  We investigate a model system for the injection of fermionic particles from
filled source sites into an empty chain. We study the ensuing dynamics for
Hermitian as well as for non-Hermitian time evolution where the particles
cannot return to the bath sites (quantum ratchet). A non-homogeneous
hybridization between bath and chain sites permits transient currents in the
chain. Non-interacting particles show decoherence in the thermodynamic limit:
the average particle number and the average current density in the chain become
stationary for long times, whereas the single-particle density matrix displays
large fluctuations around its mean value. Using the numerical time-dependent
density-matrix renormalization group ($t$-DMRG) method we demonstrate, on the
other hand, that sizable density-density interactions between the particles
introduce relaxation which is by orders of magnitudes faster than the
decoherence processes.
