  We consider a system of dynamically-modulated photonic resonator lattice
undergoing photonic transition, and show that in the ultra-strong coupling
regime such a lattice can exhibit non-trivial topological properties, including
topologically non-trivial band gaps, and the associated topologically-robust
one-way edge states. Compared with the same system operating in the regime
where the rotating wave approximation is valid, operating the system in the
ultra-strong coupling regime results in one-way edge modes that has a larger
bandwidth, and is less susceptible to loss. Also, in the ultra-strong coupling
regime, the system undergoes a topological insulator-to-metal phase transition
as one varies the modulation strength. This phase transition has no counter
part in systems satisfying the rotating wave approximation, and its nature is
directly related to the non-trivial topology of the quasi-energy space.
