Small-tunneling-amplitude boson-Hubbard dimer. II. Dynamics

We present analytical relations for the quantum evolution of the number difference of bosons between the two sites of a double-well potential, by using perturbative results for small tunneling amplitudes in the two-mode approximation. Results are obtained for two different initial conditions: completely localized states and coherent spin states. In the former case both the short and the long time behavior is investigated and the characteristic Bohr frequencies of the energy spectrum are determined. In the latter case we calculate the short time-scale evolution of the number difference. The analytical expressions compare favorably with direct numerical solutions of the quantum problem. Finally, we discuss the corresponding Gross-Pitaevskii (i.e., mean-field) dynamics and we point out the differences between the full quantum evolution and the mean-field evolution.