-1 -0.5 0 0.5 1 1.5 2 x/ -80 -70 -60 -50 -40 -30 V(x) 0 0.2 0.4 0.6 0.8 1 time (sec) 0 0.2 0.4 0.6 0.8 1 Probability FIG. 1: Structure of the double-well V ( x ) in coordinate space for Z$_{f}$ = 0 . 05 (solid line) and Z$_{f}$ = 0 . 1 (dashed line). The other parameters are V$_{xy}$ = 36 E$_{R}$ , δθ = π/ 2, δφ = 0, and E$_{R}$ = 3 . 5 kHz. Clearly, Z$_{f}$ = 0 . 05 represents higher barrier. FIG. 2: Temporal variation of the probability that a superposition state | L 〉 remains in the initial left wells. The parameters are Z$_{f}$ = 0 . 05 (solid line) and Z$_{f}$ = 0 . 1 (dotted line). The other parameters are V$_{xy}$ = 36 E$_{R}$ , δθ = π/ 2, δφ = 0, and E$_{R}$ = 3 . 5 kHz. 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: 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 ) + φ ( x ) = ∑ n d$_{n}$e 2 πinkx . (4) 4