Feynman stated that the double-slit experiment “…has in it the heart of quantum mechanics. In reality, it contains the only mystery”. Yet, if there is an experiment more mysterious, it is a double-double-slit experiment with entangled particle pairs: In this experiment, the two most important properties of quantum theory, i.e., superposition and entanglement, combine and lead to a non-local interference pattern. The full analysis of this experiment can be even more complex because of the third quantum effect, i.e., the collapse of the wave function when the first particle is detected in the middle of the experiment at random times. This effect, in principle, can change the detection statistics, however, its analysis is not completely clear in orthodox formalism. The reason is that in quantum mechanics the time is a parameter, not a self-adjoint operator, hence, there is no agreed-upon and unambiguous way to compute the temporal probability distribution of detection events from first principles (i.e., Born rule) [1, 2]. In fact, this problem is a practical encounter with the measurement problem. To overcome this problem, we use Bohmian mechanics, a version of quantum theory that avoids the measurement problem and is experimentally equivalent to orthodox quantum mechanics [3, 4] insofar as the latter is unambiguous [5, 6, 7]: Bohmian mechanics can lead to a clear prediction for arrival time distribution using the Bohmian trajectories.

Based on this approach, we study a modified version of the double-double-slit experiment using entangled atom pairs, in the presence of gravity and with horizontal screens. In this setup, we predict a novel quantum phenomenon in the time domain: i.e., a non-local interference in the arrival time distribution, which is analogous to the non-local interference observed in the arrival position distribution [8, 9, 10]. We also numerically demonstrate that there is a complementary relationship between the one-particle and two-particle interference visibilities in the arrival time distribution, which is analogous to the complementarity observed in the position distribution [11, 12]. Thanks to current present-day single-atom interferometry technologies [13, 14], these results can be used to test the Bohmian arrival time distribution in a strict manner, i.e., where the semiclassical approximation breaks down.

Finally, it is important to remark that, the issue of arrival times has really not received the attention it deserves, perhaps partially because usual experiments have been performed in the far-field regime, where a semiclassical analysis is often sufficient. Nonetheless, due to recent progress in ultra-fast detector technology, it is now possible to investigate the near-field regime, where the semiclassical approximation breaks down and a deeper analysis is required. The main point of the present paper, then, is not merely to study a particular experiment. Instead, we are going to emphasize that there is an enormously large and largely unexplored category of (real and potential future) experimental data which surely ought to be theoretically explicable and that the Bohmian mechanics, at least in principle, can explain. In fact, our Bohmian arrival time treatment can be used for a more accurate investigation of various similar experiments beyond the double-double-slit experiment, such as atomic ghost imaging [13], interferometric gravimetry [15], atomic Hong–Ou–Mandel experiments [16], and so on [17, 18], which are usually analyzed in a semiclassical approximation. Can other candidate quantum theories (such as the Copenhagen interpretation, Everett’s many worlds picture, the dynamical collapse models, etc.) do at least as well or better? We don’t claim to know the answer, but we are sure it’s a question that is worth asking because it has the potential to stimulate new experiments, novel theoretical analyses, fresh debates… in short, good science.

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[11] Bergschneider, A, et al. "Experimental characterization of two-particle entanglement through position and momentum correlations." *Nature Physics *15.7 (2019): 640-644.

[12] Georgiev, D, et al. "One-particle and two-particle visibilities in bipartite entangled Gaussian states." *Physical Review A *103.6 (2021): 062211.

[13] Khakimov RI, Henson BM, Shin DK, Hodgman SS, Dall RG, Baldwin KG, Truscott AG. Ghost imaging with atoms. Nature. 2016 Dec 1;540(7631):100-3.

[14] Keller M, Kotyrba M, Leupold F, Singh M, Ebner M, Zeilinger A. Bose-Einstein condensate of metastable helium for quantum correlation experiments. Physical Review A. 2014 Dec 1;90(6):063607.

[15] Geiger R, Trupke M. Proposal for a quantum test of the weak equivalence principle with entangled atomic species. Physical Review Letters. 2018 Jan 25;120(4):043602.

[16] Lopes R, Imanaliev A, Aspect A, Cheneau M, Boiron D, Westbrook CI. Atomic hong–ou–mandel experiment. Nature. 2015 Apr 2;520(7545):66-8.

[17] Brown MO, Muleady SR, Dworschack WJ, Lewis-Swan RJ, Rey AM, Romero-Isart O, Regal CA. Time-of-flight quantum tomography of an atom in an optical tweezer. Nature Physics. 2023 Apr;19(4):569-73.

[18] Tenart, A, et al. "Observation of pairs of atoms at opposite momenta in an equilibrium interacting Bose gas." *Nature Physics *17.12 (2021): 1364-1368.

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