Hello, everyone. Welcome back to Exploring Quantum Physics. I'm Charles Clark. In this part, we're going to look at the properties of Fermi-Dirac particles. This is a subject that is the foundation of chemistry and materials physics, has been known as such for a long time. But only in the past decade or so has it become possible to explore it directly with great controllability and observability in gaseous systems, and that's led to a number of exciting new developments. Now we're going to look at the other part of the world, the fermion sector, and what happens as Fermi gases get cold. This investigation was started by Fermi just a year or so after Einstein thought of his theory of the ideal gas, and Fermi took a very similar approach. Start with an ideal gas, then construct the quantum states of motion. But now, the big difference from Einstein's theory is that Fermi suggested that one look at an alternative quantum hypothesis from many particle systems, which is that no more than one atom can occupy any given state. This is now the prescription for what we call Fermi-Dirac statistics, and the particles that obey the statistics are fermions, the electron, the proton, the neutron. In fact, ordinary matter is made up of fermions if you have pairs of fermions that are boson, but nevertheless, the fundamental particles that are most important for our lives with massive particles are fermions. And then he calculated the properties of gas in thermal equilibrium. In particular, Fermi used the construction, which is rather different to what you find in modern textbooks. He takes the atoms and puts them in a three dimensional harmonic oscillator potential. His original paper in English translation to be found in additional materials, it's very short, very beautifully written, and anyone can benefit from looking at it. Anyway, this hypothesis that no more than one atom can occupy a given quantum state thereby results in a very different low temperature limit than we have for bosons. So for bosons, we have Bose–Einstein condensation, and all atoms go into the ground state of the system when the quantum degeneracy regime is attained. That is when the phase based density is greater than 1. For fermions on the other hand, so now, this diagram taken from Marcus Griner is a cartoon of what happens for the system of fermions with two spin states, one up and one down. The way Fermi treated it, you just consider one of these families as a fermion. Let's say the spin one, spin up. So if you take the spin up, if you have spin up particles, then you have to load up all the levels of the trap until you exhaust the number of particles. And that means that the zero temperature state has a very high total energy. So, as you approach zero temperature from above, what happens is you form this hard wall of fermions. And you can't make the system any colder by removing the energetic atoms because the zero temperature state has all these levels occupied. There are so many important implications of Fermi's idea. I mean, basically all of chemistry depends upon what's called the Pauli exclusion principle, which is that you can only have one electron per quantum state or two per quantum state if you have both spins. One of the most astonishing things for me is this limit named after Chandrasekhar. It is a stunningly beautiful equation. You can read more about it in Chandrasekhar's paper, which is included in additional materials. The paper's not about this limit, but our narrative has something about it. This is a wonderful equation because it incorporates the Planck constant with the stance for quantum mechanics, the speed of light, which is relativity, and the gravitational constant, which is gravity. This is, if you remember, Victor's diagram of the different dimensions of physical theory covered in this first lecture. This has got something from each one of them. And what it is, it expresses the maximum mass that a star can have and still be stable. And when this was first published in the scientific literature, and I think Chandrasekhar was 19 years old when he wrote this expression out, it shocked the astronomy community because it implied that there would be run away collapse, an existence of black holes. Because you can always add more matter to the star just by stellar collision or something, gravitational attraction will enable you to load as much matter onto a star as you possibly can. But Chandrasekhar showed that with the Fermi statistics that provide the support against gravity, there's a critical mass beyond which one would get gravitational collapse. The actual construction, and this is by Fermi, of an ideal gas of fermions in a harmonic trap was realized in 1999, just a few years after Bose–Einstein condensation was attained. And it was done with a system about a million potassium atoms, a million atoms the isotope potassium-40 in a 3D harmonic oscillator trap, which reminds me, I've been meaning to ask you something. Right, so I hope you figured out why potassium-40 is a fermion. It's a composite particle made up of an odd number of bosons, 59 in total. And so this is a schematic of the experiment, very similar in approach to the others that we've described, additional details of a preparation chamber in a so-called science chamber. But again, the system is consist of a magneto optical trap, and then the imaging is done in the same way, so the shadow image of light coming through the atoms and being resolved by a CCD camera. And the key finding in this experiment was an evaporative cooling curve, so there's a process of cooling by evaporating the potassium atoms from the trap, that evaporative cooling is discussed in a very easily accessible way in Eric Cornell's paper, Bose–Einstein Condensation, in our additional materials. And what you find is that it becomes progressively harder to evaporatively cool as you approach the Fermi temperature. In fact, there's sort of a limit is reached. And this is, again, because in evaporative cooling, what you're doing is taking, you have in the trap, you have lots of hot atoms as well floating around. And you cut them away and then there's a re-thermalization, which means that the temperature goes down because you've taken the hot material out of the system. So that leads to cooling, but when you get down to the Fermi, near the Fermi temperature, you're just removing atoms away from the Fermi level, so they are more energetic, but they're not thermally excited. They just happen to be the high line state. So basically, it becomes very hard to cool fermion atoms compared to bosons. Well, here is an approach to cooling fermions that goes below that limit. This is an absolutely wonderful experiment from a technical standpoint. It involves trapping two isotopes of lithium in the same trap. This is, by the way, made possible by the isotope shift in atomic energy levels that we discussed on several occasions earlier in the course. Well, that reminds me, there's something I've been wanting to ask you. So here is the answer, Lithium 7 is the boson because it's a composite made out of ten fermions. Lithium 6 is the fermion that's a composite made out of nine fermions. And so you see in this sequence of images, here is the cooling that's going on from top to bottom. The two clouds are very similar at high temperatures, but as you drop the temperature, you can see that you get a much colder cloud and more dense on the bosonic side than on the fermion side. So, that shows a difference in cooling properties between fermions and bosons. And we can also take a look at the problem from another angle. For many years, there's been discussion about the connection between superconductivity and Bose–Einstein condensation. So, one simplistic view is that the Cooper pairs of electrons that are responsible for superconductivity are a composite boson and so superconductivity is sort of like the condensation of the Bose condensation of Cooper pairs. But now, the connection between these two can be mapped out in detail in ultra cold atom systems. In a superconductor, we just have the electrons and the temperature, we can't really vary the pairing interaction in a controllable way. When a system of ultra cold atoms, in this case, Lithium 6, as was discussed in the previous slide, the interaction energy between atoms can be modified by a Feshbach resonance. Effectively, it's a sort of a resonant process of increasing the strength of pairing between the atoms in the example. So, in this case, one is looking at a phase diagram, pressure, as a function of the interaction energy that's varied. And one can move between the BCS regime of superconductivity of pairs of loosely bound pairs of electrons or pairs of atoms here. But the Cooper pairs, as discussed in Victor's lecture, reside in this regime. And then by increasing the interaction potential, one can move into the BEC region, where now you have basically dimers formed of these fermionic atoms. And they become like a composite boson, and so you see Bose-Einstein condensation there. So, we'll conclude with a very brief survey of the state of the art in quantum gasses in the final part of this lecture.