 Reception was not too long, so that everybody's awake. So I come to our first lecture. Our lecture today is Francesca Fagliino, and she's going to explain us how quantum simulation works in practice. So very much looking forward to your lecture. Hello, so hi, everybody. I hope you can hear me well. Yes, so we will be together in the next couple of hours. And so I'm an experimentalist, first of all. So I think that the hi, are there any experimentalists in the audience? One, two, three. OK, really, let's say singularities, I would say. But so I was trying to go very slowly somehow and to give you some idea of what we are doing in our lab and what in general is related to, so what is really the meaning of quantum simulation, which I would say there is no strict definition in any sense. And I will let's say guide you a bit on what happens if the platform, the architecture that you want to have for quantum simulation are ultra cold atoms. And in particular, I will show you, let's say in the first part of the lecture, we will focus on the case of system, which are discrete. What do I mean? That there is a periodic structure and so make the simulation in terms of discrete Hamiltonian. But then in the second part, I will show you something which goes really a little bit beyond the simulation but also not considering the discrete but more the continuous. Because many, many systems in nature are kind of a continuous or a quasi-continuous structure. And of course, the simulation in the continuum is somehow more tough, more difficult. And the reason for that in general is that, let's say the particle move. While the idea of having a discretization is also the idea to impose, let's say, a structure in space. And if we want even to switch off the external degree of freedom, meaning the degree of freedom of the motion, this make your life easier because kinetic energy play an extremely small to negligible effect because they kind of are pinned. But there is the internal degree of freedom can evolve. For internal degree of freedom, we always means what is the internal state of a particle, like what is the spin state, this can evolve. If you go to, if you allow particle to move, then you have to consider in a discrete system tunneling but still in the discrete system, this tunneling, you can control, it's under control. While in a bulk, where there is no external ordering imposed, it's kind of much more difficult. You start to even have a particle that move in a reservoir and this reservoir can be Marconian or non-Marconian. And I think that there is much more needed new theoretical approach to find out what is going on. So we go now to the part which is the discrete part of the lecture. And as you will see, we have some introductory that are really very basic. I heard yesterday at lunch that there are even postdocs or more advanced people in the audience. So OK, we'll be boring maybe at the beginning for you. But if you resist, then at some point something will become more at your level. So OK, let's get started. And of course, this doesn't work. Yeah, so first of all, I mean, everybody is telling, OK, so what is quantum simulation? So the most general definition that you can have is still, I mean, somehow very deep if we want to think about this. But I don't know why it doesn't work. Do you have a pointer? OK, so the really zero order definition is that you want to, so you have a very complicated system made of many, many, many particles. And then you would like to mimic the behavior of a complicated system to the behavior of something easier. And now the problem is that you would say, OK, but that's you are not allowed to do because you have a complicated system why a simple one should teach you something on the complicated system. And here comes really the idea of quantum simulation because you can have complicated or easy system. But if they follow the same equation of motion, then you can simulate one of the other. If they follow the same Hamiltonian or very similar Hamiltonian, then you can simulate, mimic, reflect one system to the other. And so that's the key idea of this. And then to try also to kind of narrow down the definition of which system to really something which is physical relevant. So typically quantum simulation at the beginning, which was really almost more than 20 years ago, started with the idea to have a complex, relevant problem of physics and whose solution was unknown and to use, let's say, really laboratory experiment to mimic and to learn something. And so what I like always to show is a really stupid graphical idea of what is the concept of quantum simulation. So you have something which is complicated. And that's your target. You would like to learn about the behavior of this dragon. And the dragon is difficult. You cannot go too close to the dragon. It's too dangerous. And it's something really that you cannot approach directly. And so the simulation would look at this dragon very carefully and try to put together the ingredient that defined the dragon. And so one ingredient is like, OK, maybe that's kind of more or less the shape of the body of my dragon. And this I can study. It's not so dangerous. Then I need to give the wings to my dragon. And then I need to give the corner. And somehow this part is very important. That's the part of identifying the key properties of the dragon or the key properties of the system that we would like to simulate. And then based on these key properties, we will get a new system that maybe it's artificial. We engineer because this system does not exist in nature. So as an engineer, there is an artificial quantum matter in this sense, in the sense that we isolate the key properties. And then based on the key properties, we create an artificial quantum system that is similar to the natural one. And then we notice that maybe the equation of motion of this new object are very similar to our target. That's the philosophy of what really quantum simulation is about. And in general, which are the topics that are relevant for quantum simulation? So originally, the motivation was really to study the solid state system. I mean, in a solid state system, you know better than me, it's extremely important because there's really many devise around us are solid state. It's very important because electricity is very important, the electron. How do the electron move in a solid state? Are very important. The electron are really many, many, many. There are kind of complex dynamics because you have the ion and the ion can arrange in space, let's say, making different band structure, and the electron should move, and maybe they get trapped. And there is the electron-electron interaction. There are the ion-electron interaction. And all these made this problem quite complex. And moreover, it's very hard. I mean, you cannot really follow totally deterministically, let's say, the behavior of one electron among the many. OK. And so the original, really, and the first experiment that have been done in the direction of quantum simulation and the first proposal was really about quantum material simulation. And there you have a very important problem which can be high-temperature superconductor or even frustrated quantum magnet, strange metals, spin glasses. Those are all examples of physical problem, I would say, that are very hard to compute and they're very hard to understand in real condensed matter solid state physics. Now, so some of these have still very big open question like high-temperature superconductor is still an open question in the field of why there is high-temperature superconductivity. Progress have been made continuously, but still the really underlying reason is hidden. We don't know. As well, frustration is really a problem because, I mean, frustration is this very simple example of having a triangular lattice and you have a spin up, a spin down, which spin will have the third one. Now, that's the really most simple idea of what does frustration would mean in the sense that you get in a situation in which the system doesn't know where to evolve in some sense. Really sad from very basic point of view. And then you have the strange metal, which is one part of the phase diagram, very important as well, and the spin glasses. But of course, I mean, over the year, so requiring, let's say, quantum simulate this problem, it's challenging. And it was particularly relevant because electrons are fermions and when you try to compute many, many fermions, there is really a problem to find an efficient simulation due to the sine problem associated with the fermion statistic, but it's also challenging in general to think about the spin frustration or to compute complex gauge field, okay? But then the field also move it to other topic. One, for example, is really the quantum chemistry to simulate complex chemistry reaction, which are in the quantum regime, let's say very low temperature or single molecule and complex molecule. And then it moved also to, let's say, try to understand transport in a more broader sense in all what are the device, the quantum device for transport, a guideline, a restricted dimension, having, you know, really very thin wire that goes, I mean, so let's say that challenge a bit, the size, the point spread function of an atom with the size of the channel and having more complex structure. Of course, another very open at the moment, which is in the quantum simulation, I would say one main trend of today quantum simulation is to understand the concept of thermalization. So, or in general, how do a quantum system, so made of many particles, which follow equation of law, which are kind of Hamiltonian, would then, you know, go, if you put it out of equilibrium, what will happens to the system? Will this stay, if it's isolated, will the system be forever out of equilibrium? Will this pre-termalize? What is the, let's say, really the fate of the non-equilibrium quantum many body system? That's a very important topic of research. I mean, there was some consolidated understanding on the equilibrium phases of lattice system and now much more open question are in the non-equilibrium sector. And then we can really make really a leap, a jump, so we can go to something which is completely far away from, let's say, the very low energy. We can, I mean, we typically quantum simulate with ultra-cold atom and ultra-cold means energy in the regime of pico-electron volt. And now we could think even to, you know, to say something about cosmology, particle physics, gravity, so to go even in regime which can be mega-electron volt or regime which can be kilo-electron volt or let's say astrophysical relevant question. And about the astrophysical relevant question in the second part of my lecture, I will give you a very precise example of what we can do. What are, okay, those are kind of the big, I would say, topic that one can address or would like to address with quantum simulation. But then there is another point, I mean, which are the platform. And now for platform, it's getting very popular, the word architecture. So how do you architect many body system and what are the nature of the, you know, component that will then allow you to, you know, unlock an Hamiltonian, unlock dynamics and study. So there have been, I mean, in my, so I'm more in the field of, you know, single atom molecule, okay? And in this domain, what you have, you can make quantum simulation with a molecule with ultra-cold molecule or you can take ultra-cold, trapped ion, okay? This is a platform also very much, that very much initiated somehow part of the quantum computing, let's say pillar of science, but then you have ultra-cold atom, which are really kind of a very easy native system for quantum simulation. And more recently, there have been a new technique in, let's say, technological development of creating very tiny sing-trap which are extremely focused. Each of these trap is independent and there's a very, very, very narrow focus so you can have exactly one atom per trap and this trap is called a tweezer just to highlight the fact that it's a dipole trap highly focused. It's nothing else and tweezer is nothing else than a tightly focused dipole trap. And it's so tightly focused that you have just few atom per lattice side and then you have to know your molecular physics to go from the few atom to a single atom and do a per projection. And so, and now those are kind of the platform for which every kind of every week there are advancement of how fine the control can be but of course there are also more complex structure in which you have maybe particle and then you couple this atom together with something else. For example, you can enter in a regime of extremely, extremely strong atom-like coupling such that your final state is not an atom, is not a photon, is hybridized between atom and photon and this is all what is the physics really of studying a few atom or many atoms in cavities because the cavities amplify the effect of the light on the atom and you get this highly-dressed state or even entanglement between light and the atom. And of course then we can go more in the solid state part and you can create really nanostructure, solid state nanostructure and then photon nanostructure. And then I mean there is all the other bounds of study in quantum simulation that is really related to solid state system. For example, superconducting quantum circuit but that's not the only one you will hear more in your school. I'm not an expert. You can have quantum dot devices, color center, van der Waal center structure, moire material where you can tilt the plane and several excitons. So those are the architecture and in this talk I will mainly focus in an ultra-cold neutral atom to give you this example but also in my lab we have a Riedberg atom array but I will not really cover this part. Okay, so now if we go here let's give the keywords. The keywords are ultra-cold atom, you have an optical lattice and then the Hamiltonian. So and somehow in general I already pointed out that even a problem like this one let's say small amount of atom in a square lattice to have the exact diagonalization of the Hamiltonian governing even a small system is extremely challenging and it's very hard to make really let's say precise, let's say physical prediction because the computational time is growing exponentially with the size of the particle. So with how many particles you have. This you know very well and much better than me but this is really a problem and so the solution was okay let's make in the experiment bigger system and let's see what is the result of the experiment. Okay, and the first application of this idea of quantum simulation was accounting for a simple Hamiltonian. We will look at this Hamiltonian a bit more in the few slide in which you have actually basically a 2D lattice and then you have one atom per lattice side and this atom can tunnel and here the tunneling is J and once it tunnel you can have two atoms in one lattice side which have interaction. Okay, but in this Hamiltonian there is no off-site interaction. So one atom placed here will not interact with an atom here. Okay, so it's really let's say the minimal model that you can have to study you know transport of electron or a particle in a periodic structure. Okay, the difference from single atom is just the possibility that this particle one moving can interact with other particle and this interaction already creates even if it's weak and create very interesting new phases. So now we have these three ingredients and the aim here now is to go one by one and say just a little bit about one by one but also with a bit more fresh look on the new trend. So I will start with the optical lattices and I will give you some basic concept of optical lattices. Why it's important or particular important for you that there are theories to know something on the optical lattices because this will tell you, let's say which are the relevant problem that we in the experiment can actually benchmark with the theory, okay. So many and not everything it's possible in the experiment and it's much more possible than what we know today but first we need to know the basic, okay. So the basic concept of optical lattices are very, let's say you know it's kind of atom light interaction in a very you know moderate regime. So if you have an atom and you shine light on this atom, the atom polarize itself. So there is a quantity always indicated with alpha which is called atomic polarizability. Now depending on the wavelengths of your laser, the coupling of the atom to the light have a different strength. So the polarizability and this strength which is proportional to the electric dipole moment, okay. This quantity here is dependent on the wavelengths of your laser which now I put it in energy but it's let's say two pi divided by lambda, okay. And so this is really then the polarizability at the frequency omega and then this polarizability has to be multiplied by the electric field. Okay, that's the first important thing, y squared. So why it's not linear in electric field beside the you know dimension, y squared ratic. Propulsion to the intensity as a consequence of being quadratic but why it's quadratic is there, you know you have atom light, the couple and then you get you know the dipole potential which is really the potential see by the atom. And actually because a neutral atom doesn't have an electric dipole moment alone in the dark doesn't have an electric dipole moment, okay. Then it means that when you add light the zero order start shift is zero, okay. So this the square here tells you that this coupling is second order perturbation theory, as simple as that. So it means that it has first to induce and then to create the potential. So there is two steps, okay. The light trap but the light induce an electric dipole moment and the polarizability to polarize the electron cloud around, okay. And so that's why it's a square. There is only one possibility to have linear. Do you know one? So linear start shift for neutral atom. This is when you use degenerate perturbation theory and you have two energy level which are degenerate to each other. You need to use the degenerate perturbation theory which give you the first term in perturbation, okay. Those are neutral atom. Neutral atom don't have a permanent electric dipole moment. It's very different in other system. Let's say for example, ion. That's very different, okay. And then this electric field, why do I put time average over an electric field? Well, this because the electric field let's say is a wave. The wave oscillates very fast. I mean, you might wonder what's the value of omega? Is omega Earth, kilo Earth, do you know? How fast the electric field is oscillating? How big is this omega? This omega typically is theta. So it's oscillating extremely fast over the time scale of anything an atom can do, okay. So that's why is always the atom is single with something which is really kind of time average in omega, okay. And so what do you create? That's the dipole trap. I mean, with this simple, then it's depend on the form of the electric field, okay. If you have a simple Gaussian beam, then the potential that you will have is simply a dipole trap, an harmonic potential. Okay, as one laser beam is focused and at the focus, I mean the atom can be at the focus for blue, for red that you need a light and you get a dipole potential. But of course, you can do a little bit more. You can take your Gaussian beam, put a mirror and your retroreflect, then you have an incoming wave. It's retroreflected on a mirror, goes back. This create interference and so that's how we create standing wave. Okay, that's simply your Gaussian beam, retroreflected on a mirror and going back, create this standing wave. Now, interesting, let's say tricky, let's say nomenclature things. This one is one dimensional lattice, so you have one beam all in one direction, but here, all these disks are two dimensional system. So if you want to create a two dimensional system, you can think about, you know, a salami and cutting slice, then every slice is a two dimensional. And there you can have 10 to the four atom, many atom, not single atom, okay? So one delet is create two dimensional system. Of course, you can have not only one standing wave, you can have two and so then you can create actually tube. Or you can have three standing wave and you get more into the crystal structure. So why this is important, you see immediately the analogy, you have a real crystal, you have the ion, the ion are very, very heavy. There is a factor of four in mass between the ion and the electron, typically. And so it means that the ion all, you can consider almost as a frozen. It's not frozen, you know there is phonon, okay? But then you can see it in first approximation, a stationary ion and then the valence electron that move in this periodic potential. And in the optical lattice, you have the same. So you have a stationary standing wave that it's quite stiff and then your atom, neutral atom can move. And this atom can be either bosons or fermions because you can do ultra called atom of both isotope, of different isotope. What are the advantage, I mean, with respect to, you know, in between the students, let's say the first ingredient, we said we wanna find the ingredient and that's a very important ingredient. We have now an optical lattice is mimicking, simulating the stationary ion. And so you can have a simple lattice geometry. One thing that at the beginning was sell it as an added value and now is a limitation is that this lattice is perfect. Why it's a limitation? Because in real system, the lattice is never perfect. So you would like to, you know, add phonon excitation of your lattice. You would like to add impurities and this is getting very challenging. Okay, but at the beginning, the idea that there was no defect, no phonon, no charge were all a simplification of our, you know, basic, a minimal Hamiltonian model. And then you can control many things. You can control the lattice step. So how are we really making the tunneling zero by increasing a lot the lattice step or not just by working on the power of your laser? It's really trivial. And you can also, and we will see more, work about the geometry like, you know, in real crystal, the crystalline structure can be very different. Now there are several Brevet lattice. I think there are kind of 14 Brevet lattice in three dimension or a number like of this order of magnitude so you can create many different crystal. But also what is cool in cool cold atom which is not really possible here. So you can go now a step farther is we can tune the interaction between the atom. So this quantity you can be, and then there is all the quest of probing the state which I will tell you later. So what about, I mean, the geometry, I mean, so far I just show you the idea that you have a laser beam like Gaussian then you have a mirror, this reflect, go back and this is the type of potential we have. You see that there is also, typically, and this is not very nice in the experiment, we try to compensate. There is also, let's say, is the lattice on top of an harmonic potential because of, you know, the Gaussian profile in the other two directions. So those are kind of, the lattice spacing is the wavelength of the laser divided by two in this configuration and that's it. But actually you can change the lattice spacing because if you don't retro reflect it's really counter propagating but you have two beam at an angle that will also create interference where they cross and this lattice spacing can become bigger. You cannot make it smaller, that's the minimal lattice spacing that you can create with a given lambda but you can make it bigger and so the lattice spacing would go as, you know, independence of this angle theta here. Okay, it's cosine of theta. But also what you can do very interesting is you can make it moving. Okay, like rigidly moving the lattice. How do you do this? Well, you fix a given frequency of a laser and the other laser if you make a small change of the frequency, so you go from the first, you know, the zero or the plus, you know, you give a small shift and then this will be a traveling wave which follow this rule here. So you can make it moving. Okay, but not moving locally, moving globally which is the problem of the phonon, yes. Okay, so controlling the angle is rather simple because I can, you know, I mean, there are many different way I can do that. So there are object in the experiment which is called, which they are very useful and those are called acoustic modulator, AOM. So you can have this device which is the AOM and then you can have one beam going here. So you enter with your laser and then you can have another one which is zero order, first order which is a shift in frequency. Then I can put a lens, I make these two like this, okay? And then what I can do, I can kind of change and so this make me changing the distance of these two beam. And then, you know, I can put kind of two mirror and then they go to the atom and by this I can adjust. That's, for example, something easy. And how precise, I mean, yeah, it's really depend on, so there is no lower bound in some sense, okay? You can be certainly precise at the level of few lambda of your wavelength. So certainly you can be sub micron precision if you do the things correctly. Okay, that's about, okay, having always a square lattice and you are changing the spacing so you made, let's say, the lattice side, let's say the lattice spacing bigger or smaller in a given geometry. And, but of course there is other degree of freedom. So the complexity can always be increased. One other properties that light has is polarization, okay? So we have not yet worked on the polarization so far. We said lambda, we said what is really the geometry of the angle but not the polarization. The polarization is how the electric field oscillate with respect to a quantization axis. And then I can have my simple square lattice in 2D. That's really two dimensional. Square lattice, I just put, you know, orthogonal couple of light and then I get the simple, let's say 2D version. But actually I could also work independently with the polarization of laser one and laser two. And if I do this, I can actually have to account on the phase, okay, between the different standing wave. They kind of, I can play with kind of the phase so the delay of the single standing wave. And by playing with this, I can get many different lattice potential, okay? And some different geometry. I can have a minimum or I can have a maximum. I can create something like this. And also, so far, we only get, okay, we get same lambda can play with the polarization of each of the beam but each beam come from the same laser but I could also make interference beam with different wavelengths. And then I could create, for example, super lattice in which I combine a set beam with different lattice. And then you can see, you can create all this type of, you know, little plaquette where you have two atom here. Then they are very well separated by the next, you know, dimer system. So you have kind of two atom plaquette which are sometimes called the dimer. And then you can also, you know, play around. You can even create a 12 potential and you can play around to make this more wash born in one direction and you can juggle with the atoms. This is all what you can do if you have two wavelengths. And then there is even something more because you can have, let's say, you can combine more than two laser and if you combine more than two laser in this different direction, you can really go to different type of geometry like triangular or honeycomb or one D chain and play around as you want. And of course, let's say the triangular is a very interesting, let's say, setting for, let's say, Hamiltonian. And so here it's a other example of using three beam of the typical lattice potential that you can use like this triangular or you can even have a hexagonal lattice that you create again with three beam but playing with their polarization. Okay, now you have this nice hexagonal lattice and so suddenly you can study physics of graphene using ultra cold atom. Because that's the geometry, the underlying geometry of the graphene. Okay, and then there is also another, you know, level of tunability or level of complexity that we can go on. Now, just take in mind that the dipole potential was the atomic polarizability and the electric field. Okay, so far I told you with the electric field I can have more than one electric field. Each of these can have different polarizability, can have different wavelengths and I can send them on different geometry. So I show you example of tunability of the laser that's in the electric field. What about the atomic polarizability? Okay, so far we treated as a number and in the majority of case is a number. Let's say you say, okay, at these wavelengths the coupling of an atom with the light is 100 atomic unit. But in reality the polarizability is a much more complex function and so this open a new door and this new door is very little explored so far. Now it's starting a little bit. You will see why now it's starting. Because indeed the polarizability is not a function of a single value is a matrix, okay? Is a tensor and that's very important that one understand that that's a tensor. And so the fact that is a tensor is a new degree of tunability. And which type of tensor? Well, in general this function which is I mean a specificity relevant when you have an anisotropic media. Always in anisotropic medium the index of reflection of an anisotropic medium is a tensor. You can think about atom in the same way. Complex electronic structure, not spherical, are like anisotropic medium. Any, every atom, because the polarizability is a single atom properties. Everyone, atom are identical. But each alpha is a single atom properties. Has nothing to do with interaction. It doesn't change if it's alone or you have many atom around. Now it's tell you the index of reflection of the medium single atom. If your medium single atom is anisotropic because the electronic cloud around the core is not spherical then it's very relevant to a tensor. How can you have non-spherical electronic configuration? Or when the electronic configuration is spherical? We go back really from basic atomic physics. It's very good. Yes, wave not so much, it's orbital. S orbital, okay? So it's really, you need, if you do let's say atomic physics for quantum science, you need to know the atomic physics and the quantum science which sometimes is a little bit challenging but you need to use the in and properties of a single atom to make new architecture. So if you have a hydrogen or any alkali, this is a class of atom which has how many valence electron in the ground state? One. So you have whatever core we don't care and one valence electron. We care about this electron. This electron move around the core and it can move in different way. Ground state alkali, which is the main atomic species used in 98% of the experiment around the world has an electronic configuration which is the core of a noble gas plus something which is an S state. It's always S, okay? The orbital here depending on which one and this is normally this. It's always an S orbital. It means that the electron move in a sphere. So it's kind of completely spherical symmetric. But if you now have atom which don't have an S but have more complex than the situation change, okay? You can have maybe electron that move in a D. You see now this is an isotropic. That's the geometry of a D wave, D orbital. And so on and so forth. So this is a medium which is an isotropic. For this medium, the polarizability is actually a tensor and that's kind of very important. Okay? Now, in general, let's don't think about this. In general, how can we decompose these matrix? There are three parts. One is the scalar one. That's the only one really relevant for alkali and for what I told you so far is a number. But then there is another contribution that we call vectorial and there is another contribution that we call tensorial. Although all this is a tensor, we give this name. How does it look like? Well, unfortunately I should have used another color but somehow it's kind of a complex, start to become a complex function but we can see something. This first term here is the number. It's one number is the scalar one. Then this is let's say the vectorial contribution and that's the tensorial contribution. Now, the vectorial contribution depend on an angle. So an isotropy always bring angle into the problem and it's the angle between the quantization axis and the wave vector of the beam that propagate. And then there is this tensorial one that you see here. It's very interesting because it depends on, let's say the spin state of your atom and the demand sub-level is occupying and there is also an angle dependent and this angle dependence is, let's say, depending on the quantization axis and the polarization of your light. So this allows you this formula to create lattices which are different for different spin state. So you know one part of all the solid state is to study spin dynamics and you can create a situation in which one spin, let's say the spin up, see a lattice and moving this up, the spin down, see another lattice or maybe see no lattice and make a reservoir like a Marconian or non-Marconian reservoir. Those are all possible as a degree. And so this is something new because now you will see later that there are more and more atom which are becoming popular in our field which have a very anisotropic electronica. So there is, let's say, an orbital anisotropic. Now, basic concept of Hamiltonian. I mean, since you are theories, I will go very fast. I mean, it's all what you know much better than me. And okay, let's go. I already told you why, let's say, we want to simulate, for example, quantum material and I already told you why, so where the analogies come from. You have the localised electron and you have a localised core and here for the model you can have atoms in optical and lettuces. And so when you think about the model, I think you have one starting interesting model are the family of Hubbard model and you have the Fermi Hubbard and the Bose Hubbard. In the experiment we can access both because I mean we have fermions or bosons and they look like more or less pretty much like this for the bosons. You have the field operator and there is an interaction term which is a number in many cases. And then, okay, so that's the kinetic energy, the external potential that's in our case is just the periodic potential. And then you have a term with, let's say, field operator that tells you about the interaction. Now how, the first simplification that typically we consider from this equation is to, let's say, replace or develop the field operator. So we can, we explicitly put in the lattice potential which is, okay, periodic. You can have periodic potential plus even something else like a broad harmonic if it's needed. And then you can expand the field operator in the Vanier basis that, I mean, I don't think I need in this audience to explain a lot but every field operator can be. So those are the Vanier function. And then you can rewrite after this expansion the parameter here. You can decompose the parameter and then it come out apart which is a tunneling J which is connected to the kinetic energy and the periodic potential, not this J, in my approximation is only a single particle term. Okay, it's the kinetic energy of a single particle. The single particle see a periodic potential. It's not a two-body tunneling. It's not correlated. So the tunneling happens equally whether you have other atom or not. This is an approximation because in reality in the real Hamiltonian that's, I mean, in the real system it's much more interesting than this. But those were the minimal model accounted at the beginning. And then you can rewrite the interaction. I mean, now the one key point when you work with ultra-cold atom that define the success of the system at the beginning is that the interaction at very low temperature between ultra-cold atom it's a, especially if you have S and alkali, it's very simple. It's basically the interaction is a delta function, okay? And this delta function it's multiplied by one parameter which is called a scattering lens. How do I understand it? It's really almost mean field. It's kind of really somehow classical because if you have, so what does the contact interaction tells you? You take a two-billion ball, you make them colliding. That's the delta function. So if the billion ball are here, they don't see each other. We agree on that. But they have to do the scattering, so the collision scattering interaction are kind of full synonymous, okay? And then they go out. And so that's the meaning of the scattering lens. But then you should ask yourself, okay, if this is the intuition of the scattering lens, how the scattering lens can be different between atom? Okay, you can understand that if the billion ball have a different mass, then probably the scattering, it's different, but there is something more. How can I tune this? And actually here it comes the point of really, let's say having the quantum version of the scattering. And the quantum version is the following. I have two atoms. They don't do this, they do this, they become molecule, stay a little bit, they feel the molecular potential, they are asking themselves, should I become a molecule or should I exit as two atoms? That's what is happening and going out. These moments where they are deciding what to do, where they feel the molecular potential, give a delay. Because, I mean, maybe the two of them can be very fast. No delay, they arrive very quickly. But then there might be other that stay a lot more. These give a delay, the delay give a phase. The phase difference of the incoming and the upcoming is the scattering lens. The limit of the phase difference for low momentum, that's the definition, is the scattering lens. I wanna give you, because you are theorists, I wanna give you the experimental definition of things, the intuition of things. That's the intuition of the scattering lens. Another intuition, and how can I formalize this a little bit better? It's the following, an atom in an atomic level with energy one. I have another atom with another atomic level, two. Let's say this can be the same, two identical atoms, same energy. Then I kind of plot the energy level, which is the sum of a one plus a two. If I do this, I'm considering that the atom are very far away. They don't cross, it's just the sum of the single. But when they approach each other, they start to do this stuff. So they feel the molecular potential. This is the van der Waals, this is air, the distance between the atom where they are very different is just the sum. When they approach the field, the molecular potential, and this is the molecular potential. Okay, how it's like this? Two independent atoms, that's the sum of the two hyperfan energy. Then here, it's what is called the van der Waals part, which typically goes like one over air to the six. It goes to the minimum, and this is the hard wall that you put because the atom cannot compenetrate one to each other. Okay, so there is a repulsion, more close they cannot get. It's a hard wall. And in this potential, you see the parabola in first approximation, you have all the energy level of the molecule. This is a molecule. These are two atoms together, it's a molecule. These are two atoms. That's what is called the atomic threshold. The wave function of two atom, how is that? The wave function of two atom, no molecule, two atom, is fast oscillation with low probability. That's the wave function of the two atom. They feel here the potential. When they arrive at short range, they do what should they do, what should they do? And they feel all this. The number of oscillation, the height of oscillation of this wave function depend on the form of this potential. And the scattering lens is the two atom, enter, stay here, go out, and have a phase shift, okay? And that they stay here inside, it's depending on the potential. How do I change that? I mean, if I have only one potential, I cannot change too much. They go, that's the potential given by whatever, and that's it. It's really determined by a single atom. But if I have two potential, I can take this one and I can bring it down with magnetic field because maybe the Zeeman energy are different, the shift is different. And effectively I can do something like this. I can make that my level here, it's very close to another level. So they think, oh my God, it's really super nice to be together because now there is energy level of the molecule very close. I stay longer the phase shift increase. That's bit the point of view, okay? More or less. And yeah, and this is about the scattering lens to give you an intuition. So we need, let's say, really scattering molecular potential. We need very close to get the precise form of this potential is extremely complex. Molecular physicists, chemistry need to help us. There is ab initio calculation for the molecular potential. This is another degree of complexity. But somehow at the end of the day, we just wanna know what is the phase shift? It's a number, scattering lens. And we want to learn how to tune it when this molecular state get in resonance with the atomic level. That's the point where we call, we have a so-called flashback resonance. The phase shift diverge because they really would like to stay together. It costs no energy to become a molecule, okay? So then the phase shift, the delay increase a lot. Okay, and that's about the term. And now if we do this expansion in the vanier basis, we have this new Hamiltonian with the operator a dagger and a. So that's the single particle tunneling. That's the two body interaction. And then this is an energy, the chemical potential, okay? So you can have the tunneling, as we said. You can have the interaction when two atoms sit on the same lattice side. And we have maybe this external potential that give you a shift of the chemical potential, a shift of the low point. This model is minimal. I mean, it's really the minimal interesting model, I would say, but there are a number of approximation. First of all, I already mentioned one. There is no interaction between atoms sitting in two different lattice side. There is no tunneling going over two lattice side. So no next-near-neighbor tunneling. There are no higher band. I mean, if you think about this as a small harmonic potential, you have different harmonic level. But all this physics tells you that if you are in the lowest energy level of this kind of harmonic potential, you do tunneling and you remain there. So there is no higher block band into the problem. And then you have this simply non-regularized pseudo-potential for the interaction. Okay. So but already this minimal potential give us something very interesting. It has an hidden phase transition that you know very well, which is the superfluid to multinsulator phase transition. So if you wanna study the ground state, and this tells you that you wanna reconstruct the phase diagram, which are the possible state phase. It's a synonymous. So you will have the superfluid where you have particle. Each particle is delocalized. Each particle is identical to the other. And so they are fully phase coherent. Okay. So you can see this as a matter wave, fully coherent. Okay. So it's the particle are delocalized. There is no excitation gap. Okay. And it's a coherent phase. Here the tunneling is much higher than the interaction energy. I mean it's easy to delocalize. And you can instead of really, you have block function, delocalized over the real space. And then you have the multinsulator, which is let's say the same Hamiltonian, but if you're parameter the tunneling is much, much reduced. So you really are increasing the height of the potential. Okay. And in this case, you have a particle which are more localized. At some point you lose the phase coherence because they cannot really speak to each other anymore. And there is an excitation gap. Okay. So the mode phase is a gap at phase. And now the phase diagram look like this one. On the one hand this is the control parameter is J divided by U. As we said that if J divided by U is very big, you're always in the super fluid phase. But once you are decreasing, if you now move by decreasing J, you enter for a given number of particle, which is embedded in the chemical potential, you enter in the first lobe as it's called in which you have a multinsulator with exactly one particle per lattice site. If you increase the atom number, you go from one particle to two particle like here in this sketch. Okay. And so what you would be able to reconstruct is this type of wedding structure in which you have now this is the super fluid. Then you enter in the first multinsulator lobe or wedding cakes layer. And then in the second one here, it's where you have two atom and in between you go back to the super fluid state. And these have been observed and how this can be observed. Well, you can use, you see those are many different properties. So you can observe this super fluid to multinsulator looking at one property that dramatically change. One is phase coherent and the other one is non-phase coherence. And when I told you that's matter wave, as soon as you heard the word wave, what do you think about wave can interfere? And it's very easy for you to understand that the interference of something which is coherent, so each wave at the same phase or something in which each wave has a random phase is very different, no? And so in the case of the super fluid, then you can see each of your atoms as a little matter wave and then this little matter wave is interfering like a slit with the other matter wave. And so in momentum, so which you can do by Fourier transforming if you are a theorist or you can do by waiting a lot of this expansion, a lot of time in this expansion and you are in the momentum space, what you see, you see an interference pattern. If you repeat this measurement under time, you will always get the same interference pattern. The fact that it's same interference pattern for different realization tells you that it's the robustness of the phase coherence. It's always the same interference pattern. And then you can see that now you would have a maximum here, zero, another maximum, so if we take only this line, you have a maximum, let's say here, another maximum and this one, but since our lattice, those are experimental data, it's three-dimensional, you have also the other peak for each interference in each dimension. The periodicity that you recover here, so what's the distance here? It's given by the lattice, via the k vector of the lattice of the laser light. That's the periodicity in the reciprocal space, in momentum space. And but actually if those are incoherent or if those are really localized particle, you would not get this interference pattern. What you get is a blob. There is no peak, there is no constructive interference. And this is what you see. So here you go from, let's say, zero lattice is a normal Bose Einstein condensate, then you start to ramp up the lattice, a super fluid interference, nice interference pattern, nice interference pattern, and then at some point the interference pattern wash out and you have this type of blob, okay? Now when you see in the experiment something like this, you could also think that, okay, just particles were heated up by the lattice because now the power is very high of the lattice and there is no coherence, everything, just because of thermal. You are going out from the zero, let's say, macroscopic occupation of a quantum state, but more it's just thermal. But actually that's not the case because if you ramp back down, you start from here and then you ramp back down, you recover the interference pattern. The fact that you can ramp up and ramp down and get back the same interference pattern tells you that this is really not due to thermal but it's really due to, let's say, the multinsulator phase, okay, in which since there is a perfect number function, in one lattice side, there is, let's say, no phase, let's say, those are complex conjugate, spreading phase localization in atom number. Those are correlated by the Eisenberg principle. Now, for many, many years from, I mean, the first paper was in 2002, I mean, Markus Greiner, which now is one of the very leading person in the field, was a young PhD student in doing this. Now he has his own fantastic group, just to give you an example of what's really, some time ago, and the observable of this first quantum simulation example was something which don't tell you anything about the single particle. So these are average observable, okay, that's the interference pattern of many atom, not one, many. So then this gave you access to, let's say, global observable, not local. Now, that's not what you wanna have. You wanna have local observable, you wanna have renyanthropy, you want to have correlation, you want to be able to extract correlation locally, you wanna study, let's say, be able to subdivide in subregion your system, and this then became almost 10 years after available with the new experimental technique, and this new experimental technique is the one of quantum gas microscope. So with the quantum gas microscope, I can explain you later if there are questions, what you can do, you can really see each of the atom, so you have the quantum gas and you can project really each of the atom in a lattice side. So you have really the single atom visibility. In all this picture, the lattice is on, so you are pinning the position of the atom and you are looking, and so here, for example, that's the one wedding cake layer, this type of thing. And, but then I mean, just as a curiosity for you since the topic of the field, you get this type of system, this type of image, and kind of, if the distance between the lattice side is big enough, then you can really distinguish whether one lattice side was occupied and one was empty. But imagine that now these two lattice side are very close. How can you distinguish? How can you say if the distance between the lattice side is smaller than the point spread function of the atom, how can you distinguish? And now, one of the recent trend of the field is really to use, let's say, this deep learning assisted classification to really, let's say, give an image which is not fully resolved. Can we use, let's say, a protocol based on neural network to reconstruct the initial, let's say, system? And so actually, this seems to be really very promising and many groups are picking up on this idea and reconstructing the pattern. Okay, now I just give you, okay, an example. I mean, this has allowed many things, this new technique, and now let me give you, for example, one example is really the direct observation of what was, let's say, Eisenberg model. So to the creation of really, you know, in the spin degree of freedom, really the creation of pattern spin ups, spin ups, spin ups, you know, that was really one of the macro phases that one wanted to observe, and this is from the group of Marcus Greiner. But of course, I mean, there is much more because on the one hand you can use the quantum simulation to shed new light on problem that exists in a long time, but also, I mean, all this new technique has allowed to see things which were not even predicted in theory, especially for the, let's say, non-equilibrium dynamics because there the prediction really goes very short. It's very difficult to build up a model on non-equilibrium dynamics. Okay, this was one example, but then the field of, let's say, evolved, creating much more complex geometry, studying transport in honeycomb, graphene type lattice, or even not using one single species, but different single species where you have multi-component gas, or even, you know, having not a lattice but put disorder on the lattice and study phenomena related to many body localization, phenomena related to Anderson localization and studying transport in a disorder potential. But all this study here have been done considering that the particle, yes, interact, but that the particle interact via this billiard ball example that I told you, by just the contact interaction, this delta function. But this, as I show you also in the Hubbard model that we were considering, is an approximation. You are saying that just atom, if they are in the same lattice side are interacting, but if they are in two different, no. And that's not always true. Can we go beyond? Because we know that that's the relevant thing for electron. Okay, and so the last ingredient are the atoms. Indeed, I mean, if you say, okay, I want to use alkali atom with one balance electron, that's the interaction you can create. Just the delta function, which I will call from now on contact because they need to be in contact. And, okay, all these things have been really fantastic. I studied in lattice, in tweezer, all what I was showing you, but there is not too much, a little bit more can be done with alkali. I will show you something, but we would like also to go a bit farther. And actually, the alkali are just one narrow part in the periodic table, but the periodic table is much bigger. Can we go to an isotropic medium, an isotropic atom? The first step was to use not alkali, but alkali hurt. All this column, it's the second column of the periodic table have two balance electron, and this already open and let's say unlock other tunability, but there is even much more, and that's where, I mean, my expertise goes in, is to move to really atomic species which are highly anisotropy. Like, for example, this strange object which I'm depicting here are the lanthanide. Okay, and so I want to tell you a little bit how to go from the Bose and Fermi-Abard model to the extended Bose and Fermi-Abard model using a lanthanide atom. One question is how is with the question to know where I should stop and then I can repeat later, yeah, yeah. So now we have lanthanide. What does it mean? Orange is the core. So neutron, proton and inertial electron together. We don't care too much. And then all these black dots are the balance electron. Lanthanide are typically more than 10 balance electron. How do, and the balance electron are really very interesting because two of them are kind of feeling a spherical symmetric shell, so an S-shellar, but then there are all the other electron which are in a high anisotropic F-shellar, okay? So the typical electronic configuration of lanthanide is a Nobel gas, which is typical xenon. And then you have something which is F-shellar with a given number of electron up to 14. With 14, the F-shellar is filled. And then there is, so this is always 4F to something, okay? In the case of this prosium atom, this something is 10 and for erbium, this something is 12. And then you have a six S-square. The S-shellar is completely filled. The F-shellar has vacancy. Because of this strong anisotropy, many properties for quantum physics are emerging. So the first time people have used lanthanide in experiment was in the group of Benjamin Lefter and they use it as a lanthanide called a dysprosium. And then it was my group in Eastbrook with erbium and now we also produce the erbium dysprosium mix. So it was a kind of a very young in some sense system. And one of the key point is that the vacancy in the F-shellar determined the magnetic moment. So I don't know if you ever asked yourself where do the magnetic moment of anything come from? Well, the magnetic moment mu is really a quantum number because it's the land-day factor J multiplied by the spin state. This J is a function of the quantum number of J, L, I of all the quantum number of the atomic species. Anisotropic shell have very large L orbital. And so it means the magnetic moment is very large. You have lanthanide in your mobile phone. So a lot of lanthanide determine the vibration of your mobile phone. Lanthanide are the most magnetic atom of the periodic table, that's it. Okay, there is nothing more magnetic than this. And the magnetism come from the vacancy in the F-shellar. If it's fully fillet the F-shellar, then no magnetic moment. That's it, terbium, zero magnetic moment. Then erbium, which I mean, fillet means 14. Erbium have only two left. It's very magnetic, but less magnetic than this prosium, which have four vacancy. Okay? And since atom have a very strong magnetic moment, the consequence is that you don't have only the contact interaction. You have also the magnetic interaction. The magnetic interaction now don't think about the billiard bolt, but think about two physical magnet, the one that you attach on your fridge. Two physical magnet, you feel the force of your magnet even before the touch. Not only these that you feel the force, but it's different whether it's poles, south pole, north pole. It's different, they can attract or repulse. Okay, so anisotropy, because it's depending on which direction you do things for attraction. Anisotropy means dependent on the angle, directional. It's a directional interaction. And then it's long range is going like one or to the tree. Okay? And actually these are so new, and with so many new properties, I really like to show this graph that have been done by Tillman-Fau. Is that, I mean, the physio was started in 2011 with dysprosium and the 2012 by us with the Erbium. And for a while few here, nobody really, when there was, we were the only two, sweet spot. I mean, we found many, many fantastic new result. And then, that's how the situation is in the cold gas experiment. Almost every big group that I know have either an Erbium or a dysprosium experiment. And that's why in identifying the new ultra cold quantum technology for the future, Lanternite are one of the big trend. And there are few, two review, let's say this is about the ultra cold quantum technology and this is a long review article explaining dipolar quantum gases. And let me just tell you that, okay, so we typically consider these two outcome because a very large magnetic moment, but that's not the only properties. We have more tool that all come from the multi-electron nature. Okay? One tool is that we have a many atomic line, really many that we can study. So we can manipulate the atom with light and we can almost adopt, decide whether we want that the light is very strong acting on the atom or very weak. This means that we have atomic transition which have a strength going from kind of six order of magnitude. So we can pick almost everything we want. Okay? Including, this is one. Then the larger quantum number due to this anisotropy, not only give you the magnetic properties as I said, but also means that this is a large spin system. In the ground state, the argument is prosem since the number of spin state is two J plus one can have, you know, this is the fermionic one and this one has 20 spins. So it's a large spin system. So you can go from really quantum spin up, spin down to the, if you would populate all of them to the classical. You could study the quantum to classical transition in spin, for example. Or you can select subset of spin and just work with these two or these two. And then another important properties is this transition here which is at one kilo, one hertz. This is a clock type transition. Interesting for metrology. Interesting for precise addressability of any energy scale which is in the earth. The small correction of the Hamiltonian you can catch. Okay, you can probe. And actually this is a clock transition that we call orbital clock transition. Also interesting is that the wavelength is huge. Is in the so-called telecom window. Okay, so that's the clock transition. And it's in this so-called telecom window. So it's very much here. It's very much interesting for application because if you have a two level system emitting at the telecom, then this telecom photon, you can put it in a fiber and create quantum network. Okay. So that's because telecom wavelengths are the one you can easily transport in fiber. And so it's very relevant for communication and tanglet but also to study what is called cooperative phenomena. So the situation in which you have a discrete set of phatom one after the other, which are a small spacing but they emit at larger wavelengths. So you immediately imagine if you have antenna that they are very close to each other. All emitting at very large wavelengths you can imagine immediately think that there are cooperative effect coming where they can organize between them to kind of all emit at the same time, all suppress the emission. But even you can, this atom can be a mirror for the light. Can be completely reflective. You can use atom for creating a mirror. And that's a bit unique of the large wavelengths and the small lattice spacing. And you can moreover, this atom have many isotope. You can do boson, you can do fermions. Each of these is a little bit different molecular potential. So this phase shift is a little different. So you can also pick up the right scattering properties that you would like to have. And the polarizability that it's what I promised to you for the polarizability for this alpha, the tensorial alpha for an isotropic media. Indeed, you can just by changing the polarization angle of your light make that the atom will feel very different. So this is, you remember the cosine function, we add that, that's the function. So you can also play with this anisotropin, the polarizability. And of course you can use the clock transition as well to really manipulate the spin state, okay? And so typically instead of having an incoherent spin ensemble, I mean we have now developed a dola technique based on rabbi-palza to really populate any spin state. Those are the bosons and the bosons have 13 spin state, the fermions have 20 spin state. And, but we can really populate put all the atom in whatever spin sub with whatever spin population that we want. It's totally deterministic thanks to the clock transition. And I, but also very interesting and maybe most relevant for this talk is that now we have extended the lattice spin model or extended lattice model. Because now, because the dipole-dipole interaction is strong, you have interaction between atom which are physically sitting in a different lattice site. And I want just, okay this, the magnetic atom are not the only strongly long range interacting system. That's also important to say. There are also molecule can be even strongly, more strongly dipolar, but much more tougher to control. You can have Riedberg atom. Riedberg atom realize a situation in which the, let's say, long range interaction is so strong that overwhelm all the other energy scale. Very interesting for many application, we are in a situation in which we can create frustration with between different energies, competition of energy, competition of interaction, because those are several interaction and these frustration as I will show you later will give you new phase of matter. Then ion, Coulomb interaction, this will be one over R. And then of course a light induced interaction so that's the one over R to the cube dipole-dipole, electric dipole-dipole interaction. And I will, let's say stop now before we go to this idea of having quantum simulation with eye connectivity, I would take some question and then we go in the next part of the, thank you. Okay, so thanks. And now the mic is on so we can take questions. When you shifted over to looking at the F-shell and mentioned that the magnetization depended on the number of occupied electrons and thus we could manipulate the magnetic moment by controlling this, in a practical setting, how does one do that given that there are so many levels that you need to address? Are they all equally spaced? Do we need the same laser frequencies? Do you have to modulate the frequency of the laser with a broadband? Can you read, sorry, the first part of your question was not- So you mentioned that we could control the magnetization of the lanthanides by- The magnetic moment or the magnetization? Magnetic moment, sorry, that's my bad. Okay, you can control the magnetic moment of the lanthanide yes and no, because this really depends on the properties of the atom. As I told you that the air-beam has a different magnetic moment of this prosium. The way you can control is changing the spin state. All, every spin state have a different magnetic moment, but this goes by a step of one. You cannot really, I mean it's, you can a little bit control the magnetic moment by rotating the magnetic field quantization axis and then you can create time average potential. But typically this is really determined by the innate quantum number. They need it for the zoom. The fact that the wavelength of a single particle decreases with the mass. With that? The wavelength of a single particle decreases with increasing mass. Does it affect the simulation of electronic systems with the atoms and molecules? No, I would not say so. What are you thinking about exactly? Nothing particular. Because you have also the temperature. As a con, so you mean the de Broglie wavelength. Yeah exactly, the de Broglie wavelength is controlled by the mass and it's also controlled by the temperature. So you can always play with the temperature. Are there more questions? So I'm a bit confused about the temperature I just mentioned. So I imagine the experiment is very fast. So can you really say that the system is thermalized and how do you know the temperature? I guess the system is pumping energy or you have some relaxation. I don't understand the temperature part of the experiment. Okay, so typically, okay so you mean how we cool down and what is the final state? So we cool down in two steps. One is dramatic and one is soft. The dramatic part is really cooling down from I would say Kelvin to micro Kelvin and this is done with laser light. With so-called laser cooling. But this is then we arrive to micro Kelvin. We are not yet both condensate quantum degeneracies. Not there is still a Boltzmann type of ensemble of particles. Let's forget about this. Although it's very important because without it. Then what we do in a way or the other, we trap the atom in an harmonic trap, okay? And so in this harmonic trap, atom are kind of somehow everywhere, okay? You have a Boltzmann ensemble, okay? Then if I put the velocity and I put the function of the velocity, this is this type of function with the long queue, okay? Boltzmann distribution velocity. And then we have two or three different way of removing a high energy atom. So basically we are cutting here all this atom. The atom at high velocity. Then what you have is a non-equilibrium truncated distribution. At this point we wait. Collision is a contact interaction. Scattering due to the contact interaction are making the system retermalizing and spreading in the tree. And so from this we get this after thermalization. We cut again, again out of equilibrium. We wait and again this will become like this. At some point we reach a critical temperature for both the Einstein condensate. The system undergo a space transition, okay? From thermal to both condensed. Which is not only saying I remove one more and I make this narrower. It's really changing the statistical distribution. Going a phase transition in which there is an avalanche of this atom going all here. This avalanche is this phase transition. And at this point they go here. We kind of can wait a little bit more. Everything is thermalized. Everything is a stationary state at equilibrium filling the lower energy level. This is not correct what I said because that's a mean field approach. Mean field picture. Because all in the mean field you have all the atom in one state. So when people say Bose Einstein condensate it is the macroscopic occupation at zero temperature. All atom in one state. That's a mean field definition. In reality on top of this you have the quantum fluctuation. So there is a probability to be in higher K, right? So it's a much more complex real ground state and bending the quantum fluctuation. Correlation and quantum fluctuation. Yeah, sure. You need to because otherwise by a zoom they cannot hear without the microphone. There's two curiosities about this. What is the difference from the time scale of evolution simulations in quantum? This type of system compared to real systems. Difference. And another one is how can I, we can we simulate an attractive interactions? How can we simulate? Attractive interactions. Local for example. Like how we see the effectivity of co-models. Attractive interaction is easy. It's very easy in many ways. Even with the contact one because you have a scattering lens. But I've not told you the sign. This can be. So typically this is used with the sign plus. Which means repulsive interaction. But since we can tune the value of the scattering lens with magnetic field, we can also have negative scattering lens. This make attractive contact interaction. This have been also done in the experiment with cold atom. So this is easy. Local interactions. So for example, inter-site interactions is the same way to. The inter-site interaction it's with contact that you cannot have. Now is what we said. I mean we have all interaction because it's short range when they are in the same lattice side. With dipole-dipole interaction. Yes, you can have interaction between different lattice side. And now like magnet, this can be repulsive or attractive depending on the orientation of the dipole moment. So if I put my atom in this, in the lattice side and two different well attractive. This is attractive interaction between lattice well. So we can do it by changing the angle. Time scale that. Then the first part of your question, the time scale. So there is one time scale is the lifetime of the system. And then there is another time scale which is set by the interaction. So in the many body system typically you wanna have many body because the interesting dynamics and phenomenology is embedded in the interaction. So the question and then you can define what it's called an interaction time. Okay, so what's the time needed to make one interaction? Okay, and this interaction time can really depend on the strength of the interaction. And since you can almost go to the unitary regime where the scattering length is infinite, you can make this interaction time extremely fast. So you can have many, many cycle of interaction time with the dipole, with the contact interaction with the dipole interaction both in the Riedberg system that now are used as a platform for quantum, let's say information, the computation slash simulation, let's say the typical interaction time of the current Riedberg system which have a problem. I mean, I don't know if I have time to tell you this problem is about 10 to 15 interaction time. Let's say 20, 20 interaction time, 15, let's say. Hard to give you a precise number. Those are the interaction time. So where you can make an operation. But then you have to tell, okay, I wanna make an operation but I want still to make this operation with let's say an error on the operation very low. So it's a little bit fluffy how you define interaction time. Interaction time without error can be just few. Interaction time in general where you allow for the fidelity to go down can be higher, yeah. Okay, so let's thank Francesca once more.