Bouncing ball orbits and symmetry breaking effects in a three-dimensional chaotic billiard

Abstract

We study the classical and quantum mechanics of a three-dimensional stadium billiard. It consists of two quarter cylinders that are rotated with respect to each other by 90 degrees, and it is classically chaotic. The billiard exhibits only a few families of nongeneric periodic orbits. We introduce an analytic method for their treatment. The length spectrum can be understood in terms of the nongeneric and unstable periodic orbits. For unequal radii of the quarter cylinders the level statistics agree well with predictions from random matrix theory. For equal radii the billiard exhibits an additional symmetry. We investigated the effects of symmetry breaking on spectral properties. Moreover, for equal radii, we observe a small deviation of the level statistics from random matrix theory. This led to the discovery of stable and marginally stable orbits, which are absent for unequal radii.

The chaotic set and the cross section for chaotic scattering in three degrees of freedom

“…Yet the results are certainly useful for semi-classical considerations, such as e.g. the ones used in [27] to design and interpret a microwave experiment.…”

Systematic Determination of Eigenfields in Time and in Frequency Domain

“…10. From this figure, it is clearly observable that the different simulation scenarios yield the expected Wigner distribution for the level-spacing analysis [27].…”

“…The measurement data of the 3-D stadium billiard are kindly provided for comparison reasons by the institute for nuclear physics at the TU Darmstadt. In addition, the experimental investigations have been published in [2] and [27].…”

Universal behaviour of a wave chaos based electromagnetic reverberation chamber

“…Quantum or Wave chaos has been a very active field of research for decades, concerned with the spectral and spatial asymptotic properties of wave systems whose ray counterpart is chaotic [13,14,15]. Predictions of Random Matrix Theory (RMT) have been extensively verified, both numerically and experimentally in two-dimensional (2D) or pseudo-2D electromagnetic cavities [16,17,15,18,19] as well as in the 3D case [20,21,22,23,24]. Previous works have already called for the similitude between the expected statistical properties a well-stirred RC and the intrinsic behaviour of individual ergodic modes of a chaotic cavity, to propose strategies for improved operation of an EM RC [7,25,26].…”

“…In the present paper, we demonstrate that the description of the electromagnetic field as a continuous plane wave-spectrum [6,4] can only be justified at high frequencies where modal overlap is large, which is not necessarily the case near the LUF according to the commonly accepted definitions of the latter recalled above. Thus, the main reason why a conventional mode-stirred RC can operate satisfactorily near the LUF is that the presence of the stirrer makes it more like a pseudo-integrable system, equivalent to a barrier 2D-billiard [27], where many eigenmodes are similar to ergodic modes of a chaotic cavity [28,29,23]. Nevertheless, even at high frequencies, pseudo-integrable cavities have a non negligible number of superscarmodes [29], with non Gaussian field distributions.…”