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When we cool anything down we know it must order and the entropy go
to zero.
What about liquids and dilute gases which are inherently chaotic?
With normal materials - no problem - they solidify and the chaos
disappears (ideally anyway).
But the heliums remain liquid to absolute zero - what happens there?
To confine a helium at into a defined lattice site means that the
wavelength l must be of order the interatomic spacing d.
And thus the momentum is ~ h/l and the zero point energy p2/2m ~
h2/2l2m.
If this is higher than the attractive potential well then the solid phase
never forms.
Both heliums have NO chemistry because the filled-shell interatomic
forces are too small to stabilise the solid.
He4 is made up of bosons, so there is no problem in their all dropping into
the same ground state to create a condensate but this particular
condensate is hard to understand.
He3 on the other hand is made up of fermions and can only go into a
single ground state by forming Cooper pairs where two fermions couple
to form a boson as in superconductivity.
(The He3 condensate is thus composed of “soft” bosons, easily broken
whereas He4 is made of “hard bosons” and we would need to ionize the
atom to break it up.)
We’ll start by looking at the general properties of the condensate at absolute
zero. That is, when there are no excitations to confuse the issue.
Let us start with the simplest wavefunction:
rs, condensate density
f, phase
We take the spatial gradient to find the momentum:
The momentum then becomes just the gradient of the phase, f, which in
a condensate is a global property of the liquid.
Divide by rs to find the condensate velocity:
This is a crunch result for simple condensates. Since the superfluid
velocity is given by the gradient of a phase, then the liquid is inherently
irrotational as the curl of a gradient must be zero.
This leads us to the basic equations for simple superfluid dynamics.
If we set the liquid as incompressible, then the divergence of the velocity
must be zero (constant density thus no sources or sinks of superfluid).
From above, since the velocity is given by the gradient of a phase the curl
(or circulation) must also be zero.
Thus we get:
and
BUT, these are the equations for the electromagnetic fields in
free space:
or:
That means we get “pure potential flow” and the flow pattern can be
calculated from standard forms in say electrostatics.
Pure potential flow (
,
) around a cylinder.
Which is the same flow field as putting an opposed linear dipole in
a uniform field.
The superfluid absolutelycannot “absorb” angular momentum (or
much ordinary momentum as we shall see later).
Let us look at the more specific cases.
v
..
v
v
v
..
v
v
v
..
v
v
..
..
D
The He4 gap equal in all
directions.
D
Ditto for simple swave superconductors
(apart from anisotropy
from lattice).
However, He3 being p-wave paired (along with
unconventional superconductors) is a bit more
complicated.
We can cool the liquid to ~80mK
This gives a purity of = ~1 in 104000
The liquid us therefore absolutely pure even before we think anything
about the superfluidity aspect.
The superfluid state emerges as 3He atoms couple across the Fermi
sphere to create the Cooper pairs.
Pz
Px
Py
The superfluid state emerges as 3He atoms couple across the Fermi
sphere to create the Cooper pairs.
Pz
Px
Py
Since 3He atoms are massive, p-wave pairing is preferred, i.e. L = 1
which means S must also be 1.
The ground state thus has S = 1 and L = 1 making the Cooper pairs like
small dimers (and easier to visualise than the s-wave pairs in
superconductors).
With S = L = 1 we have a lot of free parameters and the superfluid
can exist in several phases (principally the A- and B-phases) .
With S = L = 1 we have a lot of free parameters and the superfluid
can exist in several phases (principally the A- and B-phases) .
Let us start with the A phase which has only equal spin pairs. That
is Sz= ± 1. The momenta couple to give J = 1, thus Lz= ± 1.
The directions of the S and L vectors are global properties of the
liquid as all pairs are in the same state (this is the “texture” of the
liquid).
However, that causes problems for the pairs.
Assume the global L vector lies in the z-direction -
Assume the global L vector lies in the z-direction We can easily have pairs like this:-
L-vector
That is fine as the constituent 3He fermion states can simply orbit the
“equator” of the Fermi sphere:
However, if we try to couple pairs across the “poles” of the Fermi
sphere there is no orbit that these pairs can make which gives a vertical
L.
Thus the liquid is a good superfluid in the equatorial plane and lousy at
the poles – this is reflected in the A-phase energy gap:-
D
The A-phase gap:large
round the equator, zero at the
poles.
Thus the equal-spin pairs form a torus around the equator in
momentum space, and there are no pairs at the poles.
L-vector
pairs
The A phase is thus highly anisotropic.
Also very odd excitation gas.
In the B phase we can also have opposite spin pairs (the L- and Svectors couple to give J = 0)
This now allows us to have Lz = Sz = 0 pairs which can fill in the
hole left at the poles by the A phase, giving an “isotropic” gap:
D
The B-phase gap:all directions.
equal in
(because all spin-pair species
allowed).
The equatorial equal-spin pairs torus is still there but along with the Lz
= Sz = 0 pairs which now fill the gap at the poles.
L-vector
pairs
The equatorial equal-spin pairs torus is still there but along with the Sz
= 0 pairs which now fill the gap at the poles.
L-vector
pairs
pairs
(which add up to a spherically symmetric total)
The A phase has a higher susceptibility than the B phase (because all
pairs are  or 
no non-magnetic  components).
Thus by applying a magnetic field we can stabilise the A phase.
The A phase is the preferred phase at T = 0 when the magnetic field
reaches 340 mT.
Having made the five minute trip around the superfluid the context for
what follows is:
We can cool superfluid 3He to temperatures where there is essentially no
normal fluid (1 in ~108 unpaired 3He atoms).
We can cool and manipulate both phases to these temperatures by profiled
magnetic fields.
That means we can create a phase boundary between two coherent
condensates, itself a coherent structure, at essentially T = 0. (This is the
nearest analogue we have of a cosmological brane.)
This brings us on to defects in general in the quantum fluids.
ROTATION
Since for simple superfluids  x v= 0, the condensate is completely
irrotational.
The superfluids thus provide the perfect “gyroscopic” materials since they
sit unrotating in the “frame of the fixed stars”.
Then what happens if we try to rotate a superfluid? - say by rotating the
container.
At some point we can exceed some critical velocity at the periphery and
locally destroy the condensate - at which point we can create a vortex.
Plane of equal
phase
Vortex core (where rs goes to
zero).
2p phase change around vortex
Plane of equal
phase
Vortex core (where rs goes to
zero).
2p phase change around vortex
VORTICES
Therefore particle velocity is
We can define the circulation as the loop integration of
the velocity;
but
VORTICES
Since he circulation round a vortex is quantized and the kinetic energy in
the flow field depends on vs2, that means that multiply quantized vortices
are unstable as the kinetic energy is at its lowest when the circulation is
divided into singly-quantized vortices.
This gives a big simplification in that turbulence in the superfluid is made
up of a tangle of similar singly-quantized vortices. This is an “atomic
theory” of turbulence which may throw light on the more intractable
classical turbulence.
TOPOLOGICAL DEFECTS
These simple vortices are topologically stable in the sense that they cannot
fade away as classical vortices, since the circulation is fixed. They also
cannot terminate within the liquid. Therefore they must either exist as
closed loops or join points on the cell boundaries and they can only decay
by a reduction in length.
AN INTERESTING QUESTION
Where does the rotation reside in vortex motion?
Where is the rotation in solid-body motion?
Where is the rotation in solid-body motion?
Uniformly distributed over the
whole volume (all the liquid
rotates together).
Where is the rotation in vortex motion?
Where is the rotation in vortex motion?
Where is the rotation in vortex motion?
Where is the rotation in vortex motion?
Where is the rotation in vortex motion?
Thus the liquid locally is irrotational, and only around the
vortex cores is there any circulation.
If we attempt to put a large body of superfluid into solid-body
rotation a lattice of vortices is set up which mimics solid-body
rotation on a large scale while the liquid is still locally
irrotational.
Circulation k (h/2m3)
Circulation 2k
Circulation zero
Just a word on vibrating structures in superfluids
Here is a view of a series of
experiments packed into a small
(~2 cm3) volume inside the stack
of refrigerant plates in the inner
cell.
The experiment consists of a “goal post” shaped
VWR carrying a grid which creates turbulence
when swept through the liquid.
In front of the grid are two single-wire VWRs
which act as turbulence detectors (as we shall
see later).
The experiment consists of a “goal post” shaped
VWR carrying a grid which creates turbulence
when swept through the liquid.
In front of the grid are two single-wire VWRs
which act as turbulence detectors (as we shall
see later).
The oscillating flow through the
grid induces helical kelvin waves
on the residual vortex loops pinned
to the grid.
Since the wavelength depends on
frequency only a narrow range of
loops are excited leading to the
grid emitting a cloud of similarsized vortex loops.
A vortex ring (say a smoke ring – but true of any) moves at its own
velocity through the fluid as the core on one side moves in the rotation field from
the other side.
A vortex ring (say a smoke ring – but true of any) moves at its own
velocity through the fluid as the core on one side moves in the rotation field from
the other side.
Resultant velocity
of ring
The interaction of these loops, crossing and recombining can be simulated (Tsubota’s group,
Osaka) to show how the gas of loops builds up a random tangle – quantum turbulence.
The decay of the quantum turbulence after the loop injection ceases can also be studied.
We can use a number of these oscillating devices to probe the properties of the
superfluid.
Simple vibrating wire oscillators whose damping depends on the excitation density in
the liquid and thus act as thermometers.
We are also beginning to use quartz resonators for the same purposes
We can look at the behaviour of the superfluid inside aerogel by using similar vibrating
structures. We start with a goal-post shaped wire:
Glue on an aerogel cylinder.
However, we have got a bit ahead of ourselves because we have not discussed the
excitations yet.
Excitations.
We take 3He as the example as it is more illustrative.
The Cooper pairs in superfluid 3He form across the Fermi surface. Thus an excitation is a
“ghost” pair with one of the component particles missing. This excitation is thus a paired
hole-particle.
When the ghost pair is above the Fermi surface it looks like an extra particle, with
momentum and group velocity parallel.
When the pair is below the Fermi surface, it looks like an extra hole, but with momentum
and group velocity antiparallel. (and right at the Fermi surface it doesn’t know what it is
and the group velocity is zero.
This leads to the excitation dispersion curve shown below – the standard
BCS form.
Liquid static
This dispersion curve is fixed to the rest frame of the condensate.
If the liquid is in motion then we see the dispersion curve in a moving frame
of reference. Excitations approaching will have higher energies and those
receding lower energies.
Liquid static
Liquid moving
This means that a flow field in the condensate acts as a potential barrier to
excitations moving into the region.
If the barrier is high enough they cannot continue in their trajectories.
Liquid moving
How can we use these dispersion curves to look at the excitation dynamics?
Quasiparticle dynamics with dispersion curves
Normal scattering
Andreev scattering
MomentumUp
There is some interesting physics even in these simple scattering processes.
If we make a gentle “round the minimum” scattering the excitation changes
sex from quasiparticle to quasihole or vice versa. This conserves excitation
number but what about particle number?
Liquid static
Liquid moving
Normal scattering process
♂
Normal scattering process
Venuswilliamson
♂
♂
Andreev scattering process
♂
Andreev scattering process
♂
♀
Andreev scattering process
Andreev scattering process
We make use of the Andreev scattering of
excitations by the flow fields to detect turbulence
in the superfluid.
Turbulence Detection
Bulk 3He
Moving 3He
near vortex
Bulk 3He
3He near wire
Energy gap, D
Wire motion
Finally using superfluid 3He as a model “Universe”.
Symmetries broken by the “Universe”
There is a whole dictionary comparing quantum fluids with
cosmological objects:
Vortices = cosmic strings
Phase boundaries = branes
Etc etc
Finally a bit of fun.
A superfluid 3He “horizon”.
We start with a superfluid 3He “waterfall”
which we can start with a moving plunger.
This sets up a velocity gradient in the liquid
and creates a “horizon”.
What happens to an excitation approaching
the fall?
Watch this quasihole, which starts in the
static liquid with energy only a little above
the dispersion curve minimum
In the local frame of the moving liquid it
now has an energy far above the dispersion
curve minimum
In this case with enough energy to break a
cooper pair in the condensate.
Switching back to the lab frame: Watch the
high energy hole.
It comes out with much higher energy than
it went in.
How to extract energy from a superfluid 3He Black Hole