Quantum Chaos Y2K Proceedings of Nobel Symposium 116 Backaskog Castle, Sweden, June 1 3 - 1 7 , 2000
Editors
Karl-Fredrik Berggren Sven Aberg
Physica Scripta The Royal Swedish Academy of Sciences / World Scientific
Quantum Chaos Y2K Proceedings of Nobel Symposium 116 Backaskog Castle, Sweden, June 1 3 - 1 7 , 2 0 0 0
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Karl-Fredrik Berggren Sven Aberg
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Published jointly by Physica Scripta The Royal Swedish Academy of Sciences Box 50005, SE-104 05, Stockholm, Sweden and World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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QUANTUM CHAOS Y2K — Nobel Symposium 116 Copyright © 2001 Royal Swedish Academy of Sciences All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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The contents of this volume were also published as Vol. T90 of Physica Scripta.
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0031-8949 (0281-1847) 91 -87308-93-2 981-02-4711-7
Physica Scripta, Vol. T90, 2001
Contents Committees
6
List of participants
8
Introduction
11
After-dinner speech. M. C. Gutzwiller
13
Spectral twinkling: A new example of singularity-dominated strong fluctuations (summary). M. Berry
15
Quantum chaos in G a A s / A l x G a i x As microstructures. A. M. Chang
16
Ground state spin and Coulomb blockade peak motion in chaotic quantum dots. J. A. Folk, C. M. Marcus, R. Berkovits, I. L. Kurland, I. L.AIeiner and B. L.AItshuler
26
Quantum chaos and transport phenomena in quantum dots. A. S. Sachrajda
34
Conductance of a ballistic electron billiard in a magnetic field: Does the semiclassical approach apply?. T. Blomquist and I. Zozoulenko Semiconductor billiards - a controlled environment to study fractals. R. P. Taylor, A. P. Micolich, R. Newbury, J. M. Fromhold. A. Ehlert, A. G. Davis, L. D. Macks, C. R. Tench. J. P. Bird, H. Linke, W.R. Tribe, £ H. Linfield and D. A. Ritchie
37
41
Experimental signatures of wavefunction scarring in open semiconductor billiards. J. P. Bird, R. Akis, and D. K. Ferry
50
Chaos in quantum ratchets. H. Linke, T. E. Humphrey, R. P. Taylor, A. P. Micolich and R. Newbury
54
Statistics of resonances in open billiards. H. Ishio
60
The exterior and interior edge states of magnetic billiards: Spectral statistics and correlations. K. Hornberger and U. Smilansky
64
Non-universality of chaotic classical dynamics: implications for quantum chaos. M. Wilkinson
75
Chaos and interactions in quantum dots. Y. Alhassid
80
Stochastic aspects of many-body systems: The embedded Gaussian ensembles. H. A. Weidenmii/ler
89
Quantum-classical correspondence for isolated systems of interacting particles: Localization and ergodicity in energy space. F. M. Izrailev
95
Effect of symmetry breaking on statistical distributions. G. £ Mitchell and J. F. Shriner, Jr.
105
Quantum chaos and quantum computers. D. L. Shepelyansky
112
Disorder and quantum chromodynamics — non-linear a models. T. Guhr and T. Wi/ke Correlations between periodic orbits and their role in spectral statistics. M. Sieber and K. Richter
121 128
Quantum spectra and wave functions in terms of periodic orbits for weakly chaotic systems. R. £ Prange, R. Narevich and O. Zaitsev
1 34
Bifurcation of periodic orbit as semiclassical origin of superdeformed shell structure. K. Matsuyanagi
142
Wavefunction localization and its semiclassical description in a 3-dimensional system with mixed classical dynamics. M. Brack, M. Sieber and S. M. Reimann
146
Neutron stars and quantum billiards. A. Bulgac and P. Magierski
150
Scars and other weak localization effects in classically chaotic systems. £ J. Heller
154
Tunneling and chaos. S. Tomsovic
162
Relaxation and fluctuations in quantum chaos. G. Casati
166
Rydberg electrons in crossed fields: A paradigm for nonlinear dynamics beyond t w o degrees of freedom. T. Uzer
176
Classical analysis of correlated multiple ionization in strong fields. B. Eckhardt andK. Sacha
185
Classically-forbidden processes in photoabsorption spectra. J. B. Delos, V. Kondratovich, D. M. Wang, D. Kleppner and N. Spellmeyer Quantum hall effect breakdown steps due to an instability of laminar f l o w against electron-hole pair formation. L Eaves
189 1 96
Dynamical and wave chaos in the Bose-Einstein condensate. I/I/. P. Reinhardt and S. B. McKinney Wave dynamical chaos: An experimental approach in billiards. A. Richter Acoustic chaos. C. Ellegaard, K. Schaadt and P. Bertelsen
202 212 223
Ultrasound resonances in a rectangular plate described by random matrices. K. Schaadt, G. Simon and C. Ellegaard
231
Quantum correlations and classical resonances in an open chaotic system. W. T. Lu, K. Pance, P. Pradhan and S. Sridhar
238
Why do an experiment, if theory is exact, and any experiment can at best approximate theory? H.-J. Stockmann
246
Wave-Chaotic optical resonators and lasers. A. D. Stone
248
Angular momentum localization in oval billiards. J. U. Nockel
263
Chaos and time-reversed acoustics. M. Fink
268
Single-mode delay time statistics for scattering by a chaotic cavity. K. J. H. van Bemmel, H. Schomerus and C. W. J. Beenakker
278
Nobel Symposium Quantum Chaos Y2K Backaskog Castle 7 n
June 13-17 .Vc
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COMMITTEES International Programme Committee B. Altshuler H. U. Baranger M. Berry L. Eaves M. Gutzwiller E. J. Heller
C. G. B. A. H. K.
M. Marcus Mitchel Mottelson Richter Weidenmiiller H. Welge
Organizing Committee Karl-Fredrik Berggren, Linkoping Par Omling, Lund Sven Aberg, Lund Sponsor The symposium was sponsored by the Nobel Foundation through its Nobel Symposium Fund.
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List of Participants* Yorarn Alhassid Yale University, New Haven, USA
[email protected] Boris Altshuler Princeton University, USA
[email protected] Carlo Beenakker Universiteit Leiden, The Netherlands
[email protected] Karl-Fredrik Berggren Linkoping University, Sweden
[email protected] Michael Berry University of Bristol, UK Fax:+44 0 117 925 5624 Jonathan Bird Arizona State University, Tempe, USA
[email protected] Torbjorn Blomquist Linkoping University, Sweden
[email protected] Eugene Bogomolny Institut de Physique Nucleaire d'Orsay, France
[email protected] Oriol Bohigas Institut de Physique Nucleaire d'Orsay, France
[email protected] Aurel Bulgac University of Washington, Seattle, USA
[email protected] Carlo Canali Lund University, Sweden
[email protected] Giulio Casati Universita degli Studi dell' Insubria, Como, Italy
[email protected] Albert Chang Purdue University, West Lafayette, USA
[email protected] Predrag Cvitanovic Niels Bohr Institute, Copenhagen, Denmark
[email protected] John Delos College of William and Mary, Williamsburg, USA
[email protected] Laurence Eaves University of Nottingham, UK
[email protected] Bruno Eckhardt Philipps-Universitat Marburg, Germany bruno .eckhardt @ physik. uni-marburg ,de
Clive Ellegaard Niels Bohr Institute, Copenhagen. Denmark
[email protected] Mathias Fink Universite Denis Diderot, Paris, France
[email protected] Shmuel Fishman Technion, Haifa, Israel
[email protected] Tania Montero Univerity of London, UK
[email protected] Antti Niemi Uppsala University, Sweden
[email protected] Jens Nockel Nanovation Technologies, Evanston, USA
[email protected] Thomas Guhr Lund University. Sweden
[email protected] Par Omling Lund University. Sweden
[email protected] Martin Gutzwiller New York
[email protected] Jean-Louis Pichard Centre d'Edutes de Saclay, Gif-sur-Yvette, France
[email protected] Eivind Hauge Norwegian University of Science and Technology, Trondheim, Norway
[email protected] Eric Heller Harvard University. Cambridge, USA
[email protected] Hiromu Ishio Osaka Kyoiku University, Japan
[email protected] Felix Izrailev Universidad Autononoma de Puebla, Mexico
[email protected] Mats Jonson Chalmers Institute of Technology, Gothenburg, Sweden
[email protected] Jonathan Keating University of Bristol, UK
[email protected] Daniel Kleppner Massachusetts Institute of Technology, Cambridge, USA
[email protected] Igor Lerner University of Birmingham, UK
[email protected] Heiner Linke, The University of New South Wales, Sydney, Australia
[email protected] Charles Marcus, Harvard University, Cambridge, USA marcus@ harvard.edu Ken-ichi Matsuyanagi, Kyoto University, Japan
[email protected] Gary Mitchell North Carolina State University, Raleigh, USA
[email protected] Richard Prange University of Maryland, College Park, USA prange @ physics, umd.edu Daniel Ralph Cornell University, Ithaca. USA
[email protected] Stephanie Reimann Lund University, Sweden
[email protected] William Reinhardt University of Washington, Seattle. USA
[email protected] Achim Richter Technische Universitat Darmstadt. Germany richter @ ikp.tu-darmstadt.de Klaus Richter Max-Planck-Institut fur Physik komplexer Systeme, Dresden, Germany
[email protected] Andrew Sachrajda National Research Council. Ottowa, Canada
[email protected] Almas Sadreev Kirensky Institute of Physics, Krasnoyarsk, Russia
[email protected] Alexander Saichev Nizhny Novgorod University, Russia
[email protected] Kristian Schaadt Niels Bohr Institute, Copenhagen, Denmark
[email protected] Dima Shepelyansky Universite Paul Sabatier, Toulouse, France
[email protected] "Affiliations and e-mail addresses updated in November 2000. Physica Scripta T90
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List of Participants Uzy Smilansky Weizraann Institute of Science, Rehovot, Israel
[email protected] Steven Tomsovic Washington State University, Pullman, USA tomsovic @ wsu.edu
Vladimir Zelevinsky Michigan State University, East Lansing, USA
[email protected] Srinivas Sridhar Northeastern University, Boston, USA
[email protected] Turgay Uzer Georgia Institute of Technology, Atlanta, USA
[email protected] Martin Zirnbauer Universitat zu Koln, Germany
[email protected] Douglas Stone Yale University, New Haven, USA
[email protected] Hans-Jiirgen Stockmann Phillips-Universitat Marburg, Germany
[email protected] Richard Taylor University of Oregon, Eugene, USA
[email protected] © Physica Scripta 2001
Hans Weidenmiiller Max-Planck-Institut fur Kernphysik, Heidelberg, Germany
[email protected] Michael Wilkinson The Open University, Milton Keynes, UK
[email protected] 9
Igor Zozoulenko Linkoping University, Sweden igorz@,ifm. liu.se Sven Aberg Lund University, Sweden
[email protected] Hong-qi Xu Lund University, Sweden
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Introduction The Nobel Foundation arranges on a regular basis special Nobel Symposia in scientific disciplines that are related to the Nobel prizes. Since 1965, when the series of symposia was first initiated, more than one hundred have been arranged. A Nobel symposium corresponds to a rather small meeting with leading scientists in their particular field, facilitating informal discussions, in plenum as well as in private, on critical issues and latest results. In addition to the specialists a small number graduate students, young researchers, and researchers from nearby fields are also invited to participate in the symposia. This time the Nobel Foundation has selected the field of quantum chaos for its 116th symposium. Thus "Quantum Chaos Y2K" was held in June 13-17, 2000, at the historic and beautiful Backaskogs Castle in the south of Sweden. Quantum chaos is becoming a very wide field that ranges from experiments to theoretical physics and purely mathematical issues. Because of this grand span we decided at an early stage of the planning of this symposium to focus on experiments and theory, and to encourage the interplay between them. We also wanted to stress the interdisciplinary character of the subject, and involved physics from a broad range of subjects, including condensed matter physics, nuclear physics, atomic physics, and elementary particle physics. The physics involved in quantum chaos has much in common with acoustics, microwaves, optics, etc. Therefore we choose to include also aspects of wave chaos in this broader sense. The program was structured according to the following headings: • • • • • • •
Manifestations of Classical Chaos in Quantum Systems Transport Phenomena Description of Quantal Spectra in Terms of Periodic Orbits Semiclassical and Random Matrix Approaches Quantum Chaos in Interacting Systems Chaos and Tunneling Wave-Dynamic Chaos
The meeting at Backaskog was conducted in a truly interdisciplinary atmosphere with invited talks and poster presentations. There was ample time for discussions in which the chairmen played very important roles in summarising and highlighting the different sub-fields. In addition to invited talks and poster contributions the present proceedings therefore also contain informal contributions from some of the chairmen. In this way we hope the proceedings will convey some of the enthusiasm felt at Backaskog. Finally we wish to thank the Nobel Foundation for selecting the field of quantum chaos to be the topic of the 116th Nobel symposium and for its generous support. We also wish to thank the international program committee for all its help when designing the symposium. Karl-Fredrik Berggren
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Par Omling
Sven Aberg
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Nobel Symposium Quantum Chaos Y2K Backaskog Castle, June 13-17, 2000 Conference Dinner on Friday, June 16 After-dinner speech by Martin C. Gutzwiller Received June 27, 2000
Dear Organizers, dear participants and spouses! Kalle Berggren thought that you might benefit from some of my personal thoughts concerning this marvelous Nobel symposium. Obviously, we are celebrating the first summit meeting of Quantum Chaos. The list of participants reads like a "Who's who in quantum chaos?". But I have a question: What brings us together here at the extravagant expense of the Nobel foundation? Michael Berry proposed a short answer a few years ago with the neologism "chaology". This would put our interests into the same category as psychology, cosmology, and even theology. We would claim that we were dealing with something as profound as the human soul, the universe, or God. Frankly, chaos sounds a little skimpy by comparison with these general themes. If we insist on using chaology, however, we would move down a few notches, somewhere in the region of tautology, or even worse, Scientology. The real trouble is that chaos is a phenomenon, almost an object; not an activity like philosophy, history, chemistry. Chaos is outside of us, it is an absence, not a presence, like negative space. I am a collector of old books; I go to book fairs that are organized by the International Antiquarian Book Association (IABA). Its motto, as shown on the catalogues (and price lists), is: "Amor librorum nos unit", i.e. "The love of books unites us". Maybe we should start a club with the motto "The love of chaos unites us"; but I think we should be careful. The Nobel foundation would probably not give us any money at all because we might be mistaken as anarchists, as well as unreliable, if not disreputable! In any case: Is there anybody who truly loves chaos? There are some really irritating forms of it. I have been traveling for the last month to present my views on the method of stationary phase in quantum field theory and on the not-so-stationary phases of the moon. I am sure that this kind of scientific tourism is for the benefit of truth, beauty, and all of humankind. Also, I don't mind sleeping in a different bed occasionally. But using a different shower every 3 days is definitely more than I can handle. First I step into a square cubicle with hard walls on all sides and of infinite height. Then I become the center piece of a hydrodynamic Sinai billiard. Figuring out the degrees of freedom of the faucet is bad enough, but the worst is finding the direction of warm and cold. The fundamental reason for this kind of chaos, here as elsewhere, is the time © Physica Scripta 2001
delay between cause and effect. Please, try to remember what I just said, because it is an important scientific statement as you will see right away. A Swiss weekly newspaper, DieWeltwoche, had an article last week on a new professorship at the Federal Institute of Technology in Zuerich, of Einstein fame. The occupant of this so-called chair has to do research in the "Physics of Transport and Traffic". In other words, he has to study the cause of traffic jams. This newly appointed professor has already announced his remedy: "Brake slowly, accelerate fast!" I should not really complain about ordinary, classical chaos, which is what all this is about. Only 30 years ago it was the only kind of chaos anybody ever talked about. Nobody even thought of a thing called quantum chaos, and it was certainly not considered a proper and legitimate interest for an otherwise competent and decent physicist. Although it was not called by this name, quantum chaos was the main topic of a conference for the first time in Como 1977. Giulio Casati and Joe Ford were the organizers, and it was the occasion where I met Michael Berry, John Delos, and Eric Heller. But official recognition for our topic was not granted for another 10 years. There was a big conference in Los Alamos in 1982 with the Greek title "Kosmos en Chao", which was translated as Order in Chaos. A cosmologist friend pointed out to me that Kosmos meant not only the universe, but also cosmetics, suggesting "Beauty in Chaos". After a full week of classical chaos, the last session on Friday afternoon consisted of Eric Heller and myself. An earlier speaker, Benoit Mandelbrot, said to me: "Martin, this is an insult; either you pull strings to get an earlier slot, or you have to excuse yourself!" But I explained to him that we were obviously the wave of the future; actually, our session was well attended. A similar situation arose at the Nobel Symposium in Graeftavallen 1984. Rolf Landauer had wangled an invitation for me at the last moment, and I was again granted a slot on Friday as the only quantum chaotician in "The Physics of Chaos and Related Problems". We were treated in style, although our accomodations in the north of Sweden were not palatial as here in the south. Our organizer, Stig Lundqvist, had a robust sense of humor. He had invited the chairman of the Nobel Foundation, a Stockholm banker by the name of Stig Ramel, for the banquet on Thursday night. Then as now, the participants were living off the generosity of the foundation because of Physica Scripta T90
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Nobel Symposium
Ramel's genius for good financial investments. So after dining on the meat of reindeer, beaver, and bear (no kidding!), instead of an after-dinner speech, there was a mock Nobel ceremony, with Ramel playing the King while Lundqvist presented about a dozen winners. Unfortunately, history has not taken note of this event and its prize-winners. I was among them, because I "had given the best talk at the symposium", and I was awarded a free sauna to be enjoyed in the company of any of the symposium staff. But remember that my talk was scheduled only for the next day! Quantum chaoticians were used to "eating humble pie", and the situation improved only in 1987. Rick Heller and I were asked to organize a 6-month workshop at the Institute for Theoretical Physics in Santa Barbara. We wrote a nice detailed proposal, and it was accepted for the spring of 1989 provided we changed the title. One of the board members claimed that "Quantum Chaos" does not exist. Among the participants to the "Dynamical Properties of Small Quantum Systems" I remember Bohigas, Berry, Casati, Bogomolnyi, Eckhardt, Prange, Reinhardt, Tomsovic and Wilkinson, all of them here today. This kind of circumlocution went on for several years. I recall the following: Como 1983: Chaotic Behavior in Quantum Systems Les Houches 1989: Chaos in Quantum Systems Trieste 1990: Quantum Chaos (Clear sailing for the first time!) Copenhagen 1991: Quantum Chaos - Quantum Measurement Kyoto 1993: Quantum and Chaos - How Incompatible? (I like this one!) Now we can be proud of the Nobel Foundation calling this symposium Quantum Chaos Y2K: it is like a papal benediction. But already, we have to be careful, because we have come to the attention of the philosophers, historians, and sociologists. Two Princeton philosophers of science, Norton Wise and David Brock, have just published in "Studies in the History and Philosophy of Modern Physics" an article with the title "The Culture of Quantum Chaos". They take off from the article "Post-Modern Quantum Mechanics" by Heller and Tomsovic in Physics Today, and include pictures of whispering galleries, bouncing balls, and scars. I know that Rick feels somewhat embarassed by this philosphical-historical-sociological discussion. But I find the article quite instructive and really useful in its scope. It is particularly good for our spouses who vacillate between feelings of respect for the unknown (a healthy attitude, no doubt), and of frustration in front of the incomprehensible, which can be dangerous to your health! The interdisciplinary nature of quantum chaos is emphasized: I am not sure how successful we are in this respect,
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but there is little doubt that we are trying very hard. The many people from different backgrounds are mentioned. And there are indeed many different kinds of behavior and manner noticeable at this symposium. That might be a subject of some interest, but I will leave the details of this topic to your imagination. The word "post-modern" has come into bad repute because of some excesses among multiculturalist philosophers. But that should not prevent us from using this adjective in its original sense as Rick and Steven did. Post-modern activities are meant to bring together some formerly separate tendencies in any field. Therefore, our efforts to get classical trajectories and waves closer to each other are post-modern in the same sense. The article by Wise and Brock also mentions the potential applications of quantum chaos. They range from the ridiculous to the sublime with some of the latter clearly represented at this symposium. So I can't resist showing you an example of the ridiculous that my wife discovered in the Herald-Tribune last week. It is "The Inevitable Crackle" of somebody trying to unwrap a candy without any noise during a symphony concert, a funeral service, theater or opera performance and so on. Two scientists at NIST have shown such an effort to be doomed to failure because of the chaos of small individual bursts and uncontrollable pops. With the help of my microphone I will now make an experimental demonstration of this important effect. As you can see I am not a pure theoretician, and on this happy note I come to the end of my speech. In the long run even quantum chaos has to show that it is useful in some generous sense of the word. I believe that mesoscopic devices will be important in the future; undoubtedly they will continue to find inspiration in quantum billiards, electromagnetic and optical cavities, vibrating solids and reverberating concert halls. Eventually mesoscopic physics will descend into the microscopic realm of atoms and molecules. Therefore I hope that nuclear, atomic, and molecular physics will remain in the "club" because they provided much of the original input into quantum chaos. Now on behalf of all the participants and their spouses, I want to express my gratitude to the organizers, Karl-Fredrik Berggren, Sven Aberg, and Par Omling. You have put together a very interesting scientific program, and then badgered the Nobel foundation to provide ample funds. On that basis you have provided us with possibly the most pleasant imaginable conditions for our weighty discussions. This monastery, turned into a royal castle and surrounded by the lush countryside of Southern Sweden, provided us with accomodations, daily meals, and a leisurely atmosphere of truly royal splendor. We will all remember with special fondness this marvelous occasion of scientific discourse.
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Spectral Twinkling: A New Example of Singularity-Dominated Strong Fluctuations (summary) Michael Berry H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL U K Received June 18, 2000
PACS Ref: 03.65. Sq
Abstract In quantum systems that are neither completely integrable nor completely chaotic, energy-level fluctuations are governed by asymptotic scaling laws, analogous to those for the intensity fluctuations of twinkling starlight.
The energy levels of quantum systems whose classical counterparts exhibit mixed chaology obey neither randommatrix nor Poisson statistics. J. P. Keating, H. Schomerus and I have recently suggested [1] that there are universal aspects of these quantum spectral statistics that are characteristic of mixed systems, associated with the bifurcations of stable and unstable isolated periodic orbits that distinguish the corresponding classical mechanics from the purely chaotic and purely integrable cases. We study the Planck (h) dependence of the moments Mm(h) of the fluctuating part of the level-counting function (spectral staircase), and argue that these diverge in the classical limit, according to the scaling law Mm(h) ~ h~v(m). To determine the "twinkling exponents" v{m), the spectral staircase is first represented as a sum over classical periodic orbits [2], corrected to eliminate the divergences near bifurcations [3,4]. Then, using the /j-scaling appropriate to each bifurcation, its contribution to Mm(h) is estimated. The exponents v(m) then follow from a competition over all bifurcations (with different values of repetition number and codimension). This "battle of the bifurcations" depends on new results about the hierarchy of their associated normal forms. There are several other examples of statistics falling into this class [5], where large fluctuations are dominated by geometrical singularities. In twinkling starlight [6,7], the singularities are caustics, of light focused by atmospheric turbulence, which dominate the wavelength-dependence of the light intensity, and the classification of caustics required for their competition is catastrophe theory [8,9]. The density of states n(E) of a solid, in the case where there are many van Hove singularities, also requires the catastrophe classification, and the quantity to be calculated again the result of a competition is the universal power-law decay
© Physica Scripta 2001
of the tail of the probability distribution of values of n{E) [10]. Finally, the sex life of moths is dominated by the male's search for the female by smelling the "odour plume" that she emits [11]. In a turbulent wind, this is determined by the fluctuating concentration C of a passive scalar (pheromone) that is convected by the flow while diffusing, with diffusion constant D. The small-D asymptotics of the moments of C are determined by the streak line singularity, consisting of all those fluid particles that have passed through the female [5,12]. In all these cases, the fluctuations are governed by power laws originating in geometric singularities that have no connection with fractals.
References 1. Berry, M. V., Keating, J. P. and Schomerus, H., Universal twinkling exponents for spectral fluctuations associated with mixed chaology, Proc. Roy. Soc. Lond., A456, 1659 (2000). 2. Gutzwiller, M. C , Periodic orbits and classical quantization conditions J. Math. Phys. 12. 343 (1971). 3. Ozorio de Almeida, A. M. and Hannay, J. H., Resonant periodic orbits and the semiclassical energy spectrum J. Phys. A. 20, 5873 (1987). 4. Sieber, M., Uniform approximation for bifurcations of periodic orbits with high repetition numbers J. Phys. A. 29, 4715 (1996). 5. Berry, M. V., Spectral twinkling, in Enrico Fermi School "New Directions in Quantum Chaos", (ed. U. Smilansky), (Italian Physical Society, Bologna, Course CXLIII, 45 (2000)). 6. Berry, M. V., Focusing and twinkling: critical exponents from catastrophes in non-Gaussian random short waves J. Phys. A 10, 2061 (1997). 7. Walker. J. G., Berry, M. V. and Upstill, C , Measurement of twinkling exponents of light focused by randomly rippling water Optica Acta 30, 1001 (1983). 8. Arnold. V: I„ Catastrophe Theory (Springer, Berlin, 1986). 9. Poston, T. and Stewart, I., "Catastrophe theory and its applications", (Pitman (reprinted by Dover), London, 1978). 10. Berry, M. V., Universal power-law tails for singularity-dominated strong fluctuations J. Phys. A 15, 2735 (1982). 11. Murlis, J. and Jones, C. D., Fine-scale structure of odour plumes in relation to insect orientation to distant pheromone and other attractant sources Physiol. Entomol. 6, 71 (1981). 12. Berry. M. V., Ipotesi di scala e fluttuazioni non gaussiane nella teoria catastrofica della onde (Italian translation of "Scaling and nongaussian fluctuations in the catastrophe theory of waves") in "Prometheus", (eds. P. Bisogno and A. Forti), 1985, Vol. 1, pp. 41-79.
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Physica Scripta. T90, 16-25, 2001
Quantum Chaos in GaAs/Al x Gai_ x As Microstructures A. M. Chang* Department of Physics, Purdue University, West Lafayette, IN 47907, USA Received June 6, 2000
PACS Ref: 72.20.My, 05.45.+b, 72.15.Gd, 73.40.Kp
Abstract We review clear signatures of "quantum chaos" observable in both open and closed systems realized in GaAs/Al v Gai _vAs microstructures. In open ballistic billiards where scattering dynamics determine the characteristics of quantum transport, we find a striking difference in the shape of the negative magneto-resistance peak for transport through chaotic, stadium cavitiesLorentzian line shape, versus non-chaotic, circle cavities-unusual triangular line shape. In a nearly closed system of individual quantum dot singleelectron-transistors, the distribution of the height of Coulomb blockade conductance peaks is strongly non-Gaussian and sensitive to the absence or presence of a magnetic field. Even though our findings are in substantial agreement with the prediction of Random MatrixTheory, there exist evidence that deviations can occur due to the effect of electron-electron interaction. Lastly, we briefly summarize recent observations of pairing effect in ultra-small quantum dots associated with the spin degree of freedom.
1. Introduction Chaotic dynamics is ubiquitous in nature. Arising from non-linearities either inherent in the equation of motion or from boundary conditions in otherwise linear systems, familiar examples include turbulence in fluid flow - e.g. water flow through pipes, aerodynamics, weather, etc., orbital motion of planets in a multiple planetary system, chaotic scattering from multi-spheres, population growth in species, chemical reactions, instabilities in lasers, neural networks, and even chaotic dynamics in internet communication networks. It is therefore natural to ask whether this chaotic dynamics familiar in the classical realm has its counter part in quantum systems and whether manifestations of "Quantum Chaos" can have physically observable consequences. To this end the pioneering works of Gutzwiller, [1] Berry, [2] Bohigas, Giannoni, and Schmit, [3], Weidenmuller [4] and others have paved our way to better understand and identify the key signatures of quantum chaos. Quantum Chaos, of course, refers to the quantum signatures of systems which classically exhibit chaotic dynamics. To date the phenomenon of quantum chaos has been found to manifest itself in a variety of different physical systems, from the venerable level repulsion in complex nuclei studied in the 1950's/1960's [5-7] large Rydberg atoms, [8-11] GaAs/Al x Gai_ v As microstructures [12-19] to wave dynamics in microwave cavities, [20-22] and acoustic wave cavities [23] etc. Each of these systems possesses its own uniqueness and advantages, offering complementary information to help elucidate the nature of the quantum chaos phenomenon, particularly in the presence of additional complexities such as residual non-linearities in the scattering of wave, mode-mode conversion, energy dissipation through absorption, electron-electron or * e-mail:yingshe® physics.purdue.edu Physica Scripta T90
nucleon-nucleon interactions, presence of special symmetries, etc. In this paper, I will review our contributions to recent developments of quantum chaos in the GaAs/ Al.vGai_^As microstructure system. The GaAs/Al x Gai_ v As microstructure system offers distinct advantages, [24,25] ranging from tunability, versatility, to added complexity resulting from electron-electron interaction and spin. The recognition that lithographically defined GaAs/ Al.vGai-.vAs microstructures represent an advantageous system in which to study quantum chaos [12-14,19,24,26-29] arises from several factors. First, by lithographically defining structures of varying geometries and lead configurations, it is possible to study open systems strongly coupled to external least/reservoirs [12,14,19,26], nearly isolated systems weakly coupled to leads/reservoirs [13,16,28,30-33], as well as completely closed systems [29,34]. Secondly, the high mobility and hence long elastic scattering length as well as long quantum phase coherence length of the 2dimensional electron gas (2DEG) permits the realization of essentially ballistic, micron size cavities where the dynamics is quantum coherent and determined by the shape of the cavities rather than by residual impurity scattering. Thirdly, the tunability of the 2DEG via gating enables the shape of the microstructure to be varied in a controlled and continuous manner, facilitating ensemble averaging to reduce statistical fluctuations and the measurement of correlation properties as well [35,36]. Lastly and very importantly, the fact that an electron couples to the magnetic vector potential A through its charge -e permits the direct study of a change in statistical distribution from the orthogonal ensemble to the unitary ensemble, for various physical quantities such as transmission probability in open systems [37,38], resonance line widths and energy level spacings in nearly isolated systems [39^12], and magnetization in closed systems [43]. Moreover, the presence of electron-electron interaction coupled with the spin degree of freedom can lead to a tremendous richness in the variety of manifestations of the quantum chaos phenomenon. In what follows, I will describe two significant results on the manifestation of Quantum Chaos in the transport properties of GaAs/Al v Gai_,;As microstructures and touch on recent and future directions toward understanding the interplay between interaction, electron spin degree of freedom, and chaos. The first result pertains to the signature of quantum chaos in open ballistic cavities where the dynamics is determined by specular scattering off the cavity walls. In our work, we observe a striking difference in the weak localization (WL), negative magnetoresistance line shape for chaotic, stadium cavities versus non-chaotic, circle cavities [12,44]. Our observation of a Lorentzian WL line © Physica Scripta 2001
Quantum Chaos in GaAs/AlxGa\-xAs shape for the stadium cavities and a strikingly different, cusp-like, triangular line shape for the circle cavities clearly demonstrate that distinctly different quantum signatures are observable for these systems, and that the Lorentzian in the stadium cavities arises from boundary scattering from the cavity walls rather than from impurities. Therefore it is indeed possible to observe a difference in the transport through quantum billiards with underlying chaotic versus non-chaotic dynamics. Our observation of a clear difference in line shape is made possible by the outstanding doping-well GaAs/Al^Gai-jAs crystals grown by my colleagues, Loren Pfeiffer and Ken West, leading to exceedingly long phase coherence length and small-angle elastic scattering length. Our second contribution deals with the highly nonGaussian distribution of Coulomb Blockade conductance peak heights in quantum dots [13]. Investigations of peak height statistics [13,16] are complementary to recent studies of the Coulomb blockade peak spacing distribution [30-33]. Early on Jalabert, Stone, and Alhassid [39] recognized that the peak height fluctuations observed in transport may be related to the well-known Porter-Thomas type distribution of resonance line widths observed in the elastic scattering of neutrons from complex nuclides such as U 235 , U 238 , and Th 238 , etc. [45] Using a random matrix theory (RMT) approach which neglects the influence of electron-electron interaction, Jalabert et al. computed the distribution of peak heights in the single-level tunneling limit, where electrons tunnel through an individual, thermally resolved quantum level within the quantum dot for both B = 0 (zero magnetic field) corresponding to the Gaussian Orthogonal Ensemble (GOE), and 72 ^ 0 corresponding to the Gaussian Unitary Ensemble (GUE). These distributions are also obtainable using the supersymmetry technique [42,46] In both cases, the most striking feature is the non-Gaussian nature, with a prevalence of small valued peaks. In fact the B = 0 distribution exhibits a l/^/g singularity for g > 0 where g is the dimensionless peak conductance, while the B ^ 0 distribution attains a maximum near zero. Our experimental findings are is substantial agreement with theory. Results similar to ours are also obtained by Folk et al. [16]. Whereas our observation of the B = 0 distribution is akin to the Porter-Thomas resonance width distribution previously observed in nuclear physics and microwave cavity experiments, the 5 ^ 0 distribution is previously unobserved! In spite of the semi-quantitative agreement with theoretical predictions based on the RMT, our data contains suggestive hints of deviations arising from the influence of electron-electron interaction leading to an enhancement of small peaks even in the B jt 0 case, as will be discussed below. The influence of interaction represents a new and exciting aspect and is observed in the peak spacing distribution as well. In the last section, I will describe our recent investigations of ultra-small quantum dots containing of the order of 10-40 electrons [47,48]. The devices contain multiple independent gates (5 or 6) and are of lithographic size of 160-230 nm in diameter. We investigate effects associated with the spin degree of freedom including peak spacing pairing effect, peak height pairing effect, as well as the spin status as manifested in the Kondo resonance. We find that peak spacing pairing can be observed in dots made from both high and low density GaAs/Al v Ga!_ x As crystals, but a dot of © Physica Scripla 2001
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higher density and consequently lower rs (where rs measures the ratio of the Coulomb energy to Fermi energy) exhibits more pronounced pairing than dots of lower density and higher rs. Furthermore, qualitatively different behaviors are observable depending on the coupling strength between the dots and leads. 2. Weak-location in chaotic versus non-chaotic cavities: a striking difference in the line shape To demonstrate that the phenomenon of Quantum Chaos has observable consequences, we put to test theoretical predictions of characteristics in transport which distinguish a chaotic and non-chaotic system. The idea is as follows, the conductance, G, of a mesoscopic device is given by the Landauer [49] formula, G = (e2/h)T = (e2/h)tr[rf], where T is the quantum transmission probability, T = tr[ttf], and t the transmission amplitude matrix determined by the interference between partial waves traversing different paths through the device. Since the nature of the paths is determined by the dynamics within the device, the conductance, G, should exhibit characteristics reflecting whether the dynamics is chaotic or otherwise. In the semiclassical approach of Jalabert et al. [24], and Baranger et al. [50], at B = 0 it is sufficient to sum the amplitudes of classical paths each contributing a phase factor e'[kJ+s?] where / is the path length and Sp a phase shift. At finite B, the Aharonov-Bohm phase factor, Q^aBKhcle^ must be included as well, where a is the area and B the magnetic field strength, and therefore the area distribution of classical paths becomes relevant [24,25,51]. The area refers to the area enclosed by a particular path and can be both positive or negative depending on the sense of rotation of the trajectory (alternatively represented by the the direction of the area vector). The quantum manifestation of chaotic and non-chaotic behavior is then reflected in these distributions of classical paths or areas. In the case of chaotic ballistic cavities coupled strongly to leads through entrance/exit openings in the cavity walls, the majority of electrons entering into the cavity will escape with a single time scale determined by the traversal time to cross the cavity ballistically scaled by the perimeter to opening ratio. Classically this single time scale results from the chaotic dynamics which scrambles nearly all electron trajectory regardless of initial conditions, and gives rise to an exponential distribution of paths of length /, and a corresponding exponential distribution of enclosed areas [51,25]. It is precisely this exponential area distribution which leads to a Lorentzian in the WL line shape and a quadratic decrease in resistance with magnetic field near zero field. In direct contrast to the chaotic case, a non-chaotic cavity does not scramble the classical trajectories; the presence of conserved quantities ensures that different classes of trajectories corresponding to different values of the constants of motion are characterized by different escape times. The lack of a single time scale leads to power law distributions of path length and areas. In the area distribution, a power law in the form of a" 2 is often found for large a [50,51]. Baranger et al. first pointed out this type of area distribution, obtained for non-chaotic square and rectangular cavities, could lead to a linear decrease of resistance with B yielding a triangular line shape. Physica Scripta T90
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-80
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Fig. 1. The magneto-resistance for (a) 48 stadium cavities, and (b) 48 circle cavities at T — 50 mK. The weak localization peak line shape shows a Lorentzian behavior for the chaotic, stadium cavities. In contrast, the line shape for the non-chaotic, circle cavities shows a highly unusual, triangular shape (linearly decreasing). The resistance value is normalized to a single cavity. The vertical bar indicates the equivalent change in conductance, AG. Insets show electron micrographs of the cavities which are fabricated on a high quality GaAs/AL ( Gai_. t As heterostructure crystal.
Fig. 2. The temperature evolution of the magneto-resistance for the (a) stadium cavities, and (b) circle cavities. From top to bottom, T = 50 mK, 200 mK., 400 mK, 800 mK, 1.6 K, 2.4 K, and 4.2 K. The dash-dotted lines are Lorentzian fits. For the stadium cavities, the weak localization line shape is Lorentzian at all temperatures. For the circle cavities, the line shape is Lorentzian only at higher temperatures above 2.4 K. The linearly decreasing triangular line shape develops fully below 400 mK., showing that phase-coherence is essential in producing this shape.
In Fig. 1 we present results on the WL negativemagnetoresistance for chaotic, stadium cavities (a) and non-chaotic, circle cavities (b), at 50 mK temperature. It is apparent that there is a substantial difference in the line shape which is rounded near B = 0 for the stadium and cusp-like for the circle. In fact the stadium line shape can be fitted by a Lorentzian (Fig. 2) while it is approximately triangular for the circle. These data are obtained in stadium and circle devices fabricated side by side in one lithographic step where, for each device 48 nominally identical but actually slightly different cavities are measured at one time to average out contributions from the universal conductance fluctuations (UCF). It is necessary to establish that the observed difference is quantum-mechanical in origin rather than a classical effect. To this end we present temperature evolution of the WL peak in Fig. 2. Below 4.2 K. in temperature, the classical effects should show minimal change whereas quantum effects should become more pronounced when the quantum phase coherence length becomes long as the temperature is reduced. Evidence that we are observing a quantum effect is apparent in the data in Fig. 2 where the WL peak grows with decreasing temperature. Moreover, for the stadium the line shape is Lorentzian for the entire temperature range as indicated by the fits at 50 mK and
1.6K. On the other hand, while the circle appears to be Lorentzian-like at the higher temperatures, the full linear behavior develops below ~ 400 mK, indicating that the longer trajectories which enclose larger areas and consequently contribute at the weaker B fields become phase coherent only at these lower temperatures. In contrast, short direct paths which are coherent even at high temperatures and enclose minimal areas do not contribute to the cusp-like behavior at these smaller magnetic field scales. More recently, three other groups have observed a triangular line shape for non-chaotic cavities in the shape of a square [17,18,52], confirming our results. In particular, Lee, Faini, and Mailly have obtained a WL line shape for a square nearly identical to our result for the circle [52]. The observation of a line shape difference is in agreement with theoretical expectations. It is significant for the following reason. A disordered cavity which is diffusive rather than ballistic also exhibits essentially chaotic dynamics. To ensure the observed Lorentzian line shape is due to scattering from the cavity walls and therefore governed by cavity geometry, it is imperative to be able to directly contrast chaotic versus non-chaotic behavior as in done here. To fully demonstrate that our results are intrinsic to geometry, we make detailed comparison with
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Fig. 3. Calculated magnetoconductance (x — 1) as a function of flux through the geometric area of the cavity for the (a) stadium and (b) circle shown as insets. The lineshape is Lorentzian for the ballistic stadium (solid squares) as well as for a stadium with strong surface roughness scattering (diamonds). The lineshape is more triangular for both the ballistic circle (triangles) and the circle with a weak smooth disordered potential (solid squares), but changes to Lorentzian for strong surface roughness scattering (diamonds). For the disordered potential, the total mean-free-path is approximately 5 times the diameter of the cavity. The similarity of lineshape between this calculation and the experiment (Fig. 1) is striking for both structures.
theoretical calculations. Baranger has performed a full quantum calculation for stadium and circle cavities nominally identical to those in our experiment. For the stadium cavity the top left lead is displaced slightly, a necessary modification since a cavity with strict inversion symmetry yields no magneto-resistance at all [53]! The results of the numerical computations are presented in Fig. 3. Note the line shape difference depicted in the thick line/solid squares traces-Lorentzian for the stadium and more triangular and cusp-like for the circle. These traces are obtained upon averaging over the Fermi energy to reduce the UCF contributions, and also include the effect of small angle scattering from impurities to simulate experimentally deduced residual scattering. The experimental small angle scattering length, 4a deduced below is roughly 4.5 um compared to the cavity dimension of 1 um. In addition to the qualitative line shape similarity between theory and experiment, quantitative agreement is obtained in the field scale: the full width at half maximum (FWHM), 0 I / 2 , for the stadium is 0.19 0 (theory) and 0.25 0O (experiment), and for the circle 0.18 (j)0 (theory) and 0.22 0O (experiment). Here <j>{/2 is the magnetic flux through the estimated area of the cavity and (j>0 is the flux quantum hc/e. The size of the WL peaks are also in reasonable agreement: AG ~ 0.13e2//* (theory) versus 0.2e2/h (experiment) for © Physica Scripta 2001
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the stadium, while AG ~ 0A5e2/h (theory) versus 0.1 le2//z for the circle. In addition to these traces intended for direct comparison with experiment, as points of reference the dashed line/open triangle trace in Fig. 3b represents the magneto-conductance for the circle without any disorder. Surprisingly the agreement with experiment is even better. For this trace, < £ 1 / 2 ~ 0 . 2 $ o and AG ~ 0.11 e2/h; furthermore, near ±^1/2 extra inflections is present similar to experiment! A final check is presented in the dotted line/open diamonds curves for both types of cavities. These curves demonstrate that roughened cavity walls yield chaotic, Lorentzian line shape behavior as expected. The theory of the line shape is, however, still attracting attention [54,55]. A most critical ingredient responsible for our success in distinguishing chaotic versus non-chaotic behavior is the high quality of GaAs/Al v Gai_ v As crystals. To observe the WL line shape difference, it is essential that transport through the cavities be ballistic and phase coherent. In other words, the elastic scattering length, /t, and phase coherence length, 4>, must both be much longer than the sample dimension, L. The requirement that the total elastic scattering length, 4, must be long, not just the transport scattering length which predominantly measures back scattering. For our cavities, the cavity dimensions are ~ 1 um. The quantum coherence length, 1$, exceeds 15um below 400 mK temperature. Using an estimate based on Nyquist noise from small energy electron-electron scattering as the dominant phase breaking mechanism [56,57], we find 1 > 50 um at 50 mK! This long 4/> may actually be cut off by residual spin-orbit scattering [57,58] Such a long 46 is significant for the following reason. The cavity perimeter to entrance/exit-opening ratio is roughly 7 in our cavities, so that a fully randomized trajectory will require 6 bounces off the cavity walls corresponding to 7 traversals across the cavity before escape. Therefore, when 4> > 1L, the dynamics relevant to even longer trajectories is essentially cut off by the openings. This scenario holds for the chaotic stadium cavities. On the other hand , for the circle cavities the cuspy behavior occurs in the low magnetic field region when B < 4 Gauss, and must arise from the contributions of special, non-randomized long trajectories. Given that the area of the cavities, estimated to be 0.81 um2 after accounting for depletion from the etched walls, yields a field of 51.4 Gauss for the passage of one flux quantum, trajectories contributing to the WL in the B < 4 Gauss region must be phase coherent for a minimum of 13 loops hugging the perimeter with a corresponding distance of 40 um! We also need a long total elastic scattering length, 4While the transport scattering length, 4, of our devices is ~ 17 urn > > L ~ 1 urn, 4 is determined more by the small angle scattering length, /sa < < le. To obtain a measure of /sa we perform an electron-focusing measurement for a row of cavities spaced 11.6 um apart depicted in Fig. 4 (top). The magneto-resistance is presented in Fig. 4 (bottom) for 400 mK and 4.2 K temperatures, solid and dashed lines, respectively. The classical ballistic transport feature corresponding to focusing is observable in both traces as a periodic resistance modulation with B, and results from the focusing of electron trajectories into caustics [59]. Focusing occurs at magnetic fields B where the cyclotron diameter, dc = 2vfmc/eB = 2vF/a»c is an "integral" fraction of the disPhysica Scripta T90
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istical properties of a closed chaotic system pertaining to the distribution of energy level spacings and the properties of wave functions can readily be tested in the transport through quantum dots. Early experiments demonstrated an unexpected feature of large fluctuations in the Coulomb Blockade peak height [28,60,61]. The observed height fluctuations propelled Jalabert, Stone, and Alhassid [39] to make a connection with a well-known result in nuclear physics, namely the Porter-Thomas distribution of resonance widths in the elastic scattering of neutrons from D=n[2vF^] complex nuclides [45], which is intimately related to the Wigner-Dyson distribution of energy level spacings [6,7]. These distributions have long ago been shown to be derivable from a random matrix theory (RMT) ansatz, which in a surmise put forth by Bohigas, Giannoni, and Schmit [3] is stipulated to correctly describe the quantum properties of isolated non-interacting systems exhibiting chaotic dynamics classically. These RMT results can also be derived by use of a powerful supersymmetry technique developed by Efetov [41], Prigodin [42], and colleagues [46]. In the case of the quantum dots, the presence of the Coulomb charging energy indicates that electron-electron interaction is present and it is not clear the RMT results should hold in all situations. It may be reasonable to assume that in some appropriate limit, e.g. high electron density where screening 1180 is effective, the interaction simply contributes a classical -600 -400 -200 0 200 400 600 charging energy given by Ec = e2/C where C is the capaciB (Gauss) tance of the quantum dot to its surroundings, and does not seriously affect the RMT distributions. The assumption Fig. 4. (Upper) Sample geometry for measuring electron-focusing contriproves to be largely correct in the GaAs/Al v Gai_ x As quanbution to the magneto-resistance. (Lower) Magnetoresistance for sample tum dots studied regarding peak height statistics. However, depicted in (a). Note the periodic modulation observable in both the 400 mK (solid) and 4.2 K. (dashed) curves. This electron-focusing classical ballistic we will present suggestive evidence that deviations from effect enables us to deduce a small angle scattering length, lsa, of ~ 4.5 um. the RMT universal distributions are observable in our small (< 0.25 urn) quantum dots. Recent experimental and theoretical work on peak spacing fluctuations indicate marked tance between the openings in adjacent cavities, D, i.e., deviations from the RMT result as well, with a distribution D = ndc. Here vF is the Fermi velocity, a»c = eB/mc the which is Gaussian rather than Wigner-Dyson [30-33,62-64]. cyclotron frequency, and n an integer. When this condition The specific predictions of theory which we test pertain to is met, a maximum number of electrons are able to return. the distribution of these Coulomb blockade peak heights /sa is deduced from the amplitude of electron-focusing at B = 0, corresponding to the RMT Gaussian Orthogonal oscillations relative to full resistance assuming an Ensemble (GOE), and at 5 ^ 0 , corresponding to the exponential reduction with distance traveled (nD/2). This Gaussian Unitary Ensemble (GUE). In the lattej case, estimate yields /sa of roughly 4.5 um which exceeds the magnetic field strength, B, is required to exceed some L ~ 1 um. The long 1$ and /sa of our samples in which correlation field, Bc, which characterizes the transition from the 2DEG resides only 560 A below the top surface likely GOE to GUE statistics. According to theory [39,42], in result from the special doping geometry. The silicon dopants the single level tunneling limit of thermally broadened conare placed in a doping well which is a thin (~ 20 A) quantum ductance peaks, the B = 0 distribution is given by: well embedded in AlGaAs surroundings. The advantage in doing so is that the donor levels are only slightly below the Fermi energy. Consequently the donor configuration (1) (B=0) na e is mobile down to ~ 20 K temperature rather than frozen with a square-root singularity near zero. Here, a is related to in near 77 K, thereby allowing better screening out of donor the Coulomb blockade peak conductance, Gmax, according potential spatial fluctuations which cause scattering of to: [66] the 2DEG. K-
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rLrR 3. Non-gaussian distribution of coulomb blockade peak heights in quantum dots In nearly isolated systems fabricated in GaAs/Al x Gai_ x As microstructures, any residual disorder or lithographic imperfections will render the dynamics chaotic on the long trapping time-scale before eventual escape into external leads/reservoirs takes place. Therefore predictions on statPhysica Scripta T90
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Quantum Chaos in GaAs/AlxGa\-xAs
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level spacing. In a magnetic field stronger than the correlation field, the breaking of time-reversal symmetry reduces the number of nearly zero values of Gmm. Nevertheless, the distribution [39,40] is still non-Gaussian and peaked near zero: i W o ) = 44K0(2a) + Ki(2a)] e"
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where K„ are the modified Bessel functions. For GaAs/ AlxGai-jfAs quantum dots, F can readily be tuned to be below 3jnV (corresponding to F ~ 34 mK) while typically the level spacing A > 200 nV ( F - 230mK). The required conditions of the thermally broadened single level tunneling limit is therefore accessible at dilution refrigerator temperatures ~ 70 mK. In contrast, in a typical metallic dot of size 500 A, A - 0.1 pV (~ 1.1 mK) and many levels are accessed. This leads to a convolution of independent single level distributions resulting in a Gaussian distribution of peak heights in accordance with the central limit theorem. In order to successfully observe the predicted distributions we must fully access the single level tunneling limit. The decoupling of the electrons from the lattice at low temperatures renders it difficult to reduce the electron temperature below the 50-100 mK range [61,67] even after strongly filtering any noise (thermal or pickup) from the measurement system. To fully access the single level limit, A < 5 kT is invariably needed, requiring A > 40 p,V. For GaAs/Al x Gai™ x As, this implies small quantum dots of order 0.25 pm or smaller in its largest dimension. In Fig. 5(a) we show an electron micrograph of the metal gate pattern for four dots in series used in our experiment. Each dot is roughly 0.3 p,m x 0.35 \im in lithographic size but is reduced to below 0.25 pm x 0.25 pm after gating. In our experiment, individual dots were separately measured rather than the whole series of four dots. In Fig. 6 we show a representative trace of Coulomb blockade peaks at B = 0 for F = 75 mK (bottom) and 660 mK (top) temperatures. Note the missing peaks at the gate voltages -733, -753, and -762 mV in the 75 mK curve which are observable at the higher temperature of 660 mK. The large difference in height of adjacent peaks © Physica Scripta 2001
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Fig. 6. A typical trace showing successive Coulomb blockade conductance peaks versus the center gate voltage, Vg. B = 0 and T = 75 mK (lower trace) or T = 660 mK (upper trace, displaced by 2 unite). Note that three peaks are missing out of seven, but they emerge at higher temperature. The slight shifting in peak positions is discussed in the text.
and the many small peaks are our primary experimental observation, and a qualitative demonstration the feature of a prevalence of small peaks predicted by RMT in Eq. (1). To demonstrate that we are fully in the single level tunneling limit of temperature broadened peaks, in Fig. 5(b) and 5(c), we plot G~aX versus F a n d the line shape of a representative peak fitted to the theoretical cosh~2[(J5b -yeVg)/2kT\, respectively. In Fig. 5(b) the roughly linear dependence of G~*x on F at low temperatures indicates that we are clearly in the single level regime (see Eq. (2)). In fact we find this behavior in all of the eight peaks we studied in detail. We follow the evolution of the peak height with magnetic field for each peak. Figure 7 shows magnetic field traces of GmskiL for four representative peaks. From the fastest variation we estimate a correlation field, Bc, of the order of 500 Gauss, somewhat larger then the theoretical value [39,40] of 200 Gauss. Panels (c) and (d) depict two peaks Physica Scripta T90
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A. M. Chang The experimental distribution in Fig. 8(b) for B ^ 0 appears slightly higher for the smallest height data point compared to the theoretical dashed curve. Even though the difference is within the error bars, it is suggestive. Intrigued by the possibility of deviation from the RMT theoretical result, we further split this lowest bin into two, producing the histogram in Fig. 9. The probability of small peaks continues to increase for height values approaching zero, in stark contrast to the RMT result of a maximum at ~ 0.025 e2/h\ This excess of small peaks is related to the discussion of Fig. 7d where certain peaks which are small when B — 0 remain small for large regions of B > 0. In fact, recent theoretical calculations aimed at accounting for electron-electron interactions appear to show exactly this trend [65]. The influence of interaction certainly deserves further investigation.
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which are nearly zero at B = 0. The behavior in (d) where the peak remains small for large stretches of B and only occasionally increases to a large value is observed in roughly 1 /3 of peaks which start near zero. This type of behavior is not expected in the RMT picture where fluctuations should occur on the scale of Bc. The fact that the height remains small for large regions of B suggests an enhancement of the small peak probability above the RMT prediction (Eq. (3)) for B ^ 0 and may arise from electron interaction effects. Our sample set contains 72 independent peaks, yielding 72 peak height values for 5 = 0, and 216 values for B ^ 0. In the latter case height data are taken at three different magnetic fields well separated by several fic's to triple the data set. The resulting distributions are presented as histograms in Fig. 8. The distributions are normalized to unit area as for a probability density. Both the B = 0 (a) and B ^ 0 (b) distributions are strongly non-Gaussian, and clearly peak toward zero values. In the 5 = 0 case, nearly 1.3 of the peaks fall in the lowest bin: 23 out of 72 peaks are less than 0.005 e2/h compared to a mean of ~ 0.024 e2/h. In contrast, for B / 0 only 43 out of 216 peaks are this small. Figure 8 indicates that there is a difference between the two distributions for low values as is born out by the Kolmogorov-Smirnov statistical test. The mean decay width needed for comparing to theory is not measured experimentally and is therefore a fitting parameter. This width should be nearly independent of B\ thus we introduce a single scale parameter and fit simultaneously to the B — 0 and 5 ^ 0 data. Figure 8 shows a fit to the data using both the theory for constant pincher transmission [Eqs. (1) and (3) (solid)] and this theory averaged over a variation of the pincher transmission by a factor of 3.5 (dashed). The similarity of the two curves shows that the variation in our pincher transmission can be neglected. Physica Scripta T90
4. Pairing effect in ultra-small quantum dots The difference between an even and odd-numbered finite Fermion system, known as the even-odd parity effect, is a distinct feature reflecting the unique behavior of fermionic particles in the presence of both orbital and spin degrees of freedom [68,69]. This parity effect is expected to appear in quantum dots where electrons can be added one by one. The peak spacing fluctuation in CB peaks provides unique
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(0.01 e'/h) Fig. 8. Histograms of conductance peak heights for (a) B = 0 and (b) B ^ 0. Data are scaled to unit area; there are 72 peaks for B = 0 and 216 peaks for B ^ 0; the statistical error bars are generated by bootstrap re-sampling. Note the non-Gaussian shape of both distributions and the strong spike near zero in the B = 0 distribution. Fits to the data using both the fixed pincher theory (solid) and the theory averaged over pincher variation (dashed) are excellent. The insets show fits to xl(x) — a more Gaussian distribution — averaged over the pincher variation; the fit is extremely poor. © Physica Scripta 2001
Quantum Chaos in GaAs/AlxGa\~xAs
2
4 6 Gmox (0.01e 2 /h)
8
10
Fig. 9. The B # 0 distribution of Fig. 9b replotted with the lowest bin farther split into two. The resulting histogram shows that the probability for small peaks continues to increase toward zero, in contrast to RMT predictions. This trend is likely a result of electron-electron interaction
information about single particle energy levels, many-body interaction effects and the parity of electron numbers. In irregularly shaped dots without special symmetries the CB peaks are expected to be paired as a consequence of parity with smaller spacings in the odd electron valleys and larger spacings in the even valleys, reflecting the dot spin status which is 1/2 h for an odd number of electrons and paired to zero for an even number (Fig. 10). The importance of clear observations of effects reflecting the spin status pertains directly to the desire to couple dots together both for fundamental physics reasons (competition between the Kondo effect and indirect exchange interaction in the two impurity model [70]) and for technological reasons in the implementation of the double dot system as prototype qu-bits in quantum computation [71]. The clear observation of the pairing effect as well as the elucidation of the necessary conditions are therefore of central importance. The absence of clear signatures thus far has posed a puzzle to experimentalists and theorists alike [30-32]. Most experiments so far have been performed in devices with relatively high rs (ratio of average Coulomb to Fermi energy) ~ 1. Recently, we have succeed in observing clear pairing features attributable to the even-odd effect in small GaAs/Al v Gai_ x As lateral quantum dots, some with the smallest rs value measured to date [47]. These features are observed in the CB peak spacing, peak height as well as spin. We find that a quantum dot with smaller rs shows more pronounced peak pairing than high rs dots. Furthermore, qualitatively different behaviors in peak spacings are observable depending on the coupling strength between the dots and leads. Three devices are fabricated in different rs regimes. Device 1 containing the number of electrons, N « 10 and device 2 containing N « 40 are made from a crystal of density, n = 3.5 x 10" cnr 2 (r s = 0.93) while device 3 with N «10-20 from a high density crystal of n — 9 x 10" cm~2(rs = 0.58). The geometry and size of devices 1 and 2 (lithographic diameters of 160nm and 230 nm, respectively) were chosen carefully to maximize functionality in spite of their small sizes (Fig. 10(a)). The geometry of device 3 with a lithographic diameter of 170 nm enables us to control N between © Physica Scripta 2001
Microstructures
23
10-20 by the application of 6 independent gate voltage settings. In our experiment, clear signature of an even-odd pairing can be observed in the CB peak spacing when the coupling between the quantum dot and leads is strong in all devices. On the other hand, as the dot-lead coupling is reduced a marked difference emerges between the high rs devices 1 and 2 and the low rs device 3. In device 1 when the dot is nearly open, two broad peaks are observed as shown in Fig. 11(a). A decrease of the dot-lead coupling resolves more peaks and pairing is clearly present in Fig. 11(b). However, when the coupling is further reduced, the pairing is again no longer visible (Fig. 11(c)). Even with this small size dot, there is no clear signature of even-odd effect in the weak tunneling regime. The spin status can be deduced in the valleys 3 and 5 which exhibit the Kondo resonance [72,73] as a signature of unpaired single electron spin as well as reverse temperature dependence, i.e. an increase in conductance with decreasing temperature as shown in Fig. 11(b). Similar results are obtained for device 2. In stark contrast, in the small rs high density device 3 peak pairing is preserved in all regimes of the dot-lead coupling as evidenced by the ubiquitous pairing behavior which can persist for at least 10 peaks in succession. We have now pushed our lithography capabilities even further and at present our dots are among the smallest available both in terms of lithographic size and actual conducting
w> ***W*.
Uc AE+Uc
Gate Voltage Fig. 10. Scanning electron micrograph of device 1 (Top). Schematic diagram of the peak positions as a function of gate voltage (Bottom). The narrow period corresponds to the change from odd to even numbers by adding an electron with the opposite spin into the same spin degenerate state, and the broad period to the change from even to odd numbers, occupying different dot energy levels. Physica Scripta T90
24
A. M. Chang (a)
(b)
0.6
O
Gate Voltage (V) Fig. 11. Evolution of the peak structure for different dot-lead coupling in device 1 measured at T = 300 mK. From (a) to (c), the coupling is decreased gradually. In (b), the dotted line is the T = 900 mK trace, illustrating the temperature dependence of Kondo and non-Kondo valleys. By closing the dot, the peak spacing pairing is destroyed even with this small size dot. Inset in (c): Comparison of peak spacing in (b) (filled circle) and (c) (unfilled square) in each valley (t-axis).
sizes, with the added advantage of multiple independent gates. In fact, in terms of lithography, we have succeeded in placing two dots side by side in a space of 380 nm by 120nm, where each dot is of dimension 120nmx 120nm [48], Thus far we have succeeded in making one of the two dots functional, in which Coulomb Blockade oscillations associated with the addition of single electrons onto the dot are observable up to 77 K in temperature, as shown in Fig. 12. These results should help pave the way both for the investigations of fundamental new physics of coupled dot systems and for their implementation as prototype quantum qu-bits in quantum computation which we plan on pursuing in the near future.
References 1. Gutzwiller, M. C , "Chaos in Classical and Quantum Mechanics", (Springer-Verlag, New York, 1990). 2. Berry, M. V., Proc. R. Soc. Lond. A 413, 183 (1987). 3. Bohigas, O., Giannoni, M.-J. and Schmit, C , Phys. Rev. Lett. 52, 1 (1984). Physica Scripta T90
V0(V) Fig. 12. Scanning electron micrograph of a SET with 120 nm single dot size as measured from the inside diameter for the dot. (b) CB oscillations at 4.2 K (lower trace) and 77 K (upper trace). Bias gate voltages are slightly different for optimum CB peaks in the two traces. The offset in y axis is changed and a portion of linear background conductance is subtracted for the 77 K trace.
4. Pluhar, Z., Weidenmuller. H. A., Zuk, J. A. and Lewenkopf, C. H., Phys. Rev. Lett. 73, 2115 (1994). 5. Porter, C. E., "Statistical Theory of Spectra: Fluctuations", (Academic Press, New York, 1965). 6. Wigner, E. P., Gatlinberg Conf. on Neutron Physics, Oak Ridge National Lab Rept. No. ORNL2309, p. 59. 7. Dyson, F. J., J. Math Phys. 3, 140 (1992). 8. Bayfield, J. E. and Koch, P. M., Phys. Rev. Lett. 33, 258 (1974). 9. Simons, B. D. Hashimoto, A., Courtney, M., Kleppner, D. and Alt'shuler, B. L., Phys. Rev. Lett. 71, 2899 (1993). 10. Main, J., Wiebusch, G., Holle, A. and Welge, K. H., Phys. Rev. Lett. 57, 2789 (1986). 11. Koch, P. M. and van Leeuwen, K. A. H., Phys. Rep. 255, 289 (1995). 12. Chang, A. M., Baranger, H. U., Pfeiffer, L. N. and West, K. W., Phys. Rev. Lett. 73, 2111 (1994). 13. Chang, A. M., Baranger, H. U., Pfeiffer, L. N., West, K. W. and Chang, T. Y., Phys. Rev. Lett. 76, 1695 (1996). 14. Marcus, C. M., Rimberg, A. J., Westervelt, R. M., Hopkins, R. F. and Gossard, A. C , Phys. Rev. Lett. 69, 506 (1992). 15. Chan, I. H., Clarke, R. M., Marcus, C. M., Campman, K. and Gossard, A. C , Phys. Rev. Lett. 74, 3876 (1995). 16. Folk, J. A.et at, Phys. Rev. Lett. 76, 1699 (1996). 17. Taylor, R. P. et al., Phys. Rev. Lett. 78, 1952, (1997). 18. Bird, J. P. Ishibashi, K., Aoyagi, Y„ Sugano, T. and Ochiai, Y., Phys. Rev. B 50, 18678 (1994). © Physica Scripta 2001
Quantum Chaos in GaAs/AlxGa\-xAs 19. Berry, M.J., Baskey, J.H., Westervelt, R.M., and Gossard, A.C., Phys. Rev. B 50, 8857 (1994). 20. Stockmann, H.J., and Stein, J., Phys. Rev. Lett. 64. 2215 (1990). 21. Richter, A., "Emerging Applications of Number Theory", (Edited by D. A. Hejhal, J. Friedman, M. C. Gutzwiller and A. M. Odlyzko) (Springer-Verlag, New York 1998), vol. 109, p. 479. 22. Sridhar, S., Phys. Rev. Lett. 67. 785 (1991). 23. Chinnery. P. A. and Humphrey, V. F., Phys. Rev. E 53, 272 (1996). 24. Jalabert, R. A., Baranger, H. U. and Stone, A. D., Phys. Rev. Lett. 65, 2442 (1990). 25. Baranger, H. U„ Jalabert, R. A. and Stone. A. D., Chaos 3, 665 (1993). 26. Keller, M. W., Millo, O., Mittal, A., Prober, D. E. and Sacks, R. N„ Surf. Science 305, 501 (1994). 27. Weiss, D. et al., Phys. Rev. Lett. 70. 4118 (1993). 28. Kastner, M. A., Rev. Mod. Phys. 64, 849 (1992). 29. Levy, L. P., Reich, D. H., Pfeiffer, L. N. and West, K. W.. Physica (Amsterdam) 189B, 204 (1993). 30. Sivan, U. et al., Phys. Rev. Lett. 77, 1123 (1996). 31. Simmel, F., Heinzel. T. and Wharam, D.. Europhys. Lett. 38, 123 (1997). 32. Patel, S. R. et al., Phys. Rev. Lett. 80, 4522 (1998). 33. Luescher, S.. Heinzel, T., Ensslin, K.. Wegscheider, W. and Bichler, M., condmat/0002226. 34. Mailly, D., Chapelier, C. and Benoit, A., Phys. Rev. Lett. 70. 2020 (1993). 35. Alhassid. Y. and Lewenkopf. C. H., Phys. Rev. B 55. 7749 (1997). 36. Bruus, H., Lewenkopf, C. H. and Mucciolo, E. R., Phys. Rev. B 53, 9968 (1996). 37. Baranger, H. U. and Mello. P. A., Phys. Rev. Lett. 73, 142 (1994); Phys. Rev. B 51, 4703 (1995). 38. Jalabert. R. A., Pichard, J.-L. and Beenakker, C. W. J., Europhys. Lett. 27, 255 (1994). 39. Jalabert, R. A., Stone, A. D. and Alhassid, Y„ Phys. Rev. Lett. 68, 3468 (1992). 40. Bruus, H. and Stone. A. D„ Phys. Rev. B 50, 18275 (1994). 41. Efetov, K. B., Phys. Rev. Lett. 74, 2299 (1995). 42. Prigodin, V. N., Efetov. K. B. and lida, S., Phys. Rev. Lett. 71, 1230 (1993). 43. Ullmo, D„ Richter, K. and Jalabert, R. A., Phys. Rev. Lett. 74, 383, (1995). 44. Jensen, R. V., Nature 373, 16 (1995). 45. Porter, C. E. and Thomas, R. G., Phys. Rev. 104, 483 (1956).
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46. Mucciolo, E. R., Prigodin, V. N. and Altshuler. B. L., Phys. Rev. B 51, 1714 (1995). 47. Jeong, H. J., Chang, A. M., Melloch, M. R. and Chang, T. Y„ submitted to Phys. Rev. Lett. 48. Jeong, H. J., Chang, A. M., Melloch, M. R. and Chang, T. Y„ submitted to Appl. Phys. Lett. 49. Landauer, R„ IBM J. Res. Dev. 1, 223 (1957). 50. Baranger, H. U., Jalabert, R. A. and Stone, A. D., Phys. Rev. Lett. 70, 3876 (1993). 51. Jensen, R. V., Chaos 1, 101, (1991). 52. Lee, Y„ Faini, G., Mailly, D„ Phys. Rev. B 56. 9805 (1997). 53. Baranger, H. U. and Mello, P. A., Phys. Rev. B 54, R14297 (1996). 54. Ouchterlony, T. et al., Eur. Phys. J. B 10, 361 (1999). 55. Akis. R., Ferry, D. K., Bird, J. P. and Vasileska, D.. Phys. Rev. B 60, 2680 (1999). 56. Altshuler, B. L, Aronov, A. G. and tChmelnitskii, D. E„ J. Phys. C 15, 7367 (1982). 57. Kurdak, C , Chang, A. M„ Chin, A. and Chang, T. Y., Phys. Rev. B 46, 6846 (1992). 58. Dresselhaus, P. D., Papavassiliou, C. M. A. and Wheeler, R. G., Phys. Rev. Lett. 68, 106 (1992). 59. van Houten, H. et al.. Europhys. Lett. 5, 721 (1988). 60. Scott-Thomas, J. H. F„ Field, S. B., Kastner, M. A., Antoniadis, D. A. and Smith, H. I., Phys. Rev. Lett. 62. 583 (1989). 61. Meirav, U.. Kastner. M. A. and Wind. S. J.. Phys. Rev. Lett. 65, 771 (1990). 62. Baranger, H. U„ Ullmo, D., Glazman, L. I„ unpublished. 63. Halperin. B. I., unpublished. 64. Alhassid, Y., Jacquod, Ph., Wobst, A., Phys. Rev. B 61, R13357 (2000). 65. Alhassid, Y. and Wobst, A., condmat/0003255. 66. Beenakker, C. W. J., Phys. Rev. B 44, 646 (1991). 67. Foxman, E. B. et al., Phys. Rev. B 50, 14193 (1994). 68. Tuominen, M. T.. Hergenrother, J. M., Tighe, T. S. and Tinkham, M., Phys. Rev. Lett. 69, 1997 (1992). 69. Eiles, T. M„ Martinis, J. M. and Devoret, M. H., Phys. Rev. Lett. 70, 1862 (1993). 70. Georges, A. and Meir, Y., Phys. Rev. Lett. 82, 3508 (1999). 71. Loss, D. and DiVincenzo, D. P., Phys. Rev. A 57, 120 (1998). 72. Goldhaber-Gordon, D. et al.. Nature (London) 391, 156 (1998). 73. Cronenwett, S. M., Oosterkamp, T. H. and Kouwenhoven. L. P., Science 281, 540 (1998).
Physica Scripta T90
Physica Scripta. T90, 26-33, 2001
Ground State Spin and Coulomb Blockade Peak Motion in Chaotic Quantum Dots J. A. Folk1-2, C. M. Marcus1, R. Berkovits34'5, I. L .Kurland3, I. L. Aleiner 6 and B. L. Altshuler 34 1
Department of Physics, Harvard University, Cambridge. MA 02138, USA Department of Physics, Stanford University, Stanford, CA 94305, USA 3 Physics Department, Princeton University, Princeton, NJ 08544, USA 4 NEC Research Institute, 4 Independence Way, Princeton. NJ 08540 5 Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan 52900. Israel 6 Department of Physics. SUNYat Stony Brook, Stony Brook, NY 11794, USA 2
Received October 25, 2000
Abstract We investigate experimentally and theoretically the behavior of Coulomb blockade (CB) peaks in a magnetic field that couples principally to the ground-state spin (rather than the orbital moment) of a chaotic quantum dot. In the first part, we discuss numerically observed features in the magnetic field dependence of CB peak and spacings that unambiguously identify changes in spin S of each ground state for successive numbers of electrons on the dot, N. We next evaluate the probability that the ground state of the dot has a particular spin 5, as a function of the exchange strength, J, and external magnetic field strength, B. In the second part, we describe recent experiments on gate-defined GaAs quantum dots in which Coulomb peak motion and spacing are measured as a function of in-plane magnetic field, allowing changes in spin between N and N + 1 electron ground states to be inferred.
1. Introduction In the absence of interactions, electrons populate the orbital states of a quantum dot or metallic grain in an alternating sequence of spin-up and spin-down states, in accordance with the Pauli picture. In this case, the total spin of the dot is either zero, when the number of electrons N is even, or one half at odd N. No higher spin states appear. It is well known that in the presence of interactions, this simple scheme can be violated. Such deviations from simple even-odd filling may appear as Hund's rules in atomic physics, or be as extreme as complete spin polarization leading to ferromagnetism by the mechanism of the Stoner instability. By measuring electron transport through a weakly coupled quantum dot at low temperature and bias voltage (compared to the quantum level spacing) one can effectively study differences in ground state (GS) properties of the dot at successive electron number, providing a means of investigating the actual filling scheme [1], A standard experimental approach [1] is to measure conduction through the dot via two tunneling leads as a function of a voltage, Vg, applied between the dot and a third electrically isolated electrode, known as a gate. Because the dot GS energy, EN, at a fixed electron number is influenced by Vs, the gate can be used to set the number of electrons on the dot. The fact that a large Coulomb energy is needed to add a single electron to the dot typically suppresses conduction through the dot in the tunneling regime, where whole charges must tunnel for conduction to occur. This effect is known as the Coulomb blockade (CB). However, at specific values of gate voltage, denoted V^N\ where the conPhysica Scripta T90
dition Eiv-\(Vg) = EN(Vg) is satisfied, conductance increases dramatically, clearly marking this degeneracy condition in an experimental trace. The position, ViN\ of the Nth peak in the conductance is proportional to nN = EN(0)— EN~\(0). Accordingly, the distance between successive peaks is A^ = nN — fiN_i- (We take the constant of proportionality converting gate voltage to dot energy to be unity for the theoretical discussion; experimentally, this constant can be readily measured, for instance, by comparing the influence of bias and gate voltages.) In the last few years the GS spin of a variety of nanostructures, including metallic grains [2], semiconducting quantum dots [3,4,5,6], and carbon nanotubes [7,8], have been investigated using CB peak motion in a magnetic field. If one neglects the magnetic field coupling to the orbital degrees of freedom, one expects the field to manifest itself only through the Zeeman splitting, resulting in a shift of the GS energy by gfi^SB, where ^ B is the Bohr magneton and S denotes the GS total spin. For 2D quantum dots, orbital coupling can be strongly suppressed—though not eliminated entirely—by orienting the field strictly in the 2D plane. This is the experimental approach that will be described in Section 3 of this paper. On the other hand, for ultrasmall grains and nanotubes, it is reasonable to ignore orbital coupling for any field direction since for practical magnetic fields the total flux cutting the structure is much less than the quantum of flux. For semiconducting quantum dots [3,4,5,6] the Zeeman splitting for S — 1 /2 becomes comparable to the mean single electron level spacing denotes nearest neighbor lattice site, a]a is an creation operator of an electron at site / with spin '
dSj
dpi
dpj_ dpj_ \Bsi
- ^
0.0
(5)
dpi/
where st is a coordinate along the boundary and pt is the projection of the momentum onto the boundary at the ith bounce. The total monodromy matrix for a trajectory with n bounces is given by M = lloM' + 1 '' where the monodromy matrix for a single trajectory arc between two bounces M'+lj is available in analytical form for polygonal billiards in magnetic fields. With the help of the monodromy Physica Scripta T90
Figure 2 shows the results of quantum mechanical (QM) and SC calculations for the transmission and reflection coefficients, T, R of a triangular billiard of Fig. 1. Note that because of the time-reversal invariance the conductance (and
Fig. 2. (a) Quantum mechanical transmission probability for a triangular billiard of Fig. 1. (b) Semi-classical transmission probability, (c), (d) Semiclassical reflection probability respectively from the tip and the base of the triangle. Darker regions correspond to the lower probabilities. © Physica Scripta 2001
Conductance
of a Ballistic Electron Billiard in a Magnetic Field: does the Semiclassical
Approach Apply?
39
a) ~,o.s
0.00
0,50
1.00
1.50
2.00
Areaft?)
10 Uaigth(L)
20
fig. 3. (a) area and (b) length difference distributions for pairs of transmitted and reflected trajectories from the base and the tip of a triangle of Fig. 1. The cyclotron radius rc = 0.59 L.
differences A4 ^ = 4 — 4', AA^ — A^ — Aa> for the pairs of classical trajectories connecting the entrance and the exit leads that determine the oscillations in the transmission/ reflection coefficients as a function of both the Fermi wave vector and magnetic field. For an arbitrary billiard structure these areas and length distributions are different because they are given by different sets of different classical trajectories. This is illustrated in Fig. 3 where probability distributions P(Al) and P(AA) are plotted for the transmitted and reflected trajectories injected from the base and the tip of the triangle for a representative value of magnetic field with the cyclotron radius rc = 0.59L, where L is the size of the triangle. Because P(Al) and P{AA) are essentially different for transmission and reflection and for different current directions, the corresponding SC transmission and reflection probabilities are also different as seen in Fig. 2. This obviously makes it impossible to provide a meaningful interpretation of the calculated conductance/ reflectance at non-zero magnetic field within the framework of a standard SC approach utilized here. Note that such an attempt has been made in Ref. [13] where a non-universal feature in the conductance of a stadium billiard [1] have been attributed to a particular pair of two specific trajectories connecting the leads. In light of the above results this interpretation seems to be ambiguous because these particular trajectories do not exist for opposite direction of current at same magnetic field (or for the same direction of current but in opposite field) such that the apparent agreement with the experiment may be rather superficial. T C n m = [Cml = 22W*\2\Hx\2 Finally, we apply the SC approach to calculate the WL a. (V) corrections in a triangular billiard. The semiclasical theory of Baranger et al. [10] predicts different lineshapes of the averaged magnetoresistance peak near B = 0, namely, the where the summation is performed over all families of Lorentzian lineshape for chaotic cavities and the linear trajectories a, the factor ax is related to the density of one for regular billiards. These predictions are essentially trajectories; H"m is related to the electron diffraction at based on a difference in universal long-tail behavior of the lead mouth and describe an angular electron distribution enclosed areas distributions in chaotic and regular cavities [12,13] (similar expression can be written for the reflection (exponential vs algebraic decay). Although the above theory coefficient). The first sum in Eq. (7) corresponds, in the limit is widely used in interpretation of experimental data (see e.g. of a large mode number, to the classical transmission [19] for a brief review of related experiments), we are not coefficient; the second sum represents a quantum correction aware of any detailed SC calculation for a structure of a due to interference between paths a and a! [9]. It follows from given geometry and its comparison to the corresponding the above expression that it is mainly the length and area exact QM results. Figure 4 shows numerical QM calcu-
the reflectance) of a two-terminal device has to be the same irrespective of the direction of the current [16]. In addition, the current conservation requires that T + R = N where N is the number of incoming channels in the leads. One of the most striking findings is that none of the above requirements is satisfied in the SC calculations, Fig. 2(b)-(d). (Note that because of the symmetry of the structure under consideration both SC transmission probabilities for different directions of the current are equal. In addition, the transmission and reflection coefficient are the same for +B and —B). Besides, none of the SC transmission and reflection probabilities reproduce the QM result (Fig. 2(a)). It is worth to stress that in the previous studies the SC and QM transmission/reflection amplitudes did not always show a good correspondence as well as the unitarity of the SC scattering matrix were not conserved. However, frequencies of the oscillations (i.e. the Fourier transform over kp) were typically in a very good quantitative agreement [12-14]. In contrast, in the present case of a non-zero magnetic field, the fluctuation pattern in the reflection probability for electron injected from the tip of triangle, Fig. 2(c) is qualitatively different from that one for the case of injection from a base of the triangle, Fig. 2 (d) as well as from the transmission coefficient T =\ — R, Fig. 2(b) (in Fig. 2 we limit ourselves to the case of one propagating mode in the leads). To understand an origin of the above discrepancy let us write a transmission coefficient by combining Eq. (2),(3), in the form [9,12,13]
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Physica Scripta T90
40
T. Blomquist and I. V. Zozoulenko of the system and are not necessary related to the character of the electron dynamics (chaotic vs regular). Our finding also raise the question of to what extend one can rely on numerous predictions for statistical properties of the conductance oscillations including the autocorrelation function [9], WL lineshapes [10] and the fractal conductance [11], which are essentially based on the standard SC approach. Thus, in order to restore required symmetry with respect to the current flow direction and magnetic field and to achieve an agreement with QM one has to go beyond the standard SC theory. One of the possible ways is to include non-classical "ghost paths" arising due to diffractive reflection off the lead mouths [12]. Work is currently in progress to incorporate this effect into the theory.
Fig. 4. The averaged reflection (a) and conductance (b) of the triangular billiard shown in the inset. Solid and long dashed lines correspond to exact QM and SC conductance respectively Short dash and dash-dotted lines are SC reflectance from the base and the tip of the triangle respectively. Averaging is performed in the energy interval when the number of modes in the leads varies from 1 to 3.
lations of the average conductance (G) which, as expected, display a positive differential conductance in the vicinity of B = 0. Contrary to this, the average SC conductance (Gsc> stays almost unchanged. Besides, the behavior of the SC reflectance 2 0 bounces) play only a minor role. To conclude, we demonstrate that application of the standard SC theory to the description of a transport through quantum dots with leads leads to contradictory results. Namely, SC conductance/reflectance depends on the current direction, SC current in not conserved and SC and QM calculation are in apparent disagreement. Besides, the full SC calculations of the WL corrections in the triangular billiard are apparently different from the general predictions for the WL lineshape for the regular and chaotic cavities [10]. This supports the previous conclusions [19,20] that the shape and the magnitude of the WL corrections can be strongly sensitive to the geometry-specific, non-universal features
Physica Scripta TOO
Acknowledgements We thank John Delos and Andrew Sachrajda for stimulating discussions. This work was supported by Swedish Natural Science Research Council (NFR) and Swedish National Graduate School in Scientific Computing (NGSSQ.
References 1. Marcus, C. M. et al., Phys. Rev. Lett. 69, 506 (1992). 2. Bird, J. P. et al, Phys. Rev. B 52, 8285 (1995). 3. Persson, M. et al., Phys. Rev. B 52, 8921 (1995); K.-F. Berggren et al, ibid. 54, 11612 (1996). 4. Zozoulenko. I. V. etal. Phys. Rev. B 55, R10209 (1997); Zozoulenko. I. V. and Berggren, K.-F., ibid. 56, 6931 (1997). 5. Sachrajda, A. S. et al, Phys. Rev. Lett. 80, 1948 (1998). 6. Zozoulenko, I. V. et al, Phys. Rev. Lett. 83, 1838 (1999). 7. For a review, see e.g., Beenakker, C. W. J., Rev. Mod. Phys. 69, 731 (1997). 8. Zozoulenko, I. V. et al, Phys. Rev. B 58, 10594 (1998). 9. Jalabert, R. A. et al, Phys. Rev. Lett. 65, 2442 (1990). 10. Baranger, H. U. et al, Phys. Rev. Lett. 70, 3876 (1993). 11. Ketzmerick, R., Phys. Rev. B 54, 10841 (1996). 12. Schwieters, C. D., Alford, J. A. and Delos, J. B„ Phys. Rev. B 54, 10652 (1996). 13. Wirtz, L., Tang, J.-Z. and Burgdorfer. J., Phys. Rev. B 56, 7589 (1997); ibid. 59, 2956 (1999). 14. Blomquist, T. and Zozoulenko, I. V., Phys. Rev. B 61, 1724 (2000). 15. Christensson, L. etal, Phys. Rev. B 57, 12 306(1998); Boggild, P. etal, ibid. 57, 15 408 (1998). 16. see, e.g., Datta, S.. "Electronic Transport in Mesoscopic Systems", (Cambridge University Press, Cambridge, 1995). 17. Zozoulenko, 1. V., Maao, F. A. and Hauge. E. H., Phys. Rev. B 53, 7975 (1996); ibid. 53, 7987 (1996); ibid. 56, 4710 (1997). 18. Brack, M. and Bhaduri, R. K., "Semiclassical Physics", (AddisonWesley, Reading, Mass., 1997). 19. Ouchterlony, T. et al, Eur. Phys. J. B 10, 361 (1999). 20. Akis, R. et al, Phys. Rev. B 60, 2680 (1999).
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Physica Scripta. T90, 41-49, 2001
Semiconductor Billiards - a Controlled Environment to Study Fractals R. P. Taylor12*, A. P. Micolich12, R. Newbury2, T. M. Fromhold3, A. Ehlert2, A. G. Davies4, L. D. Macks4, C. R. Tench3, J. P. Bird5, H. Linke2, W. R. Tribe4, E. H. Linfield4 and D. A. Ritchie4 'Department of Physics. University of Oregon. Eugene, OR 97403-1274, U.S.A. School of Physics, University of New South Wales, Sydney NSW, 2052. Australia 3 School of Physics and Astronomy, Nottingham University, Nottingham NG7 2RD, U.K. 4 Cavendish Laboratory, University of Cambridge, Madingley Rd, Cambridge CB3 OHE, U.K. 5 Center for Solid State Research, Arizona State University. Tempe, AZ 85287-6206, U.S.A. 2
Received May 26, 2000
PACS Ref: 05.45Df, 73.23.Ad, 72.20.My
Abstract Fractals describe the scaling properties of a spectacular variety of natural objects. In general, fractal studies in natural environments are "passive" in the sense that there is no experimental interaction with the system being observed. In contrast, in this paper we investigate fractals in an artificial environment where controlled changes in the generation process can be used to study how fractals evolve. To do this we construct micron-sized billiards in high quality semiconductor materials where the properties of chaotic electrons can be tuned with precision. By inserting a circle at the centre of a square billiard, we investigate the transition between two distinct forms of fractals observed in the billiard's conductance - from exact to statistical self-affinity.
1. Introduction The construction of intricate metallic patterns on the surface of a semiconductor forms the cornerstone of today's microelectronics industry. For commercial transistor applications, electrostatic surface gates serve as a switch to modulate the flow of electrons through the device. In this paper, we discuss a gate operation which is considerably more refined where tuning the voltage applied to the gates induces a controlled evolution in the scattering dynamics of the electrons. In the devices presented, the gates are used to guide the electrons through micron-sized billiards defined in the semiconductor. Analogous to a billiard table, each device consists of a confined region shaped by walls. Because this region is smaller than the average distance I between scattering sites in the material, electrons passing between entrance and exit openings follow ballistic trajectories shaped predominantly by the walls [1]. Semiconductor billiards have attracted considerable experimental [2,3] and theoretical [4,5] interest because the walls can support chaotic electron trajectories [6]. We induce a change in the billiard geometry by inserting a circular object at the centre of a square billiard. This is achieved by changing the bias applied to selected gates. This transition alters the contributions of chaotic trajectories to the electrical conductance. The proposed experiment is not the first study of a physical system in which the chaotic dynamics of particles can be tuned. Indeed, transitions can be achieved in relatively basic systems. For example, the chaotic flow of water has been investigated by adjusting the aperture of a dripping tap [7]. However, the gate technology of the semiconductor billiard offers a greater degree of precision than other systems [8,9]. Furthermore, by investigating the flow of elec*e-mail:
[email protected] © Physica Scripta 2001
trons rather than water it is possible to move beyond classical chaos and investigate "wave chaos" - the quantum mechanical behaviour of classically chaotic electron dynamics [10]. To observe the signature of wave chaos in experiments, we cool the semiconductor billiards down to milli-Kelvin temperatures in order to reduce electron phase-breaking scattering events. We find that the chaotic behaviour manifests itself as fractals [11] in the billiard's conductance when measured as a function of applied magnetic field. As the billiard geometry evolves, two distinct forms of fractal behaviour are observed - exact self-affinity (where the patterns observed at increasingly fine magnetic field scales repeat exactly) and statistical self-affinity (where the patterns simply follow the same statistical relationship at different scales). Statistical self-affinity describes many of Nature's patterns ranging from clouds to coastlines [11,12]. In contrast, although exact self-affinity has been the subject of mathematical studies for over one hundred years, observations of this form of fractal in physical systems remain rare [11,12]. The billiards investigated therefore represent a unique physical environment in which both forms of fractal can be observed. Furthermore, the transition between these two forms of fractal pattern can be studied in a controlled and systematic fashion. We present an experimental quantification of the fractal behaviour in the hope that these novel observations will attract theoretical interest in the relationship between the fractals and the chaotic dynamics that generate them.
2. Background: the semiconductor Sinai billiard In 1970, Sinai presented a theory that examined how the scattering dynamics of classical particle trajectories were determined by collisions with straight and curved walls [13]. In particular, the geometries shown in Fig. l(a,b) have since become a model system for the theoretical demonstration of chaos. Whereas the "empty" square billiard (Fig. 1(a)) supports stable trajectories, introduction of the circle at its centre transforms the geometry into a "Sinai" billiard (Fig. 1(b)) and this has a profound effect on the billiard's scattering dynamics - the convex surface of the circular scatterer acts as a "Sinai diffuser" producing exponentially diverging trajectories and therefore chaotic behaviour [6,13]. Until recently, it was unclear how these predictions might be realised in a physical system. Possible systems include Physica Scripta T90
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Fig. 1. (a) Two electron trajectories launched with similar initial conditions. Shaped by collisions with the square billiard, these stable trajectories do not significantly diverge, (b) In contrast to the square billiard, the same two trajectories rapidly diverge in the Sinai billiard due to scattering events with the circle, (c) A schematic representation of the surface gate technique used to define the Sinai billiard in the sheet of electrons located below the surface at the AlGaAs/GaAs interface. The light grey regions indicate the depletion regions under the gates (d) Self-consistent calculations of the soft-wall potential landscape expected for the Sinai billiard defined by surface gates. Potential energy (vertical axis) is plotted as a function of position within the billiard.
microwave cavities [14] and the environment established by semiconductor billiards. The most flexible construction technique for semiconductor billiards is shown in the schematic representation of Fig. 1(c). Within the semiconductor heterostructure shown, a two-dimensional sheet of electrons is located at the interface between the GaAs and AlGaAs layers. The billiard walls are formed in this sheet using the patterned metallic gates deposited on the heterostructure surface - a negative gate bias defines depletion regions directly below these gates [1,8]. In contrast to the physical walls of microwave cavities, these electrostatic walls can be turned on and off and their sizes varied by tuning the gate bias, allowing a continuous evolution in the billiard geometry. Fig. 2 summarises how a semiconductor Sinai billiard can be constructed using this gate technology. In particular. Fig. 2(c) shows the bridging interconnect used to establish electrical contact to the "inner" circular gate shown in Figs. 2(a,b) [8,15]. Exploiting the flexibility of the gate technique, the basic device operation is as shown in Fig. 2(d-g). A micron-sized square billiard is formed by applying a negative bias VQ to the three "outer" gates, the circle can then be introduced and its size expanded by applying an increasingly negative bias V\ to the inner gate. Because the host material's I value of 25 /mi is significantly larger than the billiard itself, the ballistic trajectories of electrons traversing the billiard are expected to be profoundly affected by this change in geometry. Furthermore, as required in Sinai's model, these ballistic electrons scatter off the walls elastically and thus it might initially be anticipated that the evolution in billiard geometry shown in Fig. 2(d-g) will be accompanied by the transition from stable to chaotic dynamics outlined in Fig. l(a,b). The semiconductor billiard is located 163 nm below the surface gates and simulations show that the electrostatic walls differ from Sinai's model in a crucial respect. Whereas Physica Scripta T90
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Fig. 2. (a) A scanning electron micrograph of the Sinai billiard device, (b) A schematic representation of the surface gate geometry. The lithographic dimensions are in microns. The dashed lines are guides indicating the relative position of the central gate, (c) A cross-section showing the bridging interconnect which traverses the billiard above a thin layer of insulator. The bridge contacts to the central circular gate through a hole etched in the insulator in the region directly above the central gate, (d)-(g) Schematic representations of four regimes of device geometry: (d) the "empty" square billiard, (e) the square with a partially depleted central region, (f) a Sinai billiard formed by the circle at the centre of the square billiard, (g) a Sinai billiard featuring a larger circle.
Sinai's model billiard is defined by "hard" walls with a vertical energy profile, the depletion regions penetrating the sheet of electrons define electrostatic walls with an approximately parabolic energy profile [16]. Fig. 1(d) shows the "soft" - wall profile for the Sinai billiard of Fig. 2. Recently, soft-wall billiards have been predicted to generate a "mixed" scattering system featuring both stable and chaotic trajectories [16-18]. Significantly, the phase-space plots mapping the velocity versus position of electrons reveal remarkably rich structure occurring at many magnifications. For example, Fig. 3 shows a Poincare section for the Sinai billiard [16]: In particular, it has been proposed that structure at the boundaries between chaotic and stable regions of the mixed phase-space might be described in terms of an infinite hierarchy of Cantori [6] and that these Cantori would strongly influence the wave chaos properties of the soft-wall billiards [17]. Because the electron Fermi wavelength of 50 nm is small compared to the dimensions of the billiard, a semi-classical model serves as an appropriate description of the wave chaos [4,5]. By calculating the phase of the electron quantum waves as they move along the classical trajectories, this model can be used to predict the wave interference between pairs of trajectories that form closed loops. This interference can be varied by applying a small magnetic field B perpendicular to the plane of the billiard [19]. The resulting fluctuations in the magneto-conductance G(B) can be viewed as "magneto-fingerprints" because they are sensitive to the precise distribution of loop areas [20]. © Physica Scripta 2001
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Fig. 3. A Poincare section for the Sinai billiard, showing that structure in the position (x) versus velocity (vv) phase-space plot occurs at many magnifications. The x coordinate is indicated in the schematic diagram of the billiard.
For soft-wall billiards, the presence of Cantori are predicted to produce a power-law distribution of loop areas and this should be detected as fractal scaling of the magnetoconductance fluctuation patterns observed at increasingly fine magnetic field scales [17]. In light of these predictions for soft-wall billiards, the transition outlined in Fig. 2(d-g) might be re-interpreted as follows. Both the empty geometry and the Sinai geometry are expected to generate mixed trajectory systems rather than the purely stable and purely chaotic systems predicted for the two equivalent hard-wall geometries. Insertion of the circle at the centre of the soft-walled billiard is expected to tune the properties of this mixed system and this might be detected as an evolution in the fundamental charactersitics of the observed fractal conductance fluctuations (FCF). Fractal conductance fluctuations self-affinity
observation of exact
To begin the experimental investigation, we focus on the Sinai geometry where the circle is inserted at the centre of the square billiard. Characterisation studies show that the presence of the circle is minimised for V\ = +0.7 V and the depletion region under the gate is well defined by 0V [9]. For negative V\, the expression R — R% — (8x 10~8)^[ (where Rg is the circle's lithographic radius of 0.15/mi) can be used to calculate /?, the radius of the depletion region [9], As the circle is activated, conductance structure clusters around two distinct magnetic field scales: "fine" ( / ) structure superimposed on "coarse" (c) structure. This is shown in Fig. 4(a) for R = 0.37 /mi and an electron temperature of T » 50 mK. This figure is generated by first measuring the c level (top) using a magnetic field resolution of 1 mT and then concentrating on a narrower magnetic field range and adopting a finer resolution of 0.008 mT to measure the /level (bottom). For each level, G(B) has been measured for both magnetic field directions (signified in the figure by a change in the sign of B) in order to distinguish between the measurement signal and measurement noise (due to the Onsager relationships, the signal is symmetric about B = 0T [1] and thus the observed minor deviations from sym© Physica Scripta 2001
metry originate from noise). We find that both the c and / structure can be suppressed by increasing V0 to a value sufficiently high to pinch off the channel around the circle, indicating that both sets of structure are generated by circulating trajectories [21]. Furthermore, both sets of structure can also be suppressed by raising the electron temperature to 4 K to remove electron phase coherence [21], indicating that both levels are generated by wave interference associated with these trajectories. However, their shared qualities extend even further - the patterns observed in the c a n d / structure appear remarkably similar. To dismiss the possibility that this similarity is simply coincidence, we can vary billiard parameters to induce a change in the c structure and check if a similar change occurs in t h e / structure [21]. This is demonstrated in Fig. 4(b) by adjusting Vo to reduce the number of conducting modes, w, in the entrance and exit openings from 7 (Fig. 4(a)) to 3 (Fig. 4(b)). Although the patterns evolve significantly during this transition, the similarity between the two different field scales is preserved. This repetition of patterns observed on different scales clearly bears a close resemblance to mathematical constructs such as the Koch Snowflake shown in Fig. 4(c) [12]. To distinguish this exact repetition of the pattern from patterns which simply follow the same statistics
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Fig. 4. (a) Coarse (top) and fine (bottom) structure observed in the magnetoconductance G(B) measured for m = 7. See text for details, (b) An identical plot for m = 3. (c) The repeating patterns of the Koch Snowflake. Physica Scripta T90
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at different scales (see later), we will label the observed behaviour as "exact" self-affinity (ESA). We will now confirm this ESA more rigorously. For the observed patterns to exhibit ESA, the / structure should be a scaled version of the c structure. If we define SGC = GC(B) - GC{B = 0) and 8Gf{B) = Gf(B) - Gf(B = 0), where GC(B) and Gj{B) are the c scale conductance and / scale conductance respectively, then it should be possible to select conductance and field scaling factors, XG and XB, such that SGJB) and XGSGf(XBB) are nominally identical traces. To quantify this ESA, we introduce a correlation function F [22]:
S
SGC(B) - XG SGf(hB))2)
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maximum. Compare this behaviour to the experimentally observed ESA shown in Fig. 5(b): although centred around the same (XG XB) point, the peak has a slightly lower value and, as indicated in Fig. 5(e), the experiment's Fpeak is less sharp. Ideal ESA is not achieved in the experiment due to contributions from a narrow range of scaling factors rather from a unique pair, and the full widths at half maximum extracted from the second derivative plot (Fig. 5(e)) quantify this range as (AXG AXB) = (2.7, 1). Having established that the two levels exhibit ESA, we now consider if they are fractal. To be fractal, the structure should not be limited to simply two magnetic field scales but instead should continue to cluster at increasingly fine and increasingly coarse scales, building up a cascade of levels spanning many field scales. As with the Koch Snowflake, there should be a constant magnification factor separating neighbouring levels, which for our data should be set by the magnetic field scale factor XB. For the data of Fig. 4(a), XB = 18.6. Zooming into t h e / level by a magnification factor of precisely 18.6, we indeed observe an additional cluster of structure, labelled as the ultra-fine level (uf) [24]. Similarly, by zooming out from the c level by precisely 18.6 we find the ultra-coarse (uc) level. The four levels (ultra-coarse uc, coarse c, fine/ and ultra-fine uf) are shown in Fig. 6(a). A crucial parameter in characterising fractal patterns is the fractal dimension Dp. Whereas F quantifies the similarity between the patterns observed at different magnifications, DF quantifies their scaling relationship [11,12]. For a smooth line, Dp has a non-fractal value of 1, whilst for a completely filled area its non-fractal value is 2. However, for a fractal pattern, the repeating structure at different magnifications causes the line to begin to occupy area. DF then has a fractional value lying between 1 and 2 and, as the complexity of the repeating structure increases, its value moves closer to 2. To confirm that the four levels observed in G(B) are fractal it is therefore necessary to show that their scaling relationship is described by a fractional value of DF.
The averaging ( ) is performed over 100 magnetic field points between ±50 mT. N is a normalisation constant calculated by averaging 1000 values of the expression ({X(B)— Y(B)}2)[/2, where X(B) and Y(B) are functions that generate random number distributions over the 100 magnetic field points. Because SGC(B) and SGj(B) are symmetric about B = 0T, X{B) and Y(B) are therefore reflected about B = 0T to ensure the same basic symmetry as the data. The amplitude ranges of X(B) and Y(B) are equated to that of SGC(B). The role of N is to set F = 0 when 8GC(B) and XGSGj{XBB) are randomly related traces and F = 1 if the two traces are mathematically identical. In this way, F identifies similarities in patterns seen at two different field scales. Fig. 5(b) is a "scale factor map", showing Fas a function o f / G and XB for the case of R = 0.37 fim and m = 3 (i.e. the G(B) data shown in Fig. 4(b)). A clear maximum of F = 0.94 is obtained for XG = 6A and XB = 26.4. All scale factor maps of the Sinai billiard reveal a single peak characterised by F as high as 0.97 [23]. These maps are characteristic of ESA, indicating a striking similarity between the c and / structure. Note, however, that this experimentally To calculate DF we construct the scaling plot shown in observed ESA is not mathematically perfect. Fig. 5(a) is a Fig. 6(b), which summarises the scaling relationship of scale factor map for the ideal case of ESA, where t h e / structure is a mathematically-generated replica of the c structure. The peak rises to a maximum of F = 1 at a single point in the (2 G / B ) map. This is demonstrated in Fig. 5(d), where the second derivative of F features a ^-function at the peak's 16b
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Fig. 5. (a), (b) and (c) are scale factor maps, F versus ( 1 & X3), for VQ = 0.55 V. (a) circle activated, ideal case; (b) circle activated, experimental case; (c) circle de-activated, experimental case. (c,d,e) are second derivatives of F versus 10 VQ = - 0.52 V. Physica Scripta T90
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Fig. 6. (a) From top - ultra-coarse, coarse, fine and ultra-fine levels observed in the experimental G(B) data measured for m = l. Arrows in the ultra-coarse data indicate the upper cut-off (see text), (b) Scaling properties of the levels observed in the experimental data based on the central peak height AGpk and full width at half maximum A5FWHM© Physica Scripta 2001
Semiconductor Billiards - a Controlled Environment to Study Fractals
Ultra-coarse "
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T(K) Fig. 7. Peak amplitude AGpit as a function of temperature showing an exponential increase with decreasing temperature.
the four levels. For simplicity the figure shows the analysis of one selected feature within each cluster of structure - the central peak. In Fig. 6(b), A 5 F W H M corresponds to the full width at half maximum and AGpk is the full peak height. Fig. 7 shows that AGP^ follows an exponential increase for each of the four levels as the temperature is lowered and the electron phase coherence length is increased. We obtain the four filled circles shown in Fig. 6(b) by extrapolating these lines to zero temperature. With the exception of the uf data point (discussed below), the data condenses onto the zero temperature power-law line shown. This line therefore describes the case of an infinite phase coherence length, where the distribution of trajectory areas for the loops contributing to the wave interference effect is determined purely by the billiard geometry. To commence our investigation of the scaling properties, we note that the four levels lie at equal increments along the power-law line, indicating that the amplitudes and periods of consecutive levels can be related by the common field and conductance scaling factors AB = AB„/ABn+l and AG = AGn/AG„+i respectively (where the indices n = 1-4 have been assigned in order of decreasing scale). Furthermore, these scaling factors are related by a power-law relationship lG = (ABf. Such a scaling behaviour is denned as fractal if the fractal dimension DF = 2 —f} lies in the range 1 < Dp < 2 [6], Allowing for uncertainties in the uf point (see below), the data points in Fig. 6(b) all lie on a line whose gradient fi gives DF = 1.55, confirming that the four observed clusters are fractal. The dashed vertical lines in Fig. 6(b) provide the experimental observation limits of this fractal behaviour and we label these the upper and lower "cut-offs". The c and / levels, which lie well within these cut-offs, reveal the ESA discussed above. In contrast, the uc and uf levels are measured at the experiment's limits, and we now discuss how this produces the observed reduction in ESA. The low field cut-off is determined by the magnetic field resolution limit for the experiment. Although the separation between measured magnetic field points is 0.008 mT, a minimum of three data points is required to observe a feature in G(B). Thus the lower magnetic field cut-off for the experiment corresponds to an interval of 0.016 mT and this value is indicated by the leftmost dashed vertical line in Fig. 6(b). The uf pattern shown in Fig. 6(a) consists of only 46 data points compared to the 788 data points within the equivalent / pattern. This reduction of resolution by 94% distorts the uf pattern, resulting in a loss of similarity between the uf a n d / © Physica Scripta 2001
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patterns. To quantify this distortion, we compare t h e / pattern before and after removing data points to artificially reduce its resolution by 94%. We find that this introduction of reduced resolution causes AGpk to fall by 49%o. This percentage distortion corresponds to 80nS for the uf level. Whereas a visual inspection confirms the resulting loss in self-similarity for the uf pattern in Fig. 6(a), the distortion can be compensated for when calculating the fractal scaling behaviour of this level in Fig. 6(b). It is the corrected AGpk value that is indicated by the filled uf circle in Fig. 6(b). The difference between this point's position and its anticipated position on the power-law line (open circle), as indicated by the 2Q. adjacent bar, is within the noise limit of the experiment (0.05% of the signal, corresponding to The high field cut-off occurs when the cyclotron radius of the electron becomes smaller than the billiard width. Above this field limit a transition to electron transport via skipping orbits occurs and the self-affinity is no longer expected to hold. The calculated upper field limit is marked by arrows in the uc trace of Fig. 6(a). Thus parts of the uc pattern (indicated by the dashed data line) lie outside this limit. For magnetic fields higher than the arrows, where the skipping orbit transport regime becomes established, the similarity between this pattern (the dotted data line) and the equivalent section of the c pattern deteriorates. As expected, for magnetic field values smaller than the arrows, the uc pattern still bears a similarity to that observed in the c level, and the height and width of this central peak are used to calculate the uc coordinate in Fig. 6(b). Unlike the uf level, because the uc peak lies within the observation limit, its coordinate lies on the fractal power-law line with no correction required. The range of observation of fractal behaviour defined by the upper and lower cut-offs shown in Fig. 6(b) is 3.7 orders of magnitude in magnetic field, well in excess of the majority of observations in other physical systems [25]. This extended range of observation makes the Sinai billiard an ideal system in which to investigate the scaling relationships of fractal phenomenon. With this in mind we will now investigate controlled transitions in the fractal behaviour.
4. Fractal conductance fluctuations - observation of statistical self-affinity We find that the ESA can be suppressed in two ways. Firstly, Fig. 8 shows that little similarity exists between the c a n d / levels when m is increased to 9. A possible explanation is that the decreased electron dwell time in the billiard reduces the number of trajectories that scatter off the circle before exiting. If correct, this picture emphasises that trajectories must scatter off the circle in order for ESA to be produced. This is confirmed by the second method of suppression removal of the circle. In Fig. 9 we examine the transition from the Sinai billiard to the "empty" square billiard. The trace is taken for m = 7, where the appropriate scaling factors (/lG, kB) are calculated to be (6.1, 26.4). The error bar in Fig. 9 is calculated as follows. We first calculate the F value obtained by correlating two nominally identical SGC(B) traces taken at 7? = 0.37 /mi in opposite field directions. The magnitude of the error bar is the difference between this F value and the expected value of F = 1. TherePhysica Scripta T90
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B(T) •0.04 0 0.04 -*-
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Fig. 8. Coarse (top) and fine (bottom) structure observed in the magnetoconductance G{B) measured for m = 9. The two levels no longer exhibit exact self-affinity.
-1
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V,(Volts) Fig. 9. The transition from the Sinai billiard to the empty square billiard, charted by F versus Vb The trace is taken for m = 1.
fore any deviations in F greater than this indicator can be regarded as significant. The decrease in Vr and hence R is accompanied by a rapid fall in F, indicating a dramatic loss in ESA as the circle is reduced. For V\ = +0.7 V (corresponding to the minimised presence of the circle) the correlation has fallen to less than F = 0.1. The scale factor map for the empty billiard is shown in Fig. 5(c,f) and this fails to yield the well-defined single maximum found in the equivalent plot of Fig. 5(b,e) for the Sinai billiard. Instead, the plot reveals a smooth contour with an average Physica Scripta T90
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Fig. 10. (a) G(B) measured for an empty billiard. Similar to the coastline shown in (b), G(B) does not reveal an exact repetition of patterns at different magnifications.
value of F well below 0.2 for all values of (^G. ^-B) [26]. This scale factor map is a signature of a G(B) trace that no longer contains an exact repetition of patterns at different field scales. This can be seen in the magneto-conductance fluctuations shown in Fig. 10(a). These fluctuations are strikingly different from those of the Sinai billiard shown in Fig. 4. Firstly, the fluctuations no longer cluster around selected magnetic field scales but are instead evident at all scales. Secondly, regardless of the magnification factor chosen, the patterns observed at different scales are clearly different. However, although no longer exhibiting ESA, this does not necessarily exclude the possibility that fractal behaviour has been preserved during the transition to the empty billiard. As shown in the coastline picture of Fig. 10(b), in general, fractals observed in Nature do not exhibit ESA [11,12]. Rather than an exact repetition, the coastline patterns observed on different scales simply follow the same statistical relationship - "statistical self-affinity" (SSA). Perhaps, then, the empty billiard exhibits this second form of fractal behaviour? To investigate if the fractal character of the magnetoconductance fluctuations has been preserved during the transition, we have to confirm that the scaling relationship of the fluctuations can still be described by a fractional value of Dp. For the ESA observed in the Sinai billiard this was done by identifying specific features in the structure and then plotting their sizes at different magnifications (see Fig. 6(b)). However, because there is no longer an exact repetition of features, this approach is no longer possible for SSA. Rather, it is necessary to calculate how the statistical qualities of the patterns scale. To do this, we employ a well-known method for fractal detection known as the box-counting technique [11,12]. The magneto-conductance © Physica Scripta 2001
Semiconductor Billiards - a Controlled Environment to Study Fractals trace is covered with a computer-generated mesh of identical squares. The statistical qualities of the fluctuation patterns can be determined by analysing which squares are occupied by the trace and which are empty. These statistics can then be compared at different magnifications by reducing the square size in the mesh. In particular, Dp can be obtained by calculating the number of occupied squares in the mesh, N(AB), as a function of square size AB. For fractal behaviour, N(AB) scales according to N(AB) ~ AB'0* where 1 < Dp < 2 [11,12]. Therefore, by constructing a scaling plot of —log N(AB) against log AB, the fractal behaviour is detected as a straight line and quantified by extracting Dp from the gradient. This scaling plot is equivalent to Fig. 6(b) used for the case of ESA [27]. The box-counting technique confirms that fractal behaviour is preserved during the transition to the empty square. Although a detailed description of the Dp dependence on other experimental parameters such as T, m, I, and billiard area A, will be presented elsewhere [28], the transition from R = 0.37 ^m to (nominally) R = O^m is accompanied by a smooth reduction in Dp by approximately 10%. We note that, although the presence of the circle is minimised at V\ = +0.7 V, we can not exclude the possibility that remnants of the circle remain. Therefore, to confirm that this second form of fractal behaviour, SSA, does not require the presence of a circle, we have constructed a billiard with an identical geometry to the one shown in Fig. 2 but without the circular gate. Fig. 11 shows a typical scaling plot for this second billiard (T = 50 mK and m = 6) and this shows that a completely empty billiard generates SSA. The dashed straight line is a guide to the eye, indicating that the data follow the fractal scaling relationship between the magnetic field scales AB2 and AB3. The values of these upper and lower cut-offs are determined using the derivative plots shown in Fig. 1 l(a,b). The field scale marked ABX represents the measurement resolution limit of 0.1 mT. However, the data remain non-fractal up to AB2 due to the dominance of measurement noise over signal for these small fluctuations in G(B). Note, therefore, that for both ESA (Fig. 6(b)) and SSA (Fig. 11)), the lower cut-off is set by noise restrictions
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F, suggesting that also the billiard symmetry is not relevant. In the light of the observed ratchet behaviour, this may seem a surprising result. It needs to be noted, however, that the experiments on ratchets and FCF used different voltage regimes. While the FCF experiments were carried out in the linear response regime (applied voltage eU < k^T/e « uV), that is near thermal equilibrium, the ratchet effect is based on the breaking of time-reversal symmetry by a finite voltage (non-linear regime). One may speculate that in the non-linear regime, where time-reversal symmetry is broken, DF might be sensitive also to the billiard symmetry. If such an effect should exist, it may then be viewed as a signature of non-equilibrium behaviour in a semi-classically chaotic system.
6. Conclusion
(b)
Fig. 7. Asymmetric, soft-walled electron billiards should allow to study ratchet behaviour in the presence of semi-classical chaos. One question to be answered by experiment is whether the fractal dimension of magnetoconductance fluctuations depends on the sign of the applied voltage - that is, whether the billiard symmetry has an effect on the quantum chaotic behaviour. In thermal equilibrium, the billiard shape does not appear to be important for fractal behaviour.
areas in the billiard, can be characterised by a fractal dimension DF, which lies between 1, the dimension of a smooth line, and 2, the dimension of a filled area (for details, see [21] and references therein). Asymmetric, soft-walled billiards are a prime candidate for studies of semi-classical chaos in ratchets, because the fractal conductance fluctuations (FCF) and the ratchet effect are observed in very similar devices. For instance, one could measure DF as a function of positive and negative bias voltage for an asymmetrically shaped billiard (Fig. 7(a)), or a device consisting of two coupled billiards with different fractal dimensions D{ and D2 (Fig. 7(b)). In this way, the Physica Scripta T90
The central question asked by the present paper is: Can signatures of classical chaos be observed in the quantum behaviour of ratchets? It appears as if the answer may be yes, because both classical chaos and quantum effects have each been found to yield qualitative changes in ratchet current. This indicates strong potential that ratchets may allow the study of quantum chaos in non-equilibrium systems. However, to date there is no theoretical or experimental study that covered a parameter range in which both, chaos and quantum effects could be observed at the same time. As candidates for experimental systems we discussed electron tunnelling ratchets [12] and quantum dot ratchets [15] and found that both could in principle be operated in a regime where chaotic effects should be important. Other experimental systems that are promising for the observation of quantum chaos in ratchet behaviour, for instance electrons in semiconductor superlattices or cold atoms in potentials formed by standing light waves [32] were not discussed here. For the design of any experiment, however, it would be highly useful to have access to a theory that is applicable to the specific experimental system in question. It is our hope that the present work will attract theoretical interest to these questions.
Acknowledgements This work was supported by the Australian Research Council.
References 1. Smoluchowski, M.. Physik. Zeitschr. 13, 1069 (1912). 2. Feynman, R. P., Leighton, R. B. and Sands, M., "The Feynman Lectures on Physics", (Addison-Wesley, Reading, 1963), Chapter 46. 3. Prost, J. et al., Phys. Rev. Lett. 72, 2652 (1994). 4. Rousselet, J. et al., Nature 370. 446 (1994). © Physica Scripta 2001
Chaos in Quantum Ratchets 5. van den Broeck, C. et at. in "Lecture Notes in Physics", (edited by D. Reguera, M Rubi and J. Vilar), (Springer, Heidelberg, 1999), Vol. 527, p. 93. 6. Reimann, P., Habilitation Thesis. Augsburg University, 2000. 7. Astumian, R. D., Science 276, 917 (1997). 8. Julicher, F., Ajdari, A. and Prost, J., Rev. Mod. Phys. 69, 1269 (1997). 9. Bader, J. S. et a!., P. Natl. Acad. Sci. USA 96, 13165 (1999). 10. Brooks, M., in New Scientist (22 January 2000), p. 28. 11. Reimann, P., Grifoni. M. and Hanggi, P.. Phys. Rev. Lett. 79, 10(1997). 12. Linke, H. et al.. Science 286, 2314 (1999). 13. Jung, P., tCissner, J. G. and Hanggi, P.. Phys. Rev. Lett. 76, 3436 (1996). 14. Mateos, J. L., Phys. Rev. Lett. 84, 258 (2000). 15. Linke, H.etal.. Europhys. Lett. 44, 341 (1998) and Europhys. Lett. 45, 406 (1999). 16. Linke. H. et at. Phys. Rev. B 61, 15914 (2000). 17. Marcus, C. M. et at.. Phys. Rev. Lett. 69, 506 (1992). 18. Chang, A. M. et at. Phys. Rev. Lett. 73. 2111 (1994).
© Physica Scripta 2001
19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.
59
Taylor, R. P. et at, Phys. Rev. Lett. 78, 1952 (1997). Sachrajda, A. S. et at, Phys. Rev. Lett. 80, 1948 (1998). Taylor. R. P. et at. Physica Scripta T90, 41 (2001). Lindner, B. et at, Phys. Rev. E 59, 1417 (1999). Yukawa, S. et at. J. Phys. Soc. Jpn. 66, 2953 (1997). Goychuk, I., Grifoni. M. and Hanggi, P., Phys. Rev. Lett. 80, 649 (1998). Goychuk, I. and Hanggi, P., Europhys. Lett. 43, 503 (1998). Altshuler, B. L., Aronov, A. G. and Khmelnitsky, D. E., J. Phys. C: Solid State Phys. 15, 7367 (1982). Giuliani, G. F. and Quinn, J. J., Phys. Rev. B 26, 4421 (1982). Lee, P. A. and Ramakrishnan, T. V., Rev. Mod. Phys. 57, 287 (1985). Lorke, A. et al. Physica B 251. 312 (1998). Ketzmerick, R„ Phys. Rev. B 54. 10841 (1996). Micolich, A. P. et at. submitted (2000). Mennerat-Robilliard, C. et al. Phys. Rev. Lett. 82, 851 (1999).
Physica Scripta T90
Physica Scripta: T90, 60-63, 2001
Statistics of Resonances in Open Billiards H. Ishio* Department of Physics and Measurement Technology, Linkoping University, S-581 83 Linkoping, Sweden Received July 2, 2000
PACS Ref: 05.45.+b, 05.60. +w, 73.10.-d
Abstract Statistics of resonance poles are computed for time-reversal ballistic transport through chaotic and integrable mesoscopic billiards coupled to a pair of single-channel leads in the regime of overlapping resonances. In the case of chaotic open billiard, the width distribution function shows good agreement with the random-matrix-theory prediction in all ranges of the width. In the case of integrable open billiard, however, there exists some deviation and the agreement is perceived only for the tail of the width distribution function. This is understood quantitatively in terms of classical decay-time distributions. On the other hand, the statistics of resonance positions for both chaotic and integrable open billiards show deviations from the Gaussianorthogonal-ensemble and Poisson predictions. The statistical nature known for eigenvalues of the closed counterparts of the systems is retrieved after eliminating all the broad resonances compared to the mean resonance spacing.
The transport properties of chaotic open billiards have been widely explored because fully chaotic and integrable motion of non-interacting particles inside ballistic cavities can be realized, simply depending upon the design of its boundary. It is intriguing to study the effects of the underlying classical dynamics on quantum transport through such a system, in connection with an application to mesoscopic devices. In general, quantum transport coefficients, such as conductance, in open billiards show ample oscillations as a function of external parameters, see, e.g., [1,2]. This originates from sequential overlap of resonances in cavity region lying inside the billiard. In order to better understand resonance structures, it is necessary to identify poles and analyze their properties in detail. In this study, we numerically obtain a number of resonance poles for chaotic and integrable open billiards. Then, we analyze the resonance width and position distributions statistically. The quantum open billiard we wish to study consists of a two-dimentional cavity coupled to continua by attached leads which support altogether M equivalent open channels. When the cavity is weakly and perfectly open, we commonly see inevitably overlapping resonances in quantum transport. We consider a Bunimovich stadium [3] (and its deformations) as a chaotic billiard, and a circle as an integrable billiard, respectively. The billiard is coupled to a pair of leads with a common width d and their orientations are not straightforward to avoid direct transmission, see Fig. 1. The stadium billiard is characterized by the radius of a semicircle a and the half-length of a straight section /. The aspect ratio a = Ija is continuously tunable, keeping the area of the billiard A = na2 + Aal fixed, which ensures the same degree of resonance density for each billiard. * Permanent address: Division of Natural Science, Osaka Kyoiku University, Kashiwara, Osaka 582-8582, Japan, e-mail address:
[email protected] Physica Scripta T90
For a closed stadium, the maximum Lyapunov exponent vanishes in the integrable limit (a = 0) and reaches its maximum at the fully chaotic limit ( L^. Therefore, the particle acquires chaotic and non-chaotic features through multiple scattering with specular reflections on the boundary of the cavity, and this will affect its transport properties. In quantum dynamics, the dc current passes through the leads. We choose the energy of the incoming wave so that only the first transverse mode in the leads is open. We solve the time-independent Schrodinger equation under Dirichlet boundary condition based on the plane-wave-expansion method [6], giving reflection and transmission amplitudes as a function of the energy. The resonances are due to quasibound states of the open billiard and identified with the poles of the corresponding scattering matrix S(E) occuring at complex energies E% = E\ + \El (Ef < 0) for the ath resonance. They are numerically obtained as the singular points of S(E) with E directly continued into the lower complex plane. The positions and widths of the resonance states are given by ££ and r a ( = 2\Ef\), respectively. In the following, we will assume S(E) to be a simple pole, ~ (E — is*) -1 , for E close to a resonance energy, and choose both h and the mass of particle \i as unity for simplicity. We should mention that more than half of the poles closest to the real axis or to each other (i.e., almost degenerate) in the range E\ > — 0.3 are missing because
(a)
(b)
Fig. I. Geometry of open billiards, (a) Stadium, (b) Circle. © Physica Scripta 2001
Statistics of Resonances in Open Billiards
61
0.5 (b) 0
1
—^"IH^
o
%
-1.5
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
logiof
-1
-0.8 -0.6 -0.4 -0.2 0 logio r
0.2
0.4
0.6
Fig. 2. Width distribution of resonances (cross bars or open bins), RMT prediction (solid line), and classical estimation (dotted line), (a) Open stadium billiard and its four deformations. Horizontal bar indicates the average value, (b) Open circle billiard.
of a simple numerical reason. This means that many of narrow resonances are not identified in our data. We now calculate the width distribution of resonances, where we use poles in the energy range 770 < £ R < 1800 (incl. 490 - 5 6 5 poles) and 800 < ER < 1800 (incl. 479 poles) for the stadium and its deformations, and the circle, respectively. On the other hand, the analytical expression of the width distribution for the overlapping resonances was obtained employing random matrix theory (RMT) for chaotic open systems with preserved time-reversal symmetry [7]: The distribution function of scaled resonance widths y = (nE\/A) determined by a mean resonance spacing A for perfectly open systems with M — 2 is given by P(y < 0) = (2y + 1 + (2y - 1) e 4j, )/(4^ 3 ). When we calculate the width distribution and compare it with this analytical expression, we find the following features, as we see in Fig. 2. In the case of the chaotic open stadium, P(r) follows r~2 asymptotically for r » A{~ 2) while it asymptotes to a constant for r <sc A, as is predicted in [7]. We still notice small deviations for l o g 1 0 r < —0.2 and l o g 1 0 r > 0 . 5 . As previously mentioned, many of the resonances for r = 2\E\\ A, however for the smaller T region the deviation from the analytical expression by RMT is prominent and the behavior does no longer show any universality [8]. The physical meaning of the power-law tail, ~ T~2, turns out to be due to classical processes of exponential escape typical for fully chaotic systems [9]. In weakly open billiards, however, such an exponential escape is observed not only for chaotic cases [2], but also for integrable cases [10] in a time scale corresponding to a path length up to at least a few hundreds. Using an uncertainty relation, this escape time © Physica Scripta 2001
distribution P(t)dt = (1/Td)exp(—//td)d/ with a lifetime t d can be transformed into a classical width distribution [11]
Pciiy < 0) = — exp y-
(i)
where c = nra\/{2A) and rc\ = n/td = (h/Ld)y/2ER//x. For \y\ » 1, PcX(y) oc y~2. In Fig. 2, Eq. (1) is plotted with dotted line for a typical energy £R = 1500 in the data. We see that it is successfully compared with the power-law tail of the distribution function in both chaotic and integrable cases, however a failure is evident for r —>• 0, where Pc\(y —>• —0) = 0 while purely quantum effect in resonances for weak coupling limit dominates with strongly diminishing T, leading to P{T - • 0) ~ constant. Finally, we show the statistics of resonance positions for the open stadium and circle billiards in Figs. 3 and 4, respectively. In general, the resonance spacing distribution P{s), the spectral rigidity Ai(L), and the number variance Z 2 (L) show deviations from the Gaussian-orthogonalensemble (GOE) and Poisson predictions [12]. (In the case of P(s), this deviation was pointed out for the first time in the auther's earlier work [13].) These anomalies principally originate from broad resonances compared to A, since the spectral nature of closed systems, such as the GOE statistics of RMT for chaotic systems and the Poisson statistics for integrable systems, tends to recover after eliminating the resonances with r > 0.5 ~ zJWeyi/2 — nn~/ iliA). To summarize, we numerically showed that P(r), in general, follows r~~2 for r 3> A in the case of time-reversal open billiards. This is in agreement with the RMT prediction for chaotic systems, and also with the estimation by the classical decay functions. In the case of integrable open billiard, however, some deviation from the RMT prediction is observed in P{r) for F 0.5. Horizontal bar of each cross is centered on the vertical error bar which originates from unfolding procedure of the resonance energies. Solid and dashed lines are the GOE and Poisson predictions, respectively.
(a)
*
(b)
A s
>Vr\
: y'
r
-
• ; >'r \ i I 1
ay
a. K
/
'Mi >fc
>'*
L-i
_———"" ~~~
.l-T-), unless explicitely stated otherwise (as in Section 4.1) The Schrodinger equation is solved subject to the general gauge invariant boundary condition, ijj =
±Xn(r){yrxij--A(r)^), reT, b
(5)
with A = 2/(Bb)A(r) the scaled vector potential. The mixing parameter X interpolates between Dirichlet (X = 0) and Neumann (/."' = 0) boundary conditions. Equation (5) is a generalisation of the mixed boundary conditions known 1 (-\hvr-qA(r))-^{r) = E^j{r). (2) for the Helmholtz problem [7-9]. The lower sign in (5) cor2m responds to the exterior problem. where m, q, and E denote the mass, charge, and energy of the A very efficient technique to obtain precise spectra of particle, respectively. The vector potential in the symmetric quantum billiards is known as the boundary integral gauge is A = {B/2)res. Scaling time by the Larmor fremethod. It allows to calculate ten-thousands of eigenvalues quency a> = qB/(2m) reduces the parameters in (2) to the at a modest numerical effort. In a recent communication [10] two length scales. we extended the method to the case of a finite magnetic field. For lack of space we cannot describe it here, and will make h and b1=(3) do with a few remarks. The boundary integral method 2mar reduces the Schrodinger equation (2) subject to the boundary Here, p is the cyclotron radius. Unlike this classical quantity, condition (5) to an integral equation defined on the the magnetic length b has a pure quantum meaning. It gives boundary. The (scaled) spectrum is given by the zeros of the mean radius of a bulk ground state. The scaled energy the corresponding Fredholm determinant. Also the wave may be expressed in terms of the spacing between Landau functions can be calculated from the corresponding null levels vectors by a boundary integral. The main problem of this approach at finite magnetic field (4) is the occurrence of spurious solutions. In [10] we show that 2 b Vw those can be related to physical solutions corresponding The expression for the unsealed wave number k = to a complementary problem at different boundary con^/linE/h = 2p/b2 indicates that there are two distinct ditions. This allows us to construct an integral operator short-wave limits: The high-energy limit p -* oo and the in terms of the regular free Green function which defines semiclassical limit b —> 0. The former corresponds to the spectrum uniquely. Employing special regularisation increasing the energy at fixed magnetic field while in the procedures to deal with the (hyper)-singular character of © Physica Scripta 2001
Physica Scripta T90
66
Klaus Hornberger and Uzy Smilansky 2
LI
i '
•'
•'
• ' '
"I1
' 1 '
• ' ' '
' ' '
i
I I 1 M l I .1 |l 1
; a
\
.;-
~ -~
—(T
1
E
Tw.
f
~^~
j
•
0
[
>
. \ , .
•
I r
*
•%.
• —1), variations in the curvature of the boundary may be neglected and the classical weights wp approach a value of unity. In the opposite case of an orbit which is almost detached from the boundary (- +1), the weights vanish, since the cosines approach zero at a finite denominator. This justifies the omission of cyclotron orbits - which formally have zero weights - from the trace formula above. In contrast, the periodic orbit expression for the unweighted density of states would contain the contribution of cyclotron orbits (with a different order in h) and would suffer divergencies at almost detached orbits [14]. It is instructive to compare the distributions of the quantum and classical weights. Unlike the quantum weights (6) attributed to each eigenvalue, the classical weights (17) are a property of the (periodic) orbits. In Fig. 6 we compare the phase space distribution of classical weights to the corresponding weighted quantum spectrum. The data was obtained for the interior elliptic billiard (cf. Fig. 1) and is given in both cases as a function of the classical cyclotron radius p. The distribution of classical weights p(wp) was approximated numerically by the histogram over a finite number of trajectories taken uniformly from phase space. Remarkably, one observes that the pronounced features of the distributions coincide. This finding is in some sense analogous to the known fact that wave functions tend to mimic classical phase space structures. It shows, that the quantum weights (which are defined originally in a quite formal way) do have a physical meaning, which translates to the classical dynamics. Equipped with a well-defined spectral density and the corresponding trace formula, we can now proceed with a statistical and semiclassical study of edge state spectra. Here, we shall not only consider the statistics within an edge state spectrum, but also cross correlations between different, classically related spectra. © Physica Scripta 2001
The Exterior and Interior Edge States of Magnetic Billiards: Spectral Statistics and Correlations (averaged) Fourier transform autocorrelation function *V„(V) = JCge(v'
0.30 0.33 0.36 0.39 0.42 0.45 0.48 0.51 0.54 0.57 0.60 0.63
P 2.0
of the spectral 2-point
+ \ )
«&*&£§&&**-1-.
69
(w) (w~)
•(20)
W„.
Here, the first and second moments of the weights, 0.D fc =r.,s» '. . "S.:, r oo * ii&"-^
_i
i
(w) = ] T w„ g(Nedge(v„) - v0),
(21)
X ! W« £W=dge(v„) - V0),
(22)
i_
0.30 0.33 0.36 0.39 0.42 0.45 0.48 0.51 0.54 0.57 0.60 0.63
P Fig. 6. Weighted quantum spectrum (top) and phase space distribution of the classical weights (bottom) for the interior ellipse. To facilitate comparison, also the quantum spectrum (calculated at constant 6 = 0.1, as in Fig. 4) is given in terms of the classical cyclotron radius (p = b x ^/v.) One observes that the quantum weights tend to mimic the structures in the distribution of classical weights (which are due to stable islands in phase space.)
4. Spectral analysis One of the central goals in the study of quantum chaos is to understand how the statistical properties of the quantum spectrum reflect the nature of the underlying classical dynamics. We extend this line of research to magnetic billiards, by making use of the weighted spectral measure discussed above. It was constructed to focus on the non-trivial part of phase space, which is determined by the billiard boundary map. In the first part of this section we study the standard two-point auto-correlations - albeit for weighted spectral densities - in the interior and the exterior. They serve to identify generic spectral properties of hyperbolic systems (as described by random matrix theory.) The second part, in contrast, is concerned with a new kind of cross-correlation between the exterior and the interior spectra of complementary domains. It reveals the quantum finger-prints of the classical interior-exterior duality. 4.1. Spectral auto-correlations A sensitive measure to characterise the spectral properties of a quantum system is provided by the form-factor K(T), the © Physica Scripta 2001
(w2>
n=l
are taken locally in the spectrum in terms of the window function g (a normalised Gaussian of width a.) As a result of this unfolding, both the weights and the weighted density, have unit mean. Since we are dealing with a discrete spectrum, the form factor must be averaged. The standard procedure is to take the spectral average over non-overlapping parts of the spectrum. l
K(x) =
(23)
*ft,(v)ft(v)dv
- ( / * indicated by the triangular brackets. According to the spectral ergodicity hypothesis [15] this should be equivalent to an ensemble average. The Wiener-Kinchin theorem holds in spite of the presence of window functions if we choose the widths of the Gaussians gi and g2 as )
£ w„e'
27ii(v„-5o)T„/r;
g(y„ - v0) - g(%) (24)
where the Fourier transform of g is denoted by g. Since we want to compare K(x) to the prediction of random matrix theory, the corresponding classical dynamics should be hyperbolic throughout the whole spectrum considered. A convenient way to ensure this is to define the Physica Scripta T90
70
Klaus Homberger and Uzy Smilansky
spectrum in the semiclassical direction, i.e. at fixed cyclotron radius p, which renders the underlying classical phase space unchanged. The spectral variable v is increased in this case, by decreasing the magnetic length b. This can be done by increasing energy E and magnetic field B at fixed ratio E/B2 and is equivalent to scaling down h. The spectral edge density in the semiclassical direction is denned by taking the derivatives in (8) at fixed p. The smooth part of its counting function.
is again proportional to the circumference. Using the numerical techniques described in [10] we calculated the interior and exterior weighted spectra at p = 1.2, for two different boundaries, the asymmetric stadium and the skittle billiard. These shapes (as defined in [10]) were chosen because they generate essentially hyperbolic classical motion at fairly strong magnetic fields. The asymmetric stadium has one reflection symmetry, while the skittle shape was constructed to have no symmetry. In combination with the magnetic field, the asymmetric stadium Hamiltonian therefore is invariant under an anti-unitary symmetry (reflection and time reversal) while the skittle is void of any symmetry. For both shapes, complete cyclotron orbits with radius p= 1.2 do not fit into the interior domains. (Decreasing
the cyclotron radius further would spoil hyperbolicity). As a consequence, one expects that all interior states are equal, concerning their edge state character. Indeed, the interior weights are distributed narrowly around a mean value w given by the ratio of weighted and unweighted mean densities w = Cp/(AA). The weights provide no additional information in this case, and one expects that the form factors are equal to those of the unweighted spectra, reproducing random matrix theory. In the upper row of Fig. 7 we show the interior form factors (24) of the asymmetric stadium and the skittle billiard. They indeed follow the RMT prediction of the Gaussian Orthogonal and Gaussian Unitary ensembles, respectively - as expected from the symmetry of their Hamiltonians. We note in passing, that it would have been more difficult to obtain the GUE form factor for a spectrum defined in the conventional highenergy direction, because with increasing energy the classical trajectories approach straight lines violating time-reversal invariance only minimally. In contrast to the interior problem, the standard form factor does not exist for the unweighted exterior spectrum, which is dominated by infinitely many, bulk states. Only by considering a weighted spectrum did we obtain a well defined form factor. For the billiard shapes discussed above, also the exterior classical dynamics of the skipping orbits is (essentially) hyperbolic. As a crucial test for the appropriateness of the weights one should therfore demand that also the exterior form factor follows the standard RMT
-i
Fig. 7. Form factors of the weighted interior (a,b) and exterior (c,d) edge state spi functions follow the RMT predictions of the GOE and GUE ensembles, respecl than the thin lines (ag = 10 and as = 3, resp.) Physica Scripta T90
1
;
1
r
for the asymmetric stadium (a,c) and skittle (b,d) billiard, at p = 1.2. The (dashed lines). The heavy lines correspond to stronger spectral averaging © Physica Scripta 2001
The Exterior and Interior Edge States of Magnetic Billiards: Spectral Statistics and Correlations prediction. In the bottom row of Fig. 7 we display the form factors (24) calculated from the exterior edge state spectra of the two billiard shapes. One observes that they obey closely the predictions of random matrix theory. This proves that the weights as defined in (6) succeed to remove selectively the bulk state contributions out of the spectrum. Most of the other popular measures characterising two-point spectral autocorrelations can be expressed in terms of the form factor and there is no need to discuss them any further.
71
over energy and boundary parameter C(v0) = / / < > ; *) »
;
-A)h{A)g{v
- vo) dA dv
(28)
with normalized window functions h and g. Here, h serves to restrict the integration over A to the range where the linear approximation in (27) is valid and may have a width of order one. The function g, on the other hand, is necessary since we correlate discrete spectra. It selects a narrow energy interval centered around the energy v0 and should have the width of a few effective nearest neighbour spacings. Note, that unlike in Section 4.1, there is no averaging involved in the definition of 4.2. Spectral cross correlations the correlation function. From the quantum mechanical point of view it is hard to say, Substituting (27) in (28), the cross-correlation function whether the interior edge spectrum should be related to the assumes the form exterior one, and how this connection could possibly look like. Rather, it is classical dynamics that suggests that, indeed, there should be a strong, nontrivial correlation — Ct,„ . C(v„) = J2 between the two quantum spectra. The simple observation to be made is, that under rather (29) general conditions the classical dynamics in the interior and exterior are isomorphic. This is the case, precisely, if Here, the primes label the exterior energies and weights, for any circle with the radius of a cyclotron orbit intersects the sake of brevity. The important point to note in the double the boundary at most twice. Then, due to simple geometry, sum (29) is that due to the small width of g only a few pairs of every periodic orbit has a dual partner orbit, living in the interior and exterior spectral points will contribute complementary domain. They bounce along the boundary appreciably at a given vo. It is the pairs with equal weighted at the same points, complementing each other to full cyclo- distances from the left and right, respectively, to the refertron circles. Each pair of orbits has the same stability ence energy vo. Here, the energy differences are scaled indiand their actions sum up to an integer multiple of the action vidually by the reciprocal weight attached to each of a single cyclotron orbit. spectral point. The function h, in contrast, limits the absolute energy distance. Note also, that the prefactor in (29) ensures Since it is the sets of periodic orbits that determine the that those pairs which include a bulk state do not contribute spectra asymptotically, one should expect that the correlations in the bouncing orbits carry over to the edge to the sum. spectra. In order to unravel the connection between interior The term G^g in (29) subtracts the background. It is and exterior edge state energies, a special cross-correlation approximated by function is needed. It not only involves the Dirichlet energies of the edge states but rests also on the information provided n Cbg ~fl'edge(vo)I ^ 2 h { j + ^ I ' , I - ^edge(Vo) I , by their weights. As the first step to obtain the appropriate correlator, we (30) extend formally the definition of the edge state density to finite boundary mixing parameters X. Using the scaled if we neglect the width of g and disregard the fact that the version A = X x 2^/v/b one can define higher order terms of the smooth edge densities differ for the interior and exterior. b d We turn now to the semiclassical evaluation of the corrededge(v,A) := -g-jNskip Mski P (v; X). A\=2^/vdX l^v lation function using the periodic orbit expressions discussed (26) in Section 3. One obtains a double sum over the skipping Below, the dependence of the spectral density on A will be interior and exterior periodic orbits needed only in the vicinity of the Dirichlet boundary condition A = 0. In this domain, the /1-dependence may be expanded to first order. The weighted spectral density for small but finite A can then be written only in terms of the Dirichlet energies and Dirichlet weights: i
C(V 0 ) :
7*
g(v - vo) P.P'
rpDp
pi
x j c o s ( s , + S'p, - n(np + n'pl) ~^(MP + / £ ) )
A
<W(v; A) = j-^X]"df ^ S(V"W - v) '
«—1
v
n=l
x^l£cos(0O-^£cos(0;o) + cos[Sp - S'p, - n(np - ri^) - 1 ( p p - ^ ) ) x/*(A£cos(0O + ^£cos(0;)
The cross-correlation function is now defined as an integral © Physica Scripta 2001
(31)
Here, h is the Fourier transform of the window function h Physica Scripta T90
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Klaus Hornberger and Uzy Smilansky
and the exterior quantities are again marked with a prime. The width of h is small compared to the sum over cos(0,) (which is of order np). As a result, the second term in the curly brackets of (31) is exponentially suppressed. Orbits with J^i cos(0,-) « 0 that are not excluded by h are suppressed because of their vanishing classical weights as discussed at the end of Section 3. In the first term of equation (31), h reduces the sum effectively to those pairs with approximately equal sums of angles of incidence J](cos(f?,) = X!yCOs(0'). In most cases, only pairs of dual interior and exterior periodic orbits have this property. If/7 a n d / / are dual orbits their sum of actions is just a multiple of 2nv and also the other classical quantities are simply related: n
'p = "P>
Sp + S'p, = 2nvnp, D'^Dp,
^
= rP
4 also its Fourier transform is very similar to the original one. However, the peaks below t = 4 now vanish, as can be seen from the solid line in Fig. 10. Therfore, if a more semiclassical part of the spectrum was considered - where a smaller fraction of the spectrum corresponds to transitional states - the remnant peaks would disappear. Nonetheless, the result proves that the cross-correlation function studied here succeeds to extract the relevant semiclassical information, and demonstrates unambiguously the quantum interiorexterior duality.
5. Conclusions In this article, we proposed a conceptually clear and practicable spectral measure for edge states. The edge state density was found to suppress efficiently irrelevant bulk contributions - also for the exterior problem with its infinite number of bulk states. The weighted density facilitated the statistical analysis of interior and exterior spectra. We saw, that even in the exterior the spectral autocorrelations follow the predictions of random matrix theory, if the underlying Physica Scripta T90
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Klaus Hornberger and Uzy Smilansky
classical map is hyperbolic. Nontrivial cross-correlations between interior and exterior spectra were found to reflect the classical duality between the interior and the exterior classical bounce maps. This duality is clearly stated for billiards for which a geodesic crosses the boundary at most twice. This is true for the magnetic and the field free billiards alike. However, in the field free case, one can define the interior and the exterior Birkhoff maps even for non-convex billiards, representing phase space by a manifold with several sheets. Here, the quantum duality holds for any shape. In the magnetic case, the classical problem was not yet studied in this respect, but the quantum correlations presented in Fig. 8 might indicate that the above condition is too restrictive. Finally, the present work can be placed in a more general context than the spectral theory of edge states. The identification of a spectral cross correlation, which reveals an underlying classical duality appears in various possible applications and systems. Consider e.g., the Schrodinger operator on the surface of a sphere, subject to some standard boundary conditions on a closed curve. This is a billiard on the sphere, where the geodesies are large circles, and where the interior-exterior duality holds in complete analogy with the case of magnetic billiards. A more distant relative in this family of problems is the cross correlation of Dirichlet and Neumann spectra obtained for a single quantum billiard. The periodic orbits in the two systems are exactly the same, and the trace formulae differ only in the boundary phase
Physica Scripta T90
by (—1)"'. Spectra which correspond to different representations of a symmetry groups fall in the same category.
Acknowledgements We thank B. Gutkin and M. Sieber for helpful discussions. The work was partially supported by a Minerva fellowship for KH, and by the Minerva Center for Nonlinear Physics.
References 1. Doron, E. and Smilansky, U., Nonlinearity 5. 1055 (1992). 2. Diez, B. and Smilansky, U., Chaos 3, 581 (1993). 3. Eckmann, J.-P. and Pillet. C.A., Commun. Math. Phys. 170, 283 (1995). 4. Halperin, B. I., Phys. Rev. B 25, 2185 (1982). 5. Gould. C. et al., Phys. Rev. Lett. 77. 5272 (1996). 6. Macris, N., Martin. P. A. and Pule, J. V.. J. Phys. A 33, 1985 (1999). « 7. Koshlyakov, N. S., Smirnov, M. M. and Gliner, E. B.. "Differential Equations of Mathematical Physics", (North-Holland Publishing Company. Amsterdam, 1964). 8. Balian, R. and Bloch, C , Ann. Phys. (N.Y.) 60, 401 (1970), erratum in Ann. Phys. 84 (1974), 559-563. 9. Sieber, M. et al., J. Phys. A 28, 5041 (1995). 10. Hornberger, K. and Smilansky, U„ J. Phys. A 33, 2829 (2000). 11. Robnik, M. and Berry, M. V., J. Phys. A 18, 1361 (1985). 12. Tasaki, S„ Harayama, T. and Shudo, A., Phys. Rev. E 56, R13 (1997). 13. Hornberger, K. and Smilansky, U., in preparation. 14. Blaschke, J. and Brack, M.. Phys. Rev. A 56, 182 (1997), erratum in Phys. Rev. A 57, 3136 (1997). 15. Brody, T. A. et al.. Rev. Mod. Phys. 53. 385 (1981).
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Physica Scripta. T90, 75-79, 2001
Non-Universality of Chaotic Classical Dynamics: Implications for Quantum Chaos M. Wilkinson* Faculty of Mathematics and Computing, The Open University, Walton Hall, Milton Keynes, MK7 6AA, England Received September 25, 2000
PACS Ref: 05.45.Ac, 05.45.Mt
Abstract It might be anticipated that there is statistical universality in the long-time classical dynamics of chaotic systems, corresponding to the universal agreement between their quantum spectral statistics and random matrix theory. It is argued that no such universality exists. Two statistical properties of long period orbits are considered. The distribution of the phase-space density of periodic orbits of fixed length is shown to have a log-normal distribution. Also, a correlation function of periodic-orbit actions is discussed, which has a semiclassical correspondence to the quantum spectral two-point correlation function. It is shown that bifurcations are a mechanism for creating correlations of periodic-orbit actions. They lead to a result which is non-universal, and which in general may not be an analytic function of the action difference.
1. Introduction It has been appreciated for many years that sufficiently complex quantum systems exhibit a high degree of universality [1], in that many statistical properties of their spectra usually fall into one of three classes, exemplified by the three Gaussian random matrix ensembles introduced by Dyson [2]. It is natural to ask whether an analogous degree of universality exists in classical dynamics, and if it exists whether it underlies the universality observed in the behaviour of quantum systems. This paper suggests what the appropriate classical analog of quantum spectral universality should be, and gives arguments supporting the view that there is, in general, no classical universality underlying that of quantum systems. The universality exhibited by spectral properties is confined to statistics which are sufficiently short ranged in energy. This implies that the universal features are associated with dynamics over long time scales: they may be associated with universal properties of the long-time classical dynamics, or they may be purely quantum. It is natural to anticipate that universal behaviour, if present at all, will only be manifest in properties which characterise small regions of phase space. The large scale structure of phase space can clearly be non-universal (for example, the volume of the energy shell is a non-universal function of energy). An example of a property which could show universality is the statistical characterisation of the distribution of points periodic under the Nth iterate of a chaotic area preserving maps of the form a„+i = M{jx„), with a = (x, p). These points are illustrated in Fig. 1 for the case of the 7th iterate of the * e-mail:
[email protected] © Physica Scripta 2001
standard map [3] x
n+\ — Xn + pn ,
Pn+\ = pn +
Ksm(xn+[)
with K = 6 and the 11th iterate of a modified cat map
(£:)-(! D(;:Mo) with u„ = sin(x„ + 2p„) and K = 2. The patterns displayed in Fig. 1 are complex and show very wide fluctuations in the density of points. They are clearly different, but local statistical properties of their fine-scale structure might be equivalent after scaling the coordinates to give the same mean density of points. Gutzwiller's trace formula [4] gives a relation between classical periodic orbits and quantum spectra. The formula is exact for a small number of special systems, but it cannot be exact in general because it contains no information about the choice of quantisation procedure [5]. In [6,7] Gutzwiller's formula is combined with the observation that spectral correlations are described by random matrix theory to infer that the actions of long period orbits are correlated. The correlations are a function of the action difference which was predicted to be universal within each of Dyson's symmetry classes, and an analytic form was quoted for the GUE ensemble. In systems where Gutzwiller's trace formula is exact (for a discussion of the three known examples see [4,8]) such classcial correlations must certainly exist. However, in general the trace formula is not exact and it is thus necessary to find an entirely classical mechanism for such correlations. It has been suggested that action correlations are related to the symbolic dynamics of systems [7], but this has not led to quantitative predictions, and does not indicate why there should be universality. Some numerical studies have reported that action correlations exist, and are in agreement with the universal predictions [6,9]. The paper is organized as follows. In Section 2 it is demonstrated that the long-time, local structure of phase-space of chaotic systems is non-universal, using theoretical arguments and numerical experiments. Section 3 discusses the fluctuations in the density of periodic points, such as those shown in Fig. 1. It is demonstrated that the local density has a log-normal distribution. Section 4 proposes an entirely classical mechanism for periodic-orbit correlations, based on the observation that pairs of orbits have correlated actions when they are formed as a result of a bifurcation. Combined with strong, but reasonable, assumptions about the statistics of bifurcations, this mechanism gives rise to Physica Scripta T90
76
M. Wilkinson
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Fig. /. Shows periodic points of the standard map with K = 6 and iV = 7
and for the modified cap map with K = 2 and N = \\ (right).
matrix MN may be modelled as a product of random matrices. A product of a large number of random scalars has a log-normal distribution, and it is therefore natural to expect that t r M ^ will have a log-normal distribution. 2. Non-universality of long-time dynamics The central limit theorem is only applicable sufficiently close The argument against universality is based upon considering to the maximum of the probability distribution, and the tails a particular statistic, namely the proportion P{N) of of the distribution of a sum of random variables depends trajectories which are elliptic upon N iterations of upon the distribution of the variables. Because the distriarea-preserving maps such as (1), (2). The results generalise bution of the matrices m(a) is non-universal, the tail of directly to continuous time systems. The stability of a point the distribution of tr MN is expected to be non-universal. Further support for this prediction comes from considerperiodic under N iterations of M, a. = MN(a), is described N ing moments of MN(a). Modelling the <x„ as random by the monodromy matrix MN{i) = dM {a)/da . The periN variables, odic point a = M (a) is elliptic (hyperbolic) if |tr A//v| < 2 (|trM/v| > 2). The terms elliptic and hyperbolic N will be used in the same way to describe any trajectory, (M (a)) = Y\{m((x (4) N n)) = [00 AT_1(logtr[M^(a)Myv(a)])01 is where h is the largest eigenvalue of (m(a) ® m(a)). A similar termed the Lyapunov exponent. This suggests that elliptic approach can be used to estimate the growth of T k trajectories are rare in the long-time limit, and that if a uni- ((tr M NMN) ): the result is of the form (5) with h replaced versal form exists for the function P(N), it might be expected by hk, the largest eigenvalue of the fc-fold outer product {m ® m (g> • • • m). The values of these eigenvalues depend to be exponential, upon the structure of the elementary monodromy matrices P(N)~Aexp(-a2.N) (3) m, indicating that the ratios of the eigenvalues h are non-universal. It is very difficult to reconcile this with the for N » 1, where A and a are universal constants. hypothesis that P{N) is universal. Determing the fraction of elliptic trajectories is equivalent To test these predictions, the fraction of elliptic to determining the probability that tr MN lies in the interval trajectories and the Lyapunov exponent were evaluated [-2, 2]. Because the typical value of tr MN is exponentially for the mappings defined by Eqs. (1) and (2). Fig. 2 is a plot large, this question relates to the tail of the distribution. of the fraction P(N) of elliptic trajectories as a function In the case of a chaotic map the succesive positions a„ have of kN. It clearly shows the non-universality of P(N). the characteristics of random numbers, and the monodromy a non-universal and possibly non-analytic contribution to the correlation function.
Physica Scripta T90
© Physica Scripta 2001
Non-Universality of Chaotic Classical Dynamics: Implications for Quantum Chaos 1 |
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derived in [11] which is applicable to maps. This simple, universal form is therefore a result of the cancellation of the weight |det(M - / ) | against its inverse. In general, however, the weighted density of periodic points depends upon the statistics of the monodromy matrices M. Now consider the highly non-uniform distribution of periodic points shown in Fig. 1. It is natural to attempt to characterise this by a local density V(a) defined in terms of the number of orbits inside a ball of radius, say, e. In order to construct a satisfactory definition using this approach, the local density would have to converge over a range of values of e, this range becoming broader as N increases. The fluctuations in density are so wild that it appears to be impossible to define a local density in this way. Instead, define the local density
1
V -
1 I
0
I
1 1 I
1 1 1
5
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10
77
15
AN Fig. 2. Shows P(N) as a function of XN for the standard map (A"= 15,A~log AT/2 = 2.015. circles) and for the modified cap map (K = 8 , i = 1.404, squares).
V(a) = \Sa\~d,
(9)
where Sa is the displacement from the point a to the nearest periodic point, \Sa\ is the corresponding Euclidean distance, 3. Fluctuations of periodic orbit density and d is the phase-space dimension. Assume that there is a Figure 1 indicates that the density of periodic points is highly periodic point at a small displacement Sa from the test point Considering the condition for periodicity non-uniform, however an exact result due to Hannay and a. a + Sa = M(a + Sa), and expanding the mapping to first Ozorio de Almeida [11] suggests that the periodic points order in Sa leads to the approximation might have a uniform distribution in phase space. This section discusses how this apparent contradiction is resolved, Sa = (M-I)~l(a-M) (10) presents an argument indicating that the density of periodic (where the index N designating the N-th iterate has been points has a log-normal distribution. The discussion is based upon the Kac-Rice [12,13] dropped). The fluctuations of this quantity are dominated approach to calculating the densities of point singularities by those of M which are log-normal. The distribution of the of random functions. The density V of zeros of a local density evaluated according to (9) is plotted in Fig. 3. vector-valued random function f{r) can be obtained from It is well described by a log-normal distribution. the joint probability distrbution function P[i, F] of f and of its Hessian matrix F, with elements Fy = df/drj. Adapting results from [13] to vector-valued functions, the density is 4. Bifurcations as a mechanism for periodic orbit correlation V=
fdF\dstF\P[0,F].
(6)
Argaman etal. [6] introduced a classical correlation function of periodic-orbit actions, of (essentially) the form
In the following the iterated mapping is treated as if it were a random function. The problem is to find the set of points C(T, AS) = J^ ">itfMT - Tj) SC(AS - (Sj - Sfj). (11) where a. — MN(a) = 0 for large N. For long orbits, it may be assumed that the joint distribution function P[M, M] Here, S) and 7} are the action and period of they'th periodic factorizes, and orbit, and e, r\ are the widths of broadened delta functions. In systems where Gutzwiller's trace formula is exact, and which exhibit universal spectral two-point correlations, V =P[M\ I [dMP[M] |det(M - / ) | = A~x \M=n J (7) C(T, AS) must be scale to a universal form for each of may be related in a simple way (for example, they Vw — C2/A, which constitutes the version of the sum rule may have opposite signs). This mechanism creates cor© Physica Scripta 2001
Physica Scripta T90
78
M. Wilkinson.
Fig. 3. Shows the distribution of the local density of periodic orbits V(a) for the standard map with K = 6 and N = 7 (a) and for the modified cat map with K = 2 and N = 11 (b). In both cases the distribution is log-normal ( ).
relations between actions which persist until one of the orbits undergoes a further bifurcation. Bifurcations are not expected to exhibit universal statitical properties. In the case of systems with a smooth Hamiltonian and a mixed (but predominantly hyperbolic) phase space, bifurcations occur when a periodic orbit becomes elliptic, |tr(M)| = 2. We have already seen that the fraction of elliptic trajectories is non-universal, and the frequency with which periodic orbits become elliptic must also be non-universal. The Gutzwiller trace formula is, strictly speaking, only applicable in fully hyperbolic systems. In such systems the condition |tr(M)| = 2 is never realised, and the only possible bifurcations are associated with singularities of the Hamiltonian. There is even less prospect for universality here, because the form of the bifurcations is non-generic. Some parametric families do not even exhibit bifurcations. An example is a parametric version of the Baker's map, defined for (x, y) in the unit square by if y„ < a, x„+x = ax„ , yn+x = y„/a xn+i = a + (1 - a)xn , yn+x = (y„ - a)/(l - a) if yn > a. (12) Other hyperbolic systems do exhibit bifurcations. In the Lorentz gas (scattering from a regular array of circular discs), orbits which undergo a glancing collision with a disc annihilate with orbits of nearly the same length which just miss the disc (see Fig. 4 for an illustration of this type of bifurcation). In this case the radius of scattering disc would be a suitable parameter. A contribution to the correlation function (11) at small values of AS can be related directly to the form of the bifurcations. Assume that a bifurcation occurs at Xt as a parameter X is varied. The action difference in the neighbourhood of the bifurcation is of the form IAS, I
lA*,!',
(13)
where AX, = X — Xt. The exponent /? will be determined from the type of bifurcation, and the constant v, depends on the particular bifurcation. The weights w, in (11) may be either singular or regular in the neighbourhood of the bifurcation: they may have algebraic singularities with exponents y., which may be different for the two orbits. Physica Scripta T90
Fig. 4. The parameter A" could define the shape of a billiard. In this example the circular section moves to the left as X increases, and the glancing periodic orbit annihilates the other orbit at an inverse bifurcation.
In the vicinity of the bifurcation, (14)
wi oc IAX/P.
The following discussion illustrates the effects of correlation through bifurcations by discussing a model for evaluating the contributions to (11) due to long periodic orbits. The model treats the wj as random variables, which are independent except for those pairs of orbits which are related by a bifurcation: (wj) = 0,
(wJWf)=w18j?
(15)
where in the latter case it is assumed that the orbits j a n d / are not related by a bifurcation. At a bifurcation, one or both of the weights may be singular. In this case, it is reasonable to model the behaviour of {wjWf) as follows (Wjwf) ~ ±w2
A-^+V^AX^f
(16)
where A characterizes the frequency with which bifurcations occur. The factor ±1 is included because for some types of bifurcation the weights may have opposite signs: note that the sign is not a random variable. The discussion will be simplified by assuming that all of the constants v, in (13) have the same value, v. Let SPe(AS) be the probability that a periodic orbit is connected to another orbit for which the action difference is in the interval [AS, AS + e]. According to the model above, © Physica Scripta 2001
Non-Universality of Chaotic Classical Dynamics: Implications for Quantum Chaos the only contributions to the correlation function come from pairs of periodic orbits related by a bifurcation. The contribution from these orbits may be estimated as follows
where N(T) is the number of periodic orbits with period less than T, and AX is the distance to a bifurcation. From (13), the bifurcation occurs at a displacement in parameter space of the form AX ~ (AS/v) 1//f . The probability that a periodic orbit does not undergo a bifurcation in a distance AX from an arbitrarily chosen test point is modelled by a Poisson distribution. This probability is P±{AX) = e\p[—A±\AX\] for displacements to either side of the test point. For a sufficiently small separation AX, the probability of finding a bifurcation in a small interval of size SAX on either side of XQ is SP-ASAX
(18)
with A = A+ + /1_. Combining (13) and (18), one obtains SPe(AS) ~ Avl/l>\AS\l/li-l5AS. Identifying SAS with e, and substituting into (18) produces the following result, valid for small |A5|: C(T, AS) oc \AS\(l+^+yf-p)/ls .
(19)
Evaluation of the exponent in (19) requires information about the nature of the bifurcations. As an example, consider a hyperbolic billiard system with no corners, such as the Lorentz gas. In this case inverse bifurcations occur when pairs of similar orbits, only one of which bounces off a surface, become tangential to that scattering surface (see Fig. 4). Geometrical considerations imply that ft = 2, and y- = 0 for the orbit which does not bounce tangentially but y, = 1 for that which does. In this case the exponent in (19) is zero, but for other types of bifurcation the exponent may be non-zero. 5. Conclusions concerning action correlations In systems with a smooth Hamiltonian, the statistic P{N) is very closely related to the condition for bifurcations (namely that a periodic orbit intersects the manifold where trM = ±2). The statistic P(N) was shown to be non-universal, and it is implausible that bifurcations will
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79
show universal behaviour in such systems. Totally chaotic systems have no elliptic trajectories, and bifurcations are associated with singularities of the Hamiltonian: in this case the arguments against universality are no less compelling. It follows that there is a non-universal component to the periodic orbit correlation function. Equation (19) also indicates that the contributions to the correlation function coming from bifurcations has a singularity if the exponent 1 + V/ + fy — P is not an even integer. It is difficult to conceive of a non-universal and non-analtyic contribution to the correlation function which could combine with this one to give a universal result. The arguments presented in this paper indicate that classical dynamics does not have the same degree of universality as quantum spectral statistics. It has been shown that bifurcations can be associated with action correlations, which are hypothesised to be in semiclassical correspondence with spectral correlations. Bifurcations, however, differ both in statistical properties and in form between different chaotic systems. In the case of generic chaotic systems, it is therefore unlikely that the Gutzwiller trace formula contains a complete description of spectral universality.
Acknowledgements I thank Prof. B. Mehlig for providing me with the numerical examples, and for discussions concerning the ideas contained in this paper.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Haake, F., "Quantum signatures of chaos". (Springer. Berlin 1992). Dyson. F. J.. J. Math. Phys. 3, 1199 (1962). Chirikov. B. V., Phys. Rep. 52. 263 (1979). Gutzwiller, M., "Chaos in Classical and Quantum Mechanics", (Springer, New York, 1990). Wilkinson, M., J. Phys. A 21, 1173 (1988). Argaman, N. et al. Phys. Rev. Lett. 7 1 . 4326 (1993). Smilansky, U., Cohen, D. and Primack, H., Ann. Phys (New York) 264, 108 (1997). Schanz, H. and Smilansky, U„ chao-dyn/9904007. Tanner. G., J. Phys. A: Math. Gen.. 32, 5071-85 (1999). Abrahams, A. and Stephen. M. J., J. Phys. C 13. 377 (1980). Hannay, J. H. and Ozorio de Almeida. A. M., J. Phys. A 17, 3429 (1984). Kac, M.. Bull. Am. Math. Soc. 49. 314 (1943). Rice. S. O., Bell Sys. Tech. J. 24. 46 (1945).
Physica Scripta T90
Physica Scripta. T90, 80-88, 2001
Chaos and Interactions in Quantum Dots Y. Alhassid Center for Theoretical Physics. Sloane Physics Laboratory, Yale University, New Haven, Connecticut 06520, USA Received November I, 2000
PACS Ref: 73.23.Hk, 05.45+b, 73.20.Dx, 73.23.-b
Abstract
point contacts
Quantum dots are small conducting devices containing up to several thousand electrons. We focus here on closed dots whose single-electron dynamics are mostly chaotic. The mesoscopic fluctuations of the conduction properties of such dots reveal the effects of one-body chaos, quantum coherence and electron-electron interactions.
1. Quantum dots Recent advances in materials science have made possible the fabrication of quantum dots, submicron-scale conducting devices containing up to several thousand electrons [1]. A 2D electron gas is created at the interface region of a semiconductor heterostructure (e.g., GaAs/AlGaAs) and the electrons are further confined to a small region by applying a voltage to metal gates, depleting the electrons under them. Insofar as the motion of the electrons is restricted in all three dimensions, a quantum dot may be considered a zerodimensional system. The transport properties of the dot, i.e., its conductance, can be measured by connecting it to external leads. A micrograph of a quantum dot is shown in Fig. 1(a). At low temperatures, the electron preserves its phase over distances that are longer than the system's size, i.e., L$ > L, where L$ is the coherence length and L is the linear size of the system. Such systems are called mesoscopic. Elastic scatterings of the electron from impurities generally preserve phase coherence, while inelastic scatterings, e.g., from other electrons or phonons, result in phase breaking. When the mean free path I is much smaller than L, transport across the dot is dominated by diffusion, and the system is called diffusive. In the late 1980s it became possible to fabricate devices with little disorder where £ > L. In these so-called ballistic dots, transport is dominated by scattering of the electrons from the boundaries. A schematic illustration of a ballistic dot is shown in Fig. 1(b). In small dots (with typically less than ~ 20 electrons), the confining potential is often harmonic-like, leading to regular dynamics of the electron and shell structure that can be observed in the addition spectrum (i.e., the energy required to add an electron to the dot). Maxima in the addition spectrum are seen for numbers of electrons that correspond to filled (JV = 2, 6, 12) or half-filled (Af = 4, 9, 16) valence harmonic-oscillator shells [2]. Dots with a large number of electrons (J\f > 50 - 100) often have no particular symmetry, and their irregular shape results in single-particle dynamics that are mostly chaotic. For such dots the conductance and addition spectrum display "random" fluctuations when the shape of the dot or a magnetic field are varied. This is the statistical regime, where we are interested in the statistical properties of the Physica Scripta T90
Fig. 1. Quantum dots: (a) a scanning electron micrograph of a dot used by folk et al. [14]. A 2D electron gas is formed in the interface of a GaAs/AlCaAs heterostructure (darker area). Electrostatic potentials applied to metallic gates (lighter shade) confine the electrons to a sub-micron region. The shape and area of the dot can be changed by controlling the gate voltages Kgi and Vg2', (b) a schematic drawing of a ballistic dot attached to two leads. The electron's trajectory scatters from the dot's boundaries several times before exiting.
dot's spectrum and conductance when sampled from different shapes and magnetic fields. For a recent review of the statistical theory of quantum dots see Ref. [3]. Many of the physical parameters of a quantum dot can be experimentally controlled, including its degree of coupling to the leads, shape, size, and number of electrons. When the dot is "open", i.e., strongly coupled to leads, there are generally several channels in each lead and the conductance fluctuates as a function of, e.g., the Fermi momentum of the electron in the leads (see Fig. 2(a)). As the point contacts are pinched off, the coupling becomes weaker and a barrier is effectively formed between the dot and the leads. In such "closed" dots, the charge is quantized. At low temperatures, the conductance through a closed dot displays peaks as a function of gate voltage (or Fermi energy); see, e.g.. Fig. 2(b). Each peak represents the addition of one more electron into the dot. In between the peaks, the tunneling of an electron into the dot is blocked by the Coulomb repulsion of electrons already in the dot, an effect known as Coulomb blockade. In this paper we discuss closed dots in which mesoscopic phenomena are determined by the © Physica Scripta 2001
Chaos and Interactions in Quantum Dots
Here a\\0) is a complete set of sjngle-particle eigenstates in the dot with energies Ex, and Af = Yli a\a^ l% t n e electron number operator in the dot. The quantity aVg is the confining potential written in terms of a gate voltage Vg and a — Cg/C, where Cg is the gate-dot capacitance. At low temperatures, conductance occurs by resonant tunneling through a single-particle level in the dot. Assuming energy conservation for the tunneling of the Afth electron we have, EF + £g.s.(Af - 1) = £g.s.(Af), where EP is the Fermi energy of the electron in the leads and £gs(Af) is the ground state energy of a dot with Af electrons. Using Eq. (1), we find that the effective Fermi energy Ep = Ep + ea Vg satisfies
kptunr1)
EM+\Af-
100
81
150
1 2/C
The conductance displays a series of peaks at values of Ep given by (2), with each peak corresponding to the tunneling of an additional electron into the dot. The spacings between the peaks are given by
200
V g (mV) Fig. 2. (a) Conductance vs. the electron's Fermi momentum kf in an open dot (from Ref. [5]). (b) Conductance vs gate voltage in a closed dot displaying a series of Coulomb-blockade peaks (from Ref. [14]).
interplay between single-particle chaos, quantum coherence and electron-electron interactions.
2. Transport in the Coulomb blockade regime The simplest model for describing the Coulomb blockade regime is the constant interaction (CI) model, in which the Coulomb energy is taken to be e2M~/2C, where C is the total capacitance of the dot and Af is the number of electrons. The Hamiltonian of the CI model is given by
AEp = (£V+i - £,v) + e2/C.
eaV ff = Ar
(1)
C
(3)
Since the charging energy is usually much larger than the mean-level spacing A, the Coulomb-blockade peaks are almost equidistant. Coulomb blockade is illustrated in Fig. 3. The Coulomb-blockade peak heights contain information about the wave functions. For closed dots, a typical level width r is small, and r yx,
(6)
where kc is the longitudinal channel momentum (h~k2c/2m + h~K2/2m = E), Pc is the penetration factor to tunnel through the barrier in channel c (Pc = 1 in the absence of barrier and Pc c = (