The escape of an ensemble of particles from the Bunimovich stadium via a small hole has been studied numerically. The decay probability starts out exponentially but has an algebraic tail. The weight of the algebraic decay tends to zero for vanishing hole size. This behaviour is explained by the slow transport of the particles close to the marginally stable bouncing ball orbits. It is contrasted with the decay function of the corresponding quantum system.

Two superconducting microwave billiards have been electromagnetically coupled in a variable way. The spectrum of the entire system has been measured and the spectral statistics analyzed as a function of the coupling strength. It is shown that the results can be understood in terms of a random matrix model of quantum mechanical symmetry breaking -- as e.g. the violation of parity or isospin in nuclear physics.Comment: 4 pages, 5 figure

Based on very accurate measurements performed on a superconducting microwave resonator shaped like a desymmetrized three-dimensional (3D) Sinai billiard, we investigate for the first time spectral properties of the vectorial Helmholtz, i.e. non-quantum wave equation for a classically totally chaotic and theoretically precisely studied system. We are thereby able to generalize some aspects of quantum chaos and present some results which are consequences of the polarization features of the electromagnetic waves.PACS number(s): 05.45.+b, 41.20.Bt, 41.20.Jb For nearly 20 years, billiard systems have provided a very effective tool for the investigation of semiclassical quantization of conservative chaotic systems [1,2]. This is due to the fact that even two-dimensional (2D) billiards (as opposed to billiards of higher dimensionality) are able to model a wide range of fully ergodic systems in Gutzwiller's sense of "hard chaos" [3]. As a matter of fact, properties of the wave dynamical spectra of such low-dimensional but classically non-integrable systems are fully described by the Gaussian Orthogonal Ensemble (GOE) of Random Matrix Theory (RMT) [4,5] if the underlying motion is invariant under time-reversal. On the other hand, classically regular, i.e. integrable systems lead to totally uncorrelated spectra.Up to now investigations on chaotic 3D-billiards were performed in experiments with electromagnetic [6,7] and acoustic [8,9] waves, whereas the hardly feasible numerical modelling was restricted to very special geometries of high symmetry for the pure Schrödinger problem [10]. FIG. 1. Geometry of the desymmetrized 3D-Sinai billiard (boldface line) which constitutes one sixth of the dashed cube. Eight of those cubes form the full system. As indicated in the figure, one antenna was located in the center of each plane surface of the cavity.The goal of the present paper is to provide for the first time a detailed analysis of a fully chaotic three-dimensional (3D) electromagnetic billiard with a classically well-known and theoretically precisely studied geometry: the 3D-Sinai billiard resp its desymmetrized version given by 1/48 of a cube with a sphere in its center, see Fig.1. The system has to be described by the time-independent, fully vectorial Helmholtz equation with electromagnetic boundary conditions. The same geometry was recently investigated numerically in the quantum regime [10] described by the time-independent Schrödinger equation and experimentally with acoustic waves [9]. Our analysis therefore allows a very distinct comparison of totally different wave dynamical phenomena in a system with exactly the same classical analogue. Our results should agree with those of [10] and [9] if the conjecture holds that RMT is adequate to describe spectra of arbitrary wave phenomena.As several 2D-and 3D-billiards before [7,[11][12][13], the electromagnetic resonator was made of Niobium which becomes superconducting below 9.2 K. This feature tremendously increases the resolution of the measured spectra due to quality f...

We present first measurements on a superconducting three-dimensional, partly chaotic microwave billiard shaped like a small deformed cup. We analyze the statistical properties of the measured spectrum in terms of several methods originally derived for quantum systems like eigenvalue statistics and periodic orbits and obtain according to a model of Berry and Robnik 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 new bouncing ball formula for electromagnetic billiards.PACS number(s): 05.45.+b, 41.20.Bt, 84.40.Cb * Present address: