Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect

While the Sauter-Schwinger effect describes nonperturbative electron-positron pair creation from vacuum by a strong and slowly varying electric field $E_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}}$ via tunneling, the dynamically assisted Sauter-Schwinger effect corresponds to a strong (exponential) enhancement of the pair-creation probability by an additional weak and fast electric or electromagnetic pulse $E_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}$. Using the WKB and worldline instanton method, we find that this enhancement mechanism strongly depends on the shape of the fast pulse. For the Sauter profile $1\mathrm{/}{\cosh }^2(\omega t)$ considered previously, the threshold frequency ${\omega }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ (where the enhancement mechanism sets in) is basically independent of the magnitude $E_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}$ of the weak pulse—whereas for a Gaussian pulse $\exp (-{\omega }^2{t}^2)$, an oscillating profile $\cos (\omega t)$ or a standing wave $\cos (\omega t)\cos (kx)$, the value of ${\omega }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ does depend (logarithmically) on $E_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}\mathrm{/}{E}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}}$.