Higher order Magnus expansions for two-level quantum dynamics

Abstract

We investigate the Magnus expansion for a generic time-dependent two-level system under single-axis driving. By virtue of the \(\mathfrak{su}(2)\) 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 over the whole parameter range.

Figures (8)

• Figure 1: Magnus approximations of different order (1st order: dashed blue, 2nd order: dotted orange, 3rd order: dash-dotted green) and exact results (solid black) for the LZSM problem: (a) transition probabilities (exact and 3rd order results are almost indistinguishable), (b) Stokes phase (1st order not shown, as it gives a zero result) in units of $\pi$, both as a function of $\gamma$, defined in Eq. (\ref{['eq:gamma']}).
• Figure 2: Parameter regions for which the Floquet quasienergies of the semiclassical Rabi model are investigated.
• Figure 3: Exact results for the quasienergy of the Rabi model at $g/\omega=1$ as a function of $\Delta$. Panel (b) shows a magnified view for the boxed region in panel (a), displaying an avoided crossing.
• Figure 4: Quasienergies for $g=1$: (a) exact result for $\epsilon$ and $-\epsilon$ (solid black lines) and Magnus approximations of different order based on the picture appropriate for region I (1st order: dashed blue, 2nd order: dotted orange, 3rd order: dash-dotted green) for $\epsilon$, directly computed from $U(2\pi)$; (b) same as (a) but with Magnus approximations with generalized parity symmetry explicitly maintained by using $U(\pi)$ for the determination of $\epsilon$ (see text).
• Figure 5: Quasienergy of the semiclassical Rabi model at $\Delta=\omega$ as a function of $g$: exact result for $\epsilon$ and $-\epsilon$ (solid black lines) and Magnus approximations of different order based on the picture appropriate for region II: 1st order extracted from time evolution over $2\pi$: dashed blue, 1st (equal to 2nd) order with the generalized parity symmetry explicitly maintained: dotted orange, 3rd order with the generalized parity symmetry explicitly maintained: dash-dotted green.