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

TL;DR

This paper investigates the controllability of multiple quantum particles in a potential well under laser control, revealing limitations and possibilities for control in small and arbitrary times depending on the number of particles.

Contribution

It extends controllability results to multiple particles, showing when local controllability is possible or impossible in small or arbitrary times, using Coron's return method.

Findings

For N=1, local exact controllability in arbitrary time.

For N≥2, controllability does not hold in small time without additional assumptions.

For N=2, controllability holds up to a global phase or delay in arbitrary time.

Abstract

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.