Exponentially decaying electric-field pulse ($\tilde{E}(\omega )=E_0\tau /(1-i\omega \tau )$)

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.

The step at the beginning of $E_0{e}^{-t/\tau }\Theta (t)$ retains high-frequency weight. For fixed nonzero transition frequency, $|\tilde{E}|\to E_0/|\omega |$ as $\tau \to \infty$. A slowly decaying pulse with an abrupt onset is therefore not an adiabatically switched perturbation. The diagonal accumulated phase can separately grow with duration.