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White AC, Barenghi CF, Proukakis NP. Creation and characterization of vortex

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Creation and characterization of vortex clusters in atomic Bose-Einstein condensates

Angela C. White,∗ Carlo F. Barenghi, and Nick P. ProukakisSchool of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, England, UK.

We show that a moving obstacle, in the form of an elongated paddle, can create vortices that aredispersed, or induce clusters of like-signed vortices in 2D Bose-Einstein condensates. We proposenew statistical measures of clustering based on Ripley’s K-function which are suitable to the smallsize and small number of vortices in atomic condensates, which lack the huge number of lengthscales excited in larger classical and quantum turbulent fluid systems. The evolution and decay ofclustering is analyzed using these measures. Experimentally it should prove possible to create suchan obstacle by a laser beam and a moving optical mask. The theoretical techniques we presentare accessible to experimentalists and extend the current methods available to induce 2D quantumturbulence in Bose-Einstein condensates.

PACS numbers: 03.75.Lm, 03.75Kk, 67.85.De, 67.85.Hj ,67.85.Jk, 67.10.Jn,67.25.dkKeywords: Bose-Einstein condensate, vortex clusters, Ripley’s K

I. INTRODUCTION

Turbulent superfluid helium has been shown to exhibitfeatures typical of ordinary, classical turbulence, on scaleslarger than the average inter-vortex spacing, such as thesame Kolmogorov energy spectrum in three dimensional(3D) turbulence [1–8] as well as differences, such as non-Gaussian velocity statistics [7, 9–11]. Two dimensional(2D) turbulence is very different from 3D turbulence. In3D turbulence, on scales larger than the average inter-vortex spacing, there is a flow of energy from large scalesto small scales through the Richardson cascade, wherebylarge eddies are broken up into smaller and smaller eddiesgiving a Kolmogorov energy spectrum with k−5/3 scal-ing. In 2D turbulence there is an inverse energy cascade,where energy flows from the scale of energy injection(small scales), to larger scales as like-charged vorticescluster [12]. This phenomenon is established in classicalfluids and is thought to be the mechanism responsible forJupiter’s great red spot [13, 14]. In Bose-Einstein con-densates, while there have been a number of theoreticalinvestigations [15–18] and some experimental work [19],the question of whether an inverse cascade is a feature ofa system of turbulent vortices in quasi-two dimensionalsystems remains open; this is firstly because these con-densates are relatively new and secondly (and more im-portantly) because, due to their relatively small size, theylack the large range of length scales typical of other 2Dturbulent flows (such as planetary atmospheres).

On the other hand, ultra-cold Bose condensed atomicgases are ideal to investigate the dynamics of few-vortexsystems as well as turbulent dynamics of many vortices.The dimensionality of such condensates can be easilycontrolled, allowing direct study of vortices in two andthree dimensions. Advanced experimental methods forthe imaging and detection of vortices in Bose–Einsteincondensates (BECs) have been developed [20] and new

∗Electronic address: ang.c.white@gmail.com

theoretical proposals have shown that vortices can be eas-ily manipulated in BECs [21]. There are many techniquesthat can be applied to nucleate vortices in BECs, suchas: engineering the condensate phase profile [22, 23]; stir-ring the condensate with a blue or red detuned laser (forexperiments see [24, 25] and theory [21, 26, 27]); mix-ing and merging condensates of well defined phases [28–30]; moving a condensate past a defect [25]; rotating thetrapping potential or thermal cloud [31–36]; and coolingthe condensate with a rapid quench through the phasetransition (Kibble-Zureck mechanism) [37–39]. Finally,vortices can be nucleated from dynamical instabilities,such as through the decay of the snake instability of asoliton [40, 41], the bending wave instability of a vortexring [42, 43] or surface mode excitations of the conden-sate [44, 45]. In the first experimental demonstration of3D quantum turbulence in a BEC, tangled vortices werecreated by shaking the condensate with an oscillatorytrapping potential [44, 45].

In this paper we add to the numerous existing tech-niques applied to induce vortices in BECs, and apply amoving object with an elliptical paddle shape to createvortices. We demonstrate that the trajectory of the op-tical paddle through the condensate can be controlled tocreate vortices that are both well distributed or clusteredinto groups with like-winding. To determine if vorticesare indeed clustered, and if this clustering increases ordecreases significantly with time, we develop tools whichare more suitable to the small size and the relative smallnumber of vortices which can be generated in ultracoldatom BECs. Drawing on the wealth of available statis-tical pattern recognition methods, we analyze our datausing Besag’s function [46], a modification of Ripley’sK-function [47, 48], which has been extensively appliedacross a variety of scientific fields to measure clusteringand clumping of discrete objects e.g. [49–55]. Motivatedby Besag’s function, we develop some new measures ofindependent clustering when the system is comprised oftwo unique types of discrete objects. In our case, theseare vortices with ‘+’ or ‘−’ winding in a BEC. Thesetechniques can distinguish between the cases of mixed

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clusters of vortices and independent clusters of vorticesof like-winding in the condensate. After reviewing ourtheoretical model, we analyze vortex structures obtainedby different forms of stirring by applying both Besag’sfunction and nearest neighbor techniques. Finally, wesummarize our main conclusions, that the resulting vor-tex clustering is strongly dependent on the trajectory ofthe moving paddle. We find no compelling evidence ofan inverse cascade in these small systems, in the sensethat we don’t see an increase in clustering over time.

II. NUMERICAL MODEL

We simulate the dynamics of a trapped two-dimensional Bose-Einstein condensate stirred by an opti-cal paddle by integrating the following 2D dimensionlesstime-dependent Gross-Pitaevskii equation (GPE):

i∂ψ

∂t= −1

2∇2ψ + r

2

2ψ + VPψ + κ2d|ψ|2ψ , (1)

where the interaction strength, κ2d = 2√

2πaN/az, iswritten in terms of the scattering length, a, and thetotal number of condensate atoms, N. ψ is scaled asψ = arΨ/

√N , where Ψ is the condensate wavefunc-

tion and is normalized to unity i.e.∫

dx|ψ|2 = 1.az =

√h̄/(mωz) and ar =

√h̄/(mωr) are the axial and

radial harmonic oscillator lengths, determined by the ax-ial and radial trapping frequencies, ωz and ωr respec-tively.

The potential, VP , describes a far-off-resonance blue-detuned laser beam shaped into a paddle that follows arotating and circular stirring trajectory given by

VP = V0 exp

[−η2 (x̃ cos(ωt)− ỹ sin(ωt))

2

d2

− (ỹ cos(ωt) + x̃ sin(ωt))2

d2

], (2)

where x̃ = x− v sin(t) and ỹ = y − v cos(t). V0 gives thepeak strength of the potential, η determines the paddleelongation and d the width. Experimentally, the paddlecan be shaped by shining a far-off-resonance blue detunedlaser through a mask as in [56]. For a paddle rotatingwith a frequency, ω, at the center of the condensate, x̃ =x and ỹ = y. For a paddle moving at a constant radiusfrom the center of the condensate without rotating, wetake ω = 1. In all simulations the paddle is initiallylinearly ramped up to its maximum stirring frequency,ω, after which the condensate is stirred at constant ωuntil tS = 20. The paddle is then ramped off linearlyover t = 5 by making the replacement

VP →(

1− (t− tS)5

)VP , (3)

in Eq. (2). For paddle sizes and stirring frequencies whichresult in vortex formation, the condensate dynamics areevolved for a further time of t = 20.

Numerically, Eq. (1) is solved pseudo-spectrally withperiodic boundary conditions and integrated in time byapplying an adaptive 4th-5th order Runge-Kutta methodwith the help of xmds [57]. The initial state for our simu-lations is obtained by a short propagation of the 2D GPEin imaginary time (by making the replacement τ = −itin Eq. (1)), and applying a stationary paddle potential.The condensate is parameterized by the nonlinear inter-action strength κ2d = 10399. Our simulations are run ona grid of spatial extent −20 to 20 with gridsize Ng = 512(see appendix).

III. VORTEX GENERATION

This paper looks at three stirring motions of the pad-dle, VI, VII and VIII, each of which are specific general-izations of Eq. (2), corresponding to a rotating paddle, apaddle moving on a deferent from the condensate centerand a paddle moving along a deferent while rotating, re-spectively (see figure 1, top row). The stirring motionsof the paddle studied generate contrasting vortex con-figurations, as discussed below. For each case, we trackthe total number of vortices nucleated, as shown in figure3a. In all cases, at later times, vortices are lost to theedge of the condensate. The condensate edge is selectedby identifying where the density falls to less than 30%of the maximum condensate density at that time. Theresulting profile is then smoothed to give the condensateedge. When two vortices are closer than a critical sepa-ration distance and are of opposite winding, vortex pairannihilation occurs, a mechanism which also reduces thetotal number of vortices in the condensate.

For all simulations the total angular momentum of thecondensate is also tracked, given by

〈Lz〉 = −i∫

dxψ∗(x∂

∂y− y ∂

∂x

)ψ . (4)

Case I: Paddle rotating at the condensate center

Firstly we look at a paddle rotating about its centerwith frequency ω at the center of the condensate. Thisis modeled by evolving Eq. (1) with VI = VP (x̃→ x, ỹ →y). The smallest paddle size we consider is d = 0.5,rotating at a frequency ω = 6. The rotating motion ofthe paddle produces circular spiral sound waves, which atlate times interfere with each other giving a wave interfer-ence pattern (see the first column of figure 1). Paddleswith a larger width (d = 1), rotating at frequencies ofω = 4, 6 and 8, nucleate vortices in addition to creatingspiral sound waves. The density profile for a paddle ro-tating at frequency ω = 4 is show in the second columnof figure 1.

3

FIG. 1: (Color online) Column 1: Density slices for paddle VI with parameters: d = 0.5, ω = 6 (dimensionless units); Columns2–4: Density and vortex position images for paddles VI, VII and VIII. Parameters: VI (column 2): d = 1, ω = 4; VII (column3): d = 1, ω = 1, v = 4; VIII (column 4): d = 1, ω = 2, v = 4. All paddles have V0 = 150, η = 8 (dimensionless units). Positivevortices are identified by (pink) + signs and negative vortices by (blue) ◦ signs. Time is indicated in white at the bottom lefthand side of each plot.

A greater rate of rotation increases the number of vor-tices initially nucleated, as expected. In all cases, vorticesare initially nucleated from the ends of the paddle withwinding opposite to the direction of rotation of the pad-dle, as depicted in figure 2. At subsequent times, whenthe local superfluid velocity surpasses the critical veloc-ity for vortex nucleation [58], vortices of both signs arenucleated from both the center and ends of the paddle.A centered rotating paddle imparts a small amount ofangular momentum to the condensate. The angular mo-mentum imparted is proportional to the frequency of therotating paddle (see figure 3b). Note that after paddlesof frequency ω = 6 and 8 have been ramped off, the con-densate angular momentum saturates to approximatelythe same value.

Case II: Paddle stirring at constant radius

The second stirring motion of the laser paddle we sim-ulate is a paddle stirring the condensate at a constantradius from the condensate center, or traveling along adeferent, VII = VP (ω = 1) (see figure 1, column 3, toprow, for a schematic of the paddle motion). This paddletrajectory creates vortices which tend to group initiallyin like-signed clusters as shown by the progressive timesamples of the condensate density profiles in figure 1, col-umn 3. The condensate gains a large amount of angularmomentum (refer to figure 3b), and consequently at finaltimes there is a significant imbalance in the total numberof vortices with positive and negative winding.

4

FIG. 2: (Color online)Schematic diagram of vortices nucle-ated from a paddle rotating at the condensate center, VI asthe paddle is ramped up to ω = 8. Positive vortices are identi-fied by (pink) + signs and negative vortices by (blue) ◦ signs,the paddle profile is shown in purple.

(a)Total Vortex Number

(b)Angular Momentum

FIG. 3: (Color online) Variation of a) total vortex numberand d) condensate angular momentum with time. Vorticesinduced by a rotating paddle, VI, with d = 1 and v = 4, 6 and8 are represented by orange pluses, purple crosses and pinkdiamonds respectively. Data corresponding to the paddleswith v = 4 and d = 1, moving with trajectories VII (ω = 1)and VIII (ω = 2), are given by green triangles and blue squaresrespectively.

FIG. 4: The function fij(r) in Ripley’s K-function.

Case III: Paddle rotating and stirring at constant radius

The final stirring trajectory of the laser paddle we sim-ulate, VIII = VP , is a combination of the two previous mo-tions, with the paddle moved at a constant radius fromthe condensate center while rotating at a small frequency.This motion can also be described as a paddle rotatingand moving along the deferent (see figure 1, column 4,top row, for a schematic of the paddle motion). The effectof adding the rotational motion of the paddle to its tra-jectory mixes the clusters produced, resulting in smallergroups consisting of 3 and 4 vortices (compare the vor-tex distributions in figure 1 columns 3 and 4). While alarge amount of angular momentum is transfered to thecondensate, this is again reduced in comparison to thatof paddle VII (see figure 3b).

IV. CLUSTERING ANALYSIS

A. Ripley’s K-function

We analyze the clustering of vortices formed by stirringa 2D condensate with a paddle by applying Ripley’s K-function, a statistical pattern analysis method used as ameasure of spatial clustering. In the context of clusteringof like-signed vortices, Ripley’s K-function is dependenton the total number of liked signed vortices, N , withinthe total condensate area, A, and can be expressed as

K(r) =A

N2

N∑i=1

N∑j=1

fij(r) , (5)

where fij(r) = 1 for a vortex, j, within a distance rof the reference vortex, i, with like-winding. Otherwise,fij(r) = 0 if i = j, or if the distance between vortex iand j is greater than r. That is

fij =

[1 ∀ rij < r, i 6= j0 ∀ rij > r or i = j

(6)

here rij is the distance from a reference vortex i to thecomparison vortex j with like-winding. This is depictedin figure 4.

Ripley’s K-function looks at the number of like-signedvortices within a radius, r, from the position of an arbi-trarily chosen vortex, i, at its center (see figure 4 and 5).

5

FIG. 5: Compensated Besag’s function for vortices of nega-tive winding at varying times (see legend) for a condensatewith vortices nucleated by a paddle with trajectory VII. Pa-rameters: d = 1, ω = 1 and v = 4.

If the number of vortices with like-winding per unit areawithin this radius, r, is greater than the overall numberof like-signed vortices per unit area for the whole con-densate, then the vortices are said to be clustered. Clus-tering results in K(r) increasing faster than if vortices ofeither sign are distributed in a spatially random manner,that is, if they follow a Poisson distribution. Ripley’s-Kfunction for a poisson-distributed data set takes the formK(r) = πr2. For a linear scaling of Poisson-distributeddata, it is useful to normalize Ripley’s K-function toH(r) =

√K(r)/π. Ripley’s L-function, also commonly

known as Besag’s function, is obtained from further nor-malization of Ripley’s K-function:

L(r) =√K(r)/π − r. (7)

As the condensate area, A, does not necessarily remainconstant over all times, we scale r by the characteristiccondensate radius rc =

√A/π for that time and in our

subsequent analysis evaluate

L(r/rc) =

√√√√ Aπ(Nrc)2

N∑i=1

N∑j=1

fij(r/rc)−r

rc, (8)

which simplifies to:

L(r/rc) =

√√√√ 1N2

N∑i=1

N∑j=1

fij(r/rc)−r

rc. (9)

Besag’s function is zero for like-signed vortices which arerandomly distributed, takes positive values for vorticesclustered over that spatial scale, and is negative if thevortex distribution is dispersed. That is:

L(r/rc) =

1 Clustered0 Random−1 Dispersed

(10)

FIG. 6: (Color Online) Schematic diagrams of independent(A) and co-clustering (B) in systems with two distinct typesof objects, represented by pink pluses and blue circles respec-tively.

The radius around a centered vortex containing, on aver-age, the most like-signed vortices per area, is called theradius of maximal aggregation, and is given by the valueof r which maximizes L(r) [54].

For a paddle rotating at a constant radius from thecondensate center, evaluating L(r/rc) for positive vor-tices, as seen in figure 5, shows that the clustering ofvortices decreases with time. Although the vortices areclustered, the amount of clustering is not constant or in-creasing in time, giving no evidence of an inverse cascadefor this system.

B. Measures of independent clustering oflike-signed vortices

While Besag’s function gives a measure of the clus-tering of vortices with the same winding, it does notdiscriminate between between cases where like and op-posite signed vortices are clustered in the same spatialregion and cases where like-signed vortices are clusteredin spatially independent regions (refer to figure 6 for aschematic illustration). It is necessary to make this dis-tinction when looking for a measure of an inverse cascadeprocess, as the clustering of like signed vortices is not ex-pected to occur in the same spatial region as clusteredvortices of the opposite sign. To address this issue wedefine a new measure of clustering which uses the signof nearest neighboring vortices to determine if clusteringoccurs in regions that are spatially independent.

We express this measure of clustering based on lookingat the sign of all j neighboring vortices up to the Bthnearest neighbor of an arbitrary reference vortex i as

CB(t) =1

N

N∑i=1

B∑j=1

cij(t)

B. (11)

Here cij = 1 if the vortex i and its jth nearest neigh-

bor are of the same sign and cij = 0 if vortex i and itsjth nearest neighbor are of opposite sign. If vortex i isseparated by distance greater than Rc = rc/3 to its j

th

nearest neighbor, then cij = 0. B is the maximum near-est neighbor to the reference vortex i. It is necessary that

6

FIG. 7: (Color online) Comparison of evolution second near-est neighbor data, C2(t), (top figure) and fourth nearestneighbor data, C4(t), (bottom figure) for paddle trajectoriesVII, VIII and VI with v = 8 represented by green triangles,blue squares and pink diamonds respectively.

the value of Rc chosen is greater than the average inter-vortex separation distance and on order of the largestcluster size. Vortices closer than Rc to the condensateedge will bias the calculation of CB(t) as they have anarea less than πR2c surrounding them, which their nearestneighboring vortices could inhabit. To correct for theseedge effects we omit these vortices which are less thanRc from the condensate edge from the set of referencevortices, but still include them in the set of comparisonvortices for vortices a distance greater than Rc from thecondensate edge. Systems for which CB = 0.5 are ran-domly distributed and when CB takes values greater than0.5 the objects are clustered. We note that these mea-sures can be applied generically to investigate cases of coand independent clustering of two discrete objects andcould be simply extended to look at co and independentclustering of many discrete objects.

A comparison of the evolution of CB(t) for the simula-tion runs described previously is shown in figures 7 and8. From the nearest neighbor analysis we learn:

• From figure 7, we can see that a paddle moving ata constant radius from the condensate center (VII)creates vortices that are initially very clustered, in-

FIG. 8: (Color online) Comparison of evolution second near-est neighbor data, C2(t) (top figure), and fourth nearestneighbor data, C4(t) (bottom figure), for vortices created byrotating a paddle at the condensate center (VI) with v = 4(orange +), v = 6 (purple crosses) and v = 8 (pink diamonds).

dicated by C2 and C4 taking values very near one.After the paddle is turned off the vortices remainclustered, with C2 and C4 not decreasing below 0.5.

• A paddle moving with trajectory VIII creates clus-ters that are initially smaller in size than purelymoving the paddle at constant radius from the con-densate center (VII), and clustering only slightlydecreases after the condensate paddle is turned off,with vortices remaining more clustered than ran-dom.

• When the paddle is only rotated at the condensatecenter, VI, vortices never become clustered (refer tofigure 8).

For all cases, regardless of how the vortices are initiallynucleated, evaluating CB(t) gives no evidence of a ten-dency increasing clustering of like-signed vortices over ascale of Rc in a turbulent 2D BEC after the laser paddlehas been ramped off.

7

V. CONCLUSION

In this paper we have covered two main objectives:

• We have extended the available methods for cre-ating vortices in 2D atomic Bose-Einstein conden-sates, demonstrating that a paddle can be used tostir a condensate in two quite different ways, cre-ating long-lived vortex clusters or more randomlydistributed vortices that are turbulent in two di-mensions.

• A new statistical measure of clustering based on an-alyzing nearest neighbor vortices was defined, moti-vated by a well known statistical spatial point pat-tern analysis technique, Besag’s function. Thesemeasures have been applied to analyze how vorticesare distributed in 2D condensates.

We find that a paddle moved through the condensateat a constant radius from the center creates vortices ofboth positive and negative winding in clusters. When thepaddle is rotated at the condensate center, vortices cre-ated are initially clustered co-dependently in the samelocal spatial regions and later disperse throughout thecondensate. For a combination of both moving a pad-dle at a constant radius through the condensate whilesimultaneously rotating the paddle, the vortices inducedare less clustered than if the paddle is only moved at aconstant radius from the condensate center. The extentof clustering was quantitatively measured by evaluatingtwo statistical measures of clustering; applying a modi-fied Ripley’s function and a technique based on compar-ing the sign of nearest neighboring vortices.

We did not observe an increase in clustering over time.This was despite evolution times longer than the largestturnover time τ = πr2c/2π ≈ 35, determined by the con-densate size. Our system contains too few vortices todetermine if the relevant physical process for 2D turbu-lent systems in atomic Bose-gases is an inverse cascade ofincompressible kinetic energy from small to large scalesmanifesting in a clustering of like-signed vortices. In par-ticular, it would be difficult to apply traditional methodsused for large systems (planetary atmospheres, super-fluid helium), based on Fourier transforming the velocityfield and analyzing the spectra of energy and enstrophyover many decades in wavenumber space. Our statisti-cal analysis based on Ripley’s K-function and on nearestneighbor methods provides a way to quantify an increaseor decrease in the degree of vortex clustering. As thesemethods are constructed from a knowledge of the positionand winding of vortices in the system they are readily ac-cessible experimentally. Information on vortex locationin condensates can be obtained experimentally throughstandard absorption imaging techniques, e.g. [20, 31, 32],and winding of vortices is found by analysis techniquesgiving phase information, such as condensate interferom-etry, e.g. [24, 59, 60].

Acknowledgments

A. White thanks C. J. Foster for the vortex detectionalgorithm based on the plaquette technique [61]. Wethank A. Baggaley for useful discussions. This workwas supported by EPSRC grants EP/H027777/1 andBH101785.

Appendix A: Error Checking

(a)Total Vortex Number

(b)Angular Momentum

FIG. 9: b) Evolution of angular momentum and a) total num-ber of vortices nucleated by a paddle rotating at the center ofthe condensate (VI), with v = 6, d = 1. Simulation gridsizesNg = 512 (purple) and Ng = 1024 (pink).

To check our results are independent of gridsize used,the simulation grid size was doubled from Ng = 512 toNg = 1024 for a paddle rotating at the condensate centerwith v = 6 and d = 1. In figure 9 a comparison is made of

8

the total vortex number and condensate angular momen-tum, when the grid size is doubled. The angular momen-tum is calculated by evaluating Eq (4). The reasonableagreement between rates of vortex production and elim-ination, as well as evolution of the condensate angularmomentum in both runs establishes that the gridsize ofNg = 512 applied in the simulations presented in the

body of the paper is adequate. The small variance inresults from doubling the gridsize are attributed to thecondensate edge selection routine used. A further sourceof difference is the chaotic nature of vortex dynamics inturbulent systems. A small amount of numerical noisewould be enough to seed a difference in vortex trajecto-ries.

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