PHYSICAL REVIEW FLUIDS 9, 084001 (2024)

Static Bell test in pilot-wave hydrodynamics
Konstantinos Papatryfonos*

Gulliver, CNRS, ESPCI Paris, Université PSL, 10 rue Vauquelin, 75005 Paris, France
and Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave.,
Cambridge, Massachusetts 02139, USALouis Vervoort†
Higher School of Economics, 101000 Moscow, Russian Federation

André Nachbin‡

Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester,
Massachusetts 01609, USA andInstituto de Matemática Pura e Aplicada, Est. D. Castorina 110, Rio de Janeiro, RJ, 22460-320, Brazil

Matthieu Labousse§

Gulliver, CNRS, ESPCI Paris, Université PSL, 10 rue Vauquelin, 75005 Paris, France

John W. M. Bushࢱ

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge,
Massachusetts 02139, USA(Received 1 May 2023; accepted 3 July 2024; published 1 August 2024)
Since its discovery in 2005, the hydrodynamic pilot-wave system has provided a concrete macroscopic realization of wave-particle duality and concomitant classical analogs
of a growing number of quantum effects. The question naturally arises as to how closely
particle-particle correlations achieved with this classical system can mimic those arisingon the quantum scale. We here introduce a new platform for addressing this question, anumerical model of cooperative tunneling in a bipartite pilot-wave hydrodynamic system.
We execute a static Bell test, in which the system geometry is fixed and the two subsystems
are coupled through the intervening wave field. This wave-mediated coupling is notcongruent with the assumptions made in deriving Bell’s inequality, and so allows one torationalize the reported violations. Nevertheless, these violations are elusive, and arise only
in a limited corner of parameter space.
DOI: 10.1103/PhysRevFluids.9.084001
I. INTRODUCTION

In 2005, Couder and Fort [1,2] discovered that a millimetric droplet may self-propel along
the surface of a vibrating fluid bath through a resonant interaction with its own wave field. The
resulting "walker" consists of a droplet dressed in a quasimonochromatic wave field, and represents*

Contact author: kpapatry@mit.edu
Contact author: lvervoort@hse.ru‡
Contact author: anachbin@wpi.edu§
Contact author: matthieu.labousse@espci.psl.euࢱContact author: bush@math.mit.edu†

2469-990X/2024/9(8)/084001(14)084001-1©2024 American Physical Society

KONSTANTINOS PAPATRYFONOS et al.

a concrete, macroscopic example of wave-particle duality [3]. Remarkably, this hydrodynamic
pilot-wave system exhibits many features previously thought to be exclusive to the microscopic,
quantum realm [4,5]. Notable examples include single-particle diffraction and interference [2,6,7],
quantized orbits [3,8], unpredictable tunneling [9], Friedel oscillations [10], surreal trajectories [11],spin lattices [12], and quantumlike statistics and statistical projection effects in corrals [13,14]. In
all instances, the emergent quantum behavior may be rationalized in terms of the droplet’s nonMarkovian pilot-wave dynamics [5]. Specifically, the instantaneous wave force imparted to the drop
during impact depends on the droplet’s history. Thus, the drop navigates a potential landscape of its
own making [5], and the hydrodynamic pilot-wave system is said to be endowed with "memory"[15]. The walking-droplet system has provided a framework for exploring the boundary between
classical and quantum behavior. Moreover, it has inspired investigations of other hydrodynamic
pilot-wave systems in which oscillators interact with their suspending fluids [16–19].
In several settings, long-range interactions in the walking-droplet system arise through the
influence of the pilot-wave field [5]. For example, long-range lift forces are generated when awalking droplet interacts with a submerged pillar [20] or well [10], and could be misconstrued asindicating action at a distance if the influence of the pilot-wave field were not adequately resolved[5]. Moreover, long-range correlations between distant walkers may be established through the
influence of the intervening wave field [21–23]. Recently, Papatryfonos et al. [24] and Frumkin et al.
[25] established hydrodynamic analogs of superradiance, an effect originally attributed to quantum
interference of two or more entangled atoms [26–28], but subsequently rationalized in terms ofclassical electromagnetic wave interference [29]. The accumulation of these results naturally raises
the question as to whether this pilot-wave hydrodynamic system might exhibit bipartite correlations
that violate Bell’s inequality [30], and so provide a hydrodynamic analog of quantum entanglement,
the acid test of quantumness. This question seems all the more pertinent given suggestions that
Bell’s theorem may have no bearing on systems in which particles interact with a background field[30–32].
Bell’s theorem was derived by John Bell in 1964 [33] with a view to informing the Bohr-Einsteindebate concerning the completeness of quantum theory [34]. Hidden variables are those variables
that would be required for a complete description of quantum dynamics. For example, for thetype of dynamics engendered in pilot-wave hydrodynamics, these would be the discrete variables
defining the position and momentum of the particles, as well as the continuous variables defining thebackground field with which the particles interact. Quite generally, a Bell test can be performed on
any physical system consisting of two subsystems (A and B) on which one measures a dichotomic
property X (with stochastic outcomes of +1 or −1) that depends on some "analyzer setting" (α or β).
The measurement XA made in the left two-level subsystem depends on the analyzer setting α, which
may take values a or a ; likewise, the measurement XB made in the right two-level subsystemdepends on β, which may take values b or b .
In the derivation of the Bell’s inequality, Eq. (1), two assumptions are made. Assumption (i) isthat the two subsystems undergo only local interactions. Specifically, XA depends on α and not on β;
likewise, XB depends on β and not on α. Assumption (ii) is that the hidden variables that prescribe
both XA and XB are independent of α and β. Bell’s theorem [33] implies that for any classical system
for which assumptions (i) and (ii) hold, the quantity S(α = a, β = b, α = a , β = b ) = M(a, b) +M(a , b) + M(a, b ) − M(a , b ) must satisfy the inequality|S(a, b, a , b )|  2


(1)



for any choice of measurement settings (a, a , b, b ). Here, M(α, β ) is the average product,
XA XB P(XA , XB |α, β ),M(α, β ) =(2)

XA ,XB

where P(XA , XB |α, β ) is the joint probability of measurements (XA , XB ) when the left and rightanalyzers are set, respectively, to (α, β). We note that Eq. (1) is cast in the form of the ClauserHorne-Shimony-Holt (CHSH) inequality [35]. It has been well established that quantum systems
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FIG. 1. Hydrodynamic Bell test arrangement. (a) Schematic of our hydrodynamic Bell test. The system
consists of a pair of drops (red and green) walking on the surface of a vibrating liquid bath (blue) that spans
the solid substrate (gray). Each drop is confined to its subsystem, a pair of wells separated by barriers acrosswhich they may tunnel unpredictably at a rate influenced by the barrier depths α and β, as may assume valuesof a, a or b, b , respectively. Inset: β is the distance between the unperturbed free surface and the top of thesubmerged pillar separating cavities (+) and (−) in subsystem B; α is analogously defined in subsystem A. (b)The four possibilities for the droplet configuration space, (XA , XB ).

can violate the inequality (1) for a judicious choice of (a, a , b, b ), and these Bell violations arewidely taken to be proof that the statistical behavior of quantum systems cannot be underlaid by alocal, causal dynamics [36]. Quantum Bell tests were first performed with static analyzer settings[37], and so are referred to as static tests. In subsequent dynamic tests [38–44], the detector settings
α and β were altered just prior to measurement, so that the two measurement events are spacelike
separated, so that assumption (i) is expected to hold true. Bell violations have been achieved
with classical electromagnetic [45–48] and acoustic wave [49] systems. We here demonstrate thepossibility of achieving Bell violations in a static test with a theoretical model of the walking-droplet
system.
II. MODEL AND METHODS

We consider a pair of walking droplets in the bipartite tunneling system introduced by Papatryfonos et al. [24] in their demonstration of a hydrodynamic analog of quantum superradiance[see Fig. 1(a)]. We describe this system in terms of two coupled, two-level systems, as shown
schematically in Fig. 1(b). The two subsystems, labeled A and B, contain a single, wave-generating
particle confined to a pair of identical cavities separated by a barrier across which the particles maytunnel. The measurements XA and XB in our Bell experiment indicate whether the droplets are in the
inner well (X j = −1, j = A, B) or the outer well (X j = +1, j = A, B) at the time of measurement.
Each particle generates waves and moves in response to them according to Eqs. (3)–(5). In eachsubsystem, the preferred cavity corresponds to the ground state (−) and the other to the excited state(+). The ground state may be either the inner or the outer cavity, depending on the geometry of the
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KONSTANTINOS PAPATRYFONOS et al.

outer cavities [24]. For the specific geometry considered here, the inner cavity of each subsystem isthe ground state (−) and the outer cavity, the excited state (+) [see Fig. 1(b)]. The two subsystems
are separated by a coupling cavity of fixed length Lc , and by barriers that are sufficiently highas to preclude the particles from tunneling into the coupling cavity. Waves are transmitted acrossthe central cavity, and so provide the coupling between subsystems A and B. The strength of thiscoupling is prescribed by the geometry of the central cavity: by increasing its depth dc , the coupling
may be increased, allowing the coupling cavity to serve as a nearly resonant transmission line
[21]. Transitions between ground and excited states in the subsystems correspond to individual
tunneling events, the rate of which depends on the depths of the submerged barrier, denoted α
for the subsystem A and β for the subsystem B. These barrier depths α and β thus serve as themeasurement settings in our Bell tests.
We employ the numerical simulation method developed by Nachbin [21,50] for the onedimensional motion of walking droplets over a vibrating liquid bath with complex topography. We
adapt this model in order to consider the cooperative tunneling of two identical particles in thegeometry depicted in Fig. 1(a). Two identical particles of mass m are confined to their respective
subsystem (A or B), and their positions, x j ( j = A, B), evolve according to Newton’s law:mẍ j + cF (t )ẋ j = −F (t )∂η
(x j (t ), t ).
∂x

(3)

The particles move in response to gradients of the wave elevation η(x, t ). The waves are generated
in a manner consistent with models of the vertical droplet dynamics [51,52]. The droplet impacts thefree surface with the Faraday frequency ωF = ω/2 at a prescribed time relative to the vibrational
forcing, making first contact when ωt = 0.45π . Waves are generated during the contact time, ofduration Tc = TF /4, where TF = 2π /ωF is the Faraday period, a value consistent with laboratory
experiments [51] and used in prior studies [50,52]. During impact, the droplet is accelerated by thewave slope and decelerated by a drag force proportional to its horizontal speed, a drag coefficient
c = 0.01 s/cm and the normal force imparted to the droplet by the interface, F (t ). The time dependence of these propulsive and drag forces is prescribed by F (t ) = (2π 2 mg)/(ωTc ) sin (πt/Tc )1Tc , asfollows from the linear-spring-like response of the interface in the walker system [50,52,53]. Here,g is the gravitational acceleration, and the periodic indicator function 1Tc highlights the intermittent
nature of the forcing imparted by the interface to the bouncing droplets, which acts only during
impact. By generating waves during impact, the droplets establish their own time-dependent wave
potential, which serves as the memory of the system [15]. The wave forcing enters the droplet
trajectory equation through the final term in Eq. (3), and is computed as follows.
The velocity potential of the liquid bath φ(x, z, t ) is a harmonic function satisfying Laplace’s
equation. In the bulk of the fluid, the velocity field is given by (u, v) = ∇φ. The wave model is formulated in the bath’s vibrating reference frame, where the effective gravity is g(t ) = g + γ0 sin(ωt ).
The wave field thus evolves according to [50,52]:
∂φ
∂ 2η
∂η
=
+ 2ν 2 ,∂t∂z∂x
σ ∂ 2η
∂ 2 φ  c̃F (t )∂φ= −g(t )η ++ 2ν 2 −1D .
∂t
ρ ∂x 2∂xρπ R2 j
j=A,B

(4)
(5)

The particles ( j = A, B) generate waves on the free surface by imparting, during impact, a pressurelike term over an area corresponding to the droplet’s diameter, D j = 2R = 0.07 cm, as denoted by
the spatial indicator function 1D j . c̃ = 0.95 is a dimensionless constant that prescribes the strength of
the impact-induced wave forcing. Consistent with our modeling of the droplet dynamics, the wave
forcing coefficient F (t ) is activated only during the contact time Tc = TF /4. The forcing terms ofthis wave-particle dynamical system are discontinuous in both time and space, as highlighted by
the indicator functions arising in Eqs. (3) and (5). Through the imposition of periodic wave-particle
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STATIC BELL TEST IN PILOT-WAVE HYDRODYNAMICS

coupling at the Faraday frequency, we ensure resonance between the particle and the most unstable
Faraday mode of the bath [52], a key feature of pilot-wave hydrodynamics [4,5,54].
In Eqs. (3)–(5), spatial derivatives are computed using the Fast Fourier transform (FFT) in x. The
shallow outer reaches of the fluid domain extend sufficiently far to ensure quiescent conditions in thefar field. The vertical speed at the free surface φz (x, 0, t ) is defined through a Dirichlet-to-Neumann(DtN) operator [50] and yields a Fourier integral operator computed in a straightforward fashion
through a FFT. The DtN operator mathematically reduces the two-dimensional fluid problem to
one spatial variable defined along the undisturbed free surface. To compute the DtN operator, anumerical conformal mapping is performed that maps the (x, z) fluid domain onto a (ξ , ζ ) canonicalflat strip. Details of the conformal mapping can be found elsewhere [55,56]. The mapping for agiven geometry is computed only once and provides the relation x = x(ξ ) along the undisturbed
free surface. We denote by F the FFT in the ξ coordinate, which runs along the undisturbed freesurface in the canonical domain. We denote by φ(x, 0, t ) = ϕ(x, t ) the Dirichlet data. We thus havethat
φz (x, 0, t ) = DtN[φ](x, t ) =F −1 {G(k)F[ϕ(ξ )]},M(ξ (x, 0))(6)√where G(k) = k tanh k is the Fourier multiplier [50]. The metric coefficient is M(ξ ) = |J|, where|J| is the Jacobian of the [(x, z) → (ξ , ζ )] change of variables, evaluated along the undisturbed freesurface [55,56]. To summarize, the geometrical information of the cavities and barriers is encodedin M and in ξ = ξ (x, 0), which is obtained with the inverse map. The time evolution is performed
with a second-order fractional-step Verlet method [50].
System parameters are chosen to correspond to a fluid bath of density ρ = 0.95 g/cm3 , viscosityν = 16 cS, and surface tension σ = 20.9 dyn/cm vibrating vertically in a sinusoidal fashion withpeak acceleration γ0 = A0 ω2 , peak amplitude A0 , and frequency ω/2π = 80 Hz. The resonant
bouncing of the particle at the Faraday frequency triggers a quasimonochromatic damped wave
pattern with a corresponding Faraday wavelength of λF = 4.75 mm. Each of the four cavities has
a fixed length of 1.0 cm, corresponding to approximately 2.1λF and a fixed depth of 0.5 cm. The
central cavity, coupling the right and left subsystems, has a fixed length of 0.4 cm and a fixed depthof 2 cm. Each of the barriers has a fixed width of 0.4 cm. The two barriers connecting the central
cavity to the right and left subsystems, have a fixed depth of 0.045 cm. For this geometry, thecritical vibrational acceleration above which waves are generated in the absence of the droplets, theFaraday threshold, was γF = 4.69g. We report results attained with a fixed vibrational acceleration
γ0 = A0 ω2 = 4.23g = 0.92γF .
We proceed by demonstrating that for a judicious choice of pairs of static measurement settings(α, β ), the inequality (1) may be violated in our bipartite hydrodynamic system owing to the wavemediated coupling between the two subsystems. In our numerical simulations, each run begins byplacing the two particles at random positions within their own subsystem. Their trajectories are thencalculated for 48 000 Faraday periods. During that time, we observe in which cavities the droplets
reside. XA takes the value −1 if the drop A is observed in the ground state of the left subsystem,
and +1 otherwise. Similarly, XB takes the value −1 if the drop B is observed in the ground stateof the right subsystem, and +1 otherwise. The robustness of the emergent statistics was ensured byfollowing the protocol detailed in the Appendix.
III. RESULTS

Our main result is shown in Fig. 2, which indicates a narrow parameter range in which the CHSH
inequality is violated. While the inequality can be violated in our system with four different values ofthe measurement settings (a, a , b, b ), our exploration of the (a, a , b, b ) parameter space indicatesthat a local maximum of S arises when b = a and b = a , the symmetric case in which one may
write S(a, b, a , b ) = S(a, a, a , a ) = M(a, a) − M(a , a ) + 2M(a, a ). We deduced a maximum
violation of Smax = 2.49 ± 0.04 when a = b = a∗ = 0.099 cm and a = b = a∗ = 0.1033 cm. In
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KONSTANTINOS PAPATRYFONOS et al.

FIG. 2. Violation of Bell’s inequality. Bell parameter S(a, a, a , a ) as a function of the barrier depth,
a , for the symmetric case of a = b, a = b . For the calculation of the corresponding correlation functions,M(a, a), M(a, a ), and M(a , a ), the barrier depth a = a∗ = 0.099 cm remains fixed. For each combinationof measurement settings, runs continue until statistics converge. The maximum Bell violation appears ata∗ = 0.1033 cm, where S = 2.49 ± 0.04.
Fig. 2, we plot S as a function of a for fixed a = a∗ , with the dashed line showing the limit S = 2
above which the CHSH inequality, Eq. (1), is violated.
While the inequality was found to be violated only for a narrow range of parameter settings, inthis parameter regime, the violation is clear, and the statistical confidence of the violation is above
20 standard deviations (see Fig. 6). We note that this behavior is reminiscent of the quantum case,where, without guidance from the theory, it is relatively difficult to find analyzer settings that allow
for violation of the CHSH inequality, but for judiciously chosen settings, the inequality is violated
substantially. As in our previous study of superradiance [24], the wave-mediated coupling createsa collective behavior of the droplet pairs. In particular, when one of the droplets transitions to its
excited state, the probability of the second droplet doing likewise increases substantially. Thus,
through its wave-mediated interaction with its partner, each droplet is indirectly affected by thebarrier depth of the distant station.
Since the inequality involves four different correlation functions (three for the symmetric caseconsidered here), finding the combinations of measurement settings that maximized S was notentirely straightforward. The strategy we followed in seeking violations is summarized in Fig. 3.
We first investigated the evolution of a single correlation function M(α = a, β = a) as a function of
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STATIC BELL TEST IN PILOT-WAVE HYDRODYNAMICS

FIG. 3. Strategy to optimize S(a, a, a , a ) = M(α = a, β = a) − M(α = a , β = a ) + 2M(α = a, β =a ) . S is optimized by searching a parameter regime (a, a ) near the maximum of δM(a, a ) = M(α = a,β = a) − M(α = a , β = a ) and a range of a that maximizes M(a, a ) with a fixed. (a) Evolution of M(α =a, β = a) as a function of a. The indicated values a∗ = 0.099 cm and a∗ = 0.1033 cm are the S-maximizing
values used in Fig 2. We note δa = a − a and δa∗ = a∗ − a∗ . The difference between the corresponding correlation functions δM(a, a + δa) = δM(a, a ) = M(α = a, β = a) − M(α = a , β = a ) is marked in orange.
(b) Optimization of δM(α = a, β = a + δa) as a function of barrier depth a and δa. The domain for which[maxa,a (δM ) − δM]/ maxa,a (δM ) > 0.9 is bound by the black dashed curve. (c) 2M(α = a∗ , β = a ) as afunction of depth a for fixed a = a∗ . (d) Evolution of S(a, a, a , a ) in the correlation representation space(M(a, a ); M(a , a )) with a = a∗ . The direction of increasing a is indicated by the blue arrow. The dots arecolored with respect to their S values. The gray dashed line indicates the limiting case of S = 2.

a. This gave us a good sense of parameters that maximize the difference δM(a, a ) = M(a, a) −M(a , a ) [see Fig. 3(a)]. δM(a, a ) involves two of the correlation functions of Eq. (1), in thesymmetric case of interest where a = b and a = b . Figure 3(b) shows a two-dimensional (2D) plotof the optimization of δM as a function of a and a . The black dashed lines highlight the domain
in which [maxa,a (δM ) − δM]/ maxa,a (δM ) > 0.9. The other term in the inequality, specifically2M(α = a, β = a ), represents a combination of measurements from unequal barrier depths at thetwo measurement stations. Figure 3(c) represents the dependence of 2M(α = a∗ , β = a ) on deptha for fixed a = a∗ , the S-maximizing value considered in Fig. 2. a∗ marks the end of the plateau ofhigh correlation, beyond which the term M(a∗ , a ) decreases. Finally, Fig. 3(d) shows the evolutionof the correlation functions (M(a, a ); M(a , a )) with increasing a and fixed a = a∗ . The maximum
S value occurs for moderate barrier depths, for which the droplets may become most strongly
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KONSTANTINOS PAPATRYFONOS et al.

correlated through the background wave field. In Fig. 4(a), we show typical trajectories for the threecombinations of measurement settings (α, β ) ∈ {(a∗ , a∗ ), (a∗ , a∗ ), (a∗ , a∗ )} that maximize S. For(a, a ) = (a∗ , a∗ ), S is maximized because M(a∗ , a∗ ) and M(a∗ , a∗ ) are large [see Fig. 4(a), top andmiddle panels], while M(a∗ , a∗ ) is relatively small [Fig. 4(a), lower panel]. Figure 4(b) corresponds
to a shallow barrier, a = 0.0937 cm (the leftmost value in Fig. 2) and Fig. 4(c) to a relatively deep
barrier, a = 0.11 cm (the rightmost value in Fig. 2). Figures 4(b) and 4(c) correspond to minima ofS occurring when the a barrier is either too shallow [Fig. 4(b)] or too deep [Fig. 4(c)].
The degree of synchronization in the droplet tunneling depends on the extent to which the
droplets are affected by the barrier depth in the distant station. When the barrier depth in one
station is too small, the local particle is prevented from tunneling, regardless of the barrier depth inthe other. The synchronization of states is thus reduced substantially. Conversely, when the barrier
depth is too large, the particle generally tunnels across it, unaffected by the distant particle. Thus, thesynchronization again remains relatively low. For intermediate barrier depths, each particle tunnels
with a moderate probability that is strongly affected by the behavior of its distant partner.
Finally, to ensure that the dynamics were a result of the two-droplet state rather than just the
topography, we ran several complementary single-drop simulations with the same topographies. A
typical example is presented in Fig. 5, where we compare the two extreme cases, one in which the
distant barrier depth is relatively low (b = 0.099 cm) and another in which it is relatively high (b =0.1033 cm). In these simulations we set a = 0.099 cm, so the overall topography was identical tothat reported in Fig. 4(a) for a pair of drops. We observe that the single-drop trajectories in the twocases are identical for these values of b and b , but substantially different from those arising in the
two-particle case. Thus, when only one drop is present, the variation of the measurement setting in
the distant subsystem does not influence its behavior. This indicates that, in the two-droplet case,the droplets learn about the distant cavity geometry (or "measurement setting") only through theirwave-mediated interactions with their partner drop. This result is consistent with previous findings
on superradiance [24], where for certain topographies the single-droplet tunneling rate was very low,
unless a second drop was introduced into the distant cavity, in which case the tunneling probabilities
could increase dramatically.

IV. CONCLUSION AND DISCUSSION

We have devised a platform for performing static Bell tests on a classical bipartite pilot-wave
system. The maximum violation was found to be 2.49 ± 0.04, and arose when the system geometries were chosen such that the droplet motion was marked by varying degrees of correlation
between droplets with different measurement settings. A key step in the process was recognizingthat the system geometry may serve as a proxy for analyzer settings in the quantum Bell tests.
The non-Markovian nature of the droplet dynamics is engendered in the system memory [15], thecritical feature in all hydrodynamic quantum analogs [5]. The wave field serves as the memory of thesystem, and the correlations reported here may be understood as being rooted in this wave-mediatedmemory. We have focused here on the intermediate memory regime (γ0 /γF = 0.92), where thewave-mediated interactions allow for Bell violations. We note that these violations vanish at much
lower memory, when there is insufficient wave coupling between the two subsystems, and at higher
memory, when the droplet dynamics becomes highly disordered.
We proceed by rationalizing the Bell violation achieved in our static test. Specifically, we identify
which of the assumptions made in the derivation of Bell’s inequality are not applicable in our system.
In order to do so, we find it valuable to adopt two distinct perspectives in the form of two fictional
observers. Observer 1 can observe the droplets but not the waves, while observer 2 can observeboth droplets and waves. Observer 1 would infer that the measurements at each subsystem are
influenced by the distant measurement setting, and thus that assumption (i) is invalid. Throughconsideration of the wave field, observer 2 could rationalize the violation of assumption (i) in termsof the wave-mediated coupling between the two subsystems. It would also be apparent to observer
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STATIC BELL TEST IN PILOT-WAVE HYDRODYNAMICS

FIG. 4. Trajectory analysis. (a)–(c) Droplet trajectories for the symmetric case (α, β ) ∈ (a, a ) with a =a∗ = 0.099 cm. Time evolves in the vertical direction. In (a), a = a∗ = 0.1033 cm (the maximizing value
for S); in (b), a = 0.0937 cm; in (c), a = 0.11 cm. (a) Trajectories corresponding to the three correlation
functions M(a∗ , a∗ ) = 0.94 (upper panel), M(a∗ , a∗ ) = 0.84 (middle panel), and M(a∗ , a∗ ) = 0.13 (lower
panel). The tunneling events are highly correlated only in the upper and lower panels. (b) Trajectories corresponding to M(a∗ , a ) with a = 0.0937 cm. When the barrier depth a is sufficiently small, the wave-mediatedcommunication between droplets is diminished, and droplets tend to get trapped in one cavity, leading tominima of S and M(a∗ , a ) ≈ 0 when we average over the droplet’s initial conditions. (c) Another minimum ofM(a∗ , a ) and S occurs when one of the barrier depths is too large, in which case one of the droplets tunnels
continuously, unimpeded by the barrier, as if it were in a single cavity. Averaging over all initial conditions
leads to a relatively low value of M(a∗ , a ) ≈ 0. Note that the correlation function is deduced by averaging
over all initial conditions in both subsystems.
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KONSTANTINOS PAPATRYFONOS et al.

FIG. 5. Single drop trajectory analysis. (a) Single droplet trajectory for a relatively shallow distant barrier:a = 0.099 cm; b = 0.099 cm. Inset: a close-up of the dynamics illustrates the oscillatory motion of the drop
inside the cavity (−). (b) Single droplet trajectory for a relatively deep distant barrier: a = 0.099 cm; b =0.1033 cm. The two trajectories are statistically indistinguishable, indicating that, in the absence of a partner
drop, the droplet motion is not influenced by the distant barrier depth.

2 that the wave field constituting the continuous variable hidden from observer 1 is affected by bothanalyzer settings. Observer 2 would thus conclude that both assumptions (i) and (ii) are invalid.
Quantum entanglement requires that the violations of Bell’s inequality persist even when the two
subsytems are spacelike separated. While such entanglement is generally thought to be peculiar toquantum systems, Bell-violating states have been demonstrated in both classical electromagnetic[45–47] and acoustic wave systems [49]. They have also been reported in single-particle systems,
through consideration of the internal degrees of freedom of a single neutron [57]. However, neither
the classical wave states nor the internal degrees of freedom in the single-particle system can bespatially separated [48,58]. We have here demonstrated the possibility of achieving static Bellviolations for spatially separated bipartite states in the pilot-wave hydrodynamic system. We stressthat these violations may be simply rationalized: the wave-mediated coupling between the two
subsystems ensures that both the assumptions (i) and (ii) made in the derivation of Bell’s inequality
are invalid. Nevertheless, the violations exist only in a limited region of parameter space, and so are
relatively difficult to achieve.
Finally, we note that our theoretical framework introduces the possibility of performing moresophisticated Bell tests. The feasibility of incorporating time-dependent topography into our model
has recently been demonstrated by Nachbin [22], who demonstrated the persistence of correlations
between droplet pairs in single wells even after their topographic isolation. Incorporating dynamic
topography will allow for future Bell tests in which communication between the two subsystems iseliminated by closing the coupling cavity during the course of a simulation. The dynamic-Bell-test
protocol would then involve changing the analyzer settings after topographic isolation of the two
subsystems. However unlikely, attaining Bell violations in such dynamic tests does not seem entirely
impossible given that the wave field serves as the memory of the system; thus, the mutual memoryof the initial wave-mediated droplet interaction might survive the topographic isolation of the two
subsystems.
The code and the data that generated the data are available upon request from the contact author.
ACKNOWLEDGMENTS

We acknowledge the generous financial support of the NSF through Grants No. CMMI-1727565
and No. CMMI-2154151; the European Union’s Horizon 2020 Research and Innovation Program
under the Marie Sklodowska-Curie project EnHydro, Grant No. 841417; Conselho Nacional de
Desenvolvimento Cientfico e Tecnológico (PQ1D 307078/2021-3) and Fundação Carlos Chagas
Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (CNE E-26/201.156/2021).
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STATIC BELL TEST IN PILOT-WAVE HYDRODYNAMICS

FIG. 6. Statistical convergence of the Bell parameter. (a) Typical curve showing the convergence of
Smax , the Bell parameter taken at the maximum point of violation (a = b = a∗ = 0.099 cm; a = b = a∗ =0.1033 cm). The error bars indicate ±3 standard deviations. (b) Relative error δSmax /Smax of the estimation of
the Bell parameter S with the number of runs, evaluated for the maximum point of violation (a = b = a∗ ; a =b = a∗ ). Inset: log-log scale; the dashed line indicates a −1/2 slope as expected from the convergence of an
ensemble average.

APPENDIX: STATISTICAL CONVERGENCE

To confirm the statistical significance of our results, we performed six runs with durations of
48 000 Faraday periods for each geometrical configuration. We used random initial conditions forthe particle positions, and discarded the first 10% of the runs in order to eliminate any trace ofa transient. To initialize the runs, the wave and velocity fields of the bath are set to zero, andthe particle positions are assigned random, uniformly distributed values. Then, the model runs for2000 Faraday periods, a measurement is made, and all fields are reset back to zero to initialize thesubsequent run. This cycle is repeated for each set of parameter settings until the relative error in therunning average of M(α = a, β = b) is reduced to an acceptably small value. We set this tolerance
to be 3% for parameters that violate the inequality and 7% for those that do not. While extremely
accurate, this "discrete" technique is computationally intensive; thus we have used it only for themost critical points of the parameter space, in which the maximal Bell violations occurred.
To explore the parameter space more efficiently, we adopt an alternative, relatively expedient,"continuous" approach, in which the final conditions of one run serve as the initial conditions of
the next. We demonstrated the statistical equivalence of the two approaches as follows. For specificselected data points, we performed approximately 30 different runs using the two techniques, andfound the results of the discrete and continuous runs to be in agreement to within 3%. We thenexecuted continuous runs for 48 000 Faraday periods, during which measurements are performed
frequently at uniformly distributed random times. After a sufficiently long run, the full range
of initial conditions will have been effectively explored. The consistency of the results deducedwith the discrete and continuous approaches demonstrates that the long-time emergent statistics
are independent of the initial conditions. Figures 6(a) and 6(b) show a typical example of theconvergence of the "running average" with the number of runs which determines the relative error
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of our statistics. This approach indicates when our statistics have converged for each M(α, β )calculation, specifically when the relative error has fallen below the prescribed tolerance.

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