Static analogues of driven phases of matter

Driven phases of matter are often said to arise only in nonequilibrium conditions. We here show that analogues of driven phases can be created as time-independent eigenstates of matter interacting with quantum light. These phases are characterized by a hidden geometric phase, referred to here as optical Berry phase, which is related to the average photon number of the system. We illustrate our results by studying an analogue of a driven topological state, the so called Anomalous Floquet Anderson Insulator. We show that the optical Berry phase manifests itself as a topological pumping of photons at the boundary equal to the bulk invariant. Finally, we discuss how the tools of cavity QED systems can be brought to bear in the study of these phases of light and matter.