Abstract
An exact theory for the density of a one-dimensional Bose-Einstein condensate with hard-core particle interactions is developed in second quantization and applied to the scattering of the condensate by a spatially periodic impulse potential. The boson problem is mapped onto a system of free fermions obeying the Pauli exclusion principle to facilitate the calculation. The density exhibits a spatial focusing of the probability density as well as a periodic self-imaging in time, or Talbot effect. Furthermore, the transition from single-particle to many-body effects can be measured by observing the decay of the modulated condensate density pattern in time. The connection of these results to classical and atom optical phase gratings is made explicit.
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