 tell us about plasma waves in layered superconductors. OK. Thank you, Premie, and thanks to organizers for inviting me to Trieste. I'm very glad to be back here in person after a few years. Today, we'll talk to tell you a story about plasma waves in Liar of the Metals and Superconductor. Actually, this is an old story in a sense. So I will try to make a long introduction to explain what was the puzzle that we found reasoning to that interest of us and try to address again this problem. So I really want to start with something everybody knows because we started in the undergraduate school, which is essentially the way in which we describe the propagation of electromagnetic waves in a metal. OK. So usually we start in standard textbook from Maxwell's equation for electric and magnetic potential. The way matters enters via the current, in particular the current of electrons, which is induced by the electric field. And if you use a very simple Drude model for electrons, we can solve out this equation in a very simple way, especially in the regime where the frequency is much larger than the scattering rate of the electron, which means that the electrons at this frequency are like dissipation less. OK. So you just plug everything in and you discover that you can decouple the transverse and longitudinal components of the electric field. This is for an isotropic system. And then in particular, you find out that there is an equation for the transverse component. Full solution is a wave, which propagates in the material only when the frequency is larger than the plasma frequency, because this is the regime where the electric constant becomes a positive number. And then you can find the real solution to the wave equation. On the other hand, for the longitudinal component, you can even have a solution for zero longitudinal, zero component. So now it's a self-sustained mode of the system. And this happens when the electric function goes to zero. And this is exactly the plasma frequency. OK. So what we essentially learned in standard textbook is that this is the electromagnetic wave propagating the matter. This is the transverse plasma polariton. It is often called. And then for k going to zero, the limit of the plasma polariton dispersion is the energy where you can have a longitudinal plasma in the system. The small dispersion here is just due to finite compressibility of the electron gas. And here is just a sketch of the quasi-particle continuum in a simple Fermi liquid. Of course, in an interacting system, you can also have a lot of additional state here, even for zero momentum. But this is, let's say, the paradigmatic case. OK. What is the origin of the longitudinal mode? Let me just rephrase it in a slightly different way. We said that this is a longitudinal solution. So the oscillation of the electric field is in the same direction of the electric field itself. OK. And this happens just because there is a coulomb interaction between a charge in a system. So if you try imagine to have a plasma model, you just shift the electron respect to the ions. You create a charge imbalance. This charge imbalance creates an electric field in the opposite direction and so on and so forth. This is why the longitudinal oscillation of the electric field are essentially connected to oscillations of the density. OK. And this is why, as condensed matter theories, we usually study the plasma just looking at not the solution of Maxwell equation, but the properties of the density fluctuation in the metal. So in other words, the way in which compute the longitudinal electric function is via the properties of the density correlation function in the presence of a long range coulomb interaction. So you have the standard relation between epsilon and k and k, sorry, which means that, essentially, the zero of the electric function are the poles of the density response. So somehow longitudinal plasma waves appears in conventional system. And this is how we compute usually as the poles of the density response, of course, dressed by the coulomb interaction. What does it mean that a very simple standard way to get it is to take a model of electrons interacting via the coulomb interaction, and then you dress up the susceptibility via the long range coulomb. So k and zero is the linear function. It's just the spectrum of the metal without a long range coulomb interaction. And then out of this, you can compute the plasma. Notice that this is a model of electrons interacting via coulomb potential. So the only way a electromagnetic field appears is via scalar potential. Essentially, there is no sign here of the gauge field, the transverse gauge field, or the longitudinal one. And the reason is that the scalar field is the one that gives you the longitudinal component of the electric field. So if you're only interested to study the longitudinal filtration of the longitudinal component of the electric field, a description of the scalar potential, or another way with the coulomb interaction between the electrons, it's enough. And this is why, then, as to do again with the fact that in an isotropic system, there is a complete decoupling between longitudinal and transverse degrees of freedom. So I can study the plasma polarity and the plasma wave into completely different sectors that do not talk to each other. And this is, again, just a sketch of the very standard calculation. This is the result that you get for a normal metal. You see to get real dispersion of the plasma, you have to take the linear function to expand for large k over omega. And then you get here this typical omega, typical dispersion of the plasma and the k square in the numerator, which is just the signature of gauge invariance of the theory. So the density response for finite omega and k going to 0 is to be 0. OK, so far so good. So what happens in the superconductor? In the superconductor, we know that we have a breaking of a gauge symmetry. So we have an order parameter with an amplitude interface. So the amplitude of the order parameter is what appears as a gap in the spectrum of the single particle excitations, which means that if the quasi-particle continuum, even for zero momentum, is pushed above twice the delta. But every single current in an energy scale, which is of the order of terrors, so one terrors is 4 mega electron volt, which means energy scale far smaller than the typical scale of the plasma, which is electron volt in standard metals. So there are these modifications. But actually, who cares? At the energy of the plasma, we don't expect to have anything new to the appearance of superconductivity. However, there is something very nice from the point of view of the description of the plasma that once you are in the superconducting state, it's simply simple somehow to get the dispersion of the plasma, just looking at fluctuation of the phase of the superconducting order parameter with respect to the mean field solution. So we break the symmetry, theta 0 is finite. The phase is rigid. This is what happens in the superconductor. The phase is rigid, it also means that we can measure fluctuation of the phase that we cannot do in a metal. And this fluctuation described in a very beautiful way, a simple way, if you want, technically, the plasma. Why this? Let me go again around the very standard description of that. We can make a fluctuation of the phase in space on time. In space is what we get from classical Gisbel and Dow theory. You have a Gaussian mode, which means that the energy cost to create phase fluctuation goes to 0 when momentum goes to 0. And so you have the gradient term in the phase, and the coefficient is the phase stiffness. So the ratio between superfluid density and mass of electron. So this is what we were asking to Alex before, so why we could not recollect the typical cost of making a phase gradient. As we will see later on, this phase gradient can be promoted to a cosine term if you want to take into account the discrete structure of the system, but this is more or less what happens. Now, if you want to do dynamics, we need to put time dependence to get frequency. How we do that? We need to make a phase twist of the phase in time. What is the coefficient of the phase twist is the charge compressibility, just because charge and phase are conjugated variables. Or if you want, because the time derivative of the phase is connected to the chemical potential, so derivative respect to the chemical potential are the charge compressibility. So this is what you get. And this is why in a superfluid elium, which is neutral, you get the sound mode, okay? This is the sound that way. But superconductor are charged, okay? So we cannot forget that the charge compressibility is stressed by long range coulomb interaction. And then actually the short range charge compressibility has to be replaced by the long range one, which goes like the inverse of the coulomb potential for K going to zero. And then you just make it this exercise. You take this model, you write in momentum space. You see gradient theta square is omega square. Grad theta in space is K square. But now the coefficient of the omega square goes like K square because of a long range coulomb. Then you have the K square overall, and then here you have omega square minus a finite value. And this is the way you get the plasma in the superconductor. Notice that the Gosson theorem is not violated in any way. Gosson theorem just tells you that when the phase is uniform in space, energy goes to zero. And this still goes to zero because you have K square. Gosson theorem doesn't tell you anything about dynamics. And then the dynamics of the phase in the superconductor is pushed into very large energy. And this is, again, the plasma energy in the sense that in usual superconductor stiffness for very low temperatures of the order, the density of the electron. So we get, again, the same frequency that we got in the metal. But then we also understand there is some additional degrees of freedom if you want in the superconductor because we can play with the possibility to have a very small stiffness, at least in some directions or in some material. And this is the typical case in Liar-superconductor. A Liar-superconductor whose all-marked example is given by cooperates is a system where I have very good conducting planes, which also made a very good, very large stiffness to make the phase gradient of the phase in the plane. But the planes are very weakly coupled. So the stiffness connected to the fluctuation of the phase between planes, neighboring planes, can be very small. So this energy scale becomes much smaller than this other one. You can even be in the situation where this out-of-plane plasma becomes smaller than the optical gap. And this means that we have the possibility in Liar-superconductor to have a low-energy mode, which is undamped, which is below the gap, doesn't mix to the particle or continuum, and then it measures only in the superconductive state. These are reflectivity measurements in cooperates. Notice this was already discussed many years ago. And then you see reflectivity above Tc. 45 Kb is flat. It's also very small number. These are bad metals in this direction. Then you start decreasing the temperature. You see a plasma age, which appears is better seen here. And as you lower the temperature, the plasma age is always more well-defined. The reflectivity goes to 1. So the plasma age is, again, what is the penetration of the transverse wave, which only occurs above the plasma energy. So this just shows you that there is a plasma, which is soft, 0.5 terrors, and only appears below Tc. So now you are in this situation. We have these two energy scales, large and small. And then you can ask yourself, how now the plasma, the longitudinal plasma or the transverse plasma polaritone, propagate for finite momentum? How do we go from these two energy scales to the general momentum dependence? And of course, this is a problem which is relevant for many spectroscopic and probes. I will discuss this at the end. Now, again, people asking themselves this question many years ago. And then the first question, the first answer is, OK, let's do what we also can do in a metal. OK, let's imagine that we are interested in what happens near to the large energy scale, the in-plane plasma. And then we just make the same game as before. We take electrons interacting the Coulomb. I do RPA over the electronic susceptibility. The only thing I have to do to account is the anisotropy of the Linder function, because now transport in the plane and transport out of the plane will be anisotropic. Then we plug in the typical expansion of the Linder function. What we get is an extension of the plasma dispersion for the isotropic system, where you see that now the in-plane and out-of-plane plasma frequency are somehow weighted with the ratio between in-plane momentum and total momentum. OK, and this was discussed in the context of metal. We saw discussed in the context of superconductor. And actually, this kind of expansion, and even, let's say, the generalization in the case of taking into account the full discretization of, for example, the C-direction, they work pretty well, for example, to describe the experiment by resonant inelastic x-ray scattering, which are experiments which are focused on large energy per second near to the large plasma energy and for varying momentum. So you see here, this one L equal to 2 correspond if you want Kc equal to 0. So if you put Kc equal to 0, this is 1. And then you have this. You see this is not very evident because this flat dispersion. If you said you take a Kc, which is finite, you can have acoustic-like dispersion of the plasma, and this is what is measured. So here's the question, yes. Yours asks, are you making a distinction between transverse and longitudinal plasma? Yeah, I am doing it. Actually, what is this expression here is actually the expression for the longitudinal plasma. Thanks for the question, Pisa. We'll come back to this later. And one can also write an analogous for the transverse, if you want, in the same approximation. But this is the longitudinal because this is what people measure with a density response. So in resonant inelastic x-ray scattering, measure the density response of the electron. So it's correct to take care about the density, the longitudinal plasma in the system. Now, this expression, which works pretty well, also compared to the experiment of large momentum, has a very peculiar behavior. If now you fix the angle, if you want, between all the wave vectors respect to the c-axis, you go to do momentum goes to 0, you have a continuous or possible solution. This is just a function which is non-analytic in K going to 0. So essentially for K going to 0, you can have any value of the plasma between the lower, the lower plasma and the large plasma. But this is a contradiction with Maxwell equation. Why it's a contradiction? Let's take, again, our two-faraday, low and per-low, and let's consider what happens in the anisotropic system. As I said, the matter's inter-hydration between current and electric field. In the isotropic case, the conductivity is just the diagonal matrix with a coefficient which is the stiffness, which is the same in any direction. Now we are saying that the stiffness in plane is different from the stiffness out of plane. Or if you want conductivity in the plane, it's different from the conductivity out of plane. And then the current in general is not parallel to the electric field for any direction of the electric field, while it is in the case of the isotropic system. And now what's the problem? If you want to solve now on per-lows for momentum going to 0, this time disappear. Just this is cross H is just K cross H. And then the only possible solution that we have of an per-low is that J has to be parallel to E, which means that the only possible solution for K equal to 0 is either the out-of-plane plasma or the in-plane plasma. There cannot be a continuous solution for gain going to 0. So somehow, the RPA solution that gives you a continuous or possible value of the plasma frequency cannot be correct for momentum going to 0. Yes? Can I just ask a further question? Where did the distance parallel to 0? I missed that. No, here it doesn't appear because essentially I'm when I'm writing this one, sorry, when I'm writing this expression, I'm making a long wavelength in any case. So this is the kc, much smaller 1 over d, and kib, much smaller 1 over a. But of course, this can be extended. As I said, this plot here actually has to do with the full periodic solution. Taking into account distance between that, yes. Yeah? What's the full? Yeah. What if you take the full formula for chi? And will you see from the numerator that residue will vanish except for the full formula? For chi, what do you mean? Full formula? For sensitivity. For the dress or for the bear? For the dress. For the bear. OK. You said that the red cannot be continued. Yeah. If they find a continuum of solution denominator. Yes. What happens with the revenue of this? I know, the numerator is not helping. I mean, the numerator is only giving you the spectral weight where you have the continuum. So the problem here is that you're only dealing with the Coulomb potential and chi 0 and nothing else. So you cannot get any other solution which is not this one in this model. So at the moment, this calculation cannot give you anything else than this, which is actually correct for large momentum. But it looks to be wrong for small momentum. So the residue will still vanish for the k equal to 0. Yes, yes, sorry. Yes, yes, sorry. And this is also why you see here, you have a much lower spectral weight than here, essentially. OK, this is the reason why. OK, so sorry, didn't get the question. Sorry. Yeah, so this is gauge invariant, if you want, in that respect, but it's not correct. OK, why then we expect that probably RPA is wrong? Because as I told you before, RPA is some approximation which relies on the fact we only account for the fact of the Coulomb potential. Or if you want, the scalar potential. So somehow only longitudinal degrees of freedom are included. But when the current is not parallel to E, in general, you can have a mix in between longitudinal and transverse degrees of freedom that is not there in the RPA solution. But then people also notice this. And there is a full literature of community studying the behavior of the propagation of wave near to the soft out of plane plasma. And this was motivated especially by experiment in the last year with very strong terrarium pulses. Because with the terrarium pulses, you can try to excite modes, this low energy plasma, or something else I will tell you before. And actually, the description usually goes even beyond the long wave limit. So usually people start from just as an alike model. Again, this is just as an alike model for the phase difference between planes. If you take, expand the cosine, the league in turn would be somehow a gradient out of plane. So you again recognize that this is the sickness. And then how the electromagnetic field is included in this description in the full way. Electromagnetic, you do the minimal coupling substitution. So the out of plane phase gradient couples to the gauge field, the z component. Here also the distance between plane answers. Then you write down a maximum equation for the internal component of the electric magnetic field. You make a given set of approximation. That are not always so transparent somehow in the way they appear. And then you end up with an equation for this gauging variant variable. And you see the sign here is what you get, essentially the current need to get. We should derive the cosine term in the X-Y model. And then you find out that if you take the linearized version of this, so you replace sign with psi, you get a dispersion for your plasma. This would be the transverse plasma. And this dispersion start from the out of plane plasma. You can have a dependence on the in plane out of plane momentum. And this is a perfectly analytical solution for K going to zero. But you always end up with the low plasma. Okay, there is no way here you can end up with upper plasma, which is also somehow reasonable because as I said, it is an approximate solution. So this is the point where we started to struggle. We are essentially, we had the two communities working hard for many time using two completely different description of the plasma. And these two expressions do not match each other. And you don't know how to go from one to the other. But this one works very well for high momentum. This one works very well in this region of energy and momentum. And so somehow we want to understand what is the full dispersion of the electromagnetic waves in this Li-Arab system. And how do really the limiting expression present the liters can be recovered. So what is really missing one description description or in the other? And most important, as I just mentioned all this experiment with terrestrial spectroscopy needs to include the non-linear effect for the phase degrees of freedom. So instead of the gradient term of the phase I want to have the cosine. Okay, this is what I mean with non-linear. So I would like to have a model which is essentially an extension of the XY model including non-linearity with the correct dispersion in the linear regime and play with it to study experiment in terrestrial G. And finally, of course, discussing eventually new experiment with the next generation probes in intermediate energy and momentum range. Okay, I'll try to tell you now what we did in practice. I said at the beginning the description of the plasma with the phase degrees of freedom is very simple. So let's do this again. So we start with our action for the phase. Let me start again from the isotropic phase. Instead of varying, so the matter here is described via the superconducting phase. It only enters in this way, okay? And then instead of adding external interaction between electrons, I add explicitly the electromagnetic field which is the internal electromagnetic field. So fluctuation of electrons in the system generate electromagnetic field, okay? And these I can include via the minimal coupling substitution and then I can take right down the typical sound the electromagnetic action for the scalar potential and the gauge field. Now, look here, the minimal coupling substitution tells you that the gradient of the phase is coupled to A. But this also means that when J is a constant is as a tropic space, the phase only couples to the longitudinal component of the electric field which is already written in Schiffer book, okay? Which means that if I choose, for example, Coulomb gauge, the only effect which is important is the one for the scalar potential. So I can integrate out the scalar potential in this picture and then what I get the electromagnetic action for the scalar potential is what grad five square. I integrate it out and I get the dressing of the compressibility as I did in the previous case adding Coulomb interaction, okay? So again, why playing with interacting electron which means electron in the presence of bearing and Coulomb or playing only with the scalar potential is perfectly equivalent. In the tradition, in the literate people prefer to work in this way because usually you can take this action, you can decouple that but just don't know which field both term and then you write out this way. But this is completely equivalent to this. So bottom line in the isotropic case only the coupling of the phase to the scalar potential is important. But if the system is an isotropic so the syphilis in the plane and auto plane are different, you see that now this gradient term, this is the gradient in the plane, this is the gradient out of plane do not allow me to decouple the phase degrees of freedom for the transverse component of the gauge field which means that in the isotropic system the mixing of longitudinal transverse component that they already noticed at the level of max elevation appears at the level of the behavior of the phase degrees of freedom in the way that I have to retain the couple of the phase to the, essentially to the magnetic field. I have to include the retardation effect of the magnetic field unless I'm some regime where I can make it. So this was just the main point if you want then we worked out the problem. I will not give you full details but the idea actually once that you understand this calculation are extremely simple and you can do it yourself in any few steps. So you write the action with the phase the scalar potential, Hulon potential, electromagnetic action for the electromagnetic field. This is another tricky point. You have to choose the right variables to make the problem simple and the right variables are the gauge invariant phase or the currents if you want, okay? You can introduce this variable you can take the gauge phi equal to zero in the superconducting state somehow the gauge fixing is much easier than in the normal state. And then you just write down what is the action for your, these are current fluctuation if you want in the plane and out of plane. Let me now call with let's say let's fix the momentum in such a way that the in-plane component is only along A. So you see here the fluctuation of the current in the plane just described the in-plane plasma. The fluctuation of the phase along C only described the out-of-plane plasma but for finite momentum, so for momentum sorry for a generic momentum which has both C and A component there is a coupling between the two. And this is somehow what was neglected in a before if you want. And then this means that if you want to find the solution you don't have to the couple second order equation for the in-plane out-of-plane plasma but you have a quartic equation in omega you just take the zero of the determinant you get the plasma mode. And this is simple it's analytical you don't have to do it numerically, okay? And then you find out a generic expression for the plasma mode at any momentum which has the future we want you see really the mixing between the two plasma you see a function which is analytical for K going to zero and okay let me just add that here I'm working at zero infinite compressibility so the plasma will have no dispersion you can add it to just slightly more complicated but it's still analytical. Okay let's look at the solution how do they look like? Yeah. If you want to do it a diagrammatic you have to take and talk out the density current correlation function. No this is has to do with the fact that you I mean usually when let's say you can write any density density on the current correlation function the superconductor and both of them will have you know diagrams with the normal and the normal solution, okay? Here what I'm telling you that you also have to take into account of this. Coupling between density and current. But if you start with how do we do it to power density to density somehow this should require a full time problem. Yeah I mean if you want you have to dress the density density also via the density that they're all J. Okay. How do you get it? How do you get it? You start with higher or lower. Yeah. And at some point you do try to engage. Yeah. So the question is what's the diagrammatic density to power J? What is this element? Because somehow we have to include the effect the propagator. Okay let me see how we can tell you this way. So you have to include also the propagator of the transfer is a lot of magnetic field not only the Coulomb potential. This is what you have to do. And then this guy is fine at the that's the point even if you go into zero because you have an isotropy. And this is so you can do actually everything at RPA level. And this is what we did also in the normal in the metal in the case of the metal but the key point here that you have taken to account this that you usually don't. Okay. So how does dispersion look like? So in this plot I'm just choosing a finite angle for the propagation of the light. And the color code is such that the line which is totally blue is transfers and the line which is totally red is longitudinal. So here is the dispersion. The nature or the polarization of the solution has to be computed afterwards. But if you want the color code already includes the polarization of the solution. So I have a lower energy solution which always goes to the lower energy plasma. So whatever is the value of the angle I always end up here. I have an upper solution omega plus which always end up for K going to zero to the upper plasma whatever is the value of angle. You see the dash of the line here are instead the standard RPA solution the one I showed you before which is as you can see as different value for K going to zero depending on the angle. And the blue line is the transfers if you want equivalent to this longitudinal plasma. And you see that in the RPA actually you didn't have the matching of the two for K going to zero. Okay. So what is our solution looks like for a very small moment you see that the mixing of the color means the solution have a mix of the longitudinal transfers character. When momentum increases this solution matches with the value of the RPA so it becomes a longitudinal mode and the upper solution becomes a purely transfers mode. So the point is that these are if you want the fact that for K equals to zero the solution the omega the minus solution is always polarized along C and the plus solution is always polarized in the plane. But if the momentum is finite it means that this guy here has both the transfers and longitudinal component. And this line here has both the longitudinal transfers component. So the solution K equal to zero have a mixed character. When momentum increases somehow the polarization makes a rotation the minus solution becomes a purely longitudinal mode the plus solution becomes a purely transfers mode. What is the scale which is set the crossover the scale is just the anisotropy of the stiffness. And since usually this one is much larger than the other one he essentially is the scale is set by the upper plasma. The upper plasma. So if you look at your system here you don't see the mixing and RPA works pretty well. But if you want to study the problem here you have to take into account the mixing. And actually this solution that I showed you before which is usually put in the literature of the terrarium spectroscopy is just an approximation of this omega minus branch for a very small momenta. So here again the description is very good. When momenta exceed the critical value of the momentum this approximate solution fails. Why? What is missing again this description and which by the way could include very easily no linear effect. This is again easily seen here. Let me rewrite this guy here now in real space. So omega square is time derivative K square is gradient in out of plane or in plane. And then the mixing here which is KK becomes mixed in the derivative in A and C direction. You see if I want to promote this model to a no linear model I just have to take the master which is omega C square for example and get it to a cosine. This is where I go from linear to non-linear. And then I can write down two couple equation of motion for these two variables. Again, writing the equation of motion is like solving the Maxwell, the determinant in the linear model. And then you see here where is the approximation that people use. If you're walking the frequency which are of the order of the auto plane plasma frequency which is much smaller than the in-plane plasma frequency you see here this time the derivative term can be neglected compared to the other one. So I just take it out. I can derive the fluctuation. I can express the in-plane fluctuation in terms of the auto plane. I replace it here and then I get to the question that people usually quote in the literature in the context of terrors. So the real approximation which is behind this is essentially the following. I'm looking at the problem at very low energy near to the soft plasma. At this energy scale I can neglect the time dependence of the current induced in the plane due to the coupling to the electric field. And this is the real approximation which is behind this. Okay. How much time do I have still? Perfect. Let me give you a hint. Okay. Perfect. Let me tell you about application. Okay. So here the most difficult things is to explain the experiment. I'll try to do my best just to give you a few ideas. Actually, the first was also where our interest to this problem came in. There are several experiments that are done with the use of strong terrors like pulses. Strong pulses means that you can somehow measure response in the current which is beyond the linear response. So linear response is J proportional to E. No linear response is J proportional to E to the cube. Okay. Which kind of physics you can see? For example, you can see the generation of higher harmonics. Typical experiment done in several superconductor. You come with the pulse which has a central frequency omega p in the terrors which is very strong. And what you measure in the transmitted or reflected the component of the field is a component oscillating at three times then coming frequency. So this three omega oscillation is generated inside the sample. So you come with omega, you get a three omega out of that. And this is easy to be understood with the non-linear spectroscopy. These are experiments for example, what about the linear thing D over DT term associated with all the parameters diffusion? In which term, Pierce, can you? No, Pierce is asking a question, but he didn't. Yeah, no. So in the classic work on time-dependent Landau-Ginsburg of Abraham's and Sneter, they have linear in D by DT terms in the action. Yeah. Okay, this is a diffusion term, but this is present above TC. So below TC, you get rid of dissipation. So that's the idea. Okay. At least for the soft plasma because I said the soft plasma is below two delta. This is the beauty of the superconductor because in the superconductor, you can get a mode that which is low in energies and that. Okay, so these are really in the metal, indeed I will come to that back to this in a moment. The story is slightly more complicated because you have to take an account of also tell me. Okay. Yeah, but I, if I recall those terms are still present in the superconductor as well. So if they were there, would they affect you? You can get it. I mean, if you have a lot of impurities, you can have dissipation for finite frequency, you can have dissipative term. I agree. So even an, okay. So maybe what you're asking, even in this, this, this equation. How would it affect you with that? Yeah, they also add a dumping term. You're right. Okay, but this is, so it's three elements. I mean, in the sense that if you have a dissipation, you always have a bit of dissipation channel for the mode. You will have a linear in term here. But I will say the main focus here was to understand how you go from couple equation to the couple ones. Of course you are right. If you have dissipation, you will have additional linear in terms. Sorry, I didn't get the question. Thanks. Okay, thank you. Okay, so this is an experiment for reflectivity along this axis. So you come with a very strong terrors field. And then you see when the linear response is a usual plasmage and linear response, which means very strong terrors field. And you see that apart from the plasmage, we have this bump here, which is three omega. So you have generational third harmonic. Now, how you can explain this kind of physics? For example, here it is very important to describe the plasmon, taking into account the linearity. Why? Because the non-linearity, the cosine term somehow naturally gives you no linear coupling between the face and the gauge field. In the sense that if phi now is the phase gradient out of plane, you should explain this guy here. You have a term which is a square phi square. And then you can get a non-linear current which is somehow proportional to A to the cube. And the decay, the non-linear kernel will be proportional to the phase-phase propagator. In physical words, this is a system where two photons are exciting two plasma waves, okay? And this can be done in a resonant way when the frequency of the incoming photon is near to the frequency of the plasma. And this is what people have observed experimentally. This is the problem that actually we started some year, one year before. And actually the description again is very simple. If the field is only propagating in the plane but you start to find the momentum, then you need to have a description of the plasma which accounts both for non-linear effect and for the correct dispersion. And these are the kind of experiment where one would like to look at this. This is a more recent experiment that gave from the group of Andrei Cavalleri. Actually, it has been around for a while and it's just been published now. So again, the experiment is complicated. Let me go to the main message. So you come with the mid-infrared pump and then you excite a phonon, okay? So there is an infareductive phonon that is excited by the terrorist field. Now, this phonon, the idea is that this phonon can decay in two plasmon. So this is a bilayer system. So you see instead of a single layer, you have two. So you have two soft plasmon which are this line here. And then the idea is that when the phonon decay in the two plasmon, somehow, and the signature of this is in a very complicated measurement of second harmonic generation of the probe. But then the idea is that somehow you excite a phonon, this phonon decays in two plasmon and you can synchronize this plasmon even above TC. So this has to do with the idea of field induced superconductivity. How can it be that you pump a system with the strong terrorist field and then you get superconductivity even above TC. And then the bottom line is that you do that because above TC you have a soup of phase fluctuation. So the system would be paired, but it's not phase coherent. But if you come with something that makes your face coherent, you synchronize the phase, you get superconductivity. And then in all this story, actually you have to do calculation in some way. Of course, they already did some, but they had to resort to a very pharmacological model. But actually you can do that. You just needed to couple the phonon to your degrees of freedom for the plasmon. And of course in the moment where you have a very analytical way to describe the plasmon, everything becomes easier to see and then eventually you can really try to answer the question about the relevance of this mechanism is for field induced superconductivity. These are experimented with the near field spectroscopy. This is again a complicated story, but the idea is that you see there's no signal. You make the contrast between a superconductor which is lasco and gold. When you go below, you see there is the contrast in the superconductor this become huge, which is this is no signal going up. And this can be understood with the fact that there is a soft out of plane plasmon emerging above below to see. But the interesting is the calculation with the Maxwell equation. So in the linear regime, give you a signal which is one order of magnitude smaller than what is measured. And then here is another point where no linearity can be important. And then of course there is a density response. That's the response in what you probe with the ills. So in a electronic spectroscopy in elastic electronic spectroscopy or with the ricks. And as I told you, this mode, the lower and the upper have a mix of the longitudinal and transverse character for intermediate momentum. This means they both appear in the density response. Of course, there is always the K square overall. And then if you can go somehow to this intermediate regime that requires high energy, high momentum resolution and our colleagues are trying to get this using ills and along with transmission electron microscopy, you can eventually manage to see two modes appearing in the spectrum of ills. And this will also be true in the metal. So the problem in the metal, if you really want to do the correct calculation of the density response, you have to put damping coming from quasi-particle excitation. And we are just working on this now. I say the description in the metal is largely more complicated from the formal point of view, but the bottom line is always the same. You have longitudinal transverse missing and the mode recovery purely longitudinal transverse character for large momentum. Okay, this is my take on message. Essentially, you have plasma waves in the Liard system. In general, they mix the longitudinal transverse character. Notice that the crossover energy momentum depends on the light velocity. When C goes to infinity, this is zero. So this is why also these correctional calls sometimes relativistic. Okay, so somehow if C is really infinite, you don't have anything purely longitudinal purely transverse. But unless you are along the main axis, you have the mixing, for very large momentum, anyhow, pure mode of recovery. This regime is what is relevant for terrestrial spectroscopy, especially the one that is really focused in the low energy terrestrial regime. In the high energy regime, well, it's relevant for ricks and heels in the way energy momentum resolution is there today. Actually, RPA was well. So if you're just focused on this experiment, you don't have to worry about all that I said today. But of course, the next generation experiment that will eventually access, intermediate energy momentum regime can have to really worry about this kind of mixing behavior. And then this is where I hope we will go in the near future. And with this, I close you in the texture rotation. Next, put up, okay. Sorry, very naive question. How is your model is consistent with the famous Lorentz-Doniak model or the superconductivity as it is? Okay, in the sense that they're all the same. I mean, in the sense that you can write the Lorentz-Doniak model, you start from Gisbel on the theory, you just couple layers. I mean, the issue here is not how you write the model for the phase degrees of freedom. The model, the issue, how you couple the phase to the external electromagnetic field. So what I mean is that whatever you do in, okay, let me go back. If you are only interested in describing the phase degrees of freedom, the phase-only action, and then any things you do will end up with the dynamic term and then you'll have a gradient in the plane, a gradient out of plane. So that's always, any approximation you do for long wavelength limit will give you this. Here, the problem is how you describe the coupling of this to the electromagnetic field. It's not just the issue of describing the phase degrees of freedom. So the superconducting degrees of freedom. The problem is how the superconducting degrees of freedom interfere with electromagnetic field. And even for Lorentz-Donic model, you can always end up with this. The problem is what you are computing. So what is your goal? What do you want to do with one set of the model? Okay. Other question? This is related to applications. Let me ask first of all, the Riggs. So my understanding is the vertex there is p.a, not the dense, right? And that changes the game. Because if you go out of resonance, you see nothing. If you go out of? Out of resonance. If you're incoming x-rays, not in resonance, then the response is vanish. No, okay. The point is that it depends where you are in energy and momentum. With x-rays, usually you have very large momentum and very large energy. So if you look at the problem in that regime, where is it here? So in this regime here, somewhere here on here, you don't care about the mixing. So in the regime of very large momentum, which is what you probably the Riggs and the ILS, and usually also very large energy. So here you have one electron volt at 0.5. What is the resolution here? So it's a fraction of the electron volt. So essentially you're looking at everything here. And here you only see, let's say in mode, we have a pure longitudinal transverse character because you don't appreciate the mixing between the two. So RPA works pretty well. You don't have to care about that. Okay, so then you say that your P.A vertex is same as... So somehow it's again, the coupling of the phase with the gauge field also have a one over C below. Where is it? When C is infinite, you don't care about the coupling to the gauge field. If I may, I don't have a question, it's about the answer. So over there you're coupled to surface plasmons, you know, the bulk. Yeah, yeah, okay. So this is correct. What I studied so far is just the back plasmons. In this Nome experiment, of course, what you are doing, you are copying the surface just as some plasmons. So this is also what... It didn't even write in preparation. This is the next step. But to understand how the surface plasmons connected to this layer, the plasmon will behave perfectly in the red. Excuse me. As far as I understood, the coupling between the longitudinal and transverse modes appears only by the... If anisotropy is present, right? Yes. So if you make now the anisotropy extremely large. Yeah. So basically reducing the dimensionality. Do they decouple again the transverse? Well, in a sense, yes, because if you want to, you have two energy scales here. So if you look at the analytical solution here, and this is the beauty of analytic. So you have two scale, omega A, B and omega C. So you can decide what is your small parameter here. So you can expand in K with respect to this anisotropy. You can take the, this one is much larger than the other and they will end up with solution with essentially only one of them will appear. But then it should be the kind of an optimal value of anisotropy for the largest possible coupling Well, I wouldn't look at this way in the sense that system, we have real systems. So we don't have to invent. So in the sense that we know what is the anisotropy in the material, usually the anisotropy is that the in plane is of the order of electron volt and out of plane is fractional millial electron volt. Okay. So these are the numbers. And this is the numbers that we have to play with. So... Unable problem. Not much tunable. So maybe you can play a little bit. For example, in cool place, there is some spreading of this out of plane energy scale, which depends a little on the belayer structure. So variation are still in the millial electron volt scale. And these guys see one electron volt. So there is not much. Now, if you try to apply this to transition methodical coordinate other layered materials, maybe you could achieve somehow, or you can imagine to engineer some heterostructure where you couple it and you can try to see it. For example, but you see, if the anisotropy becomes... Your analytical expression should give you the optimal coupling. Just if you... But optimal coupling is... Optimal meaning... Well... Well, in terms, okay. You have an... The response of LG, CHI-LG is largest. Yeah. I mean, the crossover scale is this one. So you know that in the formula, this is the crossover scale. So you can play with this one, you can make it smaller or larger. But okay, making it larger, I find it a bit difficult in the sense that you have to increase the in-plane plasma frequency. And this, I mean, you cannot usually do more than... I mean, even if you take a system, you do it. I mean, this is N over M. So these are the materials we have. You can a little bit, but not in a way that you can really go from inverse micron to inverse, I don't know, none of either something like that. So I don't imagine that this can be done, but maybe it's possible. Thank you. Thank you. Let's thank Laura once again. Thank you. Next, we'll have the poster presentations. Okay, so there will be one minute for the presentation, of course, so...