makoto tsubota- quantum turbulence: from superfluid helium to atomic bose-einstein condensates

8/3/2019 Makoto Tsubota- Quantum Turbulence: From Superfluid Helium to Atomic Bose-Einstein Condensates

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4XDQWXP7XUEXOHQFH4XDQWXP7XUEXOHQFH--From Superfluid Helium to AtomicFrom Superfluid Helium to Atomic

BoseBose--Einstein CondensatesEinstein Condensates--

Makoto TSUBOTADepartment of Physics,

Osaka City University, JapanThanks to: M. Kobayashi, S. Kida and many friends

Review article: M. Tsubota, arXive: 0806.2737 (J. Phys. Soc. Jpn. (in press))

Progress in Low Temperature Physics, vol.16 (Elsevier), eds. W. P. Halperin and M. Tsubota

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My message

1. Quantum turbulence (QT) has recently become one of the most important fields in

low-temperature physics.

2. QT consists of quantized vortices, which are stabletopological defects. Since each ³element´ is clear and

well-defined, QT is able to give a much simpler prototype

than classical turbulence.

3. QT has been studied in superfluid helium for more than

50 years. Since the mid 90s, QT research has tended

toward a new direction, now involving atomic Bose-

Einstein condensates (BECs) too.





Another Da Vinci code Another Da Vinci code

Leonardo Da Vinci

(1452-1519) Da Vinci observed turbulent flow

in water and found that turbulence

consisted of many vortices.

Turbulence is not a simple disordered state but

rather it consists of some structures with vortices.



Turbulence appears to have many vortices«

Turbulence in the wake of a dragonfly:

«however, these vortices are unstable

- they repeatedly appear, diffuse and disappear.

http://www.nagare.or.jp/mm/2004/gallery/iida/dragonfly.html

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A quantized vortex is a vortex of superflow in a BEC.

Any rotational motion in superfluid is sustained by

quantized vortices.(i) The circulation is quantized.

(iii) The core size is very small.

v s d s ! On´ n = 0,1,2,

A vortex with n2 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.

r

s

(r )

rot v

sThe order of the coherence

length.

O ! h / m



C lassical Turbulence ( C T) vs. Quantum Turbulence (QT)C lassical Turbulence ( C T) vs. Quantum Turbulence (QT)

Classical turbulence Quantum turbulence

- The vortices are unstable. Not easy

to identify each vortex.

-The circulation varies with time.

- it is not conserved.

- Quantized vortices are stable

topological defects.- Each vortex has the same

circulation.

- Circulation is conserved.

Motion of

vortex

cores

QT is much simpler

than CT, because eachelement of turbulence

is definite and clear.







Contents of this talk:Contents of this talk:

1.1. History of QT researchHistory of QT research

2.2. Recent QT studies in superfluid heliumRecent QT studies in superfluid helium

3.3. QT in atomic BoseQT in atomic Bose--Einstein condensatesEinstein condensates

Research into QT can make a breakthrough into one of the

great mysteries of nature.



1. History of QT research

Liquid 4He enters the superfluid state below 2.17 K (P point)

with Bose-Einstein condensation (BEC).

Its hydrodynamics are well described by the two-fluid model:

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Temperature

point

The two-fluid model

The system is a mixture of inviscid

superfluid and viscous normal fluid.

V ! V s

Vn

j ! V sv

s Vnvn

Density Velocity Viscosity Entropy

Superfluid 0 0

Normal fluid

V sT

Vn T

v sr

vnr L

nT sn T



The two-fluid model can explain various experimentally

observed phenomena of superfluidity (e.g., thethermomechanical effect, film flow, etc.)

However, «



(ii) v > vs c

vs

A tangle of quantized vortices develops. The two fluids

interact through mutual friction generated by tangling, and

the superflow decays.

v = 0s

Superfluidity breaks down in fast flow

(i) v < vs c

vs

The two fluids do not interact so that the superfluid canflow forever without decaying.

vst p g

(some critical velocity)



1955: Feynman proposed that ³superfluid turbulence´ consists

of a tangle of quantized vortices.

1955 ± 1957: Vinen observed ³superfluid turbulence´.

Mutual friction between the vortex tangle and the normalfluid causes dissipation of the flow.

Progress in Low Temperature Physics

Vol. I (1955), p.17

Such a large vortex should

break up into smaller vortices

similar to the cascade processin classical turbulence.



Many experimental studies were conducted chiefly on thermal

counterflow of superfluid 4He.

1980s K . W. Schwarz Phys. Rev. B38, 2398 (1988)Performed a direct numerical simulation of the three-dimensional

dynamics of quantized vortices and succeeded in quantitatively

explaining the observed temperature difference (T .

Vortex tangle

Heater

Normal flow Superflow



Development of a vortex tangle in a thermal counterflow

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

Schwarz obtained numerically the

statistically steady state of a vortex

tangle, which is sustained by the

competition between the appliedflow and the mutual friction. The

calculated vortex density L(vns, T )

agreed quantitatively with

experimental data.

vs vn

Counterflow turbulence has been

successfully explained.

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What is the relation between superfluid

turbulence and classical turbulence

Most studies of superfluid turbulence have focused onthermal counterflow.

No analogy with classical turbulence

When Feynman drew the above figure, he was thinking of a

cascade process in classical turbulence.



How can we study

the similarities and differences between QT and CT?

One of the best ways is to focus on the most importantstatistical law in CT, namely, Kolmogorov¶s law:

E (k ) ! C I2 / 3k 5 / 3



E nergy spectrum of fully developed turbulence

Energy-

containing

range

Inertial

range

Energy-dissipative

range

E nergy spectrum of turbulence

Kolmogorov¶s law

Energy spectrum of the velocity field

Energy-containing range

Energy is in jected into the system at .k } k 0 !1/ "0

Inertial range

Dissipation does not work. The nonlinear

interaction transfers energy from low k region

to high k region.

K olmogorov¶s law : E ( k )=C 2/3 k -5/3

Energy-dissipation range

Energy is dissipated at a rate I at the

K olmogorov wavenumber, k c = (I/R3 )1/4.

Richardson cascade process

E !1

2v´

2

d r ! E (k )d k ´



Everybody believes

that turbulence issustained by

Richardson cascade.

However, this is only

based on an image ;nobody has ever

clearly confirmed it.

One reason is that it is

too difficult toidentify each eddy in

a classical fluid.



For QT to be a simple model of turbulence,

it should show express the essence of turbulence.

Based on this motivation, QT studies have Based on this motivation, QT studies have

entered a new stage since the mid 1990s!entered a new stage since the mid 1990s!



Recent studies on energy spectra of superfluid heliumExperimental confirmation of the Kolmogorov spectra:

J. Maurer and P. Tabeling, Europhy. Lett. 43, 29 (1998) 1.4 K < T < T

S.R. Stalp, L. Skrbek and R.J. Donnely, PRL 82, 4831 (1999) 1.4 < T < 2.15 K

D.I. Bradley, D.O. Clubb, S.N. Fisher, A. M. Guenault, R.P. Haley, C.J.

Matthews, G.R. Pickett, V. Tsepelin, and K. Zaki, PRL 96, 035301 (2006) 3He

Theoretical consideration of QT at finite temperatures:

W.F. Vinen, Phys. Rev. B 61, 1410 (2000)D. Kivotides, J. C. Vassilicos, D. C. Samuels, C. F. Barenghi, Europhy. Lett. 57, 845 (2002)

What happens to QT and energy spectra at very low

temperatures?

2. Recent QT studies of superfluid helium



Decaying K olmogorov turbulence in a model of superflow

C. 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-fluidcomponent

T. Araki, M.Tsubota and S.K.Nemirovskii, Phys. Rev. Lett. 89, 145301(2002)

The vortex-filament model

K olmogorov spectrum of superfluid turbulence: Numerical analysis of the 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).

Three studies have directly investigated numerically QT

energy spectrum at zero temperature:



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

By using the GP model, they

obtained a vortex tangle with

starting from the Taylor-Green

vortices.t



In order to study the K olmogorov spectrum, it is necessary to

decompose 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

d x V´d x** 2

g

2*

2«-¬

»½¼´ *

E ! E int E q E k in

* ! V exp i U

E k in !1

d x V´d x * U ´

2

E k inc

!1

d x V´d x * U c? A´

2

E k ini

!1

d x V´d x * U i? A´

2

div * U i

! 0

rot * U c! 0

This incompressible kinetic energy E kini should

obey 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 figure

shows 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 the

sound waves resulting from vortex reconnections disturb the cascade

process of the inertial range.



K olmogorov spectrum of quantum turbulenceM. K obayashi and M. Tsubota, Phys. Rev. Lett. 94, 065302 (2005), J.

Phys. Soc. Jpn. 74, 3248 (2005)

1. We solved the GP equation in wavenumber spacein order to use fast Fourier transform.

2. We achieved steady-state turbulence by:2-1 Introducing a dissipative term that dissipateshigh-wavenumber Fourier components (i.e., short-wavelength phonons).

2-2 Exciting the system on a large scale by movingthe random potential.



To solve the GP equation numerically with high accuracy, we use the

Fourier spectral method in space with cubic periodic boundary

conditions.

GP equation in Fourier space:

(healing length giving the vortex core size)

i

x

xt * k , t ! k 2

Q * k ,t

g

V 2* k 1, t ** k 2, t * k k 1 k 2, t

k 1 ,k 2

§

\2!

1 g *

2

GP equation in real space:

i

x

x t * r, t !

2

Q g * r,t 2

? A* r,t



GP equation with small-scale dissipation

(healing length giving the vortex core size)

We introduce the dissipation that operates only in the scale smaller than \.

\ 2! 1 g *

2

{i K (k )}x

xt * k , t ! k

2 Q * k , t

g

V 2* k 1,t ** k 2, t * k k 1 k 2, t

k 1 ,k 2

§

K (k ) ! K 0 U k 2T / \

H ow to dissipate the energy at small scales?

This type of dissipation is justified by coupled analysis of the GP

and Bogoliubov- de Gennes equations.

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



This is done by moving the random potential satisfying the

space-time correlation:

V (x, t )V ( dx , dt ) ! V 02exp

x dx 2

2 X 02

t dt 2

2T 02

«

-¬¬

»

½¼¼

The variable X 0 determines thescale of the energy-containing

range.

V 0=50, X 0=4 and T 0=6.410-2

H ow to inject the energy at large scales?

X0





Thus, steady-state turbulence is obtained (1)

Time development of each energy component

Vortices Phase in a central plane Moving random potential

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Thus, steady-state 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 of E kini

(2) the energy flux of the Richardson cascade.

Quantized vortices

Phonons

Is comparable to ?



(1) the energy dissipation rate of E kini

In the steady state, we turn off

the large-scale excitation

suddenly and monitor the time

development of E kini .

Thus we obtain

from the decay.

I ! d E kin

i

dt } 12.5 s 2.3



(2) the energy flux of the Richardson cascade

Dissipation rate 12.5

1. is about constant in the

inertial range.

2. is comparable to .

They confirm the picture of

the inertial range as written

in textbooks.

Ensemble averaged over 50 states.



Energy spectrum of steady-state turbulence

The energy spectrum obeys the Kolmogorov law.

Quantum turbulence is

found to express the

essence of classicalturbulence!

2 / X0 2 /

The inertial range is

sustained by a genuine

Richardson cascade of quantized vortices.

M. K obayashi and M. Tsubota, Phys. Rev. Lett. 94,

065302 (2005), J. Phys. Soc. Jpn. 74, 3248 (2005)



Overall picture of energy spectrum of QT at 0 K

Kelvin-wave cascadeD. Kivotides, J. C. Vassllicos, D. C. Samuels,

C. F. Barenghi, PRL 86, 3080 (2001)

W. F. Vinen, M. Tsubota, A. Mitani, PRL91,

135301 (2003)

E. Kozik, B. V. Svistunov, PRL 92, 035301

(2004); PRL 94, 025301(2005); PRB 72,

172505 (2005)

Classical-quantum crossover

V. S. L¶vov, S. V. Nazarenko, O. Rudenko,

PRB 76, 024520 (2007)E. Kozik, B. V. Svistunov, PRB 77, 060502

(2008)

: mean distance between vortices

Classical Quantum



3. QT in atomic Bose-Einstein condensates(BECs)M. Kobayashi and M. Tsubota, Phys. Rev. A76, 045603 (2007)

Vortex array Vortex tangle

Superfluid He

Atomic BEC

Two principal cooperative phenomena of quantized

vortices are vortex arrays and vortex tangles.

None

Packard (Berkeley)

Ketterle (MIT)



Usual setup of atomic BECs

K.W.Madison et.al Phys.Rev Lett 84, 806 (2000)

Optical spoon

(oscillating

laser beam)

Rotation frequency

;

z

x

y 100Qm

5Qm

20Qm 16Qm

³cigar-shape´

Laser cooling



Observation of quantized vortices in atomic BECs

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

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

P. Engels, et. al

PRL 87, 210403

(2001)

ENS

MIT

JILA

E. Hodby, et. al

PRL 88, 010405

(2002)

Oxford



Dynamics of vortex lattice formation by the GP model

Time development of condensate density n0

; ! 0.7[B Experiment

M. Tsubota, K. Kasamatsu

and M. Ueda, Phys. Rev. A

65, 023603 (2002)

V trap(r ) !1

2m[

B

2r

2

=(r )! n0 (r )ei U(r )

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K.W. Madison et.al., PRL 86,

4443 (2001)



Is it possible to produce turbulence in a trapped BEC?

(1) We cannot apply any flow to this system.

(2) This is a finite-size system. Can we make

turbulence with a sufficiently wide inertial range?

The coherence length is not much smaller than the system size.

However, we could confirm Kolmogorov¶s law.

M. Kobayashi and M. Tsubota, Phys. Rev. A76, 045603 (2007)





How to produce turbulence in a trapped BEC

x

y

z 1. Trap the BEC in a weak

elliptical potential.

U x !m [

2

2

1 I1 1 I2 x 2 1 I1 1 I2 y 2 1 I2 z2? A

2. Rotate the system first

around the x-axis, then

around 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)

Rotation

around

one axis

Rotationaround

two axes

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Thanks to S. Goto



Condensate density Quantized vortices

Two precessions (xz) Three precessions (y x z)

Condensate density Quantized vortices













Energy spectra for two cases

Three precessions produce more isotropic QT, whose is closer to 5/3.

M. Kobayashi and M. Tsubota, Phys. Rev. A76, 045603 (2007)

Two precessions Three precessions



What can we learn from QT of atomic BEC?

Controlling the transition to turbulence by changingthe rotational frequency or interaction parameters, etc.

We can visualize quantized vortices. We can consider

the relation of real space cascade of vortices and the

wavenumber space cascade (Kolmogorov¶s law).

Changing the trapping potential or the rotational

frequency leads to dimensional crossover (2D3D)

in turbulence.



Controlling the transition to turbulence

x

z

Vortex lattice

V o r t e x l a

t t i c e

Vortex tangle

?

0



xz



What can we learn from quantum turbulence of atomic BEC?

Controlling the transition to turbulence by changing therotational frequency or interaction parameters, etc.

We can visualize quantized vortices. We can consider

the relation of real space cascade of vortices and the

wavenumber space cascade (Kolmogorov¶s law).

Changing the trapping potential or the rotational

frequency leads to dimensional crossover (2D3D) in

turbulence.

We can obtain the most controllable turbulence!

y z

x

y

z

x

y z

x



Summary 16th century

21st century

We discussed the recent interests

in quantum turbulence.

Quantum turbulence consists of

quantized vortices, which are

stable topological defects.

Quantum turbulence may give a

prototype of turbulence that is

much simpler than classical

turbulence.

Review article: M. Tsubota, arXive:0806.2737

(J. Phys. Soc. Jpn. (in press))

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makoto tsubota- quantum turbulence: from superfluid helium to atomic bose-einstein condensates

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