THE FRONTIERS COLLECTION THE FRONTIERS COLLECTION Series Editors: D. Dragoman M. Dragoman A.C. Elitzur M.P. Silverman J. Tuszynski H.D. Zeh The books in this collection are devoted to challenging and open problems at the forefront of modern physics and related disciplines, including philosophical debates. In contrast to typical research monographs, however, they strive to present their topics in a manner accessible also to scientifically literate non-specialists wishing to gain insight into the deeper implications and fascinating questions involved. Taken as a whole, the series reflects the need for a fundamental and interdisciplinary approach to modern science. It is intended to encourage scientists in all areas to ponder over important and perhaps controversial issues beyond their own speciality. Extending from quantum physics and relativity to entropy, time and consciousness - the Frontiers Collection will inspire readers to push back the frontiers of their own knowledge. Quantum Mechanics and Gravity ByM. Sachs Mind, Matter and Quantum Mechanics ByH. Stapp Quantum-Classical Correspondence By A.O. Bolivar Knowledge and the World: Challenges Beyond the Science Wars Edited by M. Carrier, J. Roggenhofer, G. Kiippers, P. Blanchard Quantum-Classical Analogies By D. Dragoman and M. Dragoman Quo Vadis Quantum Mechanics? Edited by A. C. Elitzur, S. Dolev, N. Kolenda Series homepage - springeronline.com D. Dragoman M. Dragoman QUANTUM CLASSICAL ANALOGIES With 78 Figures ~ Springer Prof. Daniela Dragoman University of Bucharest, Physics Faculty, Solid State Chair, PO Box MG-ll, 76900 Bucharest, Romania email: [email protected] Prof. Mircea Dragoman National Research and Development Institute in Microtechnology, PO Box 38-160, 023573 Bucharest, Romania email: [email protected] Series Editors: Prof. Daniela Dragoman University of Bucharest, Physics Faculty, Solid State Chair, PO Box MG-ll, 76900 Bucharest, Romania email: [email protected] Prof. Mircea Dragoman National Research and Development Institute in Microtechnology, PO Box 38-160, 023573 Bucharest, Romania email: [email protected] Prof. Avshalom C. Elitzur Bar-Han University, Unit of Interdisciplinary Studies, 52900 Ramat-Gan, Israel email: [email protected] Prof. Mark P. Silverman Department of Physics, Trinity College, Hartford, CT 06106, USA email: [email protected] Prof. Jack Tuszynski University of Alberta, Department of Physics, Edmonton, AB, T6G 2J1, Canada email: [email protected] Prof. H. Dieter Zeh University of Heidelberg, Institute of Theoretical Physics, Philosophenweg 19, 69120 Heidelberg, Germany email: [email protected] Cover image: Detail from 'Visualizing Complicated Dynamics'. Courtesy ofM . Dellnitz, o. Junge, M. Rumpf, and M. Spielberg. Library of Congress Control Number: 2004094689 ISSN 1612-3018 ISBN 978-3-642-05766-3 ISBN 978-3-662-09647-5 (eBook) DOI 10.1007/978-3-662-09647-5 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable to prosecution under the German Copyright Law. springeronline.com © Springer-Verlag Berlin Heidelberg 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Softcover reprint of the hardcover 1St edition 2004 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the authors using a Springer T£X macro package Final processing by LE-T£X Jelonek, Schmidt & Vdckler GbR, Leipzig Cover design by KiinkelLopka, Werbeagentur GmbH, Heidelberg Printed on acid-free paper SPIN: 10906804 57/314I/YL -5 43210 Preface Analogy is a basic concept for understanding nature, since it analyses and connects different phenomena linked by common properties or similar behav ior. In particular, analogy can to some extent apply to specific quantum phe nomena and their corresponding classical effects, although quantum physics differs from classical physics in both formalism and fundamental concepts. Quantum mechanics is considered the major achievement of XXth cen tury physics and it constitutes the foundation of almost all new branches of physics such as solid state physics, quantum optics, and the recently developed nanoscience. Many important discoveries which have dramatically changed the world, like computers or modern communication systems such as mobile tele phones, incorporate devices based entirely on quantum mechanical concepts. On the other hand, classical physics, which encompasses classical mechan ics and electromagnetic field theory, is also present in many aspects of every day life. Brilliant minds like Newton, Laplace, Fermat, Lagrange, Hamilton or Maxwell devised these theories between the XVIIth and XIXth centuries. It is common opinion nowadays, however, that classical physics is a simple and old-fashioned theory without much relevance for the more refined and modern quantum theory. Moreover, many physicists consider that there are no links between the two theories, although classical instruments located in the classical world register all outcomes of quantum processes. We consider that this artificial gap between quantum and classical physics is the origin of the lack of agreement regarding the significance of quantum mechanics. Indeed at least nine different formulations and ontological inter pretations of this theory are known today. In this context, the large number of existing analogies between classical and quantum phenomena are considered by many scholars as nothing other than a mere mathematical curiosity. In contrast with the mainstream of accepted wisdom, we consider that quantum-classical analogies are a source of understanding and further devel opment of quantum physics. Indeed, many quantum physical concepts have originated from classical notions, a striking example in this respect being the Schrodinger equation, which was formulated by starting from classical optical concepts. The book does not provide a new interpretation of quantum mechanics (this could be the subject of another book) and does not treat the thorny VI Preface problem of quantum-classical correspondence - a subject that is brilliantly treated in A.O. Bolivar's book, already published in this series. The main role of quantum-classical analogies presented in ten distinct chapters is to shed some light on the genuine significance of the quantum and classical worlds. The reader will discover that the quantum and classical physics share many common concepts, and that many quantum phenomena, devices or algorithms can easily be demonstrated and implemented using classical means. Surprisingly, however, quantum--classical analogies evidence more than any other method the striking differences between classical and quantum worlds, because the analogies cannot be extended to wave-particle duality, which is a landmark quantum concept, but only to a single aspect of it: either the particle or the wave. The way of thinking promoted by this book, i.e., equal consideration of the quantum and classical worlds as a source of new knowledge about the phys ical world, could become a powerful tool for solving the so-called quantum mysteries such as quantum interference, and quantum particle-wave duality, not to mention the quantum measurement problem. The latter is the puz zling process whereby a quantum wavefunction which can be mathematically expressed as a superposition of eigenstates of a certain operator transposes itself into a probabilistic set of outcomes of the various eigenvalues of that operator. All the formal analogies presented in this book may prove valuable in this quest, but a special place is held by the common treatment of classical and quantum phenomena in phase space. The book addresses a large category of readers, especially graduates and PhD students, who can gain additional insights into classical and quantum physics after the study of basic courses dedicated to quantum mechanics, solid state physics, classical mechanics and optics. The book should also be a useful instrument for researchers working in advanced topics such as quan tum computing, quantum devices, atom optics or quantum optics. Moreover, it could be used as an additional source for a course on quantum mechanics, wherein teachers may use it as a support for lectures. We would like to thank to Dr. Claus Ascheron and Dr. Angela Lahee, both experienced editors at Springer. We have worked hard together since 2001 to create the unusual Frontiers Collection, which now includes the present book. This series deals with issues located at the forefront of physics, controversial and exotic theories, experiments and concepts. We hope that this new Agora of physics will in time produce the changes in the physical concepts needed to understand the provocative challenges of Nature encountered every day. In particular, we hope that physics will recover its lost unity and that to gether with biology, it will offer the much desired ontological answers about humanity and Nature. Bucharest, Daniela Dragoman June 2004 Mircea Dragoman Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Analogies Between Ballistic Electrons and Electromagnetic Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Analog Parameters for Ballistic Electrons and Classical Electromagnetic Waves. . . . . . . . . . . . . . . . . . .. 10 2.2 Geometrical Electron Optics ........................... 14 2.2.1 Electron Prism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15 2.2.2 Electron Lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 2.2.3 Magnetic Electron/Hole Focusing. . . . . . . . . . . . . . .. 18 2.2.4 Electron Beam Splitter ......................... 21 2.3 Quantum/Electromagnetic Waveguides. . . . . . . . . . . . . . . . .. 22 2.3.1 Dispersion Equation. . . . . . . . . . . . . . . . . . . . . . . . . . .. 23 2.3.2 Coupled Electron Waveguides. . . . . . . . . . . . . . . . . . .. 25 2.4 Analogous Quantum/Electromagnetic Filters. . . . . . . . . . . .. 27 2.5 Analogous Quantum/Electromagnetic Thnneling Structures 29 2.5.1 Quantum/Electromagnetic Resonant Thnneling Wells . . . . . . . . . . . . . . . . . . . . . .. 30 2.5.2 Quantum/Electromagnetic Resonant Thnneling Wires ...................... 32 2.5.3 Quantum/Electromagnetic Resonant Thnneling Dots ....................... 33 2.6 Photonic Band Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36 2.7 Photonic Bloch Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40 2.8 Electron Diffraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44 2.9 Electron Interference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47 2.10 Analog Quantum/Electromagnetic Devices with Non-Uniform Cross-Section. . . . . . . . . . . . . . . . . . . . . . .. 50 2.10.1 Analog Quantum/Electromagnetic T-Shaped Devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51 2.10.2 Analog Quantum/Electromagnetic Dots with Non-Uniform Cross-Section. . . . . . . . . . . . . . . .. 54 2.11 Quantum/Electromagnetic Cavities. . . . . . . . . . . . . . . . . . . .. 56 2.12 Electromagnetic Analogs of Type II Semiconductor Heterostructures. . . . . . . . . . . . . .. 58 VIII Contents 3 Electron/Electromagnetic Multiple Scattering and Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63 3.1 Quantum/Electromagnetic Wave Propagation in Disordered Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64 3.1.1 Microscopic Theory of Transport. . . . . . . . . . . . . . . .. 65 3.1.2 Single-Scattering Matrix. . . . . . . . . . . . . . . . . . . . . . .. 67 3.1.3 Velocities of Electromagnetic Waves in Random Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 68 3.1.4 Length Scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72 3.2 Light Conductance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73 3.3 Photonic Hall and Magnetoresistance Effects . . . . . . . . . . . .. 76 3.4 Interference Effects in Multiple Scattering. . . . . . . . . . . . . . .. 79 3.4.1 Universal Conductance Fluctuations. . . . . . . . . . . . .. 79 3.4.2 Light Intensity Correlations ..................... 81 3.4.3 Weak Localization. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88 3.5 Localization Criteria .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93 3.5.1 Ioffe-Regel Criterion for Localization. . . . . . . . . . . .. 94 3.5.2 Thouless Criterion for Localization. . . . . . . . . . . . . .. 95 3.5.3 Vanishing of Transport Mean Free Path as Localization Criterion . . . . . . . . . . . . . . . . . . . . . . .. 98 3.6 Experimental Evidence of Light Localization. . . . . . . . . . . .. 98 4 Acoustic Analogies for Quantum Mechanics .............. 103 4.1 Acoustic Bloch Waves ................................. 103 4.2 Acoustic Analogs of Quasicrystals ....................... 105 4.3 Multiple Scattering and Localization of Acoustic Waves .... 106 4.4 Acoustic Wave Propagation in Nonlinear Disordered Systems ........................ 112 4.5 Acoustical Analog of Persistent Currents in Mesoscopic Normal Metal Rings ...................... 115 4.6 Acoustic Casimir Effect ................................ 116 5 Optical Analogs for Multilevel Quantum Systems ......... 119 5.1 Optical Analog for a Two-Level Quantum System ......... 119 5.2 Dynamical Behavior of the Two-Level Optical Atom ...... 124 5.2.1 Adiabatic Regime .............................. 125 5.2.2 Rabi Oscillations ............................... 125 5.2.3 Multiphoton Transitions and Bloch-Siegert Shifts .. 127 5.2.4 Zener-Tunneling Regime ........................ 128 5.3 Macroscopic Optical Band Structure .................... 129 5.4 Comparison with Spin-Resonant Phenomena ............. 132 5.5 Limitations of the Optical Analog of a Quantum Two-Level System ....................... 135 5.6 Three- and Four-Level Optical Atoms ................... 136 5.7 Nonlinear Optical Atoms .............................. 137 Contents IX 5.8 Spectral Focusing in Two-Photon Processes .............. 138 5.9 Spectral Fresnel Lens .................................. 141 6 Particle Optics ........................................... 143 6.1 Particle Optics Versus Classical Optics .................. 145 6.2 Atom Interactions with Light ........................... 147 6.3 Geometrical Particle Optics ............................ 150 6.3.1 Collimation ................................... 153 6.3.2 Focusing ...................................... 154 6.3.3 Mirrors ....................................... 159 6.3.4 Atom Channeling .............................. 162 6.3.5 Beam Splitters ................................. 163 6.4 Particle Wave Optics .................................. 165 6.4.1 Diffraction .................................... 166 6.4.2 Interference ................................... 170 6.4.3 Holography .................................... 175 6.5 Atom Waveguides ..................................... 177 6.5.1 Atom Coupler ................................. 180 6.5.2 Bloch Oscillations in Atom Optics ................ 181 6.5.3 Atomic Wavepackets ............................ 182 6.5.4 Atom Lenses and Mirrors in the Time Domain ..... 182 6.6 Quasiparticle Optics .................................. 184 7 Quantum/Classical Nonlinear Phenomena ................ 187 7.1 Nonlinear Phenomena in Resonant Tunneling Diodes ...... 187 7.2 Nonlinear Barrier Traversal ............................ 191 7.3 Classical Nonlinear Analogs of Bose-Einstein Condensates. 191 7.3.1 Four-Wave Mixing in the BEC ................... 194 7.3.2 BEC Solitons .................................. 195 7.3.3 BEC Holography ............................... 201 7.4 BECs and Gravitation ................................. 203 7.5 Nonlinear Atom Optics ................................ 206 7.6 Classical Analog of Andreev Reflection .................. 208 8 Quantum/Classical Phase Space Analogies ................ 211 8.1 Classical/Quantum Hamiltonian Formulation ............. 212 8.2 Quantum/Classical Systems with Identical Motion ........ 217 8.3 Operator Formalism in Classical Physics ................. 221 8.3.1 Differential Operators in Classical Optics .......... 221 8.3.2 Integral Operators in Classical Optics ............. 225 8.3.3 Operator Factorization .......................... 226 8.4 Quantum/Classical Chaos .............................. 227 8.5 Quantum/Classical Uncertainty Relations ................ 229 8.6 Quantum/Classical Phase Space Distribution Functions .... 232 8.7 Quantum/Classical Interference in Phase Space ........... 236 X Contents 8.8 Quantum/Classical Transitions ......................... 240 8.9 Quantum/Classical Transformations ..................... 242 9 Analogies Between Quantum and Classical Computing ... 247 9.1 Bits and Registers .................................... 249 9.2 Logical Operations .................................... 256 9.3 Computing Algorithms ................................ 261 9.4 Entangled States ...................................... 270 9.5 Teleportation ......................................... 276 10 Other Quantum/Classical Analogies ...................... 279 10.1 Transmission Line Analogy to Ballistic Electron Motion ... 279 10.2 Quantum Analog of the Electro-Optic Modulator ......... 280 10.3 Electronic Flute ...................................... 282 10.4 Quantum Anyon/Classical Vortex Analogy ............... 283 10.5 Mechanical Model of Rotational States in Even-Even Nuclei .................................. 285 10.6 Quantum/Classical Malus Law ......................... 287 10.7 Analogies Between the Wave Equation and the Time-Independent Schrodinger Equation ......... 288 10.7.1 Classical Analogs of Quantum Fock States ......... 288 10.7.2 Nonlinear Quantum/Classical Operators .......... 290 10.7.3 Optical Corrals ................................ 291 10.8 Analogies Between the Wave Equation and the Time-Independent Dirac Equation ............... 291 10.9 Analogies Between the Wave Equation and the Time-Dependent Schrodinger Equation. . . . . . . . . . . 293 10.9.1 Fractional Quantum Revivals and the Fractional Optical Talbot Effect . . . . . . . . . . 296 10.9.2 Non-Paraxial Case ............................. 298 10.10 Maxwell Equations in Spinor Form ...................... 299 10.11 Diffraction in Time ................................... 300 10.12 Quantum/Classical Tunneling .......................... 304 10.12.1 Setups for Direct Measurement of the Quantum/Classical Tunneling Time ......... 308 10.12.2 Quantum/Classical Larmor Clocks ............... 311 10.13 Classical Analogs of the Aharonov-Bohm Effect .......... 314 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Index ......................................................... 341