Diffraction at a time grating in electron-positron pair creation from the vacuum

The Sauter-Schwinger process of electron-positron pair creation from the vacuum, driven by a sequence of time-dependent electric-field pulses, is studied in the framework of the quantum-field theoretical approach. As demonstrated by our numerical results, the momentum distributions of produced pairs exhibit intra- and interpulse interference structures. We show that such structures can be observed beyond the regime of applicability of the Wentzel-Kramers-Brillouin theory, which was the focus of earlier investigations. Going beyond these developments, we perform the analysis of the time-evolution operator for an arbitrary eigenmode of the fermionic field. This shows that a perfect coherent enhancement of the interpulse peaks can never be reached. A nearly perfect coherence, on the other hand, is due to nonadiabatic transitions at avoided crossings of the phases defining the unitary time evolution. This analysis allows us to determine the conditions under which the nearly perfect coherence is lost.

Figure 1

Distributions of $e^{-}{e}^{+}$ pairs created from the vacuum, ${\mathcal{P}}_{N_{\mathrm{rep}}}$, as a function of their longitudinal momentum $p_{\parallel }$ (with ${\mathit{p}}_{\perp }=\mathbf{0}$), generated by a single pulse ($N_{\mathrm{rep}}=1$) (solid blue line), or by a train of two ($N_{\mathrm{rep}}=2$) (dashed red line) or three pulses ($N_{\mathrm{rep}}=3$) (solid green line). The shape function of the driving electric field is defined by Eqs. (46), (47), and (49) with the following parameters: $\sigma =5{\tau }_{\mathrm{C}}$, $T_0=40{\tau }_{\mathrm{C}}$, and $T=400{\tau }_{\mathrm{C}}$, where ${\tau }_{\mathrm{C}}=1/(m_{\mathrm{e}}{c}^2)$ is the Compton time. The amplitude of the electric field ${\mathcal{E}}_0$ (in units of the Sauter-Schwinger critical field, ${\mathcal{E}}_{\mathrm{S}}=m_{\mathrm{e}}^2{c}^3/|e|$) is ${\mathcal{E}}_0=-0.1{\mathcal{E}}_{\mathrm{S}}$. The results are compared with the pair momentum distribution induced by a half-pulse ${\mathcal{P}}_0$ (solid black envelope), characterized by the shape function (46).

Figure 2

The same as in Fig. 1 but for $\sigma ={\tau }_{\mathrm{C}}$, $T_0=10{\tau }_{\mathrm{C}}$, and $T=100{\tau }_{\mathrm{C}}$.

Figure 3

The top panel shows the longitudinal momentum distribution of particles created in the process driven by a single electric field pulse ($N_{\mathrm{rep}}=1$) (dashed blue line), by a train of two such pulses ($N_{\mathrm{rep}}=2$) (solid red line), and by three such pulses ($N_{\mathrm{rep}}=3$) (solid green line). This panel relates to the same parameters of the electric field as Fig. 1 except that, for visual purposes, we consider now a smaller range of the longitudinal momentum. Below we plot the phases ${\vartheta }_1\in (-\pi ,0)$ (modulo $2\pi$) (in blue) and ${\vartheta }_2\in (0,\pi )$ (modulo $2\pi$) (in red) which define the eigenvalues of the time evolution operator (64), as functions of $p_{\parallel }$. Each consecutive panel (from top to bottom) corresponds to $N_{\mathrm{rep}}=1$, 2, and 3, respectively. Crossings and avoided crossings of the ${\vartheta }_1$ and ${\vartheta }_2$ curves correspond to either zeros of the momentum distribution or to its maxima. A sample crossing and avoided crossing are indicated by the green and the black arrows, respectively.

Figure 4

Details of the avoided crossing and the crossing of the ${\vartheta }_1$ and ${\vartheta }_2$ curves indicated in Fig. 3 by the black and the green arrows, respectively. In the upper panel, the results are for different $N_{\mathrm{rep}}$. Specifically, the dashed blue curve is for a single electric field pulse ($N_{\mathrm{rep}}=1$), the solid red line is for a sequence of two such pulses ($N_{\mathrm{rep}}=2$), and the solid green curve is for three such pulses ($N_{\mathrm{rep}}=3$). We see in this panel that the gap between both phases at the avoided crossing increases linearly with $N_{\mathrm{rep}}$. In the lower panel, the true crossing is presented for $N_{\mathrm{rep}}=3$.

Figure 5

Portion of the pair creation spectra from Fig. 1 (upper panel) and ${\sin }^2(\gamma )$ (lower panel) as functions of the longitudinal momentum $p_{\parallel }$. The same color coding is used in the upper panel as in the top panel of Fig. 3.

Figure 6

The same as in Fig. 1 except that in the lower panel we plot the dependence of ${\vartheta }_1$ and ${\vartheta }_2$ [which parametrize the eigenvalues of the evolution matrix, see, Eq. (64)] on the longitudinal momentum for $N_{\mathrm{rep}}=1$. Although the momentum distribution ${\mathcal{P}}_1$ is a smooth broad curve, the avoided crossings of the phases ${\vartheta }_1$ and ${\vartheta }_2$ for $N_{\mathrm{rep}}=1$ allow us to predict at which momenta $p_{\parallel }$ the momentum distribution of created particles will peak for a train of such pulses.