  We discuss the relaxation dynamics for a bosonic tunneling junction with two
modes in the central potential well. We use a master equation description for
ultracold bosons tunneling in the presence of noise and incoherent coupling
processes into the two central modes. Whilst we cannot quantitatively reproduce
the experimental data of the setup reported in [Phys. Rev. Lett. {\bf 115},
050601 (2015)], we find a reasonable qualitative agreement of the refilling
process of the initially depleted central site. Our results may pave the way
for the control of bosonic tunneling junctions by the simultaneous presence of
decoherence processes and atom-atom interaction.
