Local exact controllability of a 1D Bose-Einstein condensate in a time-varying box

TL;DR

This paper proves that it is possible to precisely control the state of a one-dimensional Bose-Einstein condensate within a variable box potential by adjusting the box length, for most chemical potentials.

Contribution

It establishes local exact controllability of a 1D Bose-Einstein condensate in a time-varying box, a novel result in quantum control of nonlinear systems.

Findings

Controllability holds for a generic set of chemical potentials.

The proof uses linearization, inverse mapping theorem, and analytic perturbation theory.

Control is achieved by varying the box length to steer the wave function.

Abstract

We consider a one-dimensional Bose-Einstein condensate in a infinite square-well (box) potential. This is a nonlinear control system in which the state is the wave function of the Bose Einstein condensate and the control is the length of the box. We prove that local exact controllability around the ground state (associated with a fixed length of the box) holds generically with respect to the chemical potential \mu; i.e. up to an at most countable set of \mu-values. The proof relies on the linearization principle and the inverse mapping theorem, as well as ideas from analytic perturbation theory.