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We consider spin-orbit coupled Bose Einstein Condensate in presence of linearand nonlinear optical lattices within the framework of quasi-one-dimensionalGross-Pitaevskii equation. The population imbalance between the states changesperiodically with time and the oscillation amplitude depends sensitively on theinitial phase. The optical lattice is found to change phase velocity of theoscillation of population imbalance. This oscillation can also be arrestedbeyond critical values of parameters. We find that the optical lattice canefficiently be used to control the critical point.

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