Figure 3: Schematic of matter-wave momentum modes affected by the Bragg pulses and the topologically equivalent optical setups.

In panel a, the diametrically opposite vectors k₁ and k₂ = −k₁ show the targeted momentum modes on the scattering halo; their amplitude is given by the halo peak radius, k_r=|k₁|=|k₂|, which is equal to k_r≃0.95|k₀| in this part of the halo⁸. Also shown are the to-be-measured momentum components k₃ and k₄ corresponding to a rotation by θ away from the targeted modes, which couple, respectively, to k₆=k₃−2k₁ and k₅=k₄+2k₁ by the same Bragg pulses. Panels b and c show equivalent optical schemes that use a χ⁽²⁾ nonlinear crystal that is optically pumped to produce twin-photons via parametric down-conversion. In b, we depict the archetypal optical HOM setup that corresponds to the case of θ=0 in the matter-wave scheme. A twin-photon state in modes k₁ and k₂ is first selected from a broadband source, then mixed at the beam splitter (BS) after reflection from the mirror (M), and photon coincidence counts are measured between the two symmetric output ports of the interferometer. In c, we depict the optical setup that is equivalent to θ>0 in the mater-wave proposal. Two twin-photon states in (k₃, k₄) and in (k₅, k₆) are selected from the broadband source; the asymmetry of the pairs about the optical axis of the interferometer means that the correlated photons from the respective pairs will arrive at the beam splitter at spatially separate locations and will mix with photons from the other pair, which introduces distinguishability between the paths through the interferometer.