where 𝜒 eq ≡ 𝑃 ℛ /𝑃 𝒫 denotes the ratio of equilibrium population between the reactant and product, see Sec. VII of the Supplementary Material. The time derivative Ṗℛ (𝑡) in Eq. 34 is evaluated numerically. For the symmetric double potential model considered in this work, 𝜒 eq = 1. The limit 𝑡 → 𝑡 p represents that the dynamics have already entered the rate process regime (linear response regime) and 𝑡 p represents the "plateau time" of the time-dependent rate which is equivalent to a flux-side time correlation function formalism. One can also view Eq. 34 as the flux-side correlation function that provides the timedependent rate constant 𝑘(𝑡), which captures both the initial transient dynamics (the oscillatory behaviors of 𝑘(𝑡)) and the longer time rate process (plateau of 𝑘(𝑡 p )). We report the numerical value of 𝑘/𝑘 0 as a function of the cavity frequency 𝜔 c . For the rate constant predicted by FGR, we only report the value of 𝑘/𝑘 0 = 1 + 𝑘 VSC /𝑘 0 (see Eq. 11) and we directly use the numerical result of 𝑘 0 obtained from the HEOM simulation.