Quantum Density Matrix Theory for a Laser Without Adiabatic Elimination of the Population Inversion: Transition to Lasing in the Class-B Limit

Despite the enormous technological interest in micro and nanolasers, surprisingly, no class-B quantum density-matrix model is available to date, capable of accurately describing coherence and photon correlations within a unified theory. In class-B lasers--applicable for most solid-state lasers at room temperature--, the macroscopic polarization decay rate is larger than the cavity damping rate which, in turn, exceeds the upper level population decay rate. Here, a density-matrix theoretical approach for generic class-B lasers is carried out, and closed equations are provided for the photonic and atomic reduced density matrix in the Fock basis of photons. Such a relatively simple model can be numerically integrated in a straightforward way, and exhibits all the expected phenomena, from one-atom photon antibunching, to the well-known S-shaped input-output laser emission and super-Poissonian autocorrelation for many atoms ((Formula presented.)), and from few photons (large spontaneous emission factors, (Formula presented.)) to the thermodynamic limit ((Formula presented.) and (Formula presented.)). Based on the analysis of (Formula presented.), it is concluded that super-Poissonian fluctuations are clearly related to relaxation oscillations in the photon number. A strong damping of relaxation oscillations with an atom number as small as (Formula presented.) is predicted. This model enables the study of few-photon bifurcations and nonclassical photon correlations in class-B laser devices, also leveraging quantum descriptions of coherently coupled nanolaser arrays.