Statistical mechanics of Floquet systems with regular and chaotic states

TL;DR

This paper analyzes the long-term occupation probabilities of regular and chaotic Floquet states in periodically driven quantum systems coupled to a heat bath, deriving analytical expressions and revealing distinct distribution patterns.

Contribution

It introduces an analytical framework for occupations of regular states in Floquet systems, including a Boltzmann-like formula with an effective temperature, and studies various Floquet state types and avoided crossing effects.

Findings

Regular states have occupation probabilities decreasing from the center outward.

Chaotic states have nearly equal occupation probabilities.

An analytical expression for regular state occupations depends on winding numbers and effective temperature.

Abstract

We investigate the asymptotic state of time-periodic quantum systems with regular and chaotic Floquet states weakly coupled to a heat bath. The asymptotic occupation probabilities of these two types of states follow fundamentally different distributions. Among regular states the probability decreases from the state in the center of a regular island to the outermost state by orders of magnitude, while chaotic states have almost equal probabilities. We derive an analytical expression for the occupations of regular states of kicked systems, which depends on the winding numbers of the regular tori and the parameters temperature and driving frequency. For a constant winding number within a regular island it simplifies to Boltzmann-like weights $\exp(-\betaeff \Ereg_m)$, similar to time-independent systems. For this we introduce the regular energies $\Ereg_m$ of the quantizing tori and an effective winding-number-dependent temperature $1/\betaeff$, different from the actual bath temperature. Furthermore, the occupations of other typical Floquet states in a mixed phase space are studied, i.e. regular states on nonlinear resonances, beach states, and hierarchical states, giving rise to distinct features in the occupation distribution. Avoided crossings involving a regular state lead to drastic consequences for the entire set of occupations. We introduce a simplified rate model whose analytical solutions describe the occupations quite accurately.