 Okay, let's start. Welcome to all of you for this colloquium. It's wonderful to see the full room. And that shows the level of the talk we are going to listen. So we know that, well, the talk is given by Gian Dallibar, who can agree to give us colloquium being part of this activity that Marcello Dalmonte and Antonio Scardicchio have been organizing with our colleagues from Cambridge and Oxford, Paul Luzvanli and Claudio Castellanova and Roderich Mosner. And the title of the talk is Topology in Atomic Flatland, that will be given by Gian Dallibar. Let me just say a few words about Gian. I have been reading, I just met, I just met him and I have been reading about his CV. He's just very impressive, so we'll just say a few highlights. So he's a, he's an experimental research group in CNRS, ENS, in the Columns de la Superiore in Coleste, France, in Paris. And some of his notable work includes the first observation of the coastal studies transition in 2D bose gases and studies of vortex dynamics in rotating condensates. But earlier in his career, Gian was responsible for key theoretical ideas in laser cooling and trapping, including polarization gradient cooling and magneto-optical trap, MOT. And those provides in his present work the link between theoretical and experimental quantum optics that I find very, very impressive. This day is one person is mastering both experimental and theoretical part of a field in physics. And Gian is a rare example of that, which is very impressive. And well, he has many awards. So he recently was given an ERC grant and he has also received the Senior Bosse Einstein Condensation Award in 2017, the Max Born Award of the Optical Society of America and a member of the French Academy of Sciences. He, Gian is a key representative of the long-standing atomic physics tradition in Paris, whose initiator is Alfred Kassler. And so for some of you who have been in ICTP, you know that we have a Kassler room in the Atlético guest house because he was one of the key persons in the early days of ICTP as chairman of our scientific council. So I think Gian is following up the tradition for us, which is very good. And before starting the talk, I wanted to mention that a special component of our community in ICTP are the diploma students that we usually receive. 40 diploma students, 10 on each different groups of our research, contents matter, physics, high energy, mathematics and climate change. And the students have just arrived, so I cannot recognize them because I haven't seen them yet. But I think in the audience there should be 36 students from more than 20, I think 24 different countries from developing countries. So you don't mind giving a welcome to all the students who are just arrived. And having said that, the tradition of ICTP is that the diploma students at the end of the colloquium, each colloquium is essentially targeted at them. And at the end of the colloquium, we have the standard questions period from everybody in the audience. And at the end, we ask everybody to leave except for the students who can come and talk to direct questions to the speaker without being intimidated by the senior colleagues. And the students have not been told that they were going to do that so because they're just arriving. But Jan had kindly agreed to do it this time. So at the end of the colloquium, we will proceed in that way. So without further more comments, so please join me to welcome Jan. Well, thank you very much for this very kind introduction. And thank you to all of you for being here. It's really impressive to speak in front of such a broad audience and also very diverse audience. So I'm really very, very happy to be here. So I gave this title Topology in Atomic Flatland and I think I have to start by explaining what I mean by that. There are two things in this title that I think need an explanation. The first one is Flatland and the second one is Topology because it's a physics talk. So why do I am putting these words into my title? So Flatland for me means a book that I loved when I was a kid or maybe older than a kid or a teenager which is this book written at the end of the 19th century by an Englishman, a clergyman actually, Edwin Abbott, which is a wonderful book. If you have not read it, you should do it. I mean, you can download it for free on the internet. It's a book about both sociology and mathematics. In his books, what Abbott wanted to do was to imagine how a society could build if it was not living in a three-dimensional world but only in 2D or even in 1D in part of the book. But the book is essentially about 2D, that is Flatland. And Abbott actually was at the same time doing sociology because what he wanted to do was to make a criticism of the English society at the end of the 19th century. It was a society with a very big hierarchy, as you know, in England at that time. And so Abbott imagined a society where the lower members of the society would be just triangles, only three sides, geometrical figures with three sides. Then, slightly above it, you have four figures with four sides, so it could be a rectangle or it could be squares. Squares would be a bit better than rectangles. Then you had pentagons and hexagon and so on. And on top of the society, you had the best figure that you could imagine, the Grand Priest, which was a circle with an infinite number of sides. University professors were about four sides, just to tell you, so not very high in the hierarchy. Okay, so I'm not going to do sociology here, but still my theme today will be to imagine what can physics be if we restrict the motion of our particles, our atoms in this case, to two degrees of freedom, not three as usual, but only two. The second word I need to explain is topology. Topology, as you know, is a branch of mathematics, so what is topology doing? Why is it playing a role here? And actually, my duty on this respect is made very easy, but thanks to the Nobel Prize, it was awarded one year and a half ago to Thaulais, Haldane, and Costa Rites, because this Nobel Prize has directly shown that topology is now at the heart of modern physics. The participants to the summer school, of course, know this very well, but I say it for the others, and in particular, the students who just arrived, topology, which used to be very disconnected from physics 30 years ago, now is very deeply rooted in modern physics. And the Nobel Prize, the 2016 Nobel Prize, was awarded for the following theme, for theoretical discoveries of topological phase transitions and topological phases of matter. And what I will discuss today is essentially the topological phase transitions. So why and how is topology, this abstract branch of mathematics, related to physics? So topology, if you look at a dictionary, a Manciclopedia, and you look at the definition, topology is essentially a classification. It's a way of classifying objects. And you will say that two objects which can be deformed in one another by a continuous deformation, by stretching, by bending, they will belong to the same topology class. And a very simple example of that is just taking objects in 3D for the moment and looking at the number of handles that they have. So they may have no handle like a sphere or an ellipsoid, and then you put them in the same box which is the box with no handle. They may have one handle which can be the case of a torus or a coffee cup, and you put them in the box C equal one. They may have two handles like this one, double torus or strange mug, and you put them in C equal two and so on. So the key point here is that if you want to take an object from one class and put it in the other, this can occur only through some singularity, and as long as you avoid singularities, then your object, even if you deform it, will stay in the same box. So this is so-called topological protection. An object which is in a box is protected as long as you don't go through some singularities. And this is a very important concept in physics now. And for the 2D fluid that we will consider, our gas moving in flat land, we will see that the topologically protected object, what will be relevant, are quantized vortices. And I will explain in a moment what the vortex is. Okay, so this is the outline of my talk. In the first part, I would like to explain you what happens when you go from the standard of three-dimensional space to two-dimension. And I will discuss this on the example of Bose gas, gas following Bose and Schoen statistics. I will first explain you that when you go from 3D to 2D, you lose something. You lose the possibility of getting long-range coherence. This is also called Mermin-Wagner theorem. And in particular, it means that you will not get any Bose and Schoen condensation when you go from 3D to 2D. But you still have room for some unconventional phase condition. This is what Kosalitz and Tauris understood, and this is one of the reasons for which they got the Nobel Prize last year. And this unconventional phase condition, this phase condition that remains valid in 2D, even though you have reduced the number of dimension, is in this case, a superfluid condition. The gas can become superfluid. And then, once we have explained that, I will show you three examples of this topology in flat land. The first one is to test, indeed, that the gas is superfluid. The second one is discussing how sound can propagate in two dimension. And in particular, I will focus on the so-called second sound. I will explain why second, why is it called second sound when I come to that. And the last topic will be about merging independent superfluids, merging the difference 2D superfluids, and the creation of quantized currents, as you will see, currents which can flow in an annulus clockwise or anticlockwise. So this is the outline of my talk, and I will start with this idea of going from 3D to 2D. So let's first start with 3D and with some very basic slide, for those of you who are not so familiar about Bosan-Chine condensation. So Bosan-Chine condensation, which was actually proposed by Einstein, both had worked on the problem of photons, and Einstein generalized the idea of both two material particles, like atoms, and he understood the phenomenon of condensation. So this phenomenon of Bosan-Chine condensation occurs when you take a gas of particles, of material particles of boson, you put them in a box, and this particle can be non-interacting, it's an ideal gas, or weakly interacting. Einstein considered only an ideal gas, but interaction will play a role in my talk. So what you get when you put the particle in the box is actually various regimes depending on the temperature. If the temperature is very high, then essentially the particle behaves as billiard ball that is the quantum nature, which is characterized by the double wavelength lambda, is very small compared to the interparticle distance, little d, and you can neglect essentially the quantum nature of the atoms. So they behave like classical particles. Then when you cool the gas, the double wavelength, which is varying like one of the square root of t increases, and there is a moment where it becomes on the order of little d, the distance between particles, then the wave packet overlap, and this is where the condensation that was predicted by Einstein occurs. You have a phase transition where you have a macroscopic accumulation of particles in the ground state of the box. And then if you cool even further, all the particles accumulate into this ground state, and the signature of the condensation, which again is valid in three dimension, in the real space, is the following. If you look at the currents of your matter wave at one point here, and the phase of your matter wave at another place here, so you get the currents between the two points, which is characterized by this function I call g1. I destroy a particle in zero, and I create a particle in r, and I take the average value over all thermal fluctuations. This quantity g1 of r will be non-zero, will tend to a non-zero value when r tends to infinity. My bucket of superfluid here can be as big as Pacific Ocean. Still, if I know the phase on the west coast of the US, say, I will know the phase on the coast of Japan. That is, the two phases are locked together because there is this long range coherence which appears during when this phase transition occurs. This is what happens in three dimension. And a signature of that is the superfluity of liquid helium when you look at the textbook in statistical physics, very often the macroscopic coherence and the superfluity are linked. We'll see that, actually, when you go to 2D, this link is much more subtle. When you go to 2D, you lose this possibility of having long range coherence. This was understood first by Peierls in 1935 and it was later generalized by Mermin, Wagner, and Hohlberg in 1966-67. I will explain this in much more detail in my lectures tomorrow and the day after tomorrow, but here I just state the result. The result obtained by Mermin, Wagner is that if you take short range interaction, and this is the case of our atoms, then you cannot have, if you are in 1D or 2D, you cannot have any breaking of a continuous symmetry. That is, you cannot have a conventional phase transition. And in particular, you cannot have Bose and Schein condensation because the Bose and Schein condensation I described before is a breaking of a continuous phase symmetry. The phase of your wave function is a continuous variable varying between zero and two pi and saying that you have a both condensation means that your sample picks up one possible phase on the segment zero to two pi. So you cannot have a Bose and Schein condensation in low dimension. This was proven by these people, which means if I come back to my function psi dagger of r psi of zero, my function g1, it means that at any non-zero temperature, this function will turn to zero when r tends to infinity. So I cannot have in 2D a well-defined phase relation between two points, which are very far from each other. However, again, this is what was discovered by Kosolius and Tauris. There is some room for unconventional phase transition, which are not in the usual classification of phase transition, which are driven by topological defects, as we will see. And this is the so-called KT for Kosolius and Tauris, or BKT, because Berezinski, a Russian physicist had worked on that just before. This is this BKT transition, which was signaled out by the Nobel Prize. So let's see what is this topological phase transition driven by quantized vertices. And for this, first I have to explain what is a vortex in this context. So I take my 2D Bose gas, and I describe this gas by a classical field psi of r. So psi of r is a matter wave field, which is a fluctuating quantity, both in space and in time. And so I look at the given time. And I have psi of r, which I can write like the square root of rho of r, where rho will be the density, because it's the module square of the field, times some phase factor into the i theta of r. So density rho of r, phase theta of r. And what you have all learned in your first year course of quantum mechanics is that when you have some phase for a matter wave field, you have some velocity associated to this field. Or maybe you learn this as a probability current, but it's the same. The velocity field is proportional to gradient of the phase. So if I have some phase which is fluctuating in position, I have some fluctuating velocity field v of r. And our text in this context is simply a place where the density rho of r is 0. So this is this place here. And around this point, the phase winds by plus 2 pi or minus 2 pi. So if I look at the gradient of the phase theta when I follow the straight contour here, the integral of the gradient, so the phase that I accumulate by moving this way, is either plus 2 pi or minus 2 pi, which means when you plug this into this formula, that the velocity varies like 1 over r. So the velocity gets faster and faster when you're getting close to the core of the vortex. And the vortex shown like this is a topologically protected object. That is, if I have a vortex somewhere with this phase winding around this vortex, I cannot simply eliminate this vortex by a smooth transformation because of the phase winding, which is something robust. The only way to eliminate a vortex is to, with rotation plus 2 pi, as I will see later, is to bring a vortex minus 2 pi and put them together. And this is a way of eliminating the vortices. But a single vortex is a robust object. It is topologically protected. This is what topology intends. So to make a long story short, what BKT have shown, there is in Schiko-Stowless, is the following phase diagram. I remind you, we have no Bose-Einstein condensation in 2D, so G1 always stands to 0 when r tends to infinity. But they have shown that because of these vortices, you may still have a phase transition between two kinds of disordered phase. At high temperature, you have a state which is not superfluid, which is normal. And this corresponds to G1 decaying to infinity very fast, decaying to infinity exponentially fast, like exponential minus r over l. Whereas below some critical temperature, you have, again, a disordered phase, that is G1 does not tend to something which is non-zero at infinity, tends to zero at infinity. But it tends to zero in a different way. It tends to zero algebraically, like 1 over r to the power alpha. And this state is superfluid. And so, again, it's not a usual phase transition where you go from disorder to order. Here, it's a phase transition between disorder and disorder, but the two types of disorder are not the same. And this is what really Schiko-Stowless understood, and this is one of the reasons for the Nobel Prize. And the vortices precisely play this role, I mean, are key to understand what is Tc. Vortices are always present in the system as thermal excitation, but if you are at low temperature, if you are in the superfluid state, when you have a vortex somewhere with a plus 2 pi phase winding in red, you have in the immediate vicinity a vortex with a minus 2 pi phase winding so that the total, the pair total has no phase winding and so you can have some superfluidity. Whereas when you increase temperature, the average distance between the two members of the pair increases, and at Tc, this distance between the two members of the pair stands to infinity, which means that you break the vortex pairs, the positive vortices leave their lives, the negative vortices minus 2 pi also leave their lives, they proliferate like that, and there, you cannot have any more superfluid current running because the vortices are fluctuating permanently and this kills any permanent current. So these vortices are the elementary break that drive this transition. So this is a nice story, but does it have any confirmation? And the answer is yes. A few years after the prediction of BKT, BKT was in 72, 73, there was an experiment done by Bishop and Repi on superfluid on liquid helium film and it's a very nice experiment, so I'll describe it in one minute. They put some helium film absorb on a substrate and they put this substrate on a torsion pendulum. This was also in a cryostat. And they look at the oscillation frequency of this torsion pendulum. So at large temperature here, they measure the oscillation frequency of the pendulum and they find a result which is relatively large. This means that there is a large mass which is oscillating and this mass which is oscillating is the sum of the mass of the substrate plus the helium film. But when they cool the system and when they cross this temperature here, then suddenly there is a jump in the moment of inertia of the pendulum. The moment of inertia is reduced and the reason for this reduction is that the substrate is still oscillating in the lab frame like this. But because the film, helium film is now superfluid, the helium film stays at rest in the lab frame. That is the viscosity, the rigosity of the film of the substrate is not enough to drag the superfluid into motion. So the superfluid stays at rest while the substrate is still oscillating so there is a reduction of the moment of inertia of the system which corresponds to this jump here and then with that they could prove that there is indeed superfluidity in 2D. That was proven with the helium film. It was proven with colloidal particles. It was studied also a lot with arrays of tunnel junctions. So we see it has worked a lot on that. And now that we have in our labs some nicely degenerate atomic gases. I mean, Emmanuel just described that in the lecture for those of you who were present. The question was, can we realize these 2D systems with atomic gases and can we investigate this boson shine, not this boson shine, this BKT transition? And so this brings me to what we have now done in the lab and when I say, yeah, it's a community of people as you will see. And the first topic I would like to discuss is, can we prove that a 2D gas is superfluid? We are not going to repeat the experiment that was done with liquid helium. I mean, our 2D gas, which are at a nano Kelvin temperature, we cannot put them on a substrate because then they would stick to the substrate and they would bounce. So we have to do something else. But you will see what we have to do. So first of all, we have to produce 2D gases. So we start with a 3D sample that we cool and trap. So here I refer to Emmanuel's talk to, for those of you who were present, for the others I would be ready to answer questions about laser cooling and trapping, but I think it would take me too long if I wanted to repeat what Emmanuel said. So I just refer you to Emmanuel's talk. So we produce a 3D sample and then we have to put it in flat land. That is, we have to squeeze the sample so that one of the degree of freedom of the gas is completely frozen. And for that, we use light. We use the potential that is created by light on atoms. More specifically, we make a laser profile which has the following intensity profile along the vertical direction. It has two intensity low like this and a minimum of intensity here. So here we have a nodal plane, if you want. So I have a lot of light above, a lot of light below. But here I have a plane where the intensity of light is zero. And I tune the frequency of this light so that the atom wants to sit where the light is very low. Okay, so that's by properly choosing the frequency of the light. I arrange the system so that the dipole force that Emmanuel described already, the dipole force attract the atom towards this point. And indeed, when I take my 3D gas and put it into this system, when I look from the side, I see like a pancake, so a pancake seen from the side is just a line like this. And when I see from the side, this is what I see. So I really have frozen the vertical direction and the reason for which I say that is that the temperature of my system, KBT, will be very small compared to any excitation quantum along this direction. If I assume that here I have a harmonic potential, this potential is characterized by its frequency on the order of kilohertz, say, and the temperature is below H bar omega z, so the atom are really in the ground state of the vertical motion. You can forget about the third direction. So now the atom moves in the XY plane, in the horizontal plane, and the confinement that we can put in the horizontal plane is very diverse. I mean we can play with many different ways. We can put a harmonic confinement in XY. So if I take a harmonic potential in XY, I get something which is very dense at the center and which density which smoothly decrease when I go to the edge. I can take a sharp hole with arbitrary shape using the digital mirror devices that Emmanuel was mentioning in his talk. I can also make some more complicated shapes like this one, like a segmented ring. You will see in the following actually why we can do that. So actually I will eat straight now in the remaining of the talk, these three, three type of shape. So first of all, some history of this field. So I think in Paris we started to work on 2D system more than 10 years ago and we got indeed some early experiment that we done with Zoran Adibabitch on the fact that indeed there were some vortices in this system which we detected by interferometry by checking that indeed the matter wave had a phase which could change by plus or minus two pi around some points. A bit later, these vortices could be directly seen like density holds in the cloud. So here you have a picture of a 2D gas trapped in harmonic potential. This was done in Korea where you see again large density at the center and the density which is decreasing on the edge. And on the edge where the density is relatively low, so the phase space density is not very large. So this is a place where the BKT transition takes place and indeed you can see vortices. And if you look carefully actually you can see vortices by pair, I mean these two are close to each other, these two are close to each other and so on. So vortices indeed exist in this system and there have been many experiments in many labs. I'm not going to cite all of them but this is a very active topic. So now let's address the question of super fluidity. Is this system, is this 2D system really super fluid? So again we are not going to make the torsion-ponderloom experiment. So we are back to the definition of super fluidity. Super fluidity is can I move, take an object, a small defect, and move it in my system without dissipating energy. This is what super fluidity means. I can drag a little impurity and move it without dissipation. So this is something that we did. Here I'm rather going to show some pictures obtained in Hamburg a bit later in 2015, the group of enigma rates. The defect that we are going to put is not a material defect because we could do it but it's a bit more complicated. The defect that we are going to use is again a laser. So we focus a laser onto the system, a very narrow spot, and again we choose the frequency so that the laser expels the atom. So we are creating a little hole in the gas. And now with some moving mirrors, I'm not technologically not so difficult, we are going to move this hole in the sample and we are going to look after a while whether or not the temperature of the sample has increased. And we do that for values, velocities of the hole in the sample. And so here I'm showing this result obtained in the group of enigma rates. So where you have here in x-axis the velocity at which the hole is moving in the 2D gas and along the y-axis you have the temperature increase of the system. And you see that when you are above the critical temperature here, you see that even a very small velocity, one or two millimeters per second, is enough to heat up the sample. Whereas when you have cooled down the system, that is you have a system which is more degenerate, you have crossed the phase, the BKT transition, then you have a whole range of velocity here up to four millimeters per second, typically, where moving this object doesn't change at all the temperature of the system. So this is a characteristic of super fluidity. Of course, if you move too fast, then you start to heat the system. So there is actually a critical velocity above which you break the super fluidity. But below this velocity of four millimeters per second, which depends on the system, the precise value depends on the system, you have some super fluid behavior. So you can go one step further and check how this velocity depends on the parameter of the system. So here I'm showing you the result that we obtained in our labs in 2012, where here I plot as a function of the control parameter of the system, which is actually the ratio between chemical potential mu and temperature T. This is the relevant parameter for describing the system. I show you the value of the critical velocity. So for low values of mu over KT, the system is not super fluid at all. So the critical velocity that we measure is compatible with zero. Essentially, any motion heats up the system. Whereas above some threshold value here, the critical velocity is non-zero and this corresponds to really a super fluid state. And there have been some recent analysis that we did with Ludwig Maté and Baytasing, that indeed this type of behavior can be reproduced theoretically by looking at some classical field analysis. And we could even reproduce the value of the critical velocity that was measured, which shows a fraction of the sound velocity. So indeed, the gas is super fluid. Now I would like to go to a second aspect of this physics of 2D gases, which is the propagation of sound in 2D. And the reason for which we have been looking at that recently in our lab, these results I'm going to show are not yet published, but they're down on the archive, is because of one feature, one very interesting feature of the Berezinski-Kossary-Stowlett Foundation, which is a so-called universal jump. This BKT phase transition that I told you is not a standard phase transition. It's not a first-order phase transition. It is not a second-order phase transition. If you want to classify it, it is an infinite-order phase transition, which means that all thermodynamic functions, pressure, density, entropy, they are all continuous at the transition point. So again, it does not enter in the usual classification of phase transition performed by lambda, for example. But something is discontinued, discontinues at the phase transition, which is the superfluid density. And this was shown by Kosterlitz. When you, if I again take my axis of temperature here, so I told you above TC, normal state, below TC, superfluid state, when I look at the value of the superfluid density, that I call rho s, and I multiply it by lambda square, lambda is the thermal wavelengths of the system. Maybe I should write the value of the thermal wavelengths, just in case you don't know it. So lambda, which is something called lambda t, is a square root of two pi h bar over square root of the mass of the particle, Kb times temperature. Okay, so when I, so rho s is measured in atoms per meter square, lambda t is measured in meter, or in micrometer, it's a more proper unit. Anyway, rho s lambda square is a dimensionless quantity, which is the phase space density of the system, superfluid phase space density. And what Kosterlitz understood is that there is a universal jump of this quantity, rho s lambda square. Rho s lambda square is larger than four for any superfluid state, it reaches four right at this point, and then suddenly it goes to zero. So there is a discontinuity in the superfluid transition, which is independent of the interaction strength of the system. Either way, I should say it, I need here some interactions. If I take an ideal gas, completely ideal, strictly no interaction, the notion of a vortex is very poorly defined. So in order to see the BKT transition, in order to have these vortices, which are going to pair or unpair, depending on the temperature, I need some interaction in my system. But irrespective of the value of interaction, I will have the jump of rho s lambda square. So how can we study sound propagation in this system? Well, this was understood a long time ago by two physicists, first Lasotiza, and then Leve Landau, so it was in the forties. This was done for understanding liquid helium. And the idea is to have a two-fluid model. This was Tizawa introduced, this is the first one, to say that the system is really composed of two fluids which are penetrating each other. One is a superfluid component, the other one is a normal component. So the total density is the sum of rho s plus rho n, and the total current which runs in the fluid is the sum of rho s times the superfluid velocity v s plus rho n times the normal velocity v a. And as Landau understood, you don't need any microscopic description of the system. We have, of course, for our terms of microscopic description, but you don't need it to write very general equations, which are enough to describe the thermodynamic of the system. You need only two hypotheses that I've written here. You have to assume that the entropy of the fluid is attributed entirely to the normal fluid, v n and rho n. And you have to say that the superfluid flow is irrotational, which amounts to neglect the vortices, but in the superfluid part, vortices are not so important because they're always paired together, so I can indeed neglect them in first approximation. And with that, with these two fluid, what these gentlemen have written is how a propagation of density wave or temperature wave can exist in the system. You look at the propagation of waves. So as usual, the propagation of waves relates a second-order time derivative on the left and a second-order spatial derivative on the right. And because you have two fluids, you have two such equations. One, the first one, relates the total density rho to the pressure, and the other one relates the entropy per unit mass, s tilde to the temperature here. And what is very interesting for our purpose, what I want to discuss, is that here in front, you have the ratio between superfluid density and rho. And since we want to study whether you have the jump of rho s, it's nice because you have an equation where rho s enters directly. So when you play with these two equations and you say, okay, let's take some kind of plane wave with some frequency little omega, some wave vector k, and look at the ratio between omega and k to get the sound velocity, you find when you combine these two equations, so here you will have some omega squares, here some k square, here again omega square and k square, where you combine everything, you get a bisquare equation for the speed of sound, so c is omega over k, so here you have omega over k to the four, combining the two, plus something c square plus some beta, where the question of alpha and beta are function of rho s over rho n, superfluid to normal densities, and the thermodynamic quantity, pressure, entropy, which are well known. So here in principle, we have everything to calculate and to measure and see whether we have some sound propagating in our system. So the application of this formalism to the 2D BOSGAS was done very recently, actually by Sandro Singari and Ozawa, his postdoc, and what they get is the following result. When you are in the superfluid regime, when your temperature is below the critical temperature, so rho s over rho n is nonzero, so if I come back to this, excuse me, to this, when I come back to here, rho s over n is nonzero here, and so I have these two equations. Then I get two branches, two roots for this bisquare equation. One is a second sound, and one is the first sound, and one is the second sound. Here they are plotted, the velocity c of the sound is plotted, normalized by c zero, which is the speed of phonon at t equals zero, the so-called bulk volume of speed. And when you are above tc, then you expect only one branch, because then rho s is supposed to be zero, and you expect only one branch. So the jump of the superfluid density that it was mentioning is reflected in the fact that you have here a discontinuity for both sound branches here. And when you analyze more in more detail what these two branches are, this is important for the experiment I'm going to describe later, you find that when you apply density perturbation, which is what we will do experimentally, you excite quasi-exclusively the second sound. So don't expect to this one, you will see this one later in the experiment. So this is what we have done. We have tried to excite some sound waves in our system. So now we take a very uniform system, so we take a rectangle actually here, these are rubidium atoms. Again, they are in 2D because we have frozen the vertical direction perpendicular to the plane here. And what we do on this rectangle is that we start by making a small density dip here by shining a second laser, which again repels the atom. So we reduce like this density here by typically 20%. So if you look carefully, the blue here is not as bright, as dark as the blue in the lower part of the figure. And then suddenly we remove this second laser that now the trap is uniform and we look how the density dip propagates. So this is the result. So here what I'm showing you is the following thing. Here, this is the time of the experiment in millisecond. And here I'm showing you the density profile or more precisely the difference in the density respect to the average integrated over the x direction. So I've created a hole here along y. So I integrate over x and so at time t equals zero, I have a density dip in this region. So this is a signal by this red value for delta rody. Delta rody is below zero, different respect to the average, whereas it's blue here, so it's positive. And so I see a density dip and you see that this density dip propagates to reach the bottom here and then it bounces back and forth like that. So you see only a sound wave propagating into the system. Okay, so you can see indeed sound waves. You cannot hear them, but you can see them. And from that you can measure the speed of this sound wave since you know how to travel over typically 40 micrometers. You see that it takes something like 25 or 30 milliseconds. So you get a velocity for sound waves here, which is 1.49 millimeter per second. So we have done this experiment for various boxes, various temperature, various densities. And when we look at what happens for t below critical temperature, we recover exactly the same prediction as the one done by Stringary and Ozawa. That is, we observe indeed the second sound in 2D. This is the first observation of sound in a two-dimensional sample, quantum two-dimensional sample. It was not possible to study that in liquid helium film because of the substrate that was present which was completely blocking the sound motion. All the points fall on the same curve which means some universality of the 2D gas. I will not enter in the detail today. I will discuss this later in the lectures. We don't see any evidence for the upper branch, but this was expected because we excite density perturbation. So this is all understood. Now let's look what happened on the right-hand side. That is for temperature above the critical temperature. So there we were expecting this branch. When we do the experiment, we find this. So it's very strange. And at the beginning, we thought our experiment was just wrong. And actually what we realized later is that actually we are not in the regime to study this branch here. This branch corresponds to the sound that I'm using to speak to you here, which is a sound which is in the hydrodynamic regime where the mean-free path of the particle is very short of the molecule of oxygen and nitrogen. The mean-free path of this particle is very small compared to the wavelengths of the sound that I'm emitting. Whereas in our system, we are not in this regime. In other words, if I speak more in the time domain than in the length domain, the collision rate should be very large compared to the sound frequency. And it is not very large in our system. It's typically one to four. So actually we see some sound here, but it's not the usual sound. It's a sound which is different. And this was understood very recently actually by two Italian groups, one in Trento and one in Padova, the papers are on the archive, which is a collision-less sound which can be described by a Landau-Vlasov-Kintzik equation which originates from mean-field and not from collision between particles, originates from mean-field energy and with their new theory, actually they get the purple prediction, which is in very good agreement with what we see. So what I can summarize here is that if, I mean it's nice because we have seen the second sound in the superfluid regime, but depending whether you are optimistic or pessimistic, it's a story of the half-glass, of the glass which is half-field. On the optimistic side, we've been able to see the second sound in the superfluid. So it's again, a nice confirmation of what 2D superfluid should be. But we didn't see any discontinuity at the transition point whereas we did this experiment actually to see the jump of the rest, but we could not see it because our normal fraction is too dilute to be in the algorithmic regime and this has been confirmed by recent theoretical work. So we do not have yet a signature of this universal jump at the transition, delta-Rs lambda square equal four. And so we have to imagine new experiments and so we're working on that, but any idea is very welcome. We have to imagine new experiment to show this discontinuity of the superfluid density. So this will appear soon in PRL and I should say that there are similar experiments done now in 3D in Martin-Virgin group at MIT on Fermi gas, close to unitarity. So I would like now to come to the last part of this talk which is another topological aspect that is present in these 2D systems which is the physics of permanent current. So it's a different topic. I'm not now going to look at a phase transition anymore but I'm going more going to look at the dynamics of the system. They are out of Caribbean properties of the system and you will see some nice picture like this one. You will see how to produce such pictures. So the way we came to this topic was actually a very fundamental question which is valid both in 2D and in 3D which is if you suppose that you have some condensate say or superfluids which are independent of each other which this one has a phase five one, this one has a phase five two, five three, five four. Let's say they are independent, not connected. So all these phases are completely independent from each other, they are randomly drawn. It's again it's a breaking of phase transition of a conduit symmetry if you're in 3D. And then suddenly suppose that you connect them. You put some links between them and you ask yourself what is the choice of the system for the common state once they are in contact with each other. So this is the question we had and it's an important problem. It's a problem which is relevant for transport experiments. It's a problem which is relevant also for the dynamics of phase condition. You may have heard about the Kibble-Zurek mechanism. The Kibble-Zurek mechanism is precise as that. When you have a phase condition and when you cool the system across the phase condition if you cool at a finite rate, your system cannot adiabatically follow the cooling. So you will have the phase condition which will occur in some patches in the system. And then if you wait longer, you have some so-called coarsely dynamics where these various patches start to speak to each other and the final state of the system is the result of this talking between them of these various patches. So it's a problem which is important in the description of the dynamics of phase condition. And the result I'm going to show were published last year in this paper. So we wanted a model system and the model system we picked up was the following one. It was what we call a segmented ring. So we use our digital mirror devices to print a matter wave field which has the following form which is made of some segments and segments. Here you have something like nine segments on this pattern like this. So what you see in yellow are patches of atoms which have been cooled very deeply. So each patch nearly forms a Bose-Einstein condensate even if it is in 2D because it's a small sample. And at a given time, we are going to reconnect this. So we are going to erase the barrier which is here, here and here. So the patches start to speak to each other. We lay weight a little bit for that and then at the end we ask ourselves, what is happening? And what is happening in this context is how did the various phases here reconnect? And as I told you before, a gradient of phase is related to a velocity or to a current. So the question is, is there a current in this interconnected ring? And if yes, what is the statistics of this current? How big is it? So just a word about time constant. We have chosen here a reconnection time of typically of a few milliseconds which when you compare this to the velocity of some velocity that we just measured correspond to something relatively long not to excite too much a system but it's short in the sense that there is no information which is allowed to travel in this reconnection time. Whereas if I take this 500 millisecond then this is long enough for having sound wave providing back and forth on this ring that the system has time to really find its equilibrium. The round trip time is 80 millisecond. And so what we want to do is to monitor as a function of the number of segments and the time we want to monitor the distribution of this current. So question, how do we see whether there is a current flowing here? I mean, when I take a picture like this I see the density but I don't see the velocity of the atoms. So how can I measure the velocity of the atoms in this ring here? So the way we did that was to do some matter wave interference and say similar device has been developed also in the group of Getschen combo in the US. Matter wave interference between our segmented ring which is on the outside and some inner ring here where we are very careful with this inner ring to have no permanent current in the inner ring. We are preparing very gently so that we know that the phase is uniform on this inner ring. So we start with that. We reconnect the outer ring and we ask ourselves is there some permanent current in the outer ring? So after the coarsening dynamics and in order to see that what we are going to do is to do interference between the outer and the inner rings. So we are going to remove the barriers which confine the atom in a ring. So the atom instead of staying in the ring will expand in the plane. This is true for the outer ring but this is true also for the inner ring. So the inner ring and the outer ring will overlap. They will interfere with each other and from the interference pattern we can see whether we have or not some permanent current. And these are the typically the interference pattern we get. So you see there are nice pictures. We can make t-shirts out of that. But we also extract some physical information. So quite often we see this pattern which is a bit boring. That is this pattern here is simply concentrated rings like this or a central dot and then a ring here and then another ring here. And this just tells us that the outer ring had a uniform phase. So we interfere the inner ring which has a uniform phase and the outer ring with a uniform phase. I get some concentric ring interference ranges. No surprise. But what is more interesting are these patterns here like spirals. When you see a spiral like this you can infer immediately that there was a plus two pi phase winding in the outer ring. The spiral may rotate like this clockwise but it may rotate also anticlockwise which means that then you have minus two pi phase winding in the outer ring. And sometimes you see spirals with three arms which means that you had not two pi but you had six pi phase rotation into the outer ring. So with that we can measure really the topological index of the current in this outer ring whether with its side we know whether it's plus two pi or minus two pi plus four pi or minus four pi and so on. So we can make some histograms, some statistics. So I'm not going to go in many details about that but just to give you a hint of what is expected. Let's take a three segment thing. Okay, so I have three random phases, theta one, theta two and theta three. By convention I can always put theta one equals zero and then two others, theta two and theta three are random respect to this theta one. And in order to understand what is the phase, the possible phase of this system the question, the best thing to do is to draw these phases on a trigonometric circle for theta one which is zero is always here and I put theta two and theta three randomly on this circle. And when there is a coarsening dynamics the question is what is the current that the system will pick up during this dynamics and the system will pick up a current such that it finds a shorter path from theta one to theta two to theta three to theta one when the current is established here. So if theta two and theta three are like that the shortest path is going from theta one to theta two like that, theta three like this and back to theta one on the same side of the circle. This corresponds to no phase winding at all. Okay, so here in this case I do not expect any current flowing in the outer ring. Whereas if I take theta two and theta three like this then the shortest path is going from theta one to theta two like this, then theta three, then theta one and now I have a plus two pi phase winding. Now you can make this a bit more quantitative. Again, taking theta one equals zero. So theta two and theta three I can represent them without loss of generality as a random number between minus pi and plus pi for theta three same thing for theta two. And so given realization of the experiment is a given point in this square of side two pi here. And you can easily understand when you think a little bit about that that this configuration that I've drawn here corresponding to a shortest path with a phase winding correspond to the slower triangle here. The reverse rotating the other way would be the upper triangle and the rest here the gray zone correspond to this configuration where I will expect no phase winding. And so if you want to look at the probability to have a plus two pi it's just the area of this triangle here and this is one eighth of the total area of the square. This is again one eighth and I have six eighths or three fourths probability of not getting anything. When we do the experiment so we do the experiment 238 times for this particular run and what we get exactly correspond to this result that is we find in three fourths of the case no winding and in one eighths and one eighths of the fourth minus one winding and plus one winding and never plus two or minus two as expected. So this is an interesting verification on a simple case that we understand in this physics. We have done the same experiment with n segments with n going up to 12. In this case what we are using is a so-called geodesic rule which is a rule which was invented in quantum field theory but we are going to verify it experimentally so to speak which is that the phase difference that we have to assign to a given link between say one and two is such that it takes its minimal value so it's a number between minus pi and pi. So we have to according to this geodesic rule we find the value of delta pi one, delta pi two, delta pi three for random choice of the phases one, two, three and with that we can make some statistics and it's relatively easy to show that we expect a zero average value and we expect a standard deviation of the current which should scale like square root of n if we have n patches and more precisely it should be square root of n over 12 and when we do the experiment so we did the experiment for three segments this is the one I already showed for six, for nine, for 12 we see indeed a very good verification of this geodesic rule and I should say again that this is the first time that this rule was verified there had been some nice experiment that at the Technion some 10 years ago with Josephson arrays of Josephson arrays but they could not check this geodesic rule they had a very strong deviation with respect to the theoretical prediction was here within our error bar we can indeed check that geodesic rule applies. Okay, so I am coming to the end of this talk I should, before concluding, saying who has done really the work and so this is a very nice team of people and what I discussed is essentially the work of Monica Hiddelberger who was a postdoc with us and the PhD work of Jean-Louville who is here and Raphael Saint-Jalme who is here and now I just summarize in one slide what I discussed today is really a part of a big game now which is this relation between topology and atomic and molecular and optical physics so I discussed topological phase transitions topology project current but topology plays a role in many other aspects Emmanuel in his lab is working a lot on topological bands in optical lattices looking for some characterization of the band in terms of topological indices. This physics is not restricted to atoms you can also have very nice topological effects with either pure photons this is for example a photonic topological insulator done in multi-seged group or you can play with hybrid objects hybrid that is objects which are half atoms half excitons I should say and half light these are so-called polaritons this is a picture from Jacqueline's blog group so we have a larger variety of platform in AMO physics where we can study these topological effects and I would say there is as a general trend a common background between all these experiments which is again to go beyond the standard classification of the states of matter via the usual geometric order now we are trying to find another type of order which is topological order like the big 80 transition I have shown before and in the long term I think most of us what we are hoping to reach are systems which where the interaction will play a very strong role a role which goes beyond mean field effect what I discussed like big 80 physics is essentially mean field effect or at least it's classical field physics but we are looking now for interactions interaction effect which go beyond mean field and then if we can reach that this will open the door to fractional multimole physics churn insulators, physics of Major and Affirmion which are very well known to people here and so it will be indeed more fascinating that what we have already seen thank you very much for your attention Thank you very much, Jan, it was a great talk and perfectly timed so it's very much appreciated so any questions from the audience? Going back to this collisionless sound could you please elaborate on the mechanism behind that? So collisionless sound in the thermal part I mean, so yes, so it's, sorry, this one here so it's essentially the fact that even though we are not degenerate density is relatively large and so there is a mean field energy so an atom sees the other atoms not because of collision but because of the mean field energy so when you have a density wave it modifies the mean field energy and you can get some sound for that so it's a mean field effect in the non-condensed gas which we are not so used to in cold atom physics but this is the idea so this is what I mentioned this Landau-Vlasov kinetic equation so it's a kinetic equation where you don't if you're used to kinetic equation you know that you have usually the Boltzmann collision term so here you forget about the Boltzmann collision term because we are not in the regime where the collision plays a lot of role but you still keep the mean field term and this mean field term is enough to propagate some sound which is again different from the sound I'm providing to you if you want it does not involve any more kappa S that is which is the compressibility isotropic compressibility but it's more kappa T the isothermal compressibility which plays which is important here the question is actually related to Jean I mean if you start putting impurities or just barriers do they actually impede in a stronger way this non-superfluid nature of the sound or not so can they actually suppress the signal that you see above Tc? that's a good question I don't know you mean you would say disorder or something like this? yeah disorder or homogeneous barrier that sounds over the same it would be easier to get a theoretical answer than an experimental answer I would say it's you could ask the same question for the superfluid in the superfluid case because the equation it's also mean field which is arriving the superfluid sound so it's a good question but I think I don't see any major differences between the two cases just a very naive question is it also possible to absorb the sound by multiplying some temperature gradient instead of density? yes in principle yes and actually when second sound was observed in 3D it was performed by Rudy Grimm in Innsbruck in a 3D thermigas at Unitarity he could indeed locally depose some heat and so look at the effect of not density perturbation but a heat perturbation in our case it's a bit more complicated for technical reason in principle we should be able to do it just that we are not able to do it for the moment and doing so maybe it's possible to excite also the other branch there by changing the nature of the excitation yes again for technical reason we didn't do it but it was done in a different system the 3D thermigas at Unitarity and there they could excite the two branches Mr. Niel last slide you mentioned that you choose the phase between minus pi and pi because of energetic considerations something like that yes yes well really since the wave function is something real part times exponential i theta at that point theta is an arbitrary modulo 2 pi it has nothing to do with the energy so it has nothing to do with the energy no the question is what will be the current when I recombine everything no I fully agree theta 1, theta 2 and theta 3 are fully random on the circle so I've chosen to put always theta 1 in 0 by convention before you break the barriers at that point each of the theta is arbitrary modulo 2 pi yes that's why I say I assume that here in this minus in this square here all points are equally probable and my point is that if I draw a point which is located in this triangle here I will end up with some current because when I remove the barrier this will be the minimal way of recombining the phases together but now I have removed the barrier now when you remove the barrier you have to much continuity one phase to the other and so the ambiguities are so yes exactly so my argument was after removing the barrier but before removing them I fully agree with you sir so is it possible by applying kind of bias to study the breaking of the vortex anti vortex pairs below the Tc is it possible to do this type of experiment so you would like to to do a kind of statistics over the distance between the member of the pairs superconductor if you apply a current basically plus and minus are pulled apart and then you get because in this way you can measure the jump through the expanse through the expanse and I don't know if similar okay I mean there are other ways to measure the jump which is one way is to measure try to measure the momentum distribution because it's a Fourier transform of g1 and so we could see a transition from g1 algebraic to g1 exponential now breaking the pair the only thing I can see which goes in your direction is rotating the system because if I rotate the system positive vortex will tend to go to center and negative vortex will be expelled so this would be the equivalent of your current we have done a lot of rotation but then when we are doing that we are doing it as low as possible and who's more with the 3d system than 2d rotating 2d systems is a bit tricky but it's a challenge that's true and then I would need to discuss with you to understand how you relate this to the exponent then but or to the jump of the superfluid density but questions from other side of the room sorry just a naive question what do you mean by fractional quantum holes yeah is it a dimension or something like fractional derivative so what I mean by fractional quantum or effect is just quantum or effect can come and become integer can come with integer values or fractional values integer values are ideal gas physics so to speak it's how you you feel on the whole level fractional quantum role effect is more tricky it's not a naive question it's a very difficult question but fractional quantum physics comes because of interactions so with fermions for example you have a simplest configuration is to have one third feeling factor for the lowest level which correspond to some some nice wave functions that were proposed by Laughlin so this is the idea that I I have in mind with fractional quantum effect is either with boson or fermions I have sufficiently strong interaction so that you can provide these strongly correlated states which are the ground state of the system for some particular feeling factor but it's not a naive question it's a difficult and fundamental question in the previous slide there are patterns of three or four different winding numbers yeah next one yeah that one how do you how do you precisely count that three do you see it from the static density profile how do I count here three branches yeah I follow this one so I see that I skipped those two I end up here so I have to count how many branches I have right okay so we have a pattern recognition algorithm to do it but we check that we check also by hand from time to time and when the pattern recognition is not happy and but if you really count here you will see you just have to follow take a pen and follow it and you will see that you have three three spirals implicated more questions it was a naive question from my side in your question do you have for the speed of sound you have parameters alpha and beta yes what are they oh okay so I didn't give them but they are fun I mean it's as when as when you derive a standard sound in the air you know that you will involve some some thermodynamic quantity I mean it will be the derivative of pressure with respect to volume with respect to temperature I mean it's just thermodynamic relation so compressibility is essentially the one which inter so the alpha and beta are these parameters I didn't write them but I have them on my computer if you want it's thermodynamic relation so it's derivative of pressure you have four variables you have pressure entropy density and chemical potential I think and so you derive one respect to the other choose two of them and take the derivative of two others very good so so as I announced before now the students will come down everybody will leave there will be some refreshments for everybody and the students are promised that something will be left they come up so let's thank Jan again