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Introduction to classical chaos and quantum chaos.

Faure, Frédéric

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

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.

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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.
Using the key properties of chaos, i.e. ergodicity and exponential instability, as a resource to control classical dynamics has a long and considerable history. However, in the context of controlling "chaotic" quantum unitary dynamics, the situation is far more tenuous. The classical concepts of exponential sensitivity to trajectory initial conditions and ergodicity do not directly translate into quantum unitary evolution. Nevertheless properties inherent to quantum chaos can take on those roles: i) the dynamical sensitivity to weak perturbations, measured by the fidelity decay, serves a similar purpose as the classical sensitivity to initial conditions; and ii) paired with the fact that quantum chaotic systems are conjectured to be statistically described by random matrix theory, implies a method to translate the ergodic feature into the control of quantum dynamics. With those two properties, it can be argued that quantum chaotic dynamical systems, in principle, allow for full controllability beyond a characteristic time that scales only logarithmically with system size and ℏ−1\hbar^{-1}ℏ−1. In the spirit of classical targeting, it implies that it is possible to fine tune the immense quantum interference with weak perturbations and steer the system from any initial state into any desired target state, subject to constraints imposed by conserved quantities. In contrast, integrable dynamics possess neither ergodicity nor exponential instability, and thus the weak perturbations apparently must break the integrability for control purposes. The main ideas are illustrated with the quantum kicked rotor. The production of revivals, cat-like entangled states, and the transition from any random state to any other random state is possible as demonstrated.
We introduce aspects of quantum chaos by analyzing the eigenvalues and the eigenstates of quantum many-body systems. The properties of quantum systems whose classical counterparts are chaotic differ from those whose classical counterparts are not chaotic. The spectrum of the first exhibits repulsion of the energy levels. This is one of the main signatures of quantum chaos. We show how level repulsion develops in one-dimensional systems of interacting spins 1/2 which are devoid of random elements and involve only two-body interactions. In addition to the statistics of the eigenvalues, we analyze how the structure of the eigenstates may indicate chaos. The programs used to obtain the data are available online.
In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between phase-space structures, reflecting the distinction between chaotic, mixed or integrable classical dynamics, and the corresponding quantum properties. Well established techniques dealing with ergodic wave interference in the usual semiclassical limit ℏ→0\hbar \to 0ℏ→0, where the classical limit is given by Hamiltonian mechanics of particles, constitute a now standard part of the toolkit of theoretical physics. During the last years, these ideas have been extended into the field theoretical domain of systems composed of NNN indistinguishable particles, aka quantum fields, displaying a different type of semiclassical limit ℏeff=1/N→0\hbar_{\rm eff}=1/N \to 0ℏeff​=1/N→0 and accounting for genuine many-body quantum interference. The foundational concept behind this idea of many-body interference, the many-body version of the van Vleck-Gutzwillers semiclassical propagator, is explained in detail. Based on this the corresponding semiclassical many-body theory is reviewed. It provides a unified framework for understanding a variety of quantum chaotic phenomena addressed, including random-matrix spectral correlations in many-body systems, the universal morphology of many-body eigenstates, interference effects kin to mesoscopic weak localization, and the key to the scrambling of many-body correlations characterized by out-of-time-order correlators.
We introduce aspects of quantum chaos by analyzing the eigenvalues and the eigenstates of quantum many-body systems. The properties of quantum systems whose classical counterparts are chaotic differ from those whose classical counterparts are not chaotic. The spectrum of the first exhibits repulsion of the energy levels, which is one of the main signatures of quantum chaos. We show how level repulsion develops in one-dimensional systems of interacting spins 1/2 which are devoid of random elements and involve only two-body interactions. In addition to the statistics of the eigenvalues, we analyze how the structure of the eigenstates may indicate chaos. The programs used to obtain the data are available online.
Quantum chaos concerns the emergence of universal statistical and dynamical prop- erties in quantum systems whose classical counterparts exhibit chaotic behavior. Unlike classical chaos, which can be characterized by sensitivity to initial conditions and exponential divergence of trajectories, quantum chaos manifests itself through a variety of indirect diagnostics, including spectral correlations, operator growth, and information scrambling. Understanding how these signatures arise from microscopic dynamics, and how they can be accessed in realistic many-body systems, is a central challenge in modern physics. This thesis investigates quantum chaos from complementary theoretical and practical perspectives. On the theoretical side, it develops effective descriptions of chaotic dy- namics in many-body systems with approximate symmetries, focusing on how weak symmetry breaking controls the exploration of Hilbert space and the crossover to random-matrix universality. Using perturbative arguments, numerical simulations, and a nonlinear sigma-model formulation, we identify a regime of Hilbert-space dif- fusion in which chaotic dynamics unfolds through a slow, collective spreading across nearly decoupled symmetry sectors. On the practical side, the thesis addresses the question of how paradigmatic chaotic models can be realized and probed in controlled experimental platforms. In partic- ular, it explores quantum simulation schemes for Sachdev–Ye–Kitaev (SYK) models using cavity quantum electrodynamics. By employing time-dependent disorder and Trotterized evolution in single-mode optical cavities, we demonstrate how low-rank interactions can be systematically promoted to effective all-to-all randomness, re- producing the spectral and dynamical signatures of SYK physics. Extensions to open-system dynamics are discussed, including Lindbladian realizations and the role of dissipation. The thesis thus highlights the connections between effective descriptions of quantum chaos and their implementation in controlled quantum simulation settings.
This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, the fluctuation–dissipation relation, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. In some cases, it is possible to extend their regimes of applicability beyond the standard thermal equilibrium domain. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We present results from numerical experiments and analytical studies of quantum quenches at integrability. We introduce the concept of the generalized Gibbs ensemble and discuss its connection with ideas of prethermalization in weakly interacting systems.