Exponential perturbative expansions and coordinate transformations

TL;DR

The paper presents a unified framework in which exponential perturbation techniques for time-dependent linear systems are derived from a single coordinate-change strategy x(t) = exp(Ω(t)) X(t). By selecting appropriate constant or time-dependent matrices F and applying recursive constructions of Ω and F, it recovers Magnus, Floquet--Magnus, quantum averaging, Lie--Deprit, and standard perturbation theory while preserving unitarity. The approach provides explicit recursion relations and convergence criteria, and demonstrates its utility through the three-lambda and Bloch--Siegert examples, highlighting when Floquet--Magnus outperforms other methods and how quasi-periodicity introduces small-divisor considerations. The framework also accommodates perturbation-removal and quasi-periodic transformations, offering a flexible tool to tailor perturbation methods to specific time dependences. Overall, this unification clarifies connections among exponential perturbation techniques and guides the selection of the most effective method for a given problem with periodic or quasi-periodic driving.

Abstract

We propose a unified approach for different exponential perturbation techniques used in the treatment of time-dependent quantum mechanical problems, namely the Magnus expansion, the Floquet--Magnus expansion for periodic systems, the quantum averaging technique and the Lie--Deprit perturbative algorithms. Even the standard perturbation theory fits in this framework. The approach is based on carrying out an appropriate change of coordinates (or picture) in each case, and can be formulated for any time-dependent linear system of ordinary differential equations. All the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schrödinger equation.

Figures (8)

• Figure 1: Transition probability for the three-lambda system with $f(t) = \mathrm{e}^{i \omega t}$ for $\omega=10 \, (1+\frac{\sqrt{2}}{2})^{-\frac{1}{2}}$ obtained with Floquet--Magnus (FM) with $n=3$ (left) and $n=4$ (right) terms in the expansion. Blue line corresponds to FM, orange (Ef) line stands for the effective Hamiltonian $H_{\mathrm{ef}} \equiv i F$ and black (Ex) line is the exact solution.
• Figure 2: Top: transition probability for the three-lambda system with $f(t) = \mathrm{e}^{i \omega t}$ obtained with Floquet--Magnus (blue line) and Magnus expansion (red line), both with $n=3$ terms. Left diagram is for $\omega = 6$, right diagram is for $\omega=12$. Bottom: error (in logarithmic scale) as a function of time for Floquet--Magnus with $n=3$ and $n=7$ terms, and the effective solution with $n=7$ terms (purple).
• Figure 3: Error (in logarithmic scale) the transition probability (in logarithmic scale) for the quasi-periodic case (\ref{['3lambda.33']}) as a function of time for FM (left) and Magnus (right) with $n=2$ and $n=3$ terms in the expansion. In both cases $\omega = 12$ in (\ref{['3lambda.33']}).
• Figure 4: Detail of Figure \ref{['figu33']} right in the interval $\omega t \in [0, 30]$. We have included up to $n=6$ terms in the Magnus series.
• Figure 5: Transition probability obtained with FM (left, blue line) and LD (right, green line) with $n=3$ terms in comparison with the exact result for the Bloch--Siegert Hamiltonian with $\varepsilon = 0.2$ (top) and $\varepsilon = 1$ (bottom). Black line corresponds to the exact solution, and orange to the result achieved with the effective Hamiltonian. On the right panels the results achieved by the algorithm removing the perturbation are also depicted ($F = A_0$).