††thanks: A calibration issue with the acceleration measurement in our system
led to acceleration amplitudes ten times higher than the actual values. In
this version, correct acceleration values have been updated. A correction note
has also been published alongside the journal version [Phys. Rev. Fluids 5,
053603 (2020)].††thanks: The first two authors contributed equally

# Quantized orbital-chasing liquid metal heterodimers directed by an
integrated pilot-wave field

Jianbo Tang<EMAIL_ADDRESS>Department of Biomedical Engineering,
School of Medicine, Tsinghua University, Beijing 100084, China  Current
address: University of New South Wales, Sydney, NSW 2052, Australia  Xi Zhao
Key Laboratory of Cryogenics, Technical Institute of Physics and Chemistry,
Chinese Academy of Sciences, Beijing 100190, China  School of Future
Technology, University of Chinese Academy of Sciences, Beijing 100049, China
Jing Liu<EMAIL_ADDRESS>Department of Biomedical Engineering, School
of Medicine, Tsinghua University, Beijing 100084, China  Key Laboratory of
Cryogenics, Technical Institute of Physics and Chemistry, Chinese Academy of
Sciences, Beijing 100190, China  School of Future Technology, University of
Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

A millimetric bouncing droplet sustained on a vibrating bath becomes a moving
wave source (particle) through periodically interacting with the local wave
field it generates during the droplet-bath impact. By virtue of such particle-
wave duality, the macroscopic hydrodynamic system imitates enigmatic behaviors
of the quantum realm. Here we show that it is possible to create an integrated
pilot-wave field to better prescribe the droplet trajectories, via amplified
bath capillarity. This is demonstrated with a liquid metal droplet-bath system
in which the local wave field generated by droplet bouncing is superposed by
the global wave field induced by bath meniscus oscillation. The resulting dual
pilot-wave configuration enables a class of directional chasing motions of two
bound dissimilar droplets (heterodimers) in multilevel hydrodynamic traps
(orbits), featuring two quantized regime parameters, namely the interdroplet
binding level and the orbit level. We investigate the dynamics of the
vibrating liquid metal bath, with its level-split ring wave field and its
peculiar vortex field being highlighted. We also rationalize the exotic
droplet motions by considering the interdroplet particle-wave interactions
mediated by the integrated pilot-wave field. It is revealed that a temporal
bouncing phase shift between the two droplets in the heterodimers, due to size
mismatch, gives rise to their horizontal propulsion, while their spatial
binding regime exclusively determines the collective chasing direction. It is
further evidenced that the horizontal in-orbit chasing motion is directly
related to vertical droplet bouncing. Our findings unveil the integrated
pilot-wave field as a trail towards improved droplet guiding, thereby
extending the hydrodynamic particle-wave analogy to optical systems and
beyond.

##  I Introduction

A hydrodynamic pilot-wave system features millimetric bouncing droplets self-
propelling on a vibrating fluid bath. The droplet behaviors emerging from the
system strongly depend on the dynamics of both the bath and the droplets [1,
2]. The study of the former, which dates back to Michael Faraday [3], has been
focusing on the nonlinearity and threshold instability of the periodically
forced bath [4]. This classic problem has been greatly enriched by the
introducing of discrete wave sources (bouncing droplets of the same fluid) to
the system by Yves Couder, Emmanuel Fort and co-workers [5, 6]. It has been
shown in their pioneering works that, below the Faraday threshold
$\gamma_{\textit{F}}$, a vertically bouncing droplet can be long-term
sustained and also be carefully tuned to walk horizontally through a particle-
wave association, that is, the droplet interacts with the waves it generates
on the bath. The creation of particle-wave duality in a macroscopic
hydrodynamic system has profound influences and meaningful implications since
wave-mediated motions are frequently encountered in various physical systems,
from the classical interpretation of the electron-phonon wave interactions of
Cooper pairs in the Bardeen–Cooper–Schrieffer theory of superconductivity [7]
to biological locomotions on water surface [8] and to astronomical scale
planetary/stellar motions in the gravitational wave field [9], which
themselves might not be readily accessible. Considerable efforts have been
made ever since to explore both individual [10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20] and collective droplet behaviors [21, 22, 23, 24, 25] in the
droplet-bath system, in particular for their quantum analogies [10, 12, 14,
16, 18].

Despite the diversity of previously observed droplet behaviors, they mostly
share the same particle-wave association regime in which the horizontal
droplet motions are solely piloted by the droplet-generated local wave field,
either by the droplets themselves or by their partners. To a large extent,
such a single pilot-wave configuration restricts the search for informative
droplet motions to a narrow acceleration range close to but below
$\gamma_{\textit{F}}$, where the walking conditions are satisfied [1, 26]. One
exception is the ratcheting droplet pairs reported by Eddi  _et al._ [21] and
Galeano-Rios  _et al._ [25], a configuration that enables switchable
horizontal motions of dissimilar droplet pairs at accelerations well below
$\gamma_{\textit{F}}$.

It has been demonstrated that tuning bath dynamics favors better prescribing
droplet trajectories. However, it is typically realized through either
auxiliary bath modifications (e.g., engineered bath structures [17, 18, 24,
27, 28, 29]) or bath movement control (e.g., a rotating bath frame [13, 15]).
In particular, Sungar  _et al._ [27, 28] has demonstrated that protruding
pillars above the vibrating bath surface can lead to a hydrodynamic effect
reminiscent of the optical Talbot effect. To enrich the content of the pilot-
wave hydrodynamics as well as to extend its implications, strategies that
enable new droplet motion regimes and at the same time avoid previous
formalities will be highly desirable. To this end, an attempt is made here to
achieve improved droplet guiding without compromising the simplicity and
elegance of this classical system.

The superposition nature of waves implies that foreign wave fields other than
the droplet-generated local wave field can be introduced to provide further
confinements for the droplets. One mechanism to produce such an additional
wave field in the droplet-bath system could be dynamic meniscus, since a
vibrating fluid bath is always attended by an oscillating meniscus at its
boundary region due to capillarity [30]. Generally, the oscillating meniscus
is considered as a troublesome feature which previous studies tried to
circumvent, since it brings both experimentation difficulties and theoretical
assessment complexities to the system [4, 31]. By contrast, here we
significantly magnify the capillary effect by selecting a liquid metal
droplet-bath system. We show that the previously believed problematic
oscillating meniscus can produce a global pilot-wave field, which is indeed
advantageous for droplet guiding. Directed by the dual pilot-wave field,
dissimilar droplet pairs in our system are able to exhibit well-defined
collective orbital motions with double quantized states.

In one-to-one correspondence, we show that the hydrodynamic particle-wave
association directed by an integrated pilot-wave field is reminiscent of the
wave-matter interaction in optical systems, and the chasing droplet pairs
become counterparts of optical heterodimers$-$two optically bound particles of
unequal sizes [32, 33]. It has been demonstrated that a perpendicular incident
light (electromagnetic waves) can trigger transverse motions of optical
heterodimers, since the size mismatch between the two particles leads to their
asymmetric interactions (scattering) in the presence of the external
electromagnetic wave field. When a second electromagnetic field (a circular
optical trap) is further applied to create a global confinement, the
collective motions of the heterodimers are guided along the optical trap,
resulting in optically directed orbital chasing motion [32]. In the optical
system, the spatial interparticle distance and the motion trajectory of the
heterodimers are modulated by the interparticle electrodynamic interaction
(optical binding) and the predefined global electromagnetic wave field
(optical trap), respectively [32]. We show that directed by an integrated
hydrodynamic pilot-wave field, the particle-wave association between the
droplets and the local wave field generated by their partners provides the
interdroplet binding, whereas the global wave field acts as a multilevel
hydrodynamic trap which leads to orbital motion. Strikingly, despite the fact
that the optical heterodimers and the hydrodynamic heterodimers interact with
wave fields of different natures, their motion regimes (e.g., configuration
and direction) are, however, surprisingly alike. In addition, the directional
transverse motions in both systems stem from the symmetry breakdown of their
respective particle-wave association, which itself is a result of the
breakdown of size symmetry. Therefore, it is attempting that the theoretical
frameworks that describe the two systems, which feature different physical
interactions and distinct scales, may share a similar form. Encouragingly, the
results reported here demonstrate the possibility to extend the hydrodynamic
pilot-wave analogy to other wave-mediated systems as well as the feasibility
to guide droplet motions with an integrated pilot-wave field.

Figure 1:  (a) Schematic side view of the liquid metal droplet-bath system.
The bath is filled with a bottom liquid metal layer and a top alkaline
electrolyte layer. The vibrating bath and the bouncing droplet induce a global
wave field $\mathcal{W}^{G}$ and a local wave field $\mathcal{W}^{D}$,
respectively. (b) Schematic top view of the directional orbital-chasing motion
of the liquid metal heterodimers and the definition of the regime parameters.
${\lambda{}}_{G}$ and ${\lambda{}}_{D}$ represent the wavelength of
$\mathcal{W}^{G}$ and $\mathcal{W}^{D}$, respectively. (c) The regime diagram
of the orbital-chasing motion. ($\Circle$) The large droplet $\mathcal{D}_{L}$
chases the small droplet $\mathcal{D}_{S}$; ($\Box$) $\mathcal{D}_{S}$ chases
$\mathcal{D}_{L}$; ($\Diamond$) $\mathcal{D}_{S}$ chases $\mathcal{D}_{L}$ but
the heterodimers either cease or derail before a full-circle orbiting is
completed (partial orbiting regimes); ($\triangle$) High-orbit levels in which
the confinement becomes too weak to sustain the heterodimers. Regimes
requiring $\textit{$d_{l}$}>2\textit{$r_{n}$}$ are geometrically invalid and
left blank. (d) Top-view snapshots of the orbiting heterodimers with different
binding levels l (n = 2 fixed). (e) Top-view snapshots of the orbiting
heterodimers in different orbit levels n (l = 1 fixed). The arrows indicate
the direction of orbiting and chasing.

##  II System configuration

The system configuration used in this study is sketched in Fig. 1(a). A
cylindrical glass container (inner radius $\textit{r}_{B}$ = 38 mm) is fixed
on an electromagnetic shaker (not shown) to form the vibration platform [34].
The container is filled with a liquid metal layer of thickness h1 = 10.0
$\pm{}$ 0.5 mm. The liquid metal used here is a eutectic alloy of gallium and
indium (EGaIn, 75.5% Ga and 24.5% In by weight percentage) which has the
lowest melting point of the Ga-In binary system $T_{m}\sim$15 ℃, density
${\rho{}}_{1}$ = 6280 kg m-3 and viscosity ${\nu{}}_{1}=5.13\times{}{10}^{-7}$
m2 s-1. The liquid metal is covered by an alkaline electrolyte top layer (0.5
mol L-1 NaOH aqueous solution, h2 = 10.0 $\pm{}$ 0.5 mm, ${\rho{}}_{2}=1012$
kg m-3, ${\nu{}}_{2}=9.32\times{}{10}^{-7}$ m2 s-1) to prevent the oxidation
of the liquid metal surface so that it behaves as a simple fluid.
Consequently, the droplet bouncing ‘trampoline’ in the current case is a
liquid metal-electrolyte interface with an exceedingly high interfacial
tension $\sigma{}$ (0.456 N m-1 measured with the pendant drop method). Before
each experiment, the platform is checked and adjusted carefully with a level
gauge to minimize off-axis vibration.

During the experiment, the bath is forced vertically with a sinusoidal
acceleration
$\gamma{}\left(t\right)={\gamma{}}_{0}\cos{\left(2\pi{}ft\right)}$, where
${\gamma{}}_{0}$ is the peak driving acceleration, $f$ the driving frequency
(fixed at 40 Hz), and t denotes time. ${\gamma{}}_{0}$ can be continuously
adjusted from zero to well beyond the Faraday threshold ${\gamma{}}_{F}$ =
0.32g, (the liquid metal-electrolyte interface) of the current system with an
accuracy of $\pm{}$0.005g, where g is the gravitational acceleration. Note
that the Faraday threshold ${\gamma{}}_{F}$ may be altered under the influence
of the meniscus waves in our system. In the current study, ${\gamma{}}_{F}$ is
measured as the threshold value when the meniscus wave pattern breaks. In
addition, the Faraday threshold of the upper electrolyte/air interface is
found to be higher than that of the liquid metal/electrolyte interface located
below. Therefore, the Faraday waves of the electrolyte/air interface do not
emerge within the acceleration range in which the droplet motions are
investigated. A plastic pipette is used to generate millimetric liquid metal
droplets which can be long-term sustained and readily manipulated on the bath.
The extrusion volume of the pipette is adjusted to control the size of
droplets and the actual droplet size is measured by post-image analysis. The
uncertainty of droplet size measurement is within 5%. To place the droplets on
the liquid metal-electrolyte interface, the tip of the pipette is inserted
near the interface to avoid any air entrainment. The vertical/horizontal
droplet motion and the evolution of the bath wave field are captured by a
high-speed camera with recording frequency up to 3200 Hz. For the vertical-
view recordings, a LED light source is illuminated from the opposite side of
the bath to the camera, while for the horizontal-view recordings, the light
source is illuminated from above the bath, atop the camera.

##  III Horizontal droplet motion

###  III.1 The orbital-chasing motion and regime diagram

Provided that suitable droplet size and forcing conditions are satisfied, two
dissimilar liquid metal droplets autonomously form a locked pair (heterodimer)
and start revolving collectively yet stably around the bath center, following
the annular rings of the global wave field $\mathcal{W}^{G}$ (Figs. 1b$-$1e
and Video 1). As will be detailed later, the so-called simple bouncing mode is
excited in the current system, where the droplets impact the bath once per
bath vibration [1]. Noticeably, the chasing liquid metal heterodimers take
discrete interdroplet binding distance dl (Fig. 1d) and orbit radius rn(Fig.
1e), where l and n represent the binding level and the orbit level,
respectively. As presented in the regime diagram in Fig. 1c, modulating the
variable set (l, n) produces highly diversified orbital motion regimes of the
heterodimers. See Video 2 of the Supplemental Material for a demonstration of
different regimes.

Among all the regimes, we observe two types of interdroplet binding which lead
to chasing motions in opposite directions. Note that the chasing direction
here is defined by the size of the leading droplet of a heterodimer rather
than the direction of its angular velocity, as the orbiting can always be
changed from clockwise to anticlockwise or vice versa by switching droplet
position. In the short-range binding regime ($l=0$, $\Circle$ in Fig. 1c), the
heterodimers advance with the large droplet ($\mathcal{D}_{L}$) chasing behind
the small droplet ($\mathcal{D}_{S}$), while in the long-range binding regimes
($l\geq{}1$, $\Box$ and $\Diamond$ in Fig. 1c), $\mathcal{D}_{L}$ takes the
lead ($\mathcal{D}_{S}$ becomes the follower) and thus the chasing reverses.
The motion of the long-range binding liquid metal heterodimers is particularly
similar to that of the optical heterodimers [32], but the achievable regimes
in the hydrodynamic system appear to be more diversified than its optical
counterpart. It should be pointed out that two dissimilar droplets placed in
adjacent orbits can also form a special type of long-range binding
heterodimers (mostly realized when l = 1 and 2). However, this kind of cross-
orbit motion will not be detailed in the current study.

We note that the center of the global orbital motion of the liquid metal
heterodimers is located at the bath center rather than along the two-droplet
alinement. The latter is the case for the orbiting walker pairs as
demonstrated in Ref. [1], [6] and [20]. We shall also point out that,
different from the ratcheting droplet pairs [21, 25], the chasing direction of
the heterodimers is exclusively determined by the binding regime l, and it is
not observed to reverse as we progressively sweep ${\gamma{}}_{0}$ from the
bouncing threshold ${\gamma{}}_{B}$ all the way to the Faraday threshold
${\gamma{}}_{F}$. These distinct features indicate the emergence of new
particle-wave association mechanisms in the current system.

###  III.2 The parametrics and quantization of the orbital motion

We combine high-speed imaging and post visual tracking to analyze the
horizontal orbital motion of individual droplets to shed light on their
collective behaviors. The motions of the heterodimers are captured from above
the bath at the same frequency as the driving acceleration ($\textit{f}=40$
Hz). In doing so, no out-of-plane motion of the bath will be seen, although it
is constantly vibrating, thereby improving the tracking accuracy. Using the
method described in Ref. [35] and [36], the typical error for horizontal
motion tracking can be kept less than $1/10$ of the characteristic droplet
diameter. Examples of the horizontal droplet motion tracking are presented in
Video 3 of the Supplemental Material. The orbiting trajectories of individual
droplets in the $x-y$ plane in different orbit levels n (l = l fixed) are
plotted against t in Fig. 2(a), which shows well-defined trajectories and
long-term stability of the orbital motions. Further analyses reveal a
definitive feature of the collective droplet motions in the current system
that the orbital motion is accompanied by cyclically repeated instantaneous
variations. For instance, calculating the horizontal orbiting velocity of the
droplets shows that, while the magnitude of their cycle-averaged velocity
$\bar{v}$ remains constant, that of their instant velocity $v$ changes
constantly yet periodically (Fig. 2b). Interestingly, when comparing Fig. 2(a)
and Fig. 2(b), it is evident that $v$ changes precisely at the same frequency
(i.e., the same number of cycles) as the orbiting motion and even the
fluctuations are well repeated for all cases. Also, we note that there exist
noticeable differences in the velocity as well as that the velocity changes
between $\mathcal{D}_{S}$ and $\mathcal{D}_{L}$.

Figure 2:  (a) The orbiting trajectories of the large droplet
$\mathcal{D}_{L}\ $(red) and the small droplet $\mathcal{D}_{S}$ (blue)
featuring different orbit levels n (l = 1 fixed). (b) The magnitudes of the
instant velocity $v$ and the average velocity $\bar{v}$ of the heterodimers in
(a) as a function of time. The errors in the measurement of v and $\bar{v}$
are $\textless$ 0.2 mm s-1 and $\textless$ 0.04 mm s-1, respectively.

As shown in Fig. 3(a), time-dependent $v$ directly results in instantaneous
variations of $d_{l}$, even when the two droplets are close to each other
($\textit{l}=0$, small binding distance). Again, two locked droplets always
experience changes in $d_{l}$, but the cycle-averaged value ${\bar{d}}_{l}$
remains constant. It is confirmed that $d_{l}$ also changes at the orbiting
frequency of each heterodimer. In addition, we find that ${\bar{d}}_{l}$ is
exclusively determined by the binding level l and it is independent of both
the orbit level n and the driving acceleration ${\gamma{}}_{0}$ (Fig. 3b).
This evidence suggests that the orbital-chasing motion of the heterodimers in
the current system is affected locally in the orbits. As will be discussed in
the next section, the perturbations the heterodimers experience during their
orbital motion indeed originate from a peculiar vortex structure formed on the
vibrating liquid metal bath. Nevertheless, as shown in Fig. 3(c), the discrete
values of ${\bar{d}}_{l}$ and $r_{n}$ both fall onto a linear curve, leading
to the quantization of the orbital chasing motions:

$\displaystyle\bar{d}_{l}$
$\displaystyle=\left(l-\varepsilon_{l}\right)k_{l},$ $\displaystyle l$
$\displaystyle=1,2,3,\ldots$ (1) $\displaystyle r_{n}$
$\displaystyle=\left(n-\varepsilon_{n}\right)k_{n},$ $\displaystyle
n=1,2,3,\ldots$ (2)

One exception is the short-range binding regime $l=0$ where
${\bar{d}}_{0}\sim\left(d_{L}+d_{S}\right)/2$ with $d_{L}$ and $d_{S}$ being
the diameters of $\mathcal{D}_{L}$ and $\mathcal{D}_{S}$, respectively. The
physical implications of the slopes ($k_{l}=6.6$ mm, $k_{n}=6.5$ mm) and
offsets (${\epsilon{}}_{l}=0.2$, ${\epsilon{}}_{n}=0.4$) of the fitting curves
of Eq.(1) and Eq.(2) can be related to the wave parameters of different pilot-
wave fields, a point that will be addressed later.

Figure 3:  (a) Binding distance dl variations of the heterodimers at different
binding levels l (n = 2 fixed). (b) Cycle-averaged binding distance
${\bar{d}}_{l}$ in different orbiting regimes as a function of the normalized
driving acceleration ${\gamma{}}_{0}$/g. (c) Quantization of the orbital
motion of the liquid metal heterodimers. In all figures, the solid lines and
scatters represent experimental data and the dashed lines are their linear
fittings.

##  IV Bath dynamics

###  IV.1 The profile of the bath wave field

One important observation from the orbital motions of the liquid metal
heterodimers is that their trajectories are always confined in the concentric
rings (orbits) formed on the bath (Figs. 1 and 2). Such circular confinements
are provided by a global annular wave field $\mathcal{W}^{G}$, the formation
of which is revealed at the bath vibrating onset. As shown in Fig. 4(a), the
bath side views (slightly oblique to show the wave field) captured at its
vibration onset show unambiguously that the annular ring patterns are formed
by the traveling wave fronts emitted from the bath boarder (see Video 4 of the
Supplemental Material). When a single droplet is deposited on the vibrating
bath, the local wave field $\mathcal{W}^{D}$ generated by the droplet impact
superposes to $\mathcal{W}^{G}$, resulting in the formation of the dual pilot-
wave field (Fig. 4b). We note, however, that no orbital motion is observed in
the single droplet scenarios (see Video 5 of the Supplemental Material).

The wave fronts constructing $\mathcal{W}^{G}$ are indeed generated by a
significant meniscus of the liquid metal bath oscillating at the driving
frequency f (harmonic)[4]. A quantitative assessment of the scale of the
meniscus-influenced region as well as the surface profile of both the
quiescent bath and the vibrating bath is further performed using a customized
micro positioning stage. A sharp-tip stainless-steel probe, which will not be
wetted by the liquid metal, is used to detect its surface (liquid metal-
electrolyte interface). Since the liquid metal surface is highly reflective,
small distortions on its surface upon contact become readily distinguishable,
which is recognized as the detecting of the surface during the measurement.
With the assistance of a micro positioning stage, both the vertical (z) and
radial (r) positions of the probe tip are determined with micrometer
resolution. The bath vibrating amplitude $\Delta{}z$ in Fig. 4(c) is obtained
by subtracting the measured bath vertical profile under vibration by its
quiescent profile (Fig. 4d). Note that the current method only picks up the
vibrating maxima of the bath surface and therefore the profile curves in Fig.
4(c) represent the vibrating amplitudes (maximum displacement in the z
direction) along the radial direction over a whole vibrating period rather
than the instantaneous profiles of the bath. The measured vibrating amplitude
indicates the total energy of the waves since on the curves the kinetic energy
is fully converted into potential energy.

Figure 4:  (a) Time-lapse side view snapshots (slightly oblique) of the liquid
metal bath at the vibrating onset. From top to bottom, t = 0.00 ms, 22.23 ms,
41.67 ms, and 86.12 ms. (b) Side view snapshots of a single droplet bouncing
on the vibrating bath (with no horizontal motion). The superposition of the
droplet-induced local wave field $\mathcal{W}^{D}$ and the global wave field
$\mathcal{W}^{G}$ can be seen from the bottom panel. (c) Bath vibrating
amplitude ${\Delta{}}z$ measured along the radial direction r at different
driving accelerations as indicated. (d) Surface profile of the quiescent
liquid metal surface. (e), (f) Top-view snapshots featuring the maximum bath
deformation at the bath center due to meniscus oscillating (e) and the maximum
localized bath deformation induced by droplet impact (f). The maximum size of
the dark region (deformed liquid metal surface) at the bath center (e) and the
droplet impact site (f) is indicated. ${\gamma{}_{0}}$ = 0.22g.

As shown in Fig. 4(d), a giant ($\sim$10 mm) meniscus-influenced region can be
seen from the bath profile. The formation of such an anomalous meniscus can be
explained by Zisman’s rule [30]: The high-energy liquid metal surface in
contact with the relatively low-energy surface of the glass container adopts a
nonwetting regime, a configuration resulting in a large contact angle (large
meniscus), despite that the capillary length of the liquid metal
$l_{C}=\sqrt{\sigma{}/({\rho{}}_{1}-{\rho{}}_{2})\textsl{{g}}}\sim 3.0$ mm is
comparable to that of common fluids (e.g., water $l_{C}\sim 2.7$ mm). The fact
that the meniscus which introduces the global wave field is governed by the
relative surface energy of the contacting phases implies that the realization
of an integrated pilot-wave field is not limited to the liquid metal droplet-
bath system demonstrated here. Systems using common working fluids can also
reproduce similar effects by choosing non-wetting (super-hydrophobic or
oleophobic) materials, or by reducing the size scale (equivalently, the Bond
number), for instance, using sessile drop configuration, to amplify
capillarity and generate a significant boundary curvature.

Emitted by the oscillating meniscus, the amplitude of the propagating wave
fronts is affected by two competing processes. The wave amplitude is damped by
viscous dissipation while amplified due to the shrinking geometry (diameter)
of the waveform when propagating towards the bath center. As shown in Fig.
3(c), the measured surface profiles increase radially towards the bath center
in a near-periodic fashion, indicating a stronger influence of the latter.
This is guaranteed by the insignificant influence of viscous dissipation
(relative to inertial and surface energy) in the current system, as can be
expected from its marginal Ohnesorge number
$Oh={\nu{}}_{1}\sqrt{({\rho{}}_{1}-{\rho{}}_{2})/(\sigma{}{\lambda{}}_{G})}\sim
7\times{}{10}^{-4}$, where ${\lambda{}}_{G}=6.4\pm{}0.3$ mm is the measured
wavelength of $\mathcal{W}^{G}$. We note that ${\lambda{}}_{G}$ can also be
deduced from the dispersion relation of surface waves interfacing two finite
fluid layers [37, 34]

$\displaystyle\omega{}^{2}$
$\displaystyle=\frac{\left(\rho{}_{1}-\rho{}_{2}\right)gk+\sigma
k^{3}}{\rho{}_{1}\coth{\left(kh_{1}\right)}+\rho{}_{2}\coth{\left(kh_{2}\right)}}$
(3)

where $\omega=2\pi\textit{f}$ is the angular velocity and
$\textit{k}=2\pi/{\lambda{}_{G}}$ is the wavenumber. Solving Eq. (3) for the
current system gives k = 0.96 mm-1 and therefore $\lambda{}_{G}$ = 6.5 mm,
which matches well with the measured value. As discussed earlier, the surface
profile measurement picks up only the maxima of the bath vibrating amplitude,
so the periodic surface profile indicates a standing wave nature of
$\mathcal{W}^{G}$. This argument can be understood given the superposition of
the inward-propagating wave fronts by the outward-propagating ones reflected
at the bath center. The traveling-to-standing-wave transition can also be
inferred from the vibrating onset process of the bath (Video 4 of the
Supplemental Material), which shows that the transition is completed after the
first few bath vibrations.

We proceed to discuss the vibration modes of the liquid metal bath in a
cylindrical coordinate (r, $\phi$), where $\phi$ is the azimuthal angle.
Considering a circular bath under vertical vibration and excluding the
influence of the dynamic meniscus, one would expect its instantaneous surface
profile to be modulated by

$\displaystyle\Delta z(r,\phi,t)$ $\displaystyle=A_{0}\cos\left(\omega
t\right)\cos\left(i\phi\right)J_{i}\left(2\pi r_{i,j}/\lambda{}_{G}\right)$
(4)

where A0 is the vibration amplitude at r = 0, $\textit{J}_{i}$(x) is the order
i Bessel function of the first kind, i and j are integers indicating the
vibration modes, and $\textit{r}_{i,j}$ = $\mu^{\text{i}}_{\text{j}}$/$r_{B}$
with $\mu^{\text{i}}_{\text{j}}$ being the root of the equation
$J_{\text{0}}^{\prime}$(x) = 0 [38]. The formation of the axisymmetric global
wave field $\mathcal{W}^{G}$ (concentric rings) requires the vibration to be
independent of $\phi$, which leads to i = 0. Eq. (4) then becomes

$\displaystyle\Delta z(r,t)$ $\displaystyle=A_{0}\cos\left(\omega
t\right)J_{0}\left(2\pi r_{0,j}/\lambda{}_{G}\right)$ (5)

Eq. (5) suggests that the surface profile is a zero order Bessel function
$\textit{J}_{0}$(2$\pi$r0,j/$\lambda{}_{G}$). This axial excitation mode
(independent of $\phi$) is the predominant mode in our system according to our
experimental observation. Eq. (5) also means that the actual orbit radius
$\textit{r}_{n}$ should satisfy
$\textit{J}_{0}$(2$\pi$$\textit{r}_{n}$/$\lambda{}_{G}$) = 0. For the sake of
simplicity, we use a linear correlation to approximate $\textit{r}_{n}$, which
should still provide sufficient accuracy to describe the current system (Fig.
2c).

The formation of the global wave field $\mathcal{W}^{G}$ provides an
additional confinement which locks the horizontal motion of the liquid metal
heterodimers into its circular orbits. This implies that the droplets acquire
the centripetal force needed for their orbital motion from their impact with
$\mathcal{W}^{G}$ and the impact should therefore be made at an inclined
inward-facing wave front. The increasing vibrating magnitude of
$\mathcal{W}^{G}$ towards the bath center (Fig. 4c and Eq. 5) implies
different effective vibrating accelerations among different orbits, which can
be recognized as a level splitting effect [23]. Due to such level splitting,
when the bouncing droplets are located in different orbits, both their
vertical and horizontal motions are expected to be altered when traveling in
different orbits even under the same driving acceleration.

###  IV.2 The integrated pilot-wave field

Before accessing droplet motion, it is worthwhile to characterize the
vibration amplitude of the two components of the integrated pilot-wave field,
$\mathcal{W}^{G}$ and $\mathcal{W}^{D}$. With the radial profile of
$\mathcal{W}^{G}$ known, we roughly estimate the relative magnitude of the two
wave fields by comparing their maximum deformation. Since the liquid metal
surface is smooth yet highly reflective, the sufficiently deformed regions on
the bath (including the meniscus region at the bath border, the deformed bath
center and the droplet impact site) deviate illuminating light away from the
recording camera on top, rendering these regions low brightness (Figs. 4e, 4f,
and 5a). The size of the dark regions outlined at the bath center and the
droplet impact site gives an indication of the vibration amplitude of
$\mathcal{W}^{G}$ and $\mathcal{W}^{D}$, respectively (note that the scale of
the meniscus-influenced region also can be estimated by measuring the width of
the dark annular band surrounding the bright region). Provided that both
$\mathcal{W}^{G}$ and $\mathcal{W}^{D}$ follow the similar profile of the
zero-order Bessel function [25], the deformed (dark) region revealed from the
top views can be directly related to the vibration amplitude of the two wave
fields. Given the small deformation (on the scale of $\sim$ 0.1 mm) in the
current case, a linear approximation can be assumed, which suggests that the
magnitude of $\mathcal{W}^{D}$ is of the same order of magnitude but
significantly smaller (about half) in reference to that of $\mathcal{W}^{G}$.

###  IV.3 The bath vortex field

Using a digital particle-imaging-velocimetry (PIV) technique, we further
demonstrate the formation of a stable counter vortex pair at the liquid metal-
electrolyte interface (Fig. 5 and Video 6 of the Supplemental Material). The
flow at the interface is visualized by adding boron tracer particles (average
size 50 $\mu{}$m) to the bath. Boron particles are selected due to their good
material compatibility with the liquid metal and the alkaline solution, as
well as their desirable density to settle at the interface. The vortex motion
is again captured at the driving frequency (40 Hz) to filter out bath motion
(Fig. 5b). In our experiments, rotating the container to different azimuthal
positions can change the orientation of the vortex pair and increasing the
driving acceleration also leads to the increase of the vortex current
magnitude. However, the vortex pair is always found to exist even though care
is taken to level the bath. Based on our observations, we propose that the
vortex pair emerges as a result of the azimuthal excitation of the system.
Apart from driving the radial excitation mode, which is responsible for the
formation of $\mathcal{W}^{G}$ and independent of the azimuth angle $\phi$
(Eq. 5), periodic forcing can also give rise to $\phi$$-$dependent azimuthal
excitation, when higher order modes, other than the zero-order mode, are
excited (Eq. 4). When i $\geq$ 0, the term $\cos$(i$\phi$) in Eq. (4) breaks
the axisymmetric vibration and the vibration begins to show $\phi$ dependence.
Since periodic displacement with a gradient along the vibration direction has
been shown to induce transverse flows [39, 40], the azimuthal excitation
should be primarily responsible for the formation of the vortex field in the
current system. The structure of the visualized vortex field suggests that the
vortex pair is formed through the
$\textit{J}_{1}$(2$\pi$r1,j/$\lambda{}_{G}$)) mode excitation.

Being aware of the existence of the counter vortex pair, the instantaneous yet
periodic variations of $v$ and $d_{l}$ in Fig. 2(b) and Fig. 3(a) can be
rationalized. Regardless of their binding regimes, the two droplets of the
heterodimers are accelerated by one of the vortices, with which they share the
same horizontal rotating direction during half of the orbiting cycle, and are
decelerated by the counter vortex during the other half cycle. Due to the
mirror symmetry of the vortex pair, the two competing processes cancel each
other out after each cycle, rendering $\bar{v}$ and ${\bar{d}}_{l}$ constant.
Therefore, although the orbits are geometrically symmetric, the orbital motion
of the heterodimers is disturbed locally by the vortex current. It thus can be
seen that the level splitting of the orbits and the formation of vortex pair
in the current system alter both the interorbit and in-orbit energy landscapes
of $\mathcal{W}^{G}$.

Figure 5:  (a) Top view of the vibrating liquid metal bath showing the global
annular wave field $\mathcal{W}^{G}$. (b) Visualized vortex field at the
liquid metal-alkaline interface and (c) the corresponding vorticity color map
(${\lambda{}}_{0}=0.16$g).

##  V Vertical droplet motion

###  V.1 The shifted vertical bouncing

While orbiting horizontally on the bath, the droplets are at the same time
bouncing vertically in resonance with the bath vibration (Figs. 6a, 6b), both
generating localized harmonic waves. Since a single droplet will never start
orbiting until it meets another dissimilar partner, the interactions between
individual droplets ($\mathcal{D}_{L}$ and $\mathcal{D}_{S}$) and their self-
generated local wave fields ($\mathcal{W}_{L}^{D}$ and $\mathcal{W}_{S}^{D}$),
hereafter denoted as $\mathcal{D}_{L}\leftrightarrow{}\mathcal{W}_{L}^{D}$ and
$\mathcal{D}_{S}\leftrightarrow{}\mathcal{W}_{S}^{D}$, respectively, should
not be responsible for their horizontal motions, nor should their interactions
with the meniscus-induced global wave field
(${\mathcal{D}_{L}\leftrightarrow{}\mathcal{W}}^{G}$ and
${\mathcal{D}_{S}\leftrightarrow{}\mathcal{W}}^{G}$). The two-droplet
configuration as a prerequisite for their orbital-chasing motion implies that
the association between the droplets and the wave fields generated by their
partners ($\mathcal{D}_{L}\leftrightarrow{}\mathcal{W}_{S}^{D}$ and
$\mathcal{D}_{S}\leftrightarrow{}\mathcal{W}_{L}^{D}$) should provide the
horizontal propulsion. Their collective motions further require the horizontal
propulsion acting on the two droplets to be in the same direction.

To elucidate the origin of the directional horizontal propulsion, we compare
the vertical motion of individual droplets of the heterodimers in different
regimes. For vertical droplet motion tracking, the motions are recorded at
1600 Hz to generate 40 data points for each bouncing cycle. The typical
spatiotemporal droplet bouncing trajectories for the short-range binding
regime (l = 0, Fig. 6c) and the long-range binding regime (l = 1, Fig. 6d) are
reconstructed from the high-speed images [1]. The droplet trajectory curves
are also generated by the aforementioned visual tracking method (Figs. 6e, 6f,
see also Video 7 of the Supplemental Material), which show an excellent match
with Figs. 6c and 6d, again indicating good tracking accuracy. Comparing the
vertical motion of the two droplets in the heterodimers reveals that, while
their vertical bouncing is always synchronized with the bath vibration (i.e.,
in the simple bouncing regime) [1], their bouncing phase is however different.
Independent of their binding level and orbit level, the impact of the large
droplet $\mathcal{D}_{L}$ is always found to be delayed by a time amount of
$\Delta{}t$ to that of the small $\mathcal{D}_{S}$, resulting in a normalized
bouncing phase shift $\theta{}/2\pi{}=\Delta{}t/T$ between the two [41, 26].

Figure 6: (a), (b) Oblique side views of the orbiting heterodimers in the
short-range binding regime l = 0 (a) and the long-range binding regime l = 1
(b). (c), (d) Vertical bouncing trajectories of the small droplet
$\mathcal{D}_{S}$ (upper panel) and the large droplet $\mathcal{D}_{L}$ (lower
panel) as a function of time for five consecutive bouncing periods T of a (0,
3) heterodimer (c) and a (1, 3) heterodimer (d). (e) The spatial-temporal
trajectories of the same heterodimer in (e) obtained by visual tracking. (f)
The spatial-temporal trajectories of the same heterodimer in (d) obtained by
visual tracking. The impact time difference $\Delta{}t$ of the two droplets of
the heterodimers are indicated.

###  V.2 The inter-droplet particle-wave association

Being aware of the size-dependent bouncing phase shift, we can rationalize the
horizontal propulsion as well as the directionality of the orbital-chasing
motions by considering the interdroplet particle-wave association of each
heterodimer ($\mathcal{D}_{L}\leftrightarrow{}\mathcal{W}_{S}^{D}$ and
$\mathcal{D}_{S}\leftrightarrow{}\mathcal{W}_{L}^{D}$). Under the assumptions
that the in-orbit chasing motion is pseudo one dimensional, the droplets and
their self-generated waves share the same phase [25], and the Doppler effect
is negligible ($v\ll{}{\lambda{}}_{D}f$), the problem is simplified to the
interactions between individual droplets and the waves emitted by their
partners with a phase shift $\theta{}$ from a distance $d_{l}$. If we allow
for the shifted evolution of $\mathcal{W}_{L}^{D}$ and $\mathcal{W}_{S}^{D}$
as presented in Fig. 7(a), the particle-wave association
$\mathcal{D}_{L}\leftrightarrow{}\mathcal{W}_{S}^{D}$ and
$\mathcal{D}_{S}\leftrightarrow{}\mathcal{W}_{L}^{D}$ of the heterodimers in
the l = 0 and l = 1 binding regimes are depicted in Fig. 7(b) and Fig. 7(c),
respectively. Here $z=0$ represents the zero position of the droplet-generated
local wave field $\mathcal{W}^{D}$.

For both regimes, by the time $\mathcal{D}_{L}$ lands on the bath,
$\mathcal{W}_{S}^{D}$ has already evolved for a period $\Delta{}t$ due to the
bouncing phase shift. Therefore, the impact site of $\mathcal{D}_{L}$ has been
distorted by $\mathcal{W}_{S}^{D}$ upon landing. As shown in Fig. 7(b), when l
= 0, since ${\bar{d}}_{0}=\left(d_{L}+d_{S}\right)/2<{\lambda{}}_{D}/3$ in the
current system (${\lambda{}}_{D}=6.5\pm{}0.2$ mm is the measured wavelength of
$\mathcal{W}^{D}$ which is independent of droplet size), $\mathcal{D}_{L}$
always lands on the back side of $\mathcal{W}_{S}^{D}$ (relative to the
direction of wave propagation) and experiences a propulsion p towards upper
right (i). The interaction between $\mathcal{D}_{S}$ and $\mathcal{W}_{L}^{D}$
takes place at $t=T$, by then $\mathcal{W}_{L}^{D}$ has evolved for a time
interval $T-\Delta{}t$. Therefore, the impact of $\mathcal{D}_{S}$ is made on
the front side of $\mathcal{W}_{L}^{D}$, while $\mathcal{W}_{L}^{D}$ is
returning to its zero position. Consequently, $\mathcal{D}_{S}$ is propelled
towards upper right, the same direction as $\mathcal{D}_{L}$ (ii).

For l = 1, due to the change of the interdroplet binding to the
${\bar{d}}_{l}=\left(l-{\epsilon{}}_{l}\right){\lambda{}}_{D}\sim{}0.8{\lambda{}}_{D}$
regime, the delay of the bouncing phase of $\mathcal{D}_{L}$ results in the
impact of $\mathcal{D}_{L}$ being made at the front side of
$\mathcal{W}_{S}^{D}$ (iii) and that of $\mathcal{D}_{S}$ at the back side of
$\mathcal{W}_{L}^{D}$ (iv), respectively. Under the long-range binding
regimes, both droplets feel a propulsion towards the upper left (Fig. 7c). It
thus can be seen that the propulsions resulted from the interdroplet particle-
wave association always propel the two droplets of the heterodimers to move
towards the same horizontal direction. Moreover, the direction of the
propulsion will be switched between the short-range binding regime and the
long-range binding regimes due to the spatial change between the droplets.

Figure 7:  (a) Schematics of the temporal phase shift between
$\mathcal{W}^{D}$ induced by the impact of $\mathcal{D}_{S}$ (black) and
$\mathcal{D}_{L}$ (red). (b), (c) Directional propulsion resulted from the
interaction between individual droplets and the wave field generated by their
partners in the l = 0 regime (b) and the l = 1 regime (c). The characteristic
time of impact for cases i-iv is indicated in (a). The arrows indicate the
direction of the propulsion that the local wave field exerts on its partner
droplet during impact. Here (b) and (c) are in the one-dimensional spatial
reference frame along the orbit.

##  VI Linking horizontal orbiting and vertical bouncing

The quantization of the orbital motions now can be written as:

$\bar{d}_{l}=\left\\{\begin{array}[]{ll}{\left(d_{L}+d_{S}\right)/2,}&{l=0}\\\
{\left(l-\varepsilon_{l}\right)\lambda_{D},}&{l=1,2,3,\ldots}\end{array}\right.$
(6)

$r_{n}=\left(n-\varepsilon_{n}\right)\lambda_{G},\quad n=1,2,3,\ldots$ (7)

As discussed earlier, in Eq. (7) a simplified linear correlation is used to
approximate $\textit{J}_{0}$(2$\pi$$\textit{r}_{n}$/$\lambda{}_{G}$) = 0. Eq.
(6) and Eq. (7) give clear indication that the quantization of the binding
level and the orbit level results from the interaction of the bouncing
droplets with two wave fields of different origins, which is a signature of
the integrated pilot-wave droplet-bath system. The offsets
${\epsilon{}}_{l}=0.2$ and ${\epsilon{}}_{n}=0.4$ are thought to result from
the self-tuning by the heterodimers in order to stabilize the two types of
particle-wave associations [1, 22, 24]. For instance, the association between
the droplets and the wave emitted by their partners should self-adjust to
maintain their collective motion when counteracting the locally changing
current of the vortex pair along the orbits, which manifests as the observed
instantaneous velocity and binding distance changes in Fig. 2(b) and Fig.
3(a). Moreover, the association between the heterodimers with the global wave
field should also self-adapt to orbit radii that ensure dynamically stable
correlation between their orbiting velocity and centripetal force in a given
orbit level. It is of course reasonable to expect mutual influences between
these self-tuning behaviors. Consequently, although the integrated pilot-wave
field has a complex energy landscape, the heterodimers are able to maintain
their well-defined yet dynamically-stable collective motions.

We proceed to investigate the relation between $\theta{}/2\pi{}$ and $\bar{v}$
as a function of the driving acceleration ${\gamma{}}_{0}$. Four most stable
orbiting regimes (l, n), namely the (0, 2), (0, 3), (1, 2) and (1, 3) regimes
are selected and compared. The driving acceleration is increased progressively
from the bouncing threshold ${\gamma{}}_{B}\sim{}0.10$g to the Faraday
threshold ${\gamma{}}_{F}\sim{}0.32$g, with a step size of 0.02g. To obtain
$\theta{}$ and $\bar{v}$, the trajectories of the vertical bouncing motion and
the horizontal orbiting motion of the heterodimers under each condition are
captured by high-speed imaging and the images are analyzed using visual
tracking. Nearly 500 heterodimers (1000 droplets) in total are analyzed, and
the size distribution of the droplets is presented in Fig. 8(a) and Fig. 8(b).
Due to the size filtering effect of the vibrating bath based on its effective
acceleration [42], which in the current case is determined by both
${\gamma{}}_{0}$ and n (level splitting), the droplet diameter d in our
experiment is restricted to a narrow range from $\sim$1.0 mm to $\sim$1.6 mm.
In addition, the mutual influence between the two droplets in the heterodimers
further imposes restrictions on the droplet size selection. As a result, the
size difference between $\mathcal{D}_{L}$ and $\mathcal{D}_{S}$ is found to be
small, with the average of their diameter ratio $d_{S}/d_{L}$ close to 0.9
(inset of Fig. 8b).

Figure 8: (a) A survey of droplet size distribution. (b) Relative droplet size
and its acceleration dependence (inset).

The delay of the impact of the large droplet to that of the small is found to
hold for all the surveyed heterodimers and all the droplets are in the simple
bouncing regime, even at high accelerations. As shown in Fig. 9(a) and Fig.
9(b), correlations are found between the maxima of $\theta{}/2\pi{}$ and
${\gamma{}}_{0}$/g, and the two types of binding regimes ($l=0$ and $l=1$)
lead to different acceleration dependence. In the short-range binding regime
($l=0$), the phase shift maxima increase almost linearly with
${\gamma{}}_{0}$/g (Fig. 9a), while for the long-range binding regimes
($l=1$), the maxima show a parabola-like acceleration dependence (Fig. 9b).
Note that multiple phase differences are possible for a given acceleration in
each regime as can be seen from Fig. 9(a) and Fig. 9(b). This is a result of
different droplet size of the heterodimers. The pentagrams in Fig. 9(b)
further demonstrate the gradual decrease of $\theta{}/2\pi{}$ from about 0.1
at ${\gamma{}}_{0}$/g = 0.24 to near zero at ${\gamma{}}_{0}$/g = 0.30 for a
(1, 3) regime heterodimer that we are able to maintain for longer than 1 h.
When $\theta{}/2\pi{}$ becomes near zero, the heterodimer stops its orbital
motion.

The influence of the global wave field $\mathcal{W}^{G}$ on $\theta{}/2\pi{}$
and $\bar{v}$ can be seen from the heterodimers in the same binding level l
but different orbit level n, since the effective acceleration of
$\mathcal{W}^{G}$ decreases as n increases due to level splitting (Fig. 4c).
Comparing the phase shift of the (0, 2) regime with the (0, 3) regime (Fig.
9a) or the (1, 2) regime with the (1, 3) regime (Fig. 9b), one finds that for
the same binding regime, moving from one orbit level to another will not
change the acceleration dependence of the phase shift. The main influence is a
downshift of its maxima at higher orbit level n. This is in good agreement
with the energy landscape of the orbits of $\mathcal{W}^{G}$ given level
splitting. As shown in Fig. 9(c) and Fig. 9(d), the surveyed average orbiting
velocity $\bar{v}$ in different binding regimes shares very similar
acceleration dependence with $\theta{}/2\pi{}$: The maxima of $\bar{v}$ also
show a near-linear and a parabola-like acceleration dependence for the $l=0$
and $l=1$ regimes, respectively. Such similar acceleration dependence between
$\theta{}/2\pi{}$ and $\bar{v}$ implies that the vertical bouncing of the
heterodimers and their horizontal orbiting motion is directly related, which
reflects the fact that the horizontal motion originates from the vertical
bouncing of the droplets. The different particle-wave association regimes
between the two types of inter-droplet binding regimes should be responsible
for their different acceleration dependence.

Figure 9:  (a), (b) Acceleration dependence of $\theta{}/2\pi{}$ for the (0,
2), (0, 3) regimes (a) and the (1, 2), (1, 3) regimes (b). The error in the
measurement of $\theta{}/2\pi{}$ is 0.02. The pentagrams show the acceleration
dependent phase shift of the same heterodimer. (c), (d) Acceleration
dependence of $\bar{v}$ for the (0, 2), (0, 3) regimes (c), and the (1, 2),
(1, 3) regimes (d). Error in $\bar{v}$ measurement $\textless$ 0.04 mm s-1.
The scatters represent the measured values, based on which the filled regions
are drawn as guides for the eyes. The inset of (d) depicts the geometrical
constraints which shows how the tangent projection of the droplet momentum is
influenced by the binding distance $\textit{d}_{l}$ and orbit radius
$\textit{r}_{n}$.

Noticeably, similar influence of n on the $\bar{v}$ maxima is also found to
hold for the $l=0$ regimes but breaks down for the $l=1$ regimes, that is,
instead of showing an overall downshift of $\bar{v}$ maxima in response to
their acceleration dependence in Fig. 9(b), the heterodimers in the (1, 3)
regime demonstrate higher velocity than the (1, 2) regime under certain
driving conditions (Fig. 9d). Such scenarios observed in the long-range
binding regimes are not violations of our discussions and the observation can
indeed be understood by further considering the enhanced spatial influences on
droplet motion in the long-range binding regimes. In the orbiting plane, the
direction of p’ (the horizontal component of p resulting from interdroplet
particle-wave association) is in the two-droplet-alignment, which is not the
same as v (along the tangent of the orbit, see inset of Fig. 9d). This
requires the momentum of the droplet to be projected partially to the tangent
direction and partially to the normal direction of the orbiting plane (the
$x-y$ plane). The normal projection is compensated for during the droplet-bath
impact, which results in the centripetal force $\textit{{p}}_{\perp{}}$ needed
for maintaining the orbital motion. Given the superposition of waves, we
consider the influence of the global wave field $\mathcal{W}^{G}$ and the
local wave field $\mathcal{W}^{D}$ separately. We also assume that both
$\mathcal{W}^{G}$ and $\mathcal{W}^{D}$ induce similar
$\bar{v}-{\gamma{}}_{0}$ correlations for both the short-range binding regime
and the long-range binding regimes, through modulating the effective
acceleration only (i.e., when the geometrical configurations are not
considered). The geometrical configurations determine how the heterodimers
distribute the tangent and normal projections, thereby affecting their
orbiting velocity. The tangent projection of the velocity resulted from
interdroplet particle-wave association scales as
$v=v_{0}$cos${\varphi{}}_{\left(l,n\right)}$, where
${\varphi{}}_{\left(l,n\right)}$ is the angle between p’ and v. The
geometrical constrains of the horizontal motion lead to

$\varphi_{(l,n)}=\arcsin\left(d_{l}/2r_{n}\right)$ (8)

Following Eq.(8), besides the effective acceleration which determines tangent
velocity $v_{0}$, the tangent projection is also affected by both quantized
regime parameters $d_{l}$ and $r_{n}$. Moreover, the two have opposite
influences. The tangent projection is suppressed (increase in
${\varphi{}}_{\left(l,n\right)}$) by increasing the binding distance $d_{l}$
but favored by increasing the orbit radius $r_{n}$. As can be deduced based on
Eqs.(6)$-$(8), the change between ${\varphi{}}_{\left(0,2\right)}\sim
7.5^{\circ}{}$ and ${\varphi{}}_{\left(0,3\right)}\sim 4.5^{\circ}{}$ is small
when the orbit level moves from n = 2 to n = 3 in the short-range binding
regimes. This is because in this case $d_{0}$ is small and $v$ is mainly
determined by the n$-$dependent effective acceleration. Therefore, the
achievable velocity decreases as the orbit level moves from n = 2 to n = 3,
due to a drop in the effective acceleration (Fig. 9c).

As $d_{l}$ significantly increases in the long-range binding regimes (Fig.
9d), the same orbit level change causes significant drop in
${\varphi{}}_{\left(l,\ n\right)}$, from ${\varphi{}}_{\left(1,\ 2\right)}\sim
30.5^{\circ}{}$ to ${\varphi{}}_{\left(1,3\right)}\sim 18.2^{\circ}{}$. In
this case, the geometrical influence is magnified due to increased $d_{l}$,
which becomes comparable to that of the effective acceleration. Since high
orbit level n favors the tangent projection, particularly for large $d_{l}$,
the (1, 3) regime is able to achieve a higher velocity than the (1, 2) regime
under certain driving conditions, which explains the different orbit-level
dependence between Fig. 9(b) and Fig. 9(d). It can also be inferred from
Eq.(8) that a more significant tangent velocity projection should be one of
the reasons for the short-range binding regime to have a higher velocity than
the long-range binding regimes. Another reason should be their more energetic
particle-wave association (less damping). Both of them are caused by the
change in the inter-droplet binding distance $d_{l}$. Similarly, the analyses
can also be extended to higher-level long-range binding regimes
($l=2,3,4,...$) by considering more significant influence of their regime
parameters on both ${\varphi{}}_{(\textit{l},\textit{n})}$ and damping.

##  VII Discussions

We have demonstrated the quantized orbital motion of dissimilar droplet pairs
in the liquid metal droplet-bath system where the droplet motions are directed
by a dual pilot-wave field. The quantization of the orbit radius and the
binding distance is shown to be a result of the droplets interacting with the
waves formed by meniscus oscillating and the impact of their partner droplets,
respectively. The origin of the horizontal motion lies in a temporal bouncing
phase shift between the two droplets of the chasing heterodimers due to their
size mismatch. As shown by Moláček and Bush [26] and Couchman  _et al._ [43],
this is required by the vertical dynamics for the dissimilar droplets with
different contact time and restitution coefficient to bounce in resonance with
the vibrating bath. Our experiments evidence that the result of the mismatched
bouncing is a delay of the impact of the large droplet to that of the small
one, and, more importantly, the breaking of bouncing symmetry is a mechanism
for initiating directional horizontal droplet motion. As discussed by Eddi
_et al._ [21], Galeano-Rios  _et al._ [25], from the perspective of symmetry
breakdown, the liquid metal heterodimers share similarity with the ratcheting
droplet pairs. However, there are also major differences between the two types
of motions in terms of directionality, reversibility, and binding regime. Note
that in the current study we limit the vertical phase comparison to the
bouncing phase shift between the two droplets in the heterodimers. In order to
reach a better (more comprehensive) understanding of phase evolution in the
system, it is worthwhile for future studies to further include the vibrating
phase of the shaker (driving phase) as well as the bath wave field.

The interaction of the heterodimers with the two pilot-wave fields should both
be considered to understand the observed quantized, directional, in-orbit
chasing motions. The interaction between the droplets and the local wave field
generated by their partners determines the chasing direction, through
different particle-wave association regimes characterized by the interdroplet
spatial binding. Different from the binding of walkers, the quantization of
interdroplet binding distance in our experiment is found to only take
successive integers of ${\lambda{}}_{D}$ (no half-wavelength binding regime
has been observed). The relatively small achievable bouncing phase shift of
the chasing liquid metal heterodimers, $\theta{}/2\pi{}<1/4$ throughout the
applied acceleration range (Figs. 9a, 9b), might be responsible for such a
difference, since half-wavelength binding typically occurs when the droplets
are bouncing in antiphase ($\theta{}/2\pi{}=1/2$) [1, 6, 20, 22, 23, 24]. The
vertical trajectories of the droplets show that their bouncing is synchronized
with the bath vibration as well as the meniscus oscillating, featuring the
driving frequency f. This indicates that both the waves generated by the
meniscus oscillating and by the droplet bouncing are harmonic. Furthermore,
the droplets in the current system mainly adopt the simple bouncing mode,
which coincides with our previous report [34]. Due to the dominance of the
single bouncing regime, the chasing direction of the heterodimers shows no
dependence to applied acceleration.There are three reasons that could possibly
account for the absence of the walking and other bouncing regimes in the
current system: (1) The peculiar fluid properties of the liquid metal in
comparison to the commonly used silicon oils; (2) Instead of a gaseous
environment (air), the liquid metal droplets bounce on a liquid-liquid
interface and their free flight takes place in a liquid phase, and,
consequently, the droplets are expected to experience distinct drag and
lubricating force due to significant differences in viscosity and
compressibility of the surrounding medium; and (3) Influenced by the meniscus-
induced waves, the bath becomes unstable before the transition states being
reached.

The interaction between the droplets and the annular global wave field
provides the confinement to lock the horizontal motion of the heterodimers
into circular orbits, which would otherwise be in an arbitrary direction. We
note that, although kept unchanged in the current experiments,
${\lambda{}}_{G}$ can be readily adjusted through frequency control. Allow for
the superposition nature of waves and their tunability, guiding droplets with
an integrated pilot-wave field could be a promising strategy for future works
to reveal the unexploited potentials of the droplet-bath system. In this
regard, our findings could provide helpful guidelines for advancing the pilot-
wave hydrodynamics and extending its implications to other physical systems.
The striking similarities between the heterodimers in the current hydrodynamic
system and the optical system are convincing as to the driving role of the
local wave field and the confining role of the global wave field for
particle/droplet guiding in both systems. The evidence unveils symmetry
breakdown as a universal mechanism for motion initiating in wave-mediated
systems.

###### Acknowledgements.

This work was supported by the NSFC Key Project under Grant No. 91748206,
Dean’s Research Funding and the Frontier Project of the Chinese Academy of
Sciences. The authors are in debt to Dr Junyu Gao (Institute of Automation,
Chinese Academy of Sciences) for his helpful advice on droplet tracking and
image analysis.

## References

  * Protière  _et al._ [2006] S. Protière, A. Boudaoud, and Y. Couder, Particle–wave association on a fluid interface, J. Fluid Mech. 554, 85 (2006).
  * Bush [2015] J. W. M. Bush, Pilot-wave hydrodynamics, Annu. Rev. Fluid Mech. 47, 269 (2015).
  * Faraday [1831] M. Faraday, Xvii. on a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces, Philos. Trans. R. Soc. London 121, 299 (1831).
  * Douady [1990] S. Douady, Experimental study of the faraday instability, J. Fluid Mech. 221, 383 (1990).
  * Couder  _et al._ [2005a] Y. Couder, E. Fort, C. H. Gautier, and A. Boudaoud, From bouncing to floating: noncoalescence of drops on a fluid bath, Phys. Rev. Lett. 94, 177801 (2005a).
  * Couder  _et al._ [2005b] Y. Couder, S. Protière, E. Fort, and A. Boudaoud, Dynamical phenomena: Walking and orbiting droplets, Nature 437, 208 (2005b).
  * Rohlf [1994] J. Rohlf,  _Modern Physics from $\alpha$ to $\mathcal{Z}^{0}$_ (Wiley, 1994).
  * Roh and Gharib [2019] C. Roh and M. Gharib, Honeybees use their wings for water surface locomotion, Proc. Natl. Acad. Sci. U.S.A. 116, 24446 (2019).
  * Flanagan and Hughes [2005] É. É. Flanagan and S. A. Hughes, The basics of gravitational wave theory, New J. Phys. 7, 204 (2005).
  * Couder and Fort [2006] Y. Couder and E. Fort, Single-particle diffraction and interference at a macroscopic scale, Phys. Rev. Lett. 97, 154101 (2006).
  * Dorbolo  _et al._ [2008] S. Dorbolo, D. Terwagne, N. Vandewalle, and T. Gilet, Resonant and rolling droplet, New J. Phys. 10, 113021 (2008).
  * Eddi  _et al._ [2009a] A. Eddi, E. Fort, F. Moisy, and Y. Couder, Unpredictable tunneling of a classical wave-particle association, Phys. Rev. Lett. 102, 240401 (2009a).
  * Fort  _et al._ [2010] E. Fort, A. Eddi, A. Boudaoud, J. Moukhtar, and Y. Couder, Path-memory induced quantization of classical orbits, Proc. Natl. Acad. Sci. U. S. A. 107, 17515 (2010).
  * Harris  _et al._ [2013] D. M. Harris, J. Moukhtar, E. Fort, Y. Couder, and J. W. M. Bush, Wavelike statistics from pilot-wave dynamics in a circular corral, Phys. Rev. E 88, 011001 (2013).
  * Harris and Bush [2014] D. M. Harris and J. W. M. Bush, Droplets walking in a rotating frame: from quantized orbits to multimodal statistics, J. Fluid Mech. 739, 444 (2014).
  * Andersen  _et al._ [2015] A. Andersen, J. Madsen, C. Reichelt, S. R. Ahl, B. Lautrup, C. Ellegaard, M. T. Levinsen, and T. Bohr, Double-slit experiment with single wave-driven particles and its relation to quantum mechanics, Phys. Rev. E 92, 013006 (2015).
  * Filoux  _et al._ [2017] B. Filoux, M. Hubert, P. Schlagheck, and N. Vandewalle, Walking droplets in linear channels, Phys. Rev. Fluids 2, 013601 (2017).
  * Sáenz  _et al._ [2018] P. J. Sáenz, T. Cristea-Platon, and J. W. M. Bush, Statistical projection effects in a hydrodynamic pilot-wave system, Nat. Phys. 14, 315 (2018).
  * Lieber  _et al._ [2007] S. I. Lieber, M. C. Hendershott, A. Pattanaporkratana, and J. E. Maclennan, Self-organization of bouncing oil drops: Two-dimensional lattices and spinning clusters, Phys. Rev. E 75, 056308 (2007).
  * Protière  _et al._ [2008] S. Protière, S. Bohn, and Y. Couder, Exotic orbits of two interacting wave sources, Phys. Rev. E 78, 036204 (2008).
  * Eddi  _et al._ [2008] A. Eddi, D. Terwagne, E. Fort, and Y. Couder, Wave propelled ratchets and drifting rafts, Europhys. Lett. 82, 44001 (2008).
  * Eddi  _et al._ [2009b] A. Eddi, A. Decelle, E. Fort, and Y. Couder, Archimedean lattices in the bound states of wave interacting particles, Europhys. Lett. 87, 56002 (2009b).
  * Eddi  _et al._ [2012] A. Eddi, J. Moukhtar, S. Perrard, E. Fort, and Y. Couder, Level splitting at macroscopic scale, Phys. Rev. Lett. 108, 264503 (2012).
  * Filoux  _et al._ [2015] B. Filoux, M. Hubert, and N. Vandewalle, Strings of droplets propelled by coherent waves, Phys. Rev. E 92, 041004 (2015).
  * Galeano-Rios  _et al._ [2018] C. A. Galeano-Rios, M. M. P. Couchman, P. Caldairou, and J. W. M. Bush, Ratcheting droplet pairs, Chaos 28, 096112 (2018).
  * Moláček and Bush [2013] J. Moláček and J. W. M. Bush, Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory, J. Fluid Mech. 727, 612 (2013).
  * Sungar  _et al._ [2017] N. Sungar, L. D. Tambasco, G. Pucci, P. J. Sáenz, and J. W. M. Bush, Hydrodynamic analog of particle trapping with the talbot effect, Phys. Rev. Fluids 2, 103602 (2017).
  * Sungar  _et al._ [2018] N. Sungar, J. P. Sharpe, J. J. Pilgram, J. Bernard, and L. D. Tambasco, Faraday-talbot effect: Alternating phase and circular arrays, Chaos 28, 096101 (2018).
  * Tambasco and Bush [2018] L. D. Tambasco and J. W. M. Bush, Exploring orbital dynamics and trapping with a generalized pilot-wave framework, Chaos 28, 096115 (2018).
  * De Gennes  _et al._ [2013] P. G. De Gennes, F. Brochard-Wyart, and D. Quéré,  _Capillarity and wetting phenomena: drops, bubbles, pearls, waves_ (Springer Science & Business Media, 2013).
  * Henderson and Miles [1994] D. M. Henderson and J. W. Miles, Surface-wave damping in a circular cylinder with a fixed contact line, J. Fluid Mech. 275, 285 (1994).
  * Yifat  _et al._ [2018] Y. Yifat, D. Coursault, C. W. Peterson, J. Parker, Y. Bao, S. K. Gray, S. A. Rice, and N. F. Scherer, Reactive optical matter: light-induced motility in electrodynamically asymmetric nanoscale scatterers, Light: Sci. Appl. 7, 105 (2018).
  * Sukhov  _et al._ [2015] S. Sukhov, A. Shalin, D. Haefner, and A. Dogariu, Actio et reactio in optical binding, Opt. Express 23, 247 (2015).
  * Zhao  _et al._ [2018] X. Zhao, J. Tang, and J. Liu, Electrically switchable surface waves and bouncing droplets excited on a liquid metal bath, Phys. Rev. Fluids 3, 124804 (2018).
  * Danelljan  _et al._ [2014] M. Danelljan, G. Häger, F. Khan, and M. Felsberg, Accurate scale estimation for robust visual tracking, in  _British Machine Vision Conference, Nottingham, September 1-5, 2014_ (BMVA Press, 2014).
  * Gao  _et al._ [2017] J. Gao, T. Zhang, X. Yang, and C. Xu, Deep relative tracking, IEEE Trans. Image Process. 26, 1845 (2017).
  * Kumar and Tuckerman [1994] K. Kumar and L. S. Tuckerman, Parametric instability of the interface between two fluids, Journal of Fluid Mechanics 279, 49–68 (1994).
  * Kinsler  _et al._ [1999] L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders,  _Fundamentals of Acoustics, 4th Edition_ (1999).
  * Hocking [1987] L. M. Hocking, Waves produced by a vertically oscillating plate, J. Fluid Mech. 179, 267 (1987).
  * Punzmann  _et al._ [2014] H. Punzmann, N. Francois, H. Xia, G. Falkovich, and M. Shats, Generation and reversal of surface flows by propagating waves, Nat. Phys. 10, 658 (2014).
  * Eddi  _et al._ [2011] A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi, and Y. Couder, Information stored in faraday waves: the origin of a path memory, J. Fluid Mech. 674, 433 (2011).
  * Gilet  _et al._ [2007] T. Gilet, N. Vandewalle, and S. Dorbolo, Controlling the partial coalescence of a droplet on a vertically vibrated bath, Phys. Rev. E 76, 035302 (2007).
  * Couchman  _et al._ [2019] M. M. P. Couchman, S. E. Turton, and J. W. M. Bush, Bouncing phase variations in pilot-wave hydrodynamics and the stability of droplet pairs, J. Fluid Mech. 871, 212 (2019).