 Exponentially decaying electric-field pulse

A uniform field $E(t)=E_0{e}^{-t/\tau }\Theta (t)$ couples through $V=-qEx$. Its first-order transition amplitude is $-q⟨f|x|i⟩\tilde{E}({\omega }_{fi})/(i\hslash )$, so the leading probability is $q^2|x_{fi}{|}^2{E}_0^2{\tau }^2/[{\hslash }^2(1+{\omega }_{fi}^2{\tau }^2)]$. A real pulse has equal spectral intensity at opposite frequencies; populations and dipole elements determine which upward or downward transitions are possible. Small peak amplitude does not remove the need to check total transition probability.