Resonance in 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. These facts are quite important because they permit 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, proving that resonance is optimal permits to justify some strategies used in experimental physics.