Ok well it’s a great pleasure to be here and it’s particularly an honour to be speaking to so many wonderful young researchers
and I’m looking forward to this afternoon when we have a chance to have some more detailed discussions.
And for some reason I’ve gone off the first slide.
Ok so I’m going to talk about cold atomic gasses, the intersection of condensed matter and atomic physics.
And I think everyone realises that in 30 minutes it’s essentially impossible to cover this topic to any extent at all.
And so the only thing that I hope to do is to give you a little bit of the flavour of this topic.
And if you find that you like the taste of it
then perhaps you may want to continue the discussions a little bit more in the afternoon.
Now many people in my research group at the Joint Quantum Institute
which is a joint operation of the National Institute of Standards in Technology in the University of Maryland.
A great many people have contributed to that work.
And here are some of the ones who have contributed to the things that I’m going to be talking about today.
But I don’t have time even to tell you a little bit about all of those people.
But let me show you a photograph of some of them and let me particularly point out the leaders of the group,
Ian Spielman and Trey Porto, Kris Helmerson and Paul Lett but if this projector could project red
you would see that there’s a red circle around 2 of these people, that’s Eduardo Gomez and this is Pierre Cladé
and I’ve circled them in red because they are in the audience.
And they know an awful lot about cold atoms and if you find either of these 2 young researchers
you can learn a lot about cold atoms from them.
So what do cold atomic gasses have to do with condensed matter,
well you’ve already gotten a little bit of a taste of this from Ted Haensch’s talk.
The thing that they have to do with is that we take a very cold atomic gas, a Bose-Einstein condensate
and we put it into a periodic potential, an optical lattice and we observe behaviour
that is analogous to the behaviour of electrons in the periodic potential of a crystal lattice.
And these things, the cold gas, the Bose-Einstein condensate and the optical lattice
are the key components of the experiments that we do and I’ll now try to explain a little bit more about what these are.
So what is a Bose condensate?
Well in 1923 Einstein figured out based on the new quantum statistics that Satyendra Bose had come up with,
he figured out that if you had a gas of particles that were bosons and it turns out that many atoms are bosons.
Sodium 23 and Rubidium 87, 2 of my favourite atoms are both bosons.
And if you make the gas of these atoms cold enough, typically 100’s of nano Kelvin and dense enough,
typically in the order of 10 to the 14 atoms per cubic centimetre, then there’s a phase transition.
And the nature of that phase transition is that a large fraction of the atoms in the gas go into the lowest possible energy state.
So if you imagine a gas held in some sort of a trap and I’ve represented that here by this potential well
and the gas is a normal gas, then various energy levels in this trap are occupied
and generally these energy levels are sparsely occupied.
But if you achieve the conditions for Bose-Einstein condensation
then a large fraction of the atoms will be in the ground state of this potential.
And a few of the atoms will be in excited states of that potential.
And having all the atoms in the ground state potential gives you this marvellous new state of matter
as some people have called it, this Bose-Einstein condensate which is very much as Ted Haensch pointed out,
very much for atoms in the same way that a laser or that laser light is for photons.
The reason that a laser has the properties that it has is that you have many,
many photons occupying the same mode in electromagnetic field.
And the thing that gives a Bose-Einstein condensate its wonderful properties
is that many, many atoms occupy the same quantum state.
So that’s our Bose-Einstein condensate and in many respects we can think of it as being a gas at zero temperature
and that’s our starting point for doing many of the experiments.
So how do we get there, well I can’t tell you that, we have not enough time.
But tricks like laser cooling and magnetic trapping and other tricks allow us to get cold enough and dense enough.
And finally the situation is that a typical gas, at least in our laboratory may have millions of atoms,
more than 90% of the total are in the same quantum state.
The same quantum state of both internal quantum state and centre of mass motion.
And the physical size of this gas is quite large.
On the order of 100 microns, macroscopic in many senses.
Many optical wavelengths in particular.
But it’s a gas, the atom-atom interactions can be quite small or depending on the circumstances in the experiments
they may become significant.
So that’s a Bose-Einstein condensate.
What is an optical lattice?
Well, let me talk first about the light shift.
Let’s say I have an atom and let’s simplify the atom for a moment and think of it as having 2 energy levels,
a ground state and an excited state.
If I shine light on to that atom and the light is tuned so that the energy of the photons
is slightly less than the energy separation between ground and excited state,
then there will be a perturbative change of the energy of these states.
And the ground state will go down.
It’s very much like the start effect when you apply an electric field to an atom in its ground state the energy is lowered.
Here you have an oscillating electric field and below resonance it acts very much like a static field, the energy goes down.
Now imagine 2 laser fields counter propagating against each other, they will form a standing wave
and that means there are, because of the interference between the 2 waves,
there are places where the intensity of the light is very high.
And in places where the intensity of the light is very low and where the intensity of the light is high
the energy of the atoms is shifted down and where the intensity of the light is zero the energy is not shifted at all.
So that means that you have a periodic change of the energy shift of the atoms
which means you have a periodic potential in which the atoms move.
So that periodic potential in which the atoms move is analogous to the periodic potential
in which electrons may move in a crystal and solid.
So that’s the optical lattice.
Now I’ve indicated that we want to think of the optical lattice as being analogous to the periodic potential
for atoms in a solid but there are many differences.
And of course I’m going to emphasise the differences that make an optical lattice better than a solid.
So one of those is that the optical lattice is essentially free of dislocations and defects.
The laser light that makes the optical lattice can be extremely pure, highly coherent and the lattice is almost perfect.
The atoms that move in this lattice can be either bosons or fermions.
I’ve talked so far only about bosons but it's possible to do similar experiments with fermions
and then you get to choose what kind of quantum statistics you want to use in your periodic structure.
There are no phonon, a phonon is an excitation of the lattice itself.
In our case the lattice is imposed externally,
so that the finite temperature that you may have in the system does not result in real phonons.
So that is one thing that we are completely free of.
The potential is exactly known, it’s very difficult to know exactly what the potential is
in which electrons move in a solid and people make all sorts of wonderful approximations.
We can calculate what our potential is and not only that, we can turn it on and off, we can move it,
we can modulate it, so we can do many things that it is very difficult to do in a solid system.
And there are other features that are important for these optical lattices as well.
Now how do we put these things together, how do we put a Bose condensate into an optical lattice.
Typically we make a Bose condensate in some sort of a magnetic trap.
And that’s what I’ve illustrated here.
The blue part is the Bose condensate and we have some sort of potential that’s a magnetic trap.
And then what we do is we simply turn on the laser field that creates this optical lattice
and we can then if we want turn off the magnetic trap
and we have transferred the atoms from the magnetic trap into the optical lattice.
And if we do this carefully the atoms will still be in the lowest energy state, in the ground state.
And we will have split this Bose condensate into a bunch of little tiny Bose condensates.
And if I do this with a one dimensional laser field, then I will make a series of pancakes.
And typically I’ll have a few hundred pancakes in this one dimensional optical lattice.
And we can also make 2 dimensional and 3 dimensional lattices, space lattices, things that really look like crystals.
And we can make systems that are 1, 2 or 3 dimensional as well.
This 1 dimensional lattice in a sense divides the Bose condensate into a number of 2 dimensional systems.
Now here is an important slide, so you have to pay attention to this,
because this has to do with how we learn what is going on with our atoms in this optical lattice.
If I have atoms in an optical lattice then what I’ve done is I’ve taken the Bose condensate
and I’ve divided it into a bunch of little mini condensates
and that means that the wave function that describes the Bose condensate is periodic.
At least periodic over a few hundred lattice sites.
Now if you take the Fourier transform of a periodic function which is to say if you find what the momentum wave function is,
when you take the Fourier transform of a periodic function you find that it has components at momentum values
that are themselves periodic.
It’s just the nature of the Fourier transform.
And if I then turn off the lattice, I don’t change the momentum of the atoms,
it is whatever the momentum was when the lattice was on.
And now the different momentum components which are spaced periodically in momentum space, separate.
And after a certain time I take a picture of where the atoms are
and the atoms that have lots of momentum have moved a long way and the atoms that have zero momentum have not moved at all.
And here is a picture of the atoms released from this optical lattice after a certain time.
So this is a picture of the momentum distribution.
This is the momentum wave function or the square of the momentum wave function of the atoms.
And you see that there are a lot of atoms that have zero momentum,
these are the ones that didn’t move and there are some atoms that have plus a certain amount of momentum
and minus a certain amount of momentum.
And the amount of momentum that they have is what we call the reciprocal lattice momentum
and is equal to 2 times the photon momentum.
Because the way the atoms get this momentum is they absorb and emit the momentum of the photons.
And this is in other context called diffraction, if you have a periodic structure and you shine light on it,
like a diffraction grading it breaks up into these pieces and that’s exactly what's happening with our atoms.
And analysing the momentum of the atoms in this way is one of the major tools
that we use to determine what's going on with our atoms.
Ok now since we have a periodic structure as is the case in a crystal and lattice,
we’re going to use the same mathematical tools that are used to study crystal lattices, namely Bloch states and band structure.
So if you turn to the beginning of most books on condensed matter physics you’ll see
that the eigenfunctions of a particle moving in a periodic potential are called Bloch states
and the Bloch states have this form,
the Bloch state psi(x) is the product of a function U, times an exponential factor E to the IQ X over H bar.
Now U is periodic in space as you might imagine for a periodic function
but it's multiplied by this phase factor which if you only had just this term,
it would look like an eigenstate of momentum in free space.
And the momentum would be Q.
And so we call Q the quasi momentum, it’s not really the momentum but it’s very much like the momentum
and it has properties very much like momentum.
But we multiply this momentum eigenfunction in free space times a periodic function
and what we get are the eigenfunctions for a periodic potential.
And this is a picture of what the eigenvalues look like as a function of this quasi momentum Q
and you see that there are energy gaps.
And these are the famous energy gaps that occur in condensed matter physics.
And this is a picture of what it would look like if the lattice potential is quite weak.
If we increase the strength of the lattice potential here is what the band structure looks like,
you can sort of see the traces of the parabolic shape.
You see if the particles are in free space then of course the energy
as a function of momentum is just P squared over 2 M, a parabola.
But these energy gaps open up because it’s periodic.
And here in this calculation for a deeper lattice, you can sort of see the traces of the parabolic nature.
But one of the things that happens is that the ground stage is starting to become flat, that is the ground band.
And as the depth of the lattice increases this ground band becomes increasingly flatter.
And we will soon see why it is important that the ground band is flat for a deep lattice.
But first let me say something about what happens when I put a condensate, a Bose condensate into an optical lattice,
what is its quasi momentum.
When its momentum is zero because the atoms are hardly moving,
I put it into an optical lattice and the momentum is still zero or the quasi momentum is zero.
It has other momentum components but its quasi momentum is zero.
The phase from one lattice site to the next is essentially unchanged.
How do I get to other quasi momentum?
Well, the way I do is I move the lattice and I go into the moving frame of the lattice.
And from the moving frame of the lattice it looks as if the condensate is moving.
And so the condensate has a non zero quasi momentum in the moving frame.
How do I move the lattice, it's easy, if I have 2 counterpropagating laser beams of the same frequency,
it makes a standing wave and we call it a standing wave because it’s not moving.
But if I use 2 different frequencies it’s a moving standing wave or sometimes people call it a walking wave,
so that I have, the wave is moving along like this.
And it's easy to do that and I can make the lattice move at whatever velocity I want
and from the point of view of the lattice the atoms now have a non zero quasi momentum.
Now how do the atoms behave in that lattice frame.
Well one of the things that determines how they behave is their group velocity.
The group velocity is just the velocity at which a wave packet would move
if it has the momentum or the quasi momentum that is centred on the quasi momentum that we’re describing.
And the group velocity is given by the derivative of the energy as a function of the quasi momentum with respect to that momentum.
Now for a free particle that’s easy to calculate because the energy is just P squared over 2 M
and if I take the derivative of that with respect to P I get P over M which is V, the ordinary velocity.
But if it’s in a lattice and the lattice is very strong, we saw that the band structure of a lattice is such
that the band is flat, that means the derivative of the energy with respect to quasi momentum is nearly zero.
And that means that the group velocity in a very deep lattice is nearly zero.
So the atoms are frozen in the lattice.
Well that’s not surprising if the lattice is very deep, the atoms are trapped very strongly,
they’re frozen, in the frame of a lattice.
So that means that if I go into the laboratory frame and the lattice is moving along,
what that means is the atoms are being dragged by the lattice.
So that’s what we expect to happen, that the atoms will be dragged by a lattice that is very deep.
So now I want to give you a kind of mechanical analogue of making an optical lattice and then moving it.
So my optical lattice, my moving optical lattice I think of as being a conveyor belt
and it’s a very special sort of conveyor belt that allows me to create on it corrugations
in which the atoms may sit and be dragged along.
So I imagine that I’ve got some atoms and they’re sitting on top of the conveyor belt
and the conveyor belt is at rest so the atoms are at rest with respect to the conveyor belt.
Then I turn on the modulation of this conveyor belt and the atoms become trapped in these valleys of the conveyor belt.
And then I start the wheels going on the conveyor belt and the atoms start to move.
Now if the lattice is deep then it means that the atoms will be dragged along by the lattice.
And we should be able to tell that by turning the lattice off and looking at the momentum of the atoms.
So here’s the experiment that we’re going to do.
We’re going to turn on the lattice, this is the depth of the lattice.
Now when the lattice has achieved a certain depth, deep enough to drag the atoms, then we start to increase the Q,
that is increase the velocity of the lattice.
And then we’re going to reach some final velocity, stay at that velocity
and then suddenly turn off the lattice and then look at what the momentum of the atoms is.
So here is a picture of the distribution of atoms when the lattice velocity is zero.
This is just a fraction, we’ve already seen this before, most of the atoms are at zero momentum,
some are at plus and minus 2 photon recoils.
And the average momentum is zero, which is what you would expect because the lattice is not moving.
And measuring the momentum of the atoms in the lab frame, but in the lattice frame the quasi momentum of the atoms
is equal to the negative of the velocity of the lattice in the lab frame.
And then if I ask how this changes as the velocity of the lattice changes,
you see that the average momentum of the atom increases exactly as the velocity of the lattice does.
So the lattice is indeed dragging the atoms along.
And here’s the confirmation, that the average velocity of the atoms is equal,
if you figure it out, we’ve done this carefully and the average velocity of the atoms is equal to the velocity of the lattice.
Now let’s do a different experiment, almost the same thing.
We’re going to put the atoms on to the conveyor belt, we’re going to turn on the modulations, the corrugations,
so the atoms are trapped.
We’re going to start the conveyor belt moving and we know that the atoms are dragged
and now instead of turning the laser off suddenly, I’m going to turn the laser off slowly.
And I’m going to ask you what do you think the velocity of the atoms is going to be, and let me emphasise what the experiment is.
Here are some atoms, I turn on the optical lattice represented by my fingers
and the atoms are now trapped in between my fingers and now I move the lattice along like this at some velocity
and the atoms are dragged because we know, because we’ve already measured them.
And now instead of turning the lattice off suddenly, the way I did in the last experiment,
I’m going to slowly turn the lattice off and then ask what will the velocity of the atoms be.
And it’s a multiple choice question, so how many vote that the velocity of the atoms will be the velocity of the lattice.
As it was in the previous experiment, well we have a few brave souls who are willing to express an opinion.
How many think that the velocity of the atoms will be zero, oh a few people, ok.
How many think that it will be something in between the velocity of the lattice and zero,
ah a lot of people believe it will be somewhere in between.
How many don’t know, ah ok good.
How many believe that it is none of the above.
Ok well let’s find out.
The way you do things in physics is you do the experiment.
Ok so here’s, oh gee the colours look horrible here.
So what's going on here, this just shows what the sequence of events in the experiment is.
We turn the lattice on, once it’s on to a fairly deep value, then we start to accelerate the lattice.
We reach some final value and then we slowly turn the lattice off.
Now here’s the result, first I’m going to show you the result when the velocity is zero,
here I’ve returned the lattice on slowly, I’ve turned it off slowly and I’m right back where I started
because if I do everything adiabatically, I finish where I began.
And there are the atoms sitting at zero velocity.
There’s no diffraction pattern because I turned it off slowly
and I re-collapsed all those momentum components that existed in the lattice because I adiabatically returned to where I was.
So I’ve just got one component right here.
And everybody expected that when the atoms are not moving, they’re not moving.
Now what happens as I start to accelerate the lattice?
The velocity stays zero.
Now let me emphasise how odd this is.
So those of you who said zero, you were partly right.
So remember the atoms are being dragged along like this at a constant velocity
and then you slowly turn off the light and the atoms are at rest.
How can that be, what happened to the law of inertia.
Well these atoms are quantum mechanical, they remember what their phase is.
And their phase is constant and constant phase means no velocity.
How do they go from moving to not moving, well they exchange momentum with the lattice.
So up until the velocity of the lattice is equal to one recoil velocity, that is the momentum of a photon in an atom
which for sodium atoms is about 3 centimetres per second,
it’s the edge of the Brillouin zone for those of you who are condensed matter physicists.
Until you get to the edge of the Brillouin zone the velocity atoms is zero.
And then when you get to the Brillouin zone the velocity jumps to twice the velocity of the lattice.
And continues to be that until you go to 3 times the photon recoil which is another 2 Brillouin zones further
and then the velocity jumps again.
So the velocity is never anything in between the reciprocal lattice vectors, it keeps jumping by reciprocal lattice vector.
And so the answer was none of the above, the velocity depends on the situation and it’s never, well except accidentally,
equal to the velocity of the lattice.
And almost never equal to the velocity of the lattice and it has this rather odd behaviour.
This whole procedure, this whole thing is an extremely simple application of band structure and Bloch functions.
And in another context is called Bloch oscillations.
And if you want to understand more in detail about this come to the discussion this afternoon.
So far I’ve told you about experiments that do not involve the interactions of the atoms,
the atoms are non interacting particles, an ideal gas
and we’ve seen that sort of rather interesting behaviour even for an interaction free system.
But now I want to talk about what happens when I have interactions.
Some experiments where the interactions play a role.
Well one of these was already spoken of by Ted Haensch in his talk, the MOTT insulator transition.
This is a quantum phase transition.
Now quantum phase transitions are odd things.
We typically think of them as happening at zero temperature,
they occur not because I changed some thermodynamic variable like the temperature,
they happen because I change some other thing having to do for example with the Hamiltonian of the system
but not a thermodynamic variable.
So that’s what distinguishes quantum phase transitions from thermodynamic phase transitions.
The ones we are usually familiar with.
And what it does is it changes something from being, if it’s an electron system,
from being a conductor to being an insulator or in the case of the atoms,
from being a super fluid to being a kind of an insulator.
It’s something that’s very difficult to see in condensed matter physics
but it’s responsible for determining the insulator or conductor properties of many materials.
So with atoms we would like to see this transition.
So here is a model that can exhibit the MOTT insulator transition, it’s called the Bose Hubbard model.
We imagine a periodic potential here and we just think of 2 things about this periodic potential.
How easy it is for atoms to tunnel from one side to the other and how much energy cost there is if 2 atoms occupy the same site.
Because if I have 2 atoms on the same site and they repel each other,
there’s a certain energy cost for having 2 atoms be on the same site.
So there’s a certain energy U and there’s a certain tunnelling rate T between the sites
and that’s all that goes into this Hamiltonian.
And that’s all you need in order to see the MOTT insulator transition.
So when the tunnelling is very big and the atoms can easily move between lattice sites,
then you have the super fluid phase and the atoms, every atom is in every site and freely moves between the sites,
it's very much like a Bose condensate.
On the other hand if you reduce the tunnelling so the tunnelling is smaller,
then the energy when 2 atoms occupy the same site and I certainly didn’t mean this colour to be yellow
so that you couldn’t see it, but the projector isn’t projecting red I guess,
so these things are really green on my screen, but anyway.
So there’s supposed to be one atom per lattice site.
When the tunnelling is small enough then you get one atom per lattice site.
And we can tell the difference between these systems because when the atoms are everywhere,
then you turn the lattice off you will get diffraction, just as we saw before.
But when you have one atom per lattice site, because of the number phase uncertainly relationship
you know exactly what the number is per site which means you have no idea what the phase is.
And that means you won’t get any diffraction because you don’t have the right phase relationship to see diffraction.
So the disappearance of diffraction will be an indicator of having gone into the MOTT state.
Now the first experiments of these were done in Ted Haensch’s laboratory a few years ago,
we’ve recently done experiments in a very nice 2 dimensional system
and here as we increase the depth of the lattice you can see that the diffraction pattern goes away.
So it’s doing the right thing.
But we want it to do more, we want it to nail down exactly where the phase transition was occurring.
So by looking in detail at the diffraction pattern and extracting from the diffraction pattern
the part that is sharp and the part that is broad and fuzzy.
And plotting that sharp part, which is the part that still maintains its phase, as a function of lattice depth,
we have this very nice behaviour where it comes down sharply to zero and then continues to be zero,
that is no condensate, no long range phase for deeper and deeper lattices.
And we compared the place where this thing went to zero with a quantum Monte Carlo calculation
and essentially have perfect agreement.
A few years ago that quantum Monte Carlo calculation was too difficult to be done.
So what we have here in a sense is a simulation of the Bose Hubbard model.
A model that until a few years ago was too difficult to calculate in detail at the level necessary
to see this kind of quantum phase transition.
And we have made that model a reality in the laboratory and gotten a very good agreement with the theoretical calculation.
Well there are many other places where people are doing the theory and the experiments of these kinds of systems,
atoms and optical lattices including Ted Haensch’s laboratory in Munich.
And let me just draw a few conclusions now about this kind of work.
So the obvious thing that I started with is the cold atoms in optical lattices are a simplified
analogue to electrons and crystals.
Now whenever you want to study electrons and crystals, theoretically you always make certain simplifying assumptions.
What we do here with our atoms and optical lattices is we realise those simplified assumptions in a real experiment.
So it’s not a mathematical version of the simplified optical lattice, it’s an experimental version,
of the simplified condensed matter lattice, it’s an experimental version of the simplified condensed matter lattice.
And because of the unique features of control and measurement
we can see things that are very difficult or sometimes impossible to see in a condensed matter system,
like the Bloch oscillations which are very hard to see,
like the MOTT insulator transition which is very difficult to actually undergo the transition in a solid state system.
But more importantly these simple condensed matter models that one thinks of are actually realised in the optical lattice.
And so the cold atoms are a kind of analogue quantum computer where we can compute these simplified models.
Not that we’re computing what's happening in the solid, we’re computing what the simplified model of the solid predicts.
Very often under circumstances that are very difficult to compute with real computers, we can do it with our cold atoms.
And my final point is that this intersection of cold atoms
and condensed matter is a wonderfully exciting and growing field in which the opportunities are very, very strong
for each of the disciplines, condensed matter and atomic physics to learn from each other.
And so if you’re interested in knowing more about that come this afternoon.
Now let me just end my talk with a few comments about Ivar Giaever’s talk this morning.
He gave you lots and lots of advice about how you should do science and that advice was excellent advice.
And I certainly agree that you should follow it.
And if you follow it you might have a 1/10th% change of getting a Nobel Prize.
The reason why you should follow that advice is not because you want to win a Nobel Prize,
the reason you should follow that advice is because you want to do exciting physics.
And that is an excellent prescription for doing exciting physics, is following that kind of advice.
But I do want to take issue with one thing that he said, he said that physicists are really competitive
and that’s true and then he said that physicists are not very nice people, now you know everything I know about him,
he’s a pretty nice person.
There are a lot of physicists who are really nice people, there are a lot of physicists who are not very nice people.
I don’t believe that getting a Nobel Prize has anything to do with whether you’re a nice person or not.
And I don’t think doing exciting physics and having a lot of fun doing physics,
well actually it does have something to do with being a nice person.
And so since winning a Nobel prize I don’t think has anything to do with being a nice person
and since getting exciting results out of your physics,
I don’t think has anything to do with whether or not you’re a nice person – why not be a nice person anyway?
Thank you very much.
(applause).