Revisiting the Siegert relation for the partially coherent regime of nanolasers

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.