Semiclassical Description of Tunneling in Mixed Systems: The Case of the Annular Billiard

TL;DR

This paper develops a semiclassical framework to understand quantum tunneling in mixed systems, specifically in the annular billiard, by relating state splittings to complex paths involving chaos-assisted tunneling.

Contribution

It introduces a semiclassical approach extending billiard boundaries into complex space to analytically describe tunneling splittings in the annular billiard.

Findings

Analytical expressions for tunneling splittings are derived.

Chaos-assisted paths dominate the tunneling process.

Complex ray extensions provide a semiclassical interpretation.

Abstract

We study quantum-mechanical tunneling between symmetry-related pairs of regular phase space regions that are separated by a chaotic layer. We consider the annular billiard, and use scattering theory to relate the splitting of quasi-degenerate states quantized on the two regular regions to specific paths connecting them. The tunneling amplitudes involved are given a semiclassical interpretation by extending the billiard boundaries to complex space and generalizing specular reflection to complex rays. We give analytical expressions for the splittings, and show that the dominant contributions come from {\em chaos-assisted}\/ paths that tunnel into and out of the chaotic layer.