Bound states in Andreev billiards with soft walls

TL;DR

This paper investigates the energy spectrum and eigenstates of a quantum dot with soft walls in contact with a superconductor, using both quantum mechanical and semiclassical methods, revealing classical orbit features in quantum states.

Contribution

It introduces a semiclassical Bohr--Sommerfeld quantization extension tailored for Andreev billiards with parabolic potential walls, linking classical orbits to quantum wavefunction features.

Findings

Classical periodic electron-hole orbits are identified in quantum wavefunctions.

The semiclassical extension accurately describes Andreev billiards with soft walls.

Quantum solutions show 'scar' like features corresponding to classical orbits.

Abstract

The energy spectrum and the eigenstates of a rectangular quantum dot containing soft potential walls in contact with a superconductor are calculated by solving the Bogoliubov-de Gennes (BdG) equation. We compare the quantum mechanical solutions with a semiclassical analysis using a Bohr--Sommerfeld (BS) quantization of periodic orbits. We propose a simple extension of the BS approximation which is well suited to describe Andreev billiards with parabolic potential walls. The underlying classical periodic electron-hole orbits are directly identified in terms of ``scar'' like features engraved in the quantum wavefunctions of Andreev states determined here for the first time.