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

Published online by Cambridge University Press:  15 October 2004

Ugo Boscain  and
Grégoire Charlot
Ugo Boscain
Affiliation:
SISSA-ISAS, via Beirut 2-4, 34014 Trieste, Italy; boscain@sissa.it.;charlot@sissa.it Département de Mathématiques, Analyse Appliquée et Optimisation, Université de Bourgogne, 9 avenue Alain Savary, BP 47870-21078 Dijon Cedex, France.
Grégoire Charlot
Affiliation:
SISSA-ISAS, via Beirut 2-4, 34014 Trieste, Italy; boscain@sissa.it.;charlot@sissa.it
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Abstract

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 $S^{2n-1}\subset\mathbb{C}^n$ 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 systemsoptimal controlsub-Riemannian geometryresonancepontryagin maximum principleabnormal extremalsrotating wave approximation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 4 , October 2004 , pp. 593 - 614
Copyright
© EDP Sciences, SMAI, 2004

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