Numerical Simulations of the Nonlinear Quantum Vacuum in the Heisenberg-Euler Weak-Field Expansion

Abstract

A numerical scheme for solving the nonlinear Heisenberg-Euler equations in up to three spatial dimensions plus time is presented and its properties are discussed. The algorithm is tested in one spatial dimension against a set of already known analytical results of vacuum effects such as birefringence and harmonic generation. Its power to go beyond analytically solvable scenarios is demonstrated in 2D. First parallelization scaling tests are conducted in 3D.

Advances in QED with intense background fields

“…For pump and probe fields depending on all space-time coordinates the signal can be determined with Green's functions methods (cf., e.g., [538,539,540]), within the S-matrix formalism (cf., e.g., [391,536]), and by a direct numerical solution of the non-linear wave equation (cf., e.g., [541,542,543]). A particularly convenient and easy to handle approach which was put forward in the past few years is the vacuum emission picture [544,115,117] which does not distinguish between pump and probe fields, but treats all driving electromagnetic fields A on the same footing.…”

“…Progress has been made recently in particular in solving the coupled set of equations in Eq. ( 146) numerically using nonlinear Maxwell equation solvers both in d = 1+1 [548,432] and in d = 3+1 [541,542,543] space-time dimensions. Although computationally demanding, this approach has the advantage that it captures the full time evolution of the electromagnetic fields and thereby self-consistently accounts for back-reaction effects as well as the depletion of the pump fields by signal emission.…”

“…We note that for light-by-light scattering phenomena, see Sec. 7, beam depletion is automatically taken into account by standard numerical solvers of the Heisenberg-Euler equations of motion that account for the full nonlinear dynamics of the electromagnetic fields [541,542,543].…”

“…This requires an accurate modelling of the precise field configurations available in experiment as well as accounting for any real-world complications such as the inevitable presence of shot-to-shot fluctuations and jitter [76]. The combination of the vacuum emission picture with a Maxwell solver [581,582] as well as complementary approaches based on a direct numerical solution of the non-linear wave equation in 3 + 1 dimensions [541,542,543] should allow tackling this challenge.…”

All-optical Quantum Vacuum Signals in Two-Beam Collision

“…Figure 2 shows the beam radii in the focus by using a probe with elliptical cross section. For numerical approaches beyond infinite Rayleigh range approximation see [51][52][53][54][55].…”