Measuring Experimental Eigenvalues and Eigenfunctions

In closed or open billiard geometries, Maxwell’s equations can be written as (Ѳ + k²) Ψ = 0, where in general Ψ = {Ε_i, Β_j} 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 (Β_z = 0) reduces to the time independent Schrodinger equation (Ѳ + k²) Ψ = 0, for frequencies ƒ < c/2d, where d is the plate separation. This QM-E&M mapping with Ψ = Ε_z 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.

The experiments are carried out using thin copper cavities which can be shaped in essentially arbitrary geometries. Metallic objects provide Dirichlet boundary conditions, absorbing materials can be used for absorbing boundary conditions for open systems, and potential wells can be created using dielectrics. The eigenvalues are obtained by observing the transmission amplitude of the microwaves fields as a function of frequency, and locating the peaks. Inductive coupling was used to minimize perturbing the fields inside the cavity. There is a width associated with each peak due to absorption in the copper walls. However if sufficient care is taken, more than 1000 levels can be measured accurately.

Experiments were conducted with Chaotic geometries shaped in the form of the Sinai Billiard and the Sinai-Stadium Billiard. This was done in order to study universal behavior in the eigenvalue statistics, and deviations from it, due to non-isolated periodic orbits (e.g. the bouncing ball periodic orbit in the Sinai Billiard). A pseudo-integrable geometry was studied as an example of an intermediate geometry between chaotic and integrable.

The Sinai-Stadium geometry, shows very good agreement with random matrix theory. While the Sinai Billiard geometry shows near agreement in the spacing statistics of the energy levels, clear disagreement is seen in statistical measures such as the spectral rigidity, which measure long range correlations. These deviations are attributed to the bouncing ball periodic orbits.