Note that in this approximate form of J eff (ω) in Eq, 9, there are N -1 dark states that do not appear , thus being decoupled from the reaction coordinate R0 . This is due to a mean-field-like approximation for the C j when deriving Eq. 9. In the general expression of the spectral density (Eq. 34), the dark states can still show up when having the angle and coupling disorders, but the dark-state peak magnitude does not dominate when considering a certain range of disorders. This will be discussed in detail in the Collective coupling mechanism section. Fig. presents J eff (ω) at various Ω R . At Ω R = 0 (black curve), the peak of J eff (ω) is located at the frequency of ω Q , and the RPV effect is at its maximum due to ω Q = ω 0 . With the higher Ω R , J eff (ω) split into two peaks (corresponding to the two vibrational polariton frequencies ω ± ≈ ω Q ± Ω R /2), and is moving away from ω 0 , hence reduce the RPV effect and the value of k 1 for |ν L ⟩ → |ν ′ L ⟩ transition (c.f. Eq. 7). Note that the J eff (ω) expression is only sensitive to the collective coupling strength Ω R , and for N > 1, the behavior of J eff (ω) is identical to N = 1 as long as Ω R and Λ are identical. The rest of the dark vibrational modes are decoupled from q c and will not explicitly show up in J eff (ω).