  We study the quantum dynamics of a spin-orbit (SO) coupled Bose-Einstein
condensate (BEC) in a double-well potential inspired by the experimental
protocol recently developed by NIST group. We focus on the regime where the
number of atoms is very large and perform a two-mode approximation. An
analytical solution of the two-site Bose-Hubbard-like Hamiltonian is found for
several limiting cases, which range from a strong Raman coupling to a strong
Josephson coupling, ending with the complete model in the presence of weak
nonlinear interactions. Depending on the particular limit, different approaches
are chosen: a mapping onto an SU(2) spin problem together with a
Holstein-Primakoff transformation in the first two cases and a rotating wave
approximation (RWA) when dealing with the complete model. The quantum evolution
of the number difference of bosons with equal or different spin between the two
wells is investigated in a wide range of parameters; finally the corresponding
total atomic current and the spin current are computed. We show a spin
Josephson effect which could be detected in experiments and employed to build
up realistic devices.
