Consider the semi-classical laser driving two-level atom, where the laser is treated classically and the atom is treated quantum mechanically. The effect of laser on the atom is a dipole coupling: $$ i\,\dot\Psi(t)=[-\frac{\omega_0}{2}\sigma_z+\Omega(t)\cos(\omega t)\sigma_x]\Psi(t) $$ where $\Psi(t)$ is the wavefunction, $\omega_0$ is the two level frequency, $\omega$ is the driving frequency, $\Omega(t)$ is the coupling strength. $\sigma_x,\sigma_z$ are the Pauli matrixes. In matrix form this can be written as:
$$ i \left( \begin{array}{c} \dot C_1(t) \\ \dot C_2(t) \\ \end{array} \right)=\left( \begin{array}{cc} -\frac{\omega _0}{2} & \Omega(t)\cos(\omega t)\\ \Omega(t)\cos(\omega t) & \frac{\omega _0}{2} \\ \end{array} \right)\cdot \left( \begin{array}{c} C_1(t) \\ C_2(t) \\ \end{array} \right) $$ where $C_1(t)$ and $C_2(t)$ are the state amplitude for the two states.
If the coupling $\Omega(t)$ is weak and $\omega_0\approx\omega$ then this can be solved using the Rotating-Wave Approximation. I'm intrested in the region where $\Omega$ not small compared to $\omega_0$ or $\omega$, and $\Omega(t)$ has a shape $$ \Omega(t)= \left\{ \begin{array}{ll} \Omega_0\sin^2(\pi t/\tau) & 0\leq t<\tau\\[1em] 0 & \tau \leq t \end{array} \right. $$ where $\tau>>\frac{2\pi}{\omega}$. If the coupling $\Omega(t)$ is time independent, then we can use the standard Floquet theory to solve the equation. But if $\Omega(t)$ is time-dependent, how do I use the Floquet formalism to solve this problem?