Chaotic behavior of three interacting vortices in a confined Bose-Einstein condensate

Chaotic behavior of three interacting vortices in a confined Bose-Einstein condensate

Nikos KyriakopoulosSUPA, Department of Physics and Institute for Complex Systems and Mathematical Biology,

King’s College, University of Aberdeen, Aberdeen, AB24 3UE, UK

Vassilis KoukouloyannisPhysics Department, Aristotle University of Thessaloniki, GR-54124, Thessaloniki, Greece

Charalampos SkokosDepartment of Mathematics and Applied Mathematics,

University of Cape Town, Rondebosch, 7701, South Africa andPhysics Department, Aristotle University of Thessaloniki, GR-54124, Thessaloniki, Greece

Panayotis KevrekidisDepartment of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-9305, USA

Motivated by recent experimental works, we investigate a system of vortex dynamics in an atomicBose-Einstein condensate (BEC), consisting of three vortices, two of which have the same charge.These vortices are modeled as point particles and the Hamiltonian structure and conservation lawsof the system are presented and discussed in detail. This tripole system is a prototypical model ofvortices in BECs exhibiting chaos. Reducing the study of the system to the investigation of a twodegree of freedom Hamiltonian model, we acquire quantitative results about its chaotic behavior.For this purpose we implement well-known methods like the construction of appropriate Poincaresurfaces of section and the computation of Lyapunov exponents, as well as new approaches basedon the construction of scan maps using the Smaller ALignment Index (SALI) of chaos detection.Applying the latter approach to a large number of initial conditions we manage to accurately andefficiently measure the extent of chaos in the system and its dependence on physically importantparameters like the energy and the angular momentum of the system.

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I. INTRODUCTION

The study of two-dimensional particle dynamics resulting from a logarithmic interaction potential is a theme ofbroad and diverse interest in Physics. Arguably, the most canonical example of both theoretical investigation andexperimental relevance is the exploration of fluid and superfluid vortex patterns and crystals, as is evidenced e.g. bythe review of Aref et al. [1] and the book of Newton [2]. However, numerous additional examples ranging from electroncolumns in Malmberg-Penning traps [3] to magnetized, millimeter sized disks rotating at a liquid-air interface [4, 5]are also characterized by the same underlying mathematical structure and hence present similar dynamical features.

In recent years, the field of atomic Bose-Einstein condensates (BECs) [6, 7] has offered an ideal playground for therealization of a diverse host of configurations showcasing remarkable vortex patterns and dynamics. The early effortsalong this direction principally focused on the existence and dynamical robustness/stability properties of individualvortices (including multi-charge ones that were generically identified as unstable in experiments), as well as of largescale vortex lattices created upon suitably fast rotation [8–11]. Some of the early theoretical and experimental effortsalso touched upon few-vortex crystals [12, 13]. Yet, it was not until the development of more recent experimentaltechniques, such as the minimally destructive imaging [14–16], the imaging of dragged laser beams through theBEC [17], or the quadrupolar excitations spontaneously producing multi-vortex states [18] that few-vortex dynamicsdrew a sharp focus of the research effort. It is worthwhile to note that in this BEC context, some of the standardproperties and conservation laws of the vortex system [19] still apply, including e.g. the angular momentum (i.e., thesum of the squared distances of the vortices from the trap center multiplied by their respective topological charge) orthe Hamiltonian of the vortex system. However, others such as the linear momentum are no longer preserved. Thisis due to the local vortex precession term arising in the dynamics as a result of the presence of the external (typicallyparabolic) trap [8, 9].

Motivated by the ongoing experimental developments, and perhaps especially the work of Seman et al. [18], in therecent work of Koukouloyannis et al. [20], a detailed study of the transition from regular to progressively chaoticbehavior has been performed in the tripole configuration (consisting of two vortices of one circulation and one of theopposite circulation). This has been achieved by using a sequence of Poincare sections with the angular momentumL of the vortex system as a parameter. Notice that, while this tripole system without the local BEC-trap inducedprecession is integrable (see e.g. the discussion of Aref and co-workers [19, 21]), here the absence of linear momentumconservation renders chaotic dynamics accessible at this level. In this context the main bifurcations which lead to thedestabilization of the system and the eventual appearance of chaotic behavior have been observed. Our aim in thepresent work is to provide more quantitative results about the chaotic behavior of the system for various energy levels.As a principal tool to this effect, we will employ the chaos detection method - Smaller ALignment Index (SALI) which,as we will argue, is a more efficient tool for such studies than the classical maximum Lyapunov exponent (mLE).

Our study is structured as follows. In section II, we briefly present the setup of the theoretical particle modeldeveloped earlier [15, 16, 20], which we will use in the present study. In section III, we present the numerical toolsthat we use in this work, namely the Poincare surface of section, the chaoticity index SALI and the scan maps thatcan be derived by using this index. In particular we briefly introduce SALI and show why it is more relevant for thisstudy than the maximum Lyapunov exponent (mLE) based on computational and physical arguments. In addition,by using SALI, we define appropriate scan maps and show the correspondence with the Poincare sections which havealso been calculated in [20]. After that, in section IV we acquire some quantitative results about the permitted areaof orbits of this system as well as about the percentage of the chaotic orbits among this area, by using the energyh and the angular momentum L of the system as parameters. Finally, we summarize our findings and present somedirections for future study in section V.

II. THE MODEL

In line with the work of Koukouloyannis et al. [20], we briefly present in this section the model used for the dynamicalstudy of vortices in a BEC. The equations of motion of a system of N interacting vortices in a quasi-two-dimensional(pancake shaped) BEC are given by

xi = −Siωpryi −BN∑

j=1,j 6=i

Sjyi − yj

2r2ij

(1a)

yi = Siωprxi +B

N∑j=1,j 6=i

Sjxi − xj

2r2ij

, (1b)

3

where (xi, yi) stand for the position coordinates of the i-th vortex in the condensate and rij denotes the distancebetween vortices i and j. Si is the topological charge of the ith vortex assuming here the values of Si = ±1 andit is associated with its rotational direction, while B is a constant factor, which in the realm of a homogeneousBEC is B = 2. Nevertheless, in the presence of the trap, a factor lower than the value of the homogeneous case,has been used [22] to emulate the more complex effect of the modulated density induced screening. Moreover,

ωpr = ω0pr/[1− r2

R2TF

]is the precession frequency of a single vortex around the center of the BEC, with RTF =

√2µ/Ω

being the so-called Thomas-Fermi radius, approximately characterizing the radial extent of the BEC. Here, µ isthe chemical potential characterizing the number of atoms in the BEC, while Ω is the effective frequency of thetwo-dimensional confining parabolic trap defined as the ratio of the radial and z-components of the potential. Theprecession frequency ω0

pr characterizes the vortex precession in the immediate vicinity of the trap center for which

the expression ω0pr = ln

(A µ

Ω

)/R2

TF, with A = 2√

2π, has been argued to yield good agreement with both PDE directsimulations and linearization spectral analysis via the Bogolyubov-de Gennes equations [22].

Equations (1) are valid for vortices that are sufficiently well-separated and thin-core (i.e., ‘particle-like’) so thattheir structure does not affect their inter-particle interaction. This case scenario is most directly applicable in thelarge density (the so-called Thomas-Fermi limit), since the characteristic healing length scale associated with thevortex width is inversely proportional to the square root of the BEC density.

Using the scaling t 7→ t/ω0pr, x 7→ xRTF, y 7→ yRTF, equations (1) become

xi = −Siyi

1− r2i

− cN∑

j=1,j 6=i

Sjyi − yjr2ij

(2a)

yi = Sixi

1− r2i

+ c

N∑j=1,j 6=i

Sjxi − xjr2ij

(2b)

with c = B/[2 ln

(A µ

Ω

)]. A typical value for the parameter c has been estimated e.g. by Navarro et al. [16] to be

c ≈ 0.1.The dynamics of system (2) can be described by the Hamiltonian function

HN =1

2

N∑k=1

ln(1− r2k)− c

2

N∑k=1

N∑j>k

SkSj ln(r2kj) (3)

via the canonical equations xi = Si∂H∂yi

, yi = −Si ∂H∂xi. Since Hamiltonian (3) does not explicitly depend on time t, HN

is an integral of motion. As it can be easily verified the system’s ‘angular momentum’

L =

N∑i=1

Sir2i . (4)

is an additional integral.In our study we consider the particular case of N = 3 interacting vortices with charges S1 = S3 = 1, S2 = −1,

following the experimental results of Seman et al. [18] who observed systematically in a substantial fraction of theirexperiments the formation of this configuration. Hamiltonian (3) becomes

H =1

2

3∑i=1

ln(1− r2i ) +

c

2

[ln(r2

12)− ln(r213) + ln(r2

23)]

(5)

where we dropped the index in H for simplicity, while ri =√x2i + y2

i and rij =√

(xi − xj)2+ (yi − yj)2

. Considering

q = (x1, y2, x3), p = (y1, x2, y3) we acquire the usual form of the canonical equations qi = ∂H∂pi

, pi = −∂H∂qi for the

system’s evolution.Applying two successive canonical transformations: a) (xi, yi) 7→ (wi, Ri) defined by

qi =√

2Ri sin(wi) , pi =√

2Ri cos(wi), i = 1 . . . 3, (6)

and b) (wi, Ri) 7→ (φ1,2, ϑ, J1,2, L) according to

φ1 = w1 − w3 J1 = R1

φ2 = w2 + w3 J2 = R2

ϑ = w3 L = R1 −R2 +R3,

(7)

4

the Hamiltonian (5) assumes the form

H =1

2[ln(1− 2J1) + ln(1− 2J2) + ln(1− 2(L− J1 + J2))]

+c

2

[ln(4J2 − 2J1 + 2L− 4

√J2

√L− J1 + J2 sin(φ2))

− ln(2L+ 2J2 − 4√J1

√L− J1 + J2 cos(φ1))

+ ln(2J1 + 2J2 − 2√J1

√J2 sin(φ1 + φ2))

].

(8)

Since ϑ does not appear explicitly in (8) its conjugate generalized momentum L (7), is an integral of motion. This isof course expected as L corresponds to the angular momentum (4) of the system, for the particular choice of chargesSi. Thus, Hamiltonian (8) can be considered as this of a two degree of freedom system with L as a parameter.

In what follows we use the value of the ‘energy’ h of the system and the value of the angular momentum Las the main parameters of our study. These values are both dependent on the initial configuration, i.e. the set(x10, y10, x20, y20, x30, y30) of initial conditions of each orbit. The energy of the system corresponds to the value ofthe Hamiltonian for a particular configuration, h = H(x10, y10, x20, y20, x30, y30), and as we can see from equation(5), it depends on the distance ri of the vortices from the center of rotation as well as on the distance between thevortices. Note that, in the normalized coordinates we use, it is 0 6 ri < 1 since the value ri = 1 stands for theThomas-Fermi radius. When the ri → 1, h decreases, while the dependence on the inter-vortex distance is somehowless obvious because of the sign interchange in the interaction part of the Hamiltonian. On the other hand, the valueof L = L(x10, y10, x20, y20, x30, y30) depends only on the distance of the vortices from the rotation center. This can beseen clearly from equations (7). In particular, since 0 6 Ri < 0.5, the typical range of values of L is −0.5 < L < 1.

III. NUMERICAL METHODS

A. The Poincare surface of section

Since our system can be treated as a two degree of freedom one, we can derive Poincare surfaces of section (PSSs) [23]like the ones depicted in the upper panels of FIG. 1, in order to understand the general trends of its dynamics. In orderto obtain a PSS we have first to consider a fixed value of the energy h. In this case we have considered h = −0.7475.In addition, for each panel of FIG. 1 we consider a fixed value of the angular momentum L. For the PSS to be definedwe also consider a fixed value for φ2, namely φ2 = π/2. In this way the plane (φ1, J1) is defined as the plane ofthe PSS. Note that the value of φ2 = π/2 corresponds to the configuration where the S2 and S3 vortices lie on thehalf-line having the center of the condensate on its edge as it can be seen from the transformations (6) and (7).

In general, every set of initial values (φ10, φ20, ϑ, J10, J20, L) defines an initial configuration (x10, y10, x20, y20, x30, y30)on the configuration space (x− y) of the system. By fixing the values of h, L, φ2 and ϑ, we can determine the valueof J20 through the equation of the energy H(φ10, φ20, ϑ, J10, J20, L) = h. This way we establish a correspondencebetween the points of the (φ1, J1)-plane of the section and the configurations of the system in the (x, y)-plane.

In order to acquire the PSSs of FIG. 1 we considered various values of L. As it has been already mentioned, thetypical range of the values of L is −0.5 < L < 1. But, since the energy constraint must also be fulfilled, the rangeis actually smaller. For this value of the energy the range considered is −0.45 6 L 6 0.55. For low values of Lthe system is fully organized featuring regular orbits. For a critical value of L ' −0.218 the central periodic orbitdestabilizes through a pitchfork bifurcation, and a chaotic region is subsequently created (FIG. 1(a)). This region getswider as L increases (FIG. 1(b)). For even larger values of L, the permitted area of the PSS shrinks, as we will seein detail later on, (FIG. 1(d)-(e)) and finally all the allowed configurations of the system correspond to regular orbitswhich are concentrated around an orbit involving the collision of the vortices S1 − S3. More information about theprogression of the PSSs for h = −0.7475 as the value of L varies, as well as the correspondence of the various areasof the sections with the configurations in the (x− y) plane can be found in the work of Koukouloyannis et al. [20].

Our aim in the present work is to acquire more quantitative results about the chaotic behavior of the system. Forthis task the inspection of a PSS is not particularly useful. Instead, we will scan the PSS and categorize its orbits aschaotic or regular using a chaos detection tool. In particular, we will not employ the maximum Lyapunov exponent(mLE) [24–26], which is a standard method for such tasks, but we will use the Smaller ALignment Index (SALI) asa measure of chaoticity [27–29] as it is, arguably, more efficient for this purpose, as we will explain below.

5

(a) L = −0.05 (b) L = 0.05 (c) L = 0.25

(d) L = 0.45 (e) L = 0.50

0

500

1000

1500

2000

2500

3000

FIG. 1: Poincare surfaces of section (upper panels) and the corresponding SALI scan maps (lower panels) ofHamiltonian (8) for energy h = −0.7475 and varying values of L. In the PSSs the thick black curves correspond toboundaries of motion. In the scan maps the black areas correspond to non-permitted orbits. The gray scale shownnext to panels (e) is used for coloring each permitted initial condition according to its tS0

value (see text for moredetails). So, dark colored points correspond to orbits with small tS0

and light colored points correspond to orbitswith large tS0

.

B. Chaos indicators

The mLE λ is defined as[24–26]

λ = limt→∞

Λ(t), with Λ(t) =1

tln‖v(t)‖‖v(0)‖

, (9)

where ‖v(0)‖, ‖v(t)‖, are the norms of a deviation vector from the studied orbit at times t = 0 and t > 0 respectively.Λ(t) tends to zero for regular orbits following the power law[30] Λ(t) ∼ t−1, while it tends to positive values forchaotic orbits. A basic practical problem of the numerical estimation of the above limit is that Λ(t) is influenced by

6

-2 0 2 4 6-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

logt

log

Λ

-2 0 2 4 6-12

-10

-8

-6

-4

-2

0

logt

log

SAL

I

-2 0 2 4 6-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

logt

log

Λ

-2 0 2 4 6-12

-10

-8

-6

-4

-2

0

logtlo

gSA

LI

FIG. 2: The time evolution of Λ(t) (9) (upper panels) and the SALI(t) (10) (lower panels) in log-log scales for twodifferent initial conditions: (φ1, J1) = (0.96π, 0.1187) (left panels) and (φ1, J1) = (0.6π, 0.1889) (right panels). In theupper panels ten different initial deviation vectors are considered, while three different pairs of them are used for the

computation of the SALI in the lower panels. The times when SALI < SALIthres = 10−12 are shown by verticaldashed lines.

the whole history of the deviation vector’s evolution and the time needed for its convergence to the limit value λ canbe extremely long and cannot be estimated a priori.

In general, the limit of equation (9) can be computed for any random choice of the initial deviation vector. Forchaotic orbits, this practically means that after some (not known in advance) transient time any two initially differentdeviation vectors will be aligned, leading to the eventual computation of the same limit. This can be easily seenin the upper panels of FIG. 2 where the computation of the mLE for two particular chaotic orbits of Hamiltonian(8) is illustrated for ten different choices of the initial deviation vector. As expected, the various curves eventuallycoincide indicating that, in general, the use of any randomly chosen initial deviation vector will allow the estimationof the mLE λ (9). The introduction[27] of the SALI exploits exactly this feature of the deviation vectors’ evolutionby practically determining the time that two initially different deviation vectors v1, v2 align.

For the computation of the SALI we follow the time evolution of two initially different deviation vectors v1(0),v2(0), and define SALI as [27]:

SALI(t) = min ‖v1(t) + v2(t)‖ , ‖v1(t)− v2(t)‖ (10)

where vi = vi(t)/ ‖vi(t)‖. Obviously, when the two vectors become collinear SALI→ 0. As was explained above thishappens in the case of chaotic motion[28] where the two initially different deviation vectors tend to coincide with thedirection defined by the mLE (FIG. 3). This alignment obviously happens well before the corresponding quantitiesΛ(t) defined by the two vectors coincide and reach their limiting value, as can be verified by comparing the upperand the lower panels of FIG. 2 where the evolution of Λ(t) and SALI(t) for two chaotic orbits is shown respectively.

On the other hand, a regular orbit evolves on a torus and any two random deviation vectors will eventually fallon the tangent space of the torus, and in general they will not have the same direction[29]. This means that thecorresponding Λ(t) will tend to zero as Λ(t) ∼ t−1, indicating that the mLE is zero. This is clearly seen in the upperpanel of FIG. 4 where we plot the evolution of Λ(t) for a regular orbit of Hamiltonian (8) for a random initial deviationvector. We see that the curve is parallel to the 1/t line, as the deviation vector is eventually confined on the torus’tangent space. It is evident from the definition of the SALI that in this case the index does not go to zero. Actuallyit fluctuates around some constant positive value, as we see for example in the lower panel of FIG. 4.

7

ˆ 1v (0)

ˆ 2v (0)

ˆ 2v (t)

ˆ 1v (t)

Trajectory

P(0)

P(t)

SALI(0)

SALI(t)

FIG. 3: Definition of the SALI based on the evolution of two distinct normalized deviation vectors (see also text).

-2 0 2 4 6

-6

-4

-2

0

2

logt

log

Λ

-2 0 2 4 6-12

-10

-8

-6

-4

-2

0

logt

log

SAL

I

FIG. 4: The time evolution of Λ(t) (9) (upper panel) and the SALI(t) (10) (lower panel) in log-log scales for theregular orbit with initial conditions (φ1, J1) = (0.3733π, 0.1135). The dashed line in the upper panel is a line with

slope λ = −1. We see that Λ(t) decays as ∝ 1/t.

The different behavior of the SALI for chaotic and regular orbits makes it an efficient chaos indicator, as its manyapplications to a variety of dynamical systems[31–38] illustrate. Another advantage of the method over the mLE isthat the SALI value of an orbit is sufficient to discriminate between regularity and chaoticity, in contrast to the mLEfor which one usually has to check the whole time evolution of Λ(t) to identify deviations from the ∼ t−1 decreasein order to verify chaos. Based on its definition, when SALI tends to 0 the orbit is chaotic. In practice, we requireSALI to become smaller than a very small threshold value (in our study we set SALIthres = 10−12) to characterizean orbit as chaotic. Thus, SALI constitutes an ideal numerical tool for the purposes of our study, as its computationfor a large sample of initial conditions allows the construction of phase space charts (which we will call ‘scan maps’)where regions of chaoticity and regularity are clearly depicted and identified.

8

C. The scan map

In order to construct a scan map, like the ones shown in the lower panels of FIG. 1, we select an equally spacedgrid of 300 × 300 initial conditions (φ1, J1) on the PSS and compute SALI for each orbit.[39] When the value ofSALI becomes SALI < SALIthres = 10−12 we consider SALI to practically be zero and the corresponding orbit tobe chaotic. We denote the time needed for an orbit to reach this threshold tS0

(φ10, J10). The maximum integrationtime we consider is tmax = 3000. If SALI(tmax) > SALIthres then the orbit is considered to be regular. In that case,we set tS0

to be tS0= tmax. Depending on the value of tS0

(φ10, J10), we assign a color to each point of the grid.In particular, darker colored points correspond to orbits with smaller tS0

, while lighter colored points correspond toorbits with larger tS0

. In this way we construct color charts of the PSS (lower panels of FIG. 1) based of how fast thechaotic nature of an orbit is revealed. These scan maps clearly show not only the regions where regular and chaoticmotion occurs, as the comparison with PSS plots in FIG. 1 easily verifies, but also indicate regions with differentdegrees of chaoticity. Finer grids and longer integration times were also considered, but the results they providedwere not significantly different from the ones presented in FIG. 1, while the additional computational time requiredwas extremely higher. Hence, the choice of the 300× 300 grid and the value tmax = 3000 has been deemed to be themost efficient in order to reveal the details of the dynamical behavior of this system.

IV. RESULTS

Firstly let us investigate in more detail the general behavior of the Hamiltonian (8) based on the results of FIG. 1.The black areas in the scan maps of this figure represent rejected initial conditions of the grid. There are three reasonsto reject an initial condition on the PSS. The first is that the specific point does not comply with both the energy andangular momentum constraints of the system. These are the upper and lower black areas in FIG. 1(a)–(d) and thelarge, central, black area of FIG. 1(e). The second reason to exclude an initial condition is if a particular configurationcorresponds to a collision orbit, i.e. two vortices lie at the same point of the configuration space (x− y). This state ismeaningless both physically and mathematically, since the energy of the system becomes infinite. This case is visiblein FIG. 1(c) where a black line is shown at J1 = 0.25, which corresponds to a collision between S2 and S3. The thirdreason is purely physical: if the initial condition represents a configuration in which the two co-rotating vortices S1

and S3 lie close to each other, these two vortices become ‘trapped’ in a motion where they rotate around each other.This is called the ‘satellite’ regime. Additionally, if the distance between them is too small (r13 < 0.1), our model doesnot describe the dynamics accurately, as it was constructed under the assumption that vortices behave like particlesretaining their structure unchanged, which of course is not true when they acquire this level of proximity. So in ourstudy we do not try to tackle questions related to close encounters of the vortices. This restriction corresponds tothe little black areas on the left and right end sides of the scan maps. In this consideration we have not excluded thecases where the counter-rotating vortices come close since in this case they are not trapped but instead they just passby each other and continue their motion.

The boundaries of the permitted areas in the PSSs of FIG. 1 (i.e. the thick black curves in these sections) arecalculated by the requirement that one of the vortices will pass through the origin [20] (Ri = 0). From the transfor-mation (7) we see that R2 has a negative contribution to L, while R1 and R3 contribute positively. Thus, for lowvalues of L, the S2 vortex is moving away from the origin and the boundaries are determined by the R1 = 0 andR3 = 0 constraints. In particular, the former condition provides the J1lo(R1=0) = 0 boundary, while the latter givesJ2 = J1 − L. Since in each panel of FIG. 1 we consider fixed values for h and L and in addition we set φ2 = π/2 forthe construction of the PSSs, Hamiltonian (8) provides an implicit relation J1up(R3=0) = J1up

(φ1;h, L) for the upperboundary of the permitted area. On the other hand, for high values of L the J1lo(R2=0) and J1up(R2=0) boundariesare both calculated by the constraint R2 = 0, through similar considerations.

The permitted area can now be numerically calculated by the integral

Ap =

∫ 2π

0

(J1up − J1lo

)dφ1. (11)

In (11), each point of J1upand J1lo

is also calculated numerically through the implicit functions J1up(φ1) and J1lo

(φ1)mentioned above. The obtained results are reported in FIG. 5 by solid lines.

Another way to verify the accuracy of this computation is by estimating the size of the permitted area as the sum ofthe areas of all the small rectangulars of the grid of the scan map which correspond to permitted orbits. The obtainedresults are depicted by dots in FIG. 5. Since we computed scan maps for values of h and L for which the systemexhibited significant amounts of chaotic behavior, we do not have in FIG. 5 data for L < −0.2 where no chaoticorbits were detected. We also note that by this approach we underestimate the size of the permitted area (the dots

9

(a) h = −1.1

-0.4 -0.2 0.0 0.2 0.4 0.60.0

0.5

1.0

1.5

2.0

2.5

L

Per

mitt

edA

rea

(b) h = −0.9

-0.4 -0.2 0.0 0.2 0.4 0.60.0

0.5

1.0

1.5

2.0

2.5

L

Per

mitt

edA

rea

(c) h = −0.8

-0.4 -0.2 0.0 0.2 0.4 0.60.0

0.5

1.0

1.5

2.0

2.5

L

Per

mitt

edA

rea

(d) h = −0.7475

-0.4 -0.2 0.0 0.2 0.4 0.60.0

0.5

1.0

1.5

2.0

2.5

L

Per

mitt

edA

rea

(e) h = −0.6

-0.4 -0.2 0.0 0.2 0.4 0.60.0

0.5

1.0

1.5

2.0

2.5

L

Per

mitt

edA

rea

(f) h = −0.5

-0.4 -0.2 0.0 0.2 0.4 0.60.0

0.5

1.0

1.5

2.0

2.5

L

Per

mitt

edA

rea

FIG. 5: Evolution of the permitted area of motion for varying values of h. The solid line represents the calculation ofthe area using (11). The dots represent the calculation via the scan maps. The dashed line connecting the two partsof the solid curve corresponds to the region of the values of L in which there is the ambiguity in the calculation of thearea through (11) discussed in the text. Panel (d) corresponds to the value of h for which FIG. 1 has been acquired.

are always below the continuous curves in FIG. 5) because in the computation of the scan maps we always excludedregions related to initial configurations with r13 < 0.1 as discussed earlier.

The curves in all the panels of FIG. 5 follow a similar pattern. For L . −0.4 no orbits exist as no initial conditionssatisfy both the h and L restrictions. As L increases (up to L . 0.15− 0.35 depending on h) the size of the permittedarea grows larger as the J1up(R3=0) curve moves upwards in FIG. 1. For large values of L (L & 0.25− 0.45 dependingon h) the S2 vortex moves closer to the center of rotation, so the boundaries are defined by the constraint R2 = 0. AsL increases, the permitted area shrinks because the boundaries defined by J1up(R2=0) and J1lo(R2=0) come closer. Forintermediate values of L, just after the maximum of the curve Ap = Ap(L), there is an ambiguity concerning if theboundary is determined by the constraint R3 = 0 or R2 = 0, because for some values of φ1 the boundary is definedby the former while for others it is defined by the latter relation. In this region we cannot calculate the size of thepermitted area by (11) and the calculation from the scan maps is more reliable. So, the two well computed by (11)parts of Ap(L) are connected in this region by a dashed straight line in order to obtain a continuous curve. It is worthnoting that this rough approximation is in good agreement with the results obtained by the scan maps. By examiningthe sequence of plots of FIG. 5(a)-(f) we conclude that the area of permitted orbits decreases as h increases.

Let us now turn our attention to the study of individual orbits of the Hamiltonian (8). Since the physical systemfrom which this study has been motivated is a Bose-Einstein condensate, which has a limited life time (commonly of theorder of a few seconds to a few tens of seconds), there are some associated considerations to be kept in mind. Firstly,the use of some typical parameters such as µ = 10, Ω = 0.2 and a z-component of the parabolic trap ωz = 2π100Hzsuggests that a reasonable estimate of the condensate’s lifetime in our dimensionless units would be of the order of afew hundred up to one thousand.

So let’s discuss in more detail the implications that this physically induced time limit has. For this purpose weconsider in FIG. 6 the scan map for h = −0.7475 and L = −0.03. This map was constructed similarly to the onesof FIG. 1. The connected chaotic region in the center of this plot (dark gray points) can be constructed by anyorbit starting in it. Nevertheless, depending on where we choose the initial condition of this orbit, the time in whichits chaotic behavior is revealed varies. This becomes evident from the results of FIG. 7 where the plot of FIG. 6 isdecomposed into four regions depending on the tS0

values of the initial conditions. In particular we consider pointswith tS0

∈ [140, 500] (FIG. 7a), tS0∈ (500, 1000] (FIG. 7b), tS0

∈ (1000, 1500] (FIG. 7c) and tS0∈ (1500, 2000]

(FIG. 7d). Since the typical lifetime of the BEC is a few hundred time units, a good candidate for a physically

10

meaningful integration time would be tmax = 500 and this will be used further on in our study. In this way, chaoticorbits which reveal their nature later than this time can be considered, from a practical point of view, as regular. Forinstance, in real experiments one would expect to detect chaotic motion in the limited time that the experiment lasts,only in regions with small tS0

. In our case, such orbits are the ones plotted in FIG. 7(a), whose initial conditions arelocated close to the center of the x-shaped chaotic region.

0

500

1000

1500

2000

2500

3000

FIG. 6: Scan map for h = −0.7475 and L = −0.03. Dark colored points correspond to orbits with small tS0

(chaotic) and light colored points correspond to orbits with large tS0 (regular). This correspondence is also shown inthe given legend.

Thus, the need for efficient chaos indicators, capable of determining in small, physically meaningful, time intervalsthe nature of orbits is of considerable importance. The SALI can successfully play this role, in contrast to the mLEas the results of FIG. 2 indicate. We name the orbit corresponding to the left (right) column of FIG. 2 orbit A (B).This orbit has tS0

= 320 (tS0= 1750), and its initial condition (φ1, J1) = (0.96π, 0.1187) ((φ1, J1) = (0.6π, 0.1889))

is taken from the points of FIG. 7a (FIG. 7d). Since both orbits belong to the same chaotic sea they have the samemLE. This becomes apparent from the upper panels of FIG. 2, where the time evolution of Λ is shown for ten differentinitial deviation vectors. We see that for both orbits (and of course for all initial deviation vectors) we eventuallyget log10(mLE) ' −1.25. The fact that orbit A behaves chaotically sooner than orbit B is seen from the evolutionof their SALI values in the lower panels of FIG. 2 and denoted by its smaller tS0

value. This distinction cannot beeasily obtained from their mLEs, especially if one does not want to visualize the whole evolution of the index. We seethat Λ requires, in general, at least 10 times longer time intervals (in some cases even more) in order to converge toa positive value, while SALI goes to zero in fewer than 3000 time units, even for the weakly chaotic, ‘sticky’ orbits.This ability of SALI is crucial for our BEC model, which has a limited life time. One additional important point isthat the SALI requires less CPU time than the mLE in order to determine the chaotic nature of orbits, which is veryimportant when we want to study large samples of orbits.

In the paper of Koukouloyannis et al. [20] it was shown for the case of h = −0.7475 that orbits are regular whentheir initial configuration is close to the one- or two-vortex regime. In the one-vortex regime the vortices lie far fromeach other and, consequently, do not interact strongly. The two-vortex regime is the one where the S1 and S3 vorticesrotate around each other (satellite regime) and the S2 vortex lies far from them. In these two regimes our systembehaves dynamically similarly to the cases of BECs with only one or two vortices, which are both integrable. On theother hand, strong interactions between all three vortices lead to chaotic behavior.

Let us now attempt a more global study of the system aiming to acquire quantitative results about its chaoticity.Performing extensive computations of scan maps we managed to calculate the percentage, PC , of initial conditionsleading to chaotic motion, within the set of the permitted initial conditions of the corresponding grid. We carriedout this calculation for several parameter values in the intervals −1.1 6 h 6 −0.5 and −0.2 6 L 6 0.6, settingtmax = 500. The obtained results are plotted in FIG. 8 by solid lines. Even by reseting the final integration time totmax = 3000 (dashed curves in FIG. 8), which is the value used in FIG. 1, no significant differences are observed. Thepercentages slightly increase because some ‘sticky’ chaotic orbits are now characterized as chaotic, but apart fromthat the obtained curves are very close to the ones constructed for tmax = 500.

There seems to exist a trend for all different values of the energy. For small values of the angular momentum

11

(a) 140 6 tS0 6 500 (b) 500 < tS0 6 1000

(c) 1000 < tS0 6 1500 (d) 1500 < tS0 6 2000

FIG. 7: The decomposition of FIG. 6 for various intervals of tS0 . We see that as tS0 increases the correspondingpoints of the scan lie further away from the center of the x-shaped chaotic region and their number decreases.

(L . −0.2) no chaotic orbits exist. As L increases, PC increases rather quickly and after an interval where it retainsconsiderably high values, it drops down. Finally PC vanishes for large values of L (L & 0.4 − 0.6 depending on h),while at the same time the number of permitted initial conditions in the phase space shrinks (FIG. 5). The values ofL for which the chaotic region appears (denoted Lc) and disappears (denoted Ld) depend on the particular value ofthe energy h. It is interesting to note that within this interval, there is at least one local minimum, implying a localmaximum in the fraction of regular trajectories.

Our results show that the range Ld − Lc is larger for lower energies. This is related to the size of the permittedPSS area, which is also larger when h is lower (FIG. 5). The appearance of the chaotic region seems to happen atabout the same value L ≈ − 0.1, for all h values, but the eventual shrinking and disappearance of this region variesfrom Ld ≈ 0.4 for h = −0.5 (FIG. 8(f)) to Ld ≈ 0.7 for h = −1.1 (FIG. 8(a)). Presumably, the invariance of the onsetof chaoticity is because of the weak h dependence of the critical point Lc for which the central stable periodic orbitundergoes a pitchfork bifurcation. This bifurcation generates the x-shaped chaotic region (FIG. 1(a)) which leads tothe onset of chaoticity.

In addition, we observe smaller percentages of chaotic motion altogether for lower energies. While the maximumpercentage for h = −0.5 is ≈ 100% (FIG. 8(f)), the one for h = −1.1 is just ≈ 70% (FIG. 8(a)). This behavior canbe explained as follows. As we have seen in FIG. 5, the higher the energy of the vortices, the smaller the area ofpermitted motion becomes. Consequently, vortices come closer and the interaction among them becomes stronger,which in turn leads to the enhancement of the chaotic behavior.

We also observe in the panels of FIG. 8 the existence of a ‘secondary’ local maximum between the two ‘main’ localmaxima. This maximum is more pronounced for low energies, for example it has the same height as the two ‘main’maxima for h = −1.1 (FIG. 8(a)), while it becomes less distinct as the value of h increases, and practically disappearsfor h = −0.5 (FIG. 8(f)). The values of L for which this maximum appears coincide with the values of L for whichthe maximum of the permitted area occurs (FIG. 5). This happens because at this L value the upper boundary ofthe permitted area is defined by both the R3 = 0 and R2 = 0 constraints, which means that vortices S2 and S3 are

12

(a) h = −1.1

-0.2 0.0 0.2 0.4 0.60

20

40

60

80

100

L

P C

(b) h = −0.9

-0.2 0.0 0.2 0.4 0.60

20

40

60

80

100

L

P C

(c) h = −0.8

-0.2 0.0 0.2 0.4 0.60

20

40

60

80

100

L

P C

(d) h = −0.7475

-0.2 0.0 0.2 0.4 0.60

20

40

60

80

100

L

P C

(e) h = −0.6

-0.2 0.0 0.2 0.4 0.60

20

40

60

80

100

L

P C

(f) h = −0.5

-0.2 0.0 0.2 0.4 0.60

20

40

60

80

100

L

P C

FIG. 8: Dependence of the percentages PC of initial conditions leading to chaotic motion on L for different values ofh. Solid lines represent the results acquired for tmax = 500 while the dashed lines show the results for tmax = 3000.

We observe that the main characteristics of PC are the same in both cases.

close to the center of rotation and consequently close to each other. Then strong interactions between them lead tochaotic behavior. As the value of h increases, the motion becomes more chaotic and the percentages of the chaoticorbits increase. Consequently, this phenomenon becomes less significant and eventually not observable.

We believe that this analysis, based on the SALI, offers a systematic view of the PSS and the fraction of accessibleorbits in it (as per FIG. 5), as well as of the fraction of chaotic orbits in it (as per FIG. 8) and how these change asa function of the canonical physical properties of the system, namely its energy and its angular momentum.

V. CONCLUSIONS - FUTURE DIRECTIONS

In the present work, we explored a theme of current interest within the research of atomic BECs, namely therecently realized experimentally tripoles of vortices and their associated nonlinear dynamical evolution. We foundthat this Hamiltonian system is arguably prototypical (at least within the realm of isotropic magnetic traps) in itsexhibiting chaotic dynamics as parameters or initial conditions are varied. We focused here on the variation of initialconditions, through the variation of associated conserved quantities such as the energy and the angular momentum.Our aim was to associate a technique that has been previously used in a variety of other low dimensional settings,namely the SALI diagnostic, for measuring the chaoticity of the orbits within this atomic physics realm of vortexdynamics under their mutual interactions and their individual precession within the parabolic trap. We found thatthe SALI is a very accurate diagnostic of the different levels of chaoticity of the system and enables a qualitativeunderstanding of how this chaoticity changes as the conserved quantities are varied, as well as a quantification of thechaotic fraction of the phase space of the system. We also illustrated the advantages of SALI over broadly used toolssuch as the maximum Lyapunov exponent, since it can distinguish between different degrees of chaoticity which is ofgreat importance in systems like the one presently considered which have a limited lifetime.

This work paves the way for the consideration of a wide range of additional problems within the dynamics ofcoherent structures in the realm of Bose-Einstein condensates. First of all, it would be straightforward to explorehow the dynamics of this tripole would compare/contrast to the recently explored [16] dynamics of 3 co-rotatingvortices (i.e., vortices of the same charge). Another natural extension in the vortex case would be to examine howthe chaoticity region expands as a fourth vortex of either a positive or a negative charge comes into play. The specialcases of 4 co-rotating vortices (with relevant square and rhombic etc. stable configurations), as well as the case ofthe generally fairly robust [22] vortex quadrupole would be of interest in this setting. Additionally, expanding suchconsiderations to other dimensions would present interesting possibilities as well. On the one hand, a wide range oftheoretical and experimental considerations (including particle based approaches, such as the ones utilized herein)

13

have been developed for dark solitons in 1d; see e.g. the recent review of Frantzeskakis [40]. On the other hand,generalizing to 3d and the consideration of multiple vortex rings and their dynamics [41] would be equally or evenmore exciting from the point of view of ordered vs. chaotic dynamics. Examination of these directions is currently inprogress and will be reported in future publications.

ACKNOWLEDGMENTS

The authors would like to thank Prof. Roy H. Goodman for a comment that provided useful insights toward furtherdevelopment of this direction.N.K. acknowledges support from the MC Career Integration Grant PCIG13-GA-2013-618399.V.K. and Ch.S. have been co-financed by the European Union (European Social Fund - ESF) and Greek national fundsthrough the Operational Program ”Education and Lifelong Learning” of the National Strategic Reference Framework(NSRF) - Research Funding Program: THALES. Investing in knowledge society through the European Social Fund.V.K. would also like to thank the research committee of the Aristotle University of Thessaloniki for its support throughthe postdoctoral research award “ Aristeia ”. Ch.S. was also supported by the Research Committees of the Universityof Cape Town (Start-Up Grant, Fund No 459221) and the Aristotle University of Thessaloniki (Prog. No 89317).P.G.K. acknowledges support from the National Science Foundation under grants CMMI-1000337, DMS-1312856,from the Binational Science Foundation under grant 2010239, from FP7-People under grant IRSES-606096 and fromthe US-AFOSR under grant FA9550-12-10332.

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