Time-optimal selective pulses of two uncoupled spin 1/2 particles

TL;DR

This paper derives analytical, time-optimal control pulses for selectively manipulating two uncoupled spin-1/2 particles with different offsets, using Pontryagin's Maximum Principle and elliptic integrals.

Contribution

It provides the first analytical solutions for time-optimal selective pulses in two uncoupled spins, revealing bifurcations and optimal control structures.

Findings

Optimal control fields expressed via elliptic integrals.

Bifurcation in control structure at a specific offset.

Optimal solutions include concatenations of extremals.

Abstract

We investigate the time-optimal solution of the selective control of two uncoupled spin 1/2 particles. Using the Pontryagin Maximum Principle, we derive the global time-optimal pulses for two spins with different offsets. We show that the Pontryagin Hamiltonian can be written as a one-dimensional effective Hamiltonian. The optimal fields can be expressed analytically in terms of elliptic integrals. The time-optimal control problem is solved for the selective inversion and excitation processes. A bifurcation in the structure of the control fields occurs for a specific offset threshold. In particular, we show that for small offsets, the optimal solution is the concatenation of regular and singular extremals.