Experimental Realization of SU(3) Dynamics in Classical Mechanical System by Non-Adiabatic Holonomic Control

The SO(N) groups have important applications in classical mechanics and gravity, as they describe spatial symmetry operations in classical physics, such as the rotation of rigid bodies in N-dimensional space and the spin of black holes. Recently, SO(N) dynamics have been successfully utilized to realize non-Abelian gauge fields in classical wave systems, achieving effects such as non-Abelian braiding. On the other hand, laws in quantum physics obey SU(N) group, such as the SU(2) rotation of electron spin and the tetraquark in SU(3) gauge field. Therefore, it remains challenging to emulate SU(N) dynamics in classical mechanical systems. In this work, we design a coupled system consisting of four spring oscillators with precisely modulated coupling coefficients to realize SU(3) operation. Our approach decomposes SU(3) matrix into two Householder reflections and a 3-site phase operation, which can be achieved by rotating around the z-axis in Bloch sphere for two different pairs of sites. Therefore, any SU(3) operation can be achieved by the sequential execution of four operations, with each operation governed by a time-independent fixed-form Hamiltonian. The Householder reflections decomposition significantly simplifies the operation steps and acheives faster and more robust non-adiabatic holonomic evolution. Our results open a new way for studying and simulating physical phenomena governed by non-Abelian gauge fields in classical systems such as photonics, acoustics, and mechanics.