Solving the Dirac equation on a GPU for strong-field processes in multidimensional background fields

TL;DR

This paper presents a GPU-accelerated framework for solving the dressed Dirac equation in multidimensional strong-field backgrounds using the scattered-wave-function approach, enabling 2+1 and 3+1 dimensional analyses. Implemented with JAX, the method achieves substantial speedups over CPU-based solutions and yields electron-positron production probabilities (Schwinger and nonlinear Breit-Weler) by directly evolving the scattered part of the wavefunction. The results are benchmarked against worldline-instanton approximations, showing good agreement and illustrating the value of SWF for exploring parameter spaces, especially when combined with fast semiclassical methods. The work also extends SWF to nonlinear Breit-Wheeler and discusses computational strategies and future directions, including extensions to Compton scattering and photon wave packets.

Abstract

In this paper, we show how to solve the Dirac equation, $(iγ^μ[\partial_μ+ieA_μ(t,{\bf x})]-m)ψ=0$, on a GPU. This is orders of magnitude faster than solving it on CPU and allows us to consider background fields, $A_μ(t,{\bf x})$, that depend on $2+1$ or even $3+1$ coordinates. Our approach is conveniently implemented using the computational library JAX. We show how to obtain the probabilities of Schwinger and nonlinear Breit-Wheeler pair production from these solutions using a scattered-wave-function approach and compare the results with the worldline-instanton approximations.

Figures (7)

• Figure 1: $(2+1)$D. Momentum spectrum with $p_2=p_3=q_2=q_3=0$ for \ref{['singleE']} with $E_0=1/4$, $\omega=E_0$ and $\kappa_x=\kappa_y=E_0/2$. The first row shows the (quadratic) instanton approximation and the second the SWF result. There are three relevant instantons: The one created around $x=0$ gives the dominant contribution, and the two created around $x\approx\pm2.3$ give the interference patterns in the upper-left corner.
• Figure 2: $(2+1)$D. Momentum spectrum with $p_2=p_3=q_2=q_3=0$ for \ref{['doubleE']} with $E_0=1/4$, $\omega=E_0$, $\kappa_x=\kappa_y=E_0/2$ and $\Delta x=1.75/\kappa_x$. The first row shows the (quadratic) instanton approximation and the second the SWF result. We have only included the two dominant instantons.
• Figure 3: $(3+1)$D. Momentum spectrum with $p_2=p_3=q_2=q_3=0$, $p_1=-P+\frac{\Delta}{2}$, $q_1=P+\frac{\Delta}{2}$, where $P=P_{\rm saddle}\approx0.51$, for \ref{['singleE']} in $3+1$D with $E_0=1/4$, $\omega=E_0$ and $\kappa_x=\kappa_y=\kappa_z=E_0/2$. The numerical SWF points have been computed with a $(x,y,z)$ grid of size $128\times128\times128$. This is an example where the instanton approximations are much better than what one should expect. In general, one should expect relative errors $\mathcal{O}(10\%)$ for $E_0=\mathcal{O}(0.1)$.
• Figure 4: $(3+1)$D. Same as Fig. \ref{['fig:great4Dexample']}, but with $\Delta=0$. The "grid dom" line shows the grid-instanton approximation including only the dominant instanton, i.e. the one created near $x=0$.
• Figure 5: $(3+1)$D. Same as Fig. \ref{['fig:great4Dexample']} and \ref{['fig:great4DexampleP']}. From left to right: quadratic-instanton approximation, grid-instanton approximation with only the dominant instanton, grid-instanton approximation with the dominant and the two subdominant instantons, and the SWF result.