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.
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 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.
While bouncing walking silicone oil droplets (walkers) do show many quantumlike phenomena, the original, most intriguing, double-slit experiment with walkers has been shown to be an overinterpretation of data. Several experiments and numerical simulations have proven that for at least some parameter region there is no randomness. Still, true randomness was claimed to be observed in an experiment on chaotically bouncing walkers. Also, most of the available phase space has not been investigated. The main goal of this paper is therefore to look for true interference and chaos in the entire phase space. Recently, we made an extensive investigation of drops interacting with slits, but still in a limited range. However, the outcome was always deterministic and only incidentally mimicked the statistics of the corresponding quantum case. We also showed that the extra interference, already seen by others, in the double-slit case was caused by reflection of waves from the outlet of the unused slit after passage and thus was not a true double-slit effect. A new theoretical treatment of bouncing drop dynamics has since given analytic relations for the associated wave field, leading to a proposal for criteria for the possible occurrence of true interference in the double-slit experiment. Satisfying these criteria, requires working close to the onset of the Faraday instability, with two spatial conditions favoring slow walkers, and a temporal condition favoring fast walkers. The regions of high velocity, where the walkers bounce periodically, and of very low velocity, with chaotically bouncing walkers, have not been comprehensively investigated so far. We have therefore looked at these regions, probing the limits for interaction with slits. Furthermore, noting that a short transit time is essential to fulfill the criteria, experiments were done using double-slit barriers only 0.5 and 2 mm broad. Nowhere was true interference or a chaotic response found. As the theory has implications for many of the observed quantumlike phenomena involving walkers as, e.g., tunneling and interaction between drops, we have measured the spatial and temporal decay of the wave field. A comparison with the theory shows very good agreement but leads to a reformulation of the above-mentioned criteria.
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.
In recent years, Couder and collaborators have initiated a series of studies on walking droplets. Experimentally, they found that at frequencies and amplitudes close to the onset of Faraday waves, droplets on the surface of silicone oil can survive and walk at a roughly constant speed due to resonance. Droplets excite local ripples from the Faraday instability when they bounce from the liquid surface. This tightly coupled particle-wave entity, although a complex yet entirely classical system, exhibits many phenomena that are strikingly similar to those of quantum systems, such as slit interference and diffraction, tunneling probability, and Anderson localization. In this Letter, we focus on the tunneling time of droplets. Specifically, we explore (1) how it changes with the width of an acrylic barrier, which gives rise to the potential barrier when the depth of the silicone oil is reduced to prevent the generation of ripples that can feed energy back to the droplet, and (2) the distribution of tunneling times at the same barrier width. Both results turn out to be similar to the numerical outcome of the Bohmian mechanics, which strengthens the analogy to a quantum system. Furthermore, we successfully derive analytic expressions for these properties by revising the multiple scattering theory and constructing a ``skipping stone" model. Provided that the resemblance in tunneling behavior of walking droplets to Bohmian particles is not coincidental, we discuss the lessons for the Copenhagen interpretation of quantum mechanics that so far fails to explain both characteristics adequately.
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.
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.