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arXiv:2603.13821 · The Journal of Chemical Physics

Higher order Magnus expansions for driven two-level quantum dynamics

Chen Wei, Frank Großmann

quant-phmath-phSubmitted Mar 14, 2026• Announced Mar 17, 2026• 0 citations• Revised May 13, 2026 (v2)

v1 · Mar 14, 2026 v2 · May 13, 2026

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-Stückelberg-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.

Journal reference: J. Chem. Phys. 164, 224121 (2026)
DOI: 10.1063/5.0333717

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Solution of two level system using

3^{rd}

order Magnus expansion

We derive a Magnus expansion for a frequency chirped quantum two-level system. We obtain a time-independent effective Hamiltonian which generates a stroboscopic time evolution. At lowest order the according dynamics is identical to results from using a rotating wave approximation. We determine, furthermore, also the next higher-order corrections within our expansion scheme in correspondence to the Bloch-Siegert shifts for harmonically driven systems. Importantly, our scheme can be extended to more complicated systems, i.e., even many-body systems.