# Supercondicting billiards and quantum chaos

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Dr. Barbara Dietz-Pilatus accepted the invitation on 15 July 2011

Prof. Achim Richter accepted the invitation on 14 July 2011

Short note and Related References added by Scholarpedia Editor D.Shepelyansky in May 2020

In sufficiently flat microwave resonators, Maxwell's equations reduce to the Schrodinger equation for the free particle, and the condition of classical chaos is realized by properly shaping the cavity. The first measurments using a superconducting cavity are reported in [1]. The high-quality factor $$Q \approx 10^5 - 10^7$$ of this device allows to measure the complete spectrum below 17.5 GHz. The semiclassical analysis is in good agreement with the experimental data, and provides a new scheme for the statistical analysis and comparison with predictions based on the Gaussian orthogonal ensemble. The partial widths of resonances are found to follow a Porter-Thomas distribution [2]. The experimental results for superconducting three-dimensional Sinai billiard reported in [3] show that the spectral correlations are totally consistent with the predictions of the Gaussian orthogonal ensemble. Additional experimental studies are presented in [4],[5] with the review of this research field given in [6].

## Related References

1. H.-D.Graf, H.L.Harney, H.Lengeler, C.H.Lewenkopf, C.Rangacharyulu, A.Richter, P.Schardt, H.A.Weidenmüller, "Distribution of eigenmodes in a superconducting stadium billiard with chaotic dynamics", Phys. Rev. Lett. 69: 1296 (1992)
2. H.Alt, H.-D.Graf, H.L.Harney, R.Hofferbert, H.Lengeler, A.Richter, P.Schardt, H.A.Weidenmüller, "Gaussian orthogonal ensemble statistics in a microwave stadium billiard with chaotic dynamics: Porter-Thomas distribution and algebraic decay of time correlations", Phys. Rev. Lett. 74: 62 (1995)
3. H.Alt, C.Dembowski, H.-D.Graf, R.Hofferbert, H.Rehfeld, A.Richter, R.Schuhmann, T. Weiland, "Wave dynamical chaos in a superconducting three-dimensional Sinai billiard", Phys. Rev. Lett. 79: 1026 (1997)
4. C.Dembowski, B.Dietz, H.-D.Graf, A.Heine, T.Papenbrock, A. Richter, C. Richter, "Experimental test of a trace formula for a chaotic three-dimensional microwave cavity", Phys. Rev. Lett. 89: 064101 (2002)
5. B.Dietz, T.Klaus, M.Miski-Oglu, A.Richter, M.Wunderle, "Partial time-reversal invariance violation in a flat, superconducting microwave cavity with the shape of a chaotic Africa billiard", Phys. Rev. Lett. 123: 174101 (2019)
6. B.Dietz, A.Richter, "Quantum and wave dynamical chaos in superconducting microwave billiards", Chaos: An Interdisciplinary Journal of Nonlinear Science 25: 097601 (2015)