Optimization of spatially localized electric fields for electron-positron pair creation

Using numerical solutions to the quantum field theoretical Dirac equation, we study the electron-positron pair-creation process from the vacuum due to a spatially localized supercritical electric field. By varying the spatial profile of this external field, we search for optimal field configurations that maximize the pair-creation rate in the steady state. We find that for the class of pulse shapes with a single maximum and fixed total energy, the rate depends nonmonotonically on the field's spatial width and it is insensitive to other characteristics of the pulse shape. It turns out that the Schwinger rate can be corrected such that it can provide analytical estimates for the threshold behavior as well as finite pulse effects.

Figure 1

The steady-state pair-creation rate $\mathrm{\Gamma }$ as a function of the effective width $\sigma$ of four electric field configurations with a singly peaked spatial envelope. We also denote the predictions due to the original Schwinger formula and its corrected form. (a) $H=3.63c^{5/2}$, (b) $H=4.5c^{5/2}$, (c) $H=7.5c^{5/2}$, (d) $H=10c^{5/2}$.

Figure 2

(a) Comparison of the prediction of the Schwinger formula for the steady-state pair-creation rate ${\mathrm{\Gamma }}_{\mathrm{Schwinger}}$ as a function of the width $\sigma$ of electric field and the exact rate ${\mathrm{\Gamma }}_{\mathrm{exact}}$ for $E_0=1.5c^3$. (b) The percentage error of the original Schwinger rate together with the error associated with the corrected Schwinger expression.

Figure 3

The optimal effective width ${\sigma }_{\mathrm{opt}}$ (a) and the maximal pair-creation rate ${\mathrm{\Gamma }}_{\mathrm{opt}}$ (b) as a function of the total energy $H$ of the external electric field pulse.

Figure 4

The steady-state pair-creation rate $\mathrm{\Gamma }$ as a function of the separation $d$ of the two electric fields with the width $w$ each. In the inset we sketch this spatial profile. The graphs correspond to the widths $w_1=0.48/c,w_2=0.6/c$, and $w_3=18\surd 3/c$, and $H=3.63c^{5/2}$ and total energy $H=3.63c^{5/2}$.

Figure 5

The steady-state pair-creation rate $\mathrm{\Gamma }$ as a function of the separation $d$ of the two electric fields with a Gaussian spatial envelope and width $w$ each. In the inset we sketch this spatial profile. The graphs correspond to width $w_1=0.15/c,w_2=4.65/c$, and $H=4.5c^{5/2}$.

Figure 6

The steady-state pair-creation rate $\mathrm{\Gamma }$ as a function of the separation $s$ (as well as the extension $s$ of each pulse) between the two fields. The corresponding configuration is shown in the inset. For comparison, the dashed curve is twice the rate associated with only the first pulse and denoted by $E_R\left(x\right)$. The open circles are the locations of the maxima according to $s_n=33.3c^{-1}{(n+1/2)}^{-1}$ for $n=4$, 5, 6, 7 and 8. $(H=4.5c^{5/2})$.