Three-dimensional non-Abelian quantum holonomy

Abstract When a quantum system undergoes slow changes, the evolution of its state depends only on corresponding trajectory in Hilbert space. This phenomenon, known as holonomy, brings to light geometric aspects theory. Depending number degrees freedom involved, these purely entities can be scalar or belong matrix-valued symmetry group. In their various forms, holonomies are vital elements description fundamental forces particle physics well theories beyond standard model such loop gravity topological field Yet, implementing thus far has proven challenging, being further complicated by difficulties involved identifying suitable dark states for construction bosonic systems. Here we develop representation holonomic theory founded Heisenberg picture and leverage insights experimental realization three-dimensional holonomy. Its non-Abelian phase is implemented via judicious manipulation modes constructed from indistinguishable photons obeys U(3) relevant strong interaction. Our findings could enable study higher-dimensional gauge symmetries exploration exotic photonic chip.