= Second-order optical coherence
{title2=$g^{(2)}(\tau)$}
For a source whose counting correlations are stationary, define the dimensionless correlation
$$
g^{(2)}(\tau)=
\frac{\left\langle\widehat E^{(-)}(t)\widehat E^{(-)}(t+\tau)\widehat E^{(+)}(t+\tau)\widehat E^{(+)}(t)\right\rangle}
{\left\langle\widehat E^{(-)}(t)\widehat E^{(+)}(t)\right\rangle
\left\langle\widehat E^{(-)}(t+\tau)\widehat E^{(+)}(t+\tau)\right\rangle}.
$$
Assume 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.<One Hundred Years of Light Quanta by Roy Jay Glauber>[2005 individual Nobel lecture, pp. 87–89, equations (18)–(22); Liceo Galilei Verona mirror]{ref}
For one selected mode with <number operator> $N=a^\dagger a$ and $\mu=\langle N\rangle>0$, evaluate both detections at the same time. The <canonical commutation relation> $aa^\dagger=a^\dagger a+1$ implies $N^2=(a^\dagger)^2a^2+N$, hence
$$
g^{(2)}(0)=\frac{\langle N(N-1)\rangle}{\mu^2}
=1+\frac{\operatorname{Var}N-\mu}{\mu^2}.
$$
The <factorial moment> counts ordered pairs of distinct <photons>. Using $\langle N^2\rangle$ 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.
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