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

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.