Question: Is there a transformation or simple approximation to use to cancel the time-dependence in a Hamiltonian of form $$ H_{G} = \sum_{(m,n)\in E,\ m<n}\Omega_{mn} e^{-i\Delta_{mn} t} |m\rangle\langle n|+h.c. $$ where $G=(V, E)$ is a cyclic graph (for which the rotating frame transformation typically fails)?
Edit: For example, one approximation is simply to prune the graph until it becomes acyclic, as Baker et al. (2008) Phys. Rev. A 98, 052111 do in an adaptive scheme that cleverly chooses which edges to remove in an attempt to minimize error. This is an idea first suggested by Einwohner, Wong, and Garrison (1976) Phys. Rev. A 14, 1452. But I do wonder if there are other transformations/approximations that can be used for this problem.
Context:
Say we have a Hamiltonian with on-state energies as well as oscillating interaction terms, such that $$ H=\sum_n \omega_n |n\rangle\langle n| + \sum_{(m,n)\in E,\ m<n}\left( \Omega_{mn} e^{-i\omega_{L_{mn}} t} + \tilde\Omega_{mn}e^{i\omega_{L_{mn}} t} \right)|m\rangle\langle n|+h.c. $$ where the pairs $E$ can be viewed as the edges of some graph $G=(V,E)$, the vertices in $V$ being identified with ordered state labels (e.g. $V\subset\mathbb{Z^*}$). (Note: we have set $\hbar = 1$.)
After the typical transformation into the interaction picture and the rotating wave approximation, we are left with the approximation $$ \begin{align} H_{1,I}^{RWA} &= \sum_{(m,n)\in E,\ m<n}\Omega_{mn} e^{-i(\omega_{L_{mn}}-\omega_n+\omega_m) t} |m\rangle\langle n|+h.c.\\ &= \sum_{(m,n)\in E,\ m<n}\Omega_{mn} e^{-i\Delta_{mn} t} |m\rangle\langle n|+h.c. \end{align} $$
The rotating frame transformation (RFT) (explained in Whaley & Light (1984)) can, via a unitary transformation with a diagonal matrix $U^{RFT}$ whose diagonal $U^{RFT}_{mm}=e^{iq_{m}t}$ satisfies $q_{m} - q_{n} = -\Delta_{mn}$, cancel out the remaining time dependence, but in general only when the graph $G$ is acyclic.
This is because for cyclic graphs, for each cycle we have conditions $\Delta_{ab}+\Delta_{bc}+\ldots+\Delta_{fg}+\Delta_{ga}=0$ (for the edges $(a,b),\ (b,c)\ldots(g,a)$ that contribute to the cycle) that, if violated, mean the RFT cannot cancel the time dependence for all transitions in the loop.
