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

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

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

\begin{align} 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) \end{align} 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.

• If I'm not mistaken Cohen-Tannoudji vol. 2 solves this exactly. Apr 13 '19 at 20:19
• I can see this is a very old post. In any case said equation can be solved by "going to the interaction picture". I can post a solution in case you are still interested.
– lcv
Sep 13 '19 at 12:53