We consider an optimal control problem describing a
laser-induced
population transfer on a n-level quantum system. For a convex cost depending only on the moduli
of controls (i.e. the lasers intensities),
we prove that there always exists a minimizer in
resonance. This permits to justify
some strategies used in experimental physics. It is also quite
important
because it permits to reduce remarkably
the complexity of the problem (and extend some of our previous
results
for n=2 and n=3): instead of looking for minimizers on the
sphere $S^{2n-1}\subset\mathbb{C}^n$ one is reduced to look just for
minimizers on the sphere $S^{n-1}\subset \mathbb{R}^n$. Moreover, for the reduced problem,
we investigate on the question of existence of strict
abnormal
minimizer.