Photon Correlations. R. J. Glauber in Phys. Rev. Lett. 10:84 (1963).  What the paper says!?

This is the first paper by Roy Glauber on his work on quantum optical coherence which shall eventually earn him The Nobel Prize in Physics 2005:

We have developed general quantum mechanica1 methods for the investigation of [...] correlation effects

This is despite the fact that he only considers essentially classical fields: coherent and thermal, also RPCS (Random-phase coherent states).

The central result is that descriptions of correlations (which he formalizes, introducing his famous two-photon correlation function) requires a quantum description. Refs. 2,4-6 are Refs. ,--

a number of papers have been written examining the correlations in considerably greater detail. These papers^2,4-6 retain the assumption that the electric field in a light beam can be described as a classical Gaussian stochastic process.

But, he says, this is not enough to capture all the possible quantum states of the light field:

In actuality, the behavior of the photon field is considerably more varied than such an assumption would indicate.

This is because:

Whereas a stationary Gaussian stochastic process is described completely by its frequency-dependent power spectrum, a great deal more information in the form of amplitude and phase relations between differing quantum states may be required to describe a steady light beam. Beams of identica, l spectral distributions may exhibit altogether different photon correlations or, alternatively, none at all. There is ultimately no substitute for the quantum theory in describing quanta.

In this way, despite he indeed focuses on classical states of the light field in this paper, he clearly captures the necessity of a full quantum treatment to cover the full complexity of the problem, something that is sometimes attributed to Sudarshan's "crucial extension".

He introduces the coherent state as «preferable to use an altogether different set of basis states. We take these to be of the form»:

«We shall call the $\ket{\alpha_k}$ coherent states;» Since they form a complete set (can decompose the unity matrix), Glauber immediately follows up with the observation that:

any state may be expanded linearly in terms of coherent states. The most general light beam can thus be described by a density operator of the form

which is close to the $P$ function that will become known as the Glauber‒Sudarshan function, except that here he uses two different coherent states $\ketbra{\alpha_k}{\alpha'_k}$, so this is not diagonal. The diagonal form was introduced by Sudarshan in his "later" paper, where he cites this Glauber paper but not this particular (non-diagonal) result.

Glauber also provides, from «A simple theorem » (does not elaborate which one) the $P$ function of the thermal field (Eq. (5)).

He then carries on to derive his other, even more important result, the correlation function:

with the $C$ provided in Eq. (7) as:

Coherent states are not correlated:

A correlation between photons only appears when incoherent mixtures or superpositions of the coherent states are present.

Are Fock states "superpositions of coherent states" in Glauber's mind?

In this way, he recovers the $g^{(2)}$ of thermal light, which is in agreement with stochastic models (Eqs. (8-9)).

He suggests that masers (will be lasers later on) produce coherent states, rather than thermal states. In which case, the would lead to «photon correlations only to the extent that random amplitude modulation is present in the statistically averaged beam.» In fact RPCS also produce correlations. He alludes to that in his footnote 7:

Incoherent light beams of exceedingly narrow bandwidth may, in principle, be formed by superposing the outputs of many identical but independent masers.

Here he seems to be describing the thermalization of coherent light (2 is Ref. ):

The fact that photon correlations are enhanced by narrowing the spectral bandwidth has led to a prediction² of large-scale correlations to be observed in the beam of an optical maser. We shall indicate that this prediction is misleading and follows from an inappropriate model of the maser beam.

In our theory or frequency-resolved photon correlations, this is what we call the "indistinguishability bunching" that manifests as red diagonal in two-photon spectra.

He concludes with an impromptu, a bit strange considerations on the number of photons detected in a time window $t$ for incoherent light. This might be related to an (unpublished) result we have with Eduardo (probably found in his Wolverhampton thesis).

Glauber thanks Saul Bergmann «for bringing these problems to his attention».