Josephson-type oscillations in spin-orbit coupled Bose-Einstein condensates with nonlinear optical lattices

We consider spin-orbit coupled Bose Einstein Condensate in presence of linear and nonlinear optical lattices within the framework of quasi-one-dimensional Gross-Pitaevskii equation. The population imbalance between the states changes periodically with time and the oscillation amplitude depends sensitively on the initial phase. The optical lattice is found to change phase velocity of the oscillation of population imbalance. This oscillation can also be arrested beyond critical values of parameters. We find that the optical lattice can efficiently be used to control the critical point.