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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Measurements of Bohmian Trajectories

Florian Huber, Carlotta Versmold, Jan Dziewior, Lukas Knips, Eric Meyer, Alexander Szameit, Harald Weinfurter, Jasmin D. A. Meinecke

Published Jun 26, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.1109/cleo/europe-eqec57999.2023.10232249

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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.
Weak measurement allows one to empirically determine a set of average trajectories for an ensemble of quantum particles. However, when two particles are entangled, the trajectories of the first particle can depend nonlocally on the position of the second particle. Moreover, the theory describing these trajectories, called Bohmian mechanics, predicts trajectories that were at first deemed "surreal" when the second particle is used to probe the position of the first particle. We entangle two photons and determine a set of Bohmian trajectories for one of them using weak measurements and postselection. We show that the trajectories seem surreal only if one ignores their manifest nonlocality.
To-date, the most elaborated attempt to complete quantum mechanics by the addition of hidden variables is the de Broglie-Bohm (pilot wave) theory (dBBT). It endows particles with definite positions at all times. Their evolution is governed by a deterministic dynamics. By construction, however, the individual particle trajectories generically defy detectability in principle. Of late, this lore might seem to have been called into question in light of so-called weak measurements. Due to their characteristic weak coupling between the measurement device and the system under study, they permit the experimental probing of quantum systems without essentially disturbing them. It is natural therefore to think that weak measurements of velocity in particular offer to actually observe the particle trajectories. If true, such a claim would not only experimentally demonstrate the incompleteness of quantum mechanics: it would provide support of dBBT in its standard form, singling it out from an infinitude of empirically equivalent alternative choices for the particle dynamics. Here we examine this possibility. Our result is deflationary: weak velocity measurements constitute no new arguments, let alone empirical evidence, in favour of standard dBBT; One must not naïvely identify weak and actual positions. Weak velocity measurements admit of a straightforward standard quantum mechanical interpretation, independent of any commitment to particle trajectories and velocities. This is revealed by a careful reconstruction of the physical arguments on which the description of weak velocity measurements rests. It turns out that for weak velocity measurements to be reliable, one must already presuppose dBBT in its standard form: in this sense, they can provide no new argument, empirical or otherwise, for dBBT and its standard guidance equation.
The popular impression of Bohmian mechanics is that it is standard quantum mechanics with the addition of some extra gadgets---exact particle positions and a guiding equation for particle trajectories---the advantages being that the gadgets pave the way for a resolution of the measurement problem that eschews state vector reduction while restoring the determinism lost in standard quantum mechanics. In fact, the Bohmian mechanics departs in significant ways from standard quantum mechanics. By itself this is not a basis for criticism; indeed, it makes Bohmian mechanics all the more interesting. But Bohmian mechanics is not, as the popular impression would have it, empirically equivalent to standard quantum mechanics in terms of probabilistic predictions for the outcomes of measurements of quantum observables. Indeed, in physically important applications to systems for which standard quantum mechanics delivers empirically well-confirmed probabilistic predictions, the sophisticated form of Bohmian mechanics designed to prove the global existence of Bohmian particle trajectories fails to deliver unequivocal predictions---of even a probabilistic variety---for the future behavior of said systems. Possible responses to this lacuna are discussed.
Abstract Pilot wave theory endows particles with de nite positions at all times governed by deterministic dynamics. However, individual particle trajectories are generically undetectable by experiment. This idea might seem to be contested in light of two proposals: (1) So-called ‘weak velocity measurements’, allegedly detecting Bohmian trajectories by weakly probing a quantum system without essentially disturbing it, and (2) the so-called ‘surrealistic’ trajectories experiment which supposedly establishes a conflict between the ‘actual’ position of a particle and its position derived from pilot wave theory. Although both attempts shed light on the nature of Bohmian particles, neither constitute empirical or theoretical evidence in favour or against pilot wave theory. Both instances admit a straightforward standard quantum mechanical interpretation compatible with the predictions of Bohmian theories. It is concluded that the puzzles arise from the absence of a coherent account of what quantum mechanical measurements signify.
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.
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.