  The Bloch vector's path of a two-level system exposed to a monochromatic
field exhibits, in the regime of strong coupling, complex corkscrew
trajectories. By considering the infinitesimal evolution of the two-level
system when the field is treated as a classical object, we show that the Bloch
vector's rotation speed oscillates between zero and twice the rotation speed
predicted by the rotating wave approximation. Cusps appear when the rotation
speed vanishes. We prove analytically that in correspondence to cusps the
curvature of the Bloch vector's path diverges. On the other hand, numerical
data show that the curvature is very large even for a quantum field in the deep
quantum regime with mean number of photons $\bar{n}\lesssim 1$. We finally
compute numerically the typical error size in a quantum gate when the terms
beyond rotating wave approximation are neglected.
