 Okay, so hello everybody. Sorry for the short delay, but that was even fast because I honestly almost missed my talk during to the shift. That was also why I was back there. Good, first time that that happened. But nevertheless, I will be ready. So the first thing to mention is I changed the title to be a bit more concrete because I decided to, is this position good or do you still have this feedback in the microphone? Because I hear that. Okay, so I decided to tell you about something more concrete and that is our recent experiments on spin charge separation and hidden correlations in Fermi hover chains. And before I start, just I want to make sure to set the stage because of the two talks in this session before were basically on solid state qubit systems. And I move now on to a different platform which is neutral atoms in optical lattices, where the focus is clearly not so much on the quantum computation, but more on the quantum simulation site which you saw actually by the fact that we were basically missing in that famous science table there. Okay, so what we are interested in is, we are interested in exploring with this experiment the Fermi-Habbat model. And why is it at all interesting to do quantum simulations on the Fermi-Habbat model? Well, the Fermi-Habbat model is the most prominent traded toy model for high-temperature superconductors. They probably noted many things in the details that are really unclear. And what is really unclear in the superconductors is really the exact nature and even the which phases are there in this phase diagram, somewhere in the vicinity this DSC, this d-wave superconducting, somewhere in the vicinity of this superconducting dome. So this year it's really the phase diagram that is conjectured for the superconductors and the question is not really that we want to address with our quantum simulators. Can we find these interesting phases in this actually most simplest or, well, let's say most prominent toy model which is this Fermi-Habbat model here. I'm gonna explain in the next slide the different terms there to give you a little bit of intuition in this. So how can we actually implement the Fermi-Habbat model in AMO systems? Well, the basic work calls that we have in the lab are optical lattices that we create by interfering laser beams. So what you see here is a little bit an artist's image of a three-dimensional optical lattices that is generated by interference of this beam which somehow, well, it's hard to work by this beam and this beam so that basically leaves you with a periodic array of intensity maxima that are these great dots. And then we load atoms into the system and these atoms in our course fermions, well, they can hop from side to side. They might have a spin that is when they become interesting for fermions because otherwise they do nothing too much interesting because of it's basically powerly blocked and they can have onsite interactions and that's what's depicted here in this 2D example. So again, this here is basically a hopping term. These fermions of both spins, red and blue, they can hop around in the lattice. When they sit on a single side, they interact. This is this interaction U-term and again, powerly blocking two red and two blue rays can never sit on top of each other. So this is the framework. And again, this sounds like a very simple model. I mean, I can draw a very simple toy cartoon here but again to stress the low temperature phase diagram of this model in the regime where there are also vacancies like where you have like a filling below one is really not clear and that is what we target. In this talk, I actually not talk about the two-dimensional model. It's basically the thing is in these business of quantum simulation, it's always your first stuff to understand. Your quantum simulator, you have to verify that you believe the results and so on. So it's always a good idea to first test your simulator in a regime where you basically know what you expect. So what we're doing here is we are studying basically the Fermi-Habbat model in one dimension where we know this is basically a Latin gel liquid. We know basically everything you can say but what is very interesting is actually that the physics in 1D is very universal. That goes under this keyword of Latin gel liquids and our experiments can provide, well, a fresh new help to understand and so on. The fundamentalist effects that appear there. So it's in the sense of a textbook-like setting and what I want to talk about is one of the most prominent effects that appear then that is spin charge separation. In one dimension, basically the independence of spins and charge, spin and charge degrees of freedom. The second thing is so why are we interested to explore 1D? Well, basically I explained it already. You have to calibrate your quantum simulator and be ready to explore then uncharted territory in two dimensions or develop new techniques that you can transfer to other dimensions or also develop new techniques to test, well, to get access to new observables and for that I will provide an example which is actually directly measuring a topological order parameter that can identify, for example, topological order and Haldane chains. Okay, so first to set the stage again, I want to link to solid state observations of spin charge separation. So these are three prominent experiments. Oh, here this is actually wrong. This is 2005, I think. This was basically the first one. Where they basically, in solid state, you have very different observables than we have with the atomic systems and typical measurements are some incarnation of spectroscopy. Something is wrong with this. Okay, maybe I should speak not too loud to let the sound waves not shake the connector or whatever. Okay, so, well, we have three spectroscopies, basically two tunnel current, tunnel spectroscopy measurements and one angle result for the emission measurement. Well, the thing is I want to highlight this so the typical results here is you change some energy and you see some spectrum and then you see spin charge separation in these experiments very indirectly, I would say, but what you see is basically you see a splitting of the excitation spectrum and you can extract two modes and then one is said to be the spin on mode and one is said to be the hole on mode. So that is basically what has been seen in solid state in this context. In ultra cold atoms, again, we have very different observables and that before I come, what we actually did, I want to show you two very prominent proposals in this context, one is from 2003 and the idea is basically now with the ultra cold atoms, we can directly track the dynamics of the system. We can take images and we can basically ask questions like, given we made an excitation at point A, where is this excitation at a later point in time and space? So this is basically related to both these proposals here. So the idea is you make an excitation initially at some point in space zero and then you look basically two wave fronts traveling outwards, one in the spin, one in the charge, degree of freedom against spin charge separation. You saw the spectrum basically means you get a spin and the charge velocity, both were linear spectra before and that means you see two different velocities directly and that means after some time, you really see directly, well, spin and charge, they spatially separated and you can even fit in the velocity and extract the story directly. So this was how, well, many proposals also around this and many experimental attempts were driven by these proposals basically to have some way of seeing this directly, but again, I will switch again in this talk, I will not talk about these dynamical aspects of spin charge separation, but I want to show you how we can actually measure, well, basically spin charge separation directly based on the measurement of correlation functions between spin and charge degrees of freedom that we have access to with our quantum gas microscopes. So what you see here is a single shot image that we took in our experiment of lithium atoms, so each red dot here is a lithium atom in a two-dimensional optical lattice and the point is this is really the density distribution of one quantum realization of the system. That means we can calculate now very complicated correlation function based on these shots. So I can ask for example the question, what is the correlation of having a hole here and atom here, a hole here, a hole here, wherever you wish. So we can make a non-local, we can analyze non-local correlation functions and that is basically where our new analysis basically is based on. Okay, but I don't want to be completely abstract, I was also as an experimentalist want to give you a flavor here how this actually looks. It's actually a pretty complicated machine that we have set up there in the experiment, so it's actually consisting of three optical tables, I show only two here, so you see this one is basically our preparation table to prepare all the laser beams, so like hands on work means you have to have this all under control, keep it stable and so on and then we guide the light by favors to the main table and the actual ultra cold atoms, they live on this table and I should say this is actually a photograph on the side so this is not a top view, so we become three-dimensional in terms of optics around the microscope, so where are the atoms? I could make the guessing game, maybe you see that there's something, there's more space here in the center and that's actually where the atoms are, let's zoom in. So what you see here is basically a class cell, it's roughly four by four centimeters and the atoms they live in the center there in an ultra high vacuum environment and that makes them actually such a nice isolated quantum system and then we absorb them with a microscope, what is the microscope? Well, that doesn't look very spectacular, it's this piece of plastic from below that we use to detect the atoms, it's actually a resolution of roughly one micrometer, so we are not so much in the typical microscope domain but I want to give you an idea why that actually works and the idea is we have actually the possibility to generate pretty complicated lattice structures. What we use in this experiment here is basically a two-dimensional super lattice, how does a two-dimensional super lattice, well, it can look very differently depending on how you choose its parameters but what is shown here is basically a light intensity distribution for one specific parameter setting of the system, the important thing to remember is we have two length scales here, we have a large well separation of 2.4 micron and we have a small well separation of 1.2 micron. Remember I told you the microscope we have a resolution of one micron, so even the smallest one we have no problems to observe that. Now that you might ask the question, okay, but typical ultra cold atom lattice experiments they typically work with 500 nanometers. That is actually something that is usually even required to have long enough tumbling time scales for the, let's say, usual atoms with usual atoms, I mean, rubidium. What we use is lithium, lithium is much lighter, it's something like 12, roughly 12 times lighter I think and that means it can tunnel over much larger distances and that is the trick actually that let us get away with these large lattice distances and that is at the root, really at the heart of the experiments I'm gonna present you in a minute. So another thing, just to give you an idea how these quantum gas microscopes work, so how did I actually, or how did we take this image with the red dots that were all these lithium atoms? Well, what you have to do is you have to keep your atoms at one point in space and then somehow make them scatter photons. How many photons they scatter? They typically scatter 10,000 photons per atom. 10,000 photons per atom means basically you heat them so strongly that they will hop out of these lattice sites very easily and with basically unity probability, meaning we cannot take these images because they start to move. So what do we do is we overlay an extra lattice, these are these black dots, so the red dots, let's say it's the super lattice that I showed you in a minute before, the red, the black dots are now an extra lattice that we call the pinning lattice and this we can make very, very deep. So what we do is we shine in just for detection this pinning lattice, keep the atoms at their position and then use so-called Raman-Seidman cooling, so this is what is called R1, R2 and the repumper, so these are just the lasers you need for Raman-Seidman cooling and what this does is this allows you to scatter photons while keeping the atoms in the individual wells. So they try to escape, they try to heat up but then this cooling mechanism brings them down to the close to the ground site again so they cannot escape and with that we can scatter enough photons and obtain these images. So why this long story about all this technology because of here's really what enables the experiments on spin charge separations and that is a specific detection technique with which we can locally, locally means on each and every lattice site can detect the spin state of the atom simultaneously with the charge degrees of freedom. Basically it means we can detect locally is the atom spin up, is the atom spin down, is the atom not there at all or do we have a spin up and a spin down atom on one side. So how does that work? First of all, as I said, we do experiments in 1D. So this is actually an image of such 1D chains so this is one 1D chain, another one, another one, another one, so this is five and they are completely separated concerning the dynamics in the vertical direction so that's what's illustrated by this thick lines. The point is now you see that the separation of the lattice sites in the horizontal direction is half as in the vertical direction. So we use at this one here only one of our super lattice is basically this direction while here we have the long component on but now prior to imaging, we can use this extra super lattice in the vertical direction as something like a local mini-stand-gelach to get a solution for the spins. So what we do is we have first this large lattice spacing and then we switch on prior to imaging, we switch on a magnetic field gradient and that means for the spin ups, the situation looks like that if we ramp up this short scale lattice component and for the spin downs, the situation looks like that. That means the spin down goes from your side right well, the spin up goes to the left well or on the image here, this left and right is up and down. So let's see, this one here basically was a spin down atom, it moved down. This one here was a spin up atom, it's moved up. This one here was a doubloon. So we had a spin up and a spin down atom and well, these are the, well, you won't say boring sites, but they actually not because if you want to see the interplay between the holes and spins, this is an empty site. And we get full resolution as I said, site-wise in this way. And then we can also calculate, answer question like what is the probability for or what is, how are the spins predominantly if I have, do I have a hole here? For example, here I have a basically a missing atom, so now I can ask questions, what is the spin alignment around such holes or what do holes in generally do to the spin order in a longer chain? And that is actually the type of question we're gonna ask. So before coming to the, coming to the doped, how about physics and doped means in a situation where holes are important in the system, I want to show your first results in the simple case, basically in the case where the system is at half filling and just to, what half filling means basically we have an equal amount of spin ups and spin downs in the system and no holes. So there's an atom at each site and that means the charge degree of freedom is not there at all because of the charge, there is nothing where the charge can move, the only motion that you can have is basically spin motion. And in this situation, the Habat model actually maps to a Heisenberg antiferromagnet and of course, given the technology I just described you, we can just check, do we see antiferromagnetic correlation in the system when we load and prepare these chains? And the answer you see here, so if we measure a standard two point correlator at the distance of one side, two side, three side, four sides we see basically we have anti-aligned spins, aligned spins, anti-aligned spins, aligned spins over these four sides. So we really see these systems, if we cool them as cold as we can at the moment, they show antiferromagnetic correlation and at the moment what we get is something like a correlation length of 1.3 site corresponding to an entropy of 0.5 KB. And that's again an example why it's interesting to benchmark your system in 1D because of these numbers, how do we get these numbers? Well, this number we can measure but how do we get this number? We cannot really measure. What we have to do is we have to ask our theory colleagues, in this case, Jacobo Nespolo and Lodopole and they do QMC with all the experimental stuff in there and we can then compare and get this number. So that's like this benchmarking idea. We are by far not the only ones working on this problem. Actually, that's a very prominent problem that is attacked in the ultra cold community and very prominently in two dimensions in the other experiment and I want to highlight this 2007 nature of the Marcus Griners group where they actually implemented a new way of cooling and they can reach actually antiferromagnetic order. In this case, I would not even call it correlations anymore, I think it's justified to call it there even antiferromagnetic order because of they see basically that they're full sample of roughly 100 sites actually orders. So that is very prominent or very promising results for these ultra cold atom systems because it shows basically that we really can access the low energy physics of these complex models. And just to mention, a similar experiments are going on in Martin Thierline's group at MIT, Michelle Curls' group in Bonn and Wasim Bacchres' group in Princeton. All right, so now let's add holes to the system. So what do we expect if we add holes? So let's keep it simple and understand this intuitively first. So we have here a kinetic energy part in the Fermi-Habat model. So this comes as usual with a negative sign that means basically the holes, they want to maximize this hopping term. They want to be delocalized as much as possible. That's what this cartoon says that if you go from fulfilling, which is black to a hole, you don't know where the hole is, it's completely delocalized. So that's what the holes want to do. The spins, they still want to be magnetically ordered in this case. And that means they want to show something like in this cartoonish way, this up-down, up-down, up-down structure. The thing in 1D is now, we can make both degrees of freedom simultaneously happy. And that is really what shows you that, well, basically this works because if you have spin charge separation, it basically means the spins, they don't care for the holes. The holes can freely delocalize even if you have anti-ferromagnetic order. And that is again in a cartoonish way shown here. So let's assume we have a situation where we have a spin up, where we have a spin up, a hole and another spin up. Now, if you would move the hole, the hole moves here, then it would cost one J of energy. J is the spin exchange energy here because if you have two spin ups next to each other, instead of a spin up and a spin down. On the other hand, if the situation is such that you have a spin up and a spin down around the hole, this hole can basically freely delocalize as much as it wants and the spin order is fully intact. So if this is the ground state basically, that means the spins and the holes, well, they don't couple, they don't care for each other. And that is what we tested in our experiment. So again, what is shown here is what I showed you already before, is basically the up, down, down, up, down, down and so on order between the spins. And now we analyze our data and look specifically for events where we have a hole and now ask the question, what is the alignment of the spins around that hole? And what we find is indeed, compared to the case where you have full filling, all atoms there basically where you have up, down, up. In this case, we have up, hole, down. That means this two point correlator, it really flips its sign, compared to the case where if you have a hole there or if you have not a hole there. And that is something we can again directly test. We can make this test also a bit more on longer length scale. We can sit for example on a distance of four sides and ask now the question, what happens if we have no holes in between the four sides? Well, then you can look here, we expect basically positive correlation. So this one. Now we add one hole between the four sides. This thing flips, we get negative correlation. We add another hole, we get positive correlations. If we add three holes, so basically two spins here and only holes in between, well they are direct neighbors again and we see here, sorry, we see negative correlations here again. So that works, we can directly study a single hole, what it does basically to spin correlators. So what I was describing you so far was basically only direct environment around the hole and very close distance physics. And I can tell you, I stay mostly short distance in this talk, but I want to give you an idea actually of how you can analyze also a longer range physics in the system based on string correlators. And for that I want first you to understand what actually these holes do. So let's assume we don't have a hole. So we have the perfect order up down, up down and so on and so forth. Now if you have a hole, I showed you already that if you have a hole, basically the order here from down down goes to form down and then up. But that means now all the spins in this block until the next hole basically have been flipped from here to here. So now if I have a fluctuating hole number and just calculate normal two point correlators, let's say around between these two sides, then this flip of what we call the anti-ferromagnetic parity of this block and not only of this block basically leads to an averaging of up spins and down spins and then basically to a vanishing of the correlator. So the question, so that is kind of a trivial impact of the holes to the spin order because you see basically the spin order here still looks pretty good. So we have up, we have down, we have up, we have down, we have up. It's just that the holes displaced basically disordering. So now how can we get that back? Well one obvious way is we just kick out all the holes and we shift everything together. We go to so-called squeeze space. So that is the idea of came out of Jan Zanens group. So you see basically we basically remove the holes and post-analysis and basically shift together, shift everything together, we get squeeze space and then you get perfect anti-ferromagnetic order. And this is now completely independent of how many holes we had in this chain. And another way which is largely equivalent but it's very interesting because of this is why my basically comment on topological order parameters, that is basically instead of squeezing out the holes, we basically go through the chain and for each and every hole, we basically multiply the next sites by minus one. Basically we flip their spin in the analysis to basically have to correlate the right again that it respects this spin flip around the hole. And that is basically then amounts to measuring of a string order correlator where you have this charge string, a string of the product of minus ones in there. And again this is exactly the thing you also would have to measure to detect order in Haldane chains where you have instead of holes spin ups and downs, you have basically spin zeros, spin ones and spin minus ones. And I want to stress this is really something that until the advent of these microscopes, you could, there was no known way of how to measure it because if you want to measure that correlator between site one and eight, you have to know everything in between. So this is a project, this is a really a non-local order parameter. Okay, let's see how that works in the experiment. So we have now a fluctuating hole number and measure the standard two-point correlator. I said already that basically amounts of averaging this ups and downs because if we don't know which hole number is there and you see basically we have next neighbor correlations which is basically an Pauliplucking effect and then basically we see no correlations. Now analyze the same data in terms of the string order parameter. And we said, you see we say basically get fully, we get back the undiferromagnetic correlation. Yeah, which shows you basically that the only effect that the holes have is this trivial effect of displacing the order. Another example in squeeze space, this time we basically bin our data into sectors of fixed density. So the green data here is around unity filling. It's a density of 0.95 to 1.5. And you see basically we see this typical string, typical two-point correlations as you've seen already before. But if you go to the very low density sector where you have very many holes, the blue one here, we basically do not see anything of these staggered correlations anymore. Now we analyze the same thing in squeeze space. We squeeze everything together and you see basically it's basically completely independent what we get there of the hole number. So all cases show very nice six-second behavior and again that emphasizes the spins. They don't really care how many holes you have there. You just have to analyze the system in the right way. And now if this is really true, we should be able to describe the system as an effective Heisenberg model. Remember the spins are described by a Heisenberg model at unity filling. And what I showed you basically that if we analyze it correctly, we don't care for the filling. The spins always behave the same. So we should be able to describe that by an effective Heisenberg model. But now, of course, if we find a temperature, the strength of this coupling here becomes important because believe me, if I have one spin there or this side of the room, the other spin there and holes only in between, they will hardly speak to each other. So they will have a very reduced coupling. And that means basically the coupling strength in this effective model depends on the filling. How many atoms you have there? Actually it depends also on the correlations that are modified by the temperature and that is why you see basically this curvy behavior for different temperatures. But let's say the main effect for now is really that we see a strongly reduced coupling if we set every very low filling. So which means once been there, once been there, then we hardly have any coupling. So now how can we test this? I should say, I mean, this is basically all this theory that is backing up this, you know, this has been done in Eugene Demler's group by Fabian Kost. Yeah, so how can we test this experimentally? Well, we can try if we can model, and now it's important our finite temperature data on the spins using this effective squeeze space Heisenberg model. So the idea here is again, we've been our data versus density that you've seen already before. And now we analyze here only the next neighbor correlator and the next neighbor correlator. And now what happens if this effective coupling goes down but we still have the same temperature? That means we increase effectively the entropy. You know, we have just some temperature and we shift now more and more quantum states basically in this range that is set by the temperature. So that happens from here to here. So what you expect if in the Heisenberg model entropy increases, then you expect obviously that correlations go down. So that is what we see in the experiment. We see strong correlations here. So this is on point three, the next neighbor. This one is on point one, the next neighbor. And then they go down and they go to zero and actually they don't do that in an arbitrary way, but they basically follow the predictions of this effective model here for temperatures around point eight J. So this matches very nicely and confirms again that the spins and charges are independent. And this actually, at least in the very large U limit. Sorry, well, to introduce the slide. You know, I basically started the whole talk with emphasizing some universality of physics in one dimension. But the point is, Latinx and liquid physics in general in 1D is only limited in the very long wavelength flow energy limit. What we tested here was basically correlations at one side, two sides, something like this distance. So this is not really in the realm of the Latinx and liquid. But the point is the Heisenberg model, 1D Heisenberg model, at least in the infinite U limit, has been shown actually to fully separate basically the wave function. Fully separates into a free fermion wave function for the charges and an effective Heisenberg wave function at all energies, basically. So that is also why that is basically explaining why we see spin charge separation. Point is now, we do not have infinite U. So can we see something that there is a remaining coupling between the spin and charges? And the answer is in the experiment, not fully. But if we ask our theory colleagues, we actually can get this information. So what you see here is basically, again, what you've seen before, the staggered correlator and the correlations around the holes. So now, if this model, if the effective Heisenberg model is completely, or if the spins and the charges are completely separated, then you would say at t equals 0, these points here should be completely equivalent. So because of these spins, this here is a direct neighbor, like here, these two. And this one here is also a neighbor, basically a direct neighbor. It's just a hole in between. But if you are t equals 0, the energy scale doesn't matter. So now if we ask this question, what are the correlations here at t equals 0, compared in this case, then if everything separates, it should be the same. That means this difference here, what we plot here is the length of this green arrow, should go to 0. But the theory says it doesn't do it, actually. It stays finite. And that is basically because of we have corrections due to non-infinite U. OK, with this, I want to basically finish and highlight the people that did this work in the lab. And that is, first of all, the postdoc, Guillaume Salomon, and then these three PhD students here, Timon Hilger, Ahmed Omran, and Martin Boll. And then, I think I can skip the summary. And I want to first give you a teaser of what we can do, actually, with this data analysis. So when you see here, we can do post-selection tricks and basically measure full order of these chains. And we think we can access low temperature physics by that. And well, what we want to do in the future is clearly we want to extend this all and bring these nice techniques to 2D. Yeah, thank you.