Wavefunctions in Chaotic and Disordered Geometries

The Sinai-Stadium billiard is a prototype example of chaotic geometries. Representative eigenfunctions of this geometry are shown here. The influence of the classical dynamical structures such as periodic orbits can be seen in some of the wavefunctions. The Sinai-Stadium was designed to have no non-isolated periodic orbits, which leads to deviation from universality.
integratability, chaos and disorder in wavefunctions

Integrability, Chaos and Disorder in Wavefunctions

A unique feature of our experiments is the ability to directly measure eigenfunctions, i.e. the spatial distribution of the waves, in essentially arbitrary geometries. The wavefunctions are obtained using a cavity perturbation technique developed by Sridhar. In this technique, a small metal bead is introduced inside the cavity. If the bead is sufficiently small compared to the wavelength, the resultant shift in frequency due to the perturbation is proportional to the square of the Electric field (hence the wavefunction), at the location of the bead. By moving the bead with a magnet, the wavefunction can be mapped out.

Not Hearing the Shape of Drums

While the theorem of isospectral domains was proved on abstract mathematical grounds, the actual resonance frequencies and wave functions of the geometries were not known. The microwave experiments were able to make an unique contribution by exploiting the the fact that below a certain frequency, the equation obeyed in thin microwave cavities is the same as the Helmholtz wave equation. The resonances were obtained by actually constructing the shapes using copper. The wavefunctions are obtained using a cavity perturbation technique. Therefore these experiments were not only able to physically realize drums which sounded alike, but also were able to show that there were no degeneracies in the first 54 resonances.
Quantum Chaos

Measuring Experimental Eigenvalues and Eigenfunctions

In closed or open billiard geometries, Maxwell's equations can be written as (Ñ2 + k2) Ψ = 0, where in general Ψ = {Εi, Βj} is a vector field. In thin 2-D geometries bounded by parallel metallic plates in the x-y plane, the wave equation for the TM modes (Βz = 0) reduces to the time independent Schrodinger equation (Ñ2 + k2) Ψ = 0, for frequencies ƒ < c/2d, where d is the plate separation. This QM-E&M mapping with Ψ = Εz allows us to study 2-D problems in Quantum Chaos by suitably constructing classically chaotic geometries. Eigenvalues, eigenfunctions, scattering resonances and widths are measured and analyzed, yielding insights towards the quantum classical correspondence.