We study the non-equilibrium dynamics of a zero temperature Mott insulator-superfluid quantum phase transition in a lattice of weakly coupled Bose-Einstein condensates. We show that crossing the critical point from the insulating to the superfluid phase at a finite rate results in a finite dispersion of current. Viceversa, crossing the critical point from the superfluid phase to the insulating one, results in a finite dispersion in the number of atoms per lattice site. We develop a microscopic dynamical model of the transition, and make quantitative predictions for realistic experiments.
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