Theory of photon condensation in a spatially varying electromagnetic field

The realization of equilibrium superradiant quantum phases (photon condensates) in a spatially uniform quantum cavity field is forbidden by a "no-go"theorem stemming from gauge invariance. We here show that the no-go theorem does not apply to spatially varying quantum cavity fields. We find a criterion for its occurrence that depends solely on the static, nonlocal orbital magnetic susceptibility χorb(q), of the electronic system (ES) evaluated at a cavity photon momentum ℏq. Only 3DESs satisfying the Condon inequality χorb(q)>1/(4π) can harbor photon condensation. For the experimentally relevant case of two-dimensional (2D) ESs embedded in quasi-2D cavities the criterion again involves χorb(q) but also the vertical size of the cavity. We use these considerations to identify electronic properties that are ideal for photon condensation. Our theory is nonperturbative in the strength of electron-electron interaction and therefore applicable to strongly correlated ESs.