Small-tunneling-amplitude boson-Hubbard dimer: Stationary states

Analytical expressions for the energy eigenstates of the boson-Hubbard (or quantum discrete-nonlinear Schrödinger) dimer are obtained by applying perturbation theory in the small-tunneling-amplitude limit. The results are relevant for a Bose-Einstein condensate trapped in a double-well potential. A detailed comparison with the numerical solutions arising from the direct diagonalization of the Hamiltonian (for even or odd numbers of bosons) is presented, and the limits of validity of the perturbative results are determined. For fixed values of the tunneling amplitude, the accuracy of the perturbative calculation varies with the total number of bosons in different ways, depending on the particular energy level. Results for the fluctuations of the boson-number difference in each eigenstate are also presented.