Visualization of quantum turbulence in superfluid 3He-B: Combinednumerical/experimental study of Andreev reflection

V. Tsepelin,1, ∗ A.W. Baggaley,2 Y.A. Sergeev,2, † C.F. Barenghi,2

S.N. Fisher,1, ‡ G.R. Pickett,1 M.J. Jackson,3 and N. Suramlishvili4

1Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom,2Joint Quantum Centre Durham-Newcastle, and School of Mathematics, Statistics and Physics,

Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom,3Faculty of Mathematics and Physics, Charles University, Karlovu 3, 121 16, Prague 2, Czech Republic,

4Department of Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom.(Dated: July 19, 2017)

We present a combined numerical and experimental study of Andreev scattering from quantumturbulence in superfluid 3He-B at ultralow temperatures. We simulate the evolution of modera-tely dense, three-dimensional, quasiclassical vortex tangles and the Andreev reflection of thermalquasiparticle excitations by these tangles. This numerical simulation enables us to generate the two-dimensional map of local Andreev reflections for excitations incident on one of the faces of a cubiccomputational domain, and to calculate the total coefficient of Andreev reflection as a function ofthe vortex line density. Our numerical simulation is then compared with the experimental measure-ments probing quantum turbulence generated by a vibrating grid. We also address the question ofwhether the quasiclassical and ultraquantum regimes of quantum turbulence can be distinguishedby their respective total Andreev reflectivities. We discuss the screening mechanisms which maystrongly affect the total Andreev reflectivity of dense vortex tangles. Finally, we present combi-ned numerical-experimental results for fluctuations of the Andreev reflection from a quasiclassicalturbulent tangle and demonstrate that the spectral properties of the Andreev reflection reveal thenature and properties of quantum turbulence.

PACS numbers: 67.40.Vs, 67.30.em, 67.30.hb, 67.30.he

I. INTRODUCTION

A pure superfluid in the zero temperature limit has noviscosity. The superfluid is formed by atoms which havecondensed into the ground state. The condensate atomsbehave collectively and are described by a coherent ma-croscopic wavefunction. In liquid 3He and dilute Fermigases, the condensate is formed by pairs of atoms knownas Cooper pairs. For most superfluids, including 3He-B,the superfluid velocity v is proportional to the gradientof the phase of the wavefunction, ∇θ. Thus the super-fluid flow is irrotational ∇×v = 0 (more complex super-fluids such as 3He-A are not considered here). Superfluidscan however support line defects on which ∇×v is singu-lar. These defects are quantum vortices. The phase θ ofthe wavefunction changes by 2π around the vortex coreand gives rise to a circulating flow. Each vortex carriesa single quantum of circulation κ = 2πh/m where m isthe mass of the constituent atoms or atom-pairs (for su-perfluid 3He-B, κ = πh/2m3 = 0.662× 10−7 m2/s, wherem3 is the mass of a 3He atom).

At low temperatures, quantum vortices move with thelocal superfluid velocity1. Instabilities and reconnectionsof quantized vortices result in the formation of a vortextangle which displays complex dynamics, as each vortexmoves according to the collective velocity field of all ot-her vortices2,3. This complex, disordered flow is known asquantum turbulence. A property that characterizes theintensity of quantum turbulence is the vortex line den-sity, L (m−2), that is, the total length of vortex lines per

unit volume. At higher temperatures the thermal excita-tions form a normal fluid component which may coupleto the superfluid via mutual friction. The resulting beha-vior is quite complex, especially in 4He where the normalfluid may also become turbulent. Here we only consi-der the low temperature regime in 3He-B, T <∼ 0.17Tc,where Tc ≈ 0.9mK, is the critical temperature of su-perfluid transition at zero pressure. In this temperatureregime, thermal excitations no longer form the normalfluid, but a few remaining excitations, whose mean freepaths greatly exceed the typical size of the experimentalcell, form a ballistic gas which has no influence on thevortex dynamics4.

Even in this low temperature case, the properties ofquantum turbulence are not yet fully understood; besi-des, it is interesting to compare the properties of purequantum turbulence with those of classical turbulence.Understanding their relationship may eventually lead toa better understanding of turbulence in general.

During the last two decades there has been signifi-cant progress in the experimental studies of quantumturbulence. Experimental techniques are being deve-loped to measure properties of quantum turbulence atlow temperatures, both in 4He (see e.g. Refs.5–11) andin 3He-B12–16. The most promising technique which isbeing developed for superfluid 3He-B utilizes the An-dreev reflection of quasiparticle excitations from super-fluid flow17. (This technique was first developed to studyquantum turbulence in stationary samples12–14, and hassince been extended to rotating samples16.)

2

Andreev reflection arises in a Fermi superfluid as fol-lows. The energy-momentum dispersion curve E(p) forexcitations has a minimum at the Fermi momentum,pF , corresponding to the Cooper pair binding energy ∆.Quasiparticle excitations have |p| > pF whereas qua-siholes have |p| < pF . On moving from one side ofthe minimum to the other, the excitation’s group velo-city reverses: quasiholes and quasiparticles with similarmomenta travel in opposite directions. In the referenceframe of a superfluid moving with velocity v, the disper-sion curve tilts17 to become E(p) + p · v. Quasiparticleswhich move into a region where the superfluid is flowingalong their momentum direction will experience a poten-tial barrier. Quasiparticles with insufficient energy arereflected as quasiholes. The momentum transfer is verysmall so there is very little force exerted whilst the out-going quasiholes almost exactly retrace the path of theincoming quasiparticles18. Andreev reflection thereforeoffers an ideal passive probe for observing vortices at verylow temperatures.

Numerical simulations of Andreev reflection for two-dimensional vortex configurations have shown that thetotal reflection from a configuration of vortices may besignificantly less than the sum of reflections from in-dividual vortices19,20. This phenomenon is called par-tial screening. The Andreev scattering cross-sections ofthree-dimensional isolated vortex rings and the role ofscreening effects from various configurations of vortexrings in 3He-B were studied in Ref.21.

In this work we analyze numerically the Andreev re-flection of quasiparticles by three-dimensional vortextangles in 3He-B. We use moderate tangle densities whichare comparable to those typically observed in the expe-riments. The plan of the paper is as follows. In Sec. IIwe introduce the basic equations governing the dynamicsof the vortex tangle and the associated flow field, theequations governing the propagation of thermal quasi-particle excitations, and the method for the calculationof the Andreev-reflected fraction of quasiparticle excitati-ons incident on the vortex tangle. In Sec. III we describeour method for the numerical generation of vortex tang-les and discuss their properties. In Sec. IV we presentour results for the simulation of the Andreev reflectionfrom saturated, quasiclassical vortex tangles and analyzethe dependence of the total reflection coefficient and thereflection scattering length on the vortex line density. Insubsection IVC we describe the experimental techniquesand the results for the experimental measurements of theAndreev reflection from quantum turbulence for compa-rison with our simulations.

In Sec. V we compare these results with those obtai-ned for the Andreev reflection from unstructured, ‘ultra-quantum’ turbulence. In Sec. VI we discuss the mecha-nisms of screening in the Andreev reflection from vortextangles. In Sec. VII we present our numerical results forthe spectral properties of the Andreev reflection and ofthe vortex line density, analyze the correlation betweenthese two statistical properties and compare them with

the statistical properties of the experimentally-observedAndreev-retroreflected signal. We discuss our findingsand their interpretation in Sec. VIII.

II. DYNAMICS OF THE VORTEX TANGLEAND PROPAGATION OF THERMAL

QUASIPARTICLE EXCITATIONS: GOVERNINGEQUATIONS AND NUMERICAL METHODS

Since our spatial resolution scale is much larger thanthe coherence length, we will model the vortex filamentsas infinitesimally thin vortex lines. At low temperatures,T <∼ 0.17Tc such that mutual friction can be ignored, thesuperflow field v(r, t) and the dynamics of the vortextangle are governed by the coupled equations22

v(s, t) = − κ

4π

∮L

(s− r)

|s− r|3× dr ,

ds

dt= v(s, t) , (1)

where the Biot-Savart line integral extends over the en-tire vortex configuration L, with s = s(t) identifying apoint on a vortex line. In the first equation, the singula-rity at r = s is removed in a standard way23. The second,kinematic equation states that the vortex line moves withthe local superfluid velocity. This is valid at low tempe-ratures where mutual friction is absent3. Our numericalsimulation of Eqs. (1) is made in a periodic cubic box forvortex tangles of line densities up to L ∼ 108 m−2, seeSec. III for detail.In Ref.18, based on the approach developed by Grea-

ves and Leggett24, we derived the following semiclassi-cal Hamiltonian equations governing the propagation ofthermal quasiparticle excitations in 3He-B:

dr

dt=

ϵp√ϵp2 +∆2

p

m∗ + v,dp

dt= − ∂

∂r(p · v) , (2)

where r(t) represents the position of a quasiparticle exci-tation taken as a compact object, p(t) is the momen-tum of excitation, ϵp = p2/(2m∗) − ϵF is the “kine-tic” energy of a thermal excitation relative to the Fermienergy, m∗ ≈ 3.01m3 is the effective mass of an excita-tion, with m3 being the bare mass of a 3He atom, and ∆is the superfluid energy gap. Equations (2) were derivedby considering the propagation of excitations at distan-ces from the vortex core large compared to the coherencelength (at zero temperature and pressure the coherencelength is ξ0 ≈ 50 nm).In principle, interactions of thermal excitations with

the vortex tangle can be analyzed by simultaneously sol-ving Eqs. (2), governing the propagation of quasiparti-cles, and Eqs. (1), governing the dynamics and the su-perflow field of the tangle. This is exactly how someof the authors of this paper earlier analyzed the An-dreev reflection from two-dimensional vortex configurati-ons19,20 and three-dimensional quantized vortex rings21.

3

However, Eqs. (2) are stiff, and solving them simultane-ously with Eqs. (1) for saturated three-dimensional vor-tex tangles requires considerable computing time and re-sources. For tangles of moderate line densities (up toL ∼ 108 m−2) considered in this work, an alternative,much simpler method for evaluating the flux of quasipar-ticles Andreev-reflected by the tangle can be developedbased on the properties18 of solution of Eqs. (2) and onthe arguments25,26 formulated below.

The alternative method makes use of the facts18; first,that at moderate tangle densities, the Andreev reflectedexcitation practically retraces the trajectory of the inci-dent quasiparticle, and secondly that the time scale ofpropagation of the quasiparticle excitation within thecomputational box (of 1 mm size in our simulations)is much shorter than the characteristic timescale, τf ∼1/(κL) of the fluid motion in the tangle. In analyzingthe quasiparticle trajectories, these results allow us toassume a ‘frozen’ configuration of vortices and the asso-ciated flow field.

Assuming that the statistically uniform and isotropicvortex tangle occupies the cubic computational box ofsizeD, consider the propagation of a flux of quasiparticlesat normal incidence on one side of the box in (say) thex-direction. Also assume that the quasiparticle flux isuniformly distributed over the (y, z) plane and coversthe full cross section of the computational box.

Ignoring angular factors, the incident flux of quasipar-ticles, as a function of the position (y, z), can be writtenas25

⟨nvg⟩i(y,z) =∫ ∞

∆

g(E)f(E)vg(E)dE , (3)

where g(E) is the density of states, and f(E) is theFermi distribution function which we can approximate atlow temperatures to the Boltzmann distribution, f(E) =exp(−E/kBT ). The average energy of a quasiparticle inthis flux is ⟨E⟩ = ∆+ kBT .

Since the typical quasiparticle group velocity is largecompared to a typical superfluid flow velocity v, the in-tegral is simplified by noting that g(E)vg(E) = g(pF ) isthe constant density of momentum states at the Fermisurface. Equation (3) therefore reduces to

⟨nvg⟩iy,z = g(pF )

∫ ∞

∆

exp(−E/kBT ) dE

= g(pF )kBT exp(−∆/kBT ) .

(4)

In the presence of a tangle, and hence regions of va-rying superfluid velocity v, a quasiparticle moving intoa region where the superfluid is flowing along their mo-mentum direction experiences a force p = −∇(p · v),which decreases its momentum towards pF and reducesits group velocity. If the flow is sufficiently large thequasiparticle is pushed around the minimum, becominga quasihole with a reversed group velocity. Consequently,the flux of quasiparticles transmitted through a tangle,⟨nvg⟩t is calculated by the integral (4) in which the lower

limit is replaced by ∆+max(p ·v) ≈ ∆+ pF vmaxx , where

vmaxx is the maximum value of the x-component of super-fluid velocity, found from the solution of Eqs. (1), alongthe quasiparticle’s rectilinear trajectory [that is, for fixed(y, z)]:

⟨nvg⟩ty,z = g(pF )

∫ ∞

∆+pF vmaxx

exp(−E/kBT ) dE

= g(pF )kBT exp[−(∆ + pF vmaxx )/kBT ] .

(5)

For the position (y, z) on the face of computationalbox, the Andreev reflected fraction of quasiparticles in-cident on a tangle along the x-direction is, therefore,

fy,z = 1−⟨nvg⟩ty,z⟨nvg⟩iy,z

= 1− exp

[−pF v

maxx

kBT

]. (6)

The total Andreev reflection fx is the average of the An-dreev reflections for all positions of the (y, z) plane. Thesame calculation is repeated for the flux of quasiholes.The results are then combined to yield the reflection coef-ficient for the full beam of thermal excitations.In our simulation, the initial positions of quasiparticle

excitations on the (y, z) plane were uniformly distribu-ted with spatial resolution ≈ 6× 6µm2. More details ofnumerical simulation will be given below in Sec. III.In the following sections, to directly compare our simu-

lations with experimental observations, we fix the tempe-rature at T = 0.15Tc and use parameters appropriate to3He-B at low pressure: Fermi energy ϵF = 2.27×10−23 J,Fermi momentum pF = 8.28× 10−25 kgm s−1, superfluidenergy gap ∆0 = 1.76kBTc, and quasiparticle effectivemass m∗ ≈ 3.01×m3 = 1.51× 10−26 kg.

III. NUMERICAL GENERATION ANDPROPERTIES OF VORTEX TANGLES

Motivated by experimental studies27, we numericallysimulate the evolution of a vortex tangle driven by vortexloop injection26. Two rings of radius Ri = 240µm areinjected at opposite corners of the numerical domain (seeFig. 1(A), where the two injected vortex rings, markedin red, can be seen entering from the left and right sidesof the box repeatedly with a frequency fi = 10Hz. Thevortex loops injected into the simulation box collide andreconnect to rapidly generate a vortex tangle. The timeevolution of the vortex tangle and the superfluid velocityv(r, t) were simulated by means of the vortex filamentmethod, that is by solving Eqs. (1), in a cubic box of sizeD = 1mm with periodic boundary conditions23. (As canbe seen from Eq. (6), the accuracy of simulation of thesuperflow field is crucially important for the calculationof the Andreev reflection.) To ensure good isotropy, theplane of injection of the loops is switched at both cornerswith a frequency of fs = 3.3Hz. We integrate Eqs. (1)for a period of 380 s.For dense vortex tangles, we use the tree-method with

a critical opening angle of 0.4 rad23. For small line

4

40

Time t [s]

0

0.2

0.4

0.6

0.8

1V

ort

ex lin

e den

sity

L [

10

8 m-2

]

3020100

A) B)

FIG. 1: Evolution of the vortex line density of the tanglegenerated by injection of vortex rings. Insets (A) and (B)demonstrate snapshots of the tangle at the moment of ringinjection from the opposite corners of the computational dom-ain for initial and steady state of simulation correspondingly(see text for details).

densities, of the order of few mm−2 or less, since thetree-method loses accuracy, calculations were performedinstead by the direct evaluation of the Biot-Savart inte-grals. The evolution of the vortex line density for thefirst 40 s of the simulation is shown in Fig. 1.

After an initial transient, lasting for about 5 s, theenergy injected into the box is balanced by the numeri-cal dissipation, which arises from the spatial resolution(≈6 µm) of the numerical routine (small scale structuressuch as small vortex loops and high frequency Kelvin wa-ves of wavelength shorter than this spatial resolution aretherefore lost), and the tangle saturates to reach the equi-librium state. In this statistically steady state the vortexline density, L(t), fluctuates around its equilibrium value⟨L⟩ = 9.7 × 107 m−2 which corresponds to the charac-teristic intervortex distance of ℓ ≈ ⟨L⟩−1/2 = 102 µm.A snapshot of this tangle is shown in the inset (B) ofFig. 1. In Fig. 2 we show the energy spectrum, E(k)of this tangle (here k is the wavenumber). The energyspectrum, representing the distribution of kinetic energyover the length scales, is an important property of homo-geneous isotropic turbulence. In the statistically steadystate the spectrum is consistent, at moderately large sca-les (kD < k < kℓ, where kD = 2π/D ≈ 6× 103 m−1 andkℓ = 2π/ℓ ≈ 6 × 104 m−1), with the Kolmogorov −5/3spectrum shown in Fig. 2 by the red dashed line. In clas-sical turbulence the Kolmogorov −5/3 scaling is the sig-nature of a ‘cascade’ mechanism which transfers energyfrom large length scales to small length scales. The bluedotted line shows the k−1 power spectrum expected forsmall scales (k > kℓ) arising from the superfluid flowaround individual vortex cores. Fig. 2 also shows thatmost of the energy of the turbulence is at the smallestwavenumbers (the largest length scales). This property iscaused by partial polarization of the vortex tangle: paral-

103

104

105

106

10-6

10-5

10-4

Wavenumber k [m-1]

k-5/3

k-1

Ener

gy s

pec

trum

E(k

) [a

. u.]

FIG. 2: The energy spectrum26, E(k) of the steady statevortex tangle vs the wavenumber k. The slope of the spectrumis consistent with the k−5/3 Kolmogorov scaling (red dashedline) for small wavenumbers and k−1 scaling (blue dotted line)for large wavenumbers.

lel vortex lines locally come together, creating metastablevortex bundles, whose net rotation represents relativelyintense large-scale flows which are responsible28 for thepile-up of energy near k ≈ kD.

IV. ANDREEV REFLECTION FROM AQUASICLASSICAL TANGLE

A. Two-dimensional map of local Andreevreflections

We start with illustrating the numerical calculation,based on the method described above in Sec. II, of theAndreev reflection from an isolated, rectilinear vortexshown in Fig. 3(A). The vortex’s circulating flow velo-city is v(r) = κ/(2πr), where r is the distance from thevortex core (for theoretical analysis and analytical cal-culation of Andreev reflection from the rectilinear vortexsee Refs.18,25,29).The velocity field around the vortex was computed

using Biot-Savart integrals. To ensure the homogeneityof the velocity flow profile across the simulation volume,the actual vortex extends another four lengths above andbelow the cube shown in the figure.We calculate the Andreev reflection of a beam of exci-

tations which propagate in the y-direction orthogonallyto the (x, z) face of the computational cube (the blackarrow shows direction of the incoming excitation). Thiscalculation gives a measure of the Andreev reflection as atwo-dimensional contour map across the full cross sectionof the incident quasiparticle beam. In Fig. 3, panels (B)and (C) show, respectively, the reflection of quasiparti-cles (p > pF ) and quasiholes (p < pF ) from the rectili-near, quantized vortex that has a direction of vorticitypointing up. The darker (red/yellow) regions correspon-

5

−0.50

0.5 −0.50

0.5

−0.5

0

0.550%

0%

A)B)

C) D)

x [mm]y [mm]

z [m

m]

x

x x

z

zz

FIG. 3: Representation of the reflection coefficient for exci-tations incident on the flow field of an isolated, rectilinearvortex line. (A) Velocity flow field around a rectilinear vor-tex. The red arrow shows the direction of vorticity. The blackarrow indicated the direction of injection of quasiparticles andquasiholes. (B) and (C) show the quasiparticle and quasiholereflection, respectively. (D) The total reflection of incomingexcitation flux. Darker (red/yellow) regions correspond to thehigher reflection coefficient.

ding to the higher reflectivity clearly show the “Andreevshadows” and are located symmetrically on the immedi-ate right of the vortex line for quasiparticles and on theimmediate left for quasiholes. Note also that for a ‘real’thermal beam consisting of excitations of both sorts theAndreev shadow of a single vortex should be symmetricwith respect to the vortex line and can be imagined asa superposition of panels (B) and (C). The panel (D) il-lustrates the total Andreev reflection from the rectilinearvortex.

We use the same method of evaluating Andreev re-flection for our simulated tangle. The velocity field ofthe vortex tangle was computed using a slightly modifiedtree-method with a critical opening angle of 0.2, compa-red to the previously published result26 where a criti-cal opening angle of 0.4 was used. The modified tree-method algorithm continues propagation along the treewhen the desired critical opening angle is reached butthe local tangle polarization (defined in Refs.30,31) ex-ceeds 3.8. The current simulation shows a slightly betteragreement with the direct Biot-Savart calculation, butoverall results obtained using the modified tree-methodreplicate previous conclusions.

Figure 4 shows a distribution of the local Andreev re-flection coefficient fx,z [calculated using Eq. (6)] on the(x, z) plane for a tangle with the saturated equilibrium

0%

100%

A B

FIG. 4: Two-dimensional representation of the reflection coef-ficient for excitations incident on the (x, z) plane of the com-putational cubic domain. The vortex cores are visible as thindark lines. (A): reflection of quasiparticles (p > pF ); the ex-tended darker and brighter regions correspond to the largescale flow into and out of the page, respectively. (B): re-flection of quasiholes (p < pF ); darker and brighter regionsindicate the large scale flow in the directions opposite to thosein panel (A).

line density ⟨L⟩ = 9.7 × 107 m−2 (Fig.1 inset (B)). Thedarker (red/yellow) regions correspond to the higher An-dreev reflection; clearly visible, very dark, thin “threads”correspond to a high/complete Andreev reflection whichis expected in the vicinity of vortex cores. The over-all distribution of dark threads corresponding to vortexcores is roughly uniform and indicates that our simu-lated turbulent tangle is practically homogeneous. Pa-nel (A) illustrates the reflection of quasiparticle excita-tions with p > pF , and (B) shows the reflection of qua-siholes (p < pF ). The shadows of large vortex rings in-stantaneously injected into the computational domain tosustain the statistically steady state are also visible.The individual maps of Andreev reflection for two spe-

cies of excitations [quasiparticles in Fig. 4 (A) and quasi-holes in Fig. 4 (B)] show the presence of large scale flow,consistent with the quasiclassical (Kolmogorov) charac-ter of the simulated vortex tangle whose energy spectrumscales as k−5/3 in the “inertial” range of moderately smallwavenumbers, see Fig. 2. The extended darker regionsin Fig. 4 (A) correspond to large scale flows whose y-component of velocity is directed into the page; the brig-hter regions show large scale flows directed out of thepage. For quasiholes reflection shown in panel (B) themeaning of darker and brighter regions is opposite tothat of the panel (A). Consequent direct calculation ofthe superflow patterns within the simulation volume hasconfirmed the presence of large scale flows in the corre-sponding regions.We should emphasize here that in the numerical ex-

periment described above the direction of the large scaleflow could be determined only because the separate si-mulations, for the same tangle, were performed for twodifferent species of excitations, that is for quasiparticleswith p > pF and quasiholes with p < pF . In the realphysical experiment the thermal beam consists of exci-tations of both sorts whose separation is, of course, not

6

possible. For such a beam the Andreev reflection shouldbe the same both for receding and approaching flows.While the presence of large scale flow may still be de-tected by the enhancement of the Andreev reflection, anidentification of its direction would hardly be possible.

A two-dimensional representation of the reflectioncoefficient for the full beam of excitations Andreev-reflected by the simulated tangle is illustrated below inFig. 5 by the insets (A) and (B) for the initial and steadystate of simulation, respectively.

B. Total Andreev reflection vs vortex line density

We now turn to the results of our calculation of thetotal Andreev reflection from the vortex tangle. Our aimis to find the total reflection coefficient, fR, that is, thefraction of incident excitations Andreev-reflected by thetangle, as a function of the vortex line density. Herethe total reflection coefficient is defined as fR = (fx +fy + fz)/3, where fx, fy, and fz are the total reflectioncoefficients of the excitation beams incident, respectively,in the x, y, and z directions on the (y, z), (x, z), and(x, y) planes of the cubic computational domain.For the tangle numerically generated by injected rings

of radius Ri = 240µm with the injection and switchingfrequencies fi = 10Hz and fi = 3.3Hz, respectively (seeabove Sec. III for details), we monitor simultaneously theevolution of the vortex line density (see Fig. 1) and of thereflection coefficient calculated using the method descri-bed in Sec. II [see Eq. (6) and the text that follows] duringthe transient period32 and for some time after the tanglehas saturated at the statistically steady-state, equili-brium value of the vortex line density, ⟨L⟩ = 9.7 × 107

m−2, see Fig. 5. After a very short initial period of less

Time t [s]

0

0.1

0.2

0.3

0.4

Andre

ev r

efle

ctio

n c

oef

ficie

nt

f R

403020100

A) B)

FIG. 5: Evolution of the total Andreev reflection for thetangle generated by injection of vortex rings (see text fordetails). Insets (A) and (B) show a two-dimensional repre-sentation for the full beam of excitations for the initial andsteady state of simulation, respectively.

Andre

ev r

efle

ctio

n c

oef

ficie

nt

f R

0.2

0.1

0.3

0.4

0.5

00

Grid velocity amplitude [mm s-1]

wire #1

wire #2

wire #3

4.3bar, 0.17 Tc

1 2 3 4 5 6 7 8 9

Onset ofQuantumTurbulence

#2 #1#3

Grid

0 0.2 0.4 0.6 0.8 1

Vortex line density L [108 m

-2]

0

0.1

0.2

0.3

0.4

Andre

ev r

efle

ctio

n c

oef

ficie

nt

f R

FIG. 6: Top: the total reflection coefficient obtained from thenumerical simulations, plotted against the line density of thevortex tangle. Bottom: experimental measurements of thefraction of Andreev scattering from vortices generated by agrid plotted versus grid’s velocity26. Measurements are shownfor three vibrating wires at different distances from the gridas shown in the inset, see text.

than 1 s, the vortex configuration becomes, and remainsduring the transient and afterwards practically homoge-neous and isotropic. This enables us to re-interpret theresults shown in Figs. 1 and 5 in the form of the Andreev-reflected fraction, fR as a function of the vortex line den-sity. This function33 is shown by the top panel of Fig. 6.

As can be seen from Fig. 6, at small line densities,L <∼ 2 × 107 m−2 the reflection coefficient rises rapidlyand linearly with L. During the initial stage of the ge-neration of the tangle, the injected rings form a dilutegas and are virtually non-interacting. As the computa-tion time advances, more rings enter the simulation box,start to interact, collide, reconnect, and eventually forma tangle which absorbs all injected rings. However, atsmall times, even after the tangle has been formed, thevortex line density remains relatively small so that theaverage distance between vortex lines is relatively large.In this case the flow around each vortex is practicallythe same as the flow field of a single, isolated vortex34.

7

In a sufficiently dilute vortex configuration a screeningof Andreev reflection (discussed below in Sec. VI) canalso be neglected so that the total reflection is a sum ofreflections from individual vortex lines; this explains alinear growth with L of the reflection coefficient at lowline densities.

C. Comparison with experiment

We compare the simulation with the experiment25,27

which has the configuration similar to that shown inthe inset of the lower panel of Fig. 6. The vorticity isproduced by the oscillating grid and surrounds the vi-brating wire detectors. The tangle’s flow field Andreev-reflects ambient thermal excitations arriving from “infi-nity”, shielding the wires and reducing the damping. Thereduction in damping on each wire (placed at 1.5mm,2.4mm and 3.5mm from the grid) provides the measureof the Andreev reflection by the tangle, see Ref.25,26. Thelower panel of Fig. 6 shows the fractional reflection ofquasiparticles incident on each wire.

The numerical and experimental data plots in Fig. 6have similar shapes. While the vortex line density L can-not be obtained directly from the measurements, we ex-pect the local line density of the quantum turbulence toincrease steadily with increasing grid velocity. However,the onset of turbulence is rather different in the simula-tions and in the experiment. In the simulations, approa-ching injected vortex ring pairs are guaranteed to collideand form a tangle, whereas the vibrating grid emits onlyoutward-going vortex rings, with the ring flux increasingsteadily with increasing grid velocity. At low grid veloci-ties, rings propagate ballistically with few collisions13,35.At higher velocities, ring collisions increase giving rise tothe vortex tangle. For the data of Fig. 6, this occurs whenthe grid velocity exceeds ∼ 3mm/s. The data for lowervelocities correspond to reflection from ballistic vortexrings and can be ignored for the current comparison.

At higher grid velocities/tangle densities, the fractionof Andreev-reflected excitations rises at an increasingrate, finally reaching a plateau. The plateau region isprominent in the experiments when the grid reaches velo-cities approaching a third of the Landau critical velo-city36,37 and may be an artifact of data analysis modelthat ignores a local flux of quasiparticles produced bypair-breaking.

For the wire closest to the grid, the experimentallyobserved absolute value of the reflectivity is almost iden-tical to that obtained from our simulation. This excellentagreement is perhaps fortuitous given that in the experi-ments quasiparticles travel through 1.5–2.5mm of turbu-lence to reach the wire, compared with 1mm in the simu-lation, thus larger screening might be expected. A bettercomparison will require high-resolution experiments toseparate the effects caused by variations of tangle den-sity from those caused by the increase of quasiparticleemission by the grid.

D. The Andreev scattering length

To compare the amount of Andreev scattering fromvortex tangles with that of isolated vortices, it might beuseful to define an Andreev scattering length. In Ref.21

the cross section, σ, for Andreev scattering was definedas

σ =N

⟨nvg⟩, (7)

where N is the number of excitations reflected per unittime, and ⟨nvg⟩ is the number flux of incident excitations.From the cross section σ we define the Andreev scatteringlength, b, as

σ = bL, (8)

where L is the total length of vortex line within the com-putational box or the experimental cell. For the simu-lation box with side length D, L = LD3. From defini-tion (7) of the cross section, the total reflectivity of avortex configuration is f = σ/A, where A is the crosssectional area of the incident excitation beam. For oursimulations, the Andreev scattering length is thus givenby

b = f/(DL) . (9)

The Andreev scattering length in the vortex tangles,obtained from Eq. (9) and our numerical simulation, isplotted in Fig. 7. For the isolated, rectilinear quanti-zed vortex line, the theoretical analysis and analyticalcalculation of the Andreev scattering length were givenin Ref.29. A simpler estimate for the scale of Andreevscattering length, used for the initial interpretation ofexperiments12–14,27, can be obtained as follows. Consi-der quasiparticle excitations incident on a straight vortexline perpendicular to the excitation flux. On one side of

0 0.2 0.4 0.6 0.8 1

4

6

8

10

12

Andre

ev s

catt

erin

g le

ngt

h b

[ m

]

Vortex line density L [108 m

-2]

FIG. 7: Andreev scattering length, b, vs vortex line density,L.

8

the vortex, an incident quasiparticle will experience apotential barrier pF v = h/(2m3r), where r is the impactparameter. The average thermal quasiparticle energy is∆ + kBT . An approximate value for the Andreev scat-tering length b is the impact parameter b0 at which anexcitation with the average energy is Andreev reflected.Equating the potential barrier with kBT yields

b0 =pF h

2m3kBT. (10)

For the parameters used in the simulations, appropriateto 3He-B at low pressure and T = 150 µK, estimate (10)gives b0 = 4.2 µm. This value is in a very good agreementwith the Andreev scattering length of the vortex tangle atthe highest line densities shown in Fig. 7. This validatesthe analysis previously used in Refs.12–14,27 to interpretthe experimental data for 3He tangles.We note, however, that while yielding the correct esti-

mate for the scale (10) of the Andreev scattering length,this simple model might be somewhat misleading since,even for an isolated vortex, a significant amount of An-dreev reflection occurs at distances larger than b0 fromthe vortex core, see Ref.29 for details.

For dilute tangles, the Andreev scattering length isconsistent with the value for large vortex rings, brings =10 µm, inferred from the results of numerical simulationsof Ref.21. It is clear that Andreev scattering from verydilute vortex tangles is dominated by reflection from in-dividual vortices.

V. TOTAL REFLECTION FROM THEUNSTRUCTURED VORTEX TANGLE

In the previous section we investigated Andreev re-flection from quasiclassical quantum turbulence whoseenergy spectrum obeys the Kolmogorov−5/3 scaling law.However, there is also evidence for the existence of a se-cond quantum turbulence regime, consisting of a tangleof randomly oriented vortex lines whose velocity fieldstend to cancel each other out. This second form of tur-bulence, named ‘ultraquantum’ or ‘Vinen’ turbulence todistinguish it from the ‘quasiclassical’ or ‘Kolmogorov’turbulence38,39, has been identified both numerically40

and experimentally (in low temperature 4He6,8 and in3He-B14), and (only numerically) in high temperatures4He driven by a uniform heat current41,42 (thermal coun-terflow). Physically, ultraquantum turbulence is a stateof disorder without an energy cascade43. Recently, it hasbeen argued that ultraquantum turbulence has also beenobserved in the thermal quench of a Bose gas44, and insmall trapped atomic condensates45.

In the ultraquantum regime the vortex tangle is un-correlated: There is no separation between scales (andhence no large-scale motion), and the only length scale isthat of the mean intervortex distance, so that it can beexpected that the total Andreev reflectivity of the ultra-quantum tangle should differ from that of the quasiclas-

102 103

10

10

10-5

10-6

104 105

10-3

10-4

Wavenumber k [m-1]

Ener

gy s

pec

trum

E(k

) [a

.u.]

k-1

k-5/3

FIG. 8: Superfluid energy spectra (arbitrary units) obtainedfrom numerical simulation. The black and red lines corre-spond to thermal counterflow42,43 and ultraquantum regimein the zero temperature limit40, respectively. The red dottedand blue dashed lines are guides to the eye for the k−5/3 andk−1 scaling respectively.

sical quantum turbulence. However, we will demonstratebelow that this is not the case.

To compare the Andreev reflection from two diffe-rent types of quantum turbulence, we should, in prin-ciple, have generated numerically the ultraquantum vor-tex tangle whose line density would match that of thequasiclassical tangle described above in Secs. III and IV.However, it turns out that a numerical generation of theultraquantum tangle with prescribed properties presentssubstantial difficulties.

To overcome these difficulties, we will use an alterna-tive approach. In Refs.41–43 we numerically simulatedvortex tangles in 4He driven by a uniform normal fluid(to model thermally driven counterflow) and by synthe-tic turbulence in the normal fluid (to model mechanicaldriving, such as grids and propellers). Although thesecalculations model quantum turbulence in 4He at ele-vated temperatures (1K <∼ T < Tλ), the vortex tanglein the counterflow turbulence shares many features withthe ultraquantum tangle in the zero temperature limit.Apart from small anisotropy in the streamwise vs trans-versal directions, the counterflow tangle is practically un-structured. The only length scale present in this tangleis the intervortex distance. Furthermore, the energyspectrum42,43 of the superfluid component presented inFig. 8 nowhere conforms to the −5/3 scaling. Instead ithas a broad peak at large wavenumbers that is followedby the k−1 dependence numerically observed40 in ultra-quantum turbulence.

On the other hand, the tangle simulated for the me-chanically driven turbulence41,42 shares many featurescommon with the quasiclassical quantum turbulence inthe zero temperature regime. In particular, in both ca-ses large-scale superflows are present, and the energy

9

0 0.5 1 1.5 2Vortex line density L [108 m-2 ]

0

0.1

0.2

0.3

0.4

0.5A

ndre

ev r

efle

ctio

n c

oef

ficie

nt

f Rquasiclassical (3He)

ultraquantum

quasiclassical (synthetic)

large scale flow

FIG. 9: Total Andreev reflection vs vortex line density. Bluecircles: 3He tangle simulation; red squares: unstructured (“ul-traquantum”) tangle; green diamonds: structured (“quasi-classical”) tangle. Solid black line shows large-scale contribu-tion to the total Andreev reflection, see Eq. (11).

spectrum obeys Kolmogorov scaling for medium wave-numbers. Ignoring the normal fluid, the vortex tanglessimulated for the thermally and mechanically driven tur-bulence in 4He can be regarded as modeling, at leastqualitatively, the ultraquantum and quasiclassical regi-mes, respectively, of quantum turbulence in the low tem-perature 3He-B46.

In Refs.41,42, both the counterflow turbulence and thevortex tangle driven by the synthetic turbulence in thenormal fluid were simulated starting from an arbitraryseeding initial condition and then letting L grow andsaturate to a statistically steady state. An importantadvantage is that in both cases the vortex line densitywas changing within the same interval from L = 0 untilsaturation at L ≈ 2.2 × 108 m−2. For three realizati-ons of the counterflow turbulence with different values ofthe counterflow velocity, and for three realizations of themechanically driven turbulence with different values ofthe rms normal fluid velocity (see Refs.41,42 for detailsand values of parameters), we calculated the Andreevreflection from the unstructured (“ultraquantum”) andstructured (“quasiclassical”) tangles. These calculationswere performed for different values of the vortex line den-sity corresponding to different stages of the tangle’s evo-lution from L = 0 to the saturated, statistically steadystate. The coefficient of the total Andreev reflection wasthen calculated by averaging Eq. (6) with the values ofparameters corresponding to T = 150µK.

The results are shown in Fig. 9. Surprisingly, for thesame vortex line density the coefficients of the total An-dreev reflection from the unstructured (“ultraquantum”)and the structured (“quasiclassical”) quantum turbu-lence appear to be practically indistinguishable, so thatthe measurements of the total Andreev reflection cannotbe used to identify the regime of quantum turbulence47.

This result is consistent with experimental observationsof Andreev reflection as a function of the grid velocity,see Fig. 6 and Refs.25,27,48 which show that the fully for-med vortex tangle can be distinguished from the “gas” ofindividual vortex rings either by the rate of decay of thevortex line density or by the cross-correlation betweendifferent detectors.The results obtained in this and the previous sections

show that the total reflection coefficient is mainly deter-mined by the vortex line density, and that the large scaleflows emerging in the quasiclassical quantum turbulencehave little effect on the total Andreev reflection from thevortex tangle. This conclusion can be further strengthe-ned by estimating the amount of scattering from largescale flows in the quasiclassical regime as follows. Weassume that the largest structures generated by the si-mulations have a size comparable to the box length D.For classical turbulence with a Kolmogorov spectrum thecharacteristic velocity of the flow on a length scale D is

v(D) ≈ C1/2K ϵ1/3(D/2π)1/3, where CK ≈ 1.6 is the Kol-

mogorov constant and ϵ is the dissipation per unit mass.For quantum turbulence, the dissipation is normally as-sumed49 to be given by ϵ = ζκ3L2, where the dimensi-onless constant ζ ≃ 0.08 is inferred from experimentaldata8. The flow generates an energy barrier pF · v(D)for incident excitations. Assuming that the barrier is re-latively small compared to kBT , the corresponding totalreflectivity is fl ≃ pfv(D)/(4kBT ) where the factor of1/4 arises from taking an angular average of the energybarrier. The fraction of quasiparticles reflected by thelarge scale flow is thus estimated as

fl ≃√CK

4

pFκ

kBT

(ζD

2π

)1/3

L2/3 . (11)

This is shown by the lower black line in Fig. 9. Evenat high line densities the contribution of the large scaleflow to the total Andreev reflection is small (only about15% at the highest line density, L ≈ 2.2× 108 m−2; note,however, that this simple model overestimates the valueof fl).

VI. SCREENING MECHANISMS IN ANDREEVREFLECTION FROM VORTEX TANGLES

In the numerical simulations described in previoussections, as the vortex line density becomes larger than∼ 2× 107 m−2 the growth of the total Andreev reflectionwith the vortex line density becomes more gradual, seeFig. 6. The nonlinearity of the growth of the reflectioncoefficient with L is due to the screening effects whichbecome more important as the density of vortex confi-guration increases. We use the term ‘screening’ to iden-tify processes which change the overall reflectivity of thetangle for a given line density. Screening, whose me-chanisms are discussed below in this section, can occurbetween different vortices, or between different segmentsof the same vortex.

10

Analyzed in detail in Refs.18,25,29, the Andreev re-flection from the flow field of an isolated, rectilinear vor-tex line is determined by the 1/r behavior of the super-fluid velocity, where r is the distance from the vortexcore50. For dilute vortex configurations and tangles thetotal Andreev reflection is a sum of reflections from in-dividual vortices. However, as the line density increa-ses the total reflection no longer remains additive due toscreening effects which may either reduce or enhance theAndreev reflectivity of the vortex configuration.

We can identify two main mechanisms of screening.The first, which was called “partial screening” in ear-lier publications19,20, may significantly reduce the totalreflectivity of the dense vortex configuration due to a mo-dification of the 1/r superflow field in the vicinity of thevortex line. This type of screening may occur betweendifferent vortices in a dense vortex configuration, or bet-ween different segments of the same vortex. In particular,the reflectivity may be reduced by partial cancellation ofthe flow produced by neighboring vortices of opposite po-larity. For example, the flow field at some distance fromtwo, sufficiently close, (nearly) antiparallel vortex linesis that of a vortex dipole, v ∼ 1/r2; this, faster decayof the superflow velocity significantly reduces the totalreflectivity of the vortex pair.The 1/r velocity field may also be modified by the

effect of large curvature of the vortex line itself. Therole of curvature effects has been analyzed in Ref.21 for agas of vortex rings of the same size but random positionsand orientations. It was shown that the contribution topartial screening arising from curvature effects may leadto a noticeable reduction of Andreev reflection only forrings whose radii are smaller than R∗ ≈ 2.4µm.

It can, therefore, be expected that partial screeninghas another, small contribution from effects of large linecurvatures arising due to high frequency Kelvin wavespropagating along filaments. We may assume that insaturated vortex tangles the additional contribution topartial screening will arise due to Kelvin waves genera-ting line curvatures, C = |d2s/dξ2| (where s identifies apoint on the vortex line, and ξ is the line’s arclength)larger than C∗ = 1/R∗ ≈ 4×105 m−1. From our numeri-cal simulation, described above in Sec. III, we calculatedthe probability distribution function, F(C) of line curva-tures normalized such that

∫∞0

F(C) dC = 1. Then thefraction of contribution of high-frequency Kelvin wavesto Andreev reflection from the tangle can be estimatedby

∞∫C∗

F(C) dC . (12)

Numerical estimates of integral (12) for the consideredsaturated tangle of the density L = 9.7 × 107 m−2 showthat the contribution of high frequency Kelvin waves topartial screening and hence the reduction of the totalAndreev reflection from the tangle is negligibly small (lessthan 1%)51.

The other mechanism is that of ‘fractional’ screening(also called ‘geometric’ screening in Ref.25). This mecha-nism is of particular importance for dense or/and largetangles: Vortices located closer to the source of excita-tions screen those located at the rear. This screeningmechanism will also be important in the case where, indense tangles, vortex bundles may form28 producing re-gions of high Andreev reflectivity surrounded by thosewhose reflectivity is much lower. This will generate asignificant ‘geometric’ screening since few excitations canreach vortices at the rear of a large vortex bundle, andadding a vortex to a region where there is already an in-tense reflection will not have a noticeable effect on thetotal reflectivity.Earlier a simple model14 of the fractional screening

was proposed to infer a rough estimate for the vortexline density from the experimentally measured Andreevreflection. Assuming that all vortex lines are orthogo-nal to the beam of excitations, and modeling the An-dreev reflection from each vortex as that from an isola-ted, rectilinear vortex line, the total fraction of excitati-ons, Andreev-reflected by the tangle whose thickness inthe direction of quasiparticles’ beam is d, was found toincrease with d as fR = 1 − exp(−d/λ), where λ = P−1

is the lengthscale of exponential decay of the flux trans-mitted through the tangle and P = b0L is the probabilityof reflection per unit distance within the tangle, with b0given by Eq. (10) being the scale of Andreev scatteringlength by a rectilinear vortex. Then the vortex line den-sity can be inferred from the experimentally measuredvalue of fR as

L = −γ ln(1− fR) , (13)

where, for the considered model14, γ = (b0d)−1. As noted

in Ref.17, formula (13) gives only rough estimates for Lwhich are correct within a factor of 2. This is not surpri-sing since the model14 ignores an influence of geometryand orientation of vortex lines on the Andreev reflection.This model can be improved by modeling the tangle

as a collection of vortex rings of the same radius, R,but of random orientations and positions. Andreev re-flection from such vortex configurations was analyzed inmore detail in our earlier work21. Following Ref.14, con-sider a thin slab of superfluid, of thickness ∆x and crosssection S, but consisting now of N randomly orientedand positioned vortex rings of radius R, see Fig. 10. Thetotal cross section of Andreev scattering is σtot = ⟨σ⟩N ,where ⟨σ⟩ is the Andreev cross section of the vortex ring,ensemble-averaged over all possible ring orientations, andN is the total number of vortex rings within the consi-dered slab. In the case where the ring is not too small,R > R∗ ≈ 2.4µm, it was found21 that ⟨σ⟩ ≈ 2πKlR,with Kl = 10µm at temperature T = 100µK. The totalnumber of vortex rings within the slab can be calcula-ted as N = Λ/(2πR), where Λ = LS∆x is the total linelength. The probability that an incident excitation willbe Andreev-reflected is ∆p = σtot/S = ⟨σ⟩L∆x/(2πR),so that the lengthscale of decay the quasiparticle flux is

11

FIG. 10: Modeling the tangle by a configuration of randomlypositioned and oriented vortex rings of the same radius. Themodel is used for estimating the line density from the me-asurements of Andreev reflection. Black arrows sketch thetrajectories of incident excitations.

λ = ∆x/∆p = 1/(KlL). Then the line density inferredfrom the fraction of excitations Andreev-reflected by thetangle of thickness d is given by formula (13), where now

γ = (Kld)−1 . (14)

For the considered temperature, we find b0 ≈ 6.3µmand Kl ≈ 10µm, so that formula (13) with γ definedby Eq. (14) provides a more accurate estimate yieldingvalues of L which, for the same value of Andreev re-flection fR, are about twice smaller than those predictedby the original model of Bradley et al.14 (however, noteour comment in Ref.52).

VII. SPECTRAL PROPERTIES OF ANDREEVREFLECTION VS THOSE OF QUANTUM

TURBULENCE

In the simulation of quasiclassical turbulence usingring injection (see Fig. 1 in Sec. III), once the tanglehas reached the statistically steady state, the vortex linedensity and the Andreev reflection coefficient fluctuatearound their equilibrium, time-averaged values, ⟨L⟩ =9.7×107 m−2 and ⟨fR⟩ = 0.34, respectively. These fluctu-ations contain important information about the quantumturbulence. In order to compare the spectral characte-ristics of fluctuations of the Andreev reflection and thevortex line density,

δfR(t) = fR(t)− ⟨fR⟩ , δL(t) = L(t)− ⟨L⟩ , (15)

respectively, we monitor a steady state of simulatedtangle for a period of approximately 380 s or 7500 snaps-

hots. Taking the Fourier transform δfR(f) of the timesignal δfR(t), where f is frequency, we compute the po-

wer spectral density |δfR(f)|2 (PSD) of the Andreev

10-2

10-1

100

101

Frequency f [Hz]

10-2

100

102

104

106

108

PSD

of A

ndre

ev r

efle

ctio

n [

arb. units]

PSD

of vo

rtex

lin

e den

sity

[ar

b. units]

quasiclassical (3He)

ultraquantum

quasiclassical (synthetic)

AR

AR

AR

VLD

VLD

VLD

f -3

f -5/3

FIG. 11: Power spectral densities of simulated Andreev re-flection and simulated vortex line density corresponding to(from the top to bottom): quasiclassical turbulence in 3He(light blue: vortex line density, dark blue: Andreev re-flection); synthetic turbulence (light green: vortex line den-sity, dark green: Andreev reflection); counterflow turbulence(orange: vortex line density, red: Andreev reflection). The

red and black dashed lines are guides to the eye for the f−5/3

and f−3 scaling respectively.

reflection fluctuations. Similarly we compute the PSD

|δL(f)|2 of the vortex line density fluctuations and plotboth of them on Fig. 11.

Figure 11 also shows the PSDs of Andreev reflectionfluctuations and of vortex line density fluctuations formechanically driven (synthetic quasiclassical) and coun-terflow (ultraquantum) tangles. For each of the simu-lations, the PSDs of Andreev reflection and vortex linedensity fluctuations look very similar. A difference bet-ween the PSDs for our 3He tangle simulated using vortexring injection is somewhat larger and may be caused bythe presence of large scale flows that affect the spectralproperties of the fluctuations, but not the time-averagedvalues of the Andreev reflection and vortex line densities.

Our observations on the similarities of the PSD of An-dreev reflection and vortex line density fluctuations aresupported by the normalized cross-correlations betweenthe fluctuations of the vortex line density, δL(t), and thefluctuations of the Andreev reflection, δfR(t), which is

12

-10 -5 10Time t [s]

-0.2

0

0.2

0.4

0.6

0.8

1C

ross

-corr

elat

ion o

f L

and

fR

quasiclassical (3He)

quasiclassical (synthetic)

ultraquantum

50

FIG. 12: Cross-correlation between the fluctuations of An-dreev reflection and vortex line density for various simulatedtangles. The value of the cross-correlation peak exceeds 0.9and shows strong link between the fluctuations of Andreevreflection and vortex line density.

computed as

FLR(τ) =⟨δL(t)δfR(t+ τ)⟩√⟨δL2(0)⟩

√⟨δf2

R(0)⟩, (16)

where the angle brackets indicate averaging over time, t,in the saturated regime, and τ is the time-lag. Figure 12shows the cross-correlation between the vortex line den-sity and the Andreev reflection for the simulated tang-les. The cross-correlation values FLR(0) for ultraquan-tum and synthetically generated tangles approach unity,while for 3He tangle is slightly lower, FLR(0) ≈ 0.9. Thehigh values of the cross-correlation clearly demonstrate astrong link between the fluctuations of Andreev reflectionand vortex line density.We will now analyze the frequency dependence of the

PSDs of the vortex line density fluctuations shown inFig. 11. At high frequencies, the vortex line density fluc-tuation spectrum for all of the simulated tangles exhibitsf−5/3 scaling and is dominated by the contribution ofunpolarized, random vortex lines28. In the intermedi-ate frequency range, only the quasiclassical 3He fluctua-tion spectrum shows a clear f−3 scaling, which is gover-ned by the large scale flows indicating polarized (thatis, cooperatively correlated) vortex lines in agreementwith the recent numerical simulations28. If we reasona-bly assume that a crossover between f−3 to the f−5/3

behavior should occur at around the frequency corre-sponding to the intervortex distance, ℓ, then using thevalue of 102µm for ℓ calculated above for the equili-brium tangle, the crossover should occur at frequencyfℓ ≈ v/ℓ = κ/(2πℓ) ≈ 1Hz. As seen in Fig. 11, this is ina very fair agreement with the frequency ≈ 2Hz of thecrossover between the two regimes.

Figure 13 shows and compares the PSDs of fluctuati-ons of Andreev reflection for the simulation (top, blue)

10−2

10−1

100

10110

1

103

105

107

Frequency f [Hz]

PSD

Andre

ev r

efle

ctio

n [

arb. units]

f -5/3

Simulation

Vortex rings

Tangle

f -3

FIG. 13: Power spectral density of the Andreev reflectionfor numerical simulations (top, blue) and experimentalobservations26 (middle and bottom, gray) versus frequency.The red and blue dashed lines are guides to the eye for thef−5/3 and f−3 scaling respectively. For the details, see text.

and for the experiments26 (middle and bottom, gray).The experimental data are shown for a fully-developedtangle (dark gray) and ballistic vortex rings (light gray)corresponding to the grid velocities of 6.3mms−1 and1.9mms−1 respectively. (The prominent peaks in thenumerical data are artifacts of the discrete vortex-ringinjection process.) The power spectrum of δfR(t) of thesimulation and of the experimental data for the develo-ped tangle (reported here and in Refs.26,27) are in overallagreement showing reduction of the signal with increa-sing frequency.

A detailed comparison of experimental observations athigh grid velocity and simulations can be based on twoalternative approaches to the interpretation of numericaland experimental data. Both of these approaches are vi-able, mostly due to a rather noisy character of the fluctu-ating Andreev reflection’s spectra for simulated tangles,but they lead to somewhat different conclusions. In thefirst approach26 we could identify the same f−5/3 scalingbehavior at intermediate frequencies. This scaling, obser-ved experimentally in high temperature 4He53 and repro-duced by numerical simulations28, has been interpreted54

as that corresponding to the fluctuations of passive (ma-terial) lines in Kolmogorov turbulence. At high frequen-cies the experimental data develop a much steeper scaling(≈ f−3), not seen in the numerical spectrum, probablyowing to the finite numerical resolution. However, thisfrequency dependence is observed in the experiment inthe case where only microscopic vortex rings are propa-gating through the active region and there are no large-scale flows or structures. Thus, we can argue that thef−3 scaling for the vortex tangles corresponds to An-dreev reflection from superflows on length scales smallerthan the intervortex distance.

In the second, alternative approach we assume that

13

the PSD of Andreev reflection fluctuations mimics thefunctional dependence of the PSD of the vortex line den-sity in accordance with Fig. 13. In this case the Andreev-reflection fluctuation spectra exhibit a f−5/3 dependenceat high frequency and a steeper (≈ f−3) dependenceat intermediate frequencies, which is almost opposite tothe scaling observed in the experiment. Figure 14 showsfluctuation spectra of the vortex line density28 calcula-ted for the total line density (black upper solid line) andthe polarized fraction of the quasiclassical tangle (red lo-wer solid line). The observed frequency dependencies aresimilar, and suggest that in our experiments the vorti-ces probed by quasiparticles are polarized and hence thetangle generated by the vibrating grid is much more po-larized than expected. It is feasible that vortex ringsmoving predominantly in the same direction polarize theresulting tangle, and that quasiparticles accompanyingturbulence production further assist the polarization ofthe tangle in the direction of grid motion.

Frequency f [Hz]100 101 102

10-1

10-3

10-5

10-7

PSD

of vo

rtex

lin

e den

sity

[a.

u.]

FIG. 14: Power spectral density of fluctuations of the totalvortex line density (black upper solid line) and of the polarizedvortex line density (red lower solid line)28. The light bluedotted and green dashed lines are guides to the eye for thef−5/3 and f−3 scaling respectively. For the details, see text.

VIII. DISCUSSION AND CONCLUSIONS

In this study we combined the three-dimensional nu-merical simulations of dynamics and evolution of turbu-lent vortex tangles, numerical analysis of Andreev scatte-ring of thermal quasiparticle excitations by the simulatedtangles, and experimental measurements of Andreev re-flection by quantum turbulence in 3He-B at ultralow tem-peratures. Our aim was to show that a combined studyis a powerful tool that might reveal important propertiesof pure quantum turbulence.

We have shown that, with sufficient resolution of the

Andreev retroreflected signal, the Andreev scatteringtechnique can generate a map of local reflections whichreveals a structure of the vortex tangle. Furthermore,in the numerical experiment it is possible to separatethe quasiparticle and quasihole excitations. The indivi-dual maps of Andreev reflection for each of these speciesof excitations also show the presence and the structureof large scale flows arising in the quasiclassical (Kolmo-gorov) regime of quantum turbulence. However, in thereal physical experiment the thermal beam consists ofexcitations of both sorts whose separation is impossible.We found that, while a presence of large scale flows maystill be detected by the enhancement of local Andreev re-flections, a direct identification of the detailed large scaleflow structure would hardly be possible.

For the quasiclassical vortex tangle, we calculated thetotal Andreev reflection as a function of the vortex linedensity. We found that the results of our numerical simu-lation reproduce rather well the experimental measure-ments of the total Andreev reflection, not only qualitati-vely but also quantitatively, although this good quantita-tive agreement should be taken with some caution for thereason explained in Sec. IVC. We also found that the to-tal reflection is strongly affected by the screening effects.Here we use the term “screening” to identify mechanismswhich, for a given line density, change the overall reflecti-vity of the tangle. In particular, screening mechanismsare responsible for the slowing of growth of the total re-flectivity with the vortex line density when the latter rea-ches, in our simulation, the value about 2×107 m−2. Weidentified two main mechanisms of screening. The first,so-called “partial screening” may significantly reduce thetotal reflectivity of the dense vortex configuration dueto a modification of the 1/r superflow in the vicinity ofthe vortex line. The second is the ‘fractional’ (or ‘geo-metric’) screening and is of importance for dense or/andlarge tangles: Vortices located closer to the source of ex-citations screen those at the rear. We investigated scree-ning mechanisms in some detail and conclude that theyoften should be taken into consideration in interpretationof experimental observations and numerical simulations.

While this paper is mostly concerned with the Andreevreflection from quasiclassical vortex tangles, there alsoexists another regime of quantum turbulence of a verydifferent spectral nature in which the energy is contai-ned in the intermediate range of scales and the energyspectrum nowhere conforms to the Kolmogorov −5/3scaling. Unlike quasiclassical quantum turbulence, thisform of turbulence, named ‘ultraquantum’ or ‘Vinen’ tur-bulence38, is unstructured: the only length scale in theultraquantum tangle is the mean intervortex separation.At very low temperatures the ultraquantum regime ofquantum turbulence has been identified experimentallyin 4He6,8 and 3He-B14. Considering that the Andreevreflection can reveal the presence of the large scale flowsexisting in quasiclassical turbulence, it could be expectedthat the ultraquantum tangle can be easily distinguishedfrom the quasiclassical one by their total Andreev reflecti-

14

vities. However, our modeling study in Sec. V of Andreevreflection from the structured (Kolmogorov) and unstruc-tured (ultraquantum) vortex tangles for the same vortexline density has demonstrated that this is not the case,and that the total reflection is determined by the vortexline density alone. This, in particular, means that thecontribution of the large scale flow to the total Andreevreflection is small in our tangles. The latter is consistentwith the experimental observations showing an identicalsignal from vortex rings and a turbulent tangle at similargrid velocities and hence vortex line densities.

Finally, we investigated the spectral characteristics ofAndreev reflection from various tangles and comparedthem with the spectral characteristics of the quantumturbulence. We found that the fluctuations of the vor-tex line density and of the Andreev reflection are verystrongly correlated and also have similar fluctuation po-wer spectra. The Andreev-reflection fluctuation spectraand of the vortex line density fluctuations look practicallyidentical in the case of ultraquantum turbulence but maydiffer for the quasiclassical tangle, that is when a largescale flow is present. Therefore, some caution should betaken in interpreting the f−5/3 scaling of the Andreev re-troreflected signal’s frequency spectrum, observed in theexperiment of Bradley et al.27, as a direct representationof the spectral properties of the vortex-line density fluc-tuations in turbulent 3He-B. Our results show that thisscaling could be attributed to the fluctuations of passive(material) lines in Kolmogorov turbulence, or to the fluc-tuations of the vortex line density of a strongly polarizedtangle. The latter possibility suggests that a turbulenttangle produced by the vibrating grid at high velocities

is more polarized than has usually been assumed ear-lier. It is very clear that the Andreev reflection techniquehas great potential for elucidating the properties of purequantum turbulence, but requires further work.

It is clear that a useful study could be made of a tanglecreated from rings emitted by two grids facing each ot-her. Such a tangle should have a vortex line density si-milar to that of the fully-developed tangle studied in ourwork but with a much smaller polarization. Numerically,it would be interesting to investigate fluctuations of theAndreev reflection and vortex line density for the showerof ballistic vortex rings using a higher spatial resolutionand whether the f−3 spectrum would reproduced expe-rimentally. The influence of large scale flows on Andreevreflections should also be studied in superfluid 3He underrotation since this scenario uniquely offers a vortex lineensemble with inherent strong polarization.

Acknowledgments

This work was supported by the Leverhulme Trustgrants No. F/00 125/AH and F/00 125/AD, the Eu-ropean FP7 Program MICROKELVIN Project 228464,and the UK EPSRC grants No. EP/L000016/1, No.EP/I028285/1 and EP/I019413/1. MJJ acknowledgesthe support from Czech Science Foundation Project No.GACR 17-03572S.

All data used in this paper are available athttp://dx.doi.org/10.17635/lancaster/researchdata/xxx,including descriptions of the data sets.

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15

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27 D. I. Bradley, S. N. Fisher, A. M. Guenault, R. P. Haley,S. O’Sullivan, G. R. Pickett, and V. Tsepelin, Fluctuationsand Correlations of Pure Quantum Turbulence in Super-fluid 3He-B, Phys. Rev. Lett. 101, 065302 (2008).

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32 Because the time scale of quasiparticle motion is muchsmaller than both the time scale of the evolution of thetangle and the time scale of the fluid motion within thetangle, τf ∼ 1/

√κL, calculation of the Andreev reflection

at any time during the transient were performed for thetangle frozen at this moment of time.

33 Changing the injection and switching frequencies, as wellas the injected ring’s size, we also generated the tang-les whose time-averaged, saturated line densities were thesame as some of the line densities during the transient.Calculations of the fraction of Andreev-reflected excita-tions for such tangles yielded the results consistent withthose shown in the top panel of Fig. 6.

34 The injected rings are relatively large so that the effectsof high curvatures on the Andreev reflection can be neg-lected21.

35 S. Fujiyama, A. Mitani, M. Tsubota, D. I. Bradley,S. N. Fisher, A. M. Guenault, R. P. Haley, G. R. Pickett,and V. Tsepelin, Generation, evolution, and decay of purequantum turbulence: A full Biot-Savart simulation, Phys.Rev. B 81, 180512 (2010).

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46 Some dissimilarity between these tangles and the tanglessimulated for the truly zero temperature limit is that atelevated temperatures the Kelvin waves are damped bythe mutual friction. However, as will be noted in Sec. VIKelvin waves have a negligible effect on the total Andreev

16

reflection.47 The question whether the regime of quantum turbu-

lence, quasiclassical or ultraquantum, can be identifiedby spectral properties of Andreev reflection, still remainsopen. For quasiclassical regime a correspondence betweenthe spectral properties of Andreev reflection and those ofquantum turbulence are discussed in Ref.26 and in Sec. VIIof this paper.

48 M. J. Jackson, D. I. Bradley, A. M. Guenault, R. P. Haley,G. R. Pickett and V. Tsepelin, Observation of quantumturbulence in superfluid 3He-B using reflection and trans-mission of ballistic thermal excitations, Phys. Rev. B 95,094518 (2017).

49 W. F. Vinen and J. J. Niemela, Quantum Turbulence, J.Low Temp. Phys. 138, 167 (2002).

50 In the cited papers the analysis of Andreev reflection wasgiven for the case where the beam of incident excitati-ons is orthogonal to the vortex line. In the general casethe dependence of Andreev reflection on the angle θ be-tween the direction of the incident beam and the vortexline should be accounted for by a factor sin θ in the formu-

lae for a fraction of Andreev reflected excitations and thecorresponding scattering length.

51 It should be noted that although Kelvin waves of wave-length smaller that R∗ do not contribute significantly tothe Andreev reflection, they may add to the total line den-sity of the tangle.

52 The values of the vortex line density found from for-mula (13) with γ defined by Eq. (14) may be underestima-ted for very dense tangles or configurations of rings suchthat overlapping of rings’ flow fields (and hence partialscreening) can no longer be neglected. For a more detailedanalysis of Andreev reflection from dense configurations ofinteracting vortex rings see our earlier paper21.

53 P.-E. Roche, P. Diribarne, T. Didelot, O. Francais, L.Rousseau, and H. Willaime, Vortex density spectrum ofquantum turbulence, Europhys. Lett. 77, 66002 (2007).

54 P.-E. Roche and C. F. Barenghi, Vortex spectrum in su-perfluid turbulence: Interpretation of a recent experiment,Europhys. Lett. 81, 36002 (2008).