Quantum Theory of Interference Effects in the Mixing of Light from Phase-Independent Sources. U. Fano in Am. J. Phys. 29:539 (1961). What the paper says!?
This is the treatment of path-indistinguishability that became standard to explain the HBT effect, e.g., it is described like this by Baym[1] (who doesn't cite Fano): two atoms $a$ and $b$ (sources) and two detectors $c$ and $d$ (two other atoms).
The introduction reminds the difference between incoherence of one-photon observable and higher (here two) photon observables:
statements are often made in this connection implying that light emitted by independently excited atoms cannot produce any interference phenomenon. This extended implication is incorrect. In fact, the reception of ‘incoherent’ light by two different detectors (or by equivalent methods) shows correlations in space and in time which can be properly described as interference effects.
These optical correlation phenomena may be considered from several alternative points of view, and it may be worthwhile to explore each of them, for the sake of an over-all understanding.
As compared to earlier treatments, he brings the effect directly to the atomic emitters and the detectors, bypassing the fields. He credits Dicke for the insight «that atoms of a thermally excited gas must be treated as a single quantum mechanical system when coherence matters are investigated». It is not entirely clear how genius or groundbreaking is this step from fields to atom, but it still remains technically correct that
his paper will discuss an atomic process that involves the minimum number of particles required to exhibit the phenomena observed by Brown, Twiss, and Forrester. It will focus on the atoms rather than on the radiation field.
Judging on his own approach, he comments:
This calculation consists of an application of standard techniques, with some care in the arrangement of operations to preserve a degree of transparency and formal simplicity.
Still, there is one trick, in Eq. (1), where he uses some "complementarity" of times (emission/detection) to avoid integrals. I leave details of that untouched here.
He mentions (end of Sec. 1) multiphoton generalizations. Note that this is before Glauber's earlier work on the topic.
The formalism, in a nutshell. He calls $C$ the probability amplitude of excitation, with subscripts what is excited:
$$a, b, u, v, c, d$$
where $u$ and $v$ are waves linking sources ($a$ and $b$) to the detectors ($c$ and $d). So he starts with $|C_{ab}(t=0)|^2=1$ and computes $P_{cd}(t)\equiv|C_{cd}(t)|^2$. He then computes this quantum-mechanical process:
$C_{cd}$ is of fourth order since two emission and two absorption processes are required to achieve the double photoionization.
The assumption is then
(notice the careless proofreading!)
Important for us:
This is not the assumption of independent excitation, but of ensemble averaging.
Regarding the computation, he says this could be done by perturbation theory involving all processes (it would be interesting to actually do that) but that to fourth order, diagrams simplify to the following picture:
He comes to the conclusion—«which is plausible by inspection of the problem.»—that
The gist of the argument:
The phases have disappeared, so the phase averaging cannot destroy the interference term at the two-photon level as it did at the one-photon one.
Section 3 is the "microscopic" part of the paper where he computes $D_{ca}$ explicitly, which, however, «constitutes a rather familiar problem.» He includes polarizations, wavevectors, relies on a model where «this superposition of sinusoidal waves represents a wave train with a sharp front at $t=r_{ca}$, propagating away from the atom $a$ with light velocity.»
Section 4 is the final technical touch, this time on the geometry of the terms from Sec. 3, where he computes the «Differential Probability Distribution». This is the main result:
where one can enjoy the profusion of technical factors entering the problem. In modern typesetting (and dropping the prefactor):
$$\begin{multline}p(t_c,t_d)\propto \lvert\hat p_a\times\hat R\rvert^2\lvert\hat p_b\times\hat R\rvert^2\cosh\left[\Delta\Gamma_{ab}(t_c-t_d)\right]\\ +\left\lvert\left(\hat p_a\times\hat R\right)\cdot\left(\hat p_b\times\hat R\right)\right\rvert^2\cos\left[\bar E_{ab}\left(r_{ca}-r_{cb}-r_{da}+r_{db}\right)-\Delta E_{ab}(t_c-t_d)\right]\end{multline}$$
The two arguments of the cosine reproduce the macroscopic results (HB, Purcell and others) exactly: the space variable collapses to $r_{cd}r_{ba}/\bar\lambda R$ (the coherence-area condition, visibility surviving averaging over many atom pairs only when this is of order unity or less), and the time variable is the Forrester beat $\Delta E_{ab}(t_c-t_d)$. The prefactor vanishes when the two source dipoles are mutually orthogonal and orthogonal to $\hat R$ — as it must. Only pairs excited within $\bar\Gamma_{ab}^{-1}$ of each other contribute.
where $$\bar E_{ab}\equiv\tfrac12\left(E_a+E_b\right),\quad \Delta E_{ab}\equiv E_a-E_b,\quad \bar\Gamma_{ab}\equiv\tfrac12\left(\Gamma_a+\Gamma_b\right),\quad \Delta\Gamma_{ab}\equiv\Gamma_a-\Gamma_b $$.
His Eq. (33) details «the space variable»: $$r_{ca}-r_{cb}-r_{da}+r_{db}\simeq\frac{1}{R}\left(\mathbf r_{cd}\times\hat R\right)\cdot\left(\mathbf r_{ba}\times\hat R\right)$$ which actually has a typo
Here, $\hat R$ is the common line of sight to the sources and detectors, so that the cross products keep only the components of the source separation $\mathbf r_{ba}$ and the detector separation $\mathbf r_{cd}$ transverse to $\hat R$. Since the longitudinal components cancel out of the combination, intensity interferometry measures an angular size on the sky and is blind to depth.
Fano puts energies in radian/second and distances in light-seconds, $\bar E_{ab}=\bar\omega/c=1/\bar\lambda$ with $\bar\lambda$ the reduced wavelength, $\lambda/2\pi$. Put together, the fringe phase is of the order $r_{cd}r_{ba}/(\bar\lambda R)$. Writing $\theta=r_{ba}/R$ for the angular size of the source pair as seen from the detectors, «averaging over the mutual positions of many different pairs yields a nonvanishing effect of constructive interference only for positions so confined that»:
$$ r_{cd}\lesssim\bar\lambda/\theta $$
i.e. the detectors must sit closer together than the coherence length on the ground. Beyond that, averaging over the many mutually displaced source pairs in a real extended source randomizes the cosine and the visibility dies. This is Hanbury Brown and Twiss's own criterion.
He also covered time: $\Delta E_{ab}\left(t_c-t_d\right)$ brings the difference of the two source frequencies vs the delay between the two detections. The coincidence rate beats at $E_a-E_b$, which is Forrester et al.[2]'s optical beat. If $E_a=E_b$, the term goes away, leaving pure spatial HBT.
Note the asymmetry: the mean frequency multiplies the space variable while the difference multiplies the time variable.