Supersolid light in a semiconductor microcavity

Supersolidity - simultaneous superfluid flow and crystalline order - has been realized in quantum atomic systems but remains unexplored in purely photonic platforms operating at weak light-matter coupling. We predict a supersolid phase of light in a plasma-filled optical microcavity, where photons acquire effective mass and interact via nonlocal, plasma-mediated nonlinearities. By deriving a Gross-Pitaevskii equation with a tunable photon-photon interaction kernel, we show that under coherent driving the cavity light field can spontaneously crystallize into a supersolid lattice via modulational instability. Crucially, this supersolid arises from a weak photon-electron coupling enabled by virtual electronic transitions, and it does not require hybrid polariton formation. Using doped semiconductor microcavities, we identify feasible conditions (electron densities $\sim 10^{10}- 10^{11}~\mathrm{cm}^{-2}$ and optical intensities $\sim 10^{2}-10^{4}~\mathrm{W/cm}^{2}$) for experimental realization. This work establishes plasmonic cavities as a platform for correlated photonic matter with emergent quantum order.