Spin-orbit coupled Bose-Einstein condensates in a double well

We study the quantum dynamics of spin-orbit (SO) coupled Bose-Einstein condensates (BECs) in a double-well potential inspired by the recent experiment 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 weak nonlinear interactions by a rotating wave approximation (RWA). The quantum dynamics of several observables, ranging from the population and pseudospin imbalance, to the atomic and spin current are explicitly computed. We show a spin Josephson effect which could be detected in experiments and employed in realistic devices.