Experimental Wavefunctions

Here is a gallery of eigenfunctions of Sinai (1,2) billiard, and Disordered billiards: 1, 2, 3
Chaotic geometries Localization

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.
Some of the issues addressed in these experiments are:

Scars which were predicted by Heller in 1983 were first observed by us in 1991 (Phys. Rev. Lett. , 67, 785 (1991)).

Observation of Porter-Thomas distribution and fluctuations in eigenfunctions of chaotic billiards.(Phys. Rev. Lett., 75, 822 (1995))

Experimental studies of correlations of chaotic and disordered eigenfunctions and comparison with supersymmetry nonlinear sigma models. (Phys. Rev. Lett., 85, 2360 (2000)).

The role of chaos and impurity scattering is strikingly evident in this summary picture below. The chaotic geometry (middle) shows essentially a "random" distribution of the wavefunction density, with a finite probability of large intensities, in contrast to the rectangle, where intensity distribution is "smoother". Indeed we have shown that the denisty distribution in the chaotic geometry obeys the Porter-Thomas law. Upon introduction of impurities in the disordered billiards, we see a new effect, localization (see later examples).

Click on the image to see a higher resolution JPEG image.

Chaotic Geometries

Sinai-Stadium

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.

36 disorder sites

Localization in Disordered Geometries

The basic geometry is a 44 x 21.8 cm rectangle with 1 cm square or circular tiles located randomly inside the rectangle. The tiles act as impurity scatterers (indicated by dots), and hence the geometry simulates a 2-D electron system with impurities. Sample wavefunctions for two such geometries with different amounts of ``impurities" are shown. In a disordered geometry there are three length scales, (a) the size of the geometry, L, (b) the mean free path, l, and (c) the wave length of the resonance frequency which is used to probe the system.
The most striking observation in the disordered geometries is the observation of localization, as predicted by Anderson. This effect is more pronounced in the lower eigenfunctions which are shown above.

72 disorder sites

Wavefunctions of a disordered geometry (left) and chaotic geometry.

In these set of experiments, the effects of localization and chaos on the wavefunctions of chaotic and disordered geometries were addressed. As is expected the lower wavefunctions are more localized then the higher ones.