Controlling topological phases of matter with quantum light

Controlling the topological properties of quantum matter is a major goal of condensed matter physics. A major effort in this direction has been devoted to using classical light in the form of Floquet drives to manipulate and induce states with non-trivial topology. A different route can be achieved with cavity photons. Here we consider a prototypical model for topological phase transition, the one-dimensional Su-Schrieffer-Heeger model, coupled to a single mode cavity. We show that quantum light can affect the topological properties of the system, including the finite-length energy spectrum hosting edge modes and the topological phase diagram. In particular we show that depending on the lattice geometry and the strength of light-matter coupling one can either turn a trivial phase into a topological one or viceversa using quantum cavity fields. Furthermore, we compute the polariton spectrum of the coupled electron-photon system, and we note that the lower polariton branch disappears at the topological transition point. This phenomenon can be used to probe the phase transition in the Su-Schrieffer-Heeger model.