  Quantum cascade lasers can be modeled within a hierarchy of different
approaches: Standard rate equations for the electron densities in the levels,
semiclassical Boltzmann equation for the microscopic distribution functions,
and quantum kinetics including the coherent evolution between the states. Here
we present a quantum transport approach based on nonequilibrium Green
functions. This allows for quantitative simulations of the transport and
optical gain of the device. The division of the current density in two terms
shows that semiclassical transitions are likely to dominate the transport for
the prototype device of Sirtori et al. but not for a recent THz-laser with only
a few layers per period. The many particle effects are extremely dependent on
the design of the heterostructure, and for the case considered here, inclusion
of electron-electron interaction at the Hartree Fock level, provides a sizable
change in absorption but imparts only a minor shift of the gain peak.
