and all explicit t-dependence in these correlation functions are dropped due to the fact that under the collective limit C 0 J→K → 1. Importantly, Eq. 28 is equivalent to the expression of Eq. 21, except for a pre-factor and the oscillatory term. This means that in the collective coupling limit, all transitions can be modeled as population transfer between undisplaced harmonic oscillators just like the dark-dark transition Eq. 21, as pointed out in previous studies of collective effects in cavities. Further, if we substitute Eq. 28a into Eq. 17, and integrate over ds, we have