Parafermionic states in rotating Bose-Einstein condensates

Rotating Bose-Einstein condensates in a trap are the place of a very rich physics. It has been predicted that, under appropriate conditions, they will behave like two dimensional electron systems in the fractional quantum Hall effect regime. In addition to the usual fractions, more exotic phases have also been predicted at filling factor $\nu=k/2$. These parafermionic states are described by the Read-Rezayi (RR) wave functions. We study how the system size and interaction act on the overlap between the true ground state and corresponding RR state. The quasihole excitations of the RR states are known to obey non-Abelian statistics. We numerically evaluate the degeneracy of these states and show it is in agreement with a formula given by E. Ardonne. We compute overlap between low-energy true eigenstates and quasihole ground states, and discuss in which cases such description is valid.