Observable signature of the Berezinskii-Kosterlitz-Thouless transition in a planar lattice of Bose-Einstein condensates

We investigate the possibility that Bose-Einstein condensates, loaded on a 2D optical lattice, undergo--at finite temperature--a Berezinskii-Kosterlitz-Thouless transition. We show that--in an experimentally attainable range of parameters--a planar lattice of Bose-Einstein condensates 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 Berezinskii-Kosterlitz-Thouless transition by showing that, near the critical temperature, the \skew3\vec{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 in this paper, and analogies with superconducting Josephson junction networks are stressed throughout the text.