As a result, the polariton linewidth scales as 1/N under the fast limit, which differs from the no narrowing effect prediction of Kubo-Anderson theory under the fast limit in Ref. . Note that the scaling relations with N in Eqs. 29b and 30b will not be influenced if the high temperature limit (k B T ≫ ℏω f ) is not satisfied. The scaling relations with N in Eqs. 29b and 30b under the slow and fast limits are the first main theoretical result of this paper. In this work, we are more interested in the fast limit because usually N ≫ 1 in typical experimental setups, such that κ N ≫ 1. As a consequence, the polariton lineshape is described by Lorentzian (Eq. 30). We also want to emphasize that only the two polariton states are included to obtain Eqs. 29-30. The dark states and inter-branch scattering processes have not been considered yet. This will be further discussed in the following section.