Consider a sinusoidal driving two-level system:

$$ i \left( \begin{array}{c} \dot C_1(t) \\ \dot C_2(t) \\ \end{array} \right)=\left( \begin{array}{cc} -\frac{\omega _0}{2} & \Omega(t)\cos(\omega t)\\ \Omega(t)\cos(\omega t) & \frac{\omega _0}{2} \\ \end{array} \right)\cdot \left( \begin{array}{c} C_1(t) \\ C_2(t) \\ \end{array} \right) $$ where $C_1(t)$ and $C_2(t)$ are the state amplitude for the two states.

If $\Omega(t)$ is a constant, then the hamiltonian is periodic in time and thus the above equation can be solved using the standard Floquet theory.

I'm intreated in the case of $\Omega(t)$ has a pulse 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 $\tau\gg\frac{2\pi}{\omega}$.

How to solve the above equation using the Floquet formalism? I'm not seeking the complete analytical solution, what I want is just the procedure to carried out the perturbation or numerical solution.



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