  We use the Gross-Pitaevskii equation to study Josephson tunneling between two
weakly coupled Bose-Einstein condensates, which compose spin-1 bosons. We show
that a rotating magnetic field on one side can produce a phase difference
across the junction, resulting in an oscillatory tunneling spin current.
Besides numerical calculation, we derive analytical results in two extreme
cases, namely the low- and high-frequency limits: in the low-frequency limit
(magnetic field rotates adiabatically), a non-Abelian geometric phase arises
and leads to the oscillatory spin current. By sharp contrast, the physics is
intrinsically different in the high-frequency limit, where an average Zeeman
energy difference leads to an oscillatory spin current. This proposed apparatus
should be promising for the future experimental search of spin superfluidity.
