Parafermionic states in rotating Bose-Einstein condensates

We investigate some properties of the parafermionic states, which may occur in rapidly rotating ultracold bosonic atomic gases at lowest Landau level filling factor ν=k∕2 (with k integer ⩾2). These states have manifolds of quasihole states, which are expected to display non-Abelian statistics. We use exact diagonalizations in the spherical geometry to investigate the quasihole manifolds for quantum Hall states at ν=1, 3∕2, 2, and 5∕2. We show that there is a perfect agreement between numerical findings and the prediction of conformal field theory concerning the number of orbital angular momentum multiplets as a function of added vorticity. The overlap between quasihole candidate wave functions and the corresponding states for realistic delta function interactions between bosons is computed. This allows us to discuss the possible relevance of non-Abelian degeneracy in the physical situation of ultracold atoms.