This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License

Newcastle University ePrints - eprint.ncl.ac.uk

Cidrim A, dos Santos FEA, Galantucci L, Bagnato VS, Barenghi CF.

Controlled polarization of two-dimensional quantum turbulence in atomic

Bose-Einstein condensates.

Physical Review A 2016, 93(3), 033651

Copyright:

This is the authors manuscript of an article that has been published in its final form by the American

Physical Society, 2016.

DOI link to article:

http://dx.doi.org/10.1103/PhysRevA.93.033651

Date deposited:

27/04/2016

http://creativecommons.org/licenses/by-nc/3.0/deed.en_GBhttp://eprint.ncl.ac.uk/javascript:ViewPublication(223913);javascript:ViewPublication(223913);http://dx.doi.org/10.1103/PhysRevA.93.033651

Controlled polarization of two-dimensional quantum turbulence in atomicBose-Einstein condensates

A. Cidrim,1, 2 F. E. A. dos Santos,3 L. Galantucci,2 V. S. Bagnato,1 and C. F. Barenghi2

1Instituto de F́ısica de São Carlos, Universidade de São Paulo,Caixa Postal 369, 13560-970 São Carlos, São Paulo, Brazil

2JQC (Joint Quantum Centre Durham-Newcastle) and School of Mathematics and Statistics,Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom

3Departamento de F́ısica, Universidade Federal de São Carlos, 13565-905, São Carlos, SP, Brazil

We propose a scheme for generating two-dimensional turbulence in harmonically trapped atomiccondensates with the novelty of controlling the polarization (net rotation) of the turbulence. Ourscheme is based on an initial giant (multicharged) vortex which induces a large-scale circular flow.Two thin obstacles, created by blue-detuned laser beams, speed up the decay of the giant vortexinto many singly-quantized vortices of the same circulation; at the same time, vortex-antivortexpairs are created by the decaying circular flow past the obstacles. Rotation of the obstacles againstthe circular flow controls the relative proportion of positive and negative vortices, from the limitof strongly anisotropic turbulence (almost all vortices having the same sign) to that of isotropicturbulence (equal number of vortices and antivortices). Using the new scheme, we numericallystudy the decay of 2D quantum turbulence as a function of the polarization. Finally, we present amodel for the decay rate of the vortex number which fits our numerical experiment curves, with thenovelty of taking into account polarization time-dependence.

PACS numbers: 03.75.Lm, 67.25.dk, 67.85.De

I. INTRODUCTION

The study of quantum turbulence is heavily motivatedby liquid helium (4He and 3He) experiments [1, 2]. Astriking discovery has been that, under appropriate forc-ing, quasi-classical behavior arises displaying statisticalproperties characteristic of ordinary turbulence; an ex-ample is the celebrated Kolmogorov −5/3 scaling of theenergy spectrum [3] which suggests the existence of aclassical energy cascade from large to small length scales.Under other conditions, a different kind of turbulence(called ‘ultra-quantum turbulence’ or ‘Vinen turbulence’)has also been found [4, 5], characterized by random tan-gles of vortices without large-scale, energy-containingflow structures. Quantum turbulence experiments arealso performed in atomic Bose-Einstein condensates [6–8]; the relative small size of these condensates (comparedto flows of liquid helium or of ordinary fluids) limits thestudy of scaling laws but offers opportunities to studyminimal processes that also take place in larger systems(e.g. vortex interactions, vortex reconnections, vortexclustering) with greater experimental controllability andmore direct visualization than in liquid helium.

Atomic condensates are also ideal systems to studytwo-dimensional (2D) turbulence [9], a problem with im-portant applications to oceans, planetary atmospheresand astrophysics. In classical systems, reduced dimen-sionality may arise from strong anisotropy, stratificationor rotation (via the Taylor-Proudman theorem). Fromthe physicist’s point of view, the dynamics of 2D turbu-lence is very different from 3D [10]. The existence (be-sides the kinetic energy) of a second inviscid quadraticinvariant - the enstrophy - implies that a downscale en-strophy transfer is accompanied by an upscale energy

cascade; in other words, in 2D turbulent flows the en-ergy flows from small to large length scales rather thanvice-versa as in 3D turbulence. With the possible excep-tion of soap films [11], 2D flows which can be created inthe laboratory are only approximations. However, usingsuitable trapping potentials, atomic condensates can beeasily shaped so that vortex dynamics is 2D rather than3D. Unlike liquid helium, in atomic condensates 2D quan-tum vortices can be directly imaged, and, unlike classicalsystems, the motion of such 2D vortices is not hinderedby viscous effects or friction with the substrate.

Several works have explored the generation of turbu-lence in 2D condensates. The 2D energy spectrum andscaling laws have been computed in numerical simula-tions [12–14], and the problem of what should be thequantum analogue of the classical enstrophy has beenraised. In Ref. [8] vortices were nucleated by small-scalestirring of a laser spoon, after which a persistent cur-rent was verified both experimentally and through nu-merical simulations, suggesting transfer of incompressiblekinetic energy from small to large length scales. Emer-gence of large-scale order from vortex turbulence was alsoobserved [15] as predicted by the ‘vortex gas’ theory ofOnsager. A similar set-up was used to explore vortexshedding and annihilation processes in both experiments[16, 17] and simulations [18]. The effect of stirring laserbeams with different shapes or along different paths wasinvestigated in Refs. [19–22]. However, in all cases cited,vortices have always been generated in such a way thatthe number of positive and negative vortices is approxi-mately the same; in other words, all vortex configurationswhich have been investigated had approximately zero po-larization. Since irrotational flow is a hallmark propertyof superfluidity, the polarization of the vortex configura-

2

tion (i.e. the relative proportion of positive and negativevortices) plays the role of net rotational angular veloc-ity Ω of a classical fluid, so it is important to explore itseffects on the properties of turbulence.

In this work, we propose a new scheme for generating2D quantum turbulence in atomic condensates. The nov-elty of our scheme, which is based on a giant vortex asthe initial state, is control over polarization of the turbu-lence, which can be interpreted as the classical rotationof the entire flow. One of the most important propertiesof turbulence is its decay, because the growth of the tur-bulence or its character in a steady state may depend onhow it is forced, whereas the decay is an intrinsic prop-erty of the dynamics. We shall report the decay of 2Dquantum turbulence as a function of the polarization.

II. GIANT VORTEX AND SMALL PINS

Multicharged vortices with circulations as large as 60quanta have already been produced in condensates us-ing dynamical methods, as consequences of rapid rota-tions of the confining trap [23]. Another route to achievethese highly excited states is using phase-engineeringtechniques, such as those described in Refs [24–27]. Inthese cases, quanta of angular momentum are added tothe condensate by adiabatically inverting the direction ofthe magnetic bias-field which composes the usual Ioffe-Pritchard magnetic trap. To date only charges below10 quanta were produced using their proposed set-ups.However, an improvement on the method, known as the‘vortex-pump’, has been described in Refs. [28–30]. Inpractical terms, a hexapole magnetic field is superposedto the Ioffe-Pritchard magnetic trap, allowing vorticityto be cyclically pumped into the condensate, thus gen-erating giant vortices. Progress in this direction hasbeen made in recent experiments with synthetic magneticmonopoles [31].

A giant vortex at the center of a harmonically trappedcondensate can be described by a single-particle wave-function of the form ψ(r) = f(r)eiκφ, where f(r) is thewave-function’s amplitude, r = (r, φ, z) is the positionin cylindrical coordinates, and a large winding numberκ corresponds to a large angular momentum. Such gi-ant vortices are dynamically unstable [29, 32], and splitinto singly-quantized (κ = 1) vortices. Being parallelto one another, these singly-quantized vortices imposea strongly azimuthal flow to the condensate. During thefollowing evolution, some vortices of the opposite polaritymay be generated by occasional large-amplitude densitywaves, but these events are rare, and do not change themain property of the flow resulting from the decay of agiant vortex configuration: the strong polarization of thevorticity - almost all vortices have the same sign.

The scheme that we propose uses blue-detuned lasers[17] to perturb this initial state with two diametricallyopposite laser beams, creating thin obstacles (which werefer to as pins) with width σ of the order of magnitude

of the healing length ξ (two pins are enough to homoge-nize the vortex distribution). The pins perturb the initialgiant vortex, accelerating its decay; they also deflect thelarge azimuthal flow, generating vortex-antivortex pairs[18, 33–35]. To control the effect of the pins, we movethem at constant angular velocity ω in the direction op-posite to the main azimuthal flow.

III. MODEL

The dynamics of our system is dictated by the 2DGross-Pitaevskii equation (GPE). We introduce dimen-sionless variables based on the trapping potential of fre-quency ω0, measuring times, distances, and energies inunits of ω−10 ,

√~/mω0 and ~ω0 respectively, where m

is the mass of one atom and ~ is the reduced Planck’sconstant. The resulting dimensionless GPE is

i∂ψ

∂t=

(−1

2∇2 + V + C |ψ|2 − µ

)ψ, (1)

where the time-dependent wavefunction ψ(r, t) is nor-malized so that

∫|ψ|2d2r = 1. The external poten-

tial is V (r, t) = Vtrap(r) + Vpins(r, t), where Vtrap(r) =(x2 + y2)/2 and Vpins(r, t) = V+(r, t) +V−(r, t) representrespectively the trapping potential which confines thecondensates and the pins which perturb the initial giantvortex. The terms V±(r, t) = V0 exp

{−|r− r±(t)|2/2σ2

}with r±(t) = [±x0 cos (ωt), y0 sin (ωt)] are diametrically-opposite, thin, Gaussian potentials of width σ = ξ whichrotate clockwise (against the flow of the initially imposedgiant vortex) at constant angular velocity ω. The quan-

tity C = 2√

2πN(a/az) parametrizes the two-body col-lisions between the atoms, where N is the total numberof atoms, a the scattering length, and az the axial har-monic oscillator’s length; we choose C = 17300. Thechemical potential µ is introduced to guarantee normal-ization of the wave-function, and the amplitude of thepins is V0 ≈ 1.43µ. In homogeneous systems (V = 0)the healing length is found by balancing kinetic and in-teraction energies terms in the GPE. In a harmonicallytrapped condensate, the healing length can be definedwith reference to the density at the center of the trappedcondensate in the absence of any vortex or hole. In ourdimensionless units, we obtain ξ ≈ 0.13, and rTF ≈ 74ξfor the Thomas-Fermi radius.

Our choice of dimensionless parameters correspondsto typical [16, 17] experiments with 23Na condensates(scattering length a = 2.75 nm, atom mass m = 3.82 ×10−26kg) with N = 1.3 × 106 atoms, radial and axialtrapping frequencies ω0 = 2π × 9 Hz and ωz = 2π × 400Hz, radial and axial harmonic oscillator’s lengths a0 =√~/mω0 ≈ 7.1 µm and az =

√~/mωz ≈ 1.0 µm, for

which the dimensional healing length is ξ = 0.13a0 ≈ 0.9µm; the laser beam would then have a Gaussian 1/e2 ra-dius of w0 = 2σ ≈ 1.8 µm. Blue-detuned Gaussian laserbeams have been used as pins in a series of experiments

3

with highly-oblate BECs [17, 22, 36]. Particularly in [36],a laser beam of width w0 ≈ 2 µm was used to stir a 2D87Rb condensate, similarly to what we propose, main-taining a circular motion with the help of piezo-drivenmirrors.

In order to define our initial state, a circulation of 37quanta (i.e. winding number κ = 37) is initially im-printed around the center of the Thomas-Fermi profile,thus imposing an initial counter-clockwise circular flow.Changing t into −it in Eq. (1), we shortly evolve the statefor t = 0.09 in imaginary-time description, guaranteeinga fixed phase of 2πκ in the center of the condensate andadjusting the density to the presence of the pins. Wethen compute the evolution in real time.

By substituting t → (1 − iγ)t in Eq. (1), we are leftwith a phenomenological dissipative GPE (dGPE), whereγ is a dissipation constant which models the interactionof the condensate with the surrounding thermal cloud.This equation can be used to investigate the effect of fi-nite temperature in our system. With this aim, we alsorun simulations with the same initial states using dGPEinstead of GPE. We choose γ = 3.0×10−4, a typical valueof dissipative parameter [37–39], particularly chosen forthe for experimentally realistic case found in current ex-periments [16, 18]. Summarizing beforehand, we find thesame overall behavior for both dissipation-less and thisspecific dissipative case.

All numeric simulations are performed in the 2D do-main −25 ≤ x, y ≤ 25 on a 512× 512 grid using the 4thorder Runge-Kutta method in Fourier space with the helpof XMDS [40].

IV. RESULTS

A. Creating Polarized Flow

We simulate the real-time evolution of the system fordifferent values of the pins’ angular velocity: ω = 0, π/16,π/8, π/6, and π/4. A series of snapshots for the case ofω = 0 is shown in Fig. 1 to exemplify a typical run.The initial large hole at the center of the figure is thecore of the giant vortex. The two small holes (north andsouth of the giant hole) are the two stationary pins. Thecritical velocity vc for the creation of a vortex-antivortexpair depends on the barrier’s shape [18, 41] and also oninhomogeneities of the system [17]. Typically vc/c ∼0.1 − 0.4 for infinitely high cylindrical barriers, wherec is the local speed of sound. Since our barriers (thepins) are either stationary or rotate against the mainflow, vortex shedding is a dissipative mechanism whichslows down the superfluid’s azimuthal flow and removesangular momentum.

Besides generating vortices of opposite sign, the pinsact as a perturbation to the giant vortex and acceler-ate its decay process; for example, a wave front whichperturbs the core of the giant vortex is visible at timet = 0.9 in Fig. 1. The decay of the giant vortex takes

t = 0.0 t = 0.9 t = 3.6

t = 5.6 t = 7.2 t = 8.1

t = 10.2 t = 50.8 t = 150.0

FIG. 1. (Color online). Density plots of the condensate atdifferent times t for ω = 0 (non-rotating obstacles). Regionsof large/low density are displayed in white/black respectively.Red triangles and blue circles identify positive-charged andnegative-charged vortices respectively. The giant vortex de-cays by injecting a large number of singly-quantized (positive)vortices into the condensate, whilst the pins generate vortexpairs, as can be clearly seen at time t = 0.9.

place via deformation of the core, which becomes ellip-tical before vanishing, and injecting a large number ofpositive, singly-quantized vortices into the condensate.At the same time, vortex-antivortex pairs are created bythe flow past the pins. This process continues until thelarge azimuthal flow is lower than the critical velocityvc; at that point the giant vortex has disappeared, andthe pins are practically unable to generate further vor-tices. Therefore, after this slowdown and due to theirsmall sizes, the pins are practically irrelevant to the vor-tex dynamics (apart from occasional creation of pairs inthe fast rotating case, ω = π/4). In spite of that, in or-der to study the vortex number decay, we simply removethem at t = 82 and allow for longer simulations.

We perform a phase-unwrapping procedure and, by de-tecting windings of ±2π around small closed paths (pla-quettes) on the phase-profile [19], we are able to countthe numbers N+ and N− of positive and negative singly-quantized vortices in the system (anticlockwise and clock-wise circulation respectively). This vortex detection algo-rithm uses a density-cut criterion (∼ 0.75 of |ψ|2’s mean-value) to avoid detection of ghost-vortices. Given theinitial giant vortex (which is multicharged and thereforenot detected by our vortex-detecting algorithm), depend-ing on the value of ω, there can be an imbalance of N+

and N− throughout the evolution. Vortices can be ex-

4

t

50

100

150

200

250

300

350

400N

tot(t)

(a) ω = 0ω = π/16

ω = π/8

ω = π/6

ω = π/4

0 20 40 60 80 100 120 140t

−1.0

−0.5

0.0

0.5

1.0

P(t

)

(b)

FIG. 2. (Color online). (a) total number of vortices, Ntotvs time t; (b) polarization P vs time t. The vertical dashedline marks the time (t = 82) we use to make initial statesfor longer simulation without the pins. The curves are dis-tributed in increasing value of ω from bottom to top, forthe top plot, and conversely, for the bottom plot. In (a),at t ≈ 8, from bottom to top, the curves refer respectivelyto ω = 0, π/16, π/8, π/6, and π/4. In (b), from top to bot-tom, the curves refer respectively to the same latter series ofincreasing values of ω.

pelled from the condensate due to their mutual interac-tion, spiral out of the condensate because of dissipation,or undergo vortex-pair annihilation processes. In our par-ticular finite-temperature simulation, we verify that thechosen experimentally realistic value of the dissipationparameter γ is small enough that, on the time scale ana-lyzed (and compared to the dissipation-less simulations),dissipation-induced spiraling out of individual vortices isless significant than vortex interactions or annihilations.

After the decay of the initial giant vortex, the imbal-ance of positive and negative vortices is measured by thepolarization

P =(N+ −N−)(N+ +N−)

(2)

which takes maximum/minimum values (P = ±1) if allvortices have positive/negative sign. Fig. 2 shows thetime evolution of the total number of vortices Ntot(t) =N+ +N− and of the polarization P (t) under influence ofthe obstacles (present throughout the whole evolution)with angular velocity ω. The top part (a) of the figureshows that the maximum value of Ntot(t) increases withω. It is apparent that, by choosing ω, we can control the

polarization. We use this tunable mechanism to createinitial vortex distributions (without the pins) as shownin Fig. 3, which plots Ntot(t) and P (t) for initial statestaken from instant t = 82 of Fig. 2. Clearly, by tuningω we can produce a condensate free of external holes(the giant vortex or the obstacles) with approximatelythe desired vortex polarization.

t

20

30

40

50

60

70

80

90

100

Nto

t(t)

(a) ω = 0ω = π/16

ω = π/8

ω = π/6

ω = π/4

0 100 200 300 400 500t

−1.0

−0.5

0.0

0.5

1.0

P(t

)

(b)

FIG. 3. (Color online). (a) total number of vortices Ntot;(b) polarization P vs time t, from initial states created at t =82 in the previous stirring process (Fig. 2), labeled by the an-gular velocities which generated them. The pins are removedand we evolve those states longer in time to study the vortexnumber decay. In (a), at t ≈ 150, from bottom to top, thecurves refer respectively to ω = 0, π/4, π/16, π/6, and π/8. In(b), from top to bottom, the curves are labeled as in Fig. 2(b).

By numerically detecting each vortex and its trajec-tory, we determineNtot at each time step. By subtractionfrom the initial total vortex number, N0 = Ntot(0), wecan infer the number of vortices which have drifted outof the condensate and the number of vortices which havedisappeared in annihilation events, colliding with vorticesof opposite sign. We find that such vortex-antivortex an-nihilation events generate density waves, as already re-ported [18, 42, 43], turning incompressible energy intocompressible energy. The reverse mechanism is also pos-sible [44] and in our 2D case takes the form of vortex-antivortex creation events, which we observe. Creationevents occur when the motion of the vortices induces asufficiently deep density wave, or when a large amplitudewave approaches the edge of the condensate where thelocal speed of sound c is less than in the central region.We have also observed annihilations events immediately

5

followed by creation events: this sequence happens whena vortex collides with an antivortex, producing a largesound wave, which almost immediately generates a newvortex- antivortex pair, due to the changing value of thelocal ratio vc/c; this effect happens near the condensate’sedge.

B. Polarized Turbulence Decay

Starting from t > 82 (when we remove the pins andstart a new simulation) we examine whether there is asimple law for turbulence decay in 2D condensates. Weremark that, consistently with previous work [16, 18], inthe time scales under investigations we do not observea tendency to form large-scale clusters of vortices of thesame sign - an effect called the inverse energy cascadein fluid dynamics and the negative temperature in thecase of the Onsager gas of vortex points; the reason, asexplained in a recent study [45], is the harmonic shapeof the trapping potential. It has been suggested [16, 18]that the decay rate of the total number of vortices is notexponential but can be phenomenologically described bythe logistic equation

dNtotdt

= −Γ1Ntot − Γ2N2tot, (3)

where the linear term refers to vortex drifting out ofthe condensate, the non-linear term arises from vortex-antivortex annihilation events, and the coefficients Γ1and Γ2 are rates to be determined. We find that thesolution of the logistic equation fits our decays for t > 82(after pins removal) fairly well. However, in most caseswhich we examined, the fitting parameter Γ1 is nega-tive, corresponding to positive growth. Clearly, afterthe pins are removed, no vortex generation is expected(apart from occasional creation of vortex-antivortex pairsas mentioned above); therefore a naive association of thelinear term of the logistic equation with vortex driftingout of the condensate does not seem appropriate.

In alternative to the logistic model of Eq. (3), we pro-pose a model which captures some essential physics ofthe complex vortex interaction, although only in a ideal-ized way. Firstly, we model the rate of drift of vorticesout of the condensate (attributed to the linear term ofEq. (3) in [16, 18]). Consider a positive vortex near theedge of the condensate. In the first approximation, itstrajectory is a random zig-zag caused by the other vor-tices (see Fig. 5). The azimuthal velocity component ofthe vortex, vθ, will be biased by its sign and, hence, the(opposite) sign of its image with respect to the boundaryof the condensate (in the present case of a positive vor-tex, the interaction with its negative image will give tovθ an anti-clockwise contribution). This azimuthal flow,however, gives no contribution to the vortex drift out ofthe condensate: what matters to the rate of vortex de-cay is only the radial component vr which will dependon the velocity induced by the surrounding vortices vi.

In this simple model, vi can be thought as the velocityinduced by the nearest vortex located (if we consider arandom vortex distribution) at the typical average inter-vortex distance ` ≈ n−1/2, where n = N/(πR2) is thenumber of vortices per unit area (in 2D) or the length ofvortex line per unit volume (in 3D) and N is the numberof vortices in the condensate of radius R [46].

The magnitude of the induced velocity vi will, hence,be approximately vi ≈ κ/(2π`) = κN1/2/(2π3/2R)(where κ = ±1 is the circulation in our units). The re-sulting radial velocity of the vortex vr is therefore givenby the expression vr ≈ βvi ≈ βκN1/2/(2π3/2R), where|β| ≤ 1 depends on the direction of vi and thus on thesign and the relative angular position of the nearest vor-tex.

Since collisions which take vortices out of the conden-sate are mainly with vortices of the same (positive) sign,we only use N+ to estimate ` in this term, accounting forthe polarization. In this idealized model, the number ofpositive vortices ∆N+ expelled from the condensate inthe (small) time ∆t will therefore be proportional to thenumber of positive vortices N+a lying in the small circu-lar annulus of width ∆r = vi∆t and area ∆A = 2πR∆r,next to the edge of the condensate: these are the onlypositive vortices which can potentially travel a radial dis-tance greater than their separation gap from the bound-ary of the condensate. Hence, assuming a uniform vortexdistribution we have ∆N+ ∝ N+a ≈ N+∆A/(πR2) =κN+

3/2

∆t/(π3/2R2). Taking the limit for small ∆t, weconclude that positive vortices drift out of the conden-sate as dN+/dt ∝ (N+)3/2; similarly dN−/dt ∝ (N−)3/2for negative vortices.

We turn now the attention to the nonlinear term ofEq. (3), and remark that polarization must be includedin the description. In Ref. [16], the authors added thenon-linear term on the intuitive assumption that anni-hilation rate depends on the number of vortex dipoles(composed of a positive and a negative vortex) which canbe formed, which is of order ∝ N2 in a zero-polarized sys-tem. Following a similar reasoning, in a polarized systemthe annihilation rate should then be of order ∝ N+N−,as fewer vortex-dipole pairs are formed if P 6= 0 (thisconsideration implicitly allows for any time dependenceof the polarization).

We conclude that, in alternative to Eq. (3), a morephysically realistic (although still rather idealized) modelis

dN+

dt= −Γ1(N+)3/2 − Γ2(N+N−)2,

dN−

dt= −Γ1(N−)3/2 − Γ2(N+N−)2.

(4)

Summing-up Eqs. (4), the total number of vortices,Ntot = N

+ + N−, decays non-trivially as a function ofpolarization P = P (t) according to

dNtotdt

= −Γ1f(t)N3/2tot − Γ2g(t)N4tot, (5)

6

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70

80N

(t)

Γ1 = 6× 10−4Γ2 = 3× 10−7

(a) ω = 0N+

N−

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60 Γ1 = 9× 10−4Γ2 = 9× 10−7

(b) ω = π/16N+

N−

0 50 100 150 200 250 300 350 4000

10

20

30

40

50Γ1 = 2× 10−3Γ2 = 7× 10−7

(c) ω = π/8N+

N−

0 50 100 150 200 250 300 350 400t

0

10

20

30

40

50

60

N(t

)

Γ1 = 6× 10−4Γ2 = 1× 10−6

(d) ω = π/6N+

N−

0 50 100 150 200 250 300 350 400t

0

10

20

30

40

50

60

70Γ1 = 0Γ2 = 1× 10−6

(e) ω = π/4N+

N−

0 50 100 150 200 250 300 350 400t

0

20

40

60

80Γ1 = 1× 10−3Γ2 = 2× 10−7

(f) P = 0N+

N−

FIG. 4. (Color online) Positive and negative vortex number N+ and N− decay as a function of time t for cases (a) ω = 0; (b)ω = π/16; (c) ω = π/8; (d) ω = π/6; (e) ω = π/4; and (f) phase-imprinted P = 0. The dashed lines are the respective fits forthe numerical data (full lines) with the fitting parameters Γ1 and Γ2 appearing at top part of each plot.

b

b

R

∆r = vr∆t

b

u

u

u

uu

uu

u

u

u

u

b

b

b

b

b

b bb

b

b

b

FIG. 5. (Color online) Schematic trajectories of vorticesin a thin annulus of thickness ∆r = vr∆t near the edge ofa condensate of radius R. Vortices (red triangles) and anti-vortices (blue dots) describe erratic paths (arrowed lines) dueto interaction with other vortices. Collisions, mainly withsame-signed vortices, drive vortices out of the condensate.

where the time-dependent polarization P (t) appears inthe functions

f(t) ≡ [(1 + P )/2]3/2 + [(1− P )/2]3/2, (6)

g(t) ≡ (1− P 2)2/8. (7)Notice that in our model the rates of drift and an-

nihilation have respectively dependence N3/2tot and N

4tot

(rather than Ntot and N2tot of Eq. (3). The fit to the

data is slightly better, and both coefficients Γ1 and Γ2are positive (see Fig. 4), consistently with the interpre-tation of the two terms of the equation. Above all, ourmodel is not arbitrary, but attempts to capture some ofthe physics of vortex interaction.

The same N4tot scaling for the annihilation rate wasjustified heuristically in the context of a quenched 2Dhomogeneous system in Ref. [47]. Recently, in Ref. [45],the N4tot scaling was also associated with a four-bodyprocess in simulations of trapped systems, in agreementwith our observations. The general behavior is as follows:initially, a vortex dipole interacts with a third, catalystvortex (or anti-vortex), turning into a rarefaction wavewhich the authors of Ref. [45] called a vortexonium inanalogy with the positronium (the neutral bound stateof an electron an a positron). The vortexonium trav-els through the condensate until it encounters a fourthvortex (or anti-vortex), which acts as a second catalyst.The four-body annihilation process is completed whenthe collision of the vortexonium with the fourth vortexconverts the vortexonium irreversibly into sound. Fig. 6illustrates the process, showing zoomed-in images of atypical vortex-pair annihilation time sequence from our

7

simulations. In this particular case, the third and fourthbodies are catalyst vortex and anti-vortex, respectively(see Fig. 6 (c) and (f)). It is important to notice thatthe process of vortex unbind that we have previously dis-cussed often frustrates the last step of this process (i.e.the interaction with the second catalyst vortex) and theannihilation does not happen: although the vortexoniumis formed, the local speed of sound may quickly changeand split the dipole back again.

(a) (b)

(c) (d)

(e) (f)

FIG. 6. (Color online) Vortex annihilation through a four-body process. The white circles highlight the time sequence,which shows that (a) a vortex dipole is formed; (b) the dipoleturns into a solitary wave (vortexonium) by interacting witha catalyst vortex; (c) the solitary wave deflects the catalystvortex; (d) it travels to a higher density region, becoming agrayer, rarefaction pulse; (e) the pulse is about to collide withan anti-vortex; and (f) the annihilation process is completeafter the collision, where the pulse is irreversibly convertedinto sound.

Vortex decay curves as a function of the polarizationare shown in Fig. 4. We can identify cases (b) and(e) (ω = π/16 and ω = π/4), (c) and (d) (ω = π/8and ω = π/6), respectively, as counterparts with theopposite-polarization: the decay curves are similar in be-

havior, with mirror-symmetric polarization. The maindifference between a curve and its counterpart is thesteeper number decay at initial times for (c) and (e).We found that the number of vortices lost due to an-nihilations is always considerably less than due to drift.Drift out of the condensate is strongly induced by vortexinteractions: the values of the dissipation parameter γwhich we used is too small to make vortices to spiral outof the condensate in the time-scales studied. Thereforeboth linear and non-linear terms in Eq. (5) have originsin vortex interactions. Case (e) (ω = π/4) illustrateswell the need for a steeper than quadratic term in therate equation, since it characterizes a purely non-lineardecay (i.e. Γ1 = 0).

0 100 200 300 400 500t

0

20

40

60

80

100

120

140

Nto

t(t)

P = 0Data

Eq. (3) fit:Γ1 = 0Γ2 = 1× 10−4

Eq. (5) fit:Γ1 = 1× 10−3Γ2 = 2× 10−7

FIG. 7. (Color online) Total number of vortices, Ntot vs timet for the case where an unpolarized P = 0 vortex distributionwas created through phase-imprinting, in order to comparefits given by Eq. (3) (black dot-dashed line) and Eq. (5) (blue-dashed line)

Finally, in order to compare models from Eq. (3) withEq. (5) we performed numerical experiments in which,rather than creating vorticity with the giant vortex-pinsset up here proposed, we simply numerically imprint agiven initial number of vortices uniformly at random po-sitions onto the same harmonically trapped condensate.We obtain essentially the following results: the polariza-tion approximately retains its initial value P = 0 (seeplot (f) in Fig. 4), and a reasonable fit is obtained us-ing Eq. (3). As opposed to the polarized cases, we findnon-negative rates. However, we see in the comparisonshown in Fig. 5 that Eq. (5) fits the curve better thanEq. (3), clearly showing that the ∝ N4 scaling is a betterfit than ∝ N2. Similarly to the polarized cases, drift isthe main mechanism of vortex loss. Eq. (3) has modeledwell the decays studied by [16, 18], which differs fromour P = 0 case not in polarization but rather in the ini-tial number of vortices (∼ 60 as opposed to our ∼ 140).

8

Therefore, we attribute the departure from the quadraticdecay (which is consistent with the ‘ultra-quantum’ decayobserved [4, 5] in superfluid helium, where the system’sfiniteness was not an issue) to vortex mutual interactionin a confined region. The finite-size system probably im-poses a limit to the number of vortices which can beaccommodated in the condensate.

In summary, for both polarized case and the particularunpolarized case where the vortex density is high, ourrate equation, Eq. (5) successfully describes the evolutionof the total number of vortices.

V. CONCLUSION

We have presented a new scheme for generating 2Dquantum turbulence in atomic condensates which allowscontrol over the polarization of the flow, equivalent tothe net rotation of a turbulent ordinary fluid. Usingthis experimentally feasible scheme, we have examinedthe decay of the turbulence and the vortex interactions(vortex-antivortex creation and annihilation) which takeplace in the condensate. We have modeled the decay ofthe number of vortices using a new rate equation thattakes into account the time-dependent polarization. Thenew rate equation Eq. (5) is physically more justified andgives a better fit to the numerical experiments than thelogistic equation proposed by [16]; in particular, its two

terms have clearly distinct physical meaning in terms ofdrift and annhilation. It also agrees with the recent find-ing of Ref. [45] in suggesting that vortex annihilation is afour-vortex process. The new rate equation is therefore abetter starting point to interpret the decay of 2D quan-tum turbulence in further experiments and simulationsin which turbulence is generated in different ways, whichwill help understanding the scatter of the values of Γ1and Γ2.

ACKNOWLEDGMENTS

We thank G. W. Stagg and A. J. Groszek for usefuldiscussions; W. J. Kwon and Y. Shin for insightful sug-gestion on the phenomenological model, particularly forproposing the inclusion of polarization in the vortex num-ber rate equation as a non-linear term ∝ N+N−.

This research was financially supported by CAPES(PDSE Proc. number BEX 9637/14-1), CNPq, andFAPESP. LG’s work is supported by Fonds National de laRecherche, Luxembourg, Grant n.7745104. The Núcleode Apoio a Óptica e Fotônica (NAPOF-USP) is acknowl-edged for computational resources. This work made useof the facilities of N8 HPC Centre of Excellence, providedand funded by the N8 consortium and EPSRC (GrantNo.EP/K000225/1).

[1] L. Skrbek and K. R. Sreenivasan, Physics of Fluids 24,011301 (2012).

[2] C. F. Barenghi, L. Skrbek, and K. R. Sreenivasan, Pro-ceedings of the National Academy of Sciences of theUnited States of America 111 Suppl, 4647 (2014).

[3] C. F. Barenghi, V. S. L’vov, and P.-E. Roche, Proceed-ings of the National Academy of Sciences of the UnitedStates of America 111 Suppl, 4683 (2014).

[4] P. M. Walmsley and A. I. Golov, Physical Review Letters100, 245301 (2008).

[5] A. W. Baggaley, C. F. Barenghi, and Y. A. Sergeev,Physical Review B 85, 060501 (2012).

[6] E. A. L. Henn, J. A. Seman, G. Roati, K. M. F. Mag-alhães, and V. S. Bagnato, Physical Review Letters 103,045301 (2009).

[7] E. A. L. Henn, J. A. Seman, G. Roati, K. M. F. Mag-alhães, and V. S. Bagnato, Journal of Low TemperaturePhysics 158, 435 (2009).

[8] T. W. Neely, A. S. Bradley, E. C. Samson, S. J. Rooney,E. M. Wright, K. J. H. Law, R. Carretero-González, P. G.Kevrekidis, M. J. Davis, and B. P. Anderson, PhysicalReview Letters 111, 235301 (2013), arXiv:1204.1102.

[9] A. C. White, B. P. Anderson, and V. S. Bagnato, Pro-ceedings of the National Academy of Sciences of theUnited States of America 111 Suppl, 4719 (2014).

[10] R. H. Kraichnan and D. Montgomery, Reports onProgress in Physics 43 (1980).

[11] M. Rivera, P. Vorobieff, and R. E. Ecke, Physical ReviewLetters 81, 1417 (1998).

[12] R. Numasato, M. Tsubota, and V. S. Lvov, PhysicalReview A 81, 63630 (2010).

[13] N. G. Parker and C. S. Adams, Physical Review Letters95, 145301 (2005).

[14] B. Nowak, D. Sexty, and T. Gasenzer, Physical ReviewB 84, 020506 (2011).

[15] T. Simula, M. J. Davis, and K. Helmerson, PhysicalReview Letters 113, 165302 (2014).

[16] W. J. Kwon, G. Moon, J. Y. Choi, S. W. Seo, and Y.-i.Shin, Physical Review A 90, 063627 (2014).

[17] W. J. Kwon, G. Moon, S. W. Seo, and Y. Shin, PhysicalReview A 91, 053615 (2015).

[18] G. W. Stagg, A. J. Allen, N. G. Parker, and C. F.Barenghi, Physical Review A 91, 013612 (2015).

[19] A. C. White, C. F. Barenghi, and N. P. Proukakis, Phys-ical Review A 86, 013635 (2012).

[20] M. T. Reeves, B. P. Anderson, and A. S. Bradley, Phys-ical Review A 86, 053621 (2012).

[21] A. C. White, N. P. Proukakis, and C. F. Barenghi, Jour-nal of Physics: Conference Series 544, 012021 (2014).

[22] T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis,and B. P. Anderson, Physical Review Letters 104, 160401(2010), arXiv:0912.3773.

[23] P. Engels, I. Coddington, P. C. Haljan, V. Schweikhard,and E. A. Cornell, Physical review letters 90, 170405(2003), arXiv:0301532 [cond-mat].

[24] A. E. Leanhardt, A. Görlitz, A. P. Chikkatur, D. Kielpin-ski, Y. Shin, D. E. Pritchard, and W. Ketterle, Physicalreview letters 89, 190403 (2002).

http://dx.doi.org/10.1063/1.3678335http://dx.doi.org/10.1063/1.3678335http://dx.doi.org/10.1073/pnas.1400033111http://dx.doi.org/10.1073/pnas.1400033111http://dx.doi.org/10.1073/pnas.1400033111http://dx.doi.org/10.1073/pnas.1312548111http://dx.doi.org/10.1073/pnas.1312548111http://dx.doi.org/10.1073/pnas.1312548111http://dx.doi.org/10.1103/PhysRevLett.100.245301http://dx.doi.org/10.1103/PhysRevLett.100.245301http://dx.doi.org/10.1103/PhysRevB.85.060501http://dx.doi.org/10.1103/PhysRevLett.103.045301http://dx.doi.org/10.1103/PhysRevLett.103.045301http://dx.doi.org/10.1007/s10909-009-0045-2http://dx.doi.org/10.1007/s10909-009-0045-2http://dx.doi.org/ 10.1103/PhysRevLett.111.235301http://dx.doi.org/ 10.1103/PhysRevLett.111.235301http://arxiv.org/abs/1204.1102http://dx.doi.org/10.1073/pnas.1312737110http://dx.doi.org/10.1073/pnas.1312737110http://dx.doi.org/10.1073/pnas.1312737110http://m.iopscience.iop.org/0034-4885/43/5/001http://m.iopscience.iop.org/0034-4885/43/5/001http://dx.doi.org/10.1103/PhysRevLett.81.1417http://dx.doi.org/10.1103/PhysRevLett.81.1417http://dx.doi.org/10.1103/PhysRevA.81.063630http://dx.doi.org/10.1103/PhysRevA.81.063630http://dx.doi.org/10.1103/PhysRevLett.95.145301http://dx.doi.org/10.1103/PhysRevLett.95.145301http://dx.doi.org/10.1103/PhysRevB.84.020506http://dx.doi.org/10.1103/PhysRevB.84.020506http://dx.doi.org/10.1103/PhysRevLett.113.165302http://dx.doi.org/10.1103/PhysRevLett.113.165302http://dx.doi.org/ 10.1103/PhysRevA.90.063627http://dx.doi.org/ 10.1103/PhysRevA.91.053615http://dx.doi.org/ 10.1103/PhysRevA.91.053615http://dx.doi.org/10.1103/PhysRevA.91.013612http://dx.doi.org/10.1103/PhysRevA.86.013635http://dx.doi.org/10.1103/PhysRevA.86.013635http://dx.doi.org/10.1103/PhysRevA.86.053621http://dx.doi.org/10.1103/PhysRevA.86.053621http://dx.doi.org/10.1088/1742-6596/544/1/012021http://dx.doi.org/10.1088/1742-6596/544/1/012021http://dx.doi.org/ 10.1103/PhysRevLett.104.160401http://dx.doi.org/ 10.1103/PhysRevLett.104.160401http://arxiv.org/abs/0912.3773http://dx.doi.org/ 10.1103/PhysRevLett.90.170405http://dx.doi.org/ 10.1103/PhysRevLett.90.170405http://arxiv.org/abs/0301532http://dx.doi.org/10.1103/PhysRevLett.89.190403http://dx.doi.org/10.1103/PhysRevLett.89.190403

9

[25] M. Nakahara, T. Isoshima, and K. Machida, Physica B:Condensed . . . 288, 17 (2000).

[26] T. Isoshima, M. Okano, H. Yasuda, K. Kasa, J. A. M.Huhtamäki, M. Kumakura, and Y. Takahashi, PhysicalReview Letters 99, 200403 (2007).

[27] M. Möttönen, N. Matsumoto, M. Nakahara, andT. Ohmi, Journal of Physics: Condensed Matter 14,13481 (2002).

[28] M. Möttönen, V. Pietilä, and S. M. M. Virtanen, Phys-ical Review Letters 99, 250406 (2007), arXiv:0706.1840.

[29] P. Kuopanportti and M. Möttönen, Journal of Low Tem-perature Physics 161, 561 (2010).

[30] P. Kuopanportti, B. P. Anderson, and M. Möttönen,Physical Review A 87, 033623 (2013).

[31] M. W. Ray, E. Ruokokoski, S. Kandel, M. Möttönen,and D. S. Hall, Nature 505, 657 (2014).

[32] P. Kuopanportti and M. Möttönen, Physical Review A81, 033627 (2010), arXiv:0911.4042v2.

[33] T. Frisch, Y. Pomeau, and S. Rica, Physical ReviewLetters 69, 1644 (1992).

[34] T. Winiecki, J. F. McCann, and C. S. Adams, Euro-physics Letters (EPL) 48, 475 (1999).

[35] N. G. Berloff and P. H. Roberts, Journal of Physics A:Mathematical and General 33, 4025 (2000).

[36] R. Desbuquois, L. Chomaz, T. Yefsah, J. Léonard,J. Beugnon, C. Weitenberg, and J. Dalibard, NaturePhysics 8, 645 (2012).

[37] S. Choi, S. Morgan, and K. Burnett, Physical Review A

57, 4057 (1998).[38] M. Tsubota, K. Kasamatsu, and M. Ueda, Physical Re-

view A 65, 023603 (2002).[39] A. S. Bradley and B. P. Anderson, Physical Review X 2,

041001 (2012).[40] G. R. Dennis, J. J. Hope, and M. T. Johnsson, Computer

Physics Communications 184, 201 (2013).[41] F. Pinsker and N. G. Berloff, Physical Review A 89,

053605 (2014).[42] M. Leadbeater, T. Winiecki, D. C. Samuels, C. F.

Barenghi, and C. S. Adams, Physical Review Letters86, 1410 (2001).

[43] N. G. Parker, N. P. Proukakis, C. F. Barenghi, andC. S. Adams, Physical Review Letters 92, 160403 (2004),arXiv:0312520 [cond-mat].

[44] N. G. Berloff and C. F. Barenghi, Physical Review Let-ters 93, 090401 (2004).

[45] A. J. Groszek, T. P. Simula, D. M. Paganin,and K. Helmerson, (2015), arXiv:1511.06552 [cond-mat.quant-gas].

[46] This scaling is well-known in the superfluid helium lit-erature, and has been numerically verified by D. Kiv-otides, Y.A. Sergeev and C.F. Barenghi, Phys. of Fluids20, 055105 (2008).

[47] J. Schole, B. Nowak, and T. Gasenzer, Physical ReviewA 86, 013624 (2012).

http://www.sciencedirect.com/science/article/pii/S0921452699019523http://www.sciencedirect.com/science/article/pii/S0921452699019523http://dx.doi.org/ 10.1103/PhysRevLett.99.200403http://dx.doi.org/ 10.1103/PhysRevLett.99.200403http://dx.doi.org/10.1088/0953-8984/14/49/306http://dx.doi.org/10.1088/0953-8984/14/49/306http://dx.doi.org/10.1103/PhysRevLett.99.250406http://dx.doi.org/10.1103/PhysRevLett.99.250406http://arxiv.org/abs/0706.1840http://dx.doi.org/10.1007/s10909-010-0216-1http://dx.doi.org/10.1007/s10909-010-0216-1http://dx.doi.org/10.1103/PhysRevA.87.033623http://dx.doi.org/10.1038/nature12954http://dx.doi.org/10.1103/PhysRevA.81.033627http://dx.doi.org/10.1103/PhysRevA.81.033627http://arxiv.org/abs/0911.4042v2http://dx.doi.org/10.1103/PhysRevLett.69.1644http://dx.doi.org/10.1103/PhysRevLett.69.1644http://dx.doi.org/10.1209/epl/i1999-00507-2http://dx.doi.org/10.1209/epl/i1999-00507-2http://dx.doi.org/10.1088/0305-4470/33/22/307http://dx.doi.org/10.1088/0305-4470/33/22/307http://dx.doi.org/10.1038/nphys2378http://dx.doi.org/10.1038/nphys2378http://dx.doi.org/10.1103/PhysRevA.57.4057http://dx.doi.org/10.1103/PhysRevA.57.4057http://dx.doi.org/10.1103/PhysRevA.65.023603http://dx.doi.org/10.1103/PhysRevA.65.023603http://dx.doi.org/10.1103/PhysRevX.2.041001http://dx.doi.org/10.1103/PhysRevX.2.041001http://dx.doi.org/10.1016/j.cpc.2012.08.016http://dx.doi.org/10.1016/j.cpc.2012.08.016http://dx.doi.org/10.1103/PhysRevA.89.053605http://dx.doi.org/10.1103/PhysRevA.89.053605http://dx.doi.org/10.1103/PhysRevLett.86.1410http://dx.doi.org/10.1103/PhysRevLett.86.1410http://dx.doi.org/10.1103/PhysRevLett.92.160403http://arxiv.org/abs/0312520http://dx.doi.org/10.1103/PhysRevLett.93.090401http://dx.doi.org/10.1103/PhysRevLett.93.090401http://arxiv.org/abs/arXiv:1511.06552 [cond-mat.quant-gas]http://arxiv.org/abs/arXiv:1511.06552 [cond-mat.quant-gas]http://dx.doi.org/10.1103/PhysRevA.86.013624http://dx.doi.org/10.1103/PhysRevA.86.013624

Controlled polarization of two-dimensional quantum turbulence in atomic Bose-Einstein condensatesAbstractIntroductionGiant vortex and small pinsModelResultsCreating Polarized FlowPolarized Turbulence Decay

ConclusionAcknowledgmentsReferences