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We present an approximate analytic expression for the photoluminescencespectral function of a model polariton system, which describes a quantum dot,with a finite number of fermionic levels, strongly interacting with the lowestphoton mode of a pillar microcavity. Energy eigenvalues and wavefunctions ofthe electron-hole-photon system are obtained by numerically diagonalizing theHamiltonian. Pumping and photon losses through the cavity mirrors are describedwith a master equation, which is solved in order to determine the stationarydensity matrix. The photon first-order correlation function, from which thespectral function is found, is computed with the help of the Quantum RegressionTheorem. The spectral function qualitatively describes the polariton lasingregime in the model, corresponding to pumping rates two orders of magnitudelower than those needed for ordinary (photon) lasing. The second-ordercoherence functions for the photon and the electron-hole subsystems arecomputed as functions of the pumping rate.

Polariton Lasing in a Multilevel Quantum Dot Strongly Coupled To a Single Photon Mode