  With this work we investigate the stationary nonequilibrium density matrix of
current carrying nonequilibrium steady states of in-between quantum systems
that are connected to reservoirs. We describe the analytical procedure to
obtain the explicit result for the reduced density matrix of quantum transport
when the system, the connecting reservoirs and, as well, the system-reservoir
interactions are described by quadratic Hamiltonians. Our procedure is detailed
for both, electronic transport described by the tight-binding Hamiltonian and
for phonon transport described by harmonic Hamiltonians. For the special case
of weak system-reservoir couplings, a more detailed description of the
steady-state density matrix is obtained. Several paradigm transport setups for
inter-electrode electron transport and low-dimensional phonon heat flux are
elucidated.
