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PROFESSOR: Ready to roll?
OK, good afternoon.
Count down four lectures and you know everything
you have to know about atomic and optical physics,
at least for those of you who take part two-- for those
of you who've taken part one and part two-- for the other ones,
well, there is one more semester which is the following spring.
So we have discussed light forces, first
with the optical Bloch equations,
and then we discussed the stimulated light force
using the dressed atom picture.
So this is sort of finishing the presentation and derivation
of they light forces.
Today, I want to continue to discuss for the aspects.
We've discussed dipole traps, different ways how
we can understand why there is a dipole potential
from the harmonic oscillator to the dressed atom
to the refraction of light by small spheres.
So in all cases, we realize red detuned light traps.
Today I want to discuss what is at work here,
electrical magnetic forces, and then I
want to address what several of you
have asked, where is energy conservation?
Where does the energy go when we cool with a stimulated force?
But this will only take 10 minutes.
The main part of today's lecture will
be techniques to go to very low temperature
to ultra-low temperature.
This is sub-Doppler cooling, sub-recoil cooling,
evaporative cooling, and magnetic trapping
as a technique.
So this is the menu today.
Again, a feature-picked menu.
Now talking about electric and magnetic forces,
you may think it's a trick question,
but it's rather subtle.
All optical forces were derived from the electric dipole
Hamiltonian D dot E. This is what
we used as a starting point for light forces.
So my question is, what about the Lorentz force?
The Lorentz force comes from the magnetic part
of the electromagnetic field.
Now, is there Lorentz force on the atoms?
And, if yes, does it contribute to the light forces or is
it negligibly small?
Actually, the question I'm asking you,
I don't think it's addressed in any textbook
and I think you can go to conferences
and ask some of your colleagues.
You may get a wide range of answers.
So but anyway, has anybody of you thought about it?
What about, is there Lorentz force on neutral atoms?
If no, you're done.
If yes, does it contribute, is it negligible,
or is it even dominant?
AUDIENCE: [INAUDIBLE] temperature
or is equivalent to the T minus A picture,
and the T minus A picture has the Lorentz force, so
[INAUDIBLE].
So.
PROFESSOR: So it should be included
because we are talking about two different representations.
Collin?
AUDIENCE: Yeah.
PROFESSOR: OK, so yes.
The Lorentz force is included.
Also, when you ride on D dot E, you
think it's the electric force of an electric dipole.
We come to that in a moment.
The next question is, do you have any idea
if the Lorentz force is important
or whether it's negligible?
AUDIENCE: For neutral, or?
PROFESSOR: For neutral atoms.
AUDIENCE: [INAUDIBLE]
PROFESSOR: OK, who of you is working on optical lattices
or wants to work on optical lattices?
OK, just to catch your attention,
when the atom goes up and down the optical lattice, if feels
the lattice potential.
The force is 100% the Lorentz force.
There is no contribution from the electric force.
So that's what I want to show you now.
So we have two fundamental forces, the electric force
and-- if you have two charges separated in the field,
two different electric fields-- we derive from the Coulomb
force, the dipole force.
And the force on the electric dipole
is-- that potential is-- D dot E. The gradient of it
is D times gradient E-- with correct Victorian notation--
whereas the Lorentz force is V cross B.
So these are the two forces we have to consider.
Let me show two configurations where we use dipole forces,
the stimulated light force.
One is here where we focus a laser beam,
and let's just assume for the sake of the discussion
that the laser beam-- it's propagating here
and the linear polarization goes up and down.
So now we have a dipole moment of the atom, which
oscillates up and down, which is parallel to the electric field.
So now we have D dot E, and indeed if you take the atom
and move it into the laser beam, it
will experience an attractive force
and this attractive force is purely electric.
But let's now come to the situation which many of you
work on in the laboratory, would that we have two plane waves--
so the laser beams are infinity extended--
we have two plane waves and they form an optical lattice.
Now, the dipole, the oscillating dipole, the electric dipole,
is again driven by linearly polarized light.
It points up, but the gradient of the electric field
is in the direction of the interference of the lattice,
so the gradient of the electric field
is perpendicular to the dipole moment.
So therefore, D dot E is exactly 0,
and therefore, the electric part of the potential-- the Coulomb
force which is microscopically behind it--
will not contribute anything.
So I could stop here and say, well, OK,
since we have only the Coulomb force and the Lorentz force,
I've proven to you it's not the Coulomb force,
so therefore it's a Lorentz force.
But let me just tell you why.
So what I'm really telling you is it is actually remarkable
that if you go along the Z direction
and we take the Z derivative of the AC Stock's shift potential,
and the AC Stock's shift potential is one half,
alpha is now the polarizability--
we use alpha in many places-- times E square.
So this is the AC Stock's shift potential,
and the spatial derivative of the AC Stock's shift potential
is a force, but it's the Lorentz force.
Actually, I was amazed.
I had to work it out myself because I haven't
seen it discussed anywhere, but you can simply
take the electric field of a standing wave,
perform this derivative, and what you find
is you get the Lorentz force.
If you wonder how you get it without going through all
the Victorian notation, the Lorentz force
is V cross B. The velocity of the charge
is related to the derivative of the dipole moment.
The dipole moment is charge times position,
and the derivative of the dipole moment
is charge times velocity.
So don't let yourself fool yourself.
When you have an oscillating field,
there are charges which move their currents inside the atom.
So our V cross B, the V is pretty much D dot.
The D dot is polarizability times E dot,
this is alpha polarizability-- frequency
times the electric field.
But, as Cody said, we can go form P dot A to D dot E--
or, to be precise-- the magnetic field is the curl of the vector
potential, and the curl gives us the [INAUDIBLE]--
I'm suppressing Victorian notation, here--
and you know that E, the electric field,
is the temporal derivative of A, so therefore we get that.
So if you now multiply D dot with the curl of A,
you get something which is K alpha E square.
K is for a standing wave, the spatial derivative.
So I've shown you-- at least you can
see by dimensional analysis, or by just pointing
to the different terms, that if you take the AC Stock's shift
potential through the derivative,
you get everything which you have in the Lorentz force.
It would be a 10 minute, easy homework assignment,
but we're running out of homework assignments-- we're
at the end of the semester-- to show that, explicitly.
So it is really the Lorentz force
which provides the trapping potential
in an optical lattice.
Questions?
[INAUDIBLE]
AUDIENCE: So in the simple picture
you have [INAUDIBLE] the lattice,
you're saying that the transverse confinement is
provided by the electric force, but what
about the perpendicular direction?
[INAUDIBLE] electric field like this, but the other direction?
PROFESSOR: OK, I can discuss with you two simple geometries.
In one case, in one direction it's
the pure Coulomb force at work.
In the other direction, it's the pure Lorentz force.
But in general, when you go through an AC Stock's shift
potential, you have both.
So you may wonder that the Coulomb force, when
you integrate it, would not be integrable.
It does not give rise to potential.
Also, the Lorentz force does not give rise to potential.
But if you add the Coulomb force and the Lorentz force,
you can integrate it up to the optical dipole potential.
I find it remarkable, but it is what it is.
Questions?
So energy conservation, yes.
That's a good thing to discuss now.
Let me first discuss I want to have-- energy conservation
has to be discussed with two different perspectives.
One is a transient phenomenon-- which
I'll do in a few minutes-- but let me first discuss something,
which is CW, and this is the following.
You have atoms which move over a standing wave,
and I use the dressed atom picture to show you
that-- for a strong, blue detuned standing wave--
we have a friction force.
So the atom-- this was the Sisyphus cooling--
that the atom is climbing up hills, there's some transition,
climbs up the next hill, but on average the atom
climbs up more hills than it scoots down,
and therefore there is work done.
There is a net force, and this net force provides friction.
It slows down the velocity of the atom.
My question to you is, where has this energy gone?
The kinetic energy lost by atom, where does this energy appear?
Which fields?
AUDIENCE: [INAUDIBLE] the laser beam?
PROFESSOR: Well, we have two fields, the laser fields,
and we have maybe a spontaneous emission?
But the force is only a stimulated force due to-- yeah?
Which fields?
AUDIENCE: The laser field?
Because when you stimulate into a laser field,
your [INAUDIBLE] modulator [INAUDIBLE].
PROFESSOR: Well, I have a little bit problem with that
because when the atoms-- the force
comes from distributing photons from one standing
wave to the other one, and these are photons of equal frequency.
So in that sense, the net effect is that those two laser beams--
actually, the atom moves, here.
So what should happen to the laser beams?
Photons are not disappearing.
I will later tell you, in the transient picture-- and this,
maybe you've already done your homework assignment number 10--
there is something happening to the laser beam.
But this would only happen when atoms fly in
and fly out of a laser beam.
Then we have a different situation.
But here, if the atoms stay in the laser beam the laser beam--
so to speak-- sees a constant index of refraction medium.
It just goes through and nothing has
happened to the photons except that-- occasionally-- photons
are spontaneously emitted.
Well, OK.
If it's not the laser beams, what remains now?
AUDIENCE: Spontaneous [INAUDIBLE].
PROFESSOR: Spontaneous emission.
Now how can spontaneous emission carry the energy?
What is the spectrum of spontaneous emission?
AUDIENCE: Triplets?
PROFESSOR: Yes, the mono-triplet.
We have the central carrier, the central carrier
is not carrying away any energy because it
is at the laser frequency.
Now we have two side bands.
Can the side bands carry away energy?
AUDIENCE: Interlacing them.
PROFESSOR: Huh?
AUDIENCE: Interlacing with the [INAUDIBLE].
PROFESSOR: Asymmetry in what sense?
AUDIENCE: If the side bands are [INAUDIBLE] escape.
PROFESSOR: Well, but didn't we discuss,
with a radiative cascade, that the blue detuned side
bend comes when you go from the upper dressed level
to the dressed level?
And in the lower dressed level, you
can only emit a red detuned side band, not a blue detuned side
bands.
What we said in the radiative cascade
after one blue detuned photon, there is a red detuned photon.
So the number of photons emitted on the red side band
and on the blue side band are exactly equal.
[INAUDIBLE]?
AUDIENCE: But then you have the [INAUDIBLE] cooling picture,
where the red detuned side then was emitted in the lower field
region?
PROFESSOR: Yes.
So the situation is the following.
The atom emits an equal number red, blue.
Red detuned and blue detuned, red detuned and blue detuned.
But when it emits the blue detuned,
the side bend spacing is larger because it
emits the blue detuned photon when it's on top of the hill.
And the red detuned photon is emitted
when the atom is more in the dark,
and therefore, the side bend spacing is lower.
So it alternates blue, red, blue, red.
But the blue detuned photon is emitted when the generalized RB
frequency is larger, when the atoms is at an anti-node
And the red detuned is more preferably emitted
when the photon is at a node of the standing wave.
So it is the difference of these modulations,
the side bands are modulated, and there is a preference
to emit the blue detuned side band when the side band
spacing is larger than in the red detuned case.
And this is where the energy goes.
AUDIENCE: You could make the argument and that
would give a simple picture for the maximum cooling rate?
PROFESSOR: Yes.
AUDIENCE: Where you have omega times your [? scattering ?]
rate?
PROFESSOR: Yes.
That's exactly how, last class, I calculated the cooling rate.
The cooling rate, the energy removed from the system,
is, well, yeah, I said it is the height of the lattice,
but the height of the lattice is also
the amount by which the-- the lattice is
nothing else than the generalized RB frequency.
This is the energy level of one crest level,
and therefore the lattice is the increment
by which we modulate blue and red side bands.
Exactly.
This can be quantitatively worked out.
If I had another half hour, I could do it for you.
It's in the references, but this is
the physical picture for cooling.
OK, but so you realize, this is one of the subtleties.
We can understand the force simply
from a simple picture, energy levels, Sisyphus cooling
and such, but in order to understand
where the energy comes from, we need the mono-triplet.
We need spontaneous emission.
So the force comes from the redistribution of photons.
It's the stimulated redistribution
of photons which is responsible for the force,
but the energy balance comes from,
what can be neglected for the momentum balance, namely
the few spontaneous emission events, and especially
those which involve the blue and red side bands.
And it beautifully works together
and all the equation of physics are obeyed.
OK, but that tells you now that the cooling effect
hinges on spontaneous emission.
And yes, spontaneous emission is dissipation.
And whenever you want the friction coefficient--
you want dissipation-- you need an open system
and the open system is spontaneous emission
into those nodes.
So let's now talk about an opposite situation
where we do not have any spontaneous emission.
And I think almost all the people
who work with optical lattices use infrared lasers for them,
or [INAUDIBLE] detuned laser where spontaneous emission
happens only once every 20, 30 seconds
or 100 seconds per atom, so we can really neglect it.
So now the question is the following.
Assume you have a focused laser beam
and you load a Bose-Einstein condensate
at the edge of the cloud, and now the condensate
is accelerated, sucked into the laser beam.
So now the atom has kinetic energy.
The question is, where does this kinetic energy come from?
And this time, we do not have spontaneous emission
as we have when we do frictional cooling.
So the question is now, where does the energy of the atom
come from?
AUDIENCE: But if we were talking about a dielectric medium,
it's coming from those light waves that are reflected or not
reflected.
PROFESSOR: If comes from those light rays,
but-- the atom is a dielectric medium, that's good--
but the reflection and deflection of light rays--
at least in leading order-- you would say photons of the same
energy are deflected--
AUDIENCE: Wouldn't the [INAUDIBLE] of the atom
just get absorbed [INAUDIBLE] and the [INAUDIBLE]?
PROFESSOR: I would hate to use the picture of absorption
and emission.
What really happens is it's just scattering.
It's scattering.
you scatter a photon which has one frequency
into a different direction.
It's really light scattering.
Well, you have a homework assignment on that,
and I don't need to go into details because it can easily
be worked out, but what happens is the following.
The atom is a dielectric medium.
It has an index of refraction.
And if you put an index-- if you suddenly
put an index of refraction into a laser beam--
then you change the phase of the laser beam.
The atoms act as an electro-optical modulator,
and they change the frequency of the transmitted photons.
You have a homework assignment on that.
So if you increase the number of atoms in your laser beam,
because they move in, this is-- the phase
of the transmitted photons is shifted.
If the phase of the photons is shifted during a certain time,
phase over time is frequency.
So therefore you will see, when the atoms move in and gain
kinetic energy, the transmitted photons of the laser beam
have lost some energy.
They are slightly shifted to the red.
So in other words, it's funny but if you have a dipole trap
and the laser beam is focused and atoms
slosh through the trap, if you would carefully
analyze the frequency of the transmitted beam,
you will find that it's a little bit blue, a little bit
red, a little bit blue, a little bit
red, and this compensates for the kinetic energy.
Again, I don't think you'll find it in any textbook.
It confused me for a while until I could work it out.
But the homework assignment gives you an exact dissoluble
model where you can exactly show what I told you.
So these are the two situations, here.
It's either spontaneous emission,
which is responsible for cooling,
but here we have a conservative potential.
And in this conservative potential,
the only player is the laser beam which is transmitted.
And, indeed, it is the photons which have been transmitted,
and usually you wouldn't put that into your picture.
Remember, the dipole force can be explained
by eliminating the coherent field of the laser
field-- for canonical transformation--
and then we have a C number.
We have a classical electric field.
This is your Hamiltonian, and from this Hamiltonian,
you derive that you have a conservative potential.
But if you now ask where the energy goes,
the energy-- really, in this situation--
goes into a frequency shift of the photons.
Also, in your Hamiltonian, you have
assumed that the electric field is an external classical field,
E naught times cosine omega T.
Well, if you want my advice, whenever
you use the AC Stock's shift for trapping,
just use the AC Stock's shift and regard it
as a classic potential, and just completely forget
that there are photons behind because if you want to account
what really happens to the photons-- what really happens
to the energy in this system-- it really gets ugly
and it can get really confusing.
Questions?
OK, we are done with the stimulated force.
We are now in preparation for next week
when I want to tell you about degenerate Fermi gases
and degenerate Bose gases.
Those gases require nanokelvin temperatures,
so today, in the next hour, I want
to tell you what are the techniques
to create such ultra-low temperatures,
and the techniques I want to discuss
is one addition to laser cooling.
We have so far discussed laser cooling to the Doppler limit,
but now I have to tell you that that's not
where laser cooling stops.
You can go sub-Doppler and sub-recoil.
And, at least when the Nobel Prize
was given to Bill Phillips, Steve Chu, and Claude
Cohen-Tannoudji, well, if you go to the Nobel website
and read the report of the Nobel Committee-- well,
they should have given it to those three
people for the many, many wonderful things
they have done, and I've made frequent references
to their work.
Bill Phillips invented Zeeman slower,
Claude Cohen-Tannoudji invented the dressed atom picture,
they have many, many contributions,
but the Nobel Committee tried to justify their choice
with a coherent story, and the coherent story
was Steve Chu cooled to the Doppler limit in molasses,
Bill Phillips discovered sub-Doppler cooling,
and Claude Cohen-Tannoudji realized sub-recoil cooling.
So the storyline was cold, colder, the coldest.
And that's what we want to talk today.
So sub-Doppler cooling.
Well, I could have spent-- or, 10 years ago, I
spent a whole lecture on sub-Doppler cooling,
polarization gradient cooling.
Wonderful!
The kind of epiphany of elegance in [INAUDIBLE]
a mechanical description of an atom.
But I have to say-- and then I explained,
in another lecture, sub-recoil cooling,
how you can even cool below the recoil limit.
Well, this year, 2013, I spent 15 minute on it.
The reason is the following.
When I joined atomic physics and was a post-doc
in the '90s and assistant professor,
the conference's [INAUDIBLE] of [INAUDIBLE],
[INAUDIBLE] was full on intents to find new ways to laser cool.
Ideas of how to get to lower and lower temperatures.
There were many, many different-- sub-Doppler,
sub-recoil techniques were discussed.
This was the main topic.
But then suddenly, in 1995, evaporative
cooling-- an intellectually boring cooling technique,
just have atoms collide and evaporate--
this led to the lowest temperature ever,
and it was almost a sudden transition.
Within a few months there was no research, no papers anymore,
on advanced methods of laser cooling.
Evaporative cooling it just completely wiped out
this area over atomic physics.
The reason being because all cooling schemes which
have been discussed had, in the end, some problems.
At some point, photons heat.
Heat because they excite an atom and if an excited atom collides
with another atom, there is a heating mechanism and such.
So even all the optimistic proposals for laser cooling
reached very low temperature, but never at high density.
And evaporative cooling just did everything for you.
But anyway, I think it-- not just because a Nobel Prize was
given for sub-Doppler and sub-recoil cooling,
this is really an accomplishment to understand how can you
go through conventional cooling limits,
and at least every graduate student
who happens to graduate in AMO physics at MIT
should know what are the concepts
behind sub-Doppler and sub-recoil cooling.
So what I need in order to introduce those cooling methods
for you is I have to remind you that optical pumping is
a cooling scheme, and one could actually say, to some extent,
every cooling you do with lasers is based on optical pumping.
And so let me explain that.
Optical pumping happens that-- let's
say you have a level structure with a circularly
polarized light where you can go maybe from M equals 1 to M
equals zero.
And then there can be spontaneous emission back to M
equals 1 and back to M equals minus 1, but the laser--
because it has angular momentum--
cannot excite this state to any excited state.
So what happens is, of course, pretty clear.
You have whatever distribution you have in state one and two.
You switch on your laser, and after a few cycles
everything has fallen into the dark state.
You have 100% population in stage two.
So this is the simplest example of optical
pumping using the three-level scheme.
Well, you may ask now, what has that to do with cooling?
Well, temperature is a Boltzmann factor,
and if you-- we can define a temperature by saying
the population between two levels
is given by a Boltzmann factor.
Let's just introduce an energy-splitting delta
E-- which is somewhat arbitrary here-- but in any event,
you see if you completely pump out a level,
if you have all the population in one state,
this corresponds to zero temperature.
So if you optically pump the atoms, preferentially,
into certain states, you have lowered
the entropy of the system, the atoms are no longer distributed
over as many states as before, and this
corresponds to lower temperature.
And the message I want to give you
is that you can understand laser cooling as optical pumping
in translation space in velocity space.
When you laser cool, you excite the atoms at high velocity
and then because of the mechanisms we discussed,
spontaneous emission leads to lower velocities,
and this is one form of optical pumping.
I also like the word-- optical pumping is, in some sense,
a spontaneous Raman process.
You go up with one photon, you go down
with-- you go up with the laser photon,
you go down with the spontaneously emitted photon.
So this is a spontaneous Raman process,
and laser cooling-- even if you have just a single ground
state-- can be regarded as a Raman process
where the initial and final state differ
in velocity or momentum.
You have the same internal state,
but you have a different external state.
So you can see laser cooling is spontaneous Raman scattering
between different momentum states.
OK, now how can we achieve sub-Doppler cooling?
Sub-Doppler cooling was discovered in 1988,
and I remember I was at the [? IKA ?] Conference in Paris
when Bill Phillips' group said, we've
carefully measured the temperature
and in sodium we measure a temperature which
is lower than the Doppler limit, which was rigorously derived.
I derived for you the Doppler limit of molasses.
So what went wrong here?
Well, it became clear the only loophole--
if you have a theory which predicts something
and then you find a violation, you carefully
have to check the assumption.
And the assumption which was made--
and which we have made in this course--
is that you had a two-level system.
Now, atoms have hyperfine structure--
and for pedagogical reasons, because I
want to explain to you why in sub-Doppler cooling scheme
I assume this hyperfine structure with F
equals one half, F equals three half--
but let me first tell you what the novel feature is
about a multi-level atom which has hyperfine structure.
Now, the one thing which happens is
that-- instead of just going between our single ground state
and our single excited state-- we now have transitions.
You may want to call them Raman transitions,
or optical pumping, between the hyperfine atoms.
So there are now transitions not only up to an excited
state which rapidly decays, there
are now transitions-- Raman transitions--
between ground states which have a very low widths.
The width of the excited state is gamma.
Well, what is the width of the ground state?
Well, if you don't have a laser beam,
the width is zero, but if you have
laser beams the atom doesn't stay there forever.
There will be a time, which is estimated here-- depending
on the RB frequency and the detuning-- this is the time,
this is simply nothing else than the scattering
rate which we derived before.
So that's the rate at which you scatter photons.
And then, depending what the bunching ratio is,
this is also the time in which you may optically
pump to another ground state.
So therefore, what we have to take into account
now is that we have narrower widths in our system,
and those widths correspond to another two-level system which
is driven by two photons in which connects the two ground
states.
I will not give you any derivation how this exactly
leads to a lower temperature, but I
want to give you two pictures which I have already
introduced to you.
One is, remember, when we discussed Doppler cooling,
the final temperature was proportionate to gamma,
the width of the transition.
So now I wave my hands and say, if you have transitions
between ground states which have a much, much smaller width--
which is the rate of optical pumping--
and for low laser power this width
can become very, very narrow.
But you have at least one ingredient
which can lead to a lower temperature.
OK, in all truth in advertising, for the Sisyphus--
for the polarization gradient cooling scheme--
the final temperature is not given
by gamma prime, the rate of optical pumping.
It is a proportionality factor, but there is another factor.
If you want to learn about it, you
have to read some of the classic papers.
But there is another picture which I can also
use to tell you-- and that's actually
related-- why optical pumping at least gives
the possibility for lower temperature.
I explained to you with the stimulated force in [? blue ?]
molasses-- that the only reason why we have cooling--
is that the atom has a lag time.
It cannot instantaneously adjust itself to the light field.
It lags behind.
And you can say, in Doppler cooling,
the lag time is the spontaneous emission time,
and the inverse of it is gamma.
Here, in this case, the lag time can
be the very long time to optically pump.
So therefore, if the atom is in one hyperfine state and now
moves into an area where the polarization of the laser beams
is different, it may take a long time to adjust.
And it is this lag time to which the friction coefficient
was proportional.
So this is a new feature, long delay times, narrow widths,
and resonances between ground state levels.
OK, a little bit show and tell now.
Nobody thought about polarization
creating cooling, nobody thought about Doppler cooling,
but it was discovered when people simply
did Doppler molasses and it's one
of those big violations of Murphy's law
where cooling worked much better than everybody had thought,
and the result is the following.
That-- when I drew for you the blue curve, which
is force versus velocity-- it's pretty much
the subtraction of two [? Lorentzian ?].
What happened is-- for sodium atoms, or for any alkali atom--
it's the red curve.
For large velocity you have the Doppler cooling mechanism,
but for smaller velocities you have a steepening of the slope,
and the steeper the slope, the larger is your friction
coefficient, alpha.
And this is the mechanism of polarization gradient cooling.
Let me just take this famous paper
and show you one way how polarization gradient cooling
works, which is the most famous form of sub-Doppler cooling,
and this goes as follows.
If you have molasses with two laser beams
and the two laser beams are lin-perp-lin,
linear polarization perpendicular
to the other linear polarization.
So you have two laser beams with these polarization.
When these two polarizations overlap and they
have the same phase, you get light at 45 degree,
but if these two polarization have 90 degree out of phase,
you get circularly polarized light.
So as these two laser beams counter-propagate,
you periodically go from linear polarized sigma minus,
linear polarized sigma plus.
So you have spatial-- at any given point
you have a polarization, but the polarization changes.
You have a-- you would say, naively,
lin-perp-lin don't interfere, don't form a standing wave.
Well, they do not form a standing wave in intensity.
They form a standing wave in polarization.
So now what happens is the following.
If you have a multi-level atom and I use this simple scheme
here, the different polarizations,
linear polarization drives, the pi transition,
sigma plus and sigma minus [INAUDIBLE]
different transitions, and those transitions
have different strengths.
So an atom here-- when it experience
this polarization-- reaches cycle,
but when it experiences the other circular polarization,
it will be optically pumped over here and then it cycles.
So therefore, an atom which from sigma plus polarization and was
here, it flies over to an area where you've
sigma minus polarization, it will actually be pumped over.
So the atom will constantly be pumped back and forth
between those hyperfine states.
And this actually gives rise to a beautiful form of Sisyphus
cooling that the atom experiences sigma plus
light in one ground state.
It climbs up the hill, it sees the AC Stock's shift
potential-- and the AC Stock's shift potential,
because of the, for instance, for one hyperfine state, sigma
plus drives the strongest transition--
and therefore we have sigma plus light,
one ground state has the deepest potential.
Where we have sigma minus light, it
is the other ground state which has the deepest potential.
And so what happens is that the atom is in one hyperfine state.
It climbs up the hill, and then it's
optically pumped to the other hyperfine state,
and we have exactly the same kind of Sisyphus cooling.
I just show you pictures and you sort of match it
with what you know.
This is more complicated because this
involves optical pumping between hyperfine states.
The different hyperfine states have different [INAUDIBLE]
coefficients and such.
It's not really complicated but more complex.
What I explained to you is how Sisyphus cooling
works in the dressed atom picture
just for a two-level system.
And here you find a more subtle form of Sisyphus cooling,
but this form is more important because whenever
you operate a magneto-optic trap,
you get this cooling mechanism for free.
OK, so that's all I want to tell you about sub-Doppler cooling.
OK, sub-recoil cooling, we can quickly deal with it
because I want to prove to you here that sub-recoil cooling is
impossible.
And unless you tell me what this wrong in my derivation,
I don't need to discuss sub-recoil cooling
because I've convinced you that it's impossible .
OK, so let's assume our atom has an initial kinetic energy
and then we absorb one photon.
That would mean the momentum has now changed by the laser photon
by the recoil-- and then it emits a photon,
so the momentum gets reduced by the immediate photon--
and all I do is I ask, what is the difference
between the initial kinetic energy
and the final kinetic energy after two photons-- a laser
photon and the spontaneously photon-- have been exchanged?
And what you find is-- and this is just exact--
that the change in energy is two times the recoil energy.
The recoil energy is h-bar square K square over 2m, plus--
and this is now the cross-term-- plus KL--
the K vector of the laser-- minus K spontaneous emission
times the velocity.
And now, of course, we assume-- which is correct--
that spontaneous emission goes randomly in all directions.
Therefore, if we average over many cycles,
this does not contribute.
So therefore we find now that the average energy exchanged
by an absorption and emission event is two times
the recoil energy plus h-bar K laser times [? width ?],
and we want to make it negative because we want to cool.
Well, we make it negative by arranging the laser beam--
surprise, surprise-- counter-propagating
to the velocity.
This is how we can get the best cooling.
Well, surprise, that's what we expected.
But you find, of course, that when this velocity is a very
small then K V is smaller than K squared.
So therefore, once the velocity is smaller than the recoil
velocity of a single photon, you cannot choose--
you cannot make this equation or this expression go negative.
In other words, what I've shown to you
is if there is an atom which has a velocity which is smaller
than the recoil velocity of a photon, whenever
this atom scatters a single photon,
it will be hotter than it was before.
So is it clear, what I've shown you, in energy?
When an atom has a velocity which
is smaller than the recoil velocity,
any further photon scattering will not cool.
It will lead to an energy transfer delta
E, which is large, which is possible.
So that shows you that sub-recoil cooling
is impossible.
Any idea how we can sub-recoil cool?
Any idea why Claude Cohen-Tannoudji
got the Nobel Prize?
AUDIENCE: Further down in the lattice,
you can have an effective mass that
is much heavier than [? bare ?] mass?
PROFESSOR: OK, great idea.
We put the atom in a lattice, a lattice has band structure,
in a band structure we have an effective mass,
and heavier atoms can be cooled to lower and lower temperature.
Actually, it's well known that, in Doppler cooling
and sub-Doppler cooling-- especially
in sub-Doppler cooling-- cesium an rubidium reach
a microkelvin, sodium reaches only 25 microkelvin.
So making the atom heavier is OK,
but I have to say you're only rescaling your recoil limit,
you're not breaking through the recoil limit which is now
defined with a heavier effective mass.
AUDIENCE: [INAUDIBLE] emission direction--
PROFESSOR: Yes?
AUDIENCE: --means that the atom must
have [INAUDIBLE] responding is [INAUDIBLE]?
PROFESSOR: Yes.
AUDIENCE: And [INAUDIBLE]?
PROFESSOR: That was the assumption.
AUDIENCE: [INAUDIBLE] at least for this cooling,
it needs the time much larger than the Doppler cooling
[INAUDIBLE] spontaneous emission time?
PROFESSOR: Well, that was the assumption here.
We scatter several times, a spontaneously emitted photon
do not contribute the recoil of the spontaneously emitted
photons cancels out because spontaneous emission-- and this
is correct, this is not wrong-- spontaneous emission goes
equally probable in the plus X and the minus X direction.
Well, I thought I want to show you a demonstration how--
mechanical demonstration-- how you can sub-recoil cool.
It's not a demonstration in velocity space,
it's a demonstration in position space.
What I have here is a Plexiglas tray
and it has a little hole in the middle.
And I have a bunch of ball bearings
and I put those ball bearings in.
Let me translate.
I can blindfold myself and all I do is I shake the tray.
So whenever I shake it, I kick the atoms
randomly in position space from here to here.
So the amount of position control I have over the atoms
is on the order of this size.
But the question is, can I-- without having
any control of the transfer of position I give to them--
can I steer all the atoms into an area
here into the hole in the middle, which is very, very
narrow in position space?
So the translation is, if I randomly scatter photons
and they kick the atoms around with h-bar K,
I don't have any control about momentum
transfer smaller than h-bar K, is it
possible to localize the atoms in momentum space
to a momentum around zero, which is
much, much smaller than h-bar K?
Well, let's do the experiment.
I just close my eyes and I shake it, and I shake it,
and I just shake it for a while and I continue shaking it.
And, well, zero temperature!
So without controlling the motion on the scale
I'm interested in, I manage to cool into a target region which
is much, much narrower.
So in other words, what you need is-- all you
need is-- you need some dark state which
is velocity selective.
If you scatter light but you create a situation where
once the atoms are close to zero velocity, they're not excited.
You're not steering the atoms in a deterministic way,
you just wait until-- by random chance-- one photon is emitted,
and by the random chance the atom hits the hole.
And then, it will never be re-excited again.
This was the idea of this demonstration.
And I could, yes, in the old days
I may have spent two hours on teaching it,
but by using Raman resonances between ground states--
which are terribly narrow-- or by using VSCPT-- velocity
selective coherent population trapping-- you can create
such narrow dark resonances which
have the effect that the atoms scatter, scatter, scatter,
but the moment they reach a very narrow region around zero
velocity, they stop scattering.
So what was wrong in my proof is that here we
have a situation where we stop the laser cooling exactly
at the time when the atom, by random chance,
happens to be at low velocity, at very low velocity.
And then you don't need any control,
you just accumulate in the same way
as I accumulated the ball bearings.
OK, this was sub-recoil cooling in five minutes. [? Timo ?]?
AUDIENCE: So I wasn't here last class.
But just to summarize, the Doppler cooling rate
gets us to a temperature on the order
of gamma, which is natural alignment.
PROFESSOR: Yes.
AUDIENCE: And the recoil limit is the recoil energy
which is usually tens of kilograms.
But if you have a really, really narrow line,
would you use the Doppler cooling limit?
You could, in principle, beat the recoil limit
with Doppler cooling, no?
If you have a--
PROFESSOR: No, you can't, because if you do Doppler
cooling-- [? Timo's ?] question is,
if you simply do molasses with a very, very narrow line,
we derived that the limit of Doppler cooling is Kt equals
gamma, and what you are saying now,
if you use narrower and narrower lines,
can we reach [? arbitrary ?] low temperature?
This is not the case.
In our derivation of Doppler cooling,
we made a continuum assumptions when
we plotted the force versus velocity
and we had the friction force, we
assumed that an atom, when it scatters a photon,
stays, let's say, within the linear part
of the force versus velocity.
So we had a hidden assumption which
required that the recoil energy is smaller than h-bar gamma.
But there is a lot of literature where people looked into it,
and it turns out that when you have a very narrow line,
you can go to the recoil limit, but you can't go beyond.
If you want to go-- and I think this is what I showed you
in the last five minutes-- if you want
to go below the recoil limit, you
need some velocity-selective dark state,
and you would not have that in this situation.
On the other hand, if you have a very narrow resonance,
you can probably engineer a dark state
that there is destructive interference in some excitation
in a very, very narrow velocity class,
and this narrow velocity class is simply
selected by the Doppler effect.
So you can use the narrow line to engineer sub-recoil cooling,
but the simple arrangement of just having molasses
with narrower and narrower line will not work.
[? Collin ?]?
AUDIENCE: I think the group that laser
cooled to BEC just had-- their dark state was just a Stock's
shift, [INAUDIBLE] and that was their-- and it shook.
There were resonances in kilohertz or something.
PROFESSOR: Thy used a very-- did they
go below the recoil limit in laser cooling?
I'm not sure.
They may have had a situation where
it was enough to go to the recoil limit.
I have to check out the paper, but--
AUDIENCE: I remember they did have some sort of dark state.
Maybe it was to increase density or-- they had the high--
PROFESSOR: Anyway, let me just be clear.
Sub-recoil cooling requires that you
have a dark region in velocity space.
For that, you need some narrow line widths.
That's necessary.
But I think you just don't get it
by having counter-propagating laser beams,
you have to do something more.
Other question?
OK, so I've told you about sub-Doppler cooling,
sub-recoil cooling, but what really got atomic physicists
to nanokelvin temperature were two other techniques.
Laser cooling was used as pre-cooling,
but then the final trapping and cooling
was done by magnetic trapping and evaporative cooling.
So I want to give you now, in the last half
hour, a quick overview of magnetic trapping
and evaporative cooling.
So I think any discussions of magnetic trapping starts
with a theorem, a theorem which tells you
that not everything is possible you would like to do.
And this is the following, then if you have a region without
charges and currents without-- through an empty space,
and what this theorem says-- in empty space,
you cannot have a local minimum--
you cannot have a local maximum of your electric and magnetic
fields.
So if you take the strengths of the electric field
or the strengths of the magnetic field,
you cannot have a maximum.
You can only have a minimum.
Well, and this is important when it comes to magnetic traps.
Depending on the magnetic moment-- spin up,
spin down-- we could create a trapping potential
around a maximum of the magnetic field
or the trapping potential around the minimum
of the magnetic field.
You want to be at the minimum of the potential,
but if you have a minus sign, you
may want to be at the maximum of a magnetic field.
But [? Bing's ?] theorem says that only one of them
is possible.
The proof really goes in two lines.
You really show, if you assume that there is a maximum, that
would mean that if you add a field to it, you add a field--
and no matter in which direction you add a field--
the total field strengths become smaller.
But that requires a violation of the Laplace equation.
It's easier for you-- you can probably read it at home--
but you simply assume you have this situation,
so you add-- and you show you have an incremental field,
delta E. This field, delta E, you
are at the maximum of an electric field,
and then delta E over R is the small difference
of the electric field.
And you can show that this expression is only negative.
That means you have a maximum of the electric field when
the electric field would not fulfill the Laplace equation.
So the gist of the argument is the following.
Each component of the electric and magnetic field
fulfills the Laplace equation.
And the Laplace equation-- I don't know if you've probably
heard about it-- if something fulfills the Laplace equation,
it says at any given point the function which
fulfills the Laplace equation is equal to the average
of function, averaged over a small sphere around it.
And that means-- if something fulfills the Laplace equation--
you cannot have a local maximum or a local minimum
because if the value here equals the average,
that would mean in one direction the value gets higher,
in the other direction the value gets lower.
So if something fulfills the Laplace equation,
you cannot have a local maximum or local minimum.
But now, we're not asking for maximum or minimum
in one component of the electric or magnetic field.
We ask for a maximum or minimum in the total value,
or the square of the electric field.
But then you show, when you assume that you have a maximum,
the field points along the Z direction,
that then the Z component of the field cannot fulfill
the Laplace equation.
So maxima are not possible.
Minima are possible.
Good!
So the magnetic trapping potential
comes because we have a magnetic dipole moment which
interacts with a magnetic field.
Magnetic traps are actually classical.
You can have a classical model for a magnetic trap,
and in that case, you say this potentially
is mu times B times cosine theta.
[? Crendo ?] mechanically, of course, the angle cosine theta
is quantized and we have the different M F levels,
the different orientations of the spin,
relative to the Z-axis.
OK, the fact that the magnetic field can only
have minima and not maxima means that we
want to make sure that the magnetic moment times the G
factor times M F is negative, or cosine theta
is negative in the classical picture.
And if we absorb whatever we choose for cosine theta--
or what we call mu Bohr G M F, the magnetic moment--
we want that the magnetic moment of the particle
is anti-parallel to the magnetic field.
Then we have a magnetic trapping potential,
which is-- I've taken care of the sin now-- mu times B,
and if B has a minimum this potential has a minimum,
and that is a magnetic trap.
But so the consequence of [? Bing's ?] theorem
is that we can only trap particles
which are anti-parallel with a magnetic field.
And they can always lower their energy
by flipping to the other state.
So therefore, a magnetic trap does not
allow us to trap particles in the absolute ground state.
There is always the possibility of spin-flip collisions
which lead to an anti-trap state which
is expelled from the trapping region.
Now, spin-flip collisions can happen
when you have particles at sufficiently high density
and they collide.
For the experts, there can be spin relaxation,
there can be dipolar relaxation, they
are two different kinds of spin-flip collisions.
Fortunately, in magnetic traps, they only
become relevant when you pick the wrong spin state.
But for suitable choices, magnetic traps
are very long-lived.
Dipolar relaxation is often-- can be a limiting process
for Bose-Einstein condensates, for instance,
for high [? dose ?] Bose-Einstein condensation,
dipolar relaxation was-- limited the number and density of atoms
in the Bose-Einstein condensate.
But I don't want to talk about cold collisions, here.
You should just know, based on the fundamental theorem,
you cannot trap in the ground state.
You have to trap into a state which
has more energy than other states, and, in principle,
it's possible to flip the spin and go to a lower state.
Now, what we have assumed here is--
when I wrote the trapping potential like this--
I assumed that the atom stays in a given hyperfine state.
So we put the atom in one state and it experiences potential.
Classically it means that the angle of the dipole,
with respect to the magnetic field, stays constant.
And this is the case due to rapid precession.
Classically, the magnetic dipole precesses
around the magnetic field, and when
the direction of the magnetic field changes,
the rapid precession keeps the dipole aligned.
Or, in other words, for slow changes of the magnetic field
direction, cosine theta is an adiabatic invariant.
So the word adiabatic is important, either classically
or you stay in a quantum state as long as you're adiabatic.
Well if an atom would rapidly go through a region
of very weak magnetic field, then the precession--
the [INAUDIBLE] precession-- of the atom is very slow,
and this adiabatic condition can be violated.
And this violation of an adiabaticity condition
is called Majorana flops.
So if you operate a magnetic trap
at very low magnetic field, you may destabilized the trap
because you violate adiabaticity.
OK, so these are the consequences of the fact that
we cannot create maxima, we can only create minima
of the magnetic field in free space,
and therefore we have to deal with those two possible loss
processes, but we have learned what kind of magnetic trap,
what kind of atom to pick, and in general these are not big
problems.
OK, so we have to-- so this shows here
the typical hyperfine structure of an alkali atom.
It could be rubidium 87 or sodium 23.
The fact that we have only a local minimum
of the magnetic field means we can only trap hyperfine
states-- which I've marked here in green-- where the slope is
positive, and these are the atoms where
the magnetic moment is inter-parallel
with the magnetic field.
And, well, for stability reasons we
usually pick the highest state of the lower manifold,
or the highest state of the upper manifold.
In principle, it would be also possible to trap
here-- you have the correct slope--
but often those states suffer form
spin relaxation and collision loss.
[? Mickey ?]?
AUDIENCE: [INAUDIBLE] the picture you showed before,
is it actually possible to make a magnetic trap with state
number four from the top so that they expel from the center?
From [INAUDIBLE] a--
PROFESSOR: Yeah, there is a peculiarity here,
you have the transition from the weak field to the strong field
region.
Whenever the magnetic moment is constant,
you need a local minimum of the magnetic field.
But if the magnetic moment would change
and you have a gradient of the magnetic field,
you could actually trap here.
So in this case, you have a magnetic field gradient
but you have a spatial variation of the magnetic moment.
This has been discussed in one special paper
in the literature, but it has not really
found any [? lucent ?] realization.
AUDIENCE: Wouldn't it help to put the Majorana
flops in the center?
Because they're repelled from the center.
PROFESSOR: Well, yeah, but these atoms
would undergo spin-flip collisions
so it's not a good choice.
It would help against Majorana flops,
but we have many solutions against Majorana flops.
And here, you would solve the Majorana flops problem
but you would solve it with another problem.
OK, so we need minima of the magnetic field.
This provides magnetic trapping potential,
and there are two kinds of possible trap configuration.
One is where the minima is at zero magnetic field,
and one is where the minima is at finite magnetic field.
Now, the zero magnetic field minimum one
can be simply created with anti-Helmholtz coil.
What happens is the magnetic field,
as a function of position, would actually cross through zero.
It's just a field gradient which crosses through zero,
but, of course, since we are interested
in the absolute value of the magnetic field,
we get the V-shape potential.
This V-shape potential was what was
used for the first demonstration for Bose-Einstein condensation
because a V-shape potential is much, much more
confining in this cusp than in harmonic oscillator potential.
And this had advantages for tight confinement
and rapid evaporative cooling.
So those quadrupoled traps with the V-shape potential
are the best confinement you can get,
the best confinement for the buck,
and the buck here is your power supply and your coils.
However, they have a problem when
the magnetic field is zero.
We violate adiabaticity because the different spin
configurations become degenerate,
and in degenerates you can't have adiabaticity.
So therefore, the two first demonstrations
of Bose-Einstein condensation avoided this cusp
in two different ways.
One way was to use a rotating magnetic field,
and to use a perpendicular field but which was time-dependent,
and the MIT solution was to use a blue detuned laser beam,
use the optical dipole trap-- optical dipole potential--
to push the atoms away from the dangerous [INAUDIBLE]
field region.
I should say rotating-- none of those-- most traps which
are now used are the other kind of trap, the trap which
has an harmonic oscillator potential, which doesn't have
the cusp and where the minimum is at a finite magnetic field.
Let me just make one comment.
The rotating trap was very popular
because it led to the first BEC and a lot of people built that,
but to the best of my knowledge, it's
only used in different places.
And the only real application it has
is-- since you have some rotating magnetic field--
by some modification you can actually
make of rotating potential.
And this is nice if you want to study
Bose-Einstein condensation on a rotating frame,
create vortices and things like that.
But for simply creating a Bose-Einstein condensate,
this trap is used, by far, most frequently.
But there is actually a renaissance of this trap.
One reason why people use this trap
and why my own research group immediately
switched to this trap-- and we had
the idea how to built it-- is, well, harmonic potential
is nice.
Every physicist loves an harmonic potential.
You can solve, immediately, thermodynamics
in a harmonic potential.
Who wants to deal with that potential?
Also, if the laser beam just drifts by a micrometer,
this symmetric potential becomes asymmetric.
So you characterize your potential today
and a few hours later you have a different potential.
Whereas a magnetic potential, harmonic potential,
once it's characterized-- as long as you don't change
the current in your power supplies--
it's the same week after week, months after months.
Well, there is a renaissance now because a lot of groups
now doing experiments in optical lattices,
they don't care what shape the potential is because they don't
do physics in this potential.
They just take the Bose-Einstein condensate or the cooled Fermi
gases and transfer them to an optical lattice.
So in that case, it doesn't play any role,
and then the advantage of this potential
is you get more bang for the buck.
AUDIENCE: [INAUDIBLE] you were saying before [INAUDIBLE] more
rapid evaporation in V-shape versus the [INAUDIBLE].
Where is the differences in evap time?
PROFESSOR: It depends.
I may run out of time today.
I needed-- I can probably-- I wanted
to go through evaporative cooling,
and I will talk to you that, for evaporative cooling,
there could be a threshold in confinement
where you go into a runaway regime
that evaporative cooling is speeding up.
So a little bit of confinement can make the difference
without never getting into a runaway regime
or being in that runaway regime.
So confinement, extra confinement,
can make the difference between getting BEC and not
getting a BEC.
So it really depends.
It's highly nonlinear, but I will show you
later on that when you put a system together,
the kind of threshold density at which the cloud can evaporate
to BEC is much lower here than it is there.
But-- and this is the next thing I
want to explain-- it's not really
that this is a linear potential and this is a [INAUDIBLE]
potential because those finite magnetic field--
these traps which have a minimum at a finite magnetic field--
are usually done in the following way.
Those traps are called Ioffe-Pritchard trap.
Ioffe actually suggested such a magnetic field configuration
for confinement of plasma, and Dave Pritchard
was the first to point out that such a configuration would
be a good choice for neutral atoms.
So I sometimes joke and I say, well, plasma physics
is the study of very hot matter.
Cold atoms is the study of very cold matter.
But when matter is either too hot or too cold that you cannot
put it in ordinary container, you want it confined with
magnetic fields.
And it happens that, for plasma, you
need a minimum of the magnetic field, and for neutral atoms
you need a minimum of the magnetic field.
So therefore here is something which ultra-cold atoms have
in common with plasma physics.
I don't think there is a lot the two fields have in common,
but when it comes to magnetic field configurations, yes.
A similar magnetic field configuration
can confine a plasma and can confine neutral atoms.
OK, so the generic way of how these magnetic fields are
generated is you want to have two coils, called pinch coils.
You can say each of them creates a magnetic field which decays,
a magnetic field which decays, and now in the middle
you have the parabolic minimum.
But then you have to add the so-called four Ioffe bars which
create a linear potential, and this is shown here.
So the pinch coil is simply creating a local minimum
along the Z-axis, and if you only
want to trap in one dimension, you would be done,
But you want to trap in 3 dimensions.
And so what happens is you add now
an anti-Helmholtz-- or quadrupold-- field in X and Y,
and this blue field is done via the green bars,
by the Ioffe bars.
So you add together a harmonic quadratic field in the Z
direction with the linear quadrupoled field in X and Y.
And what matters for the atoms is--
and what matters for the Zeeman energy
is-- the absolute value of the magnetic field.
So you add those things in quadrature,
and because you add it in quadrature,
you have now an harmonic oscillator potential
in X, Y, and Z, so you have an harmonic
trap in three directions.
However, if this field is not larger than this--
if these field becomes larger than this field,
so then when you add it in quadrature,
you actually get a linear potential.
So in other words, the Ioffe-Pritchard trap
is quadratic for smaller values of X and Y,
but if you go out to large X and Y,
you get the linear potential.
So in the end-- and this is maybe in response
to Matt's question-- for a high temperature,
the Ioffe-Pritchard trap is actually not
harmonic in three dimensions.
It's linear in two dimensions and
harmonic in the third dimension.
So, therefore the hit you take in evaporative cooling
is not as large as we'd initially assumed.
And the moment we realized that, we
were building a Ioffe-Pritchard trap.
And this has been the traps we' have
been using at MIT ever since.
Well, just as a warning, this is sort of the simplified
description, but if you have curvature,
the curvature has to fulfill Maxwell's equation.
And you cannot have a curvature only along Z.
Maxwell's equations are three-dimensional,
so you get sort of all sorts of curvature terms.
If you really build a magnetic trap,
you should understand those.
If you just want to understand why magnetic traps work,
the previous slide is sufficient.
The different ways to build those traps,
this is the design we invented at MIT.
It's called a clover-leaf trap.
So you have one coil package here,
to another coil package here, and what
you see is the pinch coils, and when you have two of them
you create this parabolic minimum.
We don't like Ioffe bars because if you have a vacuum chamber
you either have to put the Ioffe bars into the vector vacuum
chamber or you drill holes through your vacuum chamber
to put the Ioffe bars back and forth.
Don't laugh, some people have done it.
Of course, they put little tubes around it
so it was a very highly-engineered vacuum
chamber where they could string Ioffe bars through the chamber.
But we realized that the same field as Ioffe bars
can be generated by taking the Ioffe bars
and flipping them out, and after flipping them out,
they had the shape of clover leaves.
It's just a variant of creating the same field geometry.
Yes, that's all I want to tell you about magnetic trapping.
I think I should not start with evaporative cooling.
This will just take 10 or 15 minutes,
but I'll do that on Monday.
So since we're on time, do you have
any questions about magnetic trapping?
Different forms of magnetic trapping?
OK.
Final announcement, next week is the due date
for the term paper.
Yes, and the term paper is due on the date
of the last class, which is Friday.
Any questions about that?
OK.
Good.