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Hydrodynamic superradiance in wave-mediatedcooperative tunneling

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Konstantinos Papatryfonos 1,2 ✉, Mélanie Ruelle1, Corentin Bourdiol1, André Nachbin
John W. M. Bush 2 ✉ & Matthieu Labousse 1

3 ✉,Superradiance occurs in quantum optics when the emission rate of photons from multiple
atoms is enhanced by inter-atom interactions. When the distance between two atoms iscomparable to the emission wavelength, the atoms become entangled and their emission ratevaries sinusoidally with their separation distance due to quantum interference. We here
explore a theoretical model of pilot-wave hydrodynamics, wherein droplets self-propel on thesurface of a vibrating bath. When a droplet is conﬁned to a pair of hydrodynamic cavities
between which it may transition unpredictably, in certain instances the system constitutes a
two-level system with well-deﬁned ground and excited states. When two such two-levelsystems are coupled through an intervening cavity, the probability of transition between
states may be enhanced or diminished owing to the wave-mediated inﬂuence of its neighbour. Moreover, the tunneling probability varies sinusoidally with the coupling-cavity length.
We thus establish a classical analog of quantum superradiance.

1 Gulliver UMR CNRS 7083, ESPCI Paris, Université PSL, Paris 75005, France. 2 Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, MA
02139, USA. 3 IMPA, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil. ✉email: kpapatry@mit.edu; nachbin@impa.br; bush@math.mit.edu

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COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-022-00918-yn classical physics, particle motion is characterized in terms ofdynamical states which may provide in turn the basis for astatistical description of the system. In quantum physics there
is no such dynamical description: quantum systems are describedentirely in terms of the evolution of their statistical states1. For aquantum system with at least two subsystems, non-separability
arises when the quantum state cannot be factored into a productof states of its individual subsystems2. Many well-known bipartite
quantum phenomena arise as a result of such non-separable
states. One is the phenomenon of superradiance and subradiance,
whereby the probability of decay of two coupled systems, as
marked by photon emission, depends on the proximity of thosesystems3–6.
Superradiance is a “cooperative” spontaneous emission ofphotons from a collection of N atoms that was theoretically
predicted in 1954 by Robert Dicke6. When thermally excitedatoms emit photons incoherently with respect to each other, theemitted intensity is proportional to the number of atoms, N.
However, when the atoms radiate coherently, in phase with eachother, the net electromagnetic ﬁeld is proportional to N, and theemitted intensity thus scales as N2. As a result, the atoms may
decay at an enhanced rate that is up to N times faster than forincoherent emission, a phenomenon termed “superradiance”.
When the distance between the atoms is of the order of theemission wavelength, rationalizing this collective behavior
requires consideration of the system’s quantum nature. The mostelementary demonstration of Dicke’s superradiance is achieved
with two ions4. Experimentally, the photon emission rate Γ(R) is
characterized in terms of the inter-ion distance R, and comparedto the single-ion emission rate Γ0 in the same apparatus. The
quantum theory describes a bipartite system in which twointeracting two-level systems form a single four-level system.
When far apart, the two subsystems are independent and can beeither in the ground or excited states. When the two two-levelsystems interact, the collective system is treated as a single nonfactorable four-level system, consisting of a ground state ji,degenerate ﬁrst excited states jþi or jþi that are energeticallyindistinguishable, and a second excited (higher energy) statejþþi. Here jþi and ji denote the states of the two subsystems,
nomenclature that we adopt for our system. The decay rates Γ± toor from the degenerate excitedstates jþi andjþi are wellin the limit
approximated by Γ ± ðRÞ ¼ Γ0 1 ± 32 sinðkRÞ
kR þ ¼kR ⪆ 10, where k = 2π/λ denotes the wavenumber and λ theemission wavelength4. Speciﬁcally, Γ+ and Γ− are the rates corresponding
pﬃﬃﬃ to transitions to or from the symmetricpﬃﬃﬃðjþi þjþiÞ= 2 and antisymmetric ðjþi  jþiÞ= 2 states,
respectively. We note that Γ± can be either greater or less than thebare emission rate Γ0, corresponding to, respectively, superradiance or subradiance. The measured form of Γ±(R) was foundto be in very good agreement with Dicke’s theoretical prediction4.
Such sinusoidally modulated, enhanced, or diminished radiance isan indirect measure of the nonseparability of states, a phenomenon commonly thought to be peculiar to the quantum realm.
Superradiance is a canonical effect in quantum optics with
important applications in quantum communication7. Moreover,it was recently demonstrated that entangled free electrons may
also exhibit superradiance through an emission pattern thatcannot be accounted for by a classical mixed state8, suggesting ameans for non-destructive measurement9 of entanglement
between quantum systems. While a classical analog of superradiant emission has recently been reported in an experimental
study of interfacial fracture10, we here present an analog based on
the hydrodynamic pilot-wave system11,12.
Hydrodynamics has long served as a rich source of physicalanalogs for electromagnetic and optical phenomena13–15. More2

recently, it has yielded robust analogs of white holes16, black
holes17,18, and rotational superradiant scattering19. Until recently,hydrodynamic analogs of quantum systems were relatively rare, but
included the Aharanov–Bohm effect20 and the Casimir effect21.
The walking-droplet system discovered in 2005 by Yves Couder
and Emmanuel Fort11,22 has proven to be a remarkably rich sourceof physical analogs for both optical and quantum systems. Thispilot-wave hydrodynamic system exhibits many features previously
thought to be exclusive to the quantum realm, and so has initiatedthe burgeoning ﬁeld of hydrodynamic quantum analogs12,23,24.
Couder and Fort11 discovered a classical pilot-wave system thatconsists of a millimetric droplet bouncing on the surface of a vibrating
liquid bath, self-propelling by virtue of a resonant interaction with its
own wave ﬁeld. By virtue of this resonance, the droplet is accompanied
by a quasi-monochromatic wave ﬁeld that imposes a dynamic constraint on the droplet that gives rise to the emergence of quantized
dynamical states12. For example, quantized orbital states emerge when
the walking droplets move in the presence of either Coriolis25–27 orcentral spring forces28–30 and Bohr–Sommerfeld quantization may
emerge for walkers in a harmonic well31. Features of quantum optics
have also been captured, including single-particle diffraction andinterference22,32,33 and the Hong-Ou-Mandel effect34. The persistenceof the pilot-wave ﬁeld renders the drop dynamics non-Markovian35,36;
speciﬁcally, the instantaneous wave force imparted to the drop during
impact depends on the particle’s history. The droplet thus navigates apotential landscape of its own making. The non-Markovian feature of
the droplet dynamics gives rise to behavior that might be mistakenly
inferred to be spatially non-local if the inﬂuence of the wave ﬁeld is notadequately resolved24. Of particular interest, here is the hydrodynamic
analog of unpredictable quantum tunneling37, as has been demonstrated both experimentally38,39 and numerically40,41.
We introduce here a theoretical model of bipartite tunneling in
the hydrodynamic pilot-wave system that allows us to establish aclassical analog of quantum superradiance. We do so by satisfying
three criteria. First, we consider a classical two-level system inwhich one of the two states may be treated as the lower-energy
state, in the sense that it is more likely to arise. Second, we
produce a bipartite system, consisting of two such two-levelsystems, coupled in such a way that their collective behavior
cannot be speciﬁed in terms of a linear combination of its individual subsystems. Third, we demonstrate that the probability oftransition from state to state in each subsystem varies in asinusoidal fashion with the distance between the two subsystems.
Results
Model dynamics. Our model system comprises two subsystems,
labeled 1 and 2 in Fig. 1a, each consisting of a wave-generating
particle in an identical pair of cavities separated by a barrier
across which the particle may tunnel. Each particle generates
waves and moves in response to them in a manner detailed in“Methods”. The two subsystems are separated by a coupling
cavity of variable width Lc, and by barriers that are sufﬁciently
high as to preclude the particles from tunneling into the coupling
cavity. The particles are thus conﬁned to one of the two subsystems, but may ﬁnd themselves in either the inner or outercavities of their subsystem. Each of the four cavities has a ﬁxed
length of L = 1.2 cm, corresponding to ~2.5λF. Here, λF is the
Faraday wavelength, the most unstable wavelength of thevibrating bath, and the dominant wavelength of the droplet’s
pilot-wave ﬁeld11,22.
Each bouncing droplet, within its two-level system, generates
waves that are reﬂected and transmitted as they interact with thebottom topography. The coupling cavity is relatively deep and so
closer to the Faraday threshold; thus, waves transmitted into it are
relatively persistent and so capable of inﬂuencing the other two-

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Fig. 1 Description of the numerical model of two coupled, two-levels systems. a The model system consists of a pair of drops (red and green) walking on
the surface of a vibrating ﬂuid bath (blue) that spans the solid substrate (gray). Each drop is conﬁned to a pair of wells separated by barriers across which
they may tunnel unpredictably. The extent of coupling in this bipartite tunneling system is prescribed by the magnitude of the vertical vibrational forcing, γ0,angular frequency, ω0, and the width, Lc, of the intervening coupling cavity, a region that is forbidden to both drops. b The state of each droplet is denotedby ji or jþi according to whether the drops are in, respectively, the outer “ground” state or the inner “excited” state.

level system. The central cavity thus establishes the long-range
coupling between the two two-level systems. The efﬁciency of thiscoupling is prescribed by the geometry of the central cavity: byincreasing its depth dc, the coupling is increased, allowing thecoupling cavity to serve as a nearly resonant transmission line42.
In all simulations, we set the coupling cavity depth todc = 6.3λF which ensures strong intercavity coupling. In subsystems 1 and 2, the particle may tunnel between the outer andinner cavities, but the probability of presence in these cavities is
not generally 50%. Speciﬁcally, in the geometry considered, theparticles are more likely to be in the outer cavity. We thus
describe our bipartite tunneling system in terms of two coupled,
two-level systems, as shown in Fig. 1b.
Tunneling. When a single droplet is placed in one of the two
subsystems, the single-particle tunneling probability (Γ0) may be
deduced by counting the number of tunneling events per crossing
attempt over a sufﬁciently long time interval. When the two
droplets are placed in each of the two subsystems, the tunnelingprobabilities change substantially from Γ0, owing to the wavemediated coupling of the two subsystems. Figure 2a, b indicatesthat the probability of tunneling increases substantially when thesecond droplet is present in the neighboring system, an effect akinto superradiance. Figure 2c, d presents a conﬁguration with alonger coupling cavity in which the tunneling probability is
substantially decreased by the addition of a neighbor, corresponding to subradiance.
Having established that wave-mediated coupling alters thetunneling probability in this bipartite system, we proceed bycharacterizing the dependence of the transition probabilities on thesystem geometry. Figure 3a illustrates how the tunneling probability
depends on the length of the coupling cavity, Lc. The tunnelingprobability either increases or decreases relative to Γ0, varying
continuously with Lc in an oscillatory fashion, reminiscent of themodulation of the emission rate reported in ion-pair quantum
mechanical superradiance4. Further to this modulation, the peaks
become less intense and slightly broader as Lc increases, a trend thatis likewise apparent in quantum superradiance4. For Lc/λF > 6, thesingle-drop and droplet-pair crossing probabilities largely coincide,
indicating that the two components of this bipartite system behave
as if they were uncoupled. Speciﬁcally, the tunneling probability of aparticular drop is determined by the length of the coupling cavity,
but unaffected by its distant neighbor.

Rationalizing transition probabilities. The evolution of thetransition probability with coupling-cavity length is rationalized
in Fig. 3b, where we report the difference in oscillator action
δAtotal(Lc) = A+(Lc) − A− (as deﬁned in Eq. (4) of “Methods”)
between the excited and ground states. For the one-drop system,
the oscillator action difference is a constant which is independenton Lc (see Supplementary Fig. 1 for the details of all the contributions). For the two-drops system, the oscillator action in the
inner cavity is constant (see Supplementary Fig. 2), independentof Lc, and we take its mean value A− as a reference. Conversely,
the oscillator action in the outer cavity varies signiﬁcantly withcoupling-cavity length Lc; we denote its value by A+(Lc). A
comparison between Fig. 3a and b shows a direct correspondence
between the maxima and minima of the transition probability
and the minima and maxima of δAtotal, respectively. We conclude
that the wave-mediated lowering of the oscillator action strongly
favors the transition from the ground to the excited states. The
one-droplet system simulations reveal a difference of oscillator
action δAtotal that is approximately constant for sufﬁciently small
coupling-cavity length Lc < 6λF.
We further note that when Lc/λF < 1 the mid-cavity is arelatively deep, slender region, which poses problems for the
nonlinear solver of the numerical conformal mapping. This
limitation is presumably due to the crowding phenomenon, anumerical conformal mapping feature arising when two preimages of vertices get exponentially close in the canonicaldomain43. The gray region in Fig. 3 identiﬁes these slendercoupling cavities that could not be reliably explored numerically.
Finally, we examine the correlation between the waveﬁeld
structure and the particle dynamics. Figure 4a illustrates both theparticle positions and instantaneous wave ﬁeld. An assessment of
the wave intensity (Eq. (5) in “Methods”) in each cavity is shownin Fig. 4b–d. Each cavity is color-coded for ease of reference. The
rightmost panel shows the wave intensity of the system across all
cavities. Periods in which one particle has tunneled into an
excited state are denoted by light gray. Such intervals areaccompanied by a reversal in the relative wave intensity in the
inner and outer cavities, and an increase in the wave intensity inthe coupling cavity. Periods in which both particles are in excitedstates are denoted by dark gray, and marked by maxima in the
wave intensity in the coupling cavity and minima in the global
wave intensity. The wave intensity diagnostic thus provides
rationale for associating the outer and inner cavities with,respectively, ground and excited states.

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COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-022-00918-yFig. 2 Tunneling dynamics and trajectories of the drops. a A single droplet tunnels between its ground state (outer cavity) and excited state (innercavity). The middle cavity (marked in orange) couples the left and right subsystems. The red and green lines denote the particle paths in the left and rightsubsystems, respectively. Dashed lines indicate the edges of the various cavities. b In the bipartite system, a pair of drops tunnel between their ground andexcited states. In (a, b), the coupling-cavity length is Lc = 2.44λF and the four other cavities' lengths are L = 2.52λF. Comparing (a) and (b), makes clear thata single particle’s tunneling probability is substantially increased by the presence of a neighboring drop, corresponding to superradiance. The conﬁguration
shown in (c, d) is similar to (a) and (b) but with a larger coupling cavity Lc = 3.96λF. Comparing (c) and (d) reveals that the particle’s tunneling between
cavities is inhibited by the presence of the neighboring drop, an effect corresponding to subradiance.

Discussion
We have examined a model of bipartite tunneling in a classicalsystem. The tunneling probability within the two subsystems is
mediated by their common wave ﬁeld and shown to depend on
the length of the coupling cavity. This dependence is similar tothat of the probability of photon emission on inter-ion spacing in
the case of quantum superradiance (e.g., compare our Fig. 2 with
Figs. 2 and 6 of ref. 4). If we consider that the energy stored in oursystem is altered each time the droplet tunnels between the
ground and excited states, we can establish a direct analogy
between our system and quantum superradiance. Speciﬁcally, we
may identify the alteration of the wave ﬁeld that accompanies thetransition from a high-energy to lower-energy state with theemission of a photon in quantum mechanics. Thus, the probability of tunneling is directly analogous to the transition rate insuperradiant photon emission.
We have highlighted the similarities between our classical systemof bipartite tunneling and super- or subradiant photon emission
from ion pairs. To account for the optical superradiance in quantum mechanics, one must invoke the concept of collective, non4

separable, states that are formed between the two ions, which areviewed as absorbing or emitting each photon collectively4,7,44,45.
Quantum mechanics offers no physical picture for the origins of
non-separable states, only the mathematical tools required to calculate the relevant probability densities. Our system illustrates how
such correlations may arise in a classical, wave-mediated system.
The correlations in our system arise from the wave coupling
between the two tunneling subsystems. While the form of the waves
may be deduced by linear superposition, their inﬂuence on
the tunneling probability is nonlinear. Speciﬁcally, the discretetunneling events are unpredictable, as they depend in a subtle
fashion on the interactions between the particles, the waves and theunderlying potential wells formed by bottom topography39,40,46.
A hydrodynamic analog of superradiant emission was recently
realized experimentally in a distinct ﬂuidic system. Speciﬁcally,
drops were produced via vibration-induced interfacial fracturefrom adjacent circular cavities coupled through a thin layer, andthe rate of droplet ejection was seen to vary sinusoidally with the
distance between the cavities10. Table 1 summarizes the key
physical analogies between superradiant emission as arises in

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Fig. 3 Evolution of the transition probability with the width of the coupling cavity. a The dependence of the probability of tunneling from the outer to theinner cavity on the width of the coupling cavity, Lc, for a single drop (red squares) and for two drops (blue circles). The red dashed line is the averagetunneling probability for a single drop for coupling cavity lengths Lc/λF < 6. For 2.3< Lc/λF < 6 this average probability 〈P〉 = 1.0% (95% conﬁdence interval[0.91: 1.09]). Error bars represent the standard deviation over six independent numerical runs. b Evolution of the energy barrier to tunneling, δAtotal, for the twodroplet system (blue) and the one-droplet system (red). Blue shaded regions adjoining the lines denote the error bars. Note that the probability maxima in panela correspond to energy-barrier minima in b. For Lc/λF > 6, the tunneling probability of either drop converges to that of a single droplet, and so is evidently
unaffected by its distant neighbor. The gray region on the left denotes the region of tall, slender coupling cavities that could not be reliably explored numerically.

Fig. 4 Wave intensity. a The particle paths and associated wave ﬁelds. Light and dark regions correspond to wave minima and maxima, respectively.
b–e The associated wave intensity, as computed from Supplementary Eq. (3). Comparison of (a) and (b) reveals the correlation between extrema in the
wave intensity and the tunneling events. In (b–e), intervals in which one or both drops are in an excited state are denoted by light and dark gray,respectively, and are correlated with local extrema in the total wave intensity.

quantum optics, droplet emission via interfacial fracture10, andthe bipartite droplet tunneling considered here.
Experimental realization of the one-dimensional dynamics
considered here would require that one conﬁne the walking
droplets to a narrow channel, of width comparable to the Faraday
wavelength. Indeed, we have already used such conﬁnement tostudy the synchronization of drop-pairs in a three-cavity
geometry47 and extension to the ﬁve-cavity geometry considered here should be possible. While one does not expect
quantitative agreement between our numerical model and suchan experimental realization owing to the heightened dissipation

associated with the three-dimensional laboratory ﬂows, one does
expect to be able to capture qualitative features such as thesuperradiant tunneling reported here.
In the study of hydrodynamic quantum analogs, we are generally satisﬁed with qualitative analogs, the demonstration ofwhich is sufﬁcient to challenge the prevailing notion that quantum systems are entirely inscrutable from a classicalperspective24. While there are hydrodynamic analogs that exhibit
behavior quantitatively similar to their quantum counterparts,
including Friedel oscillations48 and quantum corrals49,50, morecommonly the analogy is at the qualitative level, as in the case of

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Table 1 Comparison of quantum superradiant photon emission from ion pairs4, the hydrodynamic analog of ref. 10 and thatdeveloped here.

Emission event
Characteristic wavelength
Origins of superradiance

Quantum optics4Interfacial fracture10

Hydrodynamic tunneling

Photon emission
Emission wavelength
Quantum (wave function) interference

Droplet emission
Faraday wavelength
Faraday wave interference

Droplet tunneling
Faraday wavelength
Faraday wave interference

the bipartite tunneling system studied here. Nevertheless, we
believe that the study of hydrodynamic quantum analogs complements more traditional studies of quantum systems in providing a perspective and platform for assessing which quantum
effects can and cannot be understood from a classical perspective.
Our bipartite tunneling system has given rise to collective
transition statistics, speciﬁcally subradiant and superradianttunneling rates between the ground and excited states, which may
be predictably controlled by altering the distance between the two
two-level systems. Notably, the precise energetic difference
between the ground and excited states may be altered and evenreversed by changing the geometry of the outer cavities (Supplementary Fig. 3 and Supplementary Note 3). Our system thus
provides a platform for characterizing wave-induced correlations
between particle pairs. Speciﬁcally, by taking the system geometry
as a proxy for measurement settings, we intend to test the viabilityof violating Bell-type inequalities with this classical pilot-wave
system51.

tthat we refer to as the “oscillator action”, a name originating from conserved
 j is calculated over thequantities used to describe wave propagation in ﬂuids55. A
 0 is readily
period T of a given cycle; the associated mean angular frequency Ω
1=2computed numerically. We note that A is a length related to the range of droplet
 is lower in the ground state than in the
excursions within a cavity. Notably, A
excited state in all geometries considered here: excited states are marked by relatively large particle excursions. We denote by δAtotal the energetic barrier to tunneling between two states.
As a diagnostic for the wave energy, we monitor the wave intensity:
Z
1=2

2
ð5Þ
jjηjjðtÞ ¼ηðx; tÞ dx:
Further details and discussion concerning the diagnostics (4) and (5) areprovided in the Supplementary Note 2.

Data availability

Methods
Numerical method. Nachbin et al.40 formulated a theoretical model for the onedimensional motion of a walking droplet over a vibrating liquid bath with complex
topography. Here, we adapt this model in order to consider two identical particles
in the multiple-cavity geometry depicted schematically in Fig. 1. The positions, Xj(j = 1, 2), of the two identical particles of mass m evolve according to Newton’s
Law:
∂η
ð1Þ
ðX ðtÞ; tÞ:∂x j
The particle moves in response to gradients of the wave elevation η(x, t), which
thus plays the role of a time-dependent potential. The particle motion is resisted bya drag force proportional to its speed. The drag constant c follows from themodeling presented in refs. 52,53. The time dependence of the propulsive and dragforces reﬂects that of the reaction force, F(t), acting on the drop during its contact
with the bath52,53. In terms of their lateral motion, the particles are viewed as
oscillators that can transition unpredictably between two neighboring cavities.
The particles serve as moving wave sources that establish their own timedependent wave potential that is computed as follows. The velocity potential of theliquid bath ϕ(x, z, t) is a harmonic function satisfying Laplace’s equation. In thebulk of the ﬂuid, the velocity ﬁeld is given by (u, v) = ∇ ϕ. The ﬂuid bath has
density 0.95 g/cm3, viscosity 16cS and surface tension 20.9 dynes/cm, values typicalof silicone oil, and oscillates vertically with frequency ω0/2π = 80 Hz. The resonant
bouncing of the particle at the Faraday frequency triggers a monochromatic
damped wave pattern with a corresponding deep-water Faraday wavelength of
λF = 4.75 mm. The wave model is formulated in the bath’s reference frame, where
the effective gravity is gðtÞ ¼ gð1 þ γ0 sinðω0 tÞÞ, where g is the acceleration due togravity, γ0 ¼ A0 ω20 =g the maximum dimensionless vibrational acceleration, and A0
the amplitude of vibration. We here ﬁx γ0 = 4.21 (90.3% of the Faraday threshold,γF = 4.66), where superradiant effects are pronounced, as indicated by an initial
parameter sweep. The wave ﬁeld thus evolves according to refs. 40,53:€ j þ c FðtÞX_ j ¼ FðtÞ
mX

∂η ∂ϕ
∂2 η
¼
þ 2ν 2 ;
∂t
∂z
∂x

ð2Þ

Pd ðx  X j ðtÞÞ∂ϕσ ∂2 η
∂2 ϕ:
þ 2ν 2  ∑¼ gðtÞη þ
2
ρ
∂t
ρ ∂x
∂x
j¼1;2

ð3Þ

The data that support the ﬁndings of our study are available upon request.

Code availability
The code that generated the data is available upon request.

Received: 4 January 2022; Accepted: 16 May 2022;

References
1.
2.
3.

4.

5.
6.
7.
8.

9.

The particles (j = 1, 2) generate waves on the free surface by applying local
pressure terms Pd. The wave forcing term Pd(x − Xj(t)) and the coefﬁcient F(t) are
activated only during a fraction of the Faraday period TF, corresponding to thecontact time Tc in the walking-droplet system and approximated by Tc = TF/4. The
particle is assumed to be in resonance with the most unstable (subharmonic)
Faraday mode of the bath53, a key feature of pilot-wave hydrodynamics12,24,54.
Further details and discussion concerning the numerical implementation of themodel system, (1) and (2)–(3) are provided in the Supplementary Note 1.
6

System’s energetics. We proceed by deﬁning the diagnostics that characterize thesystem’s energetics. For nearly periodic cycles, we characterize the particle energy
in phase space. We deﬁne a cycle-averaged energy quantity for particle j = 1, 2
Z tþT hi
 0 X 2 ðtÞ þ Ω 1 X_ 2 ðtÞ dt
j ¼
ð4Þ
Ω
A
j
j
0

10.

11.
12.
13.

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Acknowledgements
The authors acknowledge the ﬁnancial support of the NSF through grants CMMI1727565 and CMMI-2154151, the European Union’s Horizon 2020 Research andInnovation Program under the Marie Sklodowska-Curie project EnHydro, grant No.
841417, and the CNPq under (PQ-1B) 301949/2007-7, and FAPERJ Cientistas do Nosso
Estado Project No. 150.036/2018.

Author contributions
J.B., M.L., K.P., and A.N. designed and coordinated the studies. A.N. create the numericalmodel. K.P., M.R., and C.B. performed the numerical simulations. K.P., M.R., C.B., and
M.L. analyzed the results. A.N., J.B., M.L., and K.P. contributed to the writing of thepaper. All authors have read and agreed to the published version of the manuscript.

Competing interests
The authors declare no competing interests.

Additional information
Supplementary information The online version contains supplementary material
available at https://doi.org/10.1038/s42005-022-00918-y.
Correspondence and requests for materials should be addressed to Konstantinos
Papatryfonos, André. Nachbin or John W. M. Bush.
Peer review information Communications Physics thanks Germain Rousseaux,Florentino Borondo, Rahil Valani and Nazmi Burak Budanur for their contribution to
the peer review of this work. Peer reviewer reports are available.
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