  In this paper we consider the minimum time population transfer problem for a
two level quantum system driven by {\em two} external fields with bounded
amplitude. The controls are modeled as real functions and we do not use the
Rotating Wave Approximation. After projection on the Bloch sphere, we tackle
the time-optimal control problem with techniques of optimal synthesis on 2-D
manifolds. Based on the Pontryagin Maximum Principle, we characterize a
restricted set of candidate optimal trajectories. Properties on this set,
crucial for complete optimal synthesis, are illustrated by numerical
simulations. Furthermore, when the two controls have the same bound and this
bound is small with respect to the difference of the two energy levels, we get
a complete optimal synthesis up to a small neighborhood of the antipodal point
of the starting point.
