We study the evolution of density in an expanding Bose-Einstein condensate that initially has a spatially varying phase, concentrating on behavior when these phase variations are large. In this regime large density fluctuations develop during expansion. Maxima have a characteristic density that diverges with the amplitude of phase variations and their formation is analogous to that of caustics in geometrical optics. We analyze in detail caustic formation in a quasi-one-dimensional condensate, which before expansion is subject to a periodic or random optical potential, and we discuss the equivalent problem for a quasi-two-dimensional system. We also examine the influence of many-body correlations in the initial state on caustic formation for a Bose gas expanding from a strictly one-dimensional trap. In addition, we study a similar arrangement for noninteracting fermions, showing that Fermi surface discontinuities in the momentum distribution give rise in that case to sharp peaks in the spatial derivative of the density. We discuss recent experiments and argue that fringes reported in time of flight images [Chen et al., Phys. Rev. A 77, 033632 (2008)] are an example of caustic formation.
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