Details of the derivation are provided in Supplementary Note 5, section B. Due to the molecular phonon bath Ĥν , one needs to further consider the broadening effect in the vibration frequency ω 0 , described by a lineshape function A ν (ω c -ω 0 ). Under the homogeneous limit, A ν (ω c -ω 0 ) has a Lorentzian form as follows A