which is identical to Eq. 17 (with Ω R = 4g 2 c ω c ), with an additional M = g 1D (ω)dω which is the number of cavity modes. Thus, for a 1D FP cavity, the peak of the expression in Eq. 33 is located at ω c = ω 0 where k ∥ = 0, due to the presence of the van-Hove singularity. This means that VSC modification occurs only when ω c = ω 0 for a 1D FP cavity. We have also numerically evaluated Eq. 30 and compared it with Eq. 33 for the VSC-modified rate constant, presented in Fig. of Supplementary Note 9, which shows a nearly identical behavior.