 Sudden-onset spectral tail of an exponential pulse

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.