Thouless pump provides robust ways to realize quantized transport of waves and particles, and it casts the static 2D quantum Hall effect onto 1D dynamic systems where one of the momentum dimensions is replaced by the evolution time or path parameter. In the past few decades, various types of Abelian Thouless pump have been achieved theoretically and experimentally. However, the study of non-Abelian Thouless pump is scarce, which tells us that the order of two evolution loops with the same base point cannot be changed, and there has been no experimental observation of non-Abelian Thouless pump. Here we report the observation of a non-Abelian Thouless pump in coupled acoustic waveguide array. The non-Abelian property originates from the noncommutative combination of two different ${\mathbb{Z}}_3$ pump cycles that traverse across multiple band degeneracies in the parameter space in a three-band system. Moreover, we can pump a specific initial state to any state on any lattice site by applying these two ${\mathbb{Z}}_3$ pump cycles multiple times in a well-designed sequence. Our study paves the way for exploring and utilizing non-Abelian dynamical effects in classical wave systems and may offer different recipes for quantum walking, quantum optics, and quantum computation.

• Received 9 November 2021
• Revised 17 January 2022
• Accepted 16 May 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.244302