Excitation spectrum of Andreev billiards with a mixed phase space

TL;DR

This paper develops a semiclassical theory to analyze the excitation spectrum of Andreev billiards with mixed phase space, showing how classical phase space structures influence quantum energy levels.

Contribution

It introduces a novel semiclassical approach for mixed phase space Andreev billiards and validates it with quantum calculations, revealing the impact of regular regions on the excitation gap.

Findings

Good agreement between semiclassical theory and quantum calculations.

The excitation gap decreases when coupling to regular regions increases.

Quantum energy scales relate simply to phase space morphology.

Abstract

We present a semiclassical theory for the excitation spectrum of a ballistic quantum dot weakly coupled to a superconductor, for the generic situation that the classical motion gives rise to a phase space containing islands of regularity in a chaotic sea. The density of low-energy excitations is determined by quantum energy scales that are related in a simple way to the morphology of the mixed phase space. An exact quantum mechanical computation for the annular billiard shows good agreement with the semiclassical predictions, in particular for the reduction of the excitation gap when the coupling to the regular regions is maximal.