We study the diffusion of an expanding Bose-Einstein condensate released from a harmonic trap in a three-dimensional speckle disorder potential. To this end, we use the first Born approximation and examine the density profiles at short and long times. Analytical results are presented in different regimes. The spatiotemporal evolution of the density profiles is examined. We find that at long times and for a fixed disorder strength, the profile of the atomic average density decreases in power law. The time evolution of the typical size of the condensate is explored numerically.

This paper aims to study the diffusion and localization of cold atoms in three-dimensional (3D) optical speckles by using the theory of self-energy for N diffusers. In this paper, the Self-consistent Born approximation (SCBA) is used to study the diffusion and localization of cold atoms in an optical disordered speckles, while the spectral function is adopted to study the effect of the matter waves energy and the disorder amplitude on the behaviour of cold atoms. For this purpose, the models of SCBA and spectral function are computed by a numerical algorithm. The calculation of the latter quantity involves the application of Simpson’s integration methods. The diffusion and localization of a Bose–Einstein Condensate for First-order Born Approximation (FBA) and SCBA models are presented. In addition, the diffusion time and the mean free path between two diffusers are illustrated. Moreover, the effect of disorder amplitude and the scattering of matter waves in 3D disordered potentials are highlighted. In this study, the results show that the time response of cold atoms localization and metal–insulator transition in SCBA is faster and yields lower energy than the FBA approximation.