Quantum model of gain in phonon-polariton lasers

We develop a quantum model for the calculation of the gain of phonon-polariton intersubband lasers. The polaritonic gain arizes from the interaction between the electrons confined in a quantum well structure and the phonon confined in one layer of the material. Our theoretical approach is based on expressing the crucial matter excitations (intersubband electrons and phonons) in terms of polarisation densities in second quantisation, and treating all non-resonant polarisations with an effective dielectric function. The interaction between the electronic and phononic polarizations is treated perturbatively, and gives rise to stimulated emission of polartions in the case of inverted subbands. Our model provides a complete physical insight of the system and allows to determine the phonon and photon fraction of the laser gain. Moreover, it can be applied and extended to any type of designs and material systems, offering a wide set of possibilities for the optimization of future phonon-polariton lasers.