Revisiting the Siegert relation for the partially coherent regime of nanolasers

TL;DR

This paper critically assesses the generalized Siegert relation's validity in nanolasers, deriving a quantum-optical theory to evaluate its accuracy across different regimes, especially in the quantum and superradiant regimes.

Contribution

It provides a full quantum-optical framework to evaluate the Siegert relation's applicability in various nanolaser regimes, highlighting its limitations in quantum and superradiant conditions.

Findings

The Siegert relation is a good approximation when emitter correlations are negligible.

It does not hold well in the quantum regime of few-emitter nanolasers.

The relation fails in devices with sub- and superradiant emission.

Abstract

The Siegert relation connects the first- and second-order coherence properties of light. While strictly valid only in the thermal regime and in the absence of correlations, this relation is routinely extended to the partially coherent regime in the study of high-$\beta$ nanolasers, where it aids in the identification of the lasing threshold. We re-evaluate the use of a generalized Siegert relation in different device regimes. A full two-time quantum-optical theory is derived as a reference for obtaining first- and second-order correlation functions $g^{(1)}(\tau)$ and $g^{(2)}(\tau)$ in the steady state. We find that even in the partially coherent regime the generalized Siegert relation is well suited as an approximation to $g^{(2)}(\tau)$ as long as emitter correlation are negligible, but does not apply well in the quantum regime of few-emitter nanolasers, or to devices featuring sub- and superradiant emission.