Distinguish count correlations from two-photon interference

For a selected mode with nonzero mean, normalize the second factorial moment by the square of mean occupation. A coherent state gives one because its counts are Poissonian. A thermal single mode has a geometric occupation distribution and gives two. A number state gives one minus reciprocal occupation, reaching zero for one photon. These values concern a declared mode or properly resolved temporal correlation, not any histogram formed by an arbitrary detector bandwidth and gate.

Combining independent equally populated thermal modes reduces the observed zero-delay bunching to one plus inverse mode number. Unequal populations give a weighted effective mode count. A value near one therefore need not establish a pure coherent state; multimode thermal light, technical averaging, and other number distributions can approach or share that value. Likewise, second-order information does not determine phase coherence or the entire density operator. Additional moments or phase-sensitive measurements can distinguish states with matching mean and variance.

For an ideal two-level atom, the lowering operator applied twice is zero. Once one fluorescence photon is emitted, the atom is in its ground state and must be excited again before another emission. This gives a zero ideal zero-delay fluorescence correlation, even though the stationary emission rate can be nonzero under driving. The recovery at finite delay contains information about the drive and relaxation. It is not determined by the algebraic zero-delay argument alone.

Temporal antibunching usually means suppression near zero delay compared with later delays; sub-Poissonian integrated counts depend additionally on the counting window. Finite timing resolution, background light, multiple emitters, and detector artifacts can obscure or imitate features. A source claim therefore needs a model of what was resolved. A single-detector dead-time gap is insufficient evidence for a one-photon emitter. The normally ordered source correlation and the measured coincidence histogram agree only after the detector response is accounted for.

Choose a real balanced beam-splitter convention in which the two output annihilation operators are the sum and difference of the input operators divided by square root of two. The relative minus sign ensures a unitary transformation. Applying the inverse transformation to one photon in each input yields a superposition of two photons in the first output and two in the second, with no one-in-each term. The canceled alternatives are both-transmitted and both-reflected probability amplitudes.

This Hong–Ou–Mandel cancellation requires matched spectral, temporal, polarization, and other unobserved degrees of freedom. Fully distinguishable photons have a one-half coincidence probability instead. A fixed-frequency wave lab can illustrate addition of signed amplitudes, but it cannot reproduce two-photon statistics or prove indistinguishability. Classical intensity cancellation at one output and a quantum two-photon coincidence dip are different observables, even when both calculations contain interference terms.