Which notion of energy for bilinear quantum systems?

Nabile Boussaid (1) , Marco Caponigro (2) , Thomas Chambrion (3, 4)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 M2N - Modélisation mathématique et numérique
3 CORIDA - Robust control of infinite dimensional systems and applications
4 IECN - Institut Élie Cartan de Nancy

In this note we investigate what is the best L^p-norm in order to describe the relation between the evolution of the state of a bilinear quantum system with the L^p-norm of the external field. Although L^2 has a structure more easy to handle, the L^1 norm is more suitable for this purpose. Indeed for every p>1, it is possible to steer, with arbitrary precision, a generic bilinear quantum system from any eigenstate of the free Hamiltonian to any other with a control of arbitrary small L^p norm. Explicit optimal costs for the L^1 norm are computed on an example.

Mots clés: Bilinear systems; quantum systems; distributed parameters systems; averaging control; optimal control

Domaines: Optimisation et contrôle [math.OC]

Nabile Boussaid, Marco Caponigro, Thomas Chambrion. Which notion of energy for bilinear quantum systems?. LHMNLC - 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control - 2012, Aug 2012, Bertinoro, Italy. pp.226-230. ⟨hal-00784890⟩