Even colder still

In a previous post I was talking about how you can use a laser to cool atoms. By tuning the laser to just below the energy of an atomic transition you can selectively kick atoms that are moving towards the laser. If you fire six lasers in (one for each side of the cube) you can selectively kick any atom that is trying to leave the centre. So we've made a trap!

There is a hitch unfortunately. There is a minimum to which one can cool the atoms, once the atoms have an energy that is comparable to the photons coming from the laser then that's about as low as they can go. After all, there's only so much you can cool something by kicking it. We're already pretty cold - around 100 micro Kelvin - we'd like to go a bit colder if we can. Now we're into magnetic traps.

Magnetic Traps

Up to now we've been acting quite aggressively towards the atoms - kicking anything that's moving too quickly. To do better we're going try and round them up where we can control things better. Fortunately there's a neat way to do this. We can make use of an inhomogeneous magnetic field and the Zeeman effect.

If you apply a magnetic field to our gas of atoms then the magnetic dipoles of the atoms tend to line up with the field. Being quantum physics they can only do so in a discrete number of ways. What happens is that the transition that used to be a line splits and shifts into a number of different lines.

If we use a stronger field then the shift is larger. We can finely tune the energy at which our laser will interact with the atoms. So now we do this; if we put a magnetic field that is zero in the middle of the trap and gets bigger as you move away from the centre (you can do this) then we can control how hard we kick the atoms depending where they are. If we do it right then inside the trap we hardly kick them at all and outside trap we kick them back in.

Evaporation

We've managed to confine the atoms in our trap, the final step is to switch off the lasers (to stop all that noisy kicking and recoiling) and to try and use evaporation to get rid of as much energy as possible. It is understandably quite complicated to stop them all flying out once you've switched off the lasers and unfortunately it's at this point I start getting lost! The actual cooling mechanism is nothing more complicated than why your cup of tea goes cold.

After all this we're down the micro Kelvin level - a millionth of a degree above absolute zero! At these sort of temperatures the atoms can undergo a quantum phase transition and become a Bose-Einstein Condensate (BEC). This is a new state of matter, predicted by theory and finally observed in the nineties. As far as I know this is as cold as it gets anywhere in the universe.

Well I think I'm done with cooling things now. It starts off beautifully simple and then gets a bit harder! Needless to say I salute anyone that can actually do this - it's back to simulations for me.

EDIT: I over-link to wikipedia but this is a good page on Magneto-optical traps

Laser Cooling

Last semester I was helping out teaching a bit of quantum and atomic physics. It was quite fun going back to stuff I was a little hazy on the first time. I finally understand the periodic table for one thing. Another thing that I knew about but never really got the detail is laser cooling. This is really nice, I'll blast through it here. Watch out for the stat-mech bit, blink and you miss it.

In an atom electrons are not free to sit anywhere they want (more or less), they inhabit precisely defined quantum states that have well defined energies, angular momenta etc. Therefore if you give an atom a kick then it will release the energy you give it in precisely defined packets of energy. So if you take the light emitted by the atoms and put it through a spectrometer (could just be a prism) you'd see something like this, from here, for sodium.

You'll recognise the orange line from the street lamps that are slowly on their way out. I did a version of this experiment when I was an undergrad where we did the opposite, we shone white light through sodium gas and while most of it goes through the frequencies that match the right transition frequencies get absorbed and are missing from the final spectrum. Might look like this, ish

Notice that the lines aren't all that sharp whereas I said they should be precise lines. This is for a number of reasons. One is that the uncertainty principle doesn't like precise energies. There's an uncertainty attached to the lifetime of atomic transitions or collisions. Another, more important effect is Doppler shifting due to the temperature of the gas. We can assume that the atoms in the gas have a distribution of velocities that comes from the famous Boltzmann distribution

Light emitted from a moving atom will be Doppler shifted which will take our precise emission line and spread it out around the average. This property turns out to be very useful and what we'll use. First a mention about the laser.

Lasers are brilliant. With a laser you can send in a beam of photons with a highly tuned narrow band frequency. When a photon hits with a frequency that matches the absorption frequency of the atom, they collide and scatter. When it's too much or too little it will most likely just go straight through.

So finally we get to how you cool the gas. If you send in a laser pulse into a warm gas of atoms then different atoms will see different things. Thanks to the Doppler shift, an atom moving with speed, v, will see the laser frequency, f_0, Doppler shifted to (c = speed of light)

Atoms moving away from the laser see it red shifted (lower frequency), atoms moving toward the laser see it blue shifted (higher frequency). If we tune the laser to just below the absorption frequency of the atom then the only atoms that collide with the beam are those moving towards it (the ones that see the blue shift).

Were it not for the precision of the transition level the laser would equally kick atoms moving towards it and atoms moving away - adding no net energy into the system. However, if we only collide with atoms moving towards the beam then we can actually remove energy. What's even more staggering is that this actually works!

Laser cooling can make things seriously cold. You may have seen the headlines that the LHC is colder than space. Impressive given the size of the thing, but space is about 2 Kelvin. This is peanuts compared to laser cooling. This can get a gas down around 1 mK - that's a factor of a thousand. You can get even colder with new techniques but somehow laser cooling pleases me the most.

So that's laser cooling. It's beautifully simple, uses basic ideas from quantum mechanics, relativity, statistical mechanics and then makes something brilliant thanks to a laser.

Critical Point

I'm finally getting around to sharing what, for me, is the most beautiful piece of physics we have yet stumbled upon. This is the physics of the critical point. It doesn't involve enormous particle accelerators and it's introduction can border on the mundane. Once the consequences of critical behaviour are understood it becomes truly awe inspiring. First, to get everyone on the same page, I must start with the mundane - please stick with it, there's a really cool movie at the bottom...

Most people are quite familiar with the standard types of phase transition. Water freezes to ice, boils to water vapour and so on. Taking the liquid to gas transition, if you switch on your kettle at atmospheric pressure then when the temperature passes 100 degrees centigrade all the liquid boils. If you did this again at a higher pressure then the boiling point would be at a higher temperature - and the gas produced at a higher density. If you keep pushing up the pressure the boiling point goes higher and higher and the difference in density between the gas and the liquid becomes smaller and smaller. At a certain point, the critical point, that difference goes to zero and for any higher pressure/temperature the distinction between the liquid and gas becomes meaningless, you can only call it a fluid.

The picture below, taken from here, shows the standard phase diagram, with the critical point marked, for water.




Magnets also have a critical point. Above the critical temperature all the little magnetic dipoles inside the material are pointing in different directions and the net magnetisation is zero. Below the critical temperature they can line up all the in the same direction and create a powerful magnet. While the details of this transition are different from the liquid-gas case, it turns out that close to the critical point the details do not matter. The physics of the magnet and the liquid (and many other systems I won't mention) are identical. I'll now try to demonstrate how that can be true.

The pictures below are taken from a computer simulation of an Ising model. The Ising model is a simple model for a magnet. It's been used for so much more than that since its invention but I don't really want to get into it now. For the pictures below squares are coloured white or black. In the Ising model squares can change their shade at any time, white squares like to be next to white squares and black squares like to be next to black squares. Fighting against this is temperature, when there is a high temperature then squares are happier to be next to squares of a different colour. Above the critical temperature, if you could zoom out enough, the picture would just look grey (see T=3 below). Grey, in terms of a magnet, would be zero magnetisation.







If you drop the temperature then gradually larger and larger regions start to become the same colour. At a certain point, the critical point, the size of these regions diverges. Any colder and the system will become mostly white, or mostly black (as above, T=2). Precisely at the critical point (T=2.69 in these units), however, a rather beautiful thing happens. As the size of the cooperative regions diverge, so too do fluctuations. In fact at the critical point there is no sense of a length scale. If you are struggling to understand what this means then look at the four pictures below. They are snapshots of the Ising model, around the critical point, at four very different scales - see if you can guess which one is which.








Now watch this movie for the answer (recommend switching to HD and going full screen).





The full picture has 2^34 sites (little squares), that's about 17 billion. This kind of scale invariance is a bit like the fractals you get in mathematics (Mandelbrot set etc) except that this is not deterministic, it is a statistical distribution.

How does it demonstrate that the details of our system (particles, magnetic spins, voting intentions - whatever) are not important? In all these cases the interactions are short ranged and the symmetry and dimension are the same. Now imagine that you have a picture of your system (like above) at the critical point and you just keep zooming out. After a while you'll be so far away that you can't tell if it's particles or zebras interacting at the bottom as that level of detail has been coarse grained out and all the pictures look the same. This is not a rigorous proof, I just want to convey that it's sensible.

Of course the details will come into play at some point, the exact transition temperature is system dependent for example, but the important physics is identical. This is what's known as universality, and it's discovery, in my opinion, is one of the landmarks in modern physics. It means I can take information from a magnet and make sensible comments about a neural network or a complex colloidal liquid. It means that simple models like the Ising model can make exact predictions for real materials.

So there it is. If you don't get it then leave a comment. If you're a physics lecturer and you want to use any of these pictures then feel free. I'd only ask that you let me know as, well, I'd like to know if people think it's useful for teaching. For now you'd have to leave a comment as I haven't sorted out a spam-free email address.

UPDATE: Forward link to a post on universality.

Entropy

I've been meaning to post something interesting about stat-mech about once a fortnight and so far I'm not doing so well. For today I thought I'd share my perspective on entropy.

If you ask the (educated) person in the street what entropy is they might say something like "it's a measure of disorder". This is not a bad description, although it's not exactly how I think about it. As a statistical mechanition I tend to think of entropy in a slightly different way to say, my Dad. He's an engineer and as such he thinks of entropy more in terms of the second law of thermodynamics. This is also a good way of thinking about it, but here's mine.

Consider two pictures, I can't be bothered making them (EDIT: see this post, the T=2,3 pictures) so you can just imagine them. First imagine a frozen image of the static on your television, and secondly imagine a white screen. On the basis of the disorder description you might say that the static, looking more disordered, has a higher entropy. However, this is not the case. These are just pictures, and there is one of each, so who is to say which is more disordered?

Entropy does not apply to single pictures, it applies to 'states'. A state, in the thermodynamic sense, is a group of pictures that share some property. So for the static we'll say that the property is that there are roughly as many white pixels as black pixels with no significant correlations and for the white screen we'll say it's all pixels the same colour. The entropy of a state is the number of pictures (strictly it's proportional to the logarithm of this) that fit its description.

For our blank screen it's easy, there are only two pictures, all black or all white. For the static there are a bewildering number of pictures that fit the description. So many that you'll never see the same screen of static twice, for a standard 720x480 screen it'll be something like 10 to the power 100,000*.

So it's the disordered state, all those pictures of static that look roughly the same, that has the high entropy. If we assume that each pixel at any time is randomly (and independently) black or white, then it's clear why you never see a white screen in the static - it's simply out gunned by the stupidly large number of jumbled up screens.

In a similar way a liquid has a higher entropy than a crystal (most of the time, there is one exception), there are more ways for a load of jumbled up particles to look like a liquid than the structured, ordered crystal. So why then does water freeze? This, as you might guess, comes down to energy.

Water molecules like to line up in a particular way that lowers their energy. When temperature is low then energy is the most important thing and the particles will align on a macroscopic scale to make ice. When temperature is high entropy becomes more important, those nice crystalline configurations are washed out by the shear number of liquid configurations.

And this is essentially why matter exists in different phases, it's a constant battle between entropy and energy and depending which wins we will see very different results.

I'll try and update with some links to better descriptions soon.

*this number is only as accurate as my bad definition of the disordered state.

Busy Bees

The second installment of Swarm was on BBC 1 last night, I missed the first one but I highly recommend catching this before it goes off iPlayer.

The best bit was the fire ants making an ant raft to escape flooding. Ants are ridiculous.  They also had bees trying to decide where to make a new home.  The scouter bees come back with reports on possible locations, conveying the message with a dance. All the scouters sell their location and the others decide who to follow. When one of them gets enough support then they all up sticks and move - pretty smart.

On the same theme, I was at a talk recently about consensus decisions in sticklebacks. Apparently they're very reproducible in experimental terms. Again, they have to make a decision, this time about which way to swim. On their own they make the good decision the majority of the time (say 60%) but when they're in a group the almost always get it right. Each fish is pretty stupid, the group is less stupid.

I love problems like this because, while it is a biology problem, it's simple units (fish, ants, bees) that can interact with their peers in some measurable way (well, if you're really clever and patient it's measurable). From this emerges surprising a complex behaviour that didn't exist with the individual - that's what statistical mechanics is all about.

Critical-point post is still delayed, when you're debugging code at work all day it's hard to feel motivated to come home and do the same thing. It's coming though.

UPDATE: Just seen part one, those starlings are badass. They look like drops of liquid, just wait until I get my MD code working and I'm going to be simulating me some birds! (not in the weird science sense, although that would be cool as well).

Supercooling

Ok, a bit of proper science to try and establish some balance. This is a bit of a lazy post but I did mention I'm busy writing my thesis. On supercooled liquids...

Supercooled water is just water that is below its freezing point but for one reason or another didn't crystalise. It's a form of metastable equilibrium. If you give it a big enough kick then it can escape and this happens



And you can do it with beer



If you keep cooling it further then you eventually get glass, and that's all glass is. Now my stuff is more interested on the supercooled goes to glass bit rather than freezing beer but you get the idea.