Quantum Signatures of Chaos Springer-Verlag Berlin Heidelberg GmbH ONLINE LIBRARY Physics and Astronomy http://www.springer.de/phys/ Springer Series in Synergetics http://www.springer.de/physlbooks/sssyn An ever increasing number of scientific disciplines deal with complex systems. These are systems that are composed of many parts which interact with one another in a more or less complicated manner. One of the most striking features of many such systems is their ability to spontaneously form spatial or temporal structures. A great variety of these structures are found, in both the inanimate and the living world. In the inanimate world of physics and chemistry, examples include the growth of crystals, coherent oscillations of laser light, and the spiral structures formed in fluids and chemical reactions. In biology we encounter the growth of plants and animals (morphogenesis) and the evolution of species. In medicine we observe, for instance, the electromagnetic activity of the brain with its pronounced spatio-temporal structures. Psychology deals with characteristic features of human behavior ranging from simple pattern recognition tasks to complex patterns of social behavior. Examples from sociology include the formation of public opinion and cooperation or competition between social groups. In recent decades, it has become increasingly evident that all these seemingly quite different kinds of structure formation have a number of important features in common. The task of studying analogies as well as differences between structure formation in these different fields has proved to be an ambitious but highly rewarding endeavor. The Springer Series in Synergetics provides a forum for interdisciplinary research and discussions on this fascinating new scientific challenge. It deals with both experimental and theoretical aspects. The scientific community and the interested layman are becoming ever more conscious of concepts such as self-organization, instabilities, deterministic chaos, nonlinearity, dynamical systems, stochastic processes, and complexity. All of these concepts are facets of a field that tackles complex systems, namely synergetics. Students, research workers, university teachers, and interested laymen can find the details and latest developments in the Springer Series in Synergetics, which publishes textbooks, monographs and, occasionally, proceedings. As witnessed by the previously published volumes, this series has always been at the forefront of modem research in the above mentioned fields. It includes textbooks on all aspects of this rapidly growing field, books which provide a sound basis for the study of complex systems. Series Editor Hermann Haken Institut fUr Theoretische Physik and und Synergetik Center for Complex Systems der Universitiit Stuttgart Florida Atlantic University 70550 Stuttgart, Germany Boca Raton, FL 33431, USA Advisory Board Ake Andersson Luigi Lugiato Royal Institute of Technology Dipartimento di Fisica Department of Infrastructure Universita degli studi di Milano and Planning (RP) Via Celoria 16 10044 Stockholm, Sweden 20133 Milan, Italy Bernold Fiedler Jiirgen Parisi Freie Universitiit Berlin Fachbereich Physik, Abt. Energie Institut fUr MatlI I und Halbleiterforschung Arnimallee 2-6 Universitiit Oldenburg 14195 Berlin, Germany 26m Oldenburg, Germany Yoshiki Kuramoto Manuel G. Velarde Department of Physics Instituto Pluridisciplinar (USM) Graduate School of Sciences Paseo Juan XXIII, NO.1 Kyoto University 28040 Madrid, Spain Kyoto 606-8592, Japan Fritz Haake Quantum Signatures of Chaos Second Revised and Enlarged Edition With 66 Figures Springer Professor Dr. Fritz Haake Institut flir Theoretische Physik, Fachbereich 7, Physik University of Essen Postfach 10 37 46 45117 Essen, Germany The first edition appeared as Springer Series in Synergetics, Volume 54, under the title: F. Haake, Quantum Signatures of Chaos Library of Congress Cataloging-in-Publication Data Haake, Fritz. Quantum signatures of chaos / Fritz Haake.--2nd ed. p. cm. -- (Springer series in synergetics, ISSN 0172-7389) The first edition appeared as Springer series in synergetics, volume 54 under the title: F. Haake, Quantum signatures of chaos--T.p. verso. Includes bibliographical references and index. ISBN 978-3-642-08722-6 ISBN 978-3-662-04506-0 (eBook) DOI 10.1007/978-3-662-04506-0 I. Quantum chaos. I. Title. II. Series. QI72.5.C45 H33 2000 003'.857--dc21 00-063786 ISSN 0172-7389 ISBN 978-3-642-08722-6 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992, 2001 Originally published by Springer-Verlag Berlin Heidelberg New York in 2001 Softcover reprint of the hardcover 2nd edition 2001 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the author using a Springer TEX macro package. Final typesetting: LE-TEX Jelonek, Schmidt & Viickler GbR, Leipzig Cover design: design & production, Heidelberg Printed on acid-free paper SPIN: 10663193 55/3141/mf -5 4 3 2 1 0 Fur Citta und Julia Preface The warm reception of the first edition, as well as the tumultuous development of the field of quantum chaos have tempted me to rewrite this book and include some of the important progress made during the past decade. Now we know that quantum signatures of chaos are paralleled by wave signa tures. Whatever is undergoing wavy space-time variations, be it sound, electro magnetism, or quantum amplitudes, each shows exactly the same manifestations of chaos. The common origin is nonseparability of the pertinent wave equation; that latter "definition" of chaos, incidentally, also applies to classical mechanics if we see the Hamilton-Jacobi equation as the limiting case of a wave equation. At any rate, drums, concert halls, oscillating quartz blocks, microwave and opti cal oscillators, electrons moving ballistically or with impurity scattering through mesoscopic devices all provide evidence and data for wave or quantum chaos. All of these systems have deep analogies with billiards, much as the latter may have appeared of no more than academic interest only a decade ago. Of course, molec ular, atomic, and nuclear spectroscopy also remain witnesses of chaos, while the chromodynamic innards of nucleons are beginning to attract interest as methods of treatment become available. Of the considerable theoretical progress lately achieved, the book focuses on the deeper statistical exploitation of level dynamics, improved control of semiclassical periodic-orbit expansions, and superanalytic techniques for dealing with various types of random matrices. These three fields are beginning, independently and in conjunction, to generate an understanding of why certain spectral fluctuations in classically nonintegrable systems are universal and why there are exceptions. Only the rudiments of periodic-orbit theory and superanalysis appeared in the first edition. More could not have been included here had I not enjoyed the privilege of individual instruction on periodic-orbit theory by Jon Keating and on superanalysis by Hans-Jiirgen Sommers and Yan Fyodorov. Hans-Jiirgen and Yan have even provided their lecture notes on the subject. While giving full credit and expressing my deep gratitude to these three colleagues, I must bear all blame for blunders. Reasonable limits of time and space had to be respected and have forced me to leave out much interesting material such as chaotic scattering and the semiclassical art of getting spectra for systems with mixed phase spaces. Equally regrettably, no justice could be done here to the wealth of experiments that have now been performed, but I am happy to see that gap filled by my much more competent colleague Hans-Jiirgen Stockmann. VIII Preface Incomplete as the book must be, it now contains more material than fits into a single course in quantum chaos theory. In some technical respects, it digs deeper than general introductory courses would go. I have held on to my original inten tion though, to provide a self-contained presentation that might help students and researchers to enter the field or parts thereof. The number of co-workers and colleagues from whose knowledge and work I could draw has increased considerably over the years. Having already men tioned Van Fyodorov, Jon Keating, and Hans-Jurgen Sommers, I must also ex press special gratitude to my partner and friend Marek KU8 whose continuing help was equally crucial. My thanks for their invaluable influence go to Ser gio Albeverio, Daniel Braun, Peter Braun, Eugene Bogomolny, Chang-qi Cao, Dominique Delande, Bruno Eckhardt, Pierre Gaspard, Sven Gnutzmann, Peter Goetsch, Siegfried Grossmann, Martin Gutzwiller, Gregor Hackenbroich, Alan Huckleberry, Micha Kolobov, Pavel Kurasov, Robert Littlejohn, Nils Lehmann, Jorg Main, Alexander Mirlin, Jan Mostowski, Alfredo Ozorio de Almeida, Pjotr Peplowski, Ravi Puri, Jonathan Robbins, Kazik Rz~ewski, Henning Schome rus, Carsten Seeger, Thomas Seligmann, Frank Steiner, Hans-Jurgen Stockmann, Jurgen Vollmer, Joachim Weber, Harald Wiedemann, Christian Wiele, Gunter Wunner, Dmitri Zaitsev, Kuba Zakrzewski, Martin Zirnbauer, Marek Zukowski, Wojtek Zurek, and, last but not at all least, Karol Zyczkowski. In part this book is an account of research done within the Sonderforschungs bereich "Unordnung und GroJSe Fluktuationen" of the Deutsche Forschungsge meinschaft. This fact needs to be gratefully acknowledged, since coherent long term research of a large team of physicists and mathematicians could not be maintained without the generous funding we have enjoyed over the years through our Sonderforschungsbereich. Times do change. Like many present-day science authors I chose to pick up D-1EX and key all changes and extensions into my little machine myself. As usu ally happens when learning a new language, the beginning is all effort, but one eventually begins to enjoy the new mode of expressing oneself. I must thank Peter Gerwinski, Heike Haschke, and Rudiger Oberhage for their infinite patience in getting me going. Essen, July 2000 F. Haake Foreword to the First Edition The interdisciplinary field of synergetics grew out of the desire to find general principles that govern the spontaneous formation of ordered structures out of microscopic chaos. Indeed, large classes of classical and quantum systems have been found in which the emergence of ordered structures is governed by just a few degrees of freedom, the so-called order parameters. But then a surprise came with the observation that a few degrees of freedom may cause complicated behavior, nowadays generally subsumed under the title "deterministic chaos" (not to be confused with microscopic chaos, where many degrees of freedom are involved). One of the fundamental problems of chaos theory is the question of whether deterministic chaos can be exhibited by quantum systems, which, at first sight, seem to show no deterministic behavior at all because of the quantization rules. To be more precise, one can formulate the question as follows: How does the tran sition occur from quantum mechanical properties to classical properties showing deterministic chaos? Fritz Haake is one of the leading scientists investigating this field and he has contributed a number of important papers. I am therefore particularly happy that he agreed to write a book on this fascinating field of quantum chaos. I very much enjoyed reading the manuscript of this book, which is written in a highly lively style, and I am sure the book will appeal to many graduate students, teachers, and researchers in the field of physics. This book is an important addition to the Springer Series in Synergetics. Stuttgart, February 1991 H. Haken Preface to the First Edition More than sixty years after its inception, quantum mechanics is still exerting fascination on every new generation of physicists. What began as the scandal of non-commuting observables and complex probability amplitudes has turned out to be the universal description of the micro-world. At no scale of energies accessible to observation have any findings emerged that suggest violation of quantum mechanics. Lingering doubts that some people have held about the universality of quan tum mechanics have recently been resolved, at least in part. We have witnessed the serious blow dealt to competing hidden-variable theories by experiments on correlations of photon pairs. Such correlations were found to be in conflict with any local deterministic theory as expressed rigorously by Bell's inequalities. - Doubts concerning the accommodation of dissipation in quantum mechanics have also been eased, in much the same way as in classical mechanics. Quantum ob servables can display effectively irreversible behavior when they are coupled to an appropriate environmental system containing many degrees of freedom. Even in closed quantum systems with relatively few degrees of freedom, behavior re sembling damping is possible, provided the system displays chaotic motion in the classical limit. It has become clear that the relative phases of macroscopically distinguishable states tend, in the presence of damping, to become randomized in exceedingly short times; that remains true even when the damping is so weak that it is hardly noticeable for quantities with a well-defined classical limit. Consequently, a super position (in the quantum sense) of different readings of a macroscopic measuring device would, even if one could be prepared momentarily, escape observation due to its practically instantaneous decay. While this behavior was conjectured early in the history of quantum mechanics it is only recently that we have been able to see it explicitly in rigorous solutions for specific model systems. There are many intricacies of the classical limit of quantum mechanics. They are by no means confined to abrupt decay processes or infinitely rapid oscillations of probability amplitudes. The classical distinction between regular and chaotic motion, for instance, makes itself felt in the semiclassical regime that is typically associated with high degrees of excitation. In that regime quantum effects like the discreteness of energy levels and interference phenomena are still discernible while the correspondence principle suggests the onset of validity of classical mechanics. The semiclassical world, which is intermediate between the microscopic and the macroscopic, is the topic of this book. It will deal with certain universal XII Preface to the First Edition modes of behavior, both dynamical and spectral, which indicate whether their classical counterparts are regular or chaotic. Conservative as well as dissipative systems will be treated. The area under consideration often carries the label "quantum chaos". It is a rapidly expanding one and therefore does not yet allow for a definite treatment. The material presented reflects subjective selections. Random-matrix theory will enjoy special emphasis. A possible alternative would have been to make current developments in periodic-orbit theory the backbone of the text. Much as I admire the latter theory for its beauty and its appeal to classical intuition, I do not understand it sufficiently well that I can trust myself to do it justice. With more learning, I might yet catch up and find out how to relate spectral fluctuations on an energy scale of a typical level spacing to classical properties. There are other regrettable omissions. Most notable among these may be the ionization of hydrogen atoms by microwaves, for which convergence of theory and experiment has been achieved recently. Also too late for inclusion is the quantum aspect of chaotic scattering, which has seen such fine progress in the months between the completion of the manuscript and the appearance of this book. This book grew out of lectures given at the universities of Essen and Bochum. Most of the problems listed at the end of each chapter have been solved by students attending those lectures. The level aimed at was typical of a course on advanced quantum mechanics. The book accordingly assumes the reader to have a good command of the elements of quantum mechanics and statistical mechanics, as well as some background knowledge of classical mechanics. A little acquaintance with classical nonlinear dynamics would not do any harm either. I could not have gone through with this project without the help of many col leagues and coworkers. They have posed many of the questions dealt with here and provided most of the answers. Perhaps more importantly, they have, within the theory group in Essen, sustained an atmosphere of dedication and curiosity, from which I keep drawing knowledge and stimulus. I can only hope that my young coworkers share my own experience of receiving more than one is able to give. I am especially indebted to Michael Berry, Oriol Bohigas, Giulio Casati, Boris Chirikov, Barbara Dietz, Thomas Dittrich, Mario Feingold, Shmuel Fish man, Dieter Forster, Robert Graham, Rainer Grobe, Italo Guarneri, Klaus-Dieter Harms, Michael Hohnerbach, RalfHiibner, Felix Israilev, Marek KU8, Georg Lenz, Maciej Lewenstein, Madan Lal Mehta, Jan Mostowski, Akhilesh Pandey, Dirk Sa her, Rainer Scharf, Petr Seba, Dima Shepelyansky, Uzy Smilansky, Hans-Jiirgen Sommers, Dan Walls, and Karol Zyczkowski. Angela Lahee has obliged me by smoothening out some clumsy Teutonisms and by her careful editing of the manuscript. My secretary, Barbara Sacha, de serves a big thank you for keying version upon version of the manuscript into her computer. My friend and untiring critic Roy Glauber has followed this work from a dis tance and provided invaluable advice. - I am grateful to Hermann Haken for his invitation to contribute this book to his series in synergetics, and I am all the more honored since it can fill but a tiny corner of Haken's immense field. How ever, at least Chap. 8 does bear a strong relation to several other books in the