Note that C 0 J →K (t, s) in Eq. 24 is caused by the difference in phonon mode shift in the dark state (which is fully excitonic) and the polariton states (which have mixed exciton and photon character). The C 0 J →K (t, s) term can be interpreted as the non-equilibrium single-site population transfer FGR correlation function under the Condon approximation (which is equivalent to setting ⟨J | Ĥsb |K⟩ = Îb , the identity operator of the bath DOF). A similar expression is also encountered in the previous work on non-equilibrium FGR by Sun and Geva. In Eq. 23, the forward rate (|±⟩ → {|D⟩}) and reverse rate (|D⟩ → |±⟩) scale differently with the number of sites N . One can clearly see that the transition rate from |±⟩ polariton state to the dark states manifold contains an additional factor N -1. This is because there are N -1 degenerate dark states as the final state (or equivalently, N -1 identical. The reverse transition, |D⟩ → |±⟩, also consists of N -1 identical processes, but each transition is from a single dark state which has ∝ 1/(N -1) of the overall population of the dark states manifold {|D⟩}, leading to no overall scaling with the number of sites (∝ O(N 0 )). Taking the 1/N scaling of C D→D into consideration (see Eq. 21), the dark-to-polariton NE-FGR rate scales with 1/N , while the polariton-to-dark NE-FGR rate scales with (N -1)/N . These scaling in terms of N is consistent with well-known Equilibrium FGR polaritonic relaxation rates or Lindblad master Equations , and can be confirmed through the recent Mixed Quantum-Classical (MQC) simulations and exact quantum dynamics simulations. Transition Rate between Polariton states Finally, the transitions among two polariton states (for |+⟩ → |-⟩ and |-⟩ → |+⟩) consist of a minor part of the polariton relaxation process (when N is large). The inter-polariton transfer is often ignored in the study of polariton lifetime because the dark states usually outnumber the polariton states in the collective coupling regime, and one may reasonably assume that the interpolariton transition rate is much lower compared to the polariton-dark transition rate. For the sake of completeness, we include the inter-polariton transfer explicitly in the current study, such that its effect can be quantified when N is not very large.