 Okay, I think maybe we should get started as people are coming in. So welcome back everyone. So we have many nice talks today and three talks in the morning in this US morning session. So the first speaker of the day is Paola Orogero and she's going to tell us about quantum GHS. Okay, thank you. So first of all, I would like to thank the organizers for the nice invitation. And so, okay. And so, yes, today I would like to talk about my recent work on quantum generalized aerodynamics, which is in collaboration with Pasquale Calabrese, Ben Duayon and Jerome Dubai. And is in this, our recent paper. But, so even if most of people I think in this, in this conference actually know aerodynamics better than I do probably, still I would like to start with some basics of aerodynamics just to recall some basic facts. And so the key assumption underlying any hydrodynamic theory is, is the one of separation of scales. And by that, what I mean is that usually want to assume separation of land scale, meaning the, the existence, the existence of a mesoscopic scale small L, which assume to be much smaller than the mesoscopic scale so that this, this land, the system can be seen locally as homogeneous, but still a much larger than the microscopic scale usually associated the same to the inter particle distance so that still contains a large number of particles. Also one assumes a time scale separation, which basically amounts to assuming a local relaxation. And so the, the existence of such intermediate scale is actually, is actually important because it allows us to replace the complex problem of dealing with many large number of interacting particle with the particles with the simpler problems, problem of dealing instead with the fluid, just in terms of few, few local properties. And in practice, what this mean is that what one is allowed to do under the assumption is, is to replace expectation values of observables in the exact state at a given time with expectation time expectation values on on a stationary state. But this stationary state will will is taken to to be a different one for any cell in space and time so basically it will be position and time dependent in general. So what are these local feature that we want to keep track of well they are related to conservation laws that in general for a generic system are usually the mass the momentum and the energy if it's conserved them. And the other assumption of other dynamics is that they take the form of continuity equation for the associated densities. If we think to this equation as in terms of expectation values in the in the sense that I was explained before, then these are exactly what people usually mean by conventional dynamic. But now, so let's move to to integrable model instead which, which are different in that they possess an in feed number of conservation laws, and therefore for such models, conventional dynamics is not expected to apply. So first of all, what is interesting about these models is that they are exactly solvable by means of better answer. But also let me say that I would say that the answers is not what I would call a flexible technique, meaning that soon, for example, as we, as we add some in homogeneity in the system and then the, the solvability of the model breaks down. And, and therefore, this is one of the reasons why I think the these words that have been mentioned many times already now in this conference by Kassel Vared et al. and Bertine et al. are quite remarkable because so they need allow to treat these one dimensional integrable system in a more versatile way. And they do that by using a hydrodynamic approach, which is nowadays called generalized hydrodynamics. So let's see what this is about. So as before, the starting point are these conservation laws which can be shown to take the form of continuity equations also in this case. Now we have a different number of them. Let me say also that we remind also that in integrable models, eigenstates are labelled by a set of quantum numbers called the rapidities which are related to the momentum of the so called quasi particles, meaning the stable excitation of these models. And we are interested in the thermodynamic limit in which basically these states can instead be described by by now a continuous function which can be taken to be this what is called the occupation function basically is the occupation in this rapidity space. And now, I mean we play the same game basically. So we want to look at the problem from a macroscopic from a macroscopic point of view meaning in terms of expectation values which now will be given again in the stationary state but which is now described by a gg instead of a Gibbs ensemble and still preserve this, this local this, this local behavior. And if we think in this term basically what happens that in terms of these of these expectation values, this infinite number of conservation laws of continuity equation can be recast in this equation for for the occupation function. And this is what what the GHD is about. So instead of going to the details of this Paula. Yes. Paul Giuseppe. Naive question. Usually in hydrodynamics you have Navier-Stokes equation. What kind of equation you will have here in this context? What. So this I didn't get the question so if you want to this is so you basically what you are asking if I understand correctly is a relation between the usual equation of hydrodynamics meaning the one so let's say for for the density of for the other velocity and this one. Correct. I mean, well, the Navier-Stokes is the, it's about the pillars of hydrodynamics in usual sense. So I'm asking in this context, what kind of equation you will have. Well, I mean, this is the equation which described hydrodynamics in this in this case. Okay, I'm not sure I understood the question. Well, Navier-Stokes is a non-linear equation for the velocity itself. So, yes, yeah, yeah, yeah, yeah, if you want to, if you want to, okay, maybe, maybe you can come back to this question later because for a specific class of states for example, I will show that this equation become an equation for the for a set of rapidities which are if you want to relate to the moment of the particle so maybe it's closer to what you have in mind. Okay. But okay, so maybe as I was saying, instead of going to the details of this equation, I prefer to give a more intuitive picture of probably also this with help in what Giuseppe was asking for for what free models are concerned because in this case, this equation actually become very easy to understand, meaning that when we talk about free models, then these rapidities are nothing but the momentum of the particles themselves. And this occupation function becomes the occupation function in space space, also known as a Wigner function. And therefore in this case, this, this equation of GCD is nothing but the evolution equation of the function, which is something that is a routine use in some classical methods since a long time. All right. And so the other in this very brief overview, the last thing that I want to talk about is a special class of states in GHD, which are the zero, what I will call the zero entropy states. So what are they? Basically, they are states whose occupation function is of this form. So it is one within a given contour called Fermi contour, and it is zero outside. So first of all, let me mention that the states actually appear quite naturally, meaning that for example if you think to ground states within these particles of bosons within homogeneous potential, they are of this form. And what is important, what is peculiar of these states is that if we look at them locally, within a given cell, let's say, then they look like split from each other, let me see. Okay, so they basically, they can be, they can be parameterized by just a set of Fermi points, okay, that we are this theta one, theta two, two. And in this case, we see that it's clear that because of this simple form of the occupation function, basically the equation for the evolution of this occupation function can be written in terms of a finite number of equation for those Fermi points. And so in this case, we end up in a finite dimensional GHD. What I mean is that usually one can think of this occupation function as an infinite number of equation one for each theta, but now we have a finite number of them. What is also important, and I will come back to this later in the talk, is that if we are in a special case where we only have one Fermi C, meaning just two Fermi points, okay, so this J just run from one to two. Then these are those states for which generalized aerodynamics actually reduce to standard aerodynamics. This can be easily shown. Okay, so this was meant to be just really a flash of what GHD is about. Let me just mention that after the two pioneering work that I mentioned before, so a lot of activity has been going on in this field with many contribution and application from many groups in the world. Of course it is just a partial list, but just to say that many people are contributing to the development of this theory. And with that, I want to pass to the main part of this talk, which is about quantum aerodynamics. So another question which arises at this point is how to incorporate quantum effects in GHD. And by that, what I mean is that generalized aerodynamics as any theory at the hydroscale is not enough, if you want to say this way, to describe quantum phenomena as for example correlations among different fluid cells. And this is exactly what we want to do here. So again, so let me be very precise about what is the particular program that we have in mind here, in order to avoid confusion. So I mean a very common and well-known procedure in literature is that of starting from classical equations, in this case will be nothing but hydrodynamics itself and quantize them by hand in a sense. What I mean by that is that, so this path was basically the path taken by Landau for example, by the way, was the first to first quantize a liquid in his development of the theory of superfluid helium. But after that actually the very same approach was was also was also adapted to other system, including Bose-Anson condensate, where quantum fluctuation are captured by Bovellibot theory, superconductors or liquid and general liquids described by Latinger liquid theory. And so in this sense what we have in mind is nothing new I would say. Let me also emphasize that, so here we are not interested in going to higher order hydrodynamics, meaning we are not interested in correcting GHD equations for the GHD equations, which is instead a different path which was recently pursued, for example, by Maurizio. He actually mentioned that also in his talk the other day when he was talking about this Moyalex function and similar ideas also appear, for example, in this paper by Dean Ledusal Majunde Lechel. But here, this is not what we have in mind, instead what we want to do here is to look at the propagation of linear sound waves on top of hydrodynamics and quantize them. In the same sense in which all these people did for another situation. And so when here we talk about quantum fluid, so we talk about quantum fluid in the same sense in which Latinger liquid is a quantum fluid, maybe this can be more familiar to many of you. Okay, but to be more specific, to understand better what I mean, let's go to, let's be specific and let's take a model, a specific model, which is in this case the Lieblinger model describing repulsively interacting bosons. And let's focus again on zero entropy states. So again, the plan is to look at fluctuation around the GHG background at zero entropy one in this case and quantize such fluctuation in a semi-classical fashion. Right, so if we start from a zero entropy states fluctuation basically can be seen as a deformation of this of this Fermi contour. Okay. And, and from, from a local point of view, what this means is that these are nothing but fluctuation of these of the Fermi points, which in the interacting case are associated to this western number to a certain number of rapidities. But what we do next is to switch to, so if you want this rapidities are not physical quantities, so we want to switch to physical momenta associated to the to the excitation of the model. And the main reason for doing that is that this contrary to this detail is PA satisfy an exact conservation laws together with this to the associated energies. And now if we were if starting from this equation we write an equation for the fluctuation delta PA we easily arrive to this equation. And if you look to this equation, what is describing is exactly sound waves linearly propagating sound waves. So it's exactly the equation we were looking for. At this point, the main, the second point is that we want to want to quantize this fluctuation. And in the, in the, in the language of sorry, in the language of quantum fields, what this means is that one user. Why one usually assume the existence of an existence of an action as whose minimum you are called as classical basically gives back them the way the equation and the whose expansion to second order instead gives the quantum fluctuations. So if we if we switch now to the Hamiltonian formalism, the main result of our work is to write down explicitly the Hamiltonian which does this job basically in fact this Hamiltonian together with together with the this commutation relation is reproduces exactly as a question of motion them, then the way the equation that was showing you before. So let me comment a bit about this Hamiltonian so first of all, not that this Hamiltonian depends explicitly on the state meaning it depends on the on the Fermi contour for this zero entropy states. But most importantly, not that this Hamiltonian is quadratic. And so what we are saying basically is that fluctuations quantum fluctuation on top of a GHD background are captured by a quadratic theory, and this should not sound very surprising if one think in terms of a Lattinger liquid theory. And in fact, what this Hamiltonian describes is exactly a Lattinger liquid, though, I mean a more complicated one meaning that we see that is explicitly time dependent, especially in homogeneous and in general multi component. So, let me also mention that what so what we did in this work was basically generalize our previous work with the Yanis Brun and the Jerome Dubai and where basically we carried out the same pro the very same program but in a specific case, if we want, which is the limit of our core bosons. Okay, so the same model, but the limit where the bosons become become our core, which is also known as the limit them. And also in the specific for specific protocol, which was a quench from seeing from a single well to another single well with different frequency. And, and this, I mean, the reason why I call this slide the quantum conventional dynamic is because basically, this is exactly the situation where GHD actually reduced to standard aerodynamics and so the problem that we face back then was to quantize in the very same sense, a standard aerodynamics. And, but the, I mean, what I wanted to underline is that also there, I mean, as expected, at this point, we went up in a theory which was nothing but the Lattinger liquid again time dependent and in homogeneous. Time with a single component. This is due to the fact that for this particular protocol we are always in the situation where only two Fermi points appear, which, as I mentioned, different times now is exactly the case where generalized aerodynamics reduced to standard aerodynamics. And so the last part, the last thing that I want to talk about is about numerical checks to so I want to show you that indeed this theory works. And so what we did basically what I show you here is a numerical check for the Lieblinger model and where I consider a particular protocol which is the following so we start in the in the in the ground state of a quartic potential and at equal zero we quench to a quadratic to our harmonic potential. So this is, I mean, this, this protocol, if you want, is interesting also from an experimental point of view because it's a simple way of modeling, for example, the currently running experiment as the or previous experiment as the quantum Newton cradle but but mainly because from this theoretical point of view, we see that if we look now at the evolution. This data show you here is basically a movie picture. If we look to the evolution of the occupation function, we see that in this protocol what happens that sometimes develops more than two Fermi points so we are in the true regime of GHD. And what I show you are basically the comparison between our theory and 10 10 dependent the energy. So this liner is just the density profile and so the gray lines are our merits while the theory is the is the orange curve. So this line is just the density. So what you see here in orange is just GHD instead of the new prediction are at the third and fourth line, which for different for different point in space show the density density correlations. So you see that. So, okay, many, many features are captured already at this, at this number of particles which that we can reach with the with the numerics, but also stronger finite size effect are visible. You have to take into account that the year so the numerics we can reach is up to 20 particles, which is quite far from the thermodynamic limit to where we expect our theory to work better. And if you want to this can be also seen as a strength of this GHD and theory in the sense that it works directly in the in the thermodynamic limit. But if you want to have a more reliable comparison, what you can do is to again to go to the to the top share the limit. Because in that limit to you can rely on on three fermion techniques for the numerics. And this is what we are we are doing. We're doing right now. And so basically, in this case, so the same is the same protocol just in this limit. So you see that you can one can look at the gain at the density density correlation. And, okay, here, there's just a small point is that you need to average over a small window because to to get rid of this free the oscillation but apart that the prediction of our theory perfectly with the with the numerics. In this case, we're also able to go beyond that and to give an analytic expression for for the entanglement entropy as a function of space and time. And, and we are also we are also proceeding with the other checks of other quantities in this case. All right, so and with that, I would like to conclude with this short summary. So, so I repeated after several talks that just the need to provide a more efficient way to describe interacting quantum particles in one day, but still it misses important quantum effects. And I hope I convinced you that GHD gives indeed a way to construct quantum fluctuation around GHD. But on the other hand, let me emphasize once again that this is not the end of the story meaning that there are far farther contribution which one can take into account by considering correction to the GHD equation for the evolution itself. And with that, I would like to thank you for the attention. Thanks a lot. Nice talk. So we have a few minutes for questions. I have a question. Yes. And it's about okay, even the summary. Okay, I read that there are GHD misses important quantum effects. So I'm wondering that in the past I always consider just generalized dynamics and I compute the time evolution of collection functions and entanglement entropies. So which are the quantum effects that I missed. So if you want, and so there are, there are, let me say in this way, at least this is my point of view. If you want, for example, also also what you did in your talk, for example, so you use indeed quite a GHD to compute the evolution of of your of your quasi part of your quasi particle to compute entanglement or whatever, but still you need some other tool to to say what are the correlation. So in my case, for example. So let me go back to the main slide, which may be so. So here, if you want to, if you from the point of view of GHD, the, the, the, the problem that we describe would just to say that just give these as initial condition, and that GHD would tell us how these initial conditions. So as far as I can understand that the initial condition is the Fermi C. So we know that is described by conformal theory that are correction that the case is power law and so on and so forth. Exactly, what I'm saying is that. So what is the, what is that I mean, what is more than that. What I'm saying is that what, what, what I would call GHD or what I would say that GHD is enough to describe is this initial condition, okay, which, which has, which has no fluctuation and nothing like that. Plus, and the GHD equation tells you how this initial condition evolves. The main thing is that now on top of that you can consider fluctuation, but not only this fluctuation, you also have to give an Hamiltonian or an action if you want, which makes them become quantum. Okay, and this is exactly what we give here. If you want, if you want, and I think that what you what what the way you would say this, and I would perfectly agree with you if you want, is that what we are doing is to transport in time initial correlations. And this I agree, but I don't agree on the fact that I don't agree on the fact that this is not what I would call a quantum theory. I was just wondering what is what I could get more with a quantum generalized aerodynamics instead of generalized aerodynamics. Yes, okay, so for example, how would you, for example, you said that in your in your talk that your quasi particle picture which already I find that I mean, already there I would not say this just a general dynamic would say this is a general dynamics plus complemented with the quasi particle picture. I mean, with. Okay, but, but apart this, you also say that, for example, you cannot, you cannot start from, from any, from any state that that you want, you know, because of the state too much entangled the chair, etc. What I'm saying is that there are correlations that that you can compute to just be in a very simple way by by the way, just because you know you know this Hamiltonian. So once you know this Hamiltonian you have a mapping for example, for example, in the free case in the unit of the tonsure are though. What happens is that you this Hamiltonian also also requires conformal invariance and so there you can you can use all the machinery to compute in a very simple, in a very simple way. I don't know any in particle density matrix correlation function or any correlation function basically computed entanglement as I showed. So I think that you do get something more in this sense. What you do you do not get is a and I stress, I think during the talk is that you are what we are not correcting that the GHD equation, but also, for example, not that these correlations that, for example, the density density correlation in the in the tonsure are though they are ordered. They contribute to they give it a correction if you, in my perspective, a quantum correction which are order order h bar square. And as you mentioned, and the other day in the non-interacting case you expect the first correction to the to the GHD equation to be to be proportional to the third derivative, right? So Okay, it's only for GHD. Then there are correlation of the state that I don't know. Exactly. No, no, exactly. No, no, exactly what I'm saying, but, but, but to say that Okay, okay, fine. I don't want to monopolize the discussions. Oh, yeah, no, no. Okay, we can discuss a little bit later, but to say that probably many situations I would expect this kind of correlation coming from the from the initial state if you want to be more important than the correction to the dynamics itself in GHD. I mean, I don't know, but So, okay, we're like a couple minutes behind, but I see we have one more question. Hopefully we can make it quick. Yeah. Should I go? Yes, yes. Okay, so just to remark on what Giuseppe asked, so the question you're writing is the Euler equation, right? The Navier-Stokes would be by adding the viscosities and this is what's required to add the diffusion. That's what Okay, okay. Sorry. I was not sure. Navier-Stokes and GHD with diffusion. I mean, it's not the moment to talk about. So for me, the question is very simple. So you're in this slide you show entanglement entropy. Yes. So actually what is, how do you define here the density entanglement? This would be the entanglement to be of a small interval at position X. This is, yes, you know, so this is this. So, if you, if you see here in each of these of this frame, the time is fixed, you see, and this is a function of X. So, yes, it's the entropy entanglement entropy from zero to X and the rest. Okay. But this doesn't grow to, I mean, it seems to From minus infinity to X. From minus infinity to X. But it remains of order one, right? You don't generate, you don't go to a No, well, no, in this case, you don't expect because basically you are starting from a ground state. So you start from the ground state of So you're starting from the ground state in the double well. And then you are quenching just the potential. So in this case, you don't expect to generate Okay, but this, so this you cannot get from quasi particle picture. I mean, it's something that you need. No, I don't think so, because I mean, I mean, the trivial answer would be, I mean, since it doesn't grow linearly, I don't think you can get picture. I'm not sure that is everything that is not a linear cannot be captured by what you are to a picture, I mean, a particular after this idea of working in a Carved space and whatever, but I don't think so. All right. Move on to the start though, since we have few minutes behind then we can leave question for the discussion session. So let's thank Paolo again.