  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 $S^{2n-1}\subset\C^n$ one is reduced to
look just for minimizers on the sphere $S^{n-1}\subset \R^n$. Moreover, for the
reduced problem, we investigate on the question of existence of strict abnormal
minimizer.
