Two state system interaction with radiation - exact solution
Edited after comments.
I can't seem to find (or derive) the exact solution of the two quantum state system absorption problem in the following formulation. Let's consider the general harmonic wave absorption:
$$i \hbar \partial_t \Psi(\mathbf{r},t)=\left(\hat{H_0}(\mathbf{r})+\hat{H}(\mathbf{r}) e^{-\gamma |t|} \cos \omega t \right)\Psi(\mathbf{r},t) \tag{1}$$
Where the optical pulse has the exponential decay both for positive and negative time (it's the model I decided to use for simplicity).
Using the eigenstates of $\hat{H_0}$, we expand the wavefunction:
$$\Psi(\mathbf{r},t)=\sum_n C_n(t) \psi_n(\mathbf{r}) e^{-i \omega_n t} \tag{2}$$
Using orthonormality ($H_{mn}$ are matrix elements of the harmonic wave amplitude $\hat{H}(\mathbf{r})$):
$$i \hbar \dot{C_m}(t)=\sum_n C_n(t) H_{mn} \cos \omega t~ e^{i (\omega_m-\omega_n) t-\gamma |t|} \tag{3}$$
Now assume that we have a simple two state system.
$$\begin{bmatrix} \dot{C_1} \ \dot{C_2} \end{bmatrix}=\frac{e^{-\gamma | t|} \cos \omega t}{i\hbar}\begin{bmatrix} H_{11} & H_{12} e^{i \omega_{12} t} \ H_{12} e^{-i \omega_{12} t} & H_{22} \end{bmatrix} \begin{bmatrix} C_1 \ C_2 \end{bmatrix} \tag{4}$$
How to obtain the exact general solution for (4)? A reference would be enough for me, because I was only able to find either some particular case or a perturbation treatment of the problem.
Here's a related question but with a different matrix. 955