Simultaneous local exact controllability of 1D bilinear Schr\"odinger equations

We consider N independent quantum particles, in an infinite square potential well coupled to an external laser field. These particles are modelled by a system of linear Schr\"odinger equations on a bounded interval. This is a bilinear control system in which the state is the N-tuple of wave functions. The control is the real amplitude of the laser field. For N=1, Beauchard and Laurent proved local exact controllability around the ground state in arbitrary time. We prove, under an extra generic assumption, that their result does not hold in small time if N is greater or equal than 2. Still, for N=2, we prove using Coron's return method that local controllability holds either in arbitrary time up to a global phase or exactly up to a global delay. We also prove that for N greater or equal than 3, local controllability does not hold in small time even up to a global phase. Finally, for N=3, we prove that local controllability holds up to a global phase and a global delay

Simultaneous approximate tracking of density matrices for a system of Schrödinger equations

International audienceWe consider a non-resonant system of finitely many bilinear Schro¿dinger equations with discrete spectrum driven by the same scalar control.We prove that this system can approximately track any given system of trajectories of density matrices, up to the phase of the coordinates. The result is valid both for bounded and unbounded Schro¿dinger operators. The method used relies on finite-dimensional control techniques applied to Lie groups. We provide also an example showing that no approximate tracking of both modulus and phase is possible