$$A(t)={\cal E}_0f(t),
\label{a5}$$ then $$f(t)=f(+\infty)+\int_t^{+\infty}{\mathrm{d}}t'F(t').
\label{a6}$$ Here, taking into account Eq. , we conclude that $f(-\infty)=f(+\infty)$. This means that in the remote past and in the far future, $$\underset{t\rightarrow-\infty}{\lim}A(t)=\underset{t\rightarrow+\infty}{\lim}A(t).
\label{a1}$$ Without loosing the generality, we can put this constant to zero and, equivalently, $f(-\infty)=f(+\infty)=0$. In such case, the behavior of the vector potential $A(t)$ guarantees that our asymptotic “in” and “out” states will be indeed field-free states. This is particularly important for the Sauter-Schwinger pair creation and, hence, it justifies imposing the condition .

In the following, we shall consider a single pulse with the shape function in  given by $$F_1(t)=\exp\Bigl[-\Bigl(\frac{t-T_0/2}{\sigma}\Bigr)^{2M}\Bigr]-\exp\Bigl[-\Bigl(\frac{t+T_0/2}{\sigma}\Bigr)^{2M}\Bigr].
\label{a8}$$ Note that each half-pulse is given by either Gaussian ($M=1$) or super-Gaussian ($M>1$) envelope, with a bandwidth $\sigma$ and a time delay between them $T_0$. An interesting property of super-Gaussian envelopes is that, while they remain smooth functions (of class $C^{\infty}(\mathbb{R})$), they approach the step function, $$S(t)=\begin{cases} 1, & t\in [-\sigma,\sigma], \cr 0 , & t\notin (-\sigma,\sigma),\end{cases}
\label{a9}$$ for large $M$. This is illustrated in the upper panel of Fig. \[dirackleinpolav2\]. In the lower panel, we present the time-dependence of the corresponding vector potential . The difference between both columns in Fig. \[dirackleinpolav2\] is the time delay $T_0$ 