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A typical form of V ( x ) is shown in Fig. 1. The width of each double well is λ . The phase- differences δθ = θ$_{z}$ − θ$_{xy}$ and δφ = φ$_{z}$ − φ$_{xy}$ control the tilt of the optical lattice and Z$_{f}$ controls the barrier height of the double-well. 

The eigenfunctions of the lattice can be calculated using Fourier expansion method, which involves expansion of the eigenfunction in terms of its Fourier components as follows: 

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The normalization assures that ∑ n | d$_{n}$ | 2 = 1. We here consider only the states with zero quasi-momentum q = 0. To solve for the eigenfunction of the Hamiltonian H = ( p 2 $_{x}$/ 2 m ) + 

