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, non-local orbital magnetic susceptibility $\chi_{\rm orb}(q)$, of the electronic system (ES) evaluated at a cavity photon momentum $\hbar q$. Only 3DESs satisfying the Condon inequality $\chi_{\rm orb}(q)>1/(4\pi)$ can harbor photon condensation. For the experimentally relevant case of two-dimensional (2D) ESs embedded in quasi-2D cavities the criterion again involves $\chi_{\rm orb}(q)$ 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 non-perturbative in the strength of electron-electron interaction and therefore applicable to strongly correlated ESs.