A Review of Geometric Optimal Control for Quantum Systems in Nuclear Magnetic Resonance

Bernard Bonnard (1, 2) , Steffen Glaser (3) , Dominique Sugny (4)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 McTAO - Mathematics for Control, Transport and Applications
3 Department Chemie
4 ICB - Laboratoire Interdisciplinaire Carnot de Bourgogne

We present a geometric framework to analyze optimal control problems of uncoupled spin 1/2 particles occurring in nuclear magnetic resonance. According to the Pontryagin's maximum principle, the optimal trajectories are solutions of a pseudo-Hamiltonian system. This computation is completed by sufficient optimality conditions based on the concept of conjugate points related to Lagrangian singularities. This approach is applied to analyze two relevant optimal control issues in NMR: the saturation control problem, that is, the problem of steering in minimum time a single spin 1/2 particle from the equilibrium point to the zero magnetization vector, and the contrast imaging problem. The analysis is completed by numerical computations and experimental results.

Mots clés: TRAJECTORIES; SINGULARITIES; EXTREMALS

Domaines: Physique Nucléaire Théorique [nucl-th]

Bernard Bonnard, Steffen Glaser, Dominique Sugny. A Review of Geometric Optimal Control for Quantum Systems in Nuclear Magnetic Resonance. Advances in Mathematical Physics, 2012, -, pp.857493. ⟨10.1155/2012/857493⟩. ⟨hal-00750040⟩