Visualizing Pure Quantum Turbulence in Superfluid $^{3}$He: Andreev Reflection and its Spectral Properties

Superfluid $^3$He-B in the zero-temperature limit offers a unique means of studying quantum turbulence by the Andreev reflection of quasiparticle excitations by the vortex flow fields. We validate the experimental visualization of turbulence in $^3$He-B by showing the relation between the vortex-line density and the Andreev reflectance of the vortex tangle in the first simulations of the Andreev reflectance by a realistic 3D vortex tangle, and comparing the results with the first experimental measurements able to probe quantum turbulence on length scales smaller than the inter-vortex separation.

Turbulence in pure superfluid might be expected to show statistical properties very different from classical turbulence as there is no dissipative supporting medium to provide a decay mechanism at the small scales. Turbulence in superfluid 3 He can be imaged by the Andreev reflection of ambient thermal quasiparticle excitations, facilitating passive visualization near absolute zero. Here we combine experiments and numerical simulations to reveal the connection between the statistical properties of the turbulence and those of the Andreev reflectance. Classical turbulence is well-known for being simultaneously of universal impact while analytically intractablethe most important unsolved problem of classical physics as Feynman may have expressed it. One way forward is to start with a simpler system. A pure superfluid in the zero-temperature limit has no viscosity and thus can be considered an ideal fluid [1]. While the flow of bulk superfluid must be irrotational it can mimic classical turbulence by supporting singly quantised vortices. At low temperatures, a quantum vortex moves with the local superfluid velocity [2] which arises from the combined velocity fields of all the other vortices [1,3], providing a concrete example of the thin-core vortex filament of the classical fluids literature. The resulting complex flow (a vortex tangle) is known as quantum turbulence.
Despite the absence of frictional dissipation, quantum turbulence in the zero-temperature limit behaves remarkably similarly to classical turbulence [4] and exhibits a Kolmogorov-like energy spectrum [5,6]. Studies of turbulence in superfluid 3 He-B at microkelvin temperatures reveal several advantages over other systems, the most important being that a vortex tangle in this system can be visualized directly via Andreev reflection of ambient thermal excitations. Such non-invasive visualization has already demonstrated that a vortex tangle forms due to collisions of independent vortex rings [7], and allowed the study of the statistical properties of quantum turbulence [8,9].
In this letter we present the first numerical simulations of Andreev reflection by experimentally realistic, three-dimensional vortex tangles in 3 He-B, and contrast them with the latest experimental measurements of pure quantum turbulence, at improved experimental resolution which probes length scales smaller than the av-erage intervortex distance. This combined numericalexperimental approach allows us to understand the relation between the line density of the vortex tangle, L (defined as the total length of vortex lines per unit volume), the quantity which characterizes the intensity of turbulence, and the reflection coefficient of the thermal excitations, which is used to experimentally visualize the turbulence.
Andreev reflection arises in the 3 He-B Fermi superfluid as follows. The BCS dispersion curve E(p) for excitations has a minimum, E min = ∆, at the Fermi momentum p F ; here ∆ is the superfluid energy gap. Thus when an excitation moves from one side of the minimum to the other, the excitation group velocity reverses. Furthermore, in the reference frame of a superfluid moving with velocity v, the dispersion curve tilts by the Galilean transformation to become E(p) + p · v [10]. Thus quasiparticles moving into a region where the superfluid is flowing parallel to their momentum experience a potential barrier. A quasiparticle with insufficient energy to surmount this barrier must be reflected as a quasihole with negligible momentum transfer. Furthermore, most importantly, the outgoing quasihole almost exactly retraces the path of the incoming quasiparticle [11]. Similarly, quasiholes moving into a region of approaching flow will be Andreev-reflected as quasiparticles. Andreev reflection therefore offers an ideal passive probe for observing vortices at very low temperatures and can provide detailed information about the turbulent behavior.
To set the scene, we numerically take a small volume and inject a sequence of vortex rings into it. The rings collide, the cores intersect and recombine, and gradually an approximately homogeneous tangle is produced. We then illuminate the tangle with a beam of excitations and calculate the reflection probability. The simultaneous simulation of the combined evolution of the vortex configuration and of the thermal excitations is complicated and numerically expensive. Luckily, since the timescale of the quasiparticle motion is much shorter than that of the vortex line motion [12], we first obtain from Eqs. (1) shown below the vortex configuration and the associated flow field, v(r, t) at time t, and then analyze the propagation of excitations through this 'frozen' flow field.
The superflow field v(r, t) and the dynamics of the vortex tangle are determined by the coupled equations where the Biot-Savart integral extends over the entire vortex configuration, L, s = s(t) identifies a point on the vortex line, and κ = π /m 3 = 6.62 × 10 −8 m 2 /s is a quantum of circulation in superfluid 3 He with m 3 being the mass of a bare 3 He atom. The superfluid velocity and the time evolution of the vortex tangle are calculated by means of the vortexfilament method with periodic boundary conditions [13]. To reproduce the experimental situation, see e.g. Refs. [7,9], we take a cubic box of size D = 1 mm and numerically simulate the evolution of a vortex tangle generated by vortex loop injection for a period of 380 s. Two rings, radius R i = 240 µm, are injected at opposite corners of the numerical domain [14] at a frequency f i = 10 Hz. To ensure good isotropy, the loop injection plane is switched at both corners at a further slower rate f s = 3.3 Hz. The vortex loops injected into the simulation box collide and recombine, rapidly generating a vortex tangle. After an initial transient of about 5 s, the energy content of the box comes to equilibrium. The losses arise from the numerical spatial resolution (≈6 µm), meaning that small scale structures such as high frequency Kelvin waves are lost (effectively modelling the loss of kinetic energy due to sound radiation at high frequency). The resulting tangle has an equilibrium vortex line density, L = 9.7 × 10 7 m −2 corresponding to an average intervortex separation of ℓ ≈ L −1/2 = 102 µm. The energy spectrum of this tangle is consistent with the k −5/3 Kolmogorov scaling for intermediate wavenumbers, k, and with the k −1 scaling for large k, see the Supplemental Material [14].
To analyze the propagation of excitations, the incident quasiparticle flux, moving in (say) the x-direction, is applied normally to one side of the box. The quasiparticle beam is uniformly distributed in the (y, z)-plane and covers the full cross-section of the experimental "cell". Ignoring angular factors, the incident quasiparticle flux as a function of position (y, z) can be written [9] as where g(E) is the density of states, and f (E) is the Fermi distribution function, approximated by the Boltzmann distribution f (E) = exp(−E/k B T ) at T ≈ 0. Since typical quasiparticle group velocities are larger than typical superflow velocities, the quantity g(E)v g (E) in integral (2) can be replaced by g(p F ), the constant density of momentum states at the Fermi surface g(E F ) [9]. In the flow field of the tangle, a quasiparticle (quasihole) moving against (with) a superfluid velocity v experiences a force dp/dt = −∇(p · v), which pushes it towards the dispersion curve minimum where it becomes a quasihole (quasiparticle) with a reversed group velocity. Consequently, the flux of excitations transmitted through a tangle is determined by the highest superfluid velocity, v max x , encountered along the excitation's rectilinear trajectory determined by constant y and z, and is thus given by: The fraction of quasiparticles Andreev reflected by a tangle along the x-direction at position (y, z) is thus: The total Andreev reflection f x is the sum of the Andreev reflections for all positions of the (y, z)-plane. The equivalent calculation is repeated for the thermal quasihole flux and the result combined with that for quasiparticles to yield the reflection for a full thermal beam. The simulation [15] provides a large volume of information and gives a measure of the Andreev reflection as a 2D contour map across the full cross-section of the input quasiparticle beam, see Fig. 1, showing very graphically the distribution of large scale flows across the cell. Unfortunately, experiments do not provide us with similarly detailed information. Therefore, in order to make comparison between theory and experiment, we concentrate instead on the two most important physical properties: the average Andreev reflection coefficient, f R , and, most illuminating, the fluctuations of f R .
The average calculated reflection coefficient is shown in Fig. 2 (top) as a function of the vortex line density during the evolution of our tangle. For small line densities, L 2 × 10 7 m −2 the reflection coefficient rises quickly and linearly. At this stage of the tangle's evolution rings are virtually non-interacting. As the numerical simulation progresses, more rings enter the computational domain, start to interact, collide, and form a tangle which absorbs all further injected rings. At higher line densities the rise of Andreev reflection coefficient becomes more and more gradual due to screening effects. We use the term 'screening' to identify processes which reduce the overall reflectivity of the tangle for a given line density. There may be several mechanisms responsible for screening and we refer the interested reader to the Supplemental Material [14].
The lower part of Fig. 2 compares the numerical simulations with experimental measurements of Andreev reflection from vortices generated by a grid. Measurements are taken with three vibrating wire detectors, orientated as shown in the inset. The wires #1, #2 and #3 are placed at distances 1.47 mm, 2.37 mm and 3.49 mm from the grid, respectively. The figure shows the fractional reflection of quasiparticles incident on each wire. This is obtained directly from measurements of the thermal damping on the wires [9].
The numerical and experimental data plots in Fig. 2 have similar shapes. The vortex line density L cannot be obtained directly from the measurements but we expect the local line density of the quantum turbulence to increase steadily with increasing grid velocity. However the onset of turbulence is quite different in the numerical and experimental cases. In the numerical simulations, injected vortex pairs are guaranteed to collide and form a tangle. In contrast, the vibrating grid emits only outward-going vortex rings. The flux of emitted rings increases steadily with increasing grid velocity. At low grid velocities, the rings propagate ballistically with few collisions [7,16]. At higher velocities, ring collisions become much more frequent and a vortex tangle is formed. For the experimental data in Fig. 2 this occurs at a grid velocity of about 3 mm/s. The data at smaller velocities corresponds to reflection from ballistic vortex rings and so should be ignored for the current comparison.
At higher grid velocities/tangle densities, the fraction of excitations Andreev reflected increases at an increasing rate, before reaching a plateau. The plateau region is prominent in the experiments, and probably results from the extra quasiparticle creation produced when the grid reaches velocities approaching a third of the Landau critical velocity [17]. Compared to the simulations, the absolute value of the reflectivity is nearly identical for the wire closest to the grid. The excellent agreement between the experiment and the simulations is perhaps coincidental given that in the experiments quasiparticles travel through 1.5-2.5 mm of turbulence to reach the wire, compared with 1 mm in the simulation, thus larger screening would be expected. Aa better comparison will require experiments with high-resolution visualization to measure the variation of tangle density and the effect of extra quasiparticles emitted by the grid.
In our simulation, after the tangle has reached the statistically steady state, the vortex line density and the Andreev reflection coefficient fluctuate around their equilibrium, time-averaged values, L = 9.7 × 10 −7 m −2 and f R = 0.37, respectively. In order to contrast the spectral characteristics of fluctuations of the Andreev reflection δf R (t) = f R (t) − f R and vortex line density δL(t) = L(t) − L we monitor a steady state of simulated turbulent tangle for a period of approximately 380 s or 7500 snapshots. Taking the Fourier transform δf R (f ) of the time signal δf R (t), where f is frequency, we compute the power spectral density | δf R (f )| 2 (PSD or power spectrum for short) of the Andreev reflection fluctuations. Similarly we compute the PSD | δL(f )| 2 of the vortex line density fluctuations. Figure 3 shows the PSD of the Andreev reflections corresponding to the numerical simulation (top, blue) and the experiments (middle and bottom, gray). The experimental data are shown for a fully developed tangle (dark gray) and ballistic vor- tex rings (light gray) relative to the grid velocities of 6.3 mm s −1 and 1.9 mm s −1 respectively. The numerical data shows clearly visible peaks, arising from the discrete process of injection of vortex rings.
The numerically simulated power spectrum of δf R (t) and the experimental data for the developed tangle (reported here and in Ref. [8]) are in excellent agreement and show exactly the same f −5/3 scaling behavior at intermediate frequencies. At high frequencies the experimental data for the tangle develop a much steeper scaling (≈ f −3 ), which is not seen in the numerical spectrum, probably due to the finite numerical resolution. However, this frequency dependence is observed in the experiments in the case where only microscopic vortex rings propagate through the active region and no large scale flows or structures are present. Hence, we argue that the f −3 scaling for the vortex tangles corresponds to the Andreev reflection from the superflow at length scales smaller than the intervortex distance.
By using Taylor's frozen hypothesis, and the fact that at a grid velocity of 6.3 mm s −1 the tangle propagates with a mean velocity of 0.3-0.4 mm s −1 [9], we find that the crossover between the two scaling laws corresponds to a length scale of about 100-200 µm, which is consistent with the intervortex distance obtained from the inferred line density.
Finally, we study the relationship between the fluctuations of the vortex line density, δL(t), and the fluctuations of the Andreev reflection, δf R (t), by computing the normalised cross-correlations where the angle brackets indicate averaging over time, t, in the saturated regime, and τ is the time-lag. The insert of Fig. 4 shows that the cross-correlation between the vortex line density and the Andreev reflection is significant, with F LR (0) ≈ 0.9, clearly demonstrating the link between them, and validating the method of visualization based on Andreev reflection. Figure 4 highlights some differences between the spectral properties of the vortex line density and those of the Andreev reflection, revealed by the numerical simulation. At high frequencies, this spectrum is dominated by the contribution from unpolarised, random vortex lines and exhibits f −5/3 scaling [18]. In the intermediate frequency range, the fluctuation spectrum shows f −3 scaling and is governed by large scale flows that polarise vortex lines, in agreement with recent numerical simulations [18]. Using the saturated value of l = 102 µm, we obtain that the cross-over from the f −3 to the f −5/3 behaviour occurs at the intervortex frequency f ℓ ≈ v/ℓ = κ/(2πℓ) ≈ 1 Hz, in fair agreement with f ℓ ≈ 2 Hz shown in Fig. 4.
We conclude that the Andreev reflectance of a vortex tangle does reveal the nature of quantum turbulence. The f −5/3 scaling law of the frequency spectrum of the Andreev-retroreflected signal was observed earlier by Bradley et al. from their experimental data [8]. This work was supported by the Leverhulme Trust grants F/00 125/AH and F/00 125/AD, and the EP-SRC grant EP/I028285/1. His co-authors would like to dedicate this manuscript to Shaun Fisher who discovered quantum turbulence in 3 He and drove this work, but has died suddenly and unexpectedly.

(3) SCREENING MECHANISMS IN ANDREEV REFLECTION FROM VORTEX CONFIGURATIONS AND TANGLES
The analysis of Andreev reflection from an isolated vortex can be found in Ref. [1]. This has been developed further in Ref. [2] to calculate the corresponding scattering length.
The Andreev reflection from a single, rectilinear vortex line is determined by the 1/r behavior of the superflow field, where r is the distance from the vortex core. For dilute vortex configurations and tangles the total Andreev reflection is just a sum of reflections from individual vortices. However, as the line density increases this is no longer the case due to screening effects. Here by screening we mean processes which reduce the total Andreev reflectivity of the tangle.  There are two main mechanisms of screening. The first, which we called 'partial screening' in our earlier work [3][4][5], is due to a modification of the 1/r velocity field of the vortex caused, in particular, by neighbouring vortices in its close vicinity. For example, the flow field at some distance from two close, (nearly) antiparallel vortex lines is that of a vortex dipole, v ∼ 1/r 2 ; this, faster decay with r of the superflow velocity significantly reduces the total reflectivity of the vortex pair. The 1/r velocity field may also be modified by the effects of large curvature of the vortex line itself; thus, a reduction of the Andreev reflection from small vortex rings was reported in Ref. [5]. It can also be expected that the reflectivity of the vortex line may be reduced by the large-amplitude Kelvin waves.
The other mechanism is that of 'fractional' screening. This mechanism is of particular importance for large/dense tangles: vortices at the front (with respect to the incident beam of excitations) of the dense or/and large tangle will obscure those at the rear. A simple model [6] of this screening mechanism was developed to yield a rough estimate for the vortex line density from the experimentally measured Andreev reflection (this mechanism was also called 'geometric' screening in Ref. [7]). This screening mechanism will also be important in the case where, in dense tangles, vortex bundles [8] may form producing high reflectivity regions surrounded by those whose reflectivity is much lower. This will generate a significant 'geometric' screening since few excitations will reach vortices at the rear of a large vortex bundle, and adding a vortex to a region where there is already an intense reflection will not have a significant effect on the total reflectivity.