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be random. Further, because we consider an one-dimensional optical lattice in x direction, we consider the components of the recoil momentum only at x direction. Thus in spherical coordinate, the random phase can be written as exp[ − ik ′ x sin( θ ) cos( φ )], where k ′ = 2 π/λ ′ , and θ and φ represent the random direction of the outgoing photon. We write λ ′ = λ/m , where m defines the strength of the phase kick. Because, the wavelength of the standing wave in an optical lattice has to be much larger than the atomic wavelength and the detuning 

