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

TL;DR

This work extends a simple fourth-order Magnus expansion (ME) propagator to Liouville space, enabling efficient, CPTP-compatible time evolution for both closed and Lindblad open quantum systems. It provides explicit 4th-order expressions, including commutator-free variants, and validates them on a driven Λ-system to quantify the breakdown of the rotating wave approximation (RWA) across weak, moderate, and strong field regimes, with open-system baths often reducing RWA errors. The study demonstrates that 4th-order ME-based methods, particularly commutator-free versions, outperform standard RK4 and AM4 schemes and compare favorably with higher-order ME approaches, while offering practical advantages for long-time and periodic dynamics. Overall, the paper delivers a robust numerical toolkit for time-dependent quantum dynamics and supplies quantitative insights into RWA validity in realistic open-quantum-system settings.

Abstract

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.

Figures (26)

• Figure 1: Depiction of a three-level $\Lambda$-system. Solid lines represent coupling between states due to radiation fields while dashed lines represent spontaneous emission from the excited state. The probe (control) pulse coupling states $|1\rangle$ ($|3\rangle$) and $|2\rangle$ is parameterized by $\omega_p,\Omega_p$ ($\omega_c,\Omega_c$).
• Figure 2: Errors of RWA calculated according to Eq. (\ref{['eq:error_frob']}) for the six different cases listed in Table \ref{['table-1']}.
• Figure 3: Elements of the time dependent system density operators for the case B-I, closed system unitary dynamics, with full Hamiltonian (solid) and the RWA(dashed), for which the Hamiltonians are respectively $\hat{H}(t)$ and $\hat{H}^{\rm \small RWA}(t)$. Both were calculated using the 4th order ME-propagator with commutator, Eq. (\ref{['eq:us-4th']}).
• Figure 4: Elements of the time dependent system density operator for the case B-II, the open system non-unitary dynamics, with full Hamiltonian (solid) and the RWA (dashed), for which the Hamiltonians are respectively $\hat{H} (t)$ and $\hat{H}^{\rm \small RWA}(t)$. Both were calculated using the order ME-propagator with commutator, Eq. (\ref{['eq:usl-4th']}).
• Figure 5: Numerically calculated steady state average values of the open system case A-II for the full Hamiltonian, which are shown as data with "$\times$" symbols. The numerical results for the RWA Hamiltonian are shown with circle symbols, and dashed lines represent the analytical solution based on the RWA.