A simple fourth order propagator based on the Magnus expansion in the Liouville space: Application to a $Λ$-system and assessment of the rotating wave approximation

A simple 4th order propagator [Ture and Jang, {\it J. Phys. Chem. A.} {\bf 128}, 2871 (2024)] based on the Magnus expansion (ME) is extended to the Liouville space for both closed-system and Lindbladian open-system quantum dynamics. For both dynamics, commutator free versions of 4th order propagators are provided as well. These propagators are then applied to the dynamics of a driven $Λ$-system, where Lindblad terms represent the effect of a photonic bath. For both dynamics, the accuracy of the rotating wave approximation (RWA) for the matter-radiation interaction is assessed. We confirmed reasonable performance of RWA for weak and resonant fields. However, small errors appear for moderate fields and substantial errors can be found for strong fields where coherent population trapping can still be expected. We also found that the presence of bath for open system quantum dynamics consistently reduces the errors of the RWA. These results provide a quantitative information on how the RWA breaks down beyond weak field or for non-resonant cases. Major results are benchmarked against results of our 6th order ME-based propagator. We also provide numerical comparison of our algorithms with other 4th order algorithms for the $Λ$-system. These confirm reasonable performance of our simple propagators and the improvement gained through commutator-free expressions.

💡 Research Summary

The paper presents a fourth‑order propagator derived from the Magnus expansion (ME) and extends it to the Liouville space, enabling its use for both closed‑system unitary dynamics and open‑system Lindblad dynamics. The authors first recall their previously introduced simple 4th‑order ME propagator, which evaluates the Hamiltonian at three equally spaced points (the two endpoints and the midpoint) and includes a commutator term that improves accuracy at the cost of extra computation. To avoid the commutator, they construct a commutator‑free Magnus expansion (CFME) using the Baker‑Campbell‑Hausdorff formula, resulting in a product of two exponentials that retains fourth‑order accuracy while preserving time‑reversal symmetry and positivity (CPTP for open systems).

In the Liouville representation, the density matrix is vectorized, and the Liouvillian super‑operator L(t)= (1/ħ)