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We use the Gross-Pitaevskii equation to study Josephson tunneling between twoweakly coupled Bose-Einstein condensates, which compose spin-1 bosons. We showthat a rotating magnetic field on one side can produce a phase differenceacross the junction, resulting in an oscillatory tunneling spin current.Besides numerical calculation, we derive analytical results in two extremecases, namely the low- and high-frequency limits: in the low-frequency limit(magnetic field rotates adiabatically), a non-Abelian geometric phase arisesand leads to the oscillatory spin current. By sharp contrast, the physics isintrinsically different in the high-frequency limit, where an average Zeemanenergy difference leads to an oscillatory spin current. This proposed apparatusshould be promising for the future experimental search of spin superfluidity.

Geometric phase driven Josephson junction: Possible experimental scheme for the search of spin superfluidity