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Classical non-Abelian braiding of acoustic modes


Non-Abelian braiding is regarded as an essential process for realizing quantum logic. Its realizations in quantum systems often rely on the dynamic winding of anyons, which can be challenging to obtain. Implementing braiding in a classical system could, therefore, assist the experimental study of non-Abelian physics. Here we present the realization of the non-Abelian braiding of multiple degenerate acoustic waveguide modes. The dynamics of non-Abelian braiding can be captured by the non-Abelian Berry–Wilczek–Zee phase that connects the holonomic adiabatic evolutions of multiple degenerate states. The cyclic evolution of degenerate states induces a non-Abelian geometric phase, manifesting as the exchange of states. The non-Abelian characteristics are revealed by switching the order of two distinct braiding processes involving three modes. Our work demonstrates wave manipulations based on non-Abelian braiding and logic operations.

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Fig. 1: Braiding of two states.
Fig. 2: Experimental realization of acoustic two-state braiding.
Fig. 3: Non-Abelian braiding of three degenerate states.
Fig. 4: Experimental realization of three-state non-Abelian braiding.

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 author upon reasonable request.


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Z.-G.C. and G.M. thank W. Tang and Q. Wang for experimental assistance. This work was supported by Hong Kong Research Grants Council (12302420, 12300419, 22302718, C6013-18G, AoE/P-02/12 and 16303119), National Natural Science Foundation of China (11922416 and 11802256) and Hong Kong Baptist University (RC-SGT2/18-19/SCI/006). R.-Y.Z. and C.T.C. also acknowledge support from the Croucher Foundation (CAS20SC01).

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Authors and Affiliations



Z.-G.C. and R.-Y.Z. developed the theory. Z.-G.C. performed the numerical simulations. Z.-G.C. and G.M. carried out the experiments with support from C.T.C. All the authors analysed the results. Z.-G.C. and G.M. wrote the manuscript with inputs from R.-Y.Z. and C.T.C. The project was initiated and supervised by G.M.

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Correspondence to Guancong Ma.

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Chen, ZG., Zhang, RY., Chan, C.T. et al. Classical non-Abelian braiding of acoustic modes. Nat. Phys. 18, 179–184 (2022).

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