Path-memory induced quantization of classical orbits SEE COMMENTARY Emmanuel Forta,1, Antonin Eddib, Arezki Boudaoudc, Julien Moukhtarb, and Yves Couderb aInstitut Langevin, Ecole Supérieure de Physique et de Chimie Industrielles ParisTech and Université Paris Diderot, Centre National de la Recherche Scientifique Unité Mixte de Recherche 7587, 10 Rue Vauquelin, 75 231 Paris Cedex 05, France; bMatières et Systèmes Complexes, Université Paris Diderot, Centre National de la Recherche Scientifique Unité Mixte de Recherche 7057, Bâtiment Condorcet, 10 Rue Alice Domon et Léonie Duquet, 75013 Paris, France; and cLaboratoire de Physique Statistique, Ecole Normale Supérieure, 24 Rue Lhomond, 75231 Paris Cedex 05, France Edited* by Pierre C. Hohenberg, New York University, New York, NY, and approved August 4, 2010 (received for review May 26, 2010) A droplet bouncing on a liquid bath can self-propel due to its inter- a magnetic field. However, for technical reasons we chose a action with the waves it generates. The resulting “walker” is a variant that relies on the analogy first introduced by Berry et al. dynamical association where, at a macroscopic scale, a particle (5) between electromagnetic fields and surface waves. ~ ~ ~ (the droplet) is driven by a pilot-wave field. A specificity of this Its starting point is the similarity of relation B ¼ ∇ × A in elec- ~ system is that the wave field itself results from the superposition tromagnetism with 2Ω~¼ ∇~× U in fluid mechanics. In these re- ~ of the waves generated at the points of space recently visited by lations, the vorticity 2Ω~is the equivalent of the magnetic field B ~ ~ the particle. It thus contains a memory of the past trajectory of the and the velocity U that of the vector potential A. Considering a particle. Here, we investigate the response of this object to forces plane wave passing over a point-like vortex, Berry et al. (5) show orthogonal to its motion. We find that the resulting closed orbits that the phase field will exhibit defects as the wave passes on present a spontaneous quantization. This is observed only when either side of a point vortex. This is the analog of the Aharo- the memory of the system is long enough for the particle to inter- nov–Bohm effect (6) where the presence of a vector potential act with the wave sources distributed along the whole orbit. An generates a phase shift of the quantum-mechanical wave func- additional force then limits the possible orbits to a discrete set. tion. Although this effect is unintuitive in quantum mechanics, The wave-sustained path memory is thus demonstrated to gener- it receives a simple interpretation in the hydrodynamic analog ate a quantization of angular momentum. Because a quantum-like where the phase shift is due to the advection of the wave by uncertainty was also observed recently in these systems, the non- the fluid velocity. The hydrodynamic effect has been experimen- locality generated by path memory opens new perspectives. tally measured both in the propagation of acoustic waves (7) and in that of surface waves (8, 9) around a vortex. bouncing droplets ∣ Landau quantization ∣ pilot wave ∣ In order to investigate circular orbits we use the same analogy wave-particle duality differently. The associated forces also have similar expressions. A charge q moving at velocity V~in a homogeneous external mag- ~ ~ ~ ~ material particle dynamically coupled to a wave packet at netic field B experiences FB ¼ qðV × BÞ. In a system rotating at Amacroscopic scale has been discovered recently and has been an angular velocity Ω, the Coriolis force on a mass m moving with shown to have intriguing quantum-like properties (1–4). The ~ ~ ~ ~ ~ velocity V is FΩ ¼ −mðV × 2ΩÞ, where 2Ω is the vorticity due to particle is a droplet bouncing on the surface of a vibrated liquid solid-body rotation. In classical physics, both these forces lead to bath, and the wave is the surface wave it excites. Togetherthey are B~ Ω~ self-propelled on the interface and form a symbiotic object. orbiting motions in the planes perpendicular to or respec- Recent investigations have shown that this “walker” exhibits a tively. In a magnetic field the orbit has a radius form of wave-particle duality, a unique feature in a classical sys- ρ ¼ mv∕qB [1] tem. This appears because the localized and discrete droplet has a L common dynamics with the continuous and spatially extended τ ¼ m∕qB wave. Various situations [diffraction and interference (3) and tun- and a Larmor period L . On a rotating surface a mobile V neling (4)], where the wave is either bounded or split, have been particle with velocity moves on a circle of radius examined. The surprising result is that for each realization of an R ¼ V∕2Ω [2] experiment of this type the droplet has an unpredictable indivi- C dual response. However, a statistical behavior is recovered when T ¼ 1∕2Ω the experiment is repeated. The truncation of the wave was thus with a Coriolis period C . Here we place a walker in a shown to generate an uncertainty in the droplet’s motion. This rotating system. If it was a normal classical particle, it would move “ ” on a circle of radius RC decreasing continuously with increasing uncertainty , though unrelated to Planck constant, has an ana- Ω logy with the statistical behavior observed in the corresponding . Will walkers follow this classical behavior? What is the role of quantum-mechanical experiments. This characteristic has been the wave field induced particle path memory? ascribed to nonlocality. In this 2D experiment, the points of Experimental Setup the surface disturbed by the bouncing droplet keep emitting A sketch of our experimental system is given in Fig. 1B. The waves. The motion of the droplet is thus driven by its interaction fluid is contained in a cylindrical cell of diameter 150 mm, which with a superposition of waves emitted by the points it has visited can be simultaneously vibrated with a vertical acceleration in the recent past. This phenomenon, easily observed in the wave γ ¼ γ 2πf t Ω A m sin 0 and rotated at a constant angular velocity in pattern of a linearly moving walker (Fig. 1 ), generates a path the range 0 < Ω < 10 rad∕s around a central vertical axis. PHYSICS memory, a hitherto unexplored type of spatial and temporal non- locality. In the present work we show that this path-memory-driven Author contributions: E.F., A.E., A.B., J.M., and Y.C. designed research, performed research, nonlocality can lead to a form of quantization. Because the analyzed data, and wrote the paper. system is dissipative, its energy is imposed by a forcing at fixed The authors declare no conflict of interest. frequency. Therefore there is no possible quantization of energy. *This Direct Submission article had a prearranged editor. For this reason we studied a possible discretization of the angular See Commentary on page 17455. momentum of orbiting walkers similar to that of the Landau 1To whom correspondence should be addressed. E-mail: [email protected]. orbits of a charged particle in a magnetic field. One possibility This article contains supporting information online at www.pnas.org/lookup/suppl/ would have been to study the trajectory of charged droplets in doi:10.1073/pnas.1007386107/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1007386107 PNAS ∣ October 12, 2010 ∣ vol. 107 ∣ no. 41 ∣ 17515–17520 Downloaded by guest on September 25, 2021 A C λF ¼ 3.74, 4.75, 6.95, and 8.98 mm, respectively. The walker’s motion is recorded with a camera placed in the laboratory frame, and the trajectories in the rotating frame are reconstructed by image processing. Experimental Results The Path Memory. First, we address the path-memory concept. In the absence of rotation, a given droplet forms a walker in W F the interval γm < γm < γm. The bifurcation to walking being supercritical, the walker’s velocity increases near the onset as W 1∕2 V W ∝ ðγm − γm Þ , then saturates at a value of the order of a tenth of V ϕ, the phase velocity of Faraday waves. The global wave field results from the repeated collision of the B D droplet with the substrate. Each point of the surface visited by the droplet becomes the center of a localized mode of circular Fara- day waves. This wave packet damps out with a typical time scale τ. The transition to Faraday instability being a supercritical bifurca- F −1 tion, τ diverges near the threshold as τ ∝ jγm − γmj . By tuning γm we can thus control the time scale of this memory. Far from the Faraday threshold the waves are strongly damped and the wave packet has an approximately circular structure resulting from the most recent collisions of the droplet with the bath. In contrast, in the situation shown in Fig. 1A where γm is close to the Faraday threshold, τ is very large and the wave field is E widely extended and exhibits a complex interference structure. This wave field has an interesting relation to the Huygens– Fresnel theory of diffraction (13). The in-phase secondary sources left behind by the droplet can be considered as imple- menting in reality Huygens secondary sources along a wave front defined by its trajectory (14). For a rectilinear trajectory (Fig. 1A), the resulting wave pattern is thus similar to the Fresnel interfer- ence of light near the edge of a wall, the equivalent of the wall being the line of points that have not yet been visited by the walker. ’ Fig. 1. (A) Photograph of the wave field of a walker with long-term path We can now consider the droplet s motion.