When we cool anything down we know it must order and the entropy go to zero. What about liquids and...

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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 must be of order the interatomic spacing d.

And thus the momentum is ~ h/ and the zero point energy p2/2m ~ h2/22m.

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:

s, condensate density

, phase

We take the spatial gradient to find the momentum:

The momentum then becomes just the gradient of the phase, , which in a condensate is a global property of the liquid.

Divide by s 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:

BUT, these are the equations for the electromagnetic fields in free space:

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.

The He4 gap equal in all directions.

Ditto for simple s-wave 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 ~80KThis 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.

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).

Since 3He atoms are massive, p-wave pairing is preferred, i.e. L = 1 which means S must also be 1.

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 Sz1. 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 -

That is fine as the constituent 3He fermion states can simply orbit the “equator” of the Fermi sphere:

We can easily have pairs like this:-

L-vector

Assume the global L vector lies in the z-direction -

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:-

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

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 S-vectors 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:

The B-phase gap:- equal in all directions.

(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

pairspairspairspairspairspairspairspairs

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

pairspairspairspairspairspairspairspairs

(which add up to a spherically symmetric total)

The A phase has a higher susceptibility than the B phase (because all pairs are orno 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.

2 phase change around vortex

Vortex core (where s goes to zero).

Plane of equal phase

2 phase change around vortex

Plane of equal phase

Vortex core (where s goes to zero).

VORTICES

Therefore particle velocity is

We can define the circulation as the loop integration of the velocity;

VORTICES

Since he circulation round a vortex is quantized and the kinetic energy in the flow field depends on vs

2, 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?

Uniformly distributed over the whole volume (all the liquid rotates together).

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 (h/2m3)

Circulation 2Circulation 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 similar-sized 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.

Resultant velocity of ring

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.

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

Venuswilliamson

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

3He near wireBulk

Wire motion

Energy gap,

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

When we cool anything down we know it must order and the entropy go to zero. What about liquids and...

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