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

Ugo Boscain; Grégoire Charlot

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 4, page 593-614
  • ISSN: 1292-8119

Abstract

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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 2 n - 1 ⊂ ℂ n one is reduced to look just for minimizers on the sphere S n - 1 ⊂ ℝ n . Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.

How to cite

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Boscain, Ugo, and Charlot, Grégoire. "Resonance of minimizers for n-level quantum systems with an arbitrary cost." ESAIM: Control, Optimisation and Calculus of Variations 10.4 (2010): 593-614. <http://eudml.org/doc/90745>.

@article{Boscain2010,
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. },
author = {Boscain, Ugo, Charlot, Grégoire},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Control of quantum systems; optimal control; sub-Riemannian geometry; resonance; pontryagin maximum principle; abnormal extremals; rotating wave approximation.; atomic levels; molecular levels; control of quantum systems; sub-Riemannian geometry; resonance; Pontryagin maximum principle; rotating wave approximation},
language = {eng},
month = {3},
number = {4},
pages = {593-614},
publisher = {EDP Sciences},
title = {Resonance of minimizers for n-level quantum systems with an arbitrary cost},
url = {http://eudml.org/doc/90745},
volume = {10},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 4
SP - 593
EP - 614
AB - 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.
LA - eng
KW - Control of quantum systems; optimal control; sub-Riemannian geometry; resonance; pontryagin maximum principle; abnormal extremals; rotating wave approximation.; atomic levels; molecular levels; control of quantum systems; sub-Riemannian geometry; resonance; Pontryagin maximum principle; rotating wave approximation
UR - http://eudml.org/doc/90745
ER -

References

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