Intermediate statistics in singular quarter-ellipse shaped microwave billiards*

Abstract

We report on experiments with a flat, superconducting microwave billiard with the shape of a quarter ellipse simulating a singular billiard, that is, a quantum billiard containing zero-range perturbations. The pointlike scatterers were realized with long antennas. Their coupling to the microwaves inside the cavity depends on frequency. A complete sequence of 1013 eigenfrequencies was identified rendering possible the investigation of spectral properties as function of frequency. They exhibit intermediate statistics and are well described by analytical results derived by Bogomolny, Gerland, Giraud and Schmit for singular billiards with shapes that generate an integrable classical dynamics. This comparison revealed a quadratic frequency dependence of the coupling parameter. The size of the chaotic component induced by the diffractive effects of the scatterers was determined by comparison with analytical results derived by Haake and Lenz for an additive random-matrix model, which interpolates between the models applicable for quantum systems with an integrable and chaotic classical dynamics, respectively.