  We employ the perturbation series expansion for derivation of the reduced
master equations for the three-site Bose-Hubbard model subject to strong atom
losses from the central site. The model describes a condensate trapped in a
triple-well potential subject to externally controlled removal of atoms. We
find that the $\pi$-phase state of the coherent superposition between the side
wells decays via two dissipation channels, the single-boson channel (similar to
the externally applied dissipation) and the boson-pair channel. The quantum
derivation is compared to the classical adiabatic elimination within the
mean-field approximation. We find that the boson-pair dissipation channel is
not captured by the mean-field model, whereas the single-boson channel is
described by it. Moreover, there is a matching condition between the zero-point
energy bias of the side wells and the nonlinear interaction parameter which
separates the regions where either the single-boson or the boson-pair
dissipation channel dominate. Our results indicate that the $M$-site
Bose-Hubbard models, for $M>2$, subject to atom losses may require an analysis
which goes beyond the usual mean-field approximation for correct description of
their dissipative features. This is an important result in view of the recent
experimental works on the single site addressability of condensates trapped in
optical lattices.
