 I don't know what to switch it on, I think. So our next lecture, we have another brand new set of lectures beginning today. This is Professor Fabrice Gervier from LKB Laboratory. They call it Normal in Paris. And so we'll tell us everything we wanted to know and never dare to ask this time about ultra-cold atoms. OK. OK. Thank you. Thank you so much. So good morning to everyone. I'll give you a few informations about this field of ultra-cold atoms. And my talks will be mostly given on an experimental perspective. It's basically an experimental motivation and some illustration from recent experiments. So I will skip a bit the historical part I put in the slides coming to this presentation slide here, which is trying to give a kind of panorama of what people are doing with ultra-cold atoms today. And I've categorized into three main items, let's say, where people are investigating these atoms. The most, maybe most pursued one worldwide is the area of quantum degenerate cases, which I will speak at length in the following. But those two other items are also very important. So one of them is the area of precision measurements. So cold atoms are nowadays used in some of the most precise measurements, for example, for time, for gravitational forces, and so on. And so those two areas are maybe where cold atoms can be important for the field of quantum technology, which this conference is about, to perform very high precision measurements, at least in certain quantities, and to realize, using degenerate cases and a number of tools, that I will try to introduce to simulate many body systems of interest that are very hard to calculate numerically. And the third item, which I will mention very briefly, is what I call here ultra-cold chemistry, which is how to use these atoms, basically, as building blocks to build more complex systems, or molecules, essentially, and to study how the quantum level, molecules, and chemical bonds work out. So I will mostly focus on these two, actually, mostly on this one, in the following. And this slide here summarizes, basically, the important length scale to sort of locate in the whole landscape of many body systems that people can access in experiments, where ultra-cold atomic gases are special or specific. So one first important remark is that in the regime of temperature, or very low temperature, where we work, basically, you can speak of a quantum gas, as opposed to a classical gas, described by Maxwell Boltzmann statistics, when a typical length scale called the thermal-debal wavelength, which gets as one of the square root of temperature. I recall the impression here. So this length scale, when it becomes comparable, or even larger, then a typical intervalical distance, n to the power 1 third, where n is the spatial density. So when this happens, basically, within one coherent length, this thermal-debal wavelength, one atom will see some other atoms of the same species. And so quantum effect such indistinguishability will play an important role. And so this is summarized in this cartoon picture, where as t goes down, you start from a classical gas, where you can imagine each atom to be described as point-like particles being statistical, classical mechanics, sorry. And then as temperature goes down, the buoy wavelength increases. And at some point, the buoy wavelength becomes comparable to the distance, typical average distance between two particles. And quantum effects become manifest. And so in the case of bosons, as shown here, they will condense into the lowest particle ground state that is available to them. So this is purely quantum mechanical effect, but it has nothing to do with the fact that atoms can also interact when they meet each other. And so this is described by a parameter, which is called the scattering length, A, and which very roughly, you can think of as a 2A can be thought of as a diameter of heart sphere that are colliding against each other. So when the interparticle distance is comparable to this A here, you can imagine that this is a close packed system, where basically when one atom moves, it immediately sees a neighbor or several. And so basically that's a system that you expect to behave as a liquid, or eventually as a solid, depending on the parameters. But it's a system that people qualify as dense, so what people encounter in condensed matter physics typically. So with atomic gaseous, we're in a completely opposite limit, where the scattering length is very short compared to the interparticle distance in most situations. And so basically atoms fly around most of the time, and once in a while, they encounter another one, which we can interact. So mathematically, you can have this criterion to express this diluteness condition, which is the density times A to the power of 3 is much smaller than 1. And so we have this hierarchy of length scale. A is the smallest one. The interparticle distance is intermediate. And the debris-wave length is very large. So the fact that the debris-wave length is large tells us that the system behaves as a coherent system, a coherent, many-bodied system, and that you must describe its collective properties with quantum mechanics. And the fact that A is very short tells you that the system is still dilute, meaning that there are a number of theoretical approaches that you can use to describe it, which are not affordable in usual condensed matter systems, or liquid helium, or systems that are typically very dense. And I give you here a few numbers just to give a flavor. So typical scattering length is a few nanometers length. At the densities where you get both ancient condensate or degenerate fermi cases, the typical interparticle distance is on the order of 100 nanometers. And the debris-wave length is at least one micrometer, or even larger. So now this hierarchy of scales tells that also the system is dilute. Still interactions are important because of the extremely low temperatures. And so indeed, ultra-cooled atoms have been used to simulate or emulate a number of many-body systems. So it has some well-chosen characteristics to do this. First of all, there is a very large degree of tuneability in the experiment. So we can, for example, change the trapping potential almost as well. And there are also ways to change the strength of interactions, even dynamically. We can tune the density, et cetera. And maybe, most importantly, those systems are held in ultra-vacuum by laser beams or magnetic traps. And so basically, they are very, very isolated from the external world. And so you can forget about a lot of complications that typically arise in solid-state systems, coupling with phonons, impurities, and so on. Or we can eventually reintroduce them if you want to study it. But they don't come as a parasitic phenomenon when you want to focus on one particular observable. So I give you here three examples. The realization of B.C. was basically the start of the whole field of degenerate quantum gases in 1995. And then two other landmark achievements were, first, the realization of a multinsulator, the transition from a superfluid to a multinsulator in a lattice gas. So I will discuss this in the remainder, at length. And I show here a last example, which is maybe the most illustrative of the power of this approach, which is the condensation of fermionic pairs. But in a regime very far from the B.C.S. regime that is encountered, these superconductors. And where the interactions are essentially infinitely strong, the interactions between the fermions is essentially infinitely strong. And so this very strongly interacting thermionic pairs was observed to condense in several groups in the early 2000s. And it's still now this so-called unitary fermi-gas. It's still a very, very active field of research as a prototype of a very strongly interacting system. And there are also many other examples that I won't detail. So one of the goals would be to, through this example, the superfluid to multinsulator transition, to illustrate the use of ultra-cold atoms to study such many body phenomenon. OK. I did not mention, but of course, you should feel free to ask questions at any time or interrupt whenever you want to. OK, so you should recognize here the classical periodic table of elements, which I show to illustrate what atoms really are used experimentally. So the blue frameworks are showing which atoms are used in the degenerate gaseous experiment. And historically, the atoms from the first column, so alkali atoms, were used. But now an increasing number of species from the second column or from the lanthanide series are also gaining a lot of momentum. And so each of these columns has different advantages. So alkali are the easiest in a general role from an experimental point of view. Then you get atoms with two electrons, which have very, very narrow optical lines, which can be used for various purposes. And lanthanides here have very strong magnetic moments so that those behave as particle with dipole-dipole interactions or long-range interactions. And so all these different properties are used in experiments to simulate different quantum many body systems. OK. So this will be the plan that I would like to cover today. I will basically focus on the experimental point of view to introduce the essential techniques that we use to detect, trap, manipulate degenerate gaseous. I will very briefly give an overview of how we use to actually arrive to a degenerate gas and then focus on what we do to extract information from them, so through imaging. Or we trap them with optical dipole traps and give a few notions of what has been observed and studied with both ancient current states, in particular, their superfluid character. OK. So starting with basically the experimental path to obtaining, in an experiment, a degenerate quantum gas, boson of ferraments alike. So I plot in here a temperature scale. So here it's room temperature is around here. And here you get down to the nano Kelvin temperature. So extremely close to absolute zero. And this scale here is meant to illustrate that a typical experiment will proceed in two steps. OK. You will start from a room temperature vapor for this picture, for sodium atoms. And then you will capture those atoms from this vapor into a so-called major optical trap using relatively strong and close to resonant light. OK. So this constitutes the first step, which is called here laser cooling, which I will not give too much details about. Unless I only say that it basically involves what people call a magneto optical trap. And this allows to get to temperature of a few tens of micro Kelvin, typically, starting from a room temperature vapor in the beginning. OK. Which is already quite impressive. But then it's not sufficient to reach the condition for quantum degeneracy. And you have to supplement laser cooling with a second step called evaporation or evapivative cooling, which now proceeds in the absence of full resonant light as opposed to laser cooling. And it's a crucial step where you can get from this micro Kelvin or tens of micro Kelvin temperature down to the nano Kelvin range, where typically you would get a BC or degenerate fermi gas as illustrated by this picture, which I'm explaining in a moment. OK. So just to get a visual feeling, that's a picture of one of our experiments in our lab. So this is the optical table. And here you see a vacuum system. So everything happens on the ultra high vacuum, where basically the behavior of the atoms follow linearly what I've explained. So the atom starts here in an oven, which we heat to several hundred degrees. And then from that oven emerge a beam of atoms, which we can see here in that picture. This blue line here is materialism photons that are emitted by the atomic beam as it crosses the vacuum system. And then everything ends up in a metal chamber over there. And this green now cloud is a good cloud of atoms, which is also emitting photons, which we pick up with a camera. And this is a different wavelength and it's a different color. And in this chamber you get a cloud of laser cooled atoms at a few tens of micro Kelvin. So this is a picture of the same experiment at the later stage, just to show that the optical table is pretty busy and that this experiment quickly becomes relatively complex and complicated to handle. So now a bit in more details. Why do we have to do these sequences two steps? Why do we have to do laser cooling first and then evaporative cooling? So the goal, this is the same condition that I stated before. But now I'll rephrase it in terms of the so-called phase space density, d, the product of the density of the particles times the cube of the debris wavelength. So it tells you basically how many particles you get inside of a coherence volume for a single atom. If it's less than 1, then basically you can neglect quantum effects and you can treat the system as a Maxwell-Bosman gas. And now if it's bigger than 1, then you cannot neglect quantum effects anymore and you reach the quantum gas regime. So the reason why the first step here, laser cooling, is insufficient is that you have to use in order to perform this step, laser, which is near resonance. And near resonance means you get photons that are absorbed by the atoms and are remitted mostly through spontaneous emission. Now, spontaneous emission is essential because it provides a necessary dissipative mechanism that allows you to cool the emotional degrees of freedom of your atoms and to reach the temperature of a few tens of microkelvins. But the issue is that it's also intrinsically random. You cannot control the direction in which spontaneous photons will be emitted. And this randomness actually puts a limit to the temperature that you can achieve and prevents to cool the atoms to below certain limiting temperatures, which is one limit as well. But here, no, it will work with a single atom. It actually works with a single ion, for example, which has the same type of mechanism. And it's really the fact that the spontaneous emission has a random direction. And that leads to random work in momentum space. You get random momentum kicks in any directions, which on average makes zero, but will increase the dispersion of the atomic momentum. And this is balanced by the cooling force. And so that comes to a, like in Einstein's relation, it comes to an equilibrium at a certain limit temperature. And those temperatures are typically on the order of tens of microkelvins with densities in that range, meaning that the phase space density after this cooling stage is very small, typically 10 to the minus 5. So we're still very, very far from a quantum generator regime. And so to overcome these limitations due to laser cooling, we make use of this second step, which is evaporative cooling, in a conservative trap. So conservative means we get rid of all spontaneous emission, which has this adverse effect of limiting the temperature. So we still use light, but we use a frequency which is very, very far from any atomic resonance. So what people call conservative traps, typically, can be also used in magnetic traps. And then we proceed with the evaporative cooling technique, which I will say a word of in the following talk. Do we have questions at this stage? No, I will not, actually. But I will describe optical traps in a couple of minutes. Magnetic traps, well, optical traps are more used now. Magnetic traps were used historically, but they tend to be replaced more and more with optical traps, so that's where we'll focus on. And, OK, actually, to focus on that, and also to discuss another important topic, which is always extract information from such cases, I need to make a few reminders about the interaction of an atom with light, resonant or not resonant. And so typically, we tend to ignore, at least at the first level of approximation, the internal structure of the atom, and just model it as a two-level system with a ground state. Oops, I'm sorry, thank you too much. So there is a ground state where the atoms normally are, and then some excited state connected with light. And so atom can absorb a photon from some laser that you shine onto the sample and get promoted to this excited state. In general, the laser doesn't have to be strictly resonant, so the frequency omega L of the laser is offset by a quantity delta L, which calls the detuning from the exact resonance. And I call also gamma the width, or one of our gamma is a life time, the radiative lifetime, of this excited state up there. So now taking the limit of low light intensity, so you look at weak response, but you can see this problem as the driving of a system which possibly has an electric dipole moment by some electric field corresponding to the electromagnetic field of the laser. And so for low enough intensity, then the electric dipole moment of the atoms, which can be excited in this transition, will basically respond linearly to the electric field. And so you can describe this in terms of the average dipole moment by a susceptibility, kai, such as the average dipole moment is proportional to the incident electric field times the susceptibility. That's standard linear response. And as usual, the susceptibility is a complex quantity. And its real part is related to the component of the dipole, of the mean dipole, that oscillates in phase with the exciting field, whereas the imaginary part is related to the component that oscillates out of phase with the exciting field. And so I've plotted here the real part and imaginary part. And so we recognize the dispersive shape of the real part, which is very common in this type of problems. So it's 0 exactly on resonance, and then changes side across it, whereas the imaginary part now is maximal around the resonance. And falls off as a Lorentian for larger detuning, or when the frequency omega L gets away from the exact resonance. All right? So now I'm going to use these notions for two things. First, to explain how we image the atomic gases in order to extract information. And so this is a sketch of a typical setup. So this would be the cloud of atoms, which is held in a dry vacuum. And to extract information from it, we shine some laser light onto it. And then through the imaging system, we cast the image of this cloud onto a CCD camera. Well, because you can move either this camera or that lens. And then when you move it, at some point, your image gets sharper and sharper. And then when you find the sharpest point, you know that exactly when you focus your camera. So it's the same trick. And transverse 3 doesn't matter. And transverse 3, typically, the width here is much, much bigger than the size of the cloud. So what you only need to know is whether this is making an accurate image on this camera. So you need to make sure that this plane, basically, and that plane, where the atoms sit, are conjugated in the optical sense. That's because the radius is so large. Yes. OK. And so to describe what's happening to this light beam that's going through, you can essentially describe the atomic gas as dielectric medium. So we saw that when you send light, the atomic dipoles get excited. So what that means is that if the density is n-at, then in this cloud here, there will be an electric polarization, P, which is simply given by the mean dipole times the density of atoms. That gives you the density of polarization. And now this polarization will affect propagation of light through the usual Maxwell equation, or L-Molt, actually equation, which I remind here, appropriate from a dielectric media. So it means that this equation here, you can turn into an equation relating the electric polarization, P, to the electric field, E, with the proportionality coefficient, which is epsilon 0, the dielectric susceptibility of vacuum, times the susceptibility we have just introduced for the atomic electric dipole. So now to treat this type of equation, what people normally do is to assume that the E has a fast varying term, which is a traveling wave component, exponential iklz. And then a slowly varying component in terms of its modulus, E, and of the phase phi L. So from the Maxwell equation, you can turn this into an equation for the propagation of the amplitude, E, and for the phase as a function of z. And you will notice that this is a second order equation where those are first order. And the reason is that under the approximation that E and phi L vary slowly on the scale of a wavelength, then you can simplify this equation by neglecting their Laplacian, basically. So this is what's normally called the slowly varying envelope approximation in optics. OK? So now I will be mostly interested in the first of this equation, or for the two can be used to perform imaging. And when recast in terms of the intensity of the laser, so basically proportional to the square of the electric field, this gives you what's known as the billambert's law, which is very well known chemistry, for example, which tells you that the derivative of the intensity of our disease, so the variation of light intensity as it propagates through the sample, decays with the intensity with some proportionality constant kappa, which we can relate to the imaginary part of the susceptibility, psi second, that we saw before. OK? And so what that means is that when you look at the intensity, which is transmitted, what you will see is something which is proportional to the incoming intensity times a transmission factor that depends exponentially on the integral of the atomic density along the propagation direction. OK? What people called the column density, typically, with the proportionality factor that you can interpret as an absorption cross-section. All right, but its exact value is not very important. What matters is that if I now reverse this equation, I get a formula for this column density, which is the log of the incident intensity divided by the transmitted intensity once the atoms are present. OK? So that's exactly what the formula we are using in an experiment to extract the density that is actually realized. OK? We measure the incoming intensity. We measure the transmitted intensity, and then we apply this formula to extract a map of the atomic density through the sample. OK. Well, actually what we do, we make a first an image with the atoms present that gives us ET. And then we push the atom away, just measure the same incoming, but without transmission this time. So we actually measure both in succession. OK? And so that gives you an image like this. And that image will basically give you a map of the density distribution of the atoms prior to sending the light that is used for obtaining this image. OK? Ah, yes, sorry. So that's my typical trap, this thing. OK? And so one thing that is usually done, which is typically important, is that we are not doing this type of measurements while the atoms are trapped. So what we do is what we usually do is to perform what's called time of flight imaging, which basically means that we release the atoms from the trap where they are held. And then we let them expand in free flight for some period, typically milliseconds or tens of milliseconds. And at this stage, after it has expanded, we take an image of the atomic cloud. OK? So why do we need to do this? Why do we need to expand before doing this image? The reason is that often this low-intensity picture that I was describing before does not apply for a cloud that is trapped. The cloud is too dense. And there is a very high probability that if one atom emits a photon, it will be picked up by another one, leading to a lot of complications, multiple scattering, et cetera. So all these effects are not included into the Bir-Lambert's law I've been showing before. And so therefore, the method cannot be applied if the density, the spatial density, is too high. OK? So the primary reason for doing this is to reduce the particle density to put the system in a regime where we can apply the method that I've outlined before. OK? So now, an important part is that why do we do this during this time of flight sequence where we switch off the trap suddenly at t and then let the cloud expand for some time? What we get at the end is an image, not of the spatial distribution, but of the momentum distribution of the particles. OK? So it's an exact result if you can neglect interactions. And it's approximate only when you have interactions, but it's still very useful. OK? So I've put this in a set of problems which are available on the website, I believe, under the lecture four. And so there are solutions as well. So it's relatively easy to follow. But this is one of the first where I advise you to try to at least look at the problem to understand how this comes out. But I will use this result a lot, basically that for a time of flight experiment, the atomic density that you recall on your camera is basically an image up to some scale factor of the initial momentum distribution of the gas. OK? A side effect sort of of this formula is that now when you look at the radius of the cloud, it will correspond to the radius of the momentum distribution of the initial gas. OK? So what that means is that if you are in a high temperature regime, basically the radius of the, for example, for Maxwell-Boltzmann gas, the radius of the momentum distribution is simply related to the temperature of the gas through equipartition. OK? So that's actually the means to extract the temperature of the cloud and to assess it. OK? So it's slightly different when you get into degenerate gases, both are Fermi, but roughly the idea is the same. You look at the time of flight that gives you a momentum picture and you look at the momentum radius of your system. OK? So now I've explained how we extract information from the gas. And now I'm going to explain the same mechanism, basically excitation of a dipole, can be used to actually trap the atoms. OK? So that's the mechanism behind the so-called optical dipole traps, or sometimes a bit longer far-off resonance optical dipole traps. And so now basically the idea is still the same. So you send a laser, very far-off resonance. So it will still excite an atomic dipole, proportional to the electric field, E. OK? And now this dipole will have some potential energy inside the exciting field. OK? So it's an induced dipole, but so there is this extra factor of one-half here. But otherwise, the potential energy will be simply minus one-half D scalar E. And the average here is meant to make a time average over one optical period to just look at the average energy, potential energy, felt by the dipole. OK? And so this is a standard formula from electromagnetism. But when you use the expression for the susceptibility that is obtained before, you get a formula where the energy depends on the intensity of the square of the electric field, which is proportional to the intensity, on the dipole matrix element squared, and inversely proportional to the detuning in the limit where the detuning is very large. OK? So now if the intensity is uniform, this gives you a plane wave, this will give you a uniform shift. But in reality, we never work with plane wave. So the intensity profile of the lasers that we use always depends on space, which means that the potential energy will also depend on space. And that can be used to trap the atoms in some potential minimum. So whether a potential minimum occurs depends on this detuning here. So I remind that the detuning is a difference between the laser and the resonance frequency. So when it's negative, so-called red detuned case, then basically this potential is attractive. And you attract atom to the maxima of the intensity. On the other hand, when it's positive, the so-called blue detuned case, then you expel atoms from the maxima of intensity. OK? So both can be used, and both have been used in different cases. I will focus on the red detuned case in the remainder so that atoms will be attracted to the intensity maxima. And now just to justify the fact that we call this a conservative trap, that dissipation is not important. You can calculate, so the dipole is excited in a time-dependent fashion. And you can calculate also using the standard electromagnetism formula, how much energy it radiates averaged over an optical period. So this is called W dot here. And this, the rate of energy radiation, in simply the spontaneous emission rates, the rate at which atoms can re-emit photon by after being excited of resonancy, times the energy of a photon, H bar omega L. And that can also be interpreted as the energy absorbed by the dipole from the excitation field and then re-radiated inelastically, if you want. And so usually, delta is big enough. I should mention that this spontaneous emission rate using the same formula for the susceptibility. North K, that's 1 over delta squared, as opposed to 1 over delta for the potential. So if delta is big enough, you can make it such that the spontaneous emission rate and the heating rate associated with it is negligible, while the potential energy is substantial. And so this is why the name of a conservative trap, a trap without dissipation, is well justified. So this slide now shows an example of what is done experimentally. So here you have a Gaussian laser beam, which is what most laser beams look like in practice. So it comes to focus here on z equal to 0, and then diverges due to diffraction away from focus. So that leads to a potential which has complicated form like this, which is depending on the Gaussian fashion or the transverse coordinate, and with the Lorentzian dependence on the axial coordinate z. So here, the thing to notice is that typically the size w of the beam in the transverse plane is very large compared to one wavelength. And that means that the Rayleigh range, giving the range of variation along the axis, is very large as well compared to w. So when you translate this near the minimum, near the focus here, into trap frequencies, expanding this potential in terms of the coordinate, you will get an almost harmonic potential. With trap frequency, which will be strong in the transverse, the coordinate transverse to the beam propagation axis, but very, very weak along the beam propagation axis. Summarized by this condition that omega x here is very large compared to omega z. And so typical numbers will give you that the frequency along x can easily be kilohertz or several kilohertz. Whereas on axis, it's only a few hertz. So this trapping potential can be very anisotropic. And the last thing I want to mention is so this potential depth here, u naught, will basically give you the maximal energy over which, below which you can trap atoms. And if an atom has a higher energy, it will just escape from the trap and be lost for all experiments. So this is typically on the order of 1 milli Kelvin in temperature scale, and which explains why we still need to have this first step in the two step sequence I was mentioning before. In order to trap efficiently atoms in this kind of dipole trap, you already require cold enough sample. You cannot trap any meaningful sample just starting from a room temperature vapor. So basically the first step is always needed in order to efficiently load this kind of trap. And the same applies also to magnetic trap, which I will not discuss too much. OK, so let's give you the same experiment as before. This is a cartoon sketch of it. And that shows the thing is working in practice. So this is a view from the side. And what you see here is the same main optical trap as before, which is falling after it has been released because of gravity. And then here on top, oops, I'm sorry, you see this line of atoms here, which corresponds to a fraction of the atoms that were in this trap originally, that have been catch in an optical dipole trap propagating horizontally like this. And then because, as I've mentioned, the trapping along this axis is very weak, you see that this has a very anisotropic shape, looks like a needle. And so often it's very useful to instead of using only one such trap to cross them like this, so that you get atoms which are trapped in both arms. And as you cool the sample, shown by this plot, progressively atoms are basically emptied from the arms as the sample gets colder and colder and accumulate in the crossing region where they see a dense trap, strong trapping in all three directions. And so that's typically how you arrived at the Bose-Einstein condensation. So I'm showing here absorption images, also taken from the same experiment, where temperature decreases from the left to right. And this shows a typical signature for bosons of the appearance of a Bose-Einstein condensate in the system. So you start from a high temperature, relatively high temperature. Well, the cloud has a Gaussian shape and a relatively broad size. This is again after time of flight. And then as you cool down the sample, you see a dense peak here appearing in the center of the distribution and progressively becoming more populated while the background is progressively exhausted. And so at the end, this corresponds to a nearly pure Bose-Einstein condensate. So all atoms condensing into or populating is very narrow peak, which means that almost all atoms have near zero momentum. And then the background of thermal atoms being almost vanishing in the limit of very low temperatures. So that's the hallmark of Bose-Einstein condensation. As is described in statistical mechanics textbooks, in the box, you expect that atoms will accumulate into the k equal 0 momentum states as temperature is decreased below the temperature for condensation. So that's qualitatively similar. You can take into account the harmonic trap. You also find that it's qualitatively similar. But for quantitative agreement, it is impossible to neglect the fact that atoms interact as I've sketched before. And so as soon as the condensate is formed, interaction between these atoms becomes important. And it's necessary to include them in the description to understand the properties of quantum gases. But I think I will probably leave that for the next talk. And stop here for today thanking you for your attention.