Resonance of minimizers for n-level quantum systems with an arbitrary cost

Boscain, Ugo; Charlot, Grégoire

ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 4, pp. 593-614.

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.

Classification : 49J15, 81V80, 53C17, 49N50
Mots clés : control of quantum systems, optimal control, sub-riemannian geometry, resonance, pontryagin maximum principle, abnormal extremals, rotating wave approximation

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