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# 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.

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## 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.

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## 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

### 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.

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## 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

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

• Accepted:

• Published:

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

• ### Thouless pumping of light with a twist

• Laura Pilozzi
• Valentina Brosco

Nature Physics (2022)