Abstract
In spite of being a phenomenon studied over centuries, turbulence remains an intriguing phenomenon of nature. In the low temperature regime, turbulence has been investigated in superfluid helium during the last decades. Due to the quantum nature of superfluids, this phenomenon is named Quantum Turbulence and it is characterized by a particular configuration of quantized vortices in the sample. Recently, this topic started to be investigated in a different kind of superfluids, namely, trapped atomic Bose-Einstein condensates (BEC). In this text we review the first experimental evidences of Quantum Turbulence in a BEC of 87Rb. We describe our most important observations and discuss possible research perspectives.
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14.1 Introduction
Turbulent processes are everywhere on Earth and beyond. It is crucial to life support from microscopic scale, inside the human body, to the macroscopic scale of the Earth’s rotating core. There are many familiar examples of turbulent processes taking place in the everyday life as they are natural phenomena occurring on airplane flights, or on the sky’s clouds motion, forming characteristic patterns.
Despite of being commonly observed, turbulence is known to be difficult to treat and study. The interplay from small to large scale in turbulent fields makes it difficult to study due to the need to resolve spatial and temporal ranges over several orders of magnitude. This often requires large experimental samples and the need to measure fast fluctuations on microscopic (or mesoscopic) scales. The nonlinear equations (Navier-Stokes) of motion are difficult to handle because of the large span of scales involved in turbulence. As a result, many terms in the equations cannot be neglected because their contribution may change over the relevant scales.
Quantum fluids, such as superfluids, superconductors, and Bose-Einstein condensates, may present turbulent states that are different from classical due to long-range quantum order, which poses constraints to their dynamics. All vorticity, in the case of superfluids and Bose-Einstein condensates is restricted to topological defects in the order parameter of the system. The resulting linear structures are named quantized vortices because continuity in the order parameter quantizes the circulating flow around each topological defect. As such, turbulence in a quantum fluid displays a tangle of interacting quantized vortices, as first pointed by Feynman [1] and others [2–6], that is very different from the continuous distributions of vorticity observed in classical turbulence.
Since then many studies have investigated the distinct characteristics of this phenomenon (for a comprehensive and complete review, please see Ref. [7]). Recently the experimental realization of Bose-Einstein Condensation (BEC) in trapped atomic samples [8–10] and its relation with superfluidity [11–13] opened up new possibilities to investigate Quantum Turbulence (QT). A series of published papers by M. Tsubota and collaborators [14, 15] presented a possible mechanism for inducing QT in atomic superfluids. Using a model based on the Gross-Pitaevskii equation, they have successfully derived the Kolmogorov spectrum for QT in atomic superfluids, analogous to the previous observations made on superfluid 4He. See Chap. 13 for details.
There are a few advantages when using BECs of trapped atoms to investigate QT. A first clear advantage is the fact that BECs are weakly interacting systems, hence, its theoretical description is simpler. Second, the number density is much smaller than in liquid 4He, therefore, the vortex cores will be much larger, making their observation easier. Finally, these systems present a larger variety of controllable parameters such as the density, temperature, geometry and even the interatomic interactions. This controllability allows a vast exploration of the phenomena.
During last years, our group have observed the first clear evidences of QT emerging in a BEC of 87Rb atoms, produced by an external oscillating magnetic field [16]. Also, several peculiarities of QT in this system have been investigated. First, the generation of clusters of vortices and anti-vortices when the oscillatory excitation is introduced [17] represents an essential ingredient to reach QT. Second, effects due to the finite size of the sample are related to the existence of a definite boundary region in between the turbulent and non-turbulent regimes [18]. Finally, the different levels of excitation produced in the BEC can be summarized in a diagram of the excitation parameters [19]. This diagram shows the evolution of the number and distribution of vortices nucleated in the condensed sample.
In the next section we give a general introduction to the phenomenon of QT in atomic trapped BECs. In Sect. 14.3, we briefly describe our experimental setup and present our first results on the vortex nucleation in a BEC sample. The evolution from regular vortices to the turbulent state is explained in Sect. 14.4. We then present a simple model to help understanding the role of the BEC finite size in the transition to quantum turbulence. Then, in Sect. 14.5, a very important characteristic of the turbulent cloud is explained: the peculiar dynamics of the system when expands freely. In this section we also present a new analysis of the turbulent state in terms of the Reynolds number of the system. Finally, in Sect. 14.6, we discuss the current directions of our research and introduce our future perspectives.
14.2 Turbulence in Trapped Bose-Einstein Condensates
Superfluid BECs have some important characteristics. Since BEC samples comprise condensed and thermal fractions, these systems must be described by a two-fluid model. BECs must also obey certain constraints imposed by quantum mechanics. For instance, the vortices inside a condensate are quantized, with integer number of fundamental circulation (see, for instance, Ref. [20]). There are many ways to produce a collection of vortices in the condensate, Ref. [21] is a review on this topic. Rotating condensates normally results in a collection of vortices presenting the same circulation and spatially distributed forming a lattice [22], equivalent to the Abrikosov lattice in superconductors [23, 24]. Rotating condensate is not the only way to introduce vortices. A phase imprinting technique recently demonstrated [25] is another alternative.
However the simple presence of vortices in the condensate does not fulfill the conditions to obtain QT. Following the concepts introduced by R. P. Feynman, QT is a phenomenon characterized by a spatial distribution of quantized vortices in a tangled way. Therefore, simple rotation does not take the trapped superfluid to QT. To generate such a tangle configuration, one can imagine the introduction of rotation in two orthogonal axes as suggested by M. Tsubota and co-workers [14]. Another alternative is the introduction of periodic density modulation either by trap oscillations [16] or by oscillating the s-wave scattering length near Feshbach resonances [26]. We have applied in our system the oscillation of the trap by the introduction of an external oscillating magnetic field [16, 27].
14.3 Generation and Proliferation of Vortices
We start with a 87Rb condensate whose experimental setup has been described in details elsewhere [28]. The basic system is composed of a QUIC trap with frequencies given by ω z =ω 0 and ω x =ω y =9ω 0, with ω 0=2π×23 Hz, and produce a condensate with about 2×105 atoms in the |2,2〉 hyperfine state.
Superimposed to the trap coils are two extra coils as shown in Fig. 14.1. The symmetry axis of these coils is slightly tilted with respect to the symmetry axis of the trap itself. Through these coils runs a small sinusoidal current which causes a combination of rotation and translation on the trap bottom. Such motions take place for an oscillation frequency Ω close to 207 Hz, the highest trapping frequency. Amplitude and time of the excitation are the main parameters controlled for the vortices production. A typical time sequence for the experiment is presented in Fig. 14.2. Once the Bose condensate is produced, the excitation by the oscillatory field is applied during a time interval that can range from 0 to 55 ms. After this excitation period, there is a waiting time in the trap followed by a time-of-flight (TOF) measurement using the absorption on a CCD camera.
The result of this excitation is the generation of collective modes in the condensate cloud. A typical sequence of TOF images is reported in Fig. 14.3. A clear composition of dipole, quadrupole and breathing modes is present in the condensate cloud. We believe that such excitations are essential to couple energy and angular momentum into the cloud to further generate other excitations. Besides the collective modes, the long axes of the condensate (after 15 ms of TOF) shows an angular oscillation (scissors mode) due to the superfluid nature of the condensate [13, 29].
As the excitation amplitude increases we start to nucleate vortices in the cloud. As recently reported [30], those vortices are first nucleated at the edge of the cloud where a low density cloud of excited atoms (probably originated from the whole excitation process) surrounds the dense condensate cloud. Figure 14.4 highlights the vortices being nucleated at the edges. It seems that many of the nucleated vortices do not survive to the interior of the cloud. In fact, in Ref. [30] we have presented evidences that vortices and anti-vortices are all produced together, and they undergo dynamical processes after which only few of them survive.
The final number of vortices observed in the sample is strongly dependent on the combination of time and amplitude of excitation. Smaller amplitudes or shorter times produce only the bending mode already discussed. To observe vortices within the cloud, a compromise between amplitude and time is necessary. In Fig. 14.5 (extracted from Ref. [19]) we show the evolution for the average number of vortices as a function of the amplitude of excitation for two different times (17 ms and 33 ms). Figure 14.6 shows pictures for the observed vortices distribution. While for both times, the number strongly increases with the amplitude, longer excitation times take the system to higher number of vortices for a given amplitude. The strong proliferation for the number of vortices soon starts to cover the full sample with dark spots observed in the probe laser description. The peculiar way we excite the sample, through oscillations, results in vortex nucleation in different directions. We believe that this technique is similar to the combination of rotations proposed by Kobayashi and Tsubota [14]. The production of vortex filaments along different directions builds up the necessary ingredients for the evolution of the configuration of vortices to a tangle configuration characterizing QT in the system.
14.4 Observation of Tangled Vortex Configuration
From the graphic of Fig. 14.5, one can imagine that as the energy is pumped into the system part of it is coupled to the superfluid resulting in the vortex formation. Following Ref. [20] the energy of each vortex line within the cloud is given by:
where l 0 is the typical harmonic oscillator length for the trap (\(l_{0} = \sqrt{\frac{\hbar}{m \omega _{ho}}}\)), ω ho =(ω x ω y ω z )1/3, \(\xi =\frac{1}{\sqrt{8 \pi n a_{s}}} \) is the healing length, n is the peak atomic density, and a s is the s-wave scattering length. In writing (14.1), we have considered that the vortex line crosses the cloud diametrically.
When the cloud volume is saturated with vortex lines, the energy to create additional vortices starts to be very high and a turn over in the evolution of the vortices number with increasing amplitude should appear. However a change in overall cloud behavior is observed. Although the additional coupled energy to the cloud is not enough to generate more vortices, it is certainly sufficient to accelerate the dynamics, promoting a fast movement of the vortex lines. The final result is the production of a tangled configuration of vortices, characterizing the turbulent state. Considering the total rate of pumped energy into the cloud as R pump , the energy coupled to the cloud in the form of vortices can be written as
where t is the time of excitation, η is the fraction of energy converted into rotation. The time t 0 corresponds to the mean time to form the first vortex. Elapsed the excitation time t, the number of vortices accumulated must be N vort . In a first approximation where annihilations of vortex-anti-vortex pairs are not considered, one can write:
or
This shall be the number of vortices in the atomic cloud for an excitation time t. A graphic showing the evolution of the observed number of vortices in the cloud as a function of time for different energy pump rates (here produced using different amplitudes of excitation) is presented in Fig. 14.7.
Considering the simple model represented by (14.2) and (14.4) we find that in this experimental conditions t 0≈15 ms. Times shorter than that shall be very inefficient to produce vortices. Considering that the saturation number of vortices in the cloud must be on the order of N vort ≈l 0/ξ, a limiting relation between the pumped energy and the excitation time, before the occurrence of turbulence can be obtained
And since R pump is proportional to the amplitude of excitation (for a fixed excitation frequency), the relation between amplitude and time of excitation to generate QT must be on the type:
The existence of this limiting condition to generate turbulence is a consequence of the finite size of the atomic superfluid [18]. QT develops for the atomic superfluid densely filled with vortices. After this point the energy pumped into the system transforms not only into the formation of vortices but mostly into their motion, with the evolution to a tangle configuration. Such configuration is believed to produce reconnections and the formation of many oscillations into the vortex filaments. Sooner the rotation field is all distributed in the sample. At this point the absorption image becomes hazy, which can be considered as a first manifestation of the presence of turbulence. Figure 14.8 shows a typical transition between a regular cloud, proliferation and QT.
The distributed vortices in our sample do not reveal any regular pattern as reported in other experiments [22]. We strongly believe that this a consequence of the excitation process generating vortex and anti-vortex all together, and covering many spatial directions.
As stated before, the generation of QT is a consequence of a compromise between amplitude and time of the excitation. The diagram of Fig. 14.9 shows this compromise through the existence of domains in the amplitude × time space. This diagram was presented before in Ref. [19] and for the simplicity we have omitted the granulation domain of the diagram which will not be discussed in this paper. Three regimes presented in the diagram of Fig. 14.9 are useful in the understanding of the QT formation in a trapped atomic superfluid and the routes to achieve such regime.
As mentioned before, after reaching QT, increasing even further the excitation time and/or amplitude takes the sample to a distribution where pieces of condensate are spread in space characterizing a configuration named as granulation. There are many questions and many observations related to this regime which we are still considering and must be reported after we reach a conclusive understanding.
14.5 Characteristics Observed on the Turbulent Cloud
One of the fingerprints of QT in a BEC is the change of the expansion behavior during time-of-flight. Quantum degenerate Bose gas when confined by an anisotropic potential will present asymmetric velocity distribution. An excess of kinetic energy is liberated on the most confined direction causing an inversion of the cloud aspect ratio after a certain time of flight. This effect was well investigated by many authors (see Ref. [20]). Typically, for a cigar type condensate cloud, the aspect ratio \(\frac{R_{\rho}(t)}{R_{z}(t)}\) grows with time-of-flight, reaching an asymptotic value which has a strong dependence with the initial aspect ratio for the in situ cloud. For a typical elongated cloud, there will be a faster expansion along the most compressed direction, and vice-versa, which causes an aspect ratio inversion. In contrast, a cloud with only thermal atoms liberates its energy during expansion, in a way that the asymptotic behavior tends to a unitary value for the aspect ratio. The thermal cloud always ends its expansion in an isotropic way.
A quite different behavior is observed for a turbulent cloud during free expansion. The aspect ratio is observed to stay constant from the beginning to the end of the expansion. This complete suppression of the aspect ratio change was well reported in Ref. [16]. We refer to this as a self-similar expansion during the time-of-flight for the turbulent cloud [27]. Similarly to the fact that the inversion ratio is a macroscopic evidence for the condensate, the self-similar expansion seems to be a good evidence for the presence of random distribution of vortex lines within the sample, fact that is associated with QT. Figure 14.10 from Ref. [16] shows the expansion behavior comparing the evolution of the aspect ratio for a thermal cloud, the pure condensate and the turbulent condensate. To explain this observation, we have recently produced a theoretical analysis where the effect of vorticity in the behavior of free expansion was investigated using a hydrodynamic approach [31] in a rotational version. Due to the presence of vortices, we considered a rotational component in the cloud velocity field, such that \(\vert \mathbf {\boldsymbol{\nabla}}\times \mathbf {v} \vert = 2 \varOmega\), where \(\varOmega = \frac {h n_{v}}{2m}\). The vortex density, n v , is derived considering a uniform distribution of a large number of equally oriented vortices [20]. That leads to the modified Euler equation:
with \(g=\frac{4\pi\hbar^{2}a_{s}}{m}\) while a s is the s-wave scattering length. Together with the continuity equation, the free expansion of the cloud was calculated starting with an Ansatz for the density n(r,t) equivalent to the TF profile.
We have demonstrated that the extra kinetic contribution due to the vorticity not only produces a larger initial cloud, but also introduces an extra acceleration for the expansion in the plane perpendicular to Ω. For large values of Ω, this term becomes dominant and the presence of angular momenta in many directions promotes an expansion with insignificant time variation of the aspect ratio. Even being a “toy model” calculation, it provides the main physical insights towards the understanding of the experimental behavior.
As described previously, the oscillatory excitation of the condensate is always accompanied by the excitation of collective modes including the dipole mode. In this mode the condensate cloud as a whole travels inside the potential at the excitation frequency. One could think that such a motion is “like” a flow of the superfluid. Considering that this is truth, one can make a Reynolds type of analysis. For a classical fluid, flowing with velocity v under the influence of a lateral confinement of dimension D, the Reynolds number is defined as \(R_{e}=\frac {D v}{\eta}\) with η the viscosity. This non dimensional quantity indicates if the flow will remain as laminar or if the turbulence will take place. Conventionally, for R e ≈1000 turbulence starts to appear.
For superfluids, the viscosity is replaced by the quantum of circulation h/m, and the quantum Reynolds number is given by \(R^{Q}_{e}=\frac{D v}{h/m}\) [32]. For the oscillating cloud as indicated in Fig. 14.11, the condensate is always embedded in a thermal cloud, since our experiment is done at finite temperature. Considering D=2R TF , i.e., the flow dimension as twice the Thomas-Fermi radius, and v=ωA, where A is the oscillatory amplitude and ω the frequency, we can evaluate \(R^{Q}_{e}\) for many different amplitudes of oscillation, which corresponds to different velocities for the cloud center of mass. The quantification of \(R^{Q}_{e}\) for many excitation conditions shows that for \(R^{Q}_{e}<0.5\) there was no turbulence. On the other hand, for the cases where turbulence was observed, \(R^{Q}_{e}>2\). In the region going from \(0.5<R^{Q}_{e}<2\), the experimental observations shows that in those conditions of oscillation sometimes QT is observed and sometimes not.
In Ref. [32], G.E. Volovik proposes an interesting diagram relating \(R^{Q}_{e}\) with a frictional parameter (q). There, the transition between QT and laminar flow, for q≤1 (low frictional parameter for the sample), takes place for \(R^{Q}_{e} \approx1\). At this point we cannot present yet all the arguments that guarantee the validity of taking the oscillatory motion of the superfluid sample in the potential as a flow experiment. But it certainly produce numbers that are not far from the existing theoretical predictions for flow experiments with 4He-superfluid.
14.6 Present Stage of Investigation
In classical viscous fluids, the change in the viscosity distribution during the development of turbulence represents an intrinsic problem to characterize the turbulent flow. Therefore it is more convenient to determine statistical laws instead of the dynamics of individual variables. In a turbulent regime fully developed in its steady state, the injected energy is transferred to smaller scales without dissipation and the energy spectrum is given by:
known as Kolmogorov Law [33, 34]. The energy spectrum is defined as E(k), such that E=∫d k E(k), where k is the wavenumber originated from the Fourier transform of the velocity field. Kolmogorov law is well verified in classical turbulence. In QT, it was first experimentally demonstrated in 4He [35]. Many other experiments follow this one and much theoretical work was developed in order to understand the similarities between the energy spectrum of classical and quantum turbulence [7, 36–39]. In the case of atomic BECs, a work conducted by M. Tsubota and co-workers [14, 15] has demonstrated theoretically that QT can be developed in a BEC, and the originated energy spectrum is related to the Kolmogorov law. The research in this direction is quite important since it may allow us to directly observe the relation between real-space and reciprocal-space.
To obtain the energy spectrum of our experimental turbulent samples, we are performing measurements and calculations using time-of-flight images of the turbulent cloud obtained by absorption after time-of-flight. A normal cloud in time-of-flight is actually a momentum distribution. Considering n(x,y,z) as the density profile, after a time-of-flight τ, \(x=\frac{hk_{x}}{m}\tau\), \(y=\frac{hk_{y}}{m}\tau\) and \(z=\frac{hk_{z}}{m}\tau\). So the distribution n(x,y,z) turns into ρ(k x ,k y ,k z ) which can be used to determine n(k) such that \(E= \frac{\hbar^{2} k^{2}}{2m}n(k)\). The only problem is that the absorption image provide us with ∫n(x,y,z)dx rather than n(x,y,z). We are therefore processing our images in order to obtain information about the energy spectrum. The preliminary results seem very promising and may allow us to investigate n(k) which, according to the Kolmogorov discussion, is expected to be proportional to k −3.
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Bagnato, V.S. et al. (2013). Characteristics and Perspectives of Quantum Turbulence in Atomic Bose-Einstein Condensates. In: Bramati, A., Modugno, M. (eds) Physics of Quantum Fluids. Springer Series in Solid-State Sciences, vol 177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37569-9_14
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