 Symmetric exponential excitation pulse

For $b>0$, the Fourier transform of $f$ is ${\int }_{-\infty }^{\infty }f(t)e^{ixt}dt=2b/(b^2+x^2)$. A weak interaction $D_{0}f(t)\cos \omega t$ between energy levels of gap $\hslash {\omega }_0$ gives final probability amplitude $-idb\{[b^2+({\omega }_0-\omega {)}^2{]}^{-1}+[b^2+({\omega }_0+\omega {)}^2{]}^{-1}\}/\hslash$, where $d$ is the transition matrix element. Only when $b,|{\omega }_0-\omega |\ll {\omega }_0$ does the single resonant lobe give an approximately squared Lorentzian line shape excitation profile. Its full width at half maximum is $2b\sqrt{\sqrt{2}-1}$ in angular frequency.