  We study topological transport in the steady state of a quantum particle
hopping on a one-dimensional lattice in the presence of dissipation. The model
exhibits a rich phase structure, with the average particle velocity in the
steady state playing the role of a non-equilibrium order parameter. Within each
phase the average velocity is proportional to a topological winding number and
to the inverse of the average time between quantum jumps. While the average
velocity depends smoothly on system parameters within each phase, nonanalytic
behavior arises at phase transition points. We show that certain types of
spatial boundaries between regions where different phases are realized host a
number of topological bound states which is equal to the difference between the
winding numbers characterizing the phases on the two sides of the boundary.
These topological bound states are attractors for the dynamics; in cases where
the winding number changes by more than one when crossing the boundary, the
subspace of topological bound states forms a dark, decoherence free subspace
for the dissipative system. Finally we discuss how the dynamics we describe can
be realized in a simple cavity or circuit QED setup, where the topological
boundary mode emerges as a robust coherent state of the light field.
