Chaotic dynamics in a three-dimensional superconducting microwave billiard

We present measurements on a superconducting three-dimensional, partly chaotic microwave billiard shaped such as a small deformed cup. We analyze the statistical properties of the measured spectrum in terms of several methods originally derived for quantum systems such as eigenvalue statistics and periodic orbits and obtain, according to a model of Berry and Robnik [J. Phys. A 17, 2413 (1984)], a mixing parameter of about 25%. In numerical simulations of the classical motion in the cup, the degree of chaoticity has been estimated. This leads to an invariant chaotic Liouville measure of about 45%. The difference between this figure and the mixing parameter is due to the limited accuracy of the statistical analysis, caused by both the fairly small number of 286 resonances and the rather poor desymmetrization of the microwave cavity. Concerning the periodic orbits of the classical system, we present a comparison with the length spectrum of the resonator and introduce a bouncing ball formula for electromagnetic billiards.