PHYSICAL REVIEW A 106, L010203 (2022)
Letter

Editors’ Suggestion

Real surreal trajectories in pilot-wave hydrodynamics
Valeri Frumkin,* David Darrow ,† and John W. M. Bush‡

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Ward Struyve§

Instituut voor Theoretische Fysica & Centrum voor Logica en Filosofie van de Wetenschappen, KU Leuven, Leuven, Belgium(Received 23 May 2022; accepted 8 July 2022; published 25 July 2022)
In certain instances, the particle paths predicted by Bohmian mechanics are thought to be at odds with classicalintuition. A striking illustration arises in the interference experiments envisaged by Englert, Scully, Süssmann,and Walther, which lead the authors to claim that the Bohmian trajectories cannot be real and so must be “surreal.”
Through a combined experimental and numerical study, we here demonstrate that individual trajectories in thehydrodynamic pilot-wave system exhibit the key features of their surreal Bohmian counterparts. These realsurreal classical trajectories are rationalized in terms of the system’s non-Markovian pilot-wave dynamics.
Our study thus makes clear that the designation of Bohmian trajectories as surreal is based on misconceptions
concerning the limitations of classical dynamics and a lack of familiarity with pilot-wave hydrodynamics.
DOI: 10.1103/PhysRevA.106.L010203

Introduction. The standard theoretical formulation of quantum mechanics does not describe particle paths. Bohmianmechanics (also known as “de Broglie–Bohm theory” or“pilot-wave theory”) [1–3] provides such a description byasserting that particles move under the influence of the wave
function, with a velocity equal to the quantum velocity ofprobability in the standard quantum formalism. Specifically,
the dynamics of a single particle with position X is prescribedby the guidance equation,h̄∇ψ (X(t ), t )dX(t )
= Im
,
(1)
dt
m
ψ (X(t ), t )where ψ is the wave function satisfying the Schrödinger
equation with a potential V . Provided the wave function is
specified, a unique solution for the particle path may beobtained from the initial particle position. Taking the time
derivative of (1) yields a Newtonian-type trajectory equationd 2 X(t )= −∇V (X(t )) − ∇Q(X(t ), t ),
(2)
dt 2
h̄2 1
where Q = − 2m
∇ 2 |ψ|, the quantum potential, engenders|ψ|the departure from classical mechanics. Solving Eq. (2) for the
trajectory requires that one specify both the initial position andvelocity of the particle. The statistical predictions of Bohmianmechanics are in accord with those of the standard quantum
theory [1–3].
Among the successes of Bohmian mechanics is a selfconsistent description of single-particle diffraction and interference. The wave function guides the particles through them*valerafr@mit.edu
ddarrow@mit.edu‡
bush@math.mit.edu§
ward.struyve@kuleuven.be†2469-9926/2022/106(1)/L010203(6)slits according to Eq. (1), eventually leading to the familiar
diffraction and interference patterns building up over time.
The trajectories for the canonical double-slit experiment werefirst computed by Philippidis et al. [4], who demonstrated thatthe emergent diffraction pattern predicted by Bohmian mechanics is consistent with that observed, provided one chooses
an ensemble of initial positions according to the |ψ|2 distribution, which corresponds here to a Gaussian distributionof impact parameters at the slits. While direct measurement
would alter the wave function and hence the particle trajectory, the averaged trajectories revealed by weak measurement
strongly resemble the trajectories predicted by Bohmian mechanics [5].
In 1992, Englert, Scully, Süssman, and Walther(ESSW) [6] proposed an interference experiment intended
to expose the shortcomings of Bohmian mechanics. They
claimed that “the Bohm trajectory is here macroscopicallyat variance with the actual, that is: observed, track. Tersely:
Bohm trajectories are not realistic, they are surrealistic.” Suchcriticisms have been countered by several authors [7–11], on
the grounds that the type of observation envisioned by ESSW
cannot be expected to reliably reveal the Bohmian trajectory.
Nevertheless, ESSW [12] maintained that “one cannot
attribute reality to the Bohm trajectories” [12], “particles
do not follow the Bohm trajectories as we would expectfrom a classical type model” [13], and finally declared “theBohmian picture to be at variance with common sense” [13].
In the absence of experimental measurements of actual
particle paths, the designation of Bohmian trajectories
as real or surreal necessarily depends only on one’s
preconceptions as to how quantum particles should behave.
In their discussion of ESSW, Aharonov and Vaidman [14]conclude that “The examples considered in this work do notshow that Bohm’s causal interpretation is inconsistent. It
shows that Bohmian trajectories behave not as we would

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FIG. 1. (a) A variant of the interferometer setup considered by
ESSW [6]. 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 towards 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. The red
dot represents the one-bit, which-way detector. (b) Our experimental
arrangement. A walking droplet is released from a launching area
towards a beam splitter that directs the droplet towards one of two
submerged barriers with equal probability. The blue and red paths
are the possible analogs of those in the ESSW setup.

expect from a classical type model.” The aim of this Letter isto present a purely classical wave-particle system that exhibits
surreal trajectories. In so doing, we demonstrate that surreal
trajectories are not at odds with classical intuition informed
by a familiarity with pilot-wave hydrodynamics.
The ESSW setup. The arrangement considered by ESSW
is depicted in Fig. 1(a). A single particle is directed towards
a beam splitter B. The associated wave packet ψ is thus
represented by the superposition ψ+ + ψ− of packets ψ+ andψ− that propagate, respectively, towards the mirrors M1 andM2 . Note the perfect symmetry about the system’s verticalcenterline. After reflection from the mirrors, the packets crossbefore eventually reaching the detectors D1 and D2 . According to Bohmian mechanics, the particle will initially go either
left or right, with equal probability. In Fig. 1(a), the particle
initially goes right, bounces off the mirror M2 , and then enters
the interference region I. However, the Bohmian particle does
not continue its path towards D1 but instead turns around inregion I, then proceeds toward D2 . This behavior is generic:No Bohmian trajectories cross the symmetry axis; instead,they turn around in the region I. Such behavior is at odds
with the behavior anticipated by ESSW, who presumed thatthe particles should not change direction at I. If the mirror
M2 were to be removed or a detector were to be placed in
the path towards M2 , there would be no interference between
ψ+ and ψ− and the Bohmian path would correspond to that
anticipated by ESSW.
ESSW also assumed that a one-bit which-way detector
along one of the paths would reveal the path taken by theparticle after the particle passes through the interferometer.
They thus hoped to show that the Bohmian trajectory is at
odds with the inferred trajectory. A variant of the ESSW setup
using double slits was proposed by Braverman and Simon [15]and realized experimentally by Mahler et al. [16], who reported that the trajectories inferred from weak measurement

are consistent with the predictions of Bohmian mechanics. We
here demonstrate that the surreal character of the Bohmian
trajectories also arises in pilot-wave hydrodynamics.
Pilot-wave hydrodynamics. In 2005, Couder and Fort discovered that a millimetric droplet may self-propel along thesurface of a vibrating liquid bath by virtue of a resonantinteraction with its own quasimonochromatic wave field [17].
In a seminal 2006 paper, Couder and Fort used this system
to demonstrate single- and double-slit diffraction and singleparticle interference [18], phenomena previously thought tobe exclusive to the microscopic quantum realm. Their study
has been revisited several times, and the diffraction andinterference effects confirmed [19,20]. This hydrodynamic
pilot-wave system [21] has since been used to establish
many other hydrodynamic quantum analogs [22], including
unpredictable tunneling through potential barriers [23], quantization of orbital states [24], the emergence of wavelike
statistics in corrals [25,26], Friedel oscillations [27], and hydrodynamic spin lattices [28].
A key feature of the hydrodynamic pilot-wave system isthat the droplet dynamics is non-Markovian [21]. Specifically,
the propulsive force on the droplet is prescribed by the local
slope of the wave field, the form of which necessarily depends
on the droplet’s history. The wave field persists for a timeprescribed by the bath’s vibrational acceleration γ . The baththus effectively serves as the “memory” of the droplet [29],and the system memory is prescribed by the proximity of γ
to the Faraday threshold γF , the critical vibrational acceleration at which Faraday waves form on the bath surface inthe absence of the drop. Notably, the quantum features arise
exclusively in the high-memory limit, γ → γF− , when theeffects of the pilot wave and the system memory are mostpronounced.
In several instances, nonlocal features of the walkingdroplet system might be misinferred if the influence of thepilot wave is not given due consideration. For example, submerged pillars and posts give rise to long-range lift forces on
walking droplets that are mediated by the pilot wave [27,30].
In the walker double-slit experiment [18], the presence of thesecond slit was seen to have an influence on droplets passing
through the first [19,20]. Without due consideration of the
dynamical significance of the pilot-wave field and the itinerant
non-Markovian droplet dynamics, such wave-mediated forces
appear to act at a distance and so to be spatially nonlocal.
Moreover, Bush and Oza [22] discuss settings in which themean-pilot-wave potential [31] plays a role akin to the quantum potential in Bohmian mechanics.
Walking droplets exhibit nonspecular reflection from submerged planar barriers [32]. The weak dependence of thereflection angles on the angle of incidence allows one touse submerged planar barriers as reflectors, or analog mirrors. We proceed by presenting a hydrodynamic analog
of the ESSW thought experiment using submerged planar
barriers as reflectors. In so doing, we demonstrate that surreal trajectories may arise in the hydrodynamic pilot-wave
system.
Experiments. Our experimental system consists of a circular bath filled with a 7.0 ± 0.3-mm-deep layer of silicon oilwith surface tension σ = 0.0209 N/m, viscosity ν = 20 cSt,and density ρ = 0.965 × 10−3 kg/m3 . The system is vibrated

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FIG. 2. Droplet trajectories in the hydrodynamic pilot-wave system. (a) A single-particle trajectory, along with the pilot-wave field at the
instant that the drop is at the position shown. (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. Twenty such trajectories are shown. (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.

vertically by an electromagnetic shaker with forcing F (t ) =γ cos(2π f t ), with γ = 3.79 g and f = 75 Hz being the peak
vibrational acceleration and frequency, respectively. In all experiments reported here, the Faraday threshold, at which theflat free surface destabilizes to a pattern of subharmonic Faraday waves with wavelength λF = 5.6 mm, was γF = 3.82 g.
Spatial uniformity of the bath vibration was insured by connecting the shaker to the bath with a steel rod coupled toa linear air bearing [33]. The vibrational forcing was monitored with two accelerometers placed on opposite sides of thebath, insuring a constant vibrational acceleration amplitude
to within ±0.002 g. The droplet trajectories and their guiding wave field were captured through a semireflective mirrorthat was positioned at 45◦ between the bath and a chargecoupled device (CCD) camera that was mounted directly
above the setup. The bath was illuminated with a diffuse-lightlamp facing the mirror horizontally, yielding images withbright regions corresponding to horizontal parts of the surface,specifically extrema or saddle points [see Fig. 2(a)].
The topographical configuration used in our experiments
is depicted in Fig. 1(b). A walking droplet is confined to alaunching pad, where it wanders in an irregular fashion until
being ejected towards a submerged rhombus that forces thedroplet towards one of two submerged barriers with equal
probability. The launch randomizes the droplet’s initial conditions, while the rhombus serves as a beam splitter, and thesubmerged barriers act as reflectors. In the first experiment,
the droplet passes the beam splitter and is reflected awayfrom the adjacent barrier; subsequently, it changes direction
again before reaching the centerline. Figure 2(b) shows 20
such trajectories (10 from each side), obtained in a continuous
experiment with a single droplet. If the influence of the pilot
wave is not duly considered, one could only rationalize thesecond change in direction in terms of an external force. In
reality, this apparently nonlocal behavior is a manifestationof the local influence of the pilot wave that interacts withboth barriers and the droplet in such a way as to produce
the surreal trajectories depicted in Fig. 2(b). In the second
experiment, one of the barriers is removed while the restof the setup remains unaltered [Fig. 2(c)]. With this asymmetric configuration, after the droplet is reflected from thebarrier, it does not change direction appreciably. The second
experiment underscores the importance of the presence ofboth barriers for surreal droplet trajectories. A video of thetwo experiments is available in the Supplemental Material(SV1) [34].
Numerical simulations. We proceed by adopting the numerical model of Faria [35], as has been shown to provide
a robust description of walking-droplet-boundary interactions
in a number of settings, including reflection from walls [32],scattering from a submerged circular well [28], and diffraction
through slits [19]. The model synthesizes the walker wave
model of Milewski et al. [36] with the trajectory equation of
Molacek and Bush [37]. The effect of the topography on the
waves is captured through incorporating its influence on thelocal phase speed of the pilot wave. The distinct advantageof the simulations is that they enable the precise prescription of the initial conditions for the droplet trajectories,
computation of the associated wave fields, and characterization of the influence of increasing memory on the droplet
dynamics.
The results of the numerical simulations are presented inFig. 3. In the absence of a second reflector [Fig. 3(a)], thedroplets follow the expected trajectories, being reflected by
the barrier on the right, then proceeding to cross the centerline.
When a barrier on the left is added [Fig. 3(b)], the resulting
trajectories may exhibit surreal behavior, depending on theinitial conditions and system memory. Figure 3(c) illustrates
the dependence of the droplet trajectory on the impact parameter (x, the distance from the centerline) for an ensembleof drops launched downwards. Figures 3(d)–3(f) demonstratethat the surreal nature of the droplet trajectories becomes
more pronounced as the system memory is increased. At
sufficiently high memory, the pilot wave excited by the
droplets extends to the outer boundaries of the domain. Thus,
our simulations were limited to a memory value of 0.92γF ,above which the droplets were affected by the domain’s outeredges.

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FIG. 3. Simulated droplet trajectories in the hydrodynamic pilotwave system. In (a)–(c), the vibrational acceleration γ = 0.905γF .
(a) When only one reflector is present, the droplet follows the
expected trajectory, regardless of the initial conditions. (b) In thepresence of a second reflector, depending on the initial conditions
and the system memory, the droplet may follow a “surreal” trajectory, and so never cross the centerline. (c) An ensemble of initiallyvertical trajectories with different values of the impact parameter x.
(d)–(f) The dependence on memory of a trio of originally verticaltrajectories. As the memory parameter is increased from (d) 0.88 F
to (e) 0.9 F to (f) 0.92 F , all trajectories transform from expected
to surreal. Scale bar: 5λF .

Figure 4(a) shows the mean wave field that results from calculating the weighted average of the pilot-wave forms arisingover all droplet trajectories. Figure 4(b) depicts the trajectories
used in this ensemble, which were weighted according toa Gaussian distribution in the impact parameter in order tocompute the mean wave field. The relation between this mean
wave field and the quantum potential in Bohmian mechanics
will be the subject of future investigation.
Discussion. The walking-droplet system has been shown
to exhibit several features previously thought to be exclusive to the quantum realm [22]. We have shown here thatthe surreal trajectories predicted by Bohmian mechanics areanother such feature. The surreal character of both Bohmianand droplet trajectories may be attributed to wave-mediatedforces. However, a number of notable distinctions should bemade between the trajectories arising in Bohmian mechanics
and pilot-wave hydrodynamics. First, quantum trajectories
cannot be measured precisely without being disturbed. Second, according to Bohmian mechanics, particles are guided
by the wave function , whose form is uninfluenced by theparticle. Conversely, the trajectories in pilot-wave hydrody-

FIG. 4. The mean wave field generated by averaging simulated
droplet trajectories with a Gaussian distribution of initial impact
parameters, with a standard deviation of 1.4λF , (a) with and (b) without the associated droplet trajectories overlaid. The wave amplitude
is presented in units of the system’s Faraday wavelength. Scalebar: 5λF .

namics are directly observable and result from the droplet’sinteraction with its own wave field. Specifically, the droplet
navigates its pilot-wave field, a local potential of its own
making. One might thus attribute the designation of “surreal”
to Bohmian trajectories to ESSW’s lack of familiarity withthe walking-droplet system, wherein such trajectories may berationalized in terms of non-Markovian, classical, pilot-wave
dynamics.
Prior work [21,22] has shown that the walking-droplet
system is closer in form to de Broglie’s original doublesolution pilot-wave theory [38], according to which a quantum
particle has an internal vibration at the Compton frequency
that generates its own guiding wave, and the resulting pilotwave dynamics gives rise to emergent statistics describedby the standard quantum theory. Our study provides further

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motivation for the revisitation and extension of de Broglie’s
double-solution program, informed by the walking-droplet
system [39–42].
Acknowledgments. The authors gratefully acknowledge thefinancial support of the U.S. Department of State (V.F.), theNational Science Foundation Grant No. CMMI-2154151(J.B.), and the Research Foundation Flanders (Fonds
Wetenschappelijk Onderzoek, FWO), Grant No. G066918N(W.S.). We thank Boris Filoux and Nicolas Vandewalle forvaluable discussions.

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