Born-Oppenheimer adiabatic mechanism for regularity of states in the quantum stadium billiard

Coupling of bouncing-ball modes to the chaotic sea and their counting function

We study the coupling of bouncing-ball modes to chaotic modes in two-dimensional billiards with two parallel boundary segments. Analytically, we predict the corresponding decay rates using the fictitious integrable system approach. Agreement with numerically determined rates is found for the stadium and the cosine billiard. We use this result to predict the asymptotic behavior of the counting function N_{bb}(E)∼E^{δ}. For the stadium billiard we find agreement with the previous result δ=3/4. For the cosine billiard we derive δ=5/8, which is confirmed numerically and is well below the previously predicted upper bound δ=9/10.

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

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.

Localization of Floquet states along a continuous line of periodic orbits

A periodically driven particle in an infinite square well is shown to exhibit quantum localization due to a continuous line of periodic orbits in the classical system. Individual Floquet eigenstates localized along this line of periodic orbits are identified. The enhanced localization persists for field strengths beyond that at which the continuous line of orbits is destroyed in the classical dynamics. These results may be relevant to experiments involving trapping potentials with flat regions.

Experimental test of a trace formula for a chaotic three-dimensional microwave cavity

We have measured resonance spectra in a superconducting microwave cavity with the shape of a three-dimensional generalized Bunimovich stadium billiard and analyzed their spectral fluctuation properties. The experimental length spectrum exhibits contributions from periodic orbits of nongeneric modes and from unstable periodic orbits of the underlying classical system. It is well reproduced by our theoretical calculations based on the trace formula derived by Balian and Duplantier for chaotic electromagnetic cavities.

Distribution of Husimi zeros in polygonal billiards

The zeros of the Husimi function provide a minimal description of individual quantum eigenstates and their distribution is of considerable interest. We provide here a numerical study for pseudointegrable billiards which suggests that the zeros tend to diffuse over phase space in a manner reminiscent of chaotic systems but nevertheless contain a subtle signature of pseudointegrability. We also find that the zeros depend sensitively on the position and momentum uncertainties (Delta q and Delta p, respectively) with the classical correspondence best when Delta q = Delta p = square root of [Planck's constant/2]. Finally, short-range correlations seem to be well described by the Ginibre ensemble of complex matrices.