Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Unpredictable Tunneling of a Classical Wave-Particle Association

Antonin Eddi, Emmanuel Fort, Frédéric Moisy, Yves Couder

Published Jun 16, 2009• 203 citations
Publisher

Abstract

A droplet bouncing on a vibrated bath becomes a "walker" moving at constant velocity on the interface when it couples to the surface wave it generates. Here the motion of a walker is investigated when it collides with barriers of various thicknesses. Surprisingly, it undergoes a form of tunneling: the reflection or transmission of a given incident walker is unpredictable. However, the crossing probability decreases exponentially with increasing barrier width. This shows that this wave-particle association has a nonlocality sufficient to generate a quantumlike tunneling at a macroscopic scale.

DOI
10.1103/physrevlett.102.240401

Related papers

Tunneling with a hydrodynamic pilot-wave model

André Nachbin, Paul A. Milewski, John W. M. Bush

2009)] presented experimental results demonstrating the unpredictable tunneling of a classical wave-particle association as may arise when a droplet walking across the surface of a vibrating fluid bath approaches a submerged barrier.We here present a theoretical model that captures the influence of bottom topography on this wave-particle association and so enables us to investigate its interaction with barriers.The coupled wave-droplet dynamics results in unpredictable tunneling events.As reported in the experiments by Eddi et al. and as is the case in quantum tunneling [Gamow, Nature (London) 122, 805 (1928)], the predicted tunneling probability decreases exponentially with increasing barrier width.In the parameter regimes examined, tunneling between two cavities suggests an underlying stationary ergodic process for the droplet's position.

Physical Review Fluids•Mar 30, 2017•68 citations

Predictability in a hydrodynamic pilot-wave system: Resolution of walker tunneling

Loïc Tadrist, Tristan Gilet, Peter Schlagheck +1 more

A walker is a macroscopic coupling of a droplet and a capillary wave field that exhibits several quantumlike properties. In 2009, Eddi et al. [Phys. Rev. Lett. 102, 240401 (2009)PRLTAO0031-900710.1103/PhysRevLett.102.240401] showed that walkers may cross a submerged barrier in an unpredictable manner and named this behavior "unpredictable walker tunneling." In quantum mechanics, tunneling is one of the simplest arrangements where similar unpredictability occurs. In this paper, we investigate how unpredictability can be unveiled for walkers through an experimental study of walker tunneling with precision. We refine both time and position measurements to take into account the fast bouncing dynamics of the system. Tunneling is shown to be unpredictable until a distance of 2.6 mm from the barrier center, where we observe the separation of reflected and transmitted trajectories in the position-velocity phase-space. The unpredictability is unlikely to be attributable to either uncertainty in the initial conditions or to the noise in the experiment. It is more likely due to changes in the drop's vertical dynamics arising when it interacts with the barrier. We compare this macroscopic system to a tunneling quantum particle that is subjected to repeated measurements of its position and momentum. We show that, despite the different theoretical treatments of these two disparate systems, similar patterns emerge in the position-velocity phase space.

Physical review. E•Jul 9, 2020•31 citations

Tunneling in a Lorenz-like model for an active wave-particle entity

Runze Xu, Rahil N. Valani

Active wave-particle entities (WPEs) emerge as self-propelled oil droplets on the free surface of a vibrating oil bath. The particle (droplet) periodically imprints decaying waves on the liquid surface which in turn guide the particle motion, resulting in a two-way coupling between the particle and its self-generated waves. Such WPEs have been shown to exhibit hydrodynamic analogs of various quantum features. In this work, we theoretically and numerically explore a dynamical analog of tunneling by considering a simple setup of a one-dimensional WPE incident on an isolated Gaussian potential barrier. Our idealized model takes the form of a perturbed Lorenz system, which we use to explore the dynamics and statistics of barrier crossing as a function of initial conditions and system parameters. Our work highlights that velocity fluctuations of the WPE at high memories that are rooted in nonequilibrium features of the Lorenz system, such as spiraling motion toward equilibrium points and transient chaos, give rise to (i) sensitivity and unpredictability in barrier crossing, (ii) smooth variations in transmission probability as a function of system parameters, and (iii) wave-like features in the transmitted and reflected probability density profiles.

arXiv:2408.13761•Physical review. E•Mar 25, 2025•1 citation

Tunneling of walking oil drops

Mogens T. Levinsen

Walkers are oil drops surfing on a vibrated oil surface and driven by their self-generated capillary waves. Since some of the first measurements on walkers seemingly showed quantum-like behavior, walkers have been considered a model system for a hydrodynamic pilot-wave system. An early experiment showed that the passage of the walker over a submerged barrier resulted in what was coined "unpredictable walker tunneling" in analogy with quantum tunneling. Together with the double slit experiment, tunneling is one of the fundamental phenomena that show the probabilistic nature of quantum mechanics. A later experiment that refined the measurement of position and timing of the walker narrowed the region where transmission changed from 0 to 1 by a factor 10 but still found "unpredictable tunneling." However, at a certain small distance close to the rim of the barrier, predictability sets in, and separation between reflected and transmitted trajectories is observed. After careful analysis of identified noise sources, the authors find that neither noise nor insufficient knowledge of initial conditions alone could explain the unpredictability. They therefore conclude that it is more likely due to changes in the drop's vertical dynamics arising when it interacts with the barrier. They also analyzed a quantum particle impinging on a barrier by step-wise collapsing the wave function and showed a similar behavior for this. Here we revisit the "tunneling" of walkers. Enforcing a stringent control of temperature in all aspects of the experiment results in the observation of a much sharper transition, much like that of a classical system, reducing the unpredictability by another factor of 10. Apart from varying the velocity of the walker by changing the drop size, we varied it by changing the amplitude of the drive. We furthermore found that a micron-size change in fluid height changes transmission from 0 to 1. The observed sharpness of the transition suggests that in a noise-free environment the transition would be truly abrupt, at least in the regions of phase-space for the drop that have been investigated so far.

Physical review. E•Nov 12, 2025•0 citations

Dynamics and statistics of wave-particle interactions in a confined geometry

Tristan Gilet

A walker is a droplet bouncing on a liquid surface and propelled by the waves that it generates. This macroscopic wave-particle association exhibits behaviors reminiscent of quantum particles. This article presents a toy model of the coupling between a particle and a confined standing wave. The resulting two-dimensional iterated map captures many features of the walker dynamics observed in different configurations of confinement. These features include the time decomposition of the chaotic trajectory in quantized eigenstates and the particle statistics being shaped by the wave. It shows that deterministic wave-particle coupling expressed in its simplest form can account for some quantumlike behaviors.

Physical Review E•Nov 21, 2014•56 citations

The pilot-wave dynamics of walking droplets

Daniel M. Harris, John W. M. Bush

A millimetric droplet can be induced to bounce on the surface of a fluid bath by vibrating the bath near the droplet's resonant frequency (Figure 1(a) ). [superscript 1–3] The localized field of Faraday waves excited by the bouncing droplet can cause it to propel itself laterally across the surface, moving in resonance with its guiding wave field (Figure 1(b) ). [superscript 4,5] These walking droplets, or “walkers,” generally move in a straight line at constant speed; however, they can be diverted through interaction with boundaries or external forces. This hydrodynamic system represents a macroscopic realization of the pilot-wave theory of quantum dynamics proposed by Louis de Broglie, according to which microscopic particles are propelled through a resonant interaction with a wave field generated by the particle's internal vibration. [superscript 6] Coincidentally, it exhibits many behaviors once thought to be exclusive to the microscopic quantum realm, including single-particle diffraction, [superscript 7] tunneling, [superscript 8] quantized orbits, [superscript 9] and orbital-level splitting. [superscript 10]

Physics of Fluids•Sep 1, 2013•22 citations
A droplet bouncing on a vibrated bath becomes a "walker" moving at constant velocity on the interface when it couples to the surface wave it generates. Here the motion of a walker is investigated when it collides with barriers of various thicknesses. Surprisingly, it undergoes a form of tunneling: the reflection or transmission of a given incident walker is unpredictable. However, the crossing probability decreases exponentially with increasing barrier width. This shows that this wave-particle association has a nonlocality sufficient to generate a quantumlike tunneling at a macroscopic scale.
2009)] presented experimental results demonstrating the unpredictable tunneling of a classical wave-particle association as may arise when a droplet walking across the surface of a vibrating fluid bath approaches a submerged barrier.We here present a theoretical model that captures the influence of bottom topography on this wave-particle association and so enables us to investigate its interaction with barriers.The coupled wave-droplet dynamics results in unpredictable tunneling events.As reported in the experiments by Eddi et al. and as is the case in quantum tunneling [Gamow, Nature (London) 122, 805 (1928)], the predicted tunneling probability decreases exponentially with increasing barrier width.In the parameter regimes examined, tunneling between two cavities suggests an underlying stationary ergodic process for the droplet's position.
A walker is a macroscopic coupling of a droplet and a capillary wave field that exhibits several quantumlike properties. In 2009, Eddi et al. [Phys. Rev. Lett. 102, 240401 (2009)PRLTAO0031-900710.1103/PhysRevLett.102.240401] showed that walkers may cross a submerged barrier in an unpredictable manner and named this behavior "unpredictable walker tunneling." In quantum mechanics, tunneling is one of the simplest arrangements where similar unpredictability occurs. In this paper, we investigate how unpredictability can be unveiled for walkers through an experimental study of walker tunneling with precision. We refine both time and position measurements to take into account the fast bouncing dynamics of the system. Tunneling is shown to be unpredictable until a distance of 2.6 mm from the barrier center, where we observe the separation of reflected and transmitted trajectories in the position-velocity phase-space. The unpredictability is unlikely to be attributable to either uncertainty in the initial conditions or to the noise in the experiment. It is more likely due to changes in the drop's vertical dynamics arising when it interacts with the barrier. We compare this macroscopic system to a tunneling quantum particle that is subjected to repeated measurements of its position and momentum. We show that, despite the different theoretical treatments of these two disparate systems, similar patterns emerge in the position-velocity phase space.
Active wave-particle entities (WPEs) emerge as self-propelled oil droplets on the free surface of a vibrating oil bath. The particle (droplet) periodically imprints decaying waves on the liquid surface which in turn guide the particle motion, resulting in a two-way coupling between the particle and its self-generated waves. Such WPEs have been shown to exhibit hydrodynamic analogs of various quantum features. In this work, we theoretically and numerically explore a dynamical analog of tunneling by considering a simple setup of a one-dimensional WPE incident on an isolated Gaussian potential barrier. Our idealized model takes the form of a perturbed Lorenz system, which we use to explore the dynamics and statistics of barrier crossing as a function of initial conditions and system parameters. Our work highlights that velocity fluctuations of the WPE at high memories that are rooted in nonequilibrium features of the Lorenz system, such as spiraling motion toward equilibrium points and transient chaos, give rise to (i) sensitivity and unpredictability in barrier crossing, (ii) smooth variations in transmission probability as a function of system parameters, and (iii) wave-like features in the transmitted and reflected probability density profiles.
Walkers are oil drops surfing on a vibrated oil surface and driven by their self-generated capillary waves. Since some of the first measurements on walkers seemingly showed quantum-like behavior, walkers have been considered a model system for a hydrodynamic pilot-wave system. An early experiment showed that the passage of the walker over a submerged barrier resulted in what was coined "unpredictable walker tunneling" in analogy with quantum tunneling. Together with the double slit experiment, tunneling is one of the fundamental phenomena that show the probabilistic nature of quantum mechanics. A later experiment that refined the measurement of position and timing of the walker narrowed the region where transmission changed from 0 to 1 by a factor 10 but still found "unpredictable tunneling." However, at a certain small distance close to the rim of the barrier, predictability sets in, and separation between reflected and transmitted trajectories is observed. After careful analysis of identified noise sources, the authors find that neither noise nor insufficient knowledge of initial conditions alone could explain the unpredictability. They therefore conclude that it is more likely due to changes in the drop's vertical dynamics arising when it interacts with the barrier. They also analyzed a quantum particle impinging on a barrier by step-wise collapsing the wave function and showed a similar behavior for this. Here we revisit the "tunneling" of walkers. Enforcing a stringent control of temperature in all aspects of the experiment results in the observation of a much sharper transition, much like that of a classical system, reducing the unpredictability by another factor of 10. Apart from varying the velocity of the walker by changing the drop size, we varied it by changing the amplitude of the drive. We furthermore found that a micron-size change in fluid height changes transmission from 0 to 1. The observed sharpness of the transition suggests that in a noise-free environment the transition would be truly abrupt, at least in the regions of phase-space for the drop that have been investigated so far.
A walker is a droplet bouncing on a liquid surface and propelled by the waves that it generates. This macroscopic wave-particle association exhibits behaviors reminiscent of quantum particles. This article presents a toy model of the coupling between a particle and a confined standing wave. The resulting two-dimensional iterated map captures many features of the walker dynamics observed in different configurations of confinement. These features include the time decomposition of the chaotic trajectory in quantized eigenstates and the particle statistics being shaped by the wave. It shows that deterministic wave-particle coupling expressed in its simplest form can account for some quantumlike behaviors.
A millimetric droplet can be induced to bounce on the surface of a fluid bath by vibrating the bath near the droplet's resonant frequency (Figure 1(a) ). [superscript 1–3] The localized field of Faraday waves excited by the bouncing droplet can cause it to propel itself laterally across the surface, moving in resonance with its guiding wave field (Figure 1(b) ). [superscript 4,5] These walking droplets, or “walkers,” generally move in a straight line at constant speed; however, they can be diverted through interaction with boundaries or external forces. This hydrodynamic system represents a macroscopic realization of the pilot-wave theory of quantum dynamics proposed by Louis de Broglie, according to which microscopic particles are propelled through a resonant interaction with a wave field generated by the particle's internal vibration. [superscript 6] Coincidentally, it exhibits many behaviors once thought to be exclusive to the microscopic quantum realm, including single-particle diffraction, [superscript 7] tunneling, [superscript 8] quantized orbits, [superscript 9] and orbital-level splitting. [superscript 10]