Electron-positron pair production in external electric fields varying both in space and time

The Schwinger mechanism of electron-positron pair production in the presence of strong external electric fields is analyzed numerically for the case of one- and two-dimensional field configurations where the external field depends both on time and one spatial coordinate. In order to provide this analysis, a new efficient numerical approach is developed. The number of particles created is obtained numerically and also compared with the analytical results for several exactly solvable one-dimensional backgrounds. For the case of two-dimensional field configurations the effects of the spatial finiteness are examined, which confirms their importance and helps us to attest our approach further. The corresponding calculations are also performed for several more interesting and nontrivial combinations of temporal and spatial inhomogeneities. Finally, we discuss the case of a spatially periodic external field when the approach is particularly productive. The method employed is described in detail.

Figure 1

Illustration of the two-dimensional field configuration. The electric field is confined in the space layer (which has a width of $2L$) and is assumed to be switched on and off at the times $t_{\mathrm{in}}$ and $t_{\mathrm{out}}$, respectively. The electron created has a longitudinal component of its momentum ${\mathit{p}}_x$ and a transverse component ${\mathit{p}}_{\perp }$ which is orthogonal to the $x$ axis.

Figure 2

The mean number of electrons created by the “rectangular” static field ($T=2.0$, $|e|E=3.0$, ${\pi }_{\perp }=1.0$) as a function of the momentum component $p_x$ for different values of $L$.

Figure 3

The mean number of electrons created by the “rectangular” static field ($T=2.0$, $|e|E=3.0$, ${\pi }_{\perp }=1.0$), multiplied by the factor $\pi /L$, as a function of $p_x$ for different values of $L$. The numerical values tend to the analytical 1D ones (solid line) as $L$ increases.

Figure 4

The mean number of electrons created by the “rectangular” static field ($T=1.0$, $|e|E=3.0$, ${\pi }_{\perp }=1.0$) as a function of the momentum component $p_x$ for different values of $L$. The dashed lines relate to $L=\pi /3$, $\pi$, and $2\pi$ while the crosses represent the spectra for $L=1.0$, 3.0, and 6.0.

Figure 5

The mean number of electrons created by the “rectangular” static field ($T=2.0$, $|e|E=3.0$, $L=\pi$) as a function of their energy for different values of $\theta$.

Figure 6

The mean number of electrons created by the “rectangular” Sauter-like field ($|e|E=3.0$, $\tau =0.5$, ${\pi }_{\perp }=1.0$) as a function of their momentum $x$ component for different values of $L$.

Figure 7

The mean number of electrons created by the “rectangular” Sauter-like field ($|e|E=3.0$, $\tau =0.5$, ${\pi }_{\perp }=1.0$), multiplied by the factor $\pi /L$, as a function of $p_x$ for different values of $L$. The solid line represents the analytical 1D result for $L=\infty$.

Figure 8

The mean number of electrons created by the Sauter-like field with $\tau =0.5$ (blue points), the static field with $T=1.0$ (green points), and the “triangular” field with $T=2.0$ (red points). For all these dependences $|e|E=3.0$, ${\pi }_{\perp }=1.0$, and the field is assumed to be uniform inside a capacitor with $L=2\pi$.

Figure 9

The mean number of electrons created by the “triangular” static field with $L=2\pi$ (blue points), the “rectangular” static field with $L=\pi$ (green points), and the Sauter-like static field with $\alpha =\pi$ (red points). For all the spectra $|e|E=3.0$, $T=1.0$, and ${\pi }_{\perp }=1.0$.

Figure 10

The total number of pairs created as a function of $\omega$ for $T=37.56=0.002[\mathrm{au}]$, $\alpha =5.0$, $|e|E_1=0.147$, and two different values of $E_2$ [see Eq. (81)]. The solid lines represent the corresponding results taken from Ref. [22].

Figure 11

(a) The electric field profile $E(x)$ given by Eq. (85) for different values of $d$. (b) The total number of pairs created per pulse as a function of $d$ for $\alpha =4.0$, $T=4\pi$, and different values of $\omega$.

Figure 12

The mean number of negatively charged fermions (bosons) created by the space-time-dependent Sauter-like field (91) for $\tau =0.5$, $\alpha =\pi$, ${\pi }_{\perp }=1.0$, and two different values of $E$.