Level spacing statistics in a randomly-inhomogeneous acoustic waveguide

TL;DR

This paper investigates the statistical distribution of energy level spacings in a randomly-inhomogeneous acoustic waveguide, revealing a transition from Poissonian to Wigner-like statistics over time, with frequency-dependent behaviors linked to classical orbit bifurcations.

Contribution

It introduces a quantum transfer operator for analyzing level spacing statistics in a randomly-perturbed waveguide, connecting classical bifurcations to quantum spectral properties.

Findings

Level spacing distribution transitions from Poissonian to Wigner-like at 200 Hz.

At 600 Hz, the distribution becomes non-universal due to nearly-degenerate levels.

Nearly-degenerate levels are caused by bifurcations of classical periodic orbits.

Abstract

Dynamics of a randomly-perturbed quantum system with 3/2-degrees of freedom is considered. We introduce a transfer operator being the quantum analogue of the specific Poincar\'e map. This map was proposed in (Makarov, Uleysky, J. Phys. A: Math. Gen., 2006) for exploring domains of finite-time stability, which survive under random excitation. Our attention is concentrated on level spacing distribution of the transfer operator, averaged over ensemble of realizations. The problem of sound propagation in an oceanic waveguide is considered as an example. For the acoustic frequency of 200 Hz, level spacing distribution undergoes the crossover from Poissonian to Wigner-like statistics with increasing time, as it is consistent with classical predictions via the specific Poincar\'e map. For the frequency of 600 Hz, the level spacing statistics becomes non-universal due to large number of nearly-degenerate levels whose contribution grows with time. Occurrence of nearly-degenerate levels is attributed to bifurcations of classical periodic orbits, caused by fast spatial oscillations of the random perturbation.