Abstract : The back-reaction of a radiated wave on the emitting source is a general problem. In the most general case, back-reaction on moving wave sources depends on their whole history. Here we study a model system in which a pointlike source is piloted by its own memory-endowed wave field. Such a situation is implemented experimentally using a self-propelled droplet bouncing on a vertically vibrated liquid bath and driven by the waves it generates along its trajectory. The droplet and its associated wave field form an entity having an intrinsic dual particle-wave character. The wave field encodes in its interference structure the past trajectory of the droplet. In the present article we show that this object can self-organize into a spinning state in which the droplet possesses an orbiting motion without any external interaction. The rotation is driven by the wave-mediated attractive interaction of the droplet with its own past. The resulting “memory force” is investigated and characterized experimentally, numerically, and theoretically. Orbiting with a radius of curvature close to half a wavelength is shown to be a memory-induced dynamical attractor for the droplet’s motion.

Labousse, M., Perrard, S., Couder, Y., & Fort, E. (2016). Self-attraction into spinning eigenstates of a mobile wave source by its emission back-reaction. Physical Review E, 94(4), 042224.

Available on ResearchGate (Free login required)

Perrard, S., Fort, E., & Couder, Y. (2016). Wave-Based Turing Machine: Time Reversal and Information Erasing. Physical Review Letters, 117(9), 094502.

ABSTRACT

The investigation of dynamical systems has revealed a deep-rooted difference between waves and objects regarding temporal reversibility and particlelike objects. In nondissipative chaos, the dynamic of waves always remains time reversible, unlike that of particles. Here, we explore the dynamics of a wave-particle entity. It consists in a drop bouncing on a vibrated liquid bath, self-propelled and piloted by the surface waves it generates. This walker, in which there is an information exchange between the particle and the wave, can be analyzed in terms of a Turing machine with waves as the information repository. The experiments reveal that in this system, the drop can read information backwards while erasing it. The drop can thus backtrack on its previous trajectory. A transient temporal reversibility, restricted to the drop motion, is obtained in spite of the system being both dissipative and chaotic.

Available at Researchgate (requires free login)

https://www.researchgate.net/publication/307082497_Wave-Based_Turing_Machine_Time_Reversal_and_Information_Erasing

Borghesi, C., Moukhtar, J., Labousse, M., Eddi, A., Fort, E., & Couder, Y. (2014). Interaction of two walkers: Wave-mediated energy and force. Physical Review E, 90(6), 063017.

A bouncing droplet, self-propelled by its interaction with the waves it generates, forms a classical wave-particle association called a “walker.” Previous works have demonstrated that the dynamics of a single walker is driven by its global surface wave field that retains information on its past trajectory. Here, we investigate the energy stored in this wave field for two coupled walkers and how it conveys an interaction between them. For this purpose, we characterize experimentally the “promenade modes” where two walkers are bound, and propagate together. Their possible binding distances take discrete values, and the velocity of the pair depends on their mutual binding. The mean parallel motion can be either rectilinear or oscillating. The experimental results are recovered analytically with a simple theoretical framework. A relation between the kinetic energy of the droplets and the total energy of the standing waves is established.

http://arxiv.org/pdf/1412.7701.pdf

Labousse, M., Perrard, S., Couder, Y., & Fort, E. (2014). Build-up of macroscopic eigenstates in a memory-based constrained system. New Journal of Physics, 16(11), 113027.

A bouncing drop and its associated accompanying wave forms a walker. Based on previous works, we show in this article that it is possible to formulate a simple theoretical framework for the walker dynamics. It relies on a time scale decomposition corresponding to the effects successively generated when the memory effects increase. While the short time scale effect is simply responsible for the walkerʼs propulsion, the intermediate scale generates spontaneously pivotal structures endowed with angular momentum. At an even larger memory
scale, if the walker is spatially confined, the pivots become the building blocks of a self-organization into a global structure. This new theoretical framework is applied in the presence of an external harmonic potential, and reveals the underlying mechanisms leading to the emergence of the macroscopic spatial organization reported by Perrard et al (2014 Nature Commun. 5 3219).

https://hal-univ-artois.archives-ouvertes.fr/hal-01084731/document

Perrard, S. (2014). Une mémoire Ondulatoire: états propres, chaos et probabilités (Doctoral dissertation, Paris 7)

Résumé : Une goutte rebondissant sur un bain de liquide en vibration verticale peut se mettre spontanément en mouvement, sous l’action des ondes qu’elle a elle-même générées. Celles ci, appelées ondes de Faraday sont entretenues par la vibration du bain durant un temps de mémoire qui peut être contrôlé expérimentalement. Le champ d’ondes stationnaires généré par la goutte contient ainsi dans ses motifs d’interférence une mémoire de la trajectoire précédemment suivie. L’entité résultante appelée marcheur est caractérisée par cette interaction entre la goutte et les ondes qui l’entourent, via la mémoire de chemin. Cette thèse est consacrée à l’étude expérimentale et théorique de cette mémoire de chemin. Dans ce but, une goutte de liquide encapsulant un volume de ferrofluide est piégée dans un puits de potentiel harmonique d’origine magnétique. La goutte sera ainsi amenée à interagir avec les ondes qu’elle a précédemment générées. Ce confinement induit un processus d’auto-organisation entre la goutte et l’onde sous-jacente qui mène à des comportements de type ondulatoire pour une particule. Les notions de quantifications ou de probabilité de mesure d’un état propre peuvent ainsi être appliquées au cas d’un marcheur. Ces comportements révèlent que le marcheur est un exemple d’objet étendu en temps qui ne peut être réduit à une approximation ponctuelle rappelant, dans un tout autre contexte, la théorie de l’onde pilote développée par de Broglie au début du XXème siècle

A droplet bouncing on a vertically vibrated liquid bath can be self-propelled by the surface waves it generates. Theses Faraday waves are sustained by the vertical bath vibration for a memory time which can be tuned experimentally. The wave field thus contains in its interference pattern a memory of the past-trajectory. The resulting entity called a walker is characterized by the interaction between the drop and its surrounding waves through this path-memory. This thesis is devoted to an experimental and theoretical investigation of such a wave-mediated path-memory. For this purpose a bouncing drop is magnetically loaded with a droplet of ferrofluid and can then be trapped in an harmonie well. The drop is thus forced to interact with its own path. The confinement induces a self-organization process between the particle and its wave packet, leading to wave-type behavior for a particle. Notions such quantization or probability of measuring an eigenstate can thus be used for the walker dynamics description. These features originate from the temporal coherence of the walker’ s dynamics. In that sense, the walker is an entity extended in time, we cannot reduce to a point-like approximation. It reminds us, in another context, the pilot wave theory developped by de Broglie at the beginning of the XXst century.

https://tel.archives-ouvertes.fr/tel-01158368/document

Perrard, S., Labousse, M., Fort, E., & Couder, Y. (2014). Chaos driven by interfering memory. Physical review letters, 113(10), 104101.

The transmission of information can couple two entities of very different nature, one of them serving as a memory for the other. Here we study the situation in which information is stored in a wave field and serves as a memory that pilots the dynamics of a particle. Such a system can be implemented by a bouncing drop generating surface waves sustained by a parametric forcing. The motion of the resulting “walker” when confined in a harmonic potential well is generally disordered. Here we show that these trajectories correspond to chaotic regimes characterized by intermittent transitions between a discrete set of states. At any given time, the system is in one of these states characterized by a double quantization of size and angular momentum. A low dimensional intermittency determines their respective probabilities. They thus form an eigenstate basis of decomposition for what would be observed as a superposition of states if all measurements were intrusive

https://hal.archives-ouvertes.fr/hal-01061415/document

“À l’échelle macroscopique, les ondes et les particules sont des objets distincts. La découverte d’objets appelés marcheurs, constitués d’une goutte rebondissant sur un bain liquide vibré verticalement, a montré qu’il n’en était rien. La goutte est autopropulsée, guidée sur la surface du liquide par l’onde qu’elle a elle-même créée lors des rebonds précédents. Ces objets possèdent une dynamique originale dominée par le concept de mémoire de chemin. La structure du champ d’onde qui guide la goutte dépend, en effet, de la position des rebonds passés disposés le long de la trajectoire. La profondeur de cette mémoire peut, de plus, être contrôlée expérimentalement en changeant l’accélération du bain. De nombreuses réalisations expérimentales ont mis en évidence les comportements dynamiques singuliers de ces systèmes couplés goutte/onde. Cette thèse répond à la nécessité d’une compréhension théorique des effets non locaux en temps introduit par la mémoire de chemin. Pour ce faire, nous étudierons l’évolution d’un marcheur numérique en potentiel harmonique bidimensionnel. Un ensemble relativement restreint de trajectoires stables est obtenu. Nous constaterons que ces dernières sont quantifiées en extension moyenne et en moment angulaire moyen. Nous analyserons comment s’imbriquent les différentes échelles de temps de la dynamique, permettant ainsi de dissocier les termes propulsifs à temps court de l’émergence de structures ondulatoires cohérentes à temps long. Nous verrons en quoi l’expression du caractère non-local d’un marcheur permet d’en révéler les symétries internes et d’assurer la convergence du système dynamique vers un jeu d’états propres de basse dimension.”

Labousse, M. (2014). Étude d’une dynamique à mémoire de chemin: une expérimentation théorique (Doctoral dissertation, Université Pierre et Marie Curie UPMC Paris VI).

https://pastel.archives-ouvertes.fr/tel-01114815/document

Perrard, S., Labousse, M., Miskin, M., Fort, E., & Couder, Y. (2014). Self-organization into quantized eigenstates of a classical wave-driven particle.Nature communications, 5.

A growing number of dynamical situations involve the coupling of particles or singularities with physical waves. In principle these situations are very far from the wave particle duality at quantum scale where the wave is probabilistic by nature. Yet some dual characteristics were observed in a system where a macroscopic droplet is guided by a pilot wave it generates. Here we investigate the behaviour of these entities when confined in a two-dimensional harmonic potential well. A discrete set of stable orbits is observed, in the shape of successive generalized Cassinian-like curves (circles, ovals, lemniscates, trefoils and so on). Along these specific trajectories, the droplet motion is characterized by a double quantization of the orbit spatial extent and of the angular momentum. We show that these trajectories are intertwined with the dynamical build-up of central wave-field modes. These dual self-organized modes form a basis of eigenstates on which more complex motions are naturally decomposed.

http://www.nature.com/ncomms/2014/140130/ncomms4219/full/ncomms4219.html

http://arxiv.org/ftp/arxiv/papers/1402/1402.1423.pdf

Quantization of trajectories of a dotwave in a harmonic potential

Perrard, S., Labousse, M., Miskin, M., Fort, E., & Couder, Y. Effets de quantification d’une association onde-particule soumise à une force centrale.Résumés des exposés de la 16e Rencontre du Non-Linéaire Paris 2013, 68.

http://nonlineaire.univ-lille1.fr/SNL/media/2012/CR/Perrard.pdf

eigenstates in circular cavity

Harris, D. M., Moukhtar, J., Fort, E., Couder, Y., & Bush, J. W. (2013). Wavelike statistics from pilot-wave dynamics in a circular corral. Physical Review E, 88(1), 011001.

Abstract : Bouncing droplets can self-propel laterally along the surface of a vibrated fluid bath by virtue of a resonant interaction with their own wave field. The resulting walking droplets exhibit features reminiscent of microscopic quantum particles. Here we present the results of an experimental investigation of droplets walking in a circular corral. We demonstrate that a coherent wavelike statistical behavior emerges from the complex underlying dynamics and that the probability distribution is prescribed by the Faraday wave mode of the corral. The statistical behavior of the walking droplets is demonstrated to be analogous to that of electrons in quantum corrals.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2013/07/Harris-Corrals-2013.pdf

Statistical behavior of a walking droplet in a confined geometry

Eddi, A., Decelle, A., Fort, E., & Couder, Y. (2009). Archimedean lattices in the bound states of wave interacting particles. EPL (Europhysics Letters), 87(5), 56002.

http://stilton.tnw.utwente.nl/people/eddi/Papers/epl_Cristaux.pdf

Cristal lattices of bouncing droplets

Fort, E., Eddi, A., Boudaoud, A., Moukhtar, J., & Couder, Y. (2010). Path-memory induced quantization of classical orbits. Proceedings of the National Academy of Sciences, 107(41), 17515-17520.

http://stilton.tnw.utwente.nl/people/eddi/Papers/PNAS_2010.pdf

Rotating bath

Quantization of trajectories

“L’organisation du manuscrit est la suivante : dans le premier chapitre, nous rappelons les premieres experiences realisees avec les gouttes rebondissantes, en particulier les regimes ou elles sont statiques. Nous presentons egalement l’instabilite de Faraday qui permet d’expliquer le couplage entre goutte et ondes a l’origine des marcheurs. Leurs proprietes sont discutees, en particulier dans le cas des experiences de diraction. Dans un second chapitre, nous presentons le montage experimental ainsi que l’ensemble des techniques de mesure utilisees. Le troisieme chapitre est dedie a l’etude d’un analogue de l’eet tunnel pour les marcheurs. Le quatrieme est consacre a l’etude des ondes de surface generees par un marcheur, en particulier le lien avec l’instabilite de Faraday a l’origine de la memoire de chemin. Enfin, dans le chapitre cinq, nous analysons la nature des trajectoires circulaires d’un marcheur soumis a une force orthogonale au mouvement. L’in uence de la memoire de chemin sur la quantication des rayons est presentee en detail, pour comprendre comment cet analogue macroscopique des niveaux de Landau se met en place avec des marcheurs.”

Eddi, A. (2011). Marcheurs, Dualité onde-particule et Mémoire de chemin (Doctoral dissertation, Université Paris-Diderot-Paris VII).

http://tel.archives-ouvertes.fr/docs/00/57/56/26/PDF/these_A_Eddi.pdf

From Couder’s Lab

From Couder’s Lab, supplementary info of Phys. Rev. Lett. 102, 240401 (2009)

Eddi, A., Moukhtar, J., Perrard, S., Fort, E., & Couder, Y. (2012). Level Splitting at Macroscopic Scale. Physical Review Letters, 108(26), 264503.

http://stilton.tnw.utwente.nl/people/eddi/Papers/PhysRevLett_Zeeman.pdf

Protière, S., Boudaoud, A., & Couder, Y. (2006). Particle-wave association on a fluid interface. Journal of Fluid Mechanics, 554(10), 85-108.

http://www.lps.ens.fr/~boudaoud/Publis/Protiere06Bounc.pdf

Couder, Y., Fort, E., Gautier, C. H., & Boudaoud, A. (2005). From bouncing to floating: Noncoalescence of drops on a fluid bath. Physical review letters,94(17), 177801.

http://www.lps.ens.fr/~boudaoud/Publis/Couder05Bounc.pdf

Protiere, S. (2007). Gouttes rebondissantes: une association onde-particule à échelle macroscopique (Doctoral dissertation, Université Paris-Diderot-Paris VII).

http://tel.archives-ouvertes.fr/docs/00/19/26/20/PDF/these.pdf