Experimental Test of a Trace Formula for a Chaotic Three-Dimensional Microwave Cavity

Abstract

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.

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

“…5 we show the nearest-neighbor spacing distribution and the Σ 2 -statistics for a ratio of radii r 1 /r 2 = √ 2, i.e. for the case considered in the experiments with the microwave cavity of the shape of a three-dimensional stadium billiard [22]. Both curves agree well with the corresponding ones for random matrices from the GOE.…”

“…[16]. The experimental study [22] of the three-dimensional Bunimovich stadium led to a verification of the trace formula proposed by Balian and Duplantier [23] for electromagnetic systems. A self-bound three-body system with high-dimensional scars was studied theoretically in Ref.…”

“…For the evaluation of the resonance density in three-dimensional microwave resonators, Gutzwiller's trace formula does not apply, and one has instead to use the trace formula by Balian and Duplantier. The applicability of this periodic orbit sum was recently investigated and confirmed for a three-dimensional stadium billiard with chaotic dynamics [22]. It is the purpose of the present work to study the quantum mechanical aspects of the three-dimensional stadium billiard further.…”

“…First, the classical dynamics is strongly chaotic and no stable islands are known for r 1 = r 2 . In this work, we focus particularly on the case r 1 = √ 2r 2 since this geometry was studied in microwave experiments [22] and in Ref. [26].…”

“…[24]. Problems involving mode coupling in three-dimensional systems were investigated in vibrating crystals [11] and also in a microwave resonator [22].…”