Current and Vorticity Auto Correlation Functions in Open Microwave Billiards

Using the equivalence between the quantum-mechanical probability density in a quantum billiard and the Poynting vector in the corresponding microwave system, current distributions were studied in a quantum dot like cavity, as well as in a Robnik billiard with λ = 0.4, and an introduced ferrite cylinder. Spatial auto correlation functions for currents and vorticity were studied and compared with predictions from the random-superposition-of-plane-waves hypothesis. In addition different types of vortex neighbour spacing distributions were determined and compared with theory. The large majority of wave functions of chaotic billiards is chaotic, i. e. at any point in the system, not too far from the wall, the wave function may be well described by a random-superposition of plane-waves (RSPW), 1) ψ(r) = n a n exp i kn r , (1) where modulus k = | k n | of the incoming wave is fixed, but directions k n /k and amplitudes are considered as random. As an immediate consequence the wave function amplitudes are Gaussian distributed, or, equivalently, their squares ρ = |ψ| 2 are Porter-Thomas distributed, P (ρ) = A 2πρ exp (− A 2 ρ), (2) where A is the billiard area. For the spatial correlation function of the wave function amplitudes one obtains a Bessel function, C(r 1 , r 2) = ψ * (r 1)ψ(r 2) | ψ(r) | 2 = J 0 (kr), (3) where r = | r 1 − r 2 |. The brackets denote an average over all positions. All these features have been demonstrated by McDonald and Kaufman in their influential work on stadium wave functions. 2), 3) It is impossible to mention all papers which have been published hitherto on the subject. The RSPW approach is not restricted to quantum mechanics. This is why experiments using classical waves have played an important role, since for a long time they were the only ones with the ability to look