Topologically nontrivial Floquet band structure in a system undergoing photonic transitions in the ultrastrong-coupling regime

We consider a system of dynamically-modulated photonic resonator lattice undergoing photonic transition and show that in the ultrastrong-coupling regime such a lattice can exhibit nontrivial topological properties, including topologically nontrivial 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 ultrastrong-coupling regime results in a one-way edge mode that has a larger bandwidth and is less susceptible to loss. Also, in the ultrastrong-coupling regime, the system undergoes a topological insulator-to-metal phase transition as one varies the modulation strength. This phase transition has no counterpart in systems satisfying the rotating wave approximation, and its nature is directly related to the nontrivial topology of the quasienergy space.