Self-energy of cold atoms in a long-range disordered optical potential

We study the effect of correlation on the expansion of a Bose---Einstein condensate (BEC) released from a harmonic trap. We go beyond the first-order Born approximation (FBA) to use the self-consistent Born approximation (SCBA) to calculate the self-energy $$\Sigma (\varepsilon , k)$$Σ(?,k) of an atom in a speckle potential with constant amplitude of disorder. For very cold atoms ($$k= 0$$k=0), low chemical potential $$\mu \ll \varepsilon _{\xi }$$μ???, and $$U/ \varepsilon ^{2}_{\xi } = 1$$U/??2=1, the self-energy spectrum is wide when calculated in the SCBA compared with the FBA. The SCBA locates the band edge at somewhat higher energy, giving rise to many localized atoms. We focus mainly on the energy distribution at $$k = 0$$k=0 as expressed by the spectral function. Our numerical results show that the spectral function of the expanding atoms as calculated in the FBA at $$\varepsilon = 0$$?=0 has low weight and extends up to $$\varepsilon = 4 \varepsilon _{\xi }$$?=4??. When using the SCBA, the behavior of the energy distribution for low chemical potential is different, showing a continuum, while a large weight corresponding to $$\varepsilon $$? shifts the spectrum of negative energy values to $$\frac{\varepsilon }{\varepsilon _\xi } = -0.78$$???=-0.78 and the energy distribution is near unity. We show that the form of the density of states locates the mobility near $$\varepsilon = - \varepsilon _\xi $$?=-??. At $$\varepsilon = 0$$?=0, we obtain 35 % of delocalized atoms. We conclude that correlation disorder seems to help localize atoms with energy below zero. In particular, we show that the negative energy values observed in the energy spectrum imply that the probability of finding an atom with energy $$\varepsilon _\xi $$?? around $$\varepsilon $$? is conserved.