= Exponentially decaying electric-field pulse
{title2=$\widetilde E(\omega)=E_0\tau/(1-i\omega\tau)$}
A uniform field $E(t)=E_0e^{-t/\tau}\Theta(t)$ couples through $V=-qEx$. Its first-order transition amplitude is $-q\langle f|x|i\rangle\widetilde E(\omega_{fi})/(i\hbar)$, so the leading probability is $q^2|x_{fi}|^2E_0^2\tau^2/[\hbar^2(1+\omega_{fi}^2\tau^2)]$. 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.
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