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Photonic quadrupole topological phases


The topological phases of matter are characterized using the Berry phase, a geometrical phase associated with the energy-momentum band structure. The quantization of the Berry phase and the associated wavefunction polarization manifest as remarkably robust physical observables, such as quantized Hall conductivity and disorder-insensitive photonic transport1,2,3,4,5. Recently, a novel class of topological phases, called higher-order topological phases, were proposed by generalizing the fundamental relationship between the Berry phase and quantized polarization, from dipole to multipole moments6,7,8. Here, we demonstrate photonic realization of the quantized quadrupole topological phase, using silicon photonics. In our two-dimensional second-order topological phase, we show that the quantization of the bulk quadrupole moment manifests as topologically robust zero-dimensional corner states. We contrast these topological states against topologically trivial corner states in a system without bulk quadrupole moment, where we observe no robustness. Our photonic platform could enable the development of robust on-chip classical and quantum optical devices with higher-order topological protection.

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Fig. 1: Schematic of the photonic quadrupole topological system.
Fig. 2: Observation of corner states.
Fig. 3: Corner states at the quadrupole domain boundary.
Fig. 4: Corner states in a lattice with zero gauge flux.

Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.


  1. Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979).

    Article  ADS  Google Scholar 

  2. Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989).

    Article  ADS  Google Scholar 

  3. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    Article  ADS  Google Scholar 

  4. King-Smith, R. D. & Vanderbilt, D. Theory of polarization of crystalline solids. Phys. Rev. B 47, 1651–1654 (1993).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  6. Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  7. Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. Phys. Rev. B 96, 245115 (2017).

    Article  ADS  Google Scholar 

  8. Schindler, F. et al. Higher-order topological insulators. Sci. Adv. 4, eaat0346 (2018).

    Article  ADS  Google Scholar 

  9. Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

    Article  ADS  Google Scholar 

  10. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    Article  ADS  Google Scholar 

  11. Schindler, F. et al. Higher-order topology in bismuth. Nat. Phys. 14, 918–924 (2018).

    Article  Google Scholar 

  12. Cooper, N. R., Dalibard, J. & Spielman, I. B. Topological bands for ultracold atoms. Rev. Mod. Phys. 91, 015005 (2019).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  14. Kraus, Y. E., Lahini, Y., Ringel, Z., Verbin, M. & Zilberberg, O. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  Google Scholar 

  19. Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A quantized microwave quadrupole insulator with topologically protected corner states. Nature 555, 346–350 (2018).

    Article  ADS  Google Scholar 

  20. Imhof, S. et al. Topolectrical-circuit realization of topological corner modes. Nat. Phys. 14, 925–929 (2018).

    Article  Google Scholar 

  21. Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).

    Article  ADS  Google Scholar 

  22. Xue, H., Yang, Y., Gao, F., Chong, Y. & Zhang, B. Acoustic higher-order topological insulator on a kagome lattice. Nat. Mater. 18, 108–112 (2019).

    Article  ADS  Google Scholar 

  23. Ni, X., Weiner, M., Alù, A. & Khanikaev, A. B. Observation of higher-order topological acoustic states protected by generalized chiral symmetry. Nat. Mater. 18, 113–120 (2019).

    Article  ADS  Google Scholar 

  24. Noh, J. et al. Topological protection of photonic mid-gap defect modes. Nat. Photon. 12, 408–415 (2018).

    Article  ADS  Google Scholar 

  25. Li, F.-F. et al. Topological light-trapping on a dislocation. Nat. Commun. 9, 2462 (2018).

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  27. Mittal, S., Ganeshan, S., Fan, J., Vaezi, A. & Hafezi, M. Measurement of topological invariants in a 2D photonic system. Nat. Photon. 10, 180–183 (2016).

    Article  ADS  Google Scholar 

  28. Liu, T. et al. Second-order topological phases in non-Hermitian systems. Phys. Rev. Lett. 122, 076801 (2019).

    Article  ADS  Google Scholar 

  29. Jing, H., Özdemir, S. K., Lü, H. & Nori, F. High-order exceptional points in optomechanics. Sci. Rep. 7, 3386 (2017).

    Article  ADS  Google Scholar 

  30. Mittal, S., Goldschmidt, E. A. & Hafezi, M. A topological source of quantum light. Nature 561, 502–506 (2018).

    Article  ADS  Google Scholar 

  31. Peano, V., Houde, M., Marquardt, F. & Clerk, A. A. Topological quantum fluctuations and traveling wave amplifiers. Phys. Rev. X 6, 041026 (2016).

    Google Scholar 

  32. Barik, S. et al. A topological quantum optics interface. Science 359, 666–668 (2018).

    Article  ADS  MathSciNet  Google Scholar 

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This research was supported by AFOSR-MURI FA9550-14-1-0267, the Sloan Foundation and the Physics Frontier Center at the Joint Quantum Institute. A.P. and M.A.G. have been supported by the Russian Foundation for Basic Research grant no. 18-29-20037 and the Foundation for the Advancement of Theoretical Physics and Mathematics BASIS. A.P. also acknowledges partial support by the Australian Research Council. The authors thank H. Dehghani, Y. Kivshar and D. Leykam for discussions and E. Yamaguchi and J. Vannucci for experimental help.

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Authors and Affiliations



S.M. designed the chips and the experiment. S.M. and V.V.O. performed the experiments. A.P., G.Z. and M.A.G. performed the theoretical analysis. M.H. supervised the project. All authors contributed to analysing the data and writing the manuscript.

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Correspondence to Sunil Mittal.

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The authors declare no competing interests.

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Supplementary Information

Supplementary Information

Supplementary derivations, discussion and experimental data. Supplementary Figs. 1–10 and Supplementary references 1–6.

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Mittal, S., Orre, V.V., Zhu, G. et al. Photonic quadrupole topological phases. Nat. Photonics 13, 692–696 (2019).

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