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

Bosons in a double well: two-mode approximation and fluctuations