PERSPECTIVE | JULY 15 2024

Perspectives on pilot-wave hydrodynamics
John W. M. Bush ; Valeri Frumkin; Pedro J. Sáenz

Appl. Phys. Lett. 125, 030503 (2024)
https://doi.org/10.1063/5.0210055

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Applied Physics Letters

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Perspectives on pilot-wave hydrodynamics
Cite as: Appl. Phys. Lett. 125, 030503 (2024); doi: 10.1063/5.0210055
Submitted: 25 March 2024 . Accepted: 30 May 2024 .
Published Online: 15 July 2024
John W. M. Bush,1,a)Valeri Frumkin,2

 enz3and Pedro J. Sa

AFFILIATIONS
1

MIT, Department of Mathematics, 77 Massachusetts Ave., Cambridge, Massachusetts 02139, USA

2

Boston University, Department of Mechanical Engineering, Commonwealth Ave., Boston, Massachusetts 02215, USA
University of North Carolina, Department of Mathematics, 120 E Cameron Ave., Chapel Hill, North Carolina 27599, USA3a)Author to whom correspondence should be addressed: bush@math.mit.edu

ABSTRACTC 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial 4.0
VInternational (CC BY-NC) license (https://creativecommons.org/licenses/by-nc/4.0/). https://doi.org/10.1063/5.0210055

I. INTRODUCTION
1

The walking-droplet system represents a macroscopic realizationof wave-particle duality and has furnished a platform for exploring theboundary between classical and quantum behavior.2,3 The physicalpicture suggested is one rooted in classical mechanics and so local realism. Some hydrodynamic quantum analogs (HQAs) achieve statisticalbehavior that is strikingly similar to their quantum counterparts, provided the Faraday wavelength is identified with the de Broglie wavelength. For example, the emergent statistics in the walking droplet
experiments are virtually identical to those in Friedel oscillations4 andthe quantum corral.5,6 In other HQAs, including single-particle diffraction and interference7–9 and orbital dynamics,10–12 the emergent statistics is only similar in form, the connection to quantum mechanics only
qualitative. Whether the match is quantitative or qualitative, when ahydrodynamic quantum analog is achieved, it demonstrates that one
can grasp the essential physics of the quantum system of interest, rationalize the phenomenon, from a local realist perspective.
The HQA venture was launched in 2006 by Couder and Fort’s
study of walker diffraction.7 While these results were hotly contested,13,14 they have since been reproduced repeatedly in experiments
of increasing precision and sophistication.8,9,15 In particular, Pucci
et al.8 characterized the dependence of the diffraction on drop size andvibrational acceleration, while Ellegaard and Levinsen9 imposed

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Vthermal control on the bath in order to eliminate the slow drift of the
Faraday threshold during the course of experiments. The conclusion is
now clear: even though the particles pass through the slits one at atime, the emergent statistics of the diffraction angle take a wave-like
form. The diffraction pattern does not conform to the Fraunhofer pattern that arises in certain limits of the quantum problem, so one must
concede that the analog is strictly qualitative. Nevertheless, the emergent wave-like statistics and single-particle interference are plainly
apparent. After familiarizing oneself with the diffraction of walking
drops, one can only puzzle over Feynman’s insistence on the absoluteinscrutability of the electron doubles-slit experiment,16 particularly
when the basic physical picture needed to understand the effect was
furnished in the 1920s by Louis de Broglie in his double-solution
theory.17,18
The traditional view of quantum mechanics, also known as the
Copenhagen interpretation, provides no description of particles traveling along well-defined paths. A quantum system is describedcompletely by its wave function, which evolves deterministicallyaccording to the Schr€odinger equation. When a measurement is made,the system collapses to a particular outcome through a process that is
non-unitary and nonlocal, giving rise to the notorious measurement
problem. It has, thus, been said that, according to the Copenhagenview, particles are nothing more than a measurement of them, and125, 030503-1

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We present a number of fresh perspectives on pilot-wave hydrodynamics, the field initiated in 2005 by Couder and Fort’s discovery that millimetric droplets self-propelling along the surface of a vibrating bath can capture certain features of quantum systems. A recurring theme willbe that pilot-wave hydrodynamics furnishes a classical framework for reproducing many quantum phenomena and allows one to rationalize
such phenomena mechanistically, from a local realist perspective, obviating the need to appeal to quantum nonlocality. The distinction isdrawn between hydrodynamic pilot-wave theory and its quantum counterparts, Bohmian mechanics, the Bohm–Vigier stochastic pilot-wave
theory, and de Broglie’s theory of the double-solution. Each of these quantum predecessors provide a valuable touchstone as we take thephysical picture engendered in the walking droplets and extend it into the quantum realm via theoretical modeling. Emphasis is given to
recent developments in the field, both experimental and conceptual, and to forecasting potentially fruitful new directions.

Applied Physics Letterswithout measurement there is nothing but the wave function.19
Heisenberg’s uncertainty principle suggests that one may think ofparticles as localized wave packets that represent the probability ofdetecting particles spread out in space. However, a measurement may
result in the materialization of a particle at an appreciable distancefrom the center of the wave packet that represents it. Thus, at best, one
can associate the trajectory of a quantum wave packet with the average
trajectory of the quantum particle, but not with its actual trajectory. In
what follows, we argue that this representation of quantum particles interms of their associated wave packets is responsible for a number ofthe conceptual difficulties arising in the foundations of quantum
mechanics.
We proceed by considering a number of key hydrodynamic
quantum analogs. Recent experimental advances are highlighted,
including HQAs of spin lattices,20 Anderson localization,21 and new
directions in hydrodynamic interferometry.22,23 Conceptual advances
are also enumerated, with particular attention given to the relation
between the guiding wave field in the hydrodynamic system and thequantum potential in Bohmian mechanics. Potentially fruitful directions for future work are discussed.
II. QUANTUM PILOT-WAVE THEORIES

dxp ðtÞ
$wðx_ p ðtÞ;tÞhdt ¼ m Im wðx_ p ðtÞ;tÞ . While local in the sense that the particle

responds only to the local form of the wavefunction, Bohmianmechanics is nonlocal in that the wavefunction depends on the systemgeometry and configuration of all the particles in the system, andreacts to any change thereof instantaneously. This nonlocality represents the most important distinction between Bohmian mechanics andpilot-wave hydrodynamics. In the former, the guiding wave is imposedby fiat; in the latter, it is created by the particle itself, rendering theAppl. Phys. Lett. 125, 030503 (2024); doi: 10.1063/5.0210055
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theory local. In this regard, pilot-wave hydrodynamics is more closelyaligned with de Broglie’s original conception, his theory of the double
solution, which was rooted in local realism.
In response to a number of criticisms leveled at Bohm’s theory,27
only 2 years after its initial launch, Bohm was joined by Vigier28 inproposing that his trajectory equation be augmented by a stochastic
forcing component. It was, thus, posited that quantum particles movein an erratic fashion about the Bohmian trajectories, just as dust particles mean in a gas flow jiggle around streamlines in response to
Brownian motion. The precise form of this stochastic motion was not
specified and so must be considered ad hoc. The particle motion may,thus, be decomposed into a mean component guided by a nonlocal
field, plus an ad hoc stochastic component. A similar decomposition
arises in consideration of the HQA of the quantum corral.
III. HYDRODYNAMIC QUANTUM ANALOGS
A. Corrals
Hydrodynamic analogs of the quantum corral5,6,29 are among themost compelling in that the emergent statistics are strikingly similar to
those in their quantum counterparts.30–32 When a walker explores acircular corral of characteristic radius 2 cm, it executes a structured
random walk over its self-induced wave potential. After approximately
1 h, the resulting correlation between position and speed gives rise to astatistical signature that is remarkably similar in form to that arising in
the quantum corral and may be expressed in terms of the wave modesof the cavity. If one assumes that the emergent statistical form is acomplete description of the system, then one must contend with its
instantaneous collapse to a point following the act of observing thedroplet, a nonlocal effect in the quantum measurement problem that isseen here to be an unnecessary conceit.3
In an elliptical corral (Fig. 1), the emergent statistics may likewise
be expressed as the sum of the eigenmodes of the cavity. Moreover, theintroduction of topographic inhomogeneities allowed for alteration of
the emergent statistics through favoring one cavity mode over theothers. For example, introduction of a well at one of the foci of theellipse favored a mode with maxima at both foci, giving rise to a statistical projection effect analogous to the quantum mirage.32,35 It was further noted that the mean pilot-wave g was comparable in form to thatof the particle’s emergent probability distribution function, lðxÞ.6 Thisobservation motivated the theoretical deduction of the relation
between the two,36,37 for both periodic and ergodic droplet motion.
Specifically, the mean-pilot-wave potential may be expressed as the
convolution,
ð1
gB ðx  yÞlðyÞ dy;(1)g ðxÞ ¼
1where gB ðx; yÞ is the wave field of a stationary bouncer located at position y. Notably, this mean pilot-wave potential is nonlocal in the same
sense as the quantum potential in Bohmian mechanics, that is, specified by the system statistics. Decomposition of the instantaneous pilotwave field into its mean and fluctuating components, gðx; tÞ ¼ g ðxÞþ g0 ðx; tÞ suggests that the former could be seen as being analogous tothe quantum potential, the latter to the ad hoc stochastic dynamics
proposed by Bohm and Vigier.28 Exploring and clarifying the extent towhich the motion of walking droplets may be meaningfully decomposed into mean and stochastic components is the focus of ongoinginvestigations in HQA. Particular attention is being given to125, 030503-2

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Prior to the advent of the Copenhagen Interpretation or to thenotion of impossibility proofs, de Broglie proposed a model of quantum dynamics, his theory of the double solution, in which microscopic
quantum particles have an internal vibration at the Compton frequency, xc ¼ mc2 =h. This vibration excites Klein–Gordon waves thatpropel the particle along in such a way that the particle momentum isrelated to the wave number of the pilot wave through the de Broglie
relation p ¼ hkB . De Broglie proposed, but never proved, that theresulting dynamics could give rise to emergent statistics described by
the standard quantum wavefunction W. The pilot-wave hydrodynamic
system represents a macroscopic realization of this physical picture,2with the Faraday frequency playing the role of the Compton frequency,
the Faraday wavelength that of the de Broglie wavelength kB ¼ 2p=kB .
Its success in achieving several HQAs would seem to speak in favor ofde Broglie’s original proposal.
Bohmian mechanics24 is a realist theory of the quantum realm
that restores the notion of particle trajectories25 and has provided avaluable touchstone in connecting pilot-wave hydrodynamics to quantum mechanics.2,3 The theory is rooted in Madelung’s hydrodynamic
interpretation of the linear Schr€odinger equation, which emerges froma polar transformation of the wavefunction.26 If one assumes that theparticle velocity, x_ p is equal to the quantum velocity of probability,
then one obtains a trajectory equation that describes a classical particle
moving in response to both classical and quantum potentials, V and Q,
respectively: mx_ p ¼ rV  rQ, where Q is prescribed by the wavefunction W. Alternatively, the first order formulation of Bohmianmechanics allows
one ito write the trajectory equation in the formh

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FIG. 1. The wave field and emergent statistics in the elliptical corral.6 (a) The instantaneous wave field and recent droplet trajectory. (b) The build-up of the droplet trajectory
over 3.5 h eventually reveals a correlation between droplet position and speed. (c) The result is the emergent statistical form that may be expressed in terms of two ellipticalcavity modes. (d) Time-averaging the instantaneous wave field reveals that the mean pilot-wave field has the same form as the emergent statistics. Reprinted with permission
from Saenz et al., Nat. Phys. 14, 315–319 (2018).

elucidating the manner in which the breaking of synchrony between
particle and wave induces additional forces not considered in theoretical models based on the assumption of wave-particle resonance.37,38
B. Orbital dynamics

FIG. 2. Orbital quantization in a rotating frame. (a) The trajectory of a walker becomes circular (dashed line) in a rotating bath due to the Coriolis force. As the vibrational forcing
increases (c=cF ¼ 0:971), the wave decay time eventually exceeds the orbital period. The walker then interacts with its own wake, leading to the emergence of (b) quantizedcircular orbits.10,11,33 The orbits are color-coded according to their stability: blue indicates stable orbits and green and red indicate orbits that destabilize via oscillatory and nonoscillatory instabilities, respectively. (c) Blitstein et al.34 have recently demonstrated that, owing to wave interference, the force responsible for the orbital quantization originates
at the stationary points along the past trajectory, specifically points at which the distance to the droplet is not instantaneously changing. The full pilot-wave force, F W , may, thus,
be approximated by the sum of long-range quantization forces, F Q , emanating from stationary points (blue), and a local “boost” force, F B , that sets the walker speed. Reprinted
with permission.

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Orbital quantization is also a robust feature of pilot-wave hydrodynamics and has been demonstrated to arise for walker motion in arotating frame10,11,33 [Figs. 2(a)–2(c)] and a walker subjected to a simple harmonic potential, specifically a radial spring force.12,39 Orbital
quantization represents a clear example of a particle exciting, thenbeing constrained by, its own wave potential. Recent theoretical work
has provided new insight into the stability and origins of the quantizedorbits.
A hierarchy of theoretical models with different degrees of complexity have been successful in capturing the vast majority of phenomena observed in experiments.3 In the so-called “stroboscopic” model,38
perfect resonance between the bouncing droplet and its guiding wave
is assumed, and the droplet is approximated as a continuous source ofwaves. The resulting stroboscopic model has been particularly important in the theoretical description of the walking droplets owing to its

analytical tractability. Oza et al.33 demonstrated that it rationalizes the
emergent orbital quantization of walkers in a rotating frame.10,11
Specifically, they used it as the basis of a stability analysis to show thatthe discrete orbits observed in experiments correspond to stable circular solutions of the walker’s equation of motion [Fig. 2(b)]. In a recent
theoretical study, Liu et al.40 demonstrated that the quantized orbits
arise at local extrema of the mean-pilot-wave potential; moreover, theform of instability that sets in as the memory increases is prescribed by
the local form of the mean pilot-wave potential, g . Specifically, when
the drop orbits at local maxima or minima of 
g , the onset of instability
is, respectively, monotonic or wobbling. The manner in which themean pilot-wave potential influences the dynamics and emergent statistics is an area of focus as we explore its relation to the quantum
potential.
Elucidating the mechanism responsible for the walker’s orbital
quantization has also received significant attention. In the first study oforbital quantization, Fort et al.10 introduced the notion of a “virtual
walker” orbiting at a position diametrically opposite to the walker’s
position. Bush et al.41 derived a reduction of the stroboscopic model,
valid in the weak-acceleration limit, which captures analytically thewave force due to the recent past. This so-called “boost” model

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C. Spin lattices
Saenz et al.20 introduced hydrodynamic spin lattices (HSLs) as ananalog system that allows one to investigate the wave-mediated interactions and collective dynamics of hydrodynamic spin states42,43 inpubs.aip.org/aip/apl

both stationary and rotating frames. Using submerged circular wells at
the bottom of the fluid bath,4,6 the authors stabilized walker spin states,
leading to a clockwise or counterclockwise angular motion centered oneach well [Fig. 3(a)]. When a collection of such spin states is arranged
in a 1D or 2D lattice geometry, a thin fluid layer between wells enables
wave-mediated interactions between neighboring droplets [Fig. 3(a),inset]. For sufficiently strong pair-coupling, these interactions may
induce local spin flips, leading to preferred collective states of analog
antiferromagnetic [Fig. 3(b)] or ferromagnetic order [Fig. 3(c)]depending on the lattice spacing. Moreover, the authors used themathematical equivalence between the Lorentz force acting on a moving charge and the Coriolis force acting on a moving mass to demonstrate a transition from anti-ferromagnetic to ferromagnetic states
when the spin lattice is subject to uniform rotation at constant frequency [Fig. 3(d)]. Inverting the sense of bath rotation inverts the signof the magnetization, analogous to the polarization observed in antiferromagnetic materials subject to external magnetic fields. Saenz et al.
also investigated 2D lattices to demonstrate that square HSLs can sustain antiferromagnetic order in the absence of rotation [Fig. 3(e)] andundergo a polarization transition as the Coriolis force is increased[Fig. 3(f)].
To rationalize the collective order observed in HSLs, Saenz et al.20
examined the wave-mediated coupling between two spinning walkers,
which revealed four distinct modes of pairwise symmetry breaking.
Specifically, there are two ferromagnetic phases and two antiferromagnetic phases, distinguished by preferential in-phase and out-of-phaserotation, respectively, that may be adjusted by changing the latticespacing. The authors derived a generalized Kuramoto phase model todemonstrate that walkers self-organize in such a way as to minimize
the coupling potential.
HSLs present a range of interesting new directions, in particular,
in two dimensions wherein classical and quantum spin lattices display
features that are absent in their 1D counterparts. Of particular interestis exploring the ability of HSLs to solve combinatorial optimization
problems,44 wherein NP (nondeterministic polynomial time) problems

FIG. 3. Hydrodynamic spin lattices (HSLs). (a) Top view and schematic cross section of square lattice of hydrodynamic spin states. Owing to the influence of submerged circular wells, the walkers propel themselves along circular orbits in the clockwise (blue) or anti-clockwise (red) directions. A thin liquid layer between wells enables wave-mediatedinteractions that may prompt spin flips, leading to coherent collective order. Depending on the lattice spacing, collective (b) anti-ferromagnetic and (c) ferromagnetic order may
spontaneously emerge, here illustrated in one-dimensional lattices with periodic boundary conditions. (d) Rotating the system about the vertical axis with angular frequency X
prompts a transition from anti-ferromagnetic to ferromagnetic states, analogous to the global polarization produced when an anti-ferromagnetic material is subject to an external
magnetic field. The same (e) collective order and (f) rotation-induced polarization was observed in two-dimensional square HSLs. The emergent collective order oscillates withthe lattice separation between four different modes of pairwise symmetry breaking. Reprinted with permission from Saenz et al., Nature 596, 58–62 (2021).

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demonstrates that the most recent waves composing the pilot-wave
field act to (i) increase the walker mass relative to the droplet mass and(ii) produce a non-linear friction force that always drive the droplet
toward its free walking speed. The boost model was successful in rationalizing the offset between the orbital radii of walkers at low memory10,11 and those expected in the absence of the pilot-wave field.
Blitstein et al.34 have recently provided new insight into the origin
and form of the wave-mediated forces responsible for orbital quantization. The authors demonstrated that, owing to wave interference, theforce responsible for orbital quantization originates from waves excitednear specific stationary points along the walker’s path [Fig. 2(c)],which constructively interfere at the droplet position. Waves excitedelsewhere along the walker’s path interfere destructively at the droplet
position and may, thus, be safely neglected. Based on this insight, theauthors were able to approximate the quantizing force, F Q , in terms ofa sinusoidal wave potential, projected from the stationary points
toward the droplet position, that restricts the allowed radii. The
authors, thus, derived a model with the minimal ingredients required
to capture the origins of quantization in pilot-wave orbital dynamics,
as well as the quasi-periodic and chaotic orbits with preferred radii
arising at higher memory. This minimal model makes clear the distinction between the forces captured by the boost model,41 which prescribe
the walker’s wave-induced added mass and nonlinear drag, and theforces responsible for quantization, as are non-local in the sense thatthey originate at locations along the walker’s past trajectory. The quantizing force elucidated by Blitstein et al.34 arises not only in orbital
dynamics, but in other settings where walkers depart from rectilinear
motion. Notably, this force serves to drive the droplet along paths withpreferred radii of curvature, a feature that has been noted in a numberof HQAs.4,5,11

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are formulated as an Ising problem that is more easily solved. The ability to fine-tune the inter-spin correlations in the HSL system by varying its geometry may be translated to optimization algorithms inspiredby certain approaches to quantum computing, such as quantum
annealing.45 This approach would require the formulation of a
Hamiltonian for a simple HSL configuration (i.e., one for which theground state is well known), then to slowly vary the geometry until adifferent Hamiltonian is obtained, the ground state of which would
constitute a solution to a given optimization problem. Another interesting future direction would be to explore the collective order under
frustrated 2D lattice geometries, such as triangular or hexagonal spinlattices. In such lattices, competing interactions may not be simultaneously satisfied, giving rise to a large degeneracy of the system groundstate and interesting frustrated dynamics such those exhibited by spin
liquids46 and spin ice.47 Additional areas for new research with HSLs
include the development of wave-coupled metamaterials48 and spinwaves such as those used for the development of spintronics49 andmodern magnonics.50
D. Anderson localization

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energy of the background potential, U0  K. A classical particle evolving under such conditions is able to move easily through the disordered landscape, however, its trajectory is gradually deflected by small
random forces. After a sufficiently long period of time, the particle
will, thus, exhibit zigzag motion, effectively evolving diffusively in two
or higher dimensions58 (ballistic in one dimension). The motion of aquantum particle in the same settings is fundamentally different andparticularly counter-intuitive; an electron will effectively come to ahalt, or “localize,” beyond a critical amount of disorder even though
the particle’s energy is large relative to that of the underlying potential.57,59 The inability of classical particle systems to exhibit localization
similar to that of quantum particles has, thus, been regarded as anotherfundamental limit of classical mechanics. Since Anderson’s work,56 wehave known that the reason for localization is rooted in the wave-likenature of quantum particles; specifically, the disordered potential
causes the quantum states given by Schr€odinger’s equation to be exponentially localized in space. Subsequent studies have proven noiseinduced localization to be a generic wave effect that may be observedin various systems, including Bose–Einstein condensates, microwaves,light waves, and ultrasound.60 However, one should recall that, whilethese more recent examples have been realized with purely classicalwaves, the original quantum localization is fundamentally different as
it includes the notion of a particle.
To demonstrate that pilot-wave hydrodynamics may likewise
exhibit a dual particle-wave localization analogous to that of quantum
particles, Abraham et al.21 considered the erratic motion of a walker
above submerged random topography [Fig. 4(a)]. The walker is subject
to a random potential through the influence of the bottom topographyof the bath, which is composed of square tiles, each with a randomheight drawn from a uniform distribution. Combining experiments,
simulations, and theory, Abraham et al. demonstrate that after a sufficiently long period of time, the erratic walker motion at high memory
gives rise to coherent statistics wherein the histogram of the droplet
position features a prominent localization region [Fig. 4(b)]. Notably,
the authors examined the first eigenmode of Schr€odinger’s equation
with a potential of the same form as the bottom topography and

FIG. 4. Anderson localization of walking droplets. (a) A walking droplet may move erratically along the surface of fluid bath due to the influence of submerged bottom topography, which acts as a disordered potential. In the study by Abraham et al.,21 the bath bottom profile is composed of square tiles of width W, each with a random height 6DH
about a base depth H0 drawn from a uniform distribution. (b) Once the system reaches a statistically steady state, the histogram of the walker position exhibits a localization
region, which bears a strong resemblance to the first eigenmode of Schr€odinger’s equation for the same potential in the weak regime (c). In contrast, (d) the position histogramof a classical particle without the wave field in the same high kinetic energy regime is uniform, indicating a homogeneous exploration of the random domain. (e) Effective meanwave potential about the localization region. Reprinted with permission.

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A growing number of experiments4,6,20,51 have demonstrated thatpilot-wave hydrodynamics is also viable in relatively shallow liquid
layers, wherein the lower boundary influences the walking droplet’s
dynamics without entirely suppressing the guiding wave field.6
Notably, variations in bottom topography lead to spatial gradients inmemory that may be harnessed to subject walkers to spatial potentials.4,6,52 A walker is, thus, repelled from pillars53 and drawn toward
submerged wells,4 the modeling of which requires more sophisticatedpilot-wave models37,54 to capture the effect of the submerged topography on the pilot-wave.
Abraham et al.21 have recently exploited variable bottom topography to realize a hydrodynamic analog of Anderson localization.56,57
The fundamental problem underlying Anderson localization is thetransport of an electron in a metal with random impurities, which may
be modeled through a random potential, UðxÞ, in the limit when theparticle’s kinetic energy, K, is much larger than the characteristic

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Applied Physics Lettersobserved striking similarities [Fig. 4(c)]. To highlight the peculiar
behavior of walking droplets, Abraham et al.21 also considered the
motion of a classical particle without an accompanying wave field inthe same potential to demonstrate its diffusive motion, leading to a relatively homogeneous position histogram [Fig. 4(d)]. Numerical simulations further demonstrated the walker localization in domains muchlarger than those accessible in the laboratory. To rationalize the walker
g ðxÞ
localization, Abraham et al.21 investigated the mean wave field, 
where they noted a large-scale envelope acting as an effective potential
about the localization region [Fig. 4(e)], which may be computed byconvolving the position histogram with the long-wave modes with theslowest decay rates excited during the impacts. This new HQA demonstrates that Anderson localization, widely believed to be purely a wave
phenomenon, also arises for classical particles propelled by self-excitedwaves, motivating its further investigation in pursuit of connections tolandscape theory.60
E. Surreal trajectories

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line in Fig. 5(a)]. ESSW concluded that “the Bohm trajectory is here
macroscopically at variance with the actual, that is: observed, track.
Tersely: Bohm trajectories are not realistic, they are surrealistic.”55
Despite surreal trajectories being confirmed experimentally byMahler et al.61 using weak measurements, ESSW declared “theBohmian picture to be at variance with common sense.”62 Notably,
Mahler et al. invoked nonlocality to rationalize their experimental
results, stating that “the trajectories seem surreal only if one ignores
their manifest nonlocality.”61
Recently, Frumkin et al.22 demonstrated a variant of the thought
experiment proposed by ESSW, in the hydrodynamic pilot-wave system (Fig. 5). They constructed a hydrodynamic interferometer consisting of a submerged rhombus that forces a droplet with an initial
random orientation, toward one of two submerged barriers with equal
probability. The rhombus, thus, acted as a beam splitter, and the submerged barriers as reflectors. In the case of a symmetric setup[Fig. 5(b)], specifically when both barriers are present, the droplet
passed the beam splitter and was reflected away from the adjacent barrier. However, once it approached the centerline, it changed direction
again and so followed a “surreal” trajectory. The red curves in Fig. 5(b)depict 20 such trajectories, obtained in a continuous experiment with asingle droplet. In the second experiment [Fig. 5(c)], one of the barriers
was removed, resulting in an asymmetric configuration. In this case,after the droplet was reflected from the barrier, it maintained its
course, and surreal trajectories were suppressed, as is the case in
Bohmian mechanics when mirror M1 is removed.
The origin of surreal trajectories in pilot-wave hydrodynamics isthat the spatially extended pilot wave interacts with the system’s topography so as to ensure that droplet never crosses the system’s centerline
and instead follows a surreal path. Figure 5(d) shows the mean wave
field that results from numerically calculating the weighted average ofthe pilot wave forms arising over all droplet trajectories for a symmetric setup. The nature of a specific droplet trajectory is determined by
its initial condition and the topography of the experimental setup.
However, consideration of an ensemble of initial conditions would
produce mean trajectories guided by the mean wave field. The mean
wave field should, thus, act as a potential that is effectively statistical inthat it guides the mean trajectories, just as the quantum potential in

FIG. 5. Surreal trajectories in pilot-wave hydrodynamics. (a) A variant of the interferometer setup considered by ESSW.55 An incoming wave packet is split by a beam splitter B
and reflected by the mirrors M1 and M2 . The wave packets interfere in the region I and then move toward the detectors D1 and D2 . The blue path represents the particle trajectory anticipated by ESSW, while the red path is that predicted by Bohmian mechanics. (b) In a symmetric setup, the droplet enters the right or left channel with equal probability,
after which it is deflected away from the system centerline, resulting in a “surreal” trajectory. (c) When one of the barriers is removed, the symmetry of the system is broken.
The walking droplet is then reflected away from the remaining barrier, resulting in the trajectory that one might expect. (d) The mean wave-field generated by averaging simulated droplet trajectories with a Gaussian distribution of initial impact parameters, with a standard deviation of 1:4kF . Reprinted with permission from Frumkin et al., Phys. Rev.
A 106, L010203 (2022).

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Pilot-wave hydrodynamics furnishes a physical picture thatwould provide rationale for several notable quantum oddities. One
such example is the notion of “surreal trajectories” put forth byEnglert, Scully, S€
ussman, and Walther (ESSW).55 The authors proposed an interference experiment intended to expose the shortcomingsof the predictions of Bohmian mechanics. The arrangement considered by ESSW is depicted in Fig. 5(a). A single particle is directed
toward a beam splitter B, where, according to the Copenhagen interpretation, it is split into an equal superposition of packets w1 and w2that propagate, respectively, toward the mirrors M1 and M2 . Afterbeing reflected, the packets will recombine at the interference region I
before eventually reaching the detectors D1 and D2 , at which point theparticle will be detected at D1 or D2 with equal probability. However, aBohmian particle will travel either left or right, with equal probability.
If, for example, the particle goes to the right, it will be reflected by M2
and will continue toward the interference region I. ESSW showed intheir analysis that the Bohmian particle, instead of proceeding toward
D1 , changes course in region I owing to the influence of the quantum
potential, and so proceeds toward D2 . The resulting trajectories, thus,
never cross the centerline of the interferometer [the dashed vertical

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Bohmian mechanics guides the quantum velocity of probability.
Notably, in contrast to Bohmian mechanics, where nonlocality isrequired to rationalize surreal trajectories, in pilot-wave hydrodynamics, such trajectories result from a wave potential generated locally by
the droplet.
F. The quantum bomb tester

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form propagating along path 1 interacts with and is altered by thebomb without detonating it.
As in the case of surreal trajectories, the Elitzur–Vaidman bomb
tester is a result of quantum particles exhibiting wave-like statistics,
causing their detection probability to be governed by constructive anddestructive interference effects. Building on the similarity between
these two systems, Frumkin and Bush23 recently demonstrated thatsuch a detection of an object by a particle that interacts with the object
only through its associated waveform may likewise be achieved in ahydrodynamic pilot-wave system. Their setup consisted of two configurations of a hydrodynamic interferometer: one symmetric [Fig. 6(b)],identical to that used to realize surreal trajectories,22 and anotherasymmetric [Fig. 6(c)], with the upper part of the left reflector
removed. The upper part of the left reflector effectively served as the“bomb,” an object whose presence one seeks to detect without directinteraction. In the symmetric case [Fig. 6(b)], if the droplet took thechannel which contained the “bomb” (left), it would effectively detonate it, and the droplet’s trajectory was considered to be terminated.
Conversely, if the droplet took the right channel, its pilot-wave would
deflect it away from the centerline, along a surreal trajectory. Thus, if
the “bomb” is present, the droplet would either detonate it, or be
detected on the right side of the setup. In the absence of the “bomb”[Fig. 6(c)], the symmetry was broken, preventing the droplet from following a surreal trajectory. In this case, if the droplet took the right
path, it was reflected, and continued in a straight line toward the left. If
the droplet took the left path, no reflection occurred, and it again went
to the left following a straight path. Thus, in the absence of the“bomb,” the droplet was always detected on the left side of the setup.
After many realizations of the experiment in which the bomb was present 50% of the time, the experimenter has a 25% chance of detecting
the droplet on the right side. Such a detection indicated that the“bomb” was present in the left channel, even though the droplet took
the right path and so never interacted with it directly.

FIG. 6. Interaction-free measurement in pilot-wave hydrodynamics. (a) A schematic of the Elitzur–Vaidman bomb experiment. A particle emitted from a source S passes
through a beam splitter B1, at which point its associated wave is split in two. The wave is then recombined at a beam splitter B2, and the particle continues toward the detectors.
In the absence of a bomb, the particle will be detected at D1 100% of the time, while in its presence, the particle will be detected at D2 25% of the time. A detection event at D2
indicates the presence of a live bomb along a path the particle never took. Red and blue dashed lines indicate possible paths taken by a particle emitted from S in the absence
and presence of the bomb, respectively. (b) In a symmetric hydrodynamic setup, the droplet enters the right or left channel with equal probability. If the droplet goes to the leftchannel, it effectively detonates the “bomb,” and if it goes to the right, it is deflected away from the system centerline, resulting in a “surreal” trajectory.22 Thus, the droplet will
always be detected on the right side of the setup. Twenty such trajectories are shown. (c) If the “bomb” is removed, the symmetry of the system is broken, the droplet’s pilotwave does not interact with the bomb, and “surreal” trajectories are suppressed. Thus, the droplet will always be detected on the left side of the setup. If the bomb is present
50% of the time, there is 25% chance of the droplet being detected on the right, and so the bomb on the left. The scale bar represents the Faraday wavelength kF ¼ 4:84 mm.
Reprinted with permission from Frumkin and Bush, Phys. Rev. A 108, L060201 (2023).

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Interaction-free measurement is a peculiarity of quantum
mechanics that seemingly allows one to obtain information about thequantum state of an object without its being “disturbed” by the measurement process. The most famous example of this phenomenon is
the so-called Elitzur–Vaidman bomb tester,63 according to which, 25%of the time, a photon can detect a bomb present in an interferometer
50% of the time without directly interacting with it. A variant of thisthought experiment is depicted in Fig. 6(a). A photon is emitted froma source S into a Mach–Zehnder interferometer with arms of equal
length. At the first beam splitter B1 , the photon’s wave function is split
in two, with each part traveling along one arm of the interferometer,
and then recombined at the second beam splitter B2 . At this point, thewave function returns to its original state, thus the photon will be
detected at detector D1 with probability 1. Now consider a highly sensitive bomb that is placed along path 1 in the interferometer, so thatany photon encountering it will detonate it. In this case, 50% of thetime the particle follows path 1 and detonates the bomb. Otherwise,
the particle follows path 2, while path 1 remains blocked by the bomb,
preventing interference between the two wave packets at B2 . As aresult, the particle has a 50% chance of being detected at D2 , thus providing the experimenter information about the presence of a bomb
positioned along a path that it never traveled. Overall, if the bomb ispresent in the interferometer 50% of the time, the experimenter has a
25% chance of detecting the particle at D2 and so the bomb along path
1. Note that the experiment can be considered “interaction-free” onlyin the sense that the particle does not interact with the bomb: the wave

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Like a Bohmian particle, the droplet is localized at all times, while
its pilot-wave is spatially extended and, thus, influenced by the geometry of its environment. The similarity of the statistical behavior
between this hydrodynamic system and its quantum counterpart may
again be rationalized in terms of the influence of the droplet’s delocalized pilot-wave field. As was the case with surreal trajectories, theorigin of the pilot wave in the particle vibration allows one to rationalize the phenomenon without appealing to nonlocal effects.
G. Pair–pair correlations

IV. DISCUSSION
Purely wave-based hydrodynamic quantum analogs include theripple-tank diffraction experiments of Young,75 and Berry’s analog of
the Aharanov–Bohm effect.76 Another such analog has recently provided an elegant means of connecting the dynamics of surface wave
packets to Bohmian mechanics.77 The walking-droplet system has provided the first class of particle-based hydrodynamic quantum analogs.
As such, it has furnished a platform for exploring the boundary
between classical and quantum effects; moreover, it has inspired investigations of other macroscopic realizations of wave-particle duality inwhich oscillators interact with their suspending fluids. Le Gal et al.78
investigated the motion of a neutrally buoyant object suspended in astably stratified fluid. By driving the mass distribution within the object
pneumatically, they excited vertical oscillations. When the object was
excited at the Brunt–Vaisala frequency, the object attained resonanceAppl. Phys. Lett. 125, 030503 (2024); doi: 10.1063/5.0210055
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with its ambient wave field and so transitioned into a self-propagatingstate. It has been known for some time that acoustically forced bubbles
may be induced to propel in an erratic fashion.79 Baudoin andco-workers80 has recently been exploring the possibility of bubbles
self-propelling in response to the pressure field associated with theirown vibration. Harris and co-workers81 have investigated the “surferbot,” a floating object with an onboard oscillator that excites capillary
waves that propel the surfer-bot forward. Finally, Neufeld andco-workers82 have demonstrated that when a canoeist is literally up a
creek with no paddle, her best bet is “gunwale bobbing,” standing on
the gunwales of her canoe and bobbing up and down, then surfing on
the resulting waveform.
The theory of quantum mechanics is undoubtedly nonlocal.
HQA is questioning the extent to which quantum physics need be
nonlocal. Quantum nonlocality would appear to be ubiquitous, manifest everywhere from wave-function collapse to statistical projection
effects to slit diffraction, from surreal trajectories to the Elitzur–
Vaidman bomb tester. HQA has made clear that all these phenomena,at least, can be understood from a local realist perspective. The experimental violation of Bell’s inequality is generally taken as proof thatquantum mechanics is nonlocal; thus, one can feel justified in not seeking a mechanistic explanation of quantum phenomena, in appealing tononlocality as needed. Bohmian mechanics provides a viable realist,
nonlocal theory of quantum dynamics. HQA suggests the possibility ofgoing one step further in developing a local realist theory of quantum
dynamics, in which the particle is the source of its own guiding wave.
A number of such theories of quantum dynamics, informed by thewalker system, are currently being developed and explored.83–88 Suchtheories would follow the prescription of Einstein in providing a
dynamical underpinning for the theory of quantum statistics, just as
statistical mechanics underlies thermodynamics. For those insistent on
local realism being a basic tenet of science, the physical picture imagined by de Broglie,18 manifest in stochastic electrodynamics89,90 andengendered in pilot-wave hydrodynamics is an appealing one.
A number of important questions remain to be addressed by the
HQA community. What is the analog of the linear Schr€odinger equation in the hydrodynamic system? This question motivated the recent
study of Kutz et al.,91 who used dynamic mode decomposition to characterize the evolution of the walker wave field. Their study demonstrated that, as the memory is increased progressively, new wave
modes enter discretely until the wave dynamics becomes chaotic.
Using the convolution result of Durey et al.,36,37 they were then able toinfer the statistical behavior of the droplet without ever measuring theparticle position. Does the Faraday system have the potential to capture aspects of particle creation and annihilation? Frumkin et al.92 took
the first step in this direction by presenting a hydrodynamic analog ofsuperradiance from atomic pairs, which involved the correlations ofdroplet ejection events from neighboring deep wells in the hydrodynamic system. Finally, the wave-particle nature of the walking droplets
gives them unique properties that may be harnessed to process information in an unorthodox way. For example, information may be
encoded in a variety of hydrodynamic bound states, processed throughstatistical interference, and read out directly via droplet position measurements. Thus, given that implementations of quantum algorithms
do not rely on quantum nonlocality, only on quantum statistics, it is
only natural to ask whether the hydrodynamic pilot-wave system may
provide a new platform for quantum-inspired classical computing.

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Nachbin64,65 has examined the extent to which wave-mediatedcoupling can give rise to long-range correlations in pairs of bouncing
droplets. Papatryfonos et al.66 adapted the numerical methodology of
Nachbin and co-workers,67,68 in order to perform the first hydrodynamic Bell tests. Specifically, they assessed the correlations of two
droplets, each confined to wave-coupled tunneling subsystems. In thisanalog system, the cavity location (inner or outer) played the role ofthe dichotomic variable (spin up or down) in the quantum Bell tests,
and the system geometry the role of the measurement settings (polarizer angles). For specific combinations of the measurement settings,
Papatryfonos et al.66 reported violations of Bell’s inequality, as may berationalized in terms of the system memory.
Violations of Bell’s inequality have also been reported in bothclassical electromagnetic69–71 and acoustic wave systems.72 However,
the hydrodynamic pilot-wave system has yielded the first such violations with a spatially separated bipartite classical system. The result iscomparable to the violation achieved in the “static” Bell tests of
Aspect,73,74 in which the analyzer settings were unaltered during thecourse of the experiment. In the hydrodynamic pilot-wave system, theobserved violations may be rationalized in terms of the wave-mediatedcommunication between the two subsystems. Eliminating communication between the two subsystems would require the implementationof tests with dynamic topography. Specifically, a wall could be imposed
between the two subsystems during the course of the experiment, prior
to the alteration of the measurement settings.65 Successfully violating
Bell’s inequalities with such dynamic tests would rely on the violations
achieved with the static test surviving the isolation of the two subsystems. However unlikely, such a result is not entirely inconceivablegiven that the bath serves as the memory of the system, and so storesinformation concerning the initial interaction between the droplet
pair, even after isolation of the two subsystems.

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ACKNOWLEDGMENTS
J.B. and V.F. gratefully acknowledge the financial support of
the National Science Foundation through Grant No. CMMI2154151. P.J.S. gratefully acknowledges the financial support of the
National Science Foundation through the CAREER award CBET2144180, and the Alfred P. Sloan Foundation through a Sloan
Research Fellowship.
AUTHOR DECLARATIONSConflict of Interest
The authors have no conflicts to disclose.
Author Contributions
John W. M. Bush: Conceptualization (equal); Funding acquisition(equal); Project administration (equal); Writing – original draft (equal);
Writing – review & editing (equal). Valeri Frumkin: Conceptualization(equal); Writing – original draft (equal); Writing – review & editing(equal). Pedro J. Saenz: Conceptualization (equal); Funding
acquisition (equal); Writing – original draft (equal); Writing – review &editing (equal).
DATA AVAILABILITY

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The data that support the findings of this study are available fromthe corresponding author upon reasonable request.

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