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

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 TLHe 4 K. The Cavity had the shape of a chaotic Afrivca 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.Introduction.-An important aspect of quantum chaos is the understanding of the features of the classical dynamics in terms of the spectral properties of the corresponding quantum system [1][2][3]. Numerous experimental and numerical studies confirmed that for a fully chaotic classical dynamics they coincide with those of random matrices [4] from the Gaussian orthogonal ensemble (GOE) when time-reversal (T ) invariance is preserved, from the Gaussian unitary ensemble (GUE) when it is violated [5][6][7] and from an ensemble interpolating between the GOE and the GUE, when T invariance is only partially violated [8][9][10]. Most suitable for the experimental verification are flat, cylindrical microwave resonators [11,12] and microwave networks [10]. For microwave frequencies below a certain cutoff value f max , the associated Helmholtz equation is mathematically equivalent to the Schrödinger equation of the quantum billiard and the quantum graph of corresponding shape [13,14], respectively. The random-matrix theory (RMT) analysis of the spectral properties of a quantum system and the assignment to one of these ensembles requires complete sequences of several hundreds of eigenvalues [7] or an elaborate procedure to cope with missing levels [15][16][17][18][19][20][21] which hinder or render the unambiguous determination of the strength of T invariance violation unfeasible in cases where it is only partially violated. For T invariant systems complete sequences of up to 5000 eigenvalues [22][23][24] of the corresponding quantum billiard were obtained in high-precision experiments at liquidhelium temperature T LHe = 4 K with niobium and leadcoated microwave resonators which become superconducting at T c...

Application of topological resonances in experimental investigation of a Fermi golden rule in microwave networks

“…In this way microwave networks have become another important model systems to which belong flat microwave cavities [15,[26][27][28][29][30][31][32][33][34][35][36][37] and experiments using Rydberg atoms strongly driven by microwave fields [38][39][40][41][42][43][44][45][46][47][48][49][50] that are successfully used in simplifying experimental analysis of complex quantum systems.…”

The Generalized Euler Characteristics of the Graphs Split at Vertices

“…Microwave networks are the only ones that allow for the experimental simulation of quantum systems with all three types of symmetry within the framework of the random matrix theory (RMT): Gaussian orthogonal ensemble (GOE)systems with preserved time reversal symmetry (TRS) [16,21,24,25,27,[30][31][32], Gaussian unitary ensemble (GUE)-systems with broken TRS [24,28,[33][34][35][36], and Gaussian symplectic ensemble (GSE)-systems with TRS and half-spin [37]. The other model systems, which are not as versatile as microwave networks, but are often used in simulations of complex quantum systems, are flat microwave billiards [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54], and exited atoms in strong microwave fields [55][56][57][58][59][60][61][62][63][64][65][66][67].…”

Missing-level statistics in classically chaotic quantum systems with symplectic symmetry

“…This is possible, e.g., in microwave billiards or microwave networks where scattering matrix elements are available whose fluctuation properties also provide measures for the chaoticity, e.g., in terms of their correlation functions, the distributions of their cross sections [66][67][68][69] or the enhancement factor [26,70,71]. Complete spectra were obtained in experiments with superconducting microwave billiards [72,73] by choosing ei-ther resonators made from niobium or from lead-covered brass. This, however, is impossible for microwave networks constructed from coaxial cables, since they contain a dielectric medium, which prevents superconductivity, even if the cables are made from niobium instead of copper.…”