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arXiv:1408.2838 · Physical Review E

Relaxation of isolated quantum systems beyond chaos

Ignacio García-Mata, Augusto J. Roncaglia, Diego A. Wisniacki

Published Jan 23, 2015• 10 citations
arXivHTMLPDF

Abstract

In classical statistical mechanics there is a clear correlation between relaxation to equilibrium and chaos. In contrast, for isolated quantum systems this relation is--to say the least--fuzzy. In this work we try to unveil the intricate relation between the relaxation process and the transition from integrability to chaos. We study the approach to equilibrium in two different many-body quantum systems that can be parametrically tuned from regular to chaotic. We show that a universal relation between relaxation and delocalization of the initial state in the perturbed basis can be established regardless of the chaotic nature of system.

DOI
10.1103/physreve.91.010902

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In classical statistical mechanics there is a clear correlation between relaxation to equilibrium and chaos. In contrast, for isolated quantum systems this relation is--to say the least--fuzzy. In this work we try to unveil the intricate relation between the relaxation process and the transition from integrability to chaos. We study the approach to equilibrium in two different many-body quantum systems that can be parametrically tuned from regular to chaotic. We show that a universal relation between relaxation and delocalization of the initial state in the perturbed basis can be established regardless of the chaotic nature of system.
We study the onset of chaos and statistical relaxation in two isolated dynamical quantum systems of interacting spins 1/2, one of which is integrable and the other chaotic. Our approach to identifying the emergence of chaos is based on the level of delocalization of the eigenstates with respect to the energy shell, the latter being determined by the interaction strength between particles or quasiparticles. We also discuss how the onset of chaos may be anticipated by a careful analysis of the Hamiltonian matrices, even before diagonalization. We find that despite differences between the two models, their relaxation processes following a quench are very similar and can be described analytically with a theory previously developed for systems with two-body random interactions. Our results imply that global features of statistical relaxation depend on the degree of spread of the eigenstates within the energy shell and may happen to both integrable and nonintegrable systems.
We study the transition to chaos and the emergence of statistical relaxation in isolated dynamical quantum systems of interacting particles. Our approach is based on the concept of delocalization of the eigenstates in the energy shell, controlled by the Gaussian form of the strength function. We show that, although the fluctuations of the energy levels in integrable and nonintegrable systems are different, the global properties of the eigenstates are quite similar, provided the interaction between particles exceeds some critical value. In this case, the statistical relaxation of the systems is comparable, irrespective of whether or not they are integrable. The numerical data for the quench dynamics manifest excellent agreement with analytical predictions of the theory developed for systems of two-body interactions with a completely random character.
The problems of chaos and relaxation have a fundamental importance in the study of many-body classical and quantum systems. We investigate some of the issues related to these problems numerically in classical and quantum spin systems. New results reported in this thesis include: (i) A remarkably simple algorithm for discriminating chaotic from nonchaotic behavior in classical systems using a time series of one macroscopic observable. The effectiveness of this algorithm stems from the qualitative differences in the power spectra of chaotic and nonchaotic systems. (ii) A modified version of the Onsager regression relation applicable to pure quantum states. (iii) An efficient algorithm for computing the infinitetemperature time correlation functions in systems with large Hilbert spaces. (iv) Absence of exponential sensitivity to small perturbations in macroscopic nonintegrable systems of spins 1/2 . Such a behavior is contrasted with the exponential sensitivity to small perturbations in chaotic classical spin systems. (v) Accurate numerical investigations of free induction decay and spin diffusion in certain spin lattices. The consequences of these results have implications for the foundations of statistical mechanics or practical problems such as computing the long-time behavior of the free induction decay in solids.
One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (N→∞N \rightarrow \inftyN→∞) limit the density matrix emerges for small subsystems from almost all pure states of large quantum many-body systems. This notion of canonical is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. We show that quantum chaos plays a crucial role in the emergence of typicality for large but finite quantum systems. We demonstrate that the degree of typicality can be quantitatively controlled and tuned by the degree of quantum chaoticity present in the many-body system.
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.
Integrable quantum systems of finite size are generically robust against weak enough integrability-breaking perturbations, but become quantum chaotic and thermalizing if the integrability breaking is strong enough. We argue that the onset of quantum chaos can be described as a Fock-space delocalization process, with the eigenstates of the integrable system being taken as the Fock states. The integrability-breaking perturbation introduces hopping in this Fock space, and chaos sets in when this hopping delocalizes the many-body eigenstates in this space. Depending on the range of the dominant Fock-space hopping, delocalization can occur either through a crossover or via a transition that becomes sharp in the appropriate large-system dynamic limit. In either case, the perturbation strength at the onset of chaos scales to zero in the usual thermodynamic limit, with a size dependence that we estimate analytically and compute numerically for a few specific models. We also identify two intermediate finite-size-dependent regimes: There is generally an intermediate nonchaotic regime in which integrability is broken strongly enough to produce some systemwide many-body resonances but not enough to thermalize the system. In spatially extended systems (but not in quantum dots) there is also a crossover or transition between chaotic regimes where the ratio of the system size to the mean free path of the quasiparticles of the integrable system is small versus large compared to unity.