V ( x ), where p$_{x}$ is the momentum of the atom of mass m in the lattice, we use the above expression of φ ( x ) in the eigenvalue equation Hφ ( x ) = Eφ ( x ), to obtain a set of linear algebraic equations of d$_{n}$ ’s. The eigenvalues of the corresponding matrices are the same as those of H . We found that the Hamiltonian H has two near-degenerate lowest energy wave functions | 0 〉 and | 1 〉 . We next study the propagation of the wave functions using the Schrodinger equation i ˙ ψ = Hψ . This equation can be solved quite efficiently using split-operator-Fourier transform method. In our numerical simulations, we consider 20 double wells ranging from x = − 9 . 75 λ to x = 10 . 25 λ and a grid size of 512. In the following, we will consider two different initial conditions: (i) Superposition of the eigenstates | 0 〉 and | 1 〉 as | L 〉 = ( | 0 〉 + | 1 〉 ) $_{/}$√$_{2. (ii) Superposition of a few eigenstates}$ including | 0 〉 and | 1 〉 , that can be written as | ψ$_{G}$ ( x ) 〉 = ∑ l c$_{l}$ | l 〉 , where | l 〉 represents the energy eigenfunctions of the Hamiltonian H . We will investigate the evolution of these two initial states. Various eigenstates lead to a quantum interference of the corresponding time- independent amplitudes. We investigate the effect of this interference in the decay rate of the initial coherence of the state. We consider the initial states in such a way that the wavefunction resides in the left wells. We calculate the time-evolution of the probability that the wave function remains in the initial left wells which is given by P$_{L}$ ( t ) = ∑ i ∫ x$_{max,i}$ x$_{min,i}$ dxρ ( x, x ; t ) , (5) where x$_{min,i}$ and x$_{max,i}$ are the lower and upper limits of the i th left well and ρ ( x, x ; t ) represents the diagonal elements of the density matrix of the atom. We show in Fig. 2 the evolution of P$_{L}$ ( t ) for the state | L 〉 . Clearly, the wave function tunnels through the barrier back and forth between the left and the right well. The characteristic time-scale of this tunnelling is found to be ∼ 2 /δ where δ is the energy splitting of the of the states | 0 〉 and | 1 〉 . Note that the tunnelling occurs because of the coherence between the wave function at left and right well. Next, for the state | ψ$_{G}$ ( x ) 〉 , we choose the amplitude coefficients c$_{l}$ for the zero-quasi- momentum eigenstate | l 〉 as c$_{0}$ = − 0 . 6785, c$_{1}$ = − 0 . 677 i , c$_{2}$ = 0 . 0924, c$_{3}$ = 0 . 10107 i , c$_{4}$ = 0 . 15, c$_{5}$ = − 0 . 1602 i , c$_{6}$ = 0 . 0976, c$_{7}$ = 0 . 05777 i , c$_{8}$ = 0 . 0305, and c$_{9}$ = − 0 . 02056 i . The 5