Rotating Bose-Einstein condensates at the phase transition point

Here we give analytic description of the phase transitions in the rotating Bose-Einstein condensate of weakly interacting atoms in a magnetic trap [1,2]. The analytic solution allows one to classify the instabilities in the condensates which occur when the rotational speed is increased [3]. In the case of predominantly repulsive interactions, the transition corresponds to the vortex entry the condensate. The transition to the Abrikosov state has follows if the rotational speed is increased further. In the case of predominantly attractive interactions, the transition corresponds to the escape of the condensate from the trap at the critical speed. [1] O. K. Vorov, P. Van Isacker, M. S. Hussein and K. Bartschat, Phys. Rev. Lett. 95, 230406 (2005). [2] O. K. Vorov, M. S. Hussein and P. Van Isacker, Phys. Rev. Lett. 90, 200402 (2003). [3] O. K. Vorov, P. Van Isacker, M. S. Hussein and K. Bartschat, to be submitted to Nature (2007). 1supported by NSF (USA), CEA (France)