Resonance of minimizers for n-level quantum systems

We consider a laser-induced population transfer problem on a finite dimensional quantum system in the rotating wave approximation. For a convex cost depending only on the moduli of controls, we prove that there always exists a minimizer in resonance. This fact is quite important because it permits to reduce remarkably the complexity of the problem from dimension 2n-1 to dimension n-1 (and extend some of our previous results for n=2 and n=3). Moreover it justifies some strategies used in experimental physics.