Spectrum of the Andreev Billiard and Giant Fluctuations of the Ehrenfest Time

TL;DR

This paper investigates how classical chaos influences the quantum density of states in Andreev billiards, revealing giant fluctuations in the Ehrenfest time linked to rare trajectories with small Lyapunov exponents.

Contribution

It introduces a theoretical analysis connecting classical Lyapunov exponent fluctuations to quantum spectral features in Andreev billiards, highlighting the role of rare trajectories.

Findings

Density of states depends on fluctuations of classical Lyapunov exponent.

Rare trajectories with small Lyapunov exponent cause anomalous Ehrenfest time increases.

Spectrum gap varies with the position of the superconducting lead.

Abstract

The density of states in the semiclassical Andreev billiard is theoretically studied and shown to be determined by the fluctuations of the classical Lyapunov exponent $\lambda$. The rare trajectories with a small value of $\lambda$ give rise to an anomalous increase of the Ehrenfest time $\tau_E\approx |\ln\hbar|/\lambda$ and, consequently, to the appearance of Andreev levels with small excitation energy. The gap in spectrum is obtained and fluctuations of the value of the gap due to different positions of superconducting lead are considered.