Counter-rotating weak-transition amplitude ($c_2=-\Omega {\sum }_{\pm }(e^{i({\omega }_0\pm \omega )t}-1)/[2({\omega }_0\pm \omega )]$)

The first-order response to a cosine drive contains both resonant and counter-rotating terms. The rotating-wave approximation removes the rapidly oscillating contribution only under near-resonant, weak-drive and appropriate time-scale assumptions. It does not remove the separate requirement of weak excitation when the initial amplitude is held fixed.