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arXiv:cond-mat/0205042 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Current and vortex statistics in microwave billiards

Michael Barth, H.‐J. Stöckmann

Published Jun 20, 2002• 40 citations
arXivHTMLPDF

Abstract

Using the one-to-one correspondence between the Poynting vector in a microwave billiard and the probability current density in the corresponding quantum system, probability densities and currents were studied in a microwave billiard with a ferrite insert as well as in an open billiard. Distribution functions were obtained for probability densities, currents, and vorticities. In addition, the vortex pair correlation function could be extracted. For all studied quantities a complete agreement with the predictions from the approach using a random superposition of plane waves was obtained.

DOI
10.1103/physreve.65.066208

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•Oct 1, 2010•1 citation
Using the one-to-one correspondence between the Poynting vector in a microwave billiard and the probability current density in the corresponding quantum system, probability densities and currents were studied in a microwave billiard with a ferrite insert as well as in an open billiard. Distribution functions were obtained for probability densities, currents, and vorticities. In addition, the vortex pair correlation function could be extracted. For all studied quantities a complete agreement with the predictions from the approach using a random superposition of plane waves was obtained.
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.
This paper reports on a joint theoretical and experimental study of the Pauli quantum-mechanical stress tensor T_{alphabeta}(x,y) for open two-dimensional chaotic billiards. In the case of a finite current flow through the system the interior wave function is expressed as psi=u+iv . With the assumption that u and v are Gaussian random fields we derive analytic expressions for the statistical distributions for the quantum stress tensor components T_{alphabeta} . The Gaussian random field model is tested for a Sinai billiard with two opposite leads by analyzing the scattering wave functions obtained numerically from the corresponding Schrödinger equation. Two-dimensional quantum billiards may be emulated from planar microwave analogs. Hence we report on microwave measurements for an open two-dimensional cavity and how the quantum stress tensor analog is extracted from the recorded electric field. The agreement with the theoretical predictions for the distributions for T_{alphabeta}(x,y) is quite satisfactory for small net currents. However, a distinct difference between experiments and theory is observed at higher net flow, which could be explained using a Gaussian random field, where the net current was taken into account by an additional plane wave with a preferential direction and amplitude.
A basic quantum-mechanical model for wave functions and current flow in open quantum dots or billiards is investigated. The model involves non-Hertmitian quantum mechanics, parity-time (PT) symmetry, and PT-symmetry breaking. Attached leads are represented by positive and negative imaginary potentials. Thus probability densities, currents flows, etc., for open quantum dots or billiards may be simulated in this way by solving the Schrödinger equation with a complex potential. Here we consider a nominally open ballistic quantum dot emulated by a planar microwave billiard. Results for probability distributions for densities, currents (Poynting vector), and stress tensor components are presented and compared with predictions based on Gaussian random wave theory. The results are also discussed in view of the corresponding measurements for the analogous microwave cavity. The model is of conceptual as well as of practical and educational interest.
Am agnetic flux through a mesoscopic metallic ring gives rise to a persistent current which can be detected via characteristic oscillations of the magnetization depending on the applied field (Let al 1990 Phys. Rev. Lett. 64 2074). In this paper a direct visualization of such persistent currents in am icrowave analogue experiment is reported, making use of the analogy between the probability density current in the quantum-mechanical system and the Poynting vector in the corresponding electromagnetic one. To break time-reversal symmetry, a small ring of a ferrite material in a static external magnetic field was introduced into the resonator. In our analysis of the experimental data we employ the off-diagonal elements of the scattering matrix. Due to the small size of the ferrite compared to the resonator size the symmetry was partially broken in the sense that the real and the imaginary parts of the wavefunctions are not equally large on average. The statistical properties of the wavefunction's real and imaginary parts Re(ψ) ,I m(ψ) ,r espectively, as well as the distributions of the total wavefunction's amplitudes |ψ| and the statistical properties of the probability density currents were also examined and compared to the theoretical predictions.
lquotlquotlquotPersistent currentsrquotrquotrquot and eigenfunctions in microwave resonators with broken time-reversal symmetry
A magnetic flux through a mesoscopic metallic ring gives rise to a persistent current which can be detected via characteristic oscillations of the magnetization depending on the applied field (Lévy et al 1990 Phys. Rev. Lett. 64 2074). In this paper a direct visualization of such persistent currents in a microwave analogue experiment is reported, making use of the analogy between the probability density current in the quantum-mechanical system and the Poynting vector in the corresponding electromagnetic one. To break time-reversal symmetry, a small ring of a ferrite material in a static external magnetic field was introduced into the resonator. In our analysis of the experimental data we employ the off-diagonal elements of the scattering matrix. Due to the small size of the ferrite compared to the resonator size the symmetry was partially broken in the sense that the real and the imaginary parts of the wavefunctions are not equally large on average. The statistical properties of the wavefunction's real and imaginary parts Re(ψ), Im(ψ), respectively, as well as the distributions of the total wavefunction's amplitudes ¦ψ¦ and the statistical properties of the probability density currents were also examined and compared to the theoretical predictions.
The present doctoral thesis is concerned, on the one hand, with the investigation of the decay probability in a quantum billiard, and, on the other, the realization of the double-slit experiment in microwave billiards. For both purposes transmission elements of the scattering matrix are experimentally measured in microwave cavities with regular and chaotic dynamics. One or more pieces of the billiards' boundary are removed and the leaked outward eld is investigated. The general aim is to distinguish the eect of the billiard dynamics on the temporal decay behaviour and on the interference patterns. The decay probability in an open quantum billiard can be explored by means of the scattering matrix formalismus derived in the framework of nuclear physics. In a circular billiard, archetype of regular systems, the opening breaks the spatial symmetry causing a splitting of the degenerate states observable at low frequencies. One state of the doublet remains unaected contributing to the long-time behavior inside the cavity (survival probability), while the short-lived one is strongly influenced by the opening and decays much faster. The case of chaotic dynamics is studied by means of a tilted stadium billiard. For sufficiently long times the decay probability shows an essentially nonexponential behaviour. Agreement with theoretical results derived by Harney, Dittes and Muller is obtained for a few of the open channels and intermediate times. For both, the circle and the stadium,the short time decay resembles the classical behaviour. Furthermore, a double-slit experiment is performed with a wave packet starting inside a microwave billiard domain, both for regular and chaotic dynamics. Stationary and temporal patterns formed by the leaking of the wave packet are experimentally determined. The evolution of a directed initial state is constructed by means of the electromagnetic propagator. Also single-slit experiments are carried out and compared with the corresponding double-slit experiments. The present work sheds light on the fundamental question of the decay of open quantum systems and provides plentiful information about the structure of the diracted eld from microwave billards with a double-slit on the boundary connecting classical properties with the corresponding quantum features in both cases.