Sixth-order schemes for laser-matter interaction in the Schrödinger equation

Control of quantum systems via lasers has numerous applications that require fast and accurate numerical solution of the Schrödinger equation. In this paper, we present three strategies for extending any sixth-order scheme for the Schrödinger equation with time-independent potential to a sixth-order method for the Schrödinger equation with laser potential. As demonstrated via numerical examples, these schemes prove effective in the atomic regime as well as the semiclassical regime and are a particularly appealing alternative to time-ordered exponential splittings when the laser potential is highly oscillatory or known only at specific points in time (on an equispaced grid, for instance). These schemes are derived by exploiting the linear in space form of the time dependent potential under the dipole approximation (whereby commutators in the Magnus expansion reduce to a simpler form), separating the time step of numerical propagation from the issue of adequate time-resolution of the laser field by keeping integrals intact in the Magnus expansion and eliminating terms with unfavorable structure via carefully designed splittings.