Can the double-slit experiment distinguish between quantum interpretations?

The paper addresses the long-standing ambiguity in defining and predicting quantum arrival-time statistics, particularly the joint position–time probability density of detection events in interference experiments. In standard quantum theory time is a parameter rather than an operator, making the derivation of temporal probability distributions from first principles unclear. This ambiguity affects even cumulative spatial distributions in time-unresolved experiments. Semiclassical time-of-flight methods work in far-field regimes but can fail in near-field due to quantum phenomena like backflow. With advances in detector technology enabling time-resolved measurements, the authors ask whether different formulations and interpretations of quantum theory—standard POVM-based approaches, flux-based proposals, Bohmian and other trajectory models, and phenomenological absorbing-boundary models—lead to distinguishable predictions for spatiotemporal detection statistics in a double-slit setup. They propose an unconventional double-slit with a horizontal detection screen to probe these differences, arguing that near-field dynamics and potential multi-crossings make deviations observable.

The authors review several approaches to the quantum arrival-time problem: (1) Standard canonical approaches define arrival observables as POVMs (e.g., Aharonov–Bohm/Paul operators and Kijowski’s axiomatic construction), yielding a standard arrival-time distribution and joint screen observable via Born’s rule. (2) Trajectory-based interpretations (Bohmian mechanics, Nelson stochastic mechanics, many interacting worlds) infer arrival statistics from particle trajectories, motivating flux-based distributions and, when necessary, the Bohmian truncated current for first-arrival statistics. (3) Phenomenological detection models include absorbing boundary rules implemented as boundary conditions, complex potentials/absorbing boundaries in Schrödinger dynamics, and generalized Feynman path integrals with absorbing surfaces. Prior work suggests these approaches are often numerically similar in typical far-field experiments, with significant differences expected mainly in near-field regimes. Experimental progress in single-atom detection and time resolution now makes a detailed empirical comparison feasible.

The study proposes a double-slit experiment with a horizontal screen (at y = L_y) in addition to the conventional vertical screen (at x = L_x). The two slits are centered at y = ±s and produce Gaussian-profile waves to avoid Fresnel diffraction complexity. The incident state is modeled as a separable 2D Gaussian wave packet in x and y with a double-Gaussian superposition in y: ψ(x, y, t) = N [G(y−s, t) + G(y+s, t)] G(x, t), with standard Gaussian evolution parameters. Chosen particle: metastable helium (m = 6.64×10^−27 kg), parameters s = 10 µm, σ_x = 0.04 µm, σ_y = 0.5 µm, u_x = 3 m s^−1, u_y = 0, consistent with current atom-optics experiments. Joint and marginal spatiotemporal arrival distributions on the screen surface S are computed under six approaches: - Intrinsic (no detector back-action on evolution): Semiclassical (SC): arrival times from momentum distribution, joint density ∝ I_sc(t|x∈S)|ψ(x)|^2; Standard (STD): Kijowski’s POVM-based joint distribution P_STD(x,t|x∈S) via time operator eigenstates; Quantum flux (QF): P_QF ∝ |J(x,t)·n| with J the probability current; Bohmian truncated current (BTC): uses Bohmian trajectories and the truncated current to capture first-arrival statistics when the screen acts as a barrier, computed numerically. - Non-intrinsic (detector back-action): Absorbing Boundary Rule (ABR): free Schrödinger evolution with complex Robin (absorbing) boundary condition n·∇ψ = iκψ on S; detection statistics from flux, parameter κ tuned; Path Integral with Absorbing Boundary (PAB): detection modeled via elimination of paths reaching S in time slices, yielding an absorbing current and associated joint distribution, with a proportionality factor λ (length scale) tuned. Numerical simulations generate joint distributions P(x,t|x∈S), marginal arrival-time Π(t|x∈S), cumulative spatial distribution P(x|x∈S), and local conditional time distributions Π_t(t|x∈S). Bohmian simulations (up to ~10^8 trajectories) provide first- and all-arrival averages and local distributions, revealing multi-crossing effects and validating the flux approach for all-arrivals. Detection schemes considered: spot-detection (unilateral/bilateral point-like detectors allowing multi-crossings: QF applicable) and barrier-like screens (blocking return, hence first-arrivals only: BTC required). Detector feasibility is assessed using state-of-the-art single-atom delay-line detectors with ~220 ps temporal and ~177 µm spatial resolution at rates of several 10^6 s^−1.

• Joint spatiotemporal distributions on a horizontal screen (e.g., L_y = 15 µm) differ characteristically across approaches. SC shows well-separated fringes; STD exhibits more continuous fringes; QF shows grooves due to sign changes in J·n; ABR and PAB are broadly similar but differ in fine structure; BTC removes contributions from second/third crossings, leaving empty zones between fringes. - Temporal no-arrival windows between fringes are 0.01–0.2 ms (spatially ~0.3–2 mm), well within current detection capabilities. - Marginal arrival-time distributions Π(t|x∈S) at various L_y show increasing deviations as the screen approaches the slits. SC behaves distinctly from STD and QF; at L_y = 15 µm, STD and QF show a significant difference around t ≈ 0.2 ms. - Cumulative position distributions P(x|x∈S) at L_y = 15 µm differ across SC, STD, and QF; SC is clearly separated from the other two. - Average arrival time versus position on the horizontal screen reveals a pronounced deviation region for intrinsic methods: between x ≈ 16.2–19.2 mm, STD and QF differ significantly, with maximal deviation near x ≈ 19 mm, despite being in the far field in x. - Local conditional arrival-time distributions Π_t(t|x∈S) at x = 16.2, 17.4, 18.4, 19.2 mm show that QF curves are broken (reflecting current sign changes) while STD and SC are smoother; these local distributions expose the origins of average-time deviations. - Bohmian simulations confirm: all-arrivals averages coincide with QF; first-arrivals averages deviate in the identified region (x ≈ 16.2–19.2 mm). At local positions where QF shows first recursion, the first-arrival distribution drops to zero, predicting sizable temporal gaps under barrier-like detection. - Statistical feasibility: ~10^4 detection events suffice to reconstruct local distributions with ~10^−2 ms uncertainty in the local average arrival time; predicted differences between methods are ≳10^−1 ms. With ~10^8 total arrivals, ~10^4 events are expected in a narrow spatial bin near x ≈ 19.2 mm for robust local statistics. - Including detector back-effects: With parameters chosen to yield comparable absorption probabilities (e.g., κ = 1 µm^−1 for ABR and λ = 1 µm for PAB), PAB produces up to ~40% differences in marginal spatial distributions relative to ABR around x ≈ 0.8 mm; ABR and PAB average arrival times are in good mutual agreement but differ from BTC near x ≈ 6 mm. Overall, multiple observables (joint patterns, marginal times/positions, local distributions, first- vs all-arrivals) provide experimentally distinguishable signatures of the different proposals.

The results directly address whether a double-slit experiment can distinguish among arrival-time proposals associated with different formulations and interpretations of quantum mechanics. By employing a horizontal detection screen to probe near-field dynamics and trajectory recursions, the study identifies observables—such as joint spatiotemporal fringes, average arrival times versus position, and local conditional distributions—that exhibit clear, measurable differences among semiclassical, standard POVM-based, quantum flux, Bohmian truncated current, and absorbing-boundary models. The analysis shows that discrepancies are not merely due to negative current contributions (backflow): even excluding negative J·n, QF and STD remain distinguishable. Bohmian trajectory simulations clarify the role of multi-crossings (all-arrivals matching QF, first-arrivals diverging), providing operational interpretations for different detector schemes. With modern single-atom detectors offering sub-nanosecond timing and sub-millimeter spatial resolution, the proposed measurements are realistic; the study identifies specific spatial regions (e.g., x ≈ 16–19 mm) and time windows (∼0.01–0.2 ms no-arrival gaps) as practical targets. Incorporating detector back-action further distinguishes phenomenological models (ABR vs PAB) and their departure from first-arrival BTC statistics. These findings inform the measurement problem by linking operational detection models to distinct experimental signatures, offering a route to empirically test foundational differences in quantum interpretations.

The paper proposes and numerically substantiates a feasible double-slit experiment with a horizontal screen that can distinguish between several rival arrival-time proposals stemming from different interpretations and formulations of quantum mechanics. Differences manifest in joint spatiotemporal patterns, marginal time and position distributions, local conditional times, and first- versus all-arrival statistics, and are resolvable with existing single-atom detection technology. The study suggests metastable helium as a suitable source and identifies concrete parameter regimes and spatial regions for testing. Potential extensions include using heavier atoms to amplify discrepancies, atom interferometry in accelerator rings, and entangled-atom configurations (e.g., double–double-slit). Although photon-based experiments may have practical benefits, their theoretical analysis is more subtle due to relativistic localization and causality issues; adapting the proposal to photons is left for future work.

• The study is based on numerical simulations with assumed Gaussian initial states and soft-edged slits; real experimental imperfections may alter detailed patterns. - Distinguishability depends on near-field configurations and precise time-resolved detection; achieving and maintaining the required coherence and alignment is nontrivial. - Detector back-action is modeled phenomenologically (ABR and PAB) with tunable parameters (e.g., κ, λ); conclusions may vary with detector modeling details. - BTC (first-arrival) statistics require barrier-like detection and extensive trajectory simulations; experimental realization must closely match the assumed scheme. - Results are shown for specific parameter choices (metastable He, velocities, slit separation); generality across different particles or geometries requires further study. - Photonic implementations face additional theoretical complications (relativistic localization/causality), beyond the scope of the present nonrelativistic analysis.