= Finite-bin photon correlations
{title2=$\widehat g^{(2)}_j=C_j/(r_1r_2T_j\Delta\tau)$}
For stationary continuous detection, let $C_j$ count all cross-channel pairs with $t_2-t_1$ in a bin of full width $\Delta\tau$, and let $r_1,r_2$ be observed single-count rates. Let $T_j$ be their joint live observation time at that delay, approximately the common duration $T_{\rm obs}$ when delays are short. Independent streams have expected pair count $r_1r_2T_j\Delta\tau$, giving the displayed estimator. It estimates a bin average of <second-order optical coherence>, further blurred by detector timing response. For a finite uninterrupted run the exact overlap weighting is $T_{\rm obs}-|\tau|$ inside the delay integral, rather than a constant at long delays.
For an original numerical example, choose $r_1=2000\,\mathrm{s^{-1}}$, $r_2=3000\,\mathrm{s^{-1}}$, $T_j=100\,\mathrm s$, and $\Delta\tau=10\,\mathrm{ns}$. The independent reference is six pairs per bin. Chosen counts $C_j=3,6,12$ give estimates $0.5,1,2$; they are teaching inputs with large counting fluctuations, not historical measurements. For pulsed sources, compare properly normalized same-pulse and separate-pulse coincidences instead of assuming this continuous-source baseline.
<Detector dead time> suppresses nearby same-detector counts and can manufacture an apparent dip in a single-detector record. Two independent detector channels avoid that particular same-channel exclusion, but shared electronics can still reject pairs. Timing spread and wide bins wash out narrow features. A device recording only the first stop after each start measures a waiting-time distribution; interpreting it as all-pair correlations requires a sufficiently low stop probability over the relevant delay range.
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