Magnus-Zassenhaus methods for the semiclassical Schr\"odinger equation with oscillatory time-dependent potentials

TL;DR

This paper introduces high-order exponential splitting schemes for the semiclassical Schrödinger equation with oscillatory, time-dependent potentials, enabling larger time steps and improved accuracy in challenging quantum simulations.

Contribution

The paper develops a novel class of exponential splitting schemes combining Magnus expansions and Zassenhaus splittings for better numerical stability and efficiency.

Findings

Enables larger time steps in simulations with oscillatory potentials

Achieves high-order accuracy in semiclassical Schrödinger equations

Improves numerical stability over standard methods

Abstract

Schr\"odinger equations with time-dependent potentials are of central importance in quantum physics and theoretical chemistry, where they aid in the simulation and design of systems and processes at atomic scales. Numerical approximation of these equations is particularly difficult in the semiclassical regime because of the highly oscillatory nature of solution. Highly oscillatory potentials such as lasers compound these difficulties even further. Altogether, these effects render a large number of standard numerical methods less effective in this setting. In this paper we will develop a class of high-order exponential splitting schemes that are able to overcome these challenges by combining the advantages of integral-preserving simplified-commutator Magnus expansions with those of symmetric Zassenhaus splittings. This allows us to use large time steps in our schemes even in the presence of highly oscillatory potentials and solutions.