Time-optimal synthesis of SU(2) transformations for a spin-1/2 system

TL;DR

This paper derives analytical solutions for the fastest possible control protocols to implement specific SU(2) transformations in a spin-1/2 system with constrained magnetic field directions, including generalizations to biased and inhomogeneous controls.

Contribution

It provides the first explicit analytical solutions for time-optimal control of SU(2) transformations with directional constraints on the magnetic field.

Findings

Time-optimal solutions have a simple geometric interpretation.

Solutions are generalized to include bias fields and inhomogeneous control.

Analytical formulas for minimal-time control protocols are derived.

Abstract

We consider a quantum control problem involving a spin-1/2 particle in a magnetic field. The magnitude of the field is held constant, and the direction of the field, which is constrained to lie in the x-y plane, serves as a control parameter that can be varied to govern the evolution of the system. We analytically solve for the time dependence of the control parameter that will synthesize a given target SU(2) transformation in the least possible amount of time, and we show that the time-optimal solutions have a simple geometric interpretation in terms of the fiber bundle structure of SU(2). We also generalize our time-optimal solutions to a control problem that includes a constant bias field along the z axis, and to the case of inhomogeneous control, in which a single control parameter governs the evolution of an ensemble of spin-1/2 systems.