An introduction to superfluidity

1. Relaxation, Turbulence, and Non-Equilibrium Dynamics of Matter Fields - RETUNE 2012 -Heidelberg, June 2012 An introduction to superfluidity and quantum turbulence Joe Vinen School of Physics and Astronomy, University of Birmingham

2. Superfluids • Fluids that can exhibit frictionless flow: best known cases • liquid 4He below about 2.2 K • liquid 3He below about 2 mK • Ultra-cold atomic Bose gases. • Other superfluid systems will be discussed later in workshop

3. Superfluidity and Bose condensation • Superfluidity is associated with Bose condensation (or in a Fermi system with something similar to Bose condensation involving pairs of particles). • Bose condensation in an ideal gas is easy. With interacting particles, it is best understood in the form of the one-particle density matrix(Onsager & Penrose) Even with severe depletion of condensate by interactions where *(r) =  +(r) is a “classical” phase coherent matter field - the condensate wave function. (r)is also the complex order parameter associated with the phase transition to the superfluid state. Loss of global gauge symmetry at Tc. • If we write then is the local velocity associated with the Bose-condensate. (m is the mass of the particle undergoing condensation) • The velocity vs(r) must be irrotational, and any circulation in a multiply-connected volume must be quantized where is the quantum of circulation.

4. The superfluid velocity and its stability • vs(r) is called the superfluid velocity. We might associate it with a mass current (an observable quantity) • Two questions: • Can such a mass current be (meta)stable, implying superfluidity ? • What is the value of s ? • For the ideal Bose gas the current is not stable: single particle excitations destroy the moving condensate, even at T = 0: s = 0 • However, with suitable particle interactions the spectrum of single-particle excitations is modified (free particles  phonons (Goldstone modes)), and the supercurrent becomes stable against the production of these excitations (up to the Landau critical velocity). Then at T = 0, . (although condensate fraction < 1.) • At T > 0, excitations are generated thermally  • Hence two-fluid behaviour(normal fluid = excitations)

5. Quantized vortices • We have considered non-zero circulations in a multiply-connected volume. • But we can also have a topological defect: a free vortex line (Onsager, Feynman), with || = 0 along a line in the fluid, a phase change of 2 round the line, and hence a circulation of . • Can superflow be stable against the production of such a defect? • Yes, if we take account of the interaction of the vortex with its image in a boundary. (Stability against the production of free vortex rings is similar.) Stability depends on quantization of circulation. • Vortex nucleation: role in practice of remanent vortices (especially in 4He). • Role of vortex lines in allowing steady rotation.

6. Quantum turbulence • The flow of a classical fluid is often turbulent. • Turbulence necessarily involves rotational motion. • So turbulence in a superfluid must involve quantized vortex lines, in the form of an irregular tangle. Hence quantum turbulence. • Turbulent flow of a superfluid is in practice also very common. It provides us with a quantum system far from equilibrium, and we wish to understand how it can be generated, how it evolves, and how it can decay. Tsubota et al • The different superfluids behave in different ways, and comparisons provide valuable insight. Comparison with classical turbulence provides further insight.

7. Understanding quantum turbulence: the GP equation • An understanding of classical turbulence must be based ultimately on the Navier-Stokes equation + the continuity equation. • Ideally we want a corresponding equation(s) that describes how the CWF evolves in time. • For a weakly interacting Bose-condensed gas at T = 0 this equation is the non-linear Schrodinger equation (Gross-Pitaevskii equation) • Write , and (small depletion of the condensate); then • Note no viscous dissipation, and presence of quantum pressure term.

8. The G-P equation: the coherence length, the vortex core, and vortex reconnections • The G-P equation  coherence length • The coherence length determines the size of the vortex core. (Vortex core has a well-defined structure.) • If two vortices come within a coherence length they can reconnect (contrast behaviour of Euler fluid). Very important! [Barenghi] • An example of a solution of the G-P equation (Abid et al 2003) [Tsubota]

9. Applicability of G-P equation; vortex filament model for 4He • Good for Bose-condensed gases.  is relatively very large. • Quantitatively poor for real helium, in which interactions are too strong. • In 4He: Condensate fraction ~ 0.1.  ~ interatomic spacing. • Reconnections known to occur in 4He (from experiment), but no theory or detailed model. • 3He (BCS condensate) is in between: ~ 80 nm. • The vortex filament model for helium: • Vortices are regarded as thin vortex filaments behaving classically on scales >> . • They move with the local fluid velocity: the velocity due to other vortices being given by the Biot-Savart law • Reconnections (occurring on scale ~  ) are introduced artificially.

10. The normal fluid and mutual friction • So far we have ignored any normal fluid - present at a finite temperature. • The normal fluid may or may not be turbulent. • Any motion of the normal fluid relative to the vortex lines  frictional force between the two fluids, known as mutual friction. • Mutual friction acts on the core of a vortex and modifies the motion of the vortex in accord with the Magnus effect. It provides a major source of dissipation in quantum turbulence at a finite temperature, but can also lead to the generation of turbulence. • The mutual friction decreases rapidly with temperature and is effectively absent for T / TC < ~ 0.2.

11. Forms of quantum turbulence • As with classical turbulence, quantum turbulence can take many forms; many are very complicated. • Here we consider two typical and illustrative cases: • Homogeneous, isotropic turbulence (HIT) at T ~ 0 (no normal fluid) – some similarity to its classical analogue, but with important differences. Concerned mainly with the way it decays. Relevant experiments in 4He and 3He-B. • Turbulence produced by forced relative motion of the two fluids – no classical analogue. Concerned mainly with the way it is generated. Relevant experiments in 4He.

12. HIT in a classical fluid: the Richardson cascade Classical grid turbulence in a wind tunnel Richardson cascade Energy is injected into large turbulent eddies at large Reynolds number. It then decays through non-linear interactions* (local in k-space) in a cascade of smaller and smaller eddies, until the eddies are so small that they have a Reynolds number ~ 1 and there is viscous dissipation. Kolmogorov: In the inertial sub-range, no dissipation and local interactions * Exactly how?

13. Quantum HIT at T = 0 • Compared with classical case: • No viscous dissipation • A new “quantum” length scale : the vortex spacing • Significance of the quantum length scale . • Processes occurring on scales >>  involve many quanta. Therefore essentially classical behaviour, described by Euler equation. • Motion of scales <  is a dominated by quantum effects (quantized vorticity; discrete vortex lines; vortex reconnections) • Production of quantum HIT? By flow through a grid or by spin-down [Golov]. • How does quantum HIT evolve and decay? To what extent is it similar to classical HIT?

14. HIT at T = 0: theoretical expectations – two cascades? • Injection of energy on large scale (>>); • On a scale >>, we have a classical inertial range cascade with Kolmogorov spectrum (no dissipation) Energy flux   Large scale motion achieved by partial polarization of vortex tangle. Classical behaviour because local interactions ensure that system does not feel the quantum effects that dominate at scales <~. k • An inertial-range cascade requires dissipation at a high wavenumber Transition region k~-1 k  • Dissipation in an inviscid fluid? • Sound (phonon) emission (Bose superfluids)? On scales ~ , frequencies are too small. • Motion on scales < ? Phonon emission during reconnections? Yes, in cold gases, but small in 4He. • But reconnections  waves on vortices (Kelvin) • Kelvin-wave turbulence  Kelvin-wave cascade  phonon emission at scale of nm. [Golov] • On scales ~, transition region (Structure controversial - bottleneck?)(L’vov, Nazarenko & Rudenko; Kozik & Svistonov; Sonin)

15. HIT in 4He at T = 0: experimental evidence? • Evidence for the Kolmogorov spectrum from observations of pressure fluctuations in 4He, and for “classical” deviations from Kolmogorov. • Some indirect evidence for the rest. But the need for more direct evidence  major experimental challenge.

16. Turbulence in 3He and Bose-condensed atomic gases • Superfluid 3He is more complicated than superfluid 4He: spin and orbital motion of the Cooper pairs makes the order parameter more complicated. But turbulence in 3He-B (but not in 3He-A) turns out to be quite similar to that in 4He, although • The coherence length is larger and the vortex core has a more complicated structure, providing another path to dissipation. • The normal fluid is very viscous and cannot itself be turbulent. • Superfluid 3He may be more closely analogous than superfluid 4He to situations in cosmology and particle physics. • Turbulence can also be generated in Bose-condensed atomic gases. Again there can be more complicated order parameters.Phonon generation during reconnections is probably the dominant dissipative process. [Tsubota]

17. Generation of quantum turbulence by forced counterflow I • For example, thermal counterflow in 4He generates a self-sustaining homogeneous turbulence in the superfluid component. • Computer simulations (vortex filament model + reconnections), pioneered by Schwarz. Adachi, Fujiyama, Tsubota PR B 81, 104511 (2010) • Turbulence is generated by the mutual friction • Reconnections play a crucial role • No classical analogue • Prediction by Melotte & Barenghi that flow of normal fluid is unstable, and confirmation in recent experiments. Yet another new type of turbulence.

18. Generation of quantum turbulence by forced counterflow II • Quantum turbulence can also be generated by the forced counterflow of two co-existing superfluids. [Tsubota]

19. Summary and conclusions • Bose condensation in a fluid of interacting particles involves the formation of a coherent matter field and leads to superfluidity. • Phase coherence of the matter field leads to the quantization of circulation and to the existence of topological defects in the form of quantized vortices. • These vortices allow a forms of turbulent motion in the superfluid. • In some regimes this turbulent motion is similar to that in a classical fluid; in others it is quite different. • Quantum turbulence can involve new turbulent phenomena: • New forms of turbulence. • New ways of generating turbulence. • New routes to the decay of turbulence, via characteristically quantum structures • The two isotopes of helium and ultra-cold Bose gases behave in somewhat different ways, which can be instructive.

20. Thank you