Connect field moments to detector counts

The positive-frequency field is proportional to an annihilation operator for a selected mode. The excitation rate of an ideal weak absorptive detector is therefore proportional to the expectation of the negative-frequency field followed by the positive-frequency field. This ordering gives mean occupation rather than occupation plus one half. Vacuum has no photons for an ideal passive detector to absorb, despite its nonzero quadrature variance. A detector with dark counts, an active pump, or another excitation mechanism requires additional terms.

Two absorptions involve a normally ordered four-field product. At zero delay in one mode this becomes the expectation of number times number minus one, rather than the second number moment. The subtraction has an operational meaning: the same single photon cannot supply two separate absorption events. Normal ordering is consequently tied to the specified detection process. It does not imply that all possible quantum measurements ignore vacuum fluctuations; phase-sensitive quadrature measurements have a different observable and calibration.

Assume each incident photon is independently recorded with probability eta, independent of photon number and time within the chosen detection window. Conditional on exactly n incident photons, the recorded count is binomial. Its mean is eta times n and its variance includes the randomness of missed photons. For a general incident number distribution, average over n after performing this conditional calculation. The output variance then contains both source fluctuations and detection-loss fluctuations.

A coherent input remains Poissonian after this ideal loss, with reduced mean, while a number-state input becomes binomial and loses its perfect number certainty. The binomial instrument matches the conditional fixed-number case exactly. It does not model the entire coherent-state distribution, a random arrival-time process, detector dead time, or afterpulsing. Source and detector assumptions must be stated before comparing a measured variance with a shot-noise reference.

The detected second factorial moment acquires eta squared because two distinct incident photons must both be detected. Dividing by the detected mean squared cancels this efficiency factor. Thus an ideal normalized second-order correlation is unchanged by uniform independent linear loss when its mean is nonzero. This useful invariance does not mean loss leaves the count histogram unchanged, nor does it guarantee that a poorly resolved experimental correlation estimate is unbiased.

Background counts add uncorrelated events and usually pull measured correlations toward one. Finite detector timing convolves a time-dependent correlation, and dead time can imitate a deficit of nearby events on a single detector. A split-beam two-detector arrangement helps measure coincidences without assigning two counts to the same detector event, but still needs calibrated backgrounds and timing. The operational content of a correlation is therefore incomplete without its window, mode selection, and detector response.