 All right, thank you very much for the presentation. All right, so this is another contribution to this wonderful event here. And I'd say that this is partially attributed to Boris Asshulev, because this work started at the conference organized by him three years ago in Paris. And most importantly, there is a little bit of controversy in this work. And I know that Boris enjoys controversy and practically at every conference that I attended where he was, there was some sort of fight or however regarding certain physics. I can't see if Alex Kamenev is here, but I very much hope he is, because then he can actually say something about my arguments, probably confront them. Because what I'm going to do, especially in this work, is to criticize what they are doing in their group. OK, so let me say a couple of words about the affiliations. This was mostly done at Lancaster University. However, I have moved recently to the Leiden University. So therefore you have two logos here. This group of people who did the work are no longer in Lancaster. All right, so the original experimental motivation of this work is called atomic systems, where people recently were able to create one-dimensional fluid systems where they managed to embed impurities of difference. For example, in this experiment, an impurity atom was created by just flipping a hyperfine state of an atom inside an atomic cloud. And then the fall of this atom to the cloud was observed. This is another example where the impurity atom in 1D is created by adding actually a different sort of atom into the one-dimensional trap. And all sorts of observations can be done. One can look at the evolution of momentum of a falling impurity, or look at the diffusion of the impurity species inside the atomic cloud. So this opens a very intriguing possibility of actually real-time imaging of the evolution of an impurity atom in a quantum fluid. There are two different regimes. We are talking about one-dimensional physics. And there's one reasonably well understood and reasonably well-studied regime, which is diffusion of an impurity atom or low-field mobility at finite temperature studied in the 80s. A lot of work on X-ray age singularity for static or mobile impurities, including recent works, for example, by glasman and quotas. There is a different sort of physics, which is highly non-equilibrium, for example, relaxation of a large initial momentum, or non-linear response to a drag force. This is something which goes far beyond the methods I used here. Actually, in a sense, this is an orthogonal complement of the work which exploits renormalization, group, large and liquid concepts, and stuff like that. The phenomena that you observe here require a different type of approach. These have become available experimentally quite recently. I'm going to focus on these two types of experiments, theoretically. Momentum relaxation, we have an impurity created inside the one-dimensional trap. It is longer than, I assume, to be longer than this page. This is as much as I could fit. So we have an impurity. Give it some initial momentum P0, assuming temperature is close to zero. And then look at the time evolution of the impurity momentum. In particular, at the question of whether the momentum of impurity after infinite time is zero or not. The usual answer in higher dimensions is if a fluid is a normal fluid, then an impurity will eventually stop. It will pass all the momentum to the environment if the fluid is a superfluid, and the velocity of the impurity is low enough, then it won't stop at all. One-dimensional fluids are never superfluids. This is well known. However, as I'll show to you in the next slide, two of the first calculations by the Harvard group using an integrable model of a one-dimensional gas shows that the momentum of an impurity never relaxes to zero. Well, never is not a good word here, because they just looked at a system consisting of 40 particles. And they performed a summation of about 1 million form factors which contributes to the impurity dynamics. And there are lots more, of course. And then they plotted the momentum of an impurity as a function of time. Of course, finite size effects become important after a while, and there is a rebound of the state, which is pretty close to the initial one. So there is some finite time during which they can follow the evolution. It seems that the momentum of impurity relaxes rather rapidly for the first few moments, and then it saturates a certain value. Also oscillating around the zero average, they adopt this phenomenon, the oscillations quantum Flutzer. And they also speculate that if one goes to the infinite system size limits, the scaling is such that this final momentum value goes to some non-zero value. That is the phenomenon persists in the thermodynamic limits. So I'll talk about this development. I'll also show that there is a link between this momentum relaxation type of experiment and a different type of experiment where we consider an impurity under a constant force. Three years ago, there had already existed some very intriguing results about this system. Due to Gangart and Kamiev, they considered a very particular example of a quantum impurity in the one-dimensional fluid, an impurity embedded in both gas near the Bogolubov limits, the almost superfluid both gas, where the repulsion between bosons is big. And they semi-classically solved the dispersion relation of a polymer, which is formed by the impurity in the surrounding clouds of the gas. So there is some sort of solution of this problem presented in the paper. And what they find is that the dispersion relation of this object that they call the depleton for a certain reason. I'm not going into that. So they show that the dispersion relation looks like this. The energy of an impurity with increasing momentum behaves in a manner very similar to what you observe in a periodic lattice. So if you apply a constant drag force, the momentum of impurity will grow, but then it will bounce back. And the impurity momentum will oscillate as a function of time with this period, very similar to the block oscillations in the one-dimensional lattice. This is a very elegant prediction. And the Innsbruck group, for example, tried to observe this phenomenon. And the reason inside the information is that they actually have seen something similar to this prediction in their systems. So these are two unrelated works. And what we attempted to do was to consider a very simple example of a one-dimensional system where we could investigate all this phenomena within the same framework, answer certain questions that arise in the context of those observations. For example, is in complete momentum relaxation a finite size effect? Or does it persist in a thermodynamic limit? Is the quantum flat a finite size effect? How are the quantum flat oscillations, for example, related to the quasi-block oscillations? All sorts of questions that arise in this context, they require some analysis within some corner of the parameter space of the problem where you can do controllable expansions. So I'll go to a particular corner of the parameter space, which is the Tongzhe-Rodogas weekly coupled to the impurity. I'll show you some simple qualitative picture explaining how this phenomena arise within this model. I'll say a couple of words about the full quantitative theory. And new insights is probably not appropriate. Or there is actually a contradiction between our results and the Gangartan-Kamenya group results, which hasn't been resolved yet completely, which has so far resulted in exchange of comments and replies. OK, so this is perhaps the most general setting we are talking about. We have a one-dimensional system. We neglect the boundary effects here. So the red atoms of one sort, there is an impurity atom, which is shown as blue just characterized by its position in momentum. And this is the Hamiltonian with the important coupling constants are the repulsion between the atoms and the impurity, the repulsion between the red atoms here, the red atom and the mass of the blue atom. And as I have mentioned before, people investigate different corners of this parameter space. For example, if this m is equal to this m, this is an integrable model. You can actually exactly solve it in the limit of either equal coupling constants or this coupling constant going to infinity. Or if this constant goes to a very small value, then we are talking about the Pogolubov limit of this theory, which can be solved controllably. What I'm going to talk about is the limit in which C goes to infinity. The repulsion between red atoms is infinite. This is called the Tongs-Gerardot limit. And in this limit, a system of bosons is spectrally equivalence to a system of fermions. And we are taking the thermodynamic limits. Number of particles goes to infinity. The size of the system goes to infinity at a fixed density. And in this limit, one can introduce a dimensionless coupling constant, which is this mass times this coupling constant divided by the density. And we assume that this is much less than 1, which enables us to use more or less standard kinetic theory to analyze the system perturbatively. And we introduce a parameter, which is called eta. Eta is just the ratio of masses of host impurity. So a host atom and impurity atom. OK, this is just a simple statement that in the C to infinity limits, the host gas, the gas of red atoms, is spectrally equivalence to a gas of fermions. So if you look at the dispersion relation of the excitations, it essentially consists of a particle whole pairs where you excite particles out of the thermosy. And one can use the standard notion of the Fermi momentum, which is pi times. The density of the particles, the Fermi energy, the Fermi time, which in the language of original bosons is roughly the same as the collision time between the nearby bosons. OK, so that's the mapping to the fermions. Now let's go to this small coupling constant limit, a small repulsion between the host and the impurity particles. And let's look at the kinematics of a collision of a particle. So the capital, the big q and big k, denotes the in and out momentum of the impurity atom in the collision process. And this is the in and out momentum of the host atom in the host fermion, which is not an atom. But it's a particle in the spectrum, quasi-particle in the spectrum of excitations of the host. Kinematics tells us that the momentum is conserved. The energy is conserved. However, due to the spectral equivalence of bosons and fermions in the repulsion, in the repulsion limit, there are probably blocking constraints on the out momentum, the momentum of the in particle has to be less than 1. It has to be under the firmacy. And the out momentum has to be greater than 1. So we have four constraints on the collision process. And we can solve the system of algebraic constraints. And this leads to this structure of the collision phase space. On the horizontal axis here, we have the in momentum of an impurity colliding with the host. And on the vertical axis, we have the out momentum of an impurity after a collision. And it can collide with any particle. So for a given in momentum, there is a whole range of out momentum that can occur. So what we can see here is for any in momentum, the range of out momentum is finite. And the structure, the geometry of this phase space for the light impurity, the impurity must less than the host mass. It's different than for the heavy impurity. So here it is equal to 0.5. And there is one very crucial observation here. If you look at this map, which is there always exists a range of in momentum of an impurity for which there is no out momentum in the collision process. So the only forward scattering with the same momentum is possible. No collision with momentum changes possible between minus k0 and k0. And for the light impurity, this k0 is just the firming momentum times the ratio of masses. For the heavy impurity, this is just the firming momentum. This critical momentum. From this simple observation, we can quite easily understand why there is no complete momentum relaxation to the zero of the initial impurity momentum. What happens is if we create an impurity with a certain initial momentum, it can get scattered into a range of out states. And as shown in this picture, some of these out states may fall into the forbidden region, kinematically forbidden region, where no further scattering is possible. So if an impurity is scattered from here into this region, it will stay with this momentum forever. And if it is scattered into this state, it is further scattered into another possible set of states and so on until it falls into the forbidden region, and there it stops scattering. And if we look at the final momentum distribution after the sequence of scattering, it is going to be some distribution function. This is the probability to find impurity with this momentum as a function of k between minus k0 and k0. And generally, this distribution is going to be asymmetric. There is no fundamental reason as to why it should have any symmetry at all, because the initial state was not momentum symmetric. So we are going to have some momentum as a function of the initial momentum. And what is interesting, you can actually calculate this final momentum. You can write the collision integral, the Boltzmann equation, and solve it explicitly. The Boltzmann equation is solvable explicitly for this problem. And gamma, the coupling constant, drops out completely from the solution. So in the limit of vanishing coupling constant, the scattering rate between an impurity and the host atom, one obtains some non-trivial results. This is the P infinity as a function, the momentum at infinity time as a function of the initial momentum. And if the initial momentum is less, then if it lies inside the forbidden region, we have a linear curve. The out momentum is the same as the in momentum. But for the light impurity, when we reach eta times the firm momentum, scattering becomes possible. And the final momentum, as you see, bends down, goes down with increasing initial momentum, actually reverses. One of the predictions of this theory is if you shoot an impurity with a huge momentum into the system, it will come out on the same side as it came in. Its final momentum will be reversed compared to the original one. This is the light impurity case. The heavy impurity case is actually dramatically different from the light impurity case. Again, initially, the final momentum is the same as the initial momentum. However, as the momentum reaches the value of the firm momentum, there is a flip of the out momentum. So there is a discontinuity here. And then there is some continuous curve ending up at high values of initial momentum on the positive side. The reason for this difference is very simple. Here, at this point, in the light impurity case, the momentum relaxation is due to the emission of sound waves forward in the forward direction in the direction of the motion of the impurity. And the heavy impurity case emission of sound waves is not kinematically allowed before a flip or for a particle from the right firm surface to the left firm surface. This occurs if the momentum of an impurity is equal to k-firmness. So these are two very distinct cases. Now, one can also look at the equal masses case. We can see that there is a discontinuity between the light mass and the heavy mass. So one can wonder what happens at the equal masses in the equal masses case. And in the equal masses case, one can take formally a limit of one mass going to the other and get to this answer. But this answer doesn't make sense, because actually after the first scattering process, an impurity takes some value of momentum and energy and further scattering is allowed kinematically. It's the only case where after the first scattering impurity can still migrate in the momentum space as long as its momentum does not get larger than the firmer momentum. And in this case, Boltzmann equation collapses. The precise collapse of the Boltzmann equation is if you look at the quantum Boltzmann equation collision integral, this term in the collision integral becomes comparable with this one. This occurs when the difference of masses is on the order of the coupling constant. So what one can do in this case, unfortunately, the only solution is to look at the better answer solution of the problem. It does exist. There is a form factor representation of the momentum P infinity. And we actually managed to calculate asymptotically this sum over the form factors in the vanishing coupling constant limit. And we got this result for the impurity momentum, which is pretty close, actually, to what you get from the naive Boltzmann result, except you have a logarithm upstairs here and you have it downstairs here. And this result does not show any discontinuity of the momentum at P equal to the firmer. OK, I'll skip this. So let's go to the situation of a constant drag force. This is the same system. We now started with P equals 0 and apply a drag force to the impurity and look at its evolution as a function of time. This is a force normalized to some intrinsic force constant in the system. And this is time normalized to some intrinsic time scale in the system, just a technicality. This picture contains basically all you need to know about what happens to the system if you apply a constant drag force. So suppose we place an impurity at 0 momentum and a force accelerates the momentum of an impurity. Let's look at the horizontal axis. This is a momentum axis of the in momentum on a collision map. This is the out momentum. If we accelerate along this line, the collision is not possible for the out momentum is equal to the in momentum. So we can accelerate the impurity up to the value of it's in the light impurity case, after which back scattering becomes possible. But if you look at this diagram, back scattering is possible into a small vicinity of the same momentum either. So what happens is impurity accelerates to this point, is backscattered, and is trapped in this region. Physically, this means that an impurity is accelerated to the point where its velocity is equal to the velocity of sound waves. And then it emits sound waves constantly sticking around this velocity value. And quite different, quite the opposite thing happens if we look at the heavy impurity case. In this case, again, impurity accelerates to the point of the first reflection. But this reflection is by 2K Fermi. This is a huge momentum transfer. So the impurity is backscattered to the value of momentum minus 1. To this point here, accelerates back, is backscattered, accelerates backscattered. We observe huge oscillations of the impurity momentum in the range of 2K Fermi. Again, one can do the solution of the Boltzmann equation, which is responsible for this. And this is the evolution of impurity momentum in the light impurity case. It accelerates. And then the momentum, there is a slight overshoot. And then the momentum stops evolving in some steady state in which impurity emits forward the performance in the system. And there are dramatically different things occurs in the heavy impurity case. The momentum accelerates. However, after a while, it saturates at certain value. And the rate of saturation has been calculated in our work. It's not that important here. So these are our predictions. We also calculate other quantities of interest. I'm not going to focus on those here. I'll go to the Gangansen-Kameny work. The main argument in the work that followed the original prediction was that if we look at the lower edge of the system as a function of its momentum, it is a curve which looks like this, this black curve here. And it doesn't matter whether you study your system in a certain exactly solvable limit. This is a pretty universal statement. If you look at the lower edge of the energy of your system as a function of momentum, it will have this shape. So if you drive your system very slowly by the external force, it will adiabatically follow this line here. And therefore, the momentum of the impurity and goes from this point to this point will oscillate as a function of time. That was the prediction. It is obviously in contradiction with our statement that this phenomenon only occurs in the heavy impurity case. It does not occur in the light impurity case. So they wrote a comment where they proposed, their essential proposition was that there is some physics in the one-dimensional world which is not captured by the kinetic equation. And this is a bound state which is formed by an impurity. So if you have an impurity, which is shown as red here, and we have a hole somewhere under the pharmacy, after the impurity kicked out a particle from under the pharmacy, there is a bound state formed by the two. And this bound state is actually a non-perturbative phenomenon not captured by the Boltzmann picture at all. And what they did, they took a particular model where they severely truncated the Hilbert space. They only left this whole state as one family of states. A single hole created under the pharmacy and another family of states, an impurity, which is parameterized by its momentum, they solved the two-body problem, Schrodinger's equation. And they found the exact spectrum of the system which looks like this. It is shown for the light impurity case. There indeed is a state which is separated by a gap from the continuum of states. So this is the continuum obtained in this solution. This gap does not collapse as the system size goes to infinity. So it seems as if we can safely follow, if the force is reasonably small, we can safely follow this curve here and observe the oscillations of the impurity momentum. And if you solve their problem in the small applied force case, indeed, you can see this. Of course, you can criticize this model from various points of view. And one of the arguments are that, of course, in the full-many-body problem, it is not obvious whether the bound state will survive. There is actually no gap separating the lower stage of the spectrum and the rest of the excitations. Actually, there is another existential problem with their proposal, which is there is actually no operational definition of a bound state of an impurity in the whole in a full-many-body setting because there is no particular order parameter associated with it. So there are questions to their model. But the most important question that we thought we should try to answer is about this region of the momentum space. In this region of the momentum space, if you solve their model, you'll find out that the gap separating this bound state and the continuum is exponentially suppressed, an expression very similar to the BCS expression. So the question is, if we apply a constant force and make the system evolve in this direction, what is the probability that the system will survive inside this ground state and not be excited into a wave packet of containing different momentum, which is appropriate for the Boltzmann description. And we actually managed to completely solve this problem. I'll show you the answer. So what is shown here is the amplitude. So suppose we apply a constant drag force to our system and k starting from k equals 0, from the momentum equal to 0. And what is shown here is the amplitude that the system stays in this bound state after it has traveled momentum k in the reciprocal space. And as you can see, the vacuum amplitude, the amplitude that it stays inside this bound state decays as e to the power minus k squared. There are some coefficients here, which you don't have to worry about. The important one is it decays as 1 minus k squared times logarithm. And the logarithm contains the drag force times the system size, which means no matter how small the drag force is, if the system size in the term a dynamic limit, the solution does not survive. So the oscillations in the light impurity case simply do not survive because the Boltzmann approach is applicable in the term a dynamic limit. What about the heavy impurity case? In the heavy impurity case, we still seem to be in agreement because they predict oscillations and we predict oscillations. However, I wouldn't say that this is the same type of oscillations. Their oscillations are an adiabatic effect due to the formation of some weird state at the bottom of the dispersion relation. Oscillations that we predict are much more robust. They result from a semi-classical Boltzmann picture, which actually can be extended to beyond the Boltzmann approximation. And therefore, we believe that the two types of oscillations are actually different. They have different phenomenology occur under different conditions, although they are superficially similar in a particular case of heavy impurity. And finally, I have to say that we don't know what happens in the case of equal masses, which is the integrable case. But the integrable case turns out to be quite pathological, as I tried to explain. So we don't know much about the evolution of the system in this case. And this is exactly the setting that is being investigated in Innsbruck. So I believe we need to do some more work on that. All right, so this is the summary. So I tried to explain how the incomplete momentum relaxation occurs in the one-dimensional system. This is due to the kinematic constraints on the outstates in the system, which allow for the existence of stages states with fixed momentum, the non-zero momentum of the impurity. What is important in our picture, the qualitative behavior of the system depends on whether the ratio of impurity host masses greater or less than 1. The case of equal masses is pathological. However, we checked that we can reproduce at least the final momentum of the impurity using a quantum Boltzmann equation where we perform an estimation of letter diagrams. And so there is a tool to study it in our limits. And there are strong indications that the Bloch oscillations predicted by Kangart-Inkheim are rather singular observations, which does not survive in the thermodynamic limit. However, in the heavy impurity case, other type of similar oscillations can be observed in classical and semi-classical kinetic theory. All right, I think I'll stop here. And thank you for your attention. A quick question. So first, on this plot for the limiting value of the momentum for the light impurity case, the momentum changes sign. Yes. It doesn't seem to saturate. Is it just a visual effect? Yes, it seems to be quite honest with you. Up to this point, up to the thermal momentum, one can derive an explicit expression for the final momentum. This region here, in this region here, you need to solve the equation by iterations on the computer. And you can put upper bound and lower bound which are shown by green lines here on the finite value of momentum. And the upper bound does not saturate as you go to infinity, but the lower bound does. We don't know the exact answer. But assuming that the blue curve is between the green ones, it does not saturate. But can it be larger than the initial momentum? Can it be larger than the? In magnitude. No, that's quite a reasonable question. I don't suppose so. It shouldn't be. Yeah, you're quite right. Yeah, so again, as I say, we don't know the analytic expression for the final momentum here. Probably it can't. It would violate. So if the masses are equal, that's this special case, how to distinguish the impurity from the host particles? If the masses are equal, how do we? OK, it's a different hyperfine state. It's a different hyperfine state of an atom. So it interacts with the light in a different way. So you can visually image it. What about the purely classical limit of your theory? Because the answer is evident. Heavy impurity will decay. The momentum of heavy impurity will continue to decay. The momentum of light impurity will decay with oscillations. But by classical model, I mean just elementary balls and a particle which hits these balls along with them. Yes, that's what you say, is that rather than having the tongs, gerardogas, with the Fermi momentum, let's assume we are having, well, the closest quantum situation to this is probably a Bosan-Stein condensate, where all the atoms are in the same state. Because in the purely classical case, you are quite right. What happens is that an impurity, if it has the same mass as the rest, it simply hits the nearest ball. And then what happens is the same as the Newton-Cradle experiment. It stops, and the rest are moving. You are quite right. But what is interesting about this problem is that the quantum mechanics of this problem makes a solution completely different from the classical one. Any questions? Three more, then we stop, OK? Your momentum considerations seem to me very sensitive to the use of a tongs gerardogas, so infinite. Can you say something, what happens beyond this? Right, this is partially correct. So we have never completed this sort of calculation, but the idea is the following. I'll try to hand-wavingly explain what happens in other cases, in the other case. So suppose we inserted an impurity into the system, and then there are two classes of processes. One is the bracing of the impurity by the environment, the formation of a polaroid. And the other one is emission of particles, and they are associated with different time scales. So the formation of a polaroid occurs at the time scale of the order of Fermi energy. Oh, that depends on the coupling constant, because that depends on the coupling constant. But after the polaroid is formed, it can still emit real particles. So we can try and think about the system in this setting as creating an in-state in which we have addressed polaroid with a renormalized mass in the middle of a continuum spectrum, which is the sound waves or phonons of the system, and try and solve the Boltzmann equation in this case. And then the kinematics will be very similar. But we've never actually completed anything like that. That's just the answer. Madam, I'm a bit intrigued about this bound state. And I want to ask, is there any connection between these states and Liftschitz local and quasi-local states, which you get in the phonon spectrum when you have one either heavy or light. I don't think so. It is much more like a BCS pair. Are these Liftschitz states somehow manifest themselves in your picture? No, you can't see these states in the Bertrand's solution in the equal masses. So one of the fundamental problems is that the way these states are constructed is the same as the Cooper solution in the BCS case. So there is an impurity, which attracts a whole. There is a kinematic constraint on the firmament, and blah, blah, blah, you get a bound state. The problem is that in the BCS case, the state can be operationally defined in model independent terms by the order parameter. So you can introduce another parameter, and the long range correlation or response function will tell you that this state is formed. Here, you can't actually define, refailed to define any order parameter which would indicate the existence of the state beyond the model that they use. So the very statement that the state exists is model dependent, that's the problem. Alex, perhaps, can produce some answer, but refailed. The last question, which I think you can start setting up already. Oh, no, there's coffee break, right? Okay. Do I understand correctly that in your picture, the entropy grows as a function of time, and if so, how exactly the heat is removed and why you are using the distribution function in zero-temperature when you are considering the motion of the empirical time? Right, so the answer to this is the entropy is removed. We are not talking about exactly a thermodynamic system. We have an impurity, which is a local perturbation, and what happens is when it kicks, so what happens is under processes where it kicks out a particle from under the pharmacy, and then this particle just goes away. It carries away all the thermodynamic information. And this is not a strong thermodynamic perturbation because we are talking about a thermodynamically large system and an impurity, which is nothing compared to the system. So essentially the entropy information is carried to the boundaries of the system by the sound waves. Yes, yeah, you're quite right. Yeah, you're... That's correct, we go to the collision integral rather than... Yeah, quite right, yeah. So I think we should call it a day now. Have coffee break and come back at 22.04. Thank you very much.