Adiabatic quantization of Andreev levels

TL;DR

This paper studies the quantization of Andreev levels in a chaotic cavity, identifying the classical adiabatic invariant as the time between reflections, leading to a discrete energy spectrum and wave function squeezing.

Contribution

It introduces a novel approach to quantize Andreev levels using adiabatic invariants and describes the resulting energy spectrum and wave function properties.

Findings

Quantized periods form an energy ladder with specific levels.

Wave functions are squeezed to a transverse dimension much smaller than the contact width.

The largest quantized period relates to the Ehrenfest time, linking classical chaos to quantum states.

Abstract

We identify the time $T$ between Andreev reflections as a classical adiabatic invariant in a ballistic chaotic cavity (Lyapunov exponent $\lambda$), coupled to a superconductor by an $N$-mode point contact. Quantization of the adiabatically invariant torus in phase space gives a discrete set of periods $T_{n}$, which in turn generate a ladder of excited states $\epsilon_{nm}=(m+1/2)\pi\hbar/T_{n}$. The largest quantized period is the Ehrenfest time $T_{0}=\lambda^{-1}\ln N$. Projection of the invariant torus onto the coordinate plane shows that the wave functions inside the cavity are squeezed to a transverse dimension $W/\sqrt{N}$, much below the width $W$ of the point contact.