Compact schemes for laser–matter interaction in Schrödinger equation based on effective splittings of Magnus expansion

Abstract

Numerical solutions for laser–matter interaction in Schrödinger equation has many applications in theoretical chemistry, quantum physics and condensed matter physics. In this paper we introduce a methodology which allows, with a small cost, to extend any fourth-order scheme for Schrödinger equation with time-independent potential to a fourth-order method for Schrödinger equation with laser potential. This is made possible due to the highly specific form of the time dependent potential which is linear in space in the case of laser–matter interaction. This leads to a highly amenable structure of the commutators in the Magnus expansion, where, in particular, the first commutator reduces to a scalar multiple of the gradient. In turn, this special structure allows us to split the Magnus expansion effectively via a variety of fourth-order splittings. Additionally, the error constant remains tiny because we keep the integrals of the potential intact to the very last stage of computations. This is particularly important in the case of highly oscillatory potentials. As demonstrated via numerical examples, these fourth-order methods improve upon many leading schemes of order six due to their low costs and small error constants.