Harris, D. M., Liu, T., & Bush, J. W. (2015). A low-cost, precise piezoelectric droplet-on-demand generator. Experiments in Fluids, 56(4), 1-7.

We present the design of a piezoelectric droplet-on-demand generator capable of producing droplets of highly repeatable size ranging from 0.5 to 1.4 mm in diameter. The generator is low cost and simple to fabricate. We demonstrate the manner in which droplet diameter can be controlled through variation of the piezoelectric driving waveform parameters, outlet pressure, and nozzle diameter.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2015/04/Harris-DropGenerator.pdf

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

Carmigniani, R., Lapointe, S., Symon, S., & McKeon, B. J. (2013). Influence of a local change of depth on the behavior of bouncing oil drops. arXiv preprint arXiv:1310.2662.

http://arxiv.org/pdf/1310.2662.pdf

Full (Subscription required) : http://www.sciencedirect.com/science/article/pii/S0894177713003038

From caltech Mc Keon Research Group : http://www.mckeon.caltech.edu/publications/journal.html

Abstract :

The work of Couder et al [1] (see also Bush et al [3, 4]) inspired consideration of the impact of a submerged obstacle, providing a local change of depth, on the behavior of oil drops in the bouncing regime. In the linked videos, we recreate some of their results for a drop bouncing on a uniform depth bath of the same liquid undergoing vertical oscillations just below the conditions for Faraday instability, and show a range of new behaviors associated with change of depth.

This article accompanies a uid dynamics video entered into the Gallery of Fluid Motion of the 66th Annual Meeting of the APS Division of Fluid Dynamics.

And a very interesting video showing the influence of depth on the trajectory :

Source : http://arxiv.org/src/1310.2662v1/anc/V102356_InfluenceLocalChangeDepth_BouncingDrop.mp4

Chu, H. Y., & Fei, H. T. (2014). Vortex-mediated bouncing drops on an oscillating liquid. Physical Review E, 89(6), 063011

 http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.063011 (Subsrciption required)

Stunning Vizualisization of undersurface flows

Abstract :

We have investigated the behavior of bouncing drops on a liquid surface by using particle image velocimetry analysis. A drop on an oscillating liquid surface is observed to not coalesce with the liquid and to travel along the surface if the oscillation is strong enough. A streaming vortex pair, induced by the alternatively distorted liquid surface, shows up below a bouncing drop. The time-averaged flow fields of the vortices are measured. In our quasi-one-dimensional setup, there are three stable distances for the drops, which can be characterized by the Faraday wavelength. The interactions of the vortex-mediated bouncing drops are deduced from the streamlines in the liquid bulk. We further show that a three-dimensional vortex ring is induced by a bouncing drop in a square cell.

Andersen, A., Madsen, J., Reichelt, C., Ahl, S. R., Lautrup, B., Ellegaard, C., … & Bohr, T. (2014). Comment on Y. Couder and E. Fort:” Single-Particle Diffraction and Interference at a Macroscopic Scale”, Phys. Rev. Lett.(2006).arXiv preprint arXiv:1405.0466.

Where a danish team argue that Couder’sDouble Slit Experiment reported in  Single-Particle Diffraction and Interference at a Macroscopic Scale   is not convincing.

They tried to replicate the experiment but they did not manage.

Plus : a quick report of a numerical experimentation of Schrödinger equation with a “”walker-like”  source term

Abstract :

In a paper from 2006, Couder and Fort [1] describe a version of the famous double slit experiment performed with drops bouncing on a vibrated fluid surface, where interference in the particle statistics is found even though it is possible to determine unambiguously which slit the “walking” drop passes. It is one of the first papers in an impressive series, showing that such walking drops closely resemble de Broglie waves and can reproduce typical quantum phenomena like tunneling and quantized states [2–13]. The double slit experiment is, however, a more stringent test of quantum mechanics, because it relies upon superposition and phase coherence. In the present comment we first point out that the experimental data presented in [1] are not convincing, and secondly we argue that it is not possible in general to capture quantum mechanical results in a system, where the trajectory of the particle is well-defined.

http://arxiv.org/pdf/1405.0466.pdf

Keywords : Madelung-Bohm equation

Oza, A. U., Harris, D. M., Rosales, R. R., & Bush, J. W. (2014). Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization. Journal of Fluid Mechanics, 744, 404-429.

We present the results of a theoretical investigation of droplets walking on a
rotating vibrating fluid bath. The droplet’s trajectory is described in terms of an
integro-differential equation that incorporates the influence of its propulsive wave
force. Predictions for the dependence of the orbital radius on the bath’s rotation
rate compare favourably with experimental data and capture the progression from
continuous to quantized orbits as the vibrational acceleration is increased. The orbital
quantization is rationalized by assessing the stability of the orbital solutions, and may
be understood as resulting directly from the dynamic constraint imposed on the drop
by its monochromatic guiding wave. The stability analysis also predicts the existence
of wobbling orbital states reported in recent experiments, and the absence of stable
orbits in the limit of large vibrational forcing

http://math.mit.edu/~auoza/JFM_2.pdf

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

Grössing, G., Fussy, S., Pascasio, J. M., & Schwabl, H. (2014, April). Relational causality and classical probability: Grounding quantum phenomenology in a superclassical theory. In Journal of Physics: Conference Series (Vol. 504, No. 1, p. 012006). IOP Publishing.

Abstract. : By introducing the concepts of \superclassicality” and \relational causality”, it is shown here that the velocity eld emerging from an n-slit system can be calculated as an
average classical velocity eld with suitable weightings per channel. No deviation from classical probability theory is necessary in order to arrive at the resulting probability distributions.
In addition, we can directly show that when translating the thus obtained expression for said velocity eld into a more familiar quantum language, one immediately derives the basic
postulate of the de Broglie-Bohm theory, i.e. the guidance equation, and, as a corollary, the exact expression for the quantum mechanical probability density current. Some other direct
consequences of this result will be discussed, such as an explanation of Born’s rule and Sorkin’s first and higher order sum rules, respectively.

http://iopscience.iop.org/1742-6596/504/1/012006/pdf/1742-6596_504_1_012006.pdf

classical probability theory

n-slit system

guidance equation

Analysis of currrent state of the art modelling

Reflexions in a 1D Cavity

Computer simulations

(proceedings of RIO 2014  workshop)

http://www.personal.psu.edu/alm24/IMPA/talks/IMPANachbin.pdf

Shirokoff, D. (2013). Bouncing droplets on a billiard table. Chaos: An Interdisciplinary Journal of Nonlinear Science, 23(1), 013115.

http://arxiv.org/pdf/1203.2204.pdf

Discrete numerical simulation

Attractor ets

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

Where it all started …

http://www.feynmanlectures.info/docroot/III_01.html

Bach, R., Pope, D., Liou, S. H., & Batelaan, H. (2013). Controlled double-slit electron diffraction. New Journal of Physics, 15(3), 033018.

http://iopscience.iop.org/1367-2630/15/3/033018/pdf/1367-2630_15_3_033018.pdf

And some movies of the interference pattern build-up :

http://iopscience.iop.org/1367-2630/15/3/033018/media

The famous Feynman thought experiment reproduced ! ((cf. Feynman Lectures on Physics, vol III, figures 1–3,))

Crommie, M. F., Lutz, C. P., & Eigler, D. M. (1993). Confinement of electrons to quantum corrals on a metal surface. Science, 262(5131), 218-220.

http://users.phys.psu.edu/~rick/MATH/LECTURES/LECT13/PDF/eigler_circular_corral.pdf

Harris, D. M., & Bush, J. W. (2014). Droplets walking in a rotating frame: from quantized orbits to multimodal statistics. Journal of Fluid Mechanics, 739, 444-464.

We present the results of an experimental investigation of a droplet walking on the
surface of a vibrating rotating fluid bath. Particular attention is given to demonstrating
that the stable quantized orbits reported by Fort et al. (Proc. Natl Acad. Sci.,
vol. 107, 2010, pp. 17515–17520) arise only for a finite range of vibrational
forcing, above which complex trajectories with multimodal statistics arise. We first
present a detailed characterization of the emergence of orbital quantization, and
then examine the system behaviour at higher driving amplitudes. As the vibrational
forcing is increased progressively, stable circular orbits are succeeded by wobbling
orbits with, in turn, stationary and drifting orbital centres. Subsequently, there is a
transition to wobble-and-leap dynamics, in which wobbling of increasing amplitude
about a stationary centre is punctuated by the orbital centre leaping approximately
half a Faraday wavelength. Finally, in the limit of high vibrational forcing, irregular
trajectories emerge, characterized by a multimodal probability distribution that reflects
the persistent dynamic influence of the unstable orbital states.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2014/01/HB-JFM-2014.pdf

Robert Brady, Ross Anderson. Jan 16, 2014.

http://arxiv.org/pdf/1401.4356v1.pdf

review of all analogies between dotwaves mechanics and quantum mechanics.

Sbitnev, V. I. (2013). Droplets moving on a fluid surface: interference pattern from two slits. arXiv preprint arXiv:1307.6920.

http://arxiv.org/pdf/1307.6920.pdf

Feynman path integral approach

Droplet / Antidroplet

Soliton diffraction

Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of” hidden” variables. I. Physical Review, 85(2), 166.

http://www.psiquadrat.de/downloads/bohm52a.pdf

Hidden variable theory foundation

Oza, A. U., Rosales, R. R., & Bush, J. W. (2013). A trajectory equation for walking droplets: hydrodynamic pilot-wave theory. Journal of Fluid Mechanics,737, 552-570.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2013/12/ORB-JFM.pdf

Integro-differential equation describing the horizontal motion of a walking droplet

Stability to perturbations