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 for repulsive interaction between the atoms, we find that for very low values of the total angular momentum per particle, $L/\overrightarrow{N}0,$ where $Lħ$ 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 $m=1\mathrm{}$ state. For small negative values of $L/N-1,$ the states with $m=0\mathrm{}$ and $m=2\mathrm{}$ become populated, and for small positive values of $L/N-1,$ atoms in the states with $m=5\mathrm{}$ and $m=6\mathrm{}$ carry the additional angular momentum. In the whole region $0<~L/N<~1,$ we have strong analytic and numerical evidence that the interaction energy drops linearly as a function of $L/N.\mathrm{}$ 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.