Theory of Multiphoton Processes

414 Pages·1987·10.017 MB·English

by  Farhad H. M. Faisal (auth.)

THEORY OF
MULT IPHOTON
PROCESSES
PHYSICS OF ATOMS AND MOLECULES
Series Editors
P. G. Burke, The Queen's University of Belfast, Northern Ireland
H. Kleinpoppen, Atomic Physics Laboratory, University of Stirling, Scotland
Editorial Advisory Board
R. B. Bernstein (New York, U.S.A.) C. J. Joachain (Brussels, Belgium)
J. C. Cohen-Tannoudji (Paris, France) W. E. Lamb, Jr. (Tucson, U.S.A.)
R. W. Crompton (Canberra, Australia) P.-O. LOwdin (Gainesville, U.S.A.)
J. N. Dodd (Dunedin, New Zealand) H. 0. Lutz (Bielefeld, Germany)
G. F. Drukarev (Leningrad, U.S.S.R.) M. R. C. McDowell (London, U.K.)
W. Hanle (Giessen, Germany) K. Takayanagi (Tokyo, Japan)
ATOM-MOLECULE COLLISION THEORY: A Guide for the Experimentalist
Edited by Richard B. Bernstein
ATOMIC INNER-SHELL PHYSICS
Edited by Bernd Crasemann
ATOMS IN ASTROPHYSICS
Edited by P. G. Burke, W. B. Eissner, D. G. Hummer, and I. C. Percival
AUTOIONIZATION: Recent Developments and Applications
Edited by Aaron Temkin
COHERENCE AND CORRELATION IN ATOMIC COLLISIONS
Edited by H. Kleinpoppen and J. F. Williams
COLLISIONS OF ELECTRONS WITH ATOMS AND MOLECULES
G. F. Drukarev
DENSITY MATRIX THEORY AND APPLICATIONS
Karl Blum
ELECTRON AND PHOTON INTERACTIONS WITH ATOMS
Edited by H. Kleinpoppen and M. R. C. McDowell
ELECTRON -ATOM AND ELECTRON -MOLECULE COLLISIONS
Edited by Juergen Hinze
ELECTRON-MOLECULE COLLISIONS
Edited by Isao Shimamura and Kazuo Takayanagi
INNER-SHELL AND X-RAY PHYSICS OF ATOMS AND SOLIDS
Edited by Derek J. Fabian, Hans Kleinpoppen, and Lewis M. Watson
INTRODUCTION TO THE THEORY OF LASER-ATOM INTERACTIONS
Marvin H. Mittleman
ISOTOPE SHIFTS IN ATOMIC SPECTRA
W. H. King
PROGRESS IN ATOMIC SPECTROSCOPY, Parts A, B, and C
Edited by W. Hanle, H. Kleinpoppen, and H. J. Beyer
THEORY OF MULTIPHOTON PROCESSES
Farhad H. M. Faisal
VARIATIONAL METHODS IN ELECTRON-ATOM SCATTERING THEORY
R. K. Nesbet
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THEORY OF
MULTIPHOTON
PROCESSES
Farhad H. M. Faisal
University of Bielefeld
Bielefeld, Federal Republic of Germany
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library of Congress Cataloging in Publication Data
Faisal, Farhad H. M.
Theory of multiphoton processes.
(Physics of atoms and molecules)
Includes bibliographical references and index.
I. Multiphoton processes. I. Title. II. Series.
QC793.5.P422F35 1986 539.7'217 86-22674
ISBN 978-1-4899-1979-3 ISBN 978-1-4899-1977-9 (eBook)
DOI 10.1007/978-1-4899-1977-9
© 1987 Springer Science+ Business Media New York
Originally published by Plenum Press, New York in 1987
Softcover reprint of the hardcover 1st edition 1987
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording, or otherwise, without written permission from the Publisher
To my mother,
who nourished my first interests in theoretical subjects,
and
to the memory of my father
Preface
My aim in this book has been to give an account of the theoretical methods
of analysis of multiphoton processes in atomic physics. In this account I
have emphasized systematic methods as opposed to ad hoc approaches.
Both perturbative and nonperturbative methods are presented with il
lustrative results of concrete applications. The perturbation theory is the
primary tool of analysis of nonresonant multiphoton processes. It is
developed here in conjunction with a diagrammatic language and is also
renormalized to free it from the unwanted divergences which accompany
the ordinary treatment when higher-order corrections are considered. The
nonperturbative methods (i.e., methods other than that of power series ex
pansion in the field strength) become particularly important for consistent
treatments of problems involving, for example, intermediate resonances,
high field strengths, and finite pulse duration. The specifically nonpertur
bative methods for multiphoton transitions are presented in Chapters 6-11.
The methods of resolvent equations and of effective Hamiltonians are
developed for both the stationary and the time-dependent fields. The densi
ty matrix method is presented in conjunction with the problems of relaxa
tion and of fluctuating fields. The Floquet theory is presented both in the
energy domain and in the time domain. Also treated are the methods of
continued fractions, recursive iterative equations, and chain Hamiltonians.
Non-Hermitian Hamiltonian methods for multiphoton transitions are
developed using the techniques of optical potential and rotation of coor
dinates. The book closes with a chapter on the theory of multiple emission
and absorption of photons during electron-atom collisions in a laser field.
I have tried to give each chapter a somewhat autonomous character so
that the reader may use a part of the book without necessarily having to
read everything that precedes it. References to the literature sources used
are provided throughout the text, and I hope these will be useful to the
reader. I must add, however, that they are by no means meant to be ex
haustive. The interested reader will find a comprehensive bibliography on
vii
viii Preface
the subject in the excellent ongoing compilation Multiphoton Bibliography,
edited by J. H. Eberly eta/. (see, e.g., N .B.S. LP-92 suppl. 4/1984, Univer
sity of Colorado and University of Rochester, edited by J. H. Eberly, N.D.
Pitche, and J. W. Gallagher).
This book is primarily intended for those who plan to pursue active
theoretical research in the subject. This group may include research students
or more mature physicists to whom the subject may be new. Some of the ex
tensions and developments that have not appeared in print before might be
of interest to specialists in the subject.
During the period of writing this book I have benefited from numerous
exchanges with many of my colleagues and students. I am especially
thankful to Dr. J. T. Broad, Dr. M. Crance, Prof. A. Maquet, Prof. R. P.
McEachran, and Dr. D. Miller for their helpful correspondence and/or
comments on the manuscript. I am indebted to Prof. A. Maquet for sending
me the original of Figure 22, to Mr. L. Dimou for Figures 52-64, to Mr. S.
Jetzke for Appendix 2, and to Dr. D. Miller for preparing the index. I
would like to thank Mr. H. Zerhau for typing the manuscript with ex
emplary patience. It is a pleasure to thank Dr. Ken Derham and the staff of
Plenum Publishing Corporation for their excellent collaboration at various
stages of producing this book. The research leading to this work has been
supported by the Deutsche Forschungsgemeinschaft under Project No. SFB-
216-M2. Finally, my special thanks are due to Prof. H. Kleinpoppen, whose
friendly invitation and encouragements to write a monograph on the subject
initiated this work.
F. H. M. Faisal
Contents
1. Electrons and Atoms in a Radiation Field 1
1.1. Introduction . . . . . . . . . . . 1
1.2. The Minimum-Coupling Prescription . . . 1
1.3. The First Quantization and the Semiclassical Approximation 4
1.4. The Gauge Invariance and Gauge Transformation of
the Schrodinger Equation . . . . . . . . . 5
1.5. The Hamiltonian of a Light Wave . . . . . . . 6
1.6. The Dipole Approximation and Two Forms of the
Interaction Hamiltonian . . . . . . . . . . 8
1.7. The Wave Function of a Free Electron in a Laser Field 10
1.7.1. The Wave Function in the "Velocity Gauge" 10
1.7.1.1. The Linearly Polarized Laser Field 10
1.7.1.2. The Circularly Polarized Laser Field 11
1.7.2. The Wave Function in the "Length Gauge" 12
1.8. From Light Waves to Photon Fields · · · · · 12
1.8.1. The Second Quantization of Light Fields 14
1.9. The Hamiltonian of an Atom in a Monomode Laser Field in the
Dressed-Oscillator Representation · · . . . . 18
1.10. The Dressed-Oscillator Atom-Field Hamiltonian
with Multimode Laser Fields 22
1.11. Dressed Frequencies and Polarizations of a
Two-Mode Field . . . . . · · · · 25
1.12. The "Length-Form" Atom-Field Hamiltonian 27
2. The Perturbation Theory . . . . . 29
2.1. Introduction . . . . · · · · · 29
2.2. Time-Dependent Perturbation Theory 29
2.3. The Diagrammatic Method . . . . 34
2.4. Rules for Construction of the Diagrams 34
2.4.1. Two-Photon Absorption . 36
2.4.2. Three-Photon Absorption 39
2.4.3. Four-Photon Absorption . 39
ix
X Contents
2.4.4. Single-Photon Absorption . . . . . . . 40
2.4.5. Two-Photon Absorption (Two-Mode Field) . . 41
2.4.6. Third-Harmonic Generation (Two-Mode Field) 42
2.4.7. Stimulated Emission of One Photon and Absorption
of Two Photons in a Three-Mode Interaction . 44
2.5. Rules for Writing Down the Transition Amplitude
from a Diagram . . . . . . . 45
2.5.1. Two-Photon Absorption . . . . . 46
2.6. The Rate of Multiphoton Transition and the
Generalized Cross Section . . . . . . 47
3. Renormalization of Perturbation Theory 53
3.1. Breakdown of Conventional Perturbation Theory . 53
3.2. The Resolvent and the Wave Function . . . . 55
3.3. The T-Matrix and the "Linked Cluster" Expansion ofT . 57
3.4. Relation between True and Reduced T-Matrices 58
3.5. The Renormalized T-Matrix . 59
3.6. Energy Renormalization of External States . 60
3.7. Lagrange Expansion of the Energy Shift . . 61
3.8. Energy Shift from the Effective Hamiltonian 62
3.9. The Renormalized Rate of Multiphoton Transitions 63
3.9.1. The Singular 4" -Function 63
3.9.2. The Transition Rate . . . . . . 64
4. Methods for Evaluation of Generalized
Cross Sections 67
4.1. Introduction . . . . . 67
4.2. The Method of Green's Function 68
4.2.1. The Radial Coulomb Green's Function 69
4.2.1.1. The Closed-Form Representation 69
4.2.1.2. The Integral Representation . . 70
4.2.1.3. The Sturmian Representation 71
4.2.2. The Quantum-Defect Radial Green's Function 73
4.2.2.1. The Closed-Form Representation . . 73
4.2.2.2. The Pseudopotential Green's Function 77
4.3. The Method of Inhomogeneous Differential Equations 79
4.4. The Pseudocontinuum-Basis Method 82
4.5. The Truncated Summation Method. 85
5. Nonresonant Multiphoton Ionization 89
5.1. Introduction . . . . . . . . . 89
5.2. Multiphoton Ionization of the Hydrogen Atom . 90
5.2.1. Two-Photon Ionization of the Hydrogen Atom. 92
5.2.2. Three-and Many-Photon Ionization of Hydrogen . 97
5.3. Multiphoton Ionization of Nonhydrogenic Atoms 102