  Fast quantum spin manipulation is needed to design spin-based quantum logic
gates and other quantum applications. Here, we construct exact evolution
operator of the nitrogen-vacancy-center (NV) spin in diamond under external
magnetic fields and investigate the nonadiabatic geometric phases, both cyclic
and non-cyclic, in these fast-manipulated NV spin systems. It is believed that
the nonadiabatic geometric phases can be measured in future experiments and
these fast quantum manipulations can be useful in designing spin-based quantum
applications.
