The structure of the paper is as follows. In Sec. II, we describe the double-well optical lattice. We discuss how different external initial states of the atoms trapped in the lattice evolve with time. In Sec. III, we investigate the effect of the decoherence. We discuss the control mechanism by choice of initial states. We conclude the paper in Sec. IV. We start with a double-well optical lattice as described in [6]. A single laser beam initially propagating in the x -direction intersects with itself at the position of the cold atom four times through reflection in suitably positioned mirrors. This prepares two pairs of counter-propagating beams in x and y direction. One can have 2-dimensional square lattice if the polarization of the beams are in x − y plane (”in-plane lattice”) or in perpendicular plane (parallel to z -axis; ”out-of-plane” lattice). The double-well lattice in x − y plane can be generated by combining the ”in-plane” and ”out-of-plane” polarizations. The two- dimensional lattice potential thus constructed can be written as II. TUNNELLING IN DOUBLE-WELL OPTICAL LATTICE V ( x, y ) = − V$_{xy}$ 4 [(1 − Z$_{f}$ ) V$_{1}$ ( x, y ) + Z$_{f}$V$_{2}$ ( x, y )] , (1) V$_{1}$ ( x, y ) = 4 + 2 cos(2 kx − 2 θ$_{xy}$ − 2 φ$_{xy}$ ) +2 cos(2 ky + 2 φ$_{xy}$ ) , V$_{2}$ ( x, y ) = 4 + 4 cos( kx + ky − θ$_{z}$ ) +4 cos( kx − ky − θ$_{z}$ − 2 φ$_{z}$ ) (2) +2 cos(2 kx − 2 θ$_{z}$ − 2 φ$_{z}$ ) + 2 cos(2 ky + 2 φ$_{z}$ ) , where V$_{xy}$ is the potential depth, Z$_{f}$ is the ratio of the intensities of the ”in-plane” and ”out-of-plane” lattices, and where k = 2 π/λ , θ$_{xy}$ ( θ$_{z}$ ) and φ$_{xy}$ ( φ$_{z}$ ) are the phases of the in-plane (out-of-plane) lattice. We focus on the dynamics of the atomic states in one dimension, e.g., in x direction. The potential in this direction resembles a double well potential and is given by V ( x, y = 0) = − V$_{xy}$ 4 [6 + 2(1 − Z$_{f}$ ) cos(2 kx − 2 θ$_{xy}$ ) +2 cos(2 kx − 2 θ$_{z}$ ) + 8 cos( kx − θ$_{z}$ )] . (3) 3