This broadening is due to the molecular phonon bath J ν (ω). Note that the rate expression in Eq. 41 explicitly depends on both the cavity frequency ω c from J eff (ω) in Eq. 11 and the quantum vibration frequency ω 0 from G(ω -ω 0 ) in Eq. A10. In principle, one can use the convolution theorem to evaluate the expression in Eq. 41, the detailed discussions are provided in Appendix A. Unfortunately, a closed analytic formalism is not available. Instead, we numerically evaluate the expression in Eq. A21, using a spectral density discretization procedure outlined in Ref. 56. However, we find that in two special cases, analytic expressions for the approximate evaluation of k VSC in Eq. 41 are available. For both cases, we assume the Markovian limit for the effective spectral density J eff (ω) (Eq. 14). Similar expressions can be derived for the non-Markovian case. First, under the lossless limit (τ c → ∞), the effective spectral density function will reduce to a single δfunction, J eff (ω) ≈ πη 2 c ω 2 c δ(ω -ω c ). As a result, the broadening is fully dictated by the variance of the Gaussian,