Quantum turbulence in atomic Bose-Einstein condensates
QQuantum turbulence in atomic Bose-Einsteincondensates
A. J. Allen, N. G. Parker, N. P. Proukakis, and C. F. Barenghi.
Joint Quantum Centre (JQC) Durham-Newcastle, School of Mathematics and Statistics,Newcastle University, Newcastle upon Tyne NE1 7RU, England, UK.E-mail: [email protected], [email protected], [email protected],[email protected]
Abstract.
Weakly interacting, dilute atomic Bose-Einstein condensates (BECs) have proved to be anattractive context for the study of nonlinear dynamics and quantum effects at the macroscopicscale. Recently, weakly interacting, dilute atomic BECs have been used to investigate quantumturbulence both experimentally and theoretically, stimulated largely by the high degree ofcontrol which is available within these quantum gases. In this article we motivate the useof weakly interacting, dilute atomic BECs for the study of turbulence, discuss the characteristicregimes of turbulence which are accessible, and briefly review some selected investigations ofquantum turbulence and recent results. We focus on three stages of turbulence - the generation ofturbulence, its steady state and its decay - and highlight some fundamental questions regardingour understanding in each of these regimes.
1. Introduction
Turbulence in ordinary fluids such as air or water consists of rotational eddies of different sizeswhich we term vortices. Vortices therefore are a hallmark signature of a turbulent flow (Barenghi et al. , 2001). In superfluids, quantum vortices differ from their classical counterparts becauseof the quantization of circulation. This means that the rotational motion of a superfluid isconstrained to discrete vortices which all have the same core structure. Turbulence in superfluidhelium has been the subject of many recent experimental and theoretical investigations (Skrbek& Sreenivasan, 2012). Recently, experimentalists have been able to visualise individual vortexlines and reconnection events using tracer particles (Fonda et al. , 2012). Weakly interacting,dilute atomic Bose-Einstein condensates (henceforth referred to as BECs) present a distinctplatform to view and probe quantum turbulence. A key feature here is the ability to directlyresolve the structure of individual vortices and in turn the dynamics of a turbulent vortextangle (Henn et al. , 2009). As a result of the quantized nature of vorticity, quantum turbulencein superfluid helium and in BECs can be viewed as a simpler, idealized analog of turbulence inordinary fluids, and opens the possibility of studying problems which may be relevant to ourgeneral understanding of turbulence.
2. Why atomic Bose-Einstein condensates?
Since their first generation in 1995 (Davis et al. , 1995; Anderson et al. , 1995), atomic BECshave been used to study a wide variety of nonlinear dynamics, for example, solitons, vortices a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p nd four-wave mixing (Kevrekidis & Carretero-Gonzalez, 2008). A merit of exploiting BECsas a testbed of nonlinear physics lies with the immense control and flexibility they offer. Forexample: • Trapping geometry, shape and dimensionality
Optical and magnetic fields can be employed to precisely create a potential landscape for theatoms in the BEC, which in turn enables control of the shape and effective dimensionalityof the system (G¨orlitz et al. , 2001). A basic requirement of these gases is confinementin space to prevent contact with hot surfaces. This is typically provided by magnetic trapswhich are harmonic in shape and have the form (Fort´agh & Zimmermann, 2007) V ext ( r ) = 12 mω ( x + y + z ) , (1)where ω is the trapping frequency and m the mass of the atom. This type of trap resultsin an atomic cloud with a radial density profile which resembles an inverted parabola.If a harmonic trap is used which is very strongly confining in one direction, for example V ext ( r ) = 12 mω ( x + y + (cid:15)z ) , (2)where (cid:15) (cid:29)
1, the dynamics in that direction is effectively inhibited and the systembecomes effectively two-dimensional (2D). By changing (cid:15) , one can easily change the effectivedimensionality, which is particularly important in turbulence (2D turbulence is very differentfrom 3D turbulence). In the same way, if the trap is very tightly confining in two directions,the dynamics is mainly in the third direction, and the system is effectively one-dimensional(1D).More complicated trapping geometries can be realised, for example a toroidal ring or aperiodic optical lattice. Traps can also be made time-dependent by rotating or shaking thetrapping potential. Furthermore, one can create localized potentials using optical fields,which can mimic an obstacle and be moved through the system on demand. • Interaction strength
Typically, the dominant atomic interaction in a BEC is the short-range and isotropic s -waveinteraction. Experimentalists can employ magnetic Feshbach resonances (Inouye et al. ,1998) to change the strength of these interactions and even their nature, i.e. whetherthey are attractive or repulsive (Roberts et al. , 1998). Furthermore, by using atoms withrelatively large magnetic dipole moments, e.g. Cr, it is possible to create a BEC wherethe atoms also experience significant dipole-dipole interactions, which are long-range andanisotropic, and greatly modify the static and dynamical properties of the system (Lahaye et al. , 2009). • Vortex core optical imaging
The healing length which characterizes the vortex core size is typically around 10 − m ina BEC (c.f. 10 − m in superfluid Helium). By expanding the BEC (following releasefrom the trapping potential), the vortex can be directly imaged and resolved via opticalabsorption (Madison et al. , 2000; Raman et al. , 2001). Advanced real time imaging ofcondensates containing vortices has also recently been developed (Freilich et al. , 2010),allowing the precession of vortices to be observed.In the limit of zero temperature and weak interactions, the evolution equation for themacroscopic condensate wavefunction, φ ( r , t ), is a form of nonlinear Schr¨odinger equation,commonly known as the Gross-Pitaevskii Equation (GPE): i (cid:126) ∂φ ( r , t ) ∂t = (cid:20) − (cid:126) m ∇ + V ext ( r , t ) + g | φ ( r , t ) | − µ (cid:21) φ ( r , t ) , (3)here r is the position in space, t is time and (cid:126) is Planck’s constant divided by 2 π . TheGPE provides a good quantitative description of the dynamics of a BEC over all lengthscalesavailable, from the vortex core to the system size, up to temperatures of approximately T (cid:39) . T c (where the critical temperature T c is of the order of mK in typical experiments). At theright-hand side we recognise the kinetic energy term, ( − (cid:126) / m ) ∇ , the trapping potential V ext ( r , t ) (which in general may be time-dependent), the interaction term, g | φ ( r , t ) | where g = 4 π (cid:126) a s /m and a s is the 3D s − wave scattering length, and the chemical potential µ .The GPE can be almost exactly mapped to the classical Euler equation; the small difference,namely the quantum stress, regularizes the solutions, preventing singularities which may arisein an Euler fluid (Barenghi, 2008). The GPE is a practically exact model in the limit of zerotemperature, where essentially all of the atoms exist in the Bose-Einstein condensate phase. Inmany experiments the condensate exists at well below the BEC transition temperature such thatthis approximation is justified. Extensions of the GPE to include the effect of thermal atomsprovide a more complete (albeit not exact) physical model of a real BEC (Jackson et al. , 2009)(see e.g. Proukakis & Jackson (2008) for an in depth review of finite temperature models).However BECs suffer an important limitation. The systems which can be currently createdin the laboratory contain a small number of atoms, typically 10 to 10 , hence do not sustainthe number of quantum vortices present in helium experiments. For example, up to a fewhundred vortices have been achieved in the largest 2D BECs (Abo-Shaeer et al. , 2002). Thisbrings to light the issue of length scales. A defining feature of classical turbulence, besidesnonlinearity, is the huge number of length scales which are excited. The range of length scalesavailable in turbulent superfluid helium at very small temperatures is perhaps even larger, sinceshort wavelength helical waves along the vortices can be generated by nonlinear interactions,producing a turbulent cascade called the Kelvin wave cascade (Vinen, 2006). A simple questionarises: can a BEC, containing a limited number of vortices, really become turbulent? Ourtentative answer is yes. Numerical results thus far (Nore et al. , 1997; Berloff & Svistunov, 2002;Kobayashi & Tsubota, 2005 b ; Yepez et al. , 2009) suggest that kinetic energy is distributed overthe length scales in agreement with the k − / Kolmogorov scaling which is observed in ordinaryturbulence (where k is the wavenumber) even over this small range of length scales. Therefore,the study of turbulence in a BEC represents an exciting opportunity to probe a new regimeresiding somewhere between chaos and turbulence.In the remainder of this paper, we aim to identify some important open questions aboutturbulence in BECs; where appropriate we will review some of the work which has been carriedout to date.
3. Quantum Turbulence in atomic BECs, where are we?
The following is an extensive, but by no means exhaustive, list of aspects yet to be understoodregarding turbulence in atomic BECs. To structure our discussion, we distinguish the evolutionof turbulent flow into three stages, namely;(i) The generation of the turbulence.
What are the most effective and efficient ways to generateturbulence? Does the way in which the turbulence is generated affect the ‘type’ of turbulencecreated? (ii) The statistical steady state.
Are there universal features of turbulence, for instance, is theKolmogorov energy spectrum present? What are the statistics of the turbulence velocityfield? (iii) The decay of the turbulence.
How does the turbulence decay? What is the best way tomeasure the decay? i) The generation of the turbulence
To understand the generation of a vortex tangle in a quantum gas, we must first understand howindividual vortices are nucleated. The very first creation of such a vortex took place in a two-component condensate and was driven by the rotation of one component around the other. Thesubsequent removal of the inner component resulted in the formation of a hollow core of a singlyquantized vortex (Anderson et al. , 2000). Further techniques for generating vortex structuressoon followed, including the creation of vortex rings following the “snake instability” decay of adark soliton (Anderson et al. , 2001), phase imprinting (Leanhardt et al. , 2002) and by a rapidquench through the transition temperature for the onset of Bose-Einstein condensation, i.e. theKibble-Zurek mechanism (Weiler et al. , 2008; Freilich et al. , 2010).However, to generate a large number of vortices in the system at any one time, two othertechniques have proved to be more effective:(i)
Rotation of an anisotropic BEC excites surface modes leading to the nucleation of vorticesat the edge which then drift into the bulk of the BEC. If the rotation is performed aboutonly one axis, a vortex lattice is created (Hodby et al. , 2001; Abo-Shaeer et al. , 2002, 2001;Madison et al. , 2000, 2001). In 3D, if the rotation is performed about more than one axis,a vortex tangle has been predicted to form (Kobayashi & Tsubota, 2007).(ii)
A moving (cylindrical) obstacle , such as that created by the potential from a blue-detunedlaser beam moving through a quantum fluid, generates pairs of vortices in its wake when itsspeed exceeds a critical value (Raman et al. , 2001). Recently, this method has been usedto generate and study a collection of vortex dipoles in a 2D BEC (Neely et al. , 2010).Both methods generate a large number of vortices, in 2D as well as in 3D. However, one canbypass the initial transient period and begin with a nonequilibrium state of vortices. Experimen-tally, this can be achieved via imprinting a phase profile onto the condensate via laser beams,as performed by Leanhardt et al. (2002) for generating single vortices of arbitrary charge. Theuse of such imprinting to generate a vortex tangle has been implemented theoretically (White et al. , 2010), with the resulting tangle similar to that depicted in Fig 3.Our preliminary results with method (ii) (laser stirring), suggest that it is possible to generatea large number of vortices; we have found qualitative evidence that, by moving the obstacle alongdifferent paths, we can change the isotropy of the resulting tangle of vortices.Fig. 1 (top row) shows the density isosurface of a 3D spherical condensate at three instants intime, after the condensate has been stirred for a time t stir , along a circular path with a Gaussian-shaped laser stirrer oriented in the z − direction. For a simple measure of the isotropy of thetangle, we plot the projected vortex lengths L x , L y and L z in each Cartesian direction (bottompart of Fig. 1). All projected lengths rapidly increase during the stirring period ( t < t stir ); afterthe laser has been removed ( t > t stirr ), L x , L y and L z all decrease for a short period of time.Later, only the vortex lengths L x and L y in the transverse ( x and y ) directions further decrease,whereas L z remains approximately constant because the vortex tangle decays into a regularlattice, as it is apparent from the final density isosurface (Top row, c)). This is as expected: instirring the condensate circularly we impart angular momentum about the z -axis, and it is wellknown that the ground state of a superfluid with sufficient angular momentum features a latticeof regularly-spaced, vortex lines aligned along the z -axis.In Fig. 2 the vortex length is shown when the stirring takes place, for the same amount oftime, along a Figure-eight path. Again, the vortex length increases over the duration of the laserstirring ( t < t stir ); however, after the laser is removed ( t > t stir ), the tangle decays isotropically,i.e. all projected lengths L x , L y and L z decay together. For the Figure-eight path, the laser alsomoves through the centre of the condensate, where the density is higher, the vortices which aregenerated are longer than those generated at the edge of the condensate; therefore, this laser) b) c) Figure 1.
Top row: Density isosurfaces of a 3D spherical BEC at times tω = a) 60, b) 100 andc) 240 after stirring the condensate along the z − direction in a circular path. We see here thatthe surface plot picks up the vortex cores as well as some of the condensate edge. The resultingvortex length in each direction is shown over time in the bottom part of this figure where it hasbeen normalised by the peak vortex length. Figure 2.
Vortex length, normalised by the peak length, in the x, y and z directions for a stirreralong the z − direction, moving in a Figure-eight path.tirring path is more efficient at creating a dense, random vortex tangle than simply stirring thecondensate in a circular fashion. (ii) Statistical state Once the turbulence state has been created (by whichever means), its steady state propertiescan be investigated. BECs offer the possibility to study 2D and 3D turbulence and the cross-overregion between the two (Parker & Adams, 2005). We now review our current understanding ofthe properties of a turbulent tangle of vortices in a BEC, first in 2D and then in 3D.
Quantum turbulence in 2D
In 2D classical turbulence, the conservation of enstrophy dictates that energy must flowfrom small scales of energy injection to large scales forming, for example, large clusters ofvortices (Kraichnan, 1967). This inverse cascade is thought to underly Jupiter’s giant Red Spotand has been experimentally examined in classical fluids (Sommeria et al. , 1988; Marcus, 1988)(for a review see Kellay & Goldburg (2002)). Attempts to observe the inverse cascade effect inquantum gases have lead to modelling vortex generation (Parker & Adams, 2005; White et al. ,2012; Fujimoto & Tsubota, 2011; Reeves et al. , 2013), and to experiments on the dynamicsof vortex dipoles created by a moving potential (Neely et al. , 2010, 2012).Numasato et al. (2010), evolved the 2D GPE to a turbulent state by initially imposing arandom phase on the wavefunction. They did not observe a reverse cascade but rather a directcascade. They argued that since the total number of vortices, and therefore the enstrophy, isnot conserved in simulations of the GPE because of vortex pair annhilation, the inverse energycascade is irrelevant for 2D quantum turbulence.Conversely, Reeves et al. (2013) reported the numerical observation of the inverse cascade.They solved the 2D damped GPE (DGPE) and generated a turbulent state by imposing thefluid to flow past 4 stationary potential barriers. The speed of the flow was sufficiently high( v (cid:39) . c , where c is the sound speed of the quantum fluid) so as to create many vortices andthereby a turbulent flow. Using a statistical algorithm they measured how prone ‘like-winding’vortices, i.e. vortices of the same charge, were to cluster together depending on the amount ofdissipation imposed. They found that an intermediate level of damping lead to small clusters oflike-winding vortices being formed and inferred an inverse energy cascade from analysis of theincompressible energy spectrum (energy associated with vortices).Similar work by White et al. (2012) implemented a rotating elliptical paddle to generate largenumbers of vortices. Again, clustering of like-winding vortices was observed, but no inversecascade was reported.A possible reason for the lack of definitive evidence for or against the inverse cascade in quan-tum gases is their relatively small size, i.e. the systems which were studied lacked enough lengthscales. However, this limitation has not prevented the observation of the direct Kolmogorov-3Dcascade in systems of similar size (Nore et al. , 1997; Kobayashi & Tsubota, 2005 a , b ; Yepez et al. ,2009; Kobayashi & Tsubota, 2007). Quantum turbulence in 3D
In their seminal work, Nore et al. (1997) used the GPE to investigate 3D quantum turbulence ina homogeneous box by evolving an initial, large scale Taylor Green vortex. By decomposing thevelocity field into divergence-free and curl-free parts, they obtained incompressible (associatedwith vortex motion) and compressible (associated with acoustic excitations) kinetic energyspectra respectively. They showed that the incompressible kinetic energy spectrum is similar tothe classical k − / Kolmogorov energy spectrum at scales down to the intervortex spacing.Similarly, Berloff & Svistunov (2002) began with a non-equilibrium state, generated bymprinting a random phase on the equilibrium wavefunction of a homogeneous box. They thenobserved the evolution of the quantum turbulence by solving the GPE and further allowing thesystem to evolve to phase coherence. Similarly, Kobayashi & Tsubota (2005 a , b ) imprinted arandom phase on the equilibrium wavefunction of a homogeneous box and evolved accordingto the GPE. They introduced a dissipative term which only acted on scales smaller than thehealing length to represent thermal dissipation in the system. They also obtained a decayingincompressible energy spectrum which has the Kolmogorov power law over the inertial range.In order to clarify the extent of this range, statistical steady turbulence was created by amoving random potential which continuously injected energy into the system at large scales;a damping term removed energy at small length scales. They found that the inertial range wasslightly narrower for the continuously forced turbulence because the moving potential sets theenergy containing range. Further to these methods, White et al. (2010) imprinted a staggeredvortex array onto a harmonically trapped BEC and evolved the system according to the GPE(see Fig. 3). In this work, they calculated the probability density function of the velocitycomponents and found that (in both 2D and 3D) it is not a Gaussian like in ordinary turbulence,in agreement with experimental results obtained in superfluid helium (Paoletti et al. , 2008). Thisis an important observation which distinguishes quantum from classical turbulence. Figure 3.
3D turbulent state of a harmonically trapped BEC. Condensate edge shown by blueshading, turbulent vortex tangle shown in purple (White et al. , 2010). Figure courtesy of AngelaC. White.Using a similar method of imprinting, (Yepez et al. , 2009) performed impressively largesimulations in a homogeneous box using a quantum lattice gas algorithm (up to 5760 gridpoints) and resolved scales smaller than the vortex core radius.Most of the methods of generating quantum turbulence discussed so far in this section havea common aim: the incompressible energy spectrum of quantum turbulence after an initial,turbulent state has been set up (with the exception of (Kobayashi & Tsubota, 2005 a , b )).However, the group of Tsubota have also carried out simulations where they dynamically createa vortex tangle by solving the DGPE with combined rotation along two axis of a harmonicallytrapped BEC. They found that by changing the ratio between the rotation frequencies in bothdirections, they could generate a vortex lattice or a more disordered array of vortices whichformed a vortex tangle in which individual vortices appear to be nucleated with no preferreddirection. They measured the incompressible energy spectrum and found it to be consistentwith Kolmogorov law.Experimentally, a small vortex tangle has been created in a harmonically trapped BECthrough the combination of rotation and an external oscillating perturbation by Bagnato’sgroup (Henn et al. , 2009, 2010; Seman et al. , 2011; Shiozaki et al. , 2011). They noticedthat upon expansion of the condensate the usual inversion of aspect ratio of the gas (Mewes t al. , 1996; Castin & Dum, 1996) did not happen. This effect could be a possible signatureof the creation of a tangle of vortices.This brings us to the final stage of the evolution of turbulence, the decay. (iii) The decay of the turbulence In classical turbulence, the cascade of kinetic energy over the length scales terminates at somevery short scale where viscosity dissipates kinetic energy into heat. The absence of viscosity inquantum fluids means there must exist other mechanisms of energy dissipation. The most likelyis acoustic emission. When two vortices reconnect, some energy is lost in the form of sound.Reconnections are also thought to create high frequency Kelvin waves on vortices. It is thoughtthat, in superfluid helium, Kelvin waves interact nonlinearly and create shorter and shorterwaves, until sound waves are emitted at high frequency (Vinen, 2006). This energy transfer iscalled the Kelvin wave cascade.Experiments in superfluid helium show that, depending on the scale at which energy isinjected, the decay of the turbulence can be one of two forms (Baggaley et al. , 2012):(i) ‘Semiclassical’ or ‘Kolmogorov’ turbulence:
The vortex tangle seems polarised andstructured over many length scales. This type of turbulence is generated when the forcingis at length scales larger than the average intervortex spacing. In this regime, the vortexlength L decays as L (cid:39) t − / , which is consistent with the decay of a Kolmogorov spectrum.(ii) ‘Ultraquantum’ or ‘Vinen’ turbulence: The scale of the forcing is less than the intervortexspacing, the vortex tangle seems random and possesses a single length scale, and the vortexlength decays as L (cid:39) t − (Walmsley & Golov, 2008).How to measure the decay in a BEC is an open question. White et al. (2010) studied thedecay of the turbulent tangle by numerically monitoring the vortex length, L , over time. Theyshowed that the line length increases initially as reconnections take place before decaying overtime. By further solving the dissipative GPE, they confirmed that thermal dissipation leads toa faster decay of line length but could not clearly distinguish between L (cid:39) t − or L (cid:39) e − αt behaviour ( α being some decay parameter). However, is vortex linelength the best measure forthis decay? Or can we again look to the incompressible energy spectrum to visualise the decayof vortices and draw some conclusions from both of these quantities? Furthermore, there is thequestion of how this is best achieved experimentally. The images taken of condensate densityare typically column-integrated over the imaging direction which means that depth informationbecomes lost and an extraction of the true 3D vortex line length is not possible. Just as theattenuation of second sound is used to measure vortex length in helium, what surrogate measuresof vortex line length are accessible experimentally in BECs? Such questions, both fundamentaland practical in nature, will provide a rich source of research in these systems in the future.
4. Summary
We have discussed weakly interacting, dilute atomic Bose-Einstein condensates as tools forunderstanding the nature of quantum turbulence, motivated largely by the huge degree of controlthey offer. Even though the range of lengthscales excited in these systems is much less than insuperfluid helium, the direct Kolmogorov energy cascade has been predicted to exist and theregimes of turbulence accessible is vast and interesting its own right. We have distinguishedthree distinct phases of quantum turbulence - its generation of turbulence, its steady state andits decay - and briefly reviewed work done in understanding these so far, whilst highlightingfundamental questions about each phase. cknowledgments
We acknowledge A. C. White for useful discussions and for providing us with an image for usein this article. This work was supported by the grant EP/I019413/1 from the EPSRC.
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