where k 0 is the rate constant outside the cavity, and k VSC is the rate constant enhancement caused by coupling to the cavity. Note that the effect of the cavity causes an enhancement of k 1 , which is the vibrational excitation process, and because this step is rate limiting, the effect of the cavity thus manifests to the entire apparent rate constant. With this observation, we have derived an analytic expression for the cavityenhanced part of the rate constant for the transition |ν L ⟩ → |ν ′ L ⟩ (with a frequency ω) expressed as follows