Experimental realization of non-Abelian permutations in a three-state non-Hermitian system

Eigenstates of a non-Hermitian system exist on complex Riemannian manifolds, with multiple sheets connecting at branch cuts and exceptional points (EPs). These eigenstates can evolve across different sheets - a process that naturally corresponds to state permutation. Here, we report the first experimental realization of non-Abelian permutations in a three-state non-Hermitian system. Our approach relies on the stroboscopic encircling of two different exceptional arcs (EAs), which are smooth trajectories of order-2 EPs appearing from the coalescence of two adjacent states. The non-Abelian characteristics are confirmed by encircling the EAs in opposite sequences. A total of five non-trivial permutations are experimentally realized, which together comprise a non-Abelian group. Our approach provides a reliable way of investigating non-Abelian state permutations and the related exotic winding effects in non-Hermitian systems.