where |E J ⟩ are the polariton states that are the eigenstate of the polariton Hamiltonian ĤGTC -TR where TR is the nuclear kinetic energy operator, ∆E J = E J -E G with E G as the ground state energy of the light-matter hybrid system, Γ c is a broadening parameter that accounts for various sources of dissipation such as cavity loss, and ⟨...⟩ represents the average over different nuclear configurations. The authors find that an initial excitation to the upper polariton branch quickly decays to the dark states which then transfer population to the lower polariton. This relaxation process is reflected in the photoluminescence (PL) spectra shown in Fig. . This is because unlike the absorption spectra in Fig. , the PL spectra depend on the populations of the polariton states and can be computed as where ρ a (t) is the steady-state (non-equilibrium photodriven condition) population of the a th polariton state, |Ψ a ⟩, at a delay time t after photoexcitation. Ref. 32 simulated the PL spectra using a similar approach and analyzed the relaxation process from the upper polariton to the lower polariton through the dark state in an ensemble of nanoplatelets coupled to cavity radiation (as schematically illustrated in Fig. ). This combined theoretical and experimental study demonstrates that at small exciton-photon detunings, the phonon-assisted nonadiabatic transitions lead to the depletion of the upper polariton population and the transfer of population to the lower polariton branch. The PL spectra obtained experimentally and theoretically are shown in Fig. -f and Fig. , respectively.