The Transition to Chaos: Conservative Classical and Quantum Systems (Fundamental Theories of Physics, 200) 9783030635336, 9783030635343, 3030635333

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Fundamental Theories of Physics 200

Linda Reichl

The Transition to Chaos Conservative Classical and Quantum Systems Third Edition

Fundamental Theories of Physics Volume 200

Series editors Henk van Beijeren, Utrecht, The Netherlands Philippe Blanchard, Bielefeld, Germany Bob Coecke, Oxford, UK Dennis Dieks, Utrecht, The Netherlands Bianca Dittrich, Waterloo, Canada Detlef Dürr, Munich, Germany Ruth Durrer, Geneva, Switzerland Roman Frigg, London, UK Christopher Fuchs, Boston, USA Domenico J. W. Giulini, Hanover, Germany Gregg Jaeger, Boston, USA Claus Kiefer, Cologne, Germany Nicolaas P. Landsman, Nijmegen, The Netherlands Christian Maes, Leuven, Belgium Mio Murao, Bunkyo-ku, Tokyo, Japan Hermann Nicolai, Potsdam, Germany Vesselin Petkov, Montreal, Canada Laura Ruetsche, Ann Arbor, USA Mairi Sakellariadou, London, UK Alwyn van der Merwe, Colorado, USA Rainer Verch, Leipzig, Germany Reinhard F. Werner, Hannover, Germany Christian Wüthrich, Geneva, Switzerland Lai-Sang Young, New York City, USA

The international monograph series “Fundamental Theories of Physics” aims to stretch the boundaries of mainstream physics by clarifying and developing the theoretical and conceptual framework of physics and by applying it to a wide range of interdisciplinary scientific fields. Original contributions in well-established fields such as Quantum Physics, Relativity Theory, Cosmology, Quantum Field Theory, Statistical Mechanics and Nonlinear Dynamics are welcome. The series also provides a forum for non-conventional approaches to these fields. Publications should present new and promising ideas, with prospects for their further development, and carefully show how they connect to conventional views of the topic. Although the aim of this series is to go beyond established mainstream physics, a high profile and open-minded Editorial Board will evaluate all contributions carefully to ensure a high scientific standard.

More information about this series at http://www.springer.com/series/6001

Linda Reichl

The Transition to Chaos Conservative Classical and Quantum Systems Third Edition

Linda Reichl Center for Complex Quantum Systems Department of Physics University of Texas Austin, TX, USA

ISSN 0168-1222 Fundamental Theories of Physics ISBN 978-3-030-63533-6

ISSN 2365-6425 (electronic) ISBN 978-3-030-63534-3 (eBook)

1st edition: © Springer Science+Business Media New York 1992 2nd edition: © Springer Science+Business Media New York 2004 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Acknowledgements

The first edition of The Transition to Chaos appeared in 1992. It was based on a series of lectures on classical and quantum chaos that I gave at the University of California, San Diego, in 1987, and then at the Guangxi Normal University in Guilin China in 1990. The first edition focused on the classical chaos theory and the manifestations of chaos in bounded quantum systems. The second edition was published in 2004 and included the manifestations of chaos in open quantum systems with a strong emphasis on random matrix theory. This third edition contains selected material from the previous editions but focuses on topics that have become important to modern research, including the influence of chaos on molecular scattering processes, lattice dynamics, gravitational structures, radiation-matter interactions, and the thermalization and control of quantum systems. The book has been influenced, over the years, by discussions and collaborations with numerous colleagues. I particularly wish to thank Katya Lindenberg of UCSD who, years ago, invited me to write this series on chaos theory. I also wish to thank Christof Jung for our collaborations on classical scattering theory. This third edition has benefited from funding by the Robert A. Welch Foundation of Texas. Their support has enabled a number of students to work on aspects of the research reported here. The University of Texas at Austin Austin, TX, USA August 2020

Linda Reichl

v

Contents

1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Historical Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Plan of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 7 8

2

Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Conventional Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Integrable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Hidden Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Poincaré Surface of Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Henon-Heiles System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 The HOCl Molecule and Birkhoff Coordinates. . . . . . . . . . 2.4.3 Lattice Surfaces of Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Nonlinear Resonance and Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Single-Resonance Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Two-Resonance Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 KAM Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 The KAM theorem (for N = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The Definition of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Lyapounov Exponent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 KS Metric Entropy and K-Flows . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 13 16 17 19 20 21 23 25 25 25 30 32 34 36 36 40 45 45

3

Area-Preserving Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Twist Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Derivation of a Twist Map from a Torus . . . . . . . . . . . . . . . . . 3.2.2 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 50 50 51 vii

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3.2.3 Birkhoff Fixed Point Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 The Tangent Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Homoclinic and Heteroclinic Points. . . . . . . . . . . . . . . . . . . . . . 3.3 The Standard Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Scaling Behavior of Noble KAM Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Renormalization in Twist Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Integrable Twist Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Nonintegrable Twist Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 The Universal Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Bifurcation of M-Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Some General Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 The Quadratic Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Scaling in the Quadratic de Vogelaere Map . . . . . . . . . . . . . . 3.7 Cantori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Renormalization Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Expression for the Renormalization Map . . . . . . . . . . . . . . . . 3.8.2 Fixed Points of the Renormalization Map . . . . . . . . . . . . . . . 3.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 54 55 57 62 65 66 69 70 71 71 72 74 80 84 87 90 94 96

4

Chaotic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Complete Ternary Horseshoe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Double Gaussian Potential Energy Peaks . . . . . . . . . . . . . . . . 4.2.2 Delta-Kicked System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Scattering Chaos in a Magnetic Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Model of Chlorine Ion in a Radiation Field . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Scattering Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Delay Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Symbolic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Chaos in the HOCl Molecular System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Homoclinic Tangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Scattering Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 99 101 101 103 110 114 116 118 119 123 126 127 130 131

5

Arnol’d Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Arnol’d Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Arnol’d Diffusion and Nekhoroshev Time. . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Graphical Evolution of the Arnol’d Web . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Arnol’d Diffusion in an Optical Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Arnol’d Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Arnol’d Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Stability of the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Colliding Beam Synchrotron Particle Accelerator . . . . . . . . . . . . . . . .

133 133 135 137 139 140 143 146 149 151

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5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6

Quantum Dynamics and Random Matrix Theory . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Invariant Measure for the GOE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Probability Density that Extremizes Information . . . . . . . . . . . . . . . . . 6.3.1 Polar Form of Probability Density . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Cluster Expansion of the Probability Density . . . . . . . . . . . 6.4 Eigenvalue Statistics: Gaussian Orthogonal Ensemble . . . . . . . . . . . 6.4.1 Eigenvalue Number Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Eigenvalue Two-Body Correlations: 3 -Statistic . . . . . . . 6.4.3 Eigenvalue Nearest Neighbor Spacing Distribution (GOE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Eigenvector Statistics: Gaussian Orthogonal Ensemble. . . . . . . . . . . 6.6 Thermalization of Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 158 161 163 164 167 167 172

7

Bounded Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Quantum Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Symmetries and Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The Quantized Baker’s Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Quantum Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 The Stadium Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 The Sinai Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 The Ripple Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Peres Test for Quantum Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 The D-Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Quantum Spin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1 The XY Models with Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . 7.7.2 The XY Model with an Applied Magnetic Field . . . . . . . . 7.8 Coupled Morse Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Signatures of Chaos in a Soft Sinai Lattice . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 195 198 200 203 208 208 214 215 220 220 221 223 223 226 227 229 234 235

8

Manifestations of Chaos in Quantum Scattering Processes. . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Random Matrix Theory and Nuclear Scattering Processes . . . . . . . 8.3 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Energy Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 The Reaction Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 The Scattering Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

239 239 241 243 244 247 248 251

178 183 187 192 192

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8.4

Wigner–Smith and Partial Delay Times . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Delay Time of a Wave Packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Delay Times for Multichannel Scattering . . . . . . . . . . . . . . . 8.4.3 Delay Times and Complex Poles . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Scattering in the Ripple Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Scattering Resonances in a Ripple Waveguide . . . . . . . . . . 8.5.2 Wigner–Smith Delay Times for a Chaotic Scattering System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 COE and GOE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Circular Orthogonal Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Lorentzian Orthogonal Ensembles . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 The Relation Between COE and OE . . . . . . . . . . . . . . . . . . 8.6.4 When Does a GOE Hamiltonian Yield a COE S-Matrix? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Experimental Observation of RMT Predictions . . . . . . . . . . . . . . . . . . 8.7.1 Experimental Nuclear Spectral Statistics . . . . . . . . . . . . . . . . 8.7.2 Experimental Molecular Spectral Statistics . . . . . . . . . . . . . . 8.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

256 256 258 260 262 263

281 285 285 287 289 290

9

Semiclassical Theory: Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Green’s Function and Density of States. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 The Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Semiclassical Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Method of Stationary Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 The Semiclassical Green’s Function . . . . . . . . . . . . . . . . . . . . . 9.4.3 Conjugate Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Energy Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 General Expression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Gutzwiller Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Stationary Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Anisotropic Kepler System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Diamagnetic Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.2 Absorption Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.4 Semiclassical Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293 293 295 296 300 300 301 305 307 308 313 317 318 323 327 328 330 333 334 337 337

10

Time-Periodic Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Floquet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Floquet Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Floquet Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

339 339 341 341 344

267 272 274 277 278

Contents

10.3

A

xi

Quantum Nonlinear Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Two Primary Resonance Model . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Floquet Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Quantum Resonance Overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.4 Floquet Eigenvalue Nearest Neighbor Spacing . . . . . . . . . . 10.3.5 Quantum Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Dynamics of a Driven Bounded Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Driven Particle in Infinite Square Well . . . . . . . . . . . . . . . . . . . 10.4.2 Avoided Crossings and High Harmonic Radiation . . . . . . 10.5 Dynamical Tunneling in Atom Optics Experiments . . . . . . . . . . . . . . 10.5.1 Hamiltonian for Atomic Center-of-Mass . . . . . . . . . . . . . . . . 10.5.2 Average Momentum of Cesium Atoms . . . . . . . . . . . . . . . . . . 10.5.3 Floquet Analysis of Tunneling Oscillations. . . . . . . . . . . . . . 10.6 Quantum Delta-Kicked Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 The Schrödinger Equation for the Delta-Kicked Rotor . 10.6.2 KAM-Like Behavior of the Quantum Delta-Kicked Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.3 The Floquet Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.4 Dynamic Anderson Localization . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Microwave-Driven Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.1 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.2 One-Dimensional Approximation . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Arnol’d Diffusion in Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.1 Arnol’d Diffusion in the Driven Optical Lattice . . . . . . . . . 10.9 Quantum Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9.1 The Model (Classical Dynamics). . . . . . . . . . . . . . . . . . . . . . . . . 10.9.2 The Model (Quantum Dynamics) . . . . . . . . . . . . . . . . . . . . . . . . 10.9.3 Floquet States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9.4 STIRAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9.5 Avoided Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

345 345 346 349 350 351 352 352 356 358 358 360 362 363 363

Classical Mechanics Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Newton’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Lagrange’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Hamilton’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 The Poisson Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Phase Space Volume Conservation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Action-Angle Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.7 Hamilton’s Principal Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

397 397 398 399 399 400 400 402 403

365 365 368 369 369 372 376 378 384 384 386 387 388 390 393 395

xii

Contents

B

Simple Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 The Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.1 Libration—Trapped Orbits (E0 < g) . . . . . . . . . . . . . . . . . . . . B.1.2 Rotation—Untrapped Orbits (E0 > g) . . . . . . . . . . . . . . . . . . . B.2 Double-Well Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.1 Below the Barrier—(Eo < 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.2 Above the Barrier—(Eo > 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Infinite Square-Well Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 One-Dimensional Hydrogen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4.1 Zero Stark Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4.2 Nonzero Stark Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

405 405 406 407 408 409 411 412 414 414 416 417

C

Symmetries and the Hamiltonian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Space-Time Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1.1 Continuous Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1.2 Discrete Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Structure of the Hamiltonian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2.1 Space-Time Homogeneity and Isotropy . . . . . . . . . . . . . . . . . C.2.2 Time Reversal Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

419 419 420 422 424 424 425 429

D

Invariant Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1 General Definition of Invariant Measure . . . . . . . . . . . . . . . . . . . . . . . . . . D.1.1 Invariant Metric (Length) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1.2 Invariant Measure (Volume) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2.1 Real Symmetric Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2.2 Complex Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2.3 Quaternion Real Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2.4 General Formula for Invariant Measure of Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3 Unitary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3.1 Symmetric Unitary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3.2 General Unitary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3.3 Symplectic Unitary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3.4 General Formula for Invariant Measure of Unitary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3.5 Orthogonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.4 Volume of Invariant Measure for Unitary Matrices . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

431 431 431 432 433 433 435 438 440 441 442 444 445 446 446 447 453

Contents

xiii

E

Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1 General Properties of Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2 Quaternions in the GOE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

455 455 460 466

F

Gaussian Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1 Gaussian Orthogonal Ensemble (GOE) . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1.1 Probability Density and Quaternion Matrices for GOE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1.2 Cluster Functions for GOE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.2 Gaussian Unitary Ensemble (GUE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.2.1 Complex Hermitian Matrices and Invariant Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.2.2 Polar Form of Measure for Complex Hermitian Matrix . F.2.3 Probability Density for GUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.2.4 Cluster Functions for GUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.2.5 Eigenvalue Number Density for GUE . . . . . . . . . . . . . . . . . . . F.2.6 3 -Statistics for GUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.2.7 Eigenvalue Nearest Neighbor Spacing Distribution for GUE (N=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.3 Gaussian Symplectic Ensemble (GSE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.3.1 Quaternion Real Matrices: GSE . . . . . . . . . . . . . . . . . . . . . . . . . F.3.2 Polar Form of Invariant Measure for N ×N Quaternion Real Matrix: GSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.3.3 Probability Density for GSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.3.4 Cluster Functions for GSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

467 467

G

Circular Ensembles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.1 Vandermonde Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.2 Circular Unitary Ensemble (CUE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.3 Circular Orthogonal Ensemble (COE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.4 Circular Symplectic Ensemble (COE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.5 Cluster Functions for Circular Ensembles . . . . . . . . . . . . . . . . . . . . . . . . G.5.1 Circular Unitary Ensemble (CUE) . . . . . . . . . . . . . . . . . . . . . . . G.5.2 Circular Orthogonal Ensemble (COE) . . . . . . . . . . . . . . . . . . . G.5.3 Circular Symplectic Ensemble (CSE) . . . . . . . . . . . . . . . . . . . G.6 3 -Statistics for Circular Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.6.1 Circular Unitary Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.6.2 Circular Orthogonal Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . G.6.3 Circular Symplectic Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

468 473 474 474 475 477 479 482 482 484 485 485 486 487 489 491 493 495 496 497 504 508 508 508 509 510 510 511 512 513

xiv

Contents

H

Maxwell’s Equations for 2-d Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.1 Transverse Magnetic Modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.2 Transverse Electric Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

515 517 518 518

I

Moyal Bracket and Wigner Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.1 The Wigner Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2 Ordering of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.3 Moyal Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

519 519 521 522 524

J

Gaussian, S.I., and Atomic Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.1 Maxwell’s Equations in Gaussian Units . . . . . . . . . . . . . . . . . . . . . . . . . . . J.1.1 Physical Quantities Expressed in Gaussian Units . . . . . . . J.1.2 Values of Fundamental Constants Expressed in Gaussian Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.1.3 The Conversion Between Gaussian and SI Units . . . . . . . . J.1.4 The Conversion Between Gaussian and Atomic Units . . J.2 Maxwell’s Equations in S.I. Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.2.1 Conversion Between S.I. Units and Atomic Units . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

525 525 526

Hydrogen in a Constant Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.1 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.1.1 Equation for Relative Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.1.2 Solution for λ0 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.2 One-Dimensional Hydrogen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

529 529 530 531 533 535

K

527 527 527 528 528 528

Author Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544

Chapter 1

Overview

Abstract The dynamical evolution of conservative chaotic systems is not deterministic because it is not possible to follow their dynamical evolution with any precision for more than a short time. In quantum systems, signatures of chaos emerge when classical chaos occupies phase space volumes greater than d ( is Planck’s constant and d is number of degrees of freedom). The phase space of most conservative nonlinear systems with three or more degrees of freedom (with a few exceptions) is permeated by an Arnol’d web consisting of a fractal set of nonlinear resonances that fill the phase space. The Arnol’d web is the source of the chaos that causes thermalization of both classical and quantum systems. The publication of Newton’s Principia in 1686 and the success and power of Newton’s laws led to the huge growth in science that we see today. Belief that Newtonian mechanics is deterministic was shaken by the work of Poincaré who showed that perturbation expansions must diverge due to nonlinear resonances, making it impossible to make long-time predictions. When chaos manifests itself in quantum systems, the information content of a quantum system is extremized (minimized). In this book, we examine in more detail the mechanisms by which chaos emerges in conservative classical and quantum systems. Keywords Classical chaos · Quantum manifestations of chaos · Determinism · History of dynamics · Symmetries · Perturbation theory divergence · KAM tori · Nonlinear resonance · Renormalization theory · Random matrix theory · Path integrals

1.1 Introduction The existence of chaos in a conservative classical system means that the dynamical evolution of the system is no longer deterministic. Most classical systems with three or more degrees of freedom are either fully chaotic or have fractal regions of chaos distributed throughout the phase space. For example, the solar system appears to have regions where planetary motion is chaotic. Classical models of molecular © Springer Nature Switzerland AG 2021 L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200, https://doi.org/10.1007/978-3-030-63534-3_1

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1 Overview

motion often show large regions of chaos. Hard sphere gases are rigorously chaotic. The foundations of statistical mechanics (and thermodynamics) are based on the assumption that the underlying dynamics is chaotic. The term quantum chaos refers to the signatures of classical chaos in the quantum dynamics of particles whose classical limit is chaotic. The quantum signatures of chaos appear wherever the classically chaotic regions have a size equal to h¯ d in phase space, where h¯ is Planck’s constant and d is the number of degrees of freedom. Signatures of chaos occur, for some parameter ranges, in most quantum systems and determine if a quantum system can thermalize. The signatures of chaos can be used to control quantum transitions. They can also destabilize quantum systems. In fact, the deeper we look into the fundamental dynamics governing the world, the more we see the profound impact of chaotic behavior. The phase space of all conservative nonlinear systems with three or more degrees of freedom (with a few exceptions) forms an Arnol’d web (Arnol’d 1963). An Arnol’d web consists of a fractal set of resonances and chaos that fill the phase space. Depending on the degree of development of the web, an initial condition (one that is not known to infinite precision), may evolve deterministically for a long but finite period of time, or may begin to exhibit random behavior after a short time. The ubiquitous Arnol’d web, in conservative dynamical systems, provides the mechanism to thermalize the world. In this regard, one of the important discoveries in quantum physics in recent years is that the information content of conservative quantum systems is extremized (minimized) when the underlying classical system undergoes a transition to chaos. The information content of the conservative quantum system approaches that of a system whose dynamics is governed by a random Hamiltonian matrix that is chosen to minimize information content. In subsequent, sections we will first give a brief historical overview of the history of conservative dynamics and chaos theory. Then we will describe the content of the remaining chapters of this book.

1.2 Historical Overview On April 28, 1686 the first of the three books that comprise Newton’s Principia was formally presented to the Royal Society and, by July 1687 the complete first edition (consisting of perhaps 300 copies) was published Newton (1686). The publication of this work was probably the most important single event in the history of science because it formulated the science of mechanics in terms of just three basic laws: • A body maintains its state of rest or uniform velocity unless a net force acts on it. • The time rate of change of momentum, p, is equal to the net force, F, acting on it. • To every action there is an equal and opposite reaction. In the Principia, Newton not only wrote the three laws but also gave a systematic mathematical framework for exploring the implications of these laws. In addition,

1.2 Historical Overview

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in the Principia Newton proposed his universal inverse square law of gravitation. He then used it to derive Kepler’s empirical laws of planetary motion, to account for the motion of the moon and the phenomenon of tides, to explain the precession of the equinoxes, and to account for the behavior of falling bodies in Earth’s gravitational field. The success and power of Newton’s laws led to a great optimism about our ability to predict the behavior of mechanical objects and, as a consequence, led to the huge growth in science that we see today. In addition, it was accompanied by a deterministic view of nature that is perhaps best exemplified in the writings of Laplace. In his Philosophical Essay on Probabilities, Laplace states (Laplace 1951): Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it— an intelligence sufficiently vast to submit these data to analysis—it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom. For it, nothing would be uncertain and the future, as the past, would be present before its eyes. This deterministic view of nature was completely natural, given the success of Newtonian mechanics, and it persists up until the present day. Newton’s three laws of motion led to a description of the motion of point masses in terms of a set of coupled second-order differential equations. The theory of extended objects can be derived from Newton’s laws by treating them as collections of point masses. If we can specify the initial velocities and positions of the point particles, then Newton’s equations for the point particles (obtained from the second law) should determine all past and future motion. However, we now know that the assumption that Newton’s equations can predict the future is a fallacy. Newton’s equations are, of course, the correct starting point of mechanics, but in general, they only allow us to determine the long-time behavior of integrable mechanical systems, few of which can be found in nature. Newton’s laws, for most systems, describe inherently random behavior and cannot determine the future evolution of any real system (except for very short times) in more than a probabilistic sense. The belief that Newtonian mechanics is a basis for determinism was formally laid to rest by Sir Lighthill (1986) in a lecture to the Royal Society on the threehundredth anniversary of Newton’s Principia. In his lecture, Lighthill says . . . I speak . . . once again on behalf of the broad global fraternity of practitioners of mechanics. We are all deeply conscious today that the enthusiasm of our forebears for the marvelous achievements of Newtonian mechanics led them to make generalizations in this area of predictability which, indeed, we may have generally tended to believe before 1960, but which we now recognize were false. We collectively wish to apologize for having misled the general educated public by spreading ideas about the determinism of systems satisfying Newton’s laws of motion that, after 1960, were to be proved incorrect . . . . In a sense, Newton (and Western science) were fortunate because the solar system has amazingly regular behavior considering its complexity, and one can predict its short-time behavior with fairly good accuracy. Part of the reason for this is the fact that the two-body Kepler system is governed by symmetries, both space-time

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1 Overview

and hidden, and is integrable. A three body gravitational system is not integrable. Newton’s derivation of Kepler’s laws was based on the properties of the two-body system. However, the dynamical interactions of the many bodies that comprise the solar system lead to deviations from the predictions of Kepler’s laws, and lead one to ask why the solar system is, in fact, so regular. Is the solar system stable (Moser 1975)? Will it maintain its present configuration into the future? These questions have not yet been fully answered. Questions concerning the stability and the future evolution of the solar system have occupied scientists and mathematicians for the past 300 years. Until computers were invented, all mathematical theories used perturbation expansions of various types. In the eighteenth century, important contributions were made by Euler, Lagrange, and Laplace on predicting the change in the geometry of orbits due to small perturbations and on determining the overall stability of orbits. In addition, Lagrange (1889) reformulated Newtonian mechanics in terms of a variational principle that vastly extended our ability to analyze the behavior of dynamical systems and allowed a straight-forward extension to continuum mechanics. In the nineteenth century, there were two very important pieces of work that laid the groundwork for our current view of mechanics. Hamilton reformulated mechanics (Hamilton 1940) so that the dynamics of a mechanical system could be described in terms of a momentum-position phase space rather than a velocityposition phase space as is the case for the Lagrangian formulation. This step is extremely important because in the Hamiltonian formulation (which describes the evolution of mechanical systems in terms of coupled first-order differential equations) the flow of trajectories in phase space is volume-preserving. Furthermore, if symmetries exist (such as the space-time symmetries), then some of the generalized momenta of the system may be conserved, thus reducing the dimension of the phase space in which we must work. The relation between the symmetries of a system and conservation laws was first clarified by Noether (1918). Noether’s work provides one of the most important tools of twentieth-century science, because the key to much of what we are able to predict in science is symmetry. Symmetries imply conservation laws, and conservation laws give conservative classical mechanics and quantum mechanics whatever predictive power they have. Conservation laws are even responsible for the existence of thermodynamics and hydrodynamics. Another extremely important piece of work in the nineteenth century was due to Poincaré (1899). Poincaré not only closed the door on an era but created the first crack in the facade of determinism. Before Poincaré, most work on dynamics, subsequent to Newton, involved computation of deviations from KepIer-type orbits for two massive bodies that are perturbed by a third body. The idea was to take a Kepler orbit as a first approximation and then compute successive corrections to it using perturbation theory. One must then show that the perturbation expansions thus obtained converge. The problem of whether or not perturbation series converge was so important that it was the subject of a prize question posed by King Oscar II of Sweden in 1885. The question read as follows: For an arbitrary system of mass points which attract each

1.2 Historical Overview

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other according to Newton’s laws, assuming that no two points ever collide, give the coordinates of the individual points for all time as the sum of a uniformly convergent series whose terms are made up of known functions (Moser 1975). Poincaré entered the contest and won the prize by showing that such series could be expected to diverge because of small denominators caused by internal resonances. We now know that resonances, that give rise to these small divisors, are associated with the onset of chaos. Because of these divergences, it appears to be impossible to make long-time predictions concerning the evolution of mechanical systems (with a few exceptions such as the two-body Kepler system) using perturbation expansions. No further progress was made on the problem of long-time prediction in mechanics until l954 when Kolmogorov (1954) outlined a proof, for systems of the type proposed in King Oscar’s question, that a majority of the trajectories are quasiperiodic and can be described in terms of a special type of perturbation expansion. In 1962, Arnol’d (1963) constructed a formal proof of Kolmogorov’s results for a three-body system with an analytic Hamiltonian, and Moser (1968) obtained a similar result for twist maps. The result of the work of Kolmogorov, Arnol’d, and Moser (KAM) is that series expansions describing the motion of some orbits in many-body systems are convergent, provided the natural frequencies associated with these orbits are not close to resonance. The work of Arnol’d, also showed that nonintegrable systems, with three or more degrees of freedom, are intrinsically unstable. They contain a dense web of resonance lines, the Arnol’d web, that allows diffusion to occur throughout the available phase space. The question of how rapid the diffusion will be depends on the parameters of the system. Shenker and Kadanoff (1982) and MacKay (1983) were able to show that at the parameter value at which a given KAM torus (with quadratic irrational winding number) is destroyed, the rational approximates have self-similar structure and the areas in phase space that they occupy are related by scaling laws. They also showed that the rational approximates play a dominant role in the destruction of KAM tori. Escande and Doveil (1981) developed a renormalization theory for the destruction of KAM tori directly from the Hamiltonian for systems with two degrees of freedom. Thus, Hamiltonian systems, much like equilibrium systems near a phase transition, can exhibit self-similar structure. Much of the behavior that occurs in classical systems also occurs in their quantum counterpart. However, because of the Heisenberg uncertainty relations, we are forced to describe classical and quantum systems from quite different perspectives. In classical systems, we can examine the evolution of individual orbits in phase space, and we can see directly the chaotic flow of trajectories in phase space. If we were to describe the evolution of the classical system in terms of the probability distribution in phase space, using the Liouville equation, we would have to search for the signatures of chaos in the behavior of the probability distributions and eigenvalues of the Liouville operator. This has been done for very simple chaotic maps (Driebe 1999), but it is a formidable task when dealing with Newtonian mechanical systems with two or more degrees of freedom.

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When we study quantum systems, we have no phase space in which to describe the evolution of individual orbits because of the Heisenberg uncertainty relations. A single quantum state occupies volume of order h¯ d in the classical phase space, where h¯ is Planck’s constant and d is the number of degrees of freedom. We are forced from the outset to study quantum systems at the level of a linear probability (probability amplitude to be more precise) equation, namely the Schrödinger equation. Most of the mechanisms at work in nonlinear classical systems are also at work in their quantum counterparts. For example, nonlinear resonances exist in quantum systems and can destroy constants of the motion (good quantum numbers) in local regions of the Hilbert space. They form self-similar structures, but only down to scales of order h¯ d and not to infinitely small scales as they do in classical systems. However, because the Schrödinger equation is an equation for probability amplitudes rather than probabilities, we will find some new phenomena that can occur in quantum systems but not in classical systems. One of the most important discoveries of quantum chaos theory is that the statistical properties of energy spectra and scattering delay times indicate that the information content of a quantum system is extremized (minimized) as its classical counterpart undergoes a transition to chaos. The idea of studying the spectral statistics of quantum systems is largely due to Wigner (1951, 1957), who in the 1950s analyzed the statistical properties of nuclear scattering resonances. It was found that the nearest neighbor spacing of scattering resonances, for some nuclear scattering processes, has a distribution that agrees with the distribution of spacings of eigenvalues of ensembles of random Hermitian matrices (the Gaussian ensemble) whose matrix elements extremize information. The work of Wigner led Dyson (1962) to study the statistical properties of ensembles of random unitary matrices (the circular ensembles) that extremize information. The connection between chaos theory and random matrix theory was made in 1979 by McDonald and Kaufman (1979), who found that classically chaotic quantum billiards have spectral spacing distributions given by the Gaussian ensembles. Comparison between statistical properties of deterministic quantum systems with underlying classical chaos and predictions of random matrix theories that extremize information is now a standard tool of quantum mechanics. In the early days of quantum mechanics, before the work of Heisenberg and Schrödinger, the quantum version of a classical system was obtained by quantizing the action variables. This is straightforward if the classical system is integrable and one can find the action variables. However, Einstein, who knew of the work of Poincaré, as early as 1917 (Einstein 1917) pointed out that there may be difficulties with this method of quantization if invariant tori do not exist in the classical phase space, as is the case with chaotic systems. Indeed, until the work of Gutzwiller in the early 1980s (Gutzwiller 1982), there was no way to link classically chaotic systems to their quantum counterparts. However, Gutzwiller showed that Feynman path integrals, in the semiclassical limit, provide such a link, and the spectral properties of a quantum system, whose classical counterpart is chaotic are determined largely in terms of an infinite sum over the unstable periodic orbits of the classical system.

1.3 Plan of the Book

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Probably the most widely studied systems, as regards to the transition to chaos, are systems driven by time-periodic external fields. With a time-periodic force, one can cause a nonlinear system, with only one degree of freedom, to undergo a transition to chaos. In such systems, energy is not conserved but due to a discrete time translation invariance, the Floquet energy (quasi-energy) is conserved. Thus all the techniques used in energy-conserving systems can be applied to these driven systems.

1.3 Plan of the Book The goal of this book is to provide a thorough grounding in classical and quantum chaos theory, with the focus on topics that impact current and future research topics. Chapters 2–5 provide a description of processes underlying chaotic classical dynamics in conservative systems. Chapter 2 lays the foundations of the relevant classical mechanics for understanding chaos, and focuses on those aspects of chaotic behavior that will be used throughout the remainder of the book. Chapter 3 deals primarily with systems that have two degrees of freedom. For these systems it is possible to visualize the processes that lead to the onset of chaos, because one can construct area preserving maps to follow the process. In Chap. 3, we also focus on the fractal nature of structures in the phase space that lead to global chaos, as parameters of the system are changed. Chapter 4 deals with classical scattering processes and the fractal nature of scattering dynamics, when the scatterer is chaotic or partially chaotic. Finally, Chap. 5 focuses on the Arnold web that exists in nonlinear, non-integrable systems with three of more degrees of freedom. The Arnol’d web provides the mechanism for the global transition to chaos in systems with three or more degrees of freedom. The remaining chapters of the book, Chaps. 6–10, examine the quantum manifestations of chaos. We start in Chap. 6 with a discussion of random matrix theory, as applied to conservative Hamiltonian systems. Random matrix theory is based on the assumption that the matrix elements of a hermitian or unitary matrix are independent random variables. This implies certain behaviors of the eigenvalues and eigenvectors of such systems that have been observed in quantum systems whose classical counterpart is chaotic. As we also show in Chap. 6, a conservative quantum system that shows random matrix-like behavior has become thermalized. In Chap. 7, we discuss the behavior of some bounded quantum systems whose classical counterparts undergo a transition to chaos. The Schrödinger equation for these systems is linear. Nonlinearities appear in the Hamiltonian. We consider chaotic billiards, spin systems, and small molecules which are anharmonic oscillators. A number of the results we describe have been realized in microwave cavity experiments. The connection between the quantum manifestations of chaos and random matrix theory was first observed in nuclear scattering experiments. In Chap. 8, we describe the theory originally developed by Wigner and Eisenbud (W-E) 1947 that allowed

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analysis of nuclear scattering processes in terms of random matrix theory. We then use these tools to define scattering phenomena such as resonance, quasibound states, and delay times for scattering processes. Finally, in Chap. 8, we show a variety of experimental and numerical data on nuclear and molecular energy-Ievel sequences and show that these systems are exhibiting the manifestations of chaos. Another connection between classically chaotic systems and their quantum counterpart involves the use of semiclassical path integrals. In Chap. 9, we use the semiclassical limit of Feynman path integrals to derive the Gutzwiller trace formula, which expresses the trace of the Green’s function of a quantum system in terms of periodic orbits of the classical system. We show that the trace formula gives very good results for the energy levels of the anisotropic Kepler system, a classically chaotic system. Finally, we conclude Chap. 9 with a numerical and experimental study of the influence of periodic orbits on the absorption spectrum of diamagnetic hydrogen. Chapter 10 is devoted to periodically driven quantum systems, which can be described using Floquet theory. We show that nonlinear resonances exist in the Hilbert space of quantum systems, and we use Floquet theory to interpret the results of dynamic tunneling experiments using cold atoms confined to optical lattices. We decribe the behavior of the quantum delta-kicked rotor, which was the first system in which dynamic Anderson localization was observed numericaly. We also describe extensive experiments on microwave-driven hydrogen that give experimental confirmation of the existence of higher-order nonlinear resonances in quantum systems. Finally, we show that the Arnol’d web exists in quantum systems and plays an important role in destablizing their dynamics, and we show the influence of chaos on quantum control. This book contains several appendices that give background on subjects of importance to this book. For example, there is a review of the effect of symmetries on the structure of Hamiltonian matrices. There is a derivation of the measures for Hermitian and unitary matrices used in random matrix theory. There is a derivation of the normalization constants and expressions for probability distributions of the Gaussian and circular ensembles in terms of quaternion matrices. There are other appendices as weIl that will aid the reader with some of the theory concepts in this book. We do not have room in this book to discuss in detail all of the interesting applications of classical and quantum chaos theory, so in the concluding section of each chapter we have given references to additional topics of interest.

References Arnol’d VI (1963) Russ Math Surv 189:1885 Driebe DJ (1999) Fully chaotic maps and broken time symmetry. Kluwer Academic Publishers, Dordrecht Dyson FJ (1962) J Math Phys 3:140

References

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Einstein A (1917) Verh Dtsch Phys Ges 19:82 Escande DF, Doveil F (1981) J Stat Phys 26:257 Gutzwiller MC (1982) Physica D 5:183 Hamilton WR (1940) Conway AW, Synge JL (eds) The mathematical papers of sir william rowan hamilton, vol II, dynamics. Cambridge University Press, Cambridge Kolmogorov AN (1954) Dokl Akad Nauk SSSR 98:527 (1954) (An English version appears in R. Abraham, Foundations of Mechanics (W.A. Benjamin, New York, 1967, Appendix D).) Lagrange JL (1889) Mechanique analytique. Gauthier-Villars, Paris Laplace PS (1951) A philosophical essay on probabilities, Translated by Truscott FW, Emory FL. Dover, New York Lighthill J (1986) Proc Roy Soc London A 407:35 MacKay RS (1983) Physica D 7:283 McDonald SW, Kaufman AN (1979) Phys Rev Lett 42:1189 Moser J (1968) Nachr Akad Wiss Goettingen II, Math Phys Kd 1:1 Moser J (1975) Is the solar system stable? Neue Zurcher Zeitung, Zürich Newton I (1686) The principia. Royal Society, London. (Available from Snowball Publishing, 2010) Noether E (1918) Nach Ges Wiss Goettingen 2:235 Poincaré H (1899) Les Méthodes nouvelles de la Mécanique Céleste, vol 3. Gauthier-Villars, Paris Shenker SJ, Kadanoff LP (1982) J Stat Phys 27:631 Wigner EP (1951) Ann Math 53:36 Wigner EP (1957) Ann Math 65:203 Wigner EP, Eisenbud LE (1947) Phys Rev 72:29

Chapter 2

Fundamental Concepts

Abstract The dynamical behavior of nonlinear conservative systems is determined by global and hidden symmetries that constrain dynamical flow to lowerdimensional surfaces in the phase space. When the number of global symmetries equals the number of degrees of freedom, the dynamical system is integrable. This rarely happens, Symmetry-breaking terms added to a Hamiltonian cause nonlinear resonances to occur on all scales in the phase space and give rise to a fractal structuring of the phase space. Chaos appears in the neighborhood of the nonlinear resonances. The Poincaré surfaces of section provide a numerical tool for testing integrability of conservative dynamical systems. Non-linear resonances may appear or disappear as mparameters of the system are varied and the overlap of nonlinear resonances leads to the onset of chaos. Kolmogorov, Arnol’d, and Moser (collectively called KAM) developed a rapidly converging perturbation theory that describes non-resonant regions of the phase space. In chaotic regions of the phase space, neighboring orbits move apart exponentially in any direction. The rate of exponential divergence of pairs of orbits is given by Lyapounov exponents. Systems with positive Lyapounov exponents also have positive KS metric entropy. Keywords Noether’s theorem · Integrability · Global symmetries · Hidden symmetries · KAM tori · Poincaré surface of section · Nonlinear resonance · Definition of chaos · Lyapounov exponents · Baker’s transformation

2.1 Introduction The dynamical behavior of nonlinear conservative systems is determined by the nature of the symmetries that govern their behavior. These dynamical symmetries can be categorized as global symmetries or hidden symmetries. Both types of symmetry constrain the dynamical flow of the system to lower-dimensional surfaces in the phase space. Global symmetries are related to the space-time symmetries of the system. The other symmetries do not have an obvious source and were © Springer Nature Switzerland AG 2021 L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200, https://doi.org/10.1007/978-3-030-63534-3_2

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first called hidden symmetries by Moser (1979). When there are as many global symmetries as numbers of degrees of freedom, the dynamical system is said to be integrable. A second concept that is important for understanding the dynamics of nonlinear systems is nonlinear resonance. As Kolmogorov (1954), Arnol’d (1963), and Moser (1962) have shown, when a small symmetry-breaking term is added to the Hamiltonian of system, most of the phase space continues to behave as if the symmetries still exist. However, in regions where the symmetry-breaking term allows resonance to occur between otherwise uncoupled degrees of freedom, the dynamics begins to change its character. When resonances do occur, they generally occur on all scales in the phase space and give rise to a fractal structuring of the phase space. The third concept that is essential for understanding conservative nonlinear dynamics is chaos or sensitive dependence on initial conditions. For the class of systems in which symmetries can be broken by adding small symmetry-breaking terms, chaos first appears in the neighborhood of the nonlinear resonances. As the strength of the symmetry-breaking term increases and the size of the resonance regions increases, ever larger regions of the phase space become chaotic. The dynamical evolution of systems with broken symmetry cannot be determined using conventional perturbation theory, because of the existence of nonlinear resonances. In Sect. 2.2, we show that nonlinear resonances cause a topological change locally in the structure of the phase space, and that conventional perturbation theory is not adequate to deal with such topological changes. In Sect. 2.3, we introduce the concept of integrability. A system is integrable if it has as many global constants of the motion as degrees of freedom. The connection between global symmetries and global constants of motion was first proven for dynamical systems by Noether (1918). We will give a simple derivation of Noether’s theorem in Sect. 2.3. It is usually impossible to tell if a system is integrable just by looking at the equations of motion. As we show in Sect. 2.4, the Poincaré surface of section provides a very useful numerical tool for testing integrability and will be used throughout the remainder of this book. We will illustrate the use of the Poincaré surface of section for the classic model of Henon and Heiles (1964) and for a model of the HOCl molecule. In Sect. 2.5, we introduce the concept of nonlinear resonances and illustrate their behavior for some simple models originally introduced by Walker and Ford (1969). These models are interesting because they show that resonances may appear or disappear as parameters of the system are varied and the overlap of nonlinear resonances leads to the onset of chaos. Conventional perturbation theory does not work when nonlinear resonances are present. But Kolmogorov, Arnol’d, and Moser (collectively called KAM) have developed a rapidly converging perturbation theory that can be used to describe non-resonant regions of the phase space, precisely because it is constructed to avoid the resonance regions. KAM perturbation theory will be described in Sect. 2.6.

2.2 Conventional Perturbation Theory

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In practice, chaos is defined in terms of the dynamical behavior of pairs of orbits that initially are close together in the phase space. If the orbits move apart exponentially in any direction in the phase space, the flow is said to be chaotic. The rate of exponential divergence of pairs of orbits is measured by the so-called Lyapounov exponents. There will be one such exponent for each dimension in the phase space. If all the Lyapounov exponents are zero, the dynamical flow is regular. If even one exponent is positive, the flow will be chaotic. A detailed discussion of the behavior of Lyapounov exponents for conservative systems is given in Sect. 2.7 and is illustrated in terms of the Henon-Heiles system. Systems with positive Lyapounov exponents also have positive KS metric entropy. The KS metric entropy is defined in Sect. 2.7 and computed for the baker’s transformation, one of the simplest known chaotic dynamical systems. Finally, in Sect. 2.8, we make some concluding remarks.

2.2 Conventional Perturbation Theory Historically the first cracks in a deterministic view of the world, and an appreciation of the difficulties in obtaining long-time predictions regarding the evolution of dynamical systems, were brought into focus with Poincaré’s proof that conventional perturbation expansions generally diverge. When they diverge they cannot be used as a tool to provide long-time predictions. In order to build some intuition concerning the origin of these divergences, let us consider a 2 DoF system from celestial mechanics, the relative motion of a moon of mass m1 , orbiting a planet of mass m2 (the Kepler system). The Hamiltonian can be written H0 =

pφ 2 pr 2 k + − = E, 2 2μ r 2μr

(2.1)

where (pr , pφ ) and (r, φ) are the relative momentum and positions, respectively, of m2 the two bodies in polar coordinates, E is the total energy of the system, μ = mm11+m 2 is the reduced mass, and k = Gm1 m2 (G is the gravitational constant). The total angular momentum, L, is conserved for this problem so the plane of motion, (r, φ), is taken to lie in the plane perpendicular to L. After a canonical transformation from coordinates (pr , pφ , r, φ) to action-angle coordinates (J1 , J2 , θ1 , θ2 ), the Hamiltonian takes the form (Goldstein 1980) H0 (J1 , J2 ) =

−μk 2 = E. 2(J1 + J2 )2

(2.2)

The motion is fairly complicated (elliptic or hyperbolic orbits) in terms of coordinates (pr , pφ , r, φ), but in terms of action-angle coordinates it is simple. Hamilton’s ∂H0 dθi ∂H0 i equations of motion yield dJ dt = − ∂θi = 0 and dt = ∂Ji = ωi (J1 , J2 ), where

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Fig. 2.1 For integrable systems with two DoF, each trajectory lies on a torus √ constructed from the action-angle variables (J1 , J2 , θ1 , θ2 ). The radii of the torus are ρi = 2Ji for i = (1, 2). i If the frequencies ωi = dθ dt (i = 1, 2) are commensurate, the trajectory will be periodic. If the frequencies are incommensurate, the trajectory will never repeat

i = (1, 2) and t is the time. Thus, we find that Ji = ci and θi = ωi t + di , where ci and di are constants determined by the initial conditions. We see immediately that the energy of this system is constant. It is useful to picture the motion of this system as lying on a torus as shown in √ Fig. 2.1. The torus will have two constant radii, which we define as ρi = 2Ji for i = (1, 2), and two angular variables (θ1 , θ2 ). A single orbit of the Kepler system will evolve on this torus according to equations Ji = ci and θi = ωi t + di , so there are two frequencies associated with this system, ω1 and ω2 . If these two frequencies are commensurate (that is, if mω1 = nω2 , where m and n are integers), then the trajectory will be periodic and the orbit will repeat itself. If the two frequencies are incommensurate (irrational multiples of one another), then the trajectory will never repeat itself as it moves around the torus and eventually will cover the entire surface of the torus. Note also that the frequencies themselves depend on the action variables and therefore on the energy of the system. This is a characteristic feature of a nonlinear system. Let us now assume that a perturbation acts in the plane of motion due to the presence of another planet. We shall treat this perturbation as an external field. In the presence of this perturbation, the Hamiltonian will take the form H = H0 (J1 , J2 ) + V (J1 , J2 , θ1 , θ2 ),

(2.3)

where is a small parameter,  1. We wish to find corrections to the unperturbed trajectories, Ji = ci , due to the perturbation. Since we cannot solve the new equations of motion exactly, we can hope to obtain approximate solutions using perturbation expansions in the small parameter . Let’s try it. First we note that since we are dealing with periodic bound state motion, we can expand the perturbation in a Fourier series, and write the Hamiltonian in Eq. (2.3) in the form H = H0 (J1 , J2 ) +

∞ 

∞ 

n1 =−∞n2 =−∞

Vn1 ,n2 (J1 , J2 ) cos(n1 θ1 + n2 θ2 ).

(2.4)

2.2 Conventional Perturbation Theory

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Next, we introduce a generating function, G(J1 , J2 , θ1 , θ2 ), which we define as G(J1 , J2 , θ1 , θ2 ) = J1 θ1 + J2 θ2 +

∞ 

∞ 

gn1 ,n2 (J1 , J2 ) sin(n1 θ1 + n2 θ2 ),

(2.5)

n1 =−∞ n2 =−∞

where gn1 ,n2 will be determined below. The generating function in Eq. (2.5) generates a canonical transformation from the set of action-angle variables, (J1 , J2 , θ1 , θ2 ), to a new set of canonical action-angle variables, (J1 , J2 , 1 , 2 ), via the following equations: ∞

∞

  ∂G = Ji +

ni gn1 ,n2 cos(n1 θ1 + n2 θ2 ) ∂θi n =−∞ n =−∞

Ji =

1

(2.6)

2

and ∞

i =

∞

  ∂gn ,n ∂G 1 2 = θi +

sin(n1 θ1 + n2 θ2 ). ∂Ji ∂J i n =−∞ n =−∞ 1

(2.7)

2

The new Hamiltonian, H  (J1 , J2 , 1 , 2 ), is obtained from Eq. (2.4) by solving Eqs. (2.6) and (2.7) for (Ji , θi ) as a function of (Ji , i ) and plugging into Eq. (2.4). If we do that and then expand H  (J1 , J2 , 1 , 2 ) in a Taylor series in the small parameter , we find H  (J1 , J2 , 1 , 2 ) = H0 (J1 , J2 ) +

+

∞ 

∞ 

∞ 

(n1 ω1 + n2 ω2 )gn1 ,n2 cos(n1 1 + n2 2 )

n1 =−∞ n2 =−∞

∞ 

Vn1 ,n2 (J1 , J2 ) cos(n1 1 + n2 2 ) + O( 2 ),

(2.8)

n1 =−∞ n2 =−∞ ∂H 

where the frequencies are defined as ωi = ∂Joi . We can now remove terms of order by choosing gn1 ,n2 = −

Vn1 ,n2 (J1 , J2 ) . (n1 ω1 + n2 ω2 )

(2.9)

Then H  (J1 , J2 , 1 , 2 ) = Ho (J1 , J2 ) + O( 2 )

(2.10)

16

2 Fundamental Concepts

and Ji = Ji −

∞  n1

∞  ni Vn1 ,n2 cos(n1 1 + n2 2 ) + O( 2 ). (n ω + n ω ) 1 1 2 2 =−∞n =−∞

(2.11)

2

To lowest order in , this is the solution to the problem. New actions, Ji , have been obtained that contain corrections due to the perturbation. If, for example, = 0.01, then by retaining only first-order corrections, we neglect terms of order 2 = 0.0001. To first order in , Ji is a constant and i varies linearly in time. At least, this is the hope. However, there is a catch! For the expansion in Eq. (2.11) to have meaning, we must have |n1 ω1 + n2 ω2 |  Vn1 ,n2 .

(2.12)

However, the condition in Eq. (2.12) breaks down when internal nonlinear resonances occur and causes the perturbation expansion to diverge. Poincaré showed that perturbation expansions of this type can generally be expected to diverge and therefore, cannot be used for long-time predictions.

2.3 Integrable Systems Integrable systems form an important reference point when discussing the behavior of dynamical systems. We define an integrable system as follows. Consider a dynamical system with N degrees of freedom. Its phase space has 2N dimensions. The system is integrable if there exist N independent isolating integrals of motion, Ii , such that Ii (p1 , . . . , pN , q1 , . . . , qN ) = Ci ,

(2.13)

for i = 1, . . . , N, where Ci is a constant and pi and qi are the canonical momentum and position associated with the ith degree of freedom. The functions Ii are independent if their differentials, dIi , are linearly independent. It is important to distinguish between isolating and non-isolating integrals (Wintner 1947). Non-isolating integrals (an example is the initial coordinates of a trajectory) generally vary from trajectory to trajectory and usually do not provide useful information about a system. On the other hand, isolating integrals of motion, by Noether’s theorem, are due to symmetries (some “hidden”) of the dynamical system and define surfaces in phase space. The condition for integrability may be put in another form. A classical system with N degrees of freedom is integrable if there exist N independent globally defined functions, Ii (p1 , . . . , pN , qi , . . . , qN ), for i = 1, . . . , N , whose mutual Poisson brackets vanish,

2.3 Integrable Systems

17

{Ii , Ij }P oisson = 0,

(2.14)

for i = 1, . . . , N and j = 1, . . . , N. Then the quantities Ii form a set of N phase space coordinates. In conservative systems, the Hamiltonian, H (p1 , . . . , pN , q1 , . . . , qN ), will be one of the constants of the motion. In general, the equation of motion of a phase function, f = f (p1 , . . . , pN , q1 , . . . , qN , t), is given by ∂f df = + {H, f }P oisson . dt ∂t

(2.15)

i Thus Eqs. (2.14) and (2.15) imply that dI dt = 0. If a system is integrable, there are no internal nonlinear resonances leading to chaos. All orbits lie on N-dimensional surfaces in the 2N -dimensional phase space.

2.3.1 Noether’s Theorem As was shown by Noether (1918), isolating integrals result from symmetries. For example, the total energy is an isolating integral (is a constant of the motion) for systems that are homogeneous in time (invariant under a translation in time). Total angular momentum is an isolating integral for systems that are isotropic in space. Noether’s theorem is generally formulated in terms of the Lagrangian (see Goldstein 1980 and Appendix A). Let us consider a dynamical system with N degrees of freedom whose state is given by the set of generalized velocities and positions ({q˙i }, {qi }). Let us consider a system whose Lagrangian, L = L({q˙i }, {qi }), is known. For simplicity, we consider a system with a time-independent Lagrangian. The equations of motion are given by the Lagrange equations ∂L d − ∂qi dt



∂L ∂ q˙i

 = 0,

(i = 1, . . . , N ).

(2.16)

For such systems, Noether’s theorem may be stated as follows. • Noether’s Theorem If a transformation t → t  = t + δt, q˙i →

q˙i (t  )

qi (t) → qi (t  ) = qi (t) + δqi (t),

and

= q˙i (t) + δ q˙i (t)

(for i = 1, . . . , N ) leaves the Lagrangian form invariant, L({q˙i (t)}, {qi (t)}) → L ({q˙i (t  )}, {qi (t  )}) = L({q˙i (t  )}, {qi (t  )}),

(2.17)

18

2 Fundamental Concepts and leaves the action integral invariant, 

t2

dt  L({q˙i (t  )}, {qi (t  )}) −

t1



t2

dtL({q˙i (t)}, {qi (t)}) = 0,

(2.18)

t1

then there exists an isolating integral of motion associated with this symmetry transformation. •

Before we proceed to show this, we must distinguish between variations of the coordinates at a fixed time, qi (t) → qi (t) = qi (t) + δQi (t), and variations at a later time (as we indicated above), qi (t) → qi (t  ) = qi (t) + δqi (t). δqi (t) is a convective variation and differs from δQi (t) by a convective term, δqi (t) = δQi (t) + q˙i δt. • Proof of Noether’s Theorem Let us write Eq. (2.18) in the form 

t2 +δt2

t1 +δt1

dt L({q˙i (t)}, {qi (t)}) −



t2

dt L({q˙i (t)}, {qi (t)}) = 0,

(2.19)

t1

where on the leftmost integral we have let the dummy variable t  → t. Next let {q˙i (t)} = ˙ i (t)} and {q  (t)} = {qi (t) + δQi (t)}, and expand the integral to first order in the {q˙i (t) + δ Q i variations. We then find      t2 +δt2  N   ∂L ˙ i + ∂L δQi δQ dt L({q˙i (t)}, {qi (t)}) + ∂ q˙i ∂qi t1 +δt1 i=1



t2

−

dtL({q˙i (t)}, {qi (t)}) = 0.

(2.20)

t1

If we keep only first-order contributions in the variations in the limits of integration, we find 

t2 t1

 N       ∂L ∂L ˙ dt δ Qi + δQi ∂ q˙i ∂qi i=1

+ δt2 L(t2 ) − δt1 L(t1 ) = 0,

(2.21)

where L(tk ) = L({q˙i (tk )}, {qi (tk )}). Equation (2.21) can now be rewritten in the form 



t2

dt t1

 d (δtL) + dt N

i=1



∂L ∂ q˙i



˙i + δQ



∂L ∂qi



 δQi

˙i = Let us now make use of Lagrange’s Eqs. (2.16) and note that δ Q some rearrangement of terms, we find 

t2

dt t1

d dt

 Lδt +

 N   ∂L i=1

∂ q˙i

(2.22)

. d dt δQi .

Then, after

δQi

= 0.

(2.23)

Let us now rewrite Eq. (2.23) in terms of our convective variations. We then find 

t2

t1

d dt dt



   N N    ∂L ∂L δt + L− q˙i δqi = 0. ∂ q˙i ∂ q˙i i=1

i=1

(2.24)

2.3 Integrable Systems

19

Thus d dt



L−

N 

 q˙i

i=1

∂L ∂ q˙i

 δt +

 N   ∂L i=1

∂ q˙i

δqi

= 0,

(2.25)

and we have obtained an isolating integral as a result of our symmetry transformation. •

To illustrate the use of Eq. (2.25), let us consider some examples. Assume that we translate the system in time by a constant amount, δt = , but let δqi = 0. Then we have    N  ∂L d dH =0 (2.26) q˙i = L− dt ∂ q˙i dt i=1

since the quantity in curly brackets is the Hamiltonian (see Appendix A). Thus homogeneity in time gives rise to the Hamiltonian as an isolating integral and to energy conservation. Suppose that we let δt = 0 but translate one coordinate, qj , by a constant amount, δqi = δi,j , where δi,j is the Kronecker delta. Then we find d dt



∂L ∂ q˙j

 =

dpj = 0. dt

(2.27)

For this case, the generalized momentum associated with the degree of freedom, qj , is an isolating integral, and the component of the momentum, pj , is conserved. The variations could, in general, be functions of space or time. Then the isolating integrals resulting from the symmetry transformation would be much more complicated. However, few such isolating integrals are known aside from the ones due to the space-time symmetries.

2.3.2 Hidden Symmetries In order for a system to be integrable, it must have as many conserved quantities as there are degrees of freedom. In general, not all of these can come from the spacetime symmetries but may come from what Moser has called hidden symmetries (Moser 1979). One notable example of such a hidden symmetry occurs for the two-body Kepler problem. Because of the homogeneity of this system in time and space, the total energy and the center-of-mass momentum are conserved. In addition, the gravitational force is a central force and therefore this system exhibits isotropy in space, which means that the total angular momentum is also conserved. These space-time symmetries are sufficient to make this system integrable since they provide six conservation laws for the six degrees of freedom. However, there is still another conserved quantity, the Runge-Lenz vector

20

2 Fundamental Concepts

A = p × L − μk

r |r|

(2.28)

Moser (1970), where p is the relative momentum, L is the total angular momentum, μ and k are as defined in Sect. 2.2, and r is the relative displacement of the two bodies. This additional symmetry is responsible for the fact that there is no precession of the perihelion (the point of closest approach of the two bodies) for the two-body Kepler system. This conservation law does not hold for any other central force problem. Hidden symmetries underlie the field of soliton physics. There is a nonlinear mechanical system with a finite number of degrees of freedom, that supports solitons, and that is the N-body Toda lattice (Toda 1967, 1981). The Toda lattice is a collection of equal-mass particles coupled in one space dimension by exponentially varying forces. It is integrable and therefore has N isolating integrals of the motion. The Toda lattice is one of the few discrete lattices for which soliton solutions are exact. The continuum limit of the Toda lattice yields the Kortewegde Vries equation, which is the classic equation describing non-topological solitons in continuum mechanics. The first real indication that the Toda lattice was integrable came from numerical experiments by Ford et al. (1973). This prompted theoretical work by Henon (1974) and Flaschka (1974), who found expressions for the N isolating integrals of the motion. The actual solution of the equations of motion was due to Date and Tanaka (1976), although significant contributions were made by Kac and van Moerbeke (1975).

2.4 Poincaré Surface of Section The Poincaré surface of section (PSS) is an area preserving map of a subspace of a 2DoF conservative dynamical system. It provides a way to determine numerically if a system is integrable. Consider a conservative system (with Hamiltonian independent of time) for which the energy is conserved. The Hamiltonian is then an isolating integral of the motion and can be written H (p1 , p2 , q1 , q2 ) = E.

(2.29)

The energy, E, is constant and restricts trajectories to lie on a three-dimensional surface in the four-dimensional phase space. If the system has a second isolating integral, I2 (p1 , p2 , q1 , q2 ) = C2 ,

(2.30)

where C2 is a constant, then it too defines a three-dimensional surface in the four-dimensional phase space. Once the initial conditions are given, E and C2 are fixed, and the trajectory is constrained to the intersection of the surfaces

2.4 Poincaré Surface of Section

21

Fig. 2.2 A Poincaré SOS for a two DoF system gives an area preserving map. (a) Plot a point each time the trajectory passes through the plane q1 = 0 with p1 ≥ 0. (b) If two isolating integrals exist, the trajectory will lie along one-dimensional curves in the two-dimensional surface. (c) If only one isolating integral exists (the energy), the trajectory will spread over a two-dimensional region whose extent is limited by energy conservation

defined by Eqs. (2.29) and (2.30); that is, to a two-dimensional surface in the four-dimensional phase space. If we combine Eqs. (2.29) and (2.30), we can write p1 = p1 (q1 , q2 , E, C2 ). If we now consider the surface q2 = 0, the trajectory lies on a one-dimensional curve. In general, if we are given the Hamiltonian, H , we do not know if an additional isolating integral, I2 , exists. We can check this numerically by solving Hamilton’s dqi ∂H ∂H i equations, dp dt = − ∂qi and dt = ∂pi , for (i = 1, 2), numerically and then plotting p2 and q2 each time q1 = 0 and p1 ≥ 0 (see Fig. 2.2a). If the system is integrable, the trajectory will lie on a one-dimensional curve (see Fig. 2.2b). If the system is nonintegrable, the trajectory will appear as a scatter of points limited to a finite area due to energy conservation (see Fig. 2.2c).

2.4.1 Henon-Heiles System The visual power of this method was demonstrated by Henon and Heiles (1964) who used it to determine if a third integral of motion existed that constrained the motion of a star in a galaxy that was known to have an axis of symmetry. Such a system has three degrees of freedom and two known isolating integrals of the motion, the energy and one component of the angular momentum. It was long thought that such systems do not have a third isolating integral because none had

22

2 Fundamental Concepts

been found analytically. However, the nonexistence of a third integral implies that the dispersion of velocities of stellar objects in the direction of the galactic center is the same as that perpendicular to the galactic plane. What was observed, however, was a 2:1 ratio in these dispersions. Henon and Heiles constructed the following Hamiltonian (with no known symmetries that can give rise to a third integral) to model the essential features of the problem, H =

1 2 1 2 (p1 + p22 ) + (q12 + q22 + 2q12 q2 − q23 ) = E. 2 2 3

(2.31)

They then studied its behavior numerically. Hamilton’s equations for this system are dp1 = −q1 − 2q1 q2 , dt

dp2 dqi = −q2 − q12 + q22 , and = pi dt dt

(2.32)

(for i = 1, 2). Note that the anharmonic terms in the potential energy give rise to nonlinear terms in the equation of motion. A sketch of the results of Henon and Heiles is shown in Fig. 2.3. At low energy (see Fig. 2.3a), there appears to be a third integral, at least to the accuracy of these plots. (Enlargement of the region around the hyperbolic fixed points would show a scatter of points.) As the energy is increased (this increases the effect of the nonlinear terms) (see Fig. 2.3b), the third integral appears to be destroyed in the neighborhood of the hyperbolic fixed points. At still higher energies (see Fig. 2.3c), the second isolating integral appears to have been totally destroyed. The scattered points in the surfaces of section for the Henon-Heiles system correspond to a single trajectory, which is chaotic. Such trajectories are chaotic in that they have sensitive dependence on initial conditions. Fig. 2.3 Poincaré SOSs for the Henon-Heiles system. (a) At energy E = 0.08333, the system appears to have two isolating integrals of the motion (at the scale of these plots). (b) At energy E = 0.12500, a chaotic trajectory appears in the neighborhood of the hyperbolic fixed points. (c) At energy E = 0.16667, the energy surface has become almost entirely chaotic (Reproduced from Henon and Heiles 1964)

2.4 Poincaré Surface of Section

23

Fig. 2.4 The HOCl molecule in body-frame coordinates (x, y, z). The vector t1 has length R. The vector t2 has length r (Reproduced from Lin et al. 2015)

2.4.2 The HOCl Molecule and Birkhoff Coordinates When dealing with systems without geometric symmetry, one can sometimes use Birkhoff coordinates to obtain useful surfaces of section. An interesting example of this concerns the internal dynamics of the HOCL molecule. The HOCl molecule consists of strongly bound hydrogen and oxygen atoms (HO), and a chlorine atom (Cl), more weakly bound to the HO system. It dissociates into a free Cl atom and a bound HO molecule at energies above E = 20,312 cm−1 = 2.518 eV. We can introduce body-frame coordinates (x, y, z) whose origin is the center of mass of the molecule. The total angular momentum of the molecule, in the absence of external fields, is conserved. We can assume that all the dynamics occurs in the (x, z) plane and angular momentum vectors due to internal motions of the molecule are parallel to the y direction. A sketch of the molecule in the body frame is given in Fig. 2.4. The vector t1 has length R and connects the center of mass of HO to Cl. The center of mass of the molecule lies along t1 . The vector t2 has length r and connects H to O. The angle between t1 and t2 is θ . When θ = 0 the molecule is in the linear configuration H–O–Cl. The potential energy surface of the molecule has been constructed by Weiss et al. (2000). In Lin et al. (2015), it was found that, above dissociation, the molecule had a 2 DoF invariant manifold, due to its symmetry with respect to θ → − θ . The molecular dynamics not only conserves energy and angular momentum, but the Hamiltonian is invariant under reflection through the point θ = 0. If the molecule is initially on the surface, pθ = 0, θ = 0, then it will remain on that surface for all subsequent motion. This is the linear configuration H–O–Cl of the molecule. The HO molecule can vibrate (change r) and the Cl molecule can move relative to the center of mass of HO (change R), but the molecule will not move out of the linear configuration. The potential energy of the molecule on the invariant manifold is shown in Fig. 2.5a. In the figure, the potential energy levels are given in cm−1 (1 cm−1 = 1.24×10−4 eV). The energy at which Cl first dissociates from the HO molecule is E = 20,312 cm−1 . However, in the linear configuration, there is a saddle point of height E = 23,961 cm−1 , at (R = 4.03 a.u., r = 1.84 a.u.) that prevents dissociation of the linear molecule for energies below the barrier height as long as it remains on the invariant manifold. There is a potential energy minimum at

24

2 Fundamental Concepts

Fig. 2.5 (a) Potential energy on invariant manifold of the HOCl molecule. Coordinate r is the distance between H and O. Coordinate R is distance between Cl and the center of mass of HO. Coordinates in atomic units and potential energy levels in cm−1 . The dark horizontal line is a line (trench) of minimum potential energy. (b) Surface of section along the potential energy minimum trench using Birkhoff coordinates (ps , s) (Reproduced from Lin et al. 2015)

(R = 3.15 a.u., r = 1.80 a.u.). The thick horizontal dark line across the figure is a line of minimum potential energy (potential energy trench). This potential energy trench can be used to form an area preserving map (SOS) of the dynamics of the linear HOCl molecule, using Birkhoff coordinates (Birkhoff 1927). Birkhoff coordinates along the potential energy trench are constructed each time the phase space trajectory crosses the trench. The distance along the trench is given by the coordinate s, measured left to right. When a trajectory crosses the trench at point s, the component of the momentum ps parallel to the trench at that point is determined. This is repeated each time the trajectory crosses the trench in a given direction. When the points (ps , s) are plotted, they form an area preserving map that gives a picture of the dynamics in that region of the phase space. In Fig. 2.5b, we show the SOS formed from the Birkhoff coordinates along the trench for a trajectory with energy E = 30,000 cm−1 . The SOS shows a region of regular motion and a region of chaos for trajectories that approach the saddle point. It also shows that there are trajectories that can cross the saddle point and dissociate the linear configuration of the molecule. It is important to note that the linear configuration is not stable at these intermediate energies above dissociation (Lin et al. 2015). For the low values of R shown in Fig. 2.5, the direction transverse to the invariant manifold (the θ direction) is unstable and once the molecule moves even a small distance (in θ ) off the invariant manifold it will rapidly return to a three degree of freedom (DoF) configuration.

2.5 Nonlinear Resonance and Chaos

25

2.4.3 Lattice Surfaces of Section It is possible to construct surfaces of section for two DoF periodic materials, when the material is composed of identical unit cells. If one can identify a line of potential minima (a trench) in the unit cell, then a surface of section can be constructed along that line of potential energy minima using Birkhoff coordinates (ps , s). However, in this case the surface of section is a lattice surface of section (LSOS). In a LSOS, points are plotted each time a trajectory crosses the line of potential energy minima, regardless of the unit cell traversed by the trajectory. Examples of LSOSs were constructed in Porter et al. (2017) and Barr et al. (2017) and are discussed in more detail in Chap. 7.

2.5 Nonlinear Resonance and Chaos Chaotic regions occur when isolating integrals of motion are destroyed locally by nonlinear resonances. Walker and Ford (1969) show this explicitly for a simple model Hamiltonian. Let us first consider the case of a nonlinear system with two degrees of freedom and with a single resonance between these two degrees of freedom.

2.5.1 Single-Resonance Hamiltonians In terms of action-angle variables, a general single-resonance Hamiltonian can be written H = H0 (J1 , J2 ) + Vn1 ,n2 (J1 , J2 ) cos(n1 θ1 − n2 θ2 ) = E,

(2.33)

where (J1 , J2 , θ1 , θ2 ) are action-angle variables. This system has a second isolating integral I = n2 J1 + n1 J2 = C2 ,

(2.34)

where C2 is a constant. It is easy to see that Eq. (2.34) is an isolating integral. We write Hamilton’s equations of motion for J1 and J2 , ∂H dJ1 =− = n1 Vn1 ,n2 sin(n1 θ1 − n2 θ2 ) dt ∂θ1

(2.35)

26

2 Fundamental Concepts

and ∂H dJ2 =− = −n2 Vn1 ,n2 sin(n1 θ1 − n2 θ2 ). dt ∂θ2

(2.36)

Using Eqs. (2.35) and (2.36), we find that dI = 0. dt

(2.37)

The system described by the Hamiltonian in Eq. (2.33) contains a single (n1 , n2 ) resonance. The presence of this resonance means that for certain values of J1 and J2 there can be a large transfer of energy between the two degrees of freedom of this system.

2.5.1.1

(2,2) Resonance

To see more clearly how a resonance works, let us consider the specific case of a (2,2) resonance. Following Walker and Ford, we write the Hamiltonian H = H0 (J1 , J2 ) + αJ1 J2 cos(2θ1 − 2θ2 ) = E,

(2.38)

H0 (J1 , J2 ) = J1 + J2 − J12 − 3J1 J2 + J22 .

(2.39)

where

Equations (2.38) and (2.39) describe a nonlinear system because of the nonlinear dependence of H0 on the action variables J1 and J2 . The isolating integrals of motion are the Hamiltonian, H , and I = 2J1 + 2J2 . It is useful to make a transformation from action-angle variables (J1 , J2 , θ1 , θ2 ) to a new set of variables (J1 , J2 , 1 , 2 ) via the canonical transformation J1 = J1 + J2 = I  = I2 , J2 = J2 , 1 = θ2 , and 2 = θ2 − θ1 . The Hamiltonian then takes the form H = J1 − J21 − J1 J2 + 3J22 + αJ2 (J1 − J2 ) cos(22 ) = E.

(2.40)

Since H is independent of 1 , in this new coordinate system J1 is constant. Hamilton’s equations in this coordinate system become dJ1 = 0, dt d1 = 1 − 2J1 − J2 + αJ2 cos(22 ), dt

(2.41) (2.42)

2.5 Nonlinear Resonance and Chaos

27

and dJ2 = 2αJ2 sin(22 )(I  − J2 ), dt d2 = −I  + 6J2 + α cos(22 )(I  − 2J2 ). dt

(2.43) (2.44)

Since J1 is constant, Eqs. (2.43) and (2.44) can be solved first for J2 (t) and 2 (t) and then substituted into Eq. (2.42) to obtain 1 (t). Let us now find the fixed points of these equations. The fixed points are points d2 nπ 2 for which dJ dt = 0 and dt = 0. Fixed points occur when 2 = 2 and J2 = Jo , where Jo is a solution of the equation − I  + 6Jo + α cos(nπ )(I  − 2Jo ) = 0.

(2.45)



Note that for α  1, Jo ≈ I6 . The nature of the fixed points can be determined by linearizing the equations of motion about points (J2 = Jo , 2 = nπ 2 ). We let J2 (t) = Jo + J(t) and 2 (t) = nπ + (t) and linearize in J(t) and (t). We find 2 d dt



J(t) (t)

The solution



 =

0 4α cos(nπ )Jo (I  − Jo ) (6 − 2α cos(nπ )) 0

J(t) (t)



 J(t) . (2.46) (t)

to Eq. (2.46) determines the manner in which trajectories flow

in the neighborhood of the fixed points. For α  1 (and therefore Jo ≈ equations reduce to d dt



J(t) (t)



 ≈

0 6

20αI 2 36

cos(nπ ) 0



 J(t) . (t)

I 6 ),

these

(2.47)

Let us assume that Eq. (2.47) has a solution of the form 

J(t) (t)



 = eλt

AJ A

 (2.48)

,

where AJ and A are independent of time. Then we can solve the resulting eigenvalue equation  λ  for both λ and

AJ A



 =

0 6

20αI  2 36

cos(nπ ) 0

 AJ . The eigenvalues are given by A



AJ A



28

2 Fundamental Concepts



20αI 2 cos(nπ ) λ± = ± 6

 12 ,

and the solution to Eq. (2.47) can be written 

J(t) (t)



 =e

λ+ t

A+

b λ+

1



 +e

λ− t

A−

b λ−

1

 ,

(2.49)

2

where b = 20αI 36 , and A+ and A− are determined by the initial conditions. For n even, λ is real and the solutions contain exponentially growing and decreasing components, while for n odd, λ is pure imaginary and the solutions are oscillatory. For n even, the fixed points are hyperbolic (trajectories approach or recede from the fixed point exponentially), while for n odd, the fixed points are elliptic (trajectories oscillate about the fixed point).  For very small α, the fixed points occur for J2 = Jo ≈ I6 and therefore for   J1 ≈ 5I6 and J2 ≈ I6 . We can also find the range of energies for which these fixed 25E 1 points exist. Plugging J1 = 5J2 into Eq. (2.38), we find J12 − 10J 13 + 39 = 0 or 1

5 3 2 (1 ± (1 − 13E J1 = 13 3 ) ) = 5J2 . Thus, the fixed points only exist for E < 13 for 3 , J1 is no longer real. very small α. For E > 13 A plot of some of the trajectories on the energy surface, E = 0.18, for coupling constant α = 0.1, is given in Fig. 2.6. In this plot, we have transformed from polar coordinates (J2 , 2 ) to Cartesian coordinates (p, q) via the canonical 1 1 transformation p = −(2J2 ) 2 sin(2 ) and q = (2J2 ) 2 cos(2 ). The elliptic and hyperbolic fixed points and the separatrix associated with them can be seen clearly. The region inside and in the immediate neighborhood outside the separatrix is called the (2,2) nonlinear resonance zone. We see that large changes in the action, J2 , occur in this region of the phase space, indicating that a strong exchange of energy is occurring between the modes of the system. Let us now attempt to compute these level curves using perturbation theory as discussed earlier. We go from action-angle variables (J1 , J2 , θ1 , θ2 ) to new variables (I1 , I2 , φ1 , φ2 ) via a canonical transformation given by the generating function

G(I1 , I2 , φ1 , φ2 ) = I1 θ1 + I2 θ2 + αg2,2 (I1 , I2 ) sin(2θ1 − 2θ2 ).

(2.50)

1 I2 , where Following the procedure outlined in Sect. 2.2, we find that g2,2 = (2ω−I 1 −2ω2 ) ω1 = 1 − 2I1 − 3I2 and ω2 = 1 − 3I1 + 2I2 . The Hamiltonian to order α 2 is H = Ho (I1 , I2 ) + O(α 2 ) and the action variables (neglecting terms of order α 2 ) are

J1 (t) = I1 −

2αI1 I2 cos(2ω1 t − 2ω2 t) (2ω1 − 2ω2 )

(2.51)

2.5 Nonlinear Resonance and Chaos

29

Fig. 2.6 Phase space trajectories for the (2,2) resonance Hamiltonian in1 Eq. (2.40) (p = −(2J2 ) 2 1 sin(2 ) and q = (2J2 ) 2 cos(2 )). For all curves, E = 0.18 and α = 0.1. The curves consist of discrete points because we have plotted points along the trajectories at discrete times

and J2 (t) = I2 +

2αI1 I2 cos(2ω1 t − 2ω2 t) . (2ω1 − 2ω2 )

(2.52)

In order for these equations to have meaning, the following condition must hold: |2ω1 − 2ω2 | = |2I1 − 10I2 |  2αI1 I2 . However, near a resonance, I1 ≈ 5I2 . Therefore this condition breaks down in the neighborhood of a resonance zone. Actually this is to be expected since the resonance introduces a topological change in the flow pattern in the phase space.

2.5.1.2

(2,3) Resonance

Walker and Ford also studied a (2,3) resonance with Hamiltonian 3

H = Ho (J1 , J2 ) + βJ1 J22 cos(2θ1 − 3θ2 ) = E.

(2.53)

This again is integrable and has two isolating integrals of the motion, the Hamiltonian, H , and I = 3J1 + 2J2 = C3 .

(2.54)

30

2 Fundamental Concepts

We can again make a canonical transformation, J1 = J1 − 23 J2 , J2 = J2 , θ1 = 1 , θ2 = 2 + 23 1 (note that I = 3J1 ). The Hamiltonian then takes the form H = J1 − J21 +

5J1 J2 23 J2 β 3 − + J22 + J22 (3J1 − 2J2 ) cos(32 ) = E 3 3 9 3 (2.55)

and the coordinate J1 is a constant of the motion since H is independent of 1 . The equations of motion for J2 and 2 are 3 dJ2 = βJ22 (3J1 − 2J2 ) sin(32 ) dt

and

  1 3 1 5J1 46J2 5 d2 2 = − + + βJ2 J1 − J2 cos(32 ). dt 3 3 9 2 3

It is easy to see that the fixed points occur for 2 = satisfies the equation

nπ 3

(2.56)

(2.57)

and J2 = Jo where Jo

  1 I 1 5I 46Jo 5 2 − + + βJo − Jo cos(nπ ) = 0. 3 9 9 2 3

(2.58)

If we again linearize the equations of motion about these fixed points and determine the form of the flow in their neighborhood as we did below Eq. (2.45), we find that for even n (n = 0, 2, 4) the fixed points are hyperbolic while for odd n (n = 1, 3, 5) the fixed points are elliptic. These fixed points are clearly seen in the plot of the phase space trajectories for the (2,3) resonance system given in Fig. 2.7. In Fig. 2.7 all curves have energy E = 0.18 and coupling constant β = 0.1. The separatrix of the (2,3) resonance zone is clearly seen, as are the three hyperbolic and elliptic fixed points.

2.5.2 Two-Resonance Hamiltonian The two single-resonance systems described above are integrable. Any systems containing two or more resonances are nonintegrable because a second isolating integral of the motion cannot be found. Therefore systems with two or more resonances can undergo a transition to chaos as parameters of the system are varied. Walker and Ford showed this for the Hamiltonian with two primary resonances, H = Ho (J1 , J2 ) + αJ1 J2 cos(2θ1 − 2θ2 ) 3

+βJ1 J22 cos(2θ1 − 3θ2 ) = E.

(2.59)

2.5 Nonlinear Resonance and Chaos

31

Fig. 2.7 A plot of some phase space trajectories obtained for the (2,3) resonance Hamiltonian in Eq. (2.53). All curves have energy E = 0.18 and coupling constant β = 0.1 but have different values of the constant of motion, I . The three hyperbolic and three elliptic fixed points as well as the separatrix of the (2,3) resonance are clearly seen. The curves consist of discrete points because we plot points along the trajectories at discrete times.1 We have set p = −(2J2 ) 2 sin(2 ) and 1 q = (2J2 ) 2 cos(2 )

The surface of section for this Hamiltonian is shown in Fig. 2.8. Hamilton’s equations for the two-resonance system can be written ∂H dJ1 =− = 2αJ1 J2 sin(2θ1 − 2θ2 ) dt ∂θ1 3

+2βJ1 J22 sin(2θ1 − 3θ2 ),

(2.60)

dJ2 ∂H =− = −2αJ1 J2 sin(2θ1 − 2θ2 ) dt ∂θ2 3

−3βJ1 J22 sin(2θ1 − 3θ2 ),

(2.61)

∂H dθ1 = = 1 − 2J1 − 3J2 + αJ2 cos(2θ1 − 2θ2 ) dt ∂J1 3

+βJ22 cos(2θ1 − 3θ2 ),

(2.62)

dθ2 ∂H = = 1 − 3J1 + 2J2 + αJ1 cos(2θ1 − 2θ2 ) dt ∂J2 1 3 + βJ1 J22 cos(2θ1 − 3θ2 ). 2

(2.63)

Walker and Ford constructed a Poincaré surface of section by solving the equations of motion (2.60)–(2.63) numerically and plotting (J2 , θ2 ) each time 1 1 2 2 θ1 = 3π 2 . (If pi = −(2Ji ) sin(θi ) and qi = (2Ji ) cos(θi ), the surface of section is similar to that of Henon and Heiles, who plotted a point (p2 , q2 ) each time q1 = 0 and p1 > 0.) A sketch of the Poincaré surface of section for several energies

32

2 Fundamental Concepts

Fig. 2.8 Poincaré surfaces of section for the double-resonance Hamiltonian in Eq. (2.59) with 1 1 p2 = −(2J2 ) 2 sin(θ2 ) and q2 = (2J2 ) 2 cos(θ2 ) and coupling constants α = β = 0.02. (a) At energy E = 0.056, only the (2,2) resonance exists. (b) At energy E = 0.180, the (2,3) resonance has emerged from the origin but is well-separated from the (2,2) resonance. (c) At energy E = 0.2000, the two primary resonances have grown in size but remain separated. The chain of five islands is a higher-order resonance. (d) At energy E = 0.2095, resonance overlap has occurred and chaos can be seen in the overlap region

is shown in Fig. 2.8. In all cases shown in this figure, the coupling constants are 3 α = β = 0.02. The (2,2) resonance is present for all energies E ≤ 13 . However, the (2,3) resonance first emerges from the origin for energy E ≈ 0.16. For energies E = 0.056 (Fig. 2.8a), only the (2,2) resonance exists. For E = 0.180 (Fig. 2.8b), both resonances are present but well-separated in the phase space. As the energy is raised, the resonances occupy larger regions of the phase space. Finally, for E = 0.2095 (Fig. 2.8d), the resonances have overlapped and a chaotic trajectory is found.

2.6 KAM Theory As we have seen in Sect. 2.2, conventional perturbation theory diverges in regions containing resonance zones because of small denominators arising from the resonances. However, Kolmogorov (1954) found a way to construct a perturbation theory that was rapidly convergent and applicable to non-resonant tori. Kol-

2.6 KAM Theory

33

mogorov’s ideas were made rigorous by Arnol’d (1963) and by Moser (1962). The nonresonant tori that have not been destroyed by resonances are called KAM tori or KAM surfaces (after Kolmogorov, Arnol’d, and Moser). Examples of KAM tori can be found in Figs. 2.7 and 2.8, and many more will be seen throughout this book. The KAM theory applies to systems with N degrees of freedom whose motion is governed by a Hamiltonian of the form H (J1 , . . . , JN , θ1 , . . . , θN ) = H0 (J1 , . . . , JN ) + V (J1 , . . . , JN , θ1 , . . . , θN ), (2.64) where H0 is integrable, is a small parameter, and the potential energy can be written in the form   ... Vn1 ,...nN (J1 , . . . , JN ) ei(n1 θ1 +···+nN θN ) , V (J1 , . . . , JN , θ1 , . . . , θN ) = n1

nN

(2.65) where ni (i = 1, . . . , N ) ranges over all integers. (Note that if V (J1 , . . . , JN , θ1 , . . . , θN ) is a smooth function of angles, {θi }, the Fourier coefficients, Vn1 ,...nN , will decrease fairly rapidly with increasing {ni }.) A further requirement that is necessary for the proof of the KAM theorem is that the determinant of the matrix formed by ∂ 2 H0 the quantities ∂J (the Hessian of H0 ) must be nonzero. i ∂Jj The Hamiltonian defined in Eq. (2.64) describes a system with a dense set of resonances in phase space. KAM showed that for such systems, the volume of phase space occupied by resonances goes to zero as → 0. The idea behind this can be illustrated by a simple example. Consider the unit line (a continuous line ranging from zero to one). This line contains an infinite number of rational fractions. However, the rational fractions form a set of measure zero. Now exclude a region m

 m m

 − 3 ≤ ≤ + 3 n n n n n about each rational fraction. This mimics resonances that have finite width, n2 3 for example, and are located in regions of the phase space for which the ratio of frequencies associated with the various degrees of freedom is a rational fraction. The total length of the line that is excluded is  n  ∞   2

n=1 m=1

n3

 ∞   1

π 2 = 2

= → 0 as 3 n2

→ 0.

n=1

Thus, for very small , only a small fraction of the total volume of phase space contains resonance zones. But they exist on all scales. We do not have space here to prove the KAM theorem (for this, one should go to the references cited above), but we will try to give the flavor of it. Let us illustrate the approach for the case of a system with two degrees of freedom. We follow the discussion by Barrar (1970), which most closely follows Kolmogorov’s original approach.

34

2 Fundamental Concepts

2.6.1 The KAM theorem (for N = 2) Consider a system described by the Hamiltonian H (J1 , J2 , θ1 , θ2 ) = Ho (J1 , J2 ) +



∞ 

∞ 



Vn1 ,n2 (J1 , J2 )ei(n1 θ1 +n2 θ2 ) ,

n1 =−∞ n2 =−∞

(2.66) where is a small parameter and H0 has nonzero Hessian. The prime on the summations indicates that we exclude the term n1 = n2 = 0 since it can be included in H0 . We shall assume that H is an analytic function of all variables and is a periodic function of angles θ1 and θ2 . On a torus (J1 = J1o , J2 = J2o ) such that o o o the frequencies ωi = ( ∂H ∂Jj )o = ωi (J1 , J2 ) satisfy the conditions |n1 ω1 + n2 ω2 | ≥

K , ||n||α

where ||n|| = |n1 | + |n2 | > 0, α ≥ 2, and K is a constant, a perturbation theory will converge. The proof of the KAM theorem proceeds as follows (see Kolmogorov 1954 and Barrar 1970 for details). Let us move the origin of the coordinates to (J1o , J2o ) via a canonical transformation, Ji − Jio = pi and θi = φi . The Hamiltonian can then be written in the form H = C (0) +

2 

ωi pi + A(0) (φ1 , φ2 ) +

i=1

2 

(0)

Bi (φ1 , φ2 )pi

i=1

2 2   (0) + Ci,j (φ1 , φ2 )pi pj + D (0) (p1 , p2 , φ1 , φ2 ),

(2.67)

i=1 j =1

where C (0) is a constant and D (0) (p1 , p2 , φ1 , φ2 ) is a function whose lowestorder dependence on pi is pi3 . Let us now introduce a generating function that takes us from coordinates (p1 , p2 , φ1 , φ2 ) to a new set of canonical coordinates (1) (P1(1) , P2(1) , (1) 1 , 2 ). We write the generating function in the form (1)

(1)

S(P1 , P2 , φ1 , φ2 ) =

2 

(1)

(Pi

+ ξi )φi + X(φ1 , φ2 )

i=1

+

2  i=1

Pi(1) Yi (φ1 , φ2 ),

(2.68)

2.6 KAM Theory

35

where ξi are constants and X and Yi are functions to be determined. Then  ∂Yj ∂S ∂X = (Pi(1) + ξi ) +

+ Pj(1) ∂φi ∂φi ∂φi 2

pi =

(2.69)

j =1

and (1)

=

i

∂S (1)

∂Pi

= φi + Yi (φ1 , φ2 ).

(2.70)

We can use Eqs. (2.69) and (2.70) to write the Hamiltonian in terms of new (1) (1) (1) (1) coordinates, (P1 , P2 , 1 , 2 ). The idea of Kolmogorov was to choose the (0) quantities X, Yi and ξi so that they cancel A(0) and Bi from the resulting Hamiltonian. (Most of Barrar’s paper is devoted to showing that this can be (1) (1) (1) (1) done.) Then, in terms of the new canonical coordinates, (P1 , P2 , 1 , 2 ), the Hamiltonian becomes H (1) = C (1) +

2 

(1) ωi Pi(1) + 2 A(1) ((1) 1 , 2 )

i=1

+ 2

2 2  2   (1) (1) (1) (1) (1) (1) (1) (1) (1) Bi (1 , 2 )Pi + Ci,j (1 , 2 )Pi Pj i=1 j =1

i=1 (1)

(1)

(1)

(1)

+D (1) (P1 , P2 , 1 , 2 ).

(2.71)

This process can be repeated. In the next step, the Hamiltonian becomes H (2) = C (2) +

2 

(2) ωi Pi(2) + 4 A(2) ((2) 1 , 2 )

i=1

+ 4

2 2  2   (2) (2) (2) (2) (2) (2) (2) (2) (2) Bi (1 , 2 )Pi + Ci,j (1 , 2 )Pi Pj i=1 j =1

i=1 (2)

(2)

(2)

(2)

+D (2) (P1 , P2 , 1 , 2 ).

(2.72)

The sequence of Hamiltonians obtained by this procedure converges very rapidly to the form H (∞) = C (∞) +

2 

(∞)

ωi Pi

(∞)

2 2   (∞) (∞) (∞) (∞) ∞ Ci,j (1 , 2 )Pi Pj i=1 j =1

i=1

+D ∞ (P1

+

(∞)

, P2

(∞)

, 1

(∞)

, 2

).

(2.73)

36

2 Fundamental Concepts (∞)

In terms of the coordinates (P1 the form (∞)

dPk dt

=−

∂H (∞) ∂k(∞)

=

(∞)

, P2

2 2  

(∞)

, 1

Pi(∞) Pj(∞)

i=1 j =1

(∞)

, 2

(∞) ∂Ci,j

∂k(∞)

), Hamilton’s equations take

+ O((P (∞) )3 )

(2.74)

and (∞)

dk dt

=

∂H (∞) ∂Pk(∞)

= ωk + O((P (∞) )3 ) (∞)

(2.75) (∞)

for (k = 1, 2). These equations have solutions Pi = 0 and i = ωi t + Ci for (i = 1, 2), where Ci is a constant. Thus, a rapidly convergent procedure has been found to obtain solutions to the equations of motion at least on KAM tori sufficiently far from resonances.

2.7 The Definition of Chaos The flow of trajectories in a given region of phase space is said to be chaotic if it has positive KS metric entropy (KS stands for Krylov, Kolmogorov, and Sinai) (Kolmogorov 1958, 1959; Sinai 1963a; Arnol’d and Avez 1968; Ornstein 1974; Chirikov 1979; Lichtenberg and Lieberman 1983). Such flows are called K-flows. The KS entropy is a measure of the degree of hyperbolic instability in the relative motion of trajectories in phase space. As we saw in Sect. 2.4, in the neighborhood of fixed points, we can determine the nature of the flow by linearizing the equations of motion about the fixed point. In the neighborhood of hyperbolic fixed points, trajectories on the eigenvectors approach (depart) the fixed point in an exponentially decreasing (increasing) manner. Trajectories in the neighborhood of the fixed point, but not on the eigenvectors, contain both types of motion. There are as many sets of eigenvectors in the neighborhood of a hyperbolic fixed point as there are degrees of freedom. Along each eigenvector, the rate of approach or departure is determined by a single eigenvalue of the transition matrix (the matrix that governs the evolution in the neighborhood of the fixed point) for the linearized problem.

2.7.1 Lyapounov Exponent Oseledec (1968) was the first to show that a procedure analogous to that used to study exponential divergence of flow in the neighborhood of hyperbolic fixed points could be used to study the nature of the flow in the neighborhood

2.7 The Definition of Chaos

37

Fig. 2.9 The 2N -dimensional vector, XN t , evolves according to Hamilton’s equations and describes the evolution of the state of the system in phase space

of a moving point in phase space. To see how this works, consider a system with N degrees of freedom (2N-dimensional phase space). We shall denote the 2N-dimensional vector describing the state of the system at time t by N XN t = X (p1 (t), . . . , pN (t); q1 (t), . . . , qN (t)) (see Fig. 2.9). This vector evolves according to Hamilton’s equations. Let us now consider two neighboring points in N N N phase space, XN t and Yt = Xt + Xt . By solving Hamilton’s equations for our system, we can determine how the displacement, XN t , evolves in time. We define the magnitude of the displacement, XN , to be t 1

N N N N 2 dt (XN o , Yo ) = |Xt | = (Xt · Xt ) ,

(2.76)

N N N where XN o and Yo are the initial values of Xt and Yt . The rate of exponential N N growth (or decrease) of dt (Xo , Yo ) is given by N λ(XN o , Yo )

 N  dt (XN 1 o , Yo ) ln . = lim N t→∞ t do (XN o , Yo )

(2.77)

N λ(XN o , Yo ) is called the Lyapounov exponent. There are 2N orthogonal directions in a 2N-dimensional phase space and therefore 2N independent Lyapounov exponents. We let the set {ei } denote the 2N unit vectors associated with these 2N orthogonal directions, where the unit vector, ei , denotes the direction in which the separation of neighboring trajectories is  characterized by λi . Then, in general, we can write Xt = 2N C (t)e i i , where the i=1 coefficient, Ci (t), denotes the component of Xt in the direction ei . The Lyapounov exponent associated with the direction ei is given by

λi =

λ(XN o , ei )

  dt (XN 1 o , ei ) ln . = lim t→∞ t do (XN o , ei )

(2.78)

N The notation dt (XN o , ei ) indicates that we choose a neighboring point, Yo , so that it N deviates from Xo only in the direction ei in phase space. In Benettin and Strelcyn (1978) it is shown that, for Hamiltonian flows, the exponents satisfy the relation

38

2 Fundamental Concepts

λi = −λ2N −i+1 .

(2.79)

On the energy surface, there are 2N − 1 exponents. One of them is zero (the one associated with motion along the direction of the flow). We can now order the exponents in order of increasing value. If we relabel the indices in Eq. (2.79), we can write −λN −1 ≤ . . . ≤ −λ1 ≤ 0 ≤ λ1 ≤ . . . ≤ λN −1 . If Xt is chosen arbitrarily, it should contain some contribution from all spatial N directions. Then we will find λ(XN o , Yo ) = λN −1 . A numerical method for computing all 2N of the Lyapounov exponents in an N degree of freedom system can be found in Benettin et al. (1979). Benettin et al. (1976) have computed λN −1 for the Henon-Heiles system. For bounded systems, the quantity defined in Eq. (2.77) can be expected to saturate after a finite time. Thus a slightly different procedure is used to obtain the exponents. One essentially computes a sequence of distances each of which is obtained after a finite length of time, τ , in the following way. Let Xo,n−1 (Xτ,n−1 ) denote the position of our reference trajectory at the beginning (end) of the nth time step, τ . Let Xo,o and Yo,o denote the positions of neighboring trajectories at the initial time. Initially, the distance between them is do = |Yo,o − Xo,o |. At the end of the first time step, their distance is d1 = |Yτ,o − Xτ,o |. Now begin the second time step. We relabel the position of our reference trajectory, Xo,1 = Xτ,o , and choose a new neighboring vector, Yo,1 , so that the vector (Yo,1 − Xo,1 ) is directed along the same direction as (Yτ,o − Xτ,o ) but has length do . We then let the system evolve and obtain a distance, d2 , at the end of the second time step (see Fig. 2.10). We continue this process for n time steps, each of length τ . In doing so, we generate a sequence of distances, {dj }, where j = 1, . . . , n. The Lyapounov exponent is then defined as

Fig. 2.10 The Lyapounov exponent, kn (τ, Xo,o , Yo,o ), is obtained by computing a sequence of distances, dn , between our reference trajectory, XN t , and a neighboring trajectory. Each distance is obtained after a finite time interval, τ . In this figure, Xo,n = XN nτ . The neighboring trajectory is adjusted at the beginning of each interval to lie a distance do from XN nτ

2.7 The Definition of Chaos

kn (τ, Xo,o , Yo,o ) =

39

  n dj 1  . ln nτ do

(2.80)

j =1

If do is not too big, the quantity kn (τ, Xo,o , Yo,o ) has been found to have the following properties Benettin et al. (1976) and Casartelli et al. (1976): limn→∞ kn (τ, Xo,o , Yo,o ) = k(τ, Xo,o , Yo,o ) exists; k(τ, Xo,o , Yo,o ) is independent of τ ; k(τ, Xo,o , Yo,o ) is independent of do ; k(τ, Xo,o , Yo,o ) = 0 if Xo,o is chosen to lie in a regular region of the energy surface; 5. k(τ, Xo,o , Yo,o ) is independent of Xo,o and is positive if Xo,o is chosen to lie in a chaotic region of the energy surface. 1. 2. 3. 4.

Therefore, in a chaotic region of the energy surface, we can write k(E) = k(τ, Xo,o , Yo,o ). The quantity k(E) obtained in this manner is the largest Lyapounov exponent, λN −1 . Benettin et al. (1979) have shown that it is possible to compute all of the Lyapounov exponents for a model Hamitonian system with N (N = 4, 5) degrees of freedom. Meyer (1986) has been able to show that for sufficiently smooth Hamiltonians there are at least 2N vanishing Lyapounov exponents if there are N independent isolating integrals of the motion. In Fig. 2.11 and Eq. (2.79), we show some of the results of Benettin et al., who computed the Lyapounov exponent and KS metric entropy for the Henon-Heiles system. In Fig. 2.11, the Lyapounov

Fig. 2.11 Plot of Lyapounov exponent kn for the Henon-Heiles system for initial conditions E = 0.125, q1 = 0, p1 > 0 and six different initial values for q2 , p2 , three chosen from the chaotic regime (black circle, diamond, and square) and three chosen from the regular regime (open circle, diamond, and square). For all initial conditions, typically do = 3 × 10−4 and τ = 0.2 (see Fig. 2.10). As n → ∞, the exponent kn approaches a positive constant value for trajectories in the chaotic regime, and approaches zero for trajectories in the regular regime (Benettin et al. 1976)

40

2 Fundamental Concepts

Fig. 2.12 A plot of k(E) = limn→∞ kn as a function of energy for trajectories in the chaotic regime (black squares) and trajectories in the regular regime (black circles) of the Henon-Heiles system. The dotted line is an estimate of the KS metric entropy as a function of energy (Benettin et al. 1976)

exponent, kn , is computed for six different initial conditions, three taken from the chaotic region and three taken from the regular region (it is useful to locate these initial conditions in the surfaces of section for the Henon-Heiles system in Fig. 2.3). For initial conditions in the chaotic regime, all three exponents approach the same final value as n → ∞, even though the initial conditions are taken from quite different regions of the phase space. For initial conditions in the regular region, the three exponents steadily decrease toward zero. In Fig. 2.12, the exponent k(E) = limn→∞ kn is plotted as a function of energy in both the chaotic and regular regimes for the Henon-Heiles system. The rate of divergence of trajectories appears to increase with increasing energy. Regions of phase space for which neighboring trajectories have positive Lyapounov exponents are said to exhibit sensitive dependence on initial conditions, which is the definition of classical chaos. Any small change in the initial trajectories can lead to quite different final states.

2.7.2 KS Metric Entropy and K-Flows There is a relation between the Lyapounov exponents and the KS metric entropy. In order to build some intuition about the KS metric entropy, let us consider the baker’s map (Arnol’d and Avez 1968; Penrose 1970), which is the simplest case of a Bernoulli shift (Moser 1973). The baker’s map consists of an alphabet with two “letters,” 0 and 1, and the set, {S}, of all possible doubly infinite sequences S = (. . . , s−2 , s−1 , s0 ; s1 , s2 , . . .)

(2.81)

2.7 The Definition of Chaos

41

that can be formed from the alphabet by selecting sk = 0 or 1, where sk is the kth entry in the sequence and −∞ ≤ k ≤ ∞. The set {S} includes sequences with random ordering and periodic ordering of elements. Each sequence, S, can be mapped to a point, (p, q), in the unit square by defining 0 

p=

sk 2k−1

(2.82)

k=−∞

and q=

∞ 

sk 2−k .

(2.83)

k=1

We can introduce dynamics into this system by means of the Bernoulli shift, T , which shifts all entries in a given sequence, S, to the right by one place. Let the sequence S be defined as in Eq. (2.81). Then T S = (. . . , s−3 , s−2 , s−1 ; s0 , s1 , . . .).

(2.84)

This shift causes the following mapping of the coordinates (p, q) on the unit square  T (p, q) =

(2p, 12 q) for 0 ≤ p < 12 (2p − 1, 12 q + 12 ) for 12 ≤ p ≤ 1

It is important to note that whenever the element, s0 , of a sequence, S, has the value s0 = 0(1), the point (p, q) will lie to the left (right) of p = 12 . Thus, for random sequences, the point (p, q) will be mapped randomly to the left or right of p = 12 by T . (0) (0) Let us now introduce the partition of the unit square α = (A1 , A2 ) as shown (0) in Fig. 2.13a, where Ai , i = 1, 2 is an element of the partition, α. The effect of successive Bernoulli shifts will be to stretch the elements of this initial partition into filaments distributed throughout the unit square, as shown in Fig. 2.13. Let us (0) next assign a measure, pi(0) = μ(A(0) i ), to the element Ai (i = 1, 2) equal to

Fig. 2.13 Behavior of the phase space of the unit square under the baker’s map. The initial partition α shown in (a) gets stretched by mappings (b) T, (c) TT, (d) TTT into finer and finer filaments by the transformation, T

42

2 Fundamental Concepts

Fig. 2.14 (a) The four elements of the partition resulting from the intersection of partitions α and T α. (b) The eight elements of the partition resulting from the intersection of partitions α, T α, and T 2 α. Each element is represented by a different pattern

 (0) the fraction of the area of the unit square that it occupies. Then 2i=1 pi = 1. Thus, the measure of an element is the area that it occupies. From Fig. 2.13, we see (n) (n) that T n α will contain 2n elements, T n α = (A1 , . . . , A2n ). Let us next introduce (0) (1) the partition α ∨ T α, which consists of elements Ai ∩ Aj (i, j = 1, 2), where (1) ∩ denotes the intersection of the elements A(0) i and Aj . The partition α ∨ T α is shown in Fig. 2.14a. Similarly. the elements of the partition α ∨T α ∨T 2 α are shown in Fig. 2.14b. The KS metric entropy can now be defined as

h(α ∨ T α ∨ . . . ∨ T n−1 α) , n→∞ n

hKS (T ) = sup h(α, T ) = sup lim

(2.85)

where  h(α) = − pi ln(pi )

(2.86)

i

and the sum is taken over all elements of partition α. The maximum value of the entropy occurs when the elements of a partition all have equal area. If we assume that our partitions do have equal area, then it is easy to see that  n  2  n  1 1 = n ln(2). h(α ∨ T α ∨ . . . ∨ T n−1 α) = − ln 2 2 n

(2.87)

i=1

Thus, for the baker’s map, hKS (t) = ln(2).

(2.88)

This analysis can be extended to Bernoulli shifts with an alphabet with k “letters.” In that case, the KS metric entropy is ln(k). Therefore, the baker’s map and Bernoulli shifts in general are K-flows. The dynamics causes contraction in the q direction

2.7 The Definition of Chaos

43

and stretching in the p direction, very much like the flow in the neighborhood of a hyperbolic fixed point. The connection between the Lyapounov exponents and the KS metric entropy was established by Piesin (1976). The KS metric entropy may be related to the Lyapounov exponents in the following way. Let ρ(XN ) =

N −1 

λi (XN ),

(2.89)

i=1

where λi (XN ) denotes the Lyapounov exponent in a region of phase space in the interval XN → XN + dXN on the energy surface. (Remember that the Lyapounov exponents are constant and nonzero throughout a stochastic region and are zero in regular regions.) The KS entropy is then Benettin et al. (1979)  h(E) =

ρ(XN )dμE ,

(2.90)

E

where dμE denotes an invariant volume element of the energy surface. Thus the KS entropy is directly related to the Lyapounov exponents. Benettin et al. (1976) have made an estimate of the KS metric entropy as a function of energy for the HenonHeiles system. Their result, the dotted line, is shown in Fig. 2.12. The KS metric entropy has an energy dependence and qualitative behavior similar to that of the largest Lyapounov exponent for this system. The Henon-Heiles system is one whose phase space contains a mixture of regular and chaotic trajectories. The fraction of the phase space occupied by each can be varied by varying parameters of the system. This is the most common type of behavior found in Hamiltonian systems and is characteristic of systems with smooth differentiable Hamiltonians. One of the few systems that is known to be a K-flow for all values of its parameters is the hard sphere gas. This was proven by Sinai (1963b) for the Sinai billiard, which consists of a particle confined in a box that has periodic boundary conditions and a hard circular barrier placed inside the box (see Fig. 2.15). The Fig. 2.15 Sinai (1963b) proved that the phase space flow of a moving particle confined to a box containing a hard circular barrier is a K-flow. The box is assumed to have periodic boundary conditions

44

2 Fundamental Concepts

Fig. 2.16 The phase space of a hard-sphere gas is a K-flow. Neighboring trajectories diverge rapidly due to collisions with the hard convex surfaces

Fig. 2.17 The stadium billiard is a two-dimensional billiard with spherical ends of radius r separated by parallel sides of length 2a. For a = 0, the motion of a billiard is integrable, but for a > 0 it is a K-flow (Bunimovich 1974)

Hamiltonian of this system is not smooth and differentiable. The convex surface of the barrier causes neighboring trajectories to exponentially diverge from one another in phase space. Let us now consider the dynamics of the Sinai billiard (Berry 1978). Since the box has periodic boundary conditions, we may also view this system as that of a particle moving through a lattice of circular barriers. Assume that the average distance traveled between collisions with the barriers is D and the radius of the pillars is R. If two neighboring trajectories (we assume they have the same velocity) strike a barrier at points a distance S0 apart, the angular distance will be θ0 = S0 /R (see Fig. 2.16). However, when they strike the next barrier a distance D away, their points of collision will be separated a distance S1 ≈ θ0 D, and the angular separation of the collision points will be θ1 ≈ D R θ0 . If we continue this process for n collisions, the approximate angular spread of points of collision

n will be θn ≈ D θ . The number of collisions, n, needed for a divergence 0 R R of one radian is n = ln(θ0 )/ ln D . It is interesting to consider an example. Let θ 0 = 0.0001 radians and R/D = 0.1. Then n = 4 and it requires only four collisions to achieve a divergence of one radian. Another system that has been proven to be a K-flow is that of a billiard moving in a planar concave region called a stadium. The stadium billiard consists of two half circles of radius r connected by equal parallel line segments of length 2a (see Fig. 2.17). When a = 0 and the system is circular, the motion of the billiard is integrable. However, for a > 0, it becomes a K-flow, as was proved by Bunimovich

References

45

(1974). It should be noted, however, that (Benettin and Strelcyn 1978) have found a transition region from regular to chaotic flow for a  r. We will return to the stadium billiard when we discuss quantum systems.

2.8 Conclusions In this chapter, we have introduced concepts and model systems that will recur repeatedly throughout the remainder of the book. For example, the stadium and the bakers map will reappear in Chap. 7, where their quantum analogs will be studied. It is interesting to note that (Ramani et al. 1989) have described a method, different from that discussed in this chapter, to determine if a system is integrable. They study the singularities of the differential equations and categorize them in terms of those singularities. They conjecture that systems of equations with the Painlevi property (the only moving singularities are poles) are integrable. In this book, we will not discuss ergodic theory, which is a theory that attempts to lay the dynamical foundations of statistical mechanics. Suffice it to say that systems, such as the Sinai billiard, that are globally K-flows are also ergodic and mixing. Excellent discussions about the relation between ergodic theory and dynamics may be found in Farquhar (1964), Arnol’d and Avez (1968), and Ornstein (1974). Shorter discussions may be found in Farquhar (1972) and Lebowitz and Penrose (1973).

References Arnol’d VI (1963) Russ Math Surv 18:9; 18:85 Arnol’d VI, Avez A (1968) Ergodic problems of classical mechanics. W.A. Benjamin, New York Barr AD, Barr AR, Porter MD, Reichl LE (2017) Chaos 27:104604 Barrar R (1970) Celestial Mech 2:494 Benettin G, Strelcyn JM (1978) Phys Rev A 17:773 Benettin G, Galgani L, Strelcyn JM (1976) Phys Rev A 14:2338 Benettin G, Froeshle C, Scheidecker JP (1979) Phys Rev A 19:2454 Berry MV (1978) AIP conference proceedings, vol. 46. American Institute of Physics, New York, p. 16. Reprinted in [MacKay and Meiss 1987] Birkhoff GD (1927) Acta Math 50:359 Bunimovich LA (1974) Funct Anal Appl 8:254 Casartelli M, Diana E, Galgani L, Scotti A (1976) Phys Rev A 13:1921 Chirikov B (1979) Phys Rep 52:263 Date E, Tanaka S (1976) Prog Theor Phys 55:457; Prog Theor Phys Suppl 59:107 Farquhar IE (1964) Ergodic theory in statistical mechanics. Wiley-Interscience, New York Farquhar IE (1972) In: Beil J, Rae J (eds) Irreversibility in the Many-Body problem. Plenum Press, New York Flaschka H (1974) Phys Rev B 9:1924 Ford J, Stoddard DS, Turner JS (1973) Prog Theor Phys 50:1547 Goldstein H (1980) Classical mechanics. Addison-Wesley, Reading Henon M (1974) Phys Rev B 9:1921

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2 Fundamental Concepts

Henon M, Heiles C (1964) Astron J 69:73 Kac M, van Moerbeke P (1975) Proc Natl Acad Sci USA 72:2879 Kolmogorov AN (1954) Dokl Akad Nauk SSSR 98:527; (An English version appears in R. Abraham, Foundations of Mechanics (W.A. Benjamin, New York, 1967, Appendix D)) Kolmogorov AN (1958) Dokl Akad Nauk SSSR 119:861 Kolmogorov AN (1959) Dokl Akad Nauk SSSR 124:754 Lebowitz JL, Penrose O (1973) Phys. Today 26:23 Lichtenberg AJ, Lieberman MA (1983) Regular and stochastic motion. Springer, New York Lin Y-D, Reichl LE, Jung C (2015) J Chem Phys 142:124304 Meyer HD (1986) J Chem Phys 84:3147 Moser J (1962) Nachr Akad Wiss Goettingen II, Math Phys Kd 1:1 Moser J (1970) Commun Pure Appl Math 23:609 Moser J (1973) Stable and random motions in dynamical systems. Princeton University Press, Princeton Moser J (1979) Am Sci 67:689 Noether E (1918) Nach Ges Wiss Goettingen 2:235 Ornstein DS (1974) Ergodic theory, randomness, and dynamical systems. Yale University Press, New Haven Oseledec VI (1968) Trans Moscow Math Soc 19:197 Penrose O (1970) Foundations of statistical mechanics. Pergamon Press, Oxford Piesin YG (1976) Math Dokl 17:196 Porter MD, Barr AD, Barr AR, Reichl LE (2017) Phys Rev E 95:052213 Ramani A, Grammaticos B, Bountis T (1989) Phys Rep 180:159 Sinai YG (1963a) Am Math Soc Transl 31:62 Sinai YG (1963b) Sov Math Dokl 4:1818 Toda M (1967) J Phys Soc Jpn 22:431; 23:501 Toda M (1981) Theory of nonlinear lattices. Springer, Berlin Walker GH, Ford J (1969) Phys Rev 188:416 Weiss J, Hauschildt J, Grebenshchikov SY, Duren R, Schinke R, Koput J, Stamatiadis S, Farantos SC (2000) J Chem Phys 112:77 Wintner A (1947) The analytical foundations of celestial mechanics. Princeton University Press, Princeton

Chapter 3

Area-Preserving Maps

Abstract Area-preserving maps can provide a picture of mechanisms causing the transition to chaos. In non-integrable area-preserving twist maps, the separatrix region of a nonlinear resonance contains a stochastic (chaotic) layer. Nonlinear resonances are separated by KAM tori that become cantori as the transition to global chaos occurs. The standard map is a twist map that describes the behavior of separatrix regions, and it shows the universal scaling behavior and self-similarity associated with the transition to chaos. This self-similarity is demonstrated explicitly by another twist map, called the universal map. In systems for which it is not possible to construct an area-preserving map analytically, the global behavior of the dynamics can be determined by working directly with the Hamiltonian and using it to locate nonlinear resonance structures. The Hamiltonian is then mapped between different spatial scales in the phase space and a renormalization map is constructed for the amplitudes and relative wave numbers of the resonances from one scale to the next. The renormalization map allows one to determine parameter ranges for which chaos has developed in local regions of the phase space. Keywords Area preserving maps · Twist map · Standard map · Nonlinear resonance · KAM tori · Universal map · Cantorus · Bifurcations · Self-similarity · Scaling · Renormalization map

3.1 Introduction Area-preserving maps provide the simplest and most accurate means to visualize and quantify the dynamical behavior of conservative systems with two degrees of freedom. Such maps can be iterated on even the smallest computers with great accuracy, and provide beautiful pictures of the mechanisms at play during the transition to chaos. The class of area-preserving maps we study in this chapter are twist maps. When an integrable twist map is rendered nonintegrable by a small perturbation, resonances can occur and degenerate lines of fixed points in the integrable map © Springer Nature Switzerland AG 2021 L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200, https://doi.org/10.1007/978-3-030-63534-3_3

47

48

3 Area-Preserving Maps

are changed to finite chains of alternating hyperbolic and elliptic fixed points surrounded by nonlinear resonance zones. As the strength of the perturbation is increased, the resonance zones can grow and overlap and form a chaotic sea. Chaos appears first in the neighborhood of hyperbolic fixed points and is due to an incredibly complex dynamics that occurs in that neighborhood. In fact, near hyperbolic fixed points it is possible to embed a Bernoulli shift with an infinite alphabet in local regions of the phase space. This means that these regions are Kflows and therefore are chaotic. For small mapping parameters, resonance zones are separated from one another by KAM tori. In area-preserving maps with two degrees of freedom, KAM tori serve to isolate one region of the phase space from another. KAM tori can be destroyed by nonlinear resonances. The mechanism by which this occurs is quite beautiful. Each KAM torus has an irrational winding number (winding numbers are defined in Sect. 3.2). Resonance zones form island chains, and each island chain contains a sequence of hyperbolic and elliptic fixed points that have a rationalfraction winding number. Greene (1979a) has shown that each KAM torus can be approximated by a unique sequence of island chains whose winding numbers are given by the continued fraction representing the irrational winding number of the KAM torus. The island chains that approximate a given KAM torus play a dominant role in its destruction. A KAM torus is destroyed suddenly, as the mapping parameter increases, and forms a cantorus. A cantorus can still partially block the flow of trajectories in phase space. As the mapping parameter is increased further, the cantorus gradually disappears and trajectories are free to diffuse more or less at random in the chaotic sea. In the subsequent sections of this chapter, we will describe in detail the intricate behavior associated with the transition to chaos in area-preserving maps. We will begin in Sect. 3.2 by describing the general behavior of twist maps. Twist maps characterize well the behavior of conservative systems with two degrees of freedom. We will use them throughout this book. In nonintegrable twist maps, the separatrix region associated with nonlinear resonances contains a stochastic (chaotic) layer. In Sect. 3.3, we derive a map, the standard map, which is a map that describes the behavior of the separatrix region. We will use the standard map to show the mechanism by which KAM tori are destroyed, and we demonstrate the scaling behavior associated with the destruction of KAM tori. In 3.4, we then focus on the scaling behavior of “noble” KAM tori, which are the last KAM tori to be destroyed in a global transition to chaos. The detailed mechanism by which a KAM torus is destroyed is determined by its winding number and not the particular twist map that it belongs to. For this reason, the destruction of a KAM torus and its change into a cantorus is associated with universal scaling behavior. The phase space associated with this process shows selfsimilarity. MacKay (1982, 1983a) has constructed a map, called the universal map, that shows explicitly this self-similar behavior. In Sect. 3.5, we describe properties of the universal map.

3.1 Introduction

49

As we vary the parameters of a conservative map, an elliptic fixed point may bifurcate and change to a hyperbolic fixed point (this is called tangent bifurcation) or may bifurcate into several new fixed points. The case when it bifurcates into one hyperbolic fixed point and two elliptic fixed points is called a period-doubling bifurcation. As the mapping parameter is varied, sequences of period-doubling bifurcations may occur. Such bifurcation sequences also exhibit universal scaling behavior. In Sect. 3.6, we will give criteria to determine at what mapping parameter values such bifurcations can occur. In Sect. 3.7, we will focus on cantori, which are the remnants of KAM tori just after they are destroyed. The mechanism by which phase space trajectories pass through cantori resembles that of turnstiles or revolving doors that allow a two-way flow of traffic. Turnstiles can be associated with the rational approximates to the KAM torus that has been destroyed. They pump an amount of phase space area that itself exhibits scaling behavior. In practice, we are often confronted with a physical system whose Hamiltonian we are given, and then we must determine as much as possible about its global dynamics. We have to ask: What regions of the phase space might undergo a transition to chaos and for what parameter values does it happen? For such systems, it is usually not possible to construct an area-preserving map analytically, but there is still a great deal we can learn about the global behavior by working directly with the Hamiltonian. The global properties of a Hamiltonian system are determined by the location and size of its nonlinear resonances. However, in systems with time-independent Hamiltonians, it is not always easy to locate the nonlinear resonances. There is some hope if the Hamiltonian separates into an integrable part, for which action-angle variables can be found, and a perturbation that renders the total Hamiltonian nonintegrable. Escande and Doveil (1981); Escande (1982, 1985) developed a renormalization scheme, based on a mapping of Hamiltonians between different spatial scales in the phase space, to describe the destruction of KAM tori. The starting point of this theory is a Hamiltonian that allows one to identify the primary nonlinear resonances of the system. We can then focus on a KAM torus that will be most strongly affected by two primary resonances that bracket it. We then approximate the full Hamiltonian by a Hamiltonian containing only those two resonances. We then repeat the process and thereby generate a mapping of Hamiltonians for the pairs of resonances that bracket the same KAM torus on each scale. It is possible to write a renormalization map for the amplitudes and relative wave numbers of the resonances from one scale to the next. This renormalization map allows us to determine if the KAM torus is destroyed or not for the given parameters of the system. In Sect. 3.8, we derive the renormalization map. Finally, in Sect. 3.8.2, we study the fixed points of the renormalization map.

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3.2 Twist Maps Twist maps are area-preserving maps that provide a clear visualization of dynamical systems with two degrees of freedom. They provide an analytic representation of a Poincaré surface of section. Twist maps may be integrable or nonintegrable. Birkhoff’s fixed point theorem describes changes that occur in an integrable map when its integrability is destroyed by a perturbation. These changes are described in general terms in this section and in more detail throughout the remainder of this chapter.

3.2.1 Derivation of a Twist Map from a Torus A twist map is an area-preserving map that can be derived from a Poincaré surface of section of the torus. To see how this is done, let us consider the torus shown in Fig. 3.1. Each time the trajectory passes a given angle θ2 = , we plot its position in the (J1√ , θ1 ) plane. For an integrable system, the plotted points will lie on a circle of ˙ radius 2J1 , and the time interval between passes will be τ = 2π ω2 , where θ2 = ω2 = ω2 (J1 , J2 ). Let us now assume that the nth passage of the trajectory occurs at angle θ1 = φn . Then the (n + 1)st passage will occur at angle θ1 = φn+1 = φn + ω1 τ = θ0 + 2π ωω12 . The quantity ω = ωω12 is called the winding number. The area enclosed by the circle is 2π J1 . Now introduce the coordinate, ρ, where 2π J1 = πρ 2 , and ρ is √ the radius of the circle, ρ = 2J1 . For a given energy E, the value of ρ determines J2 as well. Our mapping now takes the form ρn+1 = ρn , φn+1 = φn + 2π ω(ρn ),

(3.1)

where ω is assumed to be a smooth function of ρn .

Fig. 3.1 For integrable systems, the twist map consists of trajectories that densely fill a circle (irrational winding number ω) and discrete, periodic points (rational winding number ω). The rate at which a trajectory completes one revolution of the circle depends on the radius. Thus an initial line of points, a, becomes twisted, b, by the map

3.2 Twist Maps

51

We may also write the mapping in the form 

ρn+1 φn+1





 ρn , φn

= To

(3.2)

where To is a twist map and is defined in Eq. (3.1). For conservative systems, To is area-preserving. As we vary ρ, the points will lie on different concentric circles. N (N and M are When the winding number is equal to a rational fraction, ω = M relatively prime integers), the map will consist of M discrete points on a circle. Each of these points will be invariant under the mapping ToM . If we denote the (o) (o) coordinates of one such point as (ρn , φn ), then after M iterations of the map, To , (o) (o) we obtain the point φn+M = φn + 2π N = φn since φn is defined mod 2π . Thus we travel around the circle N times before the initial point repeats. For ω irrational, the points will never repeat but eventually (after many iterations of the map) will densely fill the circle (see Fig. 3.1) (Berry 1978; MacKay 1982). Let us now perturb this map and write ρn+1 = ρn + f (ρn , φn ),

(3.3)

φn+1 = φn + 2π ω(ρn ) + g(ρn , φn ),

(3.4)

or 

ρn+1 φn+1



 = T

ρn φn

 (3.5)

,

where f and g are chosen so that area is preserved under the mapping T . This is done by requiring that the Jacobian 

ρn+1 φn+1 J ρn φn

 ∂ρ

 = det

n+1

 An additional requirement for a twist map is the twist is always in the same direction.

n φn ∂φ∂ρn+1 ∂ρn φn

∂ρn+1 n ρn ∂φ∂φn+1 ∂φn ρn



∂φn+1 ∂ρn φn

 = 1.

(3.6)

=0 for all (ρn , φn ), so that

3.2.2 Generating Functions The twist map, Eq. (3.5), is area-preserving and, therefore, the map is a canonical transformation from one discrete time to the next. Let us introduce a generating function, F (φn , φn+1 ) for the map, such that

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3 Area-Preserving Maps



∂F pn = − ∂φn



 pn+1 =

and

, φn+1

∂F ∂φn+1

 .

(3.7)

φn

The area-preserving property of the map makes the differential, dF = ρn+1 dφn+1 − ρn dφn ,

(3.8)

exact. This can be seen as follows. If we rearrange the partial derivatives, we can write the Jacobian in Eq. (3.6) as  J

ρn+1 φn+1 ρn φn





 ∂φn+1 ∂ρn φn φn+1      ∂ 2F ∂ 2F = = 1. ∂φn ∂φn+1 ∂φn+1 ∂φn

=−

∂ρn+1 ∂φn





(3.9)

  2   2 ∂ F Area-preservation requires ∂φn∂∂φFn+1 = ∂φn+1 ∂φn , which is also the condition for exactness of the differential dF . The twist maps most commonly used have a simple form: ρn+1 = ρn + f (φn ),

(3.10)

φn+1 = φn + ρn + f (φn ).

(3.11)

The generating function for such maps is of the form F (φn , φn+1 ) =

1 (φn − φn+1 )2 + V (φn ), 2

(3.12)

∂V where f (φn ) = ∂φ . n The generating function, F (φn , φn+1 ), is a Lagrangian, and some properties of the map may be derived from a stationary action principle.

Action Principles for Discrete Maps (MacKay et al. 1984) I. If φn−1 , φn , φn+1 are three successive points of an orbit, then ∂ (F (φn−1 , φn ) + F (φn , φn+1 )) = 0 ∂φn and conversely. II. Let {φn } denote a sequence of points with initial and final values φr and φs , respectively, specified. This sequence defines a segment of a T orbit if and only if the action sum Wr,s =

s−1  n=r

F (φn , φn+1 )

3.2 Twist Maps

53

is stationary with respect to an arbitrary variation of intermediate points φn . An infinite sequence {φn } defines an orbit if and only if every finite segment has a stationary action.

These results will prove useful in Sect. 3.4, where we discuss a mechanism for breakup of KAM tori.

3.2.3 Birkhoff Fixed Point Theorem The behavior of fixed points under a perturbation is the subject of the Birkhoff fixed point theorem (Birkhoff 1927; Berry 1978). Consider a circle, C, with winding N number ω = M and two neighboring circles, C+ and C− , with irrational winding N N numbers ω+ > M and ω− < M , respectively (see Fig. 3.2a). Under the mapping T0M , the fixed points on circle C will not move. However, points on circle C+ will be mapped in a counterclockwise direction and points on circle C− will be mapped in a clockwise direction, relative to C. If is small enough, these relative twists will not be changed under the map T M , although the circles may be distorted. Consider a radius line drawn from the origin outward. There must be some point along the radius line that is fixed under one application of the mapping T M . These points, along all possible radius lines, make up a new curve R close to C (see Fig. 3.2b). If we let T M act on R , we obtain yet another curve, R  = T M R , which must intersect R in an even number of places, since the area enclosed must be preserved. Each intersection is a fixed point of T M . Let X(0) be one point of the intersection. Then T M X(0) = X(0) . All points mapped from X(0) by T , namely X(1) = T X(0) , X(2) = T 2 X(0) ,. . . , X(M−1) = T M−1 X(0) , are fixed points under T M . Thus, all points of intersection are fixed points of T M . The number of intersections must be an even multiple of M and, therefore, there are 2kM fixed points of T M (k is an integer). The direction of flow of phase points in the

Fig. 3.2 (a) The case = 0. C is a line of orbits with period M. C+ and C− are orbits with irrational winding number. Under ToM , the periodic orbits are fixed points, while C+ and C− are mapped in opposite directions. (b) T M maps C to orbit R and maps R to orbit T M R . By area conservation, intersections occur in an even number of places and are fixed points of T M . (c) The direction of flow shows that fixed points are alternating between elliptic and hyperbolic

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Fig. 3.3 If we magnify each elliptic fixed point, we find a mixture of elliptic and hyperbolic fixed points surrounding it. This structure repeats itself for every elliptic fixed point

neighborhood of these fixed points, indicates that half of the fixed points must be elliptic and half must be hyperbolic (see Fig. 3.2c). If we move the origin of our mapping to any elliptic point, this picture will repeat itself (see Fig. 3.3).

3.2.4 The Tangent Map  ρ (0) , where X(0) = T M X(0) , φ (0) we can determine its character by linearizing the mapping, T M , about the fixed point. The linearized mapping, ∇T M , is called the tangent map. Its eigenvalues are sometimes called the “multipliers”  ofthe fixed point. ρn about the fixed point X(0) , let Xn = X(0) + To linearize the point Xn = φn δXn , where δXn is small. Then δXn+1 = ∇T M δXn , where  ∂ρ  n+1 ∂ρn+1  ∂φn n  . ∇T M = ∂φ∂ρn+1 (3.13) ∂φn+1  

If we know the location of a fixed point X(0) =

∂ρn

∂φn

X(0)

The eigenvalues, λ, of ∇T M are given by det[λ1¯ − ∇T M ] = 0 or λ2 − λTr(∇T M ) + det(∇T M ) = 0. For area-preserving maps, det(∇T M ) = 1, so the eigenvalues are given by  t2 t − 1, λ± = ± 2 4

(3.14)

(3.15)

where t = Tr[∇T M ]. Thus, the eigenvalues come in reciprocal pairs, λ+ = λ−1 − .

3.2 Twist Maps

55

Fig. 3.4 The flow of points in the neighborhood of fixed points. For regular hyperbolic points (residue R < 0), successive points on an orbit remain on one side of the fixed point, while for inversion hyperbolic points (residue R > 1) successive points alternate across the fixed point. The numbers indicate the sequence in time of the points (the residue, R, is defined in Sect. 3.3)

For −2 < t < 2, the eigenvalues form complex conjugate pairs that lie on the unit circle, and the fixed points are elliptic. For t > 2, the fixed point is regular hyperbolic. For t < −2, the fixed point is inversion hyperbolic (subsequent points of the mapping alternate across the fixed point (see Fig. 3.4). For the special cases t = ±2, the eigenvalues are degenerate, having values +1 or −1, and the fixed point is parabolic. Parabolic fixed points are generally unstable (MacKay 1982). If the mapping is defined in terms of smooth continuous functions, the eigenvectors of ∇T M in the neighborhood of the fixed point will be smooth and continuous. For elliptic fixed points, the eigenvalues will be pure imaginary and the eigenvectors will describe motion that oscillates about the fixed point. For hyperbolic fixed points, the eigenvalues will be real and of the form λ1 = λ1 and λ2 = λ, where λ is real and λ > 1. Let us denote the eigencurve associated with the eigenvalue λ1 as W (s) and the eigencurve associated with the eigenvalue λ as W (u) . Once the eigencurves of the tangent map have been found, they can be extended away from the neighborhood of the fixed point by using the full map, T M . These extensions of the eigencurves are also denoted W (s) and W (u) , and are called stable manifolds and unstable manifolds, respectively. Points on the stable manifolds, W (s) , will be mapped toward the fixed point since (∇T M )n W (s) = ( λ1 )n W (s) , while points on the unstable manifolds, W (u) , will be mapped away from the fixed point since (∇T M )n W (u) = λn W (u) (see Fig. 3.5).

3.2.5 Homoclinic and Heteroclinic Points For integrable systems, the stable and unstable manifolds of one fixed point will join smoothly together (Fig. 3.5a) or will connect smoothly to those of another fixed point (Fig. 3.5b). For example, the point r in Fig. 3.5b will be mapped toward the fixed point P by To but toward Q by To−1 .

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3 Area-Preserving Maps

Fig. 3.5 The stable and unstable manifolds, W s and W u , of hyperbolic fixed points for integrable systems join smoothly. (a) A point r on W (s) and W (u) is mapped to the same fixed point, P , by To and To−1 . (b) A point r on W (s) and W (u) is mapped to the fixed point P by To and to the fixed point Q by To−1

Fig. 3.6 For nonintegrable systems, the stable and unstable manifolds no longer join smoothly but oscillate and intersect transversally. (a) For stable and unstable manifolds that approach the same fixed point, the intersections, r, r  , r  , etc., are called homoclinic points. (b) For manifolds that approach different hyperbolic fixed points, the points of transversal intersection, r, r  , r  , etc., are called heteroclinic points

For a nonintegrable system, a totally different behavior occurs. As one begins to perturb the map, the stable and unstable manifolds, W (s) and W (u) , respectively, begin to oscillate and intersect one another transversally at an infinite number of places (see Fig. 3.6). For the case in Fig. 3.6a, where W (s) and W (u) belong to the same hyperbolic fixed point, P , the points of intersection, r, r  , r  , etc., are called homoclinic points, while for the case in Fig. 3.6b, where W (s) and W (u) attach to separate hyperbolic fixed points, P and Q, the points of intersection, r, r  , r  , etc., are called heteroclinic points. Homoclinic points in Fig. 3.5a are mapped toward P by T and T −1 , but in opposite directions, while heteroclinic points in Fig. 3.6b are mapped toward Q by T and toward P by T −1 . The homoclinic or heteroclinic points become more and more closely spaced as one approaches the hyperbolic fixed points and therefore, since area must be preserved by the map, the oscillations must grow in amplitude as one approaches the hyperbolic fixed points. It has been shown that it is possible to embed a Bernoulli shift with an alphabet containing an infinite number of “letters” (the baker’s map has two letters) in the neighborhood of each homoclinic or heteroclinic point. A very good discussion of the embedding of the Bernoulli shift into the neighborhood of the homoclinic and heteroclinic points has been given by Moser (1973). Moser draws a picture similar to Fig. 3.7, which shows a few of the

3.3 The Standard Map

57

Fig. 3.7 In the neighborhood of each homoclinic or heteroclinic point, it is possible to embed a Bernoulli shift with an alphabet containing an infinite number of letters. Thus, the flow in the neighborhood of the homoclinic and heteroclinic points is chaotic

homoclinic points about a given isolated hyperbolic fixed point. Let us select one homoclinic point, r, and draw a small neighborhood, Ar , at the point r so that one side of Ar lies along W (s) and another side lies along W (u) . Since r is a homoclinic point, it and much of its neighborhood, Ar , will be mapped to neighborhoods of the hyperbolic fixed point, P , by T and T −1 . Let us assume that r and Ar are mapped to Ao by T and are mapped to A1 by T −1 . Because of the nature of the flow close to the hyperbolic fixed point, strips of points Ui that lie parallel to W (s) in Ao will be mapped to strips of points Vj that lie parallel to W (u) in A1 (see Fig. 3.7). The net effect of this is that under repeated mappings T , strips Ui that are parallel to W (s) in Ar get mapped back to Ar by T but arrive as strips parallel to W (u) . Thus a Bernoulli shift can be embedded in the flow in the neighborhood of each of the homoclinic points. Furthermore, some of the sequences comprising this Bernoulli shift will themselves be homoclinic points so the picture repeats itself infinitely often in the neighborhood of each homoclinic point! In Sect. 2.7, we showed that Bernoulli shifts have finite KS metric entropy and therefore are K-flows. Thus, the flow in the neighborhood of each homoclinic and heteroclinic point is chaotic. In subsequent sections, we only consider reversible twist maps. This includes all maps that may be derived from a Hamiltonian that is even in momentum (Greene 1979a,b). A reversible twist map, T , is one that can be decomposed into a product of involutions, S1 and S2 , such that T = S2 S1 , where S12 = S22 = I and I is the identity map. This decomposition is important because the fixed points of I1 and I2 form lines of symmetry of the map T , which greatly facilitate the study of fixed points of the map.

3.3 The Standard Map The standard map is a nonlinear twist map that is of special importance because it describes the local behavior of nonintegrable dynamical systems in the separatrix region of nonlinear resonances. From the standard map, we obtain a very clear

58

3 Area-Preserving Maps

Fig. 3.8 The rotor

picture of the mechanisms involved in the transition to global chaos in conservative systems. The standard map is an analytic representation of the strobe plot for a onedimensional rotor (see Fig. 3.8) subjected to repeated delta function kicks. The kicks occur with period T and have an amplitude that depends on the angular position of the rotor. The Hamiltonian can be written H =

∞  J2 + K cos(θ ) δ(t − nT ), 2I n=−∞

(3.16)

where J is the angular momentum of the rotor, θ is its angular position, I is its moment of inertia, and K is the amplitude of the “kicks”. If we note the identity ∞ 

δ(t − nT ) =

n=−∞

  ∞ 2π mt 2  1 cos + , T T T

(3.17)

m=1

then the Hamiltonian can be rewritten in the form H =

  ∞ 2π mt J2 K  . cos θ − + 2I T m=−∞ T

(3.18)

The effect of the delta function kicks is to immerse the rotor in an infinite number of cosine potential waves, each traveling at a different speed. Note that all of the waves have the same amplitude, K/T . These cosine waves give rise to nonlinear resonances. A Poincaré surface of section (strobe plot) can be obtained analytically for this system and is called the standard map. To derive the standard map, we use ∂H dθ ∂H Hamilton’s equations, dJ dt = − ∂θ and dt = ∂J and obtain ∞  J dθ dJ = K sin(θ ) = . δ(t − nT ) and dt dt I n=−∞

(3.19)

The rotor is given a delta function kick at times t = nT . However, between the kicks, no force acts so the rotor evolves freely. Therefore, between kicks, J is

3.3 The Standard Map

59

constant and θ evolves linearly in time. At the kick, J changes discontinuously so that the rate of growth of θ between different kicks will differ. Let us integrate Eqs. (3.19) from a time just before the kick at t = nT to a time just before the kick at t = (n + 1)T . The only contribution from the force comes at t = nT . If we also set I = 1 and T = 1, we obtain Jn+1 = Jn + K sin(θn ),

(3.20)

θn+1 = θn + Jn+1 ,

(3.21)

which is the standard map. We can make the change of variables, Jn = 2πpn and θn = 2π xn and write the standard map in the form 

pn+1 xn+1



 = TK

pn xn



 =

 K sin(2π xn ) pn − 2π . xn + pn+1

(3.22)

When working with the standard map, one must specify the boundary conditions on the dynamical variables, pn and xn . There are two categories of boundary conditions: (1) One can choose periodic boundary conditions such that 0 ≤ pn ≤ 1 mod(1) and 0 ≤ xn ≤ 1 mod(1); or (2) one can choose boundary conditions such that pn or xn or both have infinite range. Each of these choices of boundary conditions provides useful information, and some aspects of the dynamics will behave differently depending on the choice of boundary conditions. It is useful to introduce the winding number for the standard map. It is defined as ω(p0 ) = lim

n→∞

xn − x0 n

(3.23)

and can be used to characterize both periodic orbits and KAM tori in the standard map. The periodic orbits have a rational winding number while the KAM tori have N an irrational winding number. A periodic orbit with winding number ω(p0 ) = M (M) is called an M-cycle and has the property that xM = x0 + N (mod 1) and pM = (M) (M) (M) p0 , where (p0 , x0 ) denote the coordinates of one member of the M-cycle. As we have seen in Sect. 3.2, the M-cycles will be either elliptic or hyperbolic. For the standard map, these periodic orbits are particularly easy to find numerically because of a symmetry property (Greene 1979a). The standard map, TK , can be written as a product of two involutions, I1 and I2 , so that TK = I2 I1 and I1

   p− p = x

K 2π

sin(2π x) −x

 and I2

    p p = . x p−x

(3.24)

The products I12 = I22 give the identity map, and det I1 = det I2 = −1. Each of these   p involutions has lines of fixed points; that is, lines of points for which I1 = x

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3 Area-Preserving Maps

Fig. 3.9 Symmetry lines for the standard map. The symmetry line a is the dominant symmetry line because every elliptic M-cycle will have a point on it

      p p p and I2 = . For I1 , the lines of fixed points are x = 0 and x = 12 , x x x while, for I2 , x = p2 and x = p+1 2 are lines of fixed points. These lines are shown in Fig. 3.9 and are denoted a, b, c, and d, respectively. At least two of the M points on an M-cycle will be fixed points of I1 or I2 . There are two M-cycles for each winding N number ω = M , one elliptic and one hyperbolic. Each M-cycle has M points. Of N the 2M fixed points with winding number ω = M , four will lie on the lines a, b, c, and d, with one on each line. Elliptic M-cycles will always have a point on the a line, and therefore the a line is called the dominant symmetry line. In Figs. 3.10, 3.11, and 3.12, we show a sequence of standard map plots for increasing values of K. In Fig. 3.10a, we show a number of orbits of the standard map for K = 0.1716354. The first thing to note is that the standard map has several symmetry lines. Also, it repeats itself along both the p and x directions (mod 1), so that all relevant information about the map is contained in the interval 0 ≤ p ≤ 0.5, 0 ≤ x ≤ 1.0. However, in the literature, various authors focus on different regions of the standard map in the interval 0 ≤ p ≤ 1.0, so we show the entire interval here. For K = 0.1716354, most of the map is composed of KAM tori. However, the dominant resonances and periodic orbits are clearly seen. None of the separatrices appear to be chaotic to the scale shown here. In Fig. 3.10b, we show the standard map for K = 0.4716354. Again, the largest resonances are clearly seen. The positions of the elliptic periodic orbits with winding numbers ω = 01 and ω = 12 are unchanged as we increase K, but the other periodic orbits are moved by the growing resonance zone surrounding the fixed point with winding number ω = 01 . We note also that the separatrix for the hyperbolic orbit with winding number ω = 01 begins to be chaotic in the neighborhood of the hyperbolic fixed point. This plot is still dominated, however, by KAM tori. In Fig. 3.11c, we show the structure of the standard map for K = 0.7716354. Now the hyperbolic fixed points with winding numbers ω = 01 and ω = 12 appear to have chaotic separatrices. All the resonances are becoming increasingly distorted by the large resonance surrounding the ω = 01 elliptic point. There still are many KAM tori stretching horizontally from x = 0 to x = 1.0 in this map.

3.3 The Standard Map

61

Fig. 3.10 Some orbits of the standard map (with periodic boundary conditions): (a) K = 0.1716354; (b) K = 0.4716354 (Plots by Steve Cocke)

Figure 3.11d shows the standard map at the critical value, K ∗ = 0.9716354, of the parameter K. At this value of K, the last remaining KAM torus, which stretches horizontally from x = 0 to x = 1.0, is broken. This torus is easily seen in Fig. 3.11d. For values of K ≤ 0.9716354, it is not possible for any trajectories to diffuse vertically through this line and reach arbitrarily large values of p. We see that all of the separatrices shown appear to be chaotic. In Figs. 3.12e and f, we show the structure of the standard map for K = 1.1716354 and K = 3.9716254, respectively. For K = 1.1716354, the primary resonances are still intact, although they are surrounded by a chaotic sea. For K = 3.9716354, almost all structure is wiped out.

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3 Area-Preserving Maps

Fig. 3.11 Continuation of Fig. 3.10: (c) K = 0.7716354; (d) K = K ∗ = 0.9716354 (Plots by Steve Cocke)

3.4 Scaling Behavior of Noble KAM Tori The mechanism by which a KAM torus is destroyed and converted to a cantorus is universal to all area-preserving twist maps. The details do not depend on the particular map being studied. Each KAM torus has an irrational winding number. For this reason it is impossible to locate a given KAM torus exactly numerically. However, Greene (1979b) has shown that it is possible to get as close as we like to a KAM torus by computing periodic orbits whose winding numbers are rational approximates to the irrational winding number of the KAM tori. These rational approximates also provide a means to study self-similarity and scaling behavior in the neighborhood of certain KAM tori. In this section, we first introduce the rational approximates and then discuss scaling behavior in area-preserving twist maps. Every irrational number can be approximated by a unique sequence of fractions, given by a continued fraction, that converges to the irrational number. Therefore, we can represent every winding number in terms of a unique continued fraction,

3.4 Scaling Behavior of Noble KAM Tori

63

Fig. 3.12 Continuation of Figs. 3.10 and 3.11: (e) K = 1.1716354; (f) K = 2.9716354 (Plots by Steve Cocke)

1

ω ≡ [a0 , a1 , a2 , . . .] = a0 +

1

a1 + a2 +

,

(3.25)

1 a3 + . . .

with ai an integer and ai ≥ 1 for i ≥ 1. Each rational and irrational number may be represented uniquely by a sequence [a0 , a1 , . . .]. For irrational numbers, the sequence will contain an infinite number of entries. The rational approximates to a given continued fraction are obtained by terminating the sequence by letting ai = N1 ∞. Thus, for a given sequence, ω = [a0 , a1 , a2 , . . .], the rational approximates, M , 1 N2 N3 M2 , M3 ,

etc. are given by Ni = [a0 , a1 , . . . , ai , ∞]. Mi

(3.26)

64

3 Area-Preserving Maps

These are the best rational approximates to ω in the sense of a Diophantine approximation. That is,         ω − N  > ω − Ni     M Mi  N for all other M with M < Mi+1 (Hardy and Wright 1979). Finite-length continued fractions are unique up to an ambiguity in the last partial quotient, [a0 , a1 , . . . , aN ] = [a0 , a1 , . . . , aN −1 , 1]. For winding numbers 0 ≤ ω ≤ 1, we must have a0 = 0 and ai ≥ 1. The most irrational number is given by the sequence ai = 1 for all i. That is,

[1, 1, 1, . . . , 1, . . .] ≡ γ =

(1 +

√ (5)) . 2

(3.27)

The number γ is called the golden mean and is considered to be the most irrational number because it is hardest to approximate by rationals (Prasad 1948; MacKay 1983a). In fact, all the sequences ending in a series of 1’s are the slowest-converging sequences and represent those irrational numbers that are the hardest to approximate by rational numbers. This has important consequences for dynamics. We know that KAM tori are destroyed by resonances between degrees of freedom whose periods are rationally related. Thus, each rational approximate will be associated to a resonance region in the phase space and a corresponding island chain. As the parameter K increases, these resonance regions grow and finally destroy their neighboring KAM tori. However, those KAM tori whose winding numbers are approximated by sequences of the form ω = [a0 , a1 , . . . , ai , 1, 1, . . . , 1, . . .],

(3.28)

where all entries after ai are 1’s, will be the most irrational and the hardest to approximate by rational numbers. KAM tori with winding numbers of the type given in Eq. (3.28) are called noble KAM tori. In the standard map shown in Figs. 3.10, 3.11, and 3.12, the last KAM tori to be destroyed are those with winding numbers ω = [0, 1, 1, 1, . . . , 1, . . .] =

1 ≈ 0.618034 γ

(3.29)

and ω = [0, 2, 1, 1, . . . , 1, . . .] =

 2 1 ≈ 0.381966. γ

(3.30)

These noble KAM tori are easily seen in Fig. 3.11d. The rational approximates can also be seen in these maps. Let us consider the noble KAM torus with winding

3.5 Renormalization in Twist Maps

65

Fig. 3.13 The first five rational approximates to the golden mean KAM torus for (a) K ≈ 0 and (b) K = K ∗ = 0.9716354. The symmetry lines for the standard map have been added. The o’s denote elliptic Mi -cycles and the x’s denote hyperbolic Mi -cycles. We see that there are always two elliptic and two hyperbolic points from each Mi -cycle on the symmetry lines (one per line). The symmetry line x = 0 contains one elliptic point from each rational approximate to the golden mean KAM torus

number ω = [0, 1, 1, 1, . . . , 1, . . .]. Its rational approximates are ω0 = [0, ∞] ≡ 0 1 1 2 1 , ω1 = [0, 1, ∞] = 1 , ω2 = [0, 1, 1, ∞] = 2 , ω3 = [0, 1, 1, 1, ∞] = 3 , ω4 = [0, 1, 1, 1, 1, ∞] = 35 , ω5 = [0, 1, 1, 1, 1, 1, ∞] = 58 , etc. It is interesting to note that these rational approximates to the inverse golden mean are ratios of the Fibonacci numbers, Fi . That is, ωi = FFi−1 and limi→∞ FFi−1 = γ1 . The Fibonacci i i numbers have the important property that they can be generated from the equation Fi = Fi−1 +Fi−2 with F0 = 1 and F1 = 1. Note also that Fi = √1 [γ i+1 −γ −(i+1) ]. 5 The island chains corresponding to some of these rational approximates are shown in Fig. 3.13.

3.5 Renormalization in Twist Maps A renormalization theory for the noble KAM tori was developed by MacKay (1982, 1983a). The renormalization mapping constructed by MacKay focuses on the approximates to the noble tori. In the following, we shall examine the inverse golden mean KAM torus and study the self-similarity that occurs in the neighborhood of the dominant symmetry line.

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3 Area-Preserving Maps

3.5.1 Integrable Twist Map The rational approximates to the inverse golden mean KAM torus all have one elliptic periodic orbit on the dominant symmetry line (see Fig. 3.9). Pick one particular elliptic periodic orbit P0M of an Mi -cycle with winding Ni number ωi = M on the dominant symmetry line. Construct a map for which that i periodic orbit is a fixed point. The point P0M can be written  P0M =

p0 x0



 =

 ωi + vi (t0 ) . t0 + ui (t0 )

(3.31)

Ni . where vi and ui are periodic with period one and ωi = M i Let G denote a twist map that has an inverse golden mean KAM torus. (The standard map is just one example of many such maps.) Then, under the mapping GMi , the point P0M becomes

 GMi

p0 x0



 =

 ωi + vi (t0 + Mi ωi ) . t0 + Mi ωi + ui (t0 + Mi ωi )

(3.32)

In order to return to our original point, we must introduce the mapping R, defined as 

p R 0 x0



 =

 p0 , x0 − 1

(3.33)

which commutes with G. Then let the map R act Ni times to get  Mi

G

R

Ni

p0 x0





   ωi + vi (t0 + Mi ωi ) ωi + vi (t0 ) = = (3.34) t0 + Mi ωi − Ni + ui (t0 + Mi ωi ) t0 + ui (t0 )

Ni since vi and ui are periodic with period one and ωi = M . Thus, the mapping i M N i i G R maps a point on the Mi -cycle onto itself and maps a neighborhood of that point back to itself. Are the neighborhoods of the elliptic fixed points on the dominant symmetry line (the dominant elliptic fixed points) self-similar? If we rescale each neighborhood, do we get exactly the same picture back? In order to determine how to rescale the neighborhoods of the dominant elliptic fixed points, begin the process with an integrable twist map. Consider the standard map for K = 0, which can be written pn+1 = pn and xn+1 = xn + pn+1 , with boundary condition, 0≤x≤1 mod(1). The Mi -cycle with winding number ωi = Fi−1 Ni Mi = Fi has a dominant elliptic fixed point with coordinates pi = ωi and xi = 0. The distance along the p-axis, pi , between the fixed points at p = ωi and p = ωi+1 is pi = ωi+1 − ωi . The distance along the x-axis between the dominant elliptic fixed point at x = 0 and its closest neighboring elliptic fixed point belonging

3.5 Renormalization in Twist Maps

67

to the same Mi -cycle is xi = ± F1i . More generally, for a fixed point with ωi = Fi−1 Fi ,

pi = ωi+1 − ωi and xi = (−1)i F1i . Let us now move the origin of coordinates (p, x) to the position of the dominant elliptic fixed point with winding number ωi . Then its two nearest neighbors lie at (p = ωi+1 −ωi , x = 0) and (p = 0, x = (−1)i F1i ). We next rescale the coordinates so that the nearest neighbors lie at (p = 1, x = 0) and (p = 0, x = 1). This can be done by magnifying the neighborhood of the fixed point at the origin by a sequence of mappings,     ωi+1 − ωi 1 = B0 B1 . . . Bi−1 , (3.35) (−1)i F1i 1 where  Bi−1 =

   i−1  −  W Wi  0

0



i − FFi−1

(3.36)

and Wi = ωi − ωi+1 . With this scale change, the neighborhood of each dominant elliptic fixed point gets mapped onto the unit square. Note that in the limit i → ∞,   −γ 2 0 Bi−1 → , (3.37) 0 −γ and the scale change becomes independent of i. The mapping that reproduces the neighborhood of the ith elliptic fixed point on the unit square is −1 . . . B0−1 . G R  = B0 . . . Bi−1 GMi R Ni Bi−1

(3.38)

Let us now consider one step in the renormalization transformation from GMi R NI to GMi−1 R Ni−1 , −1 . GMi−1 R Ni−1 = Bi−1 GMi R Ni Bi−1

(3.39)

If we use the relation F i+1 = F i−1 + F i for the Fibonacci numbers, we can write Eq. (3.39) in the form −1 . GMi−1 R Ni−1 = Bi−1 GMi−1 R Ni−1 GMi−2 R Ni−2 Bi−1

(3.40)

−1 We now define Ti−1 = GMi−1 R Ni−1 and Ui−1 = GMi−2 R Ni−2 = Bi−2 Ti−1 Bi−2 . Then Eq. (3.40) can be written in the form −1 Ti−1 = Bi−1 Ti−1 Ui−1 Bi−1

(3.41)

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3 Area-Preserving Maps

and −1 . Ui−1 = Bi−2 Ti−1 Bi−2

(3.42)

If we next note that in the limit i → ∞, Bi → B ∗ , where B is defined in Eq. (3.37), then in the limit i → ∞, Eqs. (3.41) and (3.42) take the form T ∗ = B ∗ T ∗ U ∗ B ∗−1 , ∗

∗

∗

U =B T B

∗−1

(3.43) (3.44)

.

Thus, in the limit i → ∞, the mapping equations (3.41) and (3.42), approach a fixed point given by the mappings T ∗ and U ∗ , which are solutions of Eqs. (3.43) and (3.44). The fixed point mappings T ∗ and U ∗ are given by T

∗

U

∗



p0 x0



 =

p0 x0 + p0 + 1

 (3.45)

and 

p0 x0



 =



p0 x0 +

p0 γ

−γ

(3.46)

with B∗



p0 x0



(as defined in Eq. (3.37)) (the relations mapping T ∗ U ∗ gives T ∗U ∗



p0 x0



 =

 = 1 γ

−γ 2 p0 −γ x0

 (3.47)

= γ − 1 and γ 2 = γ + 1 are useful). The

 p0 , x0 + (ω + 1)p0 − ω

(3.48)

where ω = γ1 . Thus, the point at the origin, (p0 = 0, x0 = 0), lies on the golden mean torus, and points in the neighborhood of the origin undergo a twist. The discussion above of a simple integrable twist map shows that a renormalization transformation of the mappings for the dominant elliptic fixed points converges to a fixed point that corresponds to the inverse golden mean KAM torus. Thus, we have shown that for an integrable mapping, the Mi -cycles that approximate the inverse golden mean KAM torus do indeed converge to it in the limit i → ∞.

3.5 Renormalization in Twist Maps

69

3.5.2 Nonintegrable Twist Map The procedure described in Sect. 3.5.1 can be extended to nonintegrable maps such as the standard map for K =0. Shenker and Kadanoff (1982) showed that the rational approximates exhibit self-similarity when K = K ∗ = 0.9716354. MacKay (1982, 1983a) showed that a set of mappings like those of Eqs. (3.41) and (3.42) could be constructed for K =0 but with different spatial scaling, Bi . MacKay also found that the fixed point Eqs. (3.43) and (3.44) appear to have another solution, which he called the critical fixed point, which determines the behavior of the neighborhood of the inverse golden mean torus for the critical parameter K = K ∗ . MacKay (1982) determined the critical parameter, K ∗ , by finding the parameter K = Ki∗ at which the ith dominant elliptic fixed point becomes unstable. In the limit i → ∞, he finds lim Ki∗ → K ∗ = 0.9716354,

i→∞

(3.49)

while the rate of approach to the critical value is given by  δ = lim

i→∞

∗ Ki∗ − Ki−1

 ≈ 1.6280 . . . .

∗ − K∗ Ki+1 i

(3.50)

Generally Ki∗ > K ∗ . Thus, δ is a scaling parameter that characterizes the rate of approach of the parameter K to its critical value, K ∗ . MacKay also determined the position, pi , of the ith dominant elliptic fixed point when K = Ki∗ . This gives the scaling parameter, β, along the p-axis, where    pi+1 − pi    = −3.0668882. β = − lim  i→∞ pi − pi−1 

(3.51)

The scaling parameter, α, along the x-axis can be found by finding the position, xi , of the elliptic point (in the same Mi -cycle) nearest the dominant symmetry line and then taking the limit    xi−1    = −1.4148360. α = − lim  i→∞ x 

(3.52)

i

Thus, for K = K ∗ , the scaling matrix, Bi , appears to take the limiting value 

∗

B = lim Bi = i→∞

β 0 0α

 .

(3.53)

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3 Area-Preserving Maps

3.5.3 The Universal Map The renormalization procedure for the twist map shown in Eqs. (3.41) and (3.42) appears to hold when K > 0, and the fixed point Eqs. (3.43) and (3.44), for K > 0, appear to have another solution. As a result, the mapping given by Eq. (3.38) is dependent on K and approaches a limiting value, Fμ∗ , where −1 . . . B0−1 . Fμ∗ = lim B0 . . . Bi−1 FμMδi −i R Ni Bi−1 i→∞

i

(3.54)

The map Fμ∗ is called the universal map. In Eq. (3.54), δ is given by Eq. (3.50) and μi measures the value of Ki at which the ith dominant elliptic fixed point becomes unstable (μ = 1 is critical). The universal map exists in a small neighborhood of all twist maps that have an inverse golden mean KAM torus and a dominant symmetry line, and, more generally, in small neighborhoods of all noble KAM tori; that is, tori whose winding number is represented by continued fractions of the form ω = [b1 , . . . , bn , 1, 1, . . . , 1]. MacKay was actually able to plot the orbits of the universal map, and some of them are shown in Fig. 3.14 for the parameter value at which the KAM torus is critical. In Fig. 3.14, the dominant symmetry line lies at Y = 0 and the critical KAM torus crosses the dominant symmetry line at X = 0. Everything in Fig. 3.14 repeats itself in the small box but on a smaller scale and reflected about X = 0. The scaling properties of KAM tori appear to depend only on the winding number of each torus. The universal map gives a very clear and graphic picture of the selfsimilarity and scaling behavior that exists in a small neighborhood of all noble KAM tori, which are KAM tori whose winding number is given by a continued fraction with a homogeneous tail consisting of ones. However, most KAM tori have Fig. 3.14 Some orbits of the universal map. Note that the small box repeats the whole map if we reflect it about the x = 0 axis and magnify it by a factor 3.067 in the X-direction and by 1.415 in the Y -direction (MacKay 1983a)

3.6 Bifurcation of M-Cycles

71

winding numbers whose continued fractions have random entries. Thus, scaling in the neighborhood of these tori fluctuates as one goes from one level to another. This has been called “renormalization chaos” by Chirikov and is discussed in (Chirikov and Shepelyansky 1986).

3.6 Bifurcation of M-Cycles We have seen from the Walker-Ford models (Sect. 2.5) that stable (elliptic) periodic orbits can suddenly appear in the phase space as parameter values of the system (in that case, the coupling constant) are varied. Existing periodic orbits may also bifurcate and give rise to additional periodic orbits with periods that are some integer multiple of the original period. A period-doubling bifurcation can be seen in the strobe plots of the Henon-Heiles system in Fig. 2.3. In Fig. 2.3c, the elliptic periodic orbit that is located at (p2 = 0, q2 ≈ 0.3) in Fig. 2.3b has bifurcated into a hyperbolic periodic orbit, and two elliptic periodic orbits have been created along the p2 = 0 axis.

3.6.1 Some General Properties There are a number of general statements that can be made about when a periodic orbit can be created or destroyed or might collide with another periodic orbit. Most of this can be determined from properties of the tangent map. Let us consider a twist map, TK , and let us assume that it has a fixed point, x0 . That is, x0 = TK (x0 ). We can make the following statements about the neighborhood of x0 . 1. This fixed point will be isolated (will have no other fixed points in its neighborhood) if the tangent map, ∇TK (x0 ), has no eigenvalues λ = 1. To see this, let F (x) = x − TK (x) so that F (x0 ) = x0 − TK (x0 ) = 0. In the neighborhood of x0 , the tangent maps ∇F (x0 ) and ∇TK (x0 ) satisfy the equation ∇F (x0 ) = 1 − ∇TK (x0 ). If ∇F (x0 ) =0, then ∇TK (x0 ) has no eigenvalue, λ = 1. Also, if ∇F (x0 ) =0, the slope of F (x) at x0 is finite and there can be no other zeros of F (x) in the neighborhood of x0 . 2. The fixed point, x0 , is stable to small perturbations as long as ∇TK (x0 ) has no eigenvalues λ = 1. Consider a small perturbation, δT (x), to the map TK (x). Assume the fixed point of the map, TK + δT , lies at x = x0 + δx. Then x = TK (x) + δT (x). If we expand to first order in δx and δT , we find δx = (1 − ∇TK (x0 ))−1 δT (x0 ). Thus, if (1 − ∇T (x0 )) has no zero eigenvalues, the fixed point is merely shifted in position under the perturbation but is not destroyed.

72

3 Area-Preserving Maps

From the above, we conclude that a periodic orbit, x0 , can only be created or destroyed or can collide with another periodic orbit of the same period when the tangent map, ∇TK (x), has an eigenvalue λ = 1. We can generalize this statement to include fixed points of the mapping, TKM (x), for M integer. Fixed points of TKM (x) are isolated unless the tangent map, ∇TKM (x), has an eigenvalue λ = 1 or unless ∇TK (x) has an eigenvalue λ such that λM = 1. If λM = 1, then fixed points of TKM (x) can be created or destroyed or can collide with another fixed point. The case M = 2 corresponds to period-doubling bifurcations. Bifurcations also occur for M > 2 but are not as important as the period-doubling bifurcations because perioddoubling bifurcations are the last to occur before a stable periodic orbit completely loses stability and renders the region of phase space in its neighborhood totally chaotic (MacKay 1983b). We shall only consider period-doubling bifurcations here, but discussion of other types that can occur may be found in Meyer (1970); Collet et al. (1981); Greene et al. (1981).

3.6.2 The Quadratic Map It is interesting to illustrate some of these ideas for the quadratic map, Qa , defined as 

yn+1 xn+1



 = Qa

yn xn



 =

 xn , 1 − yn − axn2

(3.55)

where a is the parameter of the map. Quadratic maps are especially important because they approximate the behavior of the neighborhood of stable islands in the chaotic sea of more general conservative maps. Quadratic maps generally consist of one stable island surrounded by a chaotic sea. The quadratic map in Eq. (3.55) is reversible and can be written as the product of two involutions, Qa = S2 S1 , where S12 = S22 = I and I is the identity map. The involutions S1 and S2 are defined as 

yn+1 xn+1



 = S1

yn xn



 =

xn yn

 (3.56)

and 

yn+1 xn+1



 = S2

yn xn



 =

 yn . 1 − xn − ayn2

(3.57)

The symmetry lines for this map are the lines of fixed points of the maps S1 and S2 and are given by x = y and 2x − 1 + ay 2 = 0, respectively. Thus, the orbits of the quadratic map are symmetric about an axis making an angle of 45o with the x-axis (and the y-axis).

3.6 Bifurcation of M-Cycles

73

As was shown by Bountis (1981), the first few period-doubling bifurcations can becomputed analytically. The period 1 fixed points of this map are given by  (1)  y y (1) = Qa (1) . It is easy to show that this equation has two solutions x (1) x (1) (1) (1) (1) (we write them as row vectors for convenience), (x+ , y+ ) and (x− , y− ), where √ (1) (1) 1 x± = y± = a (−1 ± 1 + a). The stability of each of these fixed points can be determined from the tangent map  ∇Q± =

 0 1 (1) , −1 −2ax±

(3.58)

(1) which is obtained by linearizing Qa about the fixed points x± . As discussed in Sect. 3.2, the fixed point will be elliptic if |Tr(∇Q± )| < 2. We find that the (1) (1) (1) (1) fixed point (x− , y− ) is always hyperbolic, while the fixed point (x+ , y+ ) is elliptic for −1 < a < 3 and hyperbolic otherwise. When a = 3, ∇Q+ has the doubly degenerate eigenvalue λ = −1. Thus, we expect that the tangent map for the mapping Q2a will have a doubly degenerate eigenvalue λ = 1, and a perioddoubling bifurcation can occur. This bifurcation is shown in Fig. 3.15, where the (1) (1) neighborhood of the fixed point (x+ , y+ ) is shown for a = 0.95, a = 2.98, and a = 3.02. The mapping Q2a is defined as



yn+2 xn+2



 = Q2a

yn xn



 =

 1 − yn − axn2 . 1 − xn − a(1 − yn − axn2 )2 (2)

(2)

(2)

(3.59) (2)

This mapping has fixed points at (x1 = A+ , y1 = A− ), (x2 = A− , y2 = (2) (2) (2) (2) A+ ),(x3 = −B+ , y3 = −B+ ), and (x4 = −B− , y4 = −B− ), where A± = √ √ (2) (2) 1 1 a (1± a − 3) and B± = a (1± a + 1). For a < 3, the fixed points (x1 , y1 ) and (2) (2) (2) (2) (2) (2) (x2 , y2 ) don’t exist, and the fixed points (x3 , y3 ) and (x4 , y4 ) correspond to the hyperbolic and elliptic fixed points of Qa . However, for a ≥ 3, all four fixed points of Q2a exist and the original orbit has bifurcated and period-doubled. It is easy to check that when a = 3, the tangent map for Q2a has a doubly degenerate eigenvalue λ = +1. Bountis found that the symmetry curve plays a dominant role in determining the location of the orbits in the period-doubling sequence. If he let (ym , xm ) denote the location of the mth member of 2k periodic points obtained after k period-doublings of the original orbit, then he found that the orbits m = 1 and m = 2k−1 will lie on the symmetry line 2x − 1 + ay 2 = 0. He called this symmetry line the “symmetry road” of the period-doubling sequence.

74

3 Area-Preserving Maps

Fig. 3.15 Period-doubling of the period 1 elliptic fixed point of the quadratic map: (a) a = 0.95, (b) a = 2.98, (c) a = 3.02 (Bountis 1981)

3.6.3 Scaling in the Quadratic de Vogelaere Map It is somewhat easier to follow the behavior of the period-doubling bifurcation for the de Vogelaere (1958) form of the quadratic map, Vp , which is defined as

3.6 Bifurcation of M-Cycles



yn+1 xn+1

75



 = Vp

yn xn



 =

 xn − fp (xn+1 ) , fp (xn ) − yn

(3.60)

where fp (x) = px − (1 − p)x 2 . Properties of the quadratic de Vogelaere map have been studied extensively in MacKay (1982, 1983b); Greene et al. (1986). It can be written as a product, Vp = S2 S1 , of two involutions, S1 and S2 , defined as  S1

yn xn



 =

−yn xn

 (3.61)

and  S2

yn xn



 =

 xn − fp (xn+1 ) . fp (xn ) + yn

(3.62)

The line of symmetry for S1 is y = 0, while for S2 it is y = x − fp (x). The map Vp has a fixed point at x = y = 0. The tangent map, ∇V , for this fixed point is  ∇V =

 p 1 − p2 . −1 p

(3.63)

Thus we expect a period-doubling bifurcation when p = −1. The first two perioddoublings of this fixed point are shown in Figs. 3.16, 3.17 and 3.18. The “symmetry road” for this period-doubling sequence is the line y = 0. Thus it is particularly easy to study its properties.

3.6.3.1

Scaling Behavior of the Bifurcation Sequence

As we can see in Fig. 3.16, the elliptic fixed point at (y = 0, x = 0) loses stability when p = p (1) = −1.0 and gives birth to a stable period 2 orbit. At p = p(2) = −1.234067977, the period 2 orbit loses stability and gives birth to a stable period 4 orbit (see Figs. 3.17 and 3.18). This period-doubling sequence continues as p decreases, giving rise, after the kth bifurcation at p = p(k) , to a period 2k orbit. As k → ∞, the parameter values, p(k) , accumulate to the value lim p(k) = p∗ = −1.26631128

k→∞

and approach this value at a rate  δ = lim

k→∞

p(k) − p(k−1) p(k+1) − p(k)

 = 8.7210972.

76

3 Area-Preserving Maps

Fig. 3.16 Period-doubling bifurcation of the period 1 fixed point of the quadratic de Vogelaere map: (a) p = −0.98, (b) p = −1.02, (c) p = −1.05

There will always be two members of a period 2k orbit on the symmetry line y = 0. When they bifurcate, the period-doubled daughters of one of them will move off the symmetry line (the “bad” point) as we decrease p, while the daughters of the other (the “good” point) will remain on the symmetry line. The position, x = x (k) , of the good point when it bifurcates also accumulates to a fixed value

3.6 Bifurcation of M-Cycles

77

Fig. 3.17 The neighborhood of the period 2 orbits of the quadratic de Vogelaere map just before they period-double (p = −1.234) into four period 4 orbits. (a) Both the good (left) and bad (right) points on the dominant symmetry line. (b) Magnification of the neighborhood of the good point. (c) Magnification of the neighborhood of the bad point

lim x (k) = x ∗ = −0.23600609

k→∞

and does it with a convergence rate  α˜ = lim

k→∞

x (k) − x (k−1) x (k+1) − x (k)

 = −4.018076704.

78

3 Area-Preserving Maps

Fig. 3.18 The neighborhood of the period 2 orbits of the quadratic de Vogelaere map just after they period-double (p = −1.247) into four period 4 orbits. (a) The neighborhood of all four period 4 points. (b) Magnification of the neighborhood of the good and bad period 4 points (daughters of the period 2 good point). (c) Magnification of daughters of the period 2 bad point

The actual value of p(k) and x (k) at the kth bifurcation for the first eight bifurcations are shown in Table 3.1. In Fig. 3.19, we plot x (k) versus p(k) . Note that the good point alternately lies to the left or right (on the symmetry line) of the bifurcation point where it was born. MacKay also found the convergence rate for 2k -cycles perpendicular to the symmetry line. By measuring the distance off the symmetry line of one of the

3.6 Bifurcation of M-Cycles

79

Table 3.1 Values of p (k) and x (k) for k = 1 to k = 9 (MacKay 1982)

k 1 2 3 4 5 6 7 8 9

2k 2 4 8 16 32 64 128 256 512

p (k) −1.000000000 −1.234067977 −1.262841686 −1.265913483 −1.266265664 −1.266306047 −1.266310677 −1.266311208 −1.266311269

x (k) 0.000000000 −0.276393202 −0.224612022 −0.238675841 −0.235323100 −0.236173934 −0.235964076 −0.236016521 −0.236003494

Fig. 3.19 The bifurcation tree for the period 1 fixed point of the quadratic de Vogelaere map shown as a function of the parameter, p, and position, x. The bifurcations of the good point are shown. The solid line indicates stable regions (elliptic), while the dotted line indicates unstable regions (hyperbolic) (based on data from (MacKay 1982) )

daughters of the bad point, he found  β˜ = lim

k→∞

y (k) − y (k−1) y (k+1) − y (k)

 = 16.363896879.

˜ Thus x (k) converges to x ∗ at the rate α, and y (k) converges to zero at the rate β. This scaling property of the bifurcation sequence indicates that the neighborhood of the good and bad fixed points exhibits self-similarity on an ever smaller scale as k → ∞. Let us consider the neighborhood of the good point just at the kth

80

3 Area-Preserving Maps k

bifurcation, p = p(k) . It will be a fixed point of the mapping, V 2 , and will be k located at (y = 0, x = x (k) ). That is, the mapping, BV 2 B −1 , where  B=

 β˜ 0 , 0 α˜ k−1

in the neighborhood of the good and bad points looks the same as that for V 2 . The scale of the bifurcation process after the kth bifurcation has shrunk by 1/β˜ in the y direction and by 1/α˜ in the x direction. The bifurcation tree for the first four bifurcations of the good point is shown in Fig. 3.19. The bifurcation points and the stability are plotted as a function of parameter, p, and position, x, for the quadratic de Vogelaere map.

3.7 Cantori A KAM torus is destroyed when the resonance zones associated with neighboring periodic orbits begin to squeeze holes in it. The periodic orbits that play the dominant role in destroying a given KAM torus are the Mi -cycles that approximate it (see Fig. 3.20). When a KAM torus is destroyed, it is transformed from a continuous barrier to a barrier that is a Cantor set. It is then called a cantorus. The suggestion that KAM tori form a Cantor set when they are destroyed was made by Percival (1979) and Aubry (1978). A general proof of the existence of cantori has been given in Katok (1982); Mather (1982); Aubry and LeDaeron (1983). The change of a KAM torus into a cantorus occurs at some critical parameter of the mapping (for the inverse golden mean KAM torus in the standard map, the critical parameter is K ∗ = 0.9716354). The cantorus may be thought of as a torus with an infinite number of deleted gaps caused by the overlapping of nearby island chains. Once the cantorus has formed, diffusion of phase space trajectories can occur across the cantorus by means of leakage through the holes. When the mapping parameter is only slightly greater than the critical parameter, the leakage is very slow and the cantorus still serves as a substantial barrier to large-scale diffusion. In the standard map, for K > K ∗ = 0.9716354, the noble cantori (cantori formed from noble KAM tori) can still form a partial barrier to diffusion. This is clearly Fig. 3.20 A KAM torus becomes a cantorus when the resonance zones associated with its rational approximates overlap

3.7 Cantori

81

Fig. 3.21 Some orbits of the standard map for parameter K = 1.121635. Four orbits were started in the upper box and stopped when they first reached the lower box. The black dots mark orbits that are homoclinic to cantori with winding numbers ω = γ12 (1+γ ) (above) and ω = (4+3γ ) (below). These cantori act to partially impede the flow of trajectories in the vertical direction (MacKay et al. 1984)

seen in Fig. 3.21, which shows diffusion in the standard map for K = 1.121635. Although diffusion occurs across the cantorus with winding number w = γ12 , it clearly blocks the free flow of trajectories. The flux across a cantorus can be obtained as the limiting case of the flux across the rational approximates to that cantorus. It is possible to determine the flux across the rational approximates because it can be expressed in terms of an action principle involving the elliptic and hyperbolic fixed points of the Mi -cycles (Bensimon and Kadanoff 1984; MacKay et al. 1984). Let us consider the rational approximate, w = 01 , to the inverse golden mean KAM torus in the standard map, and let us draw a line through all the periodic points of this cycle. The line through the periodic points is (p0 = 0, x0 = t), where (0 ≤ t ≤ 1). After one iteration of the map, this line gets K K mapped to the line (p1 = − 2π sin(2π t), x1 = t − 2π sin(2π t)). The shaded areas in Fig. 3.22 include all points initially below the line p = 0 that get mapped above it after one iteration. Since this is an area-preserving map, an equal area gets mapped below the line p = 0. But if we are interested in the flow of particular trajectories, they need never get mapped back below the original line. Thus the Mi -cycles act as pumps to move phase space trajectories from one part of the map to another. The effectiveness of the Mi -cycles increases as we go to larger K, as shown in Fig. 3.22 for the rational approximate w = 01 . The size of the shaded areas in Fig. 3.22 can be expressed in terms of an action principle. The Lagrangian (see Sect. 3.2.2) for the standard map can be written

82

3 Area-Preserving Maps

Fig. 3.22 The shaded regions show the area pumped across the line x = 0 in the positive p direction by the rational approximate with winding number ω = 01 after one iteration of the standard map: (a) K = 0.0, (b) K = 0.47, (c) K = 0.97, (d) K = 1.97

F (xn , xn+1 ) =

1 K (xn − xn+1 )2 + cos(2π xn ). 2 (2π )2

(3.64)

If we let pn+1 = xn+1 − xn , it is easy to see that the standard map can be written in the Newtonian form as xn+1 + xn−1 − 2xn = −

K sin(2π xn ) 2π

(3.65)

(Percival 1979). Equation (3.65) can be obtained by extremizing Wn−1,n = F (xn−1 , xn ) + F (xn , xn+1 ) with respect to the intermediate variable, xn (see Sect. 3.2). Let us now compute the area (or action) pumped across our original line in Fig. 3.22a. The shaded area is  Ashaded =

1 1 2

p1 (t)

dx1 dt − dt



1 1 2

p0 (t)

dx0 dt. dt

(3.66)

3.7 Cantori

83

But from Eq. (3.8) we can write  Ashaded =

1 1 2

∂F dx1 dt + ∂x1 dt



1 1 2

1 1 ∂F dx0 dt = F (1, 1) − F ( , ). ∂x0 dt 2 2 (3.67)

K Thus, from Eq. (3.64) the shaded area is Ashaded = 2π 2 . The growth of Ashaded with increasing K can be seen in Fig. 3.22. Ni The area pumped across an Mi -cycle (with winding number ωi = M ) in one i direction is given by

W Ni = Mi

Mi 

e e h h [F (x0(j ) , x1(j ) ) − F (x0(j ) , x1(j ) )],

(3.68)

j =1

h (x e ) is the initial position of the j th hyperbolic (elliptic) fixed point of where x0(j ) 0(j ) h (x e ) is its position after one iteration of the map. Structures the Mi -cycle, and x1(j ) 1(j ) of the type shown in Fig. 3.22, which allow phase space area to be pumped across a line in phase space, have been called turnstiles because they behave like revolving doors that allow a two-way flow of traffic. Equation (3.68) is independent of the original path we took through the fixed points. It depends only on the fact that the initial line and the line obtained after one iteration of the map cross at the fixed points. The area, W Ni , has been computed Mi

for a number of different Mi -cycles by MacKay (1982) for the universal map. Some of his results are shown in Fig. 3.23, which shows plots of W Ni as a tree in the Mi

neighborhood of the critical noble KAM torus. The points on the tree are arranged using the fact that [a0 , . . . , am+1 ] = [a0 , . . . , am , 1] and adding one to each side of this equality. Note that W Ni for nonnoble tori rapidly converges to a finite value, Mi

indicating that a cantorus is present, whereas W Ni for the noble KAM tori goes Mi

to zero, indicating that no flux is present. The difference between the actions for the stable and unstable fixed points for the rational approximates to the noble KAM torus goes to zero. The scaling behavior of the rational approximates implies that the flux associated with those rational approximates also exhibits scaling behavior since the flux is basically an area per unit time pumped across the rational approximate. We will let K = K − K ∗ denote the distance of the parameter K from its critical value and describe the flow of trajectories in terms of the coordinates (p, x). Then, from Eqs. (3.50) and (3.51), we find that if we rescale K, p, and x, so that K  = K δ , p = pβ , and x  = αx , the mapping in terms of the coordinates K  , p , and x  looks exactly the same as that in terms of K, p, and x. This means that the area pumped across a rational approximate scales in a similar manner. Thus, we can write

84

3 Area-Preserving Maps

Fig. 3.23 The flux, W , associated with the periodic orbits with winding number ω = [a0 , . . . , am+1 ] = [a0 , . . . , am , 1] (values of the entries ai are indicated on the figure) plotted as a function of X, the horizontal coordinate in the universal map, when the golden mean is critical. Note that the flux associated with the rational approximates to the golden mean tends to zero as m → ∞, whereas the flux associated with rational approximates for other KAM tori tends to finite values (indicating that they are actually cantori) (based on data from (MacKay 1982))

 W

K δ

 =

1 W (K). αβ

(3.69)

From Eq. (3.69), we can find the dependence of W on K. Let W (K) = A(K)η , where A is a constant and η is an exponent to be determined. Then Eq. (3.69) implies that δ η = αβ or η = lnδ (αβ) = 3.011722.

(3.70)

This scaling behavior of the flux associated with the rational approximates has been used to develop a theory of diffusion of trajectories, in mixed phase spaces, in terms of a self-similar Markov tree.

3.8 Renormalization Map The behavior of the region of phase space between any two neighboring primary resonances is largely determined by those two resonances (as long as resonances outside the region have not overlapped with them). The effects of primary reso-

3.8 Renormalization Map

85

nances outside this region tend to average out. Therefore, in order to analyze the mechanism for the destruction of KAM tori between two primary resonances, it is often sufficient to consider a Hamiltonian composed of only those two resonances, and that is the basis of the renormalization theory we describe below. Escande and Doveil (1981); Escande (1982, 1985) built a renormalization theory based on a system with two primary resonances. The Hamiltonian for the two resonance system, which they call the paradigm Hamiltonian, has the form H (α) =

pα2 − Uα(0) cos(xα ) − Uα(1) cos[να (xα − tα )], 2

(3.71)

and depends on only three parameters: the amplitudes of two cosine waves, Uα(0) (1) Nα and Uα , and the relative wave number of the two waves να = M (Mα and Nα are α relatively prime integers). The index, α, is the iteration step of the renormalization map. This Hamiltonian has period 2π Mα . The relative wave number, να , is the relative number of oscillations of the two cosine waves during this period. The (0) resonance that results from the cosine wave with amplitude Uα has speed x˙α = 0 (0)

and half-width pα = 2 Uα , while the resonance that results from the cosine  wave with amplitude Uα(1) has speed x˙α = 1 and half-width pα = 2 Uα(1) . From ∂H the resonance condition x˙α = ∂p ≈ pα , these resonances are located at pα = 0 α and pα = 1, respectively. It is possible to use the Chirikov overlap criterion to obtain a “back of the envelope” estimate of the parameter values at which two neighboring resonances overlap. This criterion works fairly well as long as the resonances have pendulumlike structure in the surface of section. The Chirikov overlap criterion says that overlap occurs when the separatrices of the two resonance structures touch. When this happens, the last KAM torus between the two resonances is destroyed. Thus, overlap occurs when   (0) (1) S = 2 Uα + 2 Uα = 1.

(3.72)

Strobe plots obtained from a numerical solution of Hamilton’s equations for the (0) system described by the Hamiltonian in Eq. (3.71), for the case να = 1 and Uα = (1) Uα , are shown in Figs. 3.24 and 3.25. In Fig. 3.24a, we are well below the Chirikov estimate for overlap and indeed the primary resonances are well-separated. Secondary islands and KAM tori are clearly shown in this plot. In Fig. 3.24b, we are still well below the Chirikov estimate for overlap. However, the last KAM torus has been destroyed, and overlap has clearly occurred in this plot. There is a chaotic trajectory that extends from the neighborhood of one resonance to the neighborhood of the other. However, there is still a lot of structure in the chaotic sea. In Fig. 3.25, we are at the value of S where the simple Chirikov overlap criterion predicts overlap. However, there is already a

86

3 Area-Preserving Maps

Fig. 3.24 Strobe plots of the phase space for the paradigm Hamiltonian for the case (0) (1) να = 1, Uα = Uα : (a) S = 0.48; (b) S = 0.72 (Plots by Chiu Liu)

well-developed chaotic sea present in this plot. The simple Chirikov criterion does not take into account the existence of the secondary islands and their contribution. In fact, there is a whole self-similar hierarchy of resonances that contribute to the destruction of the last KAM torus, as we have seen in previous sections. The renormalization procedure developed by Escande and Doveil is able to account for them.

3.8 Renormalization Map

87

Fig. 3.25 Continuation of Fig. 3.24: S = 0.96 (Plot by Chiu Liu)

3.8.1 Expression for the Renormalization Map To obtain the renormalization map, we will assume that Uα(1) < Uα(0) and perform a canonical transformation to the action-angle variables (Jα , θα ) that describe the region outside the separatrix of the dominant term in the paradigm Hamiltonian. We obtain H0 =

pα2 − Uα(0) cos(xα ) = E0 (Jα ) 2

(3.73)

since this is also the Hamiltonian for the pendulum. From App. B, the action variable Jα is Jα =

 (0) 4 Uα π κα

 K(κα ) and xα (Jα , θα ) = 2 am

 K(κα )θα , κα , π

(3.74)

where K(κα ) is the complete elliptic integral of the first kind, am is the Jacobi elliptic amplitude function. and the modulus, κα , is defined as (0)

κα2 =

2Uα

E0 (Jα ) + Uα(0)

.

(3.75)

88

3 Area-Preserving Maps

In terms of these action-angle variables, the Hamiltonian in Eq. (3.71) takes the form H (α) = E0 (Jα ) − Uα(1) cos[να (xα (Jα , θα ) − tα )] =

E0 (Jα ) − Uα(1)

∞ 

Vnα (Jα ) cos[(να + nα )θα − να tα ],

(3.76)

nα =−∞

for να = is

Nα Mα

(Nα and Mα are relatively prime integers) and the amplitude Vnα (Jα )

1 Vnα (Jα ) = 2π



Mα π −Mα π

dθ cos[να xα (Jα , θα ) − (να + nα )θ ]

(3.77)

In Eq. (3.76) we have obtained an exact expression for the Hamiltonian in Eq. (3.71), but now written in terms of an infinite number of higher order resonances. The next step is to choose two neighboring higher order (secondary) resonances, nα and nα + 1. The Hamiltonian for these secondary resonances can be written H  (α) = E0 (Jα ) − Uα(1) Vnα (Jα ) cos[(να + nα )θα − να tα ] −Uα(1) Vnα +1 (Jα ) cos[(να + nα + 1)θα − να tα ].

(3.78)

The Hamiltonian in Eq. (3.78) can again be written in the form of a paradigm Hamiltonian by a suitable set of canonical transformations, as we show below. (We will consider a special case of this transformation, namely one that leads to the golden mean KAM torus.) Transformation to Paradigm Hamiltonian Introduce a canonical transformation to the rest frame of the largest resonance. Let us assume that the position of the (nα + 1)th resonance is Jα = Incα +1 and is given by the resonance condition     ∂Eo (Jα ) να θ˙α = = , (3.79) ∂Jα να + nα + 1 Jα =I c nα +1

where  π Uα(0) ∂E0 = . ∂Jα κα K(κα )

(3.80)

We can transform to a new set of canonical variables, (p¯ α+1 , x¯α+1 ), whose origin is located at Jα = Incα +1 , by introducing the generating function  F (Jα , x¯α+1 ) = −(x¯α+1 + να tα )

Jα − Incα +1 να + nα + 1

 .

(3.81)

3.8 Renormalization Map

89

Using this generating function, we obtain the following relation between coordinates (p¯ α+1 , x¯α+1 ) and (Jα , θα ): p¯ α+1 = −

∂F = ∂ x¯α+1



Jα − Incα +1

 (3.82)

να + nα + 1

and θα = −

∂F = ∂Jα



 x¯α+1 + να tα . να + nα + 1

(3.83)

In terms of this new set of coordinates, the Hamiltonian can be written ∂F H¯ (α + 1) = H  (α) + = E0 (Jα ) − Uα(1) Vnα +1 (Jα ) cos(x¯α+1 ) ∂tα − Uα(1) Vnα (Jα ) cos[να+1 (x¯α+1 − vα+1 tα )] − να p¯ α+1 ,

(3.84) where Jα = Incα +‘ + (να + nα + 1)p¯ α+1 , να+1 =

να + nα , να + nα + 1

(3.85)

and  vα+1 = −

να να + nα

 (3.86)

.

We now perform the “pendulum approximation” on Eq. (3.84). We expand the energy, E0 , in a Taylor series to second order in p¯ α+1 , and we evaluate the amplitudes, Vnα and Vnα +1 , of the two cosine waves at the center of the resonance created by their respective cosine waves. If we make use of the resonance condition, Eq. (3.79), to eliminate terms linear in p¯ α+1 , we find H¯ (α + 1) = E0 (Incα +1 ) −

2 p¯ α+1

2mα+1

− Uα(1) Vnα +1 (Incα +1 ) cos(x¯α+1 )

−Uα(1) Vnα (Incα ) cos[να (x¯α+1 − vα+1 tα )],

(3.87)

where the “mass”, mα+1 , is defined as   2   ∂ E0  mα+1 = (να + nα + 1)2  ∂Jα2 Jα =I

nα +1

−1    . 

(3.88)

In Eq. (3.88), ∂ 2 E0 π 2 E(κα ) = , ∂Jα2 4(1 − κα2 )(K(κα ))3 where E(κα ) is the complete elliptic integral of the second kind.

(3.89)

90

3 Area-Preserving Maps To finally write the Hamiltonian in the form of a paradigm Hamiltonian, we make the following transformation from variables (p¯ α+1 , x¯α+1 , tα ) to variables (pα+1 , xα+1 , tα+1 ): tα+1 = vα+1 tα ,

pα+1 = 

H (α + 1) = −

p¯ α+1 , mα+1 vα+1

xα+1 = x¯α+1 ,

H¯ (α + 1) − Eo (Incα +1 ) mα+1 (vα+1 )2

(3.90)



(3.91)

.

We can now use the above results to write the Hamiltonian in Eq. (3.78) in the form of a paradigm Hamiltonian 1 2 (0) (1) H (α + 1)= pα+1 +Uα+1 cos(xα+1 )+Uα+1 cos[να+1 (xα+1 −tα+1 )], (3.92) 2 with amplitudes given by  (0) Uα+1

=

(1)

Uα Vnα +1 (Incα +1 ) mα+1 (vα+1 )2



 and

(1) Uα+1

=

(1)

Uα Vnα (Incα ) mα+1 (vα+1 )2

 . (3.93)

where we assume that Vnα < Vnα +1 . The Eqs. (3.85) and (3.93) define a special case of the renormalization map between two successive scales in the phase space. Note that Eq. (3.85) which maps the relative wave number να from one scale to another is independent of Eq. (3.93), which maps the cosine wave amplitudes from one scale to another, although the converse is not true. Although we have rescaled the time and momentum (and hence energy) so that our two resonances have velocities x˙α = 0 and 1 at each level, we have not rescaled the space coordinates. Thus, on each scale the Hamiltonian will appear to oscillate with a different period, 2π Mα , and different relative number of oscillations, να = Nα Mα , over that period. The renormalization map determines when a given KAM torus is “broken” by its rational approximates. Iteration of the renormalization map moves us between different spatial scales in the phase space by mapping between rational approximates of the KAM torus that exist at different spatial scales. Iteration of the map for the relative wave number, Eq. (3.85), determines the sequence of rational approximates and therefore the KAM torus of interest.

3.8.2 Fixed Points of the Renormalization Map The renormalization map is a map in the parameter space, (ν, U, V ), and not in phase space. Our first task is to determine the flow of orbits in parameter space as the map is iterated. The nature of this flow can be largely determined by the nature of

3.8 Renormalization Map

91

the fixed points of the map. We will first find the fixed points of the map, Eq. (3.85), for the relative wave number, and then we will use these results to compute the fixed points of the amplitude map, Eq. (3.93). The map of να between successive scales is given by Eq. (3.85). We shall consider Eq. (3.85) for the special case that gives us the noble KAM tori. We assume that at each level we always take the same pair of resonances. That is, we set nα = n for all scales. Relative Wave Number Fixed Point The mapping for the relative wave number takes the form να+1 =

να + n 1 = 1 να + n + 1 1 + n+ν . α

(3.94)

 1 [−n + (n2 + 4n)], 2

(3.95)

This equation has fixed points at νn =

where n ≥ 1. If we iterate Eq. (3.94), we find that νn can be expressed as a continued fraction νn = [0, 1, n, 1, n, . . .] =

For n = 1 this is just the inverse golden mean, ν1 =

1 1+ 1 γ

=

(3.96)

1 1 n+ 1+...

√

5−1 2

 .

Let us next consider the amplitude map, Eq. (3.93), for the special case in which the rational approximates are obtained by setting nα = n. The fixed point of the relative wave number map is νn . This fixed point defines a plane in the threedimensional space formed by the variables (ν, U (0) , U (1) ). Once we determine this (0) (1) plane, we can then find the fixed points of the map of Uα and Uα in this plane. Amplitude Fixed Points Rewrite the resonance condition, Eq. (3.79), as a condition on the modulus. From Eq. (3.79), we write    π Uα(0) (καc ) να . (3.97) = καc K(καc ) να + nα + 1 To find the fixed points corresponding to the special case nα = n, evaluate να at the fixed point νn and obtain  π Uα(0) (κnc ) κnc K(κnc )

 =

 νn . νn + n + 1

(3.98)

This is the resonance condition evaluated at the fixed point for each iteration of the map. It is convenient to write the renormalization map in terms of the resonance half-widths rather than the amplitudes. Define

92

3 Area-Preserving Maps

Fig. 3.26 The surface ν = νn in the parameter space (ν, X, Y ). All fixed points lie in this plane  (0) Xα = 2 Uα

and

 (1) Yα = 2 Uα .

(3.99)

Now combine Eqs. (3.86), (3.88), (3.89), (3.93), and (3.98), to obtain  2 Xα+1 = Yα2 E(κnc )Vn+1 (κnc )

π 2 (νn + n)2 (νn + n + 1)2 4(νn )2 (1 − (κnc )2 )(K(κnc ))3



(3.100)

and  2 = Yα2 E(κnc )Vn (κnc ) Yα+1

π 2 (νn + n)2 (νn + n + 1)2 4(νn )2 (1 − (κnc )2 )(K(κnc ))3

 ,

(3.101)

where Vn (κ) is obtained from Eq. (3.77). Equations (3.100) and (3.101) correspond to a mapping in the two dimensional plane, ν = νn , in the parameter space (ν, X, Y ).

Equations (3.100) and (3.101) have five fixed points, as shown in Fig. 3.26. If we denote the coordinates in this three-dimensional parameter space by (ν, X, Y ), then there is a nontrivial fixed point at (νn , Xn , Yn ) and four trivial fixed points at (νn , 0, 0), (νn , ∞, ∞), (νn , 0, ∞), and (νn , ∞, 0). Let us now explain the physics underlying Fig. 3.26 for the system whose strobe (0) (1) plot is shown in Figs. 3.24 and 3.25. If we set U0 = U0 = U and ν0 = 1, we obtain the paradigm Hamiltonian,

3.8 Renormalization Map

H (0) =

p02 − U cos(x0 ) − U cos[(x0 − t0 )], 2

93

(3.102)

which determines the phase space structure between the two primary resonances centered at periodic orbits with winding numbers ω = 01 and ω = 11 . Let us focus on the KAM torus with inverse golden mean winding number, ωI G = γ1 . This corresponds to the fixed point ν = ν1 . For this case, the renormalization map alternately switches the largest resonance to opposite sides of the KAM torus as we map between rational approximates on ever smaller scales in the phase space. Note that να=0 = 1, να=1 = 12 , να=2 = 23 , etc. This sequence leads to the inverse golden mean KAM torus. The curve (νn 0, ∞)→(νn , Xn , Yn )←(νn , ∞, 0) in Fig. 3.26 is the intersection of a two-dimensional stable manifold with the ν = νn plane. Any trajectory that initially lies on this stable manifold will go to the fixed point (νn , Xn , Yn ). A trajectory that initially lies in the stable region (on the side of the stable manifold toward the ν-axis) will approach the fixed point (νn , 0, 0) as we iterate the renormalization map, while a trajectory that initially lies on the unstable side of the stable manifold will approach the fixed point (νn , ∞, ∞) as we iterate the map. Thus, the position of the initial values, (Xo , Yo ), with respect to the stable manifold enables us to determine whether the KAM torus exists or not. We can state this in another way. If our initial point (ν0 , X0 , Y0 ) lies inside the stable manifold (in the stable region), as we move to smaller scales in the phase space, the point (να , Xα , Yα ) will approach the point (νn , 0, 0) and the size of the resonances surrounding the rational approximates will shrink to zero on very small scales. Therefore, for this case we are below the critical parameters for destruction of that particular KAM torus. On the other hand, if our initial point (ν0 , X0 , Y0 ) lies outside the stable manifold (in the unstable region), as we move to smaller scales in the phase space, the point (να , Xα , Yα ) will approach the point (νn , ∞, ∞), and thus the size of the resonances surrounding the rational approximates will continue to grow as we go to very small scales. For this second case, the KAM torus does not exist. For each choice of (ν0 , X0 , Y0 ) there will be an infinite number of KAM tori between the two primary resonances of the paradigm Hamiltonian. The various KAM tori can be studied by choosing the proper sequence of rational approximates. Of all these choices there will be one KAM torus that is the last to be destroyed (most likely a noble KAM torus). Escande and Doveil have plotted the stable manifold (in the ν = ν0 plane) of the last KAM torus for the cases ν0 = 1, 2, 3, 4 (because of the form of the paradigm Hamiltonian, this also describes the cases ν0 = 12 , 13 , and 14 ). Their results (obtained numerically) are shown in Fig. 3.27. Once we have fixed ν0 , we can read off from Fig. 3.27 for what values of X0 and Y0 the last KAM torus will remain intact and for what values it will be destroyed, thus allowing a chaotic flow of trajectories between the two primary resonances of the paradigm Hamiltonian.

94

3 Area-Preserving Maps

Fig. 3.27 Plot of stable manifolds for ν0 = 1, 2, 3, 4 (Escande and Doveil 1981)

Let us now illustrate the use of Fig. 3.27 for the paradigm Hamiltonian with √ ν0 = 1. We pick U0(0) = U0(1) = U so that X0 = Y0 = 2 U . The Chirikov estimate predicts that the last KAM surface is destroyed for S = X0 + Y0 = 1. The renormalization theory predicts S = X0 +Y0 = 0.7, which is in excellent agreement with Figs. 3.24 and 3.25.

3.9 Conclusions In this chapter, we have used a number of area-preserving twist maps to show the behavior of nonlinear conservative systems as they undergo a transition to chaos. The maps that have figured most prominently in this chapter have been the whisker map, the standard map, the universal map, and several versions of the quadratic map. There are, however, several other area-preserving maps that historically have been important but that we do not have space to discuss. We will say a few words about some of them now. One of the oldest maps is the Fermi map, which was proposed by Fermi (1949), Ulam (1961), and Zaslavsky and Chirikov (1965) to model the acceleration of cosmic rays that might occur due to repeated collisions with cosmic clouds. The model consists of a ball bouncing between infinitely heavy walls, one of which is fixed and the other undergoing small-amplitude periodic oscillations. The question originally posed by Fermi was whether or not such a system could cause the ball to attain infinitely high energies. The mapping involves the speed of the ball, un ,

3.9 Conclusions

95

just before the nth collision with the wall and the phase of the wall, ψn , at the nth collision. A simplified version of the Fermi map (Lieberman and Lichtenberg 1972) is given by   un+1 = un + ψn − ψn+1 = ψn +

C un+1

 1  , 2 (mod1),

(3.103) (3.104)

where C is a constant that depends on the amplitude of the wall oscillations and the average distance between the walls. This system can go from quasiperiodic to chaotic behavior as the constant C is varied. However, the chaotic region is restricted to low energy so the particle cannot attain infinitely high velocities. There are several maps associated with perturbed Kepler systems. Because of the long range of the Coulomb potential, the whisker map does not adequately describe the stochastic layer of hydrogen. A generalized whisker map of the stochastic layer (including the effect of a constant field of arbitrary size) has been derived by Cocke and Reichl (1990). Casati et al. (1988) have used the fact that the singularity in the Coulomb potential at x = 0 has a kick-like effect on the motion to derive a map, the Kepler map, for driven one-dimensional hydrogen that is generally valid when no constant field (Stark field) is present and can be generalized to include a small constant field. For the case of a classical one-dimensional hydrogen atom driven by a monochromatic time-periodic external field with frequency ω0 , the Kepler map is a map of the number of photons absorbed, Nj , and the change in phase, φj , of the external field for each orbit of the electron. It is given by Nj +1 = Nj + κ sin(φj ),

(3.105)

φj +1 = φj + 2π ω0 (−2ω0 Nj +1 )−3/2 .

(3.106)

In (Casati et al. 1988) it is shown that the Kepler map gives a good description of one-dimensional microwave-driven hydrogen as long as the frequency ω0 is not too low. Another map related to a perturbed Kepler system is the cometary map derived by Petrosky (1986). This map describes the mechanism by which comets that orbit the sun may be captured or lost due to the perturbing influence of Jupiter. Petrosky’s cometary map may be written Pn+1 = Pn + K sin(gn ), gn+1 = gn −

2π , (−Pn+1 )3/2

(3.107) (3.108)

96

3 Area-Preserving Maps

where Pn is the inverse semimajor axis, and gn is the phase of the orbit of Jupiter, when the comet passes the perihelion. K is a constant determined by the mass of Jupiter and the strength of the coupling between Jupiter and the comet. When P > 0, P = 0, or P < 0, the comet orbit is hyperbolic, parabolic, or elliptic, respectively. For the parameters relevant to this system, the map shows chaos at the border P = 0, indicating that there is chaotic capture and expulsion of comets in the solar system. In this chapter, we have focused primarily on the behavior of twist maps. However, in hydrodynamic flows and plasmas, dynamics has been observed that can be understood in terms of maps that violate the twist condition. Properties of these so-called nontwist maps have been discussed by del-Castillo-Negrete and Morrison (1993); del-Castillo-Negrete et al. (1996a,b). As we have seen, when maps become fully chaotic, their trajectories begin to behave very much like simple random walks. The relation of fully chaotic maps to random walks has been shown rigorously for some fully chaotic maps. For these maps (the baker’s map is one example) the spectral properties can be obtained exactly and analytic expressions have been obtained for the evolution of probability distributions and diffusion coefficients. A very clear discussion of the spectral analysis of fully chaotic maps, and related references, can be found in Driebe (1999). In this chapter, we have studied nonlinear conservative systems with two degrees of freedom from the point of view of planar area-preserving maps. In Chap. 5 we will also show that there is a qualitatively different topology of chaos in systems with three or more degrees of freedom.

References Aubry S (1978) Solitons and condensed matter physics. In: Bishop AR, Schneider T (ed). Springer, Berlin, p 264 Aubry S, LeDaeron PY (1983) Physica D 8:381 Bensimon D, Kadanoff LP (1984) Physics D 13:82 Berry MV (1978) AIP conference proceedings, vol 46. American Institute of Physics, New York, p 16. Reprinted in [MacKay and Meiss 1987] Birkhoff GD (1927) Acta Math 50:359. Reprinted in [MacKay and Meiss 1987] Bountis T (1981) Physica D 3:577 Casati G, Guarneri I, Shepelyansky DL (1988) IEEE J Quantum Electron 24:1420 Chirikov B, Shepelyansky DL (1986) Chaos border and statistical anomalies. Preprint 86–174. Institute of Nuclear Physics, Novosibirsk Cocke S, Reichl LE (1990) Phys Rev A 41:3733 Collet P, Eckmann J-P, Koch H (1981) Physica D 3:457 del-Castillo-Negrete D, Morrison PJ (1993) Phys. Fluids A 5:948 del-Castillo-Negrete D, Greene JM, Morrison PJ (1996a) Physica D 91:1 del-Castillo-Negrete D, Greene JM, Morrison PJ (1996b) Physica D 100:311 de Vogelaere R (1958) In: Lefschetz S (ed) Contributions to the theory of nonlinear oscillations, vol. IV. Princeton University Press, Princeton, p 53 Driebe DJ (1999) Fully chaotic maps and broken time symmetry. Kluwer Academic Publishers, Dordrecht Escande DF (1982) Phys. Scr. T2/1:126

References

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Escande DF (1985) Phys. Rep. 121:165 Escande DF, Doveil F (1981) J Stat Phys 26:257 Fermi E (1949) Phys Rev 75:1169 Greene JM (1979a) J Math Phys 20:1183 Greene JM (1979b) In: Month M, Herrera JC (ed) Nonlinear orbit dynamics and the beam-beam interaction. AIP conference proceedings, vol 59. American Institute of Physics, New York, p 257 Greene JM, MacKay RS, Vivaldi F, Feigenbaum MJ (1981) Physica D 3:468 Greene JM, MacKay RS, Stark J (1986) Physica D 21:267 Hardy GH, Wright EM (1979) Introduction to the theory of numbers. Clarendon, Oxford Katok A (1982) Ergodic Theor Dyn Syst 2:185 Lieberman MA, Lichtenberg AJ (1972) Phys Rev A 5:1852 MacKay RS (1982) Renormalization in area-preserving maps, Ph.D. dissertation. Princeton University (University Microfilms, Ann Arbor) MacKay RS (1983a) Physica D 7:283 MacKay RS (1983b) Period-doubling as a Universal Route to Stochasticity. In: Horton W, Reichl LE, Szebehely V, Reichl LE, Szebehely V (eds) Long time prediction in dynamics. John Wiley and Sons, New York MacKay RS, Meiss JD, Percival IC (1984) Physica D 13:55 Mather JN (1982) Topology 21:457 Meyer KR (1970) Trans Am Math Soc 149:95 Moser J (1973) Stable and random motions in dynamical systems. Princeton University Press, Princeton Percival IC (1979) Nonlinear dynamics and the Beam-Beam interaction. In: Month M, Herrera JC (eds) AIP conference proceedings, vol 57. American Institute of Physics, New York, p 302 Petrosky TY (1986) Phys Lett A 117:328 Prasad AV 1948 J London Math Soc 23:169 Shenker SJ, Kadanoff LP (1982) J Stat Phys 27:631 Ulam SM (1961) Proceedings of the fourth berkeley symposium on mathematical statistics and probabilities, vol 3. University of California Press, Berkeley Zaslavsky GM, Chirikov BV (1965) Sov Phys Dokl 9:989

Chapter 4

Chaotic Scattering

Abstract Phase space flow in the neighborhood of unstable fixed points evolves in a horseshoe-like fractal folding pattern. When incident trajectories scatter from a localized chaotic object, the outgoing scattered trajectories are described by functions that have fractal patterns of singularities characteristic of the chaotic scattering region. Fully developed chaos can imprint fairly simple fractal patterns. Scattering regions with mixed dynamics have a much more complex asymptotic dynamics. An example of a perfect scattering system, is one whose scattering dynamics is governed by a complete ternary horseshoe. There is a complete fractal pattern of singularities in the scattering cross section and delay time, and a complete symbolic dynamics exists for the scattering process. Some examples of chaotic scattering processes that can be found in nature include scattering of a charge particle from a magnetic dipole and scattering of an electron from a negative chlorine ion Cl − . in the presence of a monochromatic radiation filed. Another example is the dynamics involved in the dissociation of molecules, such as that of the HOCl molecule above the energy for dissociation of the chlorine atom Cl from the HO complex. Experimentally obtained potential energy surfaces can be used to analyze the dissociation (and scattering) process. Above dissociation, the molecular dynamics is largely chaotic and scattering of Cl from HO has fractal structure. Keywords Chaotic scattering · Ternary horseshoe · Fractal scattering patterns · Symbolic dynamics · Delay time · Scattering map · Magnetic dipole · Chlorine ion in radiation field · HOCl dissociation dynamics · Normally hyperbolic invariant manifold · Galaxies

4.1 Introduction For scattering systems, the detailed structure of the chaotic dynamics in the reaction region determines the structure of fractal patterns of singularities imprinted on classical scattering functions in the asymptotic region. These fractal patterns are © Springer Nature Switzerland AG 2021 L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200, https://doi.org/10.1007/978-3-030-63534-3_4

99

100

4 Chaotic Scattering

a signature of the type of chaotic dynamics occurring in the reaction region. Fully developed chaos can imprint fairly simple fractal patterns and statistical properties. Mixed phase spaces in the reaction region, a more typical situation, present a much more complex relation between the dynamics in the reaction region and the asymptotic dynamics of the scattering system. A pathway for analyzing the complex scattering patterns of a chaotic system was introduced by Smale (1967), who showed that a mapping of the phase space flow, in the neighborhood of unstable fixed points of chaotic systems, evolves in a horseshoe-like fractal folding pattern. Following the work of Smale, methods for extracting information about chaotic scattering processes and the underlying fractal structure that governs the dynamics, has been developed by Jung and others for a variety of model systems (Jung and Scholz 1988; Jung and Richter 1990; Rückerl 1994; Jung et al. 1999). In subsequent sections, we first describe the behavior of a perfect scattering system, one whose scattering is governed by a complete ternary horseshoe. We then show examples of chaotic scattering processes in a variety of systems that can be found in nature. A scattering system whose chaotic reaction region is describable by a complete ternary horseshoe is described in Sect. 4.2. The system involves a 1D potential kicked periodically in time with a delta-function kick, and it contains two primary unstable fixed points. It is possible to observe a fractal pattern of singularities in the scattering cross section and to map out a complete symbolic dynamics for the scattering process. This analysis will allow us to introduce concepts of stable and unstable manifolds associated to the fixed points, heteroclinic intersections, and show how the influence of these manifolds extends far into the asymptotic region. It also allows us to obtain the fractal structure of the delay time for transmission and reflection of incident particles, and more generally the fractal pattern associated with all dynamical functions that involve the scattering process. Most real systems do not involve a complete horseshoe, because the reaction region contains a phase space that is a self-similar mixture of regular and chaotic structures. Then the pattern of singularities in the scattering functions is not as simple. When there is a mixed phase space, a mixture of chaos, periodic orbits, and KAM islands can exist in the continuum and trajectories starting on a periodic orbit or KAM island will stay in the reaction region forever. The first example of a scattering system with an incomplete horseshoe, is that of a charged particle scattering from a magnetic dipole, considered in Sect. 4.3. Jung and Scholz (1987, 1988) found a direct relation between homoclinic orbits in the reaction region and the structure of the delay time for scattered particles. A slightly more complex scattering process, considered in Sect. 4.4, involves the scattering of an electron from a negative chlorine ion Cl − in the presence of a monochromatic radiation field (Emmanouilidou et al. 2003; Jung and Emmanouilidou 2005; Emmanouilidou and Jung 2006; Lin et al. 2011). This system has a fractal distribution of delay times that can be described by a symbolic dynamics. The symbolic dynamics can be described in terms of a branching tree, which, in turn, allows construction of a transfer matrix. Eigenvectors of the transfer matrix give us information about the fraction of incident particles that are transmitted or reflected in the scattering process.

4.2 The Complete Ternary Horseshoe

101

The last topic we consider in this chapter (Sect. 4.5), is the dynamics of the HOCl molecule above the energy for dissociation of the chlorine atom Cl from the HO complex. We use an experimentally obtained potential energy to analyze the scattering process (Weiss et al. 2000). Under certain fairly realistic constraints on the angular momentum of the molecule and on its vibrational dynamics, the HO–Cl scattering system can be reduced to a 2 DoF system. Above dissociation, the molecular dynamics is largely chaotic (Barr et al. 2009) and scattering of Cl from HO has fractal structure (Lin et al. 2015). Finally in Sect. 4.6, we make some concluding remarks.

4.2 The Complete Ternary Horseshoe The complete ternary horseshoe provides one of the simplest examples of a fractal scattering process that involves two dominant unstable fixed points and a fully selfsimilar scattering process. A scattering system governed by a complete ternary horseshoe can be built from a pair of Gaussian potential energy peaks, each with one degree of freedom (DoF), to which a periodic “kick” is applied. In the sections below, we first describe this unperturbed one-dimensional system, and then describe the effect of a time-periodic delta function kick applied to the system.

4.2.1 Double Gaussian Potential Energy Peaks The Hamiltonian for a particle that scatters from double Gaussian potential energy peaks, can be written H =

p2 2 + x 2 e−x = E, 2

(4.1)

where the potential energy is constructed so that it goes to zero at x = 0. Hamilton’s equations, for this system, are given by p˙ =

∂H ∂H dp dx 2 =− = 2(x 3 − x)e−x , x˙ = =− = p. dt ∂x dt ∂p

(4.2)

The fixed points of Hamilton’s equations satisfy the conditions (p˙ = 0, x˙ = 0). There are three fixed points. One fixed point is elliptic at (p = 0, x = 0) and has energy E = 0. The two remaining fixed points are hyperbolic at (p = 0, x = ±1) and have energy E = e−1 = 0.3679. A plot of the phase space trajectories for this system is given in Fig. 4.1. It is clear that these three fixed points determine the overall structure of the phase space.

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4 Chaotic Scattering

Fig. 4.1 Phase space for unperturbed system. A stable fixed point sits at (p = 0, x = 0) and two unstable fixed points sit at (p = 0, x = ±1)

Linearization of Hamilton’s equations about the elliptic fixed point gives      p˙ 0 −2 p = . x˙ 1 0 x

(4.3)

This has eigenvalues and eigenvectors √ λ1 = i 2, φ1 =

 √   √  √ i 2 −i 2 ; λ2 = −i 2, φ1 = 1 1

(4.4)

Thus, 

p(t) x(t)



= A e+i

√

2t

√   √  √  i 2 −i 2 + Be−i 2t 1 1

(4.5)

in the neighborhood of the elliptic fixed √ p(0) = √ For example, assume that √ point. 0, x(0) = 0.02. Then p(t) = −0.02 2sin( 2t) and x(t) = 0.02cos( 2t). The phase space trajectory in the neighborhood of the elliptic√ fixed point at (p = 0, x = 0) oscillates about the fixed point with a frequency ω = 2. The hyperbolic fixed points at (p = 0, x = ±1) each have four eigen-curves, two directly approaching the fixed point exponentially fast and two directly leaving the fixed point exponentially fast (note that the fixed point itself is an independent solution of Hamilton’s equations). Linearization of Hamilton’s equations about the hyperbolic fixed point at p = 0, x = 1 gives

4.2 The Complete Ternary Horseshoe



p˙ x˙

103



 =

0 4e−1 1 0



p x

 (4.6)

,

where p(t) = p(t) and x(t) = 1 + x(t). This has eigenvalues and eigenvectors 2 λ1 = − √ , φ1 = e



− √2e



1

2 ; λ 2 = + √ , φ1 = e



+ √2e 1

 (4.7)

.

Thus, 

p(t) x(t)





− √2e t

= Ae

− √2e 1



√ +i 2t

+ Be



√  +i 2 . 1

(4.8) − √2e t

√ e If we set B = 0 and A = 0.01, the solution becomes p(t) = − 0.02 e

x(t) =

2 0.01 − √e t

e

, which describes motion along the lower righthand eigencurve + √2e t

√ e in Fig. 4.1. If we set A = 0 and B = 0.01, the solution is p(t) = + 0.02 e 2 0.01 + √e t , e

x(t) = in Fig. 4.1.

and

and

which describes motion along the upper righthand eigencurve

4.2.2 Delta-Kicked System We now add the perturbation to the system in the form of a time-periodic delta function “kick”. The Hamiltonian takes the form H = p2 /2 +

    1 T , V (x)δ t − n + 2 n=−∞ ∞ 

(4.9)

where V (x) = x 2 e−x and T is the period of the “kick”. For such a delta kicked system, we can construct a symplectic map. The map corresponds to a plot of the phase space at time intervals equal to the period of the driving force. The period T between each kick serves as a development parameter and measures the degree of advancement of the folding of stable and unstable manifolds of the map. The symplectic map is given in three steps. The initial phase space coordinate is denoted (p0 , x0 ). This point evolves freely for a time period of length T2 so that, at  time t = T2 − (just before the kick), the phase space coordinate is (p0 , xa = x0 + p0 T2 ). At time t = T2 , the system is “kicked” and the momentum receives a discrete  increment, whose magnitude is given by the force function f (xa ) = − dVdx(x) x=x = 2

a

2(xa3 − xa )e−xa , so that p1 = p0 + f (xa ). The third step is again a free flight for 2

104

4 Chaotic Scattering

time T /2. Thus, after one period T , the phase space coordinates are mapped from values (p0 , x0 ) at time t = 0 to values (p1 , x1 ) at time t = T , where p1 = p0 + f (x0 + p0 T /2) x1 = x0 + (p0 + p1 ) T /2.

(4.10)

The force exerted by the ‘kick” f (x) = 2(x 3 − x)e−x is zero at the points x = 0, ±1, so the fixed points of the unperturbed system are also fixed points of the symplectic map. The behavior of the perturbed system is very different from that of the unperturbed system. We will want to focus on the behavior of the stable and unstable manifolds of the symplectic map. Therefore, the initial set of points (p0 , x0 ) will be a special set of points. The first step in constructing the map is to locate the lowest order stable and unstable manifolds of the map. If we look at phase space points near the unstable fixed points (the fixed points are not changed by the kicks), most points escape to infinity under the action of the map. However, there is a line of points that does not escape to infinity, but remains in the neighborhood of the fixed points. These lines of points are called the stable and unstable manifolds of the kicked system and can be found by a numerical search of the phase space in the neighborhood of the fixed points. The results of the numerical search are the stable and unstable manifolds shown in Fig. 4.2a. The curves leaving the fixed points at (p = 0, x = −1) and (p = 0, x = +1) are the unstable manifolds. The curves approaching these fixed points are the stable manifolds. The important point about these manifolds is that they cross at x = 0 and thereby form a fundamental area which is the diamond shaped region enclosed by the stable and unstable manifolds in Fig. 4.2a. The crossing points at x = 0 are called the primary heteroclinic points of the map. We will take the curves in Fig. 4.2a to be level 0 of the map. One iteration of the map gives the curves in Fig. 4.2b. A second iteration of the map gives the curves in Fig. 4.2c. Figure 4.3 gives a clearer picture of the effect of the mapping. We consider all the points contained within the fundamental area (the shaded region) in Fig. 4.3a (level 0). After one iteration of the map, Fig. 4.3b, the points initially inside the fundamental area are stretched in one direction and compressed in the transverse direction (so the area is conserved), and then folded back through the fundamental area. If we apply the map a second time, Fig. 4.3c, this process repeats with increasingly complex stretching and folding of the original fundamental area. With further iterations of the map, the area of the shaded region does not change. However, the fraction of the original shaded region that returns to the fundamental region decreases. Note that in Fig. 4.3b, the middle section of the shaded region has reversed its direction as it passes back through the fundamental region. Therefore, the fixed point at (p = 0, x = 0) has become inverse hyperbolic under the action of the map. As we continue to iterate the map for this system, the tendrils that form from the stretching and folding of the initial stable and unstable manifolds stretch further and further into the asymptotic region with ever greater complexity, while maintaining their intrinsic threefold symmetry. 2

4.2 The Complete Ternary Horseshoe

105

Fig. 4.2 Mapping for T = 5. (a) Ternary horseshoe at level 0. (b) Ternary horseshoe at level 1 (after one kick). (c) Ternary horseshoe up to level 2 (after two kicks) (plots by Christof Jung)

In Fig. 4.4, we show two mappings of the stable and unstable manifolds in Fig. 4.2a, one that goes forward in time and another that goes backward in time. The mapping is completely symmetric in these two time directions. As we continue mapping the system (forward in time and backward in time), the tendrils formed by the original stable and unstable manifolds in Fig. 4.2a stretch and fold further and further out into the asymptotic region (both forward in time and backward in time).

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Fig. 4.3 Mapping of the fundamental area. (a) Initially shaded points fill the fundamental area. (b) After one kick, shaded points are stretched and folded once. (c) After two kicks, shaded points stretched and folded again (plot by Christof Jung)

This means that, if we consider an initial trajectory far out in the asymptotic region (either on the left or right) it might be in a spatial region that contains the tendrils that come from mapping the system backward in time. If we evolve such a trajectory

4.2 The Complete Ternary Horseshoe

107

Fig. 4.4 Ternary horseshoe up to level two, for two maps plotted together, one map going backward in time and another map going forward in time. The line of points A–B crosses a set of tendrils (plot by Christof Jung)

forward in time, it forever remains part of the complex fractal structure of the stable and unstable manifolds.

4.2.2.1

Delay Time

If we take a line of initial trajectories in the asymptotic regions and choose the line of initial points so that they cross a region containing backward-in-time tendrils, then as we map the line of points forward in time they can give us a picture of the nature of the fractal structure of stable and unstable manifolds. To show this, we consider the line of initial trajectories shown by the line in Fig. 4.4 labeled A–B. This line of initial points crosses backward-in-time tendrils. We will see what happens to these points as we map them forward in time. In order to analyze the flow of points in the line of points A–B in Fig. 4.4, it is useful to consider the delay time associated with those points. The delay time τ = tact − tf ree , is the difference between, (a) the actual time tact for the trajectory to scatter from the potential and then traverse to a designated point in the asymptotic region, and (b) the time tf ree a free trajectory would spend traversing between the same initial and final points. More precisely, imagine that a trajectory starts at a position x1 at time t1 with an initial momentum p1 . We measure the trajectory at a later time t2 to have position x2 and momentum p2 . The actual time of flight is tact = t2 − t1 , which depends on where we start and end observation of the trajectory. For the kicked system, tact is the number of steps of the map times the period of the map. However, the time delay, τ , of the trajectory is a quantity that is more intrinsic to the scattering process. To obtain the time delay τ , we have to subtract from tact , a time due to incoming and outgoing free motion. The part of the time used for a free particle for the incoming asymptote is −x1 /p1 . The part of the time used for a free particle along the outgoing asymptote is x2 /p2 . If we now subtract these two asymptotic contributions from the

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4 Chaotic Scattering

Fig. 4.5 The delay time τ versus phase angles χ of the incident trajectories (plot by Christof Jung)

total time, we obtain the delay time τ = tac + x1 /p1 − x2 /p2 . The delay time τ is the difference in the time it takes the actual trajectory to traverse the scattering region, and the time it takes an hypothetical asymptotic reference trajectory (that evolves without interaction) to traverse the scattering region. It is useful to define another parameter, χ , which is the relative phase shift between the particle and the clock (defined by the kick frequency). Consider the x1 incoming momentum p1 , then χ = ( 2π T ) p1 (mod2π ). Note that if we replace the initial point x1 by x1 + np1 T , for any integer n, then χ is the same, as long as the particle initially sits at a position x1 + np1 T in the same asymptotic region. In practice, we fix the values of χ by fixing the values of x1 . For an incoming beam, we define a distribution of values of χ . The most natural distribution of χ is a constant density of values of χ over an interval of length 2π . Because of the interaction, the trajectory leaves the reaction region with a final momentum p2 , whose value depends on χ . In Fig. 4.5, we plot the delay time τ as a function of χ for the initial line of points A–B in Fig. 4.4. We see that the time delay also picks up the threefold symmetry of this scattering map and it appears to have a fractal structure. Also, there are values of χ for which the delay time is infinite. These are points along the A–B in Fig. 4.4 that lie directly on the stable manifolds. Such points can never escape the scattering region. We can also plot a scattering function that shows the structure of the stable and unstable manifolds. In Fig. 4.6, we plot the outgoing momentum pout as a function of χ . This scattering function, again, picks up the threefold symmetry of the tendrils of the stable and unstable manifolds.

4.2 The Complete Ternary Horseshoe

109

Fig. 4.6 The outgoing momentum pout versus phase angles χ of the incident trajectories (plot by Christof Jung)

4.2.2.2

Symbolic Dynamics

We can assign a symbol sequence to the increasingly complex stretching and folding pattern of the fundamental region under the action of the map. In Fig. 4.7, we assign symbols A, B, and C to the regions where mapped sections of the fundamental area cross the boundaries of the fundamental area. If we compare Figs. 4.3b and 4.7a, which show the first iteration of the map, we see how the symbol pattern is assigned. If we compare Figs. 4.3c and 4.7b, the pattern of crossings is the same, but the symbol sequence is more complex and becomes harder to follow with increasing number of iterations of the map. In Fig. 4.7c, we show the symbol structure of three iterations of the map. The complexity of the symbol structure grows exponentially, although there is a very simple pattern underlying it. In Fig. 4.8, we show a symbol tree that more clearly shows the evolution of the pattern of tendrils crossing the boundary of the fundamental region. At each level, the pattern of symbols in central clusters of symbols is reversed from that of the symbol sequence of the lower level of the map. This symbol tree clearly shows that scattering from the complete ternary horseshoe forms is a self-similar fractal process.

4.2.2.3

Incomplete Ternary Horseshoes

In the previous section, we examined a scattering system whose dynamics, in certain parameter ranges, was described by a complete ternary horseshoe. However, dynamics of most scattering systems is governed by incomplete binary or ternary horseshoes. Many examples of scattering systems with incomplete horseshoes have been studied in the literature. Measures for determining the degree of “incompleteness” of the horseshoe development have been discussed by Jung et al. (1999). In subsequent sections, we will see how this horseshoe development can manifest itself in some atomic and molecular systems.

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Fig. 4.7 Symbolic form of the ternary horseshoe up to level three

A B C C B A A B C C B A A B C C B A

A

B

C A

C

B

A

A B C C B A A B C

A

B

B

C C

Fig. 4.8 Symbol tree for the complete ternary horseshoe up to level three

4.3 Scattering Chaos in a Magnetic Dipole As we have seen in the previous section, there is a direct relation between the homoclinic orbits in a chaotic reaction region and the fractal structure of the delay time. Jung and Scholz (1988) have analyzed this relationship between chaotic structure and delay times for the case of scattering off of a magnetic dipole. We describe some of their results here. The motion of a particle with mass m and electric charge e in the field of a magnetic dipole is governed by the Hamiltonian

4.3 Scattering Chaos in a Magnetic Dipole

H =

 2 e 1 p − A(r) , 2m c

111

(4.11)

where r and p are the position and momentum, respectively, of the particle, A(r) = A0 − ry3 xˆ + rx3 yˆ is the magnetic vector potential, and c is the speed of light (this system was first studied extensively in Stomer 1907 and later in Contopoulis and Vlahos 1975 and Dragt and Finn 1976). We can write the Hamiltonian in terms of cylindrical coordinates, (ρ, φ, z), where x = ρcos(φ) and y = ρsin(φ). Then, the Hamiltonian takes the form     pφ 1 αρ 2 2 2 H = , p + pz + − 3 2m ρ ρ r

(4.12)

 where α = eA0 /c and r = ρ 2 + z2 . It is easy to see that the angular momentum pφ is a constant of the motion, reflecting the fact that the system has azimuthal symmetry about the z-axis. All the interesting motion occurs in the (ρ, z) plane. We can write the Hamiltonian in terms of dimensionless momenta and coordinates (Dragt and Finn 1976). We shall assume that pφ > 0. Let ρ = ρ o α/pφ , z = zo α/pφ , pρ = pρo pφ2 /α, pz = pzo pφ2 /α, and H = H o pφ4 /mα 2 , where (pρo , pzo , ρ o , zo , H o ) are dimensionless. Then let pρo →pρ , etc. The Hamiltonian, in terms of dimensionless coordinates and momenta, takes the form H =

1 2 (p + pz2 ) + V (ρ, z) = E, 2 ρ

(4.13)

where E is the total energy, V (ρ, z) is the effective potential,   ρ 2 1 1 − 3 , V (ρ, z) = 2 ρ r

(4.14)

and all quantities are now dimensionless. The only controllable parameter is the total energy E. It is useful to look more carefully at the potential energy V (ρ, z). A contour plot showing lines of constant V (ρ, z) is given in Fig. 4.9. There is an oddly shaped potential energy bowl that lies to the left of the point (ρ = 2, z = 0). The lowest value of the potential energy, V (ρ, z) = 0, is in the bowl (and at ρ = 0) and occurs along a continuous line, ρ 2 − (ρ 2 + z2 )3/2 = 0, called the thalweg, which intersects the z-axis at ρ = 0 and ρ = 1. A saddle point occurs at point PS = (ρ = 2, z = 0). 1 The value of the potential energy at the saddle point is V (ρ, z) = 32 = 0.03125. The saddle point provides a passageway from the outer region, ρ > 2, to the interior of the bowl. As ρ→0, the potential energy tends toward infinity except along the thalweg. As ρ→∞, the potential energy tends toward zero. For large enough values of ρ, one can define an asymptotic region where particles are free from the influence of the dipole potential. A particle that is scattered by the dipole potential originates in the asymptotic region. It enters the reaction region (the

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4 Chaotic Scattering

Fig. 4.9 Contour plot of the potential energy V (ρ, z) as a function of ρ and z. Some specific contour lines of constant potential energy with values of V (ρ, z) are shown Jung and Scholz (1988)

Fig. 4.10 (a) A Poincaré surface of section, ρ˙ versus ρ, for surface z = 0, z˙ > 0 and energy E = 0.079. Some homoclinic tendrils for the stable manifold, Ws , and a short segment of the unstable manifold, Wu , are shown. (b) The time delay of a sequence of trajectories (the dotted line in (a)) with energy E = 0.079, ρ = ρ0 , and a range of initial values z = zi (Jung and Scholz 1988)

region of influence of the dipole potential) and eventually returns to the asymptotic 1 region in an altered state. For energies E < EL = 32 = 0.03125, a particle cannot enter the potential energy bowl from the asymptotic region. We can study the behavior of orbits inside the potential energy bowl by means of a Poincaré surface of section (PSS). In Fig. 4.10a, we plot the phase space coordinates (ρ, ˙ ρ) each time a trajectory crosses the z = 0 surface with z˙ > 0. If we start with a line of initial conditions in the neighborhood of the potential energy

4.3 Scattering Chaos in a Magnetic Dipole

113

bowl, they need not come back through the PSS in that region. They can rapidly escape to the asymptotic region. We expect interesting behavior to occur when periodic orbits form in the PSS. The orbits initially in the neighborhood of the potential bowl remain in that region of the phase space. For energies E > EU = 0.081 . . . , no periodic orbits exist in the PSS in the neighborhood of the potential energy bowl. As the energy is lowered to the value E = EU = 0.081 . . . , a bifurcation occurs, creating a pair of period 1 periodic orbits in the PSS, one elliptic () and the other hyperbolic (γ ) (Jung and Scholz 1988; Rückerl 1994). These points lie on the symmetry line ρ˙ = 0. As energy is lowered further, the elliptic fixed point, , changes to inverse hyperbolic and the phase space in the neighborhood of the potential energy bowl becomes more chaotic. The fixed point γ is associated with the saddle, and its position approaches the point ρ = 2, z = 0 as the energy E→ES . It corresponds to an unstable periodic orbit that oscillates back and forth along the potential energy ridge that lies to the right of the potential energy bowl. A particle incident from the asymptotic region can only experience scattering chaos if its energy lies in the interval 0.03125 < E < 0.081. In Fig. 4.10a, we show a PSS for energy E = 0.079 in the region of phase space containing the pair of period 1 periodic orbits mentioned above. The position of the elliptic periodic orbit, Pe , is surrounded by KAM tori. The location of the hyperbolic period 1 periodic orbit, Pu , lies at the crossing point of its stable and unstable manifolds, W s and W u , respectively. Only a small segment of the unstable manifold is shown, but some of the homoclinic tendrils of the stable manifold are shown. If the same number of tendrils of the unstable manifold had been drawn, the figure would have reflection symmetry about the ρ˙ = 0 axis. The homoclinic tendrils of the stable manifold are obtained by taking a small segment of the stable manifold near the fixed point and iterating the PSS backward in time. Some tendrils extend into the asymptotic region. Also shown in Fig. 4.10a is a dotted line that runs parallel to the unstable manifold and crosses some of the tendrils of the stable manifold. This dotted line is obtained by starting at the initial time, t = 0, with a line of initial conditions in the asymptotic region and following it in time until it crosses the surface of section in the neighborhood of the potential bowl. The initial conditions used to obtain the dotted line have momentum parallel to the ρ − φ plane and directed toward the origin. All initial points are the same radial distance, ρ = ρ0 , from the z-axis in the asymptotic region but have a range of values of z. Notice that, when they intersect the PSS, some of these points lie inside the region formed by the tendrils. These points can be trapped for a long time inside the reaction region as they make their way through the complex network of homoclinic tangles. If they happen to lie on the stable manifold itself, they will never escape. This can happen if a tendril crosses the line of initial points in the asymptotic region. In Fig. 4.10b, we show the delay time of scattered trajectories with E = 0.079. All the trajectories initially have momenta parallel to the ρ − φ plane and directed toward the origin. All initial conditions are in the asymptotic region with the same value of ρ = ρ0 and a range of initial values z = zi in the interval 1.3577 < zi < 1.3583. The delay time is defined as T = Ta − Th , where Ta is

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Fig. 4.11 (a) The periodic orbit, γ , for E = 0.079. (b) A trajectory with E = 0.079 and zi = 1.35823165 that is caught in the homoclinic tangles and is delayed in the reaction region (Jung and Scholz 1988)

the actual time it takes the particle to start at ρ = ρ0 , traverse the reaction region, and return to ρ = ρ0 , and Th is the time it takes to scatter from a hard wall at ρ = 0 (in the absence of the dipole) and return to ρ = ρ0 . The delay time has fractal structure for those initial points that are mapped inside the tendrils. A comparison of Fig. 4.10a, b shows that the fractal structure of the delay time directly corresponds to the fractal structure of the tendrils (the homoclinic orbits associated with the unstable fixed point). In Fig. 4.11a, we show the unstable periodic orbit, γ , and in Fig. 4.11b, we show the motion of an orbit that becomes trapped in the homoclinic tangles for a long time. Rückerl and Jung have developed a symbolic sequence to directly relate the branching structure imposed by the homoclinic tangles to the fractal behavior of the delay times (Rückerl 1994). Thus, the time delay can be used to investigate chaotic structures in the reaction region of a scattering problem, at least for scattering systems with two degrees of freedom. Scattering chaos has been observed in a number of model systems, including a linear array of scatterers (Troll and Smilansky 1989), a collection of hard disks (Eckhardt 1987; Gaspard 1989), a triple hill potential (Jung and Richter 1990), and hydrogen in a circularly polarized laser beam (Okon et al. 2002). It has also been observed in satellite motion (Petit and Henon 1986) and hydrodynamic flow (Jung et al. 1993). Some reviews discussing scattering chaos include Eckhardt (1988), Smilansky (1992), Jung and Seligman (1997), Seoane and Sanjuan (2013), and Lai (2010). Two more recent examples of scattering chaos in atomic and molecular systems are described below.

4.4 Model of Chlorine Ion in a Radiation Field Laser-atom interactions provide a particularly fruitful venue for analyzing scattering processes that are intrinsically chaotic. For the case of a linearly polarized radiation field, the scattering process can be reduced to one space dimension. One widely studied system consists of a negative chlorine ion Cl − in the presence of a radiation field (Emmanouilidou et al. 2003; Jung and Emmanouilidou 2005; Emmanouilidou

4.4 Model of Chlorine Ion in a Radiation Field

115

and Jung 2006; Lin et al. 2011). It is modeled in terms of an electron of mass m and charge q, in the presence of a one-dimensional Gaussian potential well and a monochromatic radiation field whose electric field E is linearly polarized along the direction of motion of the electron. The Hamiltonian for this system can be written H1 (p , x  , t) =

(p )2  2 − V0 e−(x /δ) − x  F sin(ωt), 2m

(4.15)

where p and x  are the momentum and displacement of the electron in the lab frame and F = qE. We can transform from the lab frame {p , x  } to a reference frame {p, x} moving with the electron via a canonical transformation whose generating function is  p F   sin(ωt) . The reference frame moving F3 (p , x, t) = −p x + ω −xcos(ωt) + mω with the electron is called the Kramers-Henneberger frame (KH) (Kramers 1956; ∂F3  3 Henneberger 1992). Then p = − ∂F ∂x and x = − ∂p and the transformed 3 Hamiltonian H2 is given by H2 = H1 + ∂F ∂t . We can further transform this time-periodic Hamiltonian to a time-independent Hamiltonian by using action-angle variables (J, φ), with φ = ωt, to describe the radiation field dynamics. We then write the Hamiltonian in the form

H3 (p, x, J, φ) =

2 p2 − V0 e−(x−α(φ)/δ) + G(φ) + ωJ, 2m

(4.16)

2

F 2 2 where α(t) = α0 sin(φ), α0 = F /ω2 , and G(φ) = 2mω 2 (2sin(φ) − cos(φ) ). The equations of motion are given by Hamilton’s equations

∂H 2V0 dp =− =− 2 dt ∂x δ

  2 F x − 2 sin(φ) e−(x−α(φ)/δ) , ω

∂H p dx = = , dt ∂p m

dφ ∂H = = ω, dt ∂J

dJ ∂H =− = f (x, φ), dt ∂φ

(4.17)

where f (x, φ) is a function of x and φ. Although the action variable J may have a complicated motion, the electron dynamics is independent of J . Also, since φ˙ = ω, φ = ωt + φ0 (φ0 is the initial phase), the phase angle φ evolves linearly with time. In the KH-frame, there is an asymptotic region where the potential energy is zero and the electron motion is that of a free electron since p˙ = 0 and x˙ = p. In the KH-frame, there is also a reaction region where the inverted Gaussian potential is nonzero and oscillates back and forth. An interesting feature of this model is that the Hamiltonian is asymmetric with respect to the left-moving and right-moving trajectories. Although the Hamiltonian is invariant under the generalized symmetry transformation x→ − x and φ→φ + π , it is not invariant under the transformation

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4 Chaotic Scattering

x→ − x alone. As we shall see below, this gives rise to an asymmetry in the scattering dynamics of this system, which depends on the initial phase φ0 . The fraction of incident particles that are transmitted and reflected will differ depending on the incident direction. In describing the behavior of this system, we use atomic units (see Appendix I). Therefore, the electron mass is m = 1 a.u. and we choose V0 = 0.27035 a.u., δ = 2 a.u., ω = 0.65 a.u. and α0 = 0.9 a.u., values that have been shown to describe the quantum behavior of a negative chlorine ion Cl − in the presence of a laser field (Yao and Chu 1992; Marinescu and Gavrila 1996; Fearnside et al. 1995). The classical scattering dynamics has been analyzed in great detail in Emmanouilidou et al. (2003), Jung and Emmanouilidou (2005), Emmanouilidou and Jung (2006), and Lin et al. (2011). We will mainly follow the approach used in Lin et al. (2011), but comment on the other very interesting works as needed.

4.4.1 Scattering Map The fractal structure of the scattering process can be seen in a Poincaré surface of section (PSS) of the dynamics. We can solve Hamilton’s equations in the KH-frame and plot p and x each time φ = π/2 + n2π , where n = 0, 1, 2, ... The inverted Gaussian system, in the absence of radiation, has three primary periodic orbits (fixed points of the PSS): a stable fixed point at (p = 0, x = 0), an unstable periodic orbit (FL ) at (p = 0, x = −∞), and an unstable periodic orbit (FR ) at (p = 0, x = +∞). We define a fundamental region R, in the PSS, whose boundaries are given by segments of the invariant manifolds of the outer fixed points of the system. The invariant manifolds and the fundamental area R are shown in Fig. 4.12a. The invariant manifolds are located by taking a line of initial points with fixed x but varying p > 0 near p = 0, which is the neighborhood of the separatrix in the absence of radiation. Then evolve the points in time and determine which points escape and which points do not escape. Points that do not escape are on the invariant manifold. This process is repeated for different values of x until a segment of the manifold is mapped out. The curves n–u and s–n, in Fig. 4.12a, are segments of the unstable and stable manifolds, respectively, of the fixed point on the left, FL . The curves r–p and p–v are segments of the stable and unstable manifolds, respectively, to the fixed point on the right, FR . The fundamental area, in Fig. 4.12a, is defined and enclosed by the curves n–q, q–p, n–m and m–p. Because the invariant manifolds approach the p = 0 axis exponentially as x→±∞, it is sufficient to consider the segments shown in the figure. In Fig. 4.12b, we show four iterations backward in time of the segments r–p and s–n of the stable manifolds. The stable manifolds cannot intersect themselves and they cannot intersect each other. However, they can intersect the unstable manifolds. The curves, as they are iterated backward in time, must enclose the same area enclosed by the stable and unstable manifolds in Fig. 4.12a.

4.4 Model of Chlorine Ion in a Radiation Field

117

Fig. 4.12 A PSS with p and x plotted each time φ = π2 . (a) The curves n − u and s − n ( r − p and p − v) are segments of the unstable and stable manifolds of FL (FR ). The fundamental region is enclosed by the curves n − u, s − n, r − p and p − v. (b) Segments of the stable manifolds r–p (solid line) and n–s (dashed line) are integrated backward in time for four iterations of the PSS (reproduced from Lin et al. 2011)

We can distinguish points associated with the asymptotic region from those associated with the fundamental region. This becomes clear in Fig. 4.13a, b, where the shaded regions with positive indices contain points that lie outside the fundamental region. Iterations of these shaded regions are shown for times ranging from t = −3 to t = +3 (seven iterations of the PSS). As the map is iterated backward in time, the shaded regions form more and more complex filaments that cut across the fundamental region. As the shaded regions are iterated forward in time, they approach their respective asymptotic regions. In Fig. 4.13a, we follow the point αn (−3≤n≤3) which is incident from the right at time n = −3, but happens initially to be inside one of the tendrils in the asymptotic region at that time. In Fig. 4.13b, we follow the point βn (−3≤n≤4), which is incident from the left at time n = −3 and is initially inside a tendril in the asymptotic region at that time. Careful inspection of Figs. 4.12b and 4.13a, b shows that the “backward in time” filaments from the two stable manifolds become interlaced in the tendrils that extend backward in time to the upper left, and backward in time to the lower right. A particle, incident from the left, that initially lies in one of these “backward in time” tendrils can either be transmitted through the scattering region toward x = +∞ or it can be reflected from the scattering region toward x = −∞, depending whether it initially lies in a filament of the type shown in Fig. 4.13a or that shown in Fig. 4.13b.

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Fig. 4.13 (a) Points labeled α−3 , . . . , α+3 show the movement of a single point αn , initially in a shaded area and incident from the right for seven iterations of the PSS. The point α takes two steps inside the fundamental region R and is reflected back to the right. (b) The points labeled β−3 , . . . , β+4 show movement of a single point βn incident from the left, for eight iterations of the PSS. The point β takes one step inside R before being reflected back to the left (reproduced from Lin et al. 2011)

4.4.2 Delay Time In Fig. 4.14a, we show the evolution in time of a line of points initially lying between x = −10.5 and x = −10.1 for a particle incident from the left. The line of initial conditions, in the distant past, was chosen to lie across a tendril far out in the asymptotic region to the left. Figure 4.14a shows the line of initial conditions as it is evolved forward in time, for four iterations of the PSS, and approaches the fundamental region R. In the next iteration, the line of points will step into the fundamental region. In Fig. 4.14b, we show the time it takes various points in this line of initial conditions to leave the fundamental region (the delay time) regardless of whether the points are transmitted or reflected. The delay time forms a fractal set characteristic of this scattering process. In Fig. 4.14c, we show a plot of the number of steps taken inside R, by the points initially lying between x = −10.5 and x = −10.1 in Fig. 4.14a. We only show results for points that take fewer than 20 steps inside R. The step numbers for all initial points take values ranging from zero to infinity and form a fractal which is an image of the fractal structure of the scattering region.

4.4 Model of Chlorine Ion in a Radiation Field

119

Fig. 4.14 (a) Four iterations of a line of initial points (initially in interval −10.5≤x≤10.1) just before the line of points enters the fundamental region R. (b) Delay time of the points before they leave (transmitted or reflected) the fundamental region. (c) Number of steps taken inside the fundamental region (only points with less than 20 steps shown) (reproduced from Lin et al. 2011)

4.4.3 Symbolic Dynamics A symbolic dynamics for this scattering process can be constructed by examining the structure of the gaps that are cut into the unstable manifolds, n–q and p–m, by the tendrils formed by the stable manifolds, r–p and s–n, as they are iterated backward in time. In Fig. 4.15a, we show the first iteration of the stable manifolds backward in time. The shaded regions contain points from the asymptotic regions that enter and cut through the fundamental region R on this first iteration of the PSS. The shaded regions cut two segments out of the unstable manifold of FL and cut one segment out of the unstable manifold of FR . We label the three remaining pieces of the unstable manifold of FL as “A”, “B”, and “C”. We label the two remaining pieces of the unstable manifold of FR as “+” and “C”. In Fig. 4.15b, we show the result of two iterations of the PSS. In this second iteration, we need to introduce two additional symbols to fully determine the symbolic dynamics. The left-most symbols of the unstable manifold of FL are again “A”, “B”, and “C” because these segments are formed in exactly that same way as in Fig. 4.15a.

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4 Chaotic Scattering

Fig. 4.15 (a) The first iteration of the stable manifolds backward in time. The shaded regions are points from the asymptotic region that cut through (into) the fundamental region on the first iteration. Symbols A, B, and C and symbols + and C denote parts of the unstable manifolds, FL and FR , respectively, left intact. (b) Second iteration of stable manifolds backward in time. Symbols A, B, C, B, −, +, and C denote regions of FL ¯ +, left intact. Symbols A, A, and C denote regions of FR left intact (reproduced from Lin et al. 2011)

The segment “B–C” in Fig. 4.15a, is now cut into four segments which we label “B”, “−”, “+”, and “C”. The symbols “+” and “C” are formed in a manner similar to their formation in Fig. 4.15a. The two segments of the unstable manifold of FR ¯ “+”, and “C”. The symbols are now cut into four segments that are labeled “A”, “A”, ¯ However, the left-right order of “−” and “+” both branch into the symbols A and A. A and A¯ is opposite for “−” and “+”. We distinguish this ordering by using the two symbols “−” and “+”, rather than one symbol “+” for both cases. When counting the number of symbols A and A¯ that emerge at each branching, the distinction between “−” and “+” is not important. Given these symbols, we can form a branching tree of symbols that show the fractal structure that emerges with successive iterations of the PSS. The branching tree for particles incident from the left is shown in Fig. 4.16a for four iterations of √ the PSS. In Fig. 4.16a, we retain both symbols “−” and “+”. Lines with symbol indicate that the particle is scattered toward FL (we call these SL type points) while lines with symbol ◦ indicate that it is scattered toward FR (these are SR type points). The branching tree for particles incident from the right is shown in Fig. 4.16b. We can now represent the branching trees in terms of a transfer matrix T6 that acts ¯ T (T denotes transpose) formed with on a column matrix S6 = {A, B, C, +, −, A} the six symbols that comprise the symbolic dynamics for this system. The transfer matrix T6 is given in Eq. (4.18),

4.4 Model of Chlorine Ion in a Radiation Field

121

Fig. 4.16 (a) The branching tree for symbolic dynamics associated with scattering from the left. (b) The branching tree for scattering from the right (reproduced from Lin et al. 2011)

⎛

1 ⎜1 ⎜ ⎜ ⎜1 T6 = ⎜ ⎜0 ⎜ ⎝0 0

0 1 0 0 1 0

0 0 1 1 0 0

1 0 0 0 0 1

1 0 0 0 0 1

⎞ 0 1⎟ ⎟ ⎟ 1⎟ ⎟, 0⎟ ⎟ 0⎠ 1

⎛ 1 ⎜1 ⎜ ⎜ T5 = ⎜1 ⎜ ⎝0 0

0 1 0 1 0

0 0 1 1 0

1 0 0 0 1

⎞ 0 1⎟ ⎟ ⎟ 1⎟ . ⎟ 0⎠ 1

(4.18)

If we do not distinguish the symbols “−” and “+” and everywhere replace the symbol “−” with “+”, then the transfer matrix becomes a 5×5 matrix T5 , which is shown in Eq. (4.18). The transfer matrix T5 acts on a column matrix S5 = ¯ T {A, B, C, +, A} The transfer matrices Tα (α = 5, 6) are not self-adjoint. Therefore, Tα will have (α) (α) (α) α left eigenvectors ψ j , α right eigenvectors φ j , and α eigenvalues λj , where (α)

(α)

j = 1, . . . , α. The eigenvectors satisfy orthonormality conditions ψ j ·φ j  = δj,j  . In terms of these eigenvalues and left and right eigenvectors, the transfer matrix α (α) (α) (α) can be written Tα = j =1 λj φ j ·ψ j . For both α = 5 and α = 6, there (α)

is one eigenvalue with value λ1 = 2.3146, while for all the other eigenvalues (α) Re[λj ]≤1, j = 2, . . . , α. Therefore, when the transfer matrix acts n times, we (α)

(α)

(α)

(α)

obtain Tnα →(λ1 )n φ 1 ·ψ 1 as n→∞. The left and right eigenvectors, ψ 1 (α) φ 1 respectively, of T6 and T5 are

and

122

4 Chaotic Scattering (6)

ψ 1 = (0.441, 0.290, 0.290, 0.381, 0.381, 0.441) (6)

φ 1 = (0.441, 0.671, 0.671, 0.290, 0.290, 0.441)T

(4.19)

and ψ (5) 1 = (0.441, 0.290, 0.290, 0.763, 0.441) (5)

φ 1 = (0.441, 0.671, 0.671, 0.580, 0.441)T ,

(4.20)

respectively. Note that, since the transfer matrices are known exactly, the accuracy of these numbers is only limited by the accuracy of the matrix solver. We can now reproduce the number of each of the symbols “A”, “B”, “C”, ¯ at level n in the branching tree for scattering from the “+”, “−”, and “A” left by allowing the transfer matrix to act n − 1 times on the initial partition T SL 1 = (1, 1, 1, 0, 0, 0) of the unstable manifold of FL (see Fig. 4.15a). We obtain (6) n−1 L T6 ·S1 = 1.021(2.315)n−1 φ 1 . Similarly, if we allow the transfer matrix to T act n − 1 times on the initial partition SR 1 = (0, 0, 1, 1, 0, 0) of the unstable R manifold of FR (see Fig. 4.15a) for scattering from the right, we obtain Tn−1 6 ·S1 = (6) 0.671(2.315)n−1 φ 1 (a similar analysis can be performed on T5 ). It only takes a few steps up the branching tree for scattering from the left, or from ¯ the right, for the fractional distribution of symbols “A”, “B”, “C”, “+”, “−”, and “A” to be determined by the fractional distribution of symbols in the right eigenvector φ (6) 1 . Thus, for both incident directions, the branching trees, as regards the fraction (6) of each type of symbol present in φ 1 , become the same. The fractions are fA = 0.157, fB = 0.239, fC = 0.239, f+ = 0.103, f− = 0.103, and fA¯ = 0.157. We now can use this information, and the information in Fig. 4.16b, to obtain a rough estimate of the likelihood that an incident particle, caught up in the fractal √ structure, gets transmitted or reflected. First, remember that (◦) indicates a gap containing trajectories that ultimately travel to −∞ and are of SL type (those that travel to +∞ and are of SR type). From Fig. 4.16b, we see that intervals of type “A” ¯ in an unstable manifold, in the next iteration of the map, will contain one and “A” SL gap and one SR gap. Intervals of type “B” and “C”, in the next iteration of the map, will contain two SL gaps. Intervals of type “+” and “−”, in the next iteration of the map, will contain two SR gaps. The fraction of SL gaps, in the next iteration of the map, is 0.636 and the fraction of SR gaps is 0.364. Thus, for particles incident from the left, 36.4% of the gaps get transmitted and 63.6% of the gaps get reflected. For particles incident from the right, 63.6% of the gaps get transmitted and 36.4% get reflected. The percentage of gaps transmitted or reflected gives an indication of the asymmetry of the overall scattering process. However, for a given line of initial points, the stepping time data, like that shown in Fig. 4.14c, can be used to distinguish which initial points are SR -type and which are SL -type. In Lin et al.

4.5 Chaos in the HOCl Molecular System

123

(2011), it is determined that, for the line of initial conditions shown in Fig. 4.16, of those crossing the tendril, 44.24% are transmitted and 55.76% are reflected. The PSS used here to analyze the scattering dynamics (we plot p and x each time φ = π/2 + n2π , with n = 0, 1, 2, . . .) corresponds to a particular phase of the radiation field at time t = 0. If the phase of the field is different at time t = 0, then the asymmetric scattering properties will differ in detail but will still be asymmetric. The procedure described here can be applied to any choice of initial phase of the radiation field. The symbolic dynamics of the scattering process, and the resulting transfer matrix, can provide an important tool for disentangling reflection from transmission in such scattering processes.

4.5 Chaos in the HOCl Molecular System One of the most active regions of a molecule involves the vibrational dynamics of its atomic constituents just above the dissociation energy of the molecule. The vibrational dynamics determine the pathways for dissociation and recombination of the constituents of the molecule. In most molecules, little is known about these dynamical processes because they have so many degrees of freedom. The nonlinear dynamics of systems with more than two degrees of freedom is not well understood. Recent studies of the classical nonlinear vibrational dynamics of small molecules, above the dissociation energy, have revealed bifurcations and dynamical structures that can have significant influence on the dissociation dynamics of the molecule (Farantos et al. 2009; Schinke et al. 2010; Schinke 2011; Mauguiere et al. 2011; Joyeux et al. 2005). Therefore, small molecules provide an important laboratory for studying the classical-quantum correspondence in systems with more than two degrees of freedom. One molecule that has provided some insight into molecular dissociation dynamics is HOCl. The HOCl molecule consists of three atoms, H, O, and Cl, with masses mH , mO and mCl , respectively, that lie in a plane. If we assume that the total angular momentum of the system Ltot = 0, then the plane remains stationary and all motion occurs in the plane of the molecule. We can introduce lab-frame coordinates (x  , y  , z ) and body-frame coordinates (x, y, z). The origin of the bodyframe coordinates is the center-of-mass of the molecule. When Ltot ≡0, we can assume that all of the dynamics occurs in the (x  , z ) and (x, z) planes and that any angular momentum vectors generated by internal rotations of the molecule (such vectors must add to zero because Ltot = 0) lie along the y  and y axes in the lab and body-fixed frames, respectively. In the body-frame, t1 is a vector that connects the center of mass of HO to Cl and t2 is a vector that connects H to O (see Fig. 4.17). The vector t1 has length R and the vector t2 has length r. The vibrational motion of HO does not contribute significantly to the HOCl dynamics at lower energies (Skokov et al. 1999; Jost 1999; Weiss et al. 2000; Joyeux et al. 2005). Therefore, we set r = ro , where ro is the ground state bond length of HO. This constraint, together with the condition that

124

4 Chaotic Scattering

Fig. 4.17 HOCl in the body-frame for zero total angular momentum of the molecule. The molecule is planar and lies in the lab frame x − z and in the body-frame x  − z plane

Ltot = 0, leads to a 2D model of HOCl dynamics. The angle between t1 and t2 is θ (note that θ = 0 for the linear configuration H–O–Cl). The center of mass of the molecule lies along t1 a distance md R/M from the Cl atom, where md = mO + mH and M = mCl + mO + mH . If we assume that the center of mass of the molecule is at rest, the kinetic energy of the molecule is given by T =

mCl md 2 mH mO 2 ˙t + ˙t . 2M 1 2md 2

(4.21)

dt where ˙t = dt . Let us assume that t1 lies along the body z-axis. We can then write t1 = Rˆz and t2 = ro sin(θ )ˆx +ro cos(θ )ˆz. We further assume that the body frame (x, z) axes make an angle β with respect to the lab frame (x  , z ) axes, so that if the two frames rotate relative to one another, the angular velocity of rotation is β˙ yˆ . The time derivatives ˙ z + β˙ yˆ × t1 and ˙t2 = ro θ˙ cos(θ )ˆx − ro θ˙ sin(θ )ˆz + of the vectors t1 and t2 are ˙t1 = Rˆ β˙ yˆ × t2 . Substituting into Eq. (4.21), we obtain

T =

μ1 ˙ 2 μ2 2 2 (R + R 2 β˙ 2 ) + (r θ˙ + 2ro2 θ˙ β˙ + ro2 β˙ 2 ), 2 2 o

(4.22)

where μ1 = mCl md /M and μ2 = mO mH /md . ˙ pθ = ∂T = μ2 ro2 (θ˙ + β), ˙ In terms of the canonical momenta pR = ∂∂TR˙ = μ1 R, ∂ θ˙ ∂T 2 2 ˙ the kinetic energy takes the form ˙ + μ1 R β, and pβ = = μ2 ro (θ˙ + β) ∂ β˙

2 pβ2 pθ2 pθ2 pR pθ pβ T = + + − + . 2μ1 2μ2 ro2 2μ1 R 2 μ1 R 2 2μ1 R 2

(4.23)

However, pβ is the total angular momentum of the molecule so pβ = 0, and the Hamiltonian for HOCl (in the center of mass frame) can be written

4.5 Chaos in the HOCl Molecular System

125

Fig. 4.18 Contour plot of the 2D HOCl potential energy surface (for the case when the HO bond is in its ground state) (plot based on Lin et al. 2015)

H =

2 pθ2 pθ2 pR + + + De V (R, θ ) = E, 2μ1 2μ2 ro2 2μ1 R 2

(4.24)

where V (R, θ ) is the potential energy of the interaction between the atoms in the HOCl molecule, and E is the total energy of the system. The experimentally obtained potential energy V (R, θ ) (Weiss et al. 2000), is shown in Fig. 4.18. We can now assign numbers to this model of HOCl. The HOCl molecule dissociates into a free Cl atom and a bound HO molecule at energy De = 20,312.3 cm−1 = 2.518 eV. The reduced masses have values μ1 = mClMmd = 595 52 u 16 O = u (u the atomic mass unit). It is useful to write the above and μ2 = mHmm 17 d quantities in terms of dimensionless units (d.u.). We parametrize all energies in units of De , lengths in units of the Bohr radius aB = 5.2917×10−11 m, and angular momenta in terms of Planck’s constant h¯ = 1.05457×10−34 J·s. The length of the H − O bond, in the ground state, is ro = 1.85 (in multiples of Bohr radii). Then  , p = h¯ p  , p = hp  , H = De H  , E = De E  , R = aB R  , r = aB r  , pR = ah¯B pR ¯ θ r θ aB r pβ = h¯ pβ , and time t = Dh¯e t  , where primed quantities are dimensionless. If we now drop the primes on dimensionless quantities, the (dimensionless) Hamiltonian takes the form H =

2 pR

22R

+

pθ2 22θ

+

pθ2 22R R 2

+ V (R, θ )

(4.25)

126

4 Chaotic Scattering μ a2 D

d 2 2 where 2R = 1 02 e = 1930.43 and 2θ = m μ1 ro R = 543.44. Thus, the HOCl h¯ system is reduced to a system with two degrees of freedom. A contour plot of the potential energy, V (R, θ ), for the case when the HO bond is in the ground state, is shown in Fig. 4.18. The potential energy is symmetric about θ = 0, has a high wall at θ = π , and a saddle point and potential hill along θ = 0. These potential barriers at θ = π and θ = 0 rise high above the HO+Cl dissociation energy and play a significant role in the scattering dynamics of the molecule. The potential energy minimum occurs at (Rm = 3.232 d.u., θm = 1.347 rad). The classical dynamics of the 2D HOCl molecule is dominated by KAM tori for energies below E = 14,000 cm−1 . However, for energies above E = 14,000 cm−1 and for energies approaching the dissociation energy E = 20,312.3 cm−1 , the molecular dynamics becomes increasingly chaotic. In Fig. 4.19, we show stretch and bend surfaces of section of the 2D model of the molecule for energies E = 14,000 cm−1 , E = 17,020 cm−1 , and E = 20,150 cm−1 . In the stretch surfaces of section (Fig. 4.19a, c, e), values of (PR , R) are plotted each time the bend angle (pθ > 0, θ = θm ). In the bend surfaces of section, values of (pθ , θ ) are plotted each time (pR > 0, R = Rm ). Below E = 14,000 cm−1 , the dynamics is dominated by nonlinear resonances and by the bifurcation of periodic orbits as energy increases (Barr et al. 2009). Above E = 14,000 cm−1 , resonances start to form overlapping self-similar structures and chaos begins to occupy ever larger regions of the phase space with increasing energy. In Fig. 4.19, some of the dominant remaining resonant orbits are indicated by black dots.

4.5.1 Homoclinic Tangles For energies above the dissociation energy E = 20,312.3 cm−1 , the dynamics of the HO–Cl complex is governed by the stable and unstable manifolds associated with an unstable periodic orbit at R = ∞. The stable and unstable manifolds of the unstable periodic orbit at R = ∞ form a homoclinic tangle in phase space. Figures 4.20a, b show the stable (mapped backward in time) and unstable (mapped forward in time) manifolds, respectively, which form the homoclinic tangle, in the stretch SOS at E = 21,000 cm−1 . Using the Hamiltonian in Eq. (4.25), the zeroth order stable (unstable) tendril (thick black line) is integrated backward (forward) in time to find each trajectory’s next intersection with the SOS to produce the first order tendril t1s (t1u ) (dashed line). Continuing in this way, we obtain the second order tendrils t2s and t2u (grey lines). Already at second order, the tangle displays a great deal of structure. The first and second order tendrils stretch into the asymptotic region with both tendrils exhibiting multiple folds. As the stable and unstable manifolds evolve towards smaller R values, they are repeatedly deflected by collisions with potential barriers at θ = π and at θ = 0. These deflections dominate the structure of the homoclinic tangle.

4.5 Chaos in the HOCl Molecular System

127

Fig. 4.19 Poincaré surfaces of section for HOCl. (a) Stretch SOS at E = 14,000 cm−1 . (b) Bend SOS at E = 14,000 cm−1 . (c) Stretch SOS at E = 17,020 cm−1 . (d) Bend SOS at E = 17,020 cm−1 . (e) Stretch SOS at E = 20,150 cm−1 . (f) Bend SOS at E = 20,150 cm−1 (based on Barr et al. 2009)

4.5.2 Scattering Dynamics For energies above the dissociation energy E = 20,312.3 cm−1 , the HO–Cl system can be analyzed using techniques from scattering theory (Lin et al. 2013). We consider the scattering of the Cl atom from the HO molecule, assuming that HO remains in its ground state configuration. Since the HO vibration is very stiff compared to that of the HO–Cl vibration, this is a reasonable assumption. When analyzing the scattering dynamics, we note that the total energy E (which p2

is conserved) is distributed between the incident energy of the Cl atom, ECl = 2δR1 , and the rotational energy of the HO molecule, Erot , so that E = ECl + Erot . The initial value of ECl determines the initial value of the momentum, pR , for a Cl atom

128 50 40

t2s t1s

20

pR

Fig. 4.20 (a) Stable and (b) unstable manifold of the periodic orbit at (R = ∞, pR = 0) that form the homoclinic tangle at energy E = 21,000 cm−1 . Zeroth-, first-, and second-order tendrils are denoted t0 , t1 , and t2 respectively (based on Barr et al. 2009)

4 Chaotic Scattering

t0s

0

-20 -40

(a) 50 40

pR

20

t2u

0

t0u

t1u

-20 -40

(b) 3

4

5

6

7

8

R incident from the asymptotic region. The remaining energy is Erot . The angular momentum of HO relative to its center of mass is given by L2 = pθ yˆ . For a given total energy E, in the asymptotic region, we start to monitor the dynamics of the incident Cl atom at R = Rin = 12 d.u.. After the Cl atom has scattered from HO, we again monitor its dynamics at R = Rout = 12 d.u.. We can specify a range of initial values of ECl . We can also specify a range of initial values for the angular position χ ≡θin of the Cl atom, relative to the HO dimer. Each initial condition can therefore be labeled uniquely by (E, pR , pθ , χ ). For a given value of total energy E, we can plot values of pR and pθ , after the scattering process has occurred, for ranges of initial phases 0≤χ ≤2π . We will here analyze scattering properties of HO–Cl for a subspace of the scattering process with total angular momentum zero and total energy E≤25,000 cm−1 . The subspace with total angular momentum zero gives a good picture of the nature of the scattering processes at play in less constrained configurations of the HO–Cl system. In Fig. 4.21, we show plots of pR versus χ for the 2D model of HOCl with the HO bond held fixed at its equilibrium displacement. Figure 4.21a shows that in the range of initial orientations 0≤χ 2π , there are two discontinuous regions where the initial condition has crossed the homoclinic tangles of the scattering system. In Fig. 4.21b, which is a magnification of the discontinuous region on the left of Fig. 4.21a, there are “mirror” points in the discontinuous region, located directly

4.5 Chaos in the HOCl Molecular System

129

Fig. 4.21 Scattering dynamics, pR versus χ, for the 2D HO–Cl system with incident total energy E≤25,000 cm−1 and pθ = 5.92 d.u.. (a) The initial conditions cross the homoclinic tangles in two intervals of angular orientation of HO relative to incident Cl. (b) Magnification of the left discontinuous region. (c) Magnification of the right discontinuous region (pR in dimensionless units and χ in radians) (based on Lin et al. 2013)

under the letters B, A, and C. The structures of local regions on either side of these mirror points are approximate mirror images of each other, although the righthand side is compressed relative to the left-hand side. In Fig. 4.21c, we show a magnification of the discontinuous region on the right of Fig. 4.21a. The mirror-like structures indicated in Fig. 4.21b repeat and can be found embedded locally in these plots as one goes to ever finer scales in the phase space, so there appears to be a fractal structure embedded in the phase space. Using the “box-counting” technique, it is possible to compute the fractal dimension of the intervals of discontinuity that appear to the left and right of Fig. 4.21a. The interval of discontinuity on the left has a fractal dimension of 0.80. The interval of discontinuity on the right has a fractal dimension of 0.86. In Fig. 4.22, we show the time it takes for Cl to leave the asymptotic region at R = 12 d.u., interact with HO, and finally return to the asymptotic region at R = 12 d.u. for a range of values of initial phases χ in the interval 0.679 rad≤χ ≤0.792 rad. in the neighborhood of the mirror point at B in Fig. 4.21b. Note that Fig. 4.22 shows the running times τ for Cl (time for Cl to “run” from its initial position at R = 12 d.u., interact with HO (delay time), and then “run” back to its initial position). The “running time” for a free particle to travel a distance 2R is approximately 3000 d.u. The running time again shows the influence of the homoclinic tangles on the HO–Cl scattering process. The analysis of the HO–Cl scattering process, in this section, has been restricted to the 2 DoF system consisting of the Cl atom scattering from the HO dimer in

130

4 Chaotic Scattering

Fig. 4.22 Running time plot for values of χ in the neighborhood of the mirror point B in Fig. 4.21b (based on Lin et al. 2013)

its ground state. However, if the vibrational dynamics of the HO dimer is taken into account, the scattering problem becomes a 3 DoF scattering problem. A comparison between the HO–Cl scattering problem for 2 DoF and that for 3 DoF has been done in Lin et al. (2013). They find that, because of the stiffness of the HO bond, as long as the dimer is initially close to its ground state, the 2 DoF model of the HO–Cl scattering process provides a surprisingly good picture of the full 3 DoF scattering problem. However, if initially the HO bond is in a fairly high excited state, a large portion of the initial conditions show fractal behavior and the HO–Cl scattering process can no longer be approximated by the 2 DoF model considered here.

4.6 Conclusions Studies of fractal scattering processes generally have focused on systems with two degrees of freedom (2 DoF), because they are easy to visualize and characterize using Poincaré surfaces of section (SOS). In 2 DoF systems, it is possible to follow the flow of stable and unstable manifolds, as they form an increasingly complex network of tendrils in the phase space. These tendrils can be categorized in terms of symbolic dynamics, and their fractal structure then becomes apparent. When dealing with scattering problems with three degrees of freedom (3 DoF), surfaces of section (SOS) become four dimensional and one cannot easily visualize the complexity of the scattering processes. In Wiggins (1994), it has been shown that in some parameter regimes, a 3 DoF system can be viewed as a 2 DoF system with a weakly coupled third degree of freedom for which an approximate conserved quantity (exact for the 2 DoF system) exists. The four dimensional surface of section can then be viewed as a continuous “stack” of two dimensional surfaces of section (Wiggins 1992; Waalkens et al. 2008; Waalkens and Wiggins 2010; Kovacs and Wiesenfeld 2001). We have seen that, for the HOCl molecule, the phase space available to the HO and Cl, above dissociation, is largely chaotic, and this affects the dissociation mechanisms. In this regard, it was recognized by Wigner long ago (1939) and Thiele (1962), that phase space structures play an important role in chemical reaction dynamics. There are a number of works that relate chemical reaction dynamics to the crossing of a key unstable manifold in the molecular phase space, called a Normally Hyperbolic Invariant Manifold (NHIM) (Wiggins 1994). Constituents of a chemical

References

131

reaction must cross the NHIM to accomplish dissociation of the molecule. There is now extensive literature investigating these structures and their impact on chemical reaction dynamics (see for example Waalkens and Wiggins 2004 and Waalkens et al. 2008 for a review). Some of the analysis used to understand molecular dynamics has now been extended to galaxies. Zotos and Jung (2019) have used NHIMs to try to understand the structure of barred galaxies. They build their model of the galaxy from two symmetrically placed NHIMs, and use these to construct a model of barred galaxies that is consistent with what is observed in astronomical observations.

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Smale S (1967) Bull Am Math Soc 73:747 Smilansky U (1992) In: Giannoni MJ, Voros A, Zinn-Justin J (eds) Chaos and quantum physics (Proceedings of the 1989 Les HouchesSummer School). North-Holland, Amsterdam, p 371 Stomer C (1907) Arch Sci Phys Nat 24:305 Thiele E (1962) J Chem Phys 36:1466 Troll G, Smilansky U (1989) Phys D 35:34 Waalkens H, Wiggens S (2004) J Phys A 37:L435 Waalkens H, Wiggins S (2010) Reg Chao Dyn 15:1 Waalkens H, Schubert R, Wiggins S (2008) Nonlinearity 21:R1 Weiss J, Hauschildt J, Grebenshchikov SY, Düren R, Schinke R, Koput J, Stamatiadis S, Farantos SC (2000) J Chem Phys 112:77 Wiggins S (1992) Chaotic transport in dynamical systems. Springer, New York Wiggins S (1994) Normally hyperbolic invariant manifolds in dynamical systems. Springer, Berlin Wigner E (1939) J Chem Phys 7:646 Yao G, Chu S-I (1992) Phys Rev A 45:6735 Zotos EE, Jung C (2019) Mon Not R Astron Soc 487:1233

Chapter 5

Arnol’d Diffusion

Abstract All conservative nonintegrable classical systems, with three or more degrees of freedom (DoF), can undergo a global transition to chaos due to a dense web of resonances (the Arnol’d web) that permeates the phase space. Each resonance line in the Arnol’d web is surrounded by a resonance region and a stochastic web. The system can diffuse throughout each energy surface, both along and across the web in a universal diffusion process called Arnol’d diffusion. The rate of diffusion depends on parameters of the system. Because Arnol’d diffusion occurs in systems with three or more degrees of freedom, it is difficult to visualize and difficult to analyze. The solar system is one of the most studied classical systems that shows the effects of Arnol’d diffusion. The motion of planets, and other objects in the solar system, is affected by the Arnol’d web that permeates the solar system phase space. It also plays an important role in dynamic astronomy (Cincotta, 2002). Another type of system that shows Arnol’d diffusion, and is amenable to experiment, is a timeperiodically driven optical lattice. A classical periodically modulated optical lattice with two space dimensions does contain a complete Arnold web and will show a global transition to chaos. When the dynamics is governed by quantum mechanics, it will also show the manifestations of Arnol’d diffusion. Keywords Arnol’d diffusion · Arnol’d web · Nekhoroshev time · Nonlinear resonance · Driven optical lattice · Solar system stability · Time-periodic optical lattice · Particle accelerators

5.1 Introduction All conservative nonintegrable classical systems, with three or more degrees of freedom (DoF), can undergo a global transition to chaos due to a dense web of resonances that permeates their phase space Arnol’d [1963, 1964]. Arnol’d showed that each energy surface of the classical phase space is covered by this web of resonance lines (the Arnol’d web). The system can diffuse throughout each energy surface, both along and across the web. This universal diffusion process is called © Springer Nature Switzerland AG 2021 L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200, https://doi.org/10.1007/978-3-030-63534-3_5

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Arnol’d diffusion. In systems with two DoF, KAM tori can block the diffusion of trajectories throughout the phase space. That is no longer the case for systems with three or more DoF. For non-integrable systems with three or more degrees of freedom, trajectories will definitely diffuse throughout the phase space. The question one must then ask is “how rapid is the diffusion”? Because Arnol’d diffusion occurs in systems with three or more degrees of freedom, it is difficult to visualize and difficult to analyze. However, there are a few systems for which the Arnol’d web has been studied in some detail, and we describe some of these systems in the following sections. We start in Sect. 5.2 with an example of an Arnol’d web that forms when two delta-kicked rotors (standard maps) are coupled (Kaneko and Bagley 1985). Kaneko and Bagley were able to locate the resonance lines for this system and construct a map of the dynamics that shows clearly how Arnol’d diffusion fundamentally alters the nature of the dynamics from that described for a single standard map in Chap. 3. An estimate of the distance in phase space that a trajectory can diffuse, in a given amount of time, in a system with an Arnol’d web, was obtained by Nekhoroshev (1971, 1977) and is discussed in Sect. 5.3. Each resonance line in the web is surrounded by a resonance region and a stochastic web. For integrable systems, that are rendered non-integrable by a small perturbation, Nekhoroshev showed that diffusion occurs along the stochastic layer of the resonance line and is a slow process (the Nekhoroshev regime). However, as the strength of the perturbation is increased, the resonance regions along the lines can begin to overlap and global diffusion can begin. This is called the Chirikov regime (Chirikov 1979). (See also (Lichtenberg and Lieberman 1991) for additional discussions regarding Arnol’d diffusion.) A beautiful graphical picture of the Arnol’d diffusion process was obtained by Froeschle et al. (2000). Some of their results are described and shown in Sect. 5.4. A real-world system that shows Arnol’d diffusion and is amenable to experiment is a time-periodically driven optical lattice. Such systems have been studied in experiments for the case of a periodically modulated optical lattice with one space dimension (Steck et al. 2001). But periodically driven lattices with one space dimension only have two DoF and do not exhibit Arnol’d diffusion. In Sect. 5.5, we show that a classical periodically modulated optical lattice with two space dimensions does contain a complete Arnold web and will show a global transition to chaos (Boretz and Reichl 2016). In Chap. 10, we show that when the dynamics is governed by quantum mechanics, it will also be affected by Arnol’d diffusion. Probably the most studied classical system, that clearly shows the effects of Arnol’d diffusion, is the solar system. In Sect. 5.6, we discuss what is known, to date, about how the planets and other objects in the solar system are affected by the Arnol’d web that permeates the solar system phase space. In the past, the Arnol’d web was also a concern for high energy particle accelerators because it could destabilize the particle beams. This issue is discussed in Sect. 5.7 and, finally in Sect. 5.8, we make some concluding remarks.

5.2 The Arnol’d Web

135

5.2 The Arnol’d Web For systems with 2 DoF, diffusion from one chaotic region to another can be blocked by KAM surfaces. For such systems, although the phase space is four-dimensional, energy conservation restricts the flow of trajectories to a three-dimensional surface, and KAM tori are two-dimensional. The two-dimensional KAM surfaces can divide a three-dimensional space into disconnected regions. Kaneko and Bagley (1985) considered a simple model with 3 DoF that can be constructed by coupling two delta-kicked rotors. They showed the existence of the Arnol’d web for this model system. The two coupled delta-kicked rotors have coordinates (I, θ ) and (J, ψ) and a Hamiltonian given by  1 2 1 2 K1 K2 H = I + J + cos(2π θ ) + cos(2π ψ) 2 2 (2π )2 (2π )2    ∞ b + cos[2π(θ + ψ)] δ(t − M) . (2π )2

(5.1)

M=−∞

The delta function can be expanded in a cosine series as ∞ 

δ(t − M) = 1 + 2

M=−∞

∞ 

cos (2π mt) .

(5.2)

m=1

If we introduce canonical variables (p, x = t), we can write Eq. (5.1) in the form of a time-independent Hamiltonian with 3 DoF, H =

∞   1 2 1 2 K1 I + J +p+ cos[2π(θ − Mx)] 2 2 (2π )2 M=−∞

 K2 b + cos[2π(ψ − Mx)] + cos[2π(θ + ψ − Mx)] = E. (2π )2 (2π )2 (5.3) For small K1 , K2 , and b, we can easily locate resonance lines. The unperturbed Hamiltonian is Ho = 12 I 2 + 12 J 2 + p = Eo . This gives rise to a partial energy surface, p = Eo − 12 I 2 − 12 J 2 , which is two-dimensional (it is plotted in Fig. 5.1). There are an infinite number of resonance conditions θ˙ − M x˙ = 0, ψ˙ − M x˙ = 0, and θ˙ + ψ˙ − M x˙ = 0 (where integer M has the range −∞ ≤ M ≤ ∞). ∂Ho ∂Ho o ˙ If we note that θ˙ ≈ ∂H ∂I = I , ψ ≈ ∂J = J , and x˙ ≈ ∂p = 1, then the resonance conditions for the primary resonances take the form I = M,

J = M,

J + I = M.

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Fig. 5.1 The partial energy surface for the three DoF coupled delta-kicked rotor system. The resonance lines I = 0, J = 0, I = M, and J = M have been sketched in. These, together with an infinity of other resonance lines (not shown), form the Arnol’d web along which trajectories can diffuse

In addition, there will be infinite families of additional higher-order resonances. In Fig. 5.1, we have drawn four resonance lines that result from resonances I = 0, J = 0, I = M, and J = M. These resonance lines intersect one another. If we could draw in all resonance lines, the partial energy surface would contain a dense network of intersecting resonance lines. This is the Arnol’d web. The system can, in principle, diffuse along this network of resonance lines and eventually come close to any point on the partial energy surface as long as it stays on a resonance line. Of course, the time it takes to reach a given region of the partial energy surface may be astronomically long. In the next section, we show some numerical results indicating that diffusion, along the resonance lines that form the web, does indeed occur. The dynamics of the coupled delta-kicked rotors can be written in terms of coupled standard maps and has been studied numerically by Kaneko and Bagley (1985). From the Hamiltonian in Eq. (5.1), it is easy to construct a four-dimensional mapping following the procedure used for the standard map (see Chap. 3). We obtain K1 b sin(2π θn ) + sin[2π(θn + ψn )], 2π 2π = θn + In+1 ,

In+1 = In +

(5.4)

θn+1

(5.5)

K2 b sin(2π ψn ) + sin[2π(θn + ψn )], 2π 2π = ψn + Jn+1 .

Jn+1 = Jn +

(5.6)

ψn+1

(5.7)

For b = 0, the two maps evolve independently of one another. Kaneko and Bagley have studied the coupled standard map model for K1 = K2 = 0.8 and b = 0.02 so that the coupling between the standard maps is weak and they do not greatly perturb one another. They started the mapping with initial conditions (I = 0.5, θ = 0.3, J = 0.4, ψ = 0.2). The (I, θ ) point starts in the stochastic layer of the ω = 12 secondary resonance. Figure 5.2 shows the behavior

5.3 Arnol’d Diffusion and Nekhoroshev Time

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Fig. 5.2 A single trajectory of the map in Eqs. (5.4)–(5.7), for K1 = K2 = 0.8 and b = 0.02. The total time interval is 0 < n < 2.0 × 105 and the initial conditions are I = 0.5, θ = 0.3, J = 0.4, ψ = 0.2. The four figures show different sections of the total time interval: (a) 0 < n ≤ 5.0 × 104 ; (b) 5.0 × 104 < n ≤ 105 ; (c) 105 < n ≤ 1.5 × 105 ; (d) 1.5 × 105 < n ≤ 2.0 × 105 (Kaneko and Bagley 1985)

of this trajectory over a very long period of time (n = 2×105 ). The figure divides the total time of the mapping into four time intervals. We see that the trajectory remains in the stochastic layer of the ω = 12 secondary resonance for a very long time and then finally during the time interval 1.5 × 105 < n < 2.0 × 105 the trajectory suddenly appears in the stochastic separatrix of the ω = 23 secondary resonance and the ω = 01 and ω = 11 primary resonances. Thus the trajectory appears to have found a path along the Arnol’d web out of the stochastic separatrix of the ω = 12 secondary resonance into other stochastic separatrices.

5.3 Arnol’d Diffusion and Nekhoroshev Time For systems near integrability, the Hamiltonian can generally be written in the form H ({Ij , φj }) = H0 ({Ij }) + V ({Ij , φj })

(5.8)

where {Ij , φj }, (j = 1, . . . , d) are action-angle variables, H0 is the Hamiltonian of a nonlinear integrable system, and is a small perturbation that breaks the integrability. We assume that the system satisfies the KAM theorem, which applies if the perturbation is smooth and the integrable system is non-degenerate. When

is very small, most of the phase space will consist of non-resonant KAM tori, although slightly deformed from the integrable case. In addition, the phase space will contain a dense set of resonance lines (the Arnol’d web) determined by the

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 resonance conditions j nj ωj ({Ij }) = 0, where ωj = ∂Ho /∂Ij and nj range over all √ integers. For small , each resonance line has a width of order (or smaller than)

(the width of a pendulum separatrix—see Appendix B) and a stochastic layer associated with it. Trajectories can diffuse along these stochastic layers. Nekhoroshev showed that, for very small , diffusion along the stochastic layers may take a very long time (Nekhoroshev 1971, 1977). He found that, if Ij (0) is the initial value of an action variable then, in time τN , it will have traveled a “distance” |Ij (τN )−Ij (0)|∼ β along a stochastic layer in a time of order τN ∼(1/ )exp(1/ α ), where 0 < α < 1 and 0 < β < 1. For small , this is indeed a very long time. As increases, the width of the resonance regions increases until the system reaches the Chirikov regime (Chirikov 1979), where resonances not only cross but also begin to overlap. Then, chaotic regions of the phase space and random diffusion through the phase space can become global. One of the systems originally considered by Arnold (1963) in describing diffusion in the phase space of systems with three or more DoF had the Hamiltonian (the Arnol’d Hamiltonian) H =

1 2 (J + J22 ) + (cos(θ1 ) − 1)(1 + μ sin(θ2 ) + μ cos(t)), 2 1

(5.9)

where (J1 , J2 , θ1 , θ2 ) are action-angle variables and and μ are small coupling parameters. This system contains six primary resonances. If we introduce the coordinates (p, x = t), we can write Eq. (5.9) in the time-independent form H =

1 2 (J + J22 ) + p + (cos(θ1 ) − 1) − μ cos(x) − μ sin(θ2 ) 2 1 μ

+ [sin(θ2 − θ1 ) + sin(θ2 + θ1 ) + cos(θ1 − x) + cos(θ1 + x)]. (5.10) 2

The locations of the six primary resonances are determined by the equations θ˙1 ≈ J1 = 0,

θ˙2 ≈ J2 = 0,

θ˙1 ± θ˙2 ≈ J1 ± J2 = 0,

θ˙1 ± x˙ ≈ J1 ± 1 = 0. (5.11) √ The resonance zone, J1 = 0, has a width proportional to , while all other √ primary resonance zones have a width proportional to μ. For and small μ, the partial energy surface, Ho = 12 (J12 + J22 ) + p, is the same as that in Fig. 5.1. The resonance zones are given approximately by the intersection of the resonance surfaces in Eq. (5.11) with the unperturbed partial energy surface. The projection of the resonance curves onto the J1 − J2 plane and the respective widths of the resonance zones are shown in Fig. 5.3. Note that for a trajectory starting in the stochastic layer of the resonance zone centered at J1 = 0, the change in J1 due to diffusion across the resonance zone (in the ±J1 direction) is constrained to the width of the resonance. However, the value of J2 can increase significantly due to diffusion along the J1 = 0 resonance (in the ±J2 direction). A trajectory can also escape the J1 = 0 resonance by moving onto one of the smaller resonances that

5.4 Graphical Evolution of the Arnol’d Web

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Fig. 5.3 Projection of the primary resonances in Arnol’d’s model onto the J1 − J2 plane. The widths of the resonances are indicated in the figure

intersect it. However, this is much less probable (as long as  μ) than diffusion along the J1 = 0 resonance. Since the J1 = 0 resonance is dominant when  μ, it is called the guiding resonance. The diffusion along the guiding resonance and onto resonances that intersect it is called Arnol’d diffusion.

5.4 Graphical Evolution of the Arnol’d Web Froeschle et al. (2000) and Lega et al. (2003) have explored, numerically, a nonlinear system that contains a complete Arnol’d web. The Hamiltonian they consider, in terms of action-angle variables, can be written   I22 I12 1 + + I3 +

, (5.12) H = 2 2 cos(φ1 ) + cos(φ2 ) + cos(φ3 ) + 4 where I1 , I2 , and I3 are action variables, and φ1 , φ2 , and φ3 are the corresponding angle variables. The parameter determines the strength of the perturbation. This Hamiltonian can be expanded in terms of an infinite number of resonance terms, ∞

∞

∞

   I2 I2 H = 1 + 2 + I3 +

Vn1 ,n2 ,n3 cos(n1 φ1 + n2 φ2 + n3 φ3 ). 2 2 n1 =−∞n2 =−∞n3 =−∞ (5.13) The resonance condition is  n=1,2,3

ni

 dφi ∂H0 ni = n1 I1 + n2 I2 + n3 = 0, = dt ∂Ii n=1,2,3

where we have used Hamilton’s equations,

dφi dt

=

∂H0 ∂Ii ,

to lowest order in .

(5.14)

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Fig. 5.4 (a) Black regions are resonance islands and KAM tori of the Arnol’d web. Light regions are chaos. = 0.001 and the computer run time is t = 1000. (b) Enlargement of a section of (a) with = 0.001 t = 4000 (from (Froeschle et al. 2000) and reprinted with permission from AAAS)

Arnol’d web can be represented in the two-dimensional plane {I1 , I2 }. All i resonances n1 ω1 + n2 ω2 + n3 = 0 (with ωi = dφ dt ) appear as straight lines n1 I1 + n2 I2 + n3 = 0 in the I1 , I2 plane. The set of all resonances is dense in the plane I1 , I2 . In Fig. 5.4a, b, = 0.001 and the resonant lines are embedded in large zones filled with KAM tori. Because the perturbation has a full Fourier spectrum (all harmonics are present at order ), a large number of resonances are visible at small

and for the computer run time used (t = 1000). As increases, the volume of invariant tori decreases and the chaotic regions become more evident at the crossing of resonances. For = 0.04 (Fig. 5.5a, b), the majority of invariant tori have disappeared because of resonance overlapping, and a chaotically connected region has replaced large regions of KAM tori. To obtain the clear differentiation between chaotic and regular orbits in Figs. 5.4 and 5.5, the authors used a version of the Lyapounov exponent called the fast Lyapounov indicator (FLI) (Froeschle et al. 1997), which allowed them to the discriminate between regular and chaotic motion more easily than the simpler form of Lyapounov exponents described in Chap. 2.

5.5 Arnol’d Diffusion in an Optical Lattice An optical lattice, in one space dimension, is formed when two counter-propagating laser beams, with the same wavelength and frequency, interfere and form a spatially periodic polarization pattern. The periodic potential that is formed can trap neutral two-level atoms. Similarly, an optical lattice, in two space dimensions, can be

5.5 Arnol’d Diffusion in an Optical Lattice

141

Fig. 5.5 (a) Black regions are resonance islands and KAM tori of the Arnol’d web. Light regions are chaos. = 0.04 t = 1000. (b) Enlargement of a section of (a) with = 0.04 t = 2000 (from (Froeschle et al. 2000) and reprinted with permission from AAAS)

formed by two pairs of counter-propagating laser beams (the pairs set at 90◦ angles to one another). If the counter-propagating laser pairs have a slightly different frequency, the resulting periodic lattice will have an amplitude that oscillates in time. Optical lattices with two space dimensions have been realized in the laboratory by several experimental groups (Hemmerich et al. 1991; Greiner et al. 2001, 2002; Bloch et al. 2012). In this section, we use parameters realizable in experiments involving rubidium atoms (Bloch et al. 2012), but introduce a time-periodic modulation (TPM) of the amplitude of the laser beams. The TPM optical lattice we consider has 3 DoF and can show Arnol’d diffusion for a range of its parameters (Boretz and Reichl 2016). The lattice we consider is also a generalization of the 2 DoF TPM optical lattice, with embedded cesium atoms, considered in the Texas experiment (Steck et al. 2001, 2002; Luter and Reichl 2002), and the 2 DoF optical lattice, with embedded sodium Bose–Einstein condensate, considered in the NIST experiment (Hensinger et al. 2001). Those experiments showed the existence of chaos-assisted tunneling in the atomic dynamics of the system. However, in both experiments, the TPM optical lattice had only 2 DoF and could not show Arnol’d diffusion. The Hamiltonian, in dimensionless units, for two-level atoms in a two dimensional time-periodic optical lattice, can be written ! " H(t) = px2 + py2 + U cos2 (ω t) cos2 (x) + cos2 (y) + b cos(x) cos(y) ,

(5.15)

where U is proportional to the intensity of the laser radiation that forms the static optical lattice, ω is the oscillation frequency, and b = 2ˆ 1 ·ˆ 2 (0≤b≤2), where ˆ1 (ˆ 2 ) is the polarization unit vector for the counter-propagating laser pair along the x-direction (y-direction). When ˆ1 and ˆ2 are parallel, there is maximum coupling between the dynamics in the x- and y-directions, and when ˆ1 and ˆ2 are perpendicular, there is no coupling between the x- and y-directions.

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We want to examine the behavior of the atoms in this lattice, both in the static case and with increasing amplitude of the time-periodic modulation of the optical lattice amplitude. To this end, we generalize this Hamiltonian and write it in the form   H (t) = px2 +py2 +(V0 +V1 cos2 (ωt))U b cos(x) cos(y) + cos2 (x) + cos2 (y) , (5.16) where V0 + V1 = 1 and we choose U = 20 and ω = 2π . The case V1 = 0 corresponds to the 2 DoF static lattice considered in Horsley et al. (2014). For the case V1 = 1, the system corresponds to a 3 DoF generalization of the 2DoF timeperiodic lattices analyzed in experiments (Steck et al. 2001, 2002; Luter and Reichl 2002; Hensinger et al. 2001). The constant b is a parameter that measures the coupling of the atomic motion in the x and y directions. For the case b = 0, the atomic dynamics in the x and y directions is decoupled. For b = 0 and V1 = 0, the atomic dynamics is integrable and equivalent of two decoupled pendulums. For b = 0 and V1 =0, the atomic dynamics in each of the x and y directions is decoupled from one another but can be locally chaotic, and each is equivalent to the 2 DoF lattices considered in Steck et al. (2001, 2002), Hensinger et al. (2001), and Luter and Reichl (2002). For the case, b =0 and V1 =0, the dynamics in the x and y directions is coupled. The potential energy has an amplitude that varies periodically in time, so the atomic motion has 3 DoF and atoms must navigate an Arnol’d web. In Fig. 5.6a, we show the potential energy in the unit cell of the time-periodic optical lattice for b = 0.2 and V1 = 1 as a function of x, y, and ωt (the third coordinate). The static version of the optical lattice considered here was analyzed in Horsley et al. (2014), and as was shown there, the static lattice for b = 0.2 and V1 = 0 has potential wells and various saddle points that localize low energy particles. Some of this structure can be seen in Fig. 5.6a, although the amplitude of the potential energy varies periodically in time. In Fig. 5.6b, we show the same plot but for b = 2.0 and V1 = 1. For this case, the potential energy for a particle traveling along the time axis looks like a soft Lorenz gas that turns on and off. As shown in Horsley et al. (2014), for b = 2.0 and V1 = 0 (the static case), the lattice is open and dominated by chaos. A similar behavior appears to occur for the case b = 2.0 and V1 = 1 but with more degrees of freedom. It is useful to rewrite the Hamiltonian in the form U U U cos2 (x) + cos2 (y) + b cos(x)cos(y) 2 2 2 U U + cos(2ωt)cos2 (x) + cos(2ωt)cos2 (y) 2 2 U +b cos(2ωt)cos(x)cos(y). 2

H (t) = px2 + py2 +

(5.17)

5.5 Arnol’d Diffusion in an Optical Lattice

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Fig. 5.6 (a) Contour plot of the potential energy in the unit cell for one period of oscillation of the driving field for (a) U = 20, V1 = 1.0, b = 0.2, and (b) U = 20, V1 = 1.0, b = 2.0 (reproduced from (Boretz and Reichl 2016))

With this form of the Hamiltonian, we can use the transformation to action-angle variables for the pendulum that was described in Appendix B. We first give the Hamiltonian for both degrees of freedom in the libration region and then for both the rotation region. For simplicity, we do not consider the case where one degree of freedom lies in the libration region, and the other in the rotation region.

5.5.1 Arnol’d Web If we perform the canonical transformation (px , x, py , y)→(Jx , θx , Jy , θy ) discussed in Appendix B and note that cos( π2 + x) = −sin(x) and sin[am(gx , κx )] = sn(gx , κx ), the Hamiltonian takes the form U Ax,y sn[fx , κx ] sn[fy , κy ] 2 U U + Bx cos(2ωt)sn2 [fx , κx ] + By cos(2ωt)sn2 [fy , κy ] 2 2 U +b cos(2ωt)Ax,y sn[fx , κx ] sn[fy , κy ]. 2 H (t) = Ex + Ey + b

(5.18)

where, for example, sn[fx , κx ] is a Jacobi sn function with modulus κx . As described in Appendix B, the various parameters in this Hamiltonian have a different functional form depending on whether Ej < U2 or Ej > U2 (for j = x, y).

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5 Arnol’d Diffusion

Below we consider two cases, Ex < U2 and Ey < U2 , which we call libration, and Ex > U2 and Ey > U2 , which we call rotation. The libration region occurs at low energy where Ej < U2 (j = x, y). Then in 2E

y 2 x Eq. (5.18), Ax,y = κx κy , Bx = κx2 , and By = κy2 , κx2 = 2E U , κy = U . The rotation region occurs for higher energy where Ej > U2 (j = x, y). Then in Eq. U U (5.18), Ax,y = 1, Bx = 1, and By = 1, κx2 = 2E , κy2 = 2E . In both cases, x y

fx = π2 K(κx )θx , fy = π2 K(κy )θy , and K(κx ) is the complete elliptic integral of the first kind. The Jacobi sn function has a series expansion (Byrd and Friedman 1971). sn[z, κ] =

∞  m=0

  πz Cm (κ) sin (2m + 1) 2K(κ)

(5.19)

   (κ) π where Cm (κ) = κK(κ) . Some values of Cm (κ) include csch (2m + 1) π2 KK(κ) C0 (0.999999) = 1.2530, C0 (0.5) = 1.0176, C0 (0.1) = 1.00063, C1 (0.999999) = 0.3687, C1 (0.5) = 0.0180, C1 (0.1) = 0.0006, C2 (0.999999) = 0.1546, C2 (0.5) = 0.0003, C2 (0.1) = 4×10−7 . The values fall off rapidly with increasing m. If we now substitute the series for the Jacobi sn function in Eq. (5.19) into the Hamiltonian in Eq. (5.18), and combine the trig functions, we can write the Hamiltonian in the form H (t) = Ex (Jx ) + Ey (Jy ) +

 ∞ ∞ U     Bj Cm1 (κj )Cm2 (κx ) 8 m1 =0m2 =0 β=±1j =x,y

# × cos[2(M− θj + βωt)] − cos[2(M+ θj + βωt)] 

+b

  ∞ ∞ U    Ax,y Cmx (κx )Cmy (κy ) β cos(Mx θx − βMy θy ) 4 mx =0my =0 β=±1

+

#  1  + γ cos[Mx θx − γ My θy + β2ωt] 2 γ =±1

(5.20) where M− = m1 − m2 , M+ = m1 + m2 + 1, Mx = 2mx + 1, and My = 2my + 1. The Hamiltonain in Eq. (5.20) contains an infinite number of primary resonances. Furthermore, the interaction between the primary resonances gives rise to infinite families of higher order resonances (Reichl 1989). We can get a rough estimate of the location of the primary resonances. From Hamilton’s equations we know that, to dEy x ˙ zeroth order in κCm (κ), we have θ˙x = dE dJx and θy = dJy . Then, the approximate location of the primary resonances is given by

5.5 Arnol’d Diffusion in an Optical Lattice

145

Fig. 5.7 The Arnol’d web for V1 = 1. (a) Libration region with Ex < 10 and Ey < 10. The resonances are labeled a, b, or c according to Eq. (5.21). Lines not labeled are type-d. All type-a (type-b) resonances lie on vertical (horizontal) lines. The diagonal line bisecting the figure contains all the type-c resonances. The type-d lines that converge toward the origin correspond to ±mx ∓my = ±1 with 0≤mx , my ≤4. The type-d lines that end along the Jx = 0 or Jy = 0 axes correspond to mx = my with 0≤mx ≤7. (b) Rotation region with 10 < Ex and 10 < Ey . The same resonance designations apply except that, in this case, there are additional type-c lines that run parallel to the diagonal line that bisects the figure (reproduced from Boretz and Reichl 2016)

(a) M± θ˙x + βω = 0, (b) M± θ˙y + βω = 0, (c) Mx θ˙x − βMy θ˙y = 0, (d) Mx θ˙x − γ My θ˙y + β2ω = 0.

(5.21)

In Fig. 5.7a, we have plotted the location of a few of the infinite number of the primary resonances that form the Arnol’d web for the libration region, for low values of m1 , m2 , Mx and My . Note that, when b = 0, all the lines disappear except for the vertical and horizontal lines. For b =0, as the amplitude of the resonance terms grows, either with increasing U or b, the region influenced by each resonance widens, and as resonances start to “overlap”, chaos appears. This process occurs at all length scales in the classical phase space. The locations of some of the lower order primary resonances for the rotation region are shown in Fig. 5.7b. Again, the figure shows only a few of the infinite number of resonance lines that form the Arnol’d web.

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5 Arnol’d Diffusion

5.5.2 Arnol’d Diffusion The Hamiltonian in Eq. (5.17) can be written in a time-independent form if we replace the time by an angle variable φ3 = ωt in Eq. (5.17) and include the corresponding action, I3 . The Hamiltonian then takes the form U U U cos2 (x) + cos2 (y) + b cos(x)cos(y) 2 2 2 U U + cos(2φ3 )cos2 (x) + cos(2φ3 )cos2 (y) 2 2 U +b cos(2φ3 )cos(x)cos(y). (5.22) 2

H = px2 + py2 + ωI3 +

If one writes Hamilton’s equations for this system, one finds that φ3 (t) = ωt and I3 = −H (t)/ω, so no new dynamics is involved. This Hamiltonian, when written in terms of action-angle variables (see Eq. (5.20)), contains all harmonics at first order in the parameter b, which is the parameter governing the character of the Arnol’d diffusion. We can use the Walker-Ford method discussed in Chap. 2 to obtain a very rough estimate of the value of b for which the Nekhoroshev regime transitions to the Chirikov regime. For the rotation-version of the Hamiltonian in Eq. (5.20), we can separate out each primary resonance and plot it. For example, if we plot the primary√resonance, mx = my = 0 (Mx = My =√1), we can measure the resonance widths and find that they are given by (b, ) = (0.002, 0.08), (0.02, 0.2), and (0.2, 0.8) (see Fig. 5.8). Therefore, (b, ) = (0.002, 0.0064), (0.02, 0.04), and (0.2, 0.64). The Nekhoroshev estimate of the time for an action variable to change by an amount β (with β < 1) is τN = (1/ )exp[1/ α ] (with α < 1). As

decreases, τN varies slowly until it reaches a value N at which τN begins to grow

Fig. 5.8 The primary resonance mx = my = 0 (Mx = My = 1) √ by itself is an integrable system. Set U=20, b = 2.0, φ = θ − θ , and J = 16 − J . (a) (b,

) = (0.002, 0.08), (b) x y y x √ √ (b, ) = (0.02, 0.2), and (c) (b, ) = (0.2, 0.8)

5.5 Arnol’d Diffusion in an Optical Lattice

147

Fig. 5.9 Strobe plots of px versus x for initial conditions x(0) = π2 , −30≤px (0)≤30 with spacing of 0.6, y(0) = π , py (0) = 0.5. (a) Strobe plot for sub-Hamiltonian Hx = px2 + U2 cos2 (x) + U 2 2 cos(2ωt)cos (x). (b) Strobe plot for full Hamiltonian in Eq. (5.17) for b = 0. (c) Strobe plot for full Hamiltonian Eq. (5.17) for b = 0.002. (d) Strobe plot for full Hamiltonian in Eq. (5.17) for b = 0.02. All plots run for 2000 periods of the driving field (reproduced from Boretz and Reichl 2016)

exponentially. When α = 0.5, the rapid growth in τN occurs for N ≈0.05 while for α = 0.9 it occurs for N ≈0.14 (we don’t have an accurate estimate for α). Therefore, extrapolating between the measured values of for the optical lattice, the transition between the Nekhoroshev and Chirikov regimes (the rapid growth of τN ) appears to occur around 0.009≤b≤0.02. The phenomenon of Arnol’d diffusion can be seen explicitly in strobe plots of the dynamics if we compare plots for 2 DoF systems and 3 DoF systems. In Fig. 5.9a, we show a strobe plot of px versus x for the 2 DoF system with Hamiltonian Hx = px2 + U2 cos2 (x) + U2 cos(2ωt)cos2 (x) and U = 20. The initial conditions are x(0) = π2 and −30≤px (0)≤30 with spacing of 0.6. Coordinates (px , x) are plotted at each period of the oscillation. This Hamiltonian is the same as that considered in (Steck et al. 2001, 2002) and (Luter and Reichl 2002). It has three primary resonances which, for the parameters used here, overlap and give rise to the chaotic region shown in the plot. Outside the chaotic region, we see a sequence of KAM tori that block the trajectories from moving to larger positive or negative values of the momentum.

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5 Arnol’d Diffusion

In Fig. 5.9b, we show a strobe plot with the same initial conditions on (px , x), but now with the full Hamiltonian in Eq. (5.17) and initial values py = 0.5, y(0) = π , but with b = 0. For b = 0, there is no coupling between the motions in the x and y degrees of freedom and the system evolves as two uncoupled 2 DoF systems. Fig. 5.9a, b are essentially indistinguishable. In Fig. 5.9c and in Fig. 5.9d, we show the same strobe plots (same initial conditions) but now with b = 0.002 and b = 0.02, respectively, so all degrees of freedom are coupled. All four plots in Fig. 5.9 are run for 2000 periods of oscillation. Weak diffusion across the KAM tori appears to have occurred in Fig. 5.9c. However, in Fig. 5.9d, diffusion dominates the dynamics. This is consistent with our estimate that the transition between the Nekhoroshev and Chirikov regimes occurs for 0.009≤b≤0.02. In Fig. 5.9, we see weak diffusion for b = 0.001 and more rapid diffusion for b = 0.01. In Fig. 5.9b, where b = 0, it is clear that the energy H (t) cannot undergo large excursions. If the initial conditions lie on a KAM torus, the energy will undergo small regular oscillations. If they lie in the chaotic region, the energy H (t) can undergo small apparently random oscillations, but they are blocked by KAM tori from large excursions in energy. However, if b =0 then, from Fig. 5.9c, d, it is clear that the energy H (t) might undergo large energy oscillations because the KAM tori no longer isolate regions of the phase space from one another. It is also clear that the diffusion is faster for larger values of b. It is useful to look at the behavior of the average energy of the system for various parameter regimes. In Fig. 5.10, we plot the energy H (t) for four different values of the lattice coupling strength b = 0.002, b = 0.02, b = 0.2, and b = 2.0. In all four cases, the initial energy is E = 30 and the trajectory is run for a time t = 2000. For each value of b, we have obtained energy plots for ten different initial conditions on the energy surface. In Fig. 5.10, we show one realization (out of ten) for each value of b. As we can see from the plots, the energy fluctuates. We obtain the average energy for each plot and then average the energy of the ten plots for each value of b. We find the following average energies: (b, Eav ) = (0.002, 35.8), (0.02, 46.0), (0.2, 79.0), (2.0, 186.3). This change in the behavior of the average energy appears to be consistent with our estimates for the transition between the Nekhoroshev and Chirikov regimes. The Chirikov regime requires resonance overlap which, for low values of b, will occur in local regions of the phase space because of the different sizes and locations of resonances. The behavior of the average energy indicates that large scale diffusion in energy, due to resonance overlap, begins to occur for b > 0.2 and steadily grows as b increases. The quantum mechanical version of this time-periodically driven optical lattice was also studied in Boretz and Reichl (2016). They found that the Arnol’d web gave rise to Floquet eigenstates of the driven system that consisted of large numbers of entangled energy eigenstates of the static lattice, and that the average energy of atoms in the lattice increased dramatically in parameter regimes where the classical system showed large fluctuations in the energy. Some of their results, regarding Arnol’d diffusion on the quantum lattice, are described in Chap. 10.

5.6 Stability of the Solar System

149

Fig. 5.10 Plots of H (t), for a single trajectory, as a function of time for four different values of b. Each trajectory has initial energy E = 30. (a) b = 0.002, (b) b = 0.02, (c) b = 0.2, and (d) b = 2.0 (reproduced from Boretz and Reichl 2016)

5.6 Stability of the Solar System The gravitational two-body problem is integrable. Both the energy and the angular momentum of two masses that attract via to the gravitational force are conserved. In addition to these global space-time symmetries, there is a hidden symmetry that leads to another constant of the motion, the Runge–Lenz vector (see Chap. 2). This additional “hidden symmetry” causes the perihelion of the orbit of the two masses to remain fixed in space. However, when three or more masses interact via the gravitational force, the gravitational system is no longer integrable. An Arnol’d web is embedded in the phase space, and the system can exhibit regions of chaos or even become unstable (come apart) after some period of time. One of the major challenges of classical mechanics during the past 300 years (since the work of Newton) has been to determine whether or not the solar system is stable. On short time scales, the periods of the orbits of the planets about the sun are fairly regular and predictable and, until recently, it was generally agreed that the solar system evolves quasi-periodically. In the nineteenth century, Kirkwood (1867) discovered a series of gaps in the distribution of asteroids that circle the sun and lie between Mars and Jupiter. There were a number of theories proposed to explain these gaps, but one of the few that remains viable (Dermott and Murray 1983) is that the gaps result from resonances in the restricted gravitational three-body problem consisting of the sun, Jupiter, and

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5 Arnol’d Diffusion

Fig. 5.11 The number of asteroids as a function of the semimajor axis a measured in astronomical units (AU) (1 AU= the mean distance between Earth and the Sun). The values of a at which the period of the asteroid is a rational fraction of the period of Jupiter are marked (from [Moser 1978])

individual asteroids (this is a restricted three-body problem because the mass of the asteroid can be neglected). In other words, Jupiter acts to perturb the asteroid’s Keplerian orbit around the sun and creates resonances in the asteroid’s phase space. A plot of the number of asteroids as a function of the semimajor axis is shown in Fig. 5.11. Positions where the period of the asteroid is a rational fraction of that of Jupiter are marked. Model studies by Wisdom (1985) of the 13 resonance and by Murray (1986) of the 12 and 23 resonances seem to explain some features of Fig. 5.11. It is still not clear whether or not Arnol’d diffusion plays a role in removing asteroids from the gap regions. Wisdom found that in the 13 resonance region some of the orbits attained large enough eccentricity that they could collide with Mars, thus eliminating them from the asteroid belt. Sussman and Wisdom (1988) have found numerical evidence that the motion of the planet Pluto is chaotic. They integrated the orbits of the outer planets (Jupiter to Pluto) for a period of 845 million years and found that the long-term motion of Pluto is chaotic due to the existence of many long-period resonances. They found that the largest Lyapounov exponent for the motion of Pluto is about 10−7.3 year−1 . Laskar (1989, 1996) has obtained an even more surprising result. He has integrated the orbits of the sun and planets of the solar system for a period of billions of years using initial conditions applicable to the solar system. He has found that the eccentricity of the orbits of the inner planets (Mercury, Venus, Earth, and Mars) varies in a chaotic manner due to nonlinear resonances between their orbits (see Fig. 5.12). The chaotic changes in the eccentricity of the orbit of Mercury are large enough that, from time to time, the orbit of Mercury can intersect the orbit of Venus. It is worth quoting from Laskar’s paper (Laskar 1996): Large scale chaos is present everywhere in the solar system. It plays a major role in the sculpting of the asteroid belt and in the diffusion of comets from the outer region of the solar system. . . . On billion years time scale, the orbits of the planets themselves present strong chaotic variations which can lead to the escape of Mercury or collision with Venus in less than 3.5 Gyr.

5.7 Colliding Beam Synchrotron Particle Accelerator

151

0.5 Mercury

eccentricity

0.4 0.3 0.2 0.1

Vonus Earth

0 0.2 Mars Saturn

0.1 Neptune

Jupiter

0 -10

-5

Uranus

5

0

10

15

Time (Gyr) Fig. 5.12 The eccentricity of the orbits of the planet, ranging from 10 Gyr in the past to 15 Gyr in the future (from Laskar 1996)

5.7 Colliding Beam Synchrotron Particle Accelerator Colliding beam accelerators consist of two beams of particles that are held in a nearly circular orbit by magnetic fields (Tennyson 1983; Gerasimov et al. 1986). The particles undergo linear vertical and horizontal oscillations about their circular orbit as they travel. When the particles of one beam cross those of the other beam, they experience a kick. If we let (px , x) and (pz , z) denote the coordinates of the horizontal and vertical oscillations, respectively, and let t denote the time of a kick, then the simplest Hamiltonian that describes one of the beams is of the form H =

∞  1 2 (px + ωx2 x 2 + pz2 + ωz2 z2 ) + δ(t − m) V (x, z). 2 m=−∞

(5.23)

One example of a potential  2 that has2been used to describe the beam-beam interaction is V (x, z) = 8π exp − x2 (1 + z2 ) (Tennyson 1983). Let us now transform to the action-angle variables of the linear oscillators. We 1 1 i 2 2 let pi = −( 2I ωi ) sin(θi ) and xi = (2Ii ωi ) cos(θi ), where i = x, y and (Ix , θx ) and (Iz , θz ) are the action-angle variables associated with the horizontal and vertical oscillations of the beam, respectively. In terms of these action-angle variables, the Hamiltonian can be written

152

5 Arnol’d Diffusion

H =ωz Ix +ωz Iz +

∞  

Amx ,mz cos(mx θx +mz θz +2π nt).

(5.24)

n=−∞ mx mz

If the term mx = mz = n = 0 (which is independent of θx and θz ) is removed from the sum and included in the kinetic part of the Hamiltonian, we have H = Ho +

∞   

Amx ,mz cos(mx θx + mz θz + 2π nt),

(5.25)

n=−∞ mx mz

where Ho (Ix , Iz ) = ωz Ix + ωz Iz + A0,0 (Ix , Iz ) and A0,0 (Ix , Iz ) is the average of V (x, z) over one period of the oscillations. From Eq. (5.25), we see that the Hamiltonian for the colliding beam system contains a dense network of nonlinear resonance zones, mx θ˙x + mz θ˙z + 2π n = 0, which will exhibit Arnol’d diffusion no matter how strong the beam-beam interaction is just so long as it is nonzero. Particle beams may be stored for as long as 1011 revolutions of the beam in the circular orbit. Therefore, Arnol’d diffusion, which can cause particles in the beam to diffuse into the walls of the accelerator, can act to significantly reduce the luminosity of the beam.

5.8 Conclusions Because the Arnol’d web occurs in systems with three or more degrees of freedom, it is hard to visualize and has proven challenging to study. The fact that the Arnol’d web can cause a global transition to chaos in classical systems, and can have a profound effect on the quantum dynamics of such systems, has been shown for the time-periodically driven optical lattice described. The effect of the Arnol’d web on the dynamics of an atomic system has been explored by von Milczewski et al. (1996). They studied the classical dynamics that resulted from an Arnol’d web induced in a Rydberg atom that had been placed in crossed static electric and magnetic fields. For that system, they described how the presence of the Arnol’d web might affect the quantum dynamics of the atom-field system. The global onset of chaos that can result in nonlinear, nonintegrable systems with three or more degrees of freedom due to the presence of the Arnol’d web, provides the key mechanism for thermalizing quantum systems. In subsequent chapters, we turn our attention to quantum systems and show how chaos manifests itself in quantum dynamics.

References

153

References Arnold VI (1963) Russ Math Surv 18:9 Arnold VI (1964) Sov Math Dokl 5: 581. (Reprinted in R.S. MacKay and J.D. Meiss, Hamiltonian Dynamical Systems, (Adam Higler, Bristol, 1987) Bloch I, Dalibard J, Nascimbene S (2012) Nat Phys 8:267 Boretz Y, Reichl LE (2016) Phys Rev E 93:032214 Byrd PF, Friedman D (1971) Handbook of elliptic integrals for engineers and scientists. Springer, Berlin Chirikov B (1979) Phys Rep 52:263 Cincotta PM (2002) New Astron Rev 46:13 Dermott SF, Murray CD (1983) Nature 301:201 Froeschle C, Lega E, Gonczi M (1997) Celest Mech Dyn Astron 67:41 Froeschle C, Guzzo M, Lega, C (2000) Science 289:2108 Gerasimov A, Izrailev FM, Tennyson JL, Temnyykh AB (1986) Springer lecture notes in physics, vol 247. Springer, Berlin, p 154. (Reprinted in [MacKay and Meiss 1987]) Greiner M, Bloch I, Mandel O, Hansch TW, Esslinger T (2001) Appl Phys B 73:769–772 Greiner M, Mandel O, Esslinger T, Hansch TW, Bloch I (2002) Nature 415:39 Hemmerich A, Schropp D, Hansch TW (1991) Phys Rev A 44:1910 Hensinger WK, Haffner H, Browaeys A, Heckenberg NR, Helmerson K, McKenzie C, Milburn GJ, Phillips WD, Rolston SL, Rubinsztein-Dunlop H, Upcroft B (2001) Nature 412:52 Horsley E, Koppell S, Reichl LE (2014) Phys Rev E 89:012917 Kaneko K, Bagley RJ (1985) Phys Lett A 110:435 Kirkwood D (1867) Meteoric astronomy. Lippincott, Philadelphia Laskar J (1989). Nature 338:237 Laskar J (1996) Celestial Mech Dyn Astron 64:115 Lega F, Guzzo M, Froeschle C (2003) Phys D 182:179 Lichtenberg AJ, Lieberman MA (1991) Regular and chaoitic dynamics, 2nd edn. Springer, New York Luter R, Reichl LE (2002) Phys Rev E 66:053615 Moser J (1978) The Mathematical Intelligencer, p 65 Murray CD (1986) Icarus 65:70 Nekhoroshev NN (1971) Funct Anal Appl 5:338 Nekhoroshev NN (1977) Russ Math Surv 32:1 Reichl LE (1989) Phys Rev A 39:4817 Steck DA, Oskay WH, Raizen MG (2001) Science 293:274 Steck DA, Oskay WH, Raizen MG (2002) Phys Rev Lett 88:120406 Sussman GJ, Wisdom J (1988) Science 241:433 Tennyson JL (1983) Resonance streaming in electron-positron colliding beam systems. In: Horton W, Reichl LE, Szebehely V (eds) Long time prediction in dynamics. Wiley, New York, p 427 von Milczewski J, Diercksen GHF, Uzer T (1996) Phys Rev Lett 76:2890 Wisdom J (1985) Icarus 63:272

Chapter 6

Quantum Dynamics and Random Matrix Theory

Abstract The energy spectrum of a quantum system, whose classical counterpart is chaotic, has statistical properties like those of random matrices that extremize information. In the 1950s, Wigner surmised that classically chaotic quantum systems have energy eigenvalue spacing distributions similar to those of a Hamiltonian matrix whose matrix elements are random numbers determined by Gaussian distributions. This gives a distribution for nearest neighbor spacings (the Wigner distribution) between eigenvalues that has been verified by experiment. The Wigner distribution predicts a very low probability of finding small spacings between nearest neighbor energy eigenvalues. The random matrix theory of Hamiltonian systems is based on the assumption that global symmetries alone determine the form of the Hamiltonian. These symmetries impose restrictions on the form of the Hamiltonian matrix. Probability distributions for matrix elements of orthogonal, complex Hermitian, and quaternion real Hamiltonian matrices can be obtained. Eigenvalue cluster functions, nearest neighbor spacing distributions, and two-body eigenvalue correlation functions can then be derived. A gas of particles, whose Hamiltonian belongs to the Gaussian orthogonal ensemble, is a thermalized system and its single particle reduced probability density is given by the Maxwell-Boltzmann distribution. This important example shows the mechanism by which quantum systems become thermalized. Keywords Random matrix theory · Nuclear scattering theory · Wigner surmise · Wigner distribution · Wigner semi-circle law · Staircase function · Delta-3 statistic · Invariant metric · Hamiltonian matrices · Gaussian orthogonal ensemble · Circular ensembles · Cluster functions · Eigenvalue nearest neighbor spacing · Eigenvector distribution · Thermalization of quantum systems

6.1 Introduction Classical conservative systems that undergo a transition to chaos have very complex dynamical behavior, as we have seen in previous chapters. How much of this © Springer Nature Switzerland AG 2021 L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200, https://doi.org/10.1007/978-3-030-63534-3_6

155

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6 Quantum Dynamics and Random Matrix Theory

complex behavior remains in the corresponding quantum systems? We will show, in much of the remainder of this book, that quantum systems, whose classical counterpart is chaotic, have spectra whose statistical properties are similar to those of random matrices that extremize information. Thus, any study of the quantum manifestations of chaos requires an analysis of information content of quantum systems, using concepts from random matrix theory (RMT). We have attempted to give a complete grounding on random matrix theory in this book. Much of our discussion of random matrix theory is in the appendices, but we give an overview of some key results in this chapter. Our analysis of quantum dynamics and the behavior of solutions of the Schrödinger equation will actually begin in Chap. 7. The use of random matrix theory as a tool to study the statistical properties of quantum systems was introduced by Wigner (1951, 1955, 1957a,b, 1958) in an attempt to understand nuclear scattering data. Wigner used it to analyze complex nuclear energy-level sequences (Wigner 1959). At that time there was a shortage of close spacings in experimentally obtained data on energy levels. This lack of close spacings was generally thought to result from the inability of experimental apparatus to resolve them. Wigner was able to give an explanation using statistical arguments. Wigner surmised (Wigner 1959) a possible energy eigenvalue spacing distribution assuming that matrix elements of the Hamiltonian matrix were random numbers with Gaussian distributions. He obtained a distribution for nearest neighbor spacings, s, between eigenvalues,   πs −π s 2 PW (s) = , exp 2D 2 4D 2

(6.1)

where D is the average spacing between nearest neighbor eigenvalues in the eigenvalue sequence being considered. Equation (6.1) is now called the Wigner distribution. The Wigner distribution predicts a very low probability of finding small spacings between nearest neighbor energy eigenvalues. This is very different from the case where the eigenvalues are randomly distributed. For random eigenvalue sequences, the nearest neighbor spacing satisfies a Poisson distribution, PP (s) =

  −s 1 exp , D D

(6.2)

where D is again the average spacing between nearest neighbor eigenvalues. In Fig. 6.1, we compare the Wigner distribution, PW (s), with the Poisson distribution, PP (s), for the case D = 1. For systems whose eigenvalues are distributed at random, there is a large probability of finding very small spacing between eigenvalues. For systems whose Hamiltonian matrix elements are distributed at random, there is a very small probability of finding close spacings between eigenvalues. Random matrix theory, as applied to Hamiltonian systems, is based on the assumption that we know very little about the Hamiltonian matrix except for certain symmetry properties. These symmetry properties impose restrictions on the form of the Hamiltonian matrix, as described in Appendix C. In this chapter, we develop

6.1 Introduction

157

Fig. 6.1 A plot of the Wigner distribution, PW (s), and the Poisson distribution, PP (s), as a function of level spacing, s, for D = 1

tools that will help us analyze the information content of quantum systems that undergo a transition to chaos. We will be concerned, primarily, with Hamiltonian matrices, which are Hermitian matrices. The first step in obtaining a probability distribution for matrix elements of a Hermitian matrix is to form a metric, (ds)2 , in the space of matrix elements such that (ds)2 is invariant under a similarity (unitary) transformation. This we do in Appendix D, for the case of real symmetric, complex Hermitian, and real quaternion Hamiltonian matrices. These three types of matrices have the symmetry properties of dynamical systems of most interest in quantum dynamics. Real symmetric Hamiltonian matrices govern the dynamics of systems that are invariant under rotation and time reversal. Complex Hermitian Hamiltonian matrices govern the dynamics of systems that are invariant under rotation but not time reversal (for example, when magnetic fields are present). Quaternion real Hamiltonian matrices govern the dynamics of systems of spin 12 particles that are invariant under time reversal but are not invariant under rotation. Since these three types of matrices have different numbers of independent matrix elements, the metric (ds)2 will be different for each. In Appendix D, we also determine the metric for each of the corresponding unitary matrices. In this chapter, we focus on the statistical properties of real symmetric random Hamiltonians. In Sect. 6.2, we obtain the invariant metric and measure for real symmetric random Hamiltonians, and in Sect. 6.3, we obtain a joint probability distribution for their matrix elements. The joint probability distribution is chosen to extremize information subject to the condition that it is normalized to 1 and has matrix elements that remain finite. This leads to a Gaussian distribution for matrix elements. For real symmetric Hamiltonians, the similarity transformation is orthogonal and the probability distribution is said to describe a Gaussian orthogonal ensemble (GOE) of Hamiltonian matrices. (For complex Hermitian and real quaternion Hamiltonians, the similarity transformations are unitary and symplectic,

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6 Quantum Dynamics and Random Matrix Theory

respectively, and the probability distributions are said to describe Gaussian unitary ensembles (GUE) and Gaussian symplectic ensembles (GSE), respectively.) When dealing with N × N-dimensional Hermitian matrices, where N is large, we have far more information in the joint probability distribution than we can possibly use. As we will see, we are often only interested in pair correlations between energy eigenvalues. Therefore, in Sect. 6.4 we introduce reduced joint probability distributions for only n of the N eigenvalues. These reduced probability distributions can themselves be written in terms of n-body cluster functions that are nonzero only when n-body (n-eigenvalue) correlations exist in the system. There are a number of statistical properties of random matrices that are commonly used in analyzing the spectral properties of systems. These are the eigenvalue density, the 3 -statistic, and the eigenvalue nearest neighbor spacing distribution. All three of these quantities will be derived in Sect. 6.4 for the Gaussian orthogonal ensemble. In Sect. 6.5, we obtain the probability distribution for eigenvectors of Hamiltonian matrices that belong to the Gaussian orthogonal ensemble. More precisely we find the distribution of components of a single eigenvector taken from the complete set of eigenvectors. This eigenvector distribution will then allow us, in Sect. 6.6, to obtain the reduced probability distribution of a “noninteracting” gas of particles, where we assume that each particle is governed by a Hamiltonian that is a member of the Gaussian orthogonal ensemble. We follow the procedure of Srednicki (1994) and, like him, we find that the gas is thermalized. The single particle reduced probability density is given by the Maxwell-Boltzmann distribution. The work of Srednicki was important because it helped to establish how quantum systems become thermalized. Finally, in Sect. 6.7, we make some concluding remarks.

6.2 Invariant Measure for the GOE Hamiltonians that govern the dynamics of systems that are invariant under time reversal and rotation have matrix representations that are real and symmetric. An N ×N real symmetric matrix, ⎛

h1,1 h1,2 ⎜ h1,2 h2,2 ⎜ H¯ R = ⎜ . .. ⎝ .. . h1,N h2,N

⎞ . . . h1,N . . . h2,N ⎟ ⎟ .. ⎟ , .. . . ⎠ . . . hN,N

(6.3)

has N + 12 (N 2 − N) = 12 N(N + 1) independent real elements. An N ×N real symmetric matrix, H¯ R , is diagonalized by an N ×N orthogonal ¯ An orthogonal matrix has the property that its transpose is equal to its matrix, O. inverse,

6.2 Invariant Measure for the GOE

159

¯ ¯ O¯ T = 1, O¯ T = O¯ −1 so O¯ T ·O¯ = O·

(6.4)

where 1¯ is an N×N unit matrix. The matrix HR can be written ¯ E¯ R ·O¯ T , H¯ R = O·

(6.5)

where E¯ R is the diagonal N×N matrix containing eigenvalues of H¯ R , ⎛

0 ... e2 . . . .. . . . . 0 0 ...

e1 ⎜0 ⎜ E¯ R = ⎜ . ⎝ ..

0 0 .. .

⎞ ⎟ ⎟ ⎟, ⎠

(6.6)

eN

and O¯ is composed of the eigenvectors of H¯ R . The matrix H¯ R contains N(N + 1)/2 independent matrix elements. The matrix E¯ R contains N independent matrix elements. Therefore, the orthogonal matrix O¯ contains only N(N − 1)/2 independent matrix elements. The matrix of differential increments of the matrix H¯ R is denoted d H¯ R and is given by ⎛

dh1,1 dh1,2 ⎜ dh1,2 dh2,2 ⎜ d H¯ R = ⎜ . .. ⎝ .. . dh1,N dh2,N

⎞ . . . dh1,N . . . dh2,N ⎟ ⎟ .. ⎟ , .. . . ⎠ . . . dhN,N

(6.7)

where dhij is the (i, j )th matrix element of d H¯ R . We next introduce a metric for this system. We will require that the metric be real and invariant under an orthogonal transformation. We know that the trace of a matrix, d H¯ R , or any power of d H¯ R is invariant under an orthogonal ¯ Therefore, the simplest choice of an invariant metric is transformation, O. (ds)2HR = Tr(d H¯ R · d H¯ RT ) =

N  (dhi,i )2 + 2



1≤i 2π nc /ω0 . Here nc is the number of external field periods required for the driving field to be fully turned on. The radiation is proportional to the Fourier transform of the average acceleration, a(t) and is given by a(t) = ψ(t)|x|ψ(t) ¨ =

 e−i(α −β )t β (t)|x| ¨ α (t) α

β

×ψ(0)|β α |ψ(0).

(10.37)

The Fourier transform of a(t) is a(ω). ˜ The power spectrum is given by χ (ω) = 2. |a(ω)| ˜ In Fig. 10.7, we show the radiation spectrum, as a function of radiated frequency, for external field strength = 320 and three different initial conditions. The turnon takes nc = 12 cycles of the external driving field. The spectrum was computed from a time series for the acceleration that ran for a time interval t = 128T0 (128 external field cycles), starting after the turn-on was complete. At time t = 0, we start the system in an energy eigenstate, |φn , for each of the following three cases: n = 35 in the region dominated by KAM tori; n = 16 in the region of the large nonlinear resonance; and n = 3 in the chaotic region. We see that in the region dominated by KAM tori (Fig. 10.7a), the radiation spectrum is typical of what might be found using perturbation theory. There is a small amount of radiation at the lower harmonics of the driving field frequency, ω0 , but nothing at higher harmonics. For cases where the initial condition lies in the chaotic region, in Figs. 10.7b and c, there is significant high harmonic radiation. In both these cases, the frequency at which the radiation cuts off is determined by the spread in energy of the chaotic sea of the underlying classical phase space (Chism et al. 1998). The power spectrum

10.4 Dynamics of a Driven Bounded Particle

357

Fig. 10.7 Radiation spectrum (power spectrum) of a time-periodically driven particle in an infinite square-well potential for

= 320 and three different initial states and turn-on time of 12 cycles. (a) Regular region n = 35. (b) Resonance region n = 16. (c) Chaotic region n = 3 (Chism et al. 1998)

for initial condition n = 16 is very complex Fig. 10.7b. This can be understood in terms of a Floquet analysis. The Fourier transform of a(t) is a(ω). ˜ The power 2 . If we take the initial time in Eq. (10.37) to spectrum is given by χ (ω) = |a(ω)| ˜ be at the end of the turn-on, it is found numerically that there are 13 Floquet states that contribute significantly to the power spectrum in Fig. 10.7b and there are two Floquet states that contribute significantly to the power spectrum in Fig. 10.7c. The large number of states for the case in Fig. 10.7b is due to the large number of Floquet states that sit in the large primary resonance and are picked up by the initial state n = 16. The power spectrum contains significant peaks at the differences between the Floquet eigenvalues associated to these states. Thus, the power spectrum in Fig. 10.7b has many more peaks than that in Fig. 10.7c.

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10.5 Dynamical Tunneling in Atom Optics Experiments Floquet theory can be used to compute the tunneling frequencies observed in cold atom optics experiments for which the underlying classical phase space has regions dominated by chaos. In the experiments of Steck et al. (2001, 2002), which we consider here, a cluster of cesium atoms in the gas phase, is cooled to a temperature T ≈ 10−7 K. The density is low enough that interactions between the atoms can be neglected. These ultracold cesium atoms are allowed to interact with two counterpropagating laser beams that create a periodically modulated standing wave of light. The atoms can be treated as two-level systems with energy spacing hω ¯ 0, and the laser beams are detuned away from resonance with these two energy levels. Interaction of the atoms with the standing wave of light stimulates absorption and then emission of a photon in one of two dominant modes. In the first mode, the photon is absorbed and then emitted in the incident direction (transmitted through the atom), causing no net recoil of the atom. In. the second mode, the photon is absorbed and then emitted in a direction 180o from the incident direction (reflected from the atom), resulting in a net atomic recoil of 2hk ¯ L , where kL is the wave vector of the standing wave of light and h¯ is Planck’s constant. It is this second process that dominates the dynamics observed in the experiments. When the laser detuning δL = ω0 − ωL is large, resonant absorption can be neglected and the atoms have a large probability of being in their ground state on time-scales of importance for the experiment. Under these conditions, the dynamics is determined by the center-of-mass motion of the atoms. It was first shown by Graham et al. (1992) that simple time-periodic Hamiltonians describe the dynamics of such systems. In the subsections below, we show that Floquet analysis gives excellent agreement with experiment (Luter and Reichl 2002).

10.5.1 Hamiltonian for Atomic Center-of-Mass In the experiments reported in Steck et al. (2001, 2002), the dynamical evolution of noninteracting cold cesium atoms in a periodically modulated standing wave of light was measured. The Hamiltonian used to model the center-of-mass motion of the cesium atoms (in S.I. units) is 12 1= p − 2Vo cos2 H 2m



ωm t 2

 cos (2kL1 x) ,

(10.38)

where p, ˆ x, ˆ and m are the momentum, position, and mass, respectively, of a cesium atom, ωm = 2π T is the modulation frequency of the standing wave of light, and h2

Vo = ¯ 8δmax . Here max = 2E0 d/h¯ is the Rabi frequency, E0 is the electric field L strength, and d is the dipole moment of cesium. As we will see below, this system has three primary resonances.

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In these experiments, because of the very low temperature and the fact that the atomic recoil is quantized in steps of 2hk ¯ L , the allowed momentum states appear to be quantized in integer multiples of 2hk ¯ L with very small spread about those values. This allows us to use Floquet analysis for this system rather than FloquetBloch analysis (Mouchet et al. 2001) as one might expect from the form of the Hamiltonian in Eq. (10.38). Because of the quantization of the recoil momentum, an atom cannot transition through the continuous range of energies in the energy bands allowed by the Hamiltonian in Eq. (10.38), at least for the timescales of the experiment. Let us now write the Hamiltonian in dimensionless form. We perform a scaling that explicitly quantizes the momentum in units of 2hk ¯ L (Luter and Reichl 2002). 1th = 1 = 2kL1 Let φ x, p 1 = 21 nh¯ kL , ω = ωm /4ωr , ωr = h¯ kL2 /2m, τ = 4ωr t, and H 2 2 1/2k h¯ to obtain mH L   1 1 αω2 1th = 1 1) + cos(φ 1 − ωτ ) + cos(φ 1 + ωτ ) , cos( φ n2 − H 2 2 8π 2

(10.39)

where α = 8ωr T 2 V0 /h. ¯ All quantities are dimensionless, and nˆ is the dimensionless momentum operator with eigenstates |n and integer eigenvalues −∞≤n≤∞. 1= Note that, in the experimental papers, the following scaling is performed: φ 2 2 2. 1 1 2kL1 x , τ = ωm t/2π = t/T , ρ 1 = 4π kL p 1/mωm , Hexp = 16π kL H /mωm 2 1exp = ρ1 − With this scaling, the Hamiltonian in Eq. (10.38) takes the form H 2 2 1). However, this choice of scaling does not allow the type of 2αcos (π τ )cos(φ analysis described below. The system described by Eq. (10.39) has three primary resonances centered at (n = 0, φ = 0) and (n = ±ω/2, φ = 0). For small values of α (α < 1.5), the primary resonanceshave pendulum-like structure, and the resonance at n = 0 2

has half-width n0 = αω , while the resonances at n± = ±ω/2 have half-width 4π 2 √ n± = n0 / 2. The primary resonance at n = 0 bifurcates at α≈7.0. The two outer primaries remain visible for 0 < α ≤ 13.0 and disappear for larger values of α. The Hamiltonian in Eq. (10.39) has a classical analog if we let the dimensionless momentum n→n, ˆ where n can take on a continuum of values, −∞≤n≤∞. The ∂Hth th ˙ classical motion is obtained from Hamilton’s equations n˙ = − ∂H ∂φ and φ = ∂n . In the experiment, ωr = 13,000 rad/s and T = 2π/ωm = 20 μs, so the dimensionless radial frequency is ω = 6.0. In Fig. 10.8a, we show a strobe plot of the classical phase space for α = 2.0. The three primary resonances are clearly visible in this plot. A strobe plot of the classical phase space for α = 9.7 is shown in Fig. 10.8b. The central primary resonance has bifurcated and is largely destroyed, and the outer primary resonances have been reduced significantly in size and are centered at momentum values n = ±4.1. Note also that the chaotic region lies in the interval −5 ≤ n ≤ +5, indicating that 11 quantized momentum states determine the dynamics in the chaotic region.

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Fig. 10.8 Strobe plot of n˜ versus φ for the classical phase space for the Hamiltonian in Eq. (10.39), taken at time intervals T = 20 μs, for ω = 6.0. (a) The case α = 2.0. (b) The case α = 9.7 used in the experiment (Luter and Reichl 2002)

10.5.2 Average Momentum of Cesium Atoms The experiment used atoms prepared initially with a minimum uncertainty wave packet peaked at n = 4.1 (on the upper island) (Steck et al. 2001, 2002). This may be represented by the coherent state  n|(0)≡ n |φo no  =

σ2 π

 14

 exp

−σ 2 (n − no )2 − i (n − no ) φo 2



(10.40)

10.5 Dynamical Tunneling in Atom Optics Experiments

361

Fig. 10.9 (a) Experimental measurement of the average momentum, n, of the cesium atoms for ω = 6.0, α = 9.7, and initial condition (no = 4.1, φo = 0). (b) Experimental measurement of oscillation frequencies that dominate the time series for the average momentum, plotted as a function of α for ω = 6.0 and initial state (no = 4.1, φo = 0) (Steck et al. 2001, 2002)

centered at (n = no , φ = φo ) with σ = 1.2. The average momentum of the atoms in the modulated standing wave of light was measured as a function of time. The experimental result for α = 9.7 is shown in Fig. 10.9a. The average momentum clearly oscillates. The fact that the average momentum does not reach the negative momentum states is due to the fact that only a part of the atoms are involved in the dynamic tunneling but all are averaged over. The experimental curve appears to have a beat frequency indicating that, at this value of α, two frequencies dominate the dynamics. Experimental measurements were also performed at other values of α. It was found that for a range of values of α about α = 9.7, two frequencies dominate the average momentum oscillations, while outside this range only a single frequency dominates. A summary of the experimental results is given in Fig. 10.9b, where the dominant frequencies observed in the experiment are plotted as a function of α. The time evolution of the system with initial state n|(0) = n|n0 , φ0  is governed by the Schrödinger equation, i ∂|(t) = Hˆ th |(t). In the momentum ∂t  basis, the Schrödinger equation reduces to a system of coupled first-order differential equations for the amplitudes, n|(t). The time variation of the momentum expectation value, n(t) = (t)|n|(t), ˆ can then be computed numerically. The result is shown in Fig. 10.10a for α = 9.7, ω = 6.0, and initial state (no = 4.1, φo = 0). This system was truncated, and 81 equations for the states n|(t) with −40 ≤ n ≤ 40 were kept. The average momentum oscillates between positive and negative momentum values and has two dominant frequencies, f1 = 2.39 kHz and f2 = 2.88 kHz, giving rise to beats. In the numerical result, the average momentum reaches negative momentum values. The numerical result has the same oscillation frequency and the same beat frequency as the experimental result in Fig. 10.9a.

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Fig. 10.10 (a) Numerical simulation of the average momentum, n, of the cesium atoms for ω = 6.0, α = 9.7, and initial condition (no = 4.1, φo = 0). (b) Oscillation frequencies that dominate the time series for the average momentum plotted as a function of α for ω = 6.0 and initial state (no = 4.1, φo = 0). These oscillation frequencies, f = (j − i )/2π , are calculated from Floquet eigenphase differences for varying field strengths, α. A threshold of Pi Pj ≥0.04 overlap probability was used to select the dominant Floquet states (Luter and Reichl 2002)

10.5.3 Floquet Analysis of Tunneling Oscillations The probability of finding the system in momentum state |n at time t, starting from the initial state |(0) = |φo no (0), can be written |n |(t) (t) |2 =

 i

 2 exp −i j − i t n j (t) i (t) |n

j

3 2 × j (0) |φo no φo no |i (0)  ,

(10.41)

where j and |j (t) are the j th Floquet eigenvalue and eigenstate, respectively. The overlap probabilities, Pj ≡|j (0)|n0 , φ0 |2 , give the contribution of the j th Floquet state to the dynamics for the initial condition n|(0) = n|n0 , φ0 . The oscillation frequencies, fexp , observed in the experiment can be equated to differences between Floquet eigenphases. The frequency differences, fexp = (j − i )/2π , for Floquet eigenstates with overlap probability Pi Pj ≥ 0.04 are plotted in Fig. 10.10b for the range of parameters, α, used in the experiment.

10.6 Quantum Delta-Kicked Rotor

363

The experimental data (Fig. 10.9b) were able to resolve the dominant frequencies, fexp < 3 kHz, in the interval α ≈ 8.7 and α ≈ 10.3. The theoretical analysis (Fig. 10.10b) reproduces those experimental results. In the amplitude range α ≈ 7.6 to α ≈ 11.6, two frequencies dominate and give rise to the beats seen in Figs. 10.9a and 10.10a. A fundamental change occurs for α > 14, where a different set of Floquet states begins to dominate the dynamics. Approximately eleven Floquet states (the number of momentum states) have probability distributions that lie in the region of phase space between n = −5 and n = 5.Only three Floquet states dominate the dynamics. The eigenphase differences of these three Floquet states, (b − a )/2π = 2.8 kHz and (a − c )/2π = 2.4 kHz, correspond to the two dominant oscillation frequencies observed in the experiment at α = 9.7. The “dynamical tunneling” between positive and negative momentum states observed in the experiment appears to be due to interference among these three Floquet states. The regions of phase space where the dominant Floquet states have their support is determined by the distribution of chaotic regions of the underlying classical phase space. A similar dynamical tunneling experiment was performed by Hensinger et al. (2001) using a dilute Bose-Einstein condensate of sodium atoms. The effective Hamiltonian governing the dynamics of the sodium atoms was slightly different from the experiment described here but also exhibited a localized chaotic region in the phase space. The sodium atoms underwent a similar type of dynamical tunneling. A Floquet analysis of that experiment is also given in Luter and Reichl (2002).

10.6 Quantum Delta-Kicked Rotor The delta-kicked rotor has been one of the most intensely studied quantum systems because its Floquet matrix can be obtained analytically and because the classical version has played such an important role in conservative chaos theory. Many of the concepts used in quantum chaos theory were developed first for the quantum deltakicked rotor. However, caution also must be used in generalizing properties of the driven rotor to other systems because some of its features are nongeneric.

10.6.1 The Schrödinger Equation for the Delta-Kicked Rotor As we have shown in Chap. 3, the classical delta-kicked rotor has a Hamiltonian of the form H =

J2 + K cos(θ )δT (t), 2I

(10.42)

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where δT (t) =

∞ 

δ(t − qT ) =

q=−∞

  ∞ 1 2 2π kt + cos T T T

(10.43)

k=1

(q and k are integers). This Hamiltonian describes the motion of a one-dimensional rotor with angular momentum J and moment of inertia I , subjected to instantaneous kicks at regular intervals of time, T . The magnitude of a given kick depends on the position, θ , of the rotor at the instant the kick occurs. This system is easily quantized. ∂ The angular momentum operator is given by Jˆ = i h¯ ∂θ . Thus the Schrödinger equation is i h¯

∂ψ(θ, t) h¯ 2 ∂ 2 ψ(θ, t) + K cos(θ )δT (t)ψ(θ, t), =− ∂t 2I ∂θ 2

(10.44)

where ψ(θ, t) is the probability amplitude to find the rotor at angle θ at time t. We can also write the Schrödinger equation in terms of the probability amplitude, ψn (t), to find the system in angular momentum state |n, (Jˆ|n = hn|n) at time t. We let ¯ ψ(θ, t) =

∞ 

ψn (t)einθ .

(10.45)

n=−∞

Then the Schrödinger equation can be written i h¯

h¯ 2 n2 ∂ψn (t) K = ψn (t) + δT (t)(ψn+1 (t) + ψn−1 (t)). ∂t 2I 2

(10.46)

The equation for ψn (t) is a differential-difference equation. The Schrödinger equation can also be written in a form that makes clear the structure of primary resonances. If we again use Eq. (10.43), we obtain i h¯

 # ∞  2π kt h¯ 2 ∂ 2 ψ ∂ψ K  cos θ − =− ψ, + ∂t 2I ∂θ 2 T T

(10.47)

k=−∞

where ψ = ψ(θ, t). In terms of the state ψn (t), the Schrödinger equation takes the form i h¯

∞  ∂ψn h¯ 2 n2 K   −ikωt = ψn + e ψn−1 + eikωt ψn+1 , ∂t 2I 2T

(10.48)

k=−∞

where ω = 2π T . The primary resonance zones are located in the Hilbert space of angular momentum states, at nk = kωI h¯ (the resonance condition) and have a half-

10.6 Quantum Delta-Kicked Rotor

365

 width nk = 2 KI . These estimates for the location and width of the primary h¯ 2 T resonances will be useful in our subsequent discussion.

10.6.2 KAM-Like Behavior of the Quantum Delta-Kicked Rotor Geisel, Radons, and Rubner considered the quantum delta-kicked rotor (Geisel et al. 1986). for the parameter value K = 0.9716354, the value at which the last KAM tori that block global diffusion of the angular momentum begin to break in the classical system. At this value of K there are two remaining horizontal KAM tori in the interval 0 < J < 2π (note that J = 2πp and θ = 2π x). There are also several cantori that provide partial blockage at that parameter value. They determined the extent to which these remnants of constants of motion in the classical system also block flow of probability in Hilbert space. They take h¯ = 0.01, T = 1, and I = 1. Since J = nh, ¯ there are 2π × 102 angular momentum eigenstates in the interval 0 < J < 2π when h¯ = 0.01. They start the system with all probabilities concentrated on the state J0 = 3.2 or n0 = 320. They then compute the asymptotic probability N −1 1 |n|ψ(t)|2 . n→∞ N

P (J |J0 ) = lim

(10.49)

t=0

Their results are shown in Fig. 10.11. The dot-dashed line indicates the last two KAM tori in the interval 0 < J < 2π , and the dotted lines indicate cantori. In Fig. 10.11a, the probability is plotted on a linear scale, while in Fig. 10.11b it is plotted on a logarithmic scale. The KAM tori and cantori form barriers to the probability. The probability appears to “tunnel” through them. Brown and Wyatt (1986) have observed similar behavior in a driven oscillator model.

10.6.3 The Floquet Map The delta-kicked rotor is particularly interesting to study quantum mechanically because its dynamical evolution can be determined analytically in terms of a Floquet matrix. The delta-kicks occur at times t = qT , where q is an integer. Between these kicks, the system evolves as a free rotor. Let ψ(θ, 0+ ) denote the state of the system at time t = 0+ (just after the kick at t = 0), and let ψ(θ, 0+ ) =

∞  n=−∞

ψn (0+ )einθ .

(10.50)

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Fig. 10.11 Asymptotic spread of probability P (J |J0 ) in the quantum delta-kicked rotor for h¯ = 0.01, J0 = 3.2, K = 0.9716354, I = 1, and T = 1. Jc+ and Jc− are the largest and smallest values of J reached by the KAM tori: (a) linear scale; (b) logarithmic scale (Geisel et al. 1986)

Then during the time interval 0+ < t < T − (T − is the time just before the kick at time t = T ), the system evolves freely and the solution is ψ(θ, t) =

  i hn ¯ 2t , ψn (0+ )einθ exp − 2I n=−∞ ∞ 

(0+ < t < T − ).

(10.51)

We want to determine the state ψ(θ, T + ) just after the kick at time t = T . Let us note that since ∂ψ ∼ δT (t), ψ will be a discontinuous function of time at each kick, & t ∂t and F (t) = dtψ will be a continuous function of time but with a discontinuous slope at each kick. If we integrate the Schrödinger equation (10.42) across the kick at time t = T ,  i h¯

T +

T −

dt

h¯ 2 ∂ψ + ∂t 2I



T +

T −

dt

∂ 2ψ −K ∂θ 2



T +

T −

dt cos(θ )δT (t)ψ = 0,

(10.52)

then as → 0, the middle term gives no contribution and the change in ψ at the kick is determined by the equation i h¯

∂ψ = K cos(θ )δT (t)ψ ∂t

(T − < t < T + ).

(10.53)

ψ(θ, T − ).

(10.54)

Equation (10.53) has the solution K

ψ(θ, T + ) = e−i h¯

cos(θ)

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367

Combining Eqs. (10.51) and (10.54), we obtain −i Kh¯ cos(θ)

+

ψ(θ, T ) = e

∞ 

+

inθ

ψn (0 )e

n=−∞

  i hn ¯ 2T exp − . 2I

(10.55)

Equation (10.55) relates the state of the rotor at time t = T + to its state at time t = 0+ . It is interesting to note that the motion does not change if T → T + 4π hI¯ . Thus we can assume that 0 < T ≤ 4π hI¯ without loss of generality. If we note that ψ(θ, t) = θ |ψ(t) and ψn (t) = n|ψ(t), we can write Eq. (10.53) in the operator form ˆ

i

i

ˆ

|ψ(t + T ) = e− h¯ V e− h¯ H0 T |ψ(t), where θ |Vˆ |θ   = K cos(θ )δ(θ − θ  ) and n|Hˆ 0 |n  = Let us now write ∞ 

ψ(θ, T + ) =

(10.56)

h¯ 2 n2  2I δn,n .

ψn (T + )einθ ,

(10.57)

(−i)n Jn (z)einφ ,

(10.58)

n=−∞

and note the identity ∞ 

e−iz cos(φ) =

n=−∞

where Jn (z) is the Bessel function. &It is easy to show (using the definition of the 1 ∞ i(n−m)φ ) that Kronecker delta function δm,n = 2π −∞ dθ e ψn (T + ) =

∞ 

Unm (T + )ψm (0+ ),

(10.59)

m=−∞

where Unm (T + ) is the Floquet matrix (or Floquet map) for the delta-kicked rotor +

Unm (T ) = (−i)

 n−m

Jn−m

K h¯





i hm ¯ 2T exp − 2I

 .

(10.60)

The Floquet matrix couples many angular momentum states at each kick. The qualitative behavior of the quantum delta-kicked rotor depends on whether the period of the kick, T , is a rational or irrational multiple of 4π hI¯ . When T = 4π hI¯ pq with 0 < pq ≤ 1 a rational fraction, the Floquet spectrum is continuous or will have continuous parts (Casati et al. 1979). For the case where 0 < pq < 1 (Izrailev and Shepelyansky 1979, 1980) find that for all cases (except pq = 12 ) the average energy grows quadratically after a long time, with some oscillation superimposed

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Fig. 10.12 Two Floquet eigenstates for the delta-kicked rotor with κ = 2.8 and ξ = 4.867 (Grempel et al. 1984)

on the quadratic growth, and the spectrum will have continuous parts but may also have a discrete component as well. For the special case pq = 12 , the average energy simply oscillates with time.

10.6.4 Dynamic Anderson Localization In parameter regimes where the classical delta-kicked rotor is chaotic, its quantum analog exhibits dynamic Anderson localization. The theoretical basis for this was first established by Grempel et al. (1984), who were able to map the Floquet states of the quantum system onto the tight-binding model for Anderson localization in condensed matter systems. In Fig. 10.12, we plot two Floquet eigenstates, uα,n , as a function of n for κ = 2.8 and ξ = 4.867. Since ξ = hI¯ T = 4π α, for ξ = 4.867, α is irrational and a discrete Floquet spectrum is possible and indeed is observed. It is interesting to note that for the parameters used in Fig. 10.12, the spacing between nonlinear resonances I is nk+1 − nk = 2π = 2π T h ξ = 1.3 and the half-width of the resonance zones is ¯  KI κ n = 2 2 = 2 ξ = 1.5. Thus we are in the regime of nonlinear resonance T h¯

overlap. Blumel et al. (1986) have made an interesting study. They considered a truncated kicked-rotor with Schrödinger equation i

ξ ∂ 2 ψ(θ, τ ) ∂ψ(θ, τ ) =− + κ cos(θ )T (τ )ψ(θ, τ ), ∂τ 2 ∂θ 2

(10.61)

where N  T (τ ) = 1 + 2 cos(kπ ) cos(2mπ τ ). k=1

(10.62)

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369

Resonance zones occur at nk = 2πξ k for k ≤ N. Thus the resonances cut off at some value n = nN + n ≡ nL . The Floquet matrix has elements Un,n ≈ |J0 (2)|2 for n < nL , Un,n ≈ 1 for n > nL , Un,n+1 ≈ |J1 (2)|2 for n < nL , and Un,n+k → 0 for n > nL . For this truncated delta-kicked rotor, they found that the Floquet eigenstates are exponentially localized for irrational kicks and extended for rational kicks.

10.7 Microwave-Driven Hydrogen It requires only one photon to ionize hydrogen as long as we choose its energy to be equal to or greater than the ionization energy. Therefore, even a very low amplitude field can ionize hydrogen if the proper frequency is used. Ionization occurs as a function of frequency and not amplitude. This is the photoelectric effect. In 1974, Bayfield and Koch (1974) published a somewhat surprising result. They found that in highly excited microwave-driven hydrogen there is a multiphoton process for which the critical parameter for ionization of hydrogen is the amplitude and not the frequency. For fixed frequency in the microwave region (such that the photon energy is well below the ionization energy), ionization starts to occur suddenly as the amplitude of the field is raised. Although they did not realize it at the time, they had observed a new multiphoton ionization mechanism due to the overlap of microwave-induced nonlinear resonances.

10.7.1 Experimental Apparatus The experimental results that we will show in this section are due to several different groups, but there are great similarities in the experimental techniques they use. We will paraphrase descriptions of the experimental procedure found in Bayfield and Pinnaduwage (1985); Koch (1988); Koch et al. (1989) (see also Koch (1983)). In the microwave experiments, hydrogen atoms are excited to high Stark states while moving in a fast beam, and the interaction of the hydrogen atoms with the microwave field takes place when the atoms pass through a microwave cavity. A schematic picture of the apparatus is shown in Fig. 10.13. Before entering the microwave cavity, the hydrogen atoms are raised to high excited states using a double-resonance excitation scheme involving two CO2 lasers. As a first step in this excitation process, electron transfer collisions of a 14 keV proton beam with Xe or Ar produce neutral hydrogen atoms in excited states, including those with principal quantum number n = 7. Then, as the atoms pass through the region with static field, F1 in Fig. 10.13 (typically F1 ≈ 104 V/cm), atoms with parabolic quantum numbers (n; n1 , n2 , m) = (8; 0, 7, 0), where n = n1 + n2 + |m| + 1 (see Appendix K), are laser-excited to the state (n; n1 , n2 , m) = (10; 0, 9, 0). The atoms in the “tagged”

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Fig. 10.13 A schematic view of the experimental apparatus (Koch 1988)

Fig. 10.14 A spectroscopic scan (as a function of F3 ) of the final states of the hydrogen atoms leaving the region with static electric field, F3 in Fig. 10.13 (Koch et al. 1989)

parabolic state, (10; 0, 9, 0), then pass through the region with static field, F3 in Fig. 10.13, and are further laser-excited via m = 0 transitions. The final state of the hydrogen atoms as they leave region F3 depends on the value of F3 . Figure 10.14 shows a spectroscopic scan of the states of hydrogen atoms leaving the region with static field F3 as a function of F3 . At higher values of F3 , transition peaks coming from neighboring n-values are intermingled. For F3 above about 22 V/cm, atoms with principal quantum number n ≥ 74 are lost because of static-field ionization. In Bayfield’s experiments, a 5 to 10 V/cm static electric field is usually present during the entire lifetime of the beam. Such a field may or may not be present in other experiments. Variations on the experimental procedure described above have produced beams of each principal quantum number in the range n = 27 to n = 90. However, the unique substate that is produced by the double-resonance laser excitation as the

10.7 Microwave-Driven Hydrogen

371

atoms leave the region F3 is altered into a statistical mixture of parabolic substates of the same n-value by stray fields before the atoms entered the microwave cavity. Classically, this corresponds to a microcanonical ensemble of trajectories filling all spatial dimensions. Thus, three-dimensional theory is needed to completely model the experiments. However, the theory of one-dimensional hydrogen does explain some qualitative features of the experimental data quite well. Two different methods, called ionization and quenching, are used to study hydrogen atoms with principal quantum numbers in the range n = 27 to n = 90. In ω most experiments, the microwave field has frequency f = 2π = 9.9233(4) GHz. This corresponds to an angular frequency in atomic units of ω = 1.51 × 10−6 a.u. The binding energy for a hydrogen atom with principal quantum number n = 66, for example, is E = −1/2n2 = −1.15 × 10−4 a.u. Therefore it requires about 76 photons from the microwave field to ionize the atom. It is interesting to note also that hydrogen atoms that are excited to level n = 66 are huge objects on the atomic scale. They have a radius of about aB n2 = 2.3 × 10−5 cm as compared to hydrogen in the ground state, which has the Bohr radius aB = 5.3 × 10−9 cm. As the hydrogen atoms enter the microwave cavity, in their own reference frame they see the microwave field turn on as they enter the cavity and turn off as they leave the cavity. The precise shape of the turnon-turnoff envelope of the microwave field varies from experiment to experiment. For example, in Koch’s experiments, turn-on and turn-off times of the field each last for about 60 oscillations of the microwave field, and in between the hydrogen atoms move in a constant-amplitude microwave field that lasts for about 300 oscillations. In Bayfield’s experiments, the envelope of the microwave field, as viewed by the atom, looks like half a sine wave. The strength of the microwave field can be determined to about ±5%. In all the experiments, the hydrogen beam is in the microwave field for only a few hundred periods of the microwave field, and any theory of microwave-driven hydrogen should take this fact into account. The ionization experiments detect one or another of the ionization products (the electron or proton) as a function of the microwave electric field amplitude. The quenching experiments detect hydrogen atoms that survive the interaction with the microwave field. Experimental data from the ionization experiments are shown in Fig. 10.15 for principal quantum numbers in the range n = 32 to n = 90 and for microwave frequency f = 9.923 GHz. One notices a number of flat regions in the data for n < 82 and a qualitative change in the data for n > 82. The isolated flat regions for n < 82 are due to isolated higher-order nonlinear resonance zones. As we shall see below, n = 82 is the lower edge of the region of overlap of the primary nonlinear resonance zones (the lower edge of the classically chaotic region). It is interesting to look more closely at the experimental data giving rise to the flat regions in Fig. 10.15. In Fig. 10.16 we show the ionization data for hydrogen atoms with principal quantum numbers in the range n = 65 to n = 74. The microwave frequency is 9.92 GHz. Notice that, in Fig. 10.15, the ionization curves for atoms with principal quantum numbers n = 66 to n = 72 appear to require the same field strength for 90% ionization. This indicates that they lie in a resonance zone.

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Fig. 10.15 The 10% and 90% microwave ionization threshold electric field amplitude, Eth (in V/cm), for microwave frequency f = 9.923 GHz and principal quantum numbers n = 32 to n = 90 (Koch 1988)

Fig. 10.16 Ionization curves for principal quantum numbers in the range n = 65 to n = 74 for microwave frequency f = 9.923 GHz (Koch 1988)

Additional experimental data can be found in Bayfield and Pinnaduwage (1985); van Leeuwen et al. (1985); Bayfield and Sokol (1988); Galvez et al. (1988).

10.7.2 One-Dimensional Approximation In the experiments described in Sect. 10.7.1, the hydrogen atoms are stretched into highly excited elongated states with parabolic quantum numbers (n; n1 , n2 , m) ≈ (n; 0, n − 1, 0) and then driven by a microwave field. To first approximation, the hydrogen atoms can be treated as one-dimensional objects, and much of their large-scale behavior can be described by a one-dimensional approximation.

10.7 Microwave-Driven Hydrogen

373

We can write the Hamiltonian operator, Hˆ , for relative motion of an electron and proton that interact with a microwave field directed along the z-axis (see Appendix K). It is Hˆ = Hˆ 0 − eˆzA(t) cos(ωt),

(10.63)

where e is the charge of the electron, ω is the frequency of the microwave field, and A(t) is the envelope function describing the turnon and turnoff of the microwave field as seen from the electron rest frame. The operator Hˆ 0 is the Hamiltonian operator describing the relative motion of the electron and proton in the absence of the microwave field and is defined as e2 pˆ 2 − , Hˆ 0 = 2μ 4π 0 rˆ

(10.64)

where pˆ is the relative momentum operator of the electron and proton, rˆ is the distance between them, 0 is the permittivity constant, and μ is the electronproton reduced mass. Since μ = 9.1034 × 10−31 kg and the mass of the electron m = 9.1083 × 10−31 kg, in the subsequent discussion we shall assume μ ≈ m. It is useful to write all quantities in Eqs. (9.112) and (9.113) in terms of atomic ˆ 0 , Hˆ = EB H, ˆ t = tB τ , ω = fB ω0 , and eE = FB λ (the units. We will let Hˆ 0 = EB H quantities EB , tB , fB , and FB are defined in Appendix K. We can write the bound ˆ 0 in terms of parabolic quantum numbers (see Appendix K) state eigenvectors of H ˆ 0 |n; n1 , n2 , m = −1 |n; n1 , n2 , m, H 2n2

(10.65)

where the principal quantum number n = n1 + n2 + |m| + 1. Thus, the spectral ˆ is given by decomposition of the total Hamiltonian, H, ˆ )= H(τ



−1 |n; n1 , n2 , mn; n1 , n2 , m| 2n2 n1 ,n2 ,m   −λA (τ ) cos(ω0 τ ) n1 ,n2 ,mn ,n ,m 1

2

 5     n ; n , n , m  1 2  B  ×|n; n1 , n2 , mn ; n1 , n2 , m | + dkEk |kk|,

  z × n; n1 , n2 , m a 4

(10.66)

where A (τ ) = A(t), k is the relative wave vector, and Ek is the relative energy of the unbound proton and electron.

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10 Time-Periodic Quantum Systems

In typical experiments, the principal quantum number n ≥ 30, and then   4 5  z  n; 0, n − 1, 0 n ; k, n − k − 1, 0 ≈ C(n − n , k) n2−k , (10.67) aB where, for example, C(±1, 0) ≈ 0.32, C(±2, 0) ≈ 0.11, and C(+1, 1) ≈ 0.6 [Bardsley and Sundaram 1985]. Thus, if the system is initially in a state |n; 0, n − 1, 0 with n ≥ 30, then to good approximation the Hamiltonian operator can be written   −1 ˆ contin , ˆ )= |nn| − λA (τ ) cos(ω0 τ ) zn,n |nn | + H H(τ 2 n  n n

(10.68) ˆ contin represents contributions where |n ≡ |n; 0, n − 1, 0, zn,n = n| azB |n , and H due to coupling to the continuum. The Schrödinger equation can be written i

∂ψn (τ ) −1 = 2 ψn (τ ) − λA (τ )zn,n cos(ω0 τ ) ψn (τ ) ∂τ 2n  1 − λA (τ ) {zn,n+m eiω0 τ ψn+m (τ ) + zn,n−m e−iω0 τ ψn−m (τ )} 2 m +continuum contributions,

(10.69)

where m ranges over positive and negative integers. In order for Eq. (9.121) to give a good description of microwave-driven hydrogen, we must have ψn (τ ) = 0 if n ≤ 30. In practice, there are several ways of modeling the effect of the continuum. Blumel and Smilansky (1987) have done it by including a “memory kernel” in the Schrödinger equation. Another method is to use a Sturmian basis (Blumel and Smilansky 1987; Casati et al. 1987) in which the continuum is discretized and included in a complete basis that includes both bound state and continuum (but discretized) effects. Further discussion of the validity of using the one-dimensional model to describe highly excited microwave-driven hydrogen can be found in Shepelyansky (1985); Blumel and Smilansky (1987); Casati et al. (1987).

10.7.2.1

Nonlinear Resonances

If we want to study the behavior of the bound states of hydrogen for small enough microwave field amplitudes such that nonlinear resonances have not overlapped, then we can remove continuum contributions. Let us now assume that A (τ ) = 1 and locate the primary resonances for that case. If we neglect coupling to the continuum, the Schrödinger equation for the mth primary resonance is given by

10.7 Microwave-Driven Hydrogen

i

375

−1 ∂ψn (τ ) 1 = 2 ψn (τ ) − λ{zn,n+m eiω0 τ ψn+m (τ ) ∂τ 2 2n +zn,n−m e−iω0 τ ψn−m (τ )}.

(10.70)

We can perform a “pendulum approximation” on Eq. (10.70). Let us expand n about some principal quantum number, n = n¯ m . That is, we write n = n¯ m + η, where η may be a positive or negative integer. We will study the behavior of Eq. (10.70) in the neighborhood of n = n¯ m and therefore will only be interested in values of η small compared to n¯ m . The “pendulum approximation” involves the following approximation. We expand the first term on the right to second order in η, and we evaluate the coefficient of the second term at the point n = n¯ m . Then we find i

∂ψη (τ ) = ∂τ

 −1 η 3η2 ψη (τ ) + − 2n¯ 2m n¯ 3m 2n¯ 4m   1 − λC(m, 0)n¯ 2m eiω0 τ ψη+m (τ ) + e−iω0 τ ψη−m (τ ) . 2



(10.71)

It is useful now to write Eq. (10.71) in the angle picture. We will let G(φ, τ ) = e−iτ/(2n¯ m ) 2

∞ 

ψη (τ )eiηφ .

(10.72)

η=−∞

As long as ψη (τ ) = 0 for n = n¯ m + η < 30, the limits on the sum can be extended to ±∞ as we have done. After some algebra, Eq. (10.71) takes in the angle picture the form i

∂G 1 ∂G 3 ∂ 2G − λC(m, 0)n¯ 2p cos(mφ − ω0 τ )G, = −i 3 + 4 ∂τ n¯ m ∂φ 2n¯ m ∂φ 2

(10.73)

where G = G(φ, τ ). We now make one more change of coordinates. Let θ = mφ − ω0 τ , T = τ , and G(θ, T ) = G(φ, τ ). Then, in terms of these new coordinates, Eq. (10.73) takes the form i

m ∂G 3m2 ∂ 2 G ∂G ∂G − iω0 = −i 3 + 4 − λC(m, 0)n¯ 2m cos(θ )G. ∂T ∂θ n¯ m ∂θ 2n¯ m ∂θ 2

(10.74)

Equation (10.74) is the Schrödinger equation for a particle with average speed n¯m3 − m ω0 moving in the presence of a stationary cosine potential. The particle will be most tightly bound if the average speed of the particle is zero so that n¯m3 = ω0 . Then m Eq. (10.74) becomes i

3m2 ∂ 2 G ∂G =+ 4 − λC(m, 0)n¯ 2m cos(θ )G. ∂T 2n¯ m ∂θ 2

(10.75)

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10 Time-Periodic Quantum Systems

 1/3 With the choice n¯m3 = ω0 or equivalently n¯ m = ωm0 , the principal quantum m number n = n¯ m lies right at the center of the mth primary resonance zone. This is the resonance condition. Equation (10.75) is the Schrödinger equation for a particle n¯ 4

of “mass” m = 3mm2 moving in the presence of the cosine potential with “amplitude”  √ V = 2 λC(m, 0)n¯ 2m . The half-width of the resonance zone for that case is 2 mV . Thus, the position and half-width of the mth primary resonance in one-dimensional hydrogen in a microwave field are given approximately by   1/3 λC(m, 0)n¯ 6m m n = n¯ m = and nm = 2 , (10.76) ω0 3m2 respectively. Let us now compare the estimates above with the experimental results in Fig. 10.15. The microwave field frequency is f = 9.923 GHz. The frequency in atomic units is ω0 = 2πf/fB = 1.51 × 10−6 a.u. From Eq. (10.76), the mth primary resonance is located at n¯ m = 87m1/3 . Thus, n¯ 1 = 87, n¯ 2 = 109, etc. The half width of the first primary resonance can be obtained from Eq. (10.76). The 10% ionization curve is at E = 0.4 V/cm and the 90% ionization curve is at E = 0.8 V/cm. Let us take E = 0.6 V/cm at n = n¯ 1 = 87. Then, λ = E/EB = 1.17 × 1010 a.u. (a.u. = atomic units) and n1 ≈ 5. According to these estimates, the first primary resonance zone is centered at n = 87 and extends down to about n = 82, which is exactly where the data in Fig. 10.15 change character. Another feature of the ionization data that is important to notice in Fig. 10.15 is the plateaus that occur for n < 82. These are due to higher-order or fractional resonances that satisfy the condition m = M N < 1. Burns and Reichl (1992) have verified this by measuring the widths of the fractional resonances (obtained by solving Hamilton’s equations for the classical one-dimensional hydrogen model H = 12 p2 − 1z +λz cos(ω0 t)) at the field frequency and field strengths used to obtain the data in Fig. 10.15. A strobe plot showing some of the fractional and primary resonances for ω0 = 1.5 × 10−6 a.u. and λ = 1.9 × 10−10 a.u. is shown in Fig. 10.17. Burns finds that for ω0 = 1.5 × 10−6 a.u., when λ = 0.62 × 10−8 a.u., the m = 13 resonance, located at n = 60, had width n ≈ 4. When λ = 1.9×10−9 a.u., the m = 1 −9 2 resonance, located at n = 69, had width n ≈ 8.9. When λ = 1.1 × 10 a.u., the m = 23 resonance, located at n = 76, had width n ≈ 1.8. Other fractional resonances in the neighborhood had either decayed into the chaotic sea or had a width n < 1.

10.8 Arnol’d Diffusion in Quantum Systems In Chap. 5, we discussed the mechanisms by which chaos spreads throughout the phase space for classical systems with three or more degrees of freedom. Arnol’d diffusion also exists in quantum systems and is the key mechanism leading to

10.8 Arnol’d Diffusion in Quantum Systems

377

Fig. 10.17 A strobe plot showing some fractional and primary resonances for classical onedimensional hydrogen with Hamiltonian H = 12 p 2 − 1z + λz cos(ω0 t). The field strength is λ = 1.9 × 10−10 a.u. and the frequency is ω0 = 1.5 × 10−6 . The fractions in the figure indicate values of m = n3 ω0 (Burns and Reichl 1992)

thermalization of quantum systems. Until now there has not been much work analyzing the mechanisms leading to Arnol’d diffusion in quantum systems. One of the earliest works (von Milczewski et al. 1996) involved the computation of the Arnol’d web for a hydrogen atom in crossed electric and magnetic fields. They related the Arnol’d web in the underlying classical system to the diffusion of the electron wavefunction in the quantum system. Arnol’d diffusion has also been reported to increase the conductance of open billiards with 3 DoF (Nakamura and Harayama 2004). More recently, Malyshev and Chizhova (2010) have studied the classical and quantum dynamics of two weakly coupled oscillators placed in a timeperiodic external field. For the system they considered, which has 2.5 DoF, they found diffusion in the quantum system that parallels that in the classical system, but a diffusion rate that was an order of magnitude slower in some parameter regimes. In classical systems, primary resonances and “daughter” resonances that occur due to interaction between primary resonances, form a self-similar network in the phase space, and the diffusion process can occur at all length scales. Quantum systems are primarily influenced by classical structures that occupy a volume in the phase space of order Planck’s constant or larger. In resonance regions that are larger than Planck’s constant, quantum states can spread throughout resonances and chaotic regions (Reichl 1989; Morrow and Reichl 1994). Therefore, it is not surprising that the analog of Arnol’d diffusion exists in quantum systems, although perhaps on a more subdued scale. The effect of Arnol’d diffusion in a quantum system can be seen clearly in the driven optical lattice, as we shall now show.

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10.8.1 Arnol’d Diffusion in the Driven Optical Lattice The quantum dynamics of particles in the optical lattice discussed in Sect. 5.5, in dimensionless units, is governed by the Schrödinger equation   ∂2 ∂ ∂2 2 i ψ(x, y, t)= − 2 − 2 +(V0 +V1 cos (ωt))V (x, y) ψ(x, y, t), ∂t ∂x ∂y

(10.77)

where   V (x, y) = U b cos(x) ˆ cos(y) ˆ + cos2 (x) ˆ + cos2 (y) ˆ

(10.78)

Boretz and Reichl (2016). We focus on the dynamics of a single unit cell of the optical lattice. The potential energy V (x, y) is invariant under reflection through the origin in both the x and y directions. The wave function ψ(x, y, t) can be expanded in terms of functions that are either symmetric or antisymmetric under reflection through the origin. Each subspace can be treated independently. In subsequent sections, we will focus on the subspace of the Hamiltonian formed by the basis set sin(nx x)sin(ny y), (1 ≤ nx ≤ ∞, 1 ≤ ny ≤ ∞). The other blocks will behave in a qualitatively similar manner. Before looking at the time-periodically driven system, it is useful to consider the time-independent case V1 = 0. The Hamiltonian is   ˆ cos(y) ˆ + cos2 (x) ˆ + cos2 (y) ˆ . Hˆ (0) = pˆ x2 + pˆ y2 + U b cos(x)

(10.79)

If we denote the nth eigenfunction of the Hamiltonian Hˆ (0) as |En  then Hˆ (0) |En  = En |En  and the solution to the Schrodinger equation can be written −i Hˆ (0) t

|ψ(t) = e

∞  |ψ(0) = e−iEn t En |ψ(0)|En .

(10.80)

n=1

For the case b = 0.2, the maximum value of the potential energy is Vmax = 44 and there are 9 anti-symmetric energy eigenstates with energies below this value. For the case b = 2.0, the maximum value of the potential energy is Vmax = 80 and there are 21 anti-symmetric energy eigenstates with energies below this value. We next consider the time-periodically driven system (V1 = 0), and expand the wave function in terms of the basis states sin(nx x)sin(ny y) to get ψ(x, y, t) =

∞ ∞ 1 bnx ,ny (t)sin(nx x)sin(ny y). π nx =1ny =1

(10.81)

10.8 Arnol’d Diffusion in Quantum Systems

379

If we substitute Eq. (10.81) into Eq. (10.77), we obtain i

∞ ∞   ∂ bmx ,my (t) = Hmx ,my ;nx ,ny (t)bnx ,ny (t) ∂t

(10.82)

nx =1ny =1

where Hmx ,my ;nx ,ny (t) = (m2x + m2y )δmx ,nx δny ,my + Vmx ,my ;nx ,ny (t)

(10.83)

and Vmx ,my ;nx ,ny (t)=

1 π2



2π  2π

dxdy sin(mx x)sin(my y)V (x, y, t)sin(nx x)sin(ny y). 0

0

(10.84) For computational purposes, it is useful to separate the wave function into its real and imagninary parts. We let bnx ,ny (t) = pnx ,ny (t)+iqnx ,ny (t), and the Schrodinger equation then takes the form ∞ ∞   d mx ,my (t) = Lmx ,my ;nx ,ny (t)nx ,ny (t) dt

(10.85)

nx =1ny =1

where  nx ,ny (t) =

pnx ,ny (t) qnx ,ny (t)

 (10.86)

and 

 Lmx ,my ;nx ,ny (t) =

0

Hmx ,my ;nx ,ny (t) −Hmx ,my ;nx ,ny (t) 0

(10.87)

In practice it is necessary to truncate the infinite Hamiltonian matrix to a finite size. Let us now introduce the 1 × N 2 (N a large integer) column matrices p¯ and q¯ which are defined (we write the transpose), p¯ T = (p1,1 , p1,2 , . . . , p1,N , p2,1 , . . . p2,N , p3,1 , . . . , pN,N ) and

q¯ T = (q1,1 , q1,2 , . . . , q1,N , q2,1 , . . . q2,N , q3,1 , . . . , qN,N ) (10.88)

¯ Orthonormality of the wave function requires that b¯ † ·b¯ = 1. such that b¯ = p¯ + i q. ¯¯ is defined 2 If the 1×2N column matrix  ¯¯ (t) =



¯ p(t) ¯ q(t)

 (10.89)

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10 Time-Periodic Quantum Systems

¯¯ then the equation of motion of (t) is given by d ¯ ¯¯ ¯¯ ¯ (t) = L(t)· (t) dt

(10.90)

¯¯ where L(t) is the real skew-symmetric 2N 2 × 2N 2 matrix ¯¯ L(t) =



 ¯ 0 H(t) . ¯ −H(t) 0

(10.91)

¯¯ Note that L(t) is periodic in time with period T0 = 2π ω . In terms of the column ¯ ¯ matrices p(t) and q(t), the Schrödinger equation can be written in terms of the coupled equations ¯ ¯ d p(t) d q(t) ¯ ¯ ¯ ¯ = H(t)· q(t) and = −H(t)· p(t). dt dt

(10.92)

We have now reduced the quantum problem to the task of solving 2N coupled first order differential equations with time-periodic coefficients. For this we need Floquet theory.

10.8.1.1

Floquet States

¯¯ Let us consider Eq. (10.90) where L(t) is time-periodic with period T0 = Assume that Eq. (10.90) has a Floquet-type solution of the form 

p¯ α (t) q¯ α (t)



 =e

iα t

α (t) ≡ e

iα t

 ¯ α (t) P ¯ α (t) , Q

2π ω .

(10.93)

where α is the αth Floquet eigenphase (also called quasienergy) and the Floquet ¯ α (t))T , is periodic with period T0 , ¯ α (t), Q eigenstate, T (t) = (P 

¯ α (T0 ) P ¯ α (T0 ) Q



 =

 ¯ α (0) P ¯ α (0) . Q

(10.94)

The Floquet eigenstate satisfies the eigenvalue equation     ¯ α (t) ¯ α (t) P P ¯ ¯ LF · ¯ = iα ¯ Qα (t) Qα (t)

(10.95)

10.8 Arnol’d Diffusion in Quantum Systems

381

  ¯¯ ¯¯ , is defined L ¯¯ = L(t) d . The solution to the where the Floquet operator, L − dt F F Schrödinger equation can be expanded in a complete set of Floquet eigenstates so that      ¯ ¯ p(t) iα t Pα (t) = (10.96) Aα e ¯ α (t) . ¯ q(t) Q α Since 

¯ p(0) ¯ q(0)

 =



 Aα

α

 ¯ α (0) P ¯ α (0) , Q

(10.97)

and the Floquet eigenstates are assumed to be othonormal so 

¯ †  (0) ¯ †α  (0), Q P α

 †  P ¯ α (0) = δα  ,α , · ¯ Qα (0)

(10.98)

and the coefficients Aα can be written 

¯ † (0) ¯ †α (0), Q Aα = P α

 †  p(0) ¯ . · ¯ q(0)

The solution then takes the form          p(0) ¯ (t) ¯ ¯ p(t) P † † ¯ ¯ = . eiα t ¯ α · Pα (0), Qα (0) ¯ ¯ q(t) q(0) Qα (t) α

(10.99)

(10.100)

At time t = T0 we can write 

  ¯ p(0) ¯ ¯ . = U(T0 )· ¯ q(0)

(10.101)

  ¯ α (0)  † P ¯ † (0) . ¯ α (0), Q P α ¯ α (0) Q

(10.102)

¯ 0) p(T ¯ 0) q(T



where ¯¯ U(T 0) =



 iα T0

e

α

is the unitary Floquet evolution matrix with eigenvalues eiα T0 . The solution to the Schrödinger equation at time nT0 is 

¯ p(nT 0) ¯ q(nT 0)





¯¯ = U(T 0)

 n  p(0) ¯ . · ¯ q(0)

(10.103)

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The spatial distribution of the αth Floquet eigenstate is given by α (x, y)=

∞ ∞

1   ¯ ¯ α (0)n ,n sin(nx x)sin(ny y). (10.104) Pα (0)nx ,ny +i Q x y π nx =1ny =1

10.8.1.2

Behavior of Quantum States

We can follow the behavior of the quantum states as we turn on the amplitude V1 of the time-periodic modulation of the optical lattice. We shall assume that the driving frequency is ω = 2π , in dimensionless units. For the case V1 = 0, the Hamiltonian is independent of time and we can write (0) L¯¯ ≡



¯ (0) 0 H (0) ¯ −H 0

 (10.105)

.

The solution to the Schrödinger equation Eq. (10.90) can be written ¯¯ (0) ¯ ¯¯ ¯ (t) = eL t (0).

(10.106)

We denote the nth and (n + 1)th eigenvectors of L¯¯

(0)

1 φ¯ 2n−1 = √ 2



i|En  |En 



1 and φ¯ 2n = √ 2



as −i|En  |En 

 (10.107)

Then L¯¯ φ¯ 2n−1 = −iEn φ¯ 2n−1 and L¯¯ φ¯ 2n = iEn φ¯ 2n . Furthermore, φ¯ 2n−1 = ∗ . For V = 0, the Floquet eigenphases occur in pairs that are equal to ±iE φ¯ 2n 1 n and the real and imaginary parts of the Floquet eigenstates are equal to the energy eigenstates. Therefore, at V1 = 0 we can identify each Floquet eigenphase and eigenstate with an energy eigenvalue and eigenstate. However, as we turn on the modulation, each Floquet state changes from a pure energy eigenstate to an energy entangled state as the amplitude of the time periodic modulation increases. Energy is no longer conserved. ¯ Tα ·L¯ 0 · ¯ Tα for the Floquet In Fig. 10.18 we show the average energy E =  eigenstates α = 1, . . . 8. At V1 = 0, the average energy is equal to the energy eigenvalue En=α and remains essentially unchanged for small non-zero values of V1 . However, for larger, non-zero values of V1 the average energy undergoes significant excursions, indicating that the Floquet eigenstates begin to consist of a superposition of a number of energy eigenstates. In Fig. 10.18a, we show the average energy of Floquet eigenstates α = 1, . . . 8 for b = 0.2. There is a slight increase of the average energy of some states as V1 increases, but not a significant increase. Two of the states, α = 3 and α = 6, are almost completely unchanged by the time periodic modulation. In Fig. 10.18b, we show the average energy of Floquet (0)

(0)

10.8 Arnol’d Diffusion in Quantum Systems

383

Fig. 10.18 Change in the average energy E = α |Hˆ |α  of the Floquet eigenstates for α = 1, . . . 8, as a function of V1 (states are labeled with value of α). (a) b = 0.2. (b) b = 2.0. For V1 = 0, |α=n  = |En  (Boretz and Reichl 2016)

eigenstates α = 1, . . . 8 for b = 2.0. The average energy now takes significant excursions well above the maximum potential energy for that value of coupling parameter, b. The average energy of the state α = 6 now gets “pumped” to very high energy. This growth in the average energy is consistent with the classical behavior of the average energy shown in Fig. 5.10. We expect that the growth in average energy will lag the classical system (as a function of b) because the quantum system does not “see” phase space structure smaller than h. ¯ However, for b = 2, the classical and quantum results for average energy are in good agreement. We can conclude

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10 Time-Periodic Quantum Systems

that this dramatic change in the behavior of the quantum system is a quantum manifestation of the transition from the Nekhoroshev regime to the Chirikov regime in the periodically driven optical lattice. Indeed, it appears that Arnold diffusion gives rise to Floquet eigenstates consisting of large numbers of entangled energy states of the nondriven quantum system and leads to energy instability of such systems. The energy instability we observe in the driven 2d optical lattice, appears to be a universal property of any periodically driven material system with two or more degrees of freedom.

10.9 Quantum Control Laser radiation provides a means to control intra-molecular processes in a robust manner. The process is called STIRAP (stimulated Raman adiabatic passage). STIRAP involves the application of short laser pulses with carefully chosen carrier frequencies to a molecular system for the purpose of exciting the molecules in a controlled manner. This technique causes the coherent change of an entire molecular population between targeted molecular states (Hioe 1983; Oreg et al. 1984; Shore 1990; Bergmann et al. 1998). In the remainder of this section, we will use Floquet theory to model the exact dynamics underlying STIRAP for the case of a particle confined to a square well potential, but driven by two laser pulses of different frequency (Na and Reichl 2004).

10.9.1 The Model (Classical Dynamics) The classical Hamiltonian for the model of STIRAP that we consider here is H = p2 + Uf (t)xcos(ωf t) + Us (t)xcos(ωs t),

for

|x| < 1.

(10.108)

All parameters in Eq. (10.108) are dimensionless. The amplitude Uf (t) of the first pulse (in time) and the amplitude Us (t) of the second pulse, have Gaussian time dependence of the form, Uf (t) = Uo exp(−β(t − tf )2 ) and Us (t) = Uo exp(−β(t − ts )2 ),

(10.109)

where tf < ts . We can control the duration of each pulse by adjusting the parameter β and we can control the amount of overlap of the two pulses by changing tf and ts , which give the peak times of the first and second pulses, respectively. For simplicity, we assume that the maximum amplitude Uo and the width β of the two pulses are the same. A schematic picture of the variation in time of the amplitudes of the two pulses is displayed in Fig. 10.19. The pulse sequence ends at the total pulse duration time

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Fig. 10.19 Schematic diagram for the two pulses. The first pulse connecting levels |E2  and |E3  is shown as a solid line. The second pulse connecting levels |E1  and |E2  is shown as a dotted line. They have maximum strength Uo at times t = tf and t = ts , respectively. The whole pulse sequence takes a time t = ttot to complete. In the figure, t1 = 1/20ttot , tc = 1/2ttot and t2 = 19/20ttot

t = ttot . For the purpose of marking time intervals in our subsequent discussion, we choose times, t1 = 1/20ttot , tc = 1/2ttot , and t2 = 19/20ttot . We can perform a canonical transformation to action-angle variables (J, θ ) defined J = 2|p|/π , and θ = ±π(x + 1)/2. The Hamiltonian then has the form, H =

∞ 4Uf (tf ix )  1 π 2J 2 − cos((2ν − 1)θ − ωf t) 2 4 π (2ν − 1)2 ν=−∞

∞ 4Us (tf ix )  1 cos((2ν − 1)θ − ωs t), for 0≤θ ≤π. 2 π (2ν − 1)2 ν=−∞ (10.110) An infinite number of nonlinear resonances are produced in the classical phase space by the external fields. The primary resonances are located at J = Jν ≡ 2ωf,s /((2ν − 1)π 2 ). As ν increases, the location in energy of higher order primary resonances decreases. In Fig. 10.20, we show strobe plots of the classical phase space for the case with Uo = 3.0. We choose the pulse carrier frequencies to be ωf = 3π 2 /4 and ωs = 5π 2 /4. For the frequencies we have chosen, the period of the Hamiltonian is To = 8/π . For the five cases shown in Figs. 10.20a–e, we fix the amplitudes Uf (t) and Us (t) of the pulses by setting (a) tf ix = t1 , (b) tf ix = tf , (c) tf ix = tc , (d) tf ix = ts , and (e) tf ix = t2 , respectively. The three largest primary resonances (ν = 1, 2, 3) induced by the first pulse are located at J = 2.5, J = 0.83 and J = 0.5, respectively. The three largest primary resonances (ν = 1, 2, 3) induced by the second pulse are located at J = 1.5, J = 0.5 and J = 0.3, respectively. In Fig. 10.20a, with Uf (t1 ) = 0.1667

−

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Fig. 10.20 Strobe plots of the action-angle variables (J, θ) for the infinite square well system with pulse amplitudes Uo = 3.0 and frequencies ωf = 5ωo and ωs = 3ωo . Strobe plots are shown at times (a) tf ix = t1 , (b) tf ix = tf , (c) tf ix = tc , (d) tf ix = ts and (e) tf ix = t2 . For each plot 0≤θ≤π . The three largest primary resonances ν = 1, 2, 3 due to the first pulse are located at J = 2.5, 0.83 and 0.5. The three largest primary resonancesν = 1, 2, 3 due to the second pulse are located at J = 1.5, 0.5 and 0.3 (Na and Reichl 2004)

and Us (t1 ) = 0.000003, the primary resonances induced by the first pulse are dominant. In Fig. 10.20e, with Uf (t2 ) = 0.000003 and Us (t2 ) = 0.1667, the primary resonances induced by the second pulse are dominant. In all cases, the first primary resonance (ν = 1) is located at the highest energy and the higher order primary resonances are located at decreasing energy as ν increases. As a result, this system will always have a chaotic region at low energy due to the overlap of higher order resonances. For energies above the region of influence of the ν = 1 primary of the first pulse, the phase space is dominated by KAM (Kolmogorov-Arnold-Moser) tori. In Fig. 10.20c, where tf ix = tc = 1/2ttot , the primary ν = 1 resonances due to the two pulses have equal amplitude and are clearly visible at J = 2.5 and J = 1.5. For this case the pulse amplitudes are Uf (tc ) = Us (tc ) = 1.103. All the higher order primary resonances have been destroyed and a large chaotic sea has formed at low energy.

10.9.2 The Model (Quantum Dynamics) The Schrödinger equation for this system can be written (in dimensionless units)   ∂2 ∂ i x|ψ(t) = − 2 + Uf (t)xcos(ωf t) + Us (t)xcos(ωs t) x|ψ(t), ∂t ∂x (10.111)

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where the momentum operator is given by pˆ = −i∂/∂x. In order to satisfy the boundary condition at x = ±1, the wavefunction should satisfy ψ(x = ±1, t) = x = ±1|ψ(t) = 0 for all times, t. For the unperturbed system, the energy eigenvalues are En = n2 π 2 /4 and the orthonormal energy eigenstates are x|En  = φn (x) = sin[nπ(x −1)/2]. The dipole matrix elements in this basis are xn,n = En |x|E ˆ n  where  xn,n =

0,

16nn

π 2 (n2 −n 2 )

[n + n ] , [n + n ] 2

(modulo 2) = 0 (modulo 2) = 1.

(10.112)

Note that integer values of J (J = n) in the classical Hamiltonian correspond to the quantized states of the quantum system. This simplifies comparison between the classical and quantum systems. Once we fix the amplitudes, Uf (t = tf ix ) = Uf (tf ix ) and Us (t = tf ix ) = Us (tf ix ), the Hamiltonian becomes time periodic and the Schrödinger equation takes the form   ∂2 ∂ i x|ψ(t) = − 2 + Uf (tf ix )xcos(ωf t) + Us (tf ix )xcos(ωs t) x|ψ(t). ∂t ∂x (10.113) If the carrier frequencies of the pulses are commensurate so ωf /ωs = nf /ns , where nf and ns are integers, then the Hamiltonian is invariant under a discrete time translation H (t) = H (t + To ), where the period To of the Hamiltonian is   nf ns . (10.114) + To = π ωf ωs For the case when the total Hamiltonian is periodic in time, we can use Floquet theory to analyze the dynamics of the driven system. We can compute the Floquet eigenvalues and eigestates for fixed Uf (tf ix ) and Us (tf ix ). Then we compute them again for slightly different amplitudes Uf (tf ix + ) and Us (tf ix + ). If

is small enough, the eigenstates for the two different times, tf ix and tf ix +

will be approximately orthonormal. We can use this to follow the evolution of the eigenstates as the amplitudes Uf (tf ix ) and Us (tf ix ) evolve in time.

10.9.3 Floquet States For the system in Eq. (10.113), Floquet eigenstates, |φα (t) (which have period To so |φα (t + To ) = |φα (t)) form a complete orthonormal basis which determines the dynamics. Furthermore, the Floquet eigenphases, α , are conserved quantities. We can obtain an eigenvalue equation for α and |φα (t). Consider the case when the system is in the αth Floquet eigenstate so that |ψ(t) = e−iα t |φα (t). Then substitution into Eq. (10.113) yields the eigenvalue equation

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  ∂ ˆ |φα (t) = α |φα (t), H (t) − i ∂t

(10.115)

where Hˆ F (t)≡Hˆ (t) − i∂/∂t is the Floquet Hamiltonian. ˆ F (To ), can be written The Floquet evolution operator, U ˆ F (To ) = U



e−iα To |φα (0)φα (0)|.

(10.116)

α

We compute matrix elements of the Floquet evolution operator in the basis of square-well energy eigenstates. Then the (n, n )th matrix element of the resulting Floquet matrix is given by ˆ F (To )|En  = Un,n (To ) = En |U



e−iα To En |φα (0)φα (0)|En .

(10.117)

α

The αth eigenvalue of the Floquet matrix Un,n (To ) is exp(−iα To ) and the αth eigenvector in the unperturbed energy basis is given by a column matrix composed of matrix elements, En |φα (0), where n = 1, . . . , ∞. The eigenvalues α (quasienergies) can be obtained from exp(−iα To ), but only modulus ωo . For the driven square-well system, the Floquet matrix has a natural truncation which is determined by the nonlinear dynamics of the system. Classically, the driven square-well has a region of mixed phase space bounded at high energies by KAM tori. If the initial state |ψ(0) is the unperturbed energy level |ψ(0) = |E1 , for example, the state |ψ(t) can not penetrate very far into the high energy KAM region. This provides a natural truncation of the size of the Floquet matrix and we need to include enough unperturbed basis states, |En , to cover adequately the region of mixed phase space. Each column of the Floquet matrix can be constructed by solving the timedependent Schrödinger equation for one period, To , with the system initially in one of the unperturbed energy eigenstates. This integration is performed using each of the unperturbed energy eigenstates as an initial state until all the columns of the Floquet matrix has been computed. Floquet eigenphases and eigenstates are then obtained by numerically diagonalizing the Floquet matrix.

10.9.4 STIRAP We now consider the case where the first pulse connects levels n = 2 and n = 3 and the second pulse connects levels n = 1 and n = 2. This is the traditional model for the STIRAP ladder process. However, as distinct from the usual discussion of STIRAP, we will deal with the exact dynamics of the system. We will take account of the fact that we have a multilevel system that can undergo a transition to chaos.

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We can determine the behavior of Floquet eigenstates at fixed times t = tf ix during which the pulses drive the system. The distribution of probability in the Floquet eigenstates is sensitive to structures in the classical phase space which are larger than Planck’s constant. The first pulse has carrier frequency ωf = (E3 − E2 ) = 5π 2 /4 and the second pulse has carrier frequency ωs = (E2 − E1 ) = 3π 2 /4. These frequencies are commensurate since ωf /ωs = 5/3. From Eq. (10.114), the period of the Hamiltonian is To = 8/π and the Floquet frequency is ωo = 2π/To = π 2 /4. Thus, ωf = 5ωo and ωs = 3ωo . We set Uo = 3.0. The classical dynamics of this system tells us that we only need to keep five unperturbed energy eigenstates as a basis to form the Floquet matrix. This can be seen from Fig. 10.20 where we show the underlying classical phase space at selected values of tf ix during the time that the pulses are on. For J > 5, the classical phase space is dominated by KAM tori with almost constant values of J and the unperturbed energy states are very weakly coupled by the dynamics for n > 5. Thus, it is sufficient to construct a 5 × 5 Floquet matrix with the five basis states |E1 , . . . , |E5 . We find that only four of the five Floquet eigenstates of are actively involved in the dynamics. To keep track of the changes that occur in the Floquet eigenstates, we will give each eigenstate a unique alphabetical label determined by its dominant dependence on unperturbed energy states at time tf ix = 0. We find that at tf ix = 0 the Floquet eigenstates have the following structure and we give them, and the corresponding eigenphases, the following labels: A = |φ1 =|E1 , D = |φ4 =|E4  and E = |φ5 =|E5 , 1 1 BC + =|φ2 = √ (|E2 +|E3 ), BC − =|φ3  = √ (|E2 −|E3 ). (10.118) 2 2 The four relevant eigenphases, α , obtained from the 5 × 5 Floquet matrix are plotted modulo ωo = π 2 /4 in Fig. 10.21a. Two of these Floquet eigenphases, A and D, are almost degenerate over the time interval that the pulses act and are not distinguishable on the scale shown in Fig. 10.21a. For tf ix = 0, the four Floquet eigenphases are approximately degenerate modulo ωo . In Fig. 10.21b, we focus on eigenphases A and D. They appear to undergo an avoided crossing at time tf ix = τavoid ≈ 10/23ttot just before tf ix = tc , and a crossing for tf ix > tc . We can follow each Floquet eigenstate during the entire process by computing the eigenstates for a sequence of values of tf ix over the interval 0 ≤ tf ix ≤ ttot . For closely spaced values of tf ix , Floquet eigenstates at different times belonging to different eigenphases will be orthogonal. This provides a means of following the evolution of each eigenphase and eigenstate as a function of tf ix . As we will see, the Floquet eigenstates can change structure when avoided crossings occur between Floquet eigenphases.

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Fig. 10.21 Floquet eigenphases, for the system with maximum pulse strength Uo = 3.0 and frequencies ωf = 5ωo and ωs = 3ωo , are plotted over the entire interval 0≤tf ix ≤ttot . (a) Floquet eigenphases for four Floquet states A, BC + , BC − and D plotted mod ωo = π 2 /4. The wide avoided crossing at tf ix = tc is clear. (b) Floquet eigenphase curves for the Floquet states A and D. The sharp avoided crossing at tf ix = τI and the crossing at tf ix = τI I are clearly seen (Na and Reichl 2004)

10.9.5 Avoided Crossings The Floquet eigenphases for states A and D in Fig. 10.21b avoid crossing at time tf ix = τI . Before time tf ix = τI Floquet state A is predominately composed of level n = 1 and Floquet state D is predominately composed of level n = 4. After the avoided crossing at time tf ix = τI the states have changed their character and Floquet state A is composed predominately of level n = 4 and Floquet state D is composed predominately of level n = 1. Because of the avoided crossing at tf ix = τI , the entire population gets shifted from level n = 1 to level n = 4 before the traditional STIRAP ladder process can take place. These transitions are clearly seen in Fig. 10.22, where we show the level compositions of the four participating Floquet states, A, BC + , BC − and D as a function of tf ix . Now let us determine how behavior of the Floquet states affects the exact behavior of the system, by solving the Schrödinger equation in Eq. (10.111). We will assume that at time t = 0 the system is in state |ψ(0) = |E1 . As we will see, the actual dynamics of this system is determined by the length of time during which the pulses are allowed to act. The pulse duration time necessary to achieve adiabatic behavior of the system is determined largely by the avoided crossings in the Floquet eigenphases. Avoided crossings of Floquet eigenphases occur as the classical phase space becomes chaotic, and a symmetry has been broken in that local region of the phase

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Fig. 10.22 Probability distribution |En |φα |2 of the unperturbed energy levels |En  which compose each of the Floquet eigenstates (a) BC − (b) BC + , (c) A and (d) D, plotted over the entire interval 0≤tf ix ≤ttot for pulse amplitude Uo = 3.0 and frequencies ωf = 5ωo and ωs = 3ωo . The probability curve for level |En  is labeled with level quantum number n (Na and Reichl 2004)

space. At a crossing of levels, the states continue through the crossing unchanged in character. For example, in Fig. 10.21b, if states “A” and “D” actually crossed at tf ix = τI then the lower curve for t < τI and the upper curve for t > τI would have the same character, they look like state “A”. Similarly, the upper curve for t < τI and the lower curve for t > τI would look like state “D”. When a symmetry is broken and the states “A” and “D” undergo an avoided crossing, a different behavior occurs. If the pulses evolve slowly (adiabatically), and the system is in eigenstate “A” for t < τI it will follow the continuous curve for state “A”, which for t > τI has the character of state “D”. If the pulses evolve rapidly, the state “A” doesn’t “see” the avoided crossing, transitions across it, and continues to maintain the character of the state “A”. The probability PLZ that a transition across the avoided crossing occurs for two Floquet eigenstates involved in an isolated avoided crossing can be computed from a formula obtained independently by Landau (1932) and Zener (1932). For our system, the Landau-Zener probability is given by   π(δ )2 , PLZ = exp − 2γ

(10.119)

where δ is the eigenphase spacing at the avoided crossing and γ is the rate of change of the Floquet eigenphases with respect to time tf ix in the neighborhood of the avoided crossing. The Landau-Zener probability PLZ for the isolated sharp avoided crossing at time tf ix = τI shown in Fig. 10.21b was computed in Na and Reichl (2004). The

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Fig. 10.23 The probability Pn (t) = |En |ψ(t)|2 to find the system in the unperturbed level |En  for the system prepared in initial state |ψ(0) = |E1  with maximum pulse strength, Uo = 3.0 and frequencies ωf = 5ωo and ωs = 3ωo . The total pulse duration times are (a) ttot = 120, (b) ttot = 21000, and (c) ttot = 270,000. The numbers attached to each curve show the components of the transition probability in terms of the unperturbed energy eigenstate basis. Case (a) is not in the adiabatic regime. Cases (b) and (c) are within the adiabatic regime and basically reproduce the structure of the single Floquet eigenstate A in Fig. 10.5c (Na and Reichl 2004)

Landau-Zener probability depends on ttot . The larger ttot , the more “stretched out” the horizontal axis in Fig. 10.21c will be relative to the vertical axis. They obtained the following results by analyzing Fig. 10.21c for different values of ttot . For ttot = 120, δ = 0.005, and γ = 0.001875 giving a Landau-Zener probability PLZ = 0.979270. For ttot = 21000, δ = 0.0063, and γ = 0.00001376 giving a LandauZener probability PLZ = 0.0108. For ttot = 270000, δ = 0.0030, and γ = 1.2×10−7 giving a Landau-Zener probability PLZ ≈0. The first case is not in the adiabatic regime, but the second two cases are in the adiabatic regime because the probability of a transition is negligible. In Fig. 10.23, we show the probability Pn (t) = |En |ψ(t)|2 (for the four levels n = 1, 2, 3, 4) to find the system in the nth unperturbed level at time t for the three cases; ttot = 120, ttot = 21000 and ttot = 270000. These results are obtained by directly solving the Schrödinger equation, Eq. (10.111). In all cases, the system is started in the initial state, |ψ(0) = |E1  with maximum pulse strength, Uo = 3.0. In Fig. 10.23a, where there is a large Landau-Zener probability for the system to jump from Floquet state A to Floquet state D, the system comes out of the sharp avoided crossing at tf ix = τI still predominately in the level |E1 . As the pulses are turned on and off, the system transitions from level |E1  to level |E3 . In Fig. 10.23b and c, the Landau-Zener probability is essentially zero and no transition occurs at the sharp avoided crossing at tf ix = τavoid . The system comes out of the sharp

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avoided crossing in level |E4 . As the laser pulses are turned on and off the system transitions from the initial state |ψ(0) = |E1  to the final state |ψ(+∞) = |E4 . Note that both Fig. 10.23b and c follow almost exactly the behavior of the Floquet state A shown in Fig. 10.5c. This is an indication that we are in the adiabatic regime in Fig. 10.23b and c. The very large oscillations in the probability in Figs. 10.23b and c have been explained by Berry (1990) in terms of a sequence of “super-adiabatic bases”. He shows that the decrease in the amplitude of these oscillations as we increase ttot is a sign that we are moving further into the adiabatic regime. The frequencies of the oscillations in Fig. 10.7b and c appear to be determined by the difference in Floquet eigenphases of the two Floquet states involved in the sharp avoided crossing. For example, at tf ix = tf the period of the oscillation is Tosc ≈ 400. The difference in the Floquet eigenphases is || = |1 − 4 | ≈ 0.016. Thus, Tosc = 2π/|| = 393. Similarly, at tf ix = (τI − tf )/2, Tosc ≈ 600. The difference in the Floquet eigenphases is || = |1 − 4 | ≈ 0.011. Thus, Tosc = 2π/|| = 571. The observed oscillation periods are the same for both Fig. 10.23b and c. As a result of the application of the two carefully chosen laser pulses, the system has been excited from the ground state energy level to the third excited state with a probability P ∼ 0.9, when the pulses have a duration ttot = 120. This is the expected, and desired, result of STIRAP. However, for much longer pulse durations, Ttot = 21000 and Ttot = 270000, the system transitions from the ground state to the fourth excited state, with probability close to one. This unexpected transition is due to an avoided crossing likely induced by the underlying chaos. The case with pulse duration Ttot = 270000 is a truly adiabatic process and the system transitions to the fourth excited state with a probability close to one. Thus, the carefully applied laser pulses have essentially allowed to us to control the transitions in the quantum system with close to 100% accuracy.

10.10 Conclusions Examination of the dynamical behavior of periodically driven quantum systems has shown us that much of the nonlinear behavior found in classical mechanics carries over to the quantum domain. Although the quantum systems we have studied in this chapter do not become chaotic, they do exhibit a change in behavior as nonlinear resonances overlap and destroy quantum numbers locally. The wave function representing the state of the system becomes extended throughout the region of nonlinear resonance overlap. However, we have the additional possibility of dynamic Anderson localization, which can limit the extension of the wave function. It is interesting that higher-order resonances also exist in quantum systems and have been observed in microwave-driven hydrogen. This gives experimental backing to the observation that nonlinear resonances form a self-similar structure in quantum dynamical systems (at least down to sizes of order h) ¯ as they do in classical systems.

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In recent years, studies have been done on kicked Rydberg-type atoms that further demonstrate some of the types of phenomena discussed in this chapter. For example, Rydberg atoms that are kicked by short periodic pulses have been found to exhibit dynamical stabilization of the electron (Reinhold et al. 1997) and transient localization of the electron (Stokely et al. 2002). Also quasi onedimensional Rydberg atoms have been created with very high quantum number (n ∼ 350) (Stokely et al. 2003). The Fermi accelerator model (see Sect. 3.9), which consists of a ball bouncing between a fixed wall and an oscillating wall, is a periodic time-dependent system, although of a slightly different type than considered so far in this chapter. Jose and Cordery (1986) have quantized the Fermi accelerator model for a particular form of wall oscillation and have studied its Floquet spectral statistics. They find a transition from Poisson-like to Wigner-like spectral statistics as they vary a parameter of the system. As we have seen in this chapter, Floquet theory provides a powerful tool for describing the photon structures that can form when time-periodic fields interact with nonlinear systems. As was shown in Li and Reichl (1999, 2000); Martinez and Reichl (2001); Emmanouilidou and Reichl (2002), a Floquet scattering matrix can be constructed to study scattering processes in the presence of strong timeperiodic fields. The construction of a Floquet scattering matrix allows one to use the concepts of scattering theory to analyze scattering processes that involve the emission and absorption of large numbers of photons. This theory allows one to analyze transmission probabilities, formation of quasibound states, effects of evanescent modes, and delay times. It has been applied to electron transport in heterostructures and the scattering of electrons from atomic systems in the presence of radiation fields. In this chapter, we have focused on the behavior of quantum systems driven by time-periodic forces. An introduction to the problem of control of quantum systems with nonperiodic external forces can be found in Bayfield (1999). In Sect. 10.3.5, we showed numerical evidence of KAM behavior in quantum systems and derived a renormalization map that had a stable manifold, again indicating quantum KAM behavior. We should note that several years ago Hose and Taylor (1983); Hose et al. (1984) proposed a criterion, based on perturbation theory, to determine if states of a perturbed nonintegrable Hamiltonian can be assigned a full set of quantum numbers (equal to the number of degrees of freedom) given that the unperturbed Hamiltonian is integrable. Such a state would be a quantum KAM state. However, the Hose-Taylor criterion is limited by the fact that it is a perturbation scheme and cannot fully account for the effect of quantum nonlinear resonances (see Ramaswamy (1984) for further discussion and Zheng and Reichl (1987) for application to the Floquet states of microwave-driven hydrogen).

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MacKay RS, Meiss JD (1988) Phys Rev A 37:4702 Malyshev AI, Chizhova LA (2010) J Exp Th Phys 110:837 Martinez DF, Reichl LE (2001) Phys Rev B 64:245315 Morrow GO, Reichl LE (1994) Phys Rev A 50:2027 Morrow GO, Reichl LE (1998) Phys Rev E 57:5266 Mouchet A, Miniatura C, Kaiser R, Gremaud B, Delande D (2001) Phys Rev E 64:16221 Na K, Reichl LE (2004) Phys Rev A 70:063405 Nakamura K, Harayama T (2004) Quantum chaos and quantum dots. Oxford University Press, Oxford Oreg J, Hioe FT, Eberly JH (1984) Phys Rev A 29:690 Radons G, Prange RE (1988) Phys Rev Lett 61:1691 Ramaswamy R (1984) J Chem Phys 80:6194 Reichl LE (1989) Phys Rev A 39:4817 Reichl LE, Li W (1990) Phys Rev A 42:4543 Reichl LE, Lin WA (1986) Phys Rev A 33:3598 Reinhold CO, Burgdörfer J, Frey MT, Dunning FB (1997) Phys Rev Lett 79:5226 Sambe H (1973) Phys Rev A 7:2203 Shepelyansky DL (1985) In: Casati G (ed) Chaotic behavior in quantum systems. Plenum Press, New York Shirley JH (1965) Phys Rev 138:979 Shore BW (1990) The theory of coherent atomic excitation. Wiley, New York Steck DA, Oskay WH, Raizen MG (2001) Science 293:274 Steck DA, Oskay WH, Raizen MG (2002) Phys Rev Lett 88:120406 Stokely CL, Dunning FB, Reinhold CO, Pattanayak AK (2002) Phys Rev A 65:21405(R) Stokely CL, Lancaster JC, Dunning FB, Arbo DG, Reinhold CO, Burgdörfer J (2003) Phys Rev A 67:13403 Timberlake T, Reichl LE (1999) Phys Rev A 59:2886 Toda M, Ikeda K (1987) J Phys A 20:3833 van Leeuwen KAH, Oppen GV, Renwick S, Bowlin JB, Koch PM, Jensen RV, Rath, O, Richards D, Leopold JG (1985) Phys Rev 55:2231 von Milczewski J, Diercksen GHF, Uzer T (1996) Phys Rev Lett 76:2890 Zeldovich YB (1967) Sov Phys JETP 24:1006 Zener C (1932) Proc R Soc London Ser A 137:696 Zheng WM, Reichl LE (1987) Phys Rev A 35:474

Appendix A

Classical Mechanics Concepts

In this appendix, we review some basic concepts from classical mechanics that are used in this book.

A.1 Newton’s Equations Newton’s equations of motion describe the behavior of collections of N point particles in a three-dimensional space (3N degrees of freedom) in terms of 3N coupled second-order differential equations (the number of equations can be reduced if constraints are present), d 2 (mα rα ) = Fα , dt 2

(A.1)

where α = 1, . . . , N , mα is the mass, rα is the displacement of the αth particle, and Fα is the net force on the αth particle due to the other particles and any external fields that might be present. Equations (A.1) only have simple structure in inertial frames of reference and for Cartesian coordinates. For noninertial frames and general orthogonal curvilinear coordinate systems, they rapidly become complicated and nonintuitive. Because Newton’s equations are second order, the state of a collection of α ˙ α and N particles at time t is determined once the velocities vα = dr dt = r displacements rα are specified at time t. Newton’s equations allow one to determine the state of the system at time t uniquely in terms of the state at time t = 0. Thus a system composed of N point particles evolves in a phase space composed of 3N velocity and 3N position coordinates.

© Springer Nature Switzerland AG 2021 L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200, https://doi.org/10.1007/978-3-030-63534-3

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A Classical Mechanics Concepts

A.2 Lagrange’s Equations Lagrange showed that it is possible to formulate Newtonian mechanics in terms of a variational principle that vastly simplifies the study of mechanical systems in curvilinear coordinates and noninertial frames and allows a straightforward extension to continuum mechanics. For an N particle system in three-dimensional space, we assume there exists a function L({q˙i }, {qi }, t) of generalized velocities q˙i and positions qi ({q˙i } denotes the set of 3N generalized velocities (q˙1 , . . . , q˙3N ) and {qi } denotes the set of 3N generalized positions (q1 , . . . , q3N )) such that when we integrate L({q˙i }, {qi }, t) between two points, {qi (t1 )} and {qi (t2 )}, in phase space, the actual physical path is the one that extremizes the integral  S=

t2

L({q˙i }, {qi }, t)dt.

(A.2)

t1

The function L({q˙i }, {qi }, t) is called the Lagrangian and the integral S has units of action. Extremization of the integral in Eq. (A.2) leads to the requirement that the Lagrangian satisfy the equations ∂L d − ∂qi dt



∂L ∂ q˙i

 = 0, (i = 1, . . . , 3N).

(A.3)

Equations (A.3) are called the Lagrange equations (Goldstein 1980). For a single particle in a potential energy field V (r), the Lagrangian is simply 2 L = mv2 − V (r). Note that Eqs. (A.3) are expressed directly in terms of curvilinear coordinates. If we write down the Lagrangian in terms of curvilinear coordinates, it is then a simple matter to obtain the equations of motion. Two important quantities obtained from the Lagrangian are the generalized momenta, pi =

∂L , ∂ q˙i

(A.4)

and the Hamiltonian, H =

3N  (q˙i pi ) − L.

(A.5)

i=1

Generalized coordinates are defined from the differential element of length ds in real space. In Cartesian coordinates, (ds)2 = (dx)2 + (dy)2 + (dz)2 so that q1 = x, q2 = y, and q3 = z. In polar coordinates, (ds)2 = (dr)2 + r 2 (dθ )2 + (dz)2 so that q1 = r, q2 = θ , and q3 = z. In spherical coordinates, (ds)2 = (dr)2 + r 2 (dθ )2 + r 2 sin2 (θ )(dφ)2 so that q1 = r, q2 = θ , and q3 = φ.

A.4 The Poisson Bracket

399

A.3 Hamilton’s Equations In the Newtonian and Lagrangian formulations of mechanics, dynamical systems are described in terms of a phase space composed of generalized velocities and positions. The Hamiltonian formulation describes such systems in terms of a phase space composed of generalized momenta, {pi }, and positions, {qi }. The Hamiltonian phase space has very special properties. If the system has some translational symmetry, then some of the momenta may be conserved quantities. In addition, for systems obeying Hamilton’s equations of motion, the size of volume elements in phase space is conserved. Thus the phase space behaves like an incompressible fluid. A Legendre transformation from coordinates {q˙i }, {qi } to coordinates {pi }, {qi } yields the following equations of motion for the Hamiltonian phase space coordinates p˙ i =

∂H dpi =− , dt ∂qi

(A.6)

q˙i =

∂H dqi = , dt ∂pi

(A.7)

∂H ∂L =− . ∂t ∂t

(A.8)

Equations (A.6)–(A.8) are called Hamilton’s equations.

A.4 The Poisson Bracket The equation of motion of any phase function (any function of phase space variables) may be written in terms of Poisson brackets. Let us consider a phase function, f ({qi }, {pi }, t). Its total time derivative is  3N   df ∂f ∂f ∂f = + q˙i + p˙ i . dt ∂t ∂qi ∂pi

(A.9)

i=1

Using Hamilton’s equations, we can write this in the form ∂f df = + {f, H }P oisson , dt ∂t

(A.10)

where {f, H }P oisson =

 3N   ∂f ∂H ∂f ∂H − ∂qi ∂pi ∂pi ∂qi i=1

(A.11)

400

A Classical Mechanics Concepts

and {f, H }P oisson = −{H, f }P oisson . The Poisson bracket of any two phase functions, f ({qi }, {pi }, t) and g({qi }, {pi }, t) is written {f, g}P oisson =

 3N   ∂f ∂g ∂f ∂g . − ∂qi ∂pi ∂pi ∂qi

(A.12)

i=1

The Poisson bracket is invariant under canonical transformations. If we make a canonical transformation from coordinates (p, q) to coordinates (P , Q) (that is, p = p(P , Q), q = q(P , Q)), the Poisson bracket is given by Eq. (A.12) but with p → P and q → Q and f = f (p(P , Q), q(P , Q)).

A.5 Phase Space Volume Conservation One of the important properties of the Hamiltonian phase space is that volume elements are conserved under the flow of points in phase space. A volume element at some initial time to can be written = dp1 (to ) . . . dp3N (to )dq1 (to ) . . . dq3N (to ). dVtN o It is related to a volume element, dVt , at time t by the Jacobian, J3N (to , t), of the transformation between phase space coordinates at time to , {pi (to )}, {qi (to )}, and coordinates at time t, {pi (t)}, {qi (t)}. Thus . dVtN = J3N (t, to )dVtN o

(A.13)

For systems obeying Hamilton’s equations (even if they have a time-dependent Hamiltonian), the Jacobian is a constant of the motion, dJ3N (t, to ) = 0, dt

(A.14)

and therefore the size of volume elements in phase space does not change in time.

A.6 Action-Angle Coordinates We can write Hamilton’s equations in terms of any convenient set of generalized coordinates. We can transform between coordinate systems and leave the form of Hamilton’s equations invariant via canonical transformations. There is, however, one set of canonical coordinates, the action-angle variables, that play a distinctive role both in terms of the analysis of chaotic behavior in classical nonlinear systems

A.6 Action-Angle Coordinates

401

and in terms of the transition between classical and quantum mechanics. The action variable is an adiabatic invariant, and therefore it is a suitable variable to quantize. In quantum systems, transitions occur in discrete units of h¯ . If an external field is applied that is sufficiently weak and slow, it is possible that no changes will occur in the quantum system because the field is unable to cause a change in the action of the system by a discrete amount, h¯ . In the transition from classical to quantum mechanics, it is the action variables that are quantized in units of h¯ because they are adiabatic invariants and have a similar behavior classically (Born 1960; Landau and Lifshitz 1976). If a slowly varying weak external field (with period much longer than and incommensurate with the natural period of the system) is applied to a classical periodic system, the action remains unchanged, whereas the rate of change of the energy is proportional to the rate of change of the applied field. Let us consider a one degree of freedom system described in terms of the usual momentum and position coordinates, (p, q), with Hamiltonian H (p, q). We introduce a generating function, S(q, J ), which allows us to transform from coordinates (p, q) to action-angle coordinates, (J, θ ), via the equations  p=

∂S ∂q

 (A.15) J

and  θ=

∂S ∂J

 (A.16)

. q

The generating function is path-independent, so 

∂p ∂J



 = q

∂ ∂J



∂S ∂q



  = J

q

∂ ∂q



∂S ∂J

 

 =

q

J

∂θ ∂q

 .

(A.17)

J

We require that  (p, q) = H(J ) so that J = constant and θ = ω(J )t + θo ,  H and θo is a constant. Now consider a differential change in S, where ω = ∂H   ∂J ∂S dS = ∂S ∂q J dq + ∂J q dJ . Find the change in S along a path of fixed J (and   dq. Then therefore fixed energy), (dS)J = ∂S ∂q J





S(q, J ) − S(q , J ) =

S(q,J ) S(q  ,J )

 dS =

q q



∂S ∂q



 dq = J

q

q

pdq.

(A.18)

We now define the action as J =

1 2π

pdq. closedpath

(A.19)

402

A Classical Mechanics Concepts

Fig. A.1 The area enclosed by a periodic orbit is proportional to the action

p

Area = 2S J

q

The integral is over a path of fixed J and therefore fixed energy. The action itself is a measure of the area in phase space enclosed by the path (see Fig. A.1).   ∂θ dq + Let us find an expression for the angle variable. We can write dθ = ∂q      J ∂θ ∂p ∂p ∂θ = ∂J . Thus, for a path of fixed J , (dθ )J = ∂J dq ∂J q dJ . But ∂q J

q

q

and we can write 

θ

∂ θ − θo = dθ = ∂J θo



q

pdq.

(A.20)

qo

Equations (A.19) and (A.20) enable us to construct the canonical transformation between coordinates (J, θ ) and (p, q). The whole discussion can easily be generalized to higher-dimensional systems.

A.7 Hamilton’s Principal Function Hamilton’s principal function for a system with one degree of freedom is defined  R(x0 , t0 ; x, t) =

t

 dτ L(x, ˙ x, τ ) =

t0

t

dτ (px˙ − H (p, x, τ )).

(A.21)

t0

We wish to compute partial derivatives of R(x0 , t0 ; x, t). Let us consider the change in R(x0 , t0 ; x, t) that results if we vary the end point and end time of the path of integration by the small amounts x and t, respectively. The change in R(x0 , t0 ; x, t) is R = R(x0 , t0 ; x + x, t + t) − R(x0 , t0 ; x, t) =

∂R ∂R x + t, ∂x ∂t

(A.22)

where it is understood that x0 and t0 are held fixed. For some intermediate time, τ , the position and momentum of the path with end point (x, t) is (x(τ ), p(τ )),

References

403

while the position and momentum of the path with end point (x + x, t + t) is (x(τ ) + ξ(τ ), p(τ ) + π(τ )). The quantities ξ(τ ) and π(τ ) are small and are of the same order as x and t. We can now write  R(x0 , t0 ; x + x, t + t) =

t+t

dτ [(p + π )(x˙ + ξ˙ )

t0

− H (p + π, x + ξ, τ )] ≈ R(x0 , t0 ; x, t) + (p x˙ − H (p, x, τ ))t       t

∂H ∂H π + ··· , (A.23) + dτ (pξ˙ + x π˙ ) − ξ− ∂x p ∂p x t0 where in Eq. (A.23) we have kept terms to first order in the small quantities. If we now use Hamilton’s equations (A.6) and (A.7), the two terms in the third line of Eq. (A.23) that involve π cancel and the two remaining terms form an exact differential. Thus we find R = (px˙ − H (p, x, τ ))t + p(t)ξ(t) − p(t0 )ξ(t0 ).

(A.24)

x = x(t + t) + ξ(t + t) − x(t) ≈ x(t)t ˙ + ξ(t) + · · · ,

(A.25)

But

where we have kept terms to first order in the small quantities. If we now combine Eqs. (A.24) and (A.25), and note that ξ(t0 ) = 0, we obtain R = px − H (p, x, t)t.

(A.26)

If we now compare Eqs. (A.22) and (A.26), we finally obtain  p=

∂R ∂x



 and

x0 ,t0 ,t

∂R ∂t

 = −H.

(A.27)

= H.

(A.28)

x0 ,t0 ,x

Similarly,  p0 = −

∂R ∂x0



 and x,t,t0

∂R ∂t0

 x,t,x0

These quantities are useful for computing semiclassical path integrals.

References Born M (1960) The mechanics of the atom. Fredrick Ungar, New York Goldstein H (1980) Classical mechanics. Addison-Wesley, Reading Mass Landau LD, Lifshitz EM (1976) Mechanics. Pergamon Press, Oxford

Appendix B

Simple Models

In this appendix, we give the transformation from momentum and position variables, (p, x), to action-angle variables, (J, ), for four one-dimensional model systems that have been widely used to study the onset of chaos in classical mechanical systems. They include the pendulum, the quartic double well, the infinite square well, and one-dimensional hydrogen both with and without a constant external field (Stark field).

B.1 The Pendulum The most important one-dimensional model for nonlinear conservative physics is the pendulum because in many cases it very accurately describes the behavior of nonlinear resonances. The pendulum can be modeled with a Hamiltonian of the form Ho =

p2 − g cos(x) = Eo . 2m

(B.1)

A plot of the potential V (x) = −g cos(x) is shown in Fig. B.1 The phase space plots can be obtained from the expression for the momentum,  p = ± 2m(Eo + g cos(x))

(B.2)

and are shown in Fig. B.2. As can be seen from Figs. B.1 and B.2, there are two types of motion, libration and rotation. They must be considered separately. Let us first consider libration.

© Springer Nature Switzerland AG 2021 L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200, https://doi.org/10.1007/978-3-030-63534-3

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B Simple Models

Fig. B.1 The pendulum: x versus V (x) = −g cos(x)

Fig. B.2 The pendulum: phase space plot for m = 1

B.1.1 Libration—Trapped Orbits (E0 < g) Libration corresponds to motion trapped in the cosine potential well. The pendulum bob never goes over the top. In this energy regime, the turning points of the orbit (the points where p = 0) are given by   Eo . x± = ±arccos − g

(B.3)

The action for this case is defined as √  x  1 2m + J = dx Eo + g cos(x) pdx = 2π π x− √ 8 mg = [E(κ) − κ 2 K(κ)], π

(B.4)

where K(κ) and E(κ) are the complete elliptic integrals of the first and second kinds, respectively, and κ is the modulus, defined as κ 2 = Eo2g+g (Byrd and Friedman 1971). We cannot explicitly write the energy, Eo , as a function of J , but we can obtain the derivative of the energy and therefore the angle variable, . We find

B.1 The Pendulum

407

˙ = 

√ π g ∂Eo = √ , ∂J 2 m K(κ)

(B.5)

and therefore √ π g (t) = √ t + (0), 2 m K(κ)

(B.6)

where (0) is the value of (t) at time t = 0. The canonical transformation between variables (p, x) and action-angle variables (J,) is easy to find (Goldstein 1980). If we remember that p = mx, ˙ then we can write ' dx 2 dt = √ . (B.7) m Eo + g cos(x) If we make the change of variables sin  t' 0

Thus sin(z) = sn



g m t, κ



g dt = m

x 2

 0

z

= κ sin(z), we find after some algebra



dz 1 − κ 2 sin2 (z)

.

(B.8)

   g or x = 2 sin−1 κsn and m t, κ

   2K(κ) ,κ , x = 2 sin−1 κ sn π

(B.9)

where sn is the Jacobi elliptic sn function (Byrd and Friedman 1971). If we plug Eq. (B.9) into Eq. (B.2) for p, we find  2K(κ) p = ±2κ mg cn ,κ , π √



(B.10)

where cn is the Jacobi elliptic cn function. Equations (B.9) and (B.10) give the canonical transformation between canonical variables (p, x) and (J , ) for E0 < 0.

B.1.2 Rotation—Untrapped Orbits (E0 > g) Orbits undergoing rotation do not have a turning point but travel along the entire x axis (mod (2π )) with oscillations in momentum (see Fig. B.2). The action variable for such an orbit may be defined as

408

B Simple Models

1 J = 2π



π −π

√ 4 mg E(κ), dx 2m(Eo + g cos(x)) = κπ 

where the modulus, κ, is now defined as κ 2 = ˙ = 

2g Eo +g .

(B.11)

The frequency is

√ π g ∂E0 = √ ∂J κK(κ) m

(B.12)

√ π g √ t + (0). κK(κ) m

(B.13)

and the angle variable is given by (t) =

The canonical transformation from variables (p, x) to (J , ) can be obtained as before. Using p = mx, ˙ we can write (after a change of variables) 1 κ

Integrating, we find sin

x 2

'

d( x2 ) g . dt =  m 1 − κ 2 sin2 ( x2 )

= sn



 √ t g √ ,κ κ m 

x = 2 am

(B.14)

or

 K(κ) ,κ , π

(B.15)

where we have made use of Eq. (B.13). In Eq. (B.15), am is the Jacobi elliptic amplitude function (Byrd and Friedman 1971). If we substitute this into Eq. (B.2) for the momentum, we find √   √ K(κ) 2g p = ± 2m dn ,κ , κ π

(B.16)

where dn is the Jacobi elliptic dn function.

B.2 Double-Well Potential The double-well system is related to the pendulum by a canonical transformation. However, it is sometimes useful to have explicit solutions for both. The double-well system has two dynamical regimes, as does the pendulum. Let us write the doublewell Hamiltonian as

B.2 Double-Well Potential

409

Fig. B.3 Quartic double-well system: x versus V (x) = −2Bx 2 + x 4

Fig. B.4 Phase space plot for the quartic double-well system

Ho =

p2 − 2Bx 2 + x 4 = Eo . 2m

(B.17)

The double-well potential, V (x) = −2Bx 2 + x 4 , is plotted in Fig. B.3 and phase space orbits are shown in Fig. B.4. Trajectories with energy Eo < 0 will be trapped in one of the two wells and cannot travel across the barrier, while trajectories with energy Eo > 0 can travel freely across the barrier. This is the analog of libration and rotation, respectively, in the pendulum. We shall call trajectories trapped or untrapped according to whether or not they can cross the barrier. It is necessary to consider the two cases separately.

B.2.1 Below the Barrier—(Eo < 0) The momentum for the case Eo < 0 can be written in the form   p = ± 2m(Eo + 2Bx 2 − x 4 ) = ± 2m(f 2 − x 2 )(x 2 − e2 ), where

(B.18)

410

B Simple Models

f2 = B +



(B 2 + Eo )

e2 = B −

and



(B 2 + Eo ).

(B.19)

It is easy to see from Eq. (B.18) that x− = e and x+ = f are the inner and outer turning points for trajectories trapped below the barrier. The action variable may be written √  f  2m 1 J = dx (f 2 − x 2 )(x 2 − e2 ) pdx = 2π π e √ 2 2m f [BE(κ) − e2 K(κ)], = (B.20) 3π where K(κ) and E(κ) are complete elliptic integrals of the first and second kind, 2 2 respectively, and the modulus κ is defined as κ 2 = f f−e (Byrd and Friedman 2 1971). From Eq. (B.20), we find that √ ∂Eo 2f π ˙ =  =√ , ∂J m K(κ)

(B.21)

and the angle variable √ (t) = √

2f π t + (0) m K(κ)

(B.22)

(Byrd and Friedman 1971). The canonical transformation between variables (p, x) and (J , ) is obtained as follows. From the relation p = mx, ˙ we can write 

f

x



'

(f 2

− x 2 )(x 2

t

'

(B.23)

  K(κ) = f dn ± ,κ , π

(B.24)

− e2 )

=

2 m



2 t. m

dx

dt =

0

We then obtain  x = f dn ±f

'

2 t, κ m



where dn is the Jacobi dn elliptic function and we have set (0) = 0 (Byrd and Friedman 1971). If we substitute Eq. (B.24) into Eq. (B.18), we find     √ K(κ) K(κ) p = ± 2mf 2 κ 2 sn , κ cn ,κ . π π where cn and sn are Jacobi cn and sn elliptic functions.

(B.25)

B.2 Double-Well Potential

411

B.2.2 Above the Barrier—(Eo > 0) The momentum for an untrapped trajectory can be written   p = ± 2m(Eo + 2Bx 2 − x 4 ) = ± 2m(h2 − x 2 )(x 2 + g 2 ),

(B.26)

where h2 = B +



(B 2 + Eo )

g 2 = −B +

and



(B 2 + Eo ).

(B.27)

The turning points are now given by x± = ±h. The action is given by √  h  2m 1 dx (h2 − x 2 )(x 2 + g 2 ) J = pdx = 2π π −h √   2 2m h 3 2 = [κ (K(κ) − E(κ)) + κ 2 E(κ)], 3 π κ where κ 2 = (1−κ 2 ) and the modulus, κ, is defined as κ 2 = we obtain πh ˙ =√  . 2m κ K(κ)

(B.28)

h2 . From Eq. (B.28), h2 +g 2

(B.29)

and thus (t) = √

πh 2m κ K(κ)

t + (0).

(B.30)

The canonical transformation is obtained in the usual manner. Since p = mx, ˙ we can write '  h 2κK(κ) dx 2  = t= . (B.31) 2 2 2 2 m eπ (h − x )(x + g ) x We therefore obtain  x = h cn

 2K(κ) ,κ . π

If we substitute Eq. (B.32) into Eq. (B.26), we find

(B.32)

412

B Simple Models

Fig. B.5 Square-well system: x versus V (x)

    √ h2 2K(κ) 2K(κ) , κ dn ,κ . p = ± 2m sn κ π π

(B.33)

B.3 Infinite Square-Well Potential The motion of a particle of mass m with kinetic energy p2 /2m in an infinite squarewell potential has some special properties that make it especially useful for studying many aspects of chaotic behavior, both classical and quantum mechanical. The infinite square-well potential is shown in Fig. B.5. The momentum, p, and position, x, as a function of time can be obtained by inspection. A plot of the momentum and position as a function of time is given in Fig. B.6. Analytic expressions are given in terms of the Fourier series     2π t p(t) = (2mEo ) sign sin τ   ∞  2π nt 4  1 sin (B.34) = (2mEo ) π n τ n=1 odd and

x(t) = −a +

4a |t| for τ

 τ τ − z∗ .

References

417

The action variable for a trajectory trapped in the well is given by 1 J = 2π

√

pdz =

2μFo π



z−

0

 dz √ z

z2 −

|Eo |z κ0 e2 + Fo Fo

√ 2(z+ ) 2μFo [(1 + κ 2 )E(κ) − (1 − κ 2 )K(κ)], = 3π 3 2

(B.56)

where K(κ) and E(κ) are complete elliptic integrals of the first and second kinds, respectively, and the modulus κ is defined as κ 2 = zz−+ (Byrd and Friedman 1971). We cannot explicitly revert Eq. (B.56) to find Eo as a function of J . However, we ˙ is given by can find the derivative of Eo . Thus, the frequency, ,

˙ = 

√ π μFo ∂|Eo | =√ , 1 ∂J 2(z+ ) 2 (E(κ) − K(κ))

(B.57)

and the angle variable is given by √ π μFo t

(t) = √ + (0). 1 2(z+ ) 2 (E(κ) − K(κ))

(B.58)

Using the relation p = m˙z, we can write '

√ 2Fo zdz dt = ±  . m (z − z− )(z − z+ )

(B.59)

If we let z(t) = z− sn2 (u, κ),

(B.60)

then we find u − E(u, κ) = ±

(E(κ) − K(κ)) , π

(B.61)

where E(u, κ) is the incomplete elliptic integral of the second kind.

References Byrd PF, Friedman D (1971) Handbook of elliptic integrals for engineers and scientists. Springer, Berlin Goldstein H (1980) Classical mechanics. Addison-Wesley, Reading

Appendix C

Symmetries and the Hamiltonian Matrix

The Hamiltonian operator, Hˆ , is Hermitian (Hˆ = Hˆ † , where † denotes complex conjugate transpose), and its matrix representations are Hermitian. This ensures that its eigenvalues, the allowed energies of the system, are real. In this appendix, we consider N-particle systems described by Hamiltonians of the form Hˆ (t) = Hˆ ({pˆ α }, {qˆ α }, {ˆsα }, t), where pˆ α and qˆ α are the momentum and position operators for the αth particle, sˆα is the spin of the αth particle, and {pˆ α }, {qˆ α }, and {ˆsα } denote the sets of momenta, positions, and spins, respectively, of the N particles. The Hamiltonian matrix is formed from the numbers obtained by evaluating the Hamiltonian operator with respect to some chosen complete orthonormal set of basis states. If the complete set of basis states is denoted as {|ai }, then the (i, j )th element of the Hamiltonian matrix is Hi,j (t) = ai |Hˆ (t)|aj . If symmetries exist, they can simplify the structure of the Hamiltonian matrix. A symmetry causes the dynamics of the system, and therefore the Hamiltonian, to be invariant under the corresponding symmetry transformation. In Sect. C.1, we discuss the types of transformations associated with the various space-time symmetries. In Sect. C.2, we then show explicitly how the various space-time symmetries affect the Hamiltonian matrix.

C.1 Space-Time Symmetries The space-time symmetries give rise to infinitesimal or discrete symmetry transformations. Symmetry transformations are “generated" by linear and antilinear operators. All of the operators that we deal with in this book are linear except for the time reversal operator, which is antilinear. Linear Operators ˆ when acting on a state | = c1 |ψ1  + c2 |ψ2 , where c1 and A linear operator, O, c2 are complex constants, gives © Springer Nature Switzerland AG 2021 L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200, https://doi.org/10.1007/978-3-030-63534-3

419

420

C Symmetries and the Hamiltonian Matrix

ˆ 1 ) + c2 (O|ψ ˆ 2 ). ˆ ˆ 1 |ψ1  + c2 |ψ2 ) = c1 (O|ψ O| = O(c

(C.1)

ˆ ˆ Scalar products involving linear operators behave as (χ |O)|ψ = χ |(O|ψ). Scalar products involving the Hermitian adjoint of a linear operator behave as ˆ ]∗ . χ |Oˆ † |ψ = [ψ|O|χ Antilinear Operators ˆ when acting on the superposition | = c1 |ψ1  + c2 |ψ2 , An antilinear operator, A, gives ˆ 1 ) + c2 ∗ (A|ψ ˆ 2 ). ˆ ˆ 1 |ψ1  + c2 |ψ2 ) = c1 ∗ (A|ψ A| = A(c

(C.2)

ˆ Scalar products involving antilinear operators behave as (χ |A)|ψ = ∗ . Scalar products involving the Hermitian adjoint of an antilinear ˆ [χ |(A|ψ)] ˆ operator behave as χ |(Aˆ † |ψ) = χ |(A|ψ). The symmetries of a system may be continuous or discrete. The transformations associated with these two types of symmetry have different behavior. Continuous symmetries are associated with transformations that make only infinitesimal changes in the system as well as transformations that make finite changes. Discrete symmetries are associated with transformations that make finite changes.

C.1.1 Continuous Symmetries Continuous symmetries are associated with infinitesimal transformations. Let us consider an infinitesimal transformation given by the unitary operator, Tˆ (δα), with the property that Tˆ (δα) → 1 as δα → 0. We can write ˆ Tˆ (δα) ≈ 1 − i δα,

(C.3)

ˆ is a Hermitian operator. The operator  ˆ is called the generator of the where  ˆ is changed to Oˆ  = infinitesimal transformation. Let us assume that an operator, O, ˆ ˆ O + δ O by the transformation. That is, ˆ ˆ O]δα + ... . Tˆ † (δα)Oˆ Tˆ (δα) = Oˆ  = Oˆ + δ Oˆ ≈ Oˆ + i[,

(C.4)

ˆ ˆ O]δα Then, to lowest order in δα, δ Oˆ = +i[, and δ Oˆ ˆ ˆ O]. = +i[, δα

(C.5)

ˆ Tˆ (δα) = e−i δα .

(C.6)

For arbitrary δα, we obtain

Let us now consider some specific examples.

C.1 Space-Time Symmetries

C.1.1.1

421

Time Translation

If we translate the operator Oˆ forward in time, then δα = δt and δ Oˆ ˆ ˆ O]. = +i[, δt

(C.7)

ˆ = 1 Hˆ , so the Hamiltonian is the generator of infinitesimal translations in Thus,  h¯ time. The time translation operator is given by ˆ Tˆ (δt) = e− h¯ H ·δt i

(C.8)

for the case when Hˆ has no explicit time dependence.

C.1.1.2

Space Translation

Let us translate the system in space through a fixed displacement, δa. Then Tˆ † (δa)qˆ α Tˆ (δa) = qˆ α + δa and δ qˆ α ˆ qˆ α ], = i[, δa

(C.9)

ˆ is to give a similar result for where qˆ α is the displacement of the αth particle. If   ˆ ˆ ˆ each qˆ α , then  = P/h, ¯ where P is the total momentum, Pˆ = α pˆ α . Thus, the total momentum is the generator of translations in space. The transformation operator is given by ˆ Tˆ (δa) = e− h¯ P·δa . i

(C.10)

For the case of a Hamiltonian Hˆ ({pˆ α }, {qˆ α,β }, {ˆsα }, t) that depends only on the relative displacements of particles, qˆ α,β = qˆ α − qˆ β , we find Tˆ † (δa)Hˆ Tˆ (δa) = Hˆ

(C.11)

ˆ = 0 and the total momentum Pˆ is a constant of the motion. so that [Hˆ , P]

C.1.1.3

Rotation

Let us consider rotations of the system through an angle δφ about an axis given by ˆ Then unit vector n. ˆ qˆ α Tˆ (δφ n) ˆ qˆ α ˆ = qˆ α + δ qˆ α = qˆ α − δφ n× Tˆ † (δφ n)

(C.12)

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C Symmetries and the Hamiltonian Matrix

and ˆ pˆ α Tˆ (δφ n) ˆ pˆ α . ˆ = pˆ α + δ pˆ α = pˆ α − δφ n× Tˆ † (δφ n) 1 For simplicity, we will consider a Hamiltonian of the form Hˆ = 2m ˆ ˆ V ({qˆ α }). Then ∂ H = 1 [Hˆ , pˆ α ] and ∂ H = − 1 [Hˆ , qˆ α ], so we can write ∂ qˆ α

∂ pˆ α

i h¯

ˆ Hˆ Tˆ (δφ n) ˆ = Hˆ + Tˆ † (δφ n)

 α

(C.13) 

ˆ 2α αp

+

i h¯

 d pˆ α d qˆ α − ·δ qˆ α + ·δ pˆ α + . . . . dt dt

(C.14)

ˆ qˆ α and δpα = −δφ n×p ˆ α in C.14, we find If we let δ qˆ α = −δφ n× i d Lˆ ˆ ˆ Hˆ Tˆ (δφ n) ˆ = Hˆ + δφ n· ˆ ˆ Hˆ , L], = Hˆ + δφ n·[ Tˆ † (δφ n) dt h¯

(C.15)

 ˆ = ˆ α ×pˆ α is the total angular momentum. Thus, from Eq. (C.4), we where L αq ˆ = Lˆ and Lˆ is the infinitesimal generator of rotations. If δ Hˆ = 0, then obtain  ˆ dL ˆ dt = 0 and L is a constant of the motion. The rotation operator is given by ˆ ˆ = e− h¯ L·δφ nˆ . Tˆ (δφ n) i

(C.16)

For the case when particles have spins, {ˆsi }, the infinitesimal generator of rotations is given by the total angular momentum Jˆ = Lˆ +

 sˆα ,

(C.17)

α

and the rotation transformation generalizes to ˆ ˆ = e− h¯ J·δφ nˆ . Tˆ (δφ n) i

(C.18)

C.1.2 Discrete Symmetries Discrete symmetries give rise to transformations that cannot occur in infinitesimal steps. We give some examples below.

C.1 Space-Time Symmetries

C.1.2.1

423

Parity

The group of inversions through a point has two elements, the identity element, Iˆ, and the reflection, Pˆo (the parity operator). Under inversion, polar vectors such as ˆ and coordinate, q, ˆ change sign, while axial vectors such as angular momentum, p, ˆ remain unchanged. Thus, momentum, J, ˆ ˆ Pˆo qˆ Pˆo† = −q, ˆ and Pˆo Jˆ Pˆo† = J. Pˆo pˆ Pˆo† = −p,

(C.19)

The parity operator commutes with the rotation operator.

C.1.2.2

Time Reversal

ˆ is an antiunitary operator. When operators are written The time reversal operator, K, in the coordinate representation, they can be written as the product of the complex conjugation operator Kˆ o (Kˆ o i Kˆ o† = −i) and a unitary operator Tˆ so that Kˆ = Kˆ o Tˆ . The effect of time reversal is ˆ ˆ Kˆ qˆ Kˆ † = q, ˆ and Kˆ Jˆ Kˆ † = −J. Kˆ pˆ Kˆ † = −p,

(C.20)

Thus, the time reversal operator commutes with the rotation operator ˆ nˆ † ˆ ˆ − h¯ J·δφ Ke Kˆ = e− h¯ J·δφ nˆ i

i

(C.21)

(since both i and J change sign). If no spins are present, then Kˆ = Kˆ o (in the coordinate representation). If particles with half-integer spin are present, then the time reversal operator is more complicated. Let us consider the effect of time reversal on Pauli spin matrices,       01 0 −i 1 0 σ¯ x = , σ¯ y = , and σ¯ z = . (C.22) 10 i 0 0 −1 We find Kˆ o σx Kˆ o† = σx , Kˆ o σy Kˆ o† = −σy , and Kˆ o σz Kˆ o† = σz .

(C.23)

ˆ which Thus, in order to have Kˆ sˆi Kˆ † = −ˆsi , we must include a unitary operator, T, has the effect on coordinate and spin operators that ˆ Tˆ qˆ Tˆ † = q, ˆ Tˆ pˆ Tˆ † = p,

(C.24)

and ˆ y Tˆ † = σy , and Tσ ˆ z Tˆ † = −σz . ˆ x Tˆ † = −σx , Tσ Tσ

(C.25)

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C Symmetries and the Hamiltonian Matrix i

This can be accomplished by choosing Tˆ = e− h¯ π sˆy . If the total spin is Sˆ = i ˆ then Tˆ = e− h¯ π Sy and the time reversal operator is given by ˆ Kˆ = e− h¯ π Sy Kˆ o . i



i sˆ i ,

(C.26)

ˆ Let us now consider the square of K, ˆ ˆ ˆ Kˆ 2 = e− h¯ π Sy Kˆ o e− h¯ π Sy Kˆ o = e− h¯ 2π Sy . i

i

i

(C.27)

We can evaluate Kˆ 2 with respect to the basis of simultaneous eigenstates of Sˆ 2 and Sˆz , |S, Sy . Then, for integer total spin, Kˆ 2 = +1, while for half-integer total spin, Kˆ 2 = −1. This has important consequences as regards the form of the Hamiltonian matrix for time reversal invariant systems.

C.2 Structure of the Hamiltonian Matrix We now consider the effect of space-time symmetries on the structure of the Hamiltonian matrix. In Sect. C.2.1 below, we first consider the effect of homogeneity in time and then the effect of both homogeneity and isotropy in space on the structure of the Hamiltonian matrix. In Sect. C.2.1.2 below, we discuss the effect of time translation invariance for the cases of systems with integer and half-integer spins.

C.2.1 Space-Time Homogeneity and Isotropy In this section, we consider the effect of space and time homogeneity and of spatial isotropy on the structure of the Hamiltonian matrix.

C.2.1.1

Time Translation Invariance

Let us consider an N particle system with a time-independent Hamiltonian of the form Hˆ = Hˆ ({pˆ α }, {qˆ α }, {ˆsα }), where {pˆ α }, {qˆ α }, and {ˆsα } denote collections of momentum, position, and spin operators of the system. The equation of motion for ˆ a quantum operator, O(t), in the Heisenberg picture is ∂ Oˆ i d Oˆ ˆ = + [Hˆ (t), O(t)]. dt ∂t h¯

(C.28)

C.2 Structure of the Hamiltonian Matrix

425

ˆ If the Hamiltonian contains no explicit dependence on time, then ddtH = 0 and Hˆ is a ˆ constant of the motion. If ddtH = 0, then the eigenvalues of Hˆ (the allowed energies of the system) are constant in time.

C.2.1.2

Space Translation Invariance

If the Hamiltonian depends only on the relative displacements of the particles, qˆ α − qˆ β = qˆ α,β , then the Hamiltonian will be invariant under a translation of the entire system in space.  When this occurs, the total momentum (the center-of-mass ˆ = 0, and therefore Pˆ momentum) Pˆ = α pˆα commutes with Hamiltonian [Hˆ , P] is a constant of the motion. In such a case, we can write the Hamiltonian in the form Pˆ 2 Hˆ = + Hˆ rel ({pˆ α }, {qˆ α,β }, {ˆsα }), 2M

(C.29)

 ˆ where M = N α=1 mα is the total mass, mα is the mass of the αth particle, and Hrel is the Hamiltonian describing relative motion of the particles.

C.2.1.3

Rotational Invariance

If the Hamiltonian is invariant under a rotation of the entire system in space, then the Hamiltonian will commute with the total angular momentum operator, Jˆ = Lˆ + ˆ where Lˆ is the total orbital angular momentum and Sˆ is the total spin. Thus S, ˆ = 0, and Jˆ is a constant of the motion. Because [Hˆ , J] ˆ = 0, one can find [Hˆ , J] 2 ˆ ˆ ˆ ˆ simultaneous eigenstates of J = J·J, Jz , and H (components of Jˆ do not commute with one another). There will be no coupling between states with different values of the angular momentum quantum number, J , and the Hamiltonian matrix can be written in block diagonal form, with each block containing matrix elements that all depend on the same value of J .

C.2.2 Time Reversal Invariance In order to discuss the effect of time reversal invariance on the Hamiltonian matrix, we must consider separately the cases of integer spin and half-integer spin.

C.2.2.1

Integer Spin

For time reversal invariant systems with integer total spin, the time reversal operator, i ˆ Kˆ = e h¯ π Sy Kˆ o , has the property Kˆ 2 = +1, where Kˆ o is the complex conjugation

426

C Symmetries and the Hamiltonian Matrix

operator. (For zero spin, Kˆ = Kˆ o .) This allows us to write real eigenstates for Kˆ (Messiah 1964; Porter 1965; Mehta 1991), ˆ 1 >, |1 >= a|φ1 > +a ∗ K|φ

(C.30)

where |φ1 > is an arbitrary state normalized to one, < φ1 |φ1 >= 1, and a is a complex constant. Then, clearly, ˆ 1 >= |1 > . K|

(C.31)

We can find another state, |φ2 >, such that < 1 |φ2 >= 0. Then we can construct ˆ |2 >= a|φ2 > +a ∗ K|φ ˆ 2 >, such that < 2 |1 >= 0. a second eigenstate of K, By repeatedly following this procedure, we can construct an orthonormal basis of ˆ real eigenstates for K. Let us now consider the implications of such eigenstates for the Hamiltonian matrix. We can write ˆ n >) Hm,n =< m |Hˆ |n >= (< m |Kˆ † )Hˆ (K| ∗ ˆ n >)]∗ = [< m |(Hˆ |n >)]∗ = Hm,n , = [< m |(Kˆ † Hˆ K|

(C.32)

where we have used Eq. (C.31) and the time reversal invariance of the Hamiltonian, Kˆ † Hˆ Kˆ = Hˆ . Thus, for time reversal invariant systems with integer spin, a basis can be found in which the elements of the Hamiltonian matrix are real. For such systems the Hamiltonian matrix is real symmetric.

C.2.2.2

Half-Integer Spin

For time reversal invariant systems with half-integer spin, we must distinguish between rotationally invariant and non-rotationally invariant systems. (a) Rotationally Invariant Systems For the special case when the system has half-integer spin but the Hamiltonian ˆ = 0, it is possible to construct a commutes with the total angular momentum, [Hˆ , J] set of real basis states for which the Hamiltonian matrix is real symmetric. However, for half-integer spin, Kˆ 2 = −1, and therefore we cannot proceed as in Sect. C.2.2.1. i ˆ ˆ It has the property Let us consider the operator Kˆ  = e− h¯ π Jy K. i

ˆ

i

ˆ

i

ˆ

i

ˆ

i

ˆ

(Kˆ  )2 = e− h¯ π Jy e h¯ π Sy Kˆ o e− h¯ π Jy e h¯ π Sy Kˆ o = e− h¯ 2π Ly = 1.

(C.33)

If we use the operator Kˆ  , we can proceed as we did in Sect. C.2.2.1. Let us introduce ˆ 1 >= |1 >, and eigenstates of Kˆ  , |1 >= a|ψ1 > +a ∗ Kˆ  |ψ1 >, such that K| construct an orthonormal basis as before. Then

C.2 Structure of the Hamiltonian Matrix

427

ˆ ˆ ˆ n >) Hm,n =< m |Hˆ |n >= (< m |Kˆ † e h¯ π Jy )Hˆ (e− h¯ π Jy K| i

i

ˆ

i

i

ˆ

ˆ − h¯ π Jy |n >]∗ = [< m |Hˆ |n >]∗ , = [< m |e h¯ π Jy Kˆ † Hˆ Ke

(C.34)

ˆ where we have used the fact that Kˆ commutes with e− h¯ π Jy , Jˆy commutes with Hˆ , and Hˆ is time reversal invariant, Kˆ † Hˆ Kˆ = Hˆ . For time reversal invariant systems with half-integer spin and rotational invariance, the Hamiltonian is real and symmetric. i

(b) Nonrotationally Invariant Systems For time reversal invariant systems with half-integer spin and with a Hamiltonian that does not commute with J, it is no longer possible to find a basis of eigenstates of Kˆ that are real. Thus, the Hamiltonian matrix is no longer real symmetric. Let us first consider some properties of the time reversal operator for this case. Let i ˆ Vˆ = e h¯ π Sy , where Vˆ is a unitary operator. The time reversal operator is Kˆ = Vˆ Kˆ o . For half-integer spin, Kˆ 2 = Vˆ Kˆ o Vˆ Kˆ o = Vˆ Vˆ ∗ = −1.

(C.35)

Thus, Vˆ is an antisymmetric unitary operator, Vˆ T = −Vˆ . Such an operator cannot be diagonalized by a unitary transformation (Hua 1963). However, a basis can be ¯ where found in which the matrix representation of Vˆ is given by matrix Z, ⎛

0 ⎜ −1 ⎜ ⎜ Z¯ = ⎜ 0 ⎜ 0 ⎝ .. .

1 0 0 0 0 0 0 −1 .. .. . .

⎞ 0 ... 0 ...⎟ ⎟ 1 ...⎟ ⎟ 0 ...⎟ ⎠ .. . . . .

(C.36)

and K¯ = Z¯ Kˆ o . We can now find a condition on an arbitrary unitary matrix, U¯ , that allows it to commute with the time reversal operator. We require K¯ U¯ − U¯ K¯ = Z¯ Kˆ o U¯ − U¯ Z¯ Kˆ o = Kˆ o Z¯ U¯ − Kˆ o U¯ ∗ Z¯ = 0. Since U¯ is unitary, U¯ † = U¯ −1 , and we can write ¯ U¯ T Z¯ U¯ = Z.

(C.37)

Equation (C.37) is the definition of a symplectic unitary matrix. Such matrices commute with the time reversal operator for systems with half-integer spin. We will ¯ where S¯ T Z¯ S¯ = Z¯ and S¯ † = S¯ −1 . generally denote symplectic unitary matrices as S, We next determine what conditions time reversal invariance places on the Hamiltonian matrix. Let us consider a 2N × 2N-dimensional Hamiltonian matrix with complex matrix elements. It can be written as an N × N quaternion (see Appendix E. For example, the (ij )th matrix element of the N × N quaternion Hamiltonian matrix is given by

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C Symmetries and the Hamiltonian Matrix

(H¯ )ij =

 3

 h

(α)

τ¯α

= ij

α=0

3 

h(α) ij τ¯α ,

(C.38)

α=0

(α)

where hij are complex numbers and the 2×2 matrices τ¯α are defined in Appendix E. The condition that the Hamiltonian matrix be self-adjoint under Hermitian conjugation, H¯ † = H¯ , is (0)

(0)

(1)

(1)∗

(2)

(2)∗

(3)

(3)∗

hj i = hij , hj i = −hij , hj i = −hij , hj i = −hij

(C.39)

(see Appendix E). Let us next determine the condition under which the Hamiltonian matrix is time ¯ can be written as an reversal invariant. We first note that the 2N×2N matrix, Z, N×N quaternion with the 2×2 matrix τ¯2 appearing along its diagonal. To be time reversal invariant, the Hamiltonian matrix must satisfy the equation K¯ H¯ K¯ −1 = Z¯ Kˆ o H¯ Kˆ o−1 Z¯ −1 = Z¯ H¯ ∗ Z¯ −1 = H¯ .

(C.40)

It is easy to see that this leads to the condition τ¯2 (H¯ ∗ )ij τ¯2−1 = (H¯ )ij or (α)∗

hij

(α)

= hij .

(C.41)

Thus, time reversal invariance requires that the N×N quaternion form of the Hamiltonian, for systems with 12 integer spin that are not rotationally invariant, must be self-dual and quaternion real (see Appendix E). We shall denote such Hamiltonians as H¯ Q . ˆ will leave the Hamiltonian time Any symplectic unitary transformation, S, reversal invariant. This can be seen as follows. If Hˆ Q = Kˆ Hˆ Q Kˆ −1 , then ˆ Sˆ Hˆ Q Sˆ −1 = Sˆ Kˆ Hˆ Q Kˆ −1 Sˆ −1 = Kˆ Sˆ Hˆ Q Sˆ −1 Kˆ −1 since Sˆ commutes with K. Eigenstates of the Hamiltonian, Hˆ Q , are twofold degenerate due to the time reversal invariance of Hˆ Q . This was first pointed out by Kramers and is called the Kramers’ degeneracy. If |En 1 is an eigenstate of Hˆ Q with eigenvalue En , so ˆ n 1 is also an eigenstate of Hˆ Q with that Hˆ Q |En 1 = En |En 1 , then |En 2 = K|E eigenvalue En because Kˆ commutes with Hˆ Q , ˆ n 1 = Hˆ Q K|E ˆ n 1 = En |En 2 . Kˆ Hˆ Q |En 1 = En K|E

(C.42)

Furthermore, |En 2 is independent of |En 1 . The independence of |En 1 and |En 2 ˆ 1 . If they are not can be seen as follows. Consider two states, |φ1 and |φ2 = K|φ ˆ 1 = eiδ |φ1 . independent, they can differ by only a phase factor, so that |φ2 = K|φ 2 −iδ ˆ ˆ ˆ ˆ Now apply K again. Then K|φ2 = K |φ1 = e K|φ1 = e−iδ e+iδ |φ1 = |φ1 . However, we know that Kˆ 2 = −1 so the equation Kˆ 2 |φ1 = |φ1 cannot be satisfied. Therefore |φ1 and |φ2 must be independent.

References

429

References Hua LK (1963) Harmonic analysis of functions of several complex variables in the classical domains. American Mathematical Society, Providence Mehta ML (1991) Random matrices and the statistical theory of energy levels, 2nd edn. Academic Press, New York Messiah A (1964) Quantum mechanics. North Holland, Amsterdam Porter CE (1965) Statistical theories of spectra: fluctuations. Academic Press, New York

Appendix D

Invariant Measures

In this appendix, we will derive invariant measures for the Hermitian and unitary matrices that determine the dynamics of the three main symmetry classes of Hamiltonian systems Hua (1963), Porter (1965), Mehta (1991). The invariant measures that we consider remain invariant under a unitary transformation.

D.1 General Definition of Invariant Measure We will consider complex square matrices, ⎛

w1,1 w1,2 ⎜ w2,1 w2,2 ⎜ W¯ = ⎜ . .. ⎝ .. . wN,1 wN,2

⎞ . . . w1,N . . . w2,N ⎟ ⎟ .. ⎟ .. . . ⎠ . . . wN,N

(D.1)

where wi,j (i = 1, . . ., N, j = 1, . . ., N) is a complex number. The matrix W¯ can be diagonalized by a unitary transformation if and only if W¯ ·W¯ † = W¯ † ·W¯ Mehta (1991). These are the types of matrices we consider in this book.

D.1.1 Invariant Metric (Length) The matrix of differential increments of the matrix W¯ is denoted d W¯ and is given by

© Springer Nature Switzerland AG 2021 L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200, https://doi.org/10.1007/978-3-030-63534-3

431

432

D Invariant Measures

⎛

dw1,1 dw1,2 ⎜ dw2,1 dw2,2 ⎜ d W¯ = ⎜ . .. ⎝ .. . dwN,1 dwN,2

⎞ . . . dw1,N . . . dw2,N ⎟ ⎟ .. ⎟ , .. . . ⎠ . . . dwN,N

(D.2)

R +dw I is the (i, j )th matrix element of d W ¯ , and dw R and dw I where dwij = dwij ij ij ij are its real and imaginary parts. We will require that the metric be real and invariant under a unitary transformation. We know that the trace of a matrix, d W¯ , or any power of d W¯ is invariant under a unitary transformation, U¯ . Therefore the simplest choice of an invariant metric is

(ds)2W = Tr(d W¯ · d W¯ † ) =

N N  

|dwij |2 ,

(D.3)

i=1 j =1

where (ds)2W is a measure of distance between W¯ and W¯ + d W¯ .

D.1.2 Invariant Measure (Volume) The detailed form of the invariant measure of the matrix W¯ depends on the symmetry properties of W¯ . Symmetries give rise to relations between matrix elements of W¯ , so not all of the differentials, dwi,j , are independent. If only N of the differentials dwi,j are independent, then we can denote those independent matrix elements of d W¯ as dxk for k = 1, . . ., N, and the invariant metric takes the form (ds)2W =

N  gi,i dxi2 ,

(D.4)

i=1

where gi,j is the (i, j )th element of the metric tensor, g, ¯ and is determined by the symmetry properties of W¯ . The invariant measure is then defined as dW =

 Det[g]dx ¯ 1 dx2 ×. . .×dxN .

(D.5)

In Sect. D.2, we derive the invariant measures for the Hermitian Hamiltonian matrices for dynamical systems with three different types of underlying symmetry: (1) Systems that are rotationally invariant and are invariant under time reversal; (2) systems that are rotationally invariant but not time reversal invariant (such as systems with applied magnetic fields); (3) systems of spin 12 particles, which have strong spin-orbit coupling and are invariant under time reversal but are not rotationally invariant. In Sect. D.3, we obtain the invariant measure for unitary

D.2 Hermitian Matrices

433

matrices with these same symmetry properties. Dyson showed that these three symmetry classes exhaust the possibilities available to us in nonrelativistic systems Dyson (1962).

D.2 Hermitian Matrices In this section, we derive the invariant measures for three different types of Hermitian matrices: real symmetric, complex Hermitian, and quaternion real. These correspond to the Hamiltonians for the three dynamical symmetry classes described above.

D.2.1 Real Symmetric Matrix Hamiltonians that govern the dynamics of systems that are invariant under time reversal and rotation have matrix representations that are real and symmetric. An N ×N real symmetric matrix, ⎛

h1,1 h1,2 ⎜ h1,2 h2,2 ⎜ H¯ R = ⎜ . .. ⎝ .. . h1,N h2,N

⎞ . . . h1,N . . . h2,N ⎟ ⎟ . ⎟ .. . .. ⎠ . . . hN,N

(D.6)

has N + 12 (N 2 − N) = 12 N(N + 1) independent real elements. The invariant metric for a real symmetric matrix can be written (ds)2HR = Tr(d H¯ R · d H¯ RT ) =

N  (dhi,i )2 + 2



(dhi,j )2 =

 gi,i dxi2 ,

1≤i