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arXiv:nlin/0003055 · Progress of Theoretical Physics Supplement

Experimental Study of Generic Billiards with Microwave Resonators

Gregor Veble, Ulrich Kuhl, Marko Robnik, H.‐J. Stöckmann, Junxian Liu, Michael Barth

Published Jan 1, 2000• 8 citations
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Abstract

In this work we study the eigenstates and the energy spectra of a generic billiard system with the use of microwave resonators.This is possible due to the exact correspondence between the Schrödinger equation and the electric field equations of the lowest modes in thin microwave resonators.We obtain a good agreement between the numerical (exact) and experimental eigenstates, while the short range experimental spectral statistics show the expected Brody-like behaviour in this energy range, as opposed to the Berry-Robnik picture which is valid only in the semiclassical region of sufficiently small effective Planck's constant.

DOI
10.1143/ptps.139.283

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In this work we study the eigenstates and the energy spectra of a generic billiard system with the use of microwave resonators.This is possible due to the exact correspondence between the Schrödinger equation and the electric field equations of the lowest modes in thin microwave resonators.We obtain a good agreement between the numerical (exact) and experimental eigenstates, while the short range experimental spectral statistics show the expected Brody-like behaviour in this energy range, as opposed to the Berry-Robnik picture which is valid only in the semiclassical region of sufficiently small effective Planck's constant.
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We determine with unprecedented accuracy the lowest 900 eigenvalues of two quantum constant-width billiards from resonance spectra measured with flat, superconducting microwave resonators. While the classical dynamics of the constant-width billiards is unidirectional, a change of the direction of motion is possible in the corresponding quantum system via dynamical tunneling. This becomes manifest in a splitting of the vast majority of resonances into doublets of nearly degenerate ones. The fluctuation properties of the two respective spectra are demonstrated to coincide with those of a random-matrix model for systems with violated time-reversal invariance and a mixed dynamics. Furthermore, we investigate tunneling in terms of the splittings of the doublet partners. On the basis of the random-matrix model we derive an analytical expression for the splitting distribution which is generally applicable to systems exhibiting dynamical tunneling between two regions with (predominantly) chaotic dynamics.
Until about 1990 only a very small number of experiments on the quantum mechanics of chaotic systems existed, apart from the early studies of nuclear spectra [Por65]. In this context the experiments on hydrogen atoms in strong microwave fields by Bayfield and Koch [Bay74], and in strong magnetic fields by Welge and his group [Hol86, Mai86] have to be mentioned in particular. The studies of irregularly shaped microwave cavities by Stöckmann and Stein [Stö90] have introduced a new type of quantum chaos research. The microwave billiards and a number of variants to be discussed in this chapter are analogue systems, as they use the equivalence of the Helmholtz equation and the time independent Schrödinger equation. Whether there is a complete correspondence with quantum mechanics or not depends on the respective boundary conditions. As most of the phenomena discussed in the following are common to all types of waves, this does not reduce the conclusiveness of the analogue experiments. Starting with a historical review, the state of the art in billiard experiments is presented with emphasis on a general survey and the technical background. The results and their quantum mechanical implications will be presented later. The hydrogen experiments, too, will be described in the proper context. The discussion of mesoscopic systems is restricted to billiard-like structures such as antidot lattices [Wei91], quantum dots [Mar92], and tunnelling devices [Fro94].
We derive an explicit expression for the coupling constants of individual eigenstates of a closed billiard that is opened by attaching a waveguide. The Wigner time delay and the resonance positions resulting from the coupling constants are compared to an exact numerical calculation. Deviations can be attributed to evanescent modes in the waveguide and to the finite number of eigenstates taken into account. The influence of the shape of the billiard and of the boundary conditions at the mouth of the waveguide are also discussed. Finally we show that the mean value of the dimensionless coupling constants tends to the critical value when the eigenstates of the billiard follow random-matrix theory.