Magnus-Lanczos methods with simplified commutators for the Schr\"odinger equation with a time-dependent potential

TL;DR

This paper introduces simplified Magnus-Lanczos methods for solving the Schrödinger equation with time-dependent potentials, enabling larger time steps and reducing computational costs, especially for highly oscillatory potentials.

Contribution

It develops Magnus expansions with simplified commutators and a novel integral simplification approach, improving efficiency and flexibility over traditional methods.

Findings

Significantly cheaper exponentiation via Lanczos iterations.

Effective handling of highly oscillatory potentials with larger time steps.

Flexible approach that avoids initial quadrature discretization.

Abstract

The computation of the Schr\"odinger equation featuring time-dependent potentials is of great importance in quantum control of atomic and molecular processes. These applications often involve highly oscillatory potentials and require inexpensive but accurate solutions over large spatio-temporal windows. In this work we develop Magnus expansions where commutators have been simplified. Consequently, the exponentiation of these Magnus expansions via Lanczos iterations is significantly cheaper than that for traditional Magnus expansions. At the same time, and unlike most competing methods, we simplify integrals instead of discretising them via quadrature at the outset -- this gives us the flexibility to handle a variety of potentials, being particularly effective in the case of highly oscillatory potentials, where this strategy allows us to consider significantly larger time steps.