  We investigate the quantum dynamics of an experimentally realized spin-orbit
coupled Bose-Einstein condensate in a double well potential. The spin-orbit
coupling can significantly enhance the atomic inter-well tunneling. We find the
coexistence of internal and external Josephson effects in the system, which are
moreover inherently coupled in a complicated form even in the absence of
interatomic interactions. Moreover, we show that the spin-dependent tunneling
between two wells can induce a net atomic spin current referred as spin
Josephson effects. Such novel spin Josephson effects can be observable for
realistically experimental conditions.
