Playing Billiards with Microwaves — Quantum Manifestations of Classical Chaos

Abstract

Quantum manifestations of classical chaos in systems with few degrees of freedom can be studied experimentally. In sufficiently flat microwave resonators, Maxwell’s equations reduce to the Schrödinger equation for a free particle, and the condition of classical chaos is realized by properly shaping the corresponding cavity. Here the physics potential of such experiments using superconducting microwave resonators with Q values of Q ≈ 105 – 107 for which the complete spectrum of eigenvalues can be measured is briefly summarized and illustrated by way of selected examples. In particular, with experimental results from the Bunimovich stadium billiard, the application of Gutzwiller’s semiclassical approach is shown to yield locations and strengths of the peaks in the Fourier-transformed spectrum in terms of the shortest unstable classical periodic orbits. In addition the usual statistical analysis of the spectrum of eigenmodes is made and the results are compared to those of the Gaussian orthogonal ensemble (GOE), the standard stochastic model for time-reversal invariant systems with classical chaotic dynamics, and the deviations which are found are successfully explained and quantitatively described by the semiclassical approach. As a further test of the spectral fluctuation properties of the system, the decay amplitudes of the resonances in the superconducting stadium billiard coupled to three antennas are discussed. Both the quality of the data and the statistical accuracy afforded by the large number of resonances makes this data set one of the most comprehensive tests yet for the GOE fluctuation properties of decay amplitudes and, hence, wave functions in a classical chaotic system of adjustable dimensions. Finally, other presently experimentally investigated billiard systems (hyperbola, Pascalian snails, circle, 3D-Sinai, coupled Bunimovich stadiums, 20-cell Bloch-lattice) for the study of various features like fully vs. partly chaotic systems, the transition from regular to chaotic behavior, the determination of wave functions and tunneling phenomena, the aspects of chaos in three-dimensional systems, the spectral features of billiards coupled with varying strength and first results on Anderson localization in a simple Bloch-like lattice, respectively, are introduced.