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Quantum chaos and breaking of all anti-unitary symmetries in Rydberg excitons

Nature Materials volume 15, pages 741–745 (2016)Cite this article

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Abstract

Symmetries are the underlying principles of fundamental interactions in nature. Chaos in a quantum system may emerge from breaking these symmetries. Compared to vacuum, crystals are attractive for studying quantum chaos, as they not only break spatial isotropy, but also lead to novel quasiparticles with modified interactions. Here we study yellow Rydberg excitons in cuprous oxide which couple strongly to the vacuum light field and interact significantly with crystal phonons, leading to inversion symmetry breaking. In a magnetic field, time-reversal symmetry is also broken and the exciton states show a complex splitting pattern, resulting in quadratic level repulsion for small splittings. In contrast to atomic chaotic systems in a magnetic field, which show only a linear level repulsion, this is a signature of a system where all anti-unitary symmetries are broken simultaneously. This behaviour can otherwise be found only for the electro-weak interaction or engineered billiards.

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Figure 1: Transmission spectrum of the Cu2O yellow series measured in magnetic fields up to 7 T, starting from the state with principal quantum number n = 4.
Figure 2: Histograms of the level spacing distribution for normalized level spacings s.
Figure 3: Statistical measures of the level spacings in Cu2O.
Figure 4: Schematic depiction of time-reversal symmetry breaking in a phonon interaction.

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References

  1. Bohigas, O., Giannoni, M. J. & Schmit, C. Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52, 1–4 (1984).

    Article  Google Scholar 

  2. Frisch, A. et al. Quantum chaos in ultracold collisions of gas-phase erbium atoms. Nature 507, 475–479 (2014).

    Article  CAS  Google Scholar 

  3. Zhou, W. et al. Magnetic field control of the quantum chaotic dynamics of hydrogen analogs in an anisotropic crystal field. Phys. Rev. Lett. 105, 024101 (2010).

    Article  Google Scholar 

  4. Vina, L., Potemski, M. & Wang, W. Signatures of quantum chaos in the magneto-excitonic spectrum of quantum wells. Phys.-Usp. 41, 153–156 (1998).

    Article  Google Scholar 

  5. Mitchell, G. E., Richter, A. & Weidenmüller, H. A. Random matrices and chaos in nuclear physics: nuclear reactions. Rev. Mod. Phys. 82, 2845–2901 (2010).

    Article  CAS  Google Scholar 

  6. Robnik, M. & Berry, M. V. False time-reversal violation and energy level statistics: the role of anti-unitary symmetry. J. Phys. A 19, 669–682 (1986).

    Article  Google Scholar 

  7. Thewes, J. et al. Observation of high angular momentum excitons in cuprous oxide. Phys. Rev. Lett. 115, 027402 (2015).

    Article  CAS  Google Scholar 

  8. Wu, C. S., Ambler, E., Hayward, R. W., Hoppes, D. D. & Hudson, R. P. Experimental test of parity conservation in beta decay. Phys. Rev. 105, 1413–1415 (1957).

    Article  CAS  Google Scholar 

  9. Ponomarenko, L. A. et al. Chaotic Dirac billiard in graphene quantum dots. Science 320, 356–358 (2008).

    Article  CAS  Google Scholar 

  10. Brandt, J. et al. Ultranarrow optical absorption and two-phonon excitation spectroscopy of Cu2O paraexcitons in a high magnetic field. Phys. Rev. Lett. 99, 217403 (2007).

    Article  Google Scholar 

  11. Kazimierczuk, T., Fröhlich, D., Scheel, S., Stolz, H. & Bayer, M. Giant Rydberg excitons in the copper oxide Cu2O. Nature 514, 343–347 (2014).

    Article  CAS  Google Scholar 

  12. Wigner, E. P. Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548–564 (1955).

    Article  Google Scholar 

  13. Dyson, F. J. & Mehta, M. L. Statistical theory of the energy levels of complex systems. iv. J. Math. Phys. 4, 701–712 (1963).

    Article  Google Scholar 

  14. Plerou, V., Gopikrishnan, P., Rosenow, B., Nunes Amaral, L. A. & Stanley, H. E. Universal and nonuniversal properties of cross correlations in financial time series. Phys. Rev. Lett. 83, 1471–1474 (1999).

    Article  CAS  Google Scholar 

  15. Seligman, T. & Verbaarschot, J. Quantum spectra of classically chaotic systems without time reversal invariance. Phys. Lett. A 108, 183–187 (1985).

    Article  Google Scholar 

  16. Brody, T. A. et al. Random-matrix physics: spectrum and strength fluctuations. Rev. Mod. Phys. 53, 385–479 (1981).

    Article  CAS  Google Scholar 

  17. Weidenmüller, H. A. & Mitchell, G. E. Random matrices and chaos in nuclear physics: nuclear structure. Rev. Mod. Phys. 81, 539–589 (2009).

    Article  Google Scholar 

  18. Stoffregen, U., Stein, J., Stöckmann, H.-J., Kuś, M. & Haake, F. Microwave billiards with broken time reversal symmetry. Phys. Rev. Lett. 74, 2666–2669 (1995).

    Article  CAS  Google Scholar 

  19. So, P., Anlage, S. M., Ott, E. & Oerter, R. N. Wave chaos experiments with and without time reversal symmetry: GUE and GOE statistics. Phys. Rev. Lett. 74, 2662–2665 (1995).

    Article  CAS  Google Scholar 

  20. Fröhlich, D. et al. Coherent propagation and quantum beats of quadrupole polaritons in Cu2O. Phys. Rev. Lett. 67, 2343–2346 (1991).

    Article  Google Scholar 

  21. Schmutzler, J., Fröhlich, D. & Bayer, M. Signatures of coherent propagation of blue polaritons in Cu2O. Phys. Rev. B 87, 245202 (2013).

    Article  Google Scholar 

  22. Beg, M. M. & Shapiro, S. M. Study of phonon dispersion relations in cuprous oxide by inelastic neutron scattering. Phys. Rev. B 13, 1728–1734 (1976).

    Article  CAS  Google Scholar 

  23. Cartarius, H., Main, J. & Wunner, G. Exceptional points in atomic spectra. Phys. Rev. Lett. 99, 173003 (2007).

    Article  Google Scholar 

  24. Cartarius, H., Main, J. & Wunner, G. Exceptional points in the spectra of atoms in external fields. Phys. Rev. A 79, 053408 (2009).

    Article  Google Scholar 

  25. Gao, T. et al. Observation of non-hermitian degeneracies in a chaotic exciton-polariton billiard. Nature 526, 554–558 (2015).

    Article  CAS  Google Scholar 

Download references

Acknowledgements

We acknowledge the support by the Deutsche Forschungsgemeinschaft and the Russian Foundation for Basic Research in the frame of ICRC TRR 160 and the support from the Russian Ministry of Science and Education (contract number 14.Z50.31.0021).

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Authors and Affiliations

  1. Experimentelle Physik 2, Technische Universität Dortmund, 44221 Dortmund, Germany

    Marc Aßmann, Johannes Thewes, Dietmar Fröhlich & Manfred Bayer

  2. Ioffe Institute, Russian Academy of Sciences, 194021 St Petersburg, Russia

    Manfred Bayer

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  1. Marc Aßmann
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  2. Johannes Thewes
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Contributions

J.T. performed the experiments. M.A. analysed the data. D.F., M.B. and M.A. designed the experiment. M.A. and M.B. wrote the manuscript.

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Correspondence to Marc Aßmann.

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Aßmann, M., Thewes, J., Fröhlich, D. et al. Quantum chaos and breaking of all anti-unitary symmetries in Rydberg excitons. Nature Mater 15, 741–745 (2016). https://doi.org/10.1038/nmat4622

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