  A quantum system exhibiting $\mathcal{PT}$ symmetry is a Bose-Einstein
condensate in a double-well potential with balanced particle gain and loss,
which is described in the mean-field limit by a Gross-Pitaevskii equation with
a complex potential. A possible experimental realization of such a system by
embedding it into a Hermitian time-dependent four-mode optical lattice was
proposed by Kreibich et al. [Phys. Rev. A 87, 051601(R) (2013)], where
additional potential wells act as reservoirs and particle exchange happens via
tunneling. Since particle influx and outflux have to be controlled explicitly,
a set of conditions on the potential parameters had to be derived. In contrast
to previous work, our focus lies on a full many-body description beyond the
mean-field approximation using a Bose-Hubbard model with time-dependent
potentials. This gives rise to novel quantum effects, such that the differences
between mean-field and many-body dynamics are of special interest. We further
present stationary analytical solutions for the embedded wells in the
mean-field limit, different approaches for the embedding into a many-body
system, and a very efficient method for the evaluation of hopping terms to
calculate exact Bose-Hubbard dynamics.
