Dynamics of exciton-polaritons in a Josephson double dimer

We study the dynamics of exciton-polaritons in a double-well configuration. The system consists of two weakly coupled Bose-Josephson junctions, each corresponding to a different circular polarization of the polaritons, forming a {\it Josephson double dimer}. We show that the Josephson oscillation between the wells is strongly coupled to the polarization rotation and that consequently Josephson excitation is periodically exchanged between the two polarizations. Linearized analysis agrees well with numerical simulations using typical experimental parameters.