Topologically protected bound states in photonic parity–time-symmetric crystals

Abstract

Parity–time (PT)-symmetric crystals are a class of non-Hermitian systems that allow, for example, the existence of modes with real propagation constants, for self-orthogonality of propagating modes, and for uni-directional invisibility at defects. Photonic PT-symmetric systems that also support topological states could be useful for shaping and routing light waves. However, it is currently debated whether topological interface states can exist at all in PT-symmetric systems. Here, we show theoretically and demonstrate experimentally the existence of such states: states that are localized at the interface between two topologically distinct PT-symmetric photonic lattices. We find analytical closed form solutions of topological PT-symmetric interface states, and observe them through fluorescence microscopy in a passive PT-symmetric dimerized photonic lattice. Our results are relevant towards approaches to localize light on the interface between non-Hermitian crystals.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Figure 1: The infinite dimer lattice and its dispersion relation.
Figure 2: Spectra of topological interface states in the dimer.
Figure 3: Realization of the PT-symmetric crystal interface.
Figure 4: Translation of the centre of mass (Δ s) of a broad Gaussian input beam whose phase front is tilted.
Figure 5: Loss of guided power in lattices with different dimerizations but fixed loss.
Figure 6: Fluorescence images of the light evolution after a single-waveguide excitation of the topological defect in different dimer configurations.

References

  1. 1

    Hasan, M. Z. & Kane, C. L. Colloqium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    CAS  Article  Google Scholar 

  2. 2

    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  Article  Google Scholar 

  3. 3

    Wang, Z., Chong, Y. D., Joannopoulos, J. D. & Soljac̆ić, M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100, 013905 (2008).

    Article  Google Scholar 

  4. 4

    Wang, Z., Chong, Y. D., Joannopoulos, J. D. & Soljac̆ić, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–776 (2009).

    CAS  Article  Google Scholar 

  5. 5

    Umucallar, R. O. & Carusotto, I. Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A 84, 043804 (2011).

    Article  Google Scholar 

  6. 6

    Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nat. Phys. 7, 907–912 (2011).

    CAS  Article  Google Scholar 

  7. 7

    Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photon. 6, 782–787 (2012).

    CAS  Article  Google Scholar 

  8. 8

    Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).

    CAS  Article  Google Scholar 

  9. 9

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

    CAS  Article  Google Scholar 

  10. 10

    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  Article  Google Scholar 

  11. 11

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

    CAS  Article  Google Scholar 

  12. 12

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

    CAS  Article  Google Scholar 

  13. 13

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

    Article  Google Scholar 

  14. 14

    Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 7, 490–495 (2011).

    CAS  Article  Google Scholar 

  15. 15

    Gu, Z., Fertig, H. A., Arovas, D. P. & Auerbach, A. Floquet spectrum and transport through an irradiated graphene ribbon. Phys. Rev. Lett. 107, 21660 (2011).

    Google Scholar 

  16. 16

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

    CAS  Article  Google Scholar 

  17. 17

    Wang, Y. H., Steinberg, H., Jarillo-Herrero, P. & Gedik, N. Observation of Floquet-Bloch states on the surface of a topological insulator. Science 342, 453–457 (2013).

    CAS  Article  Google Scholar 

  18. 18

    Lumer, Y., Plotnik, Y., Rechtsman, M. C. & Segev, M. Self-localized states in photonic topological insulators. Phys. Rev. Lett. 111, 243905 (2013).

    Article  Google Scholar 

  19. 19

    Fefferman, C. L., Lee-Thorp, J. P. & Weinstein, M. I. Topologically protected states in one-dimensional continuous systems and Dirac points. Proc. Natl Acad. Sci. USA 111, 8759–8763 (2014).

    CAS  Article  Google Scholar 

  20. 20

    Bender, C. M. & Boettcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243 (1998).

    CAS  Article  Google Scholar 

  21. 21

    Makris, K. G., El-Ganainy, R., Christodoulides, D. N. & Musslimani, Z. H. Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008).

    CAS  Article  Google Scholar 

  22. 22

    Musslimani, Z. H., Makris, K. G., El-Ganainy, R. & Christodoulides, D. N. Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100, 030402 (2008).

    CAS  Article  Google Scholar 

  23. 23

    Klaiman, S., Gunther, U. & Moiseyev, N. Visualization of branch points in PT-symmetric waveguides. Phys. Rev. Lett. 101, 080402 (2008).

    Article  Google Scholar 

  24. 24

    Guo, A. et al. Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009).

    CAS  Article  Google Scholar 

  25. 25

    Rüter, C. E. et al. Observation of parity–time symmetry in optics. Nat. Phys. 6, 192–195 (2010).

    Article  Google Scholar 

  26. 26

    Regensburger, A. et al. Parity–time synthetic photonic lattices. Nature 488, 167–171 (2012).

    CAS  Article  Google Scholar 

  27. 27

    Hodaei, H., Miri, M.-A., Heinrich, M., Christodoulides, D. N. & Khajavikhan, M. Parity-time-symmetric microring lasers. Science 346, 975–978 (2014).

    CAS  Article  Google Scholar 

  28. 28

    Feng, L., Wong, Z. J., Ma, R.-M., Wang, Y. & Zhang, X. Single-mode laser by parity-time symmetry breaking. Science 346, 972–975 (2014).

    CAS  Article  Google Scholar 

  29. 29

    Nazari, F. et al. Optical isolation via PT-symmetric nonlinear Fano resonances. Opt. Express 22, 9574–9584 (2014).

    CAS  Article  Google Scholar 

  30. 30

    Hu, Y. C. & Hughes, T. L. Absence of topological insulator phases in non-Hermitian PT-symmetric Hamiltonians. Phys. Rev. B 84, 153101 (2011).

    Article  Google Scholar 

  31. 31

    Zeuner, J. M. et al. Observation of a topological transition in the bulk of a non-Hermitian system. Phys. Rev. Lett. 115, 040402 (2015).

    Article  Google Scholar 

  32. 32

    Schomerus, H. Topologically protected midgap states in complex photonic lattices. Opt. Lett. 38, 1912–1914 (2013).

    Article  Google Scholar 

  33. 33

    Yuce, C. Topological phase in a non-Hermitian PT symmetric system. Phys. Lett. A 379, 1213–1218 (2015).

    CAS  Article  Google Scholar 

  34. 34

    Yuce, C. PT symmetric Floquet topological phase. Eur. Phys. J. D 69, 184 (2015).

    Article  Google Scholar 

  35. 35

    Harter, A. K., Lee, T. E. & Joglekar, Y. N. PT-breaking threshold in spatially asymmetric Aubry-André and Harper models: hidden symmetry and topological states. Phys. Rev. A 93, 062101 (2016).

    Article  Google Scholar 

  36. 36

    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  Article  Google Scholar 

  37. 37

    Zhao, H., Longhi, S. & Feng, L. Robust light state by quantum phase transition in non-Hermitian optical materials. Sci. Rep. 5, 17022 (2015).

    CAS  Article  Google Scholar 

  38. 38

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

    Article  Google Scholar 

  39. 39

    Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698 (1979).

    CAS  Article  Google Scholar 

  40. 40

    Malkova, N., Hromada, I., Wang, X., Bryant, G. & Chen, Z. Observation of optical Shockley-like surface states in photonic superlattices. Opt. Lett. 34, 1633–1635 (2009).

    CAS  Article  Google Scholar 

  41. 41

    Delplace, P., Ullmo, D. & Montambaux, G. Zak phase and the existence of edge states in graphene. Phys. Rev. B 84, 195452 (2011).

    Article  Google Scholar 

  42. 42

    Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747 (1989).

    CAS  Article  Google Scholar 

  43. 43

    Esaki, K., Sato, M., Hasebe, K. & Kohmoto, M. Edge states and topological phases in non-Hermitian systems. Phys. Rev. B 84, 205128 (2011).

    Article  Google Scholar 

  44. 44

    El-Ganainy, R., Makris, K. G., Christodoulides, D. N. & Musslimani, Z. H. Theory of coupled optical PT-symmetric structures. Opt. Lett. 32, 2632–2634 (2007).

    CAS  Article  Google Scholar 

  45. 45

    Longhi, S. Optical realization of relativistic non-Hermitian quantum mechanics. Phys. Rev. Lett. 105, 013903 (2010).

    Article  Google Scholar 

  46. 46

    Eichelkraut, T. et al. Mobility transition from ballistic to diffusive transport in non-Hermitian lattices. Nat. Commun. 4, 2533 (2013).

    CAS  Article  Google Scholar 

  47. 47

    Ornigotti, M. & Szameit, A. Quasi PT-symmetry in passive photonic lattices. J. Opt. 16, 065501 (2014).

    Article  Google Scholar 

  48. 48

    Eichelkraut, T., Weimann, S., Stützer, S., Nolte, S. & Szameit, A. Radiation-loss management in modulated waveguides. Opt. Lett. 39, 6831–6834 (2014).

    CAS  Article  Google Scholar 

  49. 49

    Golshani, M. et al. Impact of loss on the wave dynamics in photonic waveguide lattices. Phys. Rev. Lett. 113, 123903 (2014).

    CAS  Article  Google Scholar 

  50. 50

    Eisenberg, H. S., Silberberg, Y., Morandotti, R. & Aitchison, J. S. Diffraction management. Phys. Rev. Lett. 85, 1863 (2000).

    CAS  Article  Google Scholar 

  51. 51

    Szameit, A. et al. Quasi-incoherent propagation in waveguide arrays. Appl. Phys. Lett. 90, 241113 (2007).

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (grants SZ 276/7-1, SZ 276/9-1, NO462/6-1, BL 574/13-1, GRK 2101/1) and the German Ministry for Science and Education (grant 03Z1HN31). K.G.M. is supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement number PIOF-GA-2011- 303228 (project NOLACOME), and by the European Union Seventh Framework Programme (FP7-REGPOT-2012-2013-1) under grant agreement 316165. M.C.R. acknowledges suport from the Alfred P. Sloan Foundation, the National Science Foundation under grant number ECCS-1509546, as well as the Penn State MRSEC, Center for Nanoscale Science, under the award NSF DMR-1420620. The authors acknowledge useful discussions with D. Leykam, Y. Chong, T. Hughes and H. Schomerus.

Author information

Affiliations

Authors

Contributions

S.W. and M.K. did the experimental work and data analysis. S.W., M.K., Y.P., Y.L., K.G.M. and M.C.R. did the theoretical work. K.G.M., M.S., M.C.R., A.S. and S.W. were responsible for the project planning. All authors contributed equally to the manuscript writing.

Corresponding author

Correspondence to A. Szameit.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 919 kb)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Weimann, S., Kremer, M., Plotnik, Y. et al. Topologically protected bound states in photonic parity–time-symmetric crystals. Nature Mater 16, 433–438 (2017). https://doi.org/10.1038/nmat4811

Download citation

Further reading