Origin and evolution of the multiply-quantised vortex instability

Sam Patrick,1, 2 August Geelmuyden,3 Sebastian Erne,3, 4 Carlo F. Barenghi,5 and Silke Weinfurtner3, 6

1Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, V6T 1Z1, Canada2Institute of Quantum Science and Engineering, Texas A&M University, College Station, Texas, 77840, US3School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

4Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, 1020 Vienna, Austria5Joint Quantum Centre Durham-Newcastle, School of Mathematics, Statistics

and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK6Centre for the Mathematics and Theoretical Physics of Quantum

Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK(Dated: November 5, 2021)

We show that the dynamical instability of quantum vortices with more than a single quantum ofangular momentum results from a superradiant bound state inside the vortex core. Our conclusionis supported by an analytic WKB calculation and numerical simulations of both linearised andfully non-linear equations of motion for a doubly-quantised vortex at the centre of a circular buckettrap. In the late stage of the instability, we reveal a striking novel behaviour of the system in thenon-linear regime. Contrary to expectation, in the absence of dissipation the system never entersthe regime of two well-separated phase defects described by Hamiltonian vortex dynamics. Instead,the separation between the two defects undergoes modulations which never exceed a few healinglengths, in which compressible kinetic energy and incompressible kinetic energy are exchanged. Thissuggests that, under the right conditions, pairs of vortices may be able to form meta-stable boundstates.

Introduction.— A striking property of quantum flu-ids (superfluid helium, atomic Bose-Einstein conden-sates, polariton condensates, etc.) is that the circulationof the velocity v around a closed path C is quantized [1]in units of κ = 2π~/M ,∮

C

v · dr = `κ, (1)

where M is the mass of the relevant boson, ~ is the re-duced Planck’s constant, and the integer ` is called thewinding number. In most regions of fluid the circula-tion will be zero, but there may be points (in 2D) orlines (in 3D) where the wavefunction Ψ vanishes, henceits phase is not defined and ` 6= 0. Such topological de-fects (singularities), normally surrounded by circular (in2D) or tubular (in 3D) regions of depleted density, arecalled quantum vortices. The topological nature of thesevortices deeply affects the possible flow patterns (vortexlattices, turbulence, etc.)

Experiments [2–4] show that a multiply-quantised vor-tex (MQV), i.e. a vortex with ` > 1, will spontaneouslydecay into a cluster of singly-quantised vortices (SQVs),each with ` = 1. This tendency is usually justified on thegrounds that, for a given angular momentum, a clusterof SQVs is energetically favourable as compared with aMQV [5]. Hence, in a dissipative scenario, where thesystem relaxes into the lowest energy state, an MQVwill naturally evolve into a cluster of SQVs. In non-dissipative systems, however, the decay can still occurdue to a dynamical instability [6] arising from the cou-pling of the MQV to surrounding phonons.

The instability of MQVs acquires additional signifi-cance if we note that, under certain conditions, there

are analogies between vortices and rotating black holes[7]. It has recently been argued [8] that the dynamicalinstability is related to the existence of an ergoregion,a notion from black hole physics which implies super-radiant amplification of waves in a particular frequencyrange [9]. Superradiance arises not only around rotatingblack holes but in a wide range of systems, e.g. drain-ing vortices [10] and optical vortex beams [11]. However,unbounded growth can occur if there is a mechanism fortrapping superradiant modes in the system, eventuallydriving it into the non-linear regime, like e.g. black holebomb instabilities [12].

In this Letter we study the evolution of an ` = 2 MQVin a bucket trap. Our analytic WKB prediction confirmsthe superradiant character of the dynamical instabilityin the initial linear regime [8]. By solving the full non-linear equations, we also reveal a remarkable recurrentbehaviour of the instability at later times: a modulationin which incompressible kinetic energy and compressiblekinetic energy are periodically exchanged, and the twophase defects, which we call proto-vortices, move in andout while rotating in close proximity to each other, un-able either to form two fully-fledged separated SQVs orto merge back into an MQV. This novel time-dependentstate is the limiting configuration of two parallel quan-tum vortices at close distances comparable to the heal-ing length. We also show that the new time-dependentstate can be captured by a simple two-mode model rep-resenting the proto-vortex separation and the dynami-cally unstable phonon mode. The role of dissipation thenbecomes clear: dissipation prevents the coherent reab-sorption of phonons, allowing the proto-vortices to spiralout and separate, becoming well-defined quantum vor-

arX

iv:2

111.

0256

7v1

[co

nd-m

at.q

uant

-gas

] 4

Nov

202

1

2

tices each surrounded by their own core regions.Vortex states.— We consider the dimensionless two-

dimensional Gross-Pitaevskii equation (GPE),

i∂tΨ =(− 1

2∇2 + V (x)− 1 + |Ψ|2)

Ψ, (2)

where lengths are measured in units of ξ ≡ ~/√Mµ (the

healing length), time in units of τ ≡ ~/µ, and density|Ψ|2 in units of µ/g. Here, µ is the chemical potential, Mis the atomic mass and g is the 2D interaction strength.We work in polar coordinates x = (r, θ). The condensateis confined by a circular bucket-potential of the form,

V (r) =V0

1 + (V0 − 1)ea(rB−r), (3)

where rB is the trap size and V0 and a determine thesteepness of the bucket wall at rB . We choose a = V0 = 5,although our results are essentially independent of thischoice provided the wall at rB is steep.

Using the Madelung representation of the condensatewavefunction Ψ =

√ρeiΦ, the stationary GPE has vortex

solutions with velocity v ≡ ∇Φ = `/r #»e θ, where ∇ isthe 2D gradient operator. Here we focus on the doublywinded vortex with ` = 2. The corresponding densityprofile can then be obtained by substituting Φ = `θ in(2) and solving numerically for ρ.BdG equation.— Since V (x) is independent of t and

θ, linear fluctuations δψ of the condensate wavefunctioncan be decomposed into separate frequency ω and az-imuthal m components,(

δψδψ∗

)=

∫ ∞−∞

dω

2π

∞∑m=−∞

eimθ−iωt(u+e

+i`θ−it

u−e−i`θ+it

), (4)

where we write u± = u±(ω,m, r) for brevity. The fluctu-ations can then be described the state |U〉 = (u+, u−)T

which obeys the Bogoliubov-de Gennes (BdG) equation,

L|U〉 = ω|U〉, L =

(D+ ρ−ρ −D−

),

D± = −1

2

[∂2r+

1

r∂r −

(m± `)2

r2

]+ V (r) + 2ρ− 1.

(5)

The BdG conserves a quantity called the norm,

N =

∫d2x

(|u+|2 − |u−|2

), (6)

which is related to the mode energy by a factor of ω.Solutions to (5) are obtained by diagonalising L to ob-tain the eigenvalues ω and eigenfunctions |U〉. In theright panel of Fig. 1, we display ω as a function of thetrap size rB and indicate with colour the sign of N . Theinstability (a zero norm solution) results from the cou-pling of a positive norm mode to a negative norm one,which (for ` = 2) can only occur for m = 2. For par-ticular system sizes, this coupling is suppressed and the

instability is absent. In the limit that rB →∞, the den-sity of N > 0 states becomes a continuum and the cou-pling always occurs [8]. Note that the unstable growingmode (Im[ω] > 0) is always accompanied by its complexconjugate, which corresponds to a stable decaying mode(Im[ω] < 0). Examples of the relative density eigenfunc-tions δρ/ρ = u+ + u− are also shown on Fig. 1.WKB method.— Deeper insight into these results

can be obtained via a WKB approximation, whereinthe fluctuations are assumed to behave locally likeplane waves. This provides local scattering informa-tion about the waves. Inserting the Ansatz u± ∼A±(r) exp(i

∫p(r) dr) into (5) and neglecting derivatives

of the amplitudes, we obtain the dispersion relation,

Ω2 = ρk2 + k4/4, k = (p2 + m2/r2)12 , (7)

where Ω = ω − m`/r2 is the frequency in a frame co-moving with the vortex and the effective azimuthal num-ber m in the expression for k is given by,

m2 = m2 + `2 + 2r2(ρ+ V (r)− 1). (8)

Since Eq. (7) is quadratic in p2, the dispersion relationwill have two pairs of solutions. One of the pairs corre-sponds to solutions which are evanescent throughout thesystem and must be discarded for the solution to be reg-ular at r = 0. The second pair corresponds to radiallyin- and out-going plane waves (as determined by the signof the radial group velocity #»e r · vg ≡ ∂pω) far from thevortex core and are the relevant ones for the discussionof the instability.

When these modes are propagating, scattering occursat turning points ri (locations where #»er ·vg(ri) = 0 whichoccurs for p = 0). Substituting p = 0 back into (7) yieldstwo curves,

ω± =m`

r2±√ρm2

r2+m4

4r4, (9)

which determine the locations of the turning points rithrough the relations ω = ω±(ri), see Fig. 1 for an exam-ple. Between the two curves, the modes are evanescent,whereas above ω+ (below ω−) they are propagating andhave Ω > 0 (Ω < 0). In the WKB approximation, thesign of Ω coincides with the sign of N in that region.Hence, a frequency which intersects both ω+ and ω− willtunnel from the positive norm branch of the dispersion tothe negative norm one. Due to norm conservation, sucha mode will be amplified (superradiantly) each time itscatters with a turning point.

The possibility of having instabilities arises when am-plified modes are reflected back into the system wherethey are further amplified. Such a scenario occurs forvortices since there is a region inside the vortex core (acavity, see Fig. 1 central panel) where these modes canbecome trapped. Broadly speaking, there will be an in-stability if there is a bound state inside the vortex core

3

0 2 4 6 8

-0.5

0

0.5

1

1.5

0 2 4 6 8

-0.2

0

0.2

0.4

0.6

0.8

1

10 20 30 40 500

0.01

0.02

0.03

FIG. 1. Left panel: Oscillation frequencies and the associated eigenmodes as a function of radius. Horizontal lines are thereal part of the eigenvalues Re[ω] and the superimposed thick lines are the relative density eigenmodes δρ/ρ = u+ + u−, whichare solutions of the BdG (5) for m = ` = 2. The solid vertical line represents the trap size rB = 6.5. Dashed black curves arethe WKB potentials ω± defined in (9), which separate the grey (positive norm), white (evanescent) and pink (negative norm)regions. Modes with N > 0 (N < 0) are coloured black (red). Central panel: the same for rB = 8. The unstable mode,whose real (imaginary) part is shown as solid (broken) green line, results from the coupling of two nearby modes in the leftpanel. The complex conjugate of this mode (not shown) is a decaying solution. Also illustrated are the turning points ri definedbelow (9). Right panel: the eigenvalues ω as a function of rB , with the real (imaginary) part in the upper (lower) panel. BdGsolutions are shown as thick lines and follow the same colour scheme as the previous planels. Solutions to the WKB condition(10) are shown as grey in the background and, for high frequencies, are indistinguishable from the BdG results.

(cavity mode) which couples to one of the normal modesoutside the vortex in the bulk (phonons). Such a cou-pling is only possible in the range of frequencies whichprobe the cavity (the cavity band). In an infinite system,the spectrum of phonons is continuous, hence such a cou-pling is guaranteed. This is not the case for a finite sizedsystem since the phonon spectrum becomes discrete.

To evaluate the normal mode frequencies with theWKB method, we perform a scattering computationanalogous to finding normal mode frequencies of theSchrodinger equation for a potential containing two wells.The full procedure is detailed in our companion paper[13] and leads to the following condition for frequenciesin the cavity band,

4 cot(S01) cot(S2B + π/4) = exp(−2S12), (10)

where Sij(ω) =∫ rjri|p(ω)|dr is the phase integral between

the turning points, see Fig. 1. In deriving this condition,the frequency is assumed real. For a small imaginarypart, the extension to complex frequencies proceeds viaS(ω) ' S(Re[ω]) + iIm[ω]∂ωS|Re[ω] [14].

When the zeros of the two cotangent functions arewell separated, (10) describes two sets of solutions; cav-ity modes satisfy cosS01 ' 0, whereas phonons obeycos(S2B + π/4) ' 0. However, if members of these twosets are close enough in frequency, a coupling occurs re-sulting in two zero norm modes which are a complex con-jugate pair. Solutions to (10) are shown in light grey onthe right panels of Fig. 1. The high frequency solutions(relative to the trap size) agree exceptionally well with

the BdG results. The discrepancy of the cavity modefrequency results from the fact that this mode occupies aregion where the density varies rapidly, thereby increas-ing the error of WKB. This is also the reason why WKBfails to match quantitatively the locations of the stabil-ity windows. Nonetheless, the approximation captures allthe features of the spectrum, which is enough to validatethe interpretation gained with this method.

The connection to superradiance can be seen explicitlyby looking in the large system limit rB →∞ [15]. In thatcase, the condition in (10) reduces to,

cosS01(Re[ω]) = 0, Im[ω] = − log |R|2∂ωS01

∣∣∣∣Re[ω]

, (11)

where |R| = (1+e−2S12/4)/(1−e−2S12/4) is the local re-flection coefficient associated with the tunnelling betweenthe bulk and the cavity. The factor on the denominatoris negative since the cavity modes have N < 0 in thatregion. Hence, the reason these modes are unstable is adirect consequence of superradiance, i.e. |R| > 1.Non-linear evolution.— To verify the existence of

the instability in the full non-linear system, we simulatethe full GPE in (2). Simulations are prepared with thefollowing wave function,

Ψ(x, t = 0) =√ρ`=2(r)e2iθ(1 + δΨ). (12)

The first term corresponds to the doubly-quantised vor-tex background and δΨ = ε(u+e

2iθ + u∗−e−2iθ) is the

seed for the instability where u± are the unstable solu-tions of (5) for m = 2. The constant ε is an overall

4

−4

0

4

−4

0

4

−π/2 0 π/2

Φ0.0 0.5 1.0

ρ(ti)−0.1 0.0 0.1

ρ(t1)− ρ(t0)

0.0

0.5

1.0

1.5

2.0

2.5

Vor

tex

sep

arat

ions

0 1000 2000 3000 4000 5000 6000Time t

10−4

10−3

10−2

10−1

δψm

=2(r

=1.

5)

rB = 24.3 rB = 25.0 rB = 25.7

FIG. 2. The splitting of the initial MQV into two separate singularities (proto-vortices). Panels (a) and (b) show the phase Φfor rB = 25 at t = 50 and t = 904 respectively. Panels (c) and (d) show the density ρ at the same times; here white lines aresurfaces of ρ = 1/10. Panel (e) displays the difference between the density at t = 50 (c) and the density at t = 904 (d): thedominant m = 2 mode is apparent, whose troughs coincide with the locations of the two proto-vortex centres x1 and x2. Theseparation s = |x1 − x2| between the two singularities is shown to oscillate with t in panel (f) for three different values of thetrap size rB . Finally, panel (g) shows the time evolution amplitude of the m = 2 mode inside the cavity. The initial growthrates agree with the prediction of our linear analysis, shown as dotted lines. Note that the blue and green dotted lines areessentially overlapping since the growth rates are the same. The non-linear evolution after the initial stage (t & 1000) consistsof a cycle between growth and decay.

amplitude which keeps the linear perturbation in the ini-tial condition small; we choose ε = 10−3, although pro-vided ε is small enough, the only change to our results isthe length of time before the initial instability becomesO(1). This state is taken as the initial condition for theevolution dictated by (2), for which we use a two-splitFourier-Spectral scheme as outlined in [16], see also theSupplemental Material (SM) [17] for details. In Fig. 2,we display snapshots of Ψ for rB = 25 and evolution ofthe vortex separation for three different values of rB .

We find that the instability predicted by the linearanalysis is present at early times, see panel (g) in Fig. 2for t . 1000 [18]. Whilst the unstable mode is growing,we observe that the initial ` = 2 phase singularity splitsinto two ` = 1 singularities (panels (b) and (d) of Fig. 2),each guided by a trough of the unstable m = 2 mode.Indeed, it is precisely the growth of the negative energypart of the mode (occupying the core of the original vor-tex) which causes the two resulting singularities to spiraloutward, since this lowers the energy of the overall vortexconfiguration [5]. Surprisingly, we find that once the sep-aration between the singularities reaches approximately2 or 3 healing lengths, they begin to spiral back inwards.Since the two singularities occupy the same region of de-pleted density, we refer to them as proto-vortices, as thename vortex is usually understood as a singularity em-bedded in its own low density region (the vortex core).During the inward stage, the m = 2 waveform matchesthat of the linear decaying mode discussed in the captionof Fig. 2. This mode can be thought of as describing

the re-absorption of a phonon by the proto-vortex pair.Eventually, the decay halts and growth resumes, caus-ing a modulation of the separation. This cycle continuesfor the duration of our simulations, trading kinetic en-ergy back and forth. The kinetic energy of the system,in fact, has two contributions [19]: compressible (aris-ing from phonons) and incompressible (arising from theproto-vortices), see SM [17] for the definitions. Fig. 3clearly shows that when the former increases, the latterdecreases.

We remark that during the late-stage of the instability,the two proto-vortices remain confined inside the regionof suppressed density (see Fig. 2 panel (d)). Moreover,the maximum value of the density perturbation |δΨ| dur-ing the evolution is only about 0.2. This suggests thatthe full non-linear dynamics can indeed be described per-turbatively about the original ` = 2 vortex background,contrasting the expectation that a MQV should decaynon-perturbatively into a cluster of well-separated SQVs.

In the SM [17] we describe a simple dynamical systemconsisting of two oscillators X± with opposite sign ener-gies H± = ±( 1

2X2± + V±), which describes the essential

interaction between proto-vortices and phonons. The po-tential and interaction energies are V± = 1

2 (Ω2±σ)X2±−

14c±X

4± and Vint = gX+ ·X−. We show how this system

can be solved in the limit of large Ω to reveal the signa-ture switch between exponential growth and decay seenin our simulations, as shown in the inset of Fig. 3. Thecharacteristic feature which allows this is that the nega-tive energy cavity mode (which results in the formation

5

of proto-vortices) couples only to a single phonon in thebulk, and couplings to all other phonons are considerednegligible.

The essential physics is captured in the following qual-itative argument. Whilst the instability is growing,the non-linearity in the GPE reduces the frequency ofthe cavity mode until it can no longer couple to thephonon that rendered it unstable. After decoupling, thephonon will oscillate faster than the cavity mode andthe phase difference between the two will change. Once∫

(ωph − ωvor)dt ' π, the waveforms of the two modeswill be just right to produce the decaying mode oncethey recombine. For this argument to hold, the den-sity of phonon states in Fig. 1 must be sufficiently lowthat the cavity mode cannot couple to any phonons oflower frequency as it evolves. Hence we would not ex-pect modulations of the proto-vortex separation to occurin a very large system where the density of phonon statesis effectively continuous. This being said, we have foundthat modulations can persist for trap sizes up to at leastrB = 47.

One might assume that this behaviour is highly sensi-tive to initial conditions and that any small perturbationmight be enough to destroy the effect. Whilst this seemsto be the case when the vortex is placed far from thecentre of the trap [4], we have found that our modula-tions persist when even if the vortex is displaced fromthe origin by a few healing lengths.

Finally, we have checked what happens when dampingis added to the dynamics. We have found that energy isslowly removed from the system and, at late times, we re-cover the usual Hamiltonian point vortex behaviour. Be-fore this, however, the cavity mode (whose frequency de-creases monotonically) will couple to any available lowerfrequency phonons, causing sudden modulations of thevortex separation as the singularities drift apart (see theSM [17] for an example).

Conclusion.— We have studied the instability of adoubly-quantised ` = 2 vortex using three distinct meth-ods: a linear BdG stability analysis, a WKB approxi-mation and a fully nonlinear numerical simulation of theGPE. The WKB method allowed us to identify the causeof the instability as a superradiant bound state insidethe vortex core. We then confirmed that the instabil-ity predicted in the linear equations was also present inthe full GPE dynamics. Quite unexpectedly, we foundthat, whilst the instability is present at early times (andwill cause the singularities to separate) the non-linearityin the GPE pushes the proto-vortices back together oncethey reach a critical separation, resulting in a modulationof their separation. Whilst it was already predicted thatinstabilities can be suppressed in certain trap geometries,e.g. [4, 8], it was not known that an unstable vortex statecould do something other than decay into a well sepa-rated pair of SQVs. The observed modulations of theseparation between singularities are suggestive that, un-

0.00

0.05

0.10

0.15

0.20

Com

pre

ssib

le

0 1000 2000 3000 4000 5000Time

−0.04

−0.02

0.00

Inco

mp

ress

ible

0 2500

−0.02

0.00

0.02

FIG. 3. Time evolution of the compressible (green line)and incompressible (red line) kinetic energies associated withphonons and proto-vortices respectively. Note that the in-crease of the compressible kinetic energy corresponds to thedecrease of the incompressible kinetic energy. This supportsour interpretation of the observed modulations being drivenby a back-and-forth energy exchange between phonons andproto-vortices. The inset shows the energies of our simple os-cillator model described in the SM [17], i.e. H+ (green line)and H− (red line), which captures the behaviour of proto-vortices and phonons. For the model, we used the parametersΩ = 1, σ = 1/50, g = 1/200, c+ = 0, and c− = 2.

der the right conditions, co-rotating vortex pairs may beable to form meta-stable bound states. One possible in-terpretation of this is that each of the proto-vortices canbe trapped inside the core of the other. A consequenceof this is that our system never enters the regime whereone can apply Hamiltonian vortex dynamics [20], sincethis requires that the vortex separation be much largerthan the healing length. It would be interesting to seewhether this behaviour extends to more general scenariose.g. clusters of vortices.

Acknowledgements.— SP acknowledges supportfrom Natural Science and Engineering Research Coun-cil (Grant 5-80441 to W. Unruh) and would also like tothank the IQSE, Texas A&M University, and grants ONR(Award No. N00014-20-1-2184) and NSF (Grant No.PHY-2013771) for support and an intellectually stimu-lating environment while part of this work was done.CB and SW acknowledge support provided by the Sci-ence and Technology Facilities Council on Quantum Sim-ulators for Fundamental Physics (ST/T00584X/1 andST/T006900/1) as part of the Quantum Technologiesfor Fundamental Physics programme. SW acknowledgessupport provided by the Leverhulme Research Leader-ship Award (RL-2019 - 020), the Royal Society Univer-sity Research Fellowship (UF120112) and the Royal Soci-ety Enhancement Grant (RGF/EA/180286), and partialsupport by the Science and Technology Facilities Coun-

6

cil (Theory Consolidated Grant ST/P000703/1). AG andSW acknowledge support provided by the Royal SocietyEnhancement Grant (RGF/EA/181015). SE and SWacknowledge support from the EPSRC Project Grant(EP/P00637X/1). SE acknowledges partial supportthrough the Wiener Wissenschafts- und Technologie-Fonds (WWTF) project No MA16 - 066 (“SEQUEX”),and funding from the European Union’s Horizon 2020research and innovation programme under the MarieSklodowska-Curie grant agreement No 801110 and theAustrian Federal Ministry of Education, Science and Re-search (BMBWF) from an ESQ fellowship.

[1] R. P. Feynman, in Progress in low temperature physics,Vol. 1 (Elsevier, 1955) pp. 17–53.

[2] Y. Shin, M. Saba, M. Vengalattore, T. A. Pasquini,C. Sanner, A. Leanhardt, M. Prentiss, D. Pritchard, andW. Ketterle, Phys. Rev. Lett. 93, 160406 (2004).

[3] T. Isoshima, M. Okano, H. Yasuda, K. Kasa, J. Huh-tamaki, M. Kumakura, and Y. Takahashi, Phys. Rev.Lett. 99, 200403 (2007).

[4] M. Okano, H. Yasuda, K. Kasa, M. Kumakura, andY. Takahashi, J. Low Temp. Phys. 148, 447 (2007).

[5] C. F. Barenghi and N. G. Parker, A primer on quantumfluids (Springer, 2016).

[6] H. Pu, C. K. Law, J. H. Eberly, and N. P. Bigelow, Phys.Rev. A 59, 1533 (1999).

[7] T. Torres, S. Patrick, M. Richartz, and S. Weinfurtner,Class. Quant. Grav. 36, 194002 (2019).

[8] L. Giacomelli and I. Carusotto, Phys. Rev. Reasearch 2,033139 (2020).

[9] R. Brito, V. Cardoso, and P. Pani, Superradiance(Springer, 2020).

[10] T. Torres, S. Patrick, A. Coutant, M. Richartz, E. W.Tedford, and S. Weinfurtner, Nat. Phys. 13, 833 (2017).

[11] M. C. Braidotti, R. Prizia, C. Maitland, F. Marino,A. Prain, I. Starshynov, N. Westerberg, E. M. Wright,and D. Faccio, arXiv preprint arXiv:2109.02307 (2021).

[12] S. R. Dolan, Phys. Rev. D 76, 084001 (2007).[13] In preparation.[14] Note that since the WKB method fails close to rB where

V in (3) varies rapidly, we approximate the integral S2B

by assuming that ρ has the same form as that in theinfinite system. Imposing a Neumann boundary conditionat rB gives the π/4 term in (10).

[15] See our companion paper [13] for a comment on the dif-ference between rB → ∞ and a system which is trulyinfinite, i.e. has open boundary conditions.

[16] J. Javanainen and J. Ruostekoski, J. Phys. A: Mathemat-ical and General 39, L179–L184 (2006).

[17] Please see Supplemental Material for details of our GPEsolver, the effect of adding damping to the system and adescription of our two-oscillator model.

[18] We have also confirmed that unstable modes are absentin the stable windows predicted by Fig. 1.

[19] C. Nore, M. Abid, and M. E. Brachet, Phys. Fluids 9,2644 (1997).

[20] P. K. Newton and M. F. Platzer, Appl. Mech. Rev. 55,B15 (2002).

[21] C. Nore, M. Abid, and M. Brachet, Phys. Rev. Lett. 78,3896 (1997).

[22] C. Nore, M. Abid, and M. Brachet, Phys. Fluids 9, 2644(1997).

[23] S. P. Cockburn and N. P. Proukakis, Laser Phys. 19,558–570 (2009).

[24] N. P. Proukakis and B. Jackson, J. Phys. B: Atomic,Molecular and Optical Physics 41, 203002 (2008).

[25] N. Parker, Numerical studies of vortices and dark soli-tons in atomic Bose-Einstein condensates, Ph.D. thesis,Durham University (2004).

7

Supplemental Material

1. Simulations

The numerical simulations of vortex decay presented inFig. 2 is performed in three steps: (1) Preparation of theinitial state, (2) Time evolution using the GPE (2) and(3) mode extraction.

For a given trap potential V (r) of the form (3), the ini-tial state is prepared by first estimating the lowest energyconfiguration for a central ` = 2 vortex. This is done byfixing the phase Ψ ≡ √ρei`θ and evolving the modulus√ρ in imaginary time τ ≡ it, i.e.

∂τ√ρ =

(1

2∂2r +

1

2r∂r −

`2

2r2− V (r) + 1− ρ

)√ρ. (13)

The resulting density ρ is inserted into a finite-differencematrix formulation of the BdG equation (5). Numeri-cally solving for the eigenmodes and selecting the solu-tion |U〉 = (u+, u−)T with the largest imaginary part,allows for the construction of the initial state,

Ψ0 =√ρ0(r)ei`θ

[1 + εu+(r)eimθ + εu∗−(r)e−imθ

].(14)

Here ε 1 is the initial amplitude of the unstable mode.For the data presented in Fig. 2, the values used areε = 10−3, m = 2 and ` = 2.

The state Ψ0 serves as the initial state for the full GPEsimulation. Here, the cartesian plane is discretized intoN ×N linearly spaced mesh of pixels with separation of∆l in each dimension. The time evolution proceeds in dis-crete timesteps of duration ∆t using a Fourier split oper-ator method (see [16] for further details), which amountsto,

Ψ(r, t+ ∆t) ' e i∆t2 ∇

2

e−i∆t(V−1+|Ψ|2)Ψ(r, t). (15)

Provided that the boundary x, y = ± 12 (N − 1)∆l of the

simulation domain is well outside the potential bound-ary rB , the wavefunction Ψ is sufficiently periodic for aFourier-spectral evaluation of the exponentiated Lapla-cian in (15). In the simulation presented in Fig. 2,the values N = 768, ∆l = 1/10 and ∆t = 10−3 wereused along with a potential of the form (3) with a = 5,V0 = 5, rB = 25. Armed with the initial state andthe discretization scheme outlined above, we perform8192000 = 16384 × 500 timesteps of which every 500th

frame is stored for later processing. The result is a col-lection Ψ(xi, yj , tk) ∈ CN2×Nt .

To extract the evolution of the unstable mode, thewavefunction Ψ(xi, yj , tk) is transformed to polar coor-dinates Ψ(ri, θj , tk) and Fourier transformed in the az-imuthal direction to give Ψ(ri,mj , tk) where mj is the jth

azimuthal component. The imaginary part Im[ω] of thefrequency of the unstable mode is found by performing

a log-linear fit to |Ψ|(ri,mj , tk) at ri = 1.5 and mj = 2.The real part Re[ω] is computed (at the same points)using a temporal Fourier transform.

The energy,

E =

∫d2x

1

2|∇√ρ|2︸ ︷︷ ︸Eqnt

+1

2|√ρ∇Φ|2︸ ︷︷ ︸Ekin

+ V ρ︸︷︷︸Epot

+1

2ρ2︸︷︷︸

Eint

(16)

associated with a state Ψ =√ρeiΦ in the GPE may be

decomposed into a quantum energy Eqnt, kinetic energyEkin, trap energy Epot, and interaction energy Eint. Asfirst proposed by Nore et al. [21, 22], the kinetic energyEkin may be further split into a compressible part Eckin

and an incompressible part Eikin. Defining u ≡ √ρ∇Φand introducing u ≡ uc + ui with ∇ · ui = 0, the twocomponents of the kinetic energy takes the form Eikin =∫d2x 1

2 |ui|2 and Eckin =∫d2x 1

2 |uc|2. Numerically, sucha decomposition may be obtained from the realisationthat if F denotes a spatial Fourier transform and k thecorresponding wave vector, then uc is nothing but theprojection of u onto k, i.e.

uc = F−1

[k(k · Fu)

|k|2], (17)

where, in the absence of a mean flow, the k = 0 compo-nent may be ignored to avoid zero-division.

2. Damping

Dissipation is introduced in the GPE by using the phe-nomenological damping parameter γ [23, 24],

i∂tΨ = (1− iγ)(− 1

2∇2 + V (x)− 1 + |Ψ|2)

Ψ. (18)

The presence of dissipation damps out the oscillations ofthe two proto-vortices, which spiral away from each otherand develop separate core regions. At this point we re-cover the well-known configuration of two point-vorticesof the same sign, which rotate around each other withan orbital frequency inversely proportional to the squareof the vortex separation s, as shown in Fig. 4. The factthat the point-vortex description predicts a divergence ass→ 0, signalling the break-down of the model, is a con-sequence of the overlapping vortex cores. In this regime,the system is better described as a perturbation (cavitymode) on an ` = 2 vortex background. This mode en-ters the linear regime in the limit s→ 0 and we see thatthe orbital frequency of the singularities tends to half ofthe oscillation frequency of the instability. The half isbecause two nearby singularities make a dipole pertur-bation, i.e. m = 2, hence it takes twice the time for agiven peak of the m = 2 mode to return to it’s originalposition.

8

0 1 2 3 4 5 6 7 8Vortex separation s

0.00

0.05

0.10

0.15

0.20

0.25

0.30O

rbit

alfr

equ

ency

Ωv

Ω ' 12Re[ωinstab]

Point-vortex model Ω ' 2/s2

N. Parkers thesis

Wavelet orbital frequency

FIG. 4. The orbital frequency Ω of the two singularities asa function of distance s between them (black line). At smalls, the singularities are proto-vortices with Ω correspondingto frequency of the unstable cavity mode (blue dashed line).At large s, the singularities are true vortices which obeys theexpected Hamiltonian point vortex dynamics (red line). Wecompare our data to a function used by Parker [25] (greencrosses) who also considered the dynamics of nearby vortices.

Another interesting feature of Fig. 4 is the sudden os-cillations in s as the orbital frequency decreases. This oc-curs when the cavity mode reaches the correct frequencyto couple to another phonon in the system. There are twosuch oscillations in Fig. 4 because at rB = 25, there aretwo phonons with lower frequency than the cavity mode.Hence for a small system where the cavity mode initiallycouples to the lowest phonon, these sudden additionaloscillations would not occur.

3. Two-oscillator model

The switch from exponential growth to decay observedin our simulations is characteristic of two oscillators (withopposite sign energies) interacting under the influenceof a non-linearity. We illustrate this using a simplifiedmodel with the following Lagrangian,

L = 12X

2+ − V+ − 1

2X2− + V− − gX+ ·X−,

V± = 12 (Ω2 ± σ)X2

± − 14c±X

4±,

(19)

which describes two particles at X± = (X±, Y±) oscil-lating about the coordinate origin with energies H± =±( 1

2X2± + V±) respectively. The interaction energy be-

tween the two particles is gX+ · X−. We set c+ = 0and c− = ε > 0 so that the non-linearity only affects theparticle located at X−.

To see that this model captures the essential featuresof the vortex evolution, we consider two limiting cases.When g = 0, the two particles oscillate at fixed radii

as X± ∼ e−iω±t with frequencies ω+ = Ω2 + σ 12 and

ω− = Ω2−σ−εX2−

12 . Here X− mimics the way the or-

bital frequency of two point vortices decreases as the dis-tance between them grows, whilst X+ mimics the phononwhose frequency (determined predominantly the systemsize) stays nearly constant. Next, when ε = 0, both par-ticles oscillate about the origin as a linear superpositionof the frequencies ω = Ω2 ±

√σ2 − g2 1

2 . Notice thatwhen the oscillator spacing is smaller than the coupling,|σ| < |g|, this will have unstable solutions, mimicking thebehaviour in Fig. 1 where instabilities occur if a N > 0mode comes close to a N < 0 mode in the ω-plane.

The full model with ε, g non-zero has an elegant so-lution in the regime Ω σ, g, εX2

− (which is the rele-vant one for the vortex where Re[ω] is an order of magn-tiude larger than Im[ω]). Defining the complex variableZ± = eiΩt(X± + iY±) and rescaling such that Ω = 1 andε = 2, the Lagrangian becomes,

L = Im[Z+Z∗+ − Z−Z∗−]− 1

2σ(|Z+|2 + |Z−|2)

− gRe[Z+Z∗−]− 1

2 |Z−|4.(20)

This has a conserved charge Q = |Z+|2−|Z−|2 analogousto N for the vortex. For exponentially growing/decayingmodes, this is only conserved when Q = 0, implyingZ± = |Z±|e−iϕ± with |Z±| = R. In terms of R andthe phase difference ϑ = ϕ+ − ϕ−, we get,

L = R2(ϑ− σ − g cosϑ− 12R

2), (21)

which leads to the following equations of motion,

ϑ = σ + g cosϑ+R2, R = 12gR sinϑ. (22)

These can then be combined into a single equation for ϑ,

ϑ+W ′(ϑ) = 0, W (ϑ) = − 12 (σ + g cosϑ)2, (23)

which describes a particle on S1 moving through a po-tential W . When the linear equations are unstable, Whas two maxima ϑ± = π ± cos−1(σ/g). By consideringthe phase of the solutions when ε = 0, one finds thatthe stationary solutions ϑ = ϑ− and ϑ = ϑ+ correspondrespectively to the linearly growing and decaying modes.However, in the non-linear case, (22) tells us that a non-zero initial amplitude in the instability means ϑ 6= 0,making ϑ roll all the way over the two peaks. Onceϑ(t) is known, the amplitude can be found by solvingthe equation in (22) for R,

R(t) ∼ exp

(1

2

∫g sinϑ(t)dt

), (24)

which tells us that R switches between exponentiallygrowing and decaying solutions as ϑ changes by 2π. Thisbehaviour is illustrated in Fig. 5.

The key feature of this model that renders it so simpleis that is contains only two interacting modes. This is

9

0- +

2

0

0

2

3

4

5

0 10 20 30 40 50

-5

0

5

FIG. 5. Left panel: A schematic of the two interacting oscillators described by (19), where X+ represents a positive energyphonon and X− represents the negative energy oscillation of the vortex (cavity mode). Central panel: In the limit that thecentral oscillator frequency Ω is much larger than other scales in the problem, the dynamics can be re-expressed as a non-linearoscillator ϑ moving through periodic potential W with two wells. Right panel: When ϑ evolves as it rolls along W , the originalamplitude R = |X±| switches between exponential growth and decay, mimicking the observed behaviour in our simulations.

what leads to the periodicity of the growth-decay cyclesin Fig. 5. When other modes are taken into account (i.e.the other ω and m modes around the vortex) energy can

be dissipated into these channels and we would expectthese cycles to lose perfect periodicity. This imperfectperiodicity appears to be what we see in Fig. 2 of themain text.