Dynamics of Few Co-rotating Vortices in Bose-Einstein Condensates
R. Navarro, R. Carretero-Gonzalez, P.J. Torres, P.G. Kevrekidis, D.J. Frantzeskakis, M.W. Ray, E. Altunta, D.S. Hall
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Dynamics of Few Co-rotating Vortices in Bose-Einstein Condensates
R. Navarro , R. Carretero-Gonz´alez , P.J. Torres , P.G. Kevrekidis ,D.J. Frantzeskakis , M.W. Ray , E. Altunta¸s , and D.S. Hall Nonlinear Dynamical Systems Group, Computational Science Research Center,and Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182-7720, USA Departamento de Matem´atica Aplicada, Universidad de Granada, 18071 Granada, Spain Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-4515, USA Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 157 84, Greece Department of Physics, Amherst College, Amherst, Massachusetts, 01002-5000 USA
We study the dynamics of small vortex clusters with few (2–4) co-rotating vortices in Bose-Einsteincondensates by means of experiments, numerical computations, and theoretical analysis. All ofthese approaches corroborate the counter-intuitive presence of a dynamical instability of symmetricvortex configurations. The instability arises as a pitchfork bifurcation at sufficiently large values ofthe angular momentum that induces the emergence and stabilization of asymmetric rotating vortexconfigurations. The latter are quantified in the theoretical model and observed in the experiments.The dynamics is explored both for the integrable two-vortex system, where a reduction of the phasespace of the system provides valuable insight, as well as for the non-integrable three- (or more)vortex case, which additionally admits the possibility of chaotic trajectories.
Introduction.
The realm of atomic Bose-Einstein con-densates (BECs) [1] has offered a pristine setting for stud-ies on the dynamics of few-vortex clusters [2]. Most inves-tigations, however, have focused on either a single vortexor large scale vortex lattices [3–6]. Recently, theoreticalinvestigations on the study of clusters of 2–4 vortices [7–15], have appeared, chiefly motivated by the experimen-tal realizations of such states [16–19]. This focus has beenheretofore centered on the fundamental building block ofthe vortex dipole, i.e., a pair of counter-circulating vor-tices.Our aim in the present work, in contrast, is to ex-plore the dynamics of small vortex clusters of 2–4 vor-tices that belong to the co-rotating (same charge) va-riety. The original work of Ref. [2] and subsequent ef-forts [20] have already paved the way for an understand-ing of symmetric few-vortex configurations rotating as arigid body, and their three-dimensional generalizations,i.e., U- and S-shaped vortices, as well as vortex rings [21].In this context, our work presents a rather unexpectedtwist: we have found that, under suitable conditions, theusual symmetric, co-rotating vortex configurations (cen-tered line, triangle, and square) become dynamically un-stable . More specifically, these states become subject tosymmetry breaking, pitchfork bifurcations that lead tothe spontaneous emergence of stable asymmetric rotat-ing vortex clusters .We present our analysis of these features in the in-tegrable (at the reduced particle level) setting of a co-rotating vortex pair, and illustrate their generality byfurther considering a rigidly rotating vortex triplet andquadruplet. In the first case, we devise a theoretical for-mulation that not only explores the instability and man-ifests its growth rate, but also enables a visualizationof a two-dimensional reduced phase space of the systemin which the pitchfork bifurcation becomes transparent. In the latter cases, we suitably parametrize the system,exploring the different regimes of symmetric and asym-metric periodic orbits. Our theoretical analysis treatsvortices as classical particles, with dynamics governed byordinary differential equations (ODEs). This reduction ofthe original vortex cluster system allows for the analyticalcharacterization, numerical observation and experimen-tal confirmation of the symmetry breaking phenomena.
Theoretical Analysis.
As illustrated in Refs. [15,19], and justified by means of a variational approx-imation [22], vortex dynamics governed by the two-dimensional mean-field Gross-Pitaevskii equation, i∂ t ψ = −
12 ∆ ψ + 12 Ω ( x + y ) ψ + | ψ | ψ, (1)can be reduced to a system of ODEs for the vortex posi-tions. In the original partial differential equation (PDE)model (1), the time is measured in units of ω − z , whilethe positions are measured in units of harmonic oscilla-tor length along the z -direction and Ω = ω x /ω z = ω y /ω z ,with ω j being the harmonic trap frequency along the j -direction (see, e.g., Ref. [4]). This ODE reduction is thestarting point for our analysis in the co-rotating case.The dynamics of vortex m at position ( x m , y m ) arisesfrom two contributions: (i) a position-dependent vortexprecession about the trap center with frequency S m ω pr ,and (ii) a vortex-vortex interaction with vortex n thatinduces a velocity perpendicular to their line of sight ofmagnitude S n ω vort /ρ mn , where ρ mn is the distance be-tween vortices m and n , S m and S n are their respec-tive charges, and ω vort is a dimensionless constant; seeRef. [15, 19]. The equations governing the dynamics of N interacting vortices embedded in a condensate are there-fore ˙ x m = − S m ω pr y m − ω vort X n = m S n y m − y n ρ mn , ˙ y m = S m ω pr x m + ω vort X n = m S n x m − x n ρ mn . (2)The precession about the trap center can be approxi-mated by ω pr = ω / (1 − r /R ), where the frequencyat the trap center is ω = ln (cid:0) A µ Ω (cid:1) /R , µ is the chem-ical potential, R TF = √ µ/ Ω is the Thomas-Fermi (TF)radius, and A is a numerical constant [3, 15, 19]. Todescribe better the actual vortex dynamics in the trap,the constant ω vort in Eqs. (2) may be adjusted to ac-count for the screening of vortex interactions due to thebackground density modulation [23].We now focus on the case of two identical vortices ofunit charge S = S = 1. We proceed to adimensionalizeEqs. (2) by scaling ( x, y ) by R TF and time by 1 /ω , anduse polar coordinates ( x n , y n ) = ( r n cos( θ n ) , r n sin( θ n )).We then seek symmetric stationary states r = r = r ∗ and θ − θ = π , and find the following frequency of theco-rotating vortices: ω orb = ˙ θ = ˙ θ = c r ∗ + 11 − r ∗ , (3)where c = ( ω vort /ω ) yields a measure of the rela-tive strength of vortex interaction and spatial inhomo-geneity. The comparison of the orbital frequency be-tween the ODE and the PDE models is given in Fig. 1a.Given the co-rotating nature of this state, considerationof δ mn = θ m − θ n renders this state a stationary one; lin-earizing around it using r m = r ∗ + R m and δ mn = π + δ m yields the following equations of motion for the pertur-bations about the symmetric equilibrium:¨ R m = − ω R n − R m ) , ¨ δ m = − ω δ m − δ n ) , with ω = c r ∗ − c ( − r ∗ ) . It is then straightforward to ob-serve that this squared epitrochoidal (motion of a pointin a circle that is rotating about another circle) rela-tive precession frequency of the two vortex positions andphases changes sign at r = √ c/ ( √ c + 2). This signalsour first fundamental result, namely the destabilization ofthe symmetric 2-co-rotating vortex state for sufficientlylarge symmetric distances of the vortices from the trapcenter. A comparison of the ODE and PDE models forthe orbital and epitrochoidal precession frequencies forthese two cases is given in Fig. 1a-b.The dynamical instability of symmetric states suggeststhe potential existence of additional, asymmetric, ones.Seeking states with δ mn = π and r ∗ = r ∗ yields − r ∗ r ∗ ( r ∗ + r ∗ ) + c (cid:0) − r ∗ (cid:1) (cid:0) − r ∗ (cid:1) = 0 , which will be the condition defining our radially asym-metric co-rotating solutions. The mirror symmetry ofthe 2-vortex system predisposes towards the pitchfork,symmetry breaking nature of the relevant bifurcation, afeature verified by the diagram of Fig. 1c. This diagramis given for the angle φ = tan − r /r as a function of theangular momentum L = r + r , which is a conservedquantity for our system. It is interesting to note that ifthe single dimensionless parameter of the system is small( c < L cr for L —at whichthe bifurcation from symmetric to asymmetric periodicorbits occurs— is supercritical, while if c is sufficientlylarge ( c > L , defined above,and the Hamiltonian H , which can be written in polarcoordinates as H = 12 ln (cid:2)(cid:0) − r (cid:1) (cid:0) − r (cid:1)(cid:3) − c (cid:2) r + r − D ) (cid:3) , where D ≡ r r cos( δ ) and δ = θ − θ . Using L and theangle φ to express r and r , one can rewrite the Hamil-tonian as a function of ( φ, δ ) having thus effectively re-duced the 4-dimensional system into a 2-dimensional one.Thus, for different values of L , we can represent the or-bits in the effective phase plane of ( φ, δ ) in which the dif-ferent orbits correspond to iso-energetic H ( φ, δ )=const.contours. This is done in Fig. 2 for values that are bothbelow and above than the critical value of L at fixed c .It can then be inferred that the symmetric fixed pointwith ( φ, δ ) = ( π/ , π ) is stable in the former case, whileit destabilizes in the latter case through the emergenceof two additional asymmetric ( φ = π/
4) states along thehorizontal line δ = π of anti-diametric vortex states.Remarkably, although the properties of the system dra-matically change as we go from two vortices to three andfour, the symmetry breaking bifurcation associated withthe symmetric solutions persists. In particular, when N >
2, the persistence of the two conservation lawsdiscussed above is not sufficient to ensure integrabilityof the system, and its absence is manifested in a dra-matic form in the resulting 6 ( N = 3) and 8 ( N = 4)dimensional systems through the presence of chaotic or-bits. Nevertheless, one can still theoretically analyze thehighly symmetric co-rotating states of the system.For N = 3, this state is an equilateral triangle suchthat r = r = r = r ∗ and δ i,i +1 = 2 π/
3, with an or-bital frequency predicted as ω orb , = cr ∗ + − r ∗ . In the co-rotating frame, the linear stability analysis around thisrigidly rotating triangle can be performed giving rise toan epitrochoidal frequency ω , = c r ∗ − c (1 − r ∗ ) . In thiscase too, a critical radius exists r , = √ c/ ( √ c + √ (a) r/R TF ω o r b (b) r/R TF ω ep φ / π (c) L θ / π (d) φ / π (e) L FIG. 1: (Color online) (a) Orbital and (b) epitrochoidal frequency as a function of the radial position from the trap’s centerfor two vortices. Both frequency and radial position are in rescaled units. The solid line represents results from the ODE andthe dotted line from the PDE. The vanishing of the latter signals the onset of instability. Here, Ω = 0 .
05 and µ = 1. Panel(c) depicts the quantity φ/π , which equals 1/4 when r = r , as a function of the square root of the angular momentum for c = 0 .
1. Panels (d) and (e) depict the corresponding phenomena for N = 3 and N = 4 vortices for c = 0 .
1. Panels (c), (d)and (e) include a few configurations along the main bifurcation branches ([blue] solid and [red] dashed lines corresponding,respectively, to stable and unstable configurations) depicting the relative position of the vortices (red triangles) with respect tothe center of the condensate (green crosses). −π/2 0 π/20π2π (a) φ δ −0.4 −0.2 0 0.2 0.4−0.4−0.200.20.4 (b) x y −π/2 0 π/20π2π (c) φ δ −0.4 −0.2 0 0.2 0.4−0.4−0.200.20.4 (d) x y FIG. 2: (Color online) (a) Contours from the reduced Hamil-tonian for vortices close to the center of the trap, i.e. L = 0 . L < L cr = 0 . L = 0 . L > L cr = 0 . c = 0 . such that the symmetric state is destabilized and asym-metric orbits arise and are stable past this critical pointas can be seen in Fig. 1d. The dynamical picture is con-siderably more complicated but the conservation of theangular momentum ensures that the dynamical evolutionresides on the surface of a Bloch sphere. We thus definetwo angular variables tan φ = r /r and cos θ = r / √ L and depict the associated pitchfork bifurcation in Fig. 1dfor the subspace of solutions constrained to r = r and δ = δ . This bifurcation diagram describes a vor-tex configuration containing a stable symmetric rotat-ing triangle before the bifurcation and stable asymmet-ric rotating triangles after the bifurcation. In addition tothe equilibrium and near-equilibrium orbits, we observe chaotic orbits arising both in a more localized form, ex-ploring the vicinity of equilibrium orbits, and in a moreextended one spanning all space (not shown).While the general phenomena for N = 4 are alreadyrather complex, some basic features can still be inferredand the symmetry breaking nature of the proposed insta-bility persists —cf. Fig. 1e. Here, φ = tan − r /r , andwe have constrained the vortices to be in a cross withright angles and r = r and r = r . A general ex-pression for the orbital frequency of the rigidly rotatingstate is ω orb ,N = ( N − c r ∗ + − r ∗ , which is valid for any N . In the case of the square configuration with r i = r ∗ and δ i,i +1 = π/
2, there emerge two epitrochoidal vibra-tional motions with frequencies √− λ and √− λ , where λ = c (1 − r ∗ ) − c r ∗ , and λ = c (1 − r ∗ ) − c r ∗ . These, inturn, correspond to two critical points: one identical tothe one given above for the N = 3 case, and one thatis always higher, given by r , = √ c/ ( √ c + 2); hence,the same phenomenology persists. Experimental Observations.
We now briefly discussexperimental manifestations of the symmetry breakingevents discussed above and of the emergence of asym-metric configurations.The details of the experimental setup may be foundelsewhere [17, 19]. We begin with a magnetically-trappedBEC of N ∼ × atoms in the | F = 1 , m F = − i hyperfine level of Rb. The radial and axial trap fre-quencies are ( ω r , ω z ) / π = (35 . , .
2) Hz. Vortices areintroduced through a process of elliptical magnetic trapdistortion and rotation [24] during evaporation [25]. Interms of the trap frequencies along the major and minoraxes of the distorted potential, ω x and ω y respectively, anellipticity ǫ = ( ω x − ω y ) / ( ω x + ω y ) = 0 .
20 and a rotationfrequency of 8.5 Hz usually produces a co-rotating pair.Higher rotation frequencies are used to generate largernumbers of co-circulating vortices.A partial-transfer (5%) imaging method [17] is em-ployed to create a sequence of atomic density profiles,as shown in Fig. 3a–d. The effect of the extractions isprimarily to diminish the number of atoms in the con-densate [14, 17]. Curiously, atomic losses have little ef-fect on the calculated coupling parameter c , which scalesonly as log N ; thus c falls between 0 .
11 and 0 .
10 over therange N = 0 . . × atoms. For convenience, we take c = 0 . L and Hamilto-nian H were computed for each frame [see panels e)–h)in Fig. 3]. For each series, the averaged angular momen-tum L and Hamiltonian H are computed (see horizontaldashed lines in the middle panels in Fig. 3). Using L wecompare the experimental points representing each orbitin the ( φ, δ ) plane to the isocontour of H correspondingto H , as shown in the right column of panels in Fig. 3,and find good agreement between the two.The panels a)–d) in Fig. 3 depict a cohort of typicaltime series, together with their respective fits, that ex-emplify the different qualitative cases that we observedin the experiments. In particular, we find that the dy-namics of the vortices depends on whether the averageangular momentum is below or above the critical thresh-old L cr = 2 r . This separates cases where asymmetricorbits are, respectively, non-existent and possible. Thedifferent qualitative cases that we observe, which are isdisplayed in Fig. 3, may be grouped as follows: • For L < L cr and relatively small H , the experimentdisplays symmetric orbits. See rows a) and e) in Fig. 3. • For L > L cr and moderate H , the experiment dis-plays (i) symmetric orbits where both vortices (in theco-rotating frame) are approximately on the same side ofthe cloud chasing each other on the same path [see rowsb) and f) in Fig. 3] or (ii) asymmetric orbits [see rows c)and g) in Fig. 3]. The choice between these two orbitsis determined by the initial conditions. Initial conditionsinside the area delimited by the separatrix (red double-loop curve in the right panels of Figs. 3.f and Figs. 3.g)emanating from the saddle point ( φ, δ ) = ( π/ , π ) giverise to asymmetric orbits. • For L > L cr and large H , the experiment displays or-bits in which one vortex remains close to the center whilethe other orbits around it close to the periphery of thecloud. See rows d) and h) in Fig. 3.As is clear from these examples and the remaining 48data sets that we studied (see supplemental material),asymmetric orbits are only found when L > L cr andwhen the vortex orbits fall inside the asymmetric min-ima regions of the Hamiltonian picture in the ( φ, δ ) plane.Asymmetric solutions absent in all of the cases for which L < L cr . These results are in good agreement with −0.5 0 0.5−0.500.5 e) y δ π /4 π /20 π π L & − H −0.5 0 0.5−0.500.5 f) y δ π /4 π /20 π π L & − H −0.5 0 0.5−0.500.5 g) y δ π /4 π /20 π π L & − H −0.5 0 0.5−0.500.5 x h) y φ δ π /4 π /20 π π t (ms) L & − H FIG. 3: (Color online) a)–d) Typical experimental series forthe dynamics of two co-rotating vortices (time indicated inms). The large (green) circles and the (red) crosses represent,respectively, the fitted TF radius and center of the cloud whilethe small (yellow) dots depict the fitted vortex centers. The(red) dashed circles represent the critical radius, r cr , abovewhich symmetric orbits become unstable. e)–h) Manifesta-tion of the pitchfork bifurcation for the experimental seriesdepicted in panels a)–d) which correspond to c = 0 .
1. Leftcolumn: experimental vortex positions and their correspond-ing orbit from the reduced ODE model (solid line), in TFunits in the co-rotating frame. Middle column: correspond-ing L (blue circles) and − H (green squares) and their av-erages (horizontal dashed lines) as well as the critical valuefor L (solid horizontal line). Right column: correspondingorbits in the ( φ, δ ) plane along with isocontours for constant H (highlighted in dark gray is the isocontour correspondingto the average H and in red is the separatrix delimiting thearea containing asymmetric orbits). the theoretical prediction of the pitchfork bifurcation de-picted in Fig. 1c.To extend our considerations, we briefly present a com-parison between experiment and theory for N = 3 and N = 4 vortices. The main phenomenology is depicted L & − H te) t f) tg) t h) FIG. 4: (Color online) Experimental series for trios [a) and b)]and quartets [c) and d)] below [a) and c)] and above [b) andd)] the critical threshold. e)–h) Corresponding time series for L and − H . Same notation and units as in Fig. 3. using two examples for each case in Fig. 4. Panels a) andb) correspond to the N = 3 vortex case below and abovethe pitchfork bifurcation [see panel e) and f)]. Panels c)and d) depict the equivalent scenario for N = 4 vortices.As the figure illustrates, and is observed in all of thecases that we studied (17 data sets for N = 3 and 5 datasets for N = 4; not shown), the main phenomenology for N = 3 and N = 4 persists in that all configurations with L > L cr are not symmetric and symmetric configura-tions, or epitrochoidal oscillations about them, are onlypresent when L < L cr . Conclusions.
We have revisited the theme of co-rotating few vortex clusters in atomic Bose-Einstein con-densates. By a combination of theoretical analysis,numerical computation and experimental observation,we have illustrated a strong manifestation of symmetrybreaking through a pitchfork bifurcation, which led tothe destabilization of symmetric solidly rotating config-urations and gave rise to the the emergence of stable co-rotating but asymmetric vortex configurations. Weshowed that this analysis is fruitful not only for the inte-grable (at the reduced particle level) two-vortex setting,where a suitable parametrization of the phase space wasprovided, but also for the non-integrable cases of N = 3and N = 4 vortices where chaotic orbits exist.Naturally, it would be interesting to provide a moreglobal characterization of the dynamics of the three-bodyproblem, which is perhaps the most analytically tractableand interesting case due to its potential for chaos. An-other expansion of the present considerations involvestheir generalization to higher dimensions. In this case,it would be interesting to see, upon gradual decrease ofthe trapping frequency in the third dimension, whetherthe symmetry-breaking phenomena persist for line vor-tices and vortex rings. These aspects are presently understudy and results will be reported elsewhere. exp L H exp L H a) 0.098 0 . − . . − . . − . . − . . − . . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . L and averagedHamiltonian H for all the experimental series containing twoco-rotating vortices. The critical angular momentum for theexperimental setup is L cr = 0 . Acknowledgments: Support from NSF PHY-0855475(D.S.H.), NSF DMS-0806762 and CMMI-1000337, andAlexander von Humboldt Foundation (P.G.K.), and NSFDMS-0806762 (R.C.G.), and discussions with T.K. Lan-gin are kindly acknowledged.
SUPPLEMENTAL MATERIAL:
Here we present the results that we obtained for the52 experimental series containing two co-rotating (samecharge) vortices. The experiments are ordered by the av-erage angular momentum L . which is computed fromthe fitted centers and cloud diameter. The average isthen computed by averaging over the eight experimentalsnapshots for each series. Table I lists the average angularmomentum and average Hamiltonian for all the experi-mental series containing two co-rotating vortices. Thecorresponding snapshots for all the series are shown inFig. 5 using the same ordering as given in Table I. Finally,Fig. 6 depicts the analysis for each individual series bypresenting the orbits in the co-rotating frame (respective FIG. 5: Experimental time series corresponding to the 52 experiments containing two co-rotating vortices. The snapshotswere taken at regular times for a total time spanning 240 ms or 480 ms (for individual times for each snapshot refer to thecorresponding middle column in Fig. 6). The experiments are ordered according to average angular momentum as listed inTable I. For ease of presentation, all the images have been centered and rescaled so that they occupy approximately the samevisual area. left columns), the times series for the angular momen-tum and Hamiltonian (respective middle columns), andthe orbits in the reduced ( φ, δ ) plane (respective rightcolumns). As observed in Fig. 6, the experimental orbits in theco-rotating frame (respective left columns) match wellthe theoretical orbits from our model (see gray orbit) forall of the experimental runs. Significantly also, all ex-
FIG. 6: (Color online) Analysis for experimental series depicted in Fig. 5. For each series the respective columns represent: (i)The left column depicts the vortex orbits (green squares and blue circles), and their corresponding orbit from the reduced ODEmodel (gray solid line), in TF units in the co-rotating frame. (ii) The middle column depicts the time series for the angularmomentum L (blue circles) and negative Hamiltonian − H (green squares) and their respective averages (horizontal dashedlines) as well as the critical value for L (solid horizontal line). (iii) The right column depicts the orbits in the ( φ, δ ) planealong with the isocontours for constant H (highlighted is the isocontour corresponding to the average H ). perimental orbits in the reduced ( φ, δ ) plane (respectiveright columns) match well the theoretical orbits corre-sponding to the isocontour of constant H equal to theaverage H from the experiment —with the exception ofthe experiment L cr = 2 r where the square of the critical radiusis r = √ c/ ( √ c + 2) in units of the Thomas-Fermi ra-dius. In our experiment, the value of c , the ratio be-tween the rotation induced by vortex-vortex interactionsand the rotation induced by the precession of vorticesdue to the inhomogeneity of the cloud’s background pro-duced by the external magnetic trap, is c ≈ .
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