  We report a theoretical study of the carrier relaxation in a quantum cascade
laser (QCL) subjected to a strong magnetic field. Both the alloy (GaInAs)
disorder effects and the Frohlich interaction are taken into account when the
electron energy differences are tuned to the longitudinal optical (LO) phonon
energy. In the weak electron-phonon coupling regime, a Fermi's golden rule
computation of LO phonon scattering rates shows a very fast non-radiative
relaxation channel for the alloy broadened Landau levels (LL's). In the strong
electron-phonon coupling regime, we use a magneto-polaron formalism and compute
the electron survival probabilities in the upper LL's with including increasing
numbers of LO phonon modes for a large number of alloy disorder configurations.
Our results predict a nonexponential decay of the upper level population once
electrons are injected in this state.
