If I have a 2-level system, with the energy splitting between the 2 levels $\omega_{12}$ and an external perturbation characterized by a frequency $\omega_P$, if $\omega_{12}\gg\omega_P$ I can use the adiabatic approximation, and assume that the initial state of the system changes slowly in time while for $\omega_{12}\ll\omega_P$ I can assume that the perturbation doesn't have any effect on the system (it averages out over the relevant time scales). I was wondering if I have a 3 level system with $E_1<E_2<E_3$ such that $\omega_{12}\ll\omega_P\ll\omega_{23}$. In general, the Hamiltonian of the system would look like this:
$$\begin{pmatrix}E_1 & f_{12}(t) & f_{13}(t) \\f_{12}^*(t) & E_2 & f_{23}(t) \\f_{13}^*(t) & f_{23}^*(t) & E_3\end{pmatrix}$$
But using the intuition from the 2 level system case, can I ignore $f_{12}(t)$, as the system of these 2 levels (1 and 2) moves on time scales much slower than $\omega_P$, and assume that $f_{23}(t)$ and $f_{13}(t)$ move very slow and thus use the adiabatic approximation? In practice I would basically have:
$$\begin{pmatrix}E_1 & 0 & f_{13}(t) \\0 & E_2 & f_{23}(t) \\f_{13}^*(t) & f_{23}^*(t) & E_3\end{pmatrix}$$
Or in this case I would need to fully solve the SE, without being able to make any approximations?
