Weak-Measurement Reconstruction of the Conditional Momentum Field for Atomic Matter Waves: A Dispersive Photon-Curtain Protocol Extending Kocsis–Steinberg
Revised manuscript submitted to New Journal of Physics (NJP-120031), September 2026. We propose a weak-measurement protocol for reconstructing the conditional momentum field of atomic matter waves using transverse sheets of far-detuned light ("photon curtains"). The dispersive and elastic atom–light interaction weakly encodes the atomic transverse position in the deflection of the transmitted photons. Conditioning the position-resolved deflection record on the atom's final screen position and applying an exact free-flight reconstruction identity recovers the conditional momentum field J/ρ. This field is the Madelung hydrodynamic velocity and, by Wiseman's theorem, coincides with the de Broglie–Bohm guidance velocity; the protocol measures it as an ensemble property of the quantum state and makes no ontological claim that individual atoms follow the reconstructed trajectories. The manuscript identifies the atomic operator the photon meter reads and fixes the per-photon coupling by momentum conservation; derives the reconstruction formula and its free-propagation identities, with the residual term written and bounded; obtains the measurement-induced decoherence by an explicit partial trace over the light; introduces a dual-colour, force-balanced multipass photon-curtain configuration that preserves coherence while making the deterministic optical-dipole back-action calibratable, with an in situ single-slit calibration strategy; and gives quantitative budgets for spontaneous emission, optical imbalance, Coriolis deflection, magnetic forces, background-gas collisions, screen detection and multipass optics. A complete forward model of the estimator identifies the moment contraction required for velocity reconstruction, rather than weak-value detectability, as the statistical bottleneck: five-sigma detection of the conditional momentum field at the first curtain plane requires approximately 2.2 × 10^8 atoms, one to ten weeks at the projected slit-coherent yield, and detects the far-field ballistic component of the field; quantitative mapping of the fringe-channelling structure near the interference nodes requires one to two orders of magnitude more atoms and is identified as a future target. Four engineering elements beyond the published record are stated explicitly, together with their best demonstrated values. Version 2 corrects two errors of version 1 — a per-photon coupling normalization that conflated the per-photon phase with a per-time-slice phase, and the omission of gravity from the free-fall kinematics — with more conservative feasibility conclusions and a new two-arm force-balanced multipass operating configuration; the prior atomic programme towards this measurement is now cited, and the feasibility statements were revised after an independent literature audit. Version 1 should not be relied on for its numbers. The verification code, numerical results, figure-generation scripts and reproducibility harness are archived separately (linked under Related works) and re-derive every quoted number from the model.
Revised manuscript submitted to New Journal of Physics (NJP-120031), September 2026. We propose a weak-measurement protocol for reconstructing the conditional momentum field of atomic matter waves using transverse sheets of far-detuned light ("photon curtains"). The dispersive and elastic atom–light interaction weakly encodes the atomic transverse position in the deflection of the transmitted photons. Conditioning the position-resolved deflection record on the atom's final screen position and applying an exact free-flight reconstruction identity recovers the conditional momentum field J/ρ. This field is the Madelung hydrodynamic velocity and, by Wiseman's theorem, coincides with the de Broglie–Bohm guidance velocity; the protocol measures it as an ensemble property of the quantum state and makes no ontological claim that individual atoms follow the reconstructed trajectories. The manuscript identifies the atomic operator the photon meter reads and fixes the per-photon coupling by momentum conservation; derives the reconstruction formula and its free-propagation identities, with the residual term written and bounded; obtains the measurement-induced decoherence by an explicit partial trace over the light; introduces a dual-colour, force-balanced multipass photon-curtain configuration that preserves coherence while making the deterministic optical-dipole back-action calibratable, with an in situ single-slit calibration strategy; and gives quantitative budgets for spontaneous emission, optical imbalance, Coriolis deflection, magnetic forces, background-gas collisions, screen detection and multipass optics. A complete forward model of the estimator identifies the moment contraction required for velocity reconstruction, rather than weak-value detectability, as the statistical bottleneck: five-sigma detection of the conditional momentum field at the first curtain plane requires approximately 2.2 × 10^8 atoms, one to ten weeks at the projected slit-coherent yield, and detects the far-field ballistic component of the field; quantitative mapping of the fringe-channelling structure near the interference nodes requires one to two orders of magnitude more atoms and is identified as a future target. Four engineering elements beyond the published record are stated explicitly, together with their best demonstrated values. Version 2 corrects two errors of version 1 — a per-photon coupling normalization that conflated the per-photon phase with a per-time-slice phase, and the omission of gravity from the free-fall kinematics — with more conservative feasibility conclusions and a new two-arm force-balanced multipass operating configuration; the prior atomic programme towards this measurement is now cited, and the feasibility statements were revised after an independent literature audit. Version 1 should not be relied on for its numbers. The verification code, numerical results, figure-generation scripts and reproducibility harness are archived separately (linked under Related works) and re-derive every quoted number from the model.
This manuscript proposes a solid-state electron-interferometry protocol to reconstruct quantum flow lines in an Aharonov–Bohm ring using weak values. A triggered single-electron wave packet propagates coherently through the two arms of the interferometer. A gate-defined Rashba segment implements a tunable weak coupling between the electron’s spin (as a pointer) and the local gauge-invariant kinetic momentum. Conditioning on the detected output port (postselection) allows extraction of the real part of the weak kinetic-momentum value, from which a local velocity field associated with thepostselected subensemble is reconstructed. Integrating this velocity field yields flux-dependent flow lines while preserving interference visibility in the weak-coupling regime. The work provides an operational, gauge-consistent route to map electron probability-current flow, with a clear ontological reading in de Broglie–Bohm dynamics.
Abstract We present \code{WGQED1D}, a lightweight, validated Python toolkit for microscopic simulation of single-photon scattering by N two-level atoms in a one-dimensional bidirectional waveguide. The toolkit implements the standard non-Hermitian effective Hamiltonian on the single-excitation manifold, with bidirectional dipole--dipole interactions mediated by retarded photon exchange, together with the input--output relation needed to reconstruct the output field. A careful microscopic calibration β=OD/(2N) between optical depth and per-atom waveguide-coupling efficiency is required for quantitative agreement with the free-space Beer--Lambert limit; we verify this calibration to better than 0.1% on a representative range of optical depths. The implementation is deterministic in the single-excitation subspace, runs in O(N2) memory and O(N2) operations per timestep, and produces converged results for realistic atomic ensembles (N∼200--1000) in seconds to minutes on a laptop. We document the model, implementation, and three validations: (i) pulse-averaged Beer--Lambert transmission across $\OD\in[0.25,3]$; (ii) the Wiseman--Steinberg weak-value formula $\tauT = \langle\tauG\rangle_{\rm post}$ for the post-selected atomic excitation time; and (iii) direct comparison with recent experimental data from the Angulo et al.\ measurement of negative atomic excitation time [Phys.\ Rev.\ Lett.\ \textbf{136}, 153601 (2026)]. The toolkit provides a reproducible numerical platform for studies of weak measurement, anomalous dispersion, group delay, and post-selected observables in single-photon transmission through optically thick atomic clouds. The source code is openly available under the MIT license.
This paper asks a deliberately narrower question than its predecessor: when does a Modal Triplet Theory (MTT) description reconstruct the standard nonrelativistic de Broglie-Bohm dynamics? We first specify the complete lower pilot-wave record. In addition to a complex Hilbert space and self-adjoint Schrodinger Hamiltonian, that record contains a physical configuration manifold, an actual configuration, a selected probability current, a guidance field and maximal flow, an equilibrium measure, subsystem and instrument data, a locality contract, and source and error certificates. For a regular spinless scalar Schrodinger model the standard current obeys a continuity equation, and velocity equals current divided by density away from nodes. Equivariance then propagates an initially chosen wavefunction-squared distribution. These facts are standard lower-level mathematics: continuity alone neither selects the actual configuration nor derives the initial equilibrium distribution, and it does not uniquely select the current because divergence-free modifications preserve the same continuity equation. The MTT result is therefore conditional. If one selected upper state emits the full pilot-wave record and its state, dynamics, configuration decoder, current, instruments, and records form commuting descent diagrams, then MTT reconstructs Bohmian trajectories and their operational statistics on the declared interval. The current canonical geometry is a ten-dimensional bundle over a four-dimensional Lorentzian base with a compact six-dimensional fibre; the former literal product of four-dimensional spacetime and three independent internal factors is not used. Failure of the reconstruction means that a required current, decoder, flow, or stable continuation is unavailable. It does not follow from the absence of a right inverse, and it does not establish that Bohmian mechanics is globally impossible. Standard global-existence results cover broad Hamiltonian classes for almost every equilibrium initial configuration. Entangled guidance is explicitly nonlocal in physical configuration space. Operational no-signalling can nevertheless hold in quantum equilibrium, while an MTT claim of upper-world locality requires a separate descent theorem preserving settings, local algebras, records, and independence assumptions. Measurement is treated as an ordinary system-apparatus interaction with an effective conditional wavefunction, not automatically as an admissibility-barrier crossing. The selected q=79 binary recorder provides an exact stopped operational output measure on its declared commuting Fock domain, but it does not yet source arbitrary Bohmian equilibrium ensembles or one uniquely actual history. The paper thus establishes a precise regime-limited reconstruction and a finite completion contract, not a universal derivation or a no-hidden-variable theorem.
Collective-Coordinate Dynamics and Radiative Decoherence in a Driven-Dissipative Polariton CondensateGeometric constraints on centre-of-mass path coherence and a numerical test of self-confinement This paper asks whether a room-temperature driven-dissipative exciton-polariton condensate can supply a localized collective coordinate able to occupy a resolvable two-path superposition. For the platform considered and for ordinary radiative coupling, the answer is negative for two independent reasons, and the paper states precisely where the formal construction holds and where its physical realization fails. Radiative which-path decoherence. Treating the condensate as an open system with a mode-resolved loss channel, the coherence between two localized branch states decays at ΓCOM = Nγ[1 − Re χ(d)], where χ(d) is the overlap of the photon states emitted by the two branches. The controlling variable is environmental distinguishability rather than mechanical recoil: a photon that displaces the centre of mass by a negligible amount can still record, at full strength, which branch the emitter occupied. Where ordinary out-coupling preserves transverse-mode information, two requirements act on the same length scale and conflict — branch resolvability requires a separation of several core widths, indistinguishability requires a separation well below one — so that, for a sech profile and the working convention d ≳ 3ξ, the dimensionless ratio ΓCOM/(Nγ) is bounded below by about 0.7. The coefficient is geometric; the absolute rate retains its dependence on N and γ. Numerical test of self-confinement. The operating point inferred from the measured group velocity on the CsPbBr3 microwire platform lies in the positive-effective-mass regime, where repulsive polariton interactions do not support a bright soliton. A free-propagation integration anchored to that regime shows a packet seeded at the analytic width broadening from 2.2 µm to 25.8 µm over 6 ps, so the proposed self-confined realization is not present there. A negative-mass run is retained only as a control of the propagation solver. Corrected collective wavelength. The de Broglie wavelength of the collective coordinate follows from the aggregate mass M* = N mLP, not from the single-polariton value: λCOM = h/(M*vg) = λLP/N ≈ 0.11 nm. The corresponding Fraunhofer scale far exceeds the demonstrated 60 µm propagation length, so no far-field centre-of-mass double-slit geometry is proposed for this platform. Supporting these results, the collective-coordinate construction is retained as a conditional formal sector for a rigid bright soliton: translational symmetry protects free motion exactly, the Kohn conditions protect harmonic centre-of-mass motion, and a hydrodynamic Madelung cross-check reproduces the same statement independently. In the aperture region the adiabaticity parameter is of order unity on the platform considered, so no perturbative mass-renormalization coefficient is assigned there. The paper distinguishes first-order field coherence, which a driven condensate can maintain and which is experimentally measured, from coherence between two centre-of-mass configurations, which the radiative channel destroys. The argument is independent of any interpretation of quantum mechanics. Engineering of the output reservoir so that the two branches emit into nearly indistinguishable environmental modes is identified as the route by which radiative which-path decoherence could be suppressed. Changes in version 2.0 Version 2.0 revises the central claim of version 1.0 and should be read in place of it. The title and the subject of the paper have both changed. Retracted. The suppressed decoherence rate of version 1.0 and the 1/N estimate on which it rested. That estimate measures the back-action of a quantum jump on the mean position; it does not measure the which-path information deposited in the environment, which scales with the number of constituents rather than against it. The coherence time reported in version 1.0, and the transit-time margin derived from it, do not hold. Withdrawn. The claim that the structure traverses a double slit in a coherent superposition of paths; the claim that a transition between interpretations of quantum mechanics emerges as a necessary structure of the dynamics; the reduced fringe visibility previously advanced as the primary experimental signature, which is a property of the order parameter rather than of a path superposition; and the self-confinement property, shown numerically not to hold at the operating point of the platform. The acronym under which the construction was introduced is not used further. Removed from scope. The admissibility layer and the interpretive material associated with it. This version is self-contained within condensed-matter physics and quantum decoherence, and its results depend on no unpublished work. Corrected. Parameter provenance. Version 1.0 combined quantities measured on four distinct platforms and conflated the cavity photon lifetime, the polariton lifetime and the first-order coherence time. All platform inputs are now drawn from one system and tabulated with operational definition, origin and status as measured, derived or chosen. The de Broglie wavelength of the collective coordinate is corrected. The description of the soliton solution set as a symplectic manifold is weakened to a finite-dimensional collective-coordinate manifold, and the claim to a first explicit formulation of the mapping as an identity is withdrawn; the Ehrenfest theorem is stated as a consistency check rather than as the derivation. Retained. The strong-coupling analysis, the conditional collective-coordinate reduction, the exact symmetry sectors with their hydrodynamic cross-check, and the parametric relation between mass and nonlinearity, the last now confirmed against a measurement on the platform.
According to the orthodox interpretation of quantum physics, wave-particle duality (WPD) is the intrinsic property of all massive microscopic particles. All gedanken or realistic experiments based on atom interferometers (AI) have so far upheld the principle of WPD, either by the mechanism of the Heisenberg’s position-momentum uncertainty relation or by quantum entanglement. In this paper, we propose and make a systematic quantum mechanical analysis of several schemes of weak-measurement atom interferometer (WM-AI) and compare them with the historical schemes of strong-measurement atom interferometer (SM-AI), such as Einstein’s recoiling slit and Feynman’s light microscope. As the critical part of these WM-AI setups, a weak-measurement path detector (WM-PD) deliberately interacting with the atomic internal electronic quantum states is designed and used to probe the which-path information of the atom, while only inducing negligible perturbation of the atomic center-of-mass motion. Another instrument that is used to directly interact with the atomic center-of-mass while being insensitive to the internal electronic quantum states is used to monitor the atomic center-of-mass interference pattern. Two typical schemes of WM-PD are considered. The first is the micromaser-cavity path detector, which allows us to probe the spontaneously emitted microwave photon from the incoming Rydberg atom in its excited electronic state and record unanimously the which-path information of the atom. The second is the optical-lattice Bragg-grating path detector, which can split the incoming atom beam into two different directions as determined by the internal electronic state and thus encode the which-path information of the atom into the internal states of the atom. We have used standard quantum mechanics to analyze the evolution of the atomic center-of-mass and internal electronic state wave function by directly solving Schrödinger’s equation for the composite atom-electron-photon system in these WM-AIs . We have also compared our analysis with the theoretical and experimental studies that have been presented in the previous literature. The results show that the two sets of instruments can work separately, collectively, and without mutual exclusion to enable simultaneous observation of both wave and particle nature of the atoms to a much higher level than the historical SM-AIs, while avoiding degradation from Heisenberg’s uncertainty relation and quantum entanglement. We have further investigated the space–time evolution of the internal electronic quantum state, as well as the combined atom–detector system and identified the microscopic origin and role of quantum entanglement, as emphasized in numerous previous studies. Based on these physics insights and theoretical analyses, we have proposed several new WM-AI schemes that can help to elucidate the puzzling physics of the WPD of the atoms. The principle of WM-AI scheme and quantum mechanical analyses made in this work can be directly extended to examine the principle of WPD for other massive particles.
The wave function is the cornerstone of quantum mechanics, although it is still open to different interpretations. It has been shown that the amplitude and the phase of a wave function are manifested in the imaginary and real parts of the momentum weak value, respectively. Based on this, we develop a weak measurement scheme to reconstruct the wave function through measuring the momentum weak value. A proof-of-principle experiment is made to confirm the feasibility of our measurement scheme, in which the polarization states and the momentum of light are coupled. We believe the experimental protocol has the potential to be applied in diverse fields such as wave-front sensing, edge detection, and optical communication, and measurement of the momentum weak value may offer valuable insight to better understand quantum mechanics.