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Quantum Turbulence and and Quantum Turbulence Nonlinear Phenomena - - PowerPoint PPT Presentation

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Quantum Turbulence and and Quantum Turbulence Nonlinear Phenomena in Nonlinear Phenomena in Quantum Fluids Quantum Fluids Makoto TSUBOTA Department of Physics, Osaka City University, Japan Review article: M. Tsubota, J. Phys. Soc. Jpn.

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Quantum Turbulence Quantum Turbulence and and Nonlinear Phenomena in Nonlinear Phenomena in Quantum Fluids Quantum Fluids

Makoto TSUBOTA

Department of Physics, Osaka City University, Japan

Review article: M. Tsubota, J. Phys. Soc. Jpn. 77, 111006(2008) Progress in Low Temperature Physics, vol.16 (Elsevier, 2008), eds. W. P. Halperin and M. Tsubota

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A03 Bose Superfluids and Quantized Vortices

Studies of physics of quantized vortices and “new” superfluid turbulence

  • M. Tsubota, T. Hata, H. Yano

Public participation: M. Machida, D. Takahashi

Superfluidity of atomic gases with internal degrees of freedom

  • M. Ueda, T. Hirano, H. Saito, S. Tojo, Y. Kawaguchi

Public participation: Y. Kato

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Contents

  • 1. Why is QT (quantum turbulence) so important?
  • 2. Outputs of our group through this five-years project
  • 3. Very new results

3.1 Steady state of counterflow quantum turbulence: Vortex filament simulation with the full Biot-Savart law

  • H. Adachi, S. Fujiyama, MT, Phys. Rev. B (in press) (Editors

suggestion) 3.2 Quantum Kelvin-Helmholtz instability in two-component Bose- Einstien condensates

  • H. Takeuchi, N. Suzuki, K. Kasamatsu, H. Saito, MT, Phys. Rev.

B (in press)

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  • 1. Why is QT so important ?
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Leonardo Da Vinci (1452-1519)

Da Vinci observed turbulent flow and found that turbulence consists

  • f many vortices.

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

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SLIDE 7

Certainly turbulence looks to have many vortices.

Turbulence behind a dragonfly However, these vortices are unstable; they repeatedly appear, diffuse and disappear.

It is not so straightforward to confirm the Da Vinci message in classical turbulence.

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

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The Da Vinci message is actually realized in quantum turbulence comprised of quantized vortices.

Quantum turbulence

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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.

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 vs The order of the coherence length.

= h / m

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

・The vortices are unstable. Not easy to identify each vortex. ・The circulation differs from one to another, not conserved. ・The quantized vortices are stable topological defects. ・Every vortex has the same circulation. ・Circulation is conserved.

Motion of vortex cores

QT can be simpler than CT, because each element of turbulence is definite.

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Quantum turbulence and quantized vortices were discovered in superfluid 4He in 1950’s. This field has become a major one in low temperature physics, being now studied in superfluid 4He, 3He and even cold atoms.

Current important topics are well reviewed in Progress in Low Temperature Physics, vol.16 (Elsevier, 2008), eds. W. P. Halperin and

  • M. Tsubota
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We confirmed for the first time the Kolmogorov law from the Gross-Pitaevskii model.

Quantum turbulence is found to express the essence of classical turbulence!

2π / X0 2π / ξ

  • M. Kobayashi and MT, PRL 94, 065302 (2005), JPSJ 74, 3248 (2005)
  • 2. Outputs of our group through this five-years project
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SLIDE 13

V.B.Eltsov, A.P.Finne, R.Hänninen, J.Kopu, M.Krusius, MT and E.V.Thuneberg, PRL 96, 215302 (2006) We discovered twisted vortex state in 3He-B theoretically, numerically and experimentally.

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  • R. Hänninen, MT, W. F. Vinen, PRB 75, 064502 (2007)

How remnant vortices develop to a tangle under AC flow

R=100 µm ω=200 Hz V=150 mm/s

  • a. Kelvin waves form
  • n the bridged

vortex line.

  • b. Vortex rings

nucleate by reconnection.

  • c. Turbulence

develops.

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Parameters for the sphere : Radius 3μm, Frequency 1590 Hz

  • R. Goto, S. Fujiyama, H. Yano, Y. Nago, N. Hashimoto, K. Obara, O. Ishikawa, MT, T. Hata,

PRL 100, 045301(2008) 30mm/s 137 mm/s

We found the transition to QT by seed vortex rings.

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SLIDE 16

Condensate density Quantized vortices

Two precessions (ωx×ωz)

  • M. Kobayashi and MT, PRA76, 045603(2007)

We showed how to make QT in a trapped BEC and obtained the energy spectrum consistent with the Kolmogorov law. Next talk by Bagnato!

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SLIDE 17

Vortex stripe

  • r

Double-core vortex lattice Triangular lattice Square lattice Vortex sheet

g12/g / 1

K.Kasamatsu, MT and M.Ueda PRL 91, 150406 (2003)

g12 /g =1

  • K. Kasamatsu and MT, PRA79, 023606(2009)

We revealed vortex sheet in rotating two-component BECs.

Square lattice |ψ1|2|ψ2|2

g12: Interaction between two components

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SLIDE 18
  • K. Kasamatsu and MT, PRA79, 023606(2009)

We revealed vortex sheet in rotating two-component BECs.

Density profile for g12/g= 1.5 (a), 2.0 (b) and 3.0 (c ). Imbalanced case with g12/g= 1.1, u1=4000, and u2=3000 (a), 3500 (b) and 3900 (c ).

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Contents

  • 1. Why is QT (quantum turbulence) so important?
  • 2. Outputs of our group through this five-years project
  • 3. Very new results

3.1 Steady state of counterflow quantum turbulence: Vortex filament simulation with the full Biot-Savart law

  • H. Adachi, S. Fujiyama, MT, Phys. Rev. B (in press) (Editors

suggestion) 3.2 Quantum Kelvin-Helmholtz instability in two-component Bose- Einstien condensates

  • H. Takeuchi, N. Suzuki, K. Kasamatsu, H. Saito, MT, Phys. Rev.

B (in press)

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3.1 Steady state of counterflow quantum turbulence: Vortex filament simulation with the full Biot-Savart law Hiroyuki Adachi, Shoji Fujiyama, MT, Phys. Rev. B (in press) (Editors suggestion) arXiv:0912.4822

Vortex tangle

Heater Normal flow Super flow

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

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Vortex filament model (Schwarz)

A vortex makes the superflow of the Biot-Savart law, and moves with this local flow. At a finite temperature, the mutual friction should be considered. s r

˙ s

0 =

4

  • s
  • s +

4 s1 r

( ) ds1

s1 r

3 L '

  • + vs,a s

( )

˙ s = ˙ s

0 +

s vn ˙ s

( )

  • s
  • s vn ˙

s

( )

[ ]

The approximation neglecting the nonlocal term is called the LIA(Localized Induction Approximation).

˙ s

0 =

4

  • s
  • s + vs,a s

( )

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Schwarz’s simulation(1) PRB38, 2398(1988)

Schwarz simulated the counterflow turbulence by the vortex filament model and

  • btained the statistically steady

state. However, this simulation was unsatisfactory.

  • 1. All calculations were

performed by the LIA.

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Schwarz’s simulation(2) PRB38, 2398(1988)

However, this simulation was unsatisfactory.

  • 1. All calculation was performed

by the LIA.

  • 2. He used an artificial mixing

procedure in order to obtain the steady state.

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After Schwarz, there has been no progress on the counterflow simulation. In this work we made the steady state of counterflow turbulence by fully nonlocal simulation.

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Simulation by the full Biot-Savart law

BOX (0.1cm)3 Vns = 0.367cm/s T = 1.6(K) Periodic boundary conditions for all three directions

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Comparison between LIA and full Biot-Savart

Full Biot-Savart LIA Vortices become anisotropic, forming layer structures. We need intervortex interaction.

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! "!! #!!! #"!! $!!! $"!! ! #! $! %! &! "!

'()*+ ,)#-./ $+

Full Biot-Savart LIA

Developments of the line-length density between LIA and Full Biot-Savart T=1.6 K、 Vns=0.367 cm/s、 box=(0.2 cm)3

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SLIDE 28

!"# !"#$ !"% !"%$ !"& !"&$ !"' !"'$ ! (! )! *! +! $!

,-./0 122-.,0

Full Biot-Savart LIA

Anisotropic parameter T=1.6 K、 Vns=0.367 cm/s、 box=(0.2 cm)3

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SLIDE 29

Quantitative comparison with observations

The parameter γ agrees with the experimental observation quantitatively.

Childers and Tough, Phys. Rev. B13, 1040 (1976)

(154) 157 2.1 K 133 140 1.9 K 93 109 1.6 K 59 54 1.3 K

γ（s/cm2） Experment γ（s/cm2) Our calculation

An important criterion of the steady state is to obtain

L: Vortex density, vns:relative velocity in counterflow

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SLIDE 30

Observation of the velocity by the solid hydrogen particles in counterflow

Upward particles Downward particles

Paoletti,Fiorito,Sreenivasan,and Lathrop, J.Phys. Soc. Jpn. 77,111007(2008)

The broken line shows The downward particles should be related with the velocity of vortices!

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3.2 Quantum Kelvin-Helmholtz instability in two-component Bose-Einstien condensates Hiromitsu Takeuchi, Naoya Suzuki, Kenichi Kasamatsu, Hiroki Saito, MT,

  • Phys. Rev. B (in press): arXiv.0909.2144

Hydrodynamic instability of shear flows

KHI: KHI:

One of the most fundamental instability in classical fluid dynamics We study the KHI in two-component atomic Bose-Einstein condensates(BECs).

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Classical KHI

When the relative velocity Vd=|V1-V2| is sufficiently large, the vortex sheet becomes dynamically unstable and the interface modes with complex frequencies are amplified.

interface

V1 V2

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KHI in nature KHI in nature

http://hmf.enseeiht.fr/travaux/CD0001/travaux/optmfn/hi/01pa/hyb72/kh/kh_theo.htm

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Superfluid can flow relative to a wall even in thermal equilibrium like an inviscid fluid. wall superfluid

Two important quantum effects in superfluid

  • 1. Superfluidity

Superfluidity is broken when the relative velocity exceed a critical velocity.

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SLIDE 35
  • 2. Quantized vortex

Vortices appear as topological defects. The circulation around vortices are quantized.

A quantized vorttex

：contour

n: interger

Two important quantum effects in superfluid

MT, K. Kasamatsu, M. Ueda, PRA65, 023603(2002)

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Quantum effects play important roles in the quantum KHI. Quantum effects play important roles in the quantum KHI.

Quantum effect on the KHI

stationary shear flow interface wave different states or turbulence

nonlinear dynamics nonlinear dynamics linear stability linear stability +superfluid stability

+superfluid stability

→q →quantized vortices uantized vortices

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Two-component BEC

two order parameters (macroscopic wave functions)

Gross-Pitaevskii(GP) equation m=m1=m2 g=g11=g22

particle density phase

superfluid velocity of component j

iht

1 = h2

2m1 2 + U1 + g11

1 2 + g12 2 2

  • 1

iht2 = h2 2m2 2 + U2 + g12

1 2 + g22 2 2

  • 2
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1 2 3 4 5

  • 30
  • 20
  • 10

10 20 30 x

interface layer g12=10g

y/ξ interface component 1 component 2

y x

V1=0 V2<0 strong repulsive interaction

Phase-separated two-component BEC

Uj(y)=fj y f1 =-f2 > 0 condition for phase separation： g12>g µ=µ1-mV1

2/2= µ2-mV2 2/2 > 0

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Phase diagram of quantum KHI

stable stable

V V2

2=V

=VD

D

wave number

Vd

V V2

2=V

=VL

L

dynamic thermodynamic ω：frequency of ripplon dynamic KHI（analogue of the classical KHI） thermodynamic KHI（unique to quantum KHI）

thermodynamic KHI

( (V VL

L<|V

<|V2

2|)

|) both KHI ( (V VD

D<|V

<|V2

2|

|) )

dynamic KHI in energy conserving systems

dynamic instability superflow instability

  • Analysis by the Bogoliubov-de Gennes equations-
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SLIDE 40

V1=0 V2≠0

vortex sheet vortex sheet

Dynamic KHI in energy conserving system

x y ωeff

ji=nivi

effective super-current velocity

stable stable

| |V V2

2|

|=V =VD

D

wave number

|V |V2

2|>

|>V VD

D

relative velocity

Vd

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SLIDE 41

V1=0 V2≠0

vortex sheet vortex sheet

Dynamic KHI in energy conserving system

x y ωeff

ji=nivi

effective super-current velocity

stable stable

| |V V2

2|

|=V =VD

D

wave number

|V |V2

2|>

|>V VD

D

relative velocity

Vd

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x y z x z V1=0 V2≠0 Kelvin waves cause more complicated dynamics towards quantum turbulence.

|V |V2

2|>

|>V VD

D

Dynamic KHI in 3D system

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SLIDE 43

V1=0 V2≠0

Thermodynamic KHI in dissipative system

V VD

D>|

>|V Vd

d|>

|>V VL

L

GP model with dissipation

  • K. Kasamatsu, M. Tsubota M. Ueda, Phys. Rev. A 67, 033610 (2003).

x y ωeff

stable stable

| |V V2

2|

|=V =VDL

DL

relative velocity

Vd

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SLIDE 44

Thermodynamic KHI in dissipative system

V VD

D>|

>|V Vd

d|>

|>V VL

L

GP model with dissipation

  • K. Kasamatsu, M. Tsubota M. Ueda, Phys. Rev. A 67, 033610 (2003).

x y ωeff

stable stable

| |V V2

2|

|=V =VDL

DL

relative velocity

Vd

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Summary

  • 1. Why is QT (quantum turbulence) so important?
  • 2. Outputs of our group through this five-years project
  • 3. Very new results

3.1 Steady state of counterflow quantum turbulence: Vortex filament simulation with the full Biot-Savart law

  • H. Adachi, S. Fujiyama, MT, Phys. Rev. B (in press) (Editors

suggestion): arXiv:0912.4822 3.2 Quantum Kelvin-Helmholtz instability in two-component Bose- Einstien condensates

  • H. Takeuchi, N. Suzuki, K. Kasamatsu, H. Saito, MT,
  • Phys. Rev. B (in press): arXiv:0909.2144