  It is well-known that a rotating Bose-Einstein condensate forms vortices to
carry the angular momentum. For a first vortex to nucleate at the trap center,
the rotational frequency must become larger than a certain critical value. The
vortex nucleation process, however, is sensitive to the trap shape. It was
shown earlier that for a symmetry-breaking potential that preserves parity, at
criticality the leading natural orbitals may become degenerate, giving rise to
a maximally entangled quantum state, found from exact solutions for just a few
bosons. Developing an effective two-state model, we here show that in the limit
of large particle numbers, the many-body ground state becomes either a
so-called twin-like or a Schr\"odinger cat-like state. We corroborate this
finding by a direct comparison to the exact numerical solution of the problem,
feasible for moderate particle numbers within the lowest Landau level
approximation. We show that the nature of the quantum state at criticality can
be controlled by both the quadrupolar deformation and the flatness of the
confining potential.
