Details of the derivation are provided in Supplementary Note 5, section A. Note that there is no resonance behavior in k1 , and it becomes unbounded when τ c → 0. The resonance behavior and boundedness of the rate constant will be ensured by k2 associated with the |ν L , 1⟩ → |ν ′ L , 0⟩ transition, which can be evaluated as