  We present an algorithm which combines the quantum trajectory approach to
open quantum systems with a density-matrix renormalization group scheme for
infinite one-dimensional lattice systems. We apply this method to investigate
the long-time dynamics in the Bose-Hubbard model with local particle loss
starting from a Mott-insulating initial state with one boson per site. While
the short-time dynamics can be described even quantitatively by an equation of
motion (EOM) approach at the mean-field level, many-body interactions lead to
unexpected effects at intermediate and long times: local particle currents far
away from the dissipative site start to reverse direction ultimately leading to
a metastable state with a total particle current pointing away from the lossy
site. An alternative EOM approach based on an effective fermion model shows
that the reversal of currents can be understood qualitatively by the creation
of holon-doublon pairs at the edge of the region of reduced particle density.
The doublons are then able to escape while the holes move towards the
dissipative site, a process reminiscent---in a loose sense---of Hawking
radiation.
