Observable Signature of the Berezinskii-Kosterlitz-Thouless Transition in a Planar Lattice of Bose-Einstein Condensates

We investigate the possibility that Bose-Einstein condensates (BECs), loaded on a 2D optical lattice, undergo - at finite temperature - a Berezinskii-Kosterlitz-Thouless (BKT) transition. We show that - in an experimentally attainable range of parameters - a planar lattice of BECs is described by the XY model at finite temperature. We demonstrate that the interference pattern of the expanding condensates provides the experimental signature of the BKT transition by showing that, near the critical temperature, the k=0 component of the momentum distribution and the central peak of the atomic density profile sharply decrease. The finite-temperature transition for a 3D optical lattice is also discussed, and the analogies with superconducting Josephson junction networks are stressed through the text.