Weakly interacting Bose-Einstein condensates under rotation

We investigate the ground and low excited states of a rotating, weakly interacting Bose-Einstein condensed gas in a harmonic trap for a given angular momentum. Analytical results in various limits as well as numerical results are presented, and these are compared with those of previous studies. Within the mean-field approximation and fur repulsive interaction between the atoms, we find that for very low values of the total angular momentum per particle, L/N->0, where L (h) over bar is the angular momentum and N is the total number of particles, the angular momentum is carried by quadrupole (\m\=2) surface modes. For L/N=1, a vortexlike state is formed and all the atoms occupy the m = 1 state. For small negative values of L/N-1, the states with m = 0 and m = 2 become populated, and for small positive values of L/N-1, atoms in the states with m = 5 and m = 6 carry the additional angular momentum. In the whole region 0 less than or equal toL/N less than or equal to 1, we have strong analytic and numerical evidence that the interaction energy drops linearly as a function of LIN. We have also found that an array of singly quantized vortices is formed as L/N increases. Finally, we have gone beyond the mean-field approximation and have calculated the energy of the lowest state up to order N for small negative values of L/N-1, as well as the energy of the low-lying excited states.