Generation of a Two-Photon Singlet Beam. W. Nogueira, S. Walborn, S. Pádua and C. Monken in Phys. Rev. Lett. 92:043602 (2004).  What the paper says!?

This studies the antisymmetric singlet bosonic state $\ket{\psi^-}\equiv{1\over\sqrt{2}}(\ket{H}_1\ket{V}_2-\ket{V}_1\ket{H}_2)$ in space:

To maintain their overall bosonic symmetry, photons in the singlet polarization state also display spatial antisymmetry, and cannot occupy the same plane-wave mode [2,3]

They frame their work in the context of Cabello's supersinglet states—singlet states of $N$ particles—although here they consider the (very) particular case $N=2$ only (so singlets!), but say that this is «a first step in creating a localized multiphoton state» (their Ref. [17] states that «Experimental work in this direction is currently being conducted in our laboratory» but I couldn't find the follow up). This falls in the line of decoherence-free subspace of Kwiat et al., in which context «one can regard the HWP as a special case of a decoherence environment.» One needs however $N=4$ or more to encode at least one qubit. So their framing seems exaggerated, without necessity to do so as their work is quite subtle and rich on its own.

Here they realize a collimated singlet $\ket{\psi^-}$, which is thus such that decoherence is felt globally by both photons:

Here we show experimentally that, using multimode Hong-Ou-Mandel (HOM) interference [9], it is possible to create a localized $\ket{\psi^-}$ polarization state, in which the two photons propagate in a single beam.

We typically consider only the spatial degree of freedom, so how accurate is the "necessary" below, is unclear:

Walborn et al. [9] showed that in a multimode treatment of HOM interference, it is necessary to take into account both the polarization and transverse spatial degrees of freedom.

They can tune the polarization with a QWP, as follows:

the relative phase can be manipulated in order to change from the polarization state $\ket{\psi^+}$ to $\ket{\psi^-}$

They also have slits but here that do nothing fundamental, unlike in their previous work where they were splitting the field and supporting the birefringent polarization selection.

They pair their polarization $\ket{\psi^{\pm}}$ to the spatial (multimode) part $\mathscr{W}$, which itself can acquire an odd-parity in its $y$ variable (i.e., $\mathscr{W}(x,-y)=-\mathscr{W}(x,y)$) by using a laminate on half of the pump field:

a thin glass laminate halfway into the Gaussian profile pump beam and adjusted the angle in order to achieve a π phase difference between the two halves of the beam. This produces a transverse profile that is an odd function of the horizontal y coordinate.

They thus deal with a dipole:

In the far-field region, this beam is similar to the first-order Hermite-Gaussian beam HG₀₁.

This is also in near field since HG modes are eigenfunctions of the FT. The way they get HG$_{01}$ is technically by imposing a sharp π step with the laminate, creating a superposition of many odd HG$_{0,2n+1}$ modes but those of higher orders diffract and they're left with eventually HG$_{01}$.

From all this, at the two-photon level

the Beam Splitter changes the $y_1+y_2$ to a $y_1-y_2$ (one of the photon only gets transmitted for the interference to act, as otherwise they end up on opposite ports anyway where there is nothing to interfere), and so the amplitudes are the canonical HOM ones of both exiting in the same port—$\mathscr{W}\Big(\tfrac{x_1+x_2}{2},\tfrac{y_1-y_2}{2}\Big)\chi(1,2)$ and $\mathscr{W}\Big(\tfrac{x_1+x_2}{2},\tfrac{y_2-y_1}{2}\Big)\chi(2,1)$—which add by the BS transfer matrix, so that together, this piecing-up leads to: $$\Psi(\boldsymbol{r}_1,\boldsymbol{r}_2)\propto\mathscr{W}\Big(\tfrac{x_1+x_2}{2},\tfrac{y_1-y_2}{2},z\Big)P(1,2)$$ where $P(1,2)=\chi(1,2)\pm\chi(2,1)$ depending on the parity of $\mathscr{W}$ (the full wavefunction has to remain symmetric). For opposite-port exits, on the other hand, we have $\mathscr{W}\Big(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2},z\Big)P(1,2)$ (which subtract by the BS transfer matrix) and all together—for their setup with three detectors—

this gives us the following possibilities:

laminate in $y$ makes $\mathscr{W}$	state prepared by QWP	both photons in the same port?	$D_1D_2$ same-port coincidences	$D_1D_3$ opposite-port coincidences
even	$\ket{\psi^+}$ (triplets)	yes	peak — not measured	dip — $\mathcal V=0.92$
even	$\ket{\psi^-}$	no	dip — not measured	peak — not measured
odd	$\ket{\psi^-}$	yes	peak — Fig. 3a & b	dip — $\mathcal V=0.82$
odd	$\ket{\psi^+}$ (triplets)	no	dip — Fig. 3a	peak — not measured

That's how they phrased the above in their paper:

Subjecting the down-converted photons to a beam splitter, the observed HOM interference then depends upon the parity of the function $\mathscr{W}((x_1+x_2)/2,(y_1+y_2)/2))$. Specifically, using a pump beam that is an odd function of the y coordinate, $\mathscr{W}(x,-y,z)=-\mathscr{W}(x,y,z)$ photons in the polarization state $\ket{\psi^-}$ leave the beam splitter in the same output port.

And the physics is in the interpretation of those various cases. Their Fig. 3 shows the coincidences from the same output port, with D1/D2 detectors, in H/V (left) and $+/-$ (right) basis.

Those are rows 3 and 4 of the table: row 3 gives the peaks in panels a & b, row 4 gives the dip in 3a. The use of HWP gives one or the other basis (Fig. column). They observe interferences in both basis for $\ket{\psi^-}$ but only in the H/V basis for $\ket{\psi^+}$ (not in +/-), as expected «since it is the only antisymmetric two-photon polarization state and is invariant to bilateral rotation. In this respect, one can regard the HWP as a special case of a decoherence environment.» In the +/- basis, $\ket{\psi^+}$ becomes $\ket{++}-\ket{--}$ and thus remains symmetric (global phase doesn't matter).

They don't show a figure for that case but describe in the text the cases D1-D3 HOM dips from rows 1 and 3, with a dip $\mathcal V_{\rm HOM}=0.82$ (with laminate in, row 3), and of $\mathcal V_{\rm HOM}=0.92$ (no laminate, row 1). The presence of the plate degrades the effect.

The important case—at least the case they highlight—is row 3, of spatial antibunching, i.e., both photons emerge from the same port but are not detected together.

In their previous work, the antisymmetric spatial state was made by a birefringent double slit, and the antibunching appeared as a minimum within a fourth-order interference pattern at a particular detection plane. Here there is no diffracting aperture at all — Fig. 4 is not a fringe pattern, it is the bare transverse profile of a freely propagating collimated field, and the node is a property the beam carries with it as it propagates.

In Ref. [14], the two photons were in a singlet polarization state after the birefringent double slit, but they did not constitute a beam.

Interesting comment here:

In the monochromatic approximation considered here (it is assumed that the down-converted photons have the same wavelength), the two-photon detection amplitude can be regarded as the two-photon wave function

In which cases is the two-photon detection amplitude different from the two-photon wavefunction?

I need to understand this better:

The measurements shown above (Fig. 4), however, are not of a fourth-order interference pattern that exhibits spatial antibunching in a detection region, but rather measurements of the transverse profile of a two-photon spatially antibunched singlet beam.

It seems that in their previous work, the (spatial) antibunching was observed in some field overlap from diffraction, while here it is a built-in feature of the beam, which carries this property along with it. So they compare an interference (albeit of two photons, still a fringe of diffraction) vs, here, the transverse profile of the beam.

Also of note:

while the singlet beam is necessarily spatially antibunched, spatial antibunching can also be achieved with symmetric polarization states and an even pump beam