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Superradiance and static Bell tests in pilot-wave hydrodynamics

Konstantinos Papatryfonos, Louis Vervoort, M. Ruelle, Corentin Bourdiol, André Nachbin, Valeri Frumkin, Matthieu Labousse, John W. M. Bush

Published Jun 9, 2026• 0 citations
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Abstract

Since its discovery in 2005, the hydrodynamic pilot-wave system has emerged as a classical platform for exploring aspects of wave–particle duality. In these systems, bouncing droplets or droplets emitted from hydrodynamic cavities propagate on a vertically vibrating fluid bath. These droplets, guided by their self-generated wave fields, continue to provide a macroscopic setting in which several features commonly associated with quantum theory may be examined from a classical perspective. Recent work has extended this program beyond single-particle analogies toward cooperative emission, bipartite tunneling, and static Bell-type correlation functions, with particular emphasis on the roles of memory, wave-mediated coupling, and contextuality.

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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
Since its discovery in 2005, the hydrodynamic pilot-wave system has emerged as a classical platform for exploring aspects of wave–particle duality. In these systems, bouncing droplets or droplets emitted from hydrodynamic cavities propagate on a vertically vibrating fluid bath. These droplets, guided by their self-generated wave fields, continue to provide a macroscopic setting in which several features commonly associated with quantum theory may be examined from a classical perspective. Recent work has extended this program beyond single-particle analogies toward cooperative emission, bipartite tunneling, and static Bell-type correlation functions, with particular emphasis on the roles of memory, wave-mediated coupling, and contextuality.
Since its discovery in 2005, the hydrodynamic pilot-wave system has provided a concrete macroscopic realization of wave-particle duality and concomitant classical analogs of a growing list of quantum effects. The question naturally arises as to whether this system might support statistical states that violate Bell's inequality, and so yield a classical analog of quantum entanglement. We here introduce a new platform for addressing this question, a numerical model of coupled bipartite tunneling in the hydrodynamic pilot-wave system. We demonstrate that, under certain conditions, the Bell inequality is violated in a static Bell test owing to correlations induced by the wave-mediated coupling between the two subsystems. The establishment of non-factorizable states with two spatially separated classical particles introduces the possibility of novel forms of quantum-inspired classical computing.
We present a number of fresh perspectives on pilot-wave hydrodynamics, the field initiated in 2005 by Couder and Fort's discovery that millimetric droplets self-propelling along the surface of a vibrating bath can capture certain features of quantum systems. A recurring theme will be that pilot-wave hydrodynamics furnishes a classical framework for reproducing many quantum phenomena and allows one to rationalize such phenomena mechanistically, from a local realist perspective, obviating the need to appeal to quantum nonlocality. The distinction is drawn between hydrodynamic pilot-wave theory and its quantum counterparts, Bohmian mechanics, the Bohm–Vigier stochastic pilot-wave theory, and de Broglie's theory of the double-solution. Each of these quantum predecessors provide a valuable touchstone as we take the physical picture engendered in the walking droplets and extend it into the quantum realm via theoretical modeling. Emphasis is given to recent developments in the field, both experimental and conceptual, and to forecasting potentially fruitful new directions.
A pilot-wave hydrodynamic system consists of a small droplet of viscous fluid that is propelled across a vibrating bath of the same fluid. Bouncing vertical motion and "walking" horizontal motion of the droplet can be achieved with careful control over the frequency and amplitude of the bath's oscillation. The droplet avoids merging with the fluid from which it was generated and exhibits intricate motion across the fluid surface. Of particular importance, the observed walking motion is due to the droplet's interaction with the waves it generates as it bounces off of the vibrating liquid surface. With each rebound, it receives transverse kicks in its motion dependent on the waves generated by its previous bounces. Ultimately, it is the droplet's interaction with the superposition of these waves that produce the complex motion observed. With variations in bath geometries and subsurface structures, the droplet's trajectory can be manipulated to induce fascinating behavior. Over short timescales, a droplet will exhibit seemingly random trajectories. However, when the droplet is observed over long timescales, patterns in the cumulative motion of the droplet begin to emerge. The patterns this system maps out over long timescales represent a compelling macroscopic analog to Louis de Broglie's double-solution theory of quantum mechanics. In order to analyze this behavior, a novel imaging scheme and data acquisition system has been developed, enabling observations of droplet behavior over various timescales simultaneously. An exploration into the development of a pilot-wave hydrodynamic system, and the pilot-wave hydrodynamic experiments conducted thereafter, is presented.
The experimental violation of Bell's Theorem in quantum systems is generally taken as proof of the impossibility of local hidden variable theories, as would provide a rational dynamical underpinning for quantum mechanics. We here present a platform for executing Bell tests in pilot-wave hydrodynamics, a system known to capture many features of the quantum realm. We execute a static test in a bipartite tunneling system, and rationalize the emergent violations in terms of the wave-mediated coupling between the two subsystems.
The walking droplet system discovered by Yves Couder and Emmanuel Fort presents an example of a vibrating particle self-propelling through a resonant interaction with its own wave field. It provides a means of visualizing a particle as an excitation of a field, a common notion in quantum field theory. Moreover, it represents the first macroscopic realization of a form of dynamics proposed for quantum particles by Louis de Broglie in the 1920s. The fact that this hydrodynamic pilot-wave system exhibits many features typically associated with the microscopic, quantum realm raises a number of intriguing questions. At a minimum, it extends the range of classical systems to include quantum-like statistics in a number of settings. A more optimistic stance is that it suggests the manner in which quantum mechanics might be completed through a theoretical description of particle trajectories. We here review the experimental studies of the walker system, and the hierarchy of theoretical models developed to rationalize its behavior. Particular attention is given to enumerating the dynamical mechanisms responsible for the emergence of robust, structured statistical behavior. Another focus is demonstrating how the temporal nonlocality of the droplet dynamics, as results from the persistence of its pilot wave field, may give rise to behavior that appears to be spatially nonlocal. Finally, we describe recent explorations of a generalized theoretical framework that provides a mathematical bridge between the hydrodynamic pilot-wave system and various realist models of quantum dynamics.
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]