Imaging the Phase of an Evolving Bose-Einstein Condensate Wave Function

We demonstrate a spatially resolved autocorrelation measurement with a Bose-Einstein condensate and measure the evolution of the spatial profile of its quantum mechanical phase. Upon release of the condensate from the magnetic trap, its phase develops a form that we measure to be quadratic in the spatial coordinate. Our experiments also reveal the effects of the repulsive interaction between two overlapping condensate wave packets and we measure the small momentum they impart to each other.