  We study the interpolation from occupation number Fock states to
Schr\"odinger cat states on systems modeled by two-mode Bose-Hubbard
Hamiltonian, like, for instance, bosons in a double well or superconducting
Cooper pair boxes. In the repulsive interaction regime, by a simplified single
particle description, we calculate, analytically, energy, number fluctuations,
stability under coupling to a heat bath, entanglement entropy and Fisher
information, all in terms of hypergeometric polynomials of the single particle
overlap parameter. Our approach allows us to find how those quantities scale
with the number of bosons. In the attractive interaction regime we calculate
the same physical quantities in terms of the imbalance parameter, and find that
the symmetry breaking, occurring at interaction Uc, predicted by a
semiclassical approximation, is valid only in the limit of infinite number of
bosons. For a large but finite number, we determine a characteristic strength
of interaction, Uc*, which can be promoted as the crossover point from coherent
to incoherent regimes and can be identified as the collapse threshold.
Moreover, we find that the Fisher information is always in direct ratio to the
variance of on-site number of bosons, for both positive and negative
interactions. We finally show that the entanglement entropy is maximum close to
Uc* and exceeds its coherent value within the whole range of interaction
between 2Uc and zero.
