Three-dimensional modelling of Andreev scattering inturbulent 3He-B

N Suramlishvili1, A W Baggaley1,2, C F Barenghi1 and Y A Sergeev3

1Joint Quantum Centre (JQC) Durham-Newcastle, School of Mathematics and Statistics,Newcastle University, UK.

2School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QW, UK.3Joint Quantum Centre (JQC) Durham-Newcastle, School of Mechanical and System Engineering,

Newcastle University, UK.

IntroductionPioneered at Lancaster, the Andreev scattering technique has become a pow-

erful method for studying quantum turbulence phenomena in superfluid 3He-B inthe ultra-low temperature regime. This technique is based on the fact that theenergy dispersion curve, E = E(p) of excitations with momentum p is tied tothe reference frame of the superfluid, so, in a superfluid moving with velocity vs,the dispersion curve becomes E(p) + p · vs [1, 2]. Thus one side of a vortex linepresents a potential barrier to oncoming quasiparticles, which can be reflectedalmost exactly, becoming quasiholes; the other side of the vortex lets quasipar-ticles to go through. Quasiholes are reflected or transmitted in the opposite way.The vortex thus casts a shadow for quasiparticles at one side and for quasiholesat the other side[3], and, by measuring the flux of excitations, one detects thepresence of the vortex.

We have developed a numerical method which enables us to simulate theinteraction of a three dimensional vortex tangle of quantized vortices and thermalquasiparticles moving in the velocity field of the vortices. We present resultswhich refer to a continuously driven turbulent system in a non-equilibrium steady-state, where energy injected is balanced by dissipation. In particular we show acorrespondence between Andreev reflection and vortex line density. The resultsare compared with experimental observations[4].

Governing equations and numericalmethod

The energy of an excitation in presence of quantized vortices is

E =√

e2p +∆2o + p · vs(r, t) , vs(r, t) =

κ

4π

∫

(s− r)× ds

|s− r|3, (1)

where ep = p2

2m∗ − eF is the kinetic energy, ∆0 = 1.76kBTc is the superfluid en-ergy gap, Tc is the critical temperature, vs(r, t) is the superfluid velocity at pointr determined by the Biot-Savart law, and κ = π~/m ≈ 0.662 × 10−3cm2/s is thequantum of circulation in 3He-B. The interaction term, p · vs varies on a spatialscale larger than the coherence length ξ0 = ~vF/π∆0 and Eq. (1) can be consid-ered as an effective Hamiltonian[5], yielding the following equations of motion:

dr

dt=

∂E

∂p=

ep√

e2p +∆20

p

m∗+ vs,

dp

dt= −

∂E

∂r= −

∂

∂r[p · vs]. (2)

We solved numerically Eqs. (2) and determined the fraction of the thermal fluxreflected from the vortex configuration. The heat flux generated by the source is

δQinc =

∫ ∞

∆

NFvFE∂f (E)

∂TδT dE, (3)

where δT ≪ T is a temperature difference between the source of thermal excita-tions and the opposite side of the system, f (E) is the Fermi distribution functiontransformed to the Boltzmann function, f (E) = exp(−E/kBT ), at ultra-low tem-peratures.

Eqs. (2) are solved by a variable-step, variable-order algorithm particularlyefficient for solving stiff problems [6,7]. The superfluid velocity and time evolutionof the vortex tangle is calculated from Eqs. (2) by means of the vortex filamentmethod using periodic boundary conditions and tree-method with critical openingangle 0.4 [8].

Due to the difference in the timescales associated with the motion of thequasiparticles and the vortex dynamics we assume the vortices to be frozenwhen calculating vs in Eqs. (2) (see Ref. [9]).

Numerical experiment and resultsThe numerical simulation was carried out within a periodic cubic domain of

size D = 0.1cm at temperature T = 0.1Tc and pressure 0bar.

The FLUX is modelled by Nqp = 9× 104 quasiparticles entering from one sideof the box and moving parallel to the x-direction. Initial positions on (y, z)-planeand energies of quasiparticles, E0 (∆0 < E0 ≤ 1.1∆0) are uniformly distributed.The average energy of quasiparticles is 〈E〉 = ∆0 + kBT .

Motivated by experimental studies, we numerically simulate the evolution ofa vortex TANGLE driven by loop injection. Two rings with radius 0.024cm are in-jected at opposite corners of the numerical domain with a frequency of 22Hz; tomaintain isotropy the plane of injection of the loops is switched with a frequencyof 7.4Hz. Very quickly the energy injected into the system is balanced by dissipa-tion, and the tangle saturates at L̄ = 11900cm−2. During this statistically steadystate the energy spectrum, E(k), is consistent with the Kolmogorov spectrum,E(k) ∼ k−5/3, at large scales.

101

102

103

104

10−7

10−6

10−5

10−4

log k

log

E(k

)

k−5/3

Fig.1. Left: Snapshot of vortex tangle when two vortex ringsare injected at oppositecorners. Right: The energy spectraE(k) (arbitrary units) vs wavenumber k (cm−1)corresponding to figure on the left. The red dashed line showsthe k−5/3 scaling.

We calculated fluctuations of the vortex line density, δL(t) = L(t) − L̄, andthermal reflectance, δR(t) = R(t) − R̄ of the tangle and their power spectra fora period of 40s, averaging over 180 snapshots of the vortex configuration. Timeinterval between nearest snapshots is δt = 0.225s. We find a a good corre-spondence between the fluctuations of the vortex line density and of the thermalreflectance (see Fig. 2).

5 25 450.1

0.15

0.2

t

R

b)

5 25 4510

12

14

t

L

a)

Fig.2. a) Vortex line density,L, of the saturated turbulent tangle vst. b) Thermalreflectance,R, vs t.

The power spectral densities (PSD) of δL(t) and δR(t) exhibit f−5/3 scaling(See Fig. 3).

10−1

100

10110

−9

10−5

10−2

f

b)

PS

D(R

)

10−1

10010

−5

10−2

100

f

PS

D(L

)

a)

f −5/3

f −5/3

Fig.3. a) PSD of line density fluctuations,δL, of the saturated turbulent tangle vsf (Hz). b) PSD of thermal reflectance,δR, vs f (Hz). On both a) and b) figures thegreen dashed line shows thef−5/3 scaling.

ConclusionsWe found a remarkable correspondence between the fluctuations of the vortex

line density and the fluctuations of the thermal reflectance, in agreement withexperimental results obtained for the superfluid turbulence generated at highgrid velocities in 3He-B at ultra-low temperatures [4].

ACKNOWLEDGMENTS: This research was supported by the LeverhulmeTrust, grant numbers F/00125/AH and F/00125/AD.

References[1]S. N. Fisher, A. J. Hale, A. M. Guenault, and G. R. Pickett, Phys. Rev. Lett. 86, 244 (2001).[2] D. I. Bradley et al., Phys. Rev. Lett. 95, 035301 (2005).[3] C.F. Barenghi, Y.A. Sergeev, and N. Suramlishvili, Phys. Rev. B 77 024508 (2008).[4] D. I. Bradley et al., Phys. Rev. Lett. 101, 065302 (2008).[5] N.A. Greaves and A.J. Leggett, J. Phys. C 16 4383 (1985).

[6] J.D. Lambert, Numerical Methods for Ordinary Differential Systems:the Initial Value Problems, John Wiley and Sons, 1991.[7] L.F. Shampine and M.W. Rachelt, SIAM J. Sci. Comp. 18 1 (1997).[8] A.W. Baggaley and C.F. Barenghi, J. Low Temp. Phys. 1662 (2012).[9] N. Suramlishvili , A.W. Baggaley, C.F. Barenghi and Y.A. Sergeev, Phys. Rev. B. 85174526 (2012).