Sixth-order schemes for laser--matter interaction in Schr\"odinger equation

TL;DR

This paper develops three sixth-order numerical schemes for solving the Schrödinger equation with laser potentials, improving accuracy and efficiency in simulating laser-matter interactions in quantum systems.

Contribution

It introduces novel strategies to extend sixth-order schemes to laser potentials, especially effective for highly oscillatory or discretely known laser fields.

Findings

Effective in atomic and semiclassical regimes

Outperforms time-ordered exponential splittings in oscillatory cases

Maintains high accuracy with efficient computation

Abstract

Control of quantum systems via lasers has numerous applications that require fast and accurate numerical solution of the Schr\"odinger equation. In this paper we present three strategies for extending any sixth-order scheme for Schr\"odinger equation with time-independent potential to a sixth-order method for Schr\"odinger 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 unfavourable structure via carefully designed splittings.