⚛️ quantum physics

Observing relativistic trajectories of single photons

This paper experimentally reconstructs relativistic Bohmian trajectories of a single photon in a Michelson-Sagnac interferometer, observing predicted subluminal and superluminal features to provide the first operational access to relativistic Bohmian mechanics.

Original authors: Sayantan Das, Daniel S. Dahl, Daniel Peace, Marcelo P. Almeida, Abhishek Roy, Markus Rambach, Andrew G. White, Timothy C. Ralph, Jacquiline Romero

Published 2026-08-28

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CC BY 4.0

Original authors: Sayantan Das, Daniel S. Dahl, Daniel Peace, Marcelo P. Almeida, Abhishek Roy, Markus Rambach, Andrew G. White, Timothy C. Ralph, Jacquiline Romero

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of classical physics, the path of a moving object is a straightforward story. If you know where a ball is and how fast it is going at any single moment, you can calculate exactly where it will be a second later, or a year from now. The universe, in this view, is a clockwork machine where every particle follows a definite, predictable line. Quantum mechanics, the rulebook for the very small, shattered this certainty. In the standard way physicists have understood quantum theory for nearly a century, particles like photons do not have a single, real path. Instead, they exist as a spread-out wave of possibilities, and asking "where is the particle right now?" is considered a meaningless question until the moment it is measured. This has led to a long-standing debate about whether particles are truly wandering without a map, or if they are simply following a hidden path that our current tools cannot see.

A specific interpretation of quantum theory, known as Bohmian mechanics, argues that particles do have definite trajectories, even if they are hard to find. However, for decades, this idea struggled to fit with Einstein's theory of relativity, which governs how things move at the highest speeds. Recently, a new theoretical framework emerged that successfully merged these two worlds, proposing a way to calculate the speed and path of a single photon even when it is moving at the speed of light. This new theory predicted something strange: that a photon's path could appear to move slower than light in some places and faster than light in others, depending on how its wave-like nature interferes with itself. Until now, this remained a mathematical curiosity, untested in the real world.

A team of researchers at the University of Queensland, working with colleagues from the University of Sydney and Macquarie University, has now brought this theory into the laboratory. They set out to reconstruct the actual path of a single photon as it traveled through a specialized device called a Michelson-Sagnac interferometer. In this setup, a single photon is split into two parts that travel in opposite directions before meeting again. The researchers did not try to track the photon directly, which is impossible without destroying it. Instead, they used a clever technique involving "weak measurements." Imagine trying to determine the speed of a ghost by gently brushing past it without disturbing its flight; this is essentially what the team did. They performed a series of gentle, non-invasive checks on the photon's momentum, followed by a final, strong check on where the photon ended up. By repeating this process thousands of times with many identical photons, they were able to piece together an average path that the photons seemed to follow.

The results confirmed the predictions of the new relativistic theory. As the photons traveled through the interference pattern created by their two opposing paths, their reconstructed trajectories showed a bizarre behavior. In regions where the waves reinforced each other, the photons appeared to move slower than the speed of light. In regions where the waves canceled each other out, the photons appeared to move faster than the speed of light. This does not mean the photons broke the universal speed limit or sent information back in time; rather, it reveals a counterintuitive property of the Bohmian path itself. The researchers found that a single photon, which standard physics says takes one path, effectively alternates between these sub-light and super-light speeds as it traverses the interference fringes.

The experiment was conducted using single photons generated from a laser, carefully controlled to ensure they behaved as individual particles. The team used a series of mirrors and beam splitters to guide the light, and a specific type of half-waveplate to gently encode the photon's momentum via polarization just enough to gather the necessary data without collapsing its wave nature. The data they collected showed that the photon's behavior, while radically different from our everyday experience, fits perfectly within a classical relativistic framework when viewed through the lens of Bohmian mechanics. The study provides the first experimental access to these relativistic trajectories, proving that the unusual, counterintuitive properties predicted by this specific interpretation of quantum mechanics are not just theoretical artifacts, but real features of the physical world. By mapping these paths, the researchers have opened a new window into understanding how particles move at the very limits of speed and space, offering a concrete way to explore a version of reality where particles always have a place to be, even if that place is harder to define than we ever imagined.

Technical Summary: Observing Relativistic Trajectories of Single Photons

Problem Statement
Standard quantum mechanics posits that particles do not possess definite trajectories, describing systems instead via wavefunctions that yield probabilistic measurement outcomes. In contrast, the de Broglie-Bohm (Bohmian) interpretation assigns deterministic trajectories to particles. However, experimentally accessing these trajectories is constrained by the Heisenberg uncertainty principle, which prohibits the simultaneous precise measurement of position and momentum for a single particle. While Wiseman demonstrated that Bohmian trajectories for an ensemble of identically prepared particles could be reconstructed using weak-value measurements, previous experimental realizations were limited to non-relativistic regimes (e.g., measuring transverse velocity components in double-slit setups). Furthermore, Bohmian mechanics has historically struggled to maintain consistency with special relativity. Although Foo et al. recently proposed an operational relativistic framework for weak-value measurements, a corresponding experimental demonstration of relativistic Bohmian trajectories for a single photon had not yet been achieved.

Methodology
The authors experimentally reconstruct the relativistic Bohmian trajectories of a single photon using a Michelson-Sagnac interferometer. The core theoretical protocol, based on Foo et al., defines the relativistic Bohmian velocity field $v(x, t)$ as the ratio of the weak value of the momentum operator $\langle \hat{k}_x \rangle_w$ to the weak value of the Hamiltonian $\langle \hat{H} \rangle_w$ .

In the experimental setup:

Interferometer Configuration: A single photon (from an attenuated continuous-wave laser) is injected into the interferometer via a variable beam-splitter (controlled by a half-wave plate HWP1 and a polarizing beam-splitter PBS1), creating a superposition of right-moving ( $|\psi_R\rangle$ ) and left-moving ( $|\psi_L\rangle$ ) states. These states interfere head-on along the propagation direction $x$ .
Weak Measurement of Momentum: The photon's polarization degree of freedom serves as the pointer for the weak measurement. A small angular deviation $\theta$ of the fast axis of a half-wave plate (HWP2) in one path weakly encodes the photon's momentum information in the propagation direction.
Position and Post-Selection: The position $x$ is determined by the location of a 50:50 measurement beam-splitter (MBS). The system is post-selected using a second half-wave plate (HWP3) and a polarizing beam-splitter (PBS2).
Balancing Visibility and Information: To satisfy the requirements of weak measurement, the experiment balances the visibility of the interference fringes against the "which-path" information. This is achieved by parametrically relating the reflectivity of the momentum-encoding element ( $\epsilon$ ) and the post-selection element ( $\epsilon'$ ) via a parameter $\delta$ , derived from the physical rotation angle $\theta$ .

Theoretical analysis in natural units ( $c=\hbar=1$ ) under the optical approximation shows that for the specific conditions of this experiment (degenerate energy eigenvalues for counter-propagating states), the relativistic Bohmian velocity depends directly on the weak value of the momentum. The velocity is expressed as a function of the relative phase $\phi$ between the two paths and the amplitude parameter $\eta$ .

Key Results
The experiment successfully maps the relativistic Bohmian trajectories of the photon as it traverses the interference fringes. The observed trajectories exhibit distinct features predicted by relativistic Bohmian theory:

Velocity Variations: As the photon moves through the interference pattern, its reconstructed Bohmian velocity alternates between subluminal and superluminal values.
Dependence on Interference: These velocity fluctuations correspond to the photon traversing regions of constructive and destructive interference, respectively.
Single Path vs. Trajectory: While the photon physically follows a single path through the interferometer, the reconstructed trajectory reveals a complex velocity profile that is radically different from the standard interpretation of particle motion.

Significance and Claims
The paper demonstrates that, for the specific experimental conditions employed, Foo et al.'s protocol can be reduced to measuring the weak value of momentum in a head-on interference scenario, thereby providing experimental access to relativistic Bohmian mechanics. By successfully reconstructing these trajectories, the work enables the exploration of the unusual and counterintuitive properties of relativistic Bohmian mechanics, specifically the existence of superluminal features within a framework that remains consistent with classical relativistic constraints (as the behavior can be fitted within a classical relativistic framework despite the apparent superluminality). The study bridges the gap between non-relativistic weak-value trajectory reconstruction and the relativistic domain, offering a new operational perspective on quantum particle dynamics.

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