  We consider a bosonic Josephson junction made of $N$ ultracold and dilute
atoms confined by a quasi one-dimensional double-well potential within the
two-site Bose-Hubbard model framework. The behaviour of the system is
investigated at zero temperature by varying the inter-atomic interaction from
the strongly attractive regime to the repulsive one. We show that the
ground-state exhibits a crossover from a macroscopic Schr\"odinger-cat state to
a separable Fock state through an atomic coherent regime. By diagonalizing the
Bose-Hubbard Hamiltonian we characterize the emergence of the mascroscopic cat
states by calculating the Fisher information $F$, the coherence by means of the
visibility $\alpha$ of the interference fringes in the momentum distribution,
and the quantum correlations by using the entanglement entropy $S$. Both Fisher
information and visibility are shown to be related to the ground state energy
by employing the Hellmann-Feynman theorem. This result, together with a
perturbative calculation of the ground-state energy, makes possible to obtain
simple analytical formulas for $F$ and $\alpha$ over a range of interactions,
in excellent agreement with the exact diagonalization of the Bose-Hubbard
Hamiltonian. In the attractive regime the entanglement entropy attains values
very close to its upper limit for a specific interaction strength lying in the
region where coherence is lost and self trapping sets in.
