The results about the path memory described in the previous section led us to seek situations in which the droplet would necessarily revisit regions of space that it had already disturbed in the past. This is what happens in orbital motion and we first studied the circular motion resulting from a transverse force. During Antonin Eddis PhD [T2] we thus studied the situations where the drop is submitted to a Coriolis force. We then turned to the more general orbits due to a central force. During the PhD theses of Stéphane Perrard [T3] and Matthieu Labousse [T4] we investigated the situation where the walker is trapped in an axisymmetric potential well.

# Transverse force orbits

A natural idea to obtain circular orbits was to give the droplet an electric charge and to place it in a magnetic field. If an analogy to quantum mechanics exists we should then observe a form of Landau quantization of the orbits. However, it is difficult to control the electric charge of a droplet so that we turned to a variant. It relies on an analogy first used by Michael Berry [ref 1]. Its starting point is the similarity of relation B=×A in electromagnetism with 2Ω=×U in fluid mechanics. In these relations, the vorticity 2Ω is the equivalent of the magnetic field B, the fluid velocity U that of the vector potential A. Using this analogy M. Berry found a remarkable hydrodynamic analogue of the Aharonov-Bohm effect (1). Wishing to investigate circular orbits we use the same analogy but differently. A charge q moving at velocity V in a homogeneous external magnetic field B is submitted to FB=q(V × B) . In a system rotating at an angular velocity Ω, a mass m moving with velocity V is submitted to a Coriolis force FΩ=-m(V × 2Ω) where 2Ω is the vorticity of the solid-body rotation. In classical physics, both these forces lead to orbiting motions in the planes perpendicular to B or Ω respectively. In a magnetic field the orbit has a radius ρL=mV / qB. On a rotating surface a mobile of velocity V moves on a circle of radius RC=V / 2Ω.

Film V.1 & V.2. A walker orbiting in a rotating tank
The fluid is contained in a cylindrical cell of diameter 150 mm which is simultaneously vibrated with a vertical acceleration γ=γm cos (2π f0 t) and rotated at a constant angular velocity Ω in the range 0 < Ω < 10 Rd/s around a central vertical axis.

The two films show a walker of velocity Vw=13.7 ± 0.1 mm/s as it orbits clockwise on a counter-clockwise rotating bath : top Ω= 0.9 Rd/s, bottom Ω= 5 Rd/s.

We thus devised a system in which the walker moves in a rotating system [P1, T2]. If it was a normal classical particle it should move on a circle of radius RC decreasing continuously with increasing Ω. This is what is observed in the regime of low memory. In contrast, when the memory is long enough, so that the particle interacts with the wave originated along the whole orbit, the circular orbits present a spontaneous quantization. An additional force due to the cumulated waves then limits the possible orbits to a discrete set. The wave-sustained path-memory is thus demonstrated to generate a quantization of the orbit size and thus of the angular momentum. Fig. V.1. The Coriolis experiment
(a) Sketch of the experimental set up, (b) the radius of the orbit as a function of the rotation frequency in the low memory regime. (c) The discretization of the orbital radii in the high memory regime.
This experiment was revisited by D. Harris and J. Bush [ref 2] and by A. Oza et al. [ref 3 & ref 5]. They showed that in the very high memory range there were regimes with intermittent transitions involving several of the quantized states so that in these regimes only the probabilities of being in one state are defined.

# Central force orbits

In order to establish the analogy to quantum physics on a firm ground a basic step should be to observe a quantization of energy. Take for instance a quantum particle trapped in a harmonic potential well. It is well known that in quantum physics it can be in successive eigenstates of increasing energy and decreasing mean wavelength as sketched in the one-dimensional case in Fig. 1a. This is unfortunately an experiment that cannot be done with a walker because: the droplet velocity being approximately constant, so is the particles kinetic energy. As for the wave field, it is characterized by a fixed wavelength λF. However, it can be noted that in the quantum situation, the successive eigenstates could be obtained at a fixed energy and wave-length by tuning the width of the well (see Fig. 1b). Here we will use this idea to seek the possible emergence of eigenstates when the spatial confinement is varied. Fig. V.2. The energy levels problem
The energy of the walker being fixed, the possible eigenstates are explored by varying the width of the potential well in which it is trapped.
In order to trap the walker in a potential well, the millimetre size drop is loaded with a small amount of ferrofluid and submitted to a magnetic field. Two coils in a Helmholtz configuration provide a uniform magnetic field so that the drop becomes a small dipole of moment mB. A small cylindrical magnet placed at a distance d above the liquid bath is the source of an additional non-uniform magnetic field so that the magnetized drop is submitted to a centripetal central force. The potential well can be considered as harmonic. The width of this well can be tuned by changing the distance d of the magnet to the liquid surface. For a drop of mass m, the dynamics is then reduced to the motion of a bouncing particle propelled by a wave and submitted to an external force Fext = -κr, where κ the spring constant depends on d the distance from magnet to the bath. Fig. V.3. Sketch of the experiment showing how a centripetal force is applied on the droplet.
In the low memory limit, the droplet motion is given by the classical laws of mechanics and its orbit is circular [P18 & P20]. In the high memory limit the most generally observed trajectory, has a complex aspect with a looped structure. But a fine tuning of the potential width reveals the existence of simple stable trajectories, each existing in a narrow range of values of Λ. When these orbits are circular (film V.3) they have the same discrete radii as those observed in the Coriolis experiment. In these conditions other types of stable trajectories also show-up, having the shape of lemniscates (as in film V.4), eccentric loops or trefoils.

Film V.3. The circular orbiting motion of the drop corresponding to the eigenstate (1,1)

Film V.4. The lemniscate shaped orbiting motion corresponding to the eigenstate (2,0)
These discrete orbits observed in the experiment (P18, P21, T3) as well as in its simulation (P21 & T4) can be compared by measuring their main characteristics: their mean spatial extension R and their mean angular momentum Lz. A remarkable feature is that both these quantities are quantized. Figure shows them as plotted against the non-dimensional width of the potential well Λ. Both are related to the wavelength only: the control parameter Λ only plays a role in selecting amongst the possible Rn. For each level Rn, only a discrete set of possible angular momentum values Lz is reachable by the system. The same discrete values of Lz are observed for different Rn.

A two dimensional graph can be obtained by plotting against each other R and Lz. The various types of trajectories appear to be well defined in this 2D plot. From these results, we can define two quantization numbers n that defines the orbit size and m that defines the in a discrete set of values of the angular momentum. All the trajectories can be positioned at nodes (n,m) of a lattice with the generic rules: n is an integer, and m can only take the values m∈{- n,-n+2,...,n+2,n} (see Fig. V-5)(P18, T3 & T4) Fig. V.4. The double quantization at long memory.
(a) The mean spatial extent of the observed stable trajectories as a function of the width of the potental well. (b) The possible values of the mean angular momentum for the same set of trajectories. Fig. V.5. The double quantization of the stable trajectories
The set of stable trajectories can be characterized by plotting their angular momentum as a function of their mean spatial extent. This plot makes clear that the possible trajectories are characterized by two numbers (n,m). The simulations using our path memory model show similar results.
What is the origin of these quantization effects that are also perfectly recovered in numerical simulations [T4] shown in the films IV.6 and IV.7 ?

In the high memory situations the drop keeps emitting waves in a bounded central region. The accumulation of these waves will necessarily be related to the global wave eigenmodes of this domain. This will have a feedback effect on the droplet motion. The trajectories that eventually emerge are those for which the trajectory shape and the global wave field have achieved a mutual adaptation. We examined the quantization of the mean radii of the various trajectories in this framework. We used a general approach that consists in decomposing (using Grafs theorem) the entire wave field on the basis formed by the Bessel functions centred on the axis of the potential well. The total wave generates a virtual landscape of wave potential energy that traps the particle in specific orbits. This model [P18, T3, T4] gives an excellent interpretation of the possible radii of the circular orbits: they corresponds to the successive zeros of a J0 function centered on the magnet axis.

# The intermittency and the "wave function collapse"

We have thus shown that, as expected from the situation sketched in Figure V.2, there were specific values of the confinement that permitted stable orbital motions. These "states" are characterized by a non-quantum double quantization of the orbit size and the mean angular momentum. Naturally our system has no relation with the Planck constant but, as discussed in [P11], an analogy appears by considering that the Faraday wavelength plays here a role comparable to the de Broglie wavelength in quantum mechanics. We can now examine the motion of the walker when the width Λ of the potential well does not correspond to a pure state.

Film V.5. A chaotic orbit in a central force field
As shown in Film V.5 and Fig. V.6 the motion appears complex. However, direct observation reveals that regular motions still exist but only during limited time intervals (Fig. V.6 a). If the temporal evolution of the angular momentum Lz is recorded as a function of time, (Fig. V.6 b) the signal is not erratic, but composed of well-defined domains separated by abrupt transitions. The trajectory is thus formed of a succession of sequences of pure eigenstates with intermittent transitions between them. From long recordings probability distributions function of Lz can be obtained (Fig. V.6 c). Fig. V.6. The analysis of the intermittency of the chaotic states
(a) A chaotic trajectory in the central force field. Note the intermittent appearance of regular trajectories. (b) the temporal evolution of the angular momentum. (c) Probability distribution of the angular momentum as obtained from long recordings.
As demonstrated in [P19], this chaotic motion is characterized by an intermittency phenomenon inducing transitions between different periodic orbits. In this type of complex situation, the walker, at any given time, is in one of the possible discrete modes. This justifies the use of the term eigenstates for the states (n,m): they form a basis of decomposition on which the complex trajectories can be decomposed. In general the value of Λ defines the time spent on the average in each of the states. Only a fine-tuning of Λ permits the preparation of a "pure" state for which any time evolution is blocked.

Finally, we can note that we have in this macroscopic system the possibility of performing a continuous non-intrusive observation. It is interesting to consider a gedanken situation in which this observation would be impossible and we could only perform single intrusive measurements. Each of these destructive glimpses would lead to the observation of one of the eigenstates only but they would differ from each other. With many realizations, a probability of each result would emerge. A superposition of states would then be the best reachable description, the probabilities would seem intrinsic and the measurement would appear as a resulting from a "wave function collapse".

# Self-orbiting

When walkers were discovered, they were observed to move in straight line when undisturbed. One of the questions, raised at that time by Emmanuel Fort, was whether or not this was the only attractor for the drops motion. Could there be other states where the interaction with the field led to spontaneous orbiting ? The previously described experiments, in which orbiting is due either to a Coriolis force or to a central force, show that in the high memory regime there is the build-up of an effective wave-induced potential that adds to the trapping. Can this effect be sufficient to produce orbiting ? This was first demonstrated possible by Oza et al. [ref. 3 and 5] in their theoretical investigation of the circular orbits induced by a Coriolis force. However the experimental observation was not possible in that type of experiment. The experimental difficulty in testing this idea is that a "preparation" process is needed to force the walker into a trajectory close to that expected for the self-organized mode so that the corresponding wave field has time to build-up. For this purpose we turned to the central force experiment and used it to set the drop in rotation on a small orbit [P25]. We maintain it in this state during a time longer than the memory time and then switch off the external force. As shown in Figure V.7, the drop keeps orbiting after the switch-off for a time long as compared to the memory time (up to six orbital periods). Fig. V.7. The self-orbiting of a drop when no external force is exerted onto it.
The recent trajectory of the drop is superposed on the photograph of the wave field. In the part in red the drop had been submitted to a central force. In the part in yellow the droplets has kept orbiting, even though the external force had been switched off.
The numerical simulations [P25] show that these self-orbits can be stable if the noise is low. Since these orbits form attractors for free walkers it is natural to wonder if they play a role in their dynamics. If we investigate weakly confined trajectories either in a potential well or in a corral at very long memory (e.g. Me≥500) we observe multiple occurrences in which the droplet traps itself spontaneously in tight orbiting motion. It then escapes before getting trapped in another self-orbit motion elsewhere (see Fig V.8). The observation of the wave field reveals that its structure is then, locally, similar to that observed in the pure self-orbit shown in Fig. V.7. Fig. V.8. Spontaneous transient self-orbiting loops.
The trajectory of a walker in a wide potential well at long memory, shows the spontaneous formation of multiple transient self-orbiting loops.

# Bibliography

## [P11] Path memory induced quantization of classical orbits     (PDF)

E. Fort, A. Eddi, A. Boudaoud, J. Moukhtar, & Y. Couder
PNAS, 107, 17515-17520 (2010)

## [P14] Level splitting at macroscopic scale     (PDF)

A. Eddi, J. Moukhtar, S. Perrard, E. Fort & Y. Couder
Phys. Rev. Lett., 108, 264503 (2012)

## [P16] Trajectory eigenmodes of an orbiting wave source     (PDF)

E. Fort & Y. Couder
EPL, 102, 16005 (2013)

## [P18] Self-organization into quantized eigenstates of a classical wave-driven particle     (PDF)

S. Perrard, M. Labousse, M. Miskin, E.Fort, & Y. Couder
Nature Com, 5, 3219 (2014)

## [P19] Chaos driven by interfering memory.     (PDF)

S. Perrard, M. Labousse, E. Fort, Y. Couder
Phys Rev Lett, 113, 104101 (2014)

## [P20] Non-Hamiltonian features of a classical pilot-wave dynamics     (PDF)

M Labousse, S Perrard
Phys. Rev. E, 90 , 022913 (2014)

## [P21] Build-up of macroscopic eigenstates in a memory based constrained system     (PDF)

M. Labousse, S. Perrard, Y. Couder and E. Fort
New Journal of Physics, 16, 113027 (2014)

## [P23] Pilot wave dynamics In a harmonic potential: Quantization and stability of circular orbits.    (PDF)

M. Labousse, A.U. Oza, S. Perrard, J.W.M. Bush
Phys. Rev. E, 93, 033122 (2016)

## [P25] Self-attraction into spinning eigenstates of a mobile wave-source by its emission back-reaction

M. Labousse, S. Perrard, Y. Couder & E. Fort
Phys. Rev. E. (to appear 2016)

## [T2] Marcheurs, dualité onde-particule et Mémoire de chemin     (PDF)

Antonin Eddi, Thèse de Doctorat de lUniversité Paris-Diderot (2 Février 2011)

## [T3] Une mémoire ondulatoire : Etats propres, Chaos et Probabilités     (PDF)

Stéphane Perrard, Thèse de Doctorat de lUniversité Paris-Diderot (29 Septembre 2014)

## [T4] Etude dune dynamique à mémoire de chemin: une experimentation théorique     (PDF)

Matthieu Labousse, Thèse de Doctorat de lUniversité Pierre et Marie Curie (12 Décembre 2014)

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## [ref 6] Faraday pilot-wave dynamics: modelling and simulation

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