 And Francesca, the stage is yours. So we are now moving up to, let's say, the next step. I mean, now that I gave you an idea of quantum formulation and the very first kind of realization with the last example system, we go on with the idea of having a discrete or on with the left, but we are changing the other. And so we are going now using an atom which will have a very large magnetic moment and this allows us to create the two ends with the quantum solution with high connection. What do I intend for what I told you before, that the atoms have all, because there is an interaction between atoms on different sides. Now it's very easy to understand why, because the dipole-dipole interaction is an interaction which is one over the distance. So this, in a left, is that this value is kind of important by the left. So it will be then the dipole-dipole interaction which will be one over lambda alpha, so the left is facing this, the power. But there is an interaction, a nasty interaction. It's not a thing that, you know, it's really an explicit. Excuse me. It's a, thank you very much. It's a, a native interaction that you simply have, there is no further engineering needed. And shorter is the wavelength of the laser, of course. Shorter is the distance between the atom. This goes as a cube. Larger is the dipole-dipole interaction. Okay. So one can create lattice in the UV. So around 400 nanometer, then lambda alpha is 200 nanometer. You put it as close as 200 nanometer. The interactions start to become the off-site in the order of kilohertz. Okay. So extremely huge compared to, always compared with the other scale of the system. What is very remarkable, you know, that it's very special, is that in first approximation, now, doesn't matter for the strengths of the dipole-dipole interaction, in first approximation, doesn't matter the height of the potential. So you could put an infinite barrier, an infinite wall, almost between these two atoms, the off-site dipole-dipole interaction will still be active. Maybe a little reduced, because the Vanier function in each lattice site would be reduced, but the dipole-dipole interaction is still active. Okay. And this is really something extremely new. And also, this atom will be connected with this one, will be also connected to that one, if you have two dimensions, will be connected to the one on the diagonal, to the one in the other direction. So in this sense, it can be really, you know, highly connected. Every atom is highly connected, correlated, or interacting with all the other atoms. It means that in your Hamiltonian, you will have also two... So in your Hamiltonian, you will need now to also account for, let's say, a new term, which is, again, let's say the operator is the one of the dipolar interaction, one air to the cube, and then your field operator. Again, you can do an expansion in the Vanier basis, and you will have this type of term, which is interesting, because this term depends on atom sitting in different lattice sites, and can be further decomposed in two different terms. The one in which you allow i, j, k, l to be, you know, the same, and so this is the onsite interaction. And the other, which is off-site interaction. So you have both components. This is all called off-site interaction, or also called the nearest-neighbor interaction, okay? So the dipole one, but the dipole one also acts on the same lattice site, like it was the contact. But now you see this can be very large, because the distance is very small, and this goes as one over distance cube, okay? And now, putting all together this extended Bose Abarth model, always in the approximation to have a not higher orbit, and no excited bend, just, you know, this approximation remain, you have the single particle tunneling, then you have the onsite interaction, which is both contact and dipolar, and then you have another type of interaction, which is our long range. And now it comes a little bit, also the question of one of your colleagues, can we make it attractive or repulsive? The answer is yes, and that's very important, because it would depend on, let's say, orientation, relative orientation of two atoms. Like here, you can have a different dipolar interaction between these atoms or between these atoms, just because the relative orientation, these, you know, and these, let's say, these and these, it's different. Okay? Just to see it better this, let me put an easier angle than the one in the slide. So if I have a placate like this one, just a square placate, then I have one orientation like this. Here, it's repulsive, but here, it's attractive. So you can have simultaneously, let's say that there is some direction preferred because nature always wants to favor attraction, so these already create kind of hidden frustration into the system or constrain, let's say. And then here is the contact one, so the usual one, but then also this dipole-dipole interaction, and as I said, again, this is angle dependent. Also, the onsite one can be attractive or repulsive, and this depends on the shape of the Vanier function, the three-dimensional Vanier function that you have in the lattice side, and now you can change the power of the lattice to make the Vanier function of two atoms more elongated or more flat. And with these, let's say, the colored zone here, like the little cloud, tells you the aspect ratio, so this is an aspect ratio, this is the ratio between one to the other, and this is bigger than one or the other one smaller than one. In this configuration, the dipolar interaction onsite is repulsive, and here is attractive. Okay, so you can really shape the impact of the dipolar interaction by shaping the lattice that shape the Vanier function. The Vanier function shape the effective interaction because it's an integral over the Vanier function. One interesting configuration is the one in which you can create a rectangular lattice unit cell because you would like to study the dynamics. So you see in this type of configuration, what do you have? You have the spacing between the atom, so you have the two green planes. This is just an example. Of course, we have many more planes. There are two green planes, and these two green planes are much more separated between them. So the dipolar interaction, which depends on one over D to the cube, is weaker between planes. So you would like, and in this way, you can basically restrict the dynamics that you want to study in a two-dimensional plane and not in 3D. So now we are thinking about 2D dynamics. It's also very interesting, but I will not cover the situation in which there is also interaction coupling between the plane because in this case you can even create a dimer-bound state in different planes. That, again, like a solid state physics will kind of give a friction on the motion on the plane due to this dimer physics to the next plane. And moreover, that's our lattice. What determines the orientation of my magnetic field? The orientation of the magnetic moment of my atom, which is this little arrow here, I can change it as I want in a complete sphere just because since it's a magnet, it will always orient with the magnetic field. I can change the magnetic field in my experiment as much as I want. And so the atomic dipole will follow. And so this is another degree of freedom that my B magnetic field is completely tunable to the orientation. And then, of course, I can play with the lattice strengths and this will allow me to change the aspect ratio so the Vanier function in the lattice side. Okay, and now we have this Hamiltonian now with all this thermal and we will go one by one to see what can be observed. So the first time was the first time, yes? Yes, it's possible, but I will not cover this. It's possible the tunnel probability is smaller, but it's possible to have tunnel, nearest neighbor tunneling. That's also possible. In that case, it's dependent, let's say, on different things. I mean, we can maybe discuss later a realization that we are currently thinking about that would realize these type of things that you are mentioning. Yeah. So this was the first time this extended Bosse-Abardo Hamiltonian has been created. Okay, so then the first theme is let's see that, you know, like learning about the Hamiltonian is really true that there are these thermal. We will not have time, but actually we find out that this was in this Hamiltonian type of learning part. We found out that this is not the complete Hamiltonian. There is another term which is the many-body tunneling. It means that in this Hamiltonian, the tunneling, as I told you, is single particle. So one atom tunnel hop to the next lattice side and doesn't make any difference if the lattice side is empty or occupied. But because of the dipole-dipole interaction, now it do make a difference if the lattice side is occupied or not. It might prefer to not tunneling. So there is what it's called. It has many different names. It's kind of, let's say, it's beyond a single particle tunneling. You can call it, let's say, a density-dependent tunneling or, I mean, a different name. Okay? But these I will not discuss here. So that's, again, the minimal Hamiltonian, but there is now with this complex realization in the lab for us the new challenge is to enter in the domain which now is becoming popular, called the Hamiltonian learning. How do we certify that this Hamiltonian is the right Hamiltonian? Which are the observable that can happen? Maybe you look at one observable and then you would say, yeah, that's the Hamiltonian. But in reality, maybe there is another observable that will tell you that the Hamiltonian is different. So to find the right observer to certify the Hamiltonian is the next frontier. Now we always started from the Hamiltonian and realized in the experiment, but now the experiment goes even, you know, step forward with all this tuneability that maybe you have to go back and change your equation of motion. So Hamiltonian, Gross-Petajeski, whatever it is. Okay. So let's start with the on-site interaction. I mean, this on-site is responsible for the superfluid-tumor insulator transition. Now you add in addition the on-site dipole-dipole interaction and so you can simply repeat the superfluid-tumor insulator that I was showing here. And then you can, by probing, let's say I look at the central peak here. It's very narrow. This will be this. The width of the central peak here is very broad. So I will say that my observable for this measurement is the observable in which I look at the width of my central peak. And so when I see that it makes the transition from narrow to largest where the phase transition is happening. Now this is the width, and then you see here it's very narrow and then here start to increase and then when you are in the mod state it's very big. And now because the dipole-dipole interaction depends on the orientation of the magnetic field. If I now change the orientation the point where the phase transition happens will be different. And indeed you can see the difference on the phase transition point and you can make this difference actually much bigger than this. This is just for a moderate difference. And what does it mean? That the dipole interaction can actually protect the superfluid state. So like kind of make the mod insulator coming after or not. And you can change this and the point just with the magnetic field. And so it's kind of is a change of the if you remember this wedding cake with all the lobes of the phase diagram of a super fluid to mod insulator. This would correspond to change the size of the lobes and expand the mod insulator phase or contract it just by working on the orientation of the magnetic field. Something not possible with usual alkali atom. You can also measure the energy gap of them. I told you that the mod state is a gap state. So you can measure the energy gap that you would have toward the energy cost to create a doubloin. And so this is have been done for contact interacting system. How do you do this? You modulate the lattice step at the frequency. When the frequency of modulation is resonant with the energy cost then you have a loss feature like this one. And this is the energy cost for creating just two atom. This is for creating more atom and so on. When you have the dipole instead of normal atom this energy cost is also something depending on the orientation of the dipole. So this is let's say and so you would see that depending on the orientation you actually have really a shift of the energy so the more that you need the energy cost to create this type of double occupied mod state. And this shifter it's depending on the orientation but it's also depending on the Vanier function because you can make it more, you know, squeezing more and making more dipolar or less dipolar and so for example you can go up to a shifter of the gap so an enlargement of the gap that goes alpha kilohertz. So very big just by changing the aspect ratio of your lattice side so this shape of the blue you can really have that the dipolar interaction can change very much the spectrum of excitation of a mod insulator. Then the next thing interesting is to look at let's say near neighbor interaction. And so here the key point is that this is also depending on the density on the angle. And this is now what I was talking on the blackboard. You now have all your atom and you can see the blue here is just to say that there is a bond of attractive interaction and the red is a bond of repulsive interaction. That's the situation that we are creating in the system. And so now if we want to modulate the system and try to see the cost of creating Dublin well it's making a difference whether you modulate in this direction where you have only bond of repulsion or you modulate in this direction where you have only bond of attraction. And so that's what you can do. So you start with the situation like this one with the dipole oriented in the plane and then I modulate in this direction so I'm breaking one bond of attraction put it here but then you see I created now this four bond of repulsion and if I do in the other direction you see here you have two blue and four red and here you have four blue and two red. These two configurations correspond to a different energy just because of the dipole-dipole interaction. And so you can measure this different energy and this give you directly let's say the benchmark that indeed the nearest neighbor interaction is present in this system and it can be written with this form. And so this is what we did. We measured one configuration on the other the different of energy. All what I was showing here now, yeah exactly. So these let's say quadratic lattice as I show you is done by two beam no there is one lattice beam and the other lattice beam I have two set of beam making the lattice. Now I can if I modulate the intensity of one beam in this direction I'm breaking bond in this direction. If I modulate the intensity on so I'm giving so I have a way to by modulating the lattice beam I have a way to selectively give energy only in one direction and if I give energy at some point my you know my system goes yeah this is only one of course our system is much much bigger than this so you will break several on the same direction yeah but not all of them it's not you're not depleting 100% you see also here you know the same idea here you this is the modulation in one direction you then by modulating the lattice step you promote one atom here okay and you measure the excitation related to this process there is a critical frequency that make this but not all the atom are necessary doing this I mean the minimum for example for this set of parameter half of the atom we're doing this this is just a picture to show what happens to one bond and one atom but you have more atom in the row that will do the same okay but so far what I wanted to do is just to benchmark the Hamiltonian but another question is to say okay can the dipole-dipole interaction really bring you new phase of matter really quantitatively different now what is interesting to think it's about let's say in the phase that we are considering in the mod phase that we are considering there is no breaking of the special symmetry of the system when the system goes to what does what do I mean there is no spontaneous breaking of the translation symmetry what I mean I mean that the lattice is imposing a geometry and the wave the global wave function of the same geometry you see the periodicity of the lattice and the periodicity of the atom is the same so the wave function and the potential have the same periodicity so that means there is no breaking of the translational symmetry because the symmetry of the Hamiltonian and the symmetry of the wave function are the same but thanks to the dipole-dipole interaction you can go a step forward and the phase of matter which are called mod-crystal that break translational symmetry that means that the symmetry of the lattice so the symmetry of the Hamiltonian of the wave function and these have been predicted several years ago so this is without dipole-dipole interaction without nearest neighbor interaction that's the usual phase diagram and that's it that's the phase diagram of contact interacting particle without nearest neighbor interaction when you go to nearest neighbor interaction the phase change and you can have this type of pattern which are signature of spontaneous breaking of translational symmetry because although the lattice has always the same periodicity due to repulsion and the traction of the interaction the atom will have a periodicity which is different from the lattice so you can have stripe phase checkerboard phase in general modulated phase of matter that's a symmetry breaking phenomena spontaneously you are not imprinting this, it's the wave function going to this is the equivalent of the nil order is a nematic order as nematically in space they would organize and those are for example all the different phases with different filling factor one fourth, one third, one alpha and you have all these kind of phases and very recently in Marcos Greiner that has an air biome experiment that we collaborated in this paper together so what we could see is that actually we could see all these checkerboard phases stripe phases, diagonal stripe phases that originally arise in long range interacting Abarth model these are all realization of different which spontaneously break one order the spontaneously break of an order for people that are familiar with solid state physics what does it correspond for every spontaneous breaking there is a new phonon mode appearing in the system this phonon mode you can call it in some approximation goldstone mode for every symmetry broken there is a goldstone mode this is now breaking a new goldstone mode is appearing in the system this is the predicted and in the future this will be also studied in the experiment yes with the now what we are now moving and all what we did so far is considering that the particle are all identical so there is no spin degree of freedom in the sense they have all the same spin all identical particle but what happens now want to study something more related to quantum magnetism that you will need to spin state and now we move to the next section which is the lattice spin physics with high connectivity now for the lattice spin physics it's really an extension of the Fermi Abarth of the Abarth model let's say and is an extension in which you consider explicitly that you might have spin up and spin down particle it's just an extension and to have interesting magnetic core there what you really need is to have let's say the nearest neighbor interaction between spin up and spin down in the case of contact interacting atom there is a way actually to create this type of interaction and this is an interaction which is coming from second order perturbation theory is a second order perturbation theory which is called super exchange interaction ok and ok the spin up and spin down in the experiment can be realized just by having different Zeeman sub level and then I say one Zeeman sub level is spin up another Zeeman sub level is spin down and you can have a large manifold ok so that's what is a spin up in a neutral atom is a Zeeman energy level or two different hyper fund level or two different electronic depending on the atom two different level also have different energy they are not degenerating energy spin up and spin down ok that's really little different with respect to some solid state because electrons spin up and spin down ok and then ok how can you realize this now without dipolar interaction well you can do this using this idea of super exchange so there is a reduction of the Hubbard model to the Heisenberg model in the limit of having very very large onsite interaction u you can re-end for a feeling so what you can do you can let's say do a perturbative expansion in t over u and then you will have that in the zero order there is also the super exchange there is a high w occupied side but the second order give you this term which is a super exchange what is J actually J is nothing else that the tunneling squared divided u is the your second order perturbation theory here and that's the super exchange and the super exchange is kind of a virtual is acting as a virtual process in up spin down then there is a virtual tunneling to a state very close by with up and down and then another virtual tunneling that bring you back so effectively super exchange is doing this but passing at two let's say together and then here so there is two so here t here it's J here sorry and then you can really start to study and for example to write the isenberg model for boson in optical lattices and then you will have the tunneling in the two different directions and then the order that you have whether you have ferromagnetic or antiferromagnetic order depend on the onsite interaction in this case whether the interspin onsite interaction is bigger than the intra-spin or the contrary you can have ferromagnetic order or antiferromagnetic order and this is something which I already show you that have been observed with with atom but now the point is that the super exchange it's great that you can have without a native long range interaction but it's also painful because you know you need to have very very high barrier so you need to suppress the tunneling it's kind of a physical whether you don't have the external degree of freedom are completely frozen and you cannot tune too much it's not a native interaction second order small correction is small but if you now have really long range dipolar interaction instead of having the smaller super exchange term due to the dipole-dipole interaction okay and this is super interesting and again you can realize this lattice spin physics with nearest neighbor interaction also with the polar molecule with ridberg atom with ion and we I mean as well with magnetic atom which are the focus here and now that would be your new Hamiltonian here and so you have the off-site dipole-dipole interaction and then you can rewrite your Hamiltonian term of sigma plus sigma minus which is basically and that's a direct spin exchange interaction that allows you to do this you don't need to pass to any you know virtual state it's really direct spin exchange interaction which is magnetization conserving that's also important here the total magnetization it's conserved so the sum of the spin is conserved during so if you have one up and one down you can do this but still you have one up and one down okay so that's mean magnetization conserving and then which is also interesting is a good in this Hamiltonian that we have another term which is a single particle term it's quadratic in the spin in SZ and actually these can be controlled with magnetic field or with the tensorial you remember the tensorial and I thought polarizability that's what it is you can control this term here and what's the effect of this term well is that if you have now three spin state you can have them having exactly the same energy splitting this is the own resonant condition delta here that it's there is no difference in energy the difference in energy is zero but with the resonant light you can you know also shift and make this out of resonance or you can do the contrary you can start with an out of resonance situation and you go on resonance if you are out of resonance so that the energy energy splitting between the different spin state is different then the spin the exchange dynamic cannot occur because it will require energy to do this because you have two atom here one goes up the other will go down but energy should be conserved but if this energy splitting is different that's not not possible to conserve and yeah so that means that you can initialize you know in our case with Erbium we have many many spin state and I can initialize all the atom in a given M state let's say I choose the yellow one here so the second to last and then I put all the atom in this specific spin state they are frozen they don't move there is no tunneling there is near the dipole-dipole interaction off-site they are all identical fermions and I start in the and my question is if we prepare a Pune spear state how does it evolve under the Isenberg Hamiltonia how do the correlation evolve the spin excitation what is interesting is that they cannot move but what can spread is the spin internal degree of freedom but the atom move the spreading is only in the internal degree of freedom and now if as I told you this energy splitting is different from this one nothing happens the population that I have prepared in the yellow state stay almost constant over two seconds okay but if now I switch on light I use this quadrat this light let's say tensorial light shift and I put this level on resonance you know I'm putting this level on resonance now as soon as I do this and I can do this very fast with light I can initiate a dynamic in the system where I start to have this atom moving and changing and then you see that the spin population of the yellow is decreasing and the spin population of the other state is increasing and the interesting part is that all these dynamics in the spin domain of freedom cannot be described by mean field it's really due to quantum fluctuation the mean field prediction is this one no dynamics and also this process conserves the total spin is totally magnetization conserving okay so and it's really a phenomena completely driven by quantum correlation then there is the problem of the theory and that's where now you come into the game of course and it's how do we describe we have many atom into the system is purely quantum cannot describe it by mean field you can actually not very well do exact diagonalization I mean if you have five spin state you can have probably seven atom that's it that's already your limit of what you can or you cannot simulate and so there are several method we developed one together with Ana Maria Ray Agila and Bui Zhu that at that time she was also Agila and so and their model was about you know computing the dynamics using the truncated Wigner approximation that's a method that is an approximation and I think that this method you know it's going very well in the early time dynamic of the system but then it's kind of not anymore really reproducing the long time dynamics now the question is how can we do better can we do better and this is an open question which I leave to you because I think at the moment that's not really clear how to do better in simulating already this system is beyond what can be let's say simulated another things that we did in the experiment is to notice that the exchange interaction this one spin up and P plus depend on the initial state in the sense it depend on the initial M state that you choose you know we have this 13 actually for fermions 20 spin state we can initialize in any space and actually the strength of the exchange is you know changing depending on M and what we could do we could produce any initial spin state and we could see the dynamic or this spin cone how does it spread and we were actually really seeing observing that there is really this effective interaction is going really quadratic with the M state yeah so now I would say that these are kind of more or less it's the starting point all these long range interacting neutral atom in lattice is at this starting it's a new direction more experiment and more group are working on this so stay tuned at the moment we just made the first really small step toward this physics and the field will certainly develop further and with this I have a question because now I've finished my part of the lattice and I wonder if I have still time I have more time because I want to show you something in the bulk do you have still energy for a bit of physics in the bulk okay yes okay so this is let's say the energy this is a bit now going back to the root of the question because the lattice can do a lot of solid state Hamiltonian is very nice but it's actually also irritating because you know where the atom are you can you know the degree of freedom and so on but what is really the how much can we push quantum simulation or how much can we know about you know equation of motion or law determining the behavior of the system in the bulk in the bulk is much more painful or even you know this measuring a correlation entanglement I mean have you ever noticed this very little have been done because it's not interesting but more because it's very tough at the moment now but now what I want to also show you is how incredible this dipolar interaction is even if you have atom in the bulk and so how surprise can be happening I mean the ingredient of the game you know already we don't have a lattice anymore there is no lattice and you have a bulk system with 10 to the 5 atoms in harmonic potential they can move in this harmonic potential as they want every particle is identical that's important that's why we have quantum they are indistinguishable and now you know the two ingredients each atom can interact with the other and there are two sources of interaction one is the contact one we know this sphere things and the other is the dipole-dipole interaction going at long range now as we already mentioned the scattering lens A now we spoke about the scattering lens here the scattering lens can be positive so repulsive contact interaction or negative attractive contact interaction I will put here always repulsive it's repulsive now I have my dipolar interaction is an isotropic now I don't have a lattice that it's you know blocking the external degree of freedom they can move and organize as they want there is no pinning okay so where do I want to go okay so then the atom not high but the atom can decide how to align with respect to the other with the lattice I was imposing funny a function I increased this you have to do that then no here they are free they are free in the big container I want to notice that they can become attractive or repulsive okay and now start the first puzzle the story and the discover of the phase of dipolar quantum matter it was an escape room game for us in the last 10 years and that this lesson of how you feel in an escape room or in a quantum escape room is what I want to show with you now because I have two independent source of interaction I can you know tune I can make one stronger one weaker but really the interesting point is when I make this competition of interaction something unique to magnetic atom you cannot do in reedberg I make competition and I want to know which one is working well if you have a linear equation then it's enough you say okay this is a value of 3 this one is a value of 4 this one is linear is increasing if the interaction are non-linear so are depending on the density the game change because even if the pre-factor is the same but if there is the possibility to be attractive the atom will try tend to attract to be together and attract in this orientation by doing this they increase their local density by increasing the local density they increase the interaction they are even more favorite to do this that's the essence of collapse if you have any system in nature with attractive interaction density dependent what does it happens they will tend to you know be as much as together because attraction decrease energy and because nature want to decrease energy by doing this they increase the density as crazy at some point they explode the density is too high there is really this hard wall you know they cannot be closer than this explosion fine attractive interaction so if I have that the strength it's the same but those are density dependent there is a natural favorable to unbalance the total interaction that's what it happens for example in a collapsing molecular cloud from the protostar the collapse of stars is exactly like this they first have an increased density then boom they explode attraction is winning attraction is winning over repulsion what is the repulsion here is always kinetic energy is the stabilizing mechanism of universe in some part is kinetic energy ok and then we said ok and this type of experiment was done many years ago in chromium which is a magnetic atom but not as magnetic as hermium and this prosium they said ok let's organize this in a way that attraction will you know trigger a collapse and they saw that in less than half a millisecond a nice dipolar BC was collapsing you see now at the center the maximum intensity is dark and here there is almost no density is light and it's collapsing then you can ask why it's collapsing let's say look the similarity of this geometry with this geometry why it's collapsing in type of D wave indeed this was called a D wave column well if you now are familiar with Lagrange polynomial that's a D wave Lagrange polynomial is exactly like this is the shape of the dipole-dipole interaction ok very well established finding this one is collapsing we understand attraction is winning everything fine big surprise is only the shape but then if we think about ok it's recovery the dipolar shape it's fine because there is line of attraction and line of repulsion so it makes sense that the shape is not an isotropic because the interaction is not an isotropic then we said ok now let's repeat with Erbium and Disprosium actually with Disprosium in Stuttgart we should see the collapse in the same way because those are even more magnetic so even more attraction even faster collapse but actually this was the first big surprise because the system this is the system of Disprosium atom was not collapsing but was kind of creating a conflict this is a gas phase that starts to separate little drop of gas very dilute I should really imagine alone we are not imposing any geometry and this little drop were organizing a type of triangular lattice this was then stable living for a very long time what is going on that's crazy we repeated but in another geometry and that we saw that instead so we were kind of changing the geometry of the trap and then we were seeing that instead of having many crystal we had only one macro object that increase the density as you would expect by the collapse you see this is a usual B.C. and this is the new phase that we found and then we would expect ok this increase the density and then it will collapse and it will be stable not only this that it was stable but also if I would switch off the trapping potential this object which is a gas do not expand is self bound is a gas that without a container is not expanding and this we measure now the problem of this when we saw all this thing we could not understand what it was why because the Hamiltonian Schrodinger equation the non-linear Schrodinger equation would never give you this type of solution as a wave function so something was really puzzling this is something and this is really the first key that we were searching in our escape game here you can see that this is at the beginning not really expanding this macro droplet what is going on if now I just write the Schrodinger equation with kinetic energy my trap the contact interaction the dipole interaction this give you collapse all the time there is no way and then it's kind of the first the first time really in the field where the Grosby-Tajeski fail to describe cold atom and actually what we find out there was a lot of work what is going on I mean I remember in the lab we were repeating thousand and thousand times the same experiment we could not believe what is going on I mean it's really crazy and then we find out that actually there is another term of interaction that arise that was usually neglected kick it out and this is part of this Hamiltonian learning we do also in the Bose Abad we do a lot of approximation but then at some point those are wrong can become wrong and there was a term that we were so all the Grosby-Tajeski is a mean field it's not a quantum theory but if you now what we were neglecting is the first order correction to really quantum correlation which is called a quantum fluctuation fluctuation means really have the feeling to tell you it's moving away from mean field it's not a real fluctuation it's kind of the next order correlation or correction and this is a new term which typically is very small but since it rise very fast with density more than the other two if there is a collapse that increase density at some point this guy jump up become extremely big is a repulsive interaction and that's the function of stabilizing your system is telling you more than this peak density you cannot have increase more than this and then I mean this is really a density regulator a new term that had to be added by hand and so the experiment were driving the theory in this case and okay that's simply a constant but really the important thing is the dependence with density and then of course how to verify because this was also not very clear we then do really high precision spectroscopy of quantum fluctuation and we could really verify that this is the case you need quantum fluctuation that the first enigma were solved but then comes the second one because okay with attractive dipole-dipole interaction is not collapsing now I know why now if I put repulsive if I put repulsive dipole-dipole interaction so I have repulsive contact repulsive dipole-dipole this should be stable if it was stable with attraction will be even more stable with repulsion so it will be stabilizing we said okay that's easy we will certainly see this and indeed I mean we see a nice busy stable and so on but then what we want to do is let's say we changed a bit the contact interaction and suddenly spontaneously these two sides pick a period in this lecture in my lecture where did you saw this side pick in which figure don't you remember the Mott insulator don't we add the central pick and to side pick was it not interference by matter wave doing to a periodic structure where does the periodic structure come here we don't have any lattice I'm not imposing anything but this is signalizing a periodic structure because always in the momentum space interference pick in the momentum space signalize a short wavelength in space what is imposing this there is nothing in the Hamiltonian that give you this and bit it's like thinking about I have a fluid at rest I give a wave modulation and then I have a wavy energy that was our point of view and indeed I mean what happens is that if you have dipole-dipole interaction what happens is that the spectrum of excitation it's changing and now it's again simulation I have cold atom but I will show you a helium spectrum of excitation helium type spectrum of excitation really of a super fluid in helium the spectrum of excitation is a phonon have a maxim have a minimum that it's called by Landau-Roton and then a single particle this here it's enormously interesting because you have a high momentum a specific value of energy where it costs not much energy to create excitation so if I give a small kick nothing excited but if I give a big kick then I start to have excitation but if I give it even bigger nothing is excited again there is a minimum it's like a resonant frequency here and actually what it turned out if you change the scattering lens you can even put this rotation excitation to zero so it costs no energy to create this excitation so the system will spontaneously create this excitation because no energy and then we measure that so this is the theory and you see that in the theory with all our parameters all the discretization is due to the system and then we measure the spectrum of excitation and what we found is a spectrum which is linear no rotation if the contact interaction is very high but then if we make the dipolar interaction more and more important it starts bending and you start to have this rotation minimum but then coming back to this figure here that was the first things we saw in the experiment we thought there was a reflection from a laser maybe this is what gives the periodicity but actually not this was really the number of atom in the side peak let's say the occupation of the Roton mode was first increasing and then stabilizing which is strange I mean the Bergolubov theory tells you that if you can create excitation the number of excitation will increase exponentially diverge because it costs no energy and then you have this mod populated but here it was developing again stability some stationary state again which what does it do this and what we find out is that actually the system undergo a phase transition really the ground state of the system change there is a many body phase transition that bring you to a new state I'll just go a little faster of matter in which the ground state is a modulated state this modulated state which was observed without even us having understood this in 2017 was later understood to be a phase transition for a normal BC to a new state which is called super solid and this new state is remarkable because it has broken let's say it's fully coherent so break a gauge symmetry it's one broken symmetry and the second one broken of the translational symmetry spontaneously create periodic pattern this is really the finite size system we have in the lab but actually you could do the calculation for infinite you have I mean an infinite modulation it's not a finite size effect it's really the new ground state of the system where you have the super fluid property and the periodic density so you have a double broken symmetry state and this was observed I mean precursor in this experiment in my group in 2017 and then later also by my group the group in PISA of Modugno and Stuttgart by FAU and so how do you see the phase transition while we can really probe and you see this periodic modulation of the system you can calculate the full phase diagram and you see that the full phase one is the BC the other is the super solid where there is everything it's connected and then you have this isolated droplet where a really disconnected droplet here you have phase coherence and modulation here you have modulation but no phase coherence yes the confining potential there is an harmonic confining potential which is very high like this so this is responsible of the fact that this one has less matter than this the peak height is due to the harmonic confinement but the modulation if you it's resisting in flat potential and variation without dipolar interaction no you need to have dipolar interaction this is really an effect coming from the rotten in the spectrum of excitation the rotten touch zero there is a specific rotten momentum the rotten momentum give the rotten wavelength the wavelength is this wavelength and then I mean from the phase diagram then you see unmodulated and phase coherence modulated and phase coherence modulated and non-phase coherence now the phase you know now very well that you can extract from matter wave interferometry as I show you before from the mult insulator so when we make expanding look at the difference in both case I have an interferometric signal okay but this one is as a shape and is another shape now if I repeat another time the measurement and another time this is what I get that if you don't have phase coherence each time the phase pattern is completely different in this case and at the end you don't see anything while in the case of fully coherence you still see that the interference pattern is always the same you can build up a function which is called phasor you can extract a phase and that's the difference in the case of isolated droplet no phase coherence the phase are spread in 360 degrees and in the case of phase coherence there is really a localization of phases just in one sector and this is the proof of the phase coherence of the system since I don't want to go I want to go here and show you that so far I show you how you can break symmetry, translational symmetry only in this direction but actually the phase diagram that we have calculated is much richer than this and so you can have but now I show you only the experimental data fitting very well the theory with the quantum correction so quantum fluctuation on it you can see I can change my trap and change the atom number I have two then I can have a zigzag configuration then I can have even more extensive this is all a gas organizing and then I can have even a completely fully phase coherent let's say hexagonal system and of course when you have an hexagonal system at least for me the first thing I want to do I want to rotate it and I want to see if I can see vortices now rotation in the experiment is really difficult okay but we found a new way because as I told you the die all of this atom are very magnetic so you can see one of this maximum as a gigantic magnet and the gigantic magnet would align with the magnetic field so if I simply rotate the magnetic field each of these will rotate together and the reason why I want to see vortices is because vortices is really what determine the difference between a classical fluid and a quantum fluid and now I just would like to show you what is our idea for rotating I will first rotate for simplicity not a super solid state I want to rotate a normal BC okay so my BC is a little bit elongated along the magnetic field direction because the dipolar interaction is attractive in this direction then if I give an angle this is really the ground state that's all from our calculation then the system is rotating and if I start to rotate the magnetic field the gas is rotating together and if I rotate fast enough then vortices will apply in the system and so this is our simulation so it's the gross extended gross pitaieschi simulation this is the early time dynamic I put them in rotation I rotate the magnetic field the system start to rotate start to lose matter this kind of galaxia type of shape but you see that some vortices some black hole here are in the low density and then after a long time dynamic all these vortices are starting to see this in the experiment and now look how beautiful it is this is the simulation and this is our experiment behaving in the same and you see the vortex appearing of course not as easy in the theory but that's experiment and experiment and now we wanted to do the same with the super solid now there is a classical analog of what I'm telling you imagine that this is our super solid we would like to rotate our super solid and see if we can vortices entering inside and that's now let's say the simulation I start to rotate my super solid and then if I rotate fast enough after a while you would see that in the phase there is a phase jump of 2pi that goes from yellow to blue and you see now it's yellow to blue yellow to blue, yellow to blue vortices are starting entering to the system so you can indeed create vortices by magneto steering the system those are the experiment it took us almost one year to learn how to rotate a fragile state like a super solid it was really a difficult thing but then there is a little problem because in the theory I can access the phase map and tell you that there is a vortex here that we probe the density profile not the phase and in the density profile you see nothing because the vortex is a low density but in between the droplet there is also low density so you cannot distinguish so we had a little bit the idea and I want to close to kind of rotate a super solid I would like to produce a super solid and rotate such that there is a vortex inside and then just for visibility I want to melt the super solid and the vortex is a topological excitation remain trapped and so this is what we did we prepared the super solid we started to rotate very slowly nothing happens and then we started to rotate very fast and then melting and you see now the vortex that appear but the interesting part is predicted that the vortex enter before and that's really the fingerprint of difference between B, C and the super solid vortex and then we see here that there is an intermediate frequency where you would expect a vortex in the super solid and not here you can also see vortex as a phase winding in the interference pattern indeed if you now do a toy model in which you have three gaussian toy model you can expand this three gaussian and there is all with the same phase the let's say the interference pattern will be this type of interference pattern where you have at the center some matter but if there is a phase winding like a vortex then you would have a minimum this is just a toy model of a normal gaussian it's really basic matter wave interference okay the vortex the presence of a phase winding of 2 pi will be 0 in the density and if we do our full numerics with dipolar interaction we see exactly the same things if we rotate very slowly so that we are below the critical frequency for created a vortex there is this type of interference pattern with the matter in the middle and if there is a vortex there is this and this is now what we also observe in the experiment that this is a direct observation of actually the existence of a vortex in the interference pattern of our system okay I think that now this is a I wanted to show you something more but I think there is no time and so I'll just take the opportunity to you know thanks the team and to show you the team and that's our team in Innsbruck many of the things I show you we have done in the team we have three experimental team and one theory subgroup and so there is the Erbium the Erbium Disprosium team we have Atomine tweezers experiment which I didn't have time to show you and a really great theory sub team and yeah now we are also having you know putting together quantum gas microscope into the system and with this I would like to thank you for your attention great thanks a lot for this very interesting lecture and we have time for questions there is one over there so coming back to the phase diagram for the transition from Bose Einstein condensate to super solid could you explain again what happens as you tune the scattering length I guess okay it happens something very very peculiar actually so I think I understood the fact that as you tune the scattering length the Roton minimum softens and then you have the condensation of Roton so you have those special modulations but then I didn't understand what happens after when you start increasing continuing increasing the scattering length and you form isolated droplets so somehow you have two possibilities so there is the spectrum of excitation excitation as you said phonon, maxon the Roton softens completely so this is K and that's energy and that's the Roton minimum now at this point you have the full softening what happens if you what happens at this point you have two possibilities either it's becoming sorry where are you ah sorry either it's becoming unstable so you start to have let's say this imaginary negative energy okay so the system is unstable there is imaginary Roton mod that get populated and so the system will then be fully excited okay and of course mean field breakdown and everything this is one possibility second possibility is that so kind of you know divergence of the Roton population that's kind of instability you would see what is the number of Roton particle that you create as a function of time and you kind of they populate completely the condensate everything goes to the Roton and it's an imaginary energy the other possibility is that there is a phase transition and this is actually what happens but for the phase transition you need something that stabilizes and again is this quantum fluctuation that has the role again the one that I show you for the single for the macro droplet this additional term that again enter into play to stabilize and to avoid that the system let's say diverge so with the addition without quantum fluctuation this will simply die with quantum fluctuation you have the phase transition from a ground state which is a usual Thomas Fermi let's say BC to our super solid phase and actually then what happens on the spectrum of excitation happen what I told you we are breaking at the translational symmetry we also measure the spectrum of excitation but I had no time to show what happens is that the second band actually I should do it in the other way the second band is appearing so you have two goldstone moda two branches in the spectrum of excitation one is the crystal one and the other is the super fluid branch and we measure the two branches okay but what's happening then when you start forming isolated droplets so this holds in the red region right in the red region that's in the red region then when I do when I am in the so when I move from the red this is the spectrum of excitation of the red region more I move to the blue one this is the super fluid branch and this is the crystal branch the crystal branch starts to become harder the super fluid starts to soften when you do the crossover that's not a phase transition from super solid to crystal this branch vanishes completely and the spectrum of excitation becomes like this it's really the blue line zone repeating it's one branch you don't have any more phase coherence so you lost this symmetry breaking it's totally uncoherent so you go back to one branch and which is for the crystal moda so all what is stiff moda and we measure all the spectrum as well okay thanks a lot are there more questions so I have a question regarding the spin dynamics in your previous slide so when you are changing actually the internal degree of freedom like you are exploiting many different spins so like what is the experimental way to do like are you exploiting with the Raman sideband cooling for the internal degrees of freedom what is the bandwidth of the light which you are using and what is the energy difference in your system between two spins let me start by so we don't need any further cooling no because we don't need sideband cooling or anything like this we initialize the system in one so we produce a very cold quantum degenerate let's say lower of one spin state okay then we have different way to populate different spin for example we can do in the block sphere pipe also so like Rabi oscillation and they populate whatever I want with whatever population so like the resonant light matches with your pipels so the pipels that we do is a type of you have so there are really very different technique if the energy splitting between the Zeman level it's quadratic let's say like if your atom have hyperfan structure then the coupling with magnetic field is also quadratic term is quadratic Zeman shift then if it's quadratic the energy splitting is different between the different level and then if it's different then you can simply put radio frequency and then you know exactly where you go and you stop to do the spin preparation if it's not quadratic is linear if you put IRF signal it coupled to everything so you are not deterministic there you have a different way like two photon Raman excitation so you put up and then down in another spin state and that's for example one thing which is done or you can do stirrup or you can do two Rabi parts in our case for the boson we do two Rabi parts that coupled to the clock transition as I show you in one slide Thanks I have another question regarding how you extend your super solid phase to the 2D How did I? This is a good a very good question so let's say what is really interesting is that when you have a phase diagram typically you think to have two axes but in reality the phase diagram of a dipolar gas is extremely rich you have several control parameters the trap, the atom number the dipolar interaction, the scattering lens these are things that you can independently tune now if you fix the scattering lens and you change the atom number and the trapping confinement you can reach different phases linear, 5 droplet linear, 7 droplet zigzag, hexagonal phase or fully 2D so what we had to do in the experiment was not only taking the trap to be round but also to increase the atom number and then you enter in this fully circular super solid that I was showing the one that then we rotate Further questions? Thank you very much Thank you very much