= Quantum theory of optical coherence
Fix a detected position and <polarization of waves>. Write the <electric field> operator as $\widehat E(t)=\widehat E^{(+)}(t)+\widehat E^{(-)}(t)$, where the positive-frequency part contains <annihilation operators> and $\widehat E^{(-)}=(\widehat E^{(+)})^\dagger$. An ideal absorbing <photodetector> responds to
$$
G^{(1)}(t,t')=\operatorname{Tr}\!\left[\rho\widehat E^{(-)}(t)\widehat E^{(+)}(t')\right],
$$
with <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>.<Coherent and Incoherent States of the Radiation Field by Roy Jay Glauber>[1963, §II, equations (2.1)–(2.4), pp. 2766–2767; Universidade Estadual de Campinas mirror]{ref}
The ordering follows from a detection amplitude. If absorption takes an initial <pure state> $|i\rangle$ to an unobserved final state $|f\rangle$, its field factor is $\langle f|\widehat E^{(+)}|i\rangle$. Squaring its <complex modulus> and summing over a complete <orthonormal basis> of final states gives $\langle i|\widehat E^{(-)}\widehat E^{(+)}|i\rangle$. 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>, $\widehat E^{(+)}(t)=\mathcal E u(t)a$. All <Fock states> with nonzero occupation, <coherent states>, and <thermal light> in this same mode have
$$
\frac{G^{(1)}(t,t')}{\sqrt{G^{(1)}(t,t)G^{(1)}(t',t')}}
=\frac{u^*(t)u(t')}{|u(t)||u(t')|}
$$
where 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.
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