Stochasticity and Quantum Chaos Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 317 Stochasticity and Quantum Chaos Proceedings of the 3rd Max Boro Symposium, Sob6tka Castle, September 15-17, 1993 edited by Zbigniew Raba, Wojciech Cegla and Lech Jak6bczyk Institute ofTheoretical Physics, University ofWroclaw, Wroclaw, Poland SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. CataIogue record for this book is available from the Library of Congress. ISBN 978-94-010-4076-1 ISBN 978-94-011-0169-1 (eBook) DOI 10.1007/978-94-011-0169-1 Printed an acid-free paper AII Rights Reserved © 1995 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1995 Softcover reprint ofthe hardcover Ist edition 1995 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic Of mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright ownef. CONTENTS Foreword The quantal fattening of fractals N.L. Balazs 3 How and when quantum phenomena become real Ph. Blanchard A. ladczyk 13 Chaotic dynamics in a periodically driven anharmonic oscillator Yu.L. Bolotin V. Yu. Gonchar M. Ya. Granovsky 31 Coherent and incoherent dynamics in a periodically driven bistable system T. Dittrich F. Grossmann P. Hanggi B. Oe/schlagel R. Utermann 39 The Ehrenfest theorem for Markov diffusions P. Garbaczewski 57 The quantum state diffusion model, asymptotic solutions, thermal equilibrium and Heisenberg picture N. Gisin 63 Stochastic representation of quantum dynamics Z. Haba 73 Level repulsion and exceptional points W.D. Heiss and W.-H. Steeb 91 Type-II intermittency in the presence of additive and multiplicative noise H. Herzel F. Argoul A. Arneodo 99 Aspects of Liouville integrability in quantum mechanics 1. Hietarinta 115 vi KAM techniques for time dependent quantum systems H.R. Jauslin 123 Dissipation and noise in quantum mechanics N.G. van Kampen 131 Irregular scattering, number theory, and statistical mechanics Andreas Knauf 137 Band random matrices, kicked rotator and disordered systems Luca Molinari 149 Robust scarred states D. Richards 161 The quasiclassical statistical description of quantum dynamical systems and quantum chaos Yu. P. Virchenko 177 Quantum measurement by quantum brain Mari Jibu and Kunio Yasue 185 Time reversal and Gaussian measures in quantum physics J. C. Zambrini 195 Stochastic resonance in bistable systems with fluctuating barriers L. Gammaitoni F. Marchesoni E. Menichella-Saetta S. Santucci 209 Index 217 List of participants 219 FOREWORD These are the proceedings of the Third Max Born Symposium which took place at SobOtka Castle in September 1993. The Symposium is organized annually by the Institute of Theoretical Physics of the University of Wroclaw. Max Born was a student and later on an assistant at the University of Wroclaw (Wroclaw belonged to Germany at this time and was called Breslau). The topic of the Max Born Sympo sium varies each year reflecting the developement of theoretical physics. The subject of this Symposium "Stochasticity and quantum chaos" may well be considered as a continuation of the research interest of Max Born. Recall that Born treats his "Lectures on the mechanics of the atom" (published in 1925) as a nrst volume of a complete monograph (supposedly to be written by another person). His lectures concern the quantum mechanics of integrable systems. The quantum mechanics of non-integrable systems was the subject of the Third Max Born Symposium. It is known that classical non-integrable Hamiltonian systems show a chaotic behaviour. On the other hand quantum systems bounded in space are quasiperi odic. We believe that quantum systems have a reasonable classical limit. It is not clear how to reconcile the seemingly regular behaviour of quantum systems with the possible chaotic properties of their classical counterparts. The quantum proper ties of classically chaotic systems constitute the main subject of these Proceedings. Other topics discussed are: the quantum mechanics of dissipative systems, quantum measurement theory, the role of noise in classical and quantum systems. The Symposium came into being thanks to the enthusiasm of the lecturers and the substantial help of our colleagues. It has been supported by the Polish Research Council (KBN) and by the University of Wroclaw. We are thankful to everybody who contributed to the success of this Symposium. Finally, we would like to thank Mrs. Anna Jadczyk for her excellent work in preparing the manuscripts. Wroclaw, February 28th 1994 The Editors THE QUANTAL FATTENING OF FRACTALS N.L. BALAZS Department oj PhY6ic6 State Univenity oj New York at Stony Brook Stony Brook, NY 11794 - 9800 Abstract. Periodic points of a classical map may belong to a fractal set. It is shown here on a simple model that upon quantising this classical map the influence of the classical periodic points upon the quantal results is the same whether the periodic point belongs to a fractal set, or not. A simple intuitive explanation is given. 1. INTRODUCTION How does one quantise fractals? Is there a meaning associated with these words? Fractals were born through the iteration of functions of a complex variable giving rise to Julia sets (1918)1). They were thus intimately connected with the theory of (discrete) dynamical systems where the equations of dynamics are replaced by an iterative scheme, the mapping of the phase space onto itself. It is thus a natural step to generate fractal sets through the use of a dynamical system, and ask what happens to these fractal sets if we quantise the classical map. A large amount of numerical evidence suggests that the classical periodic points (or orbits) have far reaching effects on the quantum mechanical results. In particular, we stress three such manifestations. In the quantal description of a classical map the state of the system is specified by a vector in a finite dimensional vector space, and the time evolution is generated by a unitary operator acting on this space. This unitary operator, i.e. its eigenangles and eigenvalues contains all the information about the dynamical system. The classical periodic points will leave their imprints on the quantal results in three different places. a) There is strong numerical evidence that the presence of hyperbolic periodic points in the classical description changes the statistics of the nearest neighbour distribu tion of the eigenangles in a well defined manner2}; b) the eigenvectors may be "scarred" by the periodic points3}; c) in the coherent state representation the correlations between an initial state and its time evolved version should be large at those pq points which correspond to clas sical periodic points. What happens, however, if the classical periodic points are fractals, hence form a measure zero set? This note will discuss this problem as exemplified on a simple modeI 4}. 3 Z. Haba et at. (eds.), Stochasticity and Quantum Chaos, 3-11. © 1995 Kluwer Academic Publishers. 4 N.L. BALAZS Frg(3a) Fig. 1. Nearest neighbour spacing distribution (nnsd) for quantum maps with different number of strips. The number of phase space cells, N, vary slightly, since they must be divisible by the numb~r of strips. They are succesively 288, 290, 294, 288. 2. THE MODEL Consider a classical discrete dynamical system that maps the unit square unto itself. The horizontal axis is labeled by q, the position (in units of the maximum displace ment); the vertical axis is labeled by p, the momentum (in units of the maximum momentum). The transformation that represents the model is described geometri cally as follows. Divide the unit square into three vertical strips. The first and third strips have the same width a. These vertical strips will be now changed into hori zontal ones, and stacked on the top of each other. The first strip is made horizontal through a vertical compression of factor a, followed by a horizontal dilation of the same factor, becoming thereby a horizontal strip at the bottom. The second vertical strip is simply rotated into the horizontal position, becoming thereby the second horizontal strip. The third vertical strip is subjected to the same compression and dilation as the first, but laid down as the top horizontal strip. Thus, the evolution of the system is the iteration of the transformation