Perturbative solution approach for computing the two-photon Kapitza-Dirac effect in a Gaussian beam standing light wave

Theoretical spin properties of the Kapitza-Dirac effect beyond the plane-wave description are not known in detail. We develop a method for computing electron diffraction of the two-photon Kapitza-Dirac effect in a two-dimensional Gaussian beam standing light wave within a relativistic formulation. The solutions are computed on the basis of time-dependent perturbation theory, where a momentum space formulation with the use of a Fourier transformation of the external potential allows for the solving the perturbative time-integrals. An iteration over each possible quantum state combination leads to a quadratic scaling of our method with respect to spacial grid resolution, where time-stepping does not occur in the numeric implementation. The position- and momentum space grids are adapted to the two-photon interaction geometry at low resolution, for which our study only finds partial convergence of the simulated diffraction pattern. Further, the method has the advantage of having an easy implementable parallelization layout.