Semiclassical Limit of Resonance States in Chaotic Scattering

TL;DR

This paper demonstrates how classical dynamics can describe the multifractal structure of quantum resonance states in chaotic scattering systems in the semiclassical limit, connecting quantum and classical chaos.

Contribution

It generalizes Ulam's matrix approximation to describe resonance states with various decay rates in open quantum systems.

Findings

Resonance states converge to classical measures in the semiclassical limit

Numerical examples include dielectric cavities and open quantum maps

A criterion for selecting relevant matrix approximations is proposed

Abstract

Resonance states in quantum chaotic scattering systems have a multifractal structure that depends on their decay rate. We show how classical dynamics describes this structure for all decay rates in the semiclassical limit. This result for chaotic scattering systems corresponds to the well-established quantum ergodicity for closed chaotic systems. Specifically, we generalize Ulam's matrix approximation of the Perron-Frobenius operator, giving rise to conditionally invariant measures of various decay rates. There are many matrix approximations leading to the same decay rate and we conjecture a criterion for selecting the one relevant for resonance states. Numerically, we demonstrate that resonance states in the semiclassical limit converge to the selected measure. Example systems are a dielectric cavity, the three-disk scattering system, and open quantum maps.