Electron-positron pair production in space- or time-dependent electric fields

Treating the production of electron and positron pairs by a strong electric field from the vacuum as a quantum tunneling process we derive, in semiclassical approximation, a general expression for the pair-production rate in a $z$-dependent electric field $E(z)$ pointing in the $z$ direction. We also allow for a smoothly varying magnetic field parallel to $E(z)$. The result is applied to a confined field $E(z)\ne 0$ for $|z|\lesssim \ell$, a semiconfined field $E(z)\ne 0$ for $z\gtrsim 0$, and a linearly increasing field $E(z)\sim z$. The boundary effects of the confined fields on pair-production rates are exhibited. A simple variable change in all formulas leads to results for electric fields depending on time rather than space. In addition, we discuss tunneling processes in which empty atomic bound states are spontaneously filled by negative-energy electrons from the vacuum under positron emission. In particular, we calculate the rate at which the atomic levels of a bare nucleus of finite-size $r_{\mathrm{n}}$ and large $Z\gg 1$ are filled by spontaneous pair creation.

Figure 1

Positive- and negative-energy spectra ${\mathcal{E}}_{\pm }(z)$ of Eq. (2) in units of $m_e{c}^2$, with $p_z=p_{\perp }=0$ as a function of $\widehat{z}=z/\ell$ for the Sauter potential $V_{\pm }(z)$ (56) for $\sigma =5$.

Figure 2

We plot the slice dependence of the integrand in the tunneling rate (38) for the Sauter potential (56): left, as a function of the tunnel entrance $z$ (compare with numeric results plotted in Fig. 1 of Ref. 26); right, as a function of the associated energy $\mathcal{E}$, which is normalized by the Euler-Heisenberg rate (54). The dashed curve in the left figure shows the Euler-Heisenberg expression (54) evaluated for the $z$-dependent field $E(z)$ to illustrate the nonlocality of the production rate. The dashed curve in the right figure shows the Euler-Heisenberg expression (54) is independent of energy-level crossing $\mathcal{E}$. The dimensionless parameters are $\sigma =5$, $E_0/E_c=1$.

Figure 3

Left: ratio $R_{\mathrm{rate}}$ defined in Eq. (76) is plotted as a function of $\sigma$ in the left figure. Right: number of pairs created in slab of Compton width per area and time as functions of $\sigma$. Upper curve is for the constant field (54), lower for the Sauter field (67). Both plots are for $E_0=E_c$ and $\sigma =\ell /{\lambda }_C$.

Figure 4

Energies (2) for a soft electric field step $E(z)$ of Eq. (77) and the potentials $V_{\pm }(z)$ (77) for $\sigma =5$. Positive and negative energies ${\mathcal{E}}_{\pm }(z)$ of Eq. (2) are plotted for $p_z=p_{\perp }=0$ as functions of $\widehat{z}=z/\ell$.

Figure 5

Positive- and negative-energy spectra ${\mathcal{E}}_{\pm }(z)$ of Eq. (2) for $p_z=p_{\perp }=0$ as a function of $\widehat{z}\equiv z/{\lambda }_C$ for the linearly rising electric field $E(z)$ with the harmonic potential (87). The reduced field strengths are $ϵ=2$ (left figure) and $ϵ=-2$ (right figure). On the left, bound states are filled and positrons escape to $z=\pm \infty$. On the right, bound electrons with negative energy tunnel out of the well and escape with increasing energy to $z=\pm \infty$.

Figure 6

For a radial Coulomb field, the positive- and negative-energy spectra ${\mathcal{E}}_{\pm }(r)$ of Eq. (2) are plotted as a function of $\widehat{r}=r/{\lambda }_C$. They are found by solving condition $p_r=0$ at $l=0$ and $\widehat{\alpha }=1.27$ ($Z=174$).