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Quantum Turbulence: -From Superfluid Helium to Atomic Bose-Einstein Condensates- Makoto TSUBOTA Department of Physics, Osaka City University, Japan Thanks to: M. Kobayashi, S. Kida and many friends Review article: M. Tsubota, arXive: (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 stable topological defects. Since each “element” 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.

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Another Da Vinci code Leonardo Da Vinci ( ) 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.

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Turbulence appears to have many vortices… Turbulence in the wake of a dragonfly: …however, these vortices are unstable - they repeatedly appear, diffuse and disappear.

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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. 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)(r) rot v s The order of the coherence length.

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Classical Turbulence (CT) vs. Quantum Turbulence (QT) Classical turbulenceQuantum 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 each element of turbulence is definite and clear.

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Contents of this talk: 1.History of QT research 2.Recent QT studies in superfluid helium 3.QT in atomic Bose-Einstein condensates Research into QT can make a breakthrough into one of the great mysteries of nature.

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1. History of QT research Liquid 4 He enters the superfluid state below 2.17 K ( point) with Bose-Einstein condensation (BEC). Its hydrodynamics are well described by the two-fluid model: Temperature point The two-fluid model The system is a mixture of inviscid superfluid and viscous normal fluid. DensityVelocityViscosityEntropy Superfluid 0 0 Normal fluid

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The two-fluid model can explain various experimentally observed phenomena of superfluidity (e.g., the thermomechanical effect, film flow, etc.) However, …

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(ii) v > v sc v s A tangle of quantized vortices develops. The two fluids interact through mutual friction generated by tangling, and the superflow decays. v = 0 s Superfluidity breaks down in fast flow (i) v < v sc v s The two fluids do not interact so that the superfluid can flow forever without decaying. v s (some critical velocity)

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1955: Feynman proposed that “superfluid turbulence” consists of a tangle of quantized vortices – 1957: Vinen observed “superfluid turbulence”. Mutual friction between the vortex tangle and the normal fluid 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 process in classical turbulence.

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Many experimental studies were conducted chiefly on thermal counterflow of superfluid 4 He. 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 flowSuperflow

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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 applied flow and the mutual friction. The calculated vortex density L(v ns, T) agreed quantitatively with experimental data. v s v n 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 on thermal counterflow. ⇨ No analogy with classical turbulence When Feynman drew the above figure, he was thinking of a cascade process in classical turbulence.

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How can we study the similarities and differences between QT and CT? One of the best ways is to focus on the most important statistical law in CT, namely, Kolmogorov’s law:

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Energy spectrum of fully developed turbulence Energy- containing range Inertial range Energy-dissipative range Energy spectrum of turbulence Kolmogorov’s law Energy spectrum of the velocity field Energy-containing range Energy is injected into the system at. Inertial range Dissipation does not work. The nonlinear interaction transfers energy from low k region to high k region. Kolmogorov’s law : E(k)=Cε 2/3 k -5/3 Energy-dissipation range Energy is dissipated at a rate at the Kolmogorov wavenumber, k c = ( / 3 ) 1/4. Richardson cascade process

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Everybody believes that turbulence is sustained 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 to identify each eddy in a classical fluid.

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For QT to be a simple model of turbulence, it should show express the essence of turbulence. Based on this motivation, QT studies have entered a new stage since the mid 1990s!

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Recent studies on energy spectra of superfluid helium Experimental confirmation of the Kolmogorov spectra: J. Maurer and P. Tabeling, Europhy. Lett. 43, 29 (1998) 1.4 K < T < T λ 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, (2006) 3 He 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

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Decaying Kolmogorov 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-fluid component T. Araki, M.Tsubota and S.K.Nemirovskii, Phys. Rev. Lett. 89, (2002) The vortex-filament model Kolmogorov 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, (2005), J. Phys. Soc. Jpn.74, 3248 (2005). Three studies have directly investigated numerically QT energy spectrum at zero temperature:

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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 ＝２ ４ ６ ８ １０ １２

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In order to study the Kolmogorov 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. This incompressible kinetic energy E kin i should obey the Kolmogorov spectrum.

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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) goes through 5/3 on the way of the dynamics. n(t) tk 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.

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Kolmogorov spectrum of quantum turbulence M. Kobayashi and M. Tsubota, Phys. Rev. Lett. 94, (2005), J. Phys. Soc. Jpn. 74, 3248 (2005) 1. We solved the GP equation in wavenumber space in order to use fast Fourier transform. 2. We achieved steady-state turbulence by: 2-1 Introducing a dissipative term that dissipates high-wavenumber Fourier components (i.e., short- wavelength phonons). 2-2 Exciting the system on a large scale by moving the random potential.

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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) GP equation in real space:

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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 . How 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, (2006)

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This is done by moving the random potential satisfying the space-time correlation: The variable X 0 determines the scale of the energy-containing range. V 0 =50, X 0 =4 and T 0 =6.4×10 -2 How to inject the energy at large scales? X0X0 ξ

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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

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Picture of the cascade process Quantized vortices Phonons

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In order to confirm this picture, we calculate (1) the energy dissipation rate ε of E kin i (2) the energy flux Π of the Richardson cascade. Quantized vortices Phonons Is Π comparable to ε ?

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(1) the energy dissipation rate ε of E kin i In the steady state, we turn off the large-scale excitation suddenly and monitor the time development of E kin i. Thus we obtain from the decay.

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(2) the energy flux Π of the Richardson cascade ← Dissipation rate ε 〜 Π is about constant in the inertial range. 2. Π is comparable to ε. They confirm the picture of the inertial range as written in textbooks. Ensemble averaged over 50 states.

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Energy spectrum of steady-state turbulence The energy spectrum obeys the Kolmogorov law. Quantum turbulence is found to express the essence of classical turbulence! 2π / X 0 2π / ξ The inertial range is sustained by a genuine Richardson cascade of quantized vortices. M. Kobayashi and M. Tsubota, Phys. Rev. Lett. 94, (2005), J. Phys. Soc. Jpn. 74, 3248 (2005)

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Overall picture of energy spectrum of QT at 0 K Kelvin-wave cascade D. Kivotides, J. C. Vassllicos, D. C. Samuels, C. F. Barenghi, PRL 86, 3080 (2001) W. F. Vinen, M. Tsubota, A. Mitani, PRL91, (2003) E. Kozik, B. V. Svistunov, PRL 92, (2004); PRL 94, (2005); PRB 72, (2005) Classical-quantum crossover V. S. L’vov, S. V. Nazarenko, O. Rudenko, PRB 76, (2007) E. Kozik, B. V. Svistunov, PRB 77, (2008) ℓ: mean distance between vortices ClassicalQuantum

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3. QT in atomic Bose-Einstein condensates(BECs) M. Kobayashi and M. Tsubota, Phys. Rev. A76, (2007) Vortex arrayVortex tangle Superfluid He Atomic BEC Two principal cooperative phenomena of quantized vortices are vortex arrays and vortex tangles. None Packard (Berkeley) Ketterle (MIT)

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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 100 m 5m5m 20 m 16 m “cigar-shape” Laser cooling

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Observation of quantized vortices in atomic BECs K.W. Madison, et. al PRL 84, 806 (2000) J.R. Abo-Shaeer, et. al Science 292, 476 (2001) P. Engels, et. al PRL 87, (2001) ENS MIT JILA E. Hodby, et. al PRL 88, (2002) Oxford

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Dynamics of vortex lattice formation by the GP model Time development of condensate density Experiment M. Tsubota, K. Kasamatsu and M. Ueda, Phys. Rev. A 65, (2002) K.W. Madison et.al., PRL 86, 4443 (2001)

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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, (2007)

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How to produce turbulence in a trapped BEC x y z1. Trap the BEC in a weak elliptical potential. 2. Rotate the system first around the x-axis, then around the z-axis.

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Actually, this idea has been already used in CT. S. Goto, N. Ishii, S. Kida, and M. Nishioka, Phys. Fluids 19, (2007) Rotation around one axis Rotation around two axes Thanks to S. Goto

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Precession Precessing motion of a gyroscope Spin axis itself rotates around another axis. We consider the case where the spinning and precessing rotational axes are perpendicular to each other. Hence, the two rotations do not commute, and thus cannot be represented by their sum. ΩzΩz ΩxΩx Are these two rotations represented by their sum? No!

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Condensate density Quantized vortices Two precessions (ω x ×ω z )Three precessions (ω y × ω x × ω z ) Condensate density Quantized vortices

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Energy spectra for two cases Three precessions produce more isotropic QT, whose η is closer to 5/3. M. Kobayashi and M. Tsubota, Phys. Rev. A76, (2007) Two precessionsThree precessions

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What can we learn from QT of atomic BEC? Controlling the transition to turbulence by changing the 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 (2D↔3D) in turbulence.

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Controlling the transition to turbulence ΩxΩx ΩzΩz Vortex lattice Vortex tangle ? 0 ωx×ωzωx×ωz

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What can we learn from quantum turbulence of atomic BEC? Controlling the transition to turbulence by changing the 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 (2D↔3D) in turbulence. We can obtain the most controllable turbulence! y z x y z x y z x

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

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