 Okay, welcome back folks to the second session of this day, and the first talk of this session is by Anna Sampera, and her talk is about time crystallinity in open quantum systems. Go ahead Anna. Thank you. So let me share the screen. Yeah, thank you very much. I want to thank, first of all, the organizing of this conference, which is a very nice conference. I have to say that I'm not a person which is very expertise in time crystals, but we got interested on it because we were working in open quantum systems. So let me first tell you who is the team. So this work is mostly and has to be granted to Andrea Riera Campan, which is he's doing the PhD with me, and he's the person who was really digging into this problem. And we have a lot of help from Mariona Moreno-Cardone, which is a postdoc at that time on my group, and that she's an expert in open quantum systems. So as I say, I am not an expertize on the field, but we were wondering what was this time crystals when we were talking about open systems. So there was two questions that was as important. First of all, it's how to define a time crystal in an open system because we found nice definitions for closed systems. But when we were digging into the literature to understand what was in an open system, that was not clear if there is a clear definition. And second question that we were interested in is that at that time, most of the time crystals that they were presented in open systems, they were always using a mean field or long branch interactions. And we were wondering, first of all, if that was needed. So and also which physics can be put under the carpet when one uses mean field because which part is classical, which part is quantum. So I'm going to present this talk a little bit in the backwards direction. First, I want to give the results because the thing that we did is a little bit technical and I don't want to enter into that. So I don't know if I will reach time to explain a little bit this technical part, which is not so important. But then I want to explain and to talk, which is very related to the previous talks, very nice talks by Masahito Ueda and Heimerich about the difference between the Imbladians and Hamiltonians. And finally, I will explain, well, how do you treat this periodically driven systems and what is this battle against the coherence? So when we were searching in the literature, we found pretty nice definitions of what the discrete time crystal. I'm only going to talk about discrete time crystals in a closed system is and say that if you have a closed system, a discrete time crystal is characterized by an observable, which acts as an order parameter of your system and whose expectation value fulfills three conditions. First of all, it has a time in selection and symmetry breaking. So although the Hamiltonian is periodic on time, this order parameter, it has not the same periodicity of the Hamiltonian. It has a periodicity, which is n times the period and this n is larger than one. And then there is a rigidity of the oscillations. So this order parameter has to show fixed oscillations at the required period. Even if without a fine tune Hamiltonian parameters, because we understand that that will distinguish between having a critical point. But if this is a phase of matter, it has to be stable. So it has to have some breadth. So it cannot depend strictly on the fine tune Hamiltonian. And finally, it has to have persistence to infinite time. So these non-trivial oscillations with this periodicity have to survive in the thermodynamic limit. So if we want to go from, let me present the results and then I will explain how we reach. So our idea was, first of all, we wanted to propose because we were not satisfied with the definition. So we couldn't find a good definition of what discrete time crystal should be in an open system. So we wanted to define what it is and to characterize it using the Implant equations. Then in order to do that, what we show and we propose is a proposition that there is a key element, which is the flocate propagator, which defines what the discrete time crystal must be. And this flocate propagator obviously depends on the properties of the limb-blad. So we wanted to dig into the limb-blad physics to understand which properties can bring to a time crystal. And then finally, the other question that we wanted to show was results where there is not need to have a mean field or long-range interactions. This is not needed. However, when we talk about the limb-bladians, we saw that there is a difference between having collective jump operators or individual jump operators. So let me show what is the physics behind all the companies and how we learn it, probably it's not by many of use, but for the ones that they are not used to work in open systems, let me show these different things. So when we have a system as an environment, normally if we have the Hamiltonian dynamics, we assume that the environment, it's the couple from the system and describe all the dynamics with a Hamiltonian, which we know that they use for isolated systems and has a unitary dynamics. They generate a completely positive tree preserving map, as they should be. And we know that if we want to have physics, which is exotic, what we have to do is to engineer the Hamiltonian parameters and in this way we can drive our system to, if the Hamiltonian needs a little bit complex, or at least has two terms in order that there is some kind of competition, and if we engineer in the Hamiltonian, we can have very exotic phases of matter. And often these phases of matter are linked to the fact that we have more than one ground state, or linked to the fact that we have godless transitions and then these correlations and some of these things happen. So the generality of the ground state, it's quite an important ingredient. On the contrary, if we use Limbladians, what we are doing is we are assuming that the system is in contact with an environment and we have to describe the system as a result of the Hamiltonian dynamics of the system has plus the interaction with the path. So Limbladians are used to describe decay and decoherence in quantum systems, which are induced by an environment. Of course, they also generate a completely positive stress-preserved map, which is a very nice thing, and this is the reason that, although these Limbladians have been proposed many years ago, much before Limblad indeed, so it was later when it was shown that this is a completely positive stress-preserving map. And now we have the possibility also to engineering the environment. And if we're engineering the environment, we can change also the properties of the system. And we can also go to these exotic phases of matter, and we know that in these cases, very often, it requires that the Limbladian has more than one steady state. So there is a kind of equivalence between these two things. But there are some things which are very different, and we have a talk today that it was showing. Hamiltonians are our mission. Limbladians preserve hermiticity, but they are not a mission. They can be diagonalized, and this is the way how we know how to deal with Hamiltonians. We diagonalize the Hamiltonians, and then the evolution, we know that corresponds to the eigenstates of the Hamiltonian. If the Hamiltonian is time-independent, and we have the evolution equations, and we know that symmetries, they bring conserved quantities. In the Limbladian, this is a little bit different, because sometimes they cannot be diagonalized. But I'm not going to talk on these cases, which one can solve. And the evolution is given by this typical master equation. And they also conserve quantities of evolution, but they are not linked to the symmetries in the same way. And this is an important ingredient. And as we have seen today, Limbladians are used for a lot of things, in particularly, as it was in the talking this morning by studying on our mission systems. So how is this Limbladian? So, well, this is the master equation. And what I want to show in this master equation, of course, we know there is a first part, which is the von Neumann equation with the Hamiltonian system. So if we don't have time operators, that will be the unitary dynamics. And then we have this expression of the Limbladian, in which the important panel, I want to remark here, is this kind of sandwich jump operator. This is the thing that forces us to use density matrices. And it cannot be written for a pure state, because the phasing makes that the system evolves to a density matrix. And this Limbladian can be grouped in a part, which is an honor mission Hamiltonian. This is the physics that it was today, Masahito talking, plus the sandwich operator. So Limbladians preserve hermiticity, but they are not their mission. And they generate an evolution map, which is this epsilon of t, which is the exponential of the Limbladian. And we know that this map, if the Limbladian is Markovian, can be the compose in this way. And we want to understand the physics in the same way that we do with the Hamiltonian, diagonalizing the Limbladians. And how do we diagonalize the Limbladians? Well, there are more technical ways to do it, but the way to diagonalize a Limbladian needs to use a matrix representation of a Limbladian. I mean, if you look in this equation, you see that this is not a matrix. So we cannot diagonalize this Limbladian directly. And this is a technical thing, but a very useful and very simple once, like everything, once you learn how to do it. So what doesn't mean to do vectorization? Well, we know that the operators, they are also a vectorial space. So what we do is writing the operators, which normally are written as a matrix, as a vector. That means if I have a matrix, which is n by n, I reconvert this matrix in a single vector which has components n squared, is a single column. And what it's important is to see how the operators act in this case. And in particular, how the sandwich operator acts. And the sandwich operator acts in this recipe. So if I have an operator A acting on my density matrix time B, when I vectorize that, this is a factorial product of A times B transpose. This is important because this is how I have vectorized on my vector density matrix. Now my density matrix is a vector of components n squared. So you have a little space of dimension n. The density matrix is just a column vector which has n squared components. And the Limbladian, it's a matrix now, which has dimension n squared times n squared. This is very nice, but also makes some problems when one try to diagonalize it. So now if we want, assuming that the Limbladian can be diagonalized, we diagonalize it. But since this is non-admission and this we saw it also this morning, normally or always, it has different left and right eigenvalues. So when I diagonalize my matrix, I have an expression of the Limbladian with some eigenvalues, lambda mu and eigenvectors, which again, these are matrices written as a vector right and left. And if I'm able to diagonalize my Limbladian and the Limbladian doesn't depend on time, I can immediately see how is the time evolution because when I apply my Limbadian to an initial state, I will have that the evolution is like the projection of my Limbladian on the initial state, of course, with an exponential which will be the eigenvalues of the Limbladian. So now if we want to understand what, but if we want to understand the physics, one way to go is really to dig with a little bit of heart into this Limbladian and to see what are the characteristics of the spectrum of the Limbladian. So the eigenvalues of the Limbladian are either real or come by conjugate plates, because we know that the Limbladian is non-admission, we expect that it will be also, they will have complex eigenvalues as it has. But since the Limbladian or the associated map, it's positive because it has to be a physical map, that means that the eigenvalues that they have a real part, the real part has to be negative. And then if we form the evolution map in this way, so we have a map which is just the exponential of times the Limbladian, we see that the Limbladian and the map, the generator map have to share the same left and right eigenvalues and also the eigenvectors. And now we know also that for any time independent Limbladian, there is always one eigenvalue which is equal to zero, okay? And if we plot that in our time evolution in the map, we see that the right eigenvalue, it's a steady state. And the left eigenvector corresponds to the identity. It's a steady state, although this is a little bit wrong, it's not really a steady state because it can have a phase, but we will call it that this is an asymptotic space. And it can be other states that they don't decay in this way. And this is what we were going to call the asymptotic super space of the Limbladian. So this is formed by all the right eigenvectors of my Limbladian such that the real part of the eigenvalue is equal to zero. One has to take into account that when we do that, the span of these right eigenvectors, maybe they are not density matrices. They don't have to be physical. Some of them can be, some of them they cannot be. So if we know that, okay, we can classify, we can, I have plot here where I have taken from typical plot, how is more or less this spectrum? Okay, and this spectrum we see that we have in the real part, it starts at zero, we have some eigenvalues which they have zero real part and imaginary part. And then there is something which is very important that we call the dissipative gap, which is at which distance, okay? Which is a difference between the real part of the eigenvalues, the minimum of this distance. Because this dissipative gap, which tell us how the system decays. So if we have a dissipative gap, which is very high, very big, the system will be stable in order to decay. Okay, but if it's very small, it will decay. So the important thing, and that will be fundamental to have a time crystal is that we have to have an asymptotic space whose dimension is larger than one. So, and now I say that there is also a big difference between Limbladians and Hamiltonians, which are the conserved quantities. And what we can see is that essentially for each state of the asymptotic super space, which has only imaginary eigenvalues, there is a conserved quantity. And this conserved quantity tell us that the evolution of the state, if we plug now our Limbladian here, we see that the evolution of this state, it's of the following form. It's a linear combination of the asymptotic spaces with a phase, which depends is the imaginary phase of the eigenvalues of the asymptotic space times the conserved quantity. And the other terms, they are of the order minus the gap times the dissipative gap times the time, that means for time going to infinite, this order can be out, which these things in our hands, now we can go through the flocket propagator. And now if we have a time periodic evolution, so we have a Limbladian whose dependence on time, it's periodic, we can define the flocket propagator, which for us is going to be the most important quantity. And the flocket propagator, it's like always is expressed because now our Limbladian has a time dependent. So it has to be time order. It's the exponential of the integral of the Limbladian over a period. And this flocket propagator, it's time independent. So it doesn't depend on the properties that the Limbladian has at a given time t. And this time propagator, we can also vectorize it. And we can vectorize it and we can find also that it has left and right eigenvalues. And since this is non-animation thing, the conjugated which is here written like a double dagger because it's not what we understand by the conjugate corresponds to the left eigenvalues and the right eigenvalues gives us the right eigenvectors, gives us the right epsilon mu, which is the eigenvalue times the eigenvector. And we can define now also, which is the asymptotic space of this flocket propagator. And that will be spanned by all the right eigenvectors that they have modulus one eigenvalue. And well, if you look, this is very similar to what we found before because we were asking that asymptotic super space of the Limbladian, it's formed by those eigenvectors that they have eigenvalue real part equal to zero. So when we exponential like that, that means that the eigenvalue has to have modulus one. And with this construction, we can separate our map into two parts. There is an asymptotic part and there is the decoherence part. And these two guys, they live in different super spaces and they are not connected. So time crystals, we appear only if we have asymptotic space, which is big enough. So this has to have dimension two. So once we have that, now we can define what it's a time crystal. So if we take the flocket propagator, which is the map, which is the evolution map in a dissipative system under a periodic drive, we can define or we can identify that the many body quantum system will behave as an open time discrete time crystal if the flocket propagator fulfills the following. First of all, it has a time translational symmetry breaking. And now we are not going to talk about an operator, but we are going to talk about the eigenvalues of the Limbladian. So it must exist at least one eigenvalue, which I denoted by epsilon star, which is not one. So it's not the corresponding to the lowest eigenvalues, but has to fulfill that this eigenvalue, which when it's exponential to a time n will be equal to one. And that will give me the periodicity of the time crystal. So this is the first condition. It's equivalent or it will revert in having one observable, which is periodic on time with different periodicity than the three driving. Second, what doesn't mean rigidity of the oscillation? Be robust. And we demand that it's linearly robust. So if our flocket propagator changes by some kind of eta times an operator, then we use that the susceptibility, which is the derivative of this value of the flocket eigenvalue versus this changing perturbation has to be equal to zero. Of course, one can say that in order to have rigidity, one should be invariant under any order. So not only linear order. But here we are studying, because it's very difficult, if not what happens, what it's the linear response of our system in front of perturbation. And finally, we have to say what are persistence? What doesn't mean the persistence of the oscillations to infinite time? Well, the persistence of the oscillations to infinite time here will be that we have a flocket gap, which is of the following way. So it has to be like the logarithm of this eigenvalue of the flocket operator divided by 1 over 2. If the flocket time, the flocket gap closes, the system will be stable. So what we have done with this definition of what we believe defines what it's a time crystal in an open system, we have tried for different systems, very simple systems and very complicated one. Yes. So I will go only to the complicated one. And in the case, typically in which we have that time crystal, which has periodicity 2T. In this case, we have to have 2, because this is the periodicity, 2 states in the asymptotic space which will fulfill, first of all, the one which always happens, because n in Imladian has a stationary state, silent 0 equal to 1. And another one, the star which will be e to the i pi, which of course, when we square it, it will give us the identity. And the evolution, well, the evolution, if we know how are the stationary states and we know how are the conserved quantities, we obtain it immediately. And then if we take an operator, and this operator will be the trace of the operator over the flocket, the flocket propagator, we will find that this operator reflects all these qualities of the time crystal. So what is the way? The way is, first of all, we have a system. So we have to find which are the asymptotic states and the corresponding conserved quantities. Then we have to study how is the super harmonic response of the protocol, which will be a kick protocol. And then we have to study which is the rigidity of the oscillations. And if we do that, we will see if we have time crystal or not. So what we did is we took a system which is a short branch interactions, x, y model. This is the phase. I don't want to go into details. This is quite technical. But the first thing that we had to do is to derive the microscopic master equation. And when we derive the macroscopic master equation, we find that we have two kinds of evolutions. One corresponds to collective chance. That means the whole system reacts to the environment collectivity. And the other one has independent jump operators. That means that each of the spins that are in our chain reacts independently to the bath environment. And these two collective and independent jump operators appears in different phases of the phase diagram. Because it depends on how is the energy distribution of the many body system. But when we do that with this definition, we look for the three properties of the time crystal, symmetry breaking, rigidity, and persistence. So here are the results that I wanted to show you. So we take this x, y model. And first of all, if there is, we look for the magnetization, will be our order parameter. We know how is the asymptotic space. We found that there is in the key protocol that we put, we have two states which live in the asymptotic space. This is the first condition to have a time crystal. We look at the super harmonic response of how are the oscillations of the magnetization along the x-axis. And we see that for different perturbations. And we see that in some parts of the phase diagram, they are very stable. This is the first plot. And now we look for the robustness and for the floquette grab. And this allows us to distinguish what phases are time crystals and which phases are not. So when we look for the robustness, we put an interaction and a perturbation on our key protocol. And now we look which is the response of the system. And we see that there is some responses which is zero. So the system is very robust. And this is what we call a time crystal. And we see that when we increase the perturbation and also depending on the part of the phase diagram of the model, we find that our system evolves towards a thermal state. And this distinguishes between a time crystal and a thermal. And finally, we look for the floquette gap, which has to be quite coherent, which is what we find with the robustness. And we see that the floquette gap tends to infinite. So there is no dissipation in the time crystal while it closes and brings dissipation and thermalizes outside of the time crystal. And with that, I would like to thank you. I do finish. And thank you very much. Thanks, Anna. Yes. So are there questions? The supermonic response in your results is this interference pattern that I see? Or is it? No, it's because there is many plots put it here together. Or a period is 10 or something like that? Yes, yes. OK. It's hard to see. It's hard to see, yes, yes. But it's because we wanted to go to very long periods in order to show that this really it's robust, so it's kind of in the thermodynamic limit. I have to say the thermodynamic limit here, it's very small because the ionizing Liubilian, it's very expensive. So. Rosario. So thank you. I wanted to ask you if you think there is any relation between this type of definition of time crystal in the system and those things that were done in noiseless subsystems or decoherence free subspaces? Well, yes, yes, yes. Can you comment on this, please? There is a relation because the asymptotic super space includes the decoherence free super space. So this is clear. So although it's not necessarily the same. So but when especially in all these protocols in quantum information, you want to preserve quantum information when there is noise. You want to be in the coherence of the space. Here there is something related also because I have to say that what we have found is that short range interactions allow to have a time crystal. But the nature of the collective interactions dissipation is important. And this is important because these collective jam operators tends to preserve coherence while individual jam operators tends to kill the coherence. And this is why one has to study which have the conserved quantities. And the conserved quantities depends on the form of the new billion. So what we found it's not important to have long range interactions mean field. That doesn't matter. It's not relevant. But if you don't have it, you have to have stability is much helped by collective jam operators. OK, we have time for one short question from Shane. Yeah, I was wondering what was this condition on the gap, dissipative gap? I didn't understand that. Yeah, the dissipative gap tells you that so a priori when you have Alimbladia and you have two things. You have the coherence and dissipation. So the dissipation gaps tells you how much you dissipate towards another state. So if your dissipation gap is very small, the stability of your system is small. If the dissipation gap is weak, it takes an effort to the system to dissipate to other states. So you would like to have normally, I mean, the Alimbladians are for me very magical because if you look at the Alimbladian like Masajito was looking, so there is a part which is a non-armitian Hamiltonian. And this non-armitian Hamiltonian, it's deterministic. So it acts. So what? So it's acting there and so on and creates all this dissipation. So in some way, if you only have this part, what happens is that the map is not risk-preserving and completely positive. And then you have these jump operators, the sandwich jump operators, which makes all these crazy things that, in some way, you can understand that that helps the map to be still completely positive and a physical map. So these two things, they go together. So they are not completely separated. So this is the reason one cannot study only the first part because if you only study strictly without putting the jump operators, your dynamics will be non-armitian but it will be non-positive risk-preserving. So it will never be a physical map. So I thought the gap is. I think we are out of time and you can talk in private. We thank Anna again. And the next talk will be presented by Khamia. Disappeared a phase diagram of a periodically driven quantum many-body optics platform. Khamia, can you unmute? Or can someone unmute Khamia? Rosario? Yeah. I think I need it. Thank you so much. Now, this worked out. OK, so let me start thanking the organizers for having paved the way to my talk. So basically, I don't have to say what is a time crystal. I don't have to talk about dissipation because all the speakers did it in advance very carefully and very well. And in addition, I can rely on what Anna said before. So I'm not an expert on time crystals. I do for butter and bread study of non-equilibrium phase transitions. But in some sense, this project I'm going to talk about is something that motivated me to do my usual work inspired by dissipative time crystals that are the topic of this morning session. So when we started to look and think about this stuff, the state of the art was relatively, let's say, still at its infancy. So people were asking a very reasonable question, which is, can I counteract, for instance, in preterm all time crystals, the heating, the flocay heating by cooling the system with a cold bath? And this was a sketch that most of you know from this influential work by Bauer-Els and Nayak. And of course, at the same time, there were very reasonable statements like, OK, come on. If you exaggerate with dissipation, you are going anywhere to destroy the time crystal. So how can really dissipation help in such context? As you by now all know, or if you don't, you will discover in the course of today's talks, basically just one year later, Sao and Fernando, had this idea to suggest I was a very simple and toy model dissipation can cook up a time crystal and then later on all the machinery of dissipative state preparation came at full force. And these are a few people that, as you know, already contributed to that. So I want to say that my talk, although talks about dissipation, is going to be in between. So I'm going to drive like matter interactions. So I'm going to do a flocay drive. And I will use photon losses in order to stabilize a driven dissipative steady state. So I want, in some sense, good contractive dynamics that will bring me there, no matter from what initial condition I start with. So in this sense, it's a little bit in between of the two stories. But maybe the true outlier that you should keep in mind is that in my talk, many bad interactions will have unusual role. So normally in discrete tank crystals, they stabilize. They are essentially used in order to entail the super harmonic response. In my talk, they are going to have a destabilizing role. And that's probably one of the major difference. I have to say that if you have taken due consideration to the last slides by the development of Andreas Emmerich, essentially, I've seen already there that by increasing these two body collisions, there can start to be a displacement for the period doubling. You can start to deface from that. So you are, in some sense, already mentally prepared. So depending on the familiarity that you have with the review by Norman Company, basically that's the caranzology of tank crystals. And mostly your favorite examples lay in this part of the table. And this is what we covered yesterday. Today, we are going essentially in the bottom part of the table. If you scroll it down, you're going to see that they talk about large end models or Vicky models. And they mention a fine tuning. And I want to explore a little bit what this fine tuning means. That's, after all, the content of my talk. So the model starts with an inspiration from a paper by Masahito. And basically, it's the old fashioned Vicky model. Nothing particularly fancy so far. So you are periodically driving light matter coupling. So the Vicky model, for those that don't know it, is just the Ising model of quantum optics. You put a bunch of spins inside of a cavity. There is one just single photonic mold inside of the cavity. And the photon can be lost. And you periodically drive light matter coupling in a way that I'm going to discuss in a second. So your problem is a problem of several spins that are collected in this gigantic S. This is just the sum of a value of every individual spin. And if you don't know, you see now a strong light matter coupling. You have an SSB of a discrete symmetry, as I said, is an Ising model of quantum optics. So there is this parity operator that is a good symmetry of your Hamiltonian. But you can get to degenerate ground states if lambda is greater than lambda critical. These reflect, of course, if you look for steady states of this problem. So if you do a quench, and you have losses, or if you politically drive, this was known already, I think since 10 years now, in some pioneering work by Simons and Killing. But what we are going to do here, and this is still belonging, let's say, to the preparatory work to the core of my talk, this is what the group of weather they did, they alternate a bright and dark cycle. So basically, in every parity, you are going to switch on and off light matter interactions. You are going to pick up lambda inside the super-riding phase transition. And you are going, in particular, to select, in a first instance, your frequency of photon and of the spins to be in resonance with the frequency of the drive, such that when you switch on interactions and your dynamics is trivial because just you have the quadratic path, you can basically, when they are in resonance, implement the parity swap. And so these results in the fact that, alternating bright and dark cycles, you are going to end up in a steady state with different parity only after twice of the period. Now, what is the question that usually I pursue in my research is to ask whether there are some genuine many-bodied and equilibrium phases of matter? And there is a way of stressing many-bodied here, although several of you will still think that this model is many-bodied, is that, essentially, the DECA model, if you don't add interactions that are these ones in red, you can do it, of course, in many ways. But the DECA model, plain and simple, as you get it in a textbook, is exactly solvable of mean field. So a large N, you get a gigantic spin. Its quantum nature is completely eradicated. And you just have to solve the classical equation, motion for coherent photon amplitude and for the two angles of this large spin. In other words, if you want to do a quantum simulation of finite N, this is not so much expensive because you have this permutation asymmetry, starting from the fact that every single atom indistinguishably couples in the same way to the photon. But when you add interactions, when you break this permutation asymmetry in the Hamiltonian, and you introduce for instance, a short range of ising type interaction, then you can't benefit anymore of these exact mean fields of ability. And here you start to have troubles. You start to have troubles because you are working now with a periodically driven problem. There is dissipation. So as Anna was pointing out, numerics is expensive. And you have to deal with true many-body interactions. You have to take care of the fact that you are not driving any more classical problem, but essentially the spin waves or any other type of excitations, this channel is going to build up. They will be fitted by this continuous pump of energy. And the system can eat up, in principle, to infinite temperature, even if you have a little bit of losses. So if you want, the question basically here is, are there genuine many-body dynamical phases of matter in this problem? What happens when we add J to the decadent crystal that Masaito worked out a couple of years ago? And one reason why I decided to pick up the specific topic is that essentially, as you have seen a couple of hours ago and as I discovered in December, essentially the group of Andreas took inspiration from these works on decadent crystals and did the experiment. You have seen it recently. There is nothing to add about it in case people connected just now to the conference. All right, so I'd like to go to results first and then talk about methods. So one word about methods so you understand what I'm showing and their reliability of what I'm showing and doing a self-consistent dynamic of spin-wave theory. So I end up composing the many-body interactions in spin-waves, but equilibrium spin-wave theory will break down. So I do a self-consistent loop between the motion of the collective spin and the spin-waves that will be generated by the jator. This work I'm going to tell you later. What you need to check is that at the time of which I'm showing you the plots, the density of spin waves remains small. If this happens, not shown yet, then you can trust the results. These are your first, let's say, truth at your dynamical phases. I have on the y-axis dissipation and horizontal axis many-body interactions in units of the frequency of the drive that therefore doesn't play any role. And basically, you can see a couple of behaviors that you expect. First and foremost, I introduce you with this DKTC that essentially is, in essence, the same that Masaito was finding. And it is a very old-fashioned period doubling in the firm that all of you have seen so far. So nothing, let's say, to comment about that, there is, of course, a region in which you don't have any dynamical order. It's just a mess. These points are stroboscopic points on the block sphere for the trajectory of your collective order parameter. You have to imagine that now, though, you have a finite j. So the motion of the collective order parameter has an internal depth of excitations or spin waves generated by the j-term. And what can happen is that, at a certain point, when you exaggerate with the story, with interactions, j, but you don't have enough dissipation, you may end up heating. Essentially, you are not counterbalancing enough your diode drive and you're going to eat up at infinite temperature. To be very frank, this heating region with h here is essentially a region where we don't know where the problem is going, because our metal breaks down. So this h is more something like in late in Ixunt Leonis, you can't go there, there are lions. We don't know, but as all of you in this conference are aware of solving a tremendous dissipated many-body problem is challenging. So we will, let's say, think about other methods for the future in this part of the phase that. And of course, I agree with you that probably, also this limit is in some sense, expected if you over-damp your problem, you're just going to get against some sort of boring or adaptation, this is over-damping. Perhaps if we still, let's say, confine ourselves as Moj, observations here are that you distinguish these transitions with different order parameters. Here, what you have is a broadening of the Fourier signal that goes from being sharp like in a tank crystal to something most parse, as you would expect for a signal like that, while here, instead, I'm using essentially Sx. So what's going on by intuition? So why bother in writing a paper? There is an appetizer and then there is a main dish and the appetizer is this metastable tank crystal. This metastable tank crystal, as you're going to see, is essentially a very long-lived period doubling before collapsing into an over-damped situation. So somehow interpolates between from the phase from which it comes from into the phase where it ends. So I'm going to talk about it and let me, let's say, recapitulate therefore that if you don't have spin-spin interactions, if you don't have J, the regime in which we top a lot of spin waves is going to disappear, that's natural, but also this metastable tank crystal will not appear. And this is what I'm going to show you in the next slide. So that's an ugly picture. Oh, now your collective order parameters look like if essentially you are going a very long time. And it's ugly because I want to show you that there is a damping. There is essentially basically a very slow decaying envelope. And if you look though, essentially at some short time window, you can't figure out that the signal is damped. You're going to have this period doubling and you are going to not find any difference from what I was showing you before. That's why I'm portraying here these huge time scales. And this happens, of course, at every, let's say, each interval of time that you are portraying. So you will always see this period doubling now, questioning of what provokes the metastable dissipative tank crystal. And let's start from saying what is not first. It is not a perturbative phenomenon. It starts at finite J, small but finite J. It is destroyed by many bad interactions. So it certainly, this makes sense from the way in which I'm building my discussion, but maybe more importantly, from those of you that are familiar with protermalization in Floquet problems, this is not the result of a high frequency expansion. That's not the theorem by Dima in company. Frequency is just an overall energy scale in such a case. So if you look more closely at your key quantities that have the spin rate density, and for instance, what the photon is doing, you can see that at a certain point there's stuff to be a motion encounter phase between the photon number and the spin rate density. And as you will see later, they actually couples to each other when you solve your equations of motion. So probably the fact that they keep some sort of antiphase is at the reason that you get slowly out of resonance. And this is going to kill your metastable tank crystal. If you are wondering whether, let's say, these transitions here are sharp or not, well, again, this is not an order parameter, certainly, but spin rate density is having a sharp jump at the onset of the MTC. So probably there is really, let's say, something substantial happening there. Also, the lifetime immediately feels that you are switching on many bad interactions. So this is for the aficionados that like to ask the question how long time it lives. That's roughly what is happening as a function of j for the lifetime. OK, so let me say that if this was, again, I mean, I like to build and to drive you for the same way in which we went for this project. Again, if this phase diagram here was the end of the story, basically you could tell, well, your method maybe is not in the essence perturbative, but the results are perturbative. And what I'm going to show you now that if you play with the sign of j, if you play with the sign of your many bad interactions, you can get a strong coupling, a driven dissipative tank crystal, when j is negative. Basically, now this scale is maybe a little bit exaggerated. What I'm trying to say here is that there are regions, not fine tuned point, there are regions, a finer dissipation with j negative, where you get essentially, again, this decay tank crystal. So I'm not going to show you again the plots. They look all the same. And my point is that what is happening is you have changed from anti ferromagnetic to ferromagnetic spin waves scattering. And as I told you the beginning, the decay tank crystal would like to favor a sort of ferromagnetism in the cavity. So if you're true, many body collisions have the same notion of order, they're going actually to give you resilience to many body eating. And they will not fight it. And in some sense, right, that's probably most relevant part is a non-perturbative dissipative phase of matter that lasts very long. And if you want to see what happens to your spin wave density, that's a figure of merit. So I remember you that the collective order parameter that I'm plotting for the decay Tc is self-consistently coupled to your bath of Gaussian spin waves. And you want essentially to be sure that your density of spin waves doesn't go too much, is normalized to 1. And what you should appreciate here is that by changing a one order of magnitude from anti to ferromagnet, you can get a drop of, let's say, a significant drop of a factor of 2 in the renormalized spin wave density. So this would be essentially already a signal of the fact that you are heating up too much. This is instead telling you that you are behaving similarly to what you will get here at very tiny j, but anti-ferromagnet. So that's if you want a quantitative statement besides just showing you a cartoon of what is going on. And basically, now depending on time, I would appreciate if the host can tell me how many minutes do I have still? Seven minutes. Very good. I can talk about the method, which is relevant to understand how you can trust this entire picture. So that's just a rewriting of the model, maybe a little bit messy. What I want to clarify is that we have the coupling of our spin system into the motion of the collective order parameter. So basically it is big s. And then we decompose these short-range interactions into a bath of spin wave. As I told you, they are self-consistently coupled. And I'm going to show you the equations and details of the derivation in a moment. And they generate, you want a sort of internal bath quantum fluctuation for the order parameter. So you are getting quantum noise and spin wave scattering from this j-term. But at the same time, the order parameter has to deal with another bath. That is the photon that is called. And it talks intermittently to the photon, right? Because as I told you, this lambda t has switches from bright to dark cycles. So the order parameter has to compromise essentially between these two effects. Now, this is, again, an heuristic picture. So for a fascinados of, I want to see the details, let's go to the point. So how do you do time-dependence spin wave theory? Well, if you do ocean and permaculture equilibrium, as you know, you have a fixed quantization axis that is, for instance, z. And you can allow only small displacements around z. These are your bosonic excitations, your collective excitation of the spin texture. But of course, if your collective magnetization starts to move around, there is no chance to find a good vacuum for that. So you have to go to a co-moving vacuum. This is, of course, very well known for people doing BCEs, a time-dependent global liberal transformation. You need a moving vacuum. You need, essentially, time-by-time to define your annihilation and creation operatives. So basically, we do an ocean and permaculture low order justified by the fact that the density of spin wave would be negligible. But we do it in a co-moving frame, capital X, Y, and Z. And this is, if you want, a time-dependent frame. You're jumping time-by-time on top of the collective order parameter. You have your new vacuum. And this spin wave coordinates are in that frame. The fact that they change time-by-time, your annihilation and creation operatives, together with vacuum, will give you a self-consistent loop. The self-consistent loop translates in a mess. And I'm going, let's say, to try to guide you through the structure of the equation. So of course, if you look at them in this way, even if they'll you what are the terms, you're going to get lost. But let's do it. So we have a zymutal and polar angle of your collective spin. This is theta phi. And as you can imagine, of course, you are going to have also your dR photonic amplitude. This is not particularly shocking. There are no linear equations. This is, if you throw away all the j-dependence, this large n-exact mean field solvability I was talking about. And now you're going to have a feedback, right, from the spin waves. And they are going to couple to the collective order parameter theta phi by a discriminant small delta. Small delta is just a summation of several capital deltas that are quadratures of your Gaussian spin waves. And you get this, let's say, feedback because basically you need to solve the motion of the order parameter that will solve the motion of these guys that will feed into this equation. And you have to solve them forever and ever in a self-consistent loop. There is essentially a missing ingredient here. If this epsilon is your spin wave density, you want to be sure that, let's say, an elastic scattering between spin waves is negligible, or better to say, it happens very late in time, so that you can keep yourself with Gaussian theory of spin waves. And that's why you can solve essentially a problem for very large system sites, 100 to 100. You name it. That's not a problem. You're still solving a linear problem after. All right. So, and then, quick advertisement. This business of time-dependence spin where theory is not something we developed for this project. It's a long line of research that we started with Alessio LaRose in Trieste back in 2017. And they're still continuing to give us satisfactions. That's a recent work still on dissipation, where we get also a little bit of dissipative limit cycles, but they are transient. With Sao Fazio, the accent is completely on something else. It's on dissipative state preparation. But please check, let's say, the version that you like more, crunch, periodic drive, or dissipative state preparation if you want to see something different from time crystals. We have a little bit of time swap. So I wanted to advertise a nice poster by Shane Kelly, who essentially has found a new way to generate on demand a period doubling, a period end-plates using the linear resonances in the phase space of long-range interacting. Magnetti has a very nice analogy with the aliasing that is these efforts that you see the blades of the helicopter like static, although they are moving very fast. I didn't know it's collected in English. But the poster was yesterday, so talk to him. He is around. Just check out the paper. We are always happy to discuss and confront. Basically, I believe I'm done. This is my last slide. These are the three responses that are generally many body in decantine crystals. And I'm open for questions. Thank you, Chamiah. It's a moment. There are no questions. So let me ask a question. It's difficult for me. I'm not a quantum art. Oh, Rosario and Odette. Odette first. They saved you. You are muted, Odette. Yeah, you have to unmute, Odette. So some organizers. Was unmuted and now it's muted again. No. Oh, OK. Chamiah, I am jumping to the rescue. We're going to anyway talk a bit more on Thursday. So thank you very much for the nice talk. I mean, there is an idea that I'm also going to pitch tomorrow in my talk that at a certain point, your many body system will develop due to its interactions or whatever it is that you are going to throw at it into a new normal mode. This new normal mode, if you do ED on your system, right? Your new normal mode will have a completely different eigen frequency associated to it. And now you're going to apply time-dependent drives on it. Could it be that whatever it is that you write there is H, it just means that your new normal modes are now overlapping strongly with one another and then they start to kick each other in a different way that you're not just seeing a collective single mode that goes under period doubling by forkation? Yes. That's something that can happen there. I can kill the question in this way. I'll just tell you that, for instance, the fact that you don't just see the detectant crystal hitting, but you can see other stuff was already in the experimental data or in the TWA slide by address, where it was seen that increasing the many body interaction in Boston was seeing a defacing from the period doubling. There can be a lot of stuff there, a lot of stuff. Simply the metal can capture it. So I mean, I have also, of course, ideas to go there and to see what stuff is there. But sure, what you describe is completely possible. OK. There was Mario. Jomita, I do not understand how is it possible. I don't know if I understood correctly or not to have a crossover between the metastable time crystal and the decay time crystal. I mean, it should be a transition because otherwise there would be metastability all the way down or not. Let's take a slide again. So you are worried about which part? The transition from here into the metastable time crystal. It's a transition, not a crossover. Did I understand correctly or not? OK. So if you are a fissionado for the transition, this is what the spin rate density is showing you. It is jumping abruptly when you go from the DTC to the metastable time crystal. The question for me, which is the order parameter. And I don't know the answer. So that's why I was cautious. And I was putting a dotted line. But I can accommodate both my, let's say, cautionary words and your intuition. That's a jump if you like it. Thank you. Can you connect what you computed to the worker we heard in the morning from Andreas? That's exactly what I was trying to say in my answer to that. That basically, they have this last slide, right? When they increase, I think they call it ECE, two-body collisions. That would be probably my J term if I got it right. And of course, it's very unreasonable and unlikely to think that if you change J a little bit, you're going to quick crossover into melting. You're going to crossover in no time crystal. You're probably going to crossover into irregular, probably. And actually, this is indeed what we see in TWA. So maybe if I have to make one last remark, is please, guys, really take these H as unknown regions, region where the method failed. There can be new, let's say, collective states at strong coupling, like Odette was saying. There can be just irregular dynamics. We simply don't know. OK. So thanks again for coming up. Thank you. And we move on to the next talk, which is about is from Vido Homan. And it's about Higgs time crystal in a high-TC superconductor. Can you? Right, yes. Here we go. Now you can hear me and see me, hopefully. Well, then I may just share my slides. And all right. Can you see the first slide now? Yes. OK, perfect. OK, yeah, thanks to the organizers, first of all, for having me at this very nice conference. And as announced, I'm going to talk about our proposal to make a time crystal out of a high-TC corporate superconductor. And first of all, I would like to acknowledge my co-authors, Jason Kosner and Ludwig Matai, who made the work on this project a very enriching experience. So with this, I would like to move on to my actual talk. So first of all, I'm going to introduce the effective model that we use to describe light-driven dynamics in corporate superconductors. And then I will show you the dynamical phase diagram that we have mapped out for driving the Z component of the electric field. And then I will focus on the discussion of the time crystal that we find in this dynamical phase diagram. And in fact, this time crystal appears to our model and by their corporates. So let me just mention a few aspects of corporate super conductors which are relevant for our model and for the mechanism that we propose. So here, the electrons from tightly bound cooper pairs primarily in the copper oxide planes and characteristic excitations of these systems are Josephson plasma oscillations corresponding to cooper pairs oscillating between the copper oxide layers. And these Josephson plasma oscillations are only observed in the superconducting phase. And they are linked to the breaking of u1 symmetry. So when the superconducting phase is entered, the order parameter takes a finite value and thereby spontaneously picks a phase. And generally, then there are two modes of exciting, the order parameter. So first of all, there is the phase mode indicated in blue here. And then there is the amplitude or Higgs mode indicated by a red arrow. And while the phase mode is gap less for charged new field system in a superconductor, it acquires a gap because it couples to the electromagnetic gauge field. And therefore, in these systems, we have two collective modes which are both gapped. And as superconductors, approximately particle holds a metric and low temperatures, these two modes are linearly independent. But there is a cubic coupling term between the two modes in the Lagrangian that allows for Higgs excitations by two photon processes as sketch here. All right. So in our description of the light-driven dynamics in the Coupreds, we focus on the interplay between the Cooper pair compensate and the electromagnetic field. And for this purpose, we have formulated and effect the field theory whose static part is of the Ginsburg-Lunder form. And then we added dynamical terms for the order parameter and for the gauge field. So to compute the time evolution of the order parameter and the vector potential, we discretize our theory on three-dimensional letters where the order parameter is on the bonds and the vector potential is where the order parameter is on the side and the vector potential is on the bonds. And alternatively, you can also see this as three-dimensional letters of Jolyson junctions. And to account for the anisotropic structure of the Coupreds, we also choose anisotropic letters parameters. So in particular, the in-plane Josephson coupling is much larger than the interlayer Josephson coupling. And then we also include dissipation and thermal fluctuations in the equations of motions for both fields and such that we can simulate the dynamics at finite temperature as well. All right, so once this whole model is set up, there are now different ways to drive the system. And for the remainder of this talk, I will restrict myself to the case where we add a driving term to the Z component of the electric field, which induces plasma oscillations along the C-axis. And first, I'm going to discuss the results for a monolayer system where the C-axis parameters are uniform. So dn equals dn plus 1. The same goes for the tunneling coefficients. And in the first step, we have then scanned the dynamics for different driving frequencies and different field strength. And in doing so, we obtained the dynamical phase diagram of this system. So here, we found three phases. First of all, there's the normal phase. Then there is a heating regime where the condensate is entirely depleted. And most interestingly, there is this time-trystalline regime that I will characterize in the following to point out how the dynamics here is distinct from the dynamics in the normal phase. So what happens in the normal phase? To see this, I would like to show you an example for one particular point in the dynamical phase diagram indicated by the cross. And as you can see, the amplitude of the order parameter oscillates with twice the driving frequency here, which can be expected due to this cubic coupling term in the Lagrangian. So here, we have a super harmonic response of the order parameter. Now let's get to the time-trystalline. So first, you can note that this regime appears around some of the Higgs frequency and the Joltsman-Blasma frequency. OK, why doesn't it go to this? No, perfect. And once again, we pick one example here. Now we are at the bottom of the red time-trystalline term. And as you can see here, the amplitude of the order parameter oscillates at a frequency that is notably smaller than the driving frequency. To be more specific, the amplitude of the order parameter oscillates at the Higgs frequency so that in general, the sub-commonic response is incommensurate with the drive. And as you can see, this behavior appears for a wide range of driving parameters. So it's not just the fine-tune point in parameter space. And I will discuss its robustness against thermal fluctuations later in the context of the bilayer system. But first, let's elucidate a bit the mechanism behind this sub-commonic response. So as I've shown you, normally the Higgs mode oscillates at twice the driving frequency and the conventional resonance condition reads omega drive equals omega H over 2. And this resonance condition has been exploited in a couple of recent experiments to detect the Higgs mode. Interestingly, the time-trystalline that we propose is also a resonance phenomenon. And it originates from the same cubic coupling term in the Lagrangian. Here, the optical drive resonantly excites the Higgs mode and the Jolson plus-marmode at the very same time. And this leads to this very distinct dynamics that I've already shown you and which is further highlighted in the power spectra of the amplitude motion. So in case of the conventional Higgs resonance, there's only a single sharp peak at twice the driving frequency. And in contrast, for the time-trystalline, there are several sharp peaks below 2 omega drive. And the dominant one is at the Higgs frequency. OK, now I would like to switch gears a bit and talk about bilayer group rates. So in order to capture this geometry in our model, we now introduce a staggered set of junctions along the C axis. And therefore, there are now two Jolson plus-marmodes in the system. The lower plus-marmode corresponds to oscillating supercurrents along the weak junctions, while the upper plus-marmode is associated with currents along the strong junctions. And the typical bilayer group rates, the Higgs frequency, is in between the two plasma frequencies. All right. So the first question to ask is, obviously, can we find the Higgs time-trystal in such a bilayer system? And the answer is yes. We find it at the sum of the upper plasma frequency and the Higgs frequency. And before I discuss the results here, let me emphasize that now and in the following, we consider the numerical results for a large three-dimensional system of about 10,000 sides and at final temperature. So in this example presented on this slide, we have chosen a temperature of roughly 10% of Tc. And we chose a fixed driving strength and only vary the driving frequency and observe the dynamics for different choices of the driving frequency. And to spoil it already, the sum frequency of the upper plasma mode and the Higgs frequency, in this case, is 22.4 terahertz. And as you can see, the spectrum that we observe for the auto parameter resembles the spectrum that I've shown you for the monolayer case before. So again, there is a very sharp dominant peak at the Higgs frequency. And there are a few other characteristic peaks below twice the driving frequency. And when we detune the frequency from the resonance condition, then the peak at the Higgs frequency becomes notably smaller and broader and also the other characteristic peaks vanish. So unfortunately, the auto parameter itself is not very easily accessible as an experimental observable. Is there a question already? Oh, five minutes. OK, thank you for the sign. So therefore, we also look at the dynamics of the super currents along the C axis. And indeed, we also find a signature of the time crystal here, so which is the emergence of two side peaks at the driving frequency plus minus the Higgs frequency. And then in the following, we use the height of the blue detuned side peak to quantify the time crystalline fraction at various temperatures. And as you can see here in this plot, the temperature dependence of the time crystalline order is similar to conventional phase transition. And we find that the sub harmonic motion is robust against thermal fluctuations up to roughly 20% of Tc. And let me mention again that here we are really considering a large system with long range coulomb interactions. So what happens microscopically in this large system? To show this, I would like to play a short movie in which you see the supercurrent field within a single DC plane. And here we have used red emeralds for the currents along strong junctions and blue for the currents along weak junctions. And here I also show the integrated supercurrents in each row and also integrated along the third dimension. And for comparison, there is the drive, which in the normal phase is just in phase with the currents. All right. So let me start the movie. So you see that despite the thermal fluctuations, there is a collective motion of the supercurrents. And the sub harmonic response of the Higgs mode here manifests as a breathing motion when you compare it to the drive. So this is another nice insight into the many-body nature and the rigidity of this effect, I think, exactly. And what we also did then was to check if this unusual dynamics could be observed in a pulse experiment because continuous wave lasers in this low-terrain regime with the strong intensities needed for the time crystal are not yet available. But potentially, one could excite the sun resonance in pulse fashion. And in fact, we found that also in the transient response here on the picosecond scale, that the supercurrents have this signature of the time crystal with just the appearance of the two side peaks at omega drive plus minus omega H. All right. So I would like to briefly summarize what I've told you. So first of all, we have mapped out the dynamical phase diagram of Cooperate superconductors for driving the C-axis electric field. And in doing so, we have identified a time crystalline state that is induced via sun resonance of the Higgs mode and the Josson plasma mode. And as I've shown you, this is a robust many-body effect that survives in the presence of thermal fluctuations. And with this, I would like to thank you for your attention. And I would be happy to answer your questions. Thank you, Gudo. Other questions? If not, I have a question. Your scattering is basically between the plasma waves that are made by Cooper pairs, right? Yes. The Higgs mode, which is some sort of the strength of the density of your Cooper pairs, is it? I would say both modes are fundamental excitation of the Cooper pair condensate. And you should maybe think of one of them and more as this amplitude mode and the other as kind of a phase mode coupled to the electromagnetic field. OK. And then it's a sort of Raman scattering between the two. Yeah, I think these pros and cons are reminiscent. So I think one can see it in a Raman picture. And if you have the time crystal, they must, one frequency must be commensurate with the other one to get it or? Well, that is not really. Some of the two is a third or is a multiple of a third. Well, these peaks in between. Well, as I've shown you, the motion of the amplitude is really different in this time crystalline regime compared to the normal phase. So where it always oscillates, it tries the driving frequency. So I would argue that this is a sub-anomic response of the system. I understand that. But aren't the frequency of these modes usually intrinsically determined by some material parameters and they will be generically non commensurate with each other? Well, yeah, the material itself has a well-defined plasma frequency and a well-defined peaks frequency. So what you need to do is, in order to create a time crystal is to choose the driving frequency as the sum of the two and this is generally possible. OK, good. Are there further questions? If there are no further questions, thanks again, Guddu. And we go to the final talk of the morning session. And this is by Gidius Slavlis. And he will talk about engineering time space crystalline structures. OK, hello. Can you see me and hear me? We can see you. We can hear you. And it's great. And now I have to share the slides. OK, can you see the slides? Everything's all right. Go ahead. OK, thank you. So thanks for having me here. And today I'm going to talk about how to engineer time space crystalline structures. So this talk is based on a recent paper that we wrote with Gidius Slavlis and Krzysztof Sacha. So first of all, I need to introduce the notion of the time space crystalline structures. So these are systems that are periodic both in space and time. So the spatial periodicity arises from a periodic spatial potential. For example, it could be an optical lattice. And the temporal periodicity is engineered by choosing a proper resonant and periodic driving of this spatial lattice. So for example, if we have a one-dimensional periodic potential and we drive it in a suitable way, we effectively form a two-dimensional lattice, which we call a two-dimensional time space crystalline structure. So in other words, in this case, when we drive a spatial dimension, we equip it with another temporal degree of freedom. Now, to make this construction explicit, let us take the example of a one-dimensional driven potential and see how this temporal structure arises. So first, let's say we have the classical case because it's good to build intuition. And also, it is easier to determine the parameters from the classical case for the quantum case. So we start with Hamiltonian of the system, which has a sine squared potential that is driven side to side along the x-axis periodically with some cosine driving of amplitude lambda or another word so we can perform a canonical transformation to the oscillating frame of a lattice to separate the time-independent part and the periodic time perturbation part. Another trick that simplifies the analysis is moving to the action and angle variable. That is, action is defined in terms of a static Hamiltonian part where its value is basically the area enclosed by the classical trajectory of a particle in the momentum and position phase space at a certain energy and the angle is just a canonical conjugate coordinate of the action. So what we do here is just rewrite the Hamiltonian in some convenient form. Now, the next step what we do is we choose the driving frequency to be equal to the resonant frequency that is driving frequency is an integer multiple of the natural frequency of our system. So S determines at which resonance do we drive our system. And now what we can do is we can move to another reference frame, which is moving along a resonant orbit and we get some Hamiltonian. We time average it and we see that crystalline structure appears in this reference frame. Now, why do we call this temporal coordinate in the sense because if we take it, if we measure, well, if we fix a detector at one of the lattice sites we will measure periodic directions of particles due to this periodic motion or static lattice in the rotating frame. Now, it is important to stress that we have this structure at each of the spatial lattice sites. So effectively, we have a two-dimensional system. Now, to see that our approximation was valid, we can plot the solutions for both the initial Hamiltonian and the time average one in the action and angle phase spaces. So what we see on the left side is the solutions of the equations of motion for the first Hamiltonian. And we see that the resonant action, some resonant islands form where on average one particle spends a period in each island, a driving period in each island. And we see that the effective Hamiltonian effectively reproduces the same motion around the resonance region. So this is OK. Now, another point I need to mention is that in order to have the resonance structure that we need, we essentially need rather deep lattice potential, which means that if we have a deep potential, our tunneling can be at the ground state would be strongly suppressed. So we have to work near the top of our lattice potential. That means we have to consider, as our basis, the highly excited energy states of our system. So keeping this in mind, we can move on to the quantum case. So in the quantum case, we calculate it to obtain the hopping parameters between the neighboring spatial lattice sites and between these temporal sites. To solve this problem, we essentially take the Flake Hamiltonian, we time average it, and we obtain the eigenstate solutions of the time averaged Hamiltonian. And from all of these solutions, we pick the ones that correspond to the resonant energy of the system. So having these resonant states, we can construct a near state, a localized state, which will act as the basis of our time-space crystalline structure. Now here, we have a picture of the solutions for the s equals 3 resonance case. So on the x-axis, we have 3 spatial sites, and each picture represents a time snapshot. So the first picture is taken at some initial time t0. The second picture at time equals half the period of the driving, and the third one is after the full period of the drive. So when we plot the Flake eigenstate of this Hamiltonian, which is depicted by the black curve, we see that it respects the periodicity of the drive. That is, after one period, we get exactly the same densities. But now what happens to these localized wave packets? We see that they take three times as long to complete a full period. Now here, we only depicted a single linear state per site, but in actual neutrality, we have the number of these linear packets per site is equal to the resonance number. So if we look at the evolution of these linear wave packets at a single site, we would observe something like this. They just periodically oscillate back and forth. OK. So having these linear wave packets, we can use them to build a orthogonal basis for our time-space crystalline structure. That is, we can construct a tight binding model. So now here, we have annihilation and creation operators in the second quantization form with some hopping parameters where the Latin indices indicate the spatial direction and the Greek indices indicate the temporal direction. So in the temporal direction, we have as many sites as the resonance that we consider. Also, it is possible to include interactions in this model, but they have to be sufficiently weak to not couple the resonance states with the other states of the system. So this basically completes the effective description of our two-dimensional time-space crystalline structure, which appears due to the periodic driving of the one-dimensional system. But we can extend this to higher dimensions. So if we take a three-dimensional optical lattice and we assume that each spatial direction does not couple, our Hamiltonian of this lattice would be just a sum of the Hamiltonians of each independent direction. And of course, if we drive each of the direction properly, we in the end have a six-dimensional state which consists of a product of each of the eigenstate of a Hamiltonian. Well, so we use this when your states effectively build a six-dimensional tight binding model. Okay, so we have this high-dimensional system. What can we do more? Actually, we can engineer topological effects. So the main idea is that we can induce a tilt in that temporal direction to suppress the hoppings along the temporal direction and then restore those hoppings with a two-fold on process which allows to imprint a flux along those temporal, temporal, well, a phase along those temporal lattice sides. So now we have an effective two-dimensional system and flux piercing each of this system blockets. And what this allows to do is to create, for example, ribbon which possesses this dispersion relation with topological edge modes. And if we plot the densities that have fixed momentum for a six-dimensional crystal, we actually see that, well, since we can't really plot efficiently a six-dimensional density, we take the standard on-site density definition and we trace it over the spatial parameters. And we see that at the edge of the time direction, we have this formation of edge states. So yeah, this is basically it. We can, well, combining time and space crystal and structures is possible to realize this time, space, well, crystal and structures. The single particle and midi body can does matter phenomena demonstrated already in time crystals can be also realized in six-dimensional time space crystal and structures and these structures can be endowed with the gauge fields which allow to probe topological effects. And so thank you for your attention. Thank you. The paper is open for discussion. I have a question concerning hopping in a space direction. If I have hopping in space, I can hop right and left. There's causality in time. I can only hop forward. I didn't succeed so far in hopping backward. Does that make a difference? I mean this, okay, so essentially we have an effective structure of free wave packets oscillating back and forth on that single lattice site. And this structure, when these wave packets overlap, the particles can hop from one to the other. So this is the temporal direction in hoppings, actually. So it's not like hopping in time forward and back. Okay. Can you elaborate how that, then what is hopping in time? Hopping in time would probably be these overlaps that we see or we have these wave packets that overlap and moving around. But when we tam-average it, we have some overlaps among these wave packets, which essentially give us this hopping along the time direction. Well, temporal direction. It's not really time, but effectively it's described. Well, it appears due to this resonance driving. There are further questions. If not, let's thank all the speakers of this morning and I hand over to Rosario. Okay. So just to say that, so first of all, thank you, Thomas for sharing the session. And so we start this afternoon at half past two. So now it's lunch break or depending on the time zone and see you in roughly two hours. Okay. See you. See you. Hello, Rosario. Hi. Good. I am at, as a place. Okay. It's all, I think it's good. There was something I wanted to ask you, but then now I for, if we got. I forgot, yeah, sorry. There is still some time, you know. Yeah, sorry for you. So at the end, yeah. So the poster session worked out fine yesterday. I mean, this was my first experience of this kind. The only bad thing is that those people that have the poster, they cannot move around. This is the only, so limitation. I don't know if this can be solved. Actually this poster, first of all, the number of posters was not that big. It was good because, you know. And I was, recently I attended poster session where there were breakout rooms and inside breakout rooms, there are other breakout rooms. Total mess. Total mess, you know. When you enter, you know, the last breakout room, nobody can enter, you know, as a second person. It was disaster. But this, in this way, no problem. It was very, very, very efficient. So let's see, probably we can, Valter to say. Okay, we should, I mean. Probably you can admit people, right? So. Yes, I will admit Kuba Zakrzewski. Joe Marko. Okay, Kuba. So I think Marko Skiro is. Yes, I tried to unmute him and to unmute this. I unmuted myself. Hello, hello, Rosario. Hi, hello everyone. Thomas, hello. Hello. Okay. I tried to call you, Krzysiek, but host is not letting me in. But you didn't reply to the phone. Okay. Okay, we are neighbors. So that you can even shout. Yeah, well, I normally, I don't have to shout. It's enough that I speak normally and Krzysiek knows everything what I am saying. These are new walls. Marko, did you check the presentation? Yeah, I did the, I did the check. So it should be fine. Whenever. Kuba, you should remember that it is recorded and live streaming. So that it's recorded. So I shouldn't say my usual words, right? Sorry for interrupting. Can we make a test briefly with Professor Lesanowski? Thank you. Yeah, Paul. Hello. Hi, good morning. Hello. Oh, good afternoon. You're welcome. Hello. Let me just share the screen, okay? And then we see where this works. Just in sec. All right. Not here. Okay. All right. Do you see my screen? Yes. Yes. Do you have audio or videos in your presentation? No, no, no videos. Oh, sorry. We are done. Do you see the pointer? Yes. Perfect. Okay, so then we are good. Okay. Thank you very much. All right. Thanks a lot. See you later then. Yeah. Yes. Saros tells me I'm still meeting Professor Taheri. Okay. Okay. Walter, I would like to ask you to make Jakub Zakrzewski a co-host because then he can mute and unmute himself. Okay. Yes. Where is he in the waiting room? No, he's in the meeting. It's Jakub Zakrzewski with a... Zakrzewski. ...Zeta. I'll pronounce it in Italian. Ah, okay. Zakrzewski. Okay. You will be the chairman. Thank you. Dan. Thank you. Thank you. And the second speaker is Hosain Taheri and this guy is from U.S. and probably he will join us just before his talk. So that hopefully everything will work. Hopefully he will wake up. Yeah, this is the answer for the point. This is relatively easy, right? So it's what, eight o'clock or seven o'clock for him? May I make the test also with Professor Das? Since he's in? Sure. Professor Das, can you hear me? So don't worry, Walter. If I... He's amuting. Ah, okay. Hello, sir. Hello, excuse me. Can we make the test? Can you please share your screen? Okay, okay. One second. Do you have any audio or video in your presentation? Yeah, I have one. Okay. This is important to check. No. No. See your screen. Okay. You can see it? Yes. Yes. But I can hear you very well. You are using earphones with mic or you're using the microphone of your computer? That I'm using the microphone. Can you hear me? Now it's okay, I think. I will go on my computer. I mean, I hear well. Okay. Just talk a bit louder. Thank you. Okay, let us check the audio and the video you have. Probably. But did you have a video in your presentation? No, sir. There is no video. Ah, there is no video. Sorry. Okay, so we are done. Thank you. Okay, I am... Thank you. So may I ask Professor Fernando Yamini to... Okay, here we are. Let me unmute you if we can make the same test. Please share your screen. Okay. Now, can you hear me now? Yes. Hi. Okay. So how do I share my presentation? My slides presentation is just... You should... Are you using PowerPoint, Keynote? Just the PDF. PDF, okay. Open your PDF. Then go to the Zoom menu bar on the green button. Share screen. And then you will see you have several options. You have to choose the software you are using to show the PDF. Okay? Okay. I chose the PDF that I'm opening here. Can you see my presentation? Yes, you can see. You don't have any audio or video in your... No, no, it's just this PDF, yes. Is it possible? This is not the slideshow mode. No, no, let me just... Let me see if I can put in this slideshow. Maybe it's this one here. Okay. Okay. Fantastic. Can you also see my mouse and my pointer? Just work as a point? Okay, so that's great then. Yes. Okay. Perfect. Thank you very much. Okay, thank you very much. Okay. So Fernando, you can stop saying so that we... Yeah, I'm trying to close my PDF here. Just a second. So Professor Cabalt, can you hear me? Yes, yes, I can hear you. Hello. Sorry. Can we try, please, your screen sharing? Yes. Okay, bye-bye, everybody. Okay. Can you see it? Yes. Can you launch this slideshow mode? Okay. Perfect. Do you have any video or audio in this presentation? No, I have no. Just a second. Okay, we are done. Thank you very much. Thank you. From our side, except for Professor Taheri, we made all the tests, Shado and Frisco. Very good, I think we... For him, it is 526, so I think he's still sleeping. Yeah, I think... I mean, who gave him the talk at 6 a.m.? It must be Krzysiek Saha, because he gets up at 6, but I would not accept the talk at 6 a.m. So Lata, I think we can admit people, everybody. Okay. Are close to start. Let me do it. Done. So we have still nearly three minutes. Maybe let's wait one minute more. Oh, it's better to wait because people appear at the time and then they come in the middle of the ceremony, especially in the morning. Mm-hmm, mm-hmm, mm-hmm. Okay, I think that the number 38 is stabilized. So I think we can start the session. Share my screen. Yes, Marco, Shiro from College de France will... Do you see my screen? Before Marco starts, I congratulate you on ERC. Thank you. Consolidated, which I just found on the web. Thank you. Marco will tell us about quantum melting of a dissipative time crystal. Okay, thank you very much for the introduction. Thanks for the invitation. I am enjoying a lot of this online conference. As someone said, being interested this week would have been great, but it's great to see that things are still happening. So the topic of my talk today will be about dissipative time crystals. We had quite a lot this morning. So I will try to connect to those presentation. Let me, first of all, mention the main responsible for this work and my collaborators. So this work was done mainly by Horatio Scarlettella with a post of College de France. And we enjoy collaboration with Ash Clark in Chicago and Zaro Fazio at ICTP. So this is my outline. I will give you a bit of introduction on time crystals in open quantum system. As I said, most of this part has been covered. And then I will get to the core of my presentation. I will present a model for this SIVA quantum time crystal built out of Vanderpoll oscillators. I will first talk about the classical case and then move to the quantum domain. And I will discuss a specific limit of this array of Vanderpoll oscillators, so-called large connectivity limit in which I will discuss how this dissipative time crystal grows out of a normal phase and how it is eventually destroyed or not due to final quantum fluctuation. Okay, so the starting point, I think we are, as we've seen this morning, we're going toward a classification of time crystalline phases of matter. And we have heard and I would say we understand quite a lot about floquet time crystals or discrete time crystals which are protected by many body localization. There have been various theory works and experimental demonstration. So from this point of view, one usual app to do this to think that dissipation usually is detrimental for this kind of floquet time crystals because MBL is usually destroyed by dissipation. And an interesting question that has been asked by many people is can we have a genuinely dissipative time crystals which are not destroyed or actually need dissipation to exist? And we have heard many already few talks addressing this point, in particular Anna Sampera gave a beautiful presentation this morning that essentially covered good deal of my introduction. So thanks to her. And also we heard about experiments on the Digemodel and theory in the Digemodel. And Fernando will talk about boundary time crystals. So we have, there are several angles to which you can address this question. So let me first step back and think about dynamical instability in classical systems. And I think tomorrow there will be a talk on that which I'm looking forward to here. But of course, if you remind yourself of dynamical systems, you know that there is a rich behavior emerging out of non-linearities, noise and dissipation and a variety of instabilities that happen at final momentum, a final frequency. You can have period doubling, you can have limit cycles, you can have various sorts of flow. And so it seems natural to think that the realm of the C-body system is a good starting point for to look for this kind of phase transition in the time domain if you want. So in this talk, I will take this motivation and I'll try to move toward the quantum domain. So I will work in the framework of open quantum many body systems. So I will go very quickly on this because we have had this morning an introduction. I will work in a context of Limbler master equation. So the system is described by density matrix and its evolution is described by an equation that contains a competition between coherent dynamics and dissipative jumps. You can think of it as arising from tracing out the degrees of freedom of the environment and under the bottom mark of approximation but there are various way you can get to this Limbler evolution and this dissipative part that you see here on the right contains this called jump operators which describe the effect of the environment on the system. And so this is, enjoy some nice properties, a linear completely positive trace preserving quantum map take density matrix and gives you back a density matrix. And it's a linear evolution which is important and allows, for example, in a good variety of cases to diagonalize the Limblerian and look at the spectrum, the right and left again vectors since as we know the Limblerian is an under mission operator and you can use this basis to decompose the evolution of a density matrix which from which you give initial condition you have a superposition of decay modes with app which have eigenvalues with a negative real part which correspond to decay and zero mode which correspond to a stationary state. And under rather general hypothesis if you don't have specific symmetries if you are don't have degree of free subspaces and so on, the stationary state is unique for finite system is analytic. And so the question one could ask is what happened to this spectrum of the Limblerian in the, as the limit or thermodynamic limit is taken. Let me comment right away in terms of from this perspective how time translational symmetry breaking would appear. I didn't say it, but this Limbler operator I'm having in mind is completely time independent. So enjoys continuous time traditional symmetry if you want. And so in this context, a spontaneous scenario of time traditional symmetry would result, would appear as a vanishing of the Limbler real part of the Limbler gap and a finite imaginary part. So I think Fernando Yamini will discuss this more in detail in the context of the stock on boundary time crystals. For what concerns my presentation, I will address this spectral properties of Limbler problems from the length and the point of view of Green's function, spectral function on Markovian system. And the reason is that as I will try to convince you those objects play a rather important role for the understanding of how disability time crystals emerge. And they are also directly measurable experimentally. So let me just give you an example. There are a variety of Green's function. We will focus mostly on the spectral function or the imaginary part of the retarded Green's function, which is a sort of density of state. And we know well in equilibrium what kind of expression this density, many body density of state, single particle density of state as is a sum over eigenstates. You have delta function that are located at transition between many body eigenstates with matrix elements, which are given by the operator in these cases, creation and elation operator that connects to eigenstate. And so what can generalize this so-called lemma representation, this very general representation of spectral function for Markovian system. And we will use that and the bottom line is that you have now, you don't have any more delta peaks, which are these are broadened by the fact that the system has finite lifetime. But otherwise you have a general structure and analytic properties which are rather similar. And as I said, those objects are not that exotic in the end, they can be measured through transmission. For example, if you have a circuit with the architecture in mind, if you are more in quantum optics from the quantum optics community, this is just one example of the famous mollo triplet. So those objects are also rather easy accessible. So let me, with this introduction, go get into the somehow the main part of my talk. I would like to discuss this discovery time crystal in a race of quantum mother pole oscillator. So I thought I should introduce a wonder pole oscillator which is incredibly simple but rich example on a linear oscillator. The story is also quite interesting. I don't know if you know it, but I found it amusing. This equation was written by two engineer, electrical engineers working at Philips in the late 20s. And they were playing with electrical circuits and vacuum tubes. And they wrote this paper that they call frequency demultiplication in which they say we found out that once you drive this circuit, you can have a response which rise at frequencies which are sub-multi-volt for the frequency of the applied. So this is, if you like, already a signature of a sub-harmonic response in this context. The even more amazing thing of this work is that those guys were, to probe their system, they were coupling this circuit to a telephone, okay? So, and they were converting this electrical signal into sound. And so they write this comment at the end of their nature paper that says, often an irregular noise is heard in the telephone receivers before the frequency jumps to the next lower value. However, this is a subsidiary phenomenon. The main effect being the regular frequency demultiplication. Turns out that this noise is a demonstration of chaos that they kind of discover 30, 40 years before Lawrence and others even wrote down a strange attractor. So this is just to say that this wonder pole has a kind of a rich story. And I also added here, let's say, modern day wonder pole oscillator. There is a very nice article on the New York Times that I encourage you to read if you are interested. This kind of equation is, as I said, the simplest probably non-linear oscillator described in a natural system developing self-sustaining oscillation. And how do you see that? Well, you see this equation as the first, let's say two terms look like regular harmonic oscillator. And then you have a non-linear damping. In this non-linear damping, you see that depending on the value of X, your variable, when X is very large access and damping and brings it down, when X is very small, it changes sign and acts as a negative damping which leads to amplification. So this is a sort of a feedback that gives you oscillatory self-sustaining oscillation. The form of this oscillation can be very rich. It doesn't need to be a simple sinusoidal form and depends on various parameter. And this has been studied in various domain from electrical engineering up to biology. What I want to somehow highlight here is that in the small amplitude limit, this equation, which is a second-order differential equation reduces to a linear first-order equation. And this is what I like you to keep in mind because now I would like to try to quantize this or present a model of coupled quantum van der Poel oscillator and try to convince you that in a certain limit you recover exactly that. So in order to do that, I will write down and study a Limbler master equation as I introduced before which describe a ray, a lattice of bosons which are described by this creation and relation operator A on each lattice site. Those bosons are non-linear, are non-linear oscillators. So you see they have a frequency omega naught and some self interaction, you can call it care interaction. And they are subject to, on each side of this array, they're subject to dissipative processes which are local. And we will consider two processes which kind of are the main ingredient here. One body pump which inject bosons into the each lattice side and two body non-linear losses which are described by those two jump operators. I just let me emphasize that I'm using units in such a way that essentially the strength of the dissipation, the losses, it's eta and the pump is just R times eta. So I will study this model mainly as a function of R which is the ratio between pump and losses. So this master equation enjoys some symmetries, some symmetries. First of all, the problem is time independent. So there is a continuous time transition symmetry. And there is a UN symmetry associated to the fact that I can make a phase transformation and the jump operator and the density matrix, the master equation remains invariant. The Hamiltonian is invariant and the jump operators, the combination of L and L is also invariant. This model is not that Zotoglie theoretical exercise. It's actually something that can be realized in experiments with circuit QDR rays. And with ultra cold atoms, we have heard this morning from the talk of my site, that inelastic processes can give rise to body losses in cold atoms. And even in circuit QD, there is a way to engineer these correlated dissipative processes. Okay, so with this setup, let me first of all show you that what I said before is true in a certain limit. Let me show you that we will recover the classical arrays of bander oscillator and then we will move add from there. So this is the limit of the so-called semi-classical limit in which all those bosons are, you assume they are in a coherent state. And so in a coherent state, you can write down their question of motion and the question of motion for those boson. If you somehow this record completely denoise, reduces exactly to the small amplitude limit of couple bander oscillator. So you have dynamics for this oscillator on each side. You have a non-linear oscillation that are due to the care non-linearity. You have this non-linear damping that I mentioned before where you have again as a function of a damping that depends on the amplitude of the local bosonic field. And then you have a linear coupling between different bosons. Now, if you study this equation in the uniform limit, you realize that as somehow as we were saying a textbook example of limit cycle, you have an oscillating solution that emerge. This oscillating solution emerge for any value of the pump to loss ratio. So as soon as you start driving the system enters into this oscillatory regime, the frequency is set by a combination of different parameters including the bare frequency, the hopping and the interaction strength and the amplitude of this bosonic field is set by the drive. So the starting point is that we have an array of classical bander oscillator. If you introduce a little bit of noise due to the residual quantum information fluctuation, if you think that they are small, you can not surprisingly convince yourself that this model, this system can desynchronize. If you allow me to use this term of synchronization and limit cycle in the same context. And this was discussed, for example, in this paper by Lee and coworkers. But of course, this regime of small quantum fluctuation starting from the semi-classic is one angle. Me, I would like to present the result from a rather opposite regime in which we study this array when we have only few boson per side. So the semi-classical approximation is not really a good starting point. And so to do that, we have developed an approach that I think is very promising for open quantum system that relies on the idea of taking the large connectivity limit over Limbler master equation. So this is a little bit the sketch. I will enter more in detail of what that means. But on the left side, you see this array where you have local Hamiltonian, local jump operators and hopping between the different degrees of freedom. And then you can study this problem in the limit in which the connectivity of the lattice, Z, the number of neighbors, goes to infinity or becomes very large. And in those two limits, you reduce the problem to something that is obviously much simpler, but quite rich as I will try to convince you. In one case, that is the exact infinite connectivity limit. This corresponds to what is called Good-Svillard-Minfel theory. And the picture is that your entire lattice maps into a single site coupled to a self-consistent field. And in the limit of large but finite connectivity, this will correspond to what is called dynamic-Minfel theory. And in addition to this coherent field, you will have a non-Marcovian bath which mimic the effect of the rest of the lattice. Okay, so this approach, you can think of it as a sort of quantum-Minfel limit in the sense that retains a discrete nature of the problem is a single site coupled eventually either to a field or to a quantum bath. So there are more details on this approach that is rather general and goes beyond the one that pulled problem in this paper. Let me discuss the results and in the context of this dissipative time crystal. And I will first analyze the infinite connectivity limit that if you like is more simple but it's quite interesting. So this infinite connectivity limit, as I said corresponds to a completely factorized answers on the density matrix. This corresponds to Good-Svillard-Minfel theory. So you are reducing your problem to a single quantum-Bander-Pollister later coupled to set consistent field. And what I'm plotting here is the phase diagram as a function of pump-to-loss ratio and hopping versus interaction ratio. And you see there is a critical line that separates what we call a normal phase in which the system goes to a stationary state from an oscillating phase in which the system develops many body limit cycle. Now, let me emphasize that just to compare in the semi-classical approach that I started with, the entire region of R greater than zero would be completely oscillating, okay? So we see already that treating inducing, including quantum fluctuation by in the regime of few boson per site quite change quite dramatically the picture and introduce a large region in which the system remain normal. And now we'll discuss exactly why is that and how this comes about. So the two main issue that you think that you have to, like to highlight is that there is a critical transition and critical point as a function of the hopping and also there is a threshold drive. You don't probably see it because this scale is not particularly well chosen, but this line, this horizontal line is around R equal one, meaning that you need a threshold drive in order to enter to this oscillating phase. And I will explain exactly why is that. So this approach suggests us to look a little bit more into the details of physics of quantum under pole oscillator, which is a problem that is attracting a lot of interest in the context of synchronization, mostly for coherently driven oscillator. We will not consider this case. We'll consider the case in which, as I said, there is no coherent drive, but only incoherent pump and two body losses. This model is nevertheless quite interesting. It turns out that the steady state density matrix for this nonlinear quantum oscillator is known analytically. This is a work by Mark Dickman that goes back to the late 70s. And it's known that this boson is going to go into an inquiry and mixture. So the density matrix diagram in Fox space, there is no coherence, of course. And the stationary state density matrix is completely independent from the interaction. It only depends on the ratio between pump and losses, this parameter R, and as a function of R, as a function of the ratio between pump and losses, you already see an interesting effect, the density matrix, the probabilities of having n boson per side develops what is called a population inversion. You see that a small drive, this quantity is a decreasing function of n, a large drive instead, you have that certain, you have a non-monotonicity, which means that higher occupancy is more probable than lower occupancy. We realize that there is more to this if we actually go beyond the steady state properties. And we look at the green function of this problem. And I will show you why this is relevant for our problem. But if we look at the spectral function, so the density of state of the quantum under pole oscillator, we find, we found a rather surprising result, that weak drive on the left side of this plot, you'll see a series of peaks which are broadened by the dissipation. This is not so surprising when you don't have interaction, this is essentially Laurentian. When you have interaction, you have more peaks. However, when you increase the drive above a certain threshold, you see something that at first puzzle has quite a lot. You see that the density of state at positive frequency becomes negative. So you see there is an entire range of frequency at which the spectral function of those bosons becomes negative and vanishes at a certain frequency. So what is the physical meaning of that? The spectral function contains essentially measure the power that the system absorb from a perturbation from a weak current driver frequency omega. So typically you would expect this quantity to be positive, but when this quantity is negative, it means that the system wants to develop gain, wants to actually gain energy from the external perturbation. So this is an incident effect that lead, eventually, to energy emission. And let me guide you through this kind of interpretation of this dissipative time crystal as a sort of energy emission. So how do you understand how... Five more minutes. Yes, okay. How do you understand if a dissipative time crystal emerges? What you do, you have a single site problem. What you have to do, is to put a small, seemingly breaking field and look for a self-consistent solution. And you see that by perturbing with an oscillating field, you obtain, first, the order parameter is set by the retarded unit function that is the susceptibility to the breaking field. And then you have a self-consistent condition and you obtain this equation. This self-consistent equation tells you essentially the location of the boundary between normal and dissipative time crystal. And from this equation, you immediately see that in a generic condition, you would expect this retarded unit function to have a finite imaginary part. And so you are not able to solve and satisfy this equation, unless the system develops a negative density state and develops gain, in which case you have a frequency at which the system is essentially dissipationless. And when the system is dissipationless, it can emit and oscillate at a certain frequency. Let me skip this in the advantage of time. I want to conclude by saying what happened to this picture when we include finite connectivity. So this is done by what I said, what they call dynamic unfil theory, in which you have now to solve a single site which is coupled to a coherent field and to a non-Markovian quantum path. This is a problem which is much more challenging than the previous one. Nevertheless, we developed an approach that is powerful and non-perturbative approach in which we expand self-consistently in this non-Markovian path. I will not enter into the details, but if someone is interested yet to discuss. And this is somehow the main result of this approach. It's the same phase diagram as before, but now as a function of lattice connectivity. So the gray line is the infinite connectivity and as you decrease the connectivity, you increase the quantum fluctuation from the neighboring sites. And you see two main things that are somehow the main result. First of all, the dissipative time crystal remains. There is a still region, a high drive in which the system develops oscillation. Nevertheless, the effect of fluctuation is to shrink to manage this dissipative time crystal. And maybe the last thing that I want to say before closing is that the mechanism through which this happened is a generally non-equilibrium mechanism for the destruction of an order phase. And let me argue why this, where this come from. Again, you can add a small symmetry breaking field and now you realize that dynamical mean field theory gives you two contribution to this symmetry breaking field. One coming from the coherent field and the other coming from the self-consistent bath described the finite number of neighbors. And now this self-consistent bath enters in this equation here. And you see that the boundary between the DMFT boundary with respect to the mean field one is substantially pushed toward higher values. And this is due to the fact that the bath is able to destroy the negative density of state through a mechanism that we call hopping induced desynchronization that is somehow physically you can interpret as the bath absorbing the energy that the system was taking from the perturbation. With this, we check that this effect does not come from noise. And you might have think that the effective temperature here is increasing but as we see as a function of connectivity, this remains rather stable and weakly depending from the connectivity while the phase diagram change quite dramatically. Okay, with this maybe I should conclude because I see the chair getting ready. So my conclusion, I hope I give you an overview of the Seabury kind of time crystal classical versus quantum. The message for what concern this array of underpol is that quantum fluctuation partially melt but do not destroy completely this Seabury time crystal phase even a finite connectivity. And nevertheless, the mechanism for this Seabury destruction of another phase I think it's very interesting and we would like to look more and use more this new technique that we developed and with this I'd like to thank you for your attention. Thank you, Marco. We have, well, one minute for one short question. So Jamir Marino was first. So please turn on his mic. Hi, Marco. So rather than being a question is a comment, I'm gonna make a comment that I want to know if you commit to the statement or not. After all, because you need to generate synchronization always at damping, you need always a dissipated dynamics. Having quantum synchronization means basically always having that most a semi-classical synchronization and leading classicalizing with few quantum correlation effect. If you want, I am just rephrasing the usual old question what is quantum in this Seabury phase transition? And the answer is can be quantum only in a certain crossover in time. But asymptotically and here you need to go asymptotically will be always leading a classical behavior, whatever operation you try to do. Would you commit with that or not? No, I think the message of this talk is rather the opposite. This talk shows you that quantum synchronization is much more similar to if you like non-equilibrium super fluidity than just the classical bander pole. You need to have gain and first you have gain and then you develop if you want phase coherence between the oscillator. And this is a mechanism that you don't have in classical synchronization. That's simple. And then for what constant time scale I'm not sure because this is a completely dissipative problem, this is stationary state. So there is no, I mean this is, there is a bath. So it's a truly stationary state. Then one might wonder whether it would decay. But I would say the main message of this is that really both the synchronized phase how does it develop and how it is reduced is something that I don't think has analog with the noise or effective noise or effective temperature but it's really a matter of it's a very, really a dynamical instability like you have in lazing or like you have in super fluid accident polarity. Okay, I guess we have to stop at this moment. Thank you Marco very much. And welcome Jose Tajeri who woke up as I see, so please Jose and share your screen with us if possible and you are unmuted and the floor is yours. Thank you. So can you see my screen? Yes we can. Can you hear me well? Very well. Okay, great. So thank you very much the organizing team for inviting me to this interesting conference. It's a pleasure to be among a distinguished list of speakers. I am Jose Tajeri from the University of California at Riverside and the research faculty here. I'll be presenting some results based on recent research on time crystallinity in kernel in optical resonators and particularly about dissipative discrete time crystals. So let me start with a quick outline of my talk for that make sure that I move my own screen to the corner in case that blocks for anyone. So we'll start with a very brief introduction to time crystals because I believe it's been covered very well from different angles and other talks. And then I will present a quick primer on optical frequency combs care microcombs which are the platform we are using for our demonstration and then I will proceed to our results. So first starting with a brief introduction to time crystals. Know that time crystals were introduced as the temporal equivalent of solid state spatial crystals. The idea was introduced in 2012 by Frank Wilczek and can be considered an extension or lie in line quite well with the temporal evolution of the concept of time from Newton to Einstein. Know that Newton believed in absolute time but Einstein showed that actually it's relative it should be treated on the same footing as time, as space and combine them into space time. So the concept of time crystal fits in well here. It's also intimately related to the concept of symmetry breaking, which was suggested by Landau as a reliable criteria for classifying the different phases of matter. So this concept is now being studied as a new state of matter. And originally, of course the idea focused on equilibrium systems, but it was shown that that proposal does not work. So researchers proceeded to non-equilibrium systems, particularly periodically driven systems. So we're not going to look at continuous time symmetry breaking, but in fact discrete time translation and it's breaking. So that happens when we have a drive with periodicity say T and the response of the system, the behavior of an observable in our system as a larger integer multiple periodicity in this case shown here with two T. And when we have that, then we proceed to, we get a discrete time crystal at DTC. We can define a size as the ratio of the response to the drive time. And then in experiments particularly, we look for some signatures. One of them is period multiplication or subharmonic generation. And other properties are also required, for instance, long range of order in the temporal sense and also many body interactions in the sense that the period multiplication, which we observe should be a result of many body interactions. Now, of course, there've been some, a good number of demonstrations. So for a quick review in the first two demonstrations, one of them was in a one dimensional array of trapped ions. The other one was in a 3 dB ensemble of NC defects and NB defects in diamond. And when the magnetization property of these two systems were measured and analyzed, characteristic subharmonic generation was observed. Now, these demonstrations have been mainly focused on isolated or closed systems in which the coupling between the system and an external bath or the environment is vanishingly small. And these systems of course, very interesting, particularly because of the curious ways that these systems find to avoid or postpone thermalization or heat death. But for practical applications, it's inevitable to consider systems in which we have dissipation and heat, particularly for real-world applications. And these systems, the heat which is generated as a result of dissipation can destroy the crystalline order. So there should be an environment and there should be non-vanishing a coupling to the environment to channel out generated energy and that gives rise to the concept of a dissipative discrete-time crystal. And that is what I am going to demonstrate in our system. So that was my quick introduction to time crystals. Now let's go to care microcombs. Everything starts with care non-linearity here. So let's talk about that. It's the dominant non-linearity in centrosymmetric materials, this is to build these chi-3 or the cubic non-linearity for a single mode field. We can simply write the Hamiltonian as a sum of the Hamiltonian of the radiation and the interaction Hamiltonian. The coupling coefficient G in this Hamiltonian is proportional to the chi-3 susceptibility or with the end-toon non-linearity. And what happens in a bulk medium is that we have two photons of pump which are annihilated and instead they generate two photons. The signal and idler frequency, one of them with larger frequency and the other with smaller frequency. But the point is that in a system, especially in bulk medium, the efficiency of the process is not so high as to give rise to a lot of side bands. So we study this process, not in bulk medium, but in a micro-resonator, a high quality factor micro-resonator. And what happens is a degenerate forward mixing cascaded by non-degenerate forward mixing process and then the cascading of these two effects in which we have the pump and the two photon pumps generate side bands and then those side bands for a mixing with the pump itself and also with each other and then the cascading of the effect leads to more side bands. The threshold of this process is of course inversely proportional to the non-linearity coefficient but also inversely proportional with the quality factor of the cavity, which is a measure of the amount of energy stored in the resonator over the amount of energy which is dissipated or lost in every cycle of the radiation of that particular radiation of light. And of course it's also proportional with the volume, the mode volume of the field in that resonator. Now, these resonators have been fabricated in various shapes, in spherical, so it's the starting point, I believe with Serge Haroche and then other types of micro resonators then moved from even cylindrical and toroidal in crystalline to integrated materials and different types of materials. For instance, calcium fluoride, magnesium fluoride are the quiescent types and silicon nitride and silicon dioxide to examples of integrated resonators commonly used in this platform. And the quality factors we can get in crystal resonators, something usually between 10 to 8 to 10 to 11 and in integrated resonators, something between 1 million to 100 million, usually in a lower 100 million to 10 to seven is a better number these days. The platform we have used in our experiments is the crystalline magnesium fluoride resonator. Now, when we have this schematic micro resonator, we drive it with a CW continuous wave laser pump. It generates a frequency cone, an array of frequencies. So let me show you another schematic here and because it looks like a cone, we call it a frequency cone. Now, when you look at the frequency, the spacing between adjacent harmonics is the repetition rate or almost equal to the free spectral range, the FSR of the resonator at the position of the pump. And if the system and the generated frequency cone is self-synchronized, phase locking occurs and this frequency picture corresponds to the very short pulse. And this is what we call a essentially mode locked pulse which is also generated in a mode locked laser but the mechanism of generation is of course different. Now, these structures can be used as rulers of light especially because after synchronization, after generation of a stable pulse, the frequencies separating these harmonics are very, very exactly the same. And the repetition rate, the amount of time separating two consecutive peaks of the pulse train in the pulse train are also very accurately the same. So we can use them for very precise measurements. And that's why the 2005 Nobel Prize went to, partially of course, half of it went to frequency cones. Now, a quick point I want to mention here, synchronization process that occurs in these structures is, cannot simply be reduced to a phase model and is different from the chromata model even though it was suggested as though back in 2016. Now, to experimental demonstrations when we have normal dispersion, this is the type of spectrum we get. We get the bright soliton with the characteristic hyperbolic secant envelope of the spectrum. Then that gives rise to a pulse to a soliton which is called a dissipative care soliton in our literature. And that results from a balance between dispersion and non-linearity generally a soliton in a system, in any non-linear system arises from the balance of a non-linearity in the dispersion and a very excellent, I believe, description of how that happens appears in Butcher and Cutter. And in the case of a dissipative system, there is also another balance of gain and loss. And we have shown how that gives rise to even dark solitons in the normal dispersion regime. Now, quick point about mathematical description. I showed the Hamiltonian for a single mole field. Let's go to a multi-mode field. We can write the Hamiltonian where E is the sum of the annihilation operators. And then we can find the equations of motion simply. And when we look at the averages or expectation values, we can write these in terms of not the operators but complex variables. And by taking the discrete time-free transform where theta is the angle around the resonator, we can find this mean field equation which is called the dam, which is a damped and driven non-linear Schrodinger equation. And we often call it the Lugietto-Lefver equation. So after this, I'm gonna jump to our results. Our approach will relies on using a high quality optical micro resonator. We pump it with two lasers. The two lasers create a discrete time translation symmetry in the system. And when we look at photon count probability in the system and this discrete time translation symmetry will be spontaneously broken as a result of the interaction of the photons which generates solitons in the cavity. And then we show that this platform is very useful for demonstrating large DTCs, DTCs carrying defects and also can be used for observing phase transitions. Apart from our demonstration which appeared on our archive a few months ago, there is a highly relevant result from the University of Hamburg, I believe, Andreas Herrmann, who would present on this same conference here. And they use an atom cavity system. Now, one of the experimental tricks that we use in our demonstration is self-injection locking. Self-injection locking is essentially the locking of the laser through the cavity mode. And that happens when we do not use an isolator between our laser and our micro-resonator. And as a result, the light pumped from the laser into the resonator would be back-scattered into the laser cavity as a result of really back-scattering. And that essentially what happens that is that locks the laser to the mode of the cavity. Now, when the laser also generates a micro-comb it will also lock to the micro-comb and then the laser essentially works as an arbiter between the cavity mode and the micro-mode. And that results in very stable frequency combs. And one of the very important practical benefits is that we can get soliton formation in a deterministic and turnkey fashion in the sense that usually when we wanna get the soliton state in these systems we have to sweep the laser. But without sweeping and with simplifying the system by removing the isolator we're able actually to get these solitons in a turnkey fashion. We turn on the laser as I have shown here. We get the state, we turn it off, we lose the state again, turn it back on, we get it and the cycle can continue which was demonstrated in an integrated platform recently as well. An interesting story is that this technique was used by the industry where in the field of micro-combs for a while before academics realized its potential it was essentially a secret sauce of the industry that came into academia. Now, we go beyond a pumping with one laser we go to pumping with two lasers we lock them simultaneously to cavity resonances and also to one micro-comb and that confirms our theoretical predictions from 2017 and 2019. From there let's take a look at the experimental setup. We have a whispering gallery mode resonator which is prism coupled to the two pump lasers. The two lasers are separated by an integer multiple of the repetition rate of the cavity which is the inverse of the round trip time, the time it takes us out on the go round for one full cycle. And then when we have shown that this process can be described by a modified LLE, modified Lugietto-Lefford equation with a periodic drive. Let me emphasize this drive term again here which possesses a clear symmetry in terms of both the slow and the fast time. So we talk about the fast time tau it's like the propagation distance Z in the context of fiber solid sounds and it's proportional to the theta coefficient. Now let's see how this discrete time translation symmetry can be broken in the state. We're an example I'm showing here M equals five and without soliton generation we have a periodic background, the lattice which is generated by the beating of the two pumps. And when the soliton is generated we go from periodicity TR over five to TR which means a five time multiplication by a factor of five. And we have noticed that this particular case is very robust as I will explain in a few slides. So going from M equals five to M equals eight we'll see that we have a larger variety of possible paired multiplying states for instance two T one or four T one or eight T one where T one is the periodicity of the state. And when you compare that with M equals five you see that we have more variety and there are also possibilities of getting states which are not symmetry broken for instance the bottom right I have shown here get one soliton in every lattice strap and that means the symmetry will not be broken. For the temporal evolution of these pulses which are shown at the bottom the temporal pollution is shown at the top where the dark red lines are the peaks of the solitons. See that we have observed evolution in a stable fashion over hundreds of cavity photon lifetime and thousands of cavity round trip times. And when you look at the frequency spectrum and the characteristics of harmonic generation is highly is very clear between the two pumps in these cases marked by two red arrows. And then the hyperbolic secant envelope is also clear as shown here by the red dashed line. Now jumping to the experimental results we looked at M equals four and M equals three and we observed very good matching between theory and experiment. We observed also that hyperbolic secant envelope and then to make sure that we are dealing with a calm or mode left frequency calm we look at the so-called RF signal radio frequency signal which is the beating between adjacent harmonics. The blue curve on the top left shows the beating between the two pumps and the red curve shows the RF signal for the sub harmonic generated which is you see almost one third of the first one. And both of these signals are very narrow have their small phase noise and that is a signature of a calm or stable frequency calm. Now taking the Fourier transform of the numerical data we find a numerical wave form for the generated state. And based on the discussion I had and the examples I showed earlier we see that it is indeed in both cases they are indeed paired multiplying states. Now in this system it's possible to observe the equivalence temporal equivalence of solid state physical concept of defects. We have vacancies it's a point defect when you have a missing atom here where we are missing a salt on peak. We also have dislocations essentially misplaced solid on peak and also interstitials like pulse appears where it shouldn't be. Going beyond that solid on formation regime and the regime of looking at self-injection locking also looked at the regime of modulational instability which is a regime in which the harmonics are generated from parametric gain and they rise from vacuum fluctuations. The example shown here is for a case of m equals 18 so the two pumps are 18 FSRs away from each other. And three FSR spacing between the subharmonics shows that we can realize a 60 state. And this is a result from 2019. It also looked at the stability of the state in various initial conditions were used with random initial powers. We see that the final state is very reliably produced and we also looked at the addition of noise to the system as you see the signal to noise ratio on the right panel shows that then it reaches 10 dB which is very easily achievable in experiments. We get highly reproducible and repeatable generation of the final time crystal state. There are some relevant results in the literature theoretical and also some results including experimental results in fiber resonators which we believe are very closely related and should be able to show similar results. Now future directions particularly what we are pursuing is a demonstration of controlled phase and transitions meaning we generate a different DTC phases controllably and we transition between them. Also are interested in showing larger sizes of DTCs but as you see it's possible to sweep the frequency of the laser and demonstrate in principle large DTC sizes. Thank you. Thank you. The large degree of the high degree of coherence temporal long range order in the systems in these systems can be used for applications requiring coherence large coherence times and also be used for some photonic applications for instance frequency division with high stability. And more importantly for two point stabilization of narrow band combs which cannot be used with standard techniques such as the self-referencing and when we essentially stabilize the two pump lasers by coupling them by locking them to external transitions for instance atomic transitions we can improve the stability of the system significantly. Similar demonstration of course using external broadening of the pulse period in 2018 of course there were significant differences but that tells us that this platform indeed has the potential to demonstrate what we're looking for. So in summary we've demonstrated DTC formation in a dissipative system in a nonlinear photonic system and we have looked at some of the defining signatures and the importance, one of the important advantages of using this system is that it can leverage the well-developed platform of nonlinear photonics and a lot of optical techniques which have been used in the same way that quantum optics was useful 30, 40 years ago and still is and we have talked about some of its applications. With that I'd like to thank my friends and colleagues who have contributed to this work and in particular Shishitov Sasha and Adrey Maskov been they helpful in clarifying and developing some of these ideas. My colleagues at OEA is my former colleagues particularly with Maleky and Wayne Leon could be supportive, at least supportive of the experiments and I've also benefited from many exciting conversations with Tobias Her. That I'd be happy to answer any questions or you can reach out to me via this email here. Thank you very much for your attention. Thank you. We have time for one question again or two. Please shout. I mean everything was pretty clear and there was very nice presentation but maybe somebody didn't understand something. Okay, I guess that you will be, oh, Shane Kelly would like to ask question, please do. Please unmute, Shane Kelly. Yeah, there we go. Yeah, I was wondering if any of the light it can be squeezed if you get any squeeze light in the cavity and... Yes, it's possible and it's been demonstrated. Squeeze lights, squeeze states of light have been demonstrated in micro-resonators before particularly by Avik Dot at the, I believe, Mihal Lipson group. They were at Cornell before they're not Columbia. Do you know if that will affect the stability of the time crystal or? Not sure if they use, I don't believe they use self-injection locking but in principle, the level of stability we get is about simply on the order of hours with DTC. So with temperature control, the states we get are very stable. And I believe in those states we can have the same level of stability if not better. Okay, thank you. And Vincent, we have a last question. Yeah, thanks so much. So much exciting talk. I'm quite intrigued by the multiple factors of the periods, like five, eight times. So I was wondering what the road does the interaction play to the modification of the periods? Would you please repeat the question I think I did not hear? What does the role of the interaction play in setting the period of the district time crystal? Okay, I think you're asking you about the role of something in the street. The interaction, yeah. So I was wondering how the interaction plays any role in the emergent, the period of the time crystal that breaks the translational symmetry in the district way, right? Yeah, yeah. So you see, without the care non-linearity which governs the interaction of the photons, there will be no soliton formation. Essentially, the reason we get the side bands and the reason we get their synchronization, as we have shown, is the care non-linearity between the photons. So will the interaction strands change the period? It will not change the period in the sense that the beating between the two pumps will fix the number of pulses, the maximum number of pulses we get. So it generates, essentially, a lattice throughout around the resonator and then they will be locked, the solitons will be locked in those positions. There is some playground, but it's not a lot of data. Okay, thank you. Thank you. And thank you, Hossein. And we have to move to the next talk. The next talk will be given by Igor from... Well, I don't know, Igor, is it Nottingham or Tobingen, from where you will give the talk, because you seem to be in some coherent state. So Igor Lesanovski, he has the floor. Jakub, thanks a lot for the introduction and while I'm trying to share the screen here, let me also just thank the organizer for giving me the opportunity to present our work. I'd much rather be in Trieste right now. So I will get to your question, Jakub, in a second. I have a slide there. But first, I'd like to talk about the title of the presentation, which is on time crystal phase transition fluctuations and quantum correlations and an emitter waveguide system with feedback. It's rather lengthy, I admit, but I hope I will be able to explain all these terms to your satisfaction. And concerning my question, and concerning my state of being, let's say I've moved in 2019 from the University of Nottingham to the University of Tobingen, and my main place is now Tobingen, but I have a part-time position at Nottingham. And of course I'm very happy to be able to collaborate with the people there. Also Andreas Nuremkamp, as I understand, is giving the talk. Tomorrow he is now there. And of course, time crystal related work I've been doing in the past. It's been done together with Juan Guerra and Matteo Marcuzzi and Filippo Gambetta. And what I'm going to talk about now has actually been done by Federico Carollo, Giuseppe Buonagnuto and Beatriz Olmos. Good. This is the outline of the talk. It's an open quantum systems and the realization and analysis of stationary state phases and also transitions into time crystalline phases. So what I will do, and a lot has been said already in this conference, I will very briefly tell you about how to formulate the system, discuss very briefly the master equation, but then focus a little bit more on a new aspect that I want to talk about here, which is how to include feedback and how feedback can be used to actually control the dynamical phases of such a many-body quantum optical system. I will tell you a little bit about wave-bite quantum optics because this is the system that underlies the physics that I'm going to talk about. I'll tell you a little bit about how interactions come about, what sources of dissipation there are, what is nice about these systems and I will also tell you that in the end, the system can be actually very nicely described in terms of collective observables, which allow a very nice and actually exact treatment of the system in the thermodynamic limit. I will then look at the dynamical phases, show you the phase diagram, look at fluctuations and also finally it's been squeezing and conclude with a summary and an outlook. All right, so what's the basic setting that we are dealing with? Well, I said already it's open quantum systems and in an open quantum system, we have a system of interest. Oh, let me, this is not good, I want to switch to the pointer. We have a system of interest and this is embedded in a barf and now what the system does, well, exchange this quantum with a barf and what you can also do, of course, is we can now detect what is emitted into the barf. I mean, simple example, for instance, atoms that interact with the electromagnetic field, atoms can decay from the excited state under the emission of a photon and this photon can be of course detected and depending on the detection we can now change the parameters of the system through some kind of feedback construction and this is the basic idea of the setting that I want to consider here and in which we analyze some time crystal like dynamics so the basic idea is to detect what has been emitted and in our case this will be photons and conditioned on the emission we will control the parameters of the system such that some interesting physics, some interesting states appear in the stationary regime. All right, so how do we formulate this and this we have seen a number of times already today the workhorse here is the Lindblad master equation that describes the evolution of the density matrix and there are two parts to it so one is the coherent evolution here represented by this commutator term it's just a von Neumann equation essentially and then we have here this term that accounts for dissipative irreversible effects and this whole let's say agglomeration of operators here is also usually called the dissipator that then depends on the jump operator's LJ and just one thing I want to say because it becomes of relevance on the next slide so one term here that one can single out in this dissipator is the so-called quantum jump term which basically in an unrevealed picture where you think in terms of this average dynamics but in terms of trajectories that you can then average to get the density matrix in this kind of picture this term would correspond to quantum jumps that the instantaneous state of the system undergo so just a basic example to bring this kind of closer to something tangible for instance if you look at a very simple situation where you have just a two level atom that is subject to a coherent drive you wrote on resonance so you would write the Hamiltonian as something that is proportional to the sigma x poly matrix which just coherently changes the state from the low lying level to the high lying level at a rate that is given by this Rabi frequency omega and then you would have for instance in competition with this coherent excitation incoherent decay from the excited state to the ground state at the rate kappa and the corresponding jump operator would then be in this case just squared of kappa sigma minus where sigma minus is simply the operator that takes the excited state to the ground state so now feedback and I want to make this very brief I am not an expert and b the time is also limited so I will start from feedback scenario that I think is probably the simplest and easiest to grasp and it very much relates to the system that I just showed you on the previous slide namely in the two level atom general ensemble of atoms that undergoes spontaneous emission events and once there is such a spontaneous event then the atomic ensemble will release a photon and this photon can be detected and now you see for instance if you have this detector and you record the sequences of clicks over time then you would see something like this so every stroke here corresponds to the detection of a photon emitted from the atomic ensemble so in one very simple feedback protocol that one can do and please take into account that I am a theorist here is that after each detection of photon I perform an immediately a unitary rotation on the state of my atomic ensemble and what that would entail is just in the master equation a simple change of this quantum jump term that I mentioned on the previous slide so where you just see well what happens is that I have my state row before the detection then the quantum jump takes place so that means I have to apply L, J and L on the respected side and then you see I apply a unitary rotation on the state of the system afterwards and you see in a very transparent pictorial way how such a feedback protocol based on photon counting would work this is not what we are doing here for the purpose of this work in this work we consider something that relates to homodyne detection there one does not count the photons but one actually monitors continuously in time the electric field amplitude of the emitted light field from the photons so you can do this I do not want to go into details here there are lots of experiments that have demonstrated this and also if you now implement this kind of feedback strategy and the book by Wiseman at Merburn for instance will tell you how this can be done in practice then you will see that such a feedback protocol will let us say do a little bit more to your mass equation not just change the jump operators but will actually introduce additional terms not only in the dissipative jump part but also in the Hamiltonian part that describes the coherent evolution of your system so our protocol is such that we say well we detect this photo current that is corresponding to the electric field of the emitted photons so you can write down an expression for this this photo current X of t which is kind of the expectation values of expectation of some operators of your systems times plus some noise so this is a noisy quantity it has a deterministic part and a noisy part and depending on the value of this amplitude X we change the amplitude or the Rabi frequency of the laser field that drives our atomic ensemble so and the corresponding strength of the feedback is regulated by this parameter G so if you take this and accept this so we measure the signal here and depending on what we measure we change the strength of the Rabi frequency with which we drive the atomic ensemble then you can write down your modified mass equation and what you will find in the terms to that effect where you get an extra bit here that depends on the to your Hamiltonian that depends on the feedback strength and also here on the actual structure of the photo current that you are measuring so this is very pictorial I show you so to say the the the precise operational structure of the situation that we are considering in our work and what you also have is that you modify the jump operator so what you see is that you jump operators get shifted also by an amount that is proportional to the feedback strength G and also by something that corresponds to the operator that you are effectively measuring you are measuring effectively by monitoring this photo current good so this is the idea we perform this homodyne detection scheme on our atom waveguide system which I haven't explained yet and this will change your Hamiltonian and the jump operators and this will result in something interesting so now let's jump to the system that we are interested in applying this strategy to and the system is just formed by a waveguide nanofiber for instance and next to it sits atoms that are nicely lined up in a regular chain so this is not something we made up but this is really some system that is under active investigation in a number of labs worldwide so for most maybe to mention is the group by Arnold Rauschenbeutel the Humboldt University of Berlin but of course many people who have contributed to developing the experimental side but also the theoretical understanding of the system the dynamics of the system can be modeled again in terms of a master equation and again we have a Hamiltonian here and this Hamiltonian consists of some kind of laser driving term you have seen this before and here the lasers might not necessarily be on resonance you can also have a detuning so then you have a term that amounts to the effective interactions between the atoms because you see what happens here in such a system is that the atoms actually emit photons into the guided mode and then you see for instance that when a photon is emitted by this atom 2 here it can be propagating to the right and then it can be absorbed actually by the atom at position 3 and by this you effectively create an interaction between the atoms in the chain then the next term that we want to discuss here in the system is of dissipative nature so this is actually the very nice point about the system that you have due to the fact that your atoms interact with this waveguide and emit into the waveguide modes you can very nicely let's say attach detectors to the left and to the right and very efficiently resolve photons emitted from this atomic ensemble in this waveguide and of course once you have detected those photons you can then apply your feedback strategy so this is very neat because this setting naturally gives you let's say a reduced set of modes very similar to an atom cavity setup that allows you to apply this feedback strategy in a very controlled way and last but not least what we also have in principle is of course not only emission into the guide of the atoms into the guided modes but also of course emission of photons from the atoms into free space but I can tell you already for the purpose of this talk we sweep this under the rug and we'll pretend that this decay channel can be suppressed sufficiently well so of course it's nice that we can write down this master equation that faithfully describes our atom waveguide setting but for the purpose of the work that I'm going to present on the time crystal I'd like to simplify the situation a bit more and first of all I told you already that I do not really want to consider emission into unguided modes I really want to pretend that the system is built such that all the photons that are emitted incoherently from the atoms end up in one of the two guided modes either emitted to into the right or into the left propagating direction what I also assume is that the atomic positions are commensurate with the photon wavelength so that means the distances between the atoms are multiples of the wavelength associated to the transition on which the photons that I detect here at the end of the waveguides are emitted and once I do this this whole complicated looking master equation requires a much much simpler structure which we're even going to simplify more in the next step so you see we have here the laser driving and with the Rabi field you can see omega and this laser driving is only dependent on so called collective operators which by the way also then in this setting capture the incoherent decay into the guided modes and this is actually important because this collective nature of this excitation of the atoms and the collective nature of the coupling of the atoms to the dissipation allow a very simple and very nice to study structure of the system so what I will do in the following is also consider a situation where I do not have any detuning and I do not have any interactions between the atoms which is actually something that sounds complicated to do but it's not so this is actually a generic situation so when the emission rate into the left guided and the right propagating mode when both emission rates are the same then you can show that under those conditions where the atomic positions are commensurate with the wavelength of the emitted light you effectively have no interactions because the atoms happen to sit at positions with respect to each other such that the interaction with the light field interferes away so now in this very simple setting we really have a simple Hamiltonian so this is essentially a non-interacting Hamiltonian and we have this kind of collective jump operator we want to apply feedback and this feedback that we want to apply is very similar to the one that discussed before with the detection of the photo current so to say if you want to know more precisely what you measure we are measuring so called P quadrature this is kind of measuring the electric field which you of course can decompose into an X and P plane just measuring it in a particular angle namely along the P direction in this quadrature representation of the electric field so if you want to forget what I just said in this talk so far about the experimental motivation of this system you can do so right now because now is the part where we start the theory because with all these simplifications and with all these assumptions that I just laid out we finally end up with an equation that we can analyze and so applying the feedback strategy to the system which is composed of the atoms and the waveguide and measuring this P quadrature you find that the master equation is modified in a way that you add in your Hamiltonian part interactions that are proportional to the feedback strength G and you also modify the jump operator that accounts for the emission of photons into the right propagating waveguide mode why the right propagating waveguide mode because this is the mode on which output we perform this homodyne detection which then controls what we feedback into the system just for comparison if you look at the decay term at the dissipated corresponding to emission of photons into the left propagating modes this is not modified this is just a J minus term it's just a kind of normal collective jump operator that you would write down whereas here you have this modification which is also involving here the feedback strength G now what's nice here and I already said this in passing couple of slides back is that this whole master equation here now just depends on collective spin variables that you can write for instance like this so it's just some over all local spin variables that are represented by these Pauli matrix matrices sigma k and the respective directions alpha equals x y and z this is nice because this allows us to perform a treatment that is actually becoming exact in the thermodynamic limit that means when the number of emitters number of atoms tends to infinity what I can then show is that actually the magnetization operators which are just these operators here these collective spin operators but divided by the number of atoms that these operators in effect become classical which means that their commutator tends to zero and in this limit the mean field description of our system dynamics and I will just show you the corresponding equations on the next slide so in terms of the expectation values of the magnetization operators becomes exact just note that in order to do this in a meaningful way so that one has a well defined limit one has to also scale the decay rate so the emission rate of the photons into the waveguide accordingly so it has to go down as one over n as the number of particles n increases so this is just to ensure that there is a consistent and meaningful thermodynamic limit but once you do this you can actually now arrive at a set of three coupled mean field equations which just tell you about the time evolution of the magnetization components expectation value of mx and y and mz and just I want to say so this is of course very very reminiscent of these boundary time crystals that Fernando Iemini is going to talk about this afternoon in the other session so this is just for you also to build a connection there between this talk and Fernando's so looking at these mean field equations we of course can now analyze those and I just want to tell you that I just parameterized here this equation slightly that instead of taking this feedback strength g I introduced this parameter kappa is equal to 2g plus 1 this is simply because with this the structure looks a bit more pleasant and more compact but anyhow so we and you have done this for sure when you are listening to this talk and your theorist once you have mean field equations things become really nice and pleasant because you can start analyzing them and play around with them and you can for instance draw stationary state mean field diagram so this is just obtained by putting the left hand side of this equation to zero some conserved quantities which I don't want to talk about here but if you consider this then you find the phase diagram here as a function of the strength of the laser driving and the feedback strength and what I plot here is the order parameters just a z component of this magnetization operator in the stationary state so you see here three phases one is a stationary phase so this is a plot here now performed as a function of time in this position of this square and you see the dashed line is the mean field solution and the colored lines are the exact solution or the numerical solutions of the master equation and you see really as I kind of promised as n tends to be large I mean 50 is still far away from infinity but at least you see the trend that this is really approaching this mean field result more importantly and this is connecting now to the topic of this workshop is this emergent of the time crystal phase when you increase the driving of the atoms and you really also see very nicely that you have these persistent oscillations here and when you compare this with numerically exact finite system results that these finite system results of course also show oscillations which are damped as expected but that the amplitude grows the larger the number of atoms becomes and yes thanks I'm getting there and what you can also do is you can now really look along this cut and you see these different regimes you see here the time crystal phase where the average magnetization is zero you can look at the scaling of the damping gap you see here in this phase in the time crystal phase it goes as 1 over m so the larger the number of atoms the more persistent these oscillations become and then at this transition point you see the scaling changes in between the stationary and the time crystal phase and you see the scaling with 1 over square root of n so the last three minutes just hope there will be some questions I will just spend on going beyond this mean field approximation because I think this is actually the interesting part about this study so what we were looking at in order to go beyond the mean field is to analyze the dynamics of fluctuation variables and these fluctuation variables are just defined as 1 over square root of n times the angular momentum operators J so these macroscopic spin operators minus the expectation value and you find it when you write down the anticommutator of these fluctuation variables then you get out the covariance matrix of the macroscopic spins so now if you look at the commutation relations and you take the equations of motion for our system when you look at the equation at the commutation relations between these fluctuation variables you see okay the matrix on the right hand side is not one that you would associate so to say with canonical commutation relations of a certain class of degrees of freedom but in fact you can change this when realizing that in fact this matrix can be rotated or the system can be rotated and you find that the system can be rotated in a way that from these three operators that are parametrized by alpha equals x, y, z one operator becomes classical that we call the z component and the other two operators are fluctuating perpendicularly to it so this rotation can be accomplished analytically and what you then find is that in this plane orthogonal to this classical direction you see that these fluctuation variables actually behave like position and momentum and obey the canonical commutation relations and this immediately makes you wonder well let's see what the quantum status and characterize it in terms of squeezing, why squeezing well you know if you have these canonical conjugate variables they obey Heisenberg uncertainty principle and what would be nice for instance in the context of say quantum enhanced measurements is to see whether you can actually calculate the stationary state that shows squeezing where one direction here in this plane of fluctuations has lower variance than you would typically expect for let's say a non-squeezed state or a classical state and this is what I want to wrap up here in the last minute with you can analyze this squeezing here and you find indeed that now when you do the phase diagram that there's no squeezing in the time crystal phase and actually these fluctuations here grow as a function of time but what you find is that one of the stationary phases and not the other one in fact is a phase where you get squeezing at stationarity and this is interesting I think because if you think about the system so in principle you can integrate it you don't have to use atoms let's say that you are sitting in free space next to this fiber you can in principle implement such a system maybe with impurities or quantum dots next to a wave guide and you can also of course apply this kind of feedback strategy in principle and just by applying this simple feedback you cannot only realize this for us interesting time crystal phase but probably you would be able to also create some kind of correlated on-demand for instance such squeezed phases which could be well important for technical applications like in metrology and this is just the idea so we like this model because it's to large extent analytically solvable and of course if one really wanted to implement this in a practical setting one would have to ask some questions about how long-lived are these phases really because for instance due to the effect of the emission into unguided modes but this is something we are really keen on exploring together with experimentalists which brings me to the end of the talk here are some other works of our group that we have produced in the past year thank you very much for your time I hope there are questions thanks a lot thank you very much and other please unmute yes he is unmuted please ask the question thank you very much Igor for the very nice talk I have a question about the noise squeezing that you commented on in the end so in the time crystalline phase you expect to get the period doubling by forkation so you should have actually two solutions and you are supposed to look at the fluctuations around one of them and then so my question is are you looking at truly the covariance around the attractor or when it explodes is it because of activation by noise to hop to the other attractor so okay I have to process the question because the language is really very much orthogonal to what I am used to but let me pick up maybe this last comment on noise noise is not present here it is all deterministic evolution in this framework that we are using to describe the dynamics so therefore I don't think that this is related to noise in the sense that you talked about when you were speaking for instance about activation so I would really see the connection here with in essence what you find is that if you have your microscopic spin what you can do is you can identify a classical direction and then you can essentially create bosons perpendicular to it. I mean it's a bit like the spirit of Holstein-Primakov but okay I think our approach is a little bit better and better controlled than what you would obtain when applying this Holstein-Primakov approximation I am not entirely sure whether I answered your question to be entirely honest actually when you do Holstein-Primakov you are getting an effectuation Hamiltonian around your attractor which is good it tells you how your system would respond if you kick it away from that attractor and now technically when you look at the covariance you would actually look at the statistical correlations between these fluctuation operators that you added on top of the attractor and this would be what you drive with noise actually so hence my question I still have to process it okay so I mean if you want to discuss yeah no thank you very much but I I guess we shall move to our host Rosario who also wants to ask question and so this will be the last question before coffee break I hope that the organisers will provide coffee Rosario of course just I'm not sure this is related to other questions at some point it was written that fluctuation grows in time does it mean that there is an instability or I just missed this point yeah I think okay this is a good point so I think okay it was written and I brushed okay I don't open this right now so what I said is when you look at the fluctuations in the time crystal phase then you see you have oscillations if you look at the squeezing in the time crystal phase then you see it fluctuates but it also oscillates but it also drifts away and I say okay at some point this is certainly becoming artificial right but what it means is that okay in the end you just spread out in your XP plane in this description that we are employing indefinitely so but it means that the reference point is not a good one anymore because you are expanding around something but you are drifting away I don't know if I understood correctly so what okay so when it comes to the precise interpretation okay I would not know now from the top of my head what to say here we okay to be fair we didn't really pursue this investigation further because this was not the interesting point to make about the squeezing so to say so I mean for let's say characterizing the squeezing in let's say the useful regime it's not the time crystal phase that we were interested in but the stationary phase the squeezed phase and when approaching to the boundary of the time crystal phase the choice of coordinate system let's say it like this seems to be the appropriate one but with this same choice in the time crystal phase it seems that okay one has to think whether this is a wise thing to do okay thank you okay thank you all very much I think that we we deserve our coffee and we meet again in 21 minutes at half past four thank you and see you then hi okay I just wanted you to unmute and everything is fine so we are waiting okay thank you okay it is almost so we still have to wait a little bit maybe one minute more okay I think we can start the last session of today so Albert Cabot will tell us about surprisingly some new subject time crystal the floor is yours Albert okay thank you can you hear me there are problems can you hear me we can hear you but with some please you don't hear me well we hear you well let's try okay okay okay so let me thank the organizers that I in collaboration with and it is a work about dissipative time crystals in which dissipation plays a key we have that the interplay of dissipation interactions non-linearities and driving leads to the and they have different ways we can have discrete time crystals in this context we have the system described by a time-dependent master equation and the master equation is periodic in time in the thermodynamic limit there emerge other problems that display decaying constellations and the fundamental of these oscillations is a multiple of that of the master equation and then we can have also continuous time crystals in which the master equation is time-independent in some frame and again there emerge non decaying constellations but now the crucial difference is that the period can vary continuously with the parameters of the system and these are some works in which these ideas are studied so let me focus in this limit this end to infinity in which spontaneous time symmetry occurs so if we have a spin system this is the infinite spin limit however this phenomenon can also occur in zero-dimensional systems like non-linear optical cavities and in this case it could correspond to the infinite photon or infinite acceleration of the limit and in this example we can see the difference with our interactions as well as this zero-dimensional non-linear bosonic systems in this limit the dynamics is given by classical non-linear models which fit the needs of film and for this reason in this reference the notion of imagined semi-classical time crystal has been introduced and in this sense the results of this experiment is a type of synchronization phenomena in which a cell-sustained oscillator adjusts its oscillation frequency to that of an external periodic forcing or to a fraction or multiple of it so this is an example so we have two signals in blue the one of the cell-sustained oscillator in red the one of the forcing and we can see that in absence of synchronization frequencies which are not commensurate however there is a dynamical regime in which the oscillator can adjust its frequency to half of that of the forcing and this case is known as supermonic entrainment so there is the interesting question of whether this kind of synchronization phenomena can be observed in quantum systems and in the quantum regime and this is a question that has been addressed by a lot of people in the past 10 years and these are some of the basic questions of whether these phenomena survive quantum fluctuations which are the signatures and indicators whether there are genuine quantum aspects etc. and these are some references now a very important system in the context of quantum synchronization is the quantum bander oscillator and here I have to acknowledge the talk by Marco Shiro in which he has nicely introduced this kind of system so these are some important references and here I will focus on the squeezed quantum bander oscillator which is a model known to display supermonic entrainment so basically this is the model in the laboratory frame so we have a master equation for a driven dissipative bosonic mode and basically we have a term that gives energy, a pump and a non-linear ramping and recall this is the definition of a dissipator and then we have a Hamiltonian in which there is the squeezed forcing that introduced a discrete time periodicity in the model so we have that it has this frequency and this frequency is in general different to that of the oscillator so a nice thing is that we can define a rotating frame in which this model becomes time independent and in general in this talk I will be showing results in this frame so in this rotating frame I indicate by the subscript R the dissipative part is the same but now the Hamiltonian is time independent and instead of the intrinsic frequency we have the dead unit and of course both frames are connected laboratory and rotating frame are connected by a unitary transformation that changes the Hamiltonian and the state of the system so a nice thing about this rotating frame is that the master equation can be written in terms of a time independent Liubilian and we can try to understand the dynamics of this system by analyzing the spectral properties of the Liubilian and this is something spectral properties of the Liubilian is something has been introduced already in the talk by Anna Sampera and also by Marco Chiro so basically the Liubilian has a set of right again vectors a set of left again vectors a set of again values that appear in complex conjugate pairs and whose real part is non-positive and there is at least one stationary state so a nice thing of the model I discuss is that it displays a parity symmetry that is the master equation is invariant under this transformation and more formally we can define the parity super operator which has two possible again values and the consequences that all these again modes are either parity symmetric or parity antisymmetric and in particular the stationary state is parity symmetric now I said that this time symmetry breaking phenomena occurs in certain limits in our case this is the infinite excitation limit so we define this parameter that is the gain strength over the non-linear dissipation strength so in the limiting which this parameter goes to infinity basically the boson number diverges however the rescaled quantities tend to a well-defined quantity and this well-defined quantity comes from the mean field equation so this equation for this complex amplitude is derived basically factorizing the equations of motion for the amplitude and it has two free parameters the detuning and the squeezing strength so basically we can understand the mean field dynamics by studying the bifurcation diagram and this system displays just one bifurcation at this critical point and below this squeezing strength the stable attractor of the dynamics is a limit cycle it's a time-dependent solution with this fundamental frequency and above this critical squeezing strength the stable solution, the stable attractor are to be stable fixed points so back in the laboratory frame this limit cycle the frequency is generally not commensurate with that of the forcing so in this regime, in this dynamical regime there is no synchronization whether in this of the critical point, in this dynamical regime the stable solution requires this kind of harmonic evolution and we find, indeed super harmonic entrainment with that external forcing so a bit, this is what I am to do in this talk so we have seen that we have two dynamical regimes at the mean field level the limit cycle regime is associated to a lack of synchronization and the stable regime to entrainment and I want to discuss whether at the master equation level we find continuous time-symmetry breaking or discrete time-symmetry breaking and I will do so analyzing this limit that is the infinite excitation limit and the long-time dynamics and to do so it's a a good way to do so is a finite size analysis of the Liouvillean spectrum since we can write the state of the system at any time in terms of the eigenmodes of the system and the eigenvalues, the time dependence enters through the eigenvalues and this exponential form and from here we can obtain any observable so two possible interesting outcomes are non-decaying constellations if in this limit the real part of the eigenvalues, of some eigenvalues goes to zero while the imaginary part is different from zero and we can also have dissipative phase transitions and multiple stationary states if some of these eigenvalues go to zero so let me first analyze this regime below the critical point so this is the lowest Liouvillean eigenvalues for different ends so we go from left to right towards the infinite excitation limit so in the horizontal axis we have the real part of the eigenvalues and in the vertical axis there is the imaginary part and we can already see from these pictures that there are two groups a set of eigenvalues which in this limit the real part saturates to a finite value while there is another set of eigenvalues which in this limit the real part tends to zero and we can analyze this in more detail and we can ask how this real part vanishes and they turn out to vanish following a power law and this is what I show here so I show the real part of the eighth first of these eigenvalues and we can see how as we increase n this is in log scale all of them follow one of these power laws moreover the imaginary part tends to multiples of that mean field frequency I presented so basically this is conceptually what happens in this regime so in the long time and in the infinite excitation limit basically many eigenmodes with non-zero frequency but zero decay rate contribute to dynamics at finite m what we will observe is that oscillatory quantities such as the amplitude dynamics display oscillations with increasing lifetime as we increase and as we go to the infinite excitation limit so in this case we find in the regime continuous time symmetry rankings and we cannot focus on the other regime so in this other regime the behavior of the eigenvalues is very different so basically we can already see that the leading eigenvalues are real and all of these saturate to a finite value in the infinite excitation limit while this shows the interesting behavior of going to zero so basically in this regime we find the closure of the Liubian gap that is a dissipative phase transition so basically above the critical point this eigenvalue goes to zero and this is what I show here for three different values of n so above for a squeezing strength above the critical value we basically see how this eigenvalue changes strength and goes very fast towards zero and the larger is n the closer this happens to the bifurcation point and the faster it goes and a very important point is that the eigenmode that goes to zero is parity anti-symmetric which is different from the stationary state this is what I already said so in red this is n and in red we have this eigenvalue that goes to zero while in blue we have the other eigenvalue that we see that saturates to a finite value as we increase n so basically a very important consequence of this gap closure is that due to the anti-symmetry of this mode there is a spontaneous parity symmetry this means that we can define these two states which are stationary states in this limit and which because they are made of two states of two modes with different symmetry they do not have a well-defined parity symmetry so we can ask who are these symmetry broken states and if we compute the amplitude in these symmetry broken states it tends towards the mean-field amplitude of the V-stable solutions so this parity symmetry broken states that we find are reminiscent of the mean-field V-stability so for finite n they are not completely stationary but they have a very long lifetime and this is what I show here in the amplitude dynamics so starting from one of these parity broken states we see that as we increase n the lifetime increases dramatically and this is in lock-scale so which is the relation of these with discrete time symmetry rating so for this we have to go back to the laboratory frame and analyze the dynamics of the state so I will write down the dynamics of the state in the laboratory frame and in particularly at stroboscopic times that is at times that are multiples of the Hamiltonian and why is that because at stroboscopic times we can write this state easily in terms of this rotating frame again which is what what I been explaining before so basically we know that we have a set of parity symmetric modes and a set of parity and symmetry modes and basically the parity symmetry modes contribute the same in both frames at stroboscopic times however the parity and the symmetric modes acquire an oscillating thing in the laboratory frame and this is the key feature so basically if we have the closure of the liwiang app and the mode implied in this closure is antisymmetric we have that this is equivalent to discrete time symmetry breaking in the laboratory frame and this is because due to this eigenvalue tends to zero basically the long time dynamics in the laboratory frame can be written as this and we have this contribution from this parity antisymmetric eigenmode which has an oscillating phase which comes from the fact that it is antisymmetric so now we find that the dynamics of the state displays period doubling and due to the symmetry of this mode we find that only antisymmetric observables are sensitive to this period doubling so let me so I will end up with showing what I said for this dynamical regime in these pictures so basically the main result in this regime is that the gap of the liwiang closes this is related to discrete time symmetry breaking and basically in the rotating frame we find this symmetry broken states that show a diverging lifetime and we can see this here and if we look at the dynamics in the laboratory frame we will observe as I said period doubling so for instance here I show the amplitude dynamics and this is a zoom of this region in the laboratory frame and in blue we have the full time evolution in yellow the stroboscopic time evolution and time is given in units of the period of the Hamilton and very clearly we can see this period double dynamics which even for finite n far from this infinite excitation limit it seems almost stationary and recall that any antisymmetric observable will be sensitive to it so in conclusion in this regime we find discrete time symmetry breaking so the main result here is that the squeezed interpolosilator can be regarded as an emergent semi-classical time crystal in which we can have both continuous and discrete time symmetry breaking depending on the parameter values and importantly the transition from one kind of time symmetry breaking to the other is a dissipative phase transition with finite symmetry breaking so I hope these results will be shown in the archive with these additional results and we will be happy to answer to your questions so thank you for your attention and do we have questions we have time for the question ok there is a question coming from Oded yes hi Oded thank you very much so you're bifurcation does it actually occur exactly the same point that you have the exceptional exceptional point or are they released from one another due to the fact that you have dissipation in your problem ok so well I mean so I guess you refer to this additional result well you can find that the classical bifurcation point is manifested in the Lyubilian as a spectral signature and you find that critical and this is the exceptional point and you find that the critical squeezing strength at which this exceptional point occurs goes towards the classical critical point when you do this infinite excitation limit and this is maybe you can maybe appreciate it here so this turning point turns out to be an exceptional point and as you do this limit you can see how I mean I can show with more detail how this turning point approaches the classical bifurcation point I don't know if that answers your question yes I guess it does but do you make this getting closer to the point by reducing the dissipation actually you take the zero couple limit well the end to infinity limit corresponds to making gamma 2 over gamma 1 very small nevertheless dissipation I mean you can never kill dissipation because actually non-linear dissipation is what stabilizes these attractors when there is an instability right ok thanks ok thank you and thank you Albert ok and we now switch to Fernando Iemini from Brazil hi can you hear me? yes we can hear you ok let me share them in my screen very good and we speak ok ok so first let me thank the organizers for this very nice conference on tami crystals and also I thank for the opportunity to participate and present this recent works that we have on boundary tami crystals so I'm Fernando Iemini I am from Institute of Physics from Federal from NSI in Iteroio Rio de Janeiro Brazil and this recent work it's called boundary tami crystals in collectivity level system and it was done in collaboration with Luis Fernando dos Prazer is Donado da Silva and myself and both from the same institute here in Rio de Janeiro and we just blow the few weeks ago on archive if someone is interested to take a more careful look just go to this link so just a brief outline of this talk I'm just very briefly comment on this concept of boundary tami crystals that was introduced a few years ago and then I go directly for my results on this collectivity level systems and in particular we are going to analyze in more details the equal 2, 3 and 4 cases and then I go from my conclusions and perspectives so the idea of this boundary tami crystals it was introduced in this work in 2018 with other collaborators some of many of them are here, Angelo Marcos, Kiro, José Rufazio and also in collaboration with Jonathan Killing and Marcello Dalmonte and the idea of this boundary time crystal was looking for order time crystal phases instead of the full Hubert space with all of its degrees of freedom was just in a microscope portion of this full system so the idea was to focus on the analysis of the dynamics just at the boundary of the system and by the boundary I just mean a part but a microscope part in order that we can properly define a phase there and look for spontaneous image breaking on this boundary degrees of freedom so just to mention briefly this idea comes somehow from the surface critical phenomena that we have different examples where only at the surface of my system that represents a microscope portion of my system it is ordered while in the bulk any order and we are looking now for the ground states or thermo states of my full Hubert space but on non-equilibrium dynamics that go directly out of the you know, the area so the idea then is that we have the full Hamiltonian it is just described by the part of the my boundary degrees of freedom that they are just let me call this blue spins and I have also the bulk Hamiltonian degrees of freedom that they are just the blue one and the full Hamiltonian is just the boundary theorem the bulk Hamiltonian and an interacting term between both of them but we are only be interested in the dynamics of my boundary system and in order to analyze the dynamics of my boundary system we know that the full Hubert space it is just driven by the Schrodinger equation and if you trace out the bulk degrees of freedom we have how it is always possible to define an effective description for my boundary degrees of freedom that is always described by completely positive trace preserving map so in case that we have some order time crystal phase or boundary time crystal phase appearing in my system it means that I have a microscope order parameter such that the expectation value of this order parameter in the dynamic limit of both bulk and boundary degrees of freedom it has some non-trivial time dependence or a time dependence that is different from the Hamiltonian time symmetry itself so we would have this dependence that persists only in the dynamic limit and would be also robust to some perturbations in order to characterize our boundary time crystal phase so a very simple example of a model showing this phenomenology is just looking for instead of looking for very intricate or hard to analyze CPTP maps we just look for a Markovian system such that the effective dynamics is just driven by a lead blood master equation and in particular if you look for just collective two level systems coupled to a cavity we can see this boundary time crystal or this kind of phenomenology appearing in a very simple form we know that there are many other different examples of lead blood master equations showing the dissipative time crystals also as was presented in this work but in this particular work and in this talk I'm just going to focus on collective systems so for this type of dynamics we just have a competition between a coherent dynamics that is just a field a collective field applied along the x-direction that tends to flip my spins in my system and a competition of a collective dissipation that tends to make a collective decay on my spin degrees of freedom so I just represent this type of lead blood or this type of collective dynamics here in this rough let me say it's a rough schematic representation where we have a competition between a coherent Hamiltonian of this collective level 0 and 1 and a collective dissipation between these two it's going to be easier to show the other models that I'm going to present in this schematic representation and then for this specific model here it was shown that if the supportive part is too strong it's stronger than the coherent dynamics we don't have any non-trivial time dependence of macroscopic observables in the long time limit and the spins just tend to be a ferromagnetic trying to point down in my system otherwise if we have that dissipation is smaller than the coherent dynamics we see this boundary time in crystal appearing in the system so we see that while for finite sizes we have magnetization with finite decay time but as we increase the system size we see that this lifetime increase also until if we go to the thermodynamic limit we have this persistent oscillations of magnetization so one of the models that I'm going to show here in this talk it's actually is exactly this collective two level system but we are going to take a more careful look on different interacting terms on the Hamiltonian just like sx sx or z magnetization interactions as well as magnetic field along the z direction we're going to see that they have different effects on my modern time crystal in more details in a few slides so with basis on the the size that we have on this collective two level systems I'm also going to analyze a collective four level systems that is just defined as a pair of interacting two level systems so then the system it's locally it is very similar to this collective two level system but we also consider a collective interactions between these two pairs of collective two levels and we look for the dynamic phases that appears when we have interactions between these two a boundary time crystal phase and actually we see that some more rich phases appear there and that is very interesting and as a third model that I'm going to discuss in more details in a few slides also it's this collective three level system where now we have this that each one of my subsystem has three levels so we have this three collective three level system and each one of these pairs of collective levels here we are going to consider that they have a dissipative dynamics or a link blood dynamics very similar to the boundary time crystal for the two levels so in this case here we have somehow a competition between a pair of two level link blood channels so we have this link blood and acting this first two and this link blood and acting in the second two collective channels and different phases also can appear in the system it's just worth that we're calling here that even though this is similar it resembles similar to the four level collective system in the sense that we might have a pair of collective two levels but this is not the case because these two pairs they share a common collective level so the full algebra of collective operators that drives the dynamics here is actually an SU3 algebra that cannot be decomposed as a pair of SU2 so this is just because they have some overlap in one of the collective levels and different physics can then appear in the system so first just looking for the simplest case that is the two level collective system and for the moment I'm just focused on interacting terms on the effects of them without the magnetic field along this direction so in order to analyze the system we go directly to the macroscopic limit since we know that if you want to look for the collective magnetizations in this macroscopic limit we can use a semi-classical approach that gives us some reliable results so the semi-classical approach is just defining a macroscopic magnetization and we close at a second to the expectation of values with this closure we will be an effective Heisenberg picture for the magnetization dynamics and they have this very peculiar structure from which we can also see some symmetries in the dynamics that comes from this dynamical equation of motion one of them is just the conservation of total spin that is really the one even for finite system size that we have really is a collective operator a second symmetry is that we see that in the X and Y dynamics for the magnetization we see that the magnetization along Z it has somehow factorization structure and then we see very easily that we have also a reversibility symmetry by inverting the time and the magnetization along the Z directions the dynamic equations remain the same and as a further peculiar property of the dynamical equation of motion we also have some quasi-conservative quantities and I'm saying here quasi-conservative because since they are defined with this logarithmic of magnetizations between the three different axes this logarithm is just is just defined up to integral modulus of 2 kappa pi so this very peculiar dynamics here and if you look for different effects of the interactions we have somehow the same this general view for what happens in my system so I'm just looking here for the phase portrayed in the phase space variables where the X and Y they defined in the momentum variable here and Q is just the magnetization along Z directions so we have this general picture for the dynamics that if we have some initial conditions that have a larger momentum we have some attractors appearing in the system or some repulsors appearing in the system and we don't see really the timing crystal dynamics in the system however if for certain initial conditions with small p and close to a magnetization equals 0 we see several closed orbits appearing in the system so this closed orbits is really the boundary timing crystals appearing in the system and we see that for the specific model it always appear having fixed state-state or the state-state as a center or with magnetization equals 0 so when I say here that's the center it's just looking for the stability of the state-state here located at MZ equals 0 and we see that from Jacobian stability of the state-state they have purely imaginary eigenvalues they do not have any real term so looking then from this perspective of Jacobian stability we can actually take a look at the full phase diagram for different interactions along Z and along X direction and we will characterize this phase diagram of the model with the presence or absence of boundary timing crystal just from the stability of the state-state or fixed points of the dynamic of motion so in this case we just define this three vector here and where the first element is just the number of centers like this here in the phase diagram and single hyperbolic and bear hyperbolic is the number of tractors that do not have non-trivial time dynamics and we see that as we increase the magnetization along X direction it tends to increase the number of centers to have more boundary timing crystals appearing in the system while if we increase the magnetization the interaction of the magnetization along this Z direction it tends to increase the amount of time of the center or some of the boundary time crystals in the system and then now looking for the effects of the magnetization along the Z direction we see first that the magnetization it breaks the Z to Hamiltonian symmetry and in order to understand the effects of feet along the phase diagram of the model we just look again for the stability of the state-state from the Jacobian can be seen as this three times three matrix here and from perturbation theater from for general non-air mission matrix we find that the first order corrections for all of this center state states that used to characterize the boundary time crystal they now have a real part which means that or they become attractors or they become reposers and in this way they always destroy the boundary time crystal so in the sense magnetization that breaks the Z to see Hamiltonian tends to break all of this boundary time crystal in our system so with this I'm sorry I'm sorry four point five minutes okay so let me just run a bit more fast so for this for a level collective model we take we consider magnetization along X direction but also consider the magnetization along Z direction in both of my pair of two level systems and also some interacting some collective character between these two pairs here so now we have a two algebra and the following equations of motion that are similar but now we have some interactions between the two systems here or here also and the idea here will be to consider more general phase in the sense that we break all of those dynamical symbols of the two level system in order to generate some more complex dynamics but also possibly with richer dynamical phase and for for this reason we consider for example that we kappa physical one so we have a weak and supportive regime for each local two level system so have a strong X omega X and omega X X such that they will generate the boundary time crystal but we put also some magnetization local magnetization in both of these two that we know that destroy the boundary time crystal and then start the effects of this hybridization of these broken time crystals when we increase the interaction along the Z direction between them and we see that indeed for omega Z to Z to equal zero is just a trivial phase no boundary time crystal just looking for the magnetization along the Z direction if increase above some threshold we see some limit cycles appearing in the system further increasing the interactions we see some pure doubling oscillations in my system where I see that it has now two minimums during the periodic oscillations and further increasing we see a chaotic dynamics appearing in the magnetization or in my phase space so in order to have a more general view of what is happening due to this interaction along the Z direction of this pair of collective models we just look for the orbital diagram from my system just looking for the minimums along that dynamics of magnetization and we see that why for omega Z Z close to zero it's a trivial state no dynamics at all in the long time limit but then we have a transition to a limit cycle phase here around 0.2 and if we keep increasing and then this figure here is just a prefigure here if we keep increasing the interaction we see this series of period doubling modifications up to chaotic region so indeed we consider this more general for a level collective system we see richer dynamic phases appearing there from ferromagnetics that states limit cycles chaotic dynamics and further three cycle periodic windows appearing here in between the period doubling modifications and then for this third model that I'll analyze as I mentioned very briefly is just the collective three level system where now we have a coherent Hamiltonian just along the X direction coupling each pair of collective level and the supportive decay happening also in each one of this pair of collective levels and we want to analyze essentially the competition of these two linvaladian channels acting in the first pair and the second pair of my system so I'm just recall as I briefly mentioned that since they share a common collective level we cannot reduce this just to interacting collective spins anymore because we have a more general SU3 algebra that cannot be reduced to this case and then for simplicity I'm just going to show here the case where both the omegas 1, 2 and 2 and 3 they are equal as well as kappa 1, 2 and kappa 2, 3 they are equal as well then we want to study the effects of the competition between these two channels the delta parameter and the ratio between omegas and kappa if they are in the stronger the weak dissipative region and we will take the following phase diagram for this type of model it's very interesting because we see these three different phase along also with a critical line separating some of these phases and one of the interesting thing that we see is that as long as delta is different from zero but smaller than half so even a small competition of these two first levels to the second two levels it tends to have a trivial phase in the sense that we destroy any boundary time crystal that could be appearing in the system there so the dynamics it tends to cancel a collective level in the middle here in the end it's going to be like a dark mode without any population and it stabilizes static state just between one and third collective levels so another interesting thing that we also see is that a boundary time crystal appears in the form of limit cycles for larger coupling computation delta and also omega over kappa larger than two thirds and it's interesting from the point of view that we see that the boundary time crystal here it's appearing even for the strong supportive case that we know that for just two levels this will be just a trivial phase so somehow this competition enlarges the space of support for a boundary time crystal and also this critical line separating these three different phases here it's very interesting because in this specific line here we see boundary time crystals appearing there but with the coexistence of both closed orbits just like the two level collective system but also with limit cycles depending only on the initial condition that we start the evolution of the system and moreover if we look for the Yapunov exponents of this dynamics we see that in both these cases of closed orbits or limit cycle dynamics they are all they have a full degeneracy of new Yapunov exponents so all of the Yapunov exponents are equal to zero so just to be more clear here so here is just considered initial conditions that are close to the fixed points and we see these closed orbits the Yunos and Dashed lines they are for different initial conditions but if we start with the initial condition that's far from the state from the fixed points we see the limit cycles both of them leads to a zero Yapunov exponents so with this I just go from a conclusion that in this work we analyze the boundary time crystals in two, three and four collective levels and we see that depending on the dimensionality of the collective levels this boundary time crystals can appear in different forms providing a richer phenomenology for this phase and as interesting questions there are many different but I will just mention would be this interesting to analyze the larger the levels as well as take a more careful look on this global and dynamical symmetries for this phase so with this I thank for the attention sorry for if I pass too much time thank you for your talk and Oded he has one fast question and then we get a fast answer Oded sorry it seems that I'm the only one awake so but I was only clapping if you want I can ask a question so there is technically in the conference until now there is a nomenclature that I think all comes to to be the same and I wanted to verify that you agree with me that we have heard people talk about the Lioverian eigenvalues we heard people talk about the complex pole dynamics in Green's function responses or spectral functions of the responses and you talk now about Nall Leopunov exponents and there was of course mention of exceptional points would you say that your Nall Leopunov is what people would call exceptional points I don't know actually I don't know I'm sorry I would say yes but ok ok so with with this doubt we shall pass to the last talk of this session sorry we have to move on and the talk is by Professor Das right so he can share the screen Praloidu are you there yes I am there ok good so we have to change the screen slowly now I can see it ok very good you can share your screen and the floor is yours or the screen is yours whatever the microphone is yours now firstly I want to thank the organizers for giving me an opportunity to give a talk in this interesting conference I am Praloidu from Indian Statistical Institute India now the title of my talk is the cosmological time crystal that is the cyclic universe with a small cosmological constant in our toy model approach I would like to acknowledge my supervisor, Professor Subir Ghosh from ISI Kulkata and all my co-leaders Dr. Sukriopan and Dr. Prabhupal from India now this talk is based on our work this one the plan of my talk is that I will give a brief introduction then I will talk about the classical time crystal after that I will discuss about the non-commutative action and then I will discuss about the cosmological time crystal where we will see that it will allow us to give a cyclic universe with a small cosmological constant and finally I will summarize it now within a time crystal has created an enormous amount of interest both in the theoretical and experimental constant now the symmetry of a system is broken when the ground state of the system is less symmetrical than the equations of motion that control the system now the time crystal is bought from the familiar crystal that has specially ordered structure in its ground state and it is a manifestation of breaking of the continuous translation symmetry leaving behind a ground state with the discrete translation symmetry now the question posed by Sapri and Bunjik is the following was that can a system with time periodic ground states that breaks the continuous time translation invariance exist now the resulting system with the discrete time transition symmetry was referred to as the time crystal now after that a mathematical model for a classical time crystal was provided in this paper by Bunjik and later a more physical model was constructed where the relevant degrees of freedom undergoes the periodic cc first dynamics in its lowest energy state now what we have done in our work is that we have applied the concept of classical time crystal proposed by them and which is we have applied in an extended model of Friedman Robertson worker cosmology that is the F.R.W cosmology now this extension is induced by the non-quantitative gravity contribution with an underlying quantum gravity perspective which was derived by Fabi, Herms and Stren you can see this paper now two of our main results is that the scale factor it borrows the cc first like behavior that characterizes the classical time crystal but for more interesting it serves as a physically motivated toy model for a cyclic universe and also the minimum energy state or the ground state consists of a condensate leading to an arbitrarily small positive cosmological constant now it should be noted that our model is purely geometric in the sense that no matter degrees of freedom are added from the outside now let us discuss the classical time crystal first we consider a generic Lagrangian the Lagrangian looks like of this expression that leads to the energy of this type now we have rewrite this energy as this type of form where we have seen that this energy minimizes at 50 dot equals to plus minus of root over of a by 3 b and phi is equal to phi 0 now this is the classical time crystal ground state with mutually opposing requirements of simultaneous constant velocity and constant position now to further understand this situation we note that the Lagrangian equation of motion looks like this expression for phi double dot which diverges at the energy minima and furthermore the energy becomes a multi-valued function of momentum now this momentum p cops at the energy minima of del b by del phi dot as equals to 0 where we get phi 0 dot equals to plus minus of root over of a by 3 b now we will see that this graph structure leads to catastrophe for our classical time crystal model now at exactly ground state the system becomes singular but the interesting behavior is recovered for the system goes to the ground with some energy slightly above the minimum energy that means e is equal to e plus minimum plus delta delta is very small now we assume this energy occurs at the phi t we can solve for phi dot and from this expression we get the expression for phi dot and this equation has four independent solutions and when we interpret this behavior of the system as that phi dot tends close to square root of a by 3 b but it gets altered by the second term but since also this term cannot go beyond a certain time since phi cannot be exit fighting now although the poison changes but when it reaches near about the value that minimizes the potential the velocity falls back and this cycle is repeated and in short this is called the sisyphus dynamics and thus the system is forced to undergo sisyphus dynamics where on an average phi stays close to the constant phi 0 but periodically goes through a phase of non-zero phi dot and now we come to the part of the non-commutative action now the conventional effort of Friedman-Robertson-Walker metric reads like this one where a is the scale factor now we have taken the non-commutative corrected effort to metric that was derived in this paper but x mu and x mu as the component of only theta mu nu is non-zero and this is the non-commutative parameter so this is the expression of the metric where c is the velocity of light here now the generics form the Einstein-Hilbert action is reads like this one where r is the risk scalar r minus 2 lambda root over of minus g d4x and it reads like of this expression where the non-commutative corrections are introduced higher theta in this numerical parameters sigma beta 1, beta 2, beta 3 alpha 1 and alpha 2 now we have just written in a more compact form and in the compact form it looks like this expression where a, b and c takes up this type now we have introduced a dimensionless universal parameter nu as theta square lambda square by c to the power 4 and we have defined that the canonical momentum p is del L by del A dot and from this expression of momentum we get our cherished expression for the Hamiltonian now in this work where this Hamiltonian is defined for primary result which we will now study from the classical time crystal perspective now a very richer curve structure is developed from this p versus h diagram now following the paper of this one we have the plot of p versus h shows that because of the a dot 6 term which was present in the del p by del A dot I am just going to the previous slide this one we can see that this is the fourth order equation in A dot and having four solutions thus the dotted blue line it resembles the profile of this paper which was posted by sapper and we will check where the A dot 6 contribution is negligible whereas this solid line deep grid line this shows the new structure with some significant A dot 6 contribution now we name this new structure as the battering in the cellotel catastrophe as obtained in this this paper where the contribution of the A dot 6 term becomes negligible and also this curve structure is completely washes out if the A dot 6 term dominates even slightly leaving an inverted symmetric v shape diagram with vertex at h equal to 0 now we come to the part of the cosmological time crystal now firstly we will consider our model where we will not consider the A dot system that means it's the planar model without the A dot system from the action we can find out the Lagrangian equation of motion this is the Lagrangian equation of motion where A prime is just the derivative of A with respect to the scale factor A now in this figure we have plot the scale factor A is time derivative A dot and the total energy h this solid line shows the scale factor A and this dotted line shows the A dot and this dash dot curve shows the for the Hamiltonian for our system thus we can say that the cosmological model framed by the time crystal it allows the cyclic nature of the universe now we believe that some further investigation is still required to possibly understand the nature of entropy transfer from one cycle to another in the evolution of the universe now we follow the just we have followed this paper and that shows that the way of removing the singularity by introducing a regulator in the form of mu we have just introduced the regulator mu in the Lagrangian which looks like this expression where fx and gx looks like of this one and the quantities k and j are of this type which depends on A and B now from this Lagrangian we have derived the Lagrangian equations of motion and these two are the equations of motion now this model is similar to that of the model by we will check from the time crystal point of view since they have also only A dot square and A dot four times and we have just neglected the A dot system for our present case right now and however but still however the scale factor dependence is much more complicated in our case and this is reflected in the x profile that contains both the positive and negative values whereas in for their case it has only either positive or negative values but in for our case it has both positive and negative values with the sharp edges separated by some smooth curve thus our model describes a double dynamics now this is the profile of the scale factor A this one is for x and this one is for the now we will come to our model where we will consider the A dot six terms now from the from the action we have similarly we have previously derived we have calculated the equations of motion where A prime B prime and C prime denotes a derivative with respect to the scale factor now a very surprising and interesting result is that we have found some small parameter window in which the scale factor oscillates smoothly without any similarity and with constant energy which indicates also again a cyclic universe so this is the profile of the scale factor A this one is for the x and this one is for the h now similarly we have tried here also to construct the mu regularized model what we have done in the previous one we have kept the same form of the Lagrangian as we have taken the previous one but with here the forms of f and g are new okay we have taken new expressions and from this since the Lagrangian is the same form we will find the same set of equations of motion as we have previously derived but with some new identifications with this expression but this shows that the regularized form of for A dot 6 model is not entirely satisfactory as it fails to generate the independent form of C which was given in the accent this suggests that a more elaborate version of the mu regularized model is needed to faithfully represent this A dot 6 model now we will come to the point of the time cluster where we have derived the cosmological constant with small cosmological constant in our model we have let us react the condensate energy where we will need this condensate values a g and a g dot which will minimize the Hamiltonian now the Hamiltonian can be written in the form of this we have just written from the Lagrangian one here V effective looks like of this expression now here the Hamiltonian will be minimized for A0 dot which looks like of A0 by 6 B0 where A0 is obtained from the potential of derivative of the potential with respect to where A is equal to A0 the constant energy will looks like of this expression where A0 and B0 are the forms of A and B where the scale factor is A is equal to A0 now this is the figure which shows the quantum evolution for the variation of the effective potential that means the solid line shows the del V effective by del A and the cosmological constant is shown by the dotted line now in this figure we see that this solid line it starts the axis at A0 which is instantly greater than A equal to 1 that means it is in future where we define the effective cosmological constant as this expression where we can see that the value of lambda at A0 is equal to lambda effective which can be arbitrarily very small as we see from the figure with even a large value of lambda thus the value of cosmological constant is redonarized in a sense now lastly I will summarize what we have done in our work we have we have proposed a new form of time history and some of its consequences have been studied the model is a generalization of the Friedman-Robertson Walker cosmology which is end out with some non-combative geometric corrections we have find that the scale factor undergoes the time periodic behavior or the CC first dynamic switch allows us to interpret this cosmological time crystal as a physically motivated toy model to simulate the cyclic universe the other major success of this toy model approach is that it allows us to generate an arbitrarily small positive cosmological constant in a natural way now in future we would like to see that here we have known since our model is purely geometric in sense that no matter degrees of freedom are included no matter degrees of freedom can be the area where it can be loop since where one can conjure that more than one time crystal structure that that since the behavior will come in from the matter sector as well as the FRW sector now another interesting part is that we can analyze the other forms of time crystal that can appear naturally in cosmology in particular in FR gravity and in also some high derivative gravity series thank you for your time thank you very much and we have time unlimited time for questions Krzysztof Saha wants to ask the questions so he can unmute himself okay I have a question because in your Lagrangian or your Hamiltonian you have kinetic energy part and also potential energy part and in order to solve the equation of motion you have to I guess you have to introduce some regularization parameter I guess that this is in a similar way like in a work by Al Shaper and Frank Wilczek and then do you have something because in Shaper and Wilczek work it is possible to show that this effective model can come from so two-dimensional physical model and then this as far I understand then this regularization parameter appears naturally can you justify your regularization parameter in your model yeah in our model from the Lagrangian we have found out this equation of motion and then we have to regularize what they have done in their paper we have just followed that to introduce mu mu is the regulator which we have introduced in this Lagrangian and from this Lagrangian we have found out the equations of motion and when we have plotted this and we see that this model also describes some compared to them our model describes some double physical dynamics but what is the meaning of mu where is the meaning of the parameter mu value there is a new parameter yeah mu in the Lagrangian but what is the meaning of this parameter do you have some physical meaning in your model for this mu or this is just regularization parameter regularization and I can also say that what we have for the plot what we have given the value of mu is something like 10 to the power minus 5 just a value of this no okay thank you okay thank you very much I don't see other questions so I think thank you very much prealloy and thank you we have reached happily the end of the session the ship has reached the harbor so we can safely go home or go to the conference dinner I don't know if Rosario has some dinner prepared for us in traditional Italian style well this is something we should still work out and find out how to do okay good meanwhile thanks a lot for sharing the session I think we meet tomorrow morning at 9 right yes okay see you tomorrow thank you very much bye bye