  We study the ground state for many interacting bosons in a double-well
potential, in a joint limit where the particle number and the distance between
the potential wells both go to infinity. Two single-particle orbitals (one for
each well) are macroscopically occupied, and we are concerned with deriving the
corresponding effective Bose-Hubbard Hamiltonian. We prove (i) an energy
expansion, including the two-modes Bose-Hubbard energy and two independent
Bogoliubov corrections (one for each potential well), (ii) a variance bound for
the number of particles falling inside each potential well. The latter is a
signature of a correlated ground state in that it violates the central limit
theorem.
