Barring pre-factors and the oscillatory term e ±i(ω±-ωx)s , Eq. 23 contains two major contributions, C D→D (s) and C 0 J →K (t, s) terms. The term C D→D (s) is the contribution of νj and ν-j term that provides off-diagonal coupling between the polaritons and a dark state (through the last line in Eq. A2). This C D→D (s) expression is identical to the case of the dark-to-dark state transition expressed in Eq. 21, since the j-th and -j-th modes are unshifted in the polariton or dark states, as discussed above with the derivation of Eq. 21.