Quantum ergodicity and mixing and their classical limits with quantum kicked rotor

TL;DR

This paper investigates quantum ergodicity and mixing in the quantum kicked rotor using eigen-energy analysis and quantum Poincaré sections, establishing a link to classical behavior in the semiclassical limit.

Contribution

It introduces two approaches to study quantum ergodicity and mixing in QKR, connecting quantum properties with classical dynamics in the semiclassical limit.

Findings

Quantum ergodicity and mixing are captured by eigen-energy and quantum Poincaré section methods.

Results are consistent with classical ergodicity and mixing as the effective Planck constant diminishes.

A correspondence between quantum and classical ergodic/mixing behavior is established.

Abstract

We study the ergodicity and mixing of quantum kicked rotor (QKR) with two distinct approaches. In one approach, we use the definitions of quantum ergodicity and mixing recently proposed in [Phys. Rev. E 94, 022150 (2016)], which involve only eigen-energies (Floquet quasi-energies for QKR). In the other approach, we study ergodicity and mixing with quantum Poincar\`e section, which is plotted with a method that maps a wave function unitarily onto quantum phase space composed of Planck cells. Classical Poincar\`e section can be recovered with the effective Planck constant gradually diminishing. We demonstrate that the two approaches can capture the quantum and classical characteristics of ergodicity and mixing of QKR, and give consistent results with classical model at semiclassical limit. Therefore, we establish a correspondence between quantum ergodicity (mixing) and classical ergodicity (mixing).