Results. Fig. presents the population dynamics of the vibrational states obtained from the γ-SQC approach, for outside the cavity case (thick solid lines) and for coupling inside a resonant cavity (thin lines with open circles), with a coupling strength η c = 1.25 × 10 -3 a.u.. The cavity lifetime is τ c = 2000 fs. In panel (a), one can clearly see that when coupling to the cavity, the population of the |ν L ⟩ state (red) decays much faster compared to the outside cavity scenario, and the population of |ν R ⟩ arises accordingly faster when coupling the molecule inside the cavity. Examining the population dynamics of the excited states of |ν ′ L ⟩ (blue) and |ν ′ R ⟩ (magenta) also reviews that upon coupling to the cavity, the steady state populations of these vibrationally excited states are enhanced compared to the outside cavity case, in agreement with our recent results 9 that leads to a successful analytic rate constant in Eq. 8. Population dynamics obtained from the exact HEOM approach are also provided in the Supporting Information, Sec. II. Population dynamics obtained from other MQC methods, including spin-LSC, Ehrenfest dynamics, and GFSH are provided in the Supporting Information, Sec. III. Despite qualitatively capturing the same mechanism, all of the trajectory-based methods seem to overestimate the absolute value of the rate by ten or more times, even outside the cavity case. This is likely due to the less accurate trajectory treatment of the classical DOF, especially for the off-diagonal Peierls couplings terms i c i xi • P R0 P (from Ĥν in Eq. 3). Future investigations will focus on further improvement of the vibrational dynamics using MQC methods.