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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. 

When matter is strongly coupled to an optical cavity, new hybrid light–matter states are formed, the so-called polariton states. These polaritons can qualitatively change the physical properties of the matter coupled to the cavity by completely altering its energy eigenspectrum. Fueled by experimental innovations in recent years, much progress has been made in simulating the intrinsic quantum behavior of these hybrid states. At the heart of each simulation is the choice of Hamiltonian to represent the total light–matter hybrid system. Even at this fundamental level, there has been significant progress in developing new gauges and representations for this Hamiltonian, whether exact or under approximations. As such, this review aims to discuss several different forms of Hamiltonians for the researcher trying to enter this field by clearly and concisely deriving each different representation from the fundamental Minimal Coupling Hamiltonian. In addition, this review provides commentary on the optimal usage and extent of approximations for each individual representation to assist the reader in choosing the appropriate Hamiltonian for their work. 

We benchmark the accuracy of various trajectory-based non-adiabatic methods in simulating the polariton relaxation dynamics under the collective coupling regime. The Holstein–Tavis–Cummings Hamiltonian is used to describe the hybrid light–matter system of N molecules coupled to a single cavity mode. We apply various recently developed trajectory-based methods to simulate the population relaxation dynamics by initially exciting the upper polariton state and benchmark the results against populations computed from exact quantum dynamical propagation using the hierarchical equations of motion approach. In these benchmarks, we have systematically varied the number of molecules N, light–matter detunings, and the light–matter coupling strengths. Our results demonstrate that the symmetrical quasi-classical method with γ correction and spin-mapping linearized semi-classical approaches yield more accurate polariton population dynamics than traditional mixed quantum-classical methods, such as the Ehrenfest and surface hopping techniques. 

We perform on-the-fly non-adiabatic molecular dynamics simulations using the recently developed spin-mapping formalism. Two quantum dynamics approaches based on this mapping formalism, (i) the fully linearized Spin-LSC and (ii) the partially linearized Spin-PLDM, are explored using the quasi-diabatic propagation scheme. We have performed dynamics simulations in four ab initio molecular models for which benchmark ab initio multiple spawning (AIMS) data have been published. We find that the spin-LSC and the previously reported symmetric quasi-classical (SQC) approaches provide nearly equivalent population dynamics. While we expected the more involved spin-PLDM method to provide superior accuracy compared to the other mapping-based approaches, SQC and spin-LSC, we found that it performed with equivalent accuracy compared to the AIMS benchmark results. We further explore the underpinnings of the spin-PLDM correlation function by decomposing its N2 density matrix-focused initial conditions, where N is the number of states in the quantum subsystem. Finally, we found an approximate form of the spin-PLDM correlation function, which simplifies the simulation and reduces the computational costs from N2 to N.