 actually has a probe of unconventional superconductors and I illustrated this on cuprates and iron-based superconductors. So concerning iron-based superconductors, I didn't have time to talk about the superconduting state. So if somebody has questions, I can answer them later on. But for the second part of the lecture, I want to switch a little bit of subject. I want to talk about the Higgs mode in superconductors and essentially how to detect it. So this is going to involve some Raman measurements. But I'm going also to dwell a bit on how we can couple to terroits light in time domain measurements. OK, so first, an introduction to the Higgs mode in superconductors. So as you have seen in the first lectures, you can describe your superconductor in terms of a lambda functional. And this is very general, right? So you have this Mexican hat potential. And once you look at this Mexican hat potential, the ground state is sitting at the bottom here. And you can have, from a very general point of view, two kinds of collective excitations, one that will involve the phase of the superconduting order parameter, and the other one that will involve its amplitude. Now, this has been worked out actually in the 60s by various people. And what you can show is that if the superfluid is neutral, let's say, for example, like helium superfluid, the phase mode is linearly dispersing the function of momentum. And the amplitude mode is gapped. And it's gapped by an amount which is twice the superconduting gap. Now, if you are in a charge system, actually the phase mode couples to the Coulomb interaction, and it's pushed to the plasma energy. This is the so-called Anderson-Higgs mechanism. So it's essentially irrelevant in superconductors. It's just a plasma frequency. Now, the amplitude mode stays. And actually, in that case, you can show that this amplitude mode, in the case of a superconductor, has a strong analogy with the Higgs mode of a standard model. And essentially, to make a long story short, the idea is that in Lorentz-Nevian system, so in our case, the Lorentz-Nevian in the superconductors is associated to the fact that there is some inherent particle whole symmetry in the superconductor state, the Higgs is a true collective mode in the sense that it's decoupled from the phase mode. Now, of course, what you see here is that in this simple cartoon picture, the Higgs mode is sitting right at 2 delta. And as you know from the previous lecture, at 2 delta, it's also the boundary of the electric of the Cooper player braking continuum. So it's sitting right at the threshold of a continuum. So it's expected to be damped. Now, there has been just briefly to mention that apart from superconductors, there has been observation of such an amplitude mode, or at least fingerprint of it, in other systems. The idea is that in order to observe really this excitation, you must have some approximate Lorentz-Nevian. And this can be achieved, for example, in a cold atom gas, where you can show that in some limit of the Coulomb interaction potential, you can go through a quantum critical point from a mud phase to a super-free phase. And close to this quantum critical point, there is such a symmetry. And near this quantum critical point, you can see excitations that look like this Higgs mode. Now, what I want to briefly comment is how to actually probe this mode. So this mode is actually not trivial to probe, because it doesn't carry any charge nor any spin. So it actually, relatively, it weakly couples to external probe. So I'm going to illustrate this using the formalism of BCS theory and introducing Anderson-Suelospin. So the Anderson-Suelospin sigma is written this way. So it's essentially the polymatrices contracted by the superconducting wave function written in non-boo space, so in a C dagger up C down. Now, you can write down, actually, your BCS Hamiltonian just in terms of this Anderson-Suelospin. And this form of the BCS Hamiltonian takes a relatively remarkably compact form, where you have this Anderson-Suelospin here, sigma k, that couples to an effective field. And this effective field is a three-component vector. The first two components are essentially the imaginary and real part of the superconducting gap. So if you take, for example, the superconducting gap, we rear, one of them would be zero. And the third component will include the dispersion. And if you introduce the coupling between light and the superconductor using the pyro substitution, this will be the sum of psi k minus ea plus psi k plus ea. So this is how in this formalism, if you want a light couple to this Suelospin. Now, how do you interpret this Suelospin? So if you look, for example, at the z component of the Suelospin, you can write it this way. And what you can see is that the z component of this Anderson-Suelospin is either minus 1 half or plus 1 half, depending whether the plus k and minus k states are occupied or not. So if you want, maybe the simplest is to start with a normal metal at t equals zero. When you are below kf, they are occupied. So you have spin 1 half. And when you are above kf, those are unoccupied. So you have spin minus 1 half. Now, in the superconducting state, you have actually a mix of these two states so that you have, if you want, the spins take orientation that are combination of this plus half minus 1 half state. So if you want to rethink of the BCF wave function in terms of this Anderson-Suelospin, the BCF wave function will be a linear combination of the up and down spin with uk's and vk's as the prefactors. OK, so right now this is just a rewriting of the BCS Hamiltonian. Now, what is sort of nice in this picture is now you can have sort of an intuitive picture of how this of the BCS gradient state is going to evolve once you couple it to light. So you can think of it as a Blush equations where the dynamic of this pseudo-spin is going to follow this Blush equation where the Anderson-Suelospins are going to be coupled to this effective field, bk. The orientation of the effective field will be given by the superconducting gap here, but also by the value of the vector potential of light. So that you will have a complex dynamic where essentially the pseudo-spins are going to rotate around this effective field. And the orientation of this effective field is going to depend on k. And the resulting gap dynamics is going to be the sum over all the k of the x and y component of this pseudo-spin. And this is, of course, has to be solved self-consistently because both sigma k and bk depend on the superconducting gap. Now, what is interesting in this picture is that if you look at this effective field, and in particular the z component, which represents the coupling to light, you can expand it. And what you see is that once you write down explicitly what this term gives you, you can see that there is actually no coupling to linear order in A and that the next order comes with A squared. So what this shows you is that when you couple light to this pseudo-spin, there is actually no term that couples linearly to it, to the gap dynamics. And the only term that comes goes to A squared. So light enters the z component of the effective magnetic field only up to order A squared. So this is another way of saying that actually there is actually very weak coupling between light and the motion of this Anderson pseudo-spin, so to speak, and so to the Higgs mode. OK, so now I'm going to show you an attempt to probe these Higgs dynamics, essentially using two lines of reasoning. One is, so I'm going to go in historical order. First, I'm going to show you that in a situation where superconductivity coexist with a charge-on-CD wave state, there is actually an indirect coupling between the Higgs mode and the charge-on-CD wave state that is going to make this mode active in a Raman measurement. And in the second part, I'm going to show you another way of probing these Higgs dynamics directly, not necessarily in cases where superconductivity coexist with charge-on-CD wave order. So first, let me start with the case where super, yes. So I think if there is disorder, yes. So the question was, if there is disorder, what happens to this coupling? So I think if there is disorder, the phase and amplitude start to mix. And then you can couple here. So I think this has been studied, for example, by Lara Ben-Fato in some systems. OK, so let me first start with niobium dysenium. So this is a charge-on-CD wave superconductor. The charge-on-CD wave transition happens at around 40 Kelvin. It's barely visible here on this resistivity curve. And then it undergoes a superconducting transition at about 7 Kelvin. Now the charge-on-CD wave is a slightly incommensurate and orders along three different directions, oriented by 120 degrees. So it's still an argument in this system about the respective importance of Fermi surface testing and electron phonon coupling as to the main driver of the charge-on-CD wave state. And there is still a debate about the exact interplay between charge-on-CD wave and CDW gap. But what I'm going to focus on is, of course, this is a simple picture of the charge-on-CD wave gap opening. So this is exactly the same picture as the one I showed you for the spin-on-CD wave gap. You have a new periodicity and back-folding and opening of gaps at a specific point of the Brinoise zone. But now I'm not going to focus on this. If you're on single-particle excitation, I'm going to focus on collective excitation. So just like I showed you in the case of a superconductor, you can define an order parameter of the charge-on-CD wave. In that case, it will be the modulation of the charge-on-CD that I showed here. And you can have both. So the other parameter would have, in general, a phase and an amplitude. And you can have excitation that will involve the amplitude and the phase of this parameter. So the amplitude is going to be called the amplitude-on, or the charge-on-CD wave amplitude mode. And this is essentially a modulation of the charge-on-CD wave. So if electrons are coupled to the lattice, to this modulation of the electron density will correspond to an atomic displacement. So this will be essentially a hybrid excitation between electrons and phonons. It will be a phonon-like excitation. And while if you tune the phase, here you will have the phase mode. And this phase mode is gapless. So the amplitude-on has a gap, and the phase mode is gapless. In the case of a clean charge-on-CD wave system. So you may say, well, this is the Higgs mode also. Now, this is an amplitude mode of another parameter. But the analogy with the Higgs mode is less direct than the case of a superconductor. Because in the charge-on-CD wave state, there is no guarantee that particle-hole symmetry is preserved. In fact, in general, it's not. So the analogy with the Higgs mode does not really exist in that case. But the key point is that this amplitude mode here can be Raman-active. So it can be directly seen in the Raman measurements. So here I show you Raman-spectra as a function of temperature. And what you see is that when you cool down below the charge-on-CD wave transition, there is a new mode that appears. It's a relatively sharp mode. And this new mode corresponds to a phonon-like excitation. Yes. I think in general, they do not mix. Probably in the presence of disorder, again, they would mix. But in the clean system, they do not mix. Yes. Yeah. Yeah, it's a good question. So in the case of niobium diselenide, the CDW gap is not clearly observed in this niobium diselenide. But there is another compound, thallium diselenide, where we can really nicely see the CDW gap and the amplitude mode is inside it. So from that point of view, it would be not damped. But of course, nesting is never perfect. So you will always have a residual Fermi surface. So you expect, in general, some damping of it. It's not the gap, yeah. Yeah, so I think that's a good question. So the problem is there are two other measurements of the gap. You have photoemission measurements and you have STM measurements. And they give vastly different energy scales for these gaps. So the STM measurements give gaps of several tens of millivolts, so much higher than this energy. But they're also from photoemission measurements which show smaller gaps, which are on the order of this. Now I would, yeah. Now if I go now, we have recent measurements on a 2H thallium diselenide where we actually see the gap. And the gap is at much higher energy. And the gap is very broad. So we believe that, and we see the amplitude mode, it's well inside the gap. So we believe that by comparing with nethermthallium diselenide, we believe this is the amplitude. OK, so what happens? And historically, this was the first time the coupling to the Higgs mode in a superconductor was evoked, was that when you cool down below Tc, and this is sort of the first measurements where we could actually see the signal, the Raman signal from a superconducting phase. This was on nethermthallium diselenide, so 1990 by Suryakumar and Klein. When you cool down below Tc, so this is the amplitude mode of the CDW phase that I showed you previously. And you have right below in energy this amplitude mode of the Charzon CDW phase. You have the emergence of a sharp peak, which could be interpreted as a superconducting gap energy to delta. Now what is striking about this superconducting mode is the way it interplays with the Charzon CDW mode. And this is most easily seen by doing a careful temperature dependent measurement across the superconducting transition temperature. And what you see is that essentially when you cool down below Tc, it looks as if this mode here at low energy, the superconducting induced mode, is stealing spectral weight from the amplitude mode of the Charzon CDW phase. So there seem to be an intimate link between these two excitations. And this is actually also seen, and this was first seen actually as a function of magnetic field. So the same thing happens as a function of magnetic field. So this, yes. So that's one scenario. So one scenario, so if you were to follow the idea that this mode here probably is the CDW gap, that could be a sign that could be interpreted as a sign of competition, yes. Now we believe, based on the comparison with the sister compound tally of the SNN, that this is actually the amplitude mode and not the gap. But that could in principle be a possible scenario. But at that time, this was not the scenario that was put forward. The scenario that was put forward was the one that there is actually a coupling between the amplitude mode of the Charzon CDW phase and the amplitude mode of the superconducting phase in other words, the Higgs mode. And this was the scenario proposed by Littlewood and Varma. So this is a PRB paper of 1982. So quite right after the first experimental measurements. And what they proposed here, so I'm extracting an excerpt from this paper. So keeping in mind that U0, so U0 is the lattice distortion associated to the Charzon CDW mode, coupled to the BCS gap. So the way it is coupled was presumably thought that this distortion, by distorting the lattice, modulate the Fermi, the density of state of the Fermi energy. And by modifying the density of state of the Fermi energy, it's also modifying the gap. But at that time in this paper, it was really a phenomenological coupling between the two. We produce a time-dependent modulation of delta. There's the electron phonon coupling for this mode. We build the form. And you see they propose this coupling. So G is really a phenomenological coupling between here, this operator, which are the bosonic operator of this Charzon CDW phonon, and the BCS gap function. So this was proposed. And they could actually, so this is sort of the picture. So you modulate your Charzon CDW wave. And by modulating your Charzon CDW wave, you are modulating the gap. And this is the way these two other parameter oscillations are coupled. So in this scenario, yes. Yeah, you can think of it as an optical phonon. No, so there is a related effect. So there are some constraints to this coupling. First, you need to have a coupling that is strong enough. And this is usually achieved when the two mod energy are close enough in energy. And then the other thing is you need to have a phonon that strongly modulates your gap. And this was presumed to be the case in this Charzon CDW superconductor. But you're right that in fact, you have a very similar coupling between the phonon and the superconducting state. But instead of having tau 1 here, you would have tau 3. And this would be a more regular electron phonon coupling effect. In fact, this is a very special electron phonon coupling effect. And the question is how to derive this coupling constant. And this was done actually one year after by Brown and Levine who derived a microscopic mechanism, a microscopic picture of this coupling constant. But in the paper of Littlewood and Varma, this was taken as phenomenological. OK, so the idea in this scenario is that the Higgs mode would appear as a sort of a side band of the amplitude mode of the Charzon CDW phonon, essentially. So this is the calculation in that paper. So they calculate the phonon spectral function. And by coupling it to the Higgs mode of the superconductor, the phonon spectral function acquires a new pole. And this new pole will generically appear below to delta. So that is good because that is going to render it more long-lived. And those are calculations for varying coupling constant. You can see here this feature is the amplitude mode of the Charzon CDW state. And once you cool down below Tc, there is a very sharp mode, which in this theory is a delta function. And when you crank up the coupling constant, this goes more and more below to delta. So they actually derived an energy for these coupled excitations. And it's always sitting below to delta. So the fact that it's below to delta is good. It means it's going to be less damped than the conventional Higgs mode. It's unaffected by screening. So this is also an important aspect because the Higgs mode does not carry any charge. So Coulomb screening is not going to affect it. And this naturally explains the spectral rate transfer between the two. So you can actually do a more microscopic calculation, not just of the phonon spectral function, but actually on the total Raman response. This was done by Chehan Ben-Fato. So this is really calculating this kind of diagrams, where you have the, this would be the Raman, the regular electronic Raman bubble that I showed you in the first lecture. And that would couple to the CDW phonon. And this CDW phonon will in turn couple to the Higgs mode. And if you do that, what you see is that indeed, once you cross Tc, there is a spectral rate transfer between the CDW phonon and the Higgs mode below Tc. And this matches quite well the experiment that is observed in IEM, I said. Now, I should say that, and this is something that was sort of overlooked, since quite recently, and that goes back to the coupling, to the non-linear coupling I explained you in one of the first slide. So I showed you that actually there is a coupling that is possible to a square term. And you may say, well, Raman, after all, is a two-photon process. So why don't we see it directly? And it's actually not a stupid question. Actually, you can ask yourself, well, what if I just get rid of the charge on CDW mode and I try to couple it directly? Then I would have this kind of diagrams. And then I can show that my Raman response will involve the first term here, which I labeled k, k is zero. This is the regular pair breaking term. So this is the one that is giving me sharp peak at two delta and the continuum. And if I do, if I insert the Higgs mode here, I will have an additional term. So if you want, this term will be in the density channel, so tau three in terms of polymetrices. And this one will be in the pairing channel, tau one. And what you can see that here in the numerator, this has this form. And usually, so gamma k is the Raman vertex essentially, this coupling between light and electrons. And in the literature, and in particular, if you look at Varmas paper, he would take this essentially as a constant, k independent. And because of particle whole symmetry, actually, because there is only epsilon k here, this vanishes. But you can really clearly see that as soon as you put some k dependence in the Raman vertex, there is no reason for this to vanish. Actually, and this is, I think, that was something that was overlooked. Actually, this direct coupling, not even thinking about 1000 city waveform is actually finite. Now, the question is how big it is. Now, actually, it's a little bit, in the case of a niobium dissonant, it's a little bit smaller, but not that small. So that actually, what you can see is that in the dotted lines here, this is the discontinuation here. And this can be enhanced, actually. If you gap out part of the Fermi surface, this can become more long-lived, and this can gain a lot in strength. So that actually, the direct coupling mechanism is not a stupid idea and could actually work. Now, of course, in the case of niobium dissonant, I such a direct coupling mechanism would not explain the spectrary transfer between the 1000 city waveform and the superconductive peak, at least not as easily as in the Varma model. But still, it's not completely out of the question that we can see this Higgs mode in just a regular superconductor, yes. So in this picture, it's at two delta. So the way it becomes more long-lived is because you have CDW gap. That gaps out part of the Fermi surface and that gives it more long. But it's still at two delta, no, no. For this, you need to have the CDW phase. Okay, so let me, so let's say that the spectrary transfer favor the phononic scenario and I'm going to show you experiment and the hydrostatic pressure, which also, I think, lay credit also to this phononic scenario. So the idea is that actually the phase diagram of this compound, 2-H niobium dissonant, is interesting under pressure. What you have when you apply hydrostatic pressure is that you have essentially a suppression of your charge on city wave state. But the superconducting transition temperature is actually very weakly pressure dependent. So by applying pressure, you can actually heal your CDW state and preserve essentially your superconducting state. So this is a key test if you want for this scenario in a way that let's see if really CDW state is crucial or not for the observation of this peak. So this is Raman measurement as a function of pressure. First, showing the data above Tc. So essentially here doing a horizontal line above Tc and this is tracking the pressure evolution of this amplitude mode of the charge on city wave phase. And here I'm showing you experiment in two different polarization configuration and in both polarization configuration, you can see that the peak of the amplitude mode of the city wave state indeed vanishes and that by the time you reach for GPA, it's essentially absent. So you have killed really the charge on city wave state. Now, what happens when you cool down through Tc? So this is above and below Tc at selected pressures. So what you see is that up to 3.2 GPA, you still have the emergence of this relatively sharp peak at low frequency in the superconducting state. There is still a transfer spectral rate and what you can see is that once you move out of the coexistence between superconductivity and charge on city wave, so at 4.4 GPA, instead of a relatively sharp peak, you have a broader signal and importantly, and I'm going to show you that in the next slide, this signal here which I reproduce here is located at a higher energy. Yes, as a function of magnetic field, pressure. Okay, so of course you would need an independent measurement of the superconducting gap for that, but the Higgs mode, if you look at this sharp mode here, it softens as a function of pressure. Much, yeah. Tc is flat, so presumably, yeah, doesn't change much by 10% at most. Well, this can be explained by the fact that, okay, so there is one way of explaining it, which cannot be unique, is as you kill the charge on city wave state, the amplitude mode of the charge on city wave state becomes closer and closer to the gap, and this pushes down, and this pushes down the gap to lower frequency, and this pushes down the gap to lower frequency. If they are exactly degenerate, actually, it's often completely to zero, I think in Barma theory. Yes, yes. No, at first, actually, it's not monotonic. At first, it moves closer, so see, it's a balance between the energy of the amplitude mode and to delta, and I think once they get closer, actually, it gets pushed to lower and lower frequency, and it gets to higher frequency only when it goes on the other side. I think that would be the picture of the Barma theory. Okay, but the key, I think the key observation here is a summarize in this spectrum, yes. So here, so in this scenario, the per-breaking peak is this one, right? In the red, the red spectra here, the spectra taken outside the coexistence region, right? So at high pressure, so this is above Tc and below Tc, this is this small thing, and once you enter the coexistence region, what you see is that you have a sharp peak, and this sharp peak is below to delta, and it's much sharper, and it's clearly not, we are not clearly not talking about the same feature as here, and remember, the Tc is almost equivalent in both states. Yes, yes, yes. Well, to delta, apparently, is what's remaining here. So it's essentially hidden on the high energy side of the Higgs mode. It's overwhelmed by the spectra wave of the Higgs mode, at least in this picture, yeah. And you have a good question. Again, there is another compound, okay, I could show you the data where we can actually both to delta and the Higgs mode. But in the case of Neobium disainite, we only see sharply one mode, and it's located below to delta. So there is a simultaneous collapse of the Higgs mode and CDW order. So if we identify the Higgs mode here as a sharp picture, clearly it has collapsed, while there is little change in Tc. So this is, again, in relatively good agreement with the fact that CDW phase is crucial in bringing this very sharp peak in the coexistence region. Now this can be actually compared with theoretical calculations. So these are theoretical calculations performed by Lara Benfato and coworkers, essentially taking the little VARMA theory. And what they can see is that once you are in the coexistence phase, you have a transfer of spectral weight from the amplitude mode of the Scharzen-CZF state to the Higgs mode. And once you enter the pure superconducting phase, what is remaining is just a weak to delta per breaking feature. So this, at least qualitatively, matches the data. Okay, so now let me move on to something that is somewhat different, but the motivation is, okay, so we see that, okay, we have at least some good evidence that the Higgs mode can be visible via Raman scattering in compounds where you have superconductivity and Scharzen-CZW order. I have already told you that there is some kind of warning sign that you could actually get direct coupling. It's not completely out of the question. So the question is whether you can use other techniques to couple to the Higgs mode via this A-square term, right? So that is going to be a nonlinear technique. And now I'm going to show you some experiment that we are done in the group of Riyoshi Mano. So I had the chance to spend one year in this group and I'm going to show you a bit of kind of another approach to see the Higgs mode using Terahertz light. So to see that, it's interesting to go back a bit back in time and look at some Siri paper that we're addressing sort of a different setting in which you would have a superconducting condensate or a superfluid state and where you would quench the interaction. So for example, suddenly increase the interaction and then follow the dynamic of the superconducting order parameter. So this was done, for example, by these people. So they essentially, non-adiabactically changed the pairing interaction. And what they see is that they see an oscillation. The system oscillates between the superfluid and the normal state is what they say. They see an oscillation of the order parameter. And this oscillation, this inverse of its period, is exactly two delta is the gap. So it's kind of, if you want the analog of the Higgs mode I just described, but this time, not in the frequency domain, but in the time domain. So in the non-adiabatic regime, you can have access actually to these dynamics. In principle, if you want to follow the receipt of the Siri paper, you would need to find a way to quench your pairing interaction. So of course, this is something that can be done theoretically, relatively easily, experimentally, this is another challenge. And if you do that, and in a non-adiabatic way, you would expect actually your gap oscillation. So these people could actually write down an analytical formula for the oscillation of the gap within the BCS framework. And what you can see is that indeed, you have a cosine two delta T over H bar oscillation. And there is a square root of T decay that is essentially due to the lambda damping to the fact that two delta is sitting right at the border of the electron hole continuum. What is giving this one over square root of T decay. Yeah, so there are different settings. So this is actually, this is a, okay, I won't go to the details, I'm not an expert of this kind of calculation, but actually the dynamics depends very much on how you actually prepare your state, whether you quench it down or you quench it up. And the dynamics are quite different. So here in that case, I think what they started is by reducing by half the pairing interaction. So the initial time was reducing by half the pairing interaction. And then let's see and see how the system evolves. So now you can actually try to compute something that may be a little bit more realistic in terms of condensed matter systems. You can try to compute the response of a BCS supercomputer after an optical quench. So you shine a very short pulse on the superconductor. And then you see how it evolves. So this was calculated by these people. And what you see is that two delta, so initially the superconducting gap decreases. So this can be explained by the fact that you are creating particle hole excitations in a non-diabetic way. So this is reducing a little bit your superconducting amplitude. And what you see is that for sufficiently short pulses, so in that case, three picoseconds, you have oscillations. But if you take a very long pulse, so if you don't satisfy the non-diabetic condition, so in that case, the non-diabetic condition would have to have a pulse whose duration is shorter than the inverse of two delta, essentially. You see nothing. So how to do that? So this is another way of looking at the same physics. So if you take your Mexican hat potential, you are essentially sending an optical pulse, you are reducing the widths of your Mexican hat potential, and then the system starts to evolve around the minimum. This is your new example. So you reduce here your other parameter, and then the system is going to oscillate around, around. So this kind of settings has been actually discussed even in the 70s in the context of microwave measurements in superconductors. So one way of seeing how you can quench your superconducting gap using terrorist light or infrared light near the gap frequency is by looking at the gap equation. So if you modify the temperature of your, if you create quasi-particles out of equilibrium according to the gap equation, this is going to change your gap amplitude, and this could provide a way of quenching your superconducting. So essentially creating particle hole excitation, boguluboff excitation in another diabetic way. But for that, there is a condition is that the widths of the optical pulse has to be shorter than h bar over delta. So this can be done, for example, here by shining light that is below the gap. And so with an energy that is slightly below to delta, but the widths in time that is shorter than h bar over delta. So this requires for typical low DC superconductors, this requires a pulse which are in the mid-evolved energy range. So those are terrorist regions and which have picoseconduration. So this experiment was actually done. So here I'm showing you an experiment done on liobium nitride. So it's a terrorist pump, terrorist probe measurements. So you shine an intense terrorist pulse. So this is the spectrum of this terrorist pulse. So it's roughly centered about one terrorist. So one terrorist is roughly a three mevolved. And this is essentially a single cycle pulse in the time domain. So typical width is about one picosecond. And what you're going to do is you are going to pump your system with this. And then you are going to see its evolution with time by sending another probe pulse that is going to be also in the terrorist range to probe the dynamics of your superconducting. So this is the measurements that was performed. So this is the transmitted electric field in the terrorist range. So remember that in a BCS superconductor, if you are shining light at an energy that is below two delta, there is no transmission. So if you have a reduction of superfluid density or a reduction of two delta, you're going to have a modulation of your transmission. And this is what is observed here. So right immediately after the pump, there is an increase of the transmission. So this increase of the transmission can be interpreted as a reduction of the gap. So you are killing a little bit your superconducting gap. And then you can see oscillations. The oscillations are particularly visible here at low fluence when you exert a pump that is not too intense. If you exert a pump, a pump pulse that is too intense, they are essentially damped. And you can extract the frequency of these oscillations. And the frequency of these oscillations at low pump intensity matches very well twice the gap energy that is measured independently by equilibrium transmission measurements. So you have time domain oscillations of your transmitted electric field that have a frequency of exactly two delta. Now it's interesting to compare. Yes. Okay, I'm going to come back to that later. So right now, if you think about this quench, this theoretical ideas about quench measurement superconductors, this really looks like you have realized this quench measurement, right? You quench and you see oscillations at two delta. This really matches. But I'm going to come back to a possible other alternative explanations. Okay, so just to show you, it's interesting to compare the same measurements, but now you don't pump with low energy pulses or close to delta, but you pump at a much higher frequency. So you use visible pulse for example. Now if you do that, so this is the measurements here on the left, what you see is a large signal, but there are absolutely no oscillations. And the reason is presumably relatively simple. When you shine 1.5 electron volt light, you are creating a bunch of electron hole excitation of very high energy. When they cascade, they break completely your coherence. So there is no hope of finding these two delta oscillations. And for that, you really need to use low frequency light. But there is a loss of coherence when you pump with too high energy. Now there is something that is interesting is, so up to now I have focused to a regime here where you see oscillations that persist when the pump is off. So if you want free oscillations. Now you can be interested also, is what is going on actually when both pump and probe overlap. And here this is kind of a zoom, you can zoom in this region here. So this is the waveform of the pump and this is the signal of the probe. And what you can see actually is that the probe seemed to follow not the electric field of the pump, the square of it. So if you look this minima here correspond to this maxima here, this maxima correspond to this maxima. So it looks like when both pump and probe overlap, there is a signal in the probe that is following E square. So you can actually see that more nicely if instead of using essentially a single cycle terrarium waveform, but now you shine a multi-cycle waveform. So now you use for example a narrow bandpass filter. You narrow down in frequency opals. Of course you broaden it in the time domain and this time you have many oscillations. And if you do that you see that you see really clearly these oscillations in the probe signal and these oscillations are really nicely following the square of the pump. So people who do nonlinear optics, they know that. They immediately say, ah, this is a third order signal. So this kind of response that follows E square in the language of nonlinear optics is considered as a third order nonlinearity. And by the way, this is also how these people in nonlinear optics describe Raman also as a third order susceptibility tensor. So I'm going to come back a little bit because this is not the way we in the condensed matter community are used to think about it. But for now let's say that there is a signal that follows E square. So this actually is interesting. You can do a much simpler measurements and these guys did the simpler measurements. You can say, okay, so there is an E square signal. So I am shining E and I collect some E square signal. And actually there is another third order nonlinearity that is much simpler. Third order generation. I shine omega and I look at a signal at three omega. This is also a third order nonlinear susceptibility. So you can do that. You can do not a pump probe signal but read just a very simple nonlinear optic measurement. You shine a tear out slide and you look at its frequency domain composition upon passing through the sample. And what you see is that indeed, so the key curve I think is here, is that you have the fundamental here located right below one terahertz. And when you cool down below TC, there is indeed an harmonic of this fundamental here at three omega that is only seen below two delta. And this, the intensity of this harmonics follow the six power of the pump which is also what is expected for a third order nonlinearity. Yes, yeah. So this is omega here and this is three omega. So it's third harmonic generation. No, second harmonic generation. This is third, yeah. So second harmonic generation is only active when you have a material where inversion symmetry is broken. So it's a well-known tool to study, for example, further electricity. And third harmonic generation is in principle always possible. It's just that most of the time it's weak. So here actually it's a sizable signal and it's only activated in the superconducting state. So actually the theory of this effect was worked out by Tsuji and Aoki. And what they show that if you, the Anderson pseudo-spin formalism, you can actually compute the nonlinear current and they show that the third order nonlinear current is actually directly linked to the oscillation of the amplitude of the gap. And importantly, this oscillation of the gap is resonant when the frequency of the incoming light omega is exactly equal to delta. So you shine light at delta and you collect light at three delta. So in fact, if you think about it, it's exactly like a Raman process, right? You shine light at delta, you collect light at three delta. So if I were a Raman guy, I would say, okay, so I'm now going to plot everything in terms of Raman shift. So three delta minus delta to delta. It's my two delta peak. Okay, but then you are going to see me wait. Could this be the per-breaking peak there? I will come back to that in a moment. So anyway, so you expect a resonance signal when exactly you are matching the omega equal delta condition. So it's difficult actually to do these measurements at continuously tunable frequencies. So what these people did is actually they tune the gap frequency by tuning the temperature. So if you assume that the gap frequency follows BCS temperature dependence, this is one way of tuning the gap amplitude. And what they show is that indeed, the temperature at which the signal is resonant seem to match at different frequencies. The temperature at which you would expect omega to match delta if it was to follow BCS temperature dependence. Okay, so so far so good. We have seen sort of two manifestation of a two delta mode, one in the time domain in the form of some form of transient response that was oscillating at two delta. And another one, which was sort of a forced oscillation if you want, but that seemed to be linked directly to some kind of non-linear optical susceptibility. And both of them are two delta. So actually once you see this and you know a little bit about Raman, there is of course an alternative explanation at least to this kind of measurements. I think it's a bit less obvious about the transient measurement in the time domain, but at least for the third harmonic generation, there is an obvious alternative which is that it is the pair breaking peak. And actually these people, Tugia and Aoki, when they computed the third harmonic generation response, they worked out in an approximation where the pair breaking peak was zero. And this is because they took essentially constant vertices, so it screened out. But of course, we know from Raman's scattering point of view that this is a special case. And in fact, if you don't take this a special case and you take a very general calculation that was done by Chia and Ben-Fato, you realize that indeed once you compute the third order non-linear susceptibility, so this third harmonic generation, so this is essentially the same diagram as in Raman except that the incoming frequencies are slightly different, but it's essentially the same diagram as with Raman. So you have this first diagram here, which is the one that is known from Raman's scattering and that is going to give you the pair breaking peak energy. And then you have these other diagrams here which are essentially this direct coupling to the Higgs that I was talking to you before. Yes. No, it's a pole, I think. Yeah, it's kind of a borderline case. It's really at the bottom of the continuum here. Okay, so and if you do that, you realize that generically, this contribution is actually several orders of magnitude smaller than the pair breaking contribution. So this is essentially the intensity as a function of omega. Now the bad part of it is that actually both contributions are resonance at delta. Delta to three delta, of course, because we know from Raman that we have to have a mode at two delta. But both of them are resonant at the same frequency, but one of them is at least three orders of magnitude smaller so that this fixed contribution is actually much smaller generically in the BCS approximation of this one. Now there is one way to distinguish between the two which is in the polarization dependence. So in fact, if you take just a very simple model, square lattice with nearest neuron hopping, you can actually compute the polarization dependence of the signal. And what you realize is that the Higgs mode contribution is essentially independent of the angle of the incoming light with respect to the axis. Whereas the pair breaking contribution actually is very strongly dependent on your incoming polarization. And this we know from Raman's scattering calculation that indeed such a response is going to depend very much on the kind of light polarization that you use. So in the case of just a tight binding model with only nearest neighbor hopping, actually the pair breaking contribution vanishes when you excite at 45 degrees of the axis. Whereas the Higgs contribution is completely isotropic. So this tells you that, okay, maybe there is a way to disentangle these two contribution looking at the polarization dependence of the signal. And if you do that, and this is sort of the puzzle, so you can do that in neighbor of nitride. And what you see to make a story short is that the third harmonic generation signal is essentially independent of polarization. Completely within two percent. You hardly detect any polarization dependence. And this is the puzzle because it's very hard for any generic system to have the pair breaking contribution be so independent of polarization orientation. And only the Higgs mode actually guarantees you that. So this is the puzzle. In the BCS approximation, you would expect the Higgs contribution to be orders of magnitude weaker. But yet, in the experiment, it seemed to be the relevant one for these measurements. So I think there is still no good explanation for what reason the Higgs mode apparently seemed to dominate these measurements. So there was some attempt to go beyond the BCS approximation by including retardation effect by coupling to some phonon mode. And these measurements show that if you include retardation, you bring the contribution from the Higgs mode a bit closer to what you would expect from the pair breaking. But I think right now this is a bit still speculative. Okay, so in the remaining part of this second lecture, let me now talk about some recent measurement we did in collaboration with the group of Fuyoshi Mano in Japan, yes. Yes. In the BCS approximation, yeah. Yeah. Yeah, that's really puzzling, yeah. Yeah, that is the puzzling, yeah. And if it's correct, then it sort of also would be a main drive to sort of reconsider also Raman scattering measurements if you're going to operate. So it brings a lot of question. Okay, so now what happens if you, so up to now I discussed about S wave superconductor. So Niagara Nitride is an S wave superconductor. It's not as simple as that because it's a multiband S wave superconductor, but still we think we reasonably understand the superconductor. So now let me briefly talk to you about what's going on in D wave. Okay, so in D wave the amplitude mode would correspond to a homogeneous oscillation of the D wave gap amplitude. So that would be this form. Now it was asked by Barlas and Varma that you could have actually oscillations that break the lattice symmetry, but here let me for now just consider only the conventional Higgs mode in the sense that it's in the A1G symmetry. So it's modulating the gap in a way that it does not break lattice symmetry. Now what are the challenges in probing this Higgs mode for D wave? Well the first thing is that as we discussed in general is damped and in the D wave superconductor and in the D wave superconductor it's even worse because here we have plenty of low energy excitation because there are nodes. So you would expect the damping to be even more dramatic than in the case of an S wave. So this was calculated by Peronacci and coworkers using a quench setup. So similar to the setup I showed you before, essentially quenching the pairing interaction and seeing how it evolves. And in the dotted line this is a case of S wave. And in the red line this is D wave and you see immediately that the damping in the case of D wave is as expected much faster than in the case of the S wave superconductor. So that's one challenge. In the terms of transient dynamics you would expect these two delta oscillations to be much more short lived. Now there are other challenge but there are more technical. So first it's a bit more difficult to produce intense pulse in the spectral range of delta for high Tc cuprays for example. So the delta, the superconducting gap in high Tc cuprays is in several tenths of millivolts and it's a bit harder to produce intense pulse in this energy range. So in other words, the resonance condition is harder to fulfill. And of course the equivalent of that is that in terms of reaching non-adiabaticity it requires really ultra short pulse. Sub hundred femtosecond pulse if you want to reach the non-adiabaticity. And this is quite challenging. So what I'm going to show you is a first attempt to measure this physics but I'm going to focus on essentially probing not this transient oscillation after quench or non-adiabatic excitation of your superconducting state but looking more at this third order nonlinear response that I've been discussing in the context of third harmonic generation. So the setup is the following. So we are working on crystals. So the setup that I showed you where we were looking at the transmission measurements was on on-scene films. So of course on crystals those are bulk materials so you cannot really transmit your terrohertz wave. So what we did is actually to use a setup where you pump with terrohertz but you are going to look at your non-linear response by probing its effect on the optical reflectivity. So high energy response. So this is the form of the terrohertz electric field. So it's relatively high. So it's about one order of magnitude higher in electric field than the experiment on niobium nitride. So because this is much higher. So you really need a much stronger pulse to really affect your superconducting state. And this is centered as 0.6 terrohertz so about one millivolts. So clearly we are in the non-resonant regime. So if you look at the theoretical calculation where you see the resonance signal of the third harmonic generation picking at delta here we are really on the tail of it. So we are in the non-resonant regime. So this is so what we did is that one nice thing about using optical probe in reflectivity configuration is that you can actually tune your light polarization. So you can actually quite easily look at the response as a function of orientation of polarization with respect to the crystallographic axis. So this is experiment on b-smit 2-2-1-2. So change in reflectivity as a function of time. In red here this is e squared of the terrorist pump. And what you see here in blue and red are the change in optical reflectivity in two different configuration polarization. One where the probe electric field and the terrorist electric field are parallel. And the other one where both are perpendicular. So essentially I keep my terrorist electric field along the same direction, but I rotate by 90 degree my probe electric field. So what you see that in both polarization you can see a signal. And this signal follows the square of the pump. You can actually probe for a given configuration you can plot as a function of temperature. And what you see is that this e squared signal indeed shoots up below Tc. So it's really associated to the superconducting state. Now this kind of third order signal is actually, so you heard about the current effect in the previous talk, but actually historically the current effect is defined not in a way that was talked about by Aaron Kapitunnik. In fact, if you look at optics book the current effect more rigorously is a change in the optical constant that depends on the light intensity. So if you look at your equation, your optics equation, you have the index, the optical index, and the optical index is to linear order independent of the electric field. Now if you go to the next order, the next order is actually a change of the optical index that goes as e squared for inversion symmetric materials. And essentially the effect that we are seeing here is a change in the optical index upon radiating the sample via a terrorist electric field. So it's a change of optical index upon impacting it with a terrorist electric field. So this goes as e squared. So it's a change of the optical index upon terrorist pump. So it should follow e squared scaling. And you see here that indeed it follows e squared of the pump. And more generally you can show that the change in optical reflectivity will depend on the third rank tensor that is written here chi ajkl and that is going to be depend on the e squared of the pump. So just like you did this game of extracting the symmetry component of this tensor in the case of Raman measurements, you can actually extract irreducible representation of this tensor using light polarization. And you can actually decompose here your third order tensor. You will have an a1g term that is going to be essentially independent of the light polarization. There is going to be a b1g term that is going to go as cosine two C of the pump and cosine two C of the probe. And there is going to be a b2g term that is going to scale as sine two C of. So very similar to the decomposition that I showed you for Raman scattering. And what you see here is the angular dependence of the signal by fixing the pump and varying only the probe. And what you can see is that there is a constant term and there is a term that is also oscillating and is oscillating as cosine two C of the pump. Now you can actually extract all the components. And what you see is that the signal, the terrod-scarred signal is dominantly a1g. So this is this isotropic component. And there is a subdominant component here in green that goes at cosine two C. So there is a subdominant b1g. Yes, I'm pretty much finished here. So what we did is we did essentially the deping dependence of this feature and this terrod-scarred effect that becomes activated in the superconducting state is essentially visible in all dopings. And in all dopings, it is a1g dominant. Actually, there is an interesting doping dependence. So if you plot the ratio of the b1g to the a1g component, so essentially the b1g component is negligible for under-doped sample. But they become of the same order for over-doped sample. But throughout the doping range, the a1g component is always dominant. Now, you can actually look at diagrams to explain this third-order nonlinear response. There are essentially Raman-like diagrams, but there are more. So for example, in Raman-like diagrams, you would have this kind of diagrams. But when you do pump and probe, you can have such diagrams, for example, where small omega is the optical probe and capital omega is the optical pump. And actually in the non-resonant regime, this is this kind of diagrams that dominates. And you can compute the contribution of this diagram both to the pair-breaking channel and to the Higgs channel. And what you can see is that generically, again, the Higgs mode will only appear in a1g channel. While the pair-breaking peak or the charge density fluctuation contribution, if you want, will be predominantly b1g channel. And this is actually what is observed in the Raman measurement in cuprates. In Raman measurement in cuprates, the dominant contribution is in the b1g channel, the one that is probing the maximum of the superconducting gap. But in this kind of measurements, this is different. It's dominated by the a1g channel. So the tentative interpretation is that the a1g contribution is essentially coming from the d-wave Higgs contribution. And the b1g component likely comes from this pair-breaking contribution that dominates the Raman spectrum. So I'm going to end up here. So of course, this is by no way the last row. I think it's still very puzzling why it seems that the Higgs contribution is dominant in these superconductors while BCS calculation would expect a strongly subdominant contribution. I think it's still an open question. And beyond that, it's also an open question of why these measurements seem to yield symmetry dependence that are very different than Raman. Because after all, they seem to be controlled by the same kind of response function. So I think this is sort of an other way also to probe a superconducting state. And this also brings additional questions as to what are we actually probing with these measurements. And with this, I will conclude. Thank you very much.