 Good morning everybody. I don't see my best friends among you, al-Shuler and al-Ainir. They know practically everything about this talk and this is probably the reason that they did not come. I think he's on the way. Okay, let me check whether this works. It does. So what I will do, first of all, I have to say that I'm grateful to the organizers for inviting me to this conference. I have been here at ICTP so many times that I even do not remember how many practically every year, sometimes even twice or three times. And now I will talk about quantum gases in disorder. This is the outline of my talk most related to many by the localization, the localization transition in disorder. Quantum gases, ultra cold gases and the work, the original work that I will report on today has been done together with Boris al-Shuler, Igor al-Ainir and Vincent Michel. I do not even see Vincent Michel here. Okay, so let me now start the talk. So as all of you know, most of you much better than me, that if we have many by the system in disorder and particles don't interact with each other, this is the disorder potential, these are my particles, then the transport and localization properties are determined by Anderson localization, which was predicted in 1958 and got the Nobel Prize in 1977, I think. Yes, that there is a destructive interference in the scattering of a particle from random defects. As a consequence of that, the wave function attenuates mostly exponentially at large distances and the transport properties are just absent. There is an old question, how does the inter-particle interaction influence localization? And this is a long-standing problem. It was crucial for charge transport in electronic systems and now it appears in a new light for the so-called outer cold bosons, for example, now also fermions. And there are many experiments in Palazzo, in the group of Alena Spain, in the group of Massimo in Gushoe, Atlanta, then at Rice, Urbana in Switzerland and so on. So, but you know, before people started experiments on ultra cold atoms in disorder, so what means ultra cold? Since many of you don't work in this field, ultra cold means that the temperature is below 1 micro Kelvin from 1020 to 200 nano Kelvin. Yes, many things were known. Yes, it was, for example, known or got known almost at the same time that there is Anderson localization of light, microwave, sound waves, electrons and solids. And what was expected is Anderson localization of neutral ultra cold atoms in disorder. The disorder is usually created in these experiments by making optical speckles randomly. Yes, they can do that. And then atoms interact with these pieces of light and this is the interaction of an atom with the disorder potential. Before I start explaining things, I would like to mention why to study ultra cold atoms in disorder? Because very many interesting things have been done with light, with electrons and so on. But what happened is that, let's say, some 10 years ago or even 12 years ago, there was a question of an atom laser. What is atom laser? You have a chip. On chip, you create a one-dimensional waveguide, very narrow magnetic trap. Yes, and a correlated atomic beam is propagating in this atom waveguide. It's above the surface and the beam feels imperfections of the surface. So you get this correlated beam on top of the random defects. And that's how the investigation of ultra cold atoms in disorder has been started. But very quickly people realize that there are many more interesting things from fundamental physics, in particular how the interaction between particles influence localization. Yes, and so let me now give a very brief overview. I cannot give a complete overview, but just very brief. So within my drawing facilities, that's how people work. This is the harmonic trapping potential in which you get atoms, both Einstein condensate. I made it one-dimensional. Please don't be surprised that I say both Einstein condensate in a one-dimensional system. In a 1D system, it's a purely finite size effect. The correlation length is larger than the size of the system. And then you superimpose a disorder potential and you get distortion in the density profile. Then it was extremely simple to continue, just to repeat the Nobel experiments of 1995. What to do? Imagine you switch off the harmonic confider. Without a disorder, this was the Nobel experiment of Eric Cornell and Volgenkatery. The gas expands and the condensate expands differently compared to a thermal cloud. What happens if the disorder is still present? You switch on this green guy, but the disorder is still present. Then naively, from what we know from literature, we expect that eventually the expansion should stop. Particles, when the density becomes small, behave themselves as noninteracting and noninteracting particles in disorder in one dimension should be localized. And that's what was happening in the first experiment in Orsay in the group of Alena Spea. Eventually the tails become stationary, the expansion stops. That's the density profile. This was the first observation of Anderson localization with outer cold atoms. There were more experiments after that, in particular the experiment at Lenz in the group of Massimo Inguscio, Giovanni Modugno, but I will tell later about that. Then, so what was not immediately expected, let's say some 10, 12 years ago. As we all know that in one dimension, if we have one-dimensional bosons noninteracting in a random potential, all single particle states are localized in 1D. That's what everybody knows. And there is also a dogma, even if they interact, no finite temperature phase transitions in one dimension since all special correlations decay exponentially. But these guys, following a general theory of many-body localization, said that there is an unconventional phase transition between two distinct states, fluid and insulator, and this transition is induced by the interaction. And it's nonconventional because the behavior of thermodynamic functions is so far not exactly known, but what is known is that at the transition point, the transport properties are singular. In the fluid phase, the mass transport is possible. In the insulator phase, it is completely blocked, even at a finite temperature. So I continue. So this all comes from the theory, a general theory of many-body localization, delocalization transition. The paper which shows this is this paper from these years. Imagine you are in the localized state, yes? Then one should learn how different states of two particles alphabet hybridize each other due to the interaction and find the probability p over epsilon alpha. Epsilon is the energy. That for a given state alpha, there exist states, beta alpha prime, beta prime, such that these two particles states, alpha beta and alpha prime, beta prime, are almost at resonance, which means that the interaction induced matrix element of the transition exceeds the energy mismatch. The energy mismatch exists because I'm in the localized state, and in the localized state, the size is finite in the sense that it's the localization vector. And many-body localization, delocalization, transition criterion is that this guy, p, is of other one. Okay, here is the picture. Then the matrix element is the largest when these energies are almost equal pairwise, for example, alpha to alpha prime, beta to beta prime. Then the matrix element of the transition induced by the interaction is the Hubbard constant U, multiplied occupation number of the state, beta, lattice constant divided by the maximum value of the localization vector. If I am in the continuum, in the continuous phase, this U multiplied A is just the coupling constant G. I consider short-range interacting particles, yes? And this match, it's this formula, and it's eventually equals to the minimum value of a product, localization length multiplied by the density of states. Minimum means that for all these states, alpha, beta, alpha prime, beta prime. Then what you should do, you should calculate the probability that this matrix element exceeds the energy mismatch. If the matrix element exceeds the energy mismatch, the state becomes strongly hybridized, yes? And this is this guy, yes? Right? And then you have to make a summation over the states, beta, alpha prime, beta prime, and you get this integral, yes? So then the critical coupling strength at which the localization, the localization transition occurs is given by this formula, where this is the density of states, localization length, occupation number, and then this quantity, yes? So that's what it is. I will not explain details of this general theory, which is given in the work of Alainer-Alschuler-Bascoe, but just keep the criteria. Then if I consider a one-dimensional bosogaz in the absence of disorder, we know what happens. So we have a classical gas at temperatures larger than the temperature of quantum degeneracy, which is density squared divided by the mass. Then in the weekly interacting regime that I will consider so far, yes? We have parameter gamma, which is actually the ratio of the short-range interaction to the temperature of quantum degeneracy, Fermi energy for fermions. It's much smaller than one, and then in here there is a degenerate thermal gas. No correlations, but the gas is degenerate. And here you have what people sometimes call quasi-bozer-Einstein condensate. You have long-range correlations at zero temperature, they're algebraic, yes? And there is a Bozer-Einstein condensation on a finite size. Then what happens if there is no interaction while there is disorder? I already told you all single particles are localized at any energy, Anderson insulator. Whatever energy is, the state is localized. It's only a matter of the value of the localization length. So what is more or less one can guess is that Bozer gas in disorder when particles interact with each other shows both behaviors. And this is actually true. And here, I just say the density of states is unit divided by square root of epsilon. The localization length in 1D is proportional to epsilon. They are cut at a characteristic energy epsilon star which is related to the amplitude and correlation length of the disorder. That's the behavior. In the classical gas, when temperature is larger than the temperature of quantum degeneracy, there is an ordinary expression for the chemical potential density and using the criterion of many body localization by localization transition which was, I think it's outlined here. Yes? I easily get the result that the critical coupling strength, coupling constant G because I'm in the continuum phase is inversely proportional to temperature. In the quantum decoherent regime where the chemical potential is temperature squared divided by the degeneracy temperature. Yes? I apply the same criterion and get that it's inversely proportional to unit divided by temperature. Yes? I can also consider the quasi-BEC regime down to zero temperature and find out that the critical coupling strength is just the characteristic disorder energy. I have no time to talk about that. In the end, what we can do, we can make a phase diagram. This is the phase diagram in terms of temperature and the amplitude of the disorder. Here you have insulator. The larger is the disorder strength, the more possibility to get to the insulator phase. Don't pay attention that this is blue and this is red. It's just to separate the degeneracy quasi-BEC regime from the regime of thermal gas. Yes? What we can also do, we can plot critical coupling strength as a function of temperature. And then what we see that it's something and then it goes down with increasing temperature, which we expect naively we think that if we increase temperature, then there are more possibilities to delocalize. That's what one naively thinks. Let us now say that this was the issue which has been expounded about five years ago. Yes? And there are many more interesting things which I will say about now. I hope I will have time. Yes? And then this is the picture for one-dimensional bosons in disorder and that's what we should bear in mind. There are other things here. For example, the second experiment on localization of alter-cold atoms has been done at Lens in the group of Massimo Inguscio, Giovanni Moduglio. What they have done, they have used the so-called Aubry-Asbel Harper model. This is the following. You have a lattice, yes? Tight binding model. Yes, you have particles in this lattice. J is the hopping amplitude. And then you superimpose a shallow incommensurate lattice, this blue curve. Yes? Then, if the amplitude of the shallow lattice is larger than twice the hopping amplitude in the initial lattice, then all single particles are localized. The statement belongs to Aubry and Andre in 1980. It's a remarkable work in mathematical physics which has been published in the Journal of the Israeli Physical Society. You know, it sometimes happens that remarkable works are published in some remote places. Like the work of Tony Leggett on this BCSBC crossover also from 1980-81 has been published in a very strange Polish journal. I got extreme difficulties to get this paper, but a good friend from Israel was visiting Paris, Agamodet, and then he told me, Giora, when I go back to Israel, I can find this paper, I will email it to you. That's what he did and thanks to him. At the same time, Serge Aubry returned to France from vacation and also emailed me his paper. The two versions were the same. So, a remarkable work and then what they have done, they have done several experiments. The first experiment was just the expansion to test the localization. In the second experiment they did it with potassium-39. What you can do, you can modify the interaction with atoms using so-called Feshbach resonances. I will not tell you what this is, but that's what they did and when they change interaction they detect the insulator and fluid phase and these are experimental points that transition between them. So, at the end let me continue. So, let us now try to think what happens in this system at temperature. The thing is that in these lens experiments the temperature was not very well controlled. It was something and they had difficulties to control. Yes, at the end the localization length for all Eigen states, non-interacting, is given by this formula where V is the amplitude of the shallow lattice, J the hopping amplitude of the initial lattice then this formula A is the lattice constant and it's much larger than the lattice constant. Then if kappa kappa is the ratio of the periods if the period of this blue lattice is much larger than the period of the red one then the spectrum for non-interacting particles it consists of narrow bands yes and the width of a band increases by the way exponentially when I go from the lowest energy up and up up to here then it decreases exponentially again. Now what we do is very simple. We include the interaction between the particles yes, when they are sitting in the same lattice I just have a constant view yes, not more than that let's see what happens. Let me say that as I told you about these clusters if the number of clusters which is unit divided by the ratio of the periods is N1 the width of the cluster grows exponentially with energy and then if this N1 is smaller than the localization length then the number of states of a given cluster participating in the many body localization is simply localization length in units of the lattice constant divided by this N1 or Xi multiplied Kappa if my temperature is much lower than the width of the energy spectrum the width of the energy spectrum is 8j then these states are not relevant they are far away and only the lowest energy cluster the lowest energy cluster is this guy it is the narrowest participates in the many body localization, delocalization transition because in the criterion of this transition I have density of states in the square this is my criterion of the transition occupation number of particle states let's write this relation N2 is the Hubbard constant and epsilon is again the occupation number and this is the normalization condition for the feeling factor, yes so then what I will do consider zero temperature then at zero temperature there is a very simple relation between the occupation numbers energy and chemical potential yes, mu zero is the chemical potential at zero then I get a very simple expression for the critical coupling strength mu is the feeling factor gamma zero is the width of the lowest energy cluster the ratio of the periods, localization length and one can easily understand that this formula remains valid if temperature is much smaller than the spacing between the clusters yes so far so good continue and consider temperatures which are still larger than the spacing yes, right then for large occupation numbers my exponential formula becomes algebraic this is the formula for the occupation numbers yes and this formula remains like that if the occupation numbers are smaller than one then what I do is very simple it's arithmetic I compute this formula to the criterion of the localization transition and get a critical coupling strength the function of temperature when temperature is much larger than the spacing between the clusters and much smaller than the width of the energy spectrum nothing surprising except for the fact that if I look at the temperature dependence I see that the critical coupling strength increases with temperature so what we see that the higher is the temperature the more difficult it is to delocalize particles I need a higher coupling strength this is the anomalous temperature dependence which was not initially expected so why this is the case it's very simple to get a physical understanding you know the transition is still related to particle sitting this lowest energy cluster when the temperature is very low then the number of particles is something when I increase temperature and it becomes larger than the spacing between clusters particles from here come here, here, here and the number of particles here which participate in the localization delocalization transition becomes smaller smaller number of particles a largely required coupling strength for delocalization yes that's what it is and that's how the picture looks this is the critical coupling strength in units of the hopping amplitude multiply the feeling factor three curves are for three different feeling factors this is temperature so what we call this is freezing with heating phenomenon increase in temperature favors the insulator state I'm here in the fluid I increase temperature then in contrast to the expectations I become an insulator yes that's what it is now one can ask whether this is an artifact that I selected the period of the shallow incommensurate lattice much larger than the period of my initial lattice the answer is no for example if I take the so-called golden ratio according to people doing mathematical physics for this golden ratio this Andrea O'Brie Asbel model model's best of all the disorder system yes then for example for this V I also get the same yes the same type of curves yes and this has been done by doing exact diagonalization when Saint-Michel spent plenty of time on doing that as well as for previous examples that I have shown I always do not think that the very respected guys like al-Shuler and the guy like Shlapnikov did exact diagonalization this is not true this is Vansant who did that yes so therefore we have an anomalous temperature dependence of the critical coupling strength when I'm giving public lectures what I usually do I again pronounce the words freezing with heating and that is a very good example for general public you know if I take a glass of water and start to heat it yes then I get hot water or water vapor in this example I take a glass of water start to heat it and get a piece of ice so that's what is happening quite funny so aside from interesting things like many body localization delocalization transition at finite temperatures in 1D one may get this phenomenon and the experiment is now under way so let me I think I can even say 5 minutes for other speakers of today should I do that? 10 minutes including questions ok so let me then say according to what I showed you one-dimensional bosons is a promising system to study the many body localization delocalization transitions and atoms in quasi periodic potentials is an interesting system where increasing temperature may favor localization at this point I have to say thank you for attention but I'm not yet finished yes since I have 5 minutes yes yes thank you thank you Emy yes but you know I did not sleep very well yes I slept very little therefore I just decided in order not to say something stupid my talk should be shorter yes now what I would like to show I would like to show this yes I'm very happy that Boris Alschuler eventually came to listen to me he knows everything about what I said even more yes so this is him do you recognize Boris? then so what is written here Boris welcome to the club over 60 if I look in his passport it turns out that he got 60 years old a little bit earlier than today but this does not matter you know because over 60 Boris it is quite nice still yes it's ok I am already in this club for a long time yes those who think that I am young are not right but you know after I got 60 the life did not change you only feel yourself a little bit more comfortable people start saying ok yes and that's what it is and what I hope is that in some time we will celebrate your 70 80, 90, 100 birthday and so on yes do not know not sure that this will be here at ICTP but at least somewhere yes and you know I can tell you an interesting story here some years ago when my good friend and teacher Yuri Kagan was 60 years old he was forcing his son Maxim and Maxim's wife to do the work in his house in his country house which is a huge house and they were not happy then Yuri started to talk to them and said he was 83 you know guys you should work hard here because in 17 years this all will be yours yes so Boris thank you for attention