Finite-bin photon correlations (${\hat{g}}_j^{(2)}=C_j/(r_1{r}_2{T}_j\Delta \tau )$)

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 $T_{\mathrm{obs}}$ when delays are short. Independent streams have expected pair count $r_1{r}_2{T}_j\Delta \tau$, 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 $T_{\mathrm{obs}}-|\tau |$ 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.