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'Persistent currents' and eigenfunctions in microwave resonators with broken time-reversal symmetry

Marko Vraniÿ, M ichaelBarth, M arkoRobnik

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

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.

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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 describes experimentally measured properties of the resonance spectra of flat microwave billiards with partially broken timereversal invariance induced by an embedded magnetized ferrite. A vector network analyzer determines the complex scattering matrix elements. The data is interpreted in terms of the scattering formalism developed in nuclear physics. At low excitation frequencies the scattering matrix displays isolated resonances. At these the effect of the ferrite on isolated resonances (singlets) and pairs of nearly degenerate resonances (doublets) is investigated. The hallmark of time-reversal symmetry breaking is the violation of reciprocity, i.e. of the symmetry of the scattering matrix. One finds that reciprocity holds in singlets; it is violated in doublets. This is modeled by an effective Hamiltonian of the resonator. A comparison of the model to the data yields time-reversal symmetry breaking matrix elements in the order of the level spacing. Their dependence on the magnetization of the ferrite is understood in terms of its magnetic properties. At higher excitation frequencies the resonances overlap and the scattering matrix elements fluctuate irregularly (Ericson fluctuations). They are analyzed in terms of correlation functions. The data are compared to three models based on random matrix theory. The model by Verbaarschot, Weidenmüller and Zirnbauer describes time-reversal invariant scattering processes. The one by Fyodorov, Savin and Sommers achieves the same for systems with complete time-reversal symmetry breaking. An extended model has been developed that accounts for partial breaking of time-reversal invariance. This extended model is in general agreement with the data, while the applicability of the other two models is limited. The cross-correlation function between forward and backward reactions determines the time-reversal symmetry breaking matrix elements of the Hamiltonian to up to 0.3 mean level spacings. Finally the sensitivity of the elastic enhancement factor to time-reversal symmetry breaking is studied. Based on the data elastic enhancement factors below 2 are found which is consistent with breaking of time-reversal invariance in the regime of overlapping resonances. The present work provides the framework to probe for broken time-reversal invariance in any scattering data by a multitude of methods in the whole range between isolated and overlapping resonances.
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 reciprocity, we investigate the breaking of time-reversal (T) symmetry due to a ferrite embedded in a flat microwave billiard. Transmission spectra of isolated single resonances are not sensitive to T violation, whereas those of pairs of nearly degenerate resonances do depend on the direction of time. For their theoretical description a scattering matrix model from nuclear physics is used. The T-violating matrix elements of the effective Hamiltonian for the microwave billiard with the embedded ferrite are determined experimentally as functions of the magnetization of the ferrite.
Recently, new strongly interacting phases have been uncovered in mesoscopic systems with chaotic scattering at the boundaries. The new phases are the mesoscopic analogue of spontaneous distortions of the Fermi surface induced by interactions in bulk systems and can occur in any Fermi-liquid channel with angular momentum m. Here we show that the phase with m even has a diamagnetic persistent current (seen experimentally but mysterious theoretically), while that with m odd can be driven through a transition which spontaneously breaks time-reversal symmetry by increasing the coupling to dissipative leads. Our analysis is reliable when the dimensionless conductance of the system is large, is nonperturbative in both disorder and interactions, and applies to ballistic structures with strong interactions.
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.