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

Ugo Boscain; Grégoire Charlot

ESAIM: Control, Optimisation and Calculus of Variations (2004)

  • 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 (2004): 593-614. <http://eudml.org/doc/244717>.

@article{Boscain2004,
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; sub-Riemannian geometry; Pontryagin maximum principle},
language = {eng},
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/244717},
volume = {10},
year = {2004},
}

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
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; sub-Riemannian geometry; Pontryagin maximum principle
UR - http://eudml.org/doc/244717
ER -

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Citations in EuDML Documents

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  1. Dario Prandi, Hölder equivalence of the value function for control-affine systems
  2. Thomas Chambrion, Paolo Mason, Mario Sigalotti, Ugo Boscain, Controllability of the discrete-spectrum Schrödinger equation driven by an external field
  3. Ugo Boscain, Camille Laurent, The Laplace-Beltrami operator in almost-Riemannian Geometry
  4. Andrei Agrachev, Thomas Chambrion, An estimation of the controllability time for single-input systems on compact Lie Groups
  5. Roberta Ghezzi, On almost-Riemannian surfaces

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