Stabilization of an arbitrary profile for an ensemble of half-spin systems

TL;DR

This paper develops an explicit feedback control law to asymptotically stabilize any arbitrary profile of an ensemble of non-interacting half-spin systems described by Bloch equations, with proven convergence under specific conditions.

Contribution

It introduces a novel feedback stabilization method for arbitrary profiles in spin ensembles, extending control techniques to infinite-dimensional quantum systems.

Findings

The feedback law achieves asymptotic stabilization of the target profile.

Convergence is proven for profiles entirely in the north or south hemisphere.

Numerical simulations demonstrate effectiveness even from far initial conditions.

Abstract

We consider the feedback stabilization of a variable profile for an ensemble of non interacting half spins described by the Bloch equations. We propose an explicit feedback law that stabilizes asymptotically the system around a given arbitrary target profile. The convergence proof is done when the target profile is entirely in the south hemisphere or in the north hemisphere of the Bloch sphere. The convergence holds for initial conditions in a H^1 neighborhood of this target profile. This convergence is shown for the weak H^1 topology. The proof relies on an adaptation of the LaSalle invariance principle to infinite dimensional systems. Numerical simulations illustrate the efficiency of these feedback laws, even for initial conditions far from the target profile.