Analytic rate theory of polariton relaxation that explains long polariton lifetime

Hybridization of a molecular exciton with a quantized photon creates a polariton. Despite extensive experimental investigations, the apparent lifetime of the exciton-polariton is not well-understood. We examined the steady-state population dynamics for a Holstein-Tavis-Cumming Hamiltonian to illuminate the long-term polaritonic dynamics and lifetime of the exciton-polariton in an optical cavity. For a realistic description of polariton relaxation, cavity loss and various exciton decay channels are included in the model. We found that in the presence of weak but finite exciton loss, the apparent lifetime of the lower polariton coincides with the out-of-cavity exciton lifetime and is independent of cavity-matter detuning. This is a simple explanation for the experimentally observed lifetimes for exciton polaritons and theoretically justifies the dark state reservoir hypothesis. Furthermore, if the upper polariton is initially populated, the system reaches the steady state very quickly, leading to single-exponential polariton relaxation. Starting from the lower polariton leads to a longer pre-steady-state time period, leading to double-exponential relaxation. Finally, we considered the effect of site orientational disorders and the exciton frequency disorderers. Under the collective limit, the effects of this disorder can be included in Fermi's golden rule population dynamics without explicit sampling. For the exciton energy disorders, numerical calculations are needed. Our theoretical framework is applicable to interpret exciton-polariton experiments, especially related to the measured apparent lifetime of polaritons.