Path-memory induced quantization of classical orbitsEmmanuel 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 RechercheScientifique 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-action with the waves it generates. The resulting “walker” is adynamical association where, at a macroscopic scale, a particle(the droplet) is driven by a pilot-wave field. A specificity of thissystem is that the wave field itself results from the superpositionof the waves generated at the points of space recently visited bythe particle. It thus contains a memory of the past trajectory of theparticle. Here, we investigate the response of this object to forcesorthogonal to its motion. We find that the resulting closed orbitspresent a spontaneous quantization. This is observed only whenthe memory of the system is long enough for the particle to inter-act with the wave sources distributed along the whole orbit. Anadditional force then limits the possible orbits to a discrete set.The wave-sustained path memory is thus demonstrated to gener-ate a quantization of angular momentum. Because a quantum-likeuncertainty was also observed recently in these systems, the non-locality generated by path memory opens new perspectives.

bouncing droplets ∣ Landau quantization ∣ pilot wave ∣wave-particle duality

A material particle dynamically coupled to a wave packet atmacroscopic scale has been discovered recently and has been

shown to have intriguing quantum-like properties (1–4). Theparticle is a droplet bouncing on the surface of a vibrated liquidbath, and the wave is the surface wave it excites. Together they areself-propelled on the interface and form a symbiotic object.Recent investigations have shown that this “walker” exhibits aform of wave-particle duality, a unique feature in a classical sys-tem. This appears because the localized and discrete droplet has acommon dynamics with the continuous and spatially extendedwave. Various situations [diffraction and interference (3) and tun-neling (4)], where the wave is either bounded or split, have beenexamined. The surprising result is that for each realization of anexperiment of this type the droplet has an unpredictable indivi-dual response. However, a statistical behavior is recovered whenthe experiment is repeated. The truncation of the wave was thusshown to generate an uncertainty in the droplet’s motion. This“uncertainty”, though unrelated to Planck constant, has an ana-logy with the statistical behavior observed in the correspondingquantum-mechanical experiments. This characteristic has beenascribed to nonlocality. In this 2D experiment, the points ofthe surface disturbed by the bouncing droplet keep emittingwaves. The motion of the droplet is thus driven by its interactionwith a superposition of waves emitted by the points it has visitedin the recent past. This phenomenon, easily observed in the wavepattern of a linearly moving walker (Fig. 1A), generates a pathmemory, a hitherto unexplored type of spatial and temporal non-locality.

In the present work we show that this path-memory-drivennonlocality can lead to a form of quantization. Because thesystem is dissipative, its energy is imposed by a forcing at fixedfrequency. Therefore there is no possible quantization of energy.For this reason we studied a possible discretization of the angularmomentum of orbiting walkers similar to that of the Landauorbits of a charged particle in a magnetic field. One possibilitywould have been to study the trajectory of charged droplets in

a magnetic field. However, for technical reasons we chose avariant that relies on the analogy first introduced by Berry et al.(5) between electromagnetic fields and surface waves.

Its starting point is the similarity of relation ~B ¼ ~∇ × ~A in elec-tromagnetism with 2 ~Ω ¼ ~∇ × ~U in fluid mechanics. In these re-lations, the vorticity 2 ~Ω is the equivalent of the magnetic field ~Band the velocity ~U that of the vector potential ~A. Considering aplane wave passing over a point-like vortex, Berry et al. (5) showthat the phase field will exhibit defects as the wave passes oneither side of a point vortex. This is the analog of the Aharo-nov–Bohm effect (6) where the presence of a vector potentialgenerates a phase shift of the quantum-mechanical wave func-tion. Although this effect is unintuitive in quantum mechanics,it receives a simple interpretation in the hydrodynamic analogwhere the phase shift is due to the advection of the wave bythe fluid velocity. The hydrodynamic effect has been experimen-tally measured both in the propagation of acoustic waves (7) andin that of surface waves (8, 9) around a vortex.

In order to investigate circular orbits we use the same analogydifferently. The associated forces also have similar expressions. Acharge q moving at velocity ~V in a homogeneous external mag-netic field ~B experiences ~FB ¼ qð ~V × ~BÞ. In a system rotating atan angular velocityΩ, the Coriolis force on a massmmoving withvelocity ~V is ~FΩ ¼ −mð ~V × 2 ~ΩÞ, where 2 ~Ω is the vorticity due tosolid-body rotation. In classical physics, both these forces lead toorbiting motions in the planes perpendicular to ~B or ~Ω respec-tively. In a magnetic field the orbit has a radius

ρL ¼ mv∕qB [1]

and a Larmor period τL ¼ m∕qB. On a rotating surface a mobileparticle with velocity V moves on a circle of radius

RC ¼ V∕2Ω [2]

with a Coriolis period TC ¼ 1∕2Ω. Here we place a walker in arotating system. If it was a normal classical particle, it would moveon a circle of radius RC decreasing continuously with increasingΩ. Will walkers follow this classical behavior? What is the role ofthe wave field induced particle path memory?

Experimental SetupA sketch of our experimental system is given in Fig. 1B. Thefluid is contained in a cylindrical cell of diameter 150 mm, whichcan be simultaneously vibrated with a vertical accelerationγ ¼ γm sin 2πf 0t and rotated at a constant angular velocity Ω inthe range 0 < Ω < 10 rad∕s around a central vertical axis.

Author contributions: E.F., A.E., A.B., J.M., and Y.C. designed research, performed research,analyzed data, and wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.

See Commentary on page 17455.1To whom correspondence should be addressed. E-mail: emmanuel.fort@espci.fr.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1007386107/-/DCSupplemental.

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In the absence of rotation, this is the setup in which bouncingdroplets and walkers have already been investigated (1, 2). Whena liquid bath is vertically oscillated with an increasing accelera-tion, there is an onset γFm beyond which the Faraday instabilityappears (10, 11): The entire surface of the fluid becomes unstablewith formation of parametrically forced standing waves (of sub-harmonic frequency f 0∕2). Our experiments are all performedbelow this threshold but close to it, so that the fluid surface isflat. If a small droplet (of the same fluid as the bath) is depositedon the surface, it can survive by its sustained bouncing on the in-terface (12). As the forcing acceleration is increased, the droplet’sbouncing becomes subharmonic. It thus generates waves at thefrequency of the Faraday waves. The system being tuned below,but near the threshold of this instability, these waves are almostsustained. Correspondingly, the droplet starts moving horizon-tally on the fluid interface. This is a symmetry-breaking phenom-enon characterized by the value γWm at which the bifurcationoccurs. It is worth noting that for a given fluid, walkers are ob-served to exist only in a finite range of droplet sizes and vibrationfrequencies (2). In order to investigate the wavelength depen-dence of the observed phenomena we used several differentsilicon oils of viscosities 10.10−3, 20.10−3, 50.10−3, and100.10−3 Pa s, for which the optimum forcing frequencies are110, 80, 50, and 38 Hz corresponding to Faraday wavelengths

λF ¼ 3.74, 4.75, 6.95, and 8.98 mm, respectively. The walker’smotion is recorded with a camera placed in the laboratory frame,and the trajectories in the rotating frame are reconstructed byimage processing.

Experimental ResultsThe Path Memory. First, we address the path-memory concept.In the absence of rotation, a given droplet forms a walker inthe interval γWm < γm < γFm. The bifurcation to walking beingsupercritical, the walker’s velocity increases near the onset asVW ∝ ðγm − γWm Þ1∕2, then saturates at a value of the order of atenth of Vϕ, the phase velocity of Faraday waves.

The global wave field results from the repeated collision of thedroplet with the substrate. Each point of the surface visited by thedroplet 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-tion, τ diverges near the threshold as τ ∝ jγm − γFmj−1. By tuningγm we can thus control the time scale of this memory. Far fromthe Faraday threshold the waves are strongly damped and thewave packet has an approximately circular structure resultingfrom the most recent collisions of the droplet with the bath.In contrast, in the situation shown in Fig. 1A where γm is closeto the Faraday threshold, τ is very large and the wave field iswidely 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 secondarysources left behind by the droplet can be considered as imple-menting in reality Huygens secondary sources along a wave frontdefined 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 wallbeing the line of points that have not yet been visited by thewalker.

We can now consider the droplet’s motion. At all times it isdriven by its interaction with a superposition of waves emittedat the points it visited in the past. Through the mediation of thesewaves, the present motion is influenced by the past trajectory.This is what we call the “wave-mediated path memory” of the sys-tem. The parameter τ measures the “time depth” of this memory.

The Effect of Rotation.When the experimental cell is set into rota-tion, the free surface of the fluid becomes parabolic (Fig. 1B). It isnoteworthy that a bouncing droplet does not drift due to rotationbecause its bouncing on the slanted surface exactly compensatesthe centrifugal effect. Hence a motionless droplet remainsmotionless in the rotating frame. As for a walker, it walks in therotating frame where it experiences the Coriolis force. As shownin Fig. 1C, its wave field is curled (see Movies S1 and S2). In thelaboratory frame, as shown in Fig. 1D, the droplet’s trajectory isepicycloidal and corresponds to the composition of an advectionat an angular velocity Ω with a counterrotating Coriolis-inducedorbiting. By image processing we obtain the motion in the rotat-ing frame. It is a circular orbit of radius R and angular velocityΩexp

R (Fig. 1E). In some cases the trajectory is epicycloidal in therotating frame. In those cases we measure the amplitude of themotion having an angular velocity Ωexp

R ¼ 2Ω. When the rotationfrequency is increased, the modulus of the walker velocityremains close to the value V 0

W it had without rotation, and theradius R of the orbits decreases. Two distinct behaviors are ob-served when the system is tuned far or near the Faraday instabilitythreshold, with short- or long-term memory, respectively.

For short-term memory, the walker moves on an orbit ofdecreasing radius RC (Fig. 2A) as would be expected fromEq. 2. The measured values are, however, larger than expectedand we find an excellent fit with

RexpC ¼ aðVW∕2ΩÞ; [3]

A C

B D

E

Fig. 1. (A) Photograph of the wave field of a walker with long-term pathmemory in the absence of rotation [with Γ ¼ ðγFm − γmÞ∕γFm ¼ 0.018 andVW ¼ 13.7� 0.1 mm∕s]. (B) Sketch of the experimental setup. (C) Photo-graph of the wave field of a walker of velocity VW ¼ 13� 0.2 mm∕s movingon a bath roation clockwise at Ω ¼ 0.9 Rd∕s. The motion is counterclockwiseand the orbit n ¼ 3. (D and E) The trajectory of a walker of velocityVW ¼ 15.6� 0.1 mm∕s on a bath rotated at Ω ¼ 2.16 Rd∕s (orbit n ¼ 2) asmeasured in the laboratory (D) and in the rotating frame of reference (E).The black segments are 2 mm long.

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where a is a constant. Its value, for a given oil, is a function of thememory and ranges from 1.2 to 1.5. The origin of this factor prob-ably results from the existence of two consecutive regimes duringthe bouncing. During its free flights, the droplet is submitted to apure Coriolis effect. During the impacts, the droplet generateshollows in the interface. We believe that the interaction betweenthe bath and the droplet during near contacts tends to oppose thebending of the trajectory. The numerical simulations presentedbelow confirm that this hypothesis can account for the existenceand value of the prefactor a.

For long-termmemory, the observed orbits are totally different(Fig. 2B). When Ω is increased, the radius of the orbit, instead ofdecreasing steadily, undergoes abrupt transitions between succes-sive slanted plateaus. Simultaneously a slight increase of the mod-ulus of the walkers velocity VW ðΩÞ is observed as the orbitsbecome smaller.

The possible orbits being discrete, we label the plateaus by anorder n, with n ¼ 0 corresponding to the tightest orbit observed atlarge Ω. Each plateau crosses the continuous short path-memorycurve given by Eq. 3 at a valueΩ0

n of the angular velocity (Fig. 2B).At these crossings, the observed orbits have the same radiusR0n ¼ Rexp

n ðΩ0nÞ ¼ Rexp

C ðΩ0nÞ they would have had with short-term

memory. A plateau extends on both sides of the curve given byrelation 3 between two limiting values Ω−

n < Ωn < Ωþn . A hyster-

esis is observed: The abrupt transition from one plateau to the

next does not occur for the same value when Ω is increased ordecreased.

On each of these plateaus the wave field exhibits a coherentglobal structure (see Fig. 1C). In this long path-memory regime,the spatial extension of the wave field exceeds the size of theorbit. Because of the circularity of the motion, the wave crestscurl around and the droplet follows its own path.

We repeated the same series of experiments at four frequen-cies, using different oils. In each case we studied the trajectoriesof walkers of various velocities. The number of observed plateausdepends on the distance to onset. The closer γm to γFm, the largerthe orbits that become discrete. Each plateau also becomeslonger as hysteresis increases. Limiting ourselves to the observedplateaus we find that the radii are located on the same stepswhen they are rescaled by the Faraday wavelength Rn∕λF andplotted as a function of ðVW∕2ΩλFÞ1∕2 (see Fig. 2C).

Let us now look for an analogy between the discretization inour results and that of quantum systems. In quantum mechanics,the motion of a charge in a magnetic field is quantified in Landaulevels (15). The associated Larmor radius only takes discretevalues ρn given by

ρn ¼ffiffiffiffiffiffiffiffi1∕π

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�nþ 1

2

�hqB

s[4]

with n ¼ 0;1;2;…. This equation can alternatively be expressed asa function of the de Broglie wavelength λdB ¼ h∕mV . The radiithen satisfy

ρnλdB

¼ffiffiffiffiffiffiffiffi1∕π

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�nþ 1

2

�mqB

VλdB

s: [5]

The Landau orbits coincide with the classical ones when Eqs. 1and 5 are satisfied simultaneously. Eliminating mV∕qB gives

ρn ¼ 1

π

�nþ 1

2

�λdB; [6]

which corresponds to the Bohr–Sommerfeld quantization of theorbits perimeter.

We can now ask ourselves if our experimental results havesome analogy with this quantization by first using the previouslydiscussed correspondence where 1∕2Ω is the analogue of theLarmor period τL ¼ m∕qB. A bolder step is to assume that inour classical system the Faraday wavelength could play a rolecomparable to the de Broglie wavelength in the quantum situa-tion. We thus try fitting our data with a dependence of the typegiven by

Rn

λF¼ b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�nþ 1

2

�1

2Ω

VW

λF

s: [7]

As shown in Fig. 2C, an excellent fit of all the observed radii isgiven by this discrete set of curves. Only the dimensionlessprefactor differs. Although

ffiffiffiffiffiffiffiffi1∕π

p ¼ 0.564, we find the best fitfor b ¼ 0.89. The radii R0

n for which the classical orbits coincidewith the discrete ones are those for which both Eqs. 3 and 7 aresatisfied:

R0n ¼ b2

a

�nþ 1

2

�λF [8]

with b ¼ 0.89 and a ¼ 1.5 we have b2∕a ¼ 0.528. This means thatthe diameters of the orbits are quantized:

D0n ¼ 2R0

n ≃ ðnþ 1∕2ÞλF: [9]

A B

C

Fig. 2. (A) Plot of the radius R as a function of 2Ω∕VW in the short path-memory case. Black line: Coriolis radius RC ¼ VW∕2Ω, gray line: fit by Eq. 3.(B) Same plot for a walker in the long path-memory case (experimentperformed with μ ¼ 50.10−3 Pa s, f0 ¼ 50 Hz, λF ¼ 6.95 mm, and VW ¼18 mm∕s). (C) Plot of the nondimensional radii Rn∕λF as a function ofðVW∕2ΩλFÞ1∕2. The results of four experiments with different oils and velo-cities are shown. Black squares: μ ¼ 50.10−3 Pa s, λF ¼ 6.95 mm, and VW ¼18 mm∕s. Open squares: μ ¼ 20.10−3 Pa s, λF ¼ 4.75 mm, and VW ¼12 mm∕s. Crosses: same conditions with VW ¼ 8 mm∕s. Black dots:μ ¼ 10.10−3 Pa s, λF ¼ 3.74 mm, and VW ¼ 8 mm∕s). The gray line is the ra-dius that would be observed with short path memory (relation 3 witha ¼ 1.45). The black lines are the levels given by relation 7 with b ¼ 0.89.

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Modeling and Numerical SimulationsThe motion of a droplet endowed with path memory on a rotat-ing bath can be modeled and computed by numerical simulationsthat rely on principles previously introduced for free walkers (3).We use the simplifying approximation that the vertical and hor-izontal motions of the droplet are decoupled. Takeoff and land-ing times are thus determined only by the forcing oscillations ofthe liquid bath. The principle of the modeling and the notationswe use are defined on Fig. 3A. At time ti the 2D position of thedroplet is ~ri. The motion is iterative so that the drop will movefrom ~ri to ~riþ1, which it will reach one Faraday period later, attime tiþ1 ¼ ti þ TF . At each collision with the interface it under-goes a damping of its horizontal velocity due to viscous friction,but it is simultaneously given a momentum increment by its in-elastic shock with the slanted oscillating surface. The directionand the intensity of the kick result from the local slope of theinterface. It induces, together with the next parabolic phase, thedroplet’s horizontal displacement δ ~ri ¼ ~riþ1 − ~ri.

The local surface height hð ~r;tiÞ results from the superpositionof previously emitted waves as sketched in Fig. 3A. At eachbounce in ~ri, a circular wave is emitted and this emission is sus-tained during a typical time τ. The topography of the surface thusresults from the superposition of these in-phase waves and con-tains a memory of the droplet’s trajectory. In the simulations,each emitted wave is sinusoidal with the same wavelength λFand phase ϕ. The waves being 2D, their amplitudes decreaseas the square root of the distance to the source. The relative sur-face height hð ~r;tiÞ at position ~r and time ti is given by

hð ~r;tiÞ ¼ ∑−∞

p¼i−1

ReA

j ~r − ~rpj1∕2exp−

�ti − tpτ

�

× exp−�j ~r − ~rpj

δ

�exp i

�2πj ~r − ~rpj

λFþ ϕ

�; [10]

where ~rp is the position of a previous impact that occurred attime tp ¼ ti − ði − pÞTF . As time passes, the amplitude of anold source decreases with a typical damping time τ. The wavesare also spatially damped with a typical damping distance δ. Bothτ and δ can be estimated from the experiments and turn out tobe independent. The damping δ of traveling waves is fixed andrelated to the fluid viscosity. The damping time τ, the path-memory parameter, is tunable, being determined by the distanceto the Faraday instability threshold.

The surface slope ~Sð ~ri;tiÞ at the point of bouncing ~ri at timeti is defined by the local gradient of the height in the plane ofthe interface. It is given by the spatial derivative of Eq. 10:~Sð ~ri;tiÞ ¼ ~∇hð ~ri;tiÞ. This vector is the sum of the contributionsof previous sources and can be written

~Sð ~ri;tiÞ ¼ ∑−∞

p¼i−1

~sð ~ri − ~rp;ti − tpÞ: [11]

In the following, for simplicity we use ~sðpÞ ¼ ~sð ~ri − ~rp;ti − tpÞ and~S ¼ ~Sð ~ri;tiÞ. It had already been shown (3) that simulations usingthese hypotheses are able to reproduce the walking bifurcationfrom an initially motionless droplet. In the walking regime thesimulations also produce a map of the fluid surface height thatgives the topography of the wave field. Realistic fields can be ob-tained for all path-memory regimes. If the walker is now in a ro-tating frame of reference, its Coriolis acceleration has to be takeninto account in the simulations. Because there is negligible influ-ence of the Coriolis acceleration on the radial propagation of thewaves, only the droplet motion has to be corrected. At each boun-cing cycle, its instantaneous velocity ~vi is corrected by adding anincrement δ ~vi ¼ −2 ~Ω × ~viδti. During the contact with the bath theinstantaneous velocity is weak. Realistic results are obtained byconsidering that the Coriolis effect acts on the droplet only forthe duration δti of its free flights. A correcting prefactor a is thenobserved with values similar to those measured experimentally.These simulations where the sources are distributed on a circle(Fig. 3B) produce steady regimes in which the droplet moves at aconstant velocity on stable circular orbits. The specific phenom-ena observed experimentally are recovered: (i) With short-termpath memory (e.g., τ ¼ 10TF), the radius of the orbits varies con-tinuously with Ω as expected from the classical behavior. (ii) Withlong-term path memory (e.g., τ ¼ 30TF), the orbit radii formdiscrete plateaus. These plateaus cross the continuous curveassociated with the short memory trajectories at discrete valuesΩ0

n (Fig. 3C).It is noteworthy that when submitted to rotation the modulus

of the average velocity measured in the rotating frame varies onlyslightly as compared to the velocity of the walker in the absence ofrotation. The sign of this variation depends on the memory as willbe discussed below. The simulations permit the computing of sur-face topographies. They reproduce accurately the experimentalwave field. As an example, Fig. 4 shows the comparison betweenthe experimental and the computed wave fields for two orbitsn ¼ 1 and n ¼ 0 with strong path memory.

DiscussionWe now discuss the role of path memory on the dynamics of thewalker. The droplet moves because it bounces on the superposi-tion of the waves from sources on its past trajectory. All sourcesare synchronized by the periodic forcing. In the steady regimeswhere the droplet’s velocity is of constant modulus the wave

A

B

C

Fig. 3. (A) The computation of the path-memory effect. At the time t ¼ tithe droplet, located at position ~ri , bounces on a surface deformed by a super-position of circular waves centered in the points where the droplet has pre-viously bounced. (B) The same sketch for a circular orbit. The droplet is in D atθ ¼ 0 at time ti and has a clockwise motion. The temporally phase-lockedwave-sources are distributed on the circle. Their temporal decay is symbo-lized by the fading of the gray of the crosses. Each source generates beneaththe droplet a slope ~sðpÞ. The norm of ~sðpÞ oscillates with the distancebetween the source and the droplet. The sum of these contributions is~Sð ~ri ;tiÞ. In the case of a long memory, the sources are around the whole orbit.The circles centered in D (dashed lines) show the sources of waves reaching Dwith the same phase. (C) Evolution of the orbit radius obtained in the simula-tion with short path memory (τ ¼ 10TF , gray line) and long path memory(τ ¼ 30TF , black line), respectively. The thin and thick black lines correspondto increasing and decreasing Ω, respectively; the black diamonds are the ex-perimental data points for μ ¼ 50.10−3 Pa s, f0 ¼ 50 Hz, and VW ¼ 18 mm∕s.

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sources are distributed at equal intervals along the trajectory.When the walker has this rectilinear motion, all the contributions~sðpÞ in the summation of Eq. 11 are oriented along the axis ofpropagation. In contrast, when the droplet has a circular motionthese elementary contributions are no longer colinear as sketchedin Fig 3B. The radial Sr ¼ ~S · ~ur and azimutal Sθ ¼ ~S · ~uθ compo-nents play specific roles in the dynamics of the droplet. The latterdrives the propulsion of the droplet, whereas the former gives atransverse force that adds up with the Coriolis force, modifyingthe resulting orbit radius.

With short-term memory the active sources are located justbehind the droplet on a short arc of a circle. As in the linear tra-jectories, these sources add up to propel the droplet. With long-time memory, the situation is more complex. The sources thathave to be taken into account are distributed along the wholecircle (see Fig. 3B). We can readily see that the contribution~sðpÞ from past sources located immediately behind the dropletis, for purely geometrical reasons, opposite to the contributionof a source located immediately in front of the droplet. Alonga closed loop of the orbit, ~sðpÞ has rotated by an angle π.

We now discuss the role of this geometric phase, which hasbeen introduced by Berry (16), on the path-memory dynamicsof the walker. We define srðpÞ and sθðpÞ as the elementary con-tribution of a source located in θp to the radial and azimuthalcomponents Sr and Sθ. Fig. 5 A and B show sθðpÞ and srðpÞ asa function of θp around one orbit. They both oscillate due tothe variation of the spatial phase ϕp ¼ 2πj ~ri − ~rpj∕λF along thetrajectory. This phase increases with θp until θp ¼ π, thendecreases symmetrically. The additional effect of the radialand azimuthal projections gives specific characteristics to srðpÞand sθðpÞ. In the case of sθðpÞ, the envelope is modulated bycos θp∕2 due to the azimuthal projection. Hence, while thesources on the orbit located at θp ¼ π − θ have the same spatialphase ϕp as the one situated symmetrically at θp ¼ π þ θ, theircontributions have opposite signs. Thus, their effect partiallycancels out. The remaining contribution depends only on theirrelative temporal damping. This is driven by the parameterτ∕T, which is the relative path memory τ over one rotationperiod T.

We find experimentally that the droplet velocity VW increasesslightly with τ∕T. However, the two contributions sπ−θ and sπþθ

cancel out more efficiently and the sum over a single loop tendstoward zero. This apparent paradox is solved when integratingover all the past sources. For increasing τ∕T, the number of loopsthat must be taken into account increases and the overall integra-tion gives resulting larger Sθ values.

We will now focus on the radial component srðpÞ to understandthe additional transverse force resulting from path memory thatadds up with Coriolis force (Fig. 5B). The envelope is now modu-lated by sin θp∕2 due to the radial projection. Hence, the contri-bution of the symmetrical sources with the same spatial phase ϕplocated at θp ¼ π − θ and θp ¼ π þ θ are additive. The sign asso-ciated with each source depends only on its spatial phase. Along aclosed loop of the orbit, ϕp increases to a maximum value for θp ¼π and decreases symmetrically. The phasor representation helpsto understand the effect of ϕp. In Fig. 5C, the cumulative contri-bution to the radial slope SrðθpÞ is represented as a phasor in thecomplex plane, in a construction similar to that of Cornu forFresnel diffraction (17). In Fig. 5D, we show the evolution of thereal part as a function of θp integrated up to one orbit. The struc-ture of this phasor graph shows an S-shaped spiral structure withan inflection point at θp ¼ π. Each turn of the spiral correspondsto the spatial phase variation of 2π along an arc length of theorbit. The resulting contribution tends to cancel out. The regionof the inflection point of the spiral is related to those sourcesdiametrically opposed to the droplet on the orbit. Their contri-bution does not usually cancel out. It does so only for specificorbit diameters. In case of large τ∕T, similar contributions aregenerated by each turn of the droplet.

We now study the additional force as a function of the radius.As the radius increases, the S-shaped spiral rotates so that its in-flection point leads to successive positive and negative jumps ofSr . The oscillation has the periodicity of the spatial phase of thediametrically opposed source. It thus gives Sr a periodicity of λFfor increasing diameters. Fig. 6A shows Sr as a function of thevorticity. It is characterized by a succession of sawteeth. For adiscrete set of values of the radius, Sr ¼ 0 and the short-term

A C

B D

Fig. 4. Comparison of the experimental and computed central regions ofthe wave field. (A and B) The first two modes n ¼ 1 and n ¼ 0 as observedin the experiment. (C and D) The same orbits obtained in the simulation for awalker with long path memory (τ ¼ 30TF and δ ¼ 1.6λF ).

)

A B

C D

Fig. 5. (A) The individual contributions to the azimuthal slope sθðpÞ due tothe sources distributed on the orbit as a function of their angular position θp(increasing from 0 to 2π behind the drop). (B) Same graph for the individualradial slopes srðpÞ. (C) The complex plane construction of a Cornu-like spiralgives the summation of the srðpÞ and sθðpÞ when θp varies from 0 to 2π (firstturn). (D) The resulting variation of the slope Sr as a function of θp.

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memory orbits remain unchanged. These radii correspond to theset of values of the angular velocities Ω ¼ Ω0

n. In all the othersituations a supplementary quantization force is exerted on thedroplet so that the short-term memory orbit is modified in thecase of long memory. When the angular velocity is smaller thanone of the Ω0

n, but close to it, the Coriolis effect should result inan orbit of larger radius. However, the slope Sr is negative so thatthe droplet experiences an extra centripetal force that opposesthe increase of the radius. Conversely, when Ω > Ω0

n the slopeis positive and generates a centrifugal force that opposes theradius decrease. These effects are thus responsible for the forma-tion around Ω0

n of a slanted plateau. From this analysis, a remark-able simplification appears for the particular case of circularorbits. Because the contribution to the radial force of mostsources on the circle cancel out, the remaining effect condensesto that of a single virtual source diametrically opposed to the dro-plet. The fundamental relation of dynamics applied in the radialdirection reduces to

mV 2W

R¼ −2

maΩVW þ A sin

�2π

2RλF

þ ϕ

�: [12]

m is the mass of the droplet and A and ϕ are, respectively, theamplitude and the phase of the wave due to the virtual droplet.A follows the average characteristics of the spatial and temporal

attenuation of the sources around θp ¼ π, and ϕ is set constant atπ∕2. Fig. 6B shows the computed results of this model and the fitit provides to the experimental data of Fig. 5B. The memoryeffect is contained in the amplitude of A, which reduces to zerofor low memory. We thus recover in this simple model our resultthat quantification is related to the diameter (see Eq. 9) and notto the perimeter as in Bohr–Sommerfeld (see Eq. 6). We canfinally note that this quantization is a priori independent ofthe nature of the force that sets the walkers into orbits. If itwas possible to give a droplet a large electric charge and toput it in a large enough magnetic field, the same discretizationof its orbits would apply.

ConclusionIn our experiment the droplet is driven by the wave. For this rea-son our system could be considered as implementing at macro-scopic scale the idea of a pilot-wave considered by de Broglie(18) and Bohm (19) for elementary particles at quantum scale.However, it differs from these models by its sensitivity to historydue to its wave-supported path memory. The incentive to under-take the present experiments was to investigate the constraintsimposed by this memory on the possible trajectories. The walkers,when far from any disturbance, have rectilinear trajectories.When disturbed (be it by an interaction with a diffusive centeror by an applied force), their trajectory becomes curved and whatcould be called a feed-forward effect appears. With long-termpath memory, opened trajectories can become very complex lead-ing to the probabilistic behaviors observed in diffraction (3) andtunneling (4). For closed trajectories only a discrete set of orbits ispossible: The path memory is thus proven to be responsible for aself-trapping in quantized orbits reminiscent of the Bohr–Sommerfeld semiclassical quantization. It is a remarkable factthat a dynamics dominated by path memory is directly responsiblefor both uncertainty and quantization phenomena. We believethat the concept of path memory could be extended to Hamilto-nian systems. Such an extension could bring about a previouslyundescribed framework for an approach to quantum nonlocality.

ACKNOWLEDGMENTS. We are grateful to Mathieu Receveur and LaurentRhea for their help in setting up this experiment and to Maurice Rossiand Eric Sultan for useful discussions. The authors acknowledge the supportof the French Agence Nationale de la Recherche (ANR) under reference ANRBlanche 02 97/01.

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3. Couder Y, Fort E (2006) Single-particle diffraction and interference at macroscopicscale. Phys Rev Lett 97:154101.

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5. Berry MV, Chambers RG, Large MD, Upstill C, Walmsley JC (1980) Wavefront disloca-tions in the Aharonov-Bohm effect and its water wave analogue. Eur J Phys 1:154–162.

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7. Roux P, de Rosny J, Tanter M, Fink M (1997) The Aharonov-Bohm effect revisited by anacoustic time-reversal mirror. Phys Rev Lett 79:3170–3173.

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A B

Fig. 6. (A) The variation of the total slope Srð2πÞ (of the sources of the lastorbit) as a function of the vorticity 2Ω. (B) The fit of the experimental resultsby the simplified model (see Eq. 12). Lines are darker for increasing values ofthe wave amplitude parameter A in Eq. 12.

17520 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1007386107 Fort et al.

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