  The link between the inversion gain of quantum cascade structures and the
Bloch gain in periodic superlattices is presented. The proposed theoretical
model based on the density matrix formalism is able to treat the gain mechanism
of the Bloch oscillator and Quantum cascade laser on the same footing by taking
into account in-plane momentum relaxation. The model predicts a dispersive
contribution in addition to the (usual) population-inversion-dependent
intersubband gain in quantum cascade structures and - in the absence of
inversion - provides the quantum mechanical description for the dispersive gain
in superlattices. It corroborates the predictions of the semi-classical
miniband picture, according to which gain is predicted for photon energies
lower than the Bloch oscillation frequency, whereas net absorption is expected
at higher photon energies, as a description which is valid in the
high-temperature limit. A red-shift of the amplified emission with respect to
the resonant transition energy results from the dispersive gain contribution in
any intersubband transition, for which the population inversion is small.
