Rotating Wave Approximation (RWA)

We introduce the System Hamiltonian $H_S^{\rm RWA}(t)$ in rotating wave approximation (RWA) by writing $\cos(\omega t)=\frac{1}{2}(e^{i\omega t}+e^{-i\omega t})$ and neglecting the counter-rotating terms $\sigma_- e^{-i\omega t}$ and $\sigma_+ e^{i\omega t}$

$$H_S^{\rm RWA}(t) \equiv \frac{\hbar\omega_0}{2}\sigma_z - \left(\frac{\hbar\Omega}{2} \sigma_- e^{i\omega t} + \frac{\hbar\Omega}{2} \sigma_+ e^{-i\omega t}\right).$$ (120)

In this case, $H_S^{\rm RWA}(t)={\bf B}^{\rm RWA}(t){\vec{\sigma}}$ with

$${\bf B}^{\rm RWA}(t)=\left( \begin{array}[h]{c} -\frac{1}{2}\... ...ga\sin(\omega t)\\ \frac{1}{2}\hbar\omega_0 \end{array}\right)$$ (121)