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arXiv:2608.26778

Observing relativistic trajectories of single photons

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

quant-phSubmitted Aug 27, 2026• Announced Aug 28, 2026• 0 citations
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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.

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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.
Bohmian mechanics, also referred to as the de Broglie-Bohm pilot-wave theory, represents a deterministic and nonlocal interpretation of quantum mechanics. Central to this framework is a description of quantum motion in terms of particle trajectories, whose reconciliation with relativity remains a fundamental challenge. Although relativistic guidance laws have been proposed, their predicted dynamics has remained experimentally untested. Here we use weak measurements in a single-photon interferometer to jointly reconstruct the photon energy and momentum weak values and the associated relativistic velocity field. The reconstructed average trajectories agree with theoretical predictions. Near destructive interference, the flow exhibits local reconstructed velocities exceeding the speed of light and negative values of the effective squared-mass parameter. Our results establish an experimental basis for investigating the physical meaning of quantum trajectories and their role in describing motion in relativistic spacetime.
Bohmian mechanics is a nonlocal hidden-variable interpretation of quantum theory which predicts that particles follow deterministic trajectories in spacetime. Historically, the study of Bohmian trajectories has mainly been restricted to nonrelativistic regimes due to the widely held belief that the theory is incompatible with special relativity. Here, we present an approach for constructing the relativistic Bohmian-type velocity field of single particles. The advantage of our proposal is that it is operational in nature, grounded in weak measurements of the particle's momentum and energy. We apply our weak measurement formalism to obtain the relativistic spacetime trajectories of photons in a Michelson-Sagnac interferometer. The trajectories satisfy quantum-mechanical continuity and the relativistic velocity addition rule. We propose a modified Alcubierre metric which could give rise to these trajectories within the paradigm of general relativity.
In a recent article [Nat. Commun. 13, 2 (2022)], we devised a method of constructing the Lorentz-covariant Bohmian trajectories of single photons via weak measurements of the photon's momentum and energy. However, whether such a framework can consistently describe multiparticle interactions remains to be seen. Here we present a nontrivial generalization of our framework to describe the relativistic Bohmian trajectories of two interacting photons (this interaction arising from the symmetrization of the two-particle wave function) exhibiting nonclassical interference due to their indistinguishability. We begin by deriving the average velocity fields of the indistinguishable photons using a conditional weak measurement protocol, with detectors that are agnostic to the identity of the respective photons. We demonstrate a direct correspondence between the operationally derived trajectories and those obtained using a position- and time-symmetrized multiparticle Klein-Gordon wave function, whose dynamics are manifestly Lorentz covariant. We propose a space-time metric that depends nonlocally on the positions of both particles as a curvature-based interpretation of the resulting trajectories. Contrary to prior expectations, our results demonstrate a consistent trajectory-based interpretation of relativistic multiparticle dynamics in quantum theory.
Standard quantum mechanics famously disallows to simultaneously measure particle position and momentum with arbitrary precision. In contrast, Bohmian mechanics, a hidden-variable interpretation, at the same time ascribes reality to the positions and momenta of quantum particles at the cost of a nonlocal ontology [1]. It thus allows to conceive of definite particle trajectories, while being fully compatible with the standard theory in all empirical predictions. To record such trajectories [2] we consider two optical settings using weak measurements in order to investigate different features of particle propagation in the Bohmian picture.
Bohmian mechanics has garnered significant attention as an interpretation of quantum theory since the paradigmatic experiments by Kocsis et. al. [Science 332, 6034 (2011)] and Mahler et. al. [Sci. Adv. 2, 2 (2016)], which inferred the average trajectories of photons in the nonrelativistic regime. These experiments were largely motivated by Wiseman's formulation of Bohmian mechanics, which grounded these trajectories in weak measurements. Recently, Wiseman's framework was extended to the relativistic regime by expressing the velocity field of single photons in terms of weak values of the photon energy and momentum. Here, we propose an operational, weak value-based definition for the Bohmian "local mass" of relativistic single particles. For relativistic wavefunctions satisfying the scalar Klein-Gordon equation, this mass coincides with the effective mass defined by de Broglie in his relativistic pilot-wave theory, a quantity closely connected with the quantum potential that is responsible for Bohmian trajectory self-bending and the anomalous photoelectric effect. We demonstrate the relationship between the photon trajectories and the mass in an interferometric setup.
Bohmian mechanics has garnered significant attention as an interpretation of quantum theory since the paradigmatic experiments by [Kocsis et al., Science 332, 1170 (2011)] and [Mahler et al., Sci. Adv. 2, e1501466 (2016)], which inferred the average trajectories of photons in the nonrelativistic regime. These experiments were largely motivated by Wiseman's formulation of Bohmian mechanics, which grounded these trajectories in weak measurements. Recently, Wiseman's framework was extended to the relativistic regime by expressing the velocity field of single photons in terms of weak values of the photon energy and momentum. Here, we propose an operational, weak value-based definition for the Bohmian ``local mass'' of relativistic single particles. For relativistic wave functions satisfying the scalar Klein-Gordon equation, this mass coincides with the effective mass defined by de Broglie in his relativistic pilot-wave theory, a quantity closely connected with the quantum potential that is responsible for Bohmian trajectory self-bending and the anomalous photoelectric effect. We demonstrate the relationship between the photon trajectories and the mass in an interferometric setup.