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American Institute of Physics



Book Review

Quantum Chaos and Quantum Dots

Katsuhiro Nakamura and Takahisa Harayama
Oxford University Press, New York, 2004 199 pp., $109.50 hb ISBN 0-19-852589-3

Reviewed by Arjendu K. Pattanayak

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Quantum Chaos and Quantum Dots, written by Katsuhiro Nakamura of Osaka City University and his collaborator Takahisa Harayama, is an interesting review of some quantum-transport and related problems in solid-state systems. The text should prove useful to two categories of physicists: those in solid-state physics looking for an entrée to issues in quantum chaos, and those in quantum chaos interested in learning about this specific application.

Semiconductor quantum dots are stateof- the-art structures fabricated at semiconductor heterojunctions, and consist of a mesoscopic scattering region connected to external reservoirs. The length scales are such that electrons travel essentially ballistically, so these systems act like quantum versions of classical billiards. They are excellent playgrounds for experimentalists to explore novel transport phenomena.

One research focus since the early 1990s has been quantum chaos: given certain dot shapes, classical trajectories bounce around chaotically, resulting in the rapid separation of initially close trajectories. In regular dots, individual classical trajectories travel in bands, such that small perturbations grow slowly. These two different behaviors influence the quantum dynamics of the electrons in radically different ways, resulting in different universality classes for transport properties in such systems. For example, several groups showed that at low temperatures, the magnetoresistance of the dots exhibits reproducible fluctuations. The spectra of the fluctuations are different, depending on whether the dot is chaotic or regular.

Theoretical explanations use a semiclassical expansion for the Green’s function within the Landauer–Buttiker conductance formula, which emphasizes the role of the unstable periodic orbits of the chaotic system. Another approach uses the phenomenological technique of random matrix theory, for example. Nakamura has contributed significantly to the theory, and this monograph is a somewhat selective review of these issues, focused closely on his own research in this field.

The authors tell a compelling story, especially in the earlier chapters. They begin with the experimental evidence for the difference between chaotic and regular dots, followed by an interlude that establishes the needed ideas of classical Lyapunov exponents and escape rates (classical ideas emerge on a need-to-know basis in this book, a strategy that works well). After discussing the original results on conductance fluctuations, the authors give a short pedagogical treatment of some quantum chaos results, including the Gutzwiller trace formula for the spectra of chaotic systems.

They follow this with a tour of various more recent analyses, including treatments of orbital diamagnetism and persistent currents, quantum interference and weaklocalization issues in single open billiards, Coulomb blockade, and so on. They have some interesting comparisons of results using both the Landauer–Buttiker transmission formula and a semiclassical Kubo linear response theory.

The style will not surprise those who have read Nakamura’s now out-of-print book Quantum Chaos: A New Paradigm of Nonlinear Dynamics (Cambridge University Press, 1993), although there is little overlap in content. The present book is far more focused and could be used for a special topics course on quantum chaos in billiard systems, although it is idiosyncratic in content. It should definitely be useful for researchers, including those at the graduate level. While not definitive in any way, it is a good addition to the still-small collection of books in the fascinating and wide-open field of quantum chaos.

Arjendu K. Pattanayak is an assistant professor of physics at Carleton College in Northfield, Minnesota. He has published several papers over the past decade on quantum chaos, particularly as related to decoherence and the classical limit.