Critical Fluorescence of a Transmon at the Schmid Transition

We investigate inelastic microwave photon scattering by a transmon qubit embedded in a high-impedance circuit. The transmon undergoes a charge-localization (Schmid) transition upon the impedance reaching the critical value. Because of the unique transmon level structure, the fluorescence spectrum carries a signature of the transition point. At higher circuit impedance, quasielastic photon scattering may account for the main part of the inelastic scattering cross section; we find its dependence on the qubit and circuit parameters.