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Quantum optics studies light using quantum states of the electromagnetic field and their interaction with matter. Photon statistics asks how detections fluctuate; the quantum theory of optical coherence relates those detections to field correlations. A coherent state, thermal light, and a Fock state can have the same mean photon number yet different pair counts. This distinction makes photon counting useful for testing sources intended to emit one photon at a time.

Fix a detected position and polarization of waves. Write the electric field operator as $\hat{E}(t)={\hat{E}}^{(+)}(t)+{\hat{E}}^{(-)}(t)$, where the positive-frequency part contains annihilation operators and ${\hat{E}}^{(-)}=({\hat{E}}^{(+)}{)}^{†}$. An ideal absorbing photodetector responds towith density matrix $\rho$. The diagonal gives the single-count rate up to the detector coupling. Multiple absorptions require higher, normal-ordered field products. Roy J. Glauber used these correlations to develop the quantum account of optical coherence.^[ref]

The ordering follows from a detection amplitude. If absorption takes an initial pure state $|i⟩$ to an unobserved final state $|f⟩$, its field factor is $⟨f|{\hat{E}}^{(+)}|i⟩$. Squaring its complex modulus and summing over a complete orthonormal basis of final states gives $⟨i|{\hat{E}}^{(-)}{\hat{E}}^{(+)}|i⟩$. Averaging over initial preparations replaces this expression by a trace. Thus a detector that absorbs cannot count the Fock vacuum, although the vacuum has quantum fluctuations.

For one selected normal mode, ${\hat{E}}^{(+)}(t)=\mathcal{E}u(t)a$. All Fock states with nonzero occupation, coherent states, and thermal light in this same mode havewhere both denominators are nonzero. Its magnitude is one even when the expected value of the field vanishes. Perfect first-order optical coherence therefore does not identify a coherent state. Pair detection supplies additional information.

For one selected polarization of waves, write the detected complex fields as $E_1(t),E_2(t+\tau )$ and their light intensities as $I_j=|E_j{|}^2$, absorbing a common conversion constant. With stationary ensemble averages and nonzero ${\bar{I}}_j=⟨I_j⟩$, defineHere second-order optical coherence concerns distinct detector channels, with the corresponding normal ordering in the quantum formulation. Independent detection has $g^{(2)}=1$. Subtracting the means gives the normalized covarianceThus multiplying raw photocurrents contains a large mean-product contribution; multiplying their fluctuations isolates the small excess. Independent detector shot noise does not supply a mean cross-channel excess, although it increases measurement uncertainty.

The displayed Siegert relation requires a zero-mean, jointly circular complex Gaussian random field, the usual chaotic thermal light model. Circular means invariance under a common arbitrary complex phase rotation; in particular $⟨E_i{E}_j⟩=0$. Apply the Gaussian moment pairing theorem to the real and imaginary parts:The remaining pairing vanishes by circularity. Division proves the Siegert relation. Gaussianity is an assumption about fourth moments, not a consequence of merely measuring first-order optical coherence; a coherent state or photon antibunching cannot be interpreted by this thermal formula.

For equal-intensity beams combined in a common optical mode with matched wave polarizations, interferometric visibility is $V=|g_{12}^{(1)}|$. Consequently ideal thermal stellar intensity interferometry measures $g^{(2)}-1=V^2$, while the full $g^{(2)}$ includes the independent-count reference one. Unpolarized thermal light with two independent equally populated wave polarizations has half this excess: adding their covariances doubles the numerator while multiplying total means quadruples the denominator. Finite detector response further suppresses the peak and must be calibrated.^[ref]