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

Coherence and entanglement in the ground-state of a bosonic Josephson junction:from macroscopic Schr\"odinger cats to separable Fock states