Rotating-frame Rabi Hamiltonian ($H_R=({\Omega }_0-\omega )S_z+{\Omega }_R{S}_x$)

A circularly driven spin becomes time-independent after a rotation $R=e^{-i\omega t{S}_z/\hslash }$. The transformed Hamiltonian operator is $R^{†}HR-i\hslash R^{†}\dot{R}$, so the longitudinal frequency is shifted by $-\omega$. This is an exact change of frame for a circular drive. The rotating-wave approximation is a separate approximation needed when a drive also contains a counterrotating component.