  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.
