Floquet systems with quantized drive

Time-periodic (Floquet) drive has become a powerful tool to engineer phases of matter, both in equilibrium and far from equilibrium. In this talk, I argue that treating the degrees of freedom doing the driving – usually photons – as quantized leads to interesting phases of coupled light and matter. I motivate this idea by introducing the Floquet-Thouless energy pump [MHK et al, PRL 120, 150601 (2018)], where topological pumping naturally occurs not on the charge degrees of freedom but by a quantized backaction on the drive. I then show that non-trivial physics can occur in the even simpler case of many body localization [Ng and MHK, arXiv:1809.02606], where the localizing interactions compete against global coupling to the quantized cavity mode. Applications of this physics to many body cavity QED and superconducting circuit QED will be discussed.