Approximate controllability via adiabatic techniques for the three-inputs controlled Schr{\"o}dinger equation

TL;DR

This paper introduces a constructive adiabatic control method for the three-input Schrödinger equation, enabling approximate state steering based on spectral properties and eigenvalue intersections.

Contribution

It develops a novel adiabatic control technique for bilinear Schrödinger systems with three inputs, utilizing eigenvalue intersections and conical controls for approximate controllability.

Findings

Method achieves controllability with quantifiable error bounds.

Eigenvalue intersections are crucial for control success.

Control path optimization improves accuracy and efficiency.

Abstract

We consider a system described by a controlled bilinear Schr{\"o}dinger equation with three external inputs. We provide a constructive method to approximately steer the system from a given energy level to a superposition of energy levels corresponding to a given probability distribution. The method is based on adiabatic techniques and works if the spectrum of the Hamiltonian admits eigenvalue intersections, with respect to variations of the controls, and if the latter are conical. We provide sharp estimates of the relation between the error and the controllability time, and we show how to improve these estimates by selecting special control paths.