Quantum TurbulenceQuantum Turbulence-from superfluid helium to atomicBose-Einstein condensates-

Makoto TSUBOTADepartment of Physics,Osaka City University, Japan

Thanks to many friends in this field

Another Da Vinci code

Leonard Da Vinci(1452-1519) Da Vinci observed turbulent flow

of water and found that turbulencewas comprised of many vortices.

Turbulence is not a simple disordered statebut having some structures with vortices.

Vortices in Japanese sea

GalaxyTornado over the sea

Our nature is filled with many kinds of vortices.

Certainly turbulence looks to have many vortices.Turbulence behind a dragonfly

However, these vortices are unstable; they repeatedlyappear, diffuse and disappear.It is not so straightforward to confirm another DaVinci code in classical turbulence.

Another Da Vinci code is actually realizedin quantum turbulence comprised ofquantized vortices.

Key concept of my talk

2. Quantized vortices in atomic BECs

3. Quantum turbulence in superfluid helium

4. Quantum turbulence in atomic BECs

1. Introduction

Outline

Each atom behaves as a particle athigh temperatures.

Each atom occupies the same single particle groundstate. The matter waves become coherent, making amacroscopic wave function Ψ .

Thermal de Broglie wave length~ Distance between particles

Each atom behaves like a wave atlow temperatures.

Bose-Einstein condensation (BEC)

1. Introduction

Ψ

Duality of matter and wave

Realization of atomic gas BEC

1995 87Rb, 7Li, 23NaLaser cooling

An atom is subjected to a laser beam whose frequency is tuned to liejust below that of an atomic transition between an excited state andthe ground state.→　An atom moving toward to the laser beam absorbs it by theDoppler shift, reducing its velocity.Cold atoms are collected at the focus of six laser beams.　T~ 100μK

Magnetic trapTrapping cooled gases

Evaporation coolingAtoms with high kinetic energy arereleased from the trapping potential.

BEC 　 T~ 100 nK

Observation of BECTurning off the trapping potential,

→　the gas expands with falling feely.

→　The observation of the position of atoms determines the initial distribution of velocity.

By MIT

TheNobelPrize inPhysics2001!

Advantages of atomic BEC as a system

• We can control the system easily, even theinteraction parameters.

• We can visualize directly the condensatedensity, namely the amplitude of the orderparameter.

K.W.Madison, et.al PRL 84, 806 (2000)

Quantization of circulationBEC causes a macroscopic wave function(order parameter) .

!

" = " exp i#( )

The superfluid velocity field is given by .

!

vs=

h

m"#

A single-connected region in a fluid has , no vortices.

!

rot vs

= 0

However, if the fluid has some multiple-connectedregion (for example, singularities), the aroundcirculation is quantized.

Quantized vortexQuantized vortex

!

" = vs# dl =

h

m$% # dl =

h

mC&

C& n

!

(n: an integer) C

A quantized vortex is a vortex of superflow in a BEC.Any rotational motion in superfluid is sustained byquantized vortices.

(i) The circulation is quantized.

(iii) The core size is very small.

vs! ds ="n# n = 0,1, 2,L( )

A vortex with n≧2 is unstable.

(ii) Free from the decay mechanism of the viscous diffusion of the vorticity.Every vortex has the same circulation.

The vortex is stable. ρ

r

s~Å (r)

rot vsThe order of the coherencelength.

! = h / m

How to describe the vortex dynamics

Vortex filament formulation (Schwarz)　　Biot-Savart law

A vortex makes the superflow of the Biot-Savart law, and moves withthis local flow. At a finite temperature, the mutual friction should beconsidered.

vs r( ) =!

4"

s # r( ) $ d s

s # r3% s

r

The Gross-Pitaevskii equation for the macroscopic wave function

!

ih"#(r,t)

"t= $

h2%2

2m+V

ext(r) + g#(r,t)

2&

' (

)

* + #(r, t)

!(r ) = n0(r)e

i"( r )

Atomic BEC

Superfluid HeVortex tangleVortex array

There are two main cooperative phenomena ofquantized vortices; Vortex array under rotationand Vortex tangle (Quantum turbulence).

None

2.

3.

4.

2. Quantized vortices in atomic BECsM.Tsubota, K.Kasamatsu, M.Ueda, Phys.Rev.A65, 023603(2002)K.Kasamatsu, T.Tsubota, M.Ueda, Phys.Rev.A67, 033610(2003)

Advantages of atomic BECs as a physical system

ExperimentalWe can directly control and visualize the condensate.

We can change the atomic interaction by Feshbach resonance.

TheoreticalAt low temperatures the model of dilute Bose gas and the mean fieldapproximation theory (Gross-Pataevskii(GP)equation) are quantitativelycorrect.

!

ih"#(r,t)

"t= $

h2%2

2m+V

ext(r) + g#(r,t)

2&

' (

)

* + #(r, t)

The fluid rotates with the sameangular velocity with the vessel.

This means that there appears onevortex in the vessel. The singlevortex can make the solid-bodyrotation with any angular velocity.

What happens if we rotate a vessel having a usualviscous classical fluid inside?

This does not occur in quantum fluids!Such experiments were done in atomicBECs.

Rotating superfluid and the vortex lattice

!

" = n0ei#

Ω < Ωc Ω > Ωc

!

vs= h

m

"#

!

" # vs= 0

!

F = E "#Lz

The triangular vortexlattice sustains the solidbody rotation.Yarmchuck and Packard

Minimizing the free energyin a rotating frame.

Vortex latticeobserved in rotatingsuperfluid helium

Observation of quantized vortices in atomic BECsK.W.Madison, et.al PRL 84, 806 (2000)

J.R. Abo-Shaeer, et.alScience 292, 476 (2001)

P. Engels, et.alPRL 87, 210403(2001)

ENS

MIT

JILA

E. Hodby, et.alPRL 88, 010405(2002)

Oxford

How can we rotate the trapped BEC？K.W.Madison et.al Phys.Rev Lett 84, 806 (2000)

Non-axisymmetric potential

Optical spoon

Total potential

Rotation frequency Ω

z

x

y 100µm

5µm

20µm 16µm

!

Ustir (R) =m

2"#2($xX

2+ $yY

2)

!

Vext (R) =Vtrap(R) +Ustir (R)

!

"x # "y

Axisymmetric potential

“cigar-shape”

Direct observation of the vortex lattice formation

Snapshots of the BEC after turning on the rotation

1. The BEC becomeselliptic, then oscillating.

2. The surface becomes unstable.

3. Vortices enter the BECfrom the surface.

4. The BEC recovers theaxisymmetry, the vorticesforming a lattice.

K.W.Madison et.al. PRL 86 , 4443 (2001)

Rx

Ry

! ="Rx

2

# Ry2

Rx

2

+ Ry2

The Gross-Pitaevskii(GP) equation in a rotating frame

Wave functionInteraction

s-wavescattering length

!

ih"#

"t= $

h2

2m%2#+Vtrap#+ g#

2#

!

"(r, t)

!

g =4"h2as

m

!

as

in a rotating frame

!

ih"#

"t= $

h2

2m%2#+ (Vtrap +Ustir )#+ g#

2#$&Lz#

!

Ustir (R) =m

2"#2($xX

2+ $yY

2)Two-dimensional

simplified

Ω Ω

The GP equation with a dissipative term

!

ih"#

"t= $

h2

2m%2 + (Vtrap +Ustir ) + g#

2$µ $&Lz

'

( )

*

+ , #

!

(i " #)h$%

$t

!

" = 0.03 : dimensionless parameter

S.Choi, et.al. PRA 57, 4057 (1998)I.Aranson, et.al. PRB 54, 13072 (1996)

This dissipation comes microscopically from the interactionbetween the condensate and the noncondensate.

E.Zaremba, T. Nikuni, and A. Griffin, J. Low Temp. Phys. 116, 277 (1999)

C.W. Gardiner, J.R. Anglin, and T.I.A. Fudge, J. Phys. B 35, 1555 (2002)M. Kobayashi and M. Tsubota, PRL 97, 145301 (2006)

Profile of a single quantized vortex

!h2

2m"2# + Vtrap# + g#

2# = µ# !(r ) = n

0(r)e

i"( r )

A quantized vortex

n0

!

Velocity field 0

0.005

0.01

0.015

0.02

0 2 4 6 8 10

r

|!|2

Vortex core= healing length

!

" #h

2mgn0

A vortex

Dynamics of the vortex lattice formation (1)

Time development of the condensate density n0

!

" = 0.7#$

ExperimentTsubota et al.,Phys. Rev. A 65,023603 (2002)

Vtrap(r) =1

2m!"

2r2

!(r ) = n0(r)e

i"( r )

Dynamics of the vortex lattice formation (2)

t=0 67ms 340ms

390ms 410ms 700ms

Time-development of the condensate density n0

Are these holes actually quantized vortices?

Dynamics of the vortex lattice formation (3)Time-development of the phase !

!(r ) = n0(r)e

i"( r )

Dynamics of the vortex lattice formation (4)

t=0 67ms 340ms

390ms 410ms 700ms

Ghost vortices

Becoming real vortices

Time-development of the phase!

Dynamics of the vortex lattice formation (5)

K.W.Madison et.al. Phys. Rev. Lett. 86 , 4443 (2001)

Rx

Ry

! ="Rx

2

# Ry2

Rx

2

+ Ry2

0

0.1

0.2

0.3

0 100 200 300 400 500 600 700time (msec)

!/"

Simultaneous display of the density and the phase

The three-dimensional dynamicsK.Kasamatsu, M. Machida, N. Sasa,M.Tsubota, PRA 71, 063616 (2005)

Density　　 Superflow

Space resolution　116×116×580

γ=0.03

The dynamics is qualitatively similar tothe two-dimensional case, but we have

＊Excitation and motion of Kelvinwaves

＊Turbulence made of phase defects

Atomic BEC

Superfluid HeVortex tangleVortex array

There are two main cooperative phenomena ofquantized vortices; Vortex array under rotationand Vortex tangle (Quantum turbulence).

None

2.

3.

4.

3. Quantum turbulence in superfluid heliumLiquid 4He enters the superfluid state at 2.17K (λ point) withBose condensation.

Its hydrodynamics is well described by the two-fluid model.

Temperature

point

The two-fluid modelThe system is a mixture of inviscidsuperfluid and viscous normal fluid.! = !

s+ !

n

!

j = "svs+ "

nvn

Normal fluid

0 0Superfluid

entropyviscosityvelocitydensity

!

"sT( )

!

"nT( )

!

vsr( )

!

vnr( )

!

"nT( )

!

snT( )

The two-fluid model could explain various phenomena ofsuperfluidity which were observed experimentally.

Thermo-mechanical effect, film flow, etc.

However, …….

(ii) v > vs c

vs

A tangle of quantized vortices grows. The two fluidsinteract through the mutual friction due to the tangle, andthe superflow decays.

v =0s

The superfluidity breaks down when it flows fast.

(i) v < vs c

vs

There is no interaction between two fluids, and thesuperfluid can flow forever without decaying.

vst! "

(some critical velocity)

1955 　Feynman proposed that “superfluid turbulence”consists of a tangle of quantized vortices.

1955 - 1957 　Hall and Vinen observed “superfluidturbulence”.

The mutual friction between the vortex tangle and thenormal fluid causes the dissipation of the flow.

Progress in Low Temperature PhysicsVol.I (1955), p.17

Lots of experimental studies were done chiefly for thermalcounterflow of superfluid 4He.

1980’s　K. W. Schwarz　 Phys.Rev.B38, 2398(1988)

Made the direct numerical simulation of the three-dimensionaldynamics of quantized vortices and succeeded in explainingquantitatively the observed temperature difference △T .

Vortex tangle

Heater

Normal flow Super flow

Development of a vortex tangle in a thermal counterflow

Schwarz, Phys.Rev.B38, 2398(1988).

Schwarz obtained numerically thestatistically steady state of a vortextangle which is sustained by thecompetition between the appliedflow and the mutual friction. Theobtained vortex density L(vns, T)agreed quantitatively withexperimental data.

vs vnThe research of counterflow turbulence hasbeen successful.

What is the relation between superfluidturbulence and classical turbulence？

Most studies of superfluid turbulence were devoted tothermal counterflow.

　　　　　　　　⇨　No analogy with classical turbulence

When Feynman showed the above figure, he thought of acascade process in classical turbulence.

Classical Turbulence (CT) vs. Quantum Turbulence (QT)

Classical turbulence Quantum turbulence

・The vortices are unstable. Noteasy to identify each vortex.

・The circulation differs from oneto another, not conserved.

・The quantized vortices arestable topological defects.・Every vortex has the samecirculation.・Circulation is conserved.

Motion ofvortexcores

QT is much simplerthan CT, because eachelement of turbulenceis definite.

How can we study the similarity andthe difference between QT and CT?

Let’s focus on the most important statistical law in CT,namely, the Kolmogorov law.

!

E(k) = C"2 / 3k#5 / 3

Classical turbulence

Energy-containingrange

Inertialrange

Energy-dissipativerange

Energy spectrum of turbulence

Kolmogorov law

Energy spectrum of the velocity field

Energy-containing rangeThe energy is injected into the system at .

k ! k0 = 1/ l0

Inertial rangeDissipation does not work. The nonlinearinteraction transfers the energy from low kregion to high k region.

Kolmogorov law : E(k)=Cε2/3 k -5/3

Energy-dissipative range

The energy is dissipated with the rate ε at theKolmogorov wave number kc = (ε/ν3 )1/4.

!

E =1

2v"2

dr = E(k)dk"

Everybody believesthat turbulence issustained by thisRichardson cascade.However, this is onlya cartoon; nobody hasever confirmed itclearly.

One of the reasons isthat it is too difficultto identify each eddyin a fluid.

Quantum turbulence may give a prototype ofQuantum turbulence may give a prototype ofturbulence, much simpler than conventionalturbulence, much simpler than conventionalturbulence.turbulence.

Can such quantized vortices still produce theCan such quantized vortices still produce theessence of turbulence?essence of turbulence?

If QT can be a prototype of turbulence, itshould satisfy the essence of turbulence.

The most important test would be whetherQT satisfies the Kolmogorov law or not.

Having this sort of motivation, the studiesHaving this sort of motivation, the studiesof QT have entered a new stage since theof QT have entered a new stage since themiddle of 90middle of 90’’s !s !

New study of superfluid turbulence (1)New study of superfluid turbulence (1)• Maurer and Tabeling, Europhysics. Letters. 43, 29(1998) 1.4K < T < Tλ

Measurements of local pressure in flows 　 driven by two counterrotating disks finds　the Kolmogorov spectrum.

• Stalp, Skrbek and Donnely, Phys.Rev.Lett. 82, 4831(1999) 1.4 < T < 2.15K

Decay of grid turbulence. The data of the second sound attenuation was consistentwith a classical model with the Kolmogorov spectrum.

Energy spectra

New study of superfluid turbulence (2)New study of superfluid turbulence (2)• Vinen, Phys.Rev.B61, 1410(2000)

Considering the relation between ST and CTThe Oregon’s result is understood by the coupled dynamics of the superfluid and thenormal fluid due to the mutual friction. Length scales are important, compared withthe characteristic vortex spacing in a tangle.

• Kivotides, Vassilicos, Samuels and Barenghi, Europhysics Lett. 57, 845(2002)When superfluid is coupled with the normal-fluid turbulence that obeys theKolmogorov law, its spectrum follows the Kolmogorov law too.

What happens at very low temperatures?Is there still the similarity or not?

　Energy spectra of quantum turbulence (QT)

Decaying Kolmogorov turbulence in a model of superflowC. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644 (1997)The Gross-Pitaevskii (GP) model

Energy Spectrum of Superfluid Turbulence with No Normal-FluidComponentT. Araki, M.Tsubota and S.K.Nemirovskii, Phys.Rev.Lett.89, 145301(2002)The vortex-filament model

Kolmogorov Spectrum of Superfluid Turbulence: Numerical Analysis ofthe Gross-Pitaevskii Equation with a Small-Scale Dissipation

M. Kobayashi and M. Tsubota,Phys. Rev. Lett. 94, 065302 (2005), J. Phys. Soc. Jpn.74, 3248 (2005).

There are three works which directly study the energyspectrum of QT at zero temperature.

C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997)

By using the GP model, theyobtained a vortex tangle withstarting from the Taylor-Green vortices.

t＝２ ４

６ ８

１０ １２

In order to study the Kolmogorov spectrum, it is necessary todecompose the total energy into some components. (Nore et al., 1997)

Total energy

The kinetic energy is divided into

the compressible part with

and

the incompressible part with .

E =1

dx !"dx#* $%2 +

g

2#

2& ' (

) * + " #

E = Eint+ Eq + Ekin

! = " exp i#( )

Ekin=

1

dx!"dx #$%( )"

2

Ekin

c=

1

dx !"dx #$%( )

c

[ ]"2

Ekin

i=

1

dx !"dx #$%( )

i

[ ]"2

div !"#( )i

= 0

rot !"#( )c

= 0

This incompressible kinetic energy Ekini shouldobey the Kolmogorov spectrum.

C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997)

△: 2 < k < 12○: 2 < k < 14□: 2 < k < 16

The right figure shows the energy spectrum at a moment. The left figureshows the development of the exponent n(t). The exponent n(t) goesthrough 5/3 on the way of the dynamics.

n(t)

t k

E(k)

5/3

E(k)~ k -n(t)

In the late stage, however, the exponent deviates from 5/3, because thesound waves resulting from vortex reconnections disturb the cascadeprocess of the inertial range.

Kolmogorov spectrum of quantum turbulenceM. Kobayashi and M. Tsubota, Phys. Rev. Lett. 94, 065302 (2005),J. Phys. Soc. Jpn. 74, 3248 (2005)

1. We solved the GP equation in the wave numberspace in order to use the fast Fouriertransformation.2. We made a steady state of turbulence. In orderto do that, 2-1 We introduced a dissipative term whichdissipates the Fourier component of the high wavenumber, namely, phonons of short wave length. 2-2 We excited the system at a large scale bymoving a random potential.

To solve the GP equation numerically with high accuracy, we usethe Fourier spectral method in space with the periodic boundarycondition in a cube.

The GP equation in the Fourier space

healing length giving the vortex core size

i!

!t" k,t( ) = k 2 # µ( )" k, t( )

+g

V 2" k

1,t( )"* k2,t( )" k # k1 + k2,t( )

k1 ,k 2

$

!2

= 1 g"2

The GP equation in the real space

i!

! t" r, t( ) = #$2 # µ + g" r, t( )

2

[ ]" r, t( )

The GP equation with the small scale dissipation

:healing length giving the vortex core size

We introduce the dissipation that works only in the scale smaller than ξ.

!2

= 1 g"2

{i !" (k)}#

#t$ k,t( ) = k2 ! µ( )$ k,t( )

+g

V 2$ k

1,t( )$* k2,t( )$ k ! k1 + k2,t( )

k1 ,k 2

%

! (k ) = !0" k # 2$ / %( )

How to dissipate the energy at small scales?

Since there is no vortex motion at the scales smaller than ξ, thisdissipation must work for only short-wavelength sound waves.

cf. M. Kobayashi and M. Tsubota, PRL 97, 145301 (2006)

This is done by moving the random potential satisfying thespace-time correlation:

V(x, t)V( ! x , ! t ) = V02exp "

x " ! x ( )2

2X02 "

t " ! t ( )2

2T02

#

$ % %

&

' ( (

The variable X0 determines thescale of the energy-containingrange.V0=50, X0=4 and T0=6.4×10-2

How to inject the energy at large scales?

Thus steady turbulence is obtained.(1)Time development of each energy component

Vortices Phase in a central plane Moving random potential

Thus steady turbulence is obtained.(2)Time development of each energy component

Picture of the cascade process

Quantized vortices

Phonons

In order to confirm this picture, we calculate(1) the energy dissipation rate ε of Ekini

(2) the energy flux Π of the Richardson cascade.

Quantized vortices

Phonons

(1) the energy dissipation rate ε of Ekini

In the steady state, we turn offthe large-scale excitationsuddenly and monitor the timedevelopment of Ekini .

Thus we obtain

from the decay.

! = "d E

kin

i

dt# 12.5 ± 2.3

(2) the energy flux Π of the Richardson cascade

← Dissipation rate ε~12.51. Π is about constant in theinertial range.

2. Π is comparable to ε.

They confirm the picture ofthe inertial range as writtenin textbooks.

Ensemble averaged over 50states.

Energy spectrum of the steady turbulenceThe energy spectrum obeys the Kolmogorov form.

Quantum turbulence isfound to show the essenceof classical turbulence!

2π / X0 2π / ξ

The inertial range issustained by the genuineRichardson cascade ofquantized vortices.

4. Quantum turbulence in a trapped Bose-Einstein condensateM. Kobayashi and M. Tsubota, cond-mat/0703603

Atomic BEC

Superfluid HeVortex tangleVortex array

Two main cooperative phenomena of quantized vorticesare Vortex array and Vortex tangle.

None

Is it possible to make turbulence in a trapped BEC?

(1)We cannot apply some flow to the system.

(2) This is a finite-size system. Can we maketurbulence with enough inertial range?

Then the coherence length is not much smaller than the systemsize. However, we could confirm the Kolmogorov law.

How to make turbulence in a trapped BEC

x

y

z 1. Trap the BEC in aweakly elliptic potential.

!

U x( ) =m" 2

21#$

1( ) 1#$2( )x2 + 1+ $

1( ) 1#$2( )y2 + 1+ $

2( )z2[ ]

2. Rotate the system firstaround the x-axis, nextaround the z-axis.

!

!

" t( ) = "x, "zsin"xt,"z cos"xt( )

Actually this idea has been already used in CT. S. Goto, N. Ishii, S. Kida, and M. Nishioka, Phys. Fluids 19, 061705 (2007)

Rotationaroundone axis

Rotationaroundtwo axes

Numerical analysis of the GP equation

!

i " # x( )[ ]h$

$ t% x,t( ) = "

h2

2m&2 "µ t( ) + g% x,t( )

2

+U x( ) "' t( ) (L x( ))

* +

,

- . % x,t( )

!

U x( ) =m" 2

21#$

1( ) 1#$2( )x2 + 1+ $

1( ) 1#$2( )y2 + 1+ $

2( )z2[ ]

!

" t( ) = "x, "zsin"xt,"z cos"xt( )

For γ(x), we use the dissipation obtained in ourprevious work of the coupled system of the GP andBdG equations.

cf. M. Kobayashi and M. Tsubota, PRL 97, 145301 (2006)

Dynamics of the condensate density and the vortices

Condensate density Vortices

We obtain the steady state of quantum turbulence.

Kinetic energy Anisotropic parameter

Spectrum of the incompressible kinetic energy

1. The energy spectrum obeys the Kolmogorov law.

2. The energy flux is about constant in the inertial range.

Very recently we made QT byrotation around three axes.

The turbulence should be more isotropic.

Motion of the trapping potential Motion of the condensate density

Rotation around two axes and three axes

Rotation around two axes

Rotation around three axes makes more isotropic QT whoseη is closer to 5/3.

Rotation around three axes

-comparison of the energy spectra-

What can we learn from quantum turbulence of atomic BEC?

• Controlling the transition to turbulence bychanging the rotational frequency orinteraction parameters etc.

• The relation of the vortex-size distribution (selfsimilarity in the real space) with theKolmogorov spectrum (self similarity in the kspace).

• Changing the trapping potential or therotational frequency leads to the dimensionalcrossover (2D↔3D) in turbulence.

Summary 16th century

21st century

We discussed the recent interestsin quantum turbulence.

Quantum turbulence consists ofquantized vortices which arestable topological defects.

Quantum turbulence obeys theKolmogorov spectrum too.

Quantum turbulence may give aprototype of turbulence muchsimpler than conventionalturbulence.