# Analytical solution of two level system driving by a sinusoidal potential beyond rotating wave approximation

A quantum mechanical two-level system driving by a constant sinusoidal external potential is very useful in varies areas of physics. Although the wildly used rotating-wave approximation(RWA) is very successful in treating weak coupling and near resonance cases, sometimes a analytical solution beyond RWA is desired. Are there any special cases(for example large detuning, very strong driving, etc.) where one can get the analytical solutions beyond RWA?

In mathematics, this is to say that solve the following equation analytically for $C_1$ and $C_2$:

$$i\dot{C}_1(t)=\Omega~cos(\omega t)e^{-i\omega_0t}C_2(t)\\ i\dot{C}_2(t)=\Omega~cos(\omega t)e^{i\omega_0t}C_1(t)$$ where $C_1(t)$ and $C_2(t)$ are the two level state amplitude, $\Omega$ is the coupling strength, $\omega_0$ is the two level frequency difference, and $\omega$ is the driving frequency. $\omega$, $\omega_0$, $\Omega$ are constant and $C_1$ and $C_2$ are time dependent quantities.

Any suggestions or related literatures are appreciated.

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