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Physical Review E

Weyl formula: Experimental test of ray splitting and corner corrections

C. Vaa, Patrick M. Koch, R. Blümel

Published Nov 17, 2005• 12 citations

Abstract

The number of resonances N(f) of a resonator below frequency f is an essential concept in physics. Smooth approximations N(f) are known as Weyl formulas. An abrupt change in the properties of the wave propagation medium in a resonator was predicted by Prange [Phys. Rev. E 53, 207 (1996)] to produce a universal ray-splitting correction to N(f). We confirm this effect experimentally. Our results with a quasi-two-dimensional dielectric-loaded microwave cavity are directly relevant to the ray-splitting correction in two-dimensional quantal ray-splitting billiards. Our experimental spectra have sufficient accuracy and extent to allow, as far as we are aware, the first experimental determination of the corner correction, which we find to agree with theory. We show that our movable-bar setup enhances non-Newtonian periodic orbits, thereby providing an experimental technique for periodic-orbit spectroscopy. This technique, differential spectroscopy, will facilitate the study of non-Newtonian classical physics.

DOI: 10.1103/physreve.72.056211

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The number of resonances N(f) of a resonator below frequency f is an essential concept in physics. Smooth approximations N(f) are known as Weyl formulas. An abrupt change in the properties of the wave propagation medium in a resonator was predicted by Prange [Phys. Rev. E 53, 207 (1996)] to produce a universal ray-splitting correction to N(f). We confirm this effect experimentally. Our results with a quasi-two-dimensional dielectric-loaded microwave cavity are directly relevant to the ray-splitting correction in two-dimensional quantal ray-splitting billiards. Our experimental spectra have sufficient accuracy and extent to allow, as far as we are aware, the first experimental determination of the corner correction, which we find to agree with theory. We show that our movable-bar setup enhances non-Newtonian periodic orbits, thereby providing an experimental technique for periodic-orbit spectroscopy. This technique, differential spectroscopy, will facilitate the study of non-Newtonian classical physics.

Ray splitting is a universal phenomenon that occurs in all wave systems with sharp interfaces.Quantum implications of ray splitting are: (i) the importance of non-Newtonian orbits for the density of states in the semiclassical limit, (ii) ray-splitting corrections to the average density of states and (iii) the need to include non-Newtonian orbits in trace formulas for the oscillating part of the density of states.The signatures of non-Newtonian orbits in the density of states have recently been identified experimentally (L.

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.

Recent advances in the theory of the quasiclassical approximation for systems that are chaotic in the classical limit are extended to the case of ray splitting, in particular, to the splitting of an incident ray into a reflected and refracted component at a sharp interface. An instructive example is presented and novel results are found. These include evidence for ray split and periodic orbits in the spectral correlations and a new type of ``scarred'' eigenstate based on combining nonisolated periodic orbits whose quasiclassical contributions have a nontrivial phase from total internal reflection.

In the familiar concept of ray splitting, a ray incident on an interface splits into reflected and transmitted rays. Tunnelling adds something new to ray splitting: an incident Newtonian ray splits into a reflected Newtonian ray and a transmitted ray with complex classical action?a ghost orbit. Using novel periodic-orbit expansion techniques, we are able to compute explicitly and analytically energy eigenvalues of an energy-scaling one-dimensional step potential. Inclusion of all periodic orbits in our series expansions, in particular a class of orbits called pure ghosts with imaginary classical action, makes our formulae exact. We suggest an experiment that may be used to verify the importance of pure ghosts.

Ray splitting is the phenomenon whereby a ray incident on a boundary splits into more than one ray traveling away from the boundary. The most common example of this is the situation, originally considered by Snell in 1621, in which an incident light ray splits into reflected and transmitted rays at a discontinuity in refractive index. This paper seeks to extend techniques and results from quantum chaos to short wavelength problems in which ray splitting surfaces are present. These extensions are tested using a simple model problem for the Schr\"odinger equation in two dimensions with a finite step potential discontinuity. Numerical solutions for the energy spectrum and eigenfunctions in this system are then compared with predictions based on quasiclassical theoretical results suitably extended to include ray splitting. Among the topics treated are the ray orbits for our problem, energy level statistics, scars, trace formulas, the quasiclassical transfer operator technique, and the effect of lateral waves. It is found that these extensions of quantum chaos are very effective for treating problems with ray splitting.