Quantum Mechanics of a Rotating Billiard

TL;DR

This paper investigates how a rotating square billiard transitions from integrable to chaotic behavior both classically and quantum mechanically, revealing changes in phase space, spectral statistics, and wavefunction symmetry as rotation effects increase.

Contribution

It provides a detailed analysis of the classical and quantum transition from integrability to chaos in a rotating billiard system, highlighting the correspondence between classical phase space and quantum spectral properties.

Findings

Classical phase space transitions from regular to chaotic motion as the parameter decreases.

Quantum spectral statistics shift from Poisson to Wigner distribution with increased chaos.

Wavefunction statistics indicate breakdown of time-reversal symmetry at lower parameter values.

Abstract

Integrability of a square billiard is spontaneously broken as it rotates about one of its corners. The system becomes quasi-integrable where the invariant tori are broken with respect to a certain parameter, $\lambda = 2E/\omega^{2}$ where E is the energy of the particle inside the billiard and $\omega$ is the angular frequency of rotation of billiard. We study the system classically and quantum mechanically in view of obtaining a correspondence in the two descriptions. Classical phase space in Poincar\'{e} surface of section shows transition from regular to chaotic motion as the parameter $\lambda$ is decreased. In the Quantum counterpart, the spectral statistics shows a transition from Poisson to Wigner distribution as the system turns chaotic with decrease in $\lambda$. The wavefunction statistics however show breakdown of time-reversal symmetry as $\lambda$ decreases.