I'm wondering if there is a prescription for solving for $\hat C$, $\hat D$, etc., in the series expansion $\exp{(\hat A + \lambda \hat B)} = e^{\hat A} e^{O(\lambda)\hat C} e^{O(\lambda^2)\hat D} \cdots$?
My question stems from how Eqs. (22), (44) were found as first and second order expansions, respectively, of Eq. (21) in the following paper: https://journals.aps.org/pra/pdf/10.1103/PhysRevA.87.052317 Basically, they find that
$$\begin{align} V = \exp(-i \frac{\pi}{2}(\hat \sigma_{\phi} + f\hat Z)) &\approx \exp(-i \frac{\pi}{2} \hat \sigma_{\phi}) \exp(- i f \hat \sigma_{\phi + \pi/2} ) \text{ (first order)}\\ &\approx \exp(-i \frac{\pi}{2} \hat \sigma_{\phi}) \exp(- i f \hat \sigma_{\phi + \pi/2} ) \exp(- i \frac{\pi}{4}f^2 \hat \sigma_{\phi} ) \text{ (second order)} \end{align}$$
Here, $\hat \sigma_{\phi} = \cos(\phi)\hat X + \sin(\phi)\hat Y$, and $\hat X, \hat Y, \hat Z$ are the Pauli operators.
These terms look suspiciously like a series of commutators, and I have tried working with BCH & the Zassenhaus formula, but I was not able to reproduce these results (although I have verified that they are great approximations!)
Any help/tips would be appreciated. Thanks!