Hello everyone, welcome back to exploring quantum physics, I'm Charles Clark. In this part, we're going to look at the experimental development of Bose-Einstein condensation. As you saw in the previous lecture, the theory of BEC was developed in 1924, and it more less likely dormant for about 70 years and then explosive phase of growth which is still continuing. 70 years after Einstein first proposed Bose-Einstein condensation on ideal gas, it was finally observed quite decisively in experiments. And that's led to a real revolution in atomic and condensed matter physics. So the first production of Bose-Einstein condensate in a laboratory was done in 1995 at JILA, the joint institute of the University of Colorado and the National Institute of Science and Technology. And the key enabling technology for this work, was the development of cooling and trapping of atoms that first came on the scene in the early 1980s. This has made it possible to get to much lower temperatures and has been obtained in any other material system now down into the picokelvin machine in some cases. And the decisive evidence for Bose-Einstein condensation that was presented then is a picture of the clouds of a rubidium gas after its release from atomic trap at three different temperatures. Now, we'll discuss in a moment how these temperatures are determined but what you see are three images taken at different temperatures. Going temperatures decreasing from one frame to the next and you see a distinctive change in the density profile of this release cloud. It's more or less a spherical cloud at the high temperature and then a sharp and asymmetric peak develops as the temperature is lowered. A very good, accessible account of this work is given in the additional materials, written by Eric Cornell, who's, one of the co-authors of the original experiment. Let me say something about these images that you're looking at, just to try to make them clearer. It may take a while to understand their implications. These are just simple shadow grams you might say of the atomic density in the expanded clouds. So there's a small cloud of atoms here. Light is incident from the left and then it's imaged by a microscope which can scan in the horizontal direction there and so acquire an image of the density. It's just determined by the absorption of the light by the gas. So what you're seeing is the integrated density of the gas in the line of the light. And the coloring here is just artificial which is put in to enhance the visual contrast of the image, it's just proportional. It's just a look up table that reflect the density of the atoms. So, this asymmetry is a very interesting aspect of the formation of the quantum gas. And if you return to thinking about classical mechanics, classical statistical mechanics, there is is this universal accounting of states for momentum that's independent of the trapping potential. And that has the following implication that if you have a trap of arbitrary shape with gas held in it and then you turn off the trap, the expanding cloud eventually becomes spherical completely independent of the shape of the trap. Now for a quantum gas on the other hand, position and momentum don't commute. And if you have an asymmetric trap, when you turn it off, the cloud changes the aspect ratio of asymmetry. In other words, the most tightly bound part of the gas, because of the uncertainty relation, has the higher momentum so it expands more rapidly. There is an analogy of this in optics, and that, I'll just state it briefly as follows. If you have an aperture of arbitrary shape, here's a strangely shaped aperture, and it's being illuminated by light from a thermal source, an incoherent light source. And then if you view that light on a distant screen, what you'll see is a circularly symmetric image which is independent of the shape of the original aperture. And you might think about this in the same way as the rays coming from the various part of the aperture. Just travel at a common speed, and so they scatter onto the screen at far distances without retaining any memory of the particular part of the aperture from which they came. For radiation from a coherent source, on the other hand, like a laser where all the photons have the same frequency on the same directionality, the common mode structure, then you get a result similar to what we just mentioned for the gases. So if you have coherent radiation incident on a circular aperture, well then the far field image is circular. But, if you have the coherent light going through an asymmetric aperture then the far field image reverses the aspect ratio just as in the case of a gas, that's for very similar reasons. So, the inversion of the aspect ratio as well as a very distinctive density profile is a very strong sign that you're producing coherent matter waves, that is all atoms are in the same quantum state and with a common phase. This is illustrated even more dramatically when one looks at atoms in optical lattices. And we'll say something about, more about optical lattices in the later part of this lecture. But you can think of it, for the moment, it's a just a set of standing light waves and they form a periodic structure, not that well represented by my scribbling here which forms an array of sites. Just like the sites of a crystal, atoms in a crystal, and that can be use as a trap to form sort of a synthetic lattice of atoms that are held in a regular array by light forces. So when you have a non-condensed gas in such a trap and you turn the trap off. Then what you see is spherical expansion with a little bit of memory retention of the underlying lattice structure. But here is an actual observation such as system consisting about 150,000 rubidium atoms in a cubic lattice. So you're seeing a line of sight image showing a spherical expansion of the cloud. Well, if you go into Bose-Einstein condensate phase where all the atoms in the lattice have a common phase, then the expansion image is radically different. You can see that the gas expands really indiscrete directions that carry the underlying symmetry of the lattice. This is really an analog of brag defraction of x-rays or flight. And this is not just a fluke, the last of these images is the one that's most amazing to me. And it actually shows that when you turn off the trap the cloud falls under the influence of gravity, as you would expect, and that this cubic expansion, this highly ordered expansion phase is not just a fluke, it continues forever. So that's a powerful demonstration, of the production of coherence in the matter waves. How do we understand in detail the dynamics of the Bose-Einstein condensate? Well it turns out there's a very useful framework, originally developed for the study of liquid helium. Where it wasn't really that successful because liquid helium is a strongly interacting system. But it has proven to be of incredibly valuable use in understanding the dilute Bose-Einstein condensates that are produced in the lab today. And let me emphasize, these gases that show this remarkable quantum behavior are very dilute. They have a density about 100,000th that of air, yet they showed that remarkable collective phenomenon. So this equation describes a framework in which all the atoms in the gas are said to be characterized to be of the same wave function psi or orbital psi you may call it. The interaction between the atoms is the two body interaction between a given pair of atoms is represented by a direct delta pseudo potential. The type was discussed in a previous lecture characterized by the scattering length which is a proxy for the full two body interaction potential in a limited energy range near zero energy. So called scattered length, on that is a parameter which is determined by independent experimental measurements of the atomic collision physics. And I emphasize that, in modern work, there are no adjustable parameters in the description of a Bose-Einstein condensate. Everything is determined, some cases by great accuracy, by independent types of measurements. So the time evolution of the wave function is given by a variation on the Schrodinger equation. The first part of which you should recognize as the usual single particle Schroedinger equation that we've discussed at length in previous parts of this course. And then the new element is this nonlinear term which reflects the influence of the interaction between atoms so it's proportional to the number of atoms to the scattering length and then it has a cubic nonlinearity and the wave function and this gives rise to typo dynamic that just not seen at all in the ordinary Schrodinger equation. Here's an example of the generation of soliton in a Bose-Einstein condensate by the use of optical phase imprinting. And the way the experiment is done is that there's a cloud of gas in the Bose-Einstein condensate, then it's illuminated by near resonant light, and there's a mask that shadows half the condensate from the light and the part that's illuminated acquires a phase shift which is just due to the ac-Stark effect of the light, the light shift potential. And then one looks at the evolution of the condensate after that. Which is shown here, an experiment done the year 2000. This shows the cloud of the gas at a sequence of time intervals after this imprinting of the optical phase and what you see is the development of a density notch which propagates along the cloud. This is actually a matter-wave soliton. The top row of the images are experimental measurements and the bottom are calculations using the Gross–Pitaevskii equation with the experimental parameters. Here's some more recent work that just shows how we can generate arbitrary angular momentum states in the ring-shaped traps. Prefers experimental work done by my colleagues at the National Institute Science and Technology. And so it uses an optical trap which is schematized in this frame, it consist of a supporting light sheet and then a ring that's created from symmetric being with a core in it. And this creates a confining potential that's of a toroidal shape indicated here. And then another beam is introduced that is sort of like a paddle. It's a light sheet that cuts through the Taurus just like you might cut through a donut with a knife and then that paddle can be rotated around to stir the condensate up. And so you see images here in this lower frame of how the cloud of atoms is moved by the action of this paddle. So the general characteristics of these experiments is about half a million sodium atoms in this ring trap, toroidal trap the radius of this trap is about 20 microns. So one starts with a nice quiet ground state of the condensate stirs it up and then releases and measures, the density. And what is seen there, here's the experimental data. These again are these absorption images. It's a sort of the most common way of viewing Bose-Einstein condensates. These are labeled after the fact by the values of the angular momentum. We have a donut here and there is a conserve angular momentum LZ along the axis perpendicular to the plane of the donut. And depending on the stirring conditions one can generate the successive values of the quantized angular momentum. There will be a problem in the final exam that deals with this and just to give a hint ahead of time, you could think about how these angular momentum could be generated by thinking along the lines of the bohr model. In other words, we're going to stir the atoms with a paddle and promote them to states of higher angular momentum. So you can think about, well, if the angular momentum is quantized as proposed by Bohr. That is the integer multiple of Planck's constant, what does that imply about the rotation frequency of the paddle that you need to tune to in order to get that quantized result? And here is a you might say the smoking gun for quantization. These two frames look at the distribution of whole sizes that are seen in various experiments. The upper frame and the lower, they represent two modes of release of the condensate, one of which turns out to give a more regular set of images. But what can see is that the whole sizes come in these quantized groups which are attributed to the generation of different values in the angle momentum. And the use of the Gross–Pitaevskii equation with the parameters appropriate to the experiment shows that there's a relatively good agreement with some deviation for the various angle momentum states that observe. So this frames show images of the experimental data which is indicated by the red lines with the results of theoretical calculations for different values of the angular momentum. And they show that the whole size Increases with angular momentum. This is a reflection of the sort of repulsive centrifugal potential that we found in the discussion of angular momentum in an earlier lecture. So finally, I'll mention some more recent work with this stirring procedure in which the phenomena of hysteresis is found. So this experiment reported in nature early 2014, involves stirring and condensate up and then stirring it back to return to the original state. And it has found that there is an asymmetry in the stirring and de-stirring procedure. Much of is seen In classical magnetism, the magnetic induced is a function of applied field. And this is a theme that's pursued in this work is often referred to as atomtronics which has a goal of using coherent matter waves to implement functionality such as one has in electronics. That is to make circuits which use atoms rather than electrons as a carrier of information, and of mass, and of spin, and to explore the possibilities that are available there that would perhaps someday complement conventional electronics for new functional purposes. So I'll just conclude by mention of some accessible reviews that will show you examples of things that can be done. Open source of these articles is all found on the archives. And these are also good references to search against them on Google Scholar to find examples of more recent work. I think you'll see it's a very vibrant field.