Multi-input Schrödinger equation: controllability, tracking, and application to the quantum angular momentum

Ugo Boscain (1, 2) , Marco Caponigro (3) , Mario Sigalotti (1)

1 GECO - Geometric Control Design
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 M2N - Modélisation mathématique et numérique

We present a sufficient condition for approximate controllability of the bilinear discrete-spectrum Schrödinger equation exploiting the use of several controls. The controllability result extends to simultaneous controllability, approximate controllability in $H^s$, and tracking in modulus. The result is more general than those present in the literature even in the case of one control and permits to treat situations in which the spectrum of the uncontrolled operator is very degenerate (e.g. it has multiple eigenvalues or equal gaps among different pairs of eigenvalues). We apply the general result to a rotating polar linear molecule, driven by three orthogonal external fields. A remarkable property of this model is the presence of infinitely many degeneracies and resonances in the spectrum preventing the application of the results in the literature.

Mots clés: quantum angular momentum; Quantum control; bilinear Schrödinger equation; Galerkin approximations; quantum angular momentum.

Domaines: Optimisation et contrôle [math.OC]

Ugo Boscain, Marco Caponigro, Mario Sigalotti. Multi-input Schrödinger equation: controllability, tracking, and application to the quantum angular momentum. Journal of Differential Equations, 2014, 256 (11), pp.3524-3551. ⟨10.1016/j.jde.2014.02.004⟩. ⟨hal-01097161v2⟩