Sudden-onset spectral tail of an exponential pulse
= Sudden-onset spectral tail of an exponential pulse
The step at the beginning of $E_0e^{-t/\tau}\Theta(t)$ retains high-frequency weight. For fixed nonzero transition frequency, $|\widetilde E|\to E_0/|\omega|$ 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.