  We develop simple density-matrix models to describe the role of coherence in
resonant-tunneling (RT) transport of quantum-cascade lasers (QCLs).
Specifically, we investigate the effects of coherent coupling between the
lasing levels with other levels on the transport properties and gain spectra.
In the first part of the paper, we use a three-level density-matrix model to
obtain useful analytical expressions for current transport through the injector
barrier in a QCL. An expression for the slope discontinuity in the
current-voltage characteristics at the lasing threshold is derived. This value
is shown to be a direct measure of the population inversion at threshold, and
contradicts the previously held belief of it being indicative of ratio of the
laser level lifetimes. In the second part of the paper, we use density matrices
to compute the gain spectrum for a resonant-phonon terahertz QCL design. The
large anticrossing of the doublet of lower radiative levels is reflected in a
broad gain linewidth due to a coherent RT assisted depopulation process. At
certain bias conditions, the gain spectrum exhibits double peaks which is
supported by experimental observations.
