Weak-Measurement Reconstruction of the Conditional Momentum Field for Atomic Matter Waves: A Dispersive Photon-Curtain Protocol Extending Kocsis–Steinberg

Weak measurement enabled reconstruction of average particle-like trajectories for photons in double-slit interferometers, but extension to massive matter waves has been hindered by the Heisenberg-microscope obstruction: short-wavelength probes destroy interference, while long-wavelength probes lack spatial resolution. We propose a protocol that circumvents this obstruction with a transverse photon curtain operating in the dispersive (off-resonant) regime as a weak probe of atomic matter waves through a double-slit. The far-detuned coupling encodes atomic transverse position in the photon's deflection while the absorptive part is suppressed by Γ/Δ.

We provide an operator-level analysis distinguishing the per-photon weak coupling g_γ from the cumulative atomic phase g_a, derive a post-selection-averaged reconstruction formula relating the measured weak-value profile to the conditional momentum field — which by Wiseman's theorem equals the de Broglie–Bohm guidance velocity — and present a decoherence analysis with V² + D² = 1 as expected for pure-state dephasing.

A self-consistent operating point for ⁸⁷Rb (Configuration D: 462 μW curtain at 200 GHz detuning, L = 60 cm, d = 2.5 μm, N_a = 10⁴ atoms) yields per-atom decoherence Λ₁ ≈ 0.45, ensemble signal-to-noise ratio ≈ 67, and a forward-modeled trajectory RMS of 243 μm at the screen — 33% of the fringe spacing. The reconstruction formula is verified analytically against the exact Bohmian field at <1% mean error on a 28-point grid.

The dominant remaining experimental challenge is calibration of the curtain's deterministic dipole-force back-action at relative precision ε_cal ~ 10⁻⁵, achievable in situ via single-slit atomic measurement analogous to demonstrated trap-potential mapping. We adopt the operational position throughout: the protocol measures a vector field that several distinct interpretations identify as physically meaningful; we make no ontological claim about underlying particle trajectories.