Spatial antibunching of photons with parametric down-conversion. W. Nogueira, S. Walborn, S. Pádua and C. Monken in Phys. Rev. A 66:053810 (2002).  What the paper says!?

This is the theory companion for the previous paper by the authors.

Broadly this is a repeat (often verbatim, including the whole of Section II from their previous letter), just with horrible fonts. The incentive thus remains the same and is a good one:

most theoretical and experimental investigations [of the nonclassical behaviour of light] deal with time variables only

and here they tackle space.

They use a birefringent double slit:

two quarter-wave plates mounted in front of a typical double slit, such that each wave plate covers only one slit and their fast axes are orthogonal to one another

to bring the superposition of the input state so as to trigger wavefunction symmetry effects. This could also be done with a BS. Same highlights in both papers:

the down-converted fields are incident on some sort of aperture, so as to produce fourth-order interference in the absence of second-order interference.

But this applies to polarization:

The plate-slit apertures provide a controlled phase factor, that is, no phase will be added to a field with polarization parallel to the direction of the fast axis of the wave plate, while a field with perpendicular polarization will be modified by a phase factor of exp(-iπ/2). Thus, the phase factor depends on the polarization of the field as well as through which slit the field ‘‘passes.’’

The maths of it is given by their field-aperture functions

with e/o the extraordinary/ordinary polarization, linked to y/x axes, from the SPDC type-II, $\boldsymbol{\xi}$ is the real-space position in the aperture plane (slit), while $\boldsymbol{\rho}$ is that in the detection-plane. $A_{\sigma\sigma'}(\boldsymbol\xi)$ is the slit transmission as a function of position across the slits, and $T_{\sigma\sigma'}(\mathbf q)$ is its Fourier transform. They describe them as infinitely narrow, Delta openings, with phase conditionned to the polarization.

The following is relevant for us:

Spatial photon antibunching is a fourth-order effect in a homogeneous field, that is to say, in a field that [according to Eq. 7], does not show intensity patterns.

Along the rim of our vortices, we are certainly satisfying such conditions. They worry about homogeneity along $x$ (after Eq. (24)).

They then proceed to explain how they design their setup to produce the sought antibunching:

However, in Ref. [26], the fourth-order correlation function, which is proportional to the coincidence rate, depends on $x_1+x_2$ instead of $x_1-x_2$. In order to achieve a minimum of coincidences when $x_1=x_2$, we have to introduce a phase difference of $\pi$ between the two possibilities (photon 1 through slit 1, photon 2 through slit 2) and (photon 1 through slit 2, photon 2 through slit 1). In our experiment, the phase difference was introduced by means of birefringent elements placed in front of each slit,

There is some amount of details on their wavefunction and how it gets acted upon by the slit:

Here, they basically adapt to space Glauber's argument for his photodetection theory:

Probability amplitude and modulus square, yields the familiar form of the correlator.

They detail the derivation of the quantum state through this setup, with the polarization two-photon wavefunction:

which leads them to their central observable, the photon-coincidence function (basically $g^{(2)}$) which they had experimentally measured in their previous paper:

Note that in Eq. (24), homogeneity is ensured by focusing the pump hard, $w_0\ll d$, so that $e^{-(d/2w_0)^2}$ kills the $x_1+x_2$ term and leaves only the $x_1-x_2$ invariant one. This is an engineered, approximated one. On a pure LG rim, we have it exactly.

Their discussion of classical source is also very interesting:

Perhaps, the best classical model for type-II SPDC is a superposition of two extended light sources orthogonally polarized and correlated in intensity. After the light is diffracted by the birefringent double slit, the calculation of the classical fourth-order correlation function is quite similar to the case of the BrownTwiss intensity interferometer

With this classically-correlated model of SPDC, they find that their re-phasing through the slits don't affect the state and thus leads to correlations but of a different type altogether:

Classical intensity interferometry is known to be insensitive to phase. Therefore, the birefringent elements have no effect on the predicted fourth-order correlation, that is:

The birefringent quarter plates do nothing but the double-slit still does, as this now shows bunching at $x_1=x_2$:

The visibility v is in the range $0\le v\le{1\over2}$ and depends on the statistics of the source. It is clear from expression (28) above that C_cl(x₁,x₂) predicts spatial bunching, as expected from any classical light source.

This is again a phenomenology that is very close to ours, although here cast in a slightly different way. We don't have such strong discrimination between quantum and classical light, whose phenomenology we can get much closer the one to the other, but maybe because we remain oblivious to the Cauchy-Schwarz argument and just rotate minima to space intervals where they don't matter. That would also be interesting.

In conclusions, they provide the theory for violating Cauchy-Schwarz inequalities in space with a SPDC source impinging on a birefringence double-slit.