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Non-Hermitian topological phase transitions controlled by nonlinearity

Nature Physics volume 20, pages 101–108 (2024)Cite this article

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Abstract

Manipulating topological invariants is possible by modifying the global properties of optical devices to alter their band structures. This could be achieved by statically altering devices or dynamically reconfiguring devices with considerably different geometric parameters, even though it inhibits switching speed. Recently, optical nonlinearity has emerged as a tool for tailoring topological and non-Hermitian (NH) properties, promising fast manipulation of topological phases. In this work, we observe topologically protected NH phase transitions driven by optical nonlinearity in a silicon nanophotonic Floquet topological insulator. The phase transition occurs from forbidden bandgaps to NH conducting edge modes, which emerge at a nonlinearity-induced gain–loss junction along the boundaries of a topological insulator. We find static NH edge modes and dynamic phase transitions involving exceptional points at a speed of hundreds of picoseconds, which inherently retain topological protections against fabrication imperfections. This work shows an interplay between topology and non-Hermiticity by means of nonlinear optics, and it provides a way of manipulating multiple phase transitions at high speeds that is applicable to many other materials with strong nonlinearities, which could promote the development of unconventionally robust light-controlled devices for classical and quantum applications.

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Fig. 1: Nonlinearity-driven NH phase transitions in a photonic FTI.
Fig. 2: Calculated band structures and phase diagram of the FTI in linear and nonlinear regimes.
Fig. 3: Experimental results of topological and NH phase transitions.
Fig. 4: Probing coherence of topological non-trivial edge and NH edge modes.
Fig. 5: Fast topologically protected NH phase transitions.

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Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

References

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  Google Scholar 

  3. Wang, Z. et al. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  6. Khanikaev, A. B. & Shvets, G. Two-dimensional topological photonics. Nat. Photon. 11, 763–773 (2017).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  8. Maczewsky, L. J. et al. Observation of photonic anomalous Floquet topological insulators. Nat. Commun. 8, 13756 (2017).

    Article  ADS  Google Scholar 

  9. Hafezi, M. et al. Robust optical delay lines with topological protection. Nat. Phys. 7, 907–912 (2011).

    Article  Google Scholar 

  10. Hafezi, M. et al. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001–1005 (2013).

    Article  ADS  Google Scholar 

  11. Dong, J.-W. et al. Valley photonic crystals for control of spin and topology. Nat. Mater. 16, 298–302 (2017).

    Article  ADS  Google Scholar 

  12. Shalaev, M. I. et al. Robust topologically protected transport in photonic crystals at telecommunication wavelengths. Nat. Nanotechnol. 14, 31–34 (2019).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  14. El-Ganainy, R. et al. Non-Hermitian physics and pt symmetry. Nat. Phys. 14, 11–19 (2018).

    Article  Google Scholar 

  15. Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parity–time symmetry. Nat. Photon. 11, 752–762 (2017).

    Article  ADS  Google Scholar 

  16. Özdemir, Ş. K. et al. Parity–time symmetry and exceptional points in photonics. Nat. Mater. 18, 783–798 (2019).

    Article  ADS  Google Scholar 

  17. Weidemann, S. et al. Topological triple phase transition in non-Hermitian Floquet quasicrystals. Nature 601, 354–359 (2022).

    Article  ADS  Google Scholar 

  18. Zhao, H. et al. Non-Hermitian topological light steering. Science 365, 1163–1166 (2019).

    Article  ADS  Google Scholar 

  19. Ao, Y. et al. Topological phase transition in the non-Hermitian coupled resonator array. Phys. Rev. Lett. 125, 013902 (2020).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  22. Blanco-Redondo, A. et al. Topological protection of biphoton states. Science 362, 568–571 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  23. Dai, T. et al. Topologically protected quantum entanglement emitters. Nat. Photon. 16, 248–257 (2022).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  25. Xiao, L. Non-Hermitian bulk-boundary correspondence in quantum dynamics. Nat. Phys 16, 761–766 (2020).

    Article  Google Scholar 

  26. Wang, K. et al. Generating arbitrary topological windings of a non-Hermitian band. Science 371, 1240–1245 (2021).

    Article  ADS  Google Scholar 

  27. Boyd, R. W. Nonlinear Optics 2nd edn (Academic Press, 2008).

  28. Smirnova, D. et al. Nonlinear topological photonics. Appl. Phys. Rev. 7, 021306 (2020).

    Article  ADS  Google Scholar 

  29. Maczewsky, L. J. et al. Nonlinearity-induced photonic topological insulator. Science 370, 701–704 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  30. Kruk, S. et al. Nonlinear light generation in topological nanostructures. Nat. Nanotechnol. 14, 126–130 (2019).

    Article  ADS  Google Scholar 

  31. Hu, Z. et al. Nonlinear control of photonic higher-order topological bound states in the continuum. Light Sci. Appl. 10, 1–10 (2021).

    Article  Google Scholar 

  32. Kirsch, M. S. et al. Nonlinear second-order photonic topological insulators. Nat. Phys. 17, 995–1000 (2021).

    Article  Google Scholar 

  33. Mukherjee, S. & Rechtsman, M. C. Observation of Floquet solitons in a topological bandgap. Science 368, 856–859 (2020).

    Article  ADS  Google Scholar 

  34. Konotop, V. V., Yang, J. & Zezyulin, D. A. Nonlinear waves in PT-symmetric systems. Rev. Mod. Phys. 88, 035002 (2016).

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  36. Lumer, Y. et al. Nonlinearly induced PT transition in photonic systems. Phys. Rev. Lett. 111, 263901 (2013).

    Article  ADS  Google Scholar 

  37. Xia, S. et al. Nonlinear tuning of PT symmetry and non-Hermitian topological states. Science 372, 72–76 (2021).

    Article  ADS  Google Scholar 

  38. Minzioni, P. et al. Roadmap on all-optical processing. J. Opt. 21, 063001 (2019).

    Article  ADS  Google Scholar 

  39. Wang, J. et al. Integrated photonic quantum technologies. Nat. Photon. 14, 273–284 (2020).

    Article  ADS  Google Scholar 

  40. Zeljkovic, I. et al. Strain engineering Dirac surface states in heteroepitaxial topological crystalline insulator thin films. Nat. Nanotechnol. 10, 849–853 (2015).

    Article  ADS  Google Scholar 

  41. Kudyshev, Z. A. et al. Photonic topological phase transition on demand. Nanophotonics 8, 1349–1356 (2019).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  43. Leykam, D. et al. Reconfigurable topological phases in next-nearest-neighbor coupled resonator lattices. Phys. Rev. Lett. 121, 023901 (2018).

    Article  ADS  Google Scholar 

  44. Shalaev, M. I., Walasik, W. & Litchinitser, N. M. Optically tunable topological photonic crystal. Optica 6, 839–844 (2019).

    Article  ADS  Google Scholar 

  45. You, J. W. et al. Reprogrammable plasmonic topological insulators with ultrafast control. Nat. Commun. 12, 5468 (2021).

    Article  ADS  Google Scholar 

  46. Rudner, M. S. et al. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).

    Google Scholar 

  47. Kitagawa, T. et al. Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010).

    Article  ADS  Google Scholar 

  48. Liang, G. Q. & Chong, Y. D. Optical resonator analog of a two-dimensional topological insulator. Phys. Rev. Lett. 110, 203904 (2013).

    Article  ADS  Google Scholar 

  49. Afzal, S. et al. Realization of anomalous floquet insulators in strongly coupled nanophotonic lattices. Phys. Rev. Lett. 124, 253601 (2020).

    Article  ADS  Google Scholar 

  50. Guglielmon, J. et al. Photonic realization of a transition to a strongly driven Floquet topological phase. Phys. Rev. A 97, 031801 (2018).

    Article  ADS  Google Scholar 

  51. Foster, M. A. et al. Broad-band optical parametric gain on a silicon photonic chip. Nature 441, 960–963 (2006).

    Article  ADS  Google Scholar 

  52. Lipson, M. The revolution of silicon photonics. Nat. Mat. 21, 974–975 (2022).

    Article  Google Scholar 

Download references

Acknowledgements

We acknowledge support from the National Natural Science Foundation of China (grant nos. 12325410, 61975001, 61590933, 11734001, 91950204, 92150302, 11527901,61904196, 62274179 and 62235001), the Innovation Program for Quantum Science and Technology (grant no. 2021ZD0301500), the National Key R&D Program of China (grant nos. 2019YFA0308702, 2018YFA0704404 and 2022YFB2802400), Beijing Natural Science Foundation (grant nos. Z190005 and Z220008), the Guangdong Major Project of Basic and Applied Basic Research (grant no. 2020B0301030009) and the Key R&D Program of Guangdong Province (grant no. 2018B030329001).

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Authors and Affiliations

  1. State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, Beijing, China

    Tianxiang Dai, Yutian Ao, Jun Mao, Yun Zheng, Chonghao Zhai, Yandong Li, Jingze Yuan, Xiaoyong Hu, Qihuang Gong & Jianwei Wang

  2. Institute of Microelectronics, Chinese Academy of Sciences, Beijing, China

    Yan Yang, Bo Tang, Zhihua Li, Jun Luo & Wenwu Wang

  3. Frontiers Science Center for Nano-optoelectronics & Collaborative Innovation Center of Quantum Matter, Peking University, Beijing, China

    Xiaoyong Hu, Qihuang Gong & Jianwei Wang

  4. Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, China

    Xiaoyong Hu, Qihuang Gong & Jianwei Wang

  5. Peking University Yangtze Delta Institute of Optoelectronics, Nantong, China

    Xiaoyong Hu, Qihuang Gong & Jianwei Wang

  6. Hefei National Laboratory, Hefei, China

    Xiaoyong Hu, Qihuang Gong & Jianwei Wang

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  1. Tianxiang Dai
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Contributions

T.D. and J.W. conceived the project. T.D., J.M., Y.Y., Y.Z., C.Z. and B.T. implemented the experiment. T.D. provided the simulations and performed the theoretical analysis. T.D., Y.A., Y.L. and J.Y. discussed and improved the theoretical results. Z.L., J.L., W.W., X.H., Q.G. and J.W. managed the project. T.D. and J.W. wrote the manuscript with input from all the authors. All the authors discussed the results and contributed to the manuscript.

Corresponding author

Correspondence to Jianwei Wang.

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Nature Physics thanks Jiangbin Gong and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Dai, T., Ao, Y., Mao, J. et al. Non-Hermitian topological phase transitions controlled by nonlinearity. Nat. Phys. 20, 101–108 (2024). https://doi.org/10.1038/s41567-023-02244-8

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