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Physical Review Letters

Experimental determination of billiard wave functions

J. Stein, H.‐J. Stöckmann

Published May 11, 1992• 245 citations
Publisher

Abstract

Wave functions of a stadium billiard are determined in a microwave analog experiment. It is shown that Gutzwiller's semiclassical representation of the Green's function in terms of classical trajectories can account not only for eigenvalue spectra but also for eigenfunction patterns.
DOI
10.1103/physrevlett.68.2867

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In a microwave transmission measurement the Green function $G({r,r}^{\ensuremath{'}},k)$ of a stadium billiard was determined as a function of ${r}^{\ensuremath{'}}$ and kkk for fixed rrr. From this billiard wave functions were obtained including the sign, which was not available earlier. The propagator $K({r,r}^{\ensuremath{'}},t)$ was obtained by Fourier transform of $G({r,r}^{\ensuremath{'}},k)$. It is shown that $K({r,r}^{\ensuremath{'}},t)$ supplies a very suggestive picture of pulse propagation including pulse reconstruction by focusing effects. This can be considered as an experimental verification of a work of Tomsovic and Heller, who found that semiclassical dynamics can account for the quantum mechanical behavior of billiards over surprisingly long times.
We propose a method of measuring approximate quantum eigenfunctions in polygonalized billiard geometries, based on a quasiclassical evolution operator having a (smoothened) Perron-Frobenius kernel modulated by a phase arising from quantum considerations. Using a plane wave ansatz, we show that the condition under which this is an eigenfunction of the quasiclassical operator is identical to the condition for it to be an eigenfunction of the Schrödinger equation for polygonalized billiards. Finally, we demonstrate this technique by determining the quasiclassical eigenfunctions of the polygonalized stadium billiard using arbitrary trajectories and comparing this with the exact quantum stadium eigenfunctions.
Single and double-slit experiments are performed with two microwave billiards with the shapes of a rectangle and a quarter stadium, respectively. The classical dynamics of the former is regular, whereas that of the latter is chaotic. Microwaves can leave the billiards via slits in the boundary, forming interference patterns on a screen. The aim is to determine the effect of the billiard dynamics on their structure. For this the development of a method for the construction of a directed wave packet by means of an array of multiple antennas was crucial. The interference patterns show a sensitive dependence not only on the billiard dynamics but also on the initial position and direction of the wave packet.
Until a short time ago, the study of the quantum-mechanical behaviour of classically chaotic systems was an exclusive domain of the theoretical physicist. Model systems amongst others were irregularly shaped billiards, such as the Sinai and the stadium billiard. An alternative experimental approach to study these systems has recently been demonstrated using the fact that the time-independent Schrödinger and wave equations are mathematically equivalent (though, in general, the boundary conditions are different). This chapter concentrates on the presentation of microwave studies of billiards, but experiments with vibrating plates and water surfaces waves are also briefly discussed. Topics are the statistical properties of spectra, level dynamics with respect to geometrical changes of the billiards, and the close connection between quantum-mechanical spectra and classical trajectories. Introduction One day in February 1808 Ernst F. Chladni who was then on a long stay in Paris received an invitation to the Tuileries to give a demonstration of his famous experiments with vibrating plates in the presence of Napoleon and an illustrious audience. The Emperor was impressed by the performance and offered a prize of 3000 francs for the correct mathematical explanation of the sound figures which was still lacking at that time. The prize was paid in 1816 to the French mathematician Sophie Germain, in spite of the fact that her solution was still incomplete. The correct explanation for circular plates was not given until 1850 by Robert Kirchhoff (all the above details are taken from the book of F. Melde on the life of Chladni [1]).
Two experiments with super- and normal conducting microwave cavities which are used as an analog to two-dimensional, infinitely deep quantum potentials are presented. The eigenvalues of such a potential can be measured directly through the resonance frequencies of the cavity, while the eigenvectors can be determined by measuring the field distributions inside the cavity. In the case of open systems another information – the imaginary part of the eigenvalues – is observable by measuring the widths of the resonances. As examples for this experimental approach to quantum chaos the semiclassical reconstruction of length spectra of billiards with varying chaoticity based of various trace formulas proposed by Gutzwiller, by Ullmo, Grinberg and Tomsovic and by Berry and Tabor for systems with chaotic, mixed and regular dynamics, respectively, and the first observation of square root singularities – so called exceptional points recently proposed by Heiss – in the energy spectrum of coupled cavities are presented.
We examine semiclassical propagation of nonstationary states. The fundamental object is the 1928 time dependent Van Vleck semiclassical Green`s function, as modified by Gutzwiller in 1967. This direct approach to semiclassical quantum mechanics has received very little attention over the years. The problems of evaluating the Green`s function (finding all trajectories which go from x{prime} to x in time t, for every x, x{prime}, and t sounds foreboding) and the development of caustics at an alarming rate in chaotic systems, may have suppressed its use. We have developed several strategies for evaluating it efficiently, including for chaotic dynamics. We found it can be used to accurately propagate smooth wave functions in spite of the caustics. The smooth wave function has the effect of strongly damping the caustics and leaving an accurate propagated state. Moreover many of the contributions to the semiclassical amplitude are legal, and allow a much longer time scale for breakdown of the semiclassical propagation than had previously been thought. This has allowed us to extract accurate spectra and eigenstates for chaotic dynamics in the stadium billiard and to calculate a complete molecular electronic absorption spectrum for two nonlinearly interacting vibrational modes, including line intensities and positions, usingmore » only classical trajectory input.« less