stark states are coupled by the presence of the core electrons. The blue states can now ionize on a much faster time scale through coupling with the red continuum states. In an experiment by Littman et (11. ( 1978) the ionization rate of the sodium ( n , k , m ) = ( I 3 , 3 , 2 ) state in the region of an avoided crossing with the rapidly ionizing red (14, - 11, 2 ) state was measured by scanning the electric field strength. A sharp increase in the ionization rate was observed (from lo6 s-' to >lox s ') at the avoided crossing between the two states, showing the effect of the coupling between the slowly ionizing and the fast ionizing state.
G. M. Lmkhuijzrn arid L. D.Noordum
128
The lifetime of these “quasi continuum” states is also reflected in the width of the resonances as observed in an absorption spectrum. In Fig. 4 an absorption spectrum is shown of rubidium in an electric field of 4.3 kV/cm (Lankhuijzen and Noordam, 199%). The complex structure of observed resonances can be attributed to Stark states belonging to several rz-manifolds. Close to El we observe sharp resonances, limited by the laser resolution of 0.2 cm-‘, indicating longlived states. For increasing energy of the exciting laser we observe broader peaks in the spectrum, indicating lifetimes on the order of a few picoseconds. In a similar experiment by Freeman er u1. (1978) a resonant structure above the zero-jield ionization limit was observed in rubidium in a static field of 4.335 kV/cm. The externally applied electric field induces shape resonances above the zero-field ionization limit. These resonances have also been observed in hydrogen (Glab and Nayfeh, 198% 198%; Rottke and Welge, 1986) and are more pronounced due to the lack of couping with the red continuum states.
B.
WAVEPACKET
DECAYIN AN ELECTRIC FIELD
The ionization dynamics of Stark states above the saddle point have been studied using short optical pulses to excite a coherent superposition of the “quasi continuum” states. Using optical techniques the decay of the wavepacket is monitored by measuring the amount of wave function returning to the core region as a func-
I
parallel laser polarization
-
1
E d 297
~
298
--
-
299
Wavelength (nm)
FIG. 4. loni~ationyicld of rubidium in an electric lield of 4.3 kV/cm 21s a fuiiction of the waveluiipth. The laser polarimtioii wah chosen parallel to the electric field. The upper graph shows a inagiiificd vicw of the spectrum from 299.5-300.5 nin (froin Lankhuijzen and Noordam, 19OSb).
RYDBERG IONIZATION: FROM FIELD TO PHOTON
129
tion of time. In an experiment by Broers et al. ( I 993, 1994; Christian et nl., 1993) a technique called Ramsey interferometry (Noordam et al., 1992a) was used to measure the wavepacket decay of rubidium atoms in a static electric field. In these experiments a picosecond optical pulse excites a wavepacket above the saddle point energy. After a delay, 7,.a second identical pulse is applied to probe the amount of wave function that has returned to the core region. In this way the evolution of the wavepacket can be observed. Equivalently, in the frequency domain the absorption spectrum of rubidium in an electric field has been measured. It was shown (Lankhuijzen and Noordam, 1995b) that by Fourier transforniing the relevant part of the frequency domain spectrum, the recurrence spectrum as measured with the Rariisey technique is obtained. A disadvantage of using optical techniques to study the ionizcition dynamics is the fact that the dynamics of the Rydberg electron can only be probed near the core. In the study of the ionization dynamics of atoms one would prefer to measure time resolved at the esccipe of the electron from the ionic potential, as this defines the ionization event. Using an atomic streak camera (Lankhuijzen and Noordam, 1996a),the escape of the electron has been measured directly in the time domain (Lankhuijzen and Noordam, 1996b). The atomic streak camera nieasures the escaping electron flux from the atom with picosecond time resolution. In the experiment, rubidium atoms were excited in a static electric field of 2.0 kV/cm by a short optical pulse. The excitation energy was chosen to be above the saddle point energy, thus creating an autoionizing wavepacket. By measuring the time-dependent electron flux with an atomic streak camera the ionization dynamics were resolved. In Fig. 5(a) a time resolved ionization spectrum of rubidium excited with a 4 ps pulse around 0.67 E,.is shown, where E, is the saddle point energy. Inspection of the figure shows that instead of observing an exponential decaying type of wavepacket, the main ionization was surprisingly delayed 12 ps. The polarization of the laser used to excite the wavepacket was chosen perpendicular to the electric field. The wavepacket will therefore be located perpendicular to the electric field. Because the electron can only escape in the +z direction it is still bound in the direction perpendicular to this axis. The wavepacket needs to reorient itself in the direction of the saddle point to escape. This reorienting can occur by scattering from the core electrons. The scattering in turn depends on the average value of the angular momentum of the wavepacket. For this particular case, the excited wavepacket makes an oscillation in angular momentum in 6 ps. Excited as a low angular momentum state, the wavepacket begins lo increase its angular momentum. When the wavepacket is i n a high angular momentum state it will be located far away from the core electrons (r,,,,,,(l) I ( I + 1j j , making it impossible to scatter. From Fig. 5(a) we observe that the wavepacket ionizes dominantly at the second angular recurrence. In Fig. 5(b) the corresponding recurrence spec-
-
(JX
G. M . Lunkliuijzen and L. D. Noordam
130
Rb in 2.0 2 0 kVlcm
I excitation at 0.87 0 87 E, I
-
-m
;lo 0
-
-B-
-
a,
g05 0 00
1
10-
0
'
1
'
1
'
40
20
(b) Recurrence
A
....
-
'05
00
-20
0
20 Time (ps)
40
60
FIG. 5. Comparison between time-rcsolvetl ionization spectrum as measured by the atomic streak caniei-a (a) and recurrence spectrum (b). The excihtion is at an encrgy of 0.87 E , . The laser polarization is perpendicular to the electric field of 2.0 kV/cm. In the upper spectrum the electron is prohcd at thc saddle poini, whereas in the lower spectrum the electron is probed near the core (fromLankhuijzen and Noordam. 1996b).
tmm is plotted. This spectrum is a measure for the amount of wave function that comes back to the 2 = 1 state as a function of time. We see that due to dispersion in the wavepacket, the amplitude is rather low after the first oscillation in angular momentum at 6 ps. The second angular recurrence, however, gives a high amplitude indicating that a larger fraction of the wavepacket returns to low angular momentum states. This causes the scatter event with the core electrons leading to the large ionization at 12 ps observed in Fig. 5(a). These spectra have been reproduced by a Multilevel Quantum Defect Theory (MQDT) calculation by Robicheaux and Shaw (1996). In their calculation they could observe the scatter event by observing the transfer of population from a closed channel (bound states) into an open channel (ionizing states) when the wave function returned to the core region. From the comparison of the recurrence spectra with the streak spectra the conclusion can be drawn that the lifetime as measured by an optical technique is not the same as the time it takes the electron to leave the atom. This conclusion can be drawn by probing the Rydberg electrons at different locations in the potential well. The Rydberg electron can be far away from the core, invisible for optical techniques, but still be captured in the attractive force of the parent ion. When the electron does not pass the core before ionizing, the electron appears to be ionized for an optical technique, but in fact has not yet escaped from the atom.
-
RYDBERG IONIZATION: FROM FIELD TO PHOTON
131
111. Ramped Field Ionization A. IONIZATION BY RAMPED ELECTRIC FELIX In the previous section we saw that an externally applied electric field will lower the ionization threshold of a Rydberg atom. We will now discuss the case where the electric field is not stationary, but has a pulsed character. How Rydberg states evolve from zero-tield angular momentum states to Stark states that reach the ionization threshold can be best understood in terms of a Stark energy level diagram (see Fig. 6). In this figure the energy levels of rubidium are plotted as a function of the external electric field. Inspection of the figure shows a few important features: First, every n has its own manifold of states, composed of the so-called parabolic k-states. The upper states within a manifold, blue states, increase their energy as a function of the applied electric field and are located uphill in the potential (see Fig. 3). while the red states decrease their energy and are located on the downfield side. Second, the low angular momentum states have a different zero-field energy than the high angular momentum manifold states due to their quantum defect (see Eq. 1 ). Third, the Stark states exhibit avoided crossings due to the coupling between the states.
Rb Stark rnaD -62
-63
-64
-67
-68
-69
o
5
in
15
zn
25
30
35
40
Electric Field (V/cm)
FIG. 6. Energy levels of ruhidium around I I 41 a\ ii tbnction of lhc static elcctric lield. Thc low angular nioiiicntuiii states ( I 5 2) h a w a dilfererir .xro-lield energy than the higher angular momenturn states due to their quantum defect. The dotted trajeclory hhows the adiabatic ticld ioniiation path of the 42tl statc. 7
G. M. Laiikhuijzen and L. D. Noordam
132
When the electric field is ramped, these avoided crossings, also called LandauZener crossings, can be traversed in two different ways depending on the speed at which the crossing is taken (see Fig. 7). How the crossings are traversed determines at what field strength the atom will ionize. The probability of malung a diabatic transition is well approximated by the Landau-Zener transition probability (Landau and Lifshitz, 1977),
where AE is the energy splitting at the avoided crossing and dEldt can be written as dE/dF X dFldt where dE/dF is the field-dependent Stark shift between the two levels, and dF/dt is the slew rate of the electric field. From Eq. (6) we see that for slow field ramps (i.e., dE/dt > T,,. For a discussion on T,, I become possible.
RYDBERG IONIZATION: FROM F E L D TO PHOTON
137
tion will evolve through the Start states. For microwave fields below the critical value E,, (see Fig. 10) the population will spread out over the n = 20 manifold, but is unable to reach any other manifold: Ionization will not occur. When the microwave field amplitude is high enough to reach the intersection of the adjacent ( n + 1) manifold, population can be transferred to the higher manifold in a single cycle of the electric field (Lankhuijzen and Noordam, 1995a). The population cannot be transferred down to the n - 1 manifold because the field amplitude is not large enough to reach that intersection. On subsequent cycles of the microwave field the Rydberg population can be transferred to even higher n-manifolds. If the atom interacts with the microwave radiation for a long enough time, the population will climb up the Rydberg ladder until the field reaches the classical ionization threshold at F = 1/1611':The atom ionizes. We see that the first step in the ionization process-the traversal of the first anticrossing with the adjacent manifold-is the rate limiting step for long microwave pulses. The field at which the manifolds intersect can easily be calculated and is given by 1
F,, = 311'
(7)
For n > 6 this so-called Inglis-Teller limit is smaller than the F = 1/16n4value. In an experiment by Pillet et 01. (1984). sodium Rydberg atoms were ionized with long (-0.5 ps) pulses of 3.15 GHz microwave radiation. For the Jrnl 5 1 components the onset of ionization occurred at the F,., value, but for the Irnl = 2 states the ionization occurred at the much higher field of F = 1/9n4. The coupling between the Iml = 2 states of sodium is so small that all the avoided crossings are traversed completely diabatically, making it impossible for the population to climb the Rydberg ladder. Therefore the ionization occurred at the classical limit of the red states. This has also been studied by Mahon et (11. (1991) as a function of the microwave frequency ranging from 10 MHz-I 5 GHz. In an experiment by Hettema r t al. (1990), it was found that the ionization rate is frustrated when Stark states that lie in the middle of the manifold are excited. During the oscillations ofthe microwave field these middle Stark states do not change their energy, and the coupling with the reddest members of the higher manifold is frustrated. The diabatic ionization threshold has also been observed for hydrogen (Bayfield and Koch, 1974; van Leeuwen et L i I . , 1985). If we only take into account the first order Stark effect, the slope of the Stark states, dE/dF, is constant. When a particular Stark state is put in the microwave field the Rydberg orbit does not change because the dipole moment is constant. A middle Stark state, which will remain a middle Stark state after a reverse of the field amplitude, will ionize at a much higher field then the reddest Start state ofthat manifold due to the lack of coupling with the red continuum states. The second order Stark effect is needed to describe the observed ionization threshold at F = 1/9n4. When the second order is taken into account dEldF is not constant any more, but is decreasing as a
138
G. M. Linkhuijzen and L. D. Noordatn
function of the field. On the reversing of the microwave field each Stark state will therefore be projected onto several other Stark states. The population will be diffused through the Stark manifold and ionize once the field is large enough to ionize one of the Stark states, that is, the reddest Stark state. From these observations we see that Landau-Zener transitions are possible whenever the frequency, o,is in the range where a partially diabatic transition (see Eq. 6) is possible. The role of the frequency of the microwave radiation is rather limited, as it only determines at which of the two amplitude thresholds the ionization will occur. In order to reach the ionization limit, many oscillations of the microwave field are needed. In an experiment by Gatzeke et al. (1994), short microwave pulses consisting of as little as 25 oscillations of the electric field were used to ionize Na n = 24-33 Rydberg states. Instead of the sharp thresholds observed at F = 1/31? for the long microwave pulses, the thresholds became broader, indicating that the ionization was frustrated by the limited number of oscillations of microwave radiation field. By state selective field ionization they observed an enhanced population of higher Rydberg states after exposure to the shortest microwave pulses (25 cycles). Recently Watluns et al. (1996) used even shorter microwave pulses, down to only 5 oscillations of the microwave field at 8 GHz to ionize Na n = 32-44d Rydberg states (see Fig. 11). Not only did they observe a broader threshold in the ionization signal as a function of the microwave amplitude, but for the shortest pulses (6 cycles) the threshold shifted to the classical diabatic ionization limit at F = 1/9nJ. In this case the number of steps needed to reach the ionization threshold for a microwave ampli-
--c 3000 cycles
0
100
200
300
Microwave field amplitude (Vicm)
F a . I I . Surviving Rydberg population after expostire of the 44d state in sodium to inicrowavc pulses of various duration. For the longest pulse ionization occurs at the F = 1/3n5threshold. For the shortest pulse the ladder climbing is frustrated and ionization takes place at the F = 1/9n4 threshold (from Watkins et a/., 1996).
RYDBERG IONIZATION: FROM FIELD TO PHOTON
139
tude corresponding to F,T= 1/3n5 is not sufficient. Using state selective field ionization after the short microwave pulse was applied to the n = 24s state, they were able to determine that the population was trapped in higher and lower Rydberg states (n = 23-30) due to the limited number of cycles. We have seen that, for microwave ionization of Rydberg atoms following the mechanism of climbing the Rydberg ladder, Landau-Zener transitions are needed to transfer the population up the Rydberg ladder. The transition should lie in the intermediate regime between a fully diabatic and adiabatic transition, putting some constraints on the frequency of the microwave radiation. B. REGIME 11 : w
- 1/n3
When the frequency of the microwave radiation becomes comparable to the zerofield Rydberg spacing, the corresponding slew rate of the field becomes so large that the Landau-Zener picture is not applicable any more and microwave ionization is described as a combination of photon transitions to hgher Rydberg states followed by field ionization of this higher Rydberg state. These photon transitions have been observed in hydrogen. In an experiment by Bayfield and Koch (1974) hydrogen ti 65 (23 GHz spacing between neighboring states) atoms were exposed to microwave radiation of different frequencies. For frequencies of 30 MHz and 1.5 GHz the diabatic ionization threshold was found to occur at the field value of F = 1/9n4. Because the Stark states are not coupled in hydrogen, the population transfer to higher n-manifolds was not possible, giving rise to the classical ionization limit. However, in the case of 9.9 GHz microwave radiation the required field for ionization was lowrer than this classical limit. In an experiment by van Leeuwen rf ul. (1985), the ionization of hydrogen by 9.9 GHz was studied by varying the initial Rydberg n-state. For the case where w > AE,!,or equivalently the pulse duration, T,,. is much less than the classical round-trip time of the electron, T,!,the ionization can be described using the impulse kick model. In this model it is assumed that the Rydberg electron does not move during the pulse, but gains momentum from the THz pulse in the form of a momentum kick given by t .,
Ap
=
Fl,
,
u(t)dt,
(9)
where F,, is the field amplitude and u ( t )is the normalized temporal profile of the HCP. The classical momentum of a Rydberg electron can be calculated from its binding energy El, = - 112n' and is given by plI = = Iln. The change in total energy of the Rydberg electron after excitation with the half-cycle pulse is then given by
AE
2 To
=
($' -
'T;'@
=
. A$
+ A$'/2
(10)
= + A$. We see that for Ap >> po the change in energy is given by AE -- AS;)'/2. When the change in energy exceeds the binding energy of the Rydberg electron (AE> 1/2n'), the atom can ionize, giving the relation Ap F,, > lln. In Fig. 14 the calculated scaled fields for 10 percent ionization of hydrogen are plot-
where
G. M. Lankhuijzen and L. D. Noordam
142
loo UP
lo-'
lo'
3" loo
(Jones et al 1993)
FIG. 14. Scaled field (upper figure) and scaled inomenturn transfer (lower figure) for 10 percent ionization threshold of H(n, 1 = 2, r n = 0 ) atoms as a function of the scaled pulse duration: classical resulta for inverse parabolic (full curve) and rectangular (dotted curve) pulses, quantum mechanical result for ionization of the 10d (full triangle) and Sd (inverted full triangle) states for rectangular pulses and experimental data by Jones and Bucksbaum (1993) for Na(nd) atoms, inultiplied hy a factor 2 3 (opcn squares) (from Reinhold et u / . ?1993).
ted as a function of the scaled time q, = ?,IT,,(full curve). From this figure we clearly see the transition from the short-pulse regime to the long-pulse regime. The long-pulse regime, that is, the adiabatic regime, has been discussed in section 111, giving rise to the F = lh4 scaling law. For the short-pulse regime significantly stronger pulses are needed to ionize the Rydberg atoms. Both classical and quantum calculations give the same result, indicating that the classical impulse-kick model is valid in the short pulse regime. The characterization of these HCPs in amplitude and pulse shape is still rather difficult. Because the pulse is freely propagating through space, the integral of the electric field over time has to be zero, showing that the pulse can never be unipolar. The main HCP peak is followed by a long negative tail with low amplitude. The effect of this negative tail following the HCP can substantially alter the ionization probability (Tielking et al., 1995). To circumvent this problem an experi-
RYDBERG IONIZATION: FROM FIELD TO PHOTON
143
ment was performed by Frey et ul. ( I 996) where extremely high n-states were ionized by nanosecond unitary half-cycle pulses. The production of these nanosecond electrical pulses could be done in a much more controlled fashion giving a complete characterization of the pulses. In the experiment, n-states around n 388 and n 520 were exposed to half-cycle pulses with durations ranging from T, = 2 ns to 110 ns. The corresponding classical round trip time of these n-states is T,, - 9 ns and T,, 21 ns respectively, enabling the researchers to study the transition from the short-pulse (T, > T,J regime. They found perfect agreement with the theory by Reinhold rf al. (1993).
-
-
-
VI. Far Infrared Ionization In this section we discuss the radiation frequency regime where photon-transitions between Rydberg states with An >> 1 can occur. The transitions to higher lying states and subsequent ionization are determined by the resonance frequency of the radiation and the dipole matrix elements between the Rydberg states. The ionization with far infrared radiation (FIR) is examined using blackbody radiation sources (Beiting et al., 1979; Gallagher and Cooke, 1979; Figger et id., 1980) and CO? lasers (Burkhardt et ~ 1 . 1993). . At 300 K the energy density of the blackbody radiation peaks at a wavelength of 9.6 pm. However, the stimulated absorption and emission rates of the Rydberg states depend on the photon occupation number, which drops rapidly as a function of the photon energy. In an experiment by Beiting et (11. (1979), the redistribution of the xenon 26j Rydberg state to higher states was measured for various exposure times to the blackbody radiation. They found that after a 15.5 ps exposure time Rydberg states with n > 30 were populated. The states were detected using state-selective field ionization (see section 111). The spectra showed clear peaks in the spectrum, indicating that the populated states were low angular momentum states. This showed that photon transitions (A1 = 2 1 ) to the higher lying states were indeed responsible for the population redistribution. Ionization by blackbody radiation only occurs for very high n states and long exposure times and is strongly frustrated by de-excitation to lower lying states. In an experiment by Burkhardt et nl. (1993) the ionization from the 12s state in sodium by irradiation with laser light coming from a COz laser (a few lines around 10pm) was studied. By changing the electric field high Rydberg states that lie just below the ionization threshold shifted into resonance, giving rise to a large enhancement of the two-photon ionization yield. A disadvantage of using blackbody radiation to ionize an atom is the fact that the bandwidth of the incoherent radiation is enormous, making it very difficult to drive transitions between two specific Rydberg states. Recently an intense farinfrared free-electron laser has become operational (Oepts et d.,1995). This laser has a much narrower bandwidth (ANA = 0.1% - 10%) and a large tuneability
144
G. M.Lnnkhuijzeri aiid L. D. Noordm
(A = 6 - 100 pm), malung it possible to drive photon transitions between Rydberg states where An >> 1 (Hoogenraad, 1996). Multiphoton ionization is efficient because the excitation via intermediate states is resonant within the bandwidth of the FIR laser pulse. For instance, in rubidium a resonant four-photon ionization process is possible from the 14cl state. When this state is irradiated with 46-pm radiation, the ionization process resonantly follows the 14d + 18p + 24s -+ 6Op + e / d ladder (Hoogenraad, 1996; Hoogenraad et al., 1996). A. FARINFRARED DIPOLE MATRIX ELEMENTS
In this section we will introduce some analytical results on the matrix elements between Rydberg states. The dipole matrix elements between loosely bound states of hydrogen can be approximated semiclassically with high accuracy. The following derivation has been taken from Delone et al. (1994) and Hoogenraad and Noordam (1996). From the correspondence principle it follows that dipole matrix element equals the Fourier component at the transition frequency w of the classical radial coordinate r(t) of the Rydberg electron along its Kepler orbit, as follows:
I
111
(n’(r(n)=
T,,
r(t)cos wfdf.
(11)
I1
T,, is the period of the classical orbit time (T,! = 2 m ’ ) . For transitions between near-lying states, the transition frequency can be expressed as 0 = (n’ - n)/n’. In the half of the orbit where the electron moves away from the core, the trajectory of an out-going 1 = 0 orbit can be written as
1 r(t) = -(61)~/~ 2 Note that in this equation IZ is absent: The orbiting times of higher n states increase as n3,while the outer turning points (the maximal value of r) scale as n2. Substituting the half-orbit time in Eq. 12 yields the correct n dependence of the outer turning point. By substituting Eq. 12 into Eq. 11 and using the relation 7’,, = 2m’, the matrix element is given by
where the integral takes values between 0.5 and 1 and slowly converges to ( r ( 2 / 3 ) / d 3 The . two main characteristics of matrix elements between Rydberg states are present in this formula: First, the matrix elements are normalized to the density of states (n- I ) , so that an integral over an interval of the spectrum is normalized. Second, the matrix elements depend on the transition frequency as @-?
RYDBERG IONIZATION: FROM FIELD TO PHOTON
145
Transitions between nearby-lying states are favored over high-frequency transitions. Equation 13 can be further generalized to remove the asymmetry of exchanging the initial n and final n’. Reformulating the results of Goreslavski et (11. (1982) in terms of the binding energy of the states, w,, = 1/2n2, and the exact transition frequency w = Iqj,- q,l yields
The matrix element consists of three parts: The densities of states in the Rydberg series at states n and n‘, a general frequency dependence, and a prefactor C = 0.4108. This prefactor is only valid for 1 = 0 states. It can, however, be replaced by a frequency and angular-momentum dependent function (Hoogenraad, 1996; Berson, 1982; Goreslavski; ~t ul., 1982). In doing so the C factor stays of the same order but shows a higher value for the A1 = 1 transition compared to the A1 = - 1 transition, and for transitions where the energy increases, in agreement with the Bethe rule.
+
B. MULTIPHOTON IONIZATION OF RYDBERC ATOMSBYPASSING A COOPER MINIMUM The multiphoton ionization rate of a Rydberg state depends strongly on the dipole matrix elements between the intermediate states. In general, these dipole matrix elements are rather large, making the atom very susceptible to far-infrared radiation excitation, but there are exceptions. For instance, a Cooper minimum (Cooper, 1962) exists in the bound-bound ns - n‘p transition in lithium, giving rise to a very low excitation probability (Hoogenraad et al., 1995; Hoogenraad, 1996). In Fig. 15 the relevant energy levels of lithium are plotted. The left part of the figure shows the matrix elements for the 23s to np transition, with the Cooper minimum occurring around the 50p state. When the 23s state was exposed to farinfrared radiation of A = 62.5 pm (- 160 cm-’), the measured ionization yield was much higher than expected on the basis of the presence of the Cooper minimum. By scanning the frequency of the far-infrared radiation, an enhanced ionization yield was measured at the frequency corresponding to the resonant transition from the 23s down to the 17p state. If only the direct 2-photon ionization (23s 4 50p -+continuum) was the relevant ionization mechanism, the ionization yield would be a smooth function of the frequency, because the bandwidth of the laser pulses is much larger than the Rydberg spacing around n = 50. The resonance indicates that another path dominates the ionization process. Within the bandwidth of the laser, resonant transitions follow the 23s -+17p + 22d + 50p + edcf continuum, apparently the dominant pathway in the ionization process. Due to the Cooper minimum, two-photon ionization is frustrated and a higher-order process is favored, starting with stimulated emission of a FIR photon instead of absorbing photons.
146
G. M. Lnnkhuijzen and L. D. Noordam
Binding Energy (cm
-I)
r
-20
-30
0
-40 -50
-200
-60
-400
-600
0
1
2
3
Angular Momentum Frc. IS. (a) Squared and normalized matrix clemen~snear the Cooper minimum in the 23s -+ np series. The normalized matrix elements from the 23s to the 171, state are 3.8 X 10” ( c I J R ~ ) (b) ’ . The frustrated direct two-photon path (starting with the dashed line) and alternative higher-order multiphoton paths to the continuum, bypassing the Cooper minimum (from Hoogenraad et a/., 1995).
VII. Optical Radiation The energy of an optical photon (0.1 p m < A < 1 pm) exceeds the binding energy of Rydberg states, and ionization can occur by the absorption of a single photon. Although in lowest order perturbation theory the frequency determines the ionization cross-section, the field amplitude can still play an important role at these high frequencies. It has been found that for high amplitudes of the optical radiation the ionization cross-section actually decreases (an effect known as adiabatic stabilization, which will be discussed in section VI1.A). At high frequencies the excitation of inner electrons also becomes possible (section VI1.B). The energy exchange of the excited inner electron with the Rydberg electron can lead to autoionization. In the final section V1I.C we will see that in the process of multiphoton ionization of ground state atoms, Rydberg states can be a “dead end” en route to the ionization continuum.
RYDBERC IONIZATION: FROM FIELD TO PHOTON
147
Let us start with the ionization rate, R, and its physical implications. Starting from Eq. 14, it follows that the single-photon ionization rate for a low angular momentum Rydberg state (I