Non-Abelian braiding plays a crucial role in topological quantum computing for protecting the evolution of non-Abelian anyons. Recently, the concept of non-Abelian braiding has been introduced to classical systems for applications such as quantum logic, but the working mode was restricted to the fundamental mode. Here, we propose a scheme for realizing the classical non-Abelian braiding of coupled modes in photonic waveguides system. The Hamiltonian of the designed system is built via the interplay between two sub-Hamiltonians, where one of them enables the non-Abelian braiding, while the other one generates the coupled modes. In this way, the non-Abelian braiding of coupled modes of arbitrarily high order can be realized. We carry out numerical simulations to verify the concept for coupled modes with an order of four, including the mode switching behavior as well as the non-Abelian geometric phase. The proposed concept may lead to on-chip applications including mode division multiplexing non-Abelian braiding.

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• Geometric & topological phases
• Non-Abelian gauge theories
• Waveguides