Gaussian resonance broadening (${\mathcal{R}}_{\Gamma }(\omega )=(\sqrt{2\pi }\Gamma {)}^{-1}{e}^{-{\omega }^2/(2{\Gamma }^2)}$)

For a perturbation with amplitude envelope $e^{-(t-t_0{)}^2/(2{\tau }^2)}$, the squared Fourier transform is proportional to $e^{-{\omega }^2{\tau }^2}$. Hence pulse-integrated transport weights orbital detuning with a normalized Gaussian function of width $\Gamma =(\sqrt{2}\tau {)}^{-1}$. A longer pulse selects a narrower band of orbital resonances. This spectral broadening does not by itself assume stochastic temporal forcing.