Multimode Hong-Ou-Mandel Interference. S. P. Walborn, A. N. d. Oliveira, S. Pádua and C. H. Monken in Phys. Rev. Lett. 90:143601 (2003).  What the paper says!?

This implements the HOM effect from the spatial symmetry of the wavefunction. As such, it is related to our (much) later analysis of this effect in space. In their words:

most experiments utilizing HOM-type interference consider an ideal monomode situation. In this Letter, we consider multimode two-photon interference of photon pairs created by spontaneous parametric down-conversion (SPDC)

Although involving a full field, they condensed the effect to a global constructive or destructive interference, while we do it locally:

We show how the transverse amplitude profile of the pump beam in SPDC determines whether the down-converted fields interfere constructively or destructively

They also perform experiments.

They consider the SPDC state

where $\ket{\psi}$ is taken from their earlier work and takes the form:

There is a discussion justifying the various terms and quality of approximations (walk-off effect in crystal, etc.)

They speak of a "photonic wavefunction":

Once this impinges on the BS:

with this "awkward convention" of using two references systems for the two outputs of the BS (but this is central to the algebra, I'm just wondering if this is the most elegant way to capture it):

As a result, they need a prime for one variable when both photons are on the same output. This gives rise to this decomposition of the output wavefunction

where $\boldsymbol{\Pi}$ encodes the polarization and «$\mathscr{W}$ is the transverse field amplitude of the pump beam on the plane $z﹦Z$, which has been transferred to the two-photon wave function.» That gives them the basis for their HOM interference for the multimode field: Assuming a symmetric polarization, $\boldsymbol{\Pi}(\boldsymbol{\sigma}_1,\boldsymbol{\sigma}_2)=\boldsymbol{\Pi}(\boldsymbol{\sigma}_2,\boldsymbol{\sigma}_1)$, then if $\mathscr{W}$ is even in the transverse $y$, Eqs. (11) and (12) cancel and this is the standard HOM destructive interference. On the other hand, odd $\mathscr{W}$ leads to constructive interferences on the opposite ports. Antisymmetrizing polarization obviously does this too.

To our knowledge, all HOM-type experiments performed up until now have used a pump beam that is described by an even function of y. In order to demonstrate experimentally the possibilities of controlling the HOM interferometer with space and polarization variables, we performed a series of experiments in which coincidence counts were registered, combining symmetric and antisymmetric components of $\boldsymbol{\Psi}$.

They consider specifically HG beams (i.e., egg boxes with Cartesian symmetry):

To generate HG modes we placed a 25 µm diameter wire inside the laser cavity, forcing the laser to operate in one of the HG modes with a nodal line at the position of the wire

Then they measure the coincidences, integrated over all space (all modes):

In addition to the usual HOM depletion, they also see a (HB-type) bunching [on the opposite ports, though, so strictly speaking, HOM dip comes from coalescence while their HOM peak comes from anticoalescence].

A BS reflection is a mirror inversion, and HG modes are eigenstates of it with eigenvalue $\pm1$. Laguerre-Gaussian modes are not: a mirror sends $\ell\to-\ell$. So the Walborn parity-control logic with a vortex pump would transpose with $\ell\pm(-\ell)$ superpositions, which bring us back to the HG-like combinations. The quantum state is needed to make such a superposition nontrivial, which is our input to the problem.

Applications offered in the introduction, such as Bell-state measurements and quantum-optical logic gates thus become immediate Heisingberg objectives.