Quantum open system description of a hybrid plasmonic cavity

We present a unified quantum open system framework for lossy plasmonic cavities in which coherent dynamics, relaxation, dephasing, and irreversible absorption are treated on equal footing. The Dyson equation for the cavity photon propagator in the random-phase approximation yields a complex self-energy S that accounts for both the renormalization and the damping of hybrid plasmon-photon modes (polaritons, in a quasi-particle description). Tracing out the electronic and photonic environments leads to a Liouvillian for the upper (UP) and lower (LP) polaritonic branches, incorporating leakage through the imaginary part of the self-energy, internal UP-LP scattering rates, and dephasing. Time evolution equations for polariton populations, interbranch coherence, and driven amplitudes in closed form also provide analytic expressions for their steady-state values, the quench rate of UP-LP oscillations and polaritonic lineshapes, valid in the limit of low polaritonic density, but covering light-matter ultrastrong coupling. The theory establishes a self-consistent description of dissipative polariton dynamics in plasmonic and nanophotonic cavities, directly applicable to response spectra, time-domain measurements, and dissipation engineering.