Higher order Magnus expansions for driven two-level quantum dynamics

TL;DR

This paper develops higher order Magnus expansions for driven two-level quantum systems, improving the accuracy of non-adiabatic transition predictions and Floquet quasienergy calculations.

Contribution

It introduces a commutator-free Magnus expansion tailored for su(2) systems and demonstrates its effectiveness on Landau-Zener and Rabi models.

Findings

Third order approximation closely matches exact results.

Second order Magnus in the adiabatic picture is highly accurate.

Symmetry enforcement and picture transformations improve convergence.

Abstract

We investigate the Magnus expansion for a generic time-dependent two-level system under single-axis driving.By virtue of the su(2) Lie algebra, the expansion is decomposed into a commutator-free form. To illustrate the usefulness of the gained expression, we then revisit the Landau-Zener-St\"uckelberg-Majorana model, with a focus on non-adiabatic transitions as well as the Stokes phase. In addition, the semiclassical Rabi model is systematically treated by determining the Floquet quasienergy up to different orders. We demonstrate how to employ suitable picture transformations as well as on how to enforce the symmetry of the underlying model in order to guarantee convergence of the expansion as well as to achieve satisfactory agreement with the exact results. For both models that we studied it turns out that a third order approximation yields results that are in next to perfect agreement with exact analytical ones. Surprisingly, in the case of the semiclassical Rabi model, even the second order Magnus approximation in the adiabatic picture produces almost exact results for a large parameter range.