Quantum density matrix theory for a laser without adiabatic elimination of the population inversion: transition to lasing in the class-B limit

TL;DR

This paper develops a quantum density-matrix model for class-B lasers that accurately describes coherence and photon correlations, enabling analysis of quantum effects and dynamics in nanolaser systems without adiabatic elimination.

Contribution

It introduces a novel, simplified density-matrix theoretical framework for class-B lasers that captures key quantum phenomena and is computationally straightforward.

Findings

Model reproduces photon antibunching and super-Poissonian statistics.

Predicts damping of relaxation oscillations at small atom numbers (~10).

Enables study of quantum correlations in nanolaser arrays.

Abstract

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 we carry out a density-matrix theoretical approach for generic class-B lasers, and provide closed equations 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 ($1\leq g^{(2)}(0)\leq 2$), and from few photons (large spontaneous emission factors, $\beta\sim1$) to the thermodynamic limit ($N\gg1$ and $\beta\sim 0$). Based on the analysis of $g^{(2)}(\tau)$, we conclude that super-Poissonian fluctuations are clearly related to relaxation oscillations in the photon number. We predict a strong damping of relaxation oscillations with an atom number as small as $N\sim 10$. This model enables the study of few-photon bifurcations and non-classical photon correlations in class-B laser devices, also leveraging quantum descriptions of coherently coupled nanolaser arrays.