Quantum turbulence in trapped atomic Bose-Einstein condensates
Turbulence, the complicated fluid behavior of nonlinear and statistical nature, arises in many physical systems across various disciplines, from tiny laboratory scales to geophysical and astrophysical ones. The notion of turbulence in the quantum world was conceived long ago by Onsager and Feynman, but the occurrence of turbulence in ultracold gases has been studied in the laboratory only very recently. Albeit new as a field, it already offers new paths and perspectives on the problem of turbulence. Herein we review the general properties of quantum gases at ultralow temperatures paying particular attention to vortices, their dynamics and turbulent behavior. We review the recent advances both from theory and experiment. We highlight, moreover, the difficulties of identifying and characterizing turbulence in gaseous Bose-Einstein condensates compared to ordinary turbulence and turbulence in superfluid liquid helium and spotlight future possible directions.
keywords:Ultracold Bose gas, quantized vortex, turbulence, experimental, many-body physics
textheight=25.50cm \geometrytextwidth=18cm \makenomenclature
- 1 Introduction
- 2 Background
3 Essentials of theory
- 3.1 Brief account of classical turbulence
- 3.2 Brief account of quantum turbulence
- 3.3 Theoretical models for BECs
- 3.4 Hydrodynamic turbulence in BECs
- 3.5 Wave turbulence in BECs
- 3.6 What all this has to do with the experiments?
- 4 Energy transport and dynamics
5 Experimental emergence and characterization
- 5.1 Experimental set-up and time sequence
- 5.2 Regular vortex nucleation and proliferation
- 5.3 Mechanism of vortex formation
- 5.4 Turbulence
- 5.5 Granular phase
- 5.6 Diagram of excited structures
- 5.7 Self-similar expansion
- 5.8 Frozen Modes
- 5.9 Momentum distribution of a turbulent trapped BEC
- 6 New directions and challenges
 List of frequently used symbols and abbreviations
1.1 Why quantum turbulence?
Turbulence occupies a unique position among the disciplines of physics, at the intersection of traditional fundamental problems on one hand and modern fashionable trends on the other. Influential scientists such as S. Weinberg and R. Feynman
The “super” properties of ultracold quantum matter (superconductivity and superfluidity), discovered more than a century ago, furthered our understanding of the nature of atoms and molecules. The many-body theoretical approaches that emerged from the study of superfluidity and superconductivity became the most important tools in modern quantum physics. Gradually, new phenomena emerged from exploring new settings and configurations of the superfluid state. For example, with the introduction of rotation, a very peculiar response of the system was observed; if the liquid was thermal (i.e. in the normal state) then it would follow the rotation of its container. But if it was cooled down to temperatures smaller than a critical temperature , then it would stand still, as if the container were not rotating. Because of the existence and the uniqueness of a macroscopic wave function, the liquid in the superfluid state can only rotate by threading itself with a number of quantized vortex lines, thin mini-tornadoes aligned along the axis of rotation. Vortex lines can be created also by stirring the liquid helium with grids or propellers, or by applying heat currents. Configurations of vortex lines are either laminar (when they are regularly distributed in space, for example rotating vortex lattices) or turbulent (when the vortex lines are tangled in space). Ordinary liquid helium (He) cooled below the critical temperature (called the -point) is not the only system exhibiting quantum turbulence. The rare isotope He becomes also superfluid, but at much lower temperatures (in the region), and quantum turbulence is currently studied in its B-phase. This distinction is important, as topological defects in the A-phase are quite different.
In 2009, the first evidence (3) of quantum turbulence in trapped, dilute atomic Bose-Einstein condensates (BECs) has opened new and exciting perspectives. Atomic condensates may constitute the ideal laboratory for testing fundamental aspects of quantum turbulence, and explore similarities and differences between quantum turbulence and turbulence in ordinary (classical) fluids. The study of the life-cycle of quantum turbulence – from its nucleation and temporal evolution until its decay in a variety of excitations and phases – might suggest analogies between quantum turbulence and classical turbulence.
The aim of this review is to highlight the recent and most important advances in the field of quantum turbulence in trapped gases, focusing on both theory and experiments. We try to give a self-consistent and as complete as possible presentation of our knowledge on turbulent behavior in quantum gases, as this has been boosted by the recent experimental progress. Although not intended to substitute standard textbooks in the field, much of the underlining theory is included in the paper and most concepts are explained for the layman. Our objective is to put the new findings in context and clarify core ideas of turbulence. We shall be particularly concerned with topics such as the generation of quantized vortices and turbulence, its evolution and decay, and experimental techniques to measure turbulent properties. We include in our discussion the limitations of both theory and experiment and point out new challenges.
Bose-Einstein condensation is a general phenomenon which appears not only in liquid helium and ultracold gases, but also in a variety of other non-gaseous systems, for instance magnons (4) and exciton-polaritons (5). In the present work, when referring to a BEC, we shall exclusively mean a trapped ultracold gaseous atomic BEC.
1.2 Structure of this Review
The plan of this Review is the following. In Sec. 2 we present the background to the problem: we describe the basic physics of Bose-Einstein condensation, superfluidity, quantum vortices and quantum turbulence following a historic line of presentation. We then comment on the apparent similarities and differences between turbulence in ordinary fluids, turbulence in superfluid helium and turbulence in Bose-Einstein condensates. This section ends with our answer to the frequently-posed comment as to whether proper definitions of the words ‘turbulence’ and ‘quantum turbulence’ are necessary to make progress into our investigations.
We begin Sec. 3 by exposing the reader to the basic theoretical background needed. In Secs. 3.1 and 3.2 we briefly review the main results from classical and quantum turbulence respectively. In Sec. 3.3 we present the basic models known for theoretically treating trapped BECs and systems of vortices, followed by a simple exposition of the appearance of power spectra in BECs (Secs. 3.4 and 3.5).
Section 4 treats theoretically and experimentally the problem of energy flow and transport between different scales, a problem directly related to the problem of creation, propagation and decay of turbulence. To this end various different mechanisms are examined: vortex reconnections and Kelvin waves in three dimensions (Sec. 4.1), vortex annihilation and decay in two-dimensional gases (Sec. 4.2) and also phonon-mediated transport (Sec. 4.3). Experimental results are intertwined with results from the theory and numerical analysis.
In Sec. 5 we present the principal experimental results obtained by our group; the controlled creation of quantum turbulent and granular state in an ultracold trapped gas of rubidium. After a short presentation of the experimental setup in Sec. 5.1 and a presentation of all different types of excitations encountered (Secs. 5.2–5.5) emphasis is put on the relation of the different phases and excited structures (Secs. 5.6) and also the novel phenomenon of self-similar expansion (Sec. 5.7). An observed power-law in the momenta of the gas is discussed in Sec. 5.9.
Section 6 concludes the present work. We highlight the key findings of the research to-date in quantum turbulence (Sec. 6.1) and suggest some challenging steps in the near (or more distant) future needed to be taken in order to further our current knowledge in QT (Sec. 6.2). Related, but slightly more distant, to QT topics are discussed Sec. 6.3. Last, Sec. 6.4 gives concluding remarks of the present exposition.
Density \nomenclatureHealing length \nomenclatureInteraction strength \nomenclatureParticle number \nomenclatureHold time: time that the BEC is held inside the trap before an absorption image is taken
QTQuantum turbulence \nomenclatureBECBose-Einstein condensate \nomenclatureCTClassical turbulence \nomenclatureNSNavier-Stokes \nomenclatureTFThomas-Fermi \nomenclatureBSBiot-Savart \nomenclatureQVRQuantum vortex reconnection \nomenclatureGPGross-Pitaevskii \nomenclatureWTWave turbulence
QUICQuadrupole-Ioffe configuration \nomenclatureTOFTime of flight: process that lets the gas expand and free-fall before measuring \nomenclaturein situA measurement has been performed while the gas is inside the trap \nomenclatureIECInverse energy cascade \nomenclature2DQTTwo-dimensional quantum turbulence
2.1 Bose-Einstein condensation
Bose-Einstein condensation is the macroscopic occupation of the same quantum level by the majority of the particles of a system. Because of Pauli’s principle, these particles must be bosons. The phenomenon, known since 1924, takes place if the temperature of the system is less than a certain critical value . Initially it was believed that Bose-Einstein condensation should be limited to ideal (noninteracting) particles (7), but later it was realised that the presence of interparticle interaction can actually assist in the formation of the condensate (8). In 1995 the first experimental realizations of Bose-Einstein condensation in dilute alkali gases were separately achieved by three experimental groups that cooled ensembles of sodium (9), lithium (10) and rubidium (11) to temperatures of few nK. Nowadays, BECs are routinely produced and studied in dozens of laboratories across the world.
Upon cooling the system, Bose-Einstein condensation takes place when the mean interparticle distance becomes comparable to the de Broglie wavelength of the particles. Here is the number density of particles occupying a volume , and are the mass and the thermal velocity of the particles respectively and is Boltzmann constant. Assuming a homogeneous gas at constant number density , the condition yields
for the critical transition temperature . This simple qualitative argument (see (12) for an intuitive presentation) predicts that an ensemble of bosons, if cooled below the critical temperature , undergoes Bose-Einstein condensation. An accurate calculation of the above differs only by a factor of (13). Importantly, most experiments in BECs of trapped atoms are not done in homogeneous systems but rather in heterogeneous trapping potentials. In this case, the critical temperature depends on the external potential as the condensate is not uniformly distributed but centered in the potential minimum. For a harmonically trapped gas the critical temperature for condensation is (14)
where is the geometric mean of the trap frequencies. Note the scaling, distinct in the two cases.
The many-body coherent state that appears is shown in the sequence of pictures displayed in Fig. 2. For a gas to condense, diluteness and trapping (i.e. spatial confinement of the particles) are necessary. Typically, the number density is to particles per . Trapping and cooling can be achieved in many different ways. While the trapping potential can be created efficiently by either magnetic or optical means, cooling is normally obtained by a combination of techniques like laser cooling and evaporation.
Research on Bose-Einstein condensation – boosted by experimental accomplishments – has rapidly gained a special position in modern physics (15). This activity has allowed us to create and study states macroscopic objects (consisting of few hundreds to few millions of particles) with properties usually associated with microscopic quantum objects. Atomic BECs serve as laboratories for testing many-body theories of physics and assist research in fields as diverse as quantum computation, metrology, foundations of quantum physics and even industrial applications. What distinguishes gaseous BECs from other condensed matter or atomic systems is the amount of control one can exert on the system: the type and sign of the interatomic interactions as well as the confining potentials can be changed at will, simply by modifying externally the applied magnetic fields (16); (17).
More interestingly and due to the unique controllability of atomic ultracold gases, BECs can serve as platforms where other systems from condensed matter – and beyond – are being simulated. Examples are bosonic Josephson junctions (18); (19) the Bose-Hubbard model (20), vortex lattices, synthetic gauge fields (21), hexagonal lattices and graphene (22), collapsing attractive systems (nicknamed ‘bosenova’) (23) and more.
Interactions complicate the dynamics of Bose-Einstein condensates, and introduce effective nonlinearities in the mean-field equations of motion. In ultralow temperature environments, interparticle interactions are sufficiently well described by considering -wave scattering only. In the mean-field approach – the most commonly used – each particle of the system interacts with the mean-field generated by the other particles. As we shall see later, this approximation assumes total condensation at all times, mapping the many-body problem to a one-body problem. Nonetheless, being a gas of interacting atoms, a BEC also retains many properties typical of classical fluids. Hence, features as solitons (bright, dark or other more complicated topological excitations like skyrmions), vortices, vortex rings and other fluid properties have all been observed in atomic physics laboratories and can be nowadays routinely created and imaged. Interestingly, the above appear as stable topological excitations of the quantum system. In contrast, ordinary classical fluids do not contain well-formed ‘thin’ and long-lived vortices; the latter are rather fuzzy and its characterization as line filaments is hence not clear. In such a way, excitations that in viscous (normal) fluids are mathematical ideals only, in atomic BECs become real. The underlying property of such a condensate is a special kind of fluid called a ‘superfluid’, which is the topic of the next section.
2.2 Superfluidity and vortices
In 1908, that is long before the first experimental realizations of Bose-Einstein condensation in atomic vapours, liquid helium was produced at Leiden by the Dutch physicist Heike Kamerlingh Onnes. Between 1927 and 1932 a group of physicists also at Leiden (W. H. Keesom, M. Wolfke, A. Keesom J.N. van der Ende and K. Clusius) (24) noticed that liquid helium has two distinct phases, named He I and He II, separated by the apparent singularity that the specific heat shows as a function of the temperature - the famous “lambda point”. This singularity appears at a temperature around 2.2K at low pressure. The two liquid phases seemed very different. Liquid He I behaved like an ordinary fluid, whereas liquid He II (the low temperature phase) had unusual mechanical and thermal properties. The key step forward was the discovery of superfluidity by Kapitza (25), Allen and Misener (26), which stimulated much work on macroscopic quantum systems. In their experiments, they showed that liquid He II can flow through very thin tubes or slits without viscous dissipation. In other words, helium, the first quantum fluid explored, has no viscosity in the superfluid phase, in direct analogy to the lack of resistance displayed by a superconductor. Owing to its superfluid nature, He II apparently defies gravity by siphoning itself out of a container.
In 1938 Fritz London proposed that the unusual properties of He II are actually a manifestation of Bose-Einstein condensation (27). He, thus, realized that a collective wave function could describe the condensed atoms if there was a macroscopic occupation of the zero momentum state. London’s idea was set aside for a while, as the interest was taken by a new phenomenological theory (the two-fluid model) proposed by Tisza (28). According to the two-fluid model, He II consists of two fluid components, the normal fluid and the superfluid, each with its own density and velocity field. The normal fluid moves like an ordinary fluid with all of the conventional thermodynamic properties such as entropy, temperature and viscosity, while the superfluid has no entropy, is inviscid and flows without friction. Landau formalized the model, and produced a complete set of thermodynamic relations for the two-fluid system. The two-fluid model of Landau and Tisza turned out to be enormously successful in describing the properties of He II and its strange observed effects (the siphon effect, the fountain effect, the mechano-caloric and thermo-mechanical effects). The two-fluid model also led to the prediction of the existence of second sound, a mode of oscillations in which superfluid and normal fluid move in antiphase creating a temperature wave rather than the usual pressure wave of ordinary (first) sound.
One major problem with the two-fluid model, however, is that it assumes zero superfluid vorticity (). In fact, a zero-viscosity fluid should be dissipation-free and therefore have a conservative velocity field, , implying absence of vorticity, . Experiments with helium in a rotating vessel, however, showed the typical parabolic surface of a rotating liquid, which would imply vorticity proportional to the angular velocity of rotation, like ordinary fluids. This observed parabolic profile was independent of temperature, in disagreement with the expectation that it should be proportional to the relative density of normal fluid which is strongly temperature dependent (as only the normal fluid can have a curl, hence rotate).
At this point it is worth examining London’s association of the superfluid with a Bose-Einstein condensate where a single quantum state is collectively occupied by all particles. London realised (27) that a collective wavefunction could be used for the condensed atoms if there was a macroscopic occupation of the zero momentum state,
where is the condensate density, appropriately normalized
and is its phase. Following the usual quantum mechanical prescriptions, we define the probability current as
This is in fact a flux of the density that flows with velocity
hence the flux
takes the familiar from classical physics form. Assuming conservation of mass, we can write down the continuity equation for the current :
Interestingly, as we shall show in the following Section, such a continuity equation is easily derived from the (linear or nonlinear) Schrödinger equation by the Madelung transformation.
The identification of the velocity with the gradient of a scalar has fundamental implications on the types of motion that can occur in a condensate or superfluid. Indeed, a direct consequence of Eq. (2.6) is that
as envisaged by Landau. Note that the above result holds true when the field has continuous first and second derivatives. This is not the case if the system contains at some point a vortex line, i.e., a line singularity along which the velocity diverges. Hence the velocity field of a vortex-free field is irrotational. Recall now that the condensate wavefunction is required to be single-valued: a revolution along a closed loop should leave the value of unchanged, hence the phase can change at most by , where is an integer. In other words, the circulation around any path yields only integer multiples of the quantum of circulation :
This was the argument that led Onsager
As we shall see, there exist vortex solutions of the quantum mechanical equations of motion (see Sec. 3.3.2 and Sec. 4.1), and these constitute simple paradigms of configurations with quantized circulation. To demonstrate that, we construct the simple wavefunction which, as we shall show, contains a straight vortex at the origin, aligned with the -direction. Using cylindrical coordinates, this solution has the form
where is some prescribed real function and an integer. This ansatz assumes that the magnitude of the wavefunction, , is cylindrically symmetric; the azimuthal dependence is contained only in the phase of and has the form . A rotation about the -axis leaves the wavefunction invariant, as it owes to. The velocity field corresponding to this wavefunction is in the azimuthal direction (parallel to the unit vector ):
Hence, following the quantization of the circulation, the velocity field is also quantized, since its magnitude admits values that are integer multiples of . Note, moreover, that is irrotational as long as . Formally, we write the vorticity as a (two-dimensional) delta function to show that is localized on the cylindrical axis (13), i.e.
where is the unit vector in the z-direction.
We conclude that in a quantum system the flow can be both inviscid and irrotational as long as it is not continuous. The velocity field has a singularity on the axis at (generally, at some ); the wavefunction therefore must go to zero at exactly the same points, that is, inside the vortex core. In other words, both real and imaginary parts of go to zero at yielding a zero density at the singularity . Therefore, the fact that the velocity diverges as is not physically inconsistent: there are no atoms in the vortex core which move with infinity speed. The vortex line is a hole in the superfluid, that therefore renders its bulk a multiply-connected space.
Mathematically, the flow pattern has the characteristics of a classical vortex line, as can be found in elementary fluid dynamics textbooks: a node at and a velocity field around it which decays as away from the axis.
Note that the integer of Eq. (2.11) determines how many multiples are added to the condensate phase on a single rotation and is also called the charge of the vortex. A quantized vortex is a topological excitation of the system with higher energy than the ground state (the condensate without the vortex). It can be shown that the energy of the vortex increases with the square of the vortex charge, . Therefore, for given charge , it is energetically favourable for the system to contain singly-charged vortices rather than one multiply charged vortex (33); (34).
2.3 Quantum turbulence
For the sake of simplicity, the vortex lines described in the previous sections were considered straight. We know however that vortex lines can sustain helical deformations away from the straight position called Kelvin waves. These waves rotate at angular velocity , and propagate at phase velocity , where is the quantum of circulation, is the wavenumber and the wavelength. We also know that two lines which approach each other can reconnect, forming a cusp which then relaxes into Kelvin waves. Very disordered configurations of vortex lines are easily created in liquid helium or atomic BECs and can be manifestation of quantum turbulence. The possibility of such quantum turbulence was already touched upon by Feynman in his seminal work on quantized vorticity (31). The impact of Feynman’s – and also Hall’s and Vinen’s (35); (36) work – was such that for years research towards QT has been mainly concerned with He. The recent interest in atomic BECs has been heavily motivated by progress in understanding quantum turbulence in liquid helium (He and He) experiments (37); (38). A striking discovery has been that, under appropriate forcing, quasiclassical behavior arises in such quantum systems, displaying statistical properties that characterize ordinary turbulence; an example is the celebrated Kolmogorov scaling of the energy spectrum (39) which suggests the existence of a classical energy cascade from large to small length scales. This classical-like scaling defines the quasiclassical turbulence or Kolmogorov turbulence and can be found for a specific inertial range, provided that the energy flux inside this range is maintained fixed with constant injection of energy in large scales and dissipation at small scales at the same rate. This interpretation requires self-similarity throughout a long range of scales; large bundles of vortices transfer energy to smaller bundles, in a process which goes down to the scale of single vortices. Numerical simulations (40) determined that if the energy spectrum of the turbulent system obeys the Kolmogorov scaling, the vortex tangle contains transient regions where the vortex lines are oriented in the same direction (vortex bundles); the large scale flows generated by such parallel lines concentrate the energy in the small region of the energy spectrum. Figure 3 clarifies that such vortex bundles also contain many random vortex lines; in other words, the spatial polarization is only partial. However, although Kolmogorov energy spectra have been observed, there is yet no direct experimental observations of these vortex bundles.
Kolmogorov-like turbulence has been seen in helium at both high and low temperatures. Quasiclassical behavior in the limit of high temperature is not surprising, particularly in liquid helium driven by bellows or propellers at temperatures just below : in this regime we expect the turbulent normal fluid to dominate the dynamics due to its relative larger density ( and , where , and are respectively the normal fluid, superfluid and total density). Quasiclassical behavior in the low temperature limit is more surprising, but can be understood if we recognize that the energy transfer responsible for the cascade arises from the key nonlinearity of the Euler equation, the term. This result confirms the view that quantum turbulence contains the fundamental mathematical skeleton of classical turbulence.
Numerical simulations and, more recently, direct visualisation based on tracer particles, show that quantum turbulence consists of a disordered tangle of vortex lines. In the high temperature range ( in He) the vortex lines remain relatively smooth as friction with the normal fluid damps the Kelvin waves; kinetic energy is thus turned into heat by the normal fluid via viscous forces. In the low temperature range ( in He) the vortex lines are very wiggly, as there is not enough friction to smooth out the Kelvin waves. However, other physical processes dissipate turbulent energy (see Sec. 4). Rapidly rotating Kelvin waves of very short wavelength are created by wave interactions (a process called the Kelvin wave cascade (41); (38)), which, if their wavelength is short enough, emit phonons; the sink of turbulent kinetic energy is acoustic rather than viscous.
Under other conditions, a different kind of turbulence (called ultraquantum turbulence or Vinen turbulence) has also been found both experimentally (42) and numerically (43), characterized by random tangles of vortices without large-scale, energy-containing flow structures.
We have no direct experimental judgement as of whether turbulence in atomic BECs is Vinen-like, Kolmogorov-like or both. However, from the present analysis (see Secs. 2.4 and 2.5) it results that in a trapped BEC Vinen ultraquantum turbulence is more likely to emerge. It has been recently demonstrated (44) that in the case of only two vortices initially in an orthogonal configuration, or even in the case of one doubly-charged vortex precessing and decaying in a spherically symmetric trapped gas, there appear Kolmogorov-like spectra of the energy in the momentum space, but only over a wavenumber range of one decade. Numerical calculations based on the Gross-Pitaevskii equation (45) suggest that Kolmogorov scaling should emerge from the dynamics of vortices in a trapped Bose gas, but there is no experimental verification of the effect yet. More theoretical work and experiments are necessary to characterize the turbulence observed in BECs, particularly because of the difficulty in experimentally measuring the velocity field within a trapped condensate. What has become clear recently is that excitations and vortices in a trapped BEC display very rich dynamics, which should be fertile hunting ground for theory and experiments, see Fig. 4. In the present Review, especially in Sec. 5, we focus on features of the turbulent quantum gas that are experimentally accessible and might constitute Kolmogorov-independent criteria for turbulence.
Last, another distinction which should be made when studying turbulence is its nature regarding forcing. Forced systems can exhibit the so-called stationary turbulence that requires a constant injection of energy and also removal of it at the same rate by dissipative mechanisms. In experiments in 3D systems, energy injection happens in larger scales, which then cascades down in a fixed flux to the small scales (direct energy cascade). The opposite happens for the 2D scenario (inverse energy cascade). However, in nature, a stationary turbulent system is destined to decay at a certain point, when the system is ceased to be forced. This frames a very distinct out-of-equilibrium scenario known as decaying turbulence.
2.4 Length scales
Understanding classical turbulence involves studying processes that happen in many scales which are usually separated by orders of magnitude. In that respect, QT is no different. The ratio quantifies the available space for its development; distance is the typical largest scale of the system (e.g. size of a helium container or extension of the trap for an atomic superfluid) and is the healing length, which sets the approximate size of the vortex core (see also Sec. 3.3.2). A large value of enables the development of self-similarity throughout several scales, increasing thus the inertial range. For a quantum system to exhibit quasiclassical spectral characteristics (Kolmogorov scaling), a relatively large ratio is necessary and the quantity tells us roughly how many decades are available between and for this to happen. In that sense, systems with small ratio present few decades [( or ]. This is an intrinsic spatial limitation that forbids formation of large-scale, self-similar structures. On the other hand, in experiments involving superfluid He and this number justifies why both quasiclassical and ultraquantum limits could be observed (46). In contrast, current experiments with atomic BECs display to decades and the observation of complicated vortex tangle dynamics in such relatively small-ratio systems (3) shows evidence of ultraquantum turbulence.
In Fig. 4 we present the scales that naturally appear in a trapped BEC of rubidium particles at ultralow temperatures: from distances as low as few n (extension of the scattering length and interatomic spacing) up to the size of of an expanded gas. As we can see, there is a plethora of different physical mechanisms and phenomena manifesting over various scales. However, the relevant for turbulence scales span about two decades (shaded region of Fig. 4).
2.5 Differences between classical, helium and quantum turbulence
There are significant physical differences between classical and quantum turbulence which affect the nature of the turbulence and the experiments which can be performed.
Choice of parameters When studying turbulence in liquid helium, flow properties such as density and viscosity can be varied by changing the pressure and the temperature of the sample. The vortex core size (which is proportional to the superfluid healing length) and the quantum of circulation are fixed and cannot be changed. In atomic BECs, virtually all parameters can be varied by large amounts by tuning the number of atoms, the confining trap, and the strength of the interactions. This freedom however has a drawback, because it makes it difficult to compare the hydrodynamics regimes achieved in different experiments. In most experiments, atomic BECs are confined by harmonic external potentials, so the condensate density and other quantities such as the sound speed and the healing length depend on the position. Again, this feature makes it difficult to infer general properties about the dynamics of turbulence, more so because results are reported in harmonic oscillator units, which, unlike natural units based on the healing length, are unrelated to the vortices. Fortunately, box-like traps have been created recently (54).
Vorticity The key difference between classical turbulence and quantum turbulence is the nature of the vorticity, which is continuous and unconstrained in the former and discrete and quantized in the latter. Quantum turbulence thus offers a clear advantage over classical turbulence: the ‘building blocks’ (vortices and eddies) are clearly defined and simpler to detect (55): they are zero-density points (or lines) of the condensed gas, around which the quantum mechanical phase wraps by multiples of .
Range of length scales Another major difference between turbulence in classical fluids, turbulence in liquid helium and turbulence in atomic gases is the range of length scales available. In ordinary fluids, vorticity is a continuous field. The largest vortices are as large as the whole system, , and the size of the smallest eddies is of the order of the Kolmogorov length . Smaller eddies are destroyed by viscous forces. Here, the Reynolds number is a dimensionless velocity which quantifies the intensity of the turbulence, is the flow velocity measured at the scale , and is the kinematic viscosity of the fluid. Consider, for instance, the experiment of Grant et al. (56) in which a small vessel dragged a submarine probe which measured velocity fluctuations of water in the Seymour Narrows (a tidal channel near Vancouver) and confirmed the predictions of Kolmogorov’s turbulence theory. We estimate and , which means that vortices ranged in size over six orders of magnitude.
In superfluid helium (and also atomic BECs) vortices are discrete, not continuous, and the intensity of the turbulence is quantified by the vortex line density (length of vortex line per unit volume); typical values are to , which means that a typical vortex separation is in the range from to . The vortex core size is of the order of the superfluid healing length, which is in He and in He-B. There is hence a very large separation of length scales between vortex core size, intervortex distance and system size. Considering the experiment of Maurer and Tabeling (46), who first verified Kolmogorov’s law in helium, we get: . In atomic BECs, where typical atom numbers are to , the scales are somewhat different (see also Fig. 4). As in helium, the vortex core is of the order of the healing length , which is now larger (typically ) and the intervortex distance is only few times the healing length. The system size is still larger than the latter, typically few dozens to hundreds of . Even though the inequality still holds true, these length scales are now comparable, which has implications for the nature of the resulting turbulence. The reason is the following. Traditionally, a classical turbulent state is associated with the nucleation of vorticity (for instance, at the boundaries, by Kelvin-Helmoltz or baroclinic instability) followed by nonlinear vortex interactions. Such interaction usually involves an energy cascade from ‘parent’ to ‘daughter’ eddies (see Fig. 1) and the formation of smaller and smaller structures until the appearance of scaling laws (such as Kolmogorov’s 5/3 law). At very short length scales, energy is then dissipated. There is evidence that a similar process occurs in superfluid helium (39); (38); whether it happens in an atomic BEC gas clearly depends on the range of length scales available, which is currently debated (57); (58).
Two-fluid nature Another important difference between classical and helium turbulence is the two-fluid nature of the latter. Below the superfluid transition, liquid helium is a homogeneous, intimate mixture of two fluid components, the viscous normal fluid (which carries the entropy of the system) and the inviscid superfluid. The relative proportion of superfluid to normal depends on the relative temperature . The thermal excitations, which make up the normal fluid, interact with the superfluid vortex lines (mutual friction) thus coupling the two fluids. Only at sufficiently low temperature (typically ) in both He and He-B the normal fluid is effectively negligible and we face pure quantum turbulence. That is to say, turbulence of discrete vortex lines undamped by friction (59); (60); (61); (62). At higher temperatures () the nature of turbulence is affected by the mutual friction. In He the normal fluid has a relatively small kinematic viscosity (about sixty times less than that of water) and therefore the normal fluid is turbulent in most experiments. We face therefore the rather difficult problem of two distinct turbulent fluids – the normal fluid and the superfluid – which are dynamically coupled. Historically, this was the first problem which was studied; we refer the reader to Refs. (63); (64) and references therein. In He-B, the kinematic viscosity of the normal fluid is very large, so in most experiments the normal fluid is laminar or effectively at rest. We face therefore the simpler problem of quantum vortices moving under friction forces; indeed, there has been some success in predicting whether such forces can prevent the onset or turbulence or not (65); (66); (67).
The situation in atomic BECs is similar: pure quantum turbulence exists only if . Unlike liquid helium however, in most experiments the thermal gas is in a ballistic rather than fluid regime. Still, the idea of friction should hold, for example it is predicted (68) that an off-centre vortex will spiral out of the BEC as predicted by the vortex dynamics in helium. However, the experimental conditions are such that allow for a sufficiently good control of the temperature and hence the relative density of the thermal cloud.
Dissipation Ordinary fluids are viscous. That is, unless energy is supplied to compensate for viscous losses of kinetic energy, classical turbulence decays. In a two-fluid system, at high relative temperature the inviscid component can transfer energy to the viscous component via friction. We expect therefore that any sound waves, solitons and vortices in the condensate will be damped by the presence of the thermal cloud. As said above, at sufficiently low temperatures the thermal gas is negligible and quantum fluids are effectively pure superfluid. Surprisingly, turbulence will still decay: the kinetic energy contained in the vortices will not remain constant, but will be turned into sound waves within (and at the surface of) the condensate (see also Sec. 4.3).
2.6 A strict definition of quantum turbulence?
Quantum turbulence is a relatively young field of physics research and the question has been raised as to whether there is (or there should be) a proper definition of the term ‘quantum turbulence’. This question unavoidably triggers a second question: what is ‘turbulence’?
The term ‘quantum turbulence’ was first used in 1982 in the Ph.D. thesis of one of the present authors (69) and later in 1986 by Donnelly and Swanson (70) in an article about turbulence in superfluid helium. Until 1986, it was customary to refer to turbulence in liquid helium as ’superfluid turbulence‘ (an example is Tough’s extensive 1982 review article (71)). Donnelly and Swanson shifted the attention from the property that the turbulent fluid in question is a superfluid (i.e. that it has zero viscosity) to the property that the vorticity is quantized (a consequence of the existence of a macroscopic wavefunction). This change of point of view was almost prophetic, as later experiments demonstrated that superfluid turbulence decays despite the lack of viscosity. In 2000, Davis et al (72) used a vibrating grid to excite turbulence in He at mK temperatures (, a temperature regime in which the normal fluid is utterly negligible), and observed that, when left unforced, the turbulence quickly vanishes. The observation raised a puzzle: why does turbulence decay if the fluid has zero viscosity?
Now we know that in He, at temperatures above , the turbulent kinetic energy contained in the quantum vortices (created by stirring helium with grids or propellers) is transferred to the normal fluid by mutual friction, and then is converted into heat by viscous forces, as in ordinary fluids. Despite the normal fluid being negligible, below the kinetic energy of the vortex lines is not conserved: the acceleration of rapidly rotating Kelvin waves along the vortex lines (73) and the cusps created by vortex reconnections (74) induce phonon emission. In other words vortices decay into sound. Similar results were obtained later in superfluid He-B (60) and, more recently, also in 2D BECs (34). This scenario was also confirmed by 3D and 2D numerical simulations based on the Gross-Pitaevskii model of a Bose-Einstein condensate (75); (76).
We conclude that the key property of superfluid turbulence is not the fact that the fluid in question has no viscosity (at temperatures low enough acoustic dissipation plays the same role of viscous dissipation at higher temperatures). It is rather the quantization of vorticity that results in discrete vortex lines. Donnelly and Swanson were indeed correct: the term ‘quantum turbulence’ captures the essence of the problem. Quantum turbulence is therefore the turbulence of a system described by a macroscopic complex wavefunction - the property which implies the quantization of circulation.
We turn the attention to the second question - what is ‘turbulence’? Fluid dynamics researchers have never felt the need of a proper definition of turbulence
where is the velocity, the pressure and the density. Mathematically, the Euler equation is the skeleton of the Navier-Stokes equation which governs the motion of ordinary viscous fluids (and which can be further generalized to the presence of magnetic fields, thermal effects, non-Newtonian stresses etc). The nonlinear term arises simply because Euler equation – the manifestation of Newton’s second law for a continuum – describes the force (per unit volume) acting on a moving fluid parcel, not at a fixed point in space. It is important to notice that the Euler equation is also the skeleton of the Gross-Pitaevskii equation which models a weakly interacting condensate in the mean-field approximation at sufficiently low temperatures.
We conclude that the terms normally used in the literature are indeed justified: turbulence is the time-dependent, space-dependent state of irregular motion, characterized by a huge number of degrees of freedom which interact via the fundamental nonlinearity of the Euler equation. Since this equation is mathematically at the core of our descriptions of ordinary fluids, superfluid helium and atomic BEC the term ‘turbulence’ can be applied to all these systems, provided they are disordered enough.
Yet simple, this analysis is useful because it suggests how to improve our understanding of quantum turbulence. As said, turbulence in a classical fluid contains many degrees of freedom, which we can think of as eddies. How many degrees of freedom? Since the vorticity is a continuous field, it makes no sense to count them, but we can still give a quantitative answer to this question (see next Section) by noticing that the usual measure of the intensity of the turbulence, the Reynolds number, represents the ratio of the smallest to the largest eddy. On the contrary, in a quantum fluid the number of vortices (in 2D) or their length (in 3D) can be counted or measured because vortices are discrete entities. Then the question becomes: how many vortices are needed for turbulence?
We argue that this is a good question (cf. Sec. 5.6.1). We know that a system with as little as four point vortices moving in a 2D incompressible, inviscid fluid not restricted in space is chaotic (78) in the sense that initially close trajectories diverge in time away from each other with a positive Lyapunov exponent. The same holds true for a system of three vortices confined in a 2D BEC (79). But turbulence, according to our definition, is more than chaos. Turbulence is also governed by statistical and scaling laws (such as the celebrated Kolmogorov law of homogeneous isotropic turbulence) and metastable coherent structures. We have already connected the relative size of the smallest and largest eddies in the Kolmogorov inertial range to the range of length scales. It is natural to ask how many vortices are required to observe, for example, the emergence of Kolmogorov scaling, over how many orders of magnitude this scaling persists and how can one measure the experimental signatures of it. All of the above constitute challenging open questions in today’s research in QT; we will come back in Sec. 6.
3 Essentials of theory
In this Section we give a brief account of the theoretical models that are commonly followed in the literature of trapped BECs and summarize the main physics in the study of turbulence. Note that we chose to present the quantum and classical theoretical parts in the same chapter, as this will ease the understanding of the origin of similarities in the behavior of quantum and classical fluids. In particular, we discuss the reduction of mean-field models used for the description of quantum gases to classical equations of motion and thus highlight the success but also the weakness of these models. We stress that we do not attempt to present the theory of classical turbulence in full, but only the principal ideas and characteristics.
3.1 Brief account of classical turbulence
Turbulence – the nonregular motion of fluids, undesired in many everyday situations – has been known for centuries. The extraordinary properties of turbulent fluids have attracted the attention of great thinkers in the history of science and also arts. The idea that turbulent behavior arises from intertwined eddies in a fluid is a milestone in the historical development of the subject, which dates back to Leonardo da Vinci’s drawings of water falling into a pond, see the pattern of vortices at different scales shown in Fig. 5. Despite this precise observation of the natural world, da Vinci’s suggestion was not fertilized; neither scientists (who clearly then lacked the necessary mathematical tools to formalize it), nor artists followed up Leonardo’s observation, who continued representing turbulence as a foamy unstructured mess, see for example Turner’s “Snow Storm: Steam-boat off a harbour’s mouth” (Fig. 5).
That turbulent flow involves eddies exchanging energy has only been given a mathematical justification centuries later. The nonlinear differential equation which generalizes the Euler equation and predicts the motion of a real, viscous fluid was written in the mid 19th century. In the simple case of a fluid of constant density (e.g. water, which is essentially incompressible in typical situations), the Navier-Stokes equation describing a solenoidal field takes on the form
where is the velocity, the pressure, the density, is the kinematic viscosity of the fluid (the ratio of viscosity and density) and the external forces. Note the extra term at the right hand size which generalizes Euler’s original and idealized fluid equation.
The Navier-Stokes equation is remarkably powerful, and describes a huge range of phenomena, including turbulent flows. In the late 19th century the British physicist Osborne Reynolds studied experimentally the transition from laminar to turbulent flows in a pipe (see Fig. 6) and understood the interplay of viscous and inertial forces. In particular, Reynolds recognized that the transition requires that the parameter
is sufficiently large. Here is the average velocity of the fluid in the pipe and is the pipe’s diameter. This parameter, named Reynolds number, is a dimensionless measure of the velocity, and can be interpreted as an estimate of the ratio of inertial to viscous forces. At small , viscous forces dominate, and the flow is smooth and laminar; at large , the laminar profile is destabilized, a large number of eddies of all sizes appear, and the flow becomes turbulent.
The next seminal step was Lewis Richardson’s concept of the energy cascade in the 1920s. Richardson, a British meteorologist, understood that in the simplest possible case of turbulence away from boundaries (i.e. a case which is much simpler than Reynolds’ experiment) a statistical steady state requires a constant input of energy at large length scales and, at the same rate, the dissipation of energy by viscous forces at the small length scales. At intermediate length scales (the so called inertial range), the transfer of energy from large to small eddies is independent of viscosity. This transfer is called the energy cascade.
The idea of the cascade was formalized in the 1940s by the Soviet mathematician Andrei Kolmogorov, who associated spectral symmetries and a self-similar behavior to turbulent flow patterns. In the idealized case of a homogeneous and isotropic statistically steady-state, energy flows from the largest scales to the smallest one where it is turned into heat; within this range, the energy dissipation rate is constant. Therefore the Reynolds number, which is the dimensionless measure of the velocity at the largest scales, represents also the range of length scales available for turbulence. More precisely we have
(a relation which we have already used in Sec. 2.5). The quantity is called the Kolmogorov length scale, or the dissipation length scale, and effectively represents the smallest eddies. Kolmogorov suggested that the turbulent state has universal characteristics. A key property of turbulence is the distribution of kinetic energy over the length scales, which according to Kolmogorov is the same for all flows and does not depend on the specific details of each. It is convenient to describe the length scales as inverse wavenumbers , where is the three dimensional wavevector, and introduce the concept of energy spectrum . Kolmogorov showed that in the inertial range
where is a dimensionless constant of order one. In other words, in homogeneous isotropic turbulence most of the energy is contained in the largest eddies (small ); smaller eddies of size contain proportionally less energy according to Kolmogorov’s law, Eq. (3.5).
The above argument applies to three dimensional flow. With the exception of soap films (81), most real flows are three dimensional and two-dimensional (2D) flow models are only approximations of situations created in the laboratory. Fluid dynamics researchers are, however, interested in 2D flows as well, due to their important role as a paradigm for anisotropic 3D turbulence and not necessarily due to its physical realizability. The reduced dimensionality may arise from strong anisotropy, stratification or rotation (via the Taylor-Proudman theorem), which opens route to a new complex physics present in 2D chaotic flows. This focus by itself has essential, practical applications to oceans, planetary atmospheres and astrophysical systems as useful mathematical idealizations which approximate large scale features of geophysical flows (e.g. ocean and atmospheric currents) which have vertical extension (few km) much smaller than the horizontal extension (thousands of km).
Dynamics in 2D classical turbulence differs dramatically from 3D. In 1967 the American scientist Robert Kraichnan showed that in 2D flows energy does not cascade from large to small scales (as in 3D), but is rather in the opposite direction; from small to large scales (82); (83); (84). The same scaling as for the direct cascade however is found [Eq. (3.5)]. This inverse energy cascade (IEC) is accompanied by a direct (forward) cascade of a second inviscid quadratic invariant - the enstrophy [a measure of vorticity variance, defined by the surface integral on the xy plane ] - which makes the energy spectrum scale with for large momenta.
3.2 Brief account of quantum turbulence
In the quantum world of superfluid helium and atomic condensates, turbulence acquire new aspects. In Section 2.5 we have already listed the main differences between quantum turbulence and classical turbulence in superfluid helium and atomic condensates: discrete vorticity, quantized circulation, two-fluid nature, different range of length scales available, different energy dissipation. However, there are also similarities and these arise from the same conservation laws of mass and momentum. In particular the latter implies the same vortex interactions predicted by Euler’s equation. Moreover, an “effective” viscosity which plays a role similar to that of the viscosity in ordinary fluids, can be experimentally and numerically identified (42) in superfluid helium at very low temperatures. The similarities between superfluid helium and atomic condensates include the friction exerted by the normal fluid (86) or by the thermal cloud (87) which resists the motion of vortices.
We know that quantum turbulence takes the form of a tangle of vortex lines, which interact and reconnect with each other (see Sec. 4.1). However, in trapped condensates visualization or vortex tangles is more difficult and one resides in numerical simulations. In liquid helium it is relatively easy to determine experimentally the length of vortex lines contained in the experimental cell and therefore the vortex line density (vortex length per unit volume) is a practical measure of the intensity of quantum turbulence. From the knowledge of , one estimates that the typical distance between the vortex lines is .
Panel (c) of Fig. 1 shows a small tangle of vortex lines inside an inhomogeneous, spherically trapped atomic condensate (6); note the oscillations of the surface of the condensate induced by the motion of the vortices. Panel (a) of Fig. 7 shows a vortex tangle in a homogeneous condensate (67), computed inside a periodic domain. Panel (b) of the same figure shows the classical scaling laws of the energy specrum of the incompressible kinetic energy (see next section).
It is important to notice that not all turbulent states which have been studied display the classical Kolmogorov scaling. If the superfluid is thermally driven by a small heat flux (the normal fluid being uniform), the superfluid energy spectrum has the shape of a broad ‘bump’, lacks energy at the largest scales (88) and decays as at large ; since this is the spectral signature of individual vortices, we infer that the flow is random in character. On the contrary, if the superfluid is driven by a turbulent normal fluid, for example by mechanically exciting it with grids or propellers (46), the energy spectrum is consistent with the classical Kolmogorov scaling (88); the fact that most of the energy is at the largest length scale is consistent with a relative large-scale polarization of the vortex lines, which creates large-scale flows. Fig. 9 compares the two types of spectra: thermally-driven (right) and mechanically-driven (left). It is important to notice that each of the spectra corresponds to numerical simulations with the same turbulence intensity: the vortex line density (length of vortex line per unit volume) is the same () and so is the size of the computational box (). The difference between the results – the slope and the presence/absence of large scale, energetic eddies – is remarkable.
Similar spectra, classical and nonclassical, have been observed (42) and computed (43) in helium at very low temperatures (in the absence of normal fluid): they are usually referred to as quasiclassical (or Kolmogorov) and ultraquantum (or Vinen) turbulence (89).
What happens at very low temperatures at length scales smaller than the intervortex spacing raises theoretical controversies and still lacks experimental verification. Let us call the wavenumber which corresponds to the intervortex distance. The previous discussion referred to hydrodynamical scales . We are now concerned with length scales in the range where is the healing length (vortex lines cannot bend on scales shorter than the vortex core radius).
It is generally understood that, without friction effects, the interaction of Kelvin waves produces shorter and shorter wavelengths, until the angular frequency of the wave is fast enough to efficiently radiate sound away. This process, called the Kelvin-wave cascade, is often described in terms of the one-dimensional wavenumber and the wave amplitude, or occupation number . Two conflicting theories emerged. The first, proposed by Kozik and Svistunov (90), predicted that . To convert the one-dimensional Kelvin spectrum to the three-dimensional energy spectrum we recall that and that, for Kelvin waves, ; we obtain that (see also the discussion in Sec. 3.5). The second theory, proosed by L’vov and Nazarenko (91), results in , hence . The difference between and is small, and there have been attempts to determine the exponent numerically; the most recent work (92); (93); (94) supports the theory of L’vov and Nazarenko, but we still lack direct experimental evidence.
The following possibility has also been raised: that of a bottleneck between the Kolmogorov cascade and the Kelvin cascade, which may change the shape of the spectrum in the region (95). This effect may arise because the rate at which energy flows down the (three dimensional) Kolmogorov cascade is different by the (slower) rate at which energy can progress into the (one-dimensional) Kelvin cascade. A somewhat different approach, suggesting the smooth transition of a Kolmogorov cascade of superfluid turbulence to a Kelvin-wave cascade has been suggested in Refs. (96); (97). Again, experimental evidence is lacking.
Last, more recently, there has been discussion on the possible introduction of a ‘quantum’ Reynolds number or a ‘quantum’ Navier-Stokes equation (65); (98); (99) to distinguish between laminar and turbulent superfluid helium flows and to characterize turbulence in a quantum gas. These are discussed in Sec. 3.3.3.
3.3 Theoretical models for BECs
The foundations of a quantitative description of superfluid helium under rotation were set by Hall and Vinen in the mid 1950’s (35); (36). Some decades later, computational simulations in quantum turbulence were pioneered by Schwarz (100). Schwarz observed that the velocity flow field in a quantum gas can be described by the vortex motion which is represented by a curve in three-dimensional space, where is the arclength and is time. With these computations the final distribution of the vorticity is determined entirely by the geometry of the cell, the global flow parameters, and the initial vortex distribution. The Biot-Savart (BS) formulation gives the field a self-induced velocity along the curve in complete analogy to a magnetic field. In this approximation the vortex cores are approximated as infinitesimal, one dimensional objects driven in an inviscid background. The spatial coordinate along each vortex is given by the parametric arclength .
We can write the velocity field as the rotation of a vector potential, . The vorticity is therefore defined by the Poisson equation, , which has a Green function solution given by
where is the location of the vortex core. Since in the cartesian plane, the vorticity has a constant intensity and is concentrated in the core, a change of variables allows for the rewriting . Taking the curl of and inserting the above parametrizations the well known BS velocity form is deduced,
Note that, by construction, the above solution is an incompressible field. Indeed, as the divergence of a curl, the identity
holds. However, a general (quantum) velocity field , as we shall see in the following Section, needs not be incompressible but rather of the form:
where is the field that is incompressible (i.e. has zero divergence) and is the irrotational (and compressible) part of the field. Hence, the velocity field described by the BS model gives only the incompressible part of . Physically, this suggests that captures the vortex dynamics, as described by the BS model. The form of in Eq. (3.9) is known as Helmholtz decomposition and its importance will become obvious in the analysis that follows (see Sec. 3.4).
Except for very simple cases, complete analytical solutions with this formulation are not possible. Additionally, BS calculations are computationally expensive, since operations are required over a grid of points (101). To reduce the computational effort, the BS integral is often replaced by
where and the derivatives are taken with respect to the arclength (101). In that way the nonlocal contributions of the integral of Eq. (3.10) are neglected and thus this simplification is called the local induction approximation (LIA). LIA is commonly used both for analytical intuition and for computational tractability. With this simplification it is apparent that the vortex lines move in a direction perpendicular to both the line curvature () and tangent () with a velocity inversely proportional to the curvature.
The BS law describes individual vortices in the zero temperature limit. However, high temperature viscous effects and vortex-vortex interactions are not naturally included – in neither of the two approaches – and need to be inserted ad hoc in each simulation. A quantum vortex reconnection (QVR) occurs when two vortices approach each other and connect at one point. During a reconnection the two lines topologically change into a different configuration, as shown in Fig. 10. QVRs have been observed both in He II (102); (103) and BECs (104). To circumvent this problem, one needs to artificially insert the occurrence of QVR in the BS or LIA models (100); (105). To achieve this a cut-off vortex-vortex distance is introduced, beyond which the vortices are assumed to reconnect (see for instance (106) and references therein). The absence of QVR is a critical failing of the BS simulations as it has been shown that reconnection processes are fundamental to the dynamics of QT.
The BS approach, even extremely simple can capture several of the features of QT and give statistical insight on the behavior of driven quantum fluids, especially when studying distribution of momenta. However, the lack of reconnections in the model (or the ad-hoc insertion of reconnection events) together with the assumption of infinitesimally small vortex cores are drawbacks that one cannot ignore. Infinitesimal vortex cores, that the BS model admits, do not translate well in gaseous dilute BECs: a vortex core can have a diameter comparable even to the extension of the entire condensate! Hence a formal and more precise description of the fluid alternate methods should be employed.
The mean-field Gross-Pitaevskii (GP) theory, instead, can describe the above situations. There, hydrodynamic and quantum behaviors naturally coexist and reconnections arise, as we shall see, from its solutions.
The GP approximation is today the most used theory in the BEC community and also beyond. It was formulated originally by Eugene Gross and independently by Lev Pitaevski in 1961 in order to describe quantized vorticity in a Bose gas (107); (108). Within a semi-classical approach they derived the celebrated Gross-Pitaevskii (GP) – also known as nonlinear Schrödinger – equation. The defining assumption of this mean-field theory is that all bosons of the system reside in the same quantum state (total condensation) and hence the whole system is approximated by one single-particle state . The particle density of the system is then given by . In other words, for a system of particles of mass , according to the Gross-Pitaevskii theory, the knowledge of one them is enough to characterize the collective properties of the whole system.
The wavefunction of the relaxed state is found by solving the eigenvalue problem:
i.e. the time-independent GP equation. The eigenvalue is the chemical potential, the usual nabla operator and the Planck constant. The time evolution is given from the solution of the time-dependent GP equation:
At ultralow temperatures and densities the particles are assumed to interact only weakly via an -wave scattering and this is modelled by a contact (two-body) pseudo-potential of Dirac delta form . This assumption gives rise to the ‘self-interaction’ term in the energy of the system. The parameter measures the strength of this interaction and is proportional to the s-wave scattering length: . The parameter can be either positive or negative, depending on the sign of the s-wave scattering length, . The is the confining external potential and is – in general – function of space and time, i.e. . The wavefunction is normalized by the total number of atoms in the system,
Keeping in mind that is a complex-valued function, one can perform the following transformation:
known as Madelung transformation. and are real functions and give the (square root of) the density and phase of the condensate accordingly. Inserting this in Eq. (3.12) and separating real and imaginary parts we get the two equations (13)
where is the field velocity, as defined in Sec. 2.2, and its norm. Equation (3.15) is the continuity equation [see Eq. (2.8)]. This becomes clear if one identifies the term with the flux . In other words, the inclusion of contact interactions in the GP model does not affect the conservation of the probability .
The GP equation is the mean-field approach to calculate the properties of turbulent BECs. In the limit of a ground state condensate at zero temperature and infinite particle number, the GP equation is exact (109); (110). One feature of importance is that the lines of quantized vorticity and vortex reconnections appear as solutions of the GP model, rather than ad hoc added as in the BS approach (101).
For an infinite, time-independent system, the vortex core diameter can be estimated as follows. Like a soliton, a vortex is a topological defect whose extension at equilibrium is expected to be of the order of the healing length. The healing length of a (uniform) condensate is defined as the length scale over which the interaction energy becomes equal (or comparable) to the kinetic one. Thus,
(see also (13)). For a condensate that is locally perturbed at some point , gives the length after which the density reassumes its initial value . In other words, it is the distance that it takes for the gas to ‘heal’ the disturbance. A quantum vortex, being a local node of the density, can also be thought to have an opening of diameter .
Last, in the regime of the parameters that condensation of trapped rubidium atoms normally takes place (that is, temperature in the regime, particle number of the order of around and -wave scattering length of about ) the wavefunction of the gas at equilibrium varies smoothly and slowly. Hence the kinetic contribution to the energy (scaling as the second spatial derivative) can be neglected. This is known as the Thomas-Fermi (TF) approximation and with it the spatial extension of the cloud of the condensed atoms can be easily determined. If we neglect the kinetic energy operator in Eq. (3.11) we obtain for the density of the ground state:
We see that the density is merely the inverse of the confining potential , scaled and shifted ( is the chemical potential and is fixed for some given particle number ). Naturally, this formula does not hold for . Otherwise, it gives a satisfactory approximation to the expansion of the background density of gas, over which quantum vortices with expansion are ‘seeded’.
When in the TF regime, the gas has a well-defined extension, known as the Thomas-Fermi radius , that is found from the boundary condition . In the case of a harmonic potential we obtain:
for each direction .
Reduction of mean-field to classical hydrodynamic equations
An interesting feature of the quantum gas is the resemblance of its dynamical equation to that of a classical incompressible gas. Indeed, recasting the GP equation [Eq. (3.12)] to the form of Eqs. (3.15)-(3.16) and taking the gradient of Eq. (3.16) we obtain:
where and (see also (13)). The term is kinetic energy while the term is a pressure gradient. Note that the only term of Eq. (3.21) that depends on is the left term of the right-hand side. This term, also known as quantum pressure, originates from the kinetic energy as well. However, it corresponds to the ‘zero-point’ quantum mechanical motion and is significant only when the density of the gas varies ‘rapidly’ in space and negligible otherwise. More specifically, let us imagine that the wavefunction changes significantly in space over a distance . Then the classical pressure is of order while quantum pressure is of order . Hence, only for distances less than , which is of order of the healing length [Eq. (3.18)], the quantum pressure dominates.
By taking the classical limit the above equation becomes identical to the Euler hydrodynamic equation for a classical irrotational fluid (), which is the dissipation-free () form the of the NS equation [Eq. (3.1)]. Then, the role of external forces plays the potential . That the Schrödinger equation can be reshaped into a quantum Euler equation [Eq. (3.21)] and the identification of the quantum pressure term was already noted by Erwin Madelung in 1926. The German scientist showed, in two letters (111); (112), how one can derive both the continuity and the quantum Euler equations, sometime collectively called Madelung equations.
Note that, an effective dissipation can be included in the GP equation (see, for instance (113) and references therein). The phenomelogical modification is the addition of an imaginary term in the GP equation, making its time evolution complex: , accounting thus for thermal effects and damping. The hydrodynamic form of the dissipative GP equation now, here derived for an incompressible quantum fluid, corresponds to a quantum Navier-Stokes equation:
Beyond mean-field theories
The GP theory is a widely used and successful mean-field description for the statics and dynamics of interacting bosonic gases. It has been applied to various scenarios involving vortices, solitons and other collective excitations and the agreement to experiment has been found to be satisfactory (see for instance Ref. (115) for a review). Thus, the GP approach is a natural choice as a theoretical tool to study quantum turbulence in ultracold gases as well.
However, the turbulent gas is a largely perturbed gas, i.e. a system with an average excitation energy many times higher than the relaxed-state energy. In a typical experiment the gas is brought to turbulence by shaking and oscillating the confining trap for various different times (see Sec. 5). It could well be that the coherence and condensation of the system is lost throughout the dynamics. To check the consistency of a mean-field approach one needs to go beyond the limits of mean-field and relax the restriction that the system stays condensed throughout the dynamics. In other words, there should be no a priori reason why a single one orbital describes all of the system bosons at all times. A consistent way to go beyond the GP picture is the multi-configurational time-dependent Hartree for bosons (MCTDHB) and the equations bearing the same name (116); (117); (118). This approach is capable of describing fragmentation and loss of coherence; it includes many single-particle states that are time-dependent, the non-zero occupations of which give rise to non trivial correlations. Correlations and loss of coherence are important aspects that are experimentally measurable and also shed light into the structure of complicated and highly excited perturbed systems, such as the turbulent gas prepared in the laboratory and discussed in the following Sections.
3.4 Hydrodynamic turbulence in BECs
Turbulence, as said, is generally associated with names such as Kolmogorov and Richardson and ideas such as the local transfer of energy between eddies of similar sizes giving rise to the so-called energy cascade. Despite simple, it is important to explicitly demonstrate the dependence of the energy in the wavenumber (momentum) and the emergence of power laws. Turbulence in BECs – both the quasiclassical and ultraquantum types discussed earlier – involves the nonlinear propagation of the velocity field, as is obvious from the governing equations (Schrödinger or quantum Euler). This manifestation of turbulence is hydrodynamic in nature in the sense that it requires a nontrivial velocity field, usually containing quantized vortices, i.e. line singularities. However, this is not the only possibility for turbulence in a BEC: see the upcoming Sec. 3.5.
Due to the statistical nature of turbulence it is convenient to define the averaged energy distribution in momentum space
where stands for the statistical average. Note that the above holds for an isotropic velocity field . In that case, the latter depends only on the magnitude and can be expressed in terms of a one-dimensional distribution
In general, as discussed in Sec. 3.3.1, a velocity field can be separated into an incompressible (or solenoidal) part with and an irrotational field with . This means that the Richardson picture of big vortices decaying into smaller vortices should be looked for in the field with non-vanishing vorticity, i.e. the incompressible part of the velocity field.
A similar separation has also to be applied to a generic compressible superfluid system. In contrast to liquid helium systems, atomic Bose-Einstein condensates, due to the weak interparticle interaction in these systems, have the advantage of allowing a theoretical modelling of its dynamics in terms of the GP equation [Eq. (3.12)] which has total energy given by the functional
Note that, is the energy owing to the zero-point motion (see also Sec. 3.3.3) and shall go to zero in the classical limit, when, for instance, examining the gas at scales much larger than the healing length .
It is also convenient to define the so-called density averaged velocity field which can be then separated (again, according to the Helmholtz theorem) in its solenoidal and irrotational parts. This, in turn, allows the kinetic energy to be separated in its compressible and incompressible parts (114)
which also have a simple representation in Fourier space
Analogously to the classical case a one-dimensional energy distribution can also be defined for isotropically distributed
in three and one dimension respectively. Since the dynamics and interactions of quantum vortices are strongly nonlinear problems, analytical approaches are very restricted and almost all the literature in this subject deals with numerical simulations. In Refs. (45); (66), high-accuracy numerical simulations of the GP equation for initial states with random phase profile were performed showing the existence of a Kolmogorov-like law for , suggesting thus some similarity between the turbulent dynamics of and the velocity field in incompressible Navier-Stokes equation.
The analogy between the classical and quantum fluid can be carried even further by considering the possibility of Richardson cascades involving the big structures with many quantized vortices forming bundles of vorticity and decaying into smaller structures in a self-similar fashion. This process is believed to generate direct energy cascades in 3D as in the classical case (122). Several authors also claim the existence of inverse energy cascades, while direct enstrophy cascades are still object of dispute (58).
3.5 Wave turbulence in BECs
In addition to the turbulence associated with the motion and interaction of vortices, there are some processes that can occur in BECs which are classified under the umbrella of wave turbulence (WT). In that case the turbulence emerges due to interacting dispersive waves – in analogy to eddies in hydrodynamic turbulence. WT, the interaction of waves in nonlinear media, appears as a cascade with power laws in the wave energy spectrum (see Ref. (123) and references therein). In BECs the motion of lines of quantized vorticity and also acoustic waves are both nonlinear and fall in this classification. The system is said to exhibit weak wave turbulence, when its turbulence is described by weakly nonlinear dispersive equations of motions. The weak nonlinearities allow an analytic description as described in Refs. (124); (125).
In this description, nearly all the properties associated to the stationary turbulent states can be obtained from the knowledge of the wave dispersion relations and the form of their lowest order nonlinear term. In many applications of this type, the system can be described by a complex field whose equation of motion in Fourier representation can be written up to its lowest nonlinear term.
Phonons Sound waves are small amplitude excitations over the background macroscopic wavefunction (126). For a homogeneous background one may consider . By substituting the wavefunction and keeping only the smallest nonlinearity, i.e. neglecting any third-order terms in , into Eq. (3.12), we get
In Fourier representation, the above equation becomes
which has inverse the transformation
with and In this way the equation of motion for can be obtained
where the quantity is the interaction coefficient and depends on the interaction parameter . The latter, in the long wavelength regime scales as (124)
while the dispersion coefficient behaves as , with the sound velocity being . The -dependent terms inside the integral in Eq. (3.44) describe the three-wave interactions responsible for triggering the cascade process. The first term correspond to the decay of two waves into one () while the second term corresponds to the opposed process ().
Similarly to the hydrodynamic case, the energy is also a measure of central importance and in weak interacting case is mainly given by its kinetic part
where and is the so-called one-dimensional energy distribution.
According to the statistical theory of out-of-equilibrium waves, presented in (124); (125), such three-wave processes allow the existence of a steady state characterized by a direct energy cascade. Indeed the power-law distribution associated with the energy spectrum of the steady-state turbulent system can be entirely determined from the scaling properties of the dispersion relation and the interaction coefficient. For the special case in Eq. (3.44), where dispersion and interaction coefficients scale as and , the energy spectrum turns out to be of the Zakharov-Sagdeev type (124):
Kelvin waves Another WT process that can occur in BEC is associated with the vibratory motion of vortex lines and vortex loops, the so-called Kelvin waves (see also Sec. 2.3). The first WT theory for this system was formulated by Kozik and Svistunov in (90). In such a formulation, the relevant interaction term involves 6-wave processes. In his book (124), Nazarenko points out that two different cascade processes can simultaneously occur within a WT system lead by -wave interactions, with being an even number. A direct energy cascade is present, where the energy density in Fourier space () flows from large to small length scales. In addition to this, there can also be an inverse wave action () cascade. Such dual cascades have close analogies with the energy/enstrophy dual cascade in hydrodynamic turbulence.
In the Kozik-Svistunov theory, the dispersion relation, neglecting logarithm factors, is , which leads to the direct energy cascade spectrum
while the inverse wave action spectrum is given by (90)
Interestingly and in contrast to hydrodynamic turbulence, WT relies only in the nonlinearity induced by the two-body interactions that the (and consequently ) term captures. Note that, the -dependence of the GP interaction is not present in the quantum Euler equation [Eq. (3.21)] and hence hydrodynamic turbulence does not explicitly depend on the particle-particle interaction of the bosons of the cloud.
3.6 What all this has to do with the experiments?
In time-of-flight experiments, indirect measurements of the kinetic energy density are possible. This stems from the fact that, after a long enough time of expansion , the observed density of the gas in a standard absorption image process has the form of the distribution of the momenta in situ, i.e. inside the trap, before the ballistic expansion of the gas:
However and since, as we saw, this energy is composed from several component parts [Eqs. (3.28)–(3.31)], it is a quite challenging task to associate any observed power law spectrum with specific cascade processes. That is, an experimental absorption imaging method cannot decide what type of turbulence is there. As shown above, we have for the sound wave cascade and for the direct energy Kelvin wave cascade . These exponents are very close to the Kolmogorov from hydrodynamic turbulence which complicates even further the determination of which process is actually responsible for any observed power laws.
Vorticity in superfluids and quantum systems is probably the most important ingredient of quantum turbulence. In recent experiments (see Sec. 5) a Rb Bose condensate is used to observe and investigate quantum turbulence, by means of a weak off-axis magnetic field gradient, which perturbs the BEC and injects kinetic energy onto it. This nucleates on the condensed-thermal interface of the BEC and sets up experimental conditions to the emergence of turbulence. It is important for the development of turbulence that the nucleated vortices are not coplanar, not ordered and let to proliferate and interact. Once the turbulent regime is set, the condensate is then released and expands under free fall. The atomic density profile is acquired using resonant absorption imaging, some milliseconds (typically ) after the gas has been released from its trap (time of flight). The calibrated density images are used to determine the in situ momentum distribution of the BEC. Power laws in the experimentally measured momentum and energy distributions have been observed, making these states deviate significantly from the unperturbed gas. We present these studies in Sec. 5.9. Additional characteristics of the system, such as the finite number, size and temperature of the condensate, shall play a role in the energy injection mechanisms and are further discussed throughout Sec. 5.
4 Energy transport and dynamics
4.1 Vortex reconnections
As mentioned in Sec. 2, the quantization of vorticity is a unique feature of superfluids and condensates. The study of vortex propagation and interactions are central in the study of turbulence, at least in its hydrodynamic manifestation. The experimental settings, so far, that brought BECs to turbulence have generated a large number of vortices that eventually proliferate, reconnect and tangle with each other (3); (129). A quantum vortex reconnection (QVR) is a general phenomenon in which two vortex lines interact, approach each other, connect at a given point for some (short) time and exchange tails. This process happens repeatedly in highly excited gases, like the turbulent systems. Hence, the study of QVR is central in turbulent dynamics as reconnections are, to a large extent, responsible for the energy transfer between different scales. As such, QVR can be considered as a fundamental mechanism of QT.
Quantum vortices were first observed experimentally in trapped BECs in 1999-2000 in two different laboratories, in JILA and Paris (130); (131). The vortices were nucleated with different mechanisms (phase imprint and stirring with a laser, respectively) in the two experiments. Since then, vortices have been extensively studied in gaseous BEC and nowadays they can be routinely created (see (132); (133) for reviews on this topic). Interestingly, the existence of vortices in superfluids has, decades earlier, stimulated the formulation of the Gross-Pitaevskii theory (107); (108) (see also Sec. 3.3.2). In liquid helium QVRs have been, with impressive detail, monitored by the group of Lathrop in Ref. (102); (103) and can be seen in the video of Ref. (134).
Even though, the present experimental work pertains to trapped gases, we begin the exposition with theoretical results for homogeneous systems. This happens in order to, firstly, give a brief historic account of the principal results and, secondly, to emphasize on the fact that QVR is a behavior that can be seen across many different physical settings. Vortex reconnection is a universal phenomenon appearing in different classical and quantum systems, and also arises within different theories. In Fig. 10 we present reconnections that emerge as solutions of the (a) NS equation, (b,c) the GP equation and also (d) the many-body MCTDHB equations (see Sec. 3). As already mentioned, the vortex filament models of the Biot-Savart theory (presented in Sec. 3.3.1) cannot describe the reconnection per se; an ad hoc inclusion of it is, however, possible.
In homogeneous space
A line vortex moving in the velocity field produced by another vortex will exert a force to the latter, and vice versa. This is the mutual Magnus force:
where is the superfluid density and is the sum of the velocity fields of both vortex lines, (see Eq. (3.7)). Note that the vortex line – unless completely straight – will feel a force on it induced by its own velocity field due to its curvature. Hence, vortex lines in the three-dimensional space will feel a force by the two velocity fields combined, resulting in a mutual attraction or repulsion. depending on their relative orientation. For a generic initial configuration of the vortex lines in space this will result in parts of the lines mutually attracting and approaching until reconnection. It is only the extreme case, where the two filaments lie totally straight in free space and parallel to each other, that the force (say, attraction) does not depend on ; then the two lines will approach each other keeping their initial parallel shape until the collide. However, boundary conditions, density inhomogeneities or local random fluctuations along the vortex line will break this symmetry and reconnection at one point – i.e., the familiar tail-exchange event – will eventually occur.
Since decades, reconnections have been studied extensively both in the context of quantum and classical theories. In 1987 Ashurst and Meiron simulated reconnections using a techique that combines the BS model and NS equation (135), as seen in Fig. 10 (a). Later on, in 1993, Koplik and Levine showed that quantum vortex reconnections appear in the dynamics of the GP equation (136), shown in Figs. 10 (b) and (c). In 1994 Waele and Aarts with a BS model claimed a universal route to reconnection for all kinds of initial vortex-antivortex arrangements (137). More recent relevant work includes the study of vortex reconnections in untrapped superfluids (138), the numerical computation of the minimum distance between two approaching vortices as function of time (139) and the calculation of the energy spectra of gases with reconnecting vortices (64); (140).
In a trap
As said, experiments that involve vortices are performed in parabolically trapped gases and, hence, studies of vortex interactions in non-homogeneous systems are necessary. Lately, the interest has shifted to simulate realistic situation, where vortices are seeded in a trapped gas and let proliferate. Qualitatively, the QVR process does not change dramatically when the system density is no more homogeneous. The nonuniformity of the density induces an extra force on the vortex line. For instance, a single vortex isotropically confined will precess around the centre of the trap. The tail-exchange process between two vortex lines described above still occurs, interestingly, only as long as the trapping anisotropy does not exceed a critical value (119). That is, for isotropic and close-to-isotropic traps the QVR happens in the same familiar way, at the center of the condensate where the density is higher. For larger anisotropies the QVR tends to happen at the margins of the gas, where the density is lower (see Fig. 11).
The impact of finite temperatures on the QVR in trapped BECs has been studied in (141) using the GP approach coupled to a thermal cloud. It is found that thermal effects do not inhibit or alter the reconnection process, extending thus the universality of the QVR to both zero and finite-temperature systems.
Simulations of QVR have also been obtained with methods that go beyond the common mean-field approach (as the GP model) (119). In specific, it is possible to solve the many-body time-dependent Schrödinger equation as an initial value problem in three dimensions, for a state of the trapped gas containing two perpendicular vortices. This initial state is usually created by enforcing a desired phase field in the initial wavefunction and a density node as well, accounting for the vortex core. In Ref. (119) where QVR is studied in anisotropic traps, the time-dependent Schrödinger equation is solved with the Multi-Configurational Time-Dependent Hartree for Bosons (MCTDHB) method (116); (117); (118) and its numerical implementation (MCTDH-X)(120). There, it is found that the ‘walls’ of the trap will heavily interfere with the reconnection process, as long as the trap anisotropy exceeds some critical value. This impacts the ‘shape’ of the reconnection (see Fig. 11) as well as the time which the vortices spend during the reconnection. Furthermore, this process might also alter the familiar empirical law that gives the minimum distance of two approaching vortices as a function of time.
The QVR is a fundamental process of energy transport in quantum fluids with vorticity with important consequences on the turbulent dynamics. Not only does each individual reconnection release kinetic energy, but it nucleates vibrations on individual vortex cores. The energy release can take the form of phonon radiation and vibrations on the vortex core that are carried by helically rotating Kelvin waves. The excited by the reconnection phonons and waves can offer an explanation in one of the fundamental questions of QT: what the nature of the dissipation is, at zero-temperature frictionless fluids (144). Indeed, Vinen postulated that high frequency oscillations of a vortex core can efficiently produce phonon radiation, allowing for an eventual dissipations of energy in an inviscid fluid (145). It is natural to assume that turbulent vortex tangles decay with mechanisms like phonon radiation and helical waves excitations.
Kelvin waves are long scale perturbation along the vortex line that can exist in both classical and quantum fluids (90); (146). In superfluids it has been suggested that cascades of Kelvin waves can transfer energy from the scale of the intervortex spacing down to smaller scales, as small as the vortex core (106); (55); (144). Therefore, Kelvin modes become an important mechanism in the decay of turbulence (94) as energy is directed to the smaller scales of the system, where it is eventually dissipated.
In Ref. (106) the authors, solve numerically the BS model for an initial configuration of four vortex rings and report on a Kelvin-wave cascade owing to “individual vortex reconnection events which transfers energy to higher and higher wave numbers ”. Recalling that, simulations with the BS approach involve no compressible energy at all, the mechanisms of energy transfer studied in the works of Refs. (106); (94) and others, pertain to vortex energy exclusively. This suggests, that cascades of energy mediated by Kelvin waves assist the decay of turbulence, independently of the existence of other collective modes as phonons or others. In Ref. (49) simulations confirmed that such cascades can serve as a mechanism for dissipation of energy within the GP theory too. In specific, a gas in an axisymmetric elongated trap possessing a single vortex is studied as an initial condition problem. A time-dependent rotating potential perturbs the initial state and its time-evolution shows the emergence of helical Kelvin waves along the vortex line. The Kelvin waves eventually decay to longer wavelength excitations via emission of phonons, thus making this process a zero-temperature energy decay mechanism.
4.2 In two dimensions
General characteristics of 2DQT
In subsection 3.1 we already mentioned that 2D classical turbulence is very different from the 3D case and presents interesting features, such as the inverse energy and enstrophy cascades. The search for similar startling characteristics of the 2D classical counterpart in quantum fluids has been intense. While the scaling (also expected for the 2D classical turbulent fluids, as predicted by Kraichnan (148)) has been found in simulations of the GP equation for homogeneous systems, several points concerning energy cascade direction is still an open debate.
In contrast with the classical analogue, where assuming two-dimensionality is in most cases an approximation, in quantum flows two-dimensionality can be achieved by exploiting the controllability that experimental gaseous atomic condensates offer. Using suitable trapping potentials, atomic BECs can be easily shaped so that vortex dynamics is 2D rather than 3D making them ideal systems to study 2D turbulence (57). In these settings, one direction is so tightly confined that the dynamics along it is practically frozen and the system is quasi-2D. Precisely, the anisotropic optical trapping makes the Thomas-Fermi radius [Eq. (3.20)] on the transverse direction much smaller (factor of or more) than the of the radial direction, rendering thus transverse excitations energetically costly. The quantized nature of quantum vorticity together with this reduction of dimensionality implies that the properties of the vortex line are no longer relevant for dynamics, as the one-dimensional vortex now becomes a zero-dimensional point in the plane. Kelvin waves and their cascade, that is the fundamental process for QT decay in three-dimensional turbulence, are not applicable in two dimensions. Vortex-antivortex annihilation is now a salient feature and, moreover, clusters of vortices appear as the large-scale structures that play a role analogue to the 3D vortex bundle. Another practical advantage of using atomic condensates to explore 2DQT is that, unlike liquid helium, 2D quantum vortices can be directly imaged and, unlike classical systems, the motion of such 2D vortices is not hindered by viscous effects or friction with the substrate.
The 2D problem has been investigated in both homogeneous and trapped systems, and are being discussed in the following.
Homogeneous Systems Several numerical studies have explored the generation of turbulence in 2D homogeneous condensates. The investigation of the evolution of the system towards turbulence leads to important questions regarding vortex spatial distributions. As we have discussed, in order to show the Kolmogorov scaling, a turbulent system must also display self-similarity throughout several scales in an inertial range, which requires particular vortex distributions. For the appropriate quasiclassical (with relatively large ratio ) 2D turbulent regime, the incompressible kinetic energy is found (114) to follow a spectrum of
which is qualitatively illustrated in Fig. 13 for system where forcing takes place in small scales . In 2D classical turbulence, a direct enstrophy cascade accompanies the inverse energy cascade (IEC), which scales as and so analogues in the quantum system have been sought. As Eq. (4.2) shows, the same scaling is found in 2D quantum systems for a completely different reason, however; the problem of what the quantum analogue of enstrophy should be remains. In contrast to its classical counterpart, the velocity field profile of a quantized vortex is responsible for the scaling, therefore representing solely the internal structure of the vortex core. This same quantized nature of quantum vortices makes enstrophy to be proportional to the number of vortices (114); (57) and the possibility of vortex-antivortex annihilations forces enstrophy not to be an inviscid quantity in quantum fluids. A different interpretation is typically adopted in this case: the approximate conservation of vortex number is taken to be the analogous of the enstrophy conservation in classical systems, helped by long-lived clustering of like-signed vortices. In the context of decaying turbulence, the authors in Ref. (149) created initial states from random phases and evolved them using the GP equation, highlighting the effect of non-conservation of vortex number. Although their system also developed a statistical distribution of vorticity consistent with Kolmogorov scaling, the compressible nature of the superfluid and the absence of forcing make enstrophy in these systems a non-conserved quantity; the vortex number always decays due to pair annihilation processes and no dynamical mechanism keeps injecting pairs at the same rate of their removal. This fact prohibits the development of an IEC and the calculated energy flux becomes direct, as in the 3D case. A detailed study of energy transfer mechanism in 2D systems using the GP equation can be very challenging, since determining the energy flux depends on knowing how compressible and incompressible energies couple. A way to go around this problem is proposed in Ref. (150), where a modified point-vortex model is introduced to mimic the effects of the GP equation and dissipation in a purely incompressible fluid. Their results suggest that the IEC due to vortex interactions appears only for large systems with moderate dissipation. A transient dual energy-enstrophy cascade is also verified, validating the quasiclassical behavior for such systems.