Analytical approach to the two-site Bose-Hubbard model: from Fock states to Schr\"odinger cat states and entanglement entropy

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.