Vortex reconnections in atomic condensates at finite temperature
A. J. Allen, S. Zuccher, M. Caliari, N. P. Proukakis, N. G. Parker, C. F. Barenghi
VVortex reconnections in atomic condensates at finite temperature
A. J. Allen , ∗ S. Zuccher , M. Caliari , N. P. Proukakis , N. G. Parker , and C. F. Barenghi Joint Quantum Centre (JQC) Durham-Newcastle, School of Mathematics and Statistics,Newcastle University, Newcastle upon Tyne NE1 7RU, England, UK. Dipartimento di Informatica, Universit`a di Verona,Ca’ Vignal 2, Strada Le Grazie 15, 37134 Verona, Italy (Dated: October 8, 2018)The study of vortex reconnections is an essential ingredient of understanding superfluid turbu-lence, a phenomenon recently also reported in trapped atomic Bose–Einstein condensates. In thiswork we show that, despite the established dependence of vortex motion on temperature in suchsystems, vortex reconnections are actually temperature independent on the typical length/timescales of atomic condensates. Our work is based on a dissipative Gross-Pitaevskii equation forthe condensate, coupled to a semiclassical Boltzmann equation for the thermal cloud (the Zaremba–Nikuni–Griffin formalism). Comparison to vortex reconnections in homogeneous condensates furthershow reconnections to be insensitive to the inhomogeneity in the background density.
PACS numbers: 03.75.Lm, 03.75.Kk, 67.85.De, 67.25.dkKeywords: vortices, vortex dynamics, quantum turbulence, Bose-Einstein condensates, Superfluid He
In classical hydrodynamics, reconnections of streamlines, vortex lines and magnetic flux tubes change thetopology of the flow and contribute to energy dissipa-tion. In quantum fluids, vorticity is not a continuousfield, but is concentrated in discrete vortex lines of quan-tised circulation; their reconnections are therefore iso-lated, dramatic events. Individual quantum reconnec-tions have been recently visualized [1] in superfluid He.The role of vortex reconnections in the dynamics of quan-tum turbulence [2] in superfluid He, superfluid He andatomic Bose–Einstein condensates (BECs) is currentlydebated. For example, one would like to understand theircontribution to acoustic dissipation of kinetic energy [3],their role in the proposed bottleneck [4] between a semi–classical Kolmogorov cascade at small wavenumbers anda quantum Kelvin wave cascade at large wavenumbers,and the possibility of a cascade of vortex rings scenario[5, 6]. The increasing ability to imprint [7], generate [8–11], manipulate [12] and directly image [11, 13] vorticesmakes atomic condensates ideal systems to study vortexreconnection events [14]. This problem is of particularinterest in light of recent experiments regarding quan-tum turbulence and vortex dynamics in both two [15–17]and three dimensions [18] (for a review on progress intwo and three dimensions see, e.g. [19]).Since many experiments are performed at relativelyhigh temperatures, i.e. large fractions of the critical tem-perature, T c , a natural question to ask is whether thermalexcitations affect vortex reconnections. A recent experi-ment visualising vortex reconnections in superfluid He,suggests that this is not the case [1]. However, previousinvestigations have shown that the presence of a ther-mal cloud significantly changes the motion of vortices inharmonically trapped condensates [20–25]. ∗ Electronic address: [email protected]
In this paper we present results of an investigationof vortex reconnections in finite–temperature trappedBose–Einstein condensates. We model the problem in thecontext of the Zaremba–Nikuni–Griffin (ZNG) formalism[26, 27], where the Gross-Pitaevskii equation (GPE) isgeneralized by the inclusion of the thermal cloud meanfield, and a dissipative or source term which is associ-ated with a collision term in a semiclassical Boltzmannequation for the thermal cloud. The main feature of thismodel is that the condensate and thermal cloud interactwith each other self–consistently; for a strongly nonlin-ear dynamical event like a vortex reconnection, a simplerand less accurate approach may give misleading answers.The governing ZNG equations are i ¯ h ∂φ∂t = (cid:18) − ¯ h m ∇ + V ext + g [ n c + 2˜ n ] − iR (cid:19) φ , (1)and ∂f∂t + p m · ∇ r f − ( ∇ r U eff ) · ( ∇ p f ) = C + C . (2)In this formalism φ = φ ( r , t ) is the condensate wave-function, f = f ( r , p , t ) is the phase-space distributionfunction of thermal atoms, V ext = mω r / ω is the trap-ping frequency, m the atomic mass, and g = 4 π ¯ h a s /m ,with a s being the s -wave scattering length. Equation (1)generalises the GPE for a T = 0 condensate by theaddition of the thermal cloud mean–field potential 2 g ˜ n and the dissipation/source term − iR ( r , t ). The con-densate density is n c ( r , t ) = | φ ( r , t ) | and the thermalcloud density is recovered from f ( r , p , t ) via an integra-tion over all momenta, ˜ n ( r , t ) = (2 π ¯ h ) − (cid:82) d p f ( p , r , t ).The mean-field potential acting on the thermal cloudis U eff = V ext ( r ) + 2 g [ n c ( r , t ) + ˜ n ( r , t )]. The quanti-ties C [ f ] and C [ φ, f ] are collision integrals defined in a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y Refs. [26, 27] (which contain further details of the model). C describes the redistribution of thermal atoms asa result of two–atom collisions (as in the usual Boltz-mann equation), while C , which is related to − iR via R ( r , t ) = ¯ h/ (2 n c ( r , t )) (cid:82) d p / (2 π ¯ h ) C [ φ ( r , t ) , f ( p , r , t )],describes the change in the phase-space distribution func-tion f ( p , r , t ) resulting from particle-exchange collisionsbetween thermal atoms and condensate atoms.Before solving the ZNG equations (1) and (2) numer-ically, we write them in dimensionless form, using theharmonic oscillator’s characteristic length (cid:96) = (cid:112) ¯ h/mω as the unit of distance, the inverse trapping frequency ω − as the unit of time, and ¯ hω as the unit of en-ergy. We choose experimentally realistic parameters: ω = 2 π × µ = µ/ (¯ hω ) ≈
18 where µ is thechemical potential, and ˜ g = g/ ( (cid:96) ¯ hω ) ≈ (cid:96) = 0 . µ m, ω − = 1 . N c ≈ Rb atoms. Throughout this work, in order to facili-tate more direct comparisons, we keep this value of N c approximately constant even at temperatures above zeroso that the effect of increasing temperature is to increasethe number of thermal atoms in the system rather thandepleting the condensate. As well as solving the ZNGequations, we use another model to describe finite tem-perature effects. A simple phenomenological extensionof the GPE known as the dissipative Gross–Pitaevskiiequation (DGPE, (Eq. 3)), is used for temperatures be-low 0 . T c , which is thought to be reasonable limit for itsvalidity [28], while in the trapped case, for temperatureshigher than this we use the ZNG equations.First we consider what happens in the limit of zerotemperature, for which Eq. (1) with iR = 0 and ˜ n = 0reduces to the GPE, a model known to provide an ac-curate description of condensate dynamics for T (cid:28) T c ,including collective modes and vortex dynamics [29, 30].For the quantities considered here, we have confirmedthat the results of the ZNG equations in the absenceof any significant thermal cloud, agree with the resultsgiven by the GPE. The initial state of our simulationis the condensate containing a pair of straight line anti-parallel vortices aligned in the z –direction at locations( x /(cid:96), y /(cid:96) ) = ( − , ± π winding of the phase of φ around the location of the cores.In the GPE model, the vortex core size is of the orderof the condensate healing length, ξ = ¯ h/ √ mµ . For ourassumed parameters, ξ/(cid:96) ≈ . R TF /(cid:96) = √ µ ≈
6. To ensure thatthe vortices reconnect in the central region of the con-densate and away from its boundary, we perturb their y position along z according to A [cos(2 πz/λ )] where A/(cid:96) = 0 . λ/(cid:96) = 20. This initial condition is shownin the top left panel of Fig. 1, where we have plotteda series of snapshots of the reconnection for the case of T = 0. It is apparent that the two vortices initially moveas a pair in the xy plane, traveling in the x direction.The slight initial curvature enhances the Crow instabil- FIG. 1: Vortex reconnection in the T = 0 trapped condensate.Density isosurfaces of the dynamics of the initial anti–parallelvortex pair, at times tω = 0 (top left), tω = 1 .
19 (top right), tω = 1 .
80 (bottom left) and tω = 2 . − (cid:96) ≤ x, y, z ≤ (cid:96) . ity [31]: the vortices approach each other and reconnect,creating two U–shaped cusps which lift and move awayfrom each other above and below the xy plane (bottomright panel).Simulations at various temperatures show that the vor-tex reconnection proceeds essentially unaffected by thepresence of the thermal cloud, consistent with the find-ings of Ref. [1]. To illustrate this we focus on the rela-tively high temperature case of T /T c = 0 .
62. Figure 2compares density slices of the T = 0 condensate (leftpanel) and the T /T c = 0 .
62 condensate and thermalcloud (middle and right panels), both before reconnec-tion (top row) and after reconnection (middle and bot-tom rows). It is apparent that thermal atoms are con-centrated at the edge of the condensate and inside thevortex cores [25], an effect of the mean–field repulsionfrom the condensate. Importantly, over the time scalefor the reconnection, we observe no difference in the vor-tex dynamics between the T = 0 and the T > R TF ≈ ξ ). In this case, and as evident in Figs. 1 and 2, thereconnection region is not far from the condensate outersurface. This surface region can undergo oscillations, par-ticularly in a turbulent condensate [32], and interact withthe vortices. Moreover the surface region is where ther-mal atoms accumulate, and is likely to introduce rela-tively large frictional effects upon the vortices. In orderto more clearly extract out the role of finite temperatureon reconnections it is therefore instructive to considerreconnections in a homogeneous (boundary-free) conden-sate. T = 0 T > T = 0 (left column) and T /T c = 0 .
62 (middle and rightcolumns). 2D density plots from the ZNG model of the con-densate (left and middle) and thermal cloud (right). Top:Before the reconnection, tω = 0 .
3, on the z/(cid:96) = − . z/(cid:96) = 0 plane, in order to better distin-guish the two vortex cores). Middle and bottom: After thereconnection, tω = 2, slices through the xz plane for y/(cid:96) = 0(middle row) and the yz plane for x/(cid:96) = 0 (bottom row). Inthe density scale, white corresponds to peak density and blackto zero density. Note the vanishing condensate density in thevortex cores (left and middle column) and the concentrationof thermal atoms at the edge of the condensate and inside thecores (right column). In the interest of computational ease and efficiency,we can use a simpler extension to the GPE which sim-ulates thermal effects by the inclusion of a phenomeno-logical damping parameter, called the dissipative Gross–Pitaevskii equation (DGPE) [28, 33],( i − γ )¯ h ∂φ∂t = (cid:18) − ¯ h m ∇ + g | φ | − µ (cid:19) φ. (3)The phenomenological damping parameter γ , which weassume to be constant, has been used in a variety of sys-tems to simulate thermal effects (see e.g. [28, 34–36]). For γ = 0 this model reduces to the GPE. We solve Eq. (3) ina periodic box in the absence of an external potential. Itmust be stressed that, unlike the ZNG model, the DGPEdoes not include the dynamics of the thermal cloud.Before solving Eq. (3) we write it in dimensionlessform using natural units based on the healing length ξ = ¯ h/ √ mµ and the time unit ξ/c . As done by Zuc- FIG. 3: Vortex reconnection in the homogeneous condensateat T = 0. Density isosurfaces showing the reconnection ofthe vortex-antivortex pair, according to the GPE, at times t/ ( ξ/c ) = 0 (top left), t/ ( ξ/c ) = 20 (top right), t/ ( ξ/c ) = 40(bottom left) and t/ ( ξ/c ) = 60 (bottom right). The isosur-faces are plotted at 20% of the peak density. The range ofthe axes is − ξ ≤ x, y, z ≤ ξ . cher et al. [37], we use a Fourier-splitting scheme wherethe Laplacian part is trivially solved in spectral space,whereas the remaining part is exactly solved in physicalspace as suggested by Bao et al. [38]. We place a pair ofanti–parallel vortex lines in a computational box of size − ξ ≤ x, y, z ≤ ξ at position ( x /ξ, y /ξ ) = (10 , ± A [cos(2 πz/λ )] , now with A/ξ = 1and λ/ξ = 120 (chosen such that the the vortices areunperturbed at z max and z min ). The box size is choosensuch that the vortex length is comparable to the vortexlength in the ZNG simulation.The initial configuration and subsequent evolution ofthe vortex pair is depicted in Fig. 3 for γ = 0, corre-sponding to T = 0. The vortex reconnection proceedsin much the same way as for the trapped condensateshown in Fig. 1. We repeat the simulation for γ = 0 . . T c [28, 39, 40]) and again note that the reconnection pro-ceeds essentially unchanged despite the presence of dissi-pation in the system.To monitor the vortex reconnections more preciselythan “by eye”, we consider the minimum distance be-tween the vortex lines, δ ( t ). We extract the positionof the vortex core by finding the grid points where thedensity is a local minimum and about which the phasechanges by 2 π [41]. The time–dependence of this quan-tity (before and after the reconnection) was experimen-tally observed in superfluid He, and predicted theoreti-cally for superfluids based on the GPE ( T = 0) [37] andfor ordinary viscous fluids based on the classical Navier–Stokes equation [42]. To enable comparison of δ ( t ) be-tween the homogeneous and trapped systems, we mustconvert between harmonic trap units (based on the har-monic oscillator length and frequency) and natural units(based on the healing length and the chemical potential).The conversion for length from harmonic oscillator unitsto natural units is given by x (cid:48) = ˜ x(cid:96)/ξ and for time is t (cid:48) = c/ ( ωξ )˜ t where we used a tilde to denote the quan-tity in harmonic trap units and a prime for natural units(see footnote [46] for more details). For the remainder ofthis article we express all quantities in natural units.Figure 4 compares δ (cid:48) ( t (cid:48) ) computed using the ZNGmodel (trapped condensate) and the DGPE (homoge-neous condensate) before (top) and after the reconnec-tion (bottom). For completeness, we have also carriedout DGPE simulations, within the presence of a trap forthe value γ = 0 . . T c (as used for the homogeneouscase). We find that δ (cid:48) ( t (cid:48) ) scales as δ (cid:48) ( t (cid:48) ) = κ | t (cid:48) − t (cid:48) | ν (4)where t is the time at which the reconnection takes place(defined as when δ (cid:48) ( t (cid:48) ) = 0), and κ and ν are fitting pa-rameters. It is apparent that the results are essentiallyindependent of the model, the temperature and the pres-ence/absence of trapping. The best fit to the parameter ν before the reconnection is ν = 0 . ± .
02 and after thereconnection is ν = 0 . ± . ν = 0 . t < t ) and ν = 0 .
68 ( t > t ) reported by Zuccher et al. [37] over a wide range of initial angles betweenthe vortex lines, computed for T = 0 (GPE) in a ho-mogeneous condensate. It is interesting to remark thatviscous reconnections of the Navier–Stokes equation dis-play a similar time asymmetry [42] (the largest ν beingthat after the reconnection, as in a quantum fluid). Zuc-cher et al. [37] argued that the time asymmetry is dueto acoustic emission: part of the kinetic energy of thevortices is transformed into sound waves which radiateto infinity, in analogy with viscous dissipation in an ordi-nary fluid which turns kinetic energy into heat. Indeed, ifone uses the Biot–Savart law (an incompressible model)to monitor the behaviour of vortices just before and afterthe reconnections, one finds ν = 0 . t < t and t > t . Paoletti et al. [1] observed individual quantum re-connections in superfluid He experiments, reporting theexponent ν = 0 . t < t and t > t data.Above all, Paoletti et al. did not notice any temperaturedependence, which is consistent with our findings.We stress that, a priori, one would not apply the Biot–Savart model to a vortex in a small atomic condensate,as the vortex core is not negligible, particularly in a re-connection, when two vortex cores collide. However, wemay gain some insight to the temperature–independenceof the reconnecting behaviour from it. In the Biot–Savart model [43] the vortex is a three-dimensional spacecurve s ≡ s ( ς, t ) of infinitesimal thickness, where ς is FIG. 4: Minimum distance between vortex line, δ (cid:48) ( t (cid:48) ), be-fore (top) and after (bottom) the vortex-antivortex reconnec-tion, computed for a harmonically trapped condensate anda homogeneous condensate at different values of T /T c and γ (both the homogeneous and trapped values of γ correspondto T ≈ . T c ), including the limiting cases T = 0 and γ = 0.The dashed lines are fits according to Eq. (4) with parame-ters ν = 0 . ± .
02 (line of best fit before reconnection, red,dot-dashed) and ν = 0 . ± .
02 (line of best fit after recon-nection, indigo, dashed) with corresponding κ values, for thedata considered here, of 1 .
36 and 1 .
88 respectively. arc length. The velocity of the curve at the point s is v L = v si − α s (cid:48) × v si , where v si is the self–induced veloc-ity (determined by a Biot–Savart integral over the entirevortex configuration), s (cid:48) ≡ d s (cid:48) /dς is the unit tangentvector at s , and α is a dimensionless temperature depen-dent friction parameter. In the full expression v L thereis a second friction parameter, α (cid:48) , which we have ne-glected here since it is much smaller than α [23, 44, 45].In superfluid helium, outside the phase transition region(less than 1 percent from T c ), α is less than unity andpositive [23, 44]. In atomic condensates, numerical sim-ulations of vortex motion based on the ZNG model haveshown that [23] α < .
03 for
T /T c < .
8. The small valueof α has been confirmed by an independent calculationbased on a classical field approach [44]. The Biot–Savartmodel thus suggests that, instantaneously, the frictiongives only a small contribution to the velocity of the vor-tex. One expects the friction to be significant only onlarge enough length scales and time scales, provided thatits effects can accumulate. For example, in the case ofa single off-centered vortex in a harmonic trap precess-ing at finite temperature, the interaction of the vortexwith the thermal cloud causes it to lose energy and spiralout of the condensate, thus limiting its lifetime [20–25].However, this decay may require many orbits in the trap [21, 23, 25, 36].In conclusion, we have found that, on the typical shortlength scales and time scales relevant to a vortex re-connection in an atomic Bose–Einstein condensate, thereconnection is essentially temperature independent, de-spite the significant inhomogeneity of the thermal cloudin the vortex cores and near the boundary of the conden-sate. Since vortex reconnections are essential ingredientsof turbulence, our result suggests that at least this rapidpart of the dynamics is rather universal, and does notdepend on T , although the large scale motion of vorticesdoes depend on T .AJA, NPP and CFB gratefully acknowledge fundingfrom the EPSRC (Grant number: EP/I019413/1). [1] M. S. Paoletti, M. E. Fisher, and D. P. Lathrop, PhysicaD , 1367 (2010).[2] C. F. Barenghi, L. Skrbek, and K. R. Sreenivasan, Proc.Nat. Acad. Sci. USA (Supp. 1) , 4647 (2014).[3] M. Leadbeater, T. Winiecki, D. C. Samuels,C. F. Barenghi, and C. S. Adams, Phys. Rev. Lett. ,1410 (2001).[4] V. S. L’vov, S. V. Nazarenko, and O. Rudenko, J. LowTemp. Phys. , 140 (2008).[5] M. Kursa, K. Bajer, and T. Lipniacki, Phys. Rev. B ,014515 (2011).[6] R. M. Kerr, Phys. Rev. Lett. , 224501 (2011).[7] A. E. Leanhardt, A. G¨orlitz, A. P. Chikkatur, D. Kielpin-ski, Y. Shin, D. E. Pritchard, and W. Ketterle, Phys.Rev. Lett. , 190403 (2002).[8] C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, andW. Ketterle, Phys. Rev. Lett. , 210402 (2001).[9] B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder,L. A. Collins, C. W. Clark, and E. A. Cornell, Phys. Rev.Lett. , 2926 (2001).[10] C. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley,M. J. Davis, and B. P. Anderson, Nature , 948 (2008).[11] D. V. Freilich, D. M. Bianchi, A. M. Kaufman, T. K.Langin, and D. S. Hall, Science , 1182 (2010).[12] M. C. Davis, R. Carretero-Gonz´alez, Z. Shi, K. J. H. Law,P. G. Kevrekidis, and B. P. Anderson, Phys. Rev. A ,023604 (2009).[13] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dal-ibard, Phys. Rev. Lett. , 806 (2000).[14] A. W. Baggaley (2014), arXiv:1403.8121 [physics.flu-dyn], 1403.8121.[15] T. W. Neely, A. S. Bradley, E. C. Samson, S. J. Rooney,E. M. Wright, K. J. H. Law, R. Carretero-Gonz´alez, P. G.Kevrekidis, M. J. Davis, and B. P. Anderson, Phys. Rev.Lett. , 235301 (2013).[16] K. E. Wilson, E. C. Samson, Z. L. Newman, T. W. Neely,and B. P. Anderson, Annual Review of Cold Atoms andMolecules , pp. 261-298. (2013).[17] W. J. Kwon, G. Moon, J. Choi, S. W. Seo, andY. Shin (2014), arXiv:1403.4658 [cond-mat.quant-gas],1403.4658.[18] E. A. L. Henn, J. A. Seman, G. Roati, K. M. F. Mag-alh˜aes, and V. S. Bagnato, Phys. Rev. Lett. , 045301(2009). [19] A. C. White, B. P. Anderson, and V. S. Bagnato, Proc.Nat. Acad. Sci USA (Suppl. 1) , 4719 (2014).[20] P. O. Fedichev and G. V. Shlyapnikov, Phys. Rev. A ,R1779 (1999).[21] H. Schmidt, F. Goral, K. Floegel, M. Gajda, andK. Rzazewski, J. Opt. B: Quantum Semiclass , S96(2003).[22] R. A. Duine, B. W. A. Leurs, and H. T. C. Stoof, Phys.Rev. A , 053623 (2004).[23] B. Jackson, N. P. Proukakis, C. F. Barenghi, andE. Zaremba, Phys. Rev. A , 053615 (2009).[24] S. J. Rooney, A. S. Bradley, and P. B. Blakie, Phys. Rev.A , 023630 (2010).[25] A. J. Allen, E. Zaremba, C. F. Barenghi, and N. P.Proukakis, Phys. Rev. A , 013630 (2013).[26] E. Zaremba, T. Nikuni, and A. Griffin, Journal of LowTemperature Physics , 277 (1999).[27] A. Griffin, T. Nikuni, and E. Zaremba, Bose-condensedgases at finite temperatures (Cambridge University Press,2009).[28] S. Choi, S. A. Morgan, and K. Burnett, Phys. Rev. A ,4057 (1998).[29] L. P. Pitaevskii and S. Stringari, Bose-Einstein Conden-sation (Oxford University Press, Great Clarendon Street,Oxford, 2003).[30] C. Pethick and H. Smith,
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