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, $L/N \to 0$, where $L \hbar$ is the angular momentum and $N$ is the total number of particles, the angular momentum is carried by quadrupolar ($|m| = 2$) surface modes. For $L/N =1$ a vortex-like 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 $0 \le L/N \le 1$ we have strong analytic and numerical evidence that the interaction energy drops linearly as a function of $L/N$. 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.