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
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
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)
  r1  r2    * r1 r2  as r2  r2   
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 r   r  expiSr  then
associated with the Bose-condensate.
v s r  

S r  is the local velocity
m
(m is the mass of the particle undergoing
condensation)
 The velocity vs(r) must be irrotational, and any circulation in a multiplyconnected volume must be quantized
   v s  dr  n
where   2 m is the quantum of circulation.
The superfluid velocity and its stability
 vs(r) is called the superfluid velocity. We might
associate it with a mass current
Js r   s vs r 
(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,  s   .
(although condensate fraction < 1.)
 At T > 0, excitations are generated thermally
 0  s   if T  Tc ;  s  0 if T  Tc
 Hence two-fluid behaviour
(normal fluid = excitations)
Quantized vortices
 We have considered non-zero circulations
in a multiply-connected volume.
  2 m
 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.
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.
Understanding quantum turbulence: the GP equation
 An understanding of classical turbulence must be based ultimately on the
Navier-Stokes equation + the continuity equation.

v
1
 div v   0
 (v  )v   p   2v
t
t

 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)

2 2
2
i

   m  V0  
t
2m
 Write r   r  expiSr , vr    mSr  and    2 (small
depletion of the condensate); then
v i 1 
1 p  ij
v j v j   


t 2 x i
 x i x j

 div v   0
t
 Note no viscous dissipation, and presence of quantum pressure term.
The G-P equation: the coherence length, the vortex core,
and vortex reconnections
 The G-P equation  coherence length    2 2m 2 
 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]
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.
 ~ interatomic spacing.
•
In 4He: Condensate fraction ~ 0.1.
•
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
v s r0  
 r  r0   dr

3
4
r  r0
• Reconnections (occurring on scale ~  ) are introduced artificially.
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.
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.
HIT in a classical fluid: the Richardson cascade
Classical grid turbulence in a wind tunnel
v
1
 (v  )v   p   2v
t

Richardson cascade
Energy is injected into
large turbulent eddies at
large Reynolds number. It
then decays through nonlinear 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.
* Exactly how?
Kolmogorov: In the inertial
sub-range, no dissipation and
local interactions
 E k  ~  2 3k 5 3
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?
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)
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 <~.
Energy
flux 

k
 An inertial-range cascade requires dissipation
k
at a high wavenumber
• Dissipation in an inviscid fluid?
• Sound (phonon) emission (Bose superfluids)? On
scales ~ , frequencies are too small.
Transition
region k~-1

• 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?)
Nazarenko & Rudenko; Kozik & Svistonov; Sonin)
(L’vov,
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.
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]
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.
Generation of quantum turbulence by forced counterflow II
 Quantum turbulence can also be generated by the forced counterflow of two
co-existing superfluids. [Tsubota]
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.
Thank you