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Fermionic time-reversal symmetry in a photonic topological insulator

Nature Materials volume 19, pages 855–860 (2020)Cite this article

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Abstract

Much of the recent attention directed towards topological insulators is motivated by their hallmark feature of protected chiral edge states. In electronic (or fermionic) topological insulators, these states originate from time-reversal symmetry and allow carriers with opposite spin-polarization to propagate in opposite directions at the edge of an insulating bulk. By contrast, photonic (or bosonic) systems are generally assumed to be precluded from supporting edge states that are intrinsically protected by time-reversal symmetry. Here, we experimentally demonstrate counter-propagating chiral states at the edge of a time-reversal-symmetric photonic waveguide structure. The pivotal step in our approach is the design of a Floquet driving protocol that incorporates effective fermionic time-reversal symmetry, enabling the realization of the photonic version of an electronic topological insulator. Our findings allow for fermionic properties to be harnessed in bosonic systems, thereby offering alternative opportunities for photonics as well as acoustics, mechanical waves and cold atoms.

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Fig. 1: Conceptual idea.
Fig. 2: Schematic of the driving protocol.
Fig. 3: Experimental demonstration of TRS-protected counter-propagating edge modes.
Fig. 4: Experimental verification of fermionic TRS.

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Data availability

The data represented in Figs. 3d,e,g,h and 4b are provided with the paper as source data. All other data that support results in this Article are available from the corresponding author upon reasonable request.

Code availability

The numerical codes that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

References

  1. Kane, C. L. & Mele, E. J. Z 2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

    CAS  Google Scholar 

  2. Bernevig, B. A. & Zhang, S.-C. Quantum spin Hall effect. Phys. Rev. Lett. 96, 106802 (2006).

    Google Scholar 

  3. König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).

    Google Scholar 

  4. Hsieh, D. et al. A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970–974 (2008).

    CAS  Google Scholar 

  5. Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačić, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009).

    CAS  Google Scholar 

  6. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045 (2010).

    CAS  Google Scholar 

  7. Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).

    CAS  Google Scholar 

  8. Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001–1005 (2013).

    CAS  Google Scholar 

  9. Lu, L., Joannopoulos, J. D. & Soljačić, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).

    CAS  Google Scholar 

  10. Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015).

    Google Scholar 

  11. Süsstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47–50 (2015).

    Google Scholar 

  12. Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).

    Google Scholar 

  13. Stützer, S. et al. Photonic topological Anderson insulators. Nature 560, 461–465 (2018).

    Google Scholar 

  14. Klembt, S. et al. Exciton–polariton topological insulator. Nature 562, 552–556 (2018).

    CAS  Google Scholar 

  15. Blanco-Redondo, A., Bell, B., Oren, D., Eggleton, B. J. & Segev, M. Topological protection of biphoton states. Science 362, 568–571 (2018).

    CAS  Google Scholar 

  16. Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).

    CAS  Google Scholar 

  17. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015 (1988).

    CAS  Google Scholar 

  18. Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).

    CAS  Google Scholar 

  19. Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).

    CAS  Google Scholar 

  20. Maczewsky, L. J., Zeuner, J. M., Nolte, S. & Szameit, A. Observation of photonic anomalous Floquet topological insulators. Nat. Commun. 8, 13756 (2017).

    CAS  Google Scholar 

  21. Mukherjee, S. et al. Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice. Nat. Commun. 8, 13918 (2017).

    CAS  Google Scholar 

  22. Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).

    CAS  Google Scholar 

  23. Ningyuan, J., Owens, C., Sommer, A., Schuster, D. & Simon, J. Time- and site-resolved dynamics in a topological circuit. Phys. Rev. X 5, 021031 (2015).

    Google Scholar 

  24. Cheng, X. et al. Robust reconfigurable electromagnetic pathways within a photonic topological insulator. Nat. Mater. 15, 542–548 (2016).

    CAS  Google Scholar 

  25. Slobozhanyuk, A. P. et al. Experimental demonstration of topological effects in bianisotropic metamaterials. Sci. Rep. 6, 22270 (2016).

    CAS  Google Scholar 

  26. He, C. et al. Photonic topological insulator with broken time-reversal symmetry. Proc. Natl Acad. Sci. USA 113, 4924–4928 (2016).

    CAS  Google Scholar 

  27. Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).

    Google Scholar 

  28. Carpentier, D., Delplace, P., Fruchart, M. & Gawędzki, K. Topological index for periodically driven time-reversal invariant 2D systems. Phys. Rev. Lett. 114, 106806 (2015).

    Google Scholar 

  29. Kitagawa, T., Rudner, M. S., Berg, E. & Demler, E. Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010).

    Google Scholar 

  30. Roy, R. & Harper, F. Periodic table for Floquet topological insulators. Phys. Rev. B 96, 155118 (2017).

    Google Scholar 

  31. Wu, L.-H. & Hu, X. Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114, 223901 (2015).

    Google Scholar 

  32. Nathan, F. & Rudner, M. S. Topological singularities and the general classification of Floquet-Bloch systems. New J. Phys. 17, 125014 (2015).

    Google Scholar 

  33. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982).

    CAS  Google Scholar 

  34. Fu, L. & Kane, C. L. Time reversal polarization and a Z 2 adiabatic spin pump. Phys. Rev. B 74, 195312 (2006).

    Google Scholar 

  35. Kitagawa, T., Berg, E., Rudner, M. & Demler, E. Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010).

    Google Scholar 

  36. Höckendorf, B., Alvermann, A. & Fehske, H. Efficient computation of the W 3 topological invariant and application to Floquet-Bloch systems. J. Phys. A 50, 295301 (2017).

    Google Scholar 

  37. Höckendorf, B., Alvermann, A. & Fehske, H. Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry. Phys. Rev. B 97, 045140 (2018).

    Google Scholar 

  38. Szameit, A. & Nolte, S. Discrete optics in femtosecond-laser-written photonic structures. J. Phys. B 43, 163001 (2010).

    Google Scholar 

  39. Rachel, S. Interacting topological insulators: a review. Rep. Prog. Phys. 81, 116501 (2018).

    Google Scholar 

  40. Poli, C., Bellec, M., Kuhl, U., Mortessagne, F. & Schomerus, H. Selective enhancement of topologically induced interface states in a dielectric resonator chain. Nat. Commun. 6, 6710 (2015).

    CAS  Google Scholar 

  41. Weimann, S. et al. Topologically protected bound states in photonic parity-time-symmetric crystals. Nat. Mater. 16, 433–438 (2017).

    CAS  Google Scholar 

  42. Kremer, M. et al. Demonstration of a two-dimensional \({\cal P}{\cal T}\)-symmetric crystal. Nat. Commun. 10, 435 (2019).

    Google Scholar 

  43. Heinrich, M. et al. Supersymmetric mode converters. Nat. Commun. 5, 3698 (2014).

    CAS  Google Scholar 

  44. Queraltó, G. et al. Topological state enineering via supersymmetric transformations. Commun. Phys. 3, 49 (2020).

    Google Scholar 

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Acknowledgements

A.S. gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft (grants SZ 276/9-1, SZ 276/19-1, SZ 276/20-1, BL 574/13-1) and the Alfried Krupp von Bohlen und Halbach Foundation. We thank C. Otto for preparing the high-quality fused silica samples used in all experiments presented here.

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Author notes

  1. These authors contributed equally: Lukas J. Maczewsky, Bastian Höckendorf.

Authors and Affiliations

  1. Institut für Physik, Universität Rostock, Rostock, Germany

    Lukas J. Maczewsky, Mark Kremer, Tobias Biesenthal, Matthias Heinrich & Alexander Szameit

  2. Institut für Physik, Universität Greifswald, Greifswald, Germany

    Bastian Höckendorf, Andreas Alvermann & Holger Fehske

Authors
  1. Lukas J. Maczewsky
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Contributions

The theory was established by B.H., A.A. and H.F. The sample design and lattice implementation were developed by L.M., M.H. and A.S. The characterization of the lattice structure was carried out by L.M., M.K. and T.B. The project was supervised by H.F. and A.S. All authors discussed the results and co-wrote the paper.

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Correspondence to Andreas Alvermann or Alexander Szameit.

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Supplementary information

Supplementary Information

Detailed experimental techniques, additional measurements, extended theory, stability checking

Source data

Source Data Fig. 3

Image data of Fig. 3

Source Data Fig. 4

Experimental data points of Fig. 4b

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Maczewsky, L.J., Höckendorf, B., Kremer, M. et al. Fermionic time-reversal symmetry in a photonic topological insulator. Nat. Mater. 19, 855–860 (2020). https://doi.org/10.1038/s41563-020-0641-8

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