Spectral Properties and Dynamical Tunneling in Constant-Width Billiards

TL;DR

This study measures the quantum eigenvalues of constant-width billiards, revealing dynamical tunneling effects through resonance splittings and demonstrating their agreement with a random-matrix model.

Contribution

It provides the first precise measurement of eigenvalues in constant-width billiards and models dynamical tunneling effects using a novel random-matrix approach.

Findings

Resonance spectra show doublets due to tunneling.

Spectral fluctuation properties match a random-matrix model.

Derived an analytical distribution for tunneling splittings.

Abstract

We determine with unprecedented accuracy the lowest 900 eigenvalues of two quantum constant-width billiards from resonance spectra measured with flat, superconducting microwave resonators. While the classical dynamics of the constant-width billiards is unidirectional, a change of the direction of motion is possible in the corresponding quantum system via dynamical tunneling. This becomes manifest in a splitting of the vast majority of resonances into doublets of nearly degenerate ones. The fluctuation properties of the two respective spectra are demonstrated to coincide with those of a random-matrix model for systems with violated time-reversal invariance and a mixed dynamics. Furthermore, we investigate tunneling in terms of the splittings of the doublet partners. On the basis of the random-matrix model we derive an analytical expression for the splitting distribution which is generally applicable to systems exhibiting dynamical tunneling between two regions with (predominantly) chaotic dynamics.