Interference, spectral momentum correlations, entanglement, and Bell inequality for a trapped interacting ultracold atomic dimer: Analogies with biphoton interferometry

Elucidating similarities and differences between quantum-optics biphoton interference phenomena and the quantum physics of quasi-one-dimensional double-well optically-trapped ultracold neutral bosonic or fermionic atoms, we show that the analog of the optical biphoton joint-coincidence spectral correlations, studied with massless non-interacting biphotons emanating from EPR-Bell-Bohm single-occupancy sources, corresponds to a distinct contribution in the total second-order momentum correlations of the massive, interacting, and time-evolving ultracold atoms. This single-occupancy contribution can be extracted from the total second-order momentum correlation function measured in time-of-flight experiments, which for the trapped atomic system contains, in general, a double-occupancy, NOON, component. The dynamics of the two-particle system are modeled by a Hubbard Hamiltonian. This partial coincidence spectrum is a cosine-square quantum beating dependent on the difference of the momenta of the two particles, while the corresponding coincidence probability proper, familiar from its role in describing the Hong-Ou-Mandel coincidence dip of overlapping photons, results from an integration over the particle momenta. Because the second-order momentum correlations are mirrored in the time-of-flight spectra in space, our theoretical findings provide impetus for time-of-flight experimental protocols for emulating with (massive) ultracold atoms venerable optical interferometries that use two space-time separated and entangled (massless) photons or double-slit optical sources. The implementation of such developments will facilitate testing of fundamental aspects and enable applications of quantum physics with trapped massive ultracold atoms, that is, investigations of nonlocality and violation of Bell inequalities, entanglement, and quantum information science.