Momentum correlation in pair production by spacetime dependent fields from scattered wave functions

TL;DR

This work addresses the joint momentum correlation $P(p_z,p'_z)$ in Schwinger pair production for spacetime-dependent fields, where momentum is not conserved along the propagation direction. It develops a Dirac-equation framework that splits the wavefunction into a background and scattered part, enabling exact numerical access to the correlation function and its relation to Bogoliubov coefficients. The authors implement a robust pseudospectral numerical approach to compute $P(p_z,p'_z)$ in 2D and 2+1D field configurations, and validate the results against worldline instanton predictions, finding good agreement and revealing interference patterns tied to multi-peak field structures. They further demonstrate the method’s scalability to higher dimensions and discuss computational strategies, including GPU acceleration, to facilitate broader parameter scans. The work provides a practical, first-principles route to characterize momentum correlations in spacetime-dependent Schwinger pair production with potential relevance for multi-dimensional strong-field settings.

Abstract

We consider Sauter-Schwinger pair production by electric fields that depend on both time and space, $E(t,z)$ and $E(t,x,y)$. For space-independent fields, $E(t)$, momentum conservation, $δ({\bf p}+{\bf p}')$, fixes the positron momentum, ${\bf p}'$, in terms of the electron momentum, ${\bf p}$. For $E(t,z)$, on the other hand, $p_z$ and $p'_z$ are independent. However, previous exact-numerical studies have considered only the probability as a function of a single momentum variable, $P(p_z)$, $P(p'_z)$ or $P(p'_z-p_z)$, but not the correlation $P(p_z,p'_z)$. In this paper, we show how to obtain $P(p_z,p'_z)$ by solving the Dirac equation numerically. To do so, we split the wave function into a background and a scattered wave, $ψ(t,{\bf x})=ψ_{\rm back.}(t,{\bf x})+ψ_{\rm scat.}(t,{\bf x})$, where $ψ_{\rm back.}\propto\exp(\pm ipx+\text{gauge term})$. $ψ_{\rm scat.}$ vanishes outside a past light cone and is obtained by solving $(iγ^μD_μ-m)ψ_{\rm scat.}=-(iγ^μD_μ-m)ψ_{\rm back.}$ backwards in time starting with $ψ_{\rm scat.}(t\to+\infty,{\bf x})=0$.

Figures (6)

• Figure 1: Background wave function (left), $U_\infty$, and scattered wave function (right), $\Delta U$, in \ref{['UVsplit']}. The plots show the real part of the first component of the 2D spinors in Sec. \ref{['2D fields']}. The field is $A_3(t,z)=(E/\omega)\tanh(\omega t)\text{sech}^2(\kappa z)$ with $E=1/3$ and $\omega=\kappa=2E/3$. For this gauge choice, $A_3(\infty,z)\ne0$, so $U_\infty$ is not just a plane wave, but includes the gauge integral in the exponent. $\Delta U$ is obtained by solving \ref{['DiracDelta2D']} backwards in time, starting with $\Delta U=0$ in the asymptotic future, $t=t_{\rm out}\sim22$. The momentum, $p_3=-0.85$, enters \ref{['DiracDelta2D']} via $U_\infty$. The pair probability is obtained by Fourier transforming the $z$ dependence of $\psi(t,z)$ in the asymptotic past, $t=t_{\rm in}\sim-25$.
• Figure 2: The spectrum for $E(t,z)=E_0\text{sech}^2(\omega t)\text{sech}^2(\kappa z)$ with $E_0=1/3$ and $\gamma_\omega=\gamma_\kappa$, where $\gamma_\omega=\omega/E_0$ and $\gamma_\kappa=\kappa/E_0$. $\gamma=2/3$ in the first plot, and $\gamma=1$ in the second. The plots show a cross section of the spectrum, $\ref{['N12D']}/V_{ \perp}$, where $p_3=-P+\Delta/2$ and $p'_3=P+\Delta/2$ with $\Delta=0$. The "numerical" result has been obtained by solving \ref{['DiracDelta2D']} numerically. Each data point takes about 3 seconds to compute on a laptop, with a grid in $z$ with $\mathcal{O}(200)$ points. The "grid" and "quadratic" lines show the worldline-instanton ($E\ll1$) approximations, obtained with the methods described in DegliEsposti:2024upq.
• Figure 3: Spectrum for $E(t,z)=E_0[\text{sech}^2(\omega t)+\text{sech}^2(\omega t-5)]\text{sech}^2(\kappa z)$ with $E_0=0.3$ and $\gamma_\omega=\gamma_\kappa=2/3$. Same notation as in Fig. \ref{['fig:singlePeak']}. Each numerical point took about one minute to compute, with $\mathcal{O}(700)$ grid points. From the instanton approximations for similar examples shown in Fig. 11 in DegliEsposti:2024upq, one expects essentially no oscillations in the $\Delta$ direction for this type of superposition of two pulses separated in $t$ but not in $z$.
• Figure 4: Spectrum for $E(t,z)=E_0\text{sech}^2(\omega t)[\text{sech}^2(\kappa z+2)+\text{sech}^2(\kappa z-2)]$ with $E_0=\omega=\kappa=0.3$. The plot shows a cross section of the spectrum, $p_3=-P+\Delta/2$ and $p'_3=P+\Delta/2$ with $P=P_{\rm saddle}\approx0.59$. From the instanton approximations for similar examples shown in Fig. 6 in DegliEsposti:2024upq, one expects essentially no oscillations in the $P$ direction for this type of superposition of two pulses separated in $z$ but not in $t$.
• Figure 5: Spectrum for $E(t,z)=2E_0\kappa z\exp[-(\kappa z)^2-(\omega t)^2]$ with $E_0=\omega=\kappa=1/3$. This field has two peaks with opposite signs, $E_{\rm min}=E(0,z\approx-0.7/\kappa)<0$ and $E_{\rm max}=E(0,z\approx0.7/\kappa)>0$. The plot shows a cross section with $\Delta=0$. $|M_a+M_b|^2$ gives what we in the other plots refer to as the quadratic approximation. For this particular example, $M_a$ is the amplitude obtained by expanding around an instanton with a turning point ($\dot{t}(0)=0$) near $z(0)\approx-0.75$ and with asymptotic momentum $P_a\approx-0.28$ and $\Delta_a\approx0.11$. $M_b$ corresponds to $z(0)\approx0.75$, $P_b\approx0.28$ and $\Delta_b\approx-0.11$. $|M_a-M_b|^2$ shows what one would have obtained if one had made a mistake in calculating the relative sign of the two amplitude terms. The agreement between $|M_a+M_b|^2$ and the numerical results confirms that we have the correct sign.