  The mean-field approach to two-site Bose-Hubbard systems is well established
and leads to nonlinear classical equations of motion for the population
imbalance and the phase difference. It can, e.g., be based on the
representation of the solution of the time-dependent Schrodinger equation
either by a single Glauber state or by a single atomic (SU(2)) coherent state
[S. Wimberger et al., Phys. Rev. A 103, 023326 (2021)]. We demonstrate that
quantum effects beyond the mean-field approximation are easily uncovered if,
instead, a multi-configuration ansatz with a few time-dependent SU(2) basis
functions is used in the variational principle. For the case of plasma
oscillations, the use of just two basis states, whose time-dependent parameters
are determined variationally, already gives good qualitative agreement of the
phase space dynamics with numerically exact quantum solutions. In order to
correctly account for more non-trivial effects, like macroscopic quantum self
trapping, moderately more basis states are needed. If one is interested in the
onset of spontaneous symmetry breaking, however, a multiplicity of two gives a
big improvement towards the exact result already. In any case, the number of
variational trajectories needed for good agreement with full quantum results is
orders of magnitude smaller than in the semiclassical case, which is based on
multiple mean-field trajectories.
