  A density matrix theory of electron transport and optical gain in quantum
cascade lasers in an external magnetic field is formulated. Starting from the
general quantum kinetic treatment, we describe the intra- and inter-period
electron dynamics at the non-Markovian, Markovian and Boltzmann approximation
levels. Interactions of electrons with longitudinal optical phonons and
classical light field are included in the present description. The
non-Markovian calculation for a prototype structure reveals significantly
different gain spectra in terms of linewidth and additional polaronic features
in comparison to the Markovian and Boltzmann ones. Despite strongly opposed
interpretations of the origin of the transport processes in the non-Markovian
or Markovian and the Boltzmann approaches, they yield comparable values of the
current densities.
