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 ${S}^{2n-1}\subset {\u2102}^{n}$ 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.

Keywords: control of quantum systems, optimal control, sub-riemannian geometry, resonance, pontryagin maximum principle, abnormal extremals, rotating wave approximation

@article{COCV_2004__10_4_593_0, author = {Boscain, Ugo and Charlot, Gr\'egoire}, title = {Resonance of minimizers for n-level quantum systems with an arbitrary cost}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {593--614}, publisher = {EDP-Sciences}, volume = {10}, number = {4}, year = {2004}, doi = {10.1051/cocv:2004022}, mrnumber = {2111082}, zbl = {1072.49002}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2004022/} }

TY - JOUR AU - Boscain, Ugo AU - Charlot, Grégoire TI - Resonance of minimizers for n-level quantum systems with an arbitrary cost JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2004 SP - 593 EP - 614 VL - 10 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2004022/ DO - 10.1051/cocv:2004022 LA - en ID - COCV_2004__10_4_593_0 ER -

%0 Journal Article %A Boscain, Ugo %A Charlot, Grégoire %T Resonance of minimizers for n-level quantum systems with an arbitrary cost %J ESAIM: Control, Optimisation and Calculus of Variations %D 2004 %P 593-614 %V 10 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2004022/ %R 10.1051/cocv:2004022 %G en %F COCV_2004__10_4_593_0

Boscain, Ugo; Charlot, Grégoire. Resonance of minimizers for n-level quantum systems with an arbitrary cost. ESAIM: Control, Optimisation and Calculus of Variations, Volume 10 (2004) no. 4, pp. 593-614. doi : 10.1051/cocv:2004022. http://archive.numdam.org/articles/10.1051/cocv:2004022/

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