Selective and Robust Time-Optimal Rotations of Spin Systems

TL;DR

This paper develops analytical methods for designing time-optimal, selective, and robust control pulses for spin-1/2 particles, with applications in NMR, using geometric analysis and the Pontryagin Maximum Principle.

Contribution

It introduces a combined geometric and analytical approach to derive optimal control solutions for inhomogeneous spin systems, including new robust control strategies.

Findings

Analytical solutions for selective and robust controls are obtained.

Standard NMR control mechanisms are validated as optimal.

New robust control protocols are designed and verified.

Abstract

We study the selective and robust time-optimal rotation control of several spin-1/2 particles with different offset terms. For that purpose, the Pontryagin Maximum Principle is applied to a model of two spins, which is simple enough for analytic computations and sufficiently complex to describe inhomogeneity effects. We find that selective and robust controls are respectively described by singular and regular trajectories. Using a geometric analysis combined with numerical simulations, we determine the optimal solutions of different control problems. Selective and robust controls can be derived analytically without numerical optimization. We show the optimality of several standard control mechanisms in Nuclear Magnetic Resonance, but new robust controls are also designed.