Quantized, compact graphs are excellent paradigms for quantum chaos in bounded systems. Connecting them with leads to infinity, we show that they display all the features which characterize quantum chaotic scattering. We derive exact expressions for the scattering matrix, and an exact trace formula for the density of resonances, in terms of classical orbits, analogous to the semiclassical theory of chaotic scattering. A statistical analysis of the cross sections and resonance parameters compares well with the predictions of random matrix theory. Hence, this system is proposed as a convenient tool to study the generic behavior of chaotic scattering systems and their semiclassical description.
Quantized, compact graphs are excellent paradigms for quantum chaos in bounded systems. Connecting them with leads to infinity, we show that they display all the features which characterize quantum chaotic scattering. We derive exact expressions for the scattering matrix, and an exact trace formula for the density of resonances, in terms of classical orbits, analogous to the semiclassical theory of chaotic scattering. A statistical analysis of the cross sections and resonance parameters compares well with the predictions of random matrix theory. Hence, this system is proposed as a convenient tool to study the generic behavior of chaotic scattering systems and their semiclassical description.
We connect quantum compact graphs with infinite leads, and turn them into scattering systems.We derive an exact expression for the scattering matrix, and explain how it is related to the spectrum of the corresponding closed graph.The resulting expressions allow us to get a clear understanding of the phenomenon of resonance trapping due to over-critical coupling with the leads.Finally, we analyze the statistical properties of the resonance widths and compare our results with the predictions of Random Matrix Theory.Deviations appearing due to the dynamical nature of the system are pointed out and explained.
This book introduces the quantum mechanics of classically chaotic systems, or quantum chaos for short. The author's philosophy has been to keep the discussion simple and to illustrate theory, wherever possible, with experimental or numerical examples. The microwave billiard experiments, initiated by the author and his group, play a major role in this respect. Topics covered include the various types of billiard experiment, random matrix theory, systems with periodic time dependences, the analogy between the dynamics of a one-dimensional gas with a repulsive interaction and spectral level dynamics, where an external parameter takes the role of time, scattering theory distributions and fluctuation, properties of scattering matrix elements, semiclassical quantum mechanics, periodic orbit theory, and the Gutzwiller trace formula. This book will be of great value to anyone working in quantum chaos.
Many-body quantum chaos at ℏ∼1: Analytic connection to random matrix theory
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry (1985) within the so-called diagonal approximation of semiclassical periodic-orbit sums. Derivation of the full RMT spectral form factor K(t) from semiclassics has been completed only much later in a tour de force by Mueller et al (2004). In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed as `many-body localized phase' and `ergodic phase'. In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. Here we provide the first theoretical explanation for these observations. We compute K(t) explicitly in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.
We review random matrix approaches to chaotic wave scattering and transport in open systems. Starting from the effective non-Hermitian Hamiltonian formulation, we discuss the scattering matrix, reaction matrix, time delays, and complex resonances as complementary probes of open chaotic dynamics. We emphasize universal statistics governed by symmetry, openness, and channel coupling. Topics include the maximum-entropy description of fixed-energy scattering and its applications to quantum transport, energy correlations, resonance and eigenfunction statistics, and selected wave-chaotic phenomena induced by finite absorption. The focus throughout is on non-perturbative methods and universal structures underlying open quantum and wave chaotic systems.
Random matrix spectral correlations is a defining feature of quantum chaos. Here, we study such correlations in a minimal model of chaotic many-body quantum dynamics where interactions are confined to the system’s boundary, dubbed , in terms of the spectral form factor and its fluctuations. We exactly calculate the latter in the limit of large local Hilbert space dimension q q for different classes of random boundary interactions and find it to coincide with random matrix theory, possibly after a non-zero Thouless time. The latter effect is due to a drastic enhancement of the spectral form factor, when integer time and system size fulfill a resonance condition. We compare our semiclassical (large q q ) results with numerics at small local Hilbert space dimension ( q=2,3 q = 2 , 3 ) and observe qualitatively similar features as in the semiclassical regime.
This article gives an introduction to chaotic systems in classical mechanics and their quantum mechanical counterparts.The author assumes very few prerequisites and starts with a basic introduction to the mathematical formulation of classical and quantum mechanics and discusses the relation between classical and quantum systems by means of the mathematical procedure of quantization. Then, the article proceeds with discussing the notion of ``chaos'' in classical mechanics and particularly focuses on the concepts of hyperbolicity, ergodicity and mixing. In the last section, the quantum mechanical properties of classically chaotic systems are discussed. Here, the focus lies in particular on quantum ergodicity and the role of Egorov's theorem. Finally, the distribution of eigenvalues, according to Weyl's law, {\it M.C. Gutzwiller}'s trace formula [``Periodic orbits and classical quantization conditions'', J. Math. Phys., 12, No. 3, 343--358 (1971; \url{doi:10.1063/1.1665596})] as well as the random matrix conjecture of {\it O. Bohigas} et al. [``Characterization of chaotic quantum spectra and universality of level fluctuation laws'', Phys. Rev. Lett. 52, No. 1, 1--4 (1984; \url{doi:10.1103/PhysRevLett.52.1})] is also discussed.A particularly nice feature throughout the whole text is that it includes both mathematically rigorous definitions which can be illustrated for toy models such as the cat map, as well as phyiscally relevant sytems such as billiards. For the latter it is then outlined, where the difficulties in a mathematically rigorous description lie.