Bose-Einstein Condensates with Large Number of Vortices

We show that as the number of vortices in a three dimensional Bose-Einstein condensate increases, the system reaches a ``quantum Hall'' regime where the density profile is a Gaussian in the xy plane and an inverted parabolic profile along z. The angular momentum of the system increases as the vortex lattice shrinks. However, Coriolis force prevents the unit cell of the vortex lattice from shrinking beyond a minimum size. Although the recent MIT experiment is not exactly in the quantum Hall regime, it is close enough for the present results to be used as a guide. The quantum Hall regime can be easily reached by moderate changes of the current experimental parameters.