Hello everyone, welcome back to Exploring Quantum Physics. I'm Charles Clark, we're now beginning the last week of lectures which is going to focus on current research topics. And we'll begin with the study of the state of the art in quantum gasses. We're going to begin by looking at the problem of ultra-cold Bose-Einstein gases, which lead to the famous phenomena of Bose-Einstein condensation. So we'll start by seeing how that concept originated. And the basic idea began very simply. Albert Einstein decided to understand how to quantize many particle systems and he chose the simplest one, the ideal gas, to ask the question of what's the equation of state of an idea gas when you apply quantum mechanics? So, the simplest approach is to construct quantum states of motion for a single gas atom and then ask how do these get modified when many atoms are present. So Einstein decided to try the following assumption that for a given quantum state any number of atoms can occupy it. So, there's no restriction on the number of atoms that can be in the same quantum state of the system. So, if you think of a particle in a box the ground state has no nodes and presumably you can populate that state with as many atoms as you'd like. And then, you use statistical mechanics to calculate the properties of the gas in thermal equilibrium. So now, in modern language, this assumption that any number of atoms can occupy given a quantum state is called Bose Einstein statistics which will apply to bosons, which have integer spin zero, one, two, and so on. So photon is a boson, and a pair of fairmions is also a boson, we'll get to that momentarily. Well, in carrying out this seemingly simple program, Einstein discovered something absolutely amazing. Here is the original paper in which he announces it. And he describes the finding here in blue, I'll give you my own translation. He starts by considering the problem of compressing an ideal gas at a constant temperature. So you take a gas that has some particular value, take the air in the room, and you keep the temperature constant while condensing it. And he finds that for a given temperature, there's a saturation density of the ideal gas. So in other words, as you keep compressing it you reach a critical pressure at which suddenly a number of the molecules of the gas no longer engage in thermal motion, so its like you create by increasing the pressure, a new phase of the gas, which has zero temperature, this is very strange. And in fact Einstein, in this paper, wondered whether it was simply a mathematical artifact of the theory and had any correspondence in reality. Well when does this condensation phenomena occur and what is it? So we're going to go back and return to classical statistical mechanics which is what Einstein used. Now, you might say it's quasi-classical because it has a Planck's constant in it, but this is the accepted standard form for calculating the number of thermally accessible states of a classical system. And you will recall that in the previous lecture, we discussed the calculation of the partition function in this form. So I trust you were able to evaluate that integral again. And here's the, let me rewrite the partition function by taking the contribution from the volume out front and then isolating this contribution from the momentum divided by h bar cubed. So this first integral just has the dimensions of volume. In fact, we might just have v equals 0 inside a box and use that as a volume for the purposes of thinking about this. And the second integral also has the dimensions of volume, or the inverse dimensions of volume. You see it's, you can think about the uncertainty principle p over h is equal to 1 over x. So in fact this partition function takes a simple form, easy to remember. It's the ratio of the volume to the cube of the de Broglie wavelength, and the de Broglie wavelength is defined as the well, this integral here is an integral over three spatial dimensions. The de Broglie wavelength is defined as the inverse of the one dimensional equivalent of that integral. That's right here, and so that's the de Broglie wavelength there, inversely proportional to the square root of temperature. Now the conventional terminology in this field is to measure quantum gases in terms of their phase space density. Which is if you have a gas with a given number density, say a number of atoms per cubic centimeter, then the phase space density is a pure number. So product of the density and the cube of the de Broglie wavelength and it has the interpretation of being the number of atoms that exist in the sample per available quantum state. So let's see what that implies, here's a cartoon from a very nice paper by Eric Cornell, which you can find additional materials. It's a good introduction, an accessible introduction to Bose-Einstein Condensation. And it makes note that in an ideal, classical gas, there's basically one scale parameter that describes the gas. It's the mean distance per particle or the inverse cube root of the number density that represents the average separation of particles. In the quantum gas there is a new length scale that's introduced by the de Broglie wavelength, and this is a wave length that depends upon the temperatures. It depends upon the mean thermal energy of the gas, you might say. So the interesting physics occurs when these two length scales become comparable, in other words when the face based density of the system approaches one. And what happens then is that as the phase based density increases above the critical value needed for condensation, a value that depends on circumstances that we'll discuss a little bit later, then macroscopic number of atoms occupy the ground state of the system. So in other words, a significant fraction of the system has effectively zero temperature. And as the phase based density increases, a larger quantity of the system goes into the ground state. Well let's see how close we are to Bose-Einstein condensation in everyday life. So you see, the air in your room right now is not very close to being a Bose-Einstein condensate. And if you just take a simple consideration of what would be required to get it to a phase-based density, of one, you have two choices. You either increase number density by a factor of ten to the seventh. If you did that, you'd have a very heavy form of matter. It would be nothing like an ideal gas and these ideas of Einstein would simply be inapplicable. As an alternative you might decrease the temperature by a factor of 10 to the 5th. Well that brings you down to the milli Kelvin regime, rather hard to access and also almost everything freezes out at such a cold temperature. And so, basically, the prospects for Bose-Einstein condensation for a very long time seemed to be very improbable. So, here's a very nice book by Schroedinger, a little essay, of probably only 100 pages long or so on on statistical thermodynamics. He covers a number of interesting topics in this book, and here's a quote which represents the consensus feeling in the scientific community, that the conditions of phase-based density are so demanding, that they're just never going to apply to ordinary matter. Nevertheless, there were signs of the existence of same like the Bose-Einstein condensation and most notably the phenomena of Helium super fluidity. That was discovered in 1938 and very quickly, it was suggested by Fritz London that the super fluidity of Helium was related in some way to Bose-Einstein condensation. However, liquid helium is a dense system and very strong interactions exist between the atoms. So its not really reasonable to suggest that it has the dynamics of an ideal gas, and there are a lot of problems associated with trying to treat liquid helium as if it were a Bose-Einstein condensate, but then about 70 years later, the situation changed which is the topic of the next part.