  We study the directed transport of bosons along a one dimensional lattice in
a dissipative setting, where the hopping is only facilitated by coupling to a
Markovian reservoir. By combining numerical simulations with a field-theoretic
analysis, we investigate the current fluctuations for this process and
determine its asymptotic behavior. These findings demonstrate that dissipative
bosonic transport belongs to the KPZ universality class and therefore, in spite
of the drastic difference in the underlying particle statistics, it features
the same coarse grained behavior as the corresponding asymmetric simple
exclusion process (ASEP) for fermions. However, crucial differences between the
two processes emerge when focusing on the full counting statistics of current
fluctuations. By mapping both models to the physics of fluctuating interfaces,
we find that dissipative transport of bosons and fermions can be understood as
surface growth and erosion processes, respectively. Within this unified
description, both the similarities and discrepancies between the full counting
statistics of the transport are reconciled. Beyond purely theoretical interest,
these findings are relevant for experiments with cold atoms or long-lived
quasi-particles in nanophotonic lattices, where such transport scenarios can be
realized.
