Experimental Observation of Spatial Antibunching of Photons. W. Nogueira, S. Walborn, S. Pádua and C. Monken in Phys. Rev. Lett. 86:4009 (2001).  What the paper says!?

This is an important, thought-provoking paper reporting antibunching in space, with the emphasis that this violates Cauchy-Schwarz inequality in the same way as in time, thus is something that only genuine quantum states can achieve. The meaning of "antibunching" here is thus comparing local fluctuations to cross-correlations (not $g^{(2)}$<1).

This might still appear to be in tension with our own findings, which find that one get spatial antibunching from states with antisymmetric spatial wavefunction (say because the polarization spinor is also antisymmetric).

Here they consider a generic spatial field:

Let us now turn to space domain and consider that the transverse field profile of a given stationary light beam propagating along the z direction is described by a complex stochastic vector amplitude $\mathscr{V}(\boldsymbol{\rho},t)$ with an associated probability functional $\mathscr{P}(\mathscr{V})$.

This is an interesting analogy, which deserves more thoughts:

In the space domain, the concept analogous to stationarity is homogeneity.

Following what is essentially the same reasoning as in time, they arrive to a criterion for spatial antibunching:

which brings them to the conclusion that this cannot be violated for classical light (but can for quantum light):

for field states represented by positive nonsingular Glauber-Sudarshan distributions [...] photons are detected either spatially bunched or randomly spaced in a transverse detection screen.

On the concept of spatial antibunching, before them:

Spatial antibunching of photons has been predicted by some authors [5–9] and a possible experiment was recently proposed to observe it in squeezed states [8,9].

Those references are:

1. Propagation Effect in Resonance Fluorescence: Spatial Antibunching of Photons. M. Le Berre-Rousseau, E. Ressayre and A. Tallet in Phys. Rev. Lett. 43:1314 (1979).
2. Transverse Photon Bunching and Two-Photon Processes in the Field of Parametrically Scattered Light. D. N. Klyshko in Sov. Phys. JETP 56:753 (1982).
3. Spatial antibunching of photons. Z. Bialynicka-Birula, I. Bialynicki-Birula and G. Salamone in Phys. Rev. A 43:3696 (1991).
4. Multimode Squeezing, Antibunching in Space and Noise-Free Optical Images. M. Kolobov and I. Sokolov in Europhys. Lett. 15:271 (1991).
5. The spatial behavior of nonclassical light. M. Kolobov in Rev. Mod. Phys. 71:1539 (1999).

I wonder how homogeneity matters for such arguments to hold. The generalized Cauchy-Schwarz would yield: $$\big|G^{(2)}(\boldsymbol\rho_1,\boldsymbol\rho_2)\big|^2\ \leq\ G^{(2)}(\boldsymbol\rho_1,\boldsymbol\rho_1)\,G^{(2)}(\boldsymbol\rho_2,\boldsymbol\rho_2),$$ valid for all fields, while homogeneous ones—their result—yield: $$G^{(2)}(\boldsymbol\rho_1,\boldsymbol\rho_2) \leq\ G^{(2)}(\boldsymbol\rho,\boldsymbol\rho).$$

While the former stays a nonclassicality witness without homogeneity, it no longer implies anything about a dip. Indeed, assuming a classical beam so that $G^{(2)}(\boldsymbol\rho_1,\boldsymbol\rho_2)=I_1I_2$ with $I_1\ll I_2$, since $$\frac{G^{(2)}(\boldsymbol\rho_1,\boldsymbol\rho_1)}{G^{(2)}(\boldsymbol\rho_1,\boldsymbol\rho_2)}=\frac{I_1^2}{I_1I_2}=\frac{I_1}{I_2}$$, we would violate the homogenous CS (the inhomogeneous version saturates it into an equality and nothing is violated with classical beams).

It would be interesting to put the two above inequalities to the test in our work. If normalized, this gives us the familiar two-mode CS criterion $\big[g^{(2)}_{12}\big]^2\le g^{(2)}_{11}g^{(2)}_{22}$.

Here they demonstrate experimentally spatial antibunching (in the homogenous case) with SPDC:

In this paper we show that strong antibunching in one transverse direction can be observed in down-converted light, violating (10) by several standard deviations.

They use a double slit:

The effect is produced by fourth-order interference of a two-photon beam diffracted by a birefringent double slit.

Here they use the field-interference terminology that order counts powers of the field (not of intensity like Glauber's coherence functions). So fourth-order is $G^{(2)}$ (numerator of $g^{(2)}$). Birefringence is to bring polarization-dependent retardance so that the two photons, which travel in one beam with orthogonal polarizations (type II SPDC), acquire a phase difference.

With such alignment, the waveplates introduce a phase difference of $\pi$ between the two slits.

Still, they report, in several cases, variations of this result, which shows a drop at zero delay of the coincidences as compared to nearby points:

The theoretical coincidence rate is given as:

but is not justified here but in their next paper.

Our infamous Beam-Splitter problem: how to superimpose two fields...

In order to make possible that two detectors (D1 and D2 ) share the same transverse position without being limited by their physical dimensions, a beam splitter (BS) is inserted into the down-converted beam, with D1 and D2 placed in front of each exit port.

This is important to relate to our own work, in particular, what makes the SPDC state special to behave so differently than other states (our Fock states in superpositions produce bunching in their sense, but maybe because we are more closely implementing the HOM effect while here the single-input is split in two differently).