Observing relativistic trajectories of single photons

AI summary

Overview

Research area: Quantum foundations and quantum optics, specifically the experimental reconstruction of relativistic Bohmian (de Broglie–Bohm) trajectories using weak-value measurements.

Technical level: Intermediate. The paper combines weak-measurement theory, relativistic quantum mechanics, and a tabletop interferometry experiment; the core ideas are accessible, but the derivation of the velocity field from Klein–Gordon current and density requires some quantum optics background.

Scope (one sentence): This paper proposes and experimentally implements a protocol that reconstructs the relativistic Bohmian trajectories of single photons interfering head-on in a Michelson–Sagnac interferometer, observing both subluminal and superluminal behaviour in the fringes.

What This Paper Is About

Standard quantum mechanics does not assign definite paths to particles, but Bohmian mechanics does — and only recently has an operational method for making Bohmian mechanics consistent with special relativity been proposed (by Foo et al.). The authors build and run an experiment to reconstruct those relativistic Bohmian trajectories for a single photon, where counter-propagating probability amplitudes interfere head-on at the speed of light. The goal is to give experimental access to relativistic Bohmian mechanics and test its counterintuitive predictions directly.

Key Contributions

1. First experimental protocol for relativistic Bohmian velocity: The authors present, for the first time, a theoretical protocol for experimentally measuring the relativistic Bohmian velocity predicted by Foo et al., reducing the general requirement (weak measurement of both momentum and energy) to a weak measurement of momentum along the propagation direction under their experimental conditions.

2. A working implementation in a Michelson–Sagnac interferometer: They implement the protocol using single photons from an attenuated continuous-wave laser at 925 nm, with polarization as the pointer for the weak measurement and a translatable 50:50 measurement beam-splitter (MBS) to probe position.

3. Reconstruction of relativistic Bohmian trajectories from photon counts: They derive an explicit experimental formula, v_expt(x) = [κ_eff N1(x) − N2(x)] / [sin(2θ)(κ_eff N1(x) + N2(x))], allowing the velocity field to be extracted from detector counts alone.

4. Observation of sub- and superluminal photon motion in the fringes: They observe trajectories that curve through the interference pattern, alternating between speeds below and above c depending on whether the photon is in a region of high or low probability density.

Main Findings

• Subluminal in bright regions, superluminal in dark regions: The Bohmian velocity-vector-angle χ = tan⁻¹[v(x)] exceeds 45° (superluminal) in regions of low probability density and falls below 45° (subluminal) in regions of high probability density. Trajectories bunch together where density is high and diverge where density is low.

• No interference means speed of light: For a 0–100 splitting ratio, the trajectories are straight lines at 45° on the spacetime plot with constant probability density — the relativistic Bohmian velocity equals the velocity of light in the absence of interference.

• Interference curves the trajectories: For 5–95 and 20–80 splitting ratios, the photon trajectories curve. The authors argue these curves are a feature of the underlying physics rather than experimental noise, supported by matching theoretical trajectory plots.

• Contrast grows with interference visibility: The difference between subluminal and superluminal velocity-vector-angles is larger for the 20–80 splitting ratio than for the 5–95 splitting ratio, consistent with the fact that the probability density varies with MBS position while the probability current stays constant.

• Reversing the splitting reverses the path: If the splitting ratios are reversed (e.g., 80–20), the photon trajectories follow the opposite path through the interferometer (shown in the supplementary material).

• Constant current, varying density: The probability current j_x ∝ (1 − 2η) is constant in both x and t and depends only on the splitting ratio, while the probability density ρ ∝ (1 + 2cos(φ)√(η(1−η))) varies with the relative phase φ.

• Relativistic consistency: The velocity v(x,t) = j_x/ρ is written as the ratio of the Klein–Gordon probability current and density, which are components of the Klein–Gordon charge-current two-vector and transform covariantly under Lorentz transformation. The continuity equation ensures the density of trajectories matches the Born rule.

• Experimental parameters reported: The HWP2 offset was fixed at θ = 4° and HWP3 at 2°, giving ε ≈ 0.14. The MBS was scanned over a range of 1 μm in 20 nm increments, with counts recorded for 60 s per position across three detection channels (D1, D2, D3). The effective detection efficiency κ_eff was 0.62. Simultaneous counts within a 1 ns interval occurred below 10⁻³, confirming the low-photon regime.

Methodology in Plain English

The team sends heavily attenuated laser light — effectively one photon at a time — into a Michelson–Sagnac interferometer, a setup where two copies of the photon's wavefunction travel in opposite directions and meet head-on. A half-wave plate and polarizing beam-splitter (HWP1 and PBS1) set how much amplitude goes down each path, i.e. the splitting ratio η.

To extract the Bohmian velocity, they use the photon's polarization as a "pointer." A half-wave plate (HWP2) in one arm is rotated by a small angle θ away from its ideal 45° setting, which very weakly couples the photon's momentum information into its polarization without destroying the interference pattern. A second half-wave plate (HWP3) outside the interferometer, offset by θ′ = 2° (with θ = 2θ′), followed by a polarizing beam-splitter (PBS2), analyzes the polarization and directs light to detectors.

The key trick is a movable 50:50 beam-splitter (MBS) on a motorized translation stage. By sliding it along the propagation direction in 20 nm steps over 1 μm, the team samples both the interference fringes and the momentum weak value at many positions. From the sum and difference of counts in detectors D1 and D2 — corrected by the measured detection efficiency — they compute the velocity at each position using their derived formula, then stack the velocity vectors in a spacetime plot to draw the trajectories. A third detector, D3, does not enter the reconstruction but confirms the experiment stays in the single-photon regime.

Why This Matters

Impact on research: This is the first experimental reconstruction of relativistic Bohmian trajectories, turning a recently proposed operational framework (Foo et al.) into a laboratory technique. It gives physicists a way to test interpretations of quantum mechanics in the relativistic regime, where Bohmian mechanics has historically struggled, and connects the Bohmian picture to spacetime curvature, which here replaces the non-relativistic quantum potential.

Real-world applications (as pathways, not demonstrated technologies):

• Benchmarks and techniques for weak-value metrology and precision interferometry.
• Methods for characterizing single-photon interference in head-on counter-propagating geometries, relevant to integrated photonic circuits.
• A testbed for relativistic quantum information protocols, where notions of which-path and nonlocality must be reconciled with Lorentz covariance.
• A teaching and conceptual tool for contrasting the Copenhagen and Bohmian interpretations of the same interference data.

Industry relevance: The weak-measurement and interferometer techniques have overlap with quantum sensing, precision optical metrology, and photonic hardware development, particularly where superconducting nanowire single-photon detectors (SNSPDs) and polarization-based weak coupling are already standard tools.

Future Directions

• Entangled photons: Extending the experiment to entangled photons, which the authors describe as a pathway to exploring nonlocality in the relativistic regime.
• Boosted reference frames: Investigating the experiment from moving reference frames, which the authors suggest could lead to measurement of backwards-in-time trajectories.
• Closer to balanced splitting: Approaching the 50–50 splitting ratio makes the trajectories more sensitive to experimental errors, so improved stability and error handling would let the technique probe more extreme regimes.
• Comparison with alternative approaches: The authors note that a different experimental approach to the relativistic proposal was recently reported in a double-slit interferometer setup, opening the question of how the two methods compare.

Target Audience

This paper is most valuable to researchers in quantum foundations and quantum optics — particularly those working on Bohmian mechanics, weak-value measurements, and relativistic quantum information — as well as graduate students and experimentally minded theorists interested in testing interpretations of quantum mechanics. It is also relevant to interferometry and precision-measurement groups, who may find the weak-value protocol and head-on interference geometry useful beyond foundational questions.

Authors’ abstract

While the standard interpretation of quantum mechanics does not assign definite trajectories to particles, the Bohmian interpretation does. Only recently has an operational method for reconciling Bohmian mechanics with relativity been proposed. Here, we experimentally reconstruct relativistic Bohmian trajectories of a single photon in a Michelson-Sagnac interferometer, where counter-propagating probability amplitudes interfere head-on at the speed of light. As predicted by the relativistic Bohmian theory, we observe subluminal and superluminal features of the Bohmian trajectories of the photon traversing through the fringes. Our work provides experimental access to relativistic Bohmian mechanics and enables exploration of its unusual and counterintuitive properties.