PHYSICAL REVIEW LETTERS 130, 064002 (2023)
Featured in Physics

Superradiant Droplet Emission from Parametrically Excited Cavities
Valeri Frumkin* and John W. M. Bush†

Department of Mathematics, Massachusetts Institute of Technology

Konstantinos Papatryfonos*

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
and Gulliver UMR CNRS 7083, ESPCI Paris, Université PSL, 75005 Paris, France(Received 5 May 2022; accepted 23 December 2022; published 9 February 2023)
Superradiance occurs when a collection of atoms exhibits a cooperative, spontaneous emission ofphotons at a rate that exceeds that of its component parts. Here, we reveal a similar phenomenon in ahydrodynamic system consisting of a pair of vibrationally excited cavities, coupled through their common
wave field, that spontaneously emit droplets via interfacial fracture. We show that the droplet emission rateof two coupled cavities is higher than the emission rate of two isolated cavities. Moreover, the amplified
emission rate varies sinusoidally with distance between the cavities, as is characteristic of superradiance.
We thus present a hydrodynamic phenomenon that captures several essential features of superradiance in
optical systems.
DOI: 10.1103/PhysRevLett.130.064002

When a group of N quantum emitters (e.g., excitedatoms) interact coherently with a common electromagnetic
field, they may collectively emit photons at a rate that is
proportional to N 2 [1,2]. In quantum optics, this phenomenon is known as superradiance [3], an effect of both
fundamental and practical interest, with applications invarious fields, including quantum information technologies [4–6], cryptography [7], and narrow linewidth lasers[8–10]. When the emitters are separated by less than awavelength, this situation requires the emitters to radiatewith the same phase [3]. When they are separated bydistances comparable to or greater than the wavelength,each emitter must radiate with the local phase of the mode
into which they emit [11–13]. In the case of coherently
driven emitters, the atomic coherence, and thus the phase of
the radiated field, is set by the local phase of the drive field.
The phase-matching condition then leads to a sinusoidal
modulation of the spontaneous emission rate versus theemitter spacing [14]. This modulation was first demonstrated using a pair of trapped ions whose separation
distance d was varied gradually [15]. Experiments revealed
sinusoidal oscillations of the spontaneous emission rateΓðdÞ of the two-ion crystal, in accord with detailed
quantum mechanical theoretical analysis [16]. This typeof superradiance was first considered to be a purely
quantum phenomenon [15,16], but has since been rationalized in terms of classical electromagnetic theory [17].
Fluid mechanics has produced laboratory-scale physicalanalogs for phenomena as disparate as the wave natureof light [18], black holes [19], the Casimir effect [20],the Aharonov-Bohm effect [21], and the periodic table[22]. The relatively recent discovery of a pilot-wave
0031-9007=23=130(6)=064002(5)hydrodynamic system [23] has led to a new class ofhydrodynamic quantum analogs [24] that includes analogs
of quantized orbital states [25,26], quantum corrals [27,28],Friedel oscillations [29], and spin lattices [30].
We here show that superradiance can also be seen in ahydrodynamic setting. We consider a system of vibrationally excited hydrodynamic cavities that spontaneously
emit droplets via interfacial fracture. The cavities are deep
circular wells spanned by a thin layer of oil that allows fortheir coupling through a common wave field (see Fig. 1).
We demonstrate that the wave field in each cavity is
influenced by the presence of its neighbor. Specifically,
the neighboring cavity may amplify the local oscillation
amplitude, resulting in an increased chance of interfacial
fracture and thus an amplified droplet emission rate.
Figure 1 shows a schematic representation of ourexperimental setup. A bath of fluorinated oil has two
6-mm-deep circular wells that serve as hydrodynamic
cavities. The cavities, each with diameter 7 mm, areseparated by a center-to-center distance d that is varied
between experiments, from 8 mm to 12 mm, in 0.5-mm
increments. In the shallow layer spanning the wells, thedepth is 0.75  0.05 mm. The system is subjected tovertical vibration by an electromagnetic shaker with forcing
FðtÞ ¼ γ cosð2πftÞ, where γ ¼ 1.75 g and f ¼ 39 Hz are
the peak driving acceleration and frequency, respectively. A
more detailed description of the experimental setup isprovided in the Supplemental Material [31], which includes
Ref. [32].
A liquid bath of uniform depth subject to vertical vibration
at a fixed frequency undergoes two critical transitions as
the driving amplitude is increased progressively. The first

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PHYSICAL REVIEW LETTERS 130, 064002 (2023)

FIG. 1. The experimental setup (see also Supplemental
Material, Fig. 1 [31]). (a) A schematic illustration of a circularbath with two cavities spanned by a thin layer of fluorinated oil.
The bath is vertically oscillated by an electromagnetic shaker,
resulting in the emission of droplets from the two cavities. (b) A
rare generation event in which two droplets are about to becreated simultaneously. Scale bar, 3 mm.

FIG. 2. Images of the instantaneous wave field generated by thecavities (red circles). (a) A single cavity. Two cavities with centerto-center distances of (b) d ¼ 8 mm, (c) d ¼ 10.5 mm,(d) d ¼ 12 mm, and (e) d ¼ 87 mm.

transition occurs as the vibrational acceleration, γ ¼ 4π 2 f 2 A,
where A is the vibration amplitude, is increased beyond the
Faraday threshold, γ F , at which point the initially flat freesurface destabilizes into a pattern of standing Faraday waves[33]. As the driving amplitude is increased further, thestabilizing influence of surface tension is exceeded by thedestabilizing inertial forces associated with the bath vibration, and the interfacial fracture threshold γ B is crossed.
Above this threshold, the Faraday waves break spontaneously, and millimetric droplets are emitted from the freesurface in an irregular fashion [34,35]. Importantly, forshallow layers, both γ F and γ B depend strongly on the local
depth of the liquid. We thus define γ cF and γ sF to be the
Faraday threshold above the cavities and the shallow region,
respectively, and likewise for γ B.
With the increase of the driving acceleration γ, ourvariable-depth system undergoes the following evolution.
First, as the acceleration crosses γ cF , Faraday waves appear
above the cavities and propagate some distance into thesurrounding shallow region. When γ > γ sF , Faraday waves
emerge over the entirety of the bath surface, but are mostvigorous above the cavities. Figure 2 illustrates the instantaneous wave field near the cavities when γ sF < γ < γ cB .
Figure 2(a) shows the wave field of a single cavity, whereas
Figs. 2(b)–2(d) show the two-cavity wave field for threedifferent values of the center-to-center separation distance,d ¼ 8, 10.5, 12 mm, respectively. Figure 2(e) depicts
the resulting wave field for two distant cavities, withd ¼ 87 mm. Notably, even at such large separation distances, the perturbation wave field can reach the othercavity, allowing for long-range interactions. The perturbation wave field, recorded near the frequency of the most
unstable Faraday mode, f=2, is shown in the Supplemental
Material, Movie 1 [31].
When the acceleration is increased beyond the interfacialbreaking threshold of the cavities, γ cB < γ < γ sB , droplet
emission sets in. Movie 2 in the Supplemental Material [31]shows the spontaneous droplet emission from a pair ofhydrodynamic cavities. The emission events occur unpredictably, as indicated by Fourier analysis shown in the
Supplemental Material [31], but arise exclusively within
the cavities. We define a spontaneous emission rate Γ forthe combined two-cavity system, as the average number ofemission events per second, and the anomalous emissionrate, ΓN ðdÞ ¼ ½ΓðdÞ − 2Γ0 =2Γ0 , where Γ0 ¼ 1.47 s−1 isthe measured emission rate of a single cavity in isolation.
In Fig. 3(a), we present our experimental measurementsof the dependence of the anomalous emission rate ΓN ðdÞ on
the separation distance d. An amplification of up to 46%
relative to 2Γ0 is evident. One expects the emission rate to

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PHYSICAL REVIEW LETTERS 130, 064002 (2023)

FIG. 3. Experimental measurements of the droplet emission
rate. (a) Dependence of the anomalous emission rate, ΓN ðdÞ ¼½ΓðdÞ − 2Γ0 =2Γ0 (black dots) on center-to-center intercavity
distance d. Each data point represents an average over a timeinterval of 300 seconds, corresponding to roughly 950–1300
droplet emission events. The dashed curve represents A cos2 ð2kdÞ,with A ¼ 1.36 and k ¼ 0.85 mm−1 being the experimentallymeasured Faraday wave number in the vicinity of the cavities.
(b) Time series of the emission events from a single cavity over a
300-second interval indicates the unpredictability of a single
emission event. (c) Histogram of the interemission time intervals
and its comparison to the first passage time model [36]. The red
curve represents an inverse Gaussian distribution as given by
Eq. (2). (d) Measured correlation C of the drop emission event in
the two cavities as a function of their separation distance d. The
upper dashed line represents two uncorrelated cavities.

be a function of the kinetic energy of the fluid inside thecavity. For an isolated cavity, the average kinetic energy per
oscillation period is constant, corresponding to a steadyemission rate Γ0 . In the presence of a second, coupledcavity, the wave field inside each cavity is effected by its
neighbor, and the average kinetic energy per oscillation

period will depend on the distance between the two. As in
the trapped ion pair experiment [15], the amplified emission rate of our two-cavity system oscillates sinusoidally as
a function of the distance between the cavities. The
observed oscillatory behavior shows that the probabilityof the emission events is affected by the interference
between the waves generated by the individual cavities.
The dashed curve in Fig. 3(a) represents a simple fit
to the anomalous emission rate, ΓN ðxÞ ¼ Acos2 ð2kdÞ for
A ¼ 1.36, and k ¼ 2π=λ, with λ ¼ 6.60  0.05 mm being
the experimentally measured wavelength of the Faraday
waves in the vicinity of the cavities (Fig. 2). Owing toexperimental limitations, we did not systematically characterize the decay of the anomalous emission rate withincreasing separation distance d. Nevertheless, we note thatthe influence of the neighboring cavity’s wave field will
decay with d due to viscous damping, as will the anomalous emission rate.
Figure 3(b) depicts the time dependence of the emission
events from a single cavity, showing the unpredictability ofa single emission event, as is confirmed by the FFT analysis
presented in the Supplemental Material [31]. While thehighly nonlinear and chaotic nature of the emission events
makes theoretical or numerical modeling of the emission
phenomenon a daunting task, we proceed by demonstratingthat the problem lends itself to a stochastic approach. Let Xt
be a stochastic process representing the maximal wave field
amplitude inside the cavities, with X0 ¼ 0 representing theinitially flat state. Between two consecutive emission
events, we assume that the maximal wave field amplitude
oscillates stochastically about some mean value μðtÞ thatgrows in time as a result of the resonant interaction between
the external forcing and the cavities. Eventually, themaximal amplitude crosses a threshold α, resulting in anemission of a droplet, after which Xt relaxes back to X0 ¼ 0
and the process starts over. Thus, one may write
Xt ¼ νt þ σW t ;ν; σ > 0ð1Þ

where νt is a stochastic drift representing the increase inμðtÞ between consecutive emission events, and W t is a
Wiener process with an amplitude σ, representing thestochastic oscillations of Xt about μðtÞ. The emission
process can thus be modeled as a first passage time
problem, where we seek to find the first time that Xt
reaches the critical value α, at which point an emission
event occurs. For a Wiener process with a stochastic drift,
the probability density function for the first passage time isgiven by the inverse Gaussian distribution [36]rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃκκðT − μÞ2PðT α Þ ¼
exp −
ð2Þ
2πT 3
2μ2 T
where T α is the first time Xt crosses the threshold value α,κ ¼ ðα=σÞ is the shape parameter, and μ ¼ ðα=νÞ is themean value of T α . Figure 3(c) presents a histogram of the

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PHYSICAL REVIEW LETTERS 130, 064002 (2023)
experimentally measured interemission time intervals forthe case d ¼ 12 mm, which shows good agreement with
Eq. (2) when κ ¼ 3.3 is used for the shape parameter.
The mechanism responsible for the superradiant emissionof droplets is the wave coupling between the two cavities. We
quantify this coupling by measuring the correlations between
the emission events in the two-cavity system, as detailed in
the Supplemental Material [31]. We see that the two cavities
are strongly anticorrelated, with the correlation values varying from C ¼ −0.30 (for d ¼ 11.5 mm) to C ¼ −0.41 (ford ¼ 10 mm). Figure 4 shows the measured correlation C ofthe drop emission event in the two cavities as a function of
their separation distance d. Presumably, the anticorrelationarises because each time a droplet is emitted, a small amountof liquid is removed from the system for a brief period oftime, leading to a diminution in the probability of anotherejection. The emission of one droplet serves to delay theemission of the next.
These anticorrelations, together with the amplification of
the combined emission rate, suggest that the two-cavity
system cannot be factored into distinct states, as theprobabilities of emission events in the two cavities arecoupled. Acting on one of the cavities of this coupledsystem, by, for example, changing its position or depth,
would affect the emission rate of its neighboring cavity. The
possibility thus arises of altering the system’s global
emission rate by a local operation on one of its individualcomponents, thereby creating a new platform for probabilistic computational operations in fluid mechanics.
It is also worth considering the relation between the system
introduced here and pilot-wave hydrodynamics [24,37]. In
the latter, the notion of an analog photon is more nebulous:
when the system jumps between quantized states (e.g., thewalking droplet transitions from one orbit to another, or
tunnels between two energetically distinguishable cavities[37]), energy is dumped into or extracted from the bath. In thesystem considered here, droplets are generated by breaking
waves, their appearance representing a discrete transition
event, an analog of photon emission from an excited state. We
note that in our current experiments, we used fluorinated oilin order to facilitate the rapid reabsorption of the emitted
droplets into the bath. However, this reabsorbtion can be
minimized by using a relatively low density silicon oil, in
which case the generated particles may persist on the surface,bounce, and self-propel, thereby providing a possible link
between pilot-wave hydrodynamics and the new class ofanalog systems established here.
We proceed by enumerating several notable differences
between optical superradiance and our hydrodynamic analogy. First, in our experiments we did not characterize thestructure of the energy levels as would potentially beprescribed by the size and kinetic energy of the emitted
droplets, or the transition rates associated with these levels,
both of which are well characterized in the quantum
mechanical system. Second, in our experiments we did
not observe subradiant droplet emission. We believe thatthe latter is due to the chosen cavity geometry precluding thepossibility of robust destructive interference. Specifically, inorder to support a single oscillatory mode in each cavity, theliquid bath would need to be strongly driven at a frequency on
the order of 10 Hz, which was unreachable with our current
setup. Driving the system at 39 Hz excited higher harmonics
inside the cavities, yielding a complex 2D wave field which
could not be canceled by the ordered wave field in theshallow intercavity region. We note that in the optical case,while superradiance is readily observed in a wide variety ofsystems, subradiance is very difficult to obtain [13,38].
Another comparison can be made between the generationof droplets in our hydrodynamic system and the emission ofphotons in its optical counterpart. Both processes represent
dissipative mechanisms, the rates of which depend nonlinearly on the amplitude of the relevant field. In thehydrodynamic case, the probability of random discrete
events, specifically drop ejection, is prescribed by a continuous wave field resulting from two interfering sources.
This statistical behavior is reminiscent of the way probabilities of outcomes are obtained from Born’s rule in thestandard quantum theory.
We have introduced a new hydrodynamic system thatshares several key features with the phenomenon of superradiance in optics. In addition to the amplification of theemission rate typically associated with superradiance, oursystem exhibits sinusoidal dependence of the amplified
emission rate on separation distance [see Fig. 3(b)], as arises
from classical wave interference. Finally, our study suggeststhat droplet creation through interfacial fracture may provide
a valuable new platform for exploring hydrodynamic analogsof particle emission phenomena, and so further extend therange of hydrodynamic quantum analogs.
We thank Masha Bluvshtein for her help with analyzing
the data, and Christos Orestis Apostolidis for providing thegraphical schematics used in this Letter. We also thankMatthieu Labousse and André Nachbin for valuable discussions, as well as Shimon Rubin and Tal Kachman for
their constructive comments on the manuscript. We also
gratefully acknowledge the financial support of the UnitedStates Department of State (V. F.), the European Union’s
Horizon 2020 research and innovation program under
the Marie Sklodowska-Curie project EnHydro, Grant
Agreement No. 841417 (K. P.), and the National Science
Foundation Grant No. CMMI-1727565 (J. B.).
The authors declare no competing interests.

*

These authors contributed equally to this work.
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