Renormalized photon propagator in quantum electrodynamics of plasmonic cavities

Cavity quantum electrodynamics (cQED) is a captivating field of optical physics that explores the interactions between light and matter and is particularly fascinating in the field of ultrastrong coupling. First, considering the hybridization of electromagnetic cavity (EC) modes and surface plasmon polaritons (SPPs) in a resonant cavity, we show that classical Lagrangian electrodynamics and cQED lead to the same expressions for frequency dispersion when mode dissipation and system-bath interactions are neglected. We then expand the photon propagator to a Dyson series and derive a novel non-perturbative expression for the frequency dispersion of the hybrid modes, which is equivalent to the cQED result but has a richer content. In this context, it is shown that the SPP self-interaction generates a positive self-energy that renormalizes the hybrid SPP-EC eigenfrequencies and is responsible for the well-known blue-shift of the hybrid modes, an aspect that was previously hidden in the known expressions for frequency dispersion.