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Physical Review A

Ergodicity and mixing in quantum theory. I

Asher Peres

Published Jul 1, 1984• 136 citations
Publisher

Abstract

Quantum chaos is defined by the following property: Simple dynamical variables (position, momentum, etc.) are represented by pseudorandom matrices, when the Hamiltonian is diagonal. As a consequence, the expectation values of these variables tend to equilibrium values which are insensitive to the initial preparation, for nearly all preparations involving many energy levels. Moreover, the fluctuations around these equilibrium values are, on the average, very small.

DOI
10.1103/physreva.30.504

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•Jan 1, 2009•0 citations
Quantum chaos is defined by the following property: Simple dynamical variables (position, momentum, etc.) are represented by pseudorandom matrices, when the Hamiltonian is diagonal. As a consequence, the expectation values of these variables tend to equilibrium values which are insensitive to the initial preparation, for nearly all preparations involving many energy levels. Moreover, the fluctuations around these equilibrium values are, on the average, very small.
The approach proposed in the first part of this paper is illustrated by two examples. The first one involves the H\'enon-Heiles model. A small wave packet, initially localized in the classically chaotic region, evolves so that 〈x〉〈x〉〈x〉 and 〈x2〉〈{x}^{2}〉〈x2〉 rapidly reach their equilibrium values. The second example makes use of a pair of rotators with nonlinear coupling. The matrix elements of Jz{J}_{z}Jz​, in the energy representation, are pseudorandom in the energy range which is classically chaotic, but obey selection rules in the classically regular energy range.
We consider a quantum system with N degrees of freedom which is classically chaotic. When N is large, and both and the quantum energy uncertainty are small, quantum chaos theory can be used to demonstrate the following results: (i) given a generic observable A, the infinite time average of the quantum expectation value is independent of all aspects of the initial state other than the total energy, and equal to an appropriate thermal average of A; (ii) the time variations of are too small to represent thermal fluctuations; (iii) however, the time variations of can be consistently interpreted as thermal fluctuations, even though these same time variations would be called quantum fluctuations when N is small.
We consider a quantum system with N degrees of freedom which is classically chaotic. When N is large, and both ¯h and the quantum energy uncertainty ∆E are small, quantum chaos theory can be used to demonstrate the following results: (1) given a generic observable A, the infinite time average A of the quantum expectation value 〈A(t) 〉 is independent of all aspects of the initial state other than the total energy, and equal to an appropriate thermal average of A; (2) the time variations of 〈A(t)〉−A are too small to represent thermal fluctuations; (3) however, the time variations of 〈A 2 (t) 〉 − 〈A(t) 〉 2 can be consistently interpreted as thermal fluctuations, even though these same time variations would be called quantum fluctuations when N is small. Typeset using REVTEX
If a many-body quantum system approaches thermal equilibrium from a generic initial state, then the expectation value ⟨ψ(t)∣Ai∣ψ(t)⟩\langleψ(t)|A_i|ψ(t)\rangle⟨ψ(t)∣Ai​∣ψ(t)⟩, where ∣ψ(t)⟩|ψ(t)\rangle∣ψ(t)⟩ is the system's state vector and AiA_iAi​ is an experimentally accessible observable, should approach a constant value which is independent of the initial state, and equal to a thermal average of AiA_iAi​ at an appropriate temperature. We show that this is the case for all simple observables whenever the system is classically chaotic.
The pendulum is the simplest system having all the basic properties inherent in dynamic stochastic systems. In the present paper we investigate the pendulum with the aim to reveal the properties of a quantum analogue of dynamic stochasticity or, in other words, to obtain the basic properties of quantum chaos. It is shown that a periodic perturbation of the quantum pendulum (similarly to the classical one) in the neighborhood of the separatrix can bring about irreversible phenomena. As a result of recurrent passages between degenerate states, the system gets self-chaotized and passes from the pure state to the mixed one. Chaotization involves the states, the branch points of whose levels participate in a slow "drift" of the system along the Mathieu characteristics this "drift" being caused by a slowly changing variable field. Recurrent relations are obtained for populations of levels participating in the irreversible evolution process. It is shown that the entropy of the system first grows and, after reaching the equilibrium state, acquires a constant value.
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].