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arXiv:1910.04248 · Physical Review Letters

Partial Time-Reversal Invariance Violation in a Flat, Superconducting Microwave Cavity with the Shape of a Chaotic Africa Billiard

Barbara Dietz, T. Klaus, M. Miski-Oglu, A. Richter, M. Wunderle

Published Oct 24, 2019• 39 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.123.174101

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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.
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 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.
We report on the behavior of a superconducting microwave cavity containing levitated permanent magnets having a range of magnet strengths. By observing the changes in the cavity’s loaded quality factor and resonance frequency as functions of temperature and magnet remanence, we gain an understanding of the transient motion of the magnet as well as its steady-state levitation height when the temperature drops below 1 K. Experimental measurements of Meissner-effect levitation within a 10 GHz superconducting aluminum coaxial quarter-wave stub cavity are performed for a sequence of identically shaped millimeter-scale neodymium magnets having varying strengths. Magnet levitation within the cavity is accompanied by both gradual and abrupt shifts in the resonance frequency (with a height sensitivity as large as 400 MHz/mm) as well as changes in the total quality factor (8%-17%) as a function of temperature during the superconducting transition of the aluminum cavity. Prior to magnet motion and levitation, the Q of the cavity changes quadratically with temperature, as expected, as the walls of the cavity undergo the superconducting transition. We observe, however, a deviation from the quadratic trend which is attributable to magnet movement within the cavity. Such an electromechanical system is a transducer between mechanical and microwave oscillators enabling coupling of low-frequency mechanical motion of the magnet to other quantum objects, such as magnons and transmons, which are used for sensing and quantum information processing.
Two experiments with super- and normal conducting microwave cavities which are used as an analog to two-dimensional, infinitely deep quantum potentials are presented. The eigenvalues of such a potential can be measured directly through the resonance frequencies of the cavity, while the eigenvectors can be determined by measuring the field distributions inside the cavity. In the case of open systems another information – the imaginary part of the eigenvalues – is observable by measuring the widths of the resonances. As examples for this experimental approach to quantum chaos the semiclassical reconstruction of length spectra of billiards with varying chaoticity based of various trace formulas proposed by Gutzwiller, by Ullmo, Grinberg and Tomsovic and by Berry and Tabor for systems with chaotic, mixed and regular dynamics, respectively, and the first observation of square root singularities – so called exceptional points recently proposed by Heiss – in the energy spectrum of coupled cavities are presented.