Fundamental limitations on gain of terahertz quantum cascade lasers

We analyze the main physical processes in quantum cascade lasers with a spatial separation between the region of photon radiation and longitudinal optical (LO) phonon emission, which facilitates the depopulation of the lower level of the optical transition. Our objective is to identify the reasons for the reduction of population inversion at low photon energy and explore methods to enhance it. The expression for population inversion is derived from an equation for a simplified density matrix. This approach allows us to consider the coherence of tunneling between different levels and comprehend its influence on transition probabilities in a straightforward manner. We have found out that the energy uncertainty principle is the fundamental factor limiting population inversion in terahertz lasers. By optimizing the tunneling matrix element between the two regions and the LO phonon emission time, it is possible to significantly increase the population inversion. The optimal value for the matrix element is smaller than its maximum possible value, while the optimal LO phonon emission time exceeds the time achieved during LO phonon resonant emission.