Semiclassical gaps in the density of states of chaotic Andreev billiards

TL;DR

This paper uses semiclassical analysis to explain the formation and size of spectral gaps in chaotic Andreev billiards, highlighting the effects of classical orbit correlations and Ehrenfest time on the density of states.

Contribution

It introduces a semiclassical framework linking classical trajectories to spectral gaps and predicts a second gap caused by finite Ehrenfest time effects.

Findings

Classical orbit correlations lead to the formation of a hard spectral gap.

Finite Ehrenfest time causes the gap to shrink.

A second spectral gap appears at a specific energy related to Ehrenfest time.

Abstract

The connection of a superconductor to a chaotic ballistic quantum dot leads to interesting phenomena, most notably the appearance of a hard gap in its excitation spectrum. Here we treat such an Andreev billiard semiclassically where the density of states is expressed in terms of the classical trajectories of electrons (and holes) that leave and return to the superconductor. We show how classical orbit correlations lead to the formation of the hard gap, as predicted by random matrix theory in the limit of negligible Ehrenfest time $\tE$, and how the influence of a finite $\tE$ causes the gap to shrink. Furthermore, for intermediate $\tE$ we predict a second gap below $E=\pi\hbar /2\tE$ which would presumably be the clearest signature yet of $\tE$-effects.