where the coefficients N -1 n=0 exp -2πi nk N = 0. These Dark states also satisfy ⟨G, 0|μ|D k ⟩ = 0 due to the zero-sum property of the expansion coefficients, and as such, direct optical transition is not allowed and they are thus dark in spectra. Note that the |±⟩ polariton states and the dark states manifold {|D k ⟩} are "diabatic" states in their nature because they are the eigenstates of Ĥs (Eq. 10), and their character do not change as