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