  Non-Abelian gauge fields give rise to nontrivial topological physics. Here we
develop a scheme to create an arbitrary SU(2) lattice gauge field for photons
in the synthetic frequency dimension using an array of dynamically modulated
ring resonators. The photon polarization is taken as the spin basis to
implement the matrix-valued gauge fields. Using a non-Abelian generalization of
the Harper-Hofstadter Hamiltonian as a specific example, we show that the
measurement of the steady-state photon amplitudes inside the resonators can
reveal the band structures of the Hamiltonian, which show signatures of the
underlying non-Abelian gauge field. These results provide opportunities to
explore novel topological phenomena associated with non-Abelian lattice gauge
fields in photonic systems.
