To investigate the polariton relaxation process, we derive the expression of the non-equilibrium Fermi's golden rule (NE-FGR) theory for the HTC Hamiltonian expressed in Eq. A2. NE-FGR has been previously developed to investigate photoinduced charge transfer processes by Sun, Geva, and co-workers. Note that NE-FGR treats the off-diagonal elements of the Hamiltonian as a small perturbation, which in the polariton basis effectively assumes weak system-bath coupling (excitonphonon coupling). This assumption is expected to be valid when N becomes larger, since all off-diagonal terms in Eq. A2 scales with 1/ √ N , reducing the magnitude of the polariton-phonon coupling as N enters the collective limit (for N = 10 6 ∼ 10 12 ).