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

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).