 Zelo, da se vse zelo. Vsi počutiti, da se izvanjevamo. Andrei, Mikail, Stefano, za odvijeg. Ja in Lea moramo počutiti vse počutiti na vse. Zelo, da so vse zelo za vse. Spravno lea sva da so vse. Zelo, da so vse, da so vse. Vse zelo, da se je. sodat, da sem tudi vedno vreselom Kajser Odhnept, but this, let's say, are working on this topic also with Roben Kaiser and Romain Bashlav. As we have seen in this conferences, long-range interethning systems has very many peculiar features, like broken ergodicity, long-lasting out of equilibrium regimes, in spomečenja vsečnega stati, in je zpraviljna občas, da se počusti na to, zame je nekako in delativa na vsečnega. Čedna vsečnega je, da je bilo vzgleda, še je začeno v ljubni zepometanje, na svetu nanoskopov, na kanalji nespečnih svetu. I pa sem si nekako država, nekako ta neč to, na vsečenje, in na sejnega neskopov, where long range is important can be devised and as we've seen from the talk of Ros, the range of interaction can be even tuned in high on trap experiments for instance. On the other side as I will briefly show afterwards, long range is present even in natural system like photosynthetic complexes. So why long range is important is not only because it is there but is deeply connected v kooperativnosti in vsečenosti kvantum proprtise, kaj superkonditivnosti, superradičnosti, mikroskopi kvantum tunneling, zelo, da ima nekaj izgledaj. In tudi, kooperativne proprtise potrebno je več vsečenosti in je zelo robasno, zelo, da je zelo potrebno vsečenosti, vsečenosti zelo robasno vsečenosti. Zelo, vsečenosti, tudi je dve systemi, kaj smo vsečenosti vsečenosti, ne zelo vsečenosti, in tudi, da sem bojte, da v sestu, da je to kodatomi klad v Robin-Kaiser experimentu, imamo nekaj interakciju, nekaj nekaj interakciju v zelo sestu rubidium atomi v kladu, in to je zelo natučno, nekaj nekaj klorofila molekul, imamo dipolek interakciju in nekaj nekaj interakciju, svojo se... ... ... ... ... ... ... ... ... ... ... Is the role in long range in the spreading of perturbations. So what we mean by spreading of perturbations is, let's say you have a system and you quench, you perturb locally one part of it. Then you might ask yourself, how long does it take for this perturbation, for the information of this perturbation to reach the whole system, right? And this question is interesting and is important. For instance, it's connected in quantum computation. Say you want to, you create a locally entanglement. You want to know how fast entanglement will spread all over the system. And of course the spreading of perturbation is also related to the thermalization of a system. So to know how fast information can spread in a system is important for many different reasons. So in short range interacting system, Libra Abelson theorem states that the spreading of the perturbation is linear. Its velocity is independent of the system size and it's proportional to the short range coupling. So this is called, so it determines a sort of a non-relativistic light cone within which perturbation can spread through the system. And of course this sets also the time scale of thermalization in short range interacting systems. Of course the question that people ask themselves, it was whether using long range we can break, we can violate Libra Abelson bounds and spread correlation much faster, which would be good. It would lead to maybe faster relaxation or super fast spreading of entanglement. So some theoretical work, for instance in this PRL of Auk and Tagliacozzo, they consider a 1D chain where there is an external field along the z direction, this pin 1 half chain and the long range interaction along the x with a tunable range of interaction alpha. Of course short range is for alpha less than 1 and long range is for alpha larger than 1. So they perturb the system, they start with all the spin down along the z direction, then they flip the middle spin up and they see how fast this perturbation spreads. So what these 2D plots are, the average z component of each spin. So you see that in time, this is time, the perturbation spreads within a linear light cone, yes please. J is positive here, negative, sorry, negative, they are both negative, B and J are negative, so actually, yes, they are both, yes, they are negative. So you see that the perturbation for alpha free which is short range spreads linearly within this kind of light cone according to the Libra Abelson theorem. On the other side as you decrease alpha and you go in the long range side, you see that the spreading of perturbation is almost instantaneous, it's very, very fast and Mikal Kastner did a lot of work, determined the velocity of this spread is in long range system if you want to know more. So, yes, when you do cut restaling, they didn't do cut restaling yet, but we will talk about this rescaling, I will show you results with the rescaling and you do it on J of course, because this is the term which is non extensive. So, this model has been realized even experimentally, as again Ross showed us yesterday, in ion trap experiments and the range of interaction again can be tuned from zero, so all to all interaction to free, short range interaction. Again, what they did, they start with all spin down along the Z direction, they flip the middle spin up, you see they interact with everybody and they see how the perturbation spreads and they confirmed experimentally that long range can break the rub in some bound. At the same time, many theoretical results have shown that the opposite can happen in long range interacting system. Actually, instead of a super fast spreading of the perturbation, you can have a suppression of the spreading of the perturbation. This is a figure from Mikal Kastner work, this is the velocity of the spreading of perturbations, this is alpha, the long range, so as alpha decreases, so the interaction becomes more long range, the velocity decreases, you see. So, this is an example of a contradictory features that people found in long range interacting system. So, on one side propagation perturbation can be very fast, on the other side can be very slow and the shielding effect I wanna show you today actually can explain these contradictory features. Okay, good, so, what is this very nice cooperative shielding stuff I wanna tell you about? So, the point is this, let's say we have an Hamiltonian described by H0 plus V, okay, where H0 contains external field and short range interaction, and V long range interaction. So, I will show you that we can eliminate V from the dynamics in certain subspaces, so the propagation can occur as if long range is not present and actually in these subspaces the propagation occurs determined by an effective short range Hamiltonian which drives the perturbation propagation. So, in long range system you can have propagation within the librob in some bounds as if it would be short range, okay. And why, so, we call it shielding cause the propagation or perturbation occurs as if long range is not there, but this is true only for finite times, longer times, long range will come back again to life, okay. But this time scale increases with the system size, that's why it's comparative effect. So, let me make a very simple examples how shielding can arise. So, let us consider, so H0 plus V again, but now I make a trivial assumption, H0 and V commute. And let's say that V has a highly degenerate subspace. So, of course if our initial state belongs to the degenerate subspace of V and V and H0 commute, the evolution V only gives a global phase to the evolution, so to the dynamics. So, the dynamics is determined only by H0. So, in this trivial example, H0 is our emergent Hamiltonian which drives the dynamics if we start from and against subspaces of V and V is shielded, okay. This is a trivial case where you can talk about shielding. Of course, again you can see that you can have very different behavior if your initial state is in a superposition of two against subspaces of V, then V, of course, will be very important. So, you can have both a dynamics determined by V in this stupid example or a dynamic shielded from V according to which subspace you start with. What I will show you is that when V is long range, this kind of effect can occur even if V and H0 do not commute and even if V is not degenerate, okay. So, I will start to present the results of me, Le, and Fausto in the recent PRL. And first, we start simple, we start from a linear chain of spin one-half systems with an external field along Z and a long range interaction along X like they did in experiments, okay. And notice, in the experiments, again, they started with all the spin aligned in the Z direction, okay. And they quenched along the Z direction and they've seen superfast propagation. Yeah, we will choose another initial state condition just to make an example for you. We will start along X and we will change quenched along X, okay. So, before showing you the first results, let us discuss what is the spectrum, the structure of the spectrum of the long range interaction, which is very important to understand what's going on. So, you see, for alpha zero, then it's very easy. We can exactly diagonalize the long range interaction part. The long range is just a function of the total magnetization along X squared. And so, what are the, it's eigenstates. These are the eigenstates of Mx, basically. So, all spin down will be the ground state along X. All spin down, but one spin up, one excitation, B counts the number of excitation, B1 is one excitation, this is the first excited manifold, second excited manifold, and so on. So, you have many manifold characterized by the number of excitation along X, and these are highly degenerate, exponentially degenerate, and there are gaps between these subspaces, which increase with the system size. When alpha is larger than zero, but less than one, you still have, let's say, bands, but these bands are not degenerate anymore. Now, the degeneracy is broken, so that each band is not degenerate. So, the against subspaces are not degenerate, and at slow energy B is still a good quantum number. While for alpha, larger than one, that bands mix, there is no energy gap anymore. Okay, so, first feature of a long range is this clustering of eigenvalues around bands. Okay, let's see immediately a spreading of perturbation in this model. So, we have an external field, and the long range interaction along X. We start with all spin down along X, the middle up, this is for alpha free, so short range, and you see, as expected, the linear propagation of perturbation, which depends on the range of the interaction, the coupling strength of the interaction, okay. Now, as you increase the interaction range, so you send alpha to zero, for instance. You might naively expect that the propagation gets faster, but, yeah, you see exactly the opposite, right. The propagation gets frozen for a very long time. And this is true also for alpha zero five, not only for alpha zero. It gets frozen for a shorter time, but it gets frozen for a very long time as well, okay. So, this is a simple case I wanted to show you. Maybe, rather counterintuitive, it's very easy to explain as I will show in the next slide. So, we increase the interaction range, and the propagation spreading decreases, okay. So, how can we explain it? Let's start from alpha zero. So, these are the bands of V for alpha zero, that they generate bands, all spin down, one up, et cetera. The external field is a field along Z. It can be written as a rising and lowering operator on X. So, this field, what it does, is tries to mix the bands. It connects nearest neighbor's bands, because it changes the excitation by one. Now, as we increase the system size, this distance increases, but also the number of connections between the bands increases. But at the end, let's say it wins, the energy gap wins, and the probability to leak out the against subspaces of V as you increase the system size goes to zero. So, basically, you get stuck in an, if you start from an against subspaces of V, you get stuck there when alpha is zero. If you get stuck there, V does not connect the states inside of it, and so the dynamics, the propagation is given by an effective Hamiltonian, which is just zero in this case, okay? For alpha different from zero, it's also easy to understand what happens, because still there are bands. V is a good quantum number, the number of excitations. V connects different bands, even if the bands are not degenerate, the external field does not connect states inside the band. So, again, the energy gap increases with the system size for alpha long range, and this suppresses connection between different bands, and so you also get stuck in the against subspaces of V, and effective evolution is zero up to a certain time, which we computed, but I don't wanna enter in this detail now. Okay, this was a simple case, easy to understand. It happens anyway, I will show you at the end. It happens anyway. Indeed, so this is an important point, actually. So now I'm giving you an explanation in terms of energy bands. I will show you that even if this proves what happens here, this is not necessary, okay? This will be the last point. So, I am using it to explain you rigorously what is happening here, but it is sufficient, but not necessary for this effect to happen, okay? Okay, now we complessify the system. On the top of long range interaction, we add nearest neighbor coupling along Z, okay? What is the role of the, let's see what happens with nearest neighbor coupling now, instead of external field? So, here in the left column, I am showing you the case alpha zero. So, you see, the spreading of the perturbation before was frozen. Now, with nearest neighbor, we have a short range propagation of the perturbation. This propagation of the perturbation is independent of J of the long range coupling strength. You see, we go from, we change J by four times from one half to two and the velocity does not change. And actually, it is also independent of the range of the interaction. As we change alpha, the velocity does not change. So, when we have a propagation of perturbation, which is independent of the long range coupling strength, and it depends only on the short range coupling strength, if we change JZ, the nearest neighbor coupling strength, you will have a change in the perturbation, okay? So, this is clearly show why we talk about shielding. So, we change J, the long range coupling strength, and the velocity spreading of perturbation does not change. Now, let me focus on alpha zero, and then we discuss alpha different from zero. So, let's explain what's going on for alpha zero. So, the nearest neighbor coupling ZZ can be written as, as every Z can be decomposed in rising and local wearing operators along X. So, we can decompose this interaction is an excitation preserving part, the blue one, which connects the states inside the band, which are characterized by a fixed number of excitation, and an excitation changing part, which changes the number of excitation by two. So, it connects bands with the first by two, okay? Of course, as we increase the system size, the green part will be killed. So, you will remain in the against the spaces of V, but now the interaction will connect the states, the nearest neighbor interaction will connect the states within the band. And the effective Hamiltonian will drive the interaction, will be given by this excitation preserving part, which is the nearest neighbor Hamiltonian. So, that's why it is independent of the long range part, it given only by this effective short range part, which is the projection of the nearest neighbor part, is not exactly the nearest neighbor part on the against the spaces of V. And it depends, of course, only on the on JZ. Now, we can write down for the case alpha zero, the emerging long range Hamiltonian, making the analogy with the quantum Zeno effect. You know, the quantum Zeno effect is the freezing due to frequent observations of dynamics in invariance spaces, okay? Following Pascazio, we can make a model for a continuous measurement, where the total Hamiltonian described by H naught, which is our system, coupled with a coupling strength kappa to a measurement apparatus described by H measurement. As you increase kappa, the dynamics get, and so the rate of measurement becomes more frequent. The dynamics get stuck in the against the spaces of each measurement, okay? In our case, the long range part plays the role of the measurement apparatus, so to say. But now we don't increase, you don't need to increase the interaction strength, even when you increase the size of the system, the long range part becomes important, and the dynamics get stuck in its against the spaces. And so the Zeno-Miltonian, which is the projection of the total Hamiltonian over the against the spaces of V, becomes exactly our effective Hamiltonian. So the effective short range Hamiltonian would describe the propagation of the perturbation in our system, in the case alpha zero, so all to all interaction, is exactly the Zeno-Miltonian if you map long range in a measurement apparatus system, okay? Indeed, we computed the fidelity, the Zeno-Fidelity, so we make the system, and now we don't start with a specific initial condition. We start from an ensemble of random initial condition, okay? And then we make it evolve according to the full Hamiltonian, then we make it evolve according to the Zeno-Miltonian, we compute the overlap, if the fidelity is one, the two evolution are close by, if the fidelity decreases, the two evolution gets far apart, okay? And here we change the strength of the external field, here we change the nearest neighbor strength, black, red and green curve are increasing n. So as you can see, as we increase the system size in this direction, the fidelity decays get slower, and so the Zeno-Miltonian becomes a better, better description of the system, and the Zeno-Miltonian is either zero or contains nearest neighbor interaction. So I'm almost done. So for alpha-zero, then using the energy gaps, okay, we have proven that the dynamics can be described by a Zeno-Miltonian, which is an emergent short-range Hamiltonian, which is valid up to a time scale which increases with the system size, and that's it for the cooperative shielding case. Now I want you to think a little bit better to decays alpha not zero, because there you have a hint that the energy gap is not so important. For alpha different from zero, we remain also constrained in fixed number of excitation manifold, but this manifold now is not degenerate. So the interaction, the nearest neighbor interaction connects different states, which now do not have the same value of the long-range coupling, okay? They are not degenerate against states of V. So you might wonder, why if I couple different states which have a different value of the long-range interaction, nevertheless, the propagation does not feel the long-range interaction. So evidently, during the dynamics, you do not excite states which differ very, very little by the value of V. So even in the band, in the bands of V, we find that long-range interaction, you get stuck in subspaces in which V changes very little. This is the only way we can explain that the propagation is short-range. But to convince you that the gap is not important, I will do even more. So now we show you some preliminary results we are having with Romain Bachelard. And here I'm not considering classical systems. So nobody can doubt that there are no gaps in the classical systems, okay? So we consider a many-body spin system with an external field along Z, the nearest neighbor interaction along X and we also rescaled the long-range interaction here. So we make everybody happy. Now there are no gaps and the interaction is rescaped. So as we increase the system size, this term and this term, they weight the same, okay? And now we ask ourselves, okay, is shielding present also in these totally different case, I would say, okay? And the answer that we have is yes, we have cooperative shielding. This is the spreading of perturbation in a classical system is independent. These two cones are for two different values of the long-range interaction. You see, the propagation is linear, like in a short-range system, is independent of the long-range strength. Here j is one and here is two. So we double the j and it doesn't change the velocity. While if we change the nearest neighbor coupling, here we increase it twice and the velocity decreases, sorry, we decrease the nearest neighbor coupling and the velocity decreases. So even in classical systems with no gaps and rescaled long-range interaction, the propagation is independent of the long-range up to some time and it depends only on the nearest neighbor coupling. Finally, here we change the system size and you see that here the shielding, the linear cone goes up to above 10 and then it breaks, you see a signature of breaking of the linear propagation and here you see for 350 that it goes up to 150. So we increase the system size, now there are no gaps, the interaction is rescaled, but the time scale over which the propagation is described by an emergent short-range dynamics increases with time. So again it's cooperative and it's shielded. And that's it, I'm done. The reason why I think it happens in classical systems and in general, I think is to do with the mean field description. So long-range system, long-range description, what does it mean, long-range? It means that the particle behave as if they are independent, right? Mean field. So if you can make a mean field description that the particles do not interact and if these mean field manifolds are robust, you perturb them and the propagation of perturbation occurs as if long-range is very weak if mean field prevails, right? So here we add a short-range interaction on the top of the long-range. So mean field is very strong, and the fluctuation around mean field are very weak and gets weaker as any increases. And so the short-range interaction you add on the top of it prevails. So you have a huge collective motion which does not transmit perturbations. On the top of it, any short-range remaining part is the leading part in these many faults determined by the mean field. That's the idea we have now and thank you for your attention. All right. So I'll be talking, thank the organizers for this very interesting conference. I'll be talking here about the same quantum model that Luca just told you about, but in the regime, this limit of infinite range interaction. And what we'll be looking at is what are the consequences to the static and dynamical properties of the systems when we reach what is known as excited state quantum phase transition. So quantum phase transitions that happen at high levels. And you'll see that these results are very much connected with some experiments in ion traps, Bose-Eisen condensate and NMR. So the Hamiltonian that we are looking at is this one, the one that was already realized in experience with ion traps as we heard yesterday in Roll's talk. What is fascinating about this experiment is that they can tune the range of the interaction by varying alpha from three to something very close to zero. And in these experiments, as Luca already told you about, they were very interesting in the dynamics. How fast these systems can evolve in the presence of long range interaction. So the longer the interaction, the faster the dynamics. So they started with initial states such as this one, with all the spins pointing down in the z direction, all pointing down but one or two, which are eigenstates of the z part of the Hamiltonian. And then they let it evolve according to the whole Hamiltonian. I'll be doing pretty much the same thing, but now instead I'll be looking at alpha zero. So alpha zero is the case of infinite range interaction, all to all couplings. And as Odell already told us about, I can rewrite the Hamiltonian in terms of the total spin in the z direction and the total spin in the x direction. So this is the famous Lipkin model. So the Lipkin model has a ground state quantum phase transition when this control parameter c is 0.2. In this talk I'm mostly interested in what happens above 0.2. And well, some details about this Hamiltonian. This Hamiltonian has a U2, algebraic structure with two limiting dynamical symmetries, the U1, which is the z part, and the SO2, which is the x part, the sx squared. I will focus on the U2 Hamiltonian, but just to let you know, these results are general. They also added to U3, U4. These are Hamiltonians used to describe the vibrational aspect of molecules in 2D and 3D. Very well. So what are excited state quantum phase transitions? In a system where we have excited state quantum phase transition, the ground state quantum phase transition on the vanish between the gap and the first excited state does not happen in isolation. It happens together with the clustering of the eigenvalues around the ground state. This is what I'm showing with this plot here. I'm showing the density of states, histogram of eigenvalues, and you see this peak around the ground state right at that point of the ground state quantum phase transition. Okay, let's increase this control parameter, and what you see is that this peak moves to higher energies. And this peak, so this peak, this divergence in the density of states is what became known as excited state quantum phase transition. And the energy where this peak happens is the energy of the excited state quantum phase transition. Another way to see the same story is with this plot. So here I'm showing to you all of the eigenvalues for different values of the control parameter. And you see that above point two we have this clustering of the eigenvalues around this dashed line. So this dashed line is the separatrix, is the line that marks the excited state quantum phase transition. There is an equation for this dashed line which is obtained in the semi-classical limit. So from the point of view of eigenvalues, this subject is very well understood. You know, the subject emerged in the context of nuclear physics. And in nuclear physics, in molecular physics, you have access to the spectrum. What we are asking here is, what are the effects of the dynamics? So, and this will allow us to make a connection with the experiments where dynamics is frequently studied. So what are the effects of these transitions to the dynamics? Okay, before going straight to the dynamics, let me talk a little bit about this tract of the eigenstates. So here I have the eigenstates of the total Hamiltonian and I'm writing it in this basis which is the z part of the Hamiltonian, the u one basis. And I want to know how much is spread out, how much delocalized these states are. To do that, I'm going to compute this quantity known as participation ratio which is one over the sum of these coefficients to the fourth. So you see if the state is very spread out, I have many, many tiny c's. The participation ratio is large. If the state is localized, few c's. So the participation ratio is small. Okay, so in each panel here, I'm showing the participation ratio for all of the eigenstates. Up to point two, nothing really special. What I have is just small participation at the edges, localization at the edges. As I go above point two, you see this sudden dip. So this sudden dip happens right at the separatrix, right at the energy of the excited state quantum phase transition. So this sudden dip, this sudden localization of the eigenstates, capture this transition. To understand better this localization, let's go and have a look at a single each individual eigenstate. So in each panel here, I have just one eigenstate. So I'm showing the components of the eigenstates versus the energy of those basis vectors. So this is a state below the separatrix. This is a state above the separatrix. Nothing really special. They are pretty delocalized. Of course, this one prefers basis vectors with lower energy, this one with higher energy. But this one is the state right at the separatrix, right at the energy of the excited state quantum phase transition. And you see it's highly localized in a single basis vector. And what is this basis vector? Is the one where all the states are pointing, all the spins are pointing down in the z direction. So this basis vector is the ground state of the z part of the Hamiltonian. So let me explain what is going on here. Here I'm plotting the energies of all of the basis vectors versus the control parameter. So basis vectors, not the eigenstates. And that state with all the spins pointing down in the z direction is the one that has lowest energy up to 0.2. But as I go above 0.2, you see the state being carried up in energy and it's following the separatrix. So now let me come back to the eigenstates. What do we have? The eigenstates that are below the separatrix, these are states that have structure closer to eigenstates of the SO2 part, the x part of the Hamiltonian. Above the separatrix, the states have structure closer to the u1, the z part of the Hamiltonian. So let me fix the control parameter and go up. So I'm going up in energy, SO2, SO2, SO2, I hit the separatrix, this is the beginning of the u1. So the eigenstate there is highly localized in the ground state of the u1. Now, this localization will, of course, have consequences for the dynamics. If I start with this initial state and let it evolve according to the lifting model, even though this state may have very high energy, the dynamics will be, of course, very slow. So I'm going to look at the survival probability as a way to quantify the speed. It's just the probability for finding the initial state later in time. And what do you see? Is, of course, this state, this basis vector evolves much slower than the other ones. All right, survival probability may not be the best quantity to observe experimentally. So, of course, you could look, for example, the total magnetization, the z direction. And you will see that this observable is evolving much slower for that state. OK, so I talked about magnetization, so let me make a break and come back to static properties. This is what I'm showing here is the total magnetization in z for all of the eigenstates. Remember the eigenstate right at the separatrix is the one very close, very localized to this basis vector. So the total magnetization in z here drops. So the value of the total magnetization captures the transition very well. Natural question is, what happens to the total magnetization in x? The total magnetization in the x direction. And there again, we can capture the presence of the excited state quantum phase transition. What you see here is a bifurcation, much connected with experiments. So before going into details about that, let me tell you about some of these experiments. There is a recent experiment done by the Florence group. What they have there was a Bose-Eisen condensate in a double well in the ground state. And a Hamiltonian equivalent to the one we are studying. And what they are looking at is this imbalance. The difference in population between the left and the right well. And they are after the ground state quantum phase transition. So when the control parameter is below the critical point, that state is symmetric. So the population on the left and the right is the same, the imbalance is just zero. When they go above the critical point, then you have a very high population either on the left or on the right. So you have this bifurcation, that's how they capture the ground state quantum phase transition. Let me come back to my language of spins. And what they are calling balance there is our total magnetization in the x direction. We are talking about ground state, so let's focus on the ground state first. So here I am with my ground state. Up to point two, my ground state is closer to the u1. Symmetry is closer to the eigenstate with all the spins are pointing down in z. So my total magnetization in x is just zero. As we cross the critical point, then my ground state becomes degenerate. Now I have eigenstates degenerate and closer to the SO2 symmetry. Remember SO2 is Sx squared. So that means I have one state with positive magnetization and the other one with negative magnetization. So I have this bifurcation. Now, the bifurcation does not happen just for the ground state. It happens to all of the excited states. Once I cross the separatrix, I see pairs of degenerate states. This is what you see here, black and white. So this degenerate states. And of course, this bifurcation happens also as a function of energy. So this is what I'm showing in the next one. Here I fix the value of the control parameter point six, and I'm going up in energy. So all of the eigenstates below the separatrix, they are closer to the SO2 symmetry. So I have positive and negative magnetization x. As I go above the separatrix, closer to u1, and then we are zero. So now the bifurcation will also have consequences for the dynamics. This was already seen in some experiments, but they were mostly focusing on classical bifurcation and not making this connection with excited state quantum phase transition. But what are these two experiments? One is the one that Odell already mentioned to you, Obethala's group in Heidelberg. So again, they have their Bose-Eisen condensate with two internal modes. And the other experiment is done by the group in Rio, Oliveta's group, this is NMR, so they are thinking in terms of magnetization. So Obethala is thinking in terms of imbalance and Oliveta in terms of magnetization. What they do, they prepare an initial state, let it evolve and see how this dynamics depends on the parameter, which is the control parameter. So what do we have? This is the result from Obethala. Let me make the translation in my language of spins. So I'm starting with an initial state, which is an eigenstate of the SO2 part, so I have all the spins pointing up or down in the x direction, and I'm looking at the total magnetization in x, the imbalance for Obethala. When my control parameter is smaller than the critical point, I'm in this region. So I have oscillations around zero. When my control parameter is above the critical point, I'm in this region, so I'm trapped in one of these two branches, and you see this self trapping. This self trapping happens as a function of the control parameter, but it happens also as a function of energy. If I start with an initial state here, with low energy, you see the self trapping, okay? Anyway, so this was a very short talk. I just wanted to show to you that there are different ways to capture the presence of this excited state quantum phase transition. The special one is to capture it in terms of the dynamics. You saw here that we start with initial states that are accessible to experiments, such as those in ion traps. Now an important message of this talk, which was also an important message for Luca's talk, is that dynamics does not depend only on the Hamiltonian, of course. It depends also on the initial state. Now the dynamics is not just because we have long range interaction, but the dynamics has to be super fast. It depends on the initial state that we picked. So here I showed to you cases where the dynamics can be very slow, or we can have self trapping, and Luca showed to you the case of cooperative shielding. So thank you for your attention.