Second-order optical coherence ($g^{(2)}(\tau )$)

For a source whose counting correlations are stationary, define the dimensionless correlationAssume finite moments and nonzero denominators. The numerator is in normal ordering: two detections remove two photons. The denominator is the product of the two single-count factors, so independent counts have reference value one. This is a diagonal restriction of the four-point second-order field correlation, rather than its complete definition.^[ref]

For one selected mode with number operator $N=a^{†}a$ and $\mu =⟨N⟩>0$, evaluate both detections at the same time. The canonical commutation relation $a{a}^{†}=a^{†}a+1$ implies $N^2=(a^{†}{)}^2{a}^2+N$, henceThe factorial moment counts ordered pairs of distinct photons. Using $⟨N^2⟩$ would include pairing each photon with itself and would give the wrong coincidence prediction. At $\mu =0$ this normalized expression is undefined, rather than zero.