We notice that the Hamiltonian Eq. A2 is invariant under any permutations among the dark states index, hence all inter-dark state rates are identical to each other. The same argument can be applied to the population transfer rates between the dark and polariton states, such that the N -1 dark states are equivalent in the population dynamics. In this work, the polariton relaxation process is investigated by initially populating the polariton state (either |+⟩ or |-⟩) and then simulating the population transfer dynamics to the dark state. As a result, all dark states have zero initial population and will have the same population throughout the dynamics. Therefore, we instead study the population dynamics of the dark states manifold that have the combined population of all dark states