Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Non-Abelian Thouless pumping in photonic waveguides

Abstract

Thouless pumping enables topological transport and the direct measurement of topological invariants. So far, realizations of Thouless pumping rely on the adiabatic evolution of a physical system following a non-degenerate band, but it has been predicted that pumping can become non-Abelian in nature when degenerate bands exist. The resulting non-Abelian gauge fields and associated non-commutative operations would be promising for applications related to unitary matrices, such as photonic quantum logic. Here we propose the experimental realization of non-Abelian Thouless pumping in an on-chip photonic platform. By modulating the coupling coefficients within photonic waveguides with degenerate flat bands, we observe non-Abelian Thouless pumping in a three-step pumping device where the outcomes depend on the sequence of the pumping operations. We anticipate our versatile platform to reveal more complex non-Abelian topological physics and realize on-chip non-Abelian photonic devices in the future.

This is a preview of subscription content, access via your institution

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Fig. 1: Lieb lattice consisting of photonic waveguides with modulated coupling coefficients.
Fig. 2: Mode switching and geometric phases in Thouless pumping.
Fig. 3: Non-Abelian nature of Thouless pumping.
Fig. 4: Non-Abelian Thouless pumping with multiple pumping operations.

Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The codes used for performing the theoretical analysis and numerical simulations are available from X.-L.Z. upon reasonable request.

References

  1. Thouless, D. J. Quantization of particle transport. Phys. Rev. B 27, 6083–6087 (1983).

    ADS  MathSciNet  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    Google Scholar 

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

    ADS  Google Scholar 

  7. Nalitov, A., Solnyshkov, D. & Malpuech, G. Polariton Z topological insulator. Phys. Rev. Lett. 114, 116401 (2015).

    ADS  MathSciNet  Google Scholar 

  8. Lohse, M., Schweizer, C., Price, H. M., Zilberberg, O. & Bloch, I. Exploring 4D quantum Hall physics with a 2D topological charge pump. Nature 553, 55–58 (2018).

    ADS  Google Scholar 

  9. Privitera, L., Russomanno, A., Citro, R. & Santoro, G. E. Nonadiabatic breaking of topological pumping. Phys. Rev. Lett. 120, 106601 (2018).

    ADS  Google Scholar 

  10. Fedorova, Z. et al. Limits of topological protection under local periodic driving. Light. Sci. Appl. 8, 63 (2019).

    ADS  Google Scholar 

  11. Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162–166 (2015).

    Google Scholar 

  12. Wimmer, M., Price, H., Carusotto, I. & Peschel, U. Experimental measurement of the Berry curvature from anomalous transport. Nat. Phys. 13, 545–550 (2017).

    Google Scholar 

  13. Zilberberg, O. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553, 59–62 (2018).

    ADS  Google Scholar 

  14. Wang, Y. et al. Direct observation of topology from single-photon dynamics. Phys. Rev. Lett. 122, 193903 (2019).

    ADS  Google Scholar 

  15. Niu, Q. & Thouless, D. J. Quantised adiabatic charge transport in the presence of substrate disorder and many-body interaction. J. Phys. A 17, 2453 (1984).

    ADS  MathSciNet  MATH  Google Scholar 

  16. Lohse, M., Schweizer, C., Zilberberg, O., Aidelsburger, M. & Bloch, I. A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice. Nat. Phys. 12, 350–354 (2016).

    Google Scholar 

  17. Nakajima, S. et al. Topological Thouless pumping of ultracold fermions. Nat. Phys. 12, 296–300 (2016).

    Google Scholar 

  18. Ma, W. et al. Experimental observation of a generalized Thouless pump with a single spin. Phys. Rev. Lett. 120, 120501 (2018).

    ADS  Google Scholar 

  19. Fedorova, Z., Qiu, H., Linden, S. & Kroha, J. Observation of topological transport quantization by dissipation in fast Thouless pumps. Nat. Commun. 11, 3758 (2020).

    ADS  Google Scholar 

  20. Cerjan, A., Wang, M., Huang, S., Chen, K. P. & Rechtsman, M. C. Thouless pumping in disordered photonic systems. Light. Sci. Appl. 9, 178 (2020).

    ADS  Google Scholar 

  21. Jürgensen, M., Mukherjee, S. & Rechtsman, M. C. Quantized nonlinear Thouless pumping. Nature 596, 63–67 (2021).

    ADS  Google Scholar 

  22. Ke, Y. et al. Topological phase transitions and Thouless pumping of light in photonic waveguide arrays. Laser Photon. Rev. 10, 995–1001 (2016).

    ADS  Google Scholar 

  23. Ke, Y., Qin, X., Kivshar, Y. S. & Lee, C. Multiparticle Wannier states and Thouless pumping of interacting bosons. Phys. Rev. A 95, 063630 (2017).

    ADS  Google Scholar 

  24. Kolodrubetz, M. H., Nathan, F., Gazit, S., Morimoto, T. & Moore, J. E. Topological floquet Thouless energy pump. Phys. Rev. Lett. 120, 150601 (2018).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  26. Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).

    ADS  Google Scholar 

  27. Wu, Q., Soluyanov, A. A. & Bzdušek, T. Non-Abelian band topology in noninteracting metals. Science 365, 1273–1277 (2019).

    MathSciNet  MATH  Google Scholar 

  28. Tiwari, A. & Bzdušek, T. Non-Abelian topology of nodal-line rings in PT-symmetric systems. Phys. Rev. B 101, 195130 (2020).

    ADS  Google Scholar 

  29. Yang, E. et al. Observation of non-Abelian nodal links in photonics. Phys. Rev. Lett. 125, 033901 (2020).

    ADS  Google Scholar 

  30. Guo, Q. et al. Experimental observation of non-Abelian topological charges and edge states. Nature 594, 195–200 (2021).

    ADS  Google Scholar 

  31. Brosco, V., Pilozzi, L., Fazio, R. & Conti, C. Non-Abelian Thouless pumping in a photonic lattice. Phys. Rev. A 103, 063518 (2021).

    ADS  MathSciNet  Google Scholar 

  32. Carolan, J. et al. Universal linear optics. Science 349, 711–716 (2015).

    MathSciNet  MATH  Google Scholar 

  33. Imany, P. et al. High-dimensional optical quantum logic in large operational spaces. npj Quantum Inf. 5, 59 (2019).

    ADS  Google Scholar 

  34. Zhang, X. L. et al. Non-Abelian braiding on photonic chips. Nat. Photon. 16, 390–395 (2022).

    ADS  Google Scholar 

  35. Wu, S., Gao, Z., Wu, T., Zhang, Z. & Feng, L. Ultrafast heterodyne mode imaging and refractive index mapping of a femtosecond laser written multimode waveguide. Opt. Lett. 47, 214–217 (2022).

    ADS  Google Scholar 

  36. Yu, F., Zhang, X. L., Tian, Z. N., Chen, Q. D. & Sun, H. B. General rules governing the dynamical encircling of an arbitrary number of exceptional points. Phys. Rev. Lett. 127, 253901 (2021).

    ADS  Google Scholar 

  37. Boross, P., Asbóth, J. K., Széchenyi, G., Oroszlány, L. & Pályi, A. Poor man’s topological quantum gate based on the Su-Schrieffer-Heeger model. Phys. Rev. B 100, 045414 (2019).

    ADS  Google Scholar 

  38. Wu, Y., Liu, H., Liu, J., Jiang, H. & Xie, X. C. Double-frequency Aharonov-Bohm effect and non-Abelian braiding properties of Jackiw-Rebbi zero-mode. Natl Sci. Rev. 7, 572–578 (2020).

    Google Scholar 

  39. Wilczek, F. & Zee, A. Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111–2114 (1984).

    ADS  MathSciNet  Google Scholar 

  40. Chen, Z. G., Zhang, R. Y., Chan, C. T. & Ma, G. Classical non-Abelian braiding of acoustic modes. Nat. Phys. 18, 179–184 (2022).

    Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  43. Yu, F. et al. Resetting directional couplers for high-fidelity quantum photonic integrated chips. Opt. Lett. 46, 5181–5184 (2021).

    ADS  Google Scholar 

  44. You, O. et al. Observation of non-Abelian Thouless pump. Phys. Rev. Lett. 128, 244302 (2022).

    ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (NSFC) under grant nos. 61590930, 11974140, 61805098, 61825502 and 61827826, as well as China Postdoctoral Science Foundation grant no. 2019M651200. X.-L.Z. thanks G. Ma for fruitful discussions.

Author information

Authors and Affiliations

Authors

Contributions

X.-L.Z. conceived the idea, performed the theoretical analysis and designed the experiment. Y.-K.S. and F.Y. carried out the experimental measurements under the supervision of Z.-N.T. and Q.-D.C. The manuscript was written by X.-L.Z. with input from all the authors. The project was supervised by H.-B.S.

Corresponding authors

Correspondence to Xu-Lin Zhang, Zhen-Nan Tian or Hong-Bo Sun.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

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

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Notes 1–4, Figs. 1–16, Table 1 and refs. 1–6.

Source data

Source Data Fig. 1c–e

Source data for Fig. 1c–e.

Source Data Fig. 2a,b

Source data for Fig. 2a,b.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sun, YK., Zhang, XL., Yu, F. et al. Non-Abelian Thouless pumping in photonic waveguides. Nat. Phys. 18, 1080–1085 (2022). https://doi.org/10.1038/s41567-022-01669-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-022-01669-x

Further reading

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing