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Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard

Nature volume 526, pages 554–558 (2015)Cite this article

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Abstract

Exciton-polaritons are hybrid light–matter quasiparticles formed by strongly interacting photons and excitons (electron–hole pairs) in semiconductor microcavities1,2,3. They have emerged as a robust solid-state platform for next-generation optoelectronic applications as well as for fundamental studies of quantum many-body physics. Importantly, exciton-polaritons are a profoundly open (that is, non-Hermitian4,5) quantum system, which requires constant pumping of energy and continuously decays, releasing coherent radiation6. Thus, the exciton-polaritons always exist in a balanced potential landscape of gain and loss. However, the inherent non-Hermitian nature of this potential has so far been largely ignored in exciton-polariton physics. Here we demonstrate that non-Hermiticity dramatically modifies the structure of modes and spectral degeneracies in exciton-polariton systems, and, therefore, will affect their quantum transport, localization and dynamical properties7,8,9. Using a spatially structured optical pump10,11,12, we create a chaotic exciton-polariton billiard—a two-dimensional area enclosed by a curved potential barrier. Eigenmodes of this billiard exhibit multiple non-Hermitian spectral degeneracies, known as exceptional points13,14. Such points can cause remarkable wave phenomena, such as unidirectional transport15, anomalous lasing/absorption16,17 and chiral modes18. By varying parameters of the billiard, we observe crossing and anti-crossing of energy levels and reveal the non-trivial topological modal structure exclusive to non-Hermitian systems9,13,14,15,16,17,18,19,20,21,22. We also observe mode switching and a topological Berry phase for a parameter loop encircling the exceptional point23,24. Our findings pave the way to studies of non-Hermitian quantum dynamics of exciton-polaritons, which may uncover novel operating principles for polariton-based devices.

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Figure 1: Non-Hermitian exciton-polariton Sinai billiard and its spectrum.
Figure 2: Crossing and anti-crossing for two near-degenerate modes.
Figure 3: Eigenvalues of a two-level non-Hermitian model in the vicinity of the exceptional point.
Figure 4: Observation of the topological Berry phase acquired after circling around the exceptional point in the parameter plane.

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Acknowledgements

We thank M. Berry and O. Kirillov for comments. This research was supported by the Australian Research Council, the ImPACT Program of the Council for Science, Technology and Innovation (Cabinet Office, Government of Japan), the RIKEN iTHES Project, the MURI Center for Dynamic Magneto-Optics, a Grant-in-Aid for Scientific Research (type A), and the State of Bavaria.

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Authors and Affiliations

  1. Research School of Physics and Engineering, The Australian National University, Canberra, 2601, Australian Capital Territory, Australia

    T. Gao, E. Estrecho, K. Y. Bliokh, Y. S. Kivshar, A. G. Truscott, R. G. Dall & E. A. Ostrovskaya

  2. Center for Emergent Matter Science, RIKEN, Wako-shi, 351-0198, Saitama, Japan

    K. Y. Bliokh, M. D. Fraser & F. Nori

  3. School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    T. C. H. Liew

  4. Technische Physik and Wilhelm-Conrad-Röntgen Research Center for Complex Material Systems, Universität Würzburg, Am Hubland, D-97074, Würzburg, Germany

    S. Brodbeck, M. Kamp, C. Schneider & S. Höfling

  5. SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews, KY16 9SS, UK

    S. Höfling

  6. ImPACT Project, Japan Science and Technology Agency, Chiyoda-ku, 102-0076, Tokyo, Japan

    Y. Yamamoto

  7. Edward L. Ginzton Laboratory, Stanford University, Stanford, 94305-4085, California, USA

    Y. Yamamoto

  8. Physics Department, University of Michigan, Ann Arbor, 48109-1040, Michigan, USA

    F. Nori

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Contributions

E.A.O., T.G., E.E. and K.Y.B. conceived the idea for this research; T.G., E.E., M.D.F., R.G.D. and A.G.T. designed and built the experiment with conceptual contributions from E.A.O.; T.G., E.E. and R.G.D. collected and analysed experimental data; K.Y.B, E.E., T.C.H.L. and E.A.O. performed theoretical and numerical analysis; S.B., M.K., C.S. and S.H. fabricated and characterized the semiconductor microcavity; E.A.O. and K.Y.B. wrote the paper with input from T.G., E.E. and T.C.H.L.; F.N., M.D.F., A.G.T, S.H., Y.Y. and Y.S.K. contributed to discussions and the shaping of the manuscript.

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Correspondence to E. A. Ostrovskaya.

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Extended data figures and tables

Extended Data Figure 1 Diagram of the experimental apparatus.

See Methods for details.

Extended Data Figure 2 Schematics of the optically induced billiard potential with two different wall thicknesses.

a, Thin walls; b, thick walls. The active regions corresponding to the optical pump are shown in black, and we note that the enclosed area does not change with wall thickness.

Extended Data Figure 3 Effect of wall thickness on spectroscopic line profiles of the Sinai billiard.

a, b, Profiles are shown in the vicinity of the degeneracy for the levels highlighted in Fig. 1c, d with thick (a) and thin (b) walls. The thick lines demonstrate the principle of data extraction for anti-crossing (a) and crossing (b) of the energy levels corresponding to those shown in Fig. 2a and b, respectively

Extended Data Figure 4 Spatial density distribution of the first seven simultaneously populated lowest-energy modes of the Sinai billiard.

Spatial density distributions were obtained from the thick-wall setup (Extended Data Fig. 2b) with R/W = 0.35. Top row, experimentally imaged; middle row, calculated using the effective linear potential model; bottom row, calculated using the full dynamical model given by equation (2).

Extended Data Figure 5 Spatial modes in the hybridization regions.

a–g, Calculated spatial modes; each panel shows the modulus squared of the wavefunction (left) and the wavefunction’s phase distribution (right, colour coded). a, b, e, f, Numerically calculated pure spatial eigenstates (modes 3 (a, e) and 4 (b, f)) for the Sinai billiard with thick and thin walls in the corresponding hybridization regions shown in Fig. 2a and b, respectively. c, d, g, The superpositions of modes 3 and 4 that match the experimentally imaged modes shown in Fig. 2; c (boxed in blue) and d (boxed in red) correspond to the blue and red curves of Fig. 2a, respectively, while g (boxed in red and blue) corresponds to the crossing point in Fig. 2b. The relative populations of pure modes in the superposition states are: c, |α|2 = 0.85 and |β|2 = 0.15; d, |α|2 = 0.65 and |β|2 = 0.35; g, |α|2 = 0.60 and |β|2 = 0.40.

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Gao, T., Estrecho, E., Bliokh, K. et al. Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard. Nature 526, 554–558 (2015). https://doi.org/10.1038/nature15522

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