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Time-Reversal Symmetry Breaking in Quantum Billiards

F. Schäfer

Published Jan 1, 2009• 0 citations
Publisher

Abstract

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.

DOI
10.26083/tuprints-00001329

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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 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.
We report on the experimental realization of a flat, superconducting microwave resonator, a microwave billiard, with partially violated time-reversal (T) invariance, induced by inserting a ferrite into the cavity and magnetizing it with an external magnetic field perpendicular to the resonator plane. In order to prevent its expulsion caused by the Meissner-Ochsenfeld effect, we used a cavity of which the top and bottom plate were made from niobium, a superconductor of type II, and cooled it down to liquid-helium temperature T_{LHe}≃4 K. The cavity had the shape of a chaotic Africa billiard. Superconductivity rendered possible the accurate determination of complete sequences of the resonance frequencies and of the widths and strengths of the resonances, an indispensable prerequisite for the unambiguous detection of T invariance violation, especially when it is only partially violated. This allows for the first time the precise specification of the size of T invariance violation from the fluctuation properties of the resonance frequencies and from the strength distribution, which actually depends sensitively on it and thus provides a most suitable measure. For this purpose we derived an analytical expression for the latter which is valid for isolated resonances in the range from no T invariance violation to complete violation.
We determine with unprecedented accuracy the lowest 900 eigenvalues of two quantum constant-width billiards from resonance spectra measured with flat, superconducting microwave resonators. While the classical dynamics of the constant-width billiards is unidirectional, a change of the direction of motion is possible in the corresponding quantum system via dynamical tunneling. This becomes manifest in a splitting of the vast majority of resonances into doublets of nearly degenerate ones. The fluctuation properties of the two respective spectra are demonstrated to coincide with those of a random-matrix model for systems with violated time-reversal invariance and a mixed dynamics. Furthermore, we investigate tunneling in terms of the splittings of the doublet partners. On the basis of the random-matrix model we derive an analytical expression for the splitting distribution which is generally applicable to systems exhibiting dynamical tunneling between two regions with (predominantly) chaotic dynamics.
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.
The distributions of the diagonal elements of the Wigner's reaction [Formula: see text] matrix for open systems with violated time reversal T invariance in the case of large absorption are for the first time experimentally studied. The Wigner's reaction matrix links the properties of chaotic systems with the scattering processes in the asymptotic region. Microwave networks consisting of microwave circulators were used in the experiment to simulate quantum graphs with violated T invariance. The distributions of the diagonal elements of the reaction [Formula: see text] matrix were experimentally evaluated by measuring of the two-port scattering matrix [Formula: see text]. The violation of T invariance in the networks with large absorption was demonstrated by calculating the enhancement factor W of the matrix [Formula: see text]. Our experimental results are in very good agreement with the analytic ones attained for the Gaussian unitary ensemble in the random matrix theory. The obtained results suggest that the distributions P(ʋ) and P(u) of the imaginary and the real parts of the diagonal elements of the Wigner's reaction [Formula: see text] matrix together with the enhancement factor W can be used as a powerful tool for identification of systems with violated T symmetry and quantification of their absorption.