 Yes, sorry, thank you for pointing that out. So the experiments I will tell you about were performed in collaborations with my friends and colleagues at Harvard, Michelle Lukin, and Marcus Greiner. What unites these two themes is that we are manipulating individual quanta, making them strong interact. And as I will tell you, we use so-called Rydberg atoms to achieve this. Rydberg atoms are very highly excited atomic states. We think of hydrogen with a principal quantum number, n equals 100, where the electron is very far away from the nucleus. And it turns out that under these conditions, you have relatively long range interactions. In the first part of the talk, I would like to tell you about quasar particles that are made of photons and collective atomic excitations. We call them polaritons. And these can travel like light in the medium, but they can also strongly interact, so strongly in effect that we have, for the first time, observed two and three body bound states of photons. And then in the second part of the talk, we will use Rydberg interactions not to manipulate photons, but to manipulate atoms. And I will tell you about a 51-atom quantum simulator, a system where we have 51 individually controlled atoms on a line that interacts strong enough that we can observe a quantum phase transition in this system. So first, about photon-photon interactions. Here's the picture, the level scheme that we use. We use rubidium atoms, but if you're not familiar, think just of hydrogen. Rubidium has one electron in the outer shell. So we have an S state. This is like the ground state of hydrogen. We have a P state, an unstable P state. This is like the P state of hydrogen. It decays relatively quickly within tens of nanoseconds. And then, and we can excite this with a laser. And this is the light that we will manipulate. This is a weak laser beam. And then we take a strong, very strong laser beam to couple the P state to a very highly lying Rydberg state. This is, again, an S state, but with a principle quantum number of 100. So think we're going from the ground state of hydrogen to the P state of hydrogen to n equals 100 S state of hydrogen again. Why do you need the P state in between? So the P state in between is what allows us to control these photons. So basically, we want to control these photons, and we need a resonance. And that's what the P state is for, so that we can couple these two atoms. So the work is built on concepts for slow light or electromagnetically induced transparencies, which were pioneered in 1989 first by Steve Harris from Stanford University. So what Steve Harris pointed out, surprisingly, it only happened in 1989, although all the formulas have been known for many decades, is that if you have a strong absorption, so this is the imaginary part of the sustainability, this is absorption. If you have a strong absorption, like here on the S to P transition, and then you add another laser beam, coupling this P state to a meta-stable state. The Rydberg state is relatively long-lived, hundreds of microseconds. Then what you will see that in this strong absorption, you will induce a transmission window. You will kind of burn a hole through which light can be transmitted. So that induces a transparency induced by light. And because this is a relatively narrow now transmission feature, associated with it is a strong variation in the index of refraction. The lower part is the real part of the sustainability. Think of this as index of refraction minus 1. So what you see is that there's a steep part where the index of refraction changes very, very quickly with frequency. And if you ask yourself, like in undergraduate optics, what is the behavior of light in such a system, you can define a group velocity for pulses of light. And the group velocity will depend on the index of refraction, which is near 1, plus something that will depend on the steepness, on the slope of this index of refraction with frequency. And so what Steve Harris pointed out is that this can be used to strongly reduce the velocity of light inside a medium. Our case, for instance, by a factor of 100,000. So you can make light that is very slowly traveling in the medium. A maybe easier picture to remember is the following. When light is inside the medium, this is the free level system, ground state atoms, the metastable P state, and then the control field to the red book state, you can think of light inside the medium being a quasi-particle, which is not only a photon, traveling at the speed of light v equals c. But every now and then, or quite often, the photon is absorbed by one of these atoms that is then via the strong laser field propelled to the excited red book state like so. In this case, the probe photon lives inside the medium, not as a photon, but as an atomic excitation. It turns out it's a collective atomic excitation because you can't know which atom has absorbed a photon. And that allows the system to retain all memory of the photon direction, of the photon phase, et cetera. The photon is completely, coherently converted into a collective atomic excitation. So inside the medium, we have this quasi-particle, which has the photonic component and this spin excitation, or magnon component. And the reason why the light is so much slower than the speed of light is because most of the time, the dominant contribution to this polarity and to this quasi-particle is this atomic excitation component. It's actually, in our system, 10 to the 5 times larger than this other component. So you can think of it, the photon travels a little bit, then spends a lot of time as a collective excitation. And the atom travels a little bit more, et cetera. So that's why it travels at one kilometer per second in our situation, rather than 300 kilometers per second. So I have a silly question, but this is the high-power photon, not the S2P photon, right? No, this is the S2P photon. Oh, the S2P photon. So this is the S2P photon. This is the weak beam. And that photon disappears and then excites an atom. But because of the strong coupling, the atom is immediately propelled to this state. So you're saying that you said applies to that photon? Applies to that photon, exactly. So this is just an auxiliary laser field that is strong, that we filter out, that we're not interested in. The photons we are interested are the S2P photons that would otherwise be strongly absorbed or experience strong dispersion. So this not only slows down the speed of light. You may have heard about stopped light. Stopped light is when you change adiabatically this control field to make the photon slower and slower until ultimately there's no photonic component that's just stored in the medium. And in this way, people have stored photons coherently for up to seconds, individual photons. In our system, the lifetime of this state is more like a few hundred microseconds, but there's still a very long time escape for the things we are interested in. The other thing that this allows you to do is you can now make photons interact, because as they are polaritons and have this atomic component, if the atoms interact with one another, effectively the photons can be made to interact with one another. Because these slow light polaritons interact with one another via the atomic component, which makes the photons interact. So the photon-photon interactions are atom-mediated interactions. And that is the reason why we use these Rydberg states because they can be made to interact very, very strongly compared to ground state atoms. So Serja Roche, of course, is famous for using Rydberg state and his Nobel Prize-winning work. This is what the wave function of an n equals 35 Rydberg state looks like. We use typically n equals 100 principal quantum number. Here, the electron is about one micrometer away from the nucleus. The nucleus is tiny. Typically, the atom is one angstrom. The electron is one angstrom away. Here, it's 10,000 times further away, one micron away from the nucleus. What that means is that these atoms are very, very easily polarizable. A very small electric field will pull the electron away easily from the nucleus. And that, in particular, means that if I have two Rydberg atoms in some proximity, then quantum fluctuations in one, in the position or in the polarization of one atom will induce a dipole moment in the other atom. This is the van der Waals interaction. So these atoms interact with the C6 over R to the six van der Waals interaction, but the C6 coefficient is enormous. It scales with the principal quantum number to the 11th power. So if you go from n equals 1 to n equals 100, you have 100 to the 11th power stronger interactions than in the ground state. It's a repulsive interaction. If you mix the C6, it's a repulsive interaction. In terms of? Sorry? What is the power 11 plus? It's basically the quantum fluctuations in one, and then the polarizability of the other atom entering in there. So I forget. I can figure it out. It's the van der Waals. It's, let me think for the easiest. It's not a ground state of the atom. So usually the ground state is pushed down, but for the excited state. The denominator is in the van der Waals, you have a denominator, right? And if you start from the ground state, it's attractive. It's attractive for you. Excited state for these as repulsive in principle, depending on which other states are nearby, it can have either sign because the detuning matters from the state. I have a question. On the state of the photons? Or it's like, it can be explained with just a singular? No, so if you did this with a single atom, the atom would get excited. But when the atom radiates back down the photon, it would radiate it in a dipole pattern in all directions. So if a photon came in and there was just one atom, the photon would go out into 4 pi. When you have these many atoms, these collective interactions, you excite a superposition of any of these atoms could have absorbed the photon, but now with a definite phase. So these atoms act like an array of antennas, of little antennas. It turns out that's not necessary. You just need an ensemble. And as the photon comes in, it will be stored with the correct phase in every atom so that if the atoms don't move during the storage time, it will be re-emitted in the forward direction. So for the moment, yes. So if these atoms don't interact, you can store light and it will come in. It will be absorbed and it will be emitted in exactly the same mode with exactly the correct phase. But photons will be not interacting with one another, just like in vacuum. So the storage part and the slow light part is independent of the interactions. But that would only make the light slow. It would not create interesting effects. So for the interactions, we add the Rydberg interaction of the atoms to it. And as I will show you, that gives rise to very strong consequences. Otherwise, what you would have is you would have a pulse of light. It would travel slowly through the medium. And it would come out again on the other side eventually after some delay time. Very good question. So it's really these atoms as an area of antennas that store the photon. And you need this quantum superposition state. OK, so now the interactions. And here's a simple picture how you can understand photons attracting one another. So maybe what I should have pointed out here is you have a three level system and they are different eigenstates. The eigenstate that interests us is the superposition of either the photon being there or the atom being in the Rydberg state. This is a so-called dark state. And this state has essentially no component from the excited P state. The state that decays very quickly. So this is why the photons can travel without loss. This is the reason for this window here. Basically, this dark state polariton has essentially no component in the excited P state that would decay away very, very quickly. So as the atoms travel through the medium in this kind of system, we have the probe photons that we are interested in. We have this S2P transition. In this case, we detuned the light a little bit from the S2P transition by a few line widths. And then we have the strong laser coupling to the Rydberg state. So under these conditions where photons are far apart from one another, you have basically this transition. The P state is not populated. And you have photons traveling as slow light polaritons as a superposition of atoms in the ground. So the one probe photon and the atom being collected. Really excited. Now the situation changes when these two photons are closer together in a characteristic distance. We call this a blockade radius. It's about 10 micrometers. In this case, because of the Rydberg-Rydberg interaction, the Rydberg state, the state here, is shifted away. At a few micrometers, it can be as large as a gigahertz shift. It's shifted away far from resonance. What that means is as this state is no longer resonant, you can forget about the effect of this control laser. It's simply no longer couples resonantly to anything. So you can think of this control laser as not being there in this situation. So what do you have now? Now you have a photon traveling near the S2P transition. In this case, it is detuned by some detuning delta. The dominant effect is no longer absorption but dispersion. When I'm detuned a few lines, the dominant effect is index of refraction. So these photons experience an index of refraction from the atoms, but they only do that if the photons are close together. If the photons are far apart, the index of refraction is one. If the photons are close together, the index of refraction experienced by this pair or by these individual photons close together is something different from one. So the cartoon picture is as follows. If this is the atomic medium, in our case a few hundred micrometers long, then if you imagine one of the photons is at this location here, then if the second photon is over here, more than 10 microns or so away, it experience an index of refraction, essentially one. Travels slowly but the index of refraction is one, so no phase shift. If the photon are over here, if the photon is close to the other photon, it has a different index of refraction, which is different from one. So this looks to the second photon like a glass plate, like a glass plate with the wavelength of the light inside changes. This N2 is like a little glass plate. However, this glass plate travels and is induced by the first photon. So basically it's like one photon carries with it a region of index of refraction that is moved. What happens here where we know that from undergraduate optics or quantum physics, the wavelength has one value in this region and has a different value in this region. The wavelength is, say, shorter in our typical situation. Shorter wavelength, it turns out, can be mapped. These photons also have phase onto a difference in potential, simply means shorter wavelength. There's a potential in the equivalent system. So you can think of this as an undergraduate 1D problem where you have a scattering potential here, an attractive potential in this region. So basically you have one potential and another potential. So you can think of this as a step in the potential. One wavelength here, another wavelength, shorter wavelength here because the kinetic energy is higher. It turns out we should describe these photons approximately by a 1D Schrodinger equation because they also acquire mass. And they acquire mass, I will show in a moment because there's a curvature of the dispersion relation which is equivalent to a mass. So these photons are slowly traveling and they acquire mass and they can be described by this type of potential. So this is what the dispersion curve looks like. This is the phase as a function of the detuning. The blue curve is at very, very low photon rate where at any time you have essentially just one photon inside the medium and you can forget about these interaction effects. And so what you see here is the usual dispersion curve of the atomic resonance and then detuned here is the two photon resonance and it has this sharp dispersion curve over here and we are working there. And so here you have relatively steep slope and you also have some curvature that corresponds to mass. So the photons, individual photons we're talking about are slow, they have a group velocity about one kilometer per second and they're also massive. This curvature here corresponds to a mass of about 1000 photon energies divided by C squared if you wanted to express it in that way. So these are slow massive particles traveling and they are also interacting. And when we increase the rate of photons so that now there's a chance of having a second photon inside the medium then the blue curve changes to the green curve. The phase changes and this is the induced photon-photon interaction effect. If one photon is in the medium the second photon experiences a different phase. So how do we measure this? We send weak light pulses through this medium. There's a weak laser beam, there's also a strong laser beam, the coupling laser beam that we filter out and then we detect single photons on single photon counters. It detects single photon events. In this case we have three counters because we would like to look at two and three photon correlation functions. For now think about just these two counters for instance. And so we look at coincidence events between the two counters. If the photon, the incident beam is a laser beam so if the photons are independent the correlation function is one which means if I plot the probability or calculate the probability of having two photons at the same time, normalize to what I would expect from the average rates, then there's no, basically if the first photon is detected on this detector there's no knowledge that you can gain about when the second photon will arrive for laser light. These are independent Poisson statistics. So we plot the two photon correlation function G2 shown here or the three photon correlation function G3 for three photon events and we normalize it to the rate that we expect for independent photons. And so what you see is that at short times the two photon correlations are enhanced and the three photon correlation function are enhanced even more. This plot shows it on one axis we show the separation between two detector clicks on the other axis between three detector clicks. One is when the photons are independent so if you're down here then basically one photon comes independently, a second independently, a third photon the correlation function is one as for laser light. But then when all three photons arrive at the same time you have a strong enhancement of events. So basically these lines here are two photon events. Two photons come together and the third photon comes much later and this region here are three photon events. And you can see there's a strong bunching, a strong enhancement of probability of three photon events. We can also measure the phase. Basically we measure the phase by looking at the phase of the light conditioned on these detection events and we also see a strong change of the two and the three photon phase when the photons are coming out close together. So when we model this we can for instance model the two photon bound state it has an enhanced probability at zero the two photons are together and this is the simple theoretical model from the Schrodinger equation. What I should like to point out this is like a delta if you think of your undergraduate quantum mechanics this is like a delta function potential it's a relatively short range potential compared to the size of the wave function. In other words, the potential is not very deep. We just have one bound state and as you can see the bound state is mostly outside the potential, right? Outside this blockade radius and correspondingly the phase shift is less than two pi. Can you manage more than one bound state? We are trying to, it would be very interesting. For that you see our phase is about this is in units of pi it's about 0.75 pi to get more than one bound state you would need to push the phase towards two pi and then it would be very interesting because then you would get resonance phenomena, right? Then by tuning things a little bit around you can change the phase. So we are working towards making the phase even larger. So far we have done only attractive interactions. You might just ask yourself well if I change the sign of the detuning I should change the sign of the phase and therefore I should change the sign of the interaction. That is true but when we flip the sign of the detuning it turns out we also invert the curvature. So that means we go from attractive interactions for positive mass particles to repulsive interactions for negative mass particles. That's actually more or less equivalent to always attractive interactions for negative mass particles. So in newer results we have now added a second leg like that, a second electromagnetic induced transparency that allows us to tune the mass separately from the interaction. So this was the original system. We have added another control laser to a ground state here and now we have two laser fields that allow us to separately control the mass and the interactions. And under these conditions we see instead of enhancement of the arrival of two photons we see now a suppression of the arrival of two photons. The G2 function, the correlation function suppressed at the middle and the photons are pushed out. There's more probability to see them at the distance. These are initial results. We would like to see whether we can make a crystal of light. One of the dreams is to make strong interacting particles that act like a tongs gas, that act like a crystal of light. So here you see the same thing in a plot, arrival time between two photons, arrival time between the other two photons and now you see excluded regions where the photons push each other apart. The interaction is exponential, right? What is it? You should think of it as a delta function potential. So either an attractive delta function potential or under these conditions as a repulsive delta function potential. And then you get a corresponding physics of it. So maybe as a disclaimer, what does it mean are the photons bounded? How can we see it after all they're only bound while traveling in the medium? So what happens is the incident photons have a correlation function of one for laser light. It's Poissonian distribution, meaning you gain no information by detecting one photon when the second photon will arrive. Then while they travel in the medium, let's say that they form the bound states, while they travel in the medium, they are really bound to each other, right? And when they come out of the medium, they are of course no longer interacting for one another, but because vacuum has no dispersion, they keep the shape that they had in the medium. That's why we can detect them in vacuum. They are no longer bound to each other but they just keep exactly the shape that they had while interacting in the medium. So there is really the polaritons of the quasi-particles that are interacting. Okay, so this was about photons. Now I would like to switch subject. We will still keep the Rydberg atoms and the Rydberg atom interactions and the blockade effect, but now use them not to induce interactions between photons but between atoms. And in particular, I would like to tell you about a system where we can deterministically trap any number of atoms up to 51 now, arrange them in a line in any pattern that we want essentially and make them interact. So this is what the system now looks like. We have a laser beam. This will be now a trapping beam. We use an optical force to trap atoms that will be laser-cooled in the system. We send it through a so-called acoustic deflector to make a series of traps. Only three are shown here, but think of it up to 100 traps or so we can make. We then image these traps onto a vacuum cell where we prepare laser-cooled atoms. And then what happens is we can either trap zero or one atoms inside each of these traps. That it is possible to trap a single atom and see it inside tightly focused trap has been known for about 10 years. This focus is about one micron and you see simply the atoms by the light that it emits while it's being laser-cooled. The new thing about this is that up to now you could only or even this experiment we can only trap an atom with 50% roughly probability. So sometimes the atom is there, sometimes the atom is not there. So if you use this to make a pattern of atoms you will have a random pattern each time you try to experiment which makes things difficult. The new part about this is that what we do now is we make this pattern. We look at which atoms, which traps are filled by an atom and then in real time within a few 10 milliseconds we rearrange the pattern so that we only use the traps that are filled and remove all the traps that are empty. And we can rearrange the pattern then into an arbitrary pattern. So this will show pictures. I will show you a movie as well. This down here is the array of traps. This is about 100 traps. Each of this is a real image of the laser beams. Each spot here is about one micrometer. There's about 100 traps. The things in green are no cartoons. These are real pictures of atoms in any given shot. So this is the signal to noise that you see when you image each of these atoms onto a CCD camera in about 40 milliseconds of imaging time. So you see here for instance the first four traps are filled then I don't know three traps are empty another three are filled and so on. This is just a random pattern in one shot that we get. So what we do is we look at this picture with a computer. We decide which traps are empty. We turn off those laser beams and then we move the other laser beams that contain the atoms that have the traps that have an atom over to for instance make this pattern. So this shows an example. One time we could make this pattern because on average we had a little bit more than on average atom straps. The next time we could make this pattern, this pattern and so on. But what you can see is that if we are interested in sizes shorter than this minimum length say up to here then we can essentially terministically prepare an array which contains one atom everywhere with high probability. Instead of making a linear array, we can also make pairs of atoms like so many pairs or many, I don't know, these are 10 atoms at a spot so we can so far in 1D make an arbitrary pattern. So this shows you how long we would have to wait if we didn't have this feedback. So for instance if you wanted to make 20 traps and we just had random filling of the traps we would need to take like an hour to get one shot. But with this feedback we can make something like 70 or 60 atoms within a second with success. So let me show you a movie for this. What you will see is slightly slowed down by factor two or so is a real image of atoms. Each of these things is an atom. You can see traps containing an atom and then traps that are empty as these spots. And I will show it repeatedly as we load the atoms and then rearrange them into a pattern on the left how it's mostly deterministic. So you can see sometimes there's a hole somewhere, this was a perfect shot, this one had a hole in it because we have a small probability of losing an atom while we transport them. But almost all the time and in any case we can tell we can make a pattern that contains all the atoms. So the first part of the sequence is we trap these atoms and then we rearrange them and the whole thing takes a few hundred milliseconds or so. Now what we can do with these atoms is we can try to excite them to the Rittberg state. So we turn on a laser beam, a two photon transition and we look at the probability of the atom arriving in the Rittberg state. This is the single atom probability. Same scheme as before we go with a two photon transition from ground to Rittberg state but in this case we're not interested in the photons so both lasers are pretty strong. And so what we see for a single atom if we arrange the traps like this we see beautiful Rabi flopping, right? This is just Rabi flopping between ground and excited state. Now if we arrange a pattern where two atoms are close together, closer than a blockade radius, then it's impossible, as I will show a little bit better later to excite both atoms to the Rittberg state because in this state the atoms repel each other so strongly that the transition frequency is far, far detuned, several gigahertz. So what you can excite is only one Rittberg atom. So now you see the atom goes, even though we have two atoms in this case, one of the atoms goes to the Rittberg state but it does so at an enhanced frequency root two. The root two is understood as you have two atoms, you're trying to put one in the Rittberg state, you expect that on average in a rate limit to go twice as fast, right? One of the two atoms can go to the Rittberg state and so if you think of it as amplitude it's a root two enhancement. If we take three atoms within the blockade radius we have this root three enhancement but only one of the atoms goes to the excited state. So this shows that the collective interactions are working strongly and are controlled very well. So now what we can do is the following, we can start with a laser beam, this is a detuning axis, this is where the resonance would be. So these previous experiments basically we sit with the laser on resonance and we suddenly turn it on and we look what happens to the atoms individually. In this case what we do is now we start with a laser beam over here, detuned below resonance, we turn it on so nothing happens initially and then we sweep adiabatically the laser beam through resonance from the left detuning to the right detuning and this axis shows you how far the atoms are spaced. We make a long chain of atoms, here a few 10, like 10 atoms but we can go up to 50 but we can space them either so that they are far apart from one another or the two atoms are within the blockade radius or three or four atoms. So the simplest one is where the atom distance is smaller than the blockade radius but the two atom distance, the next neighbor distance is larger than the blockade radius. So basically only neighboring atoms influence each other the second atom to lowest order does nothing. So what do you expect as you sweep the laser? Well if you for a single atom you sweep a laser through resonance you expect what is called an adiabatic transfer. You expect the atom to go from the ground state all the way to the Rydberg state. That's what would happen for a single atom. Now when you do that for two atoms you expect also Rydberg atom to be excited but you know that out of two atoms only one can be excited. So it turns out that the lowest order state is one where the edge atom gets excited, the next atom stays in the ground state, the third atom gets excited and so on. It's effectively an anti-ferromagnetic interaction. And the outer ones are somehow favored because they have only one neighbor. So they are more likely to go into the excited state than the middle atom so that's the ground state. So as you adiabatically sweep from here to here you go from a disordered phase, all atoms are in the ground state nothing interesting is happening adiabatically to an ordered state of an anti-ferromagnetic order. And so you can see in this picture this is as the detuning is swept how every second atom in yellow is excited to the excited state. So we induce anti-ferromagnetic order here. If we place the atoms close together so that the blockade radius is two atoms but then we have this Z-free order. Basically one atom excited, two atoms in the ground state, next atom excited and so on. As you see here and we can even go to the Z-4 order. Mm-hmm. As we think velocity. Yes, we do that. So there are limitations. The main limitation is the probability to lose an atom while we do this. So the lifetime of the Rydberg state is 100 microseconds. However to do things cleanly we need to turn off the trap because the trap shifts the levels around. So what we do is we prepare the atoms, we briefly turn off the trap so we have about 10 microseconds to do the sweep. Yes, yes we have done that so we have a new paper. I don't have slides here but we are basically studying the Kibelsurik mechanism. We are looking at as a function of sweep speed. What is the number or the fraction of defects in the system? You are exactly right, you expect defects, you expect not always to be perfect. But then I assume you don't have temperature in this game, right? There is no temperature. There is no temperature. The atoms are essentially cold. There are errors, for instance, from finite sweep speed. However, when we find we can't model them thermally. You know, they are not described by temperature in the system. And the atomic temperature plays very little role. So this shows you. Now for 50, so this was done with 11 atoms here. We went all the way to 51 atoms. The reason why we use always an odd number is for an odd number, the ground state is unique. The two outer atoms are excited and then everything else arranges itself. And if you use an even number, there are two degenerate ground states. It's very interesting physics but it's more complicated. We'll do that in the future so so far we have done only odd atoms. So here we went to a system with about 50, not about exactly 51 atoms. So what is shown here now is in color is the correlations between neighboring atoms. Red means they are correlated, blue means they are anti-correlated. So you can see that there's an anti-ferromagnetic pattern. You know, there are errors in the system but in this system of about six, the correlation length is about six or so on average. We can do a little bit better than that now but you can see that throughout the crystal the atoms are correlated with their neighbors over a certain length scale. So this is showing the average populations. So you see that near the edge we are dominated by edge effects. The outer Rydberg atom is always excited but then in the middle we have really a bulk situation that for at least for the large change that knows nothing about the edge effects. On average, this atom is half the time in the ground state or in the Rydberg state but there as you can see from this plot still very strongly correlated in this region. You can see that for shorter crystal like 13 we are dominated by edge effects and so we are not getting kind of the bulk phase of the system. So we can ask ourselves how likely is it to reach the ground state, right? For 51 atoms let's say or for any atom number how often are we making the true ground state the anti-ferromagnetic state without any boundaries, without any dislocations in the system. And so this shows the ground state probably logarithmically as a function of system size up to 51 atom. So in these experiments that were published we could make the ground state about 1% of the time. If it doesn't impress you I would like to point out that for 51 atoms you have 10 to the 15 quantum states. Okay so whatever that is 1 million billion quantum states but we reach the ground state of the system. In newer experiments we can reach the ground state for the same system size about 10% of the time. Also here are the statistics of the number of occurrences. So for instance looking at how many states are available so we can see that we often get some states with one defect, two defects and so on but the ground state is reached clearly the most often. The ground state is special here, right? It's reached 20 times out of, I forget now, 2000 or so trials. So the ground state is reached much more often than any other state, one state that has these locations. So this shows the same thing before. This is 2013 results from the iron trap experiments where they saw a much larger decay of system size. And as I said we've pushed this up to about 10% at the moment. So the upshot is we think we can control in the near future maybe 102, 200 qubit states in the system. Here are when we prepare crystals at finite speed. Then we have these arrows, right? So in this case we have two ground state atoms next to each other. In this case we have sometimes two red book atoms next to each other. So there are these dislocations which have to do with the fact that when you're going finite speed you're not fully adiabatic and you prepare sometimes these excited states. And we see also the quantum phase transition occurring. So the domain wall density as we detune goes down and near this critical point, near the phase transition point there's a large variance because various states are being occupied if we stop at this point. So the crystal is not perfect, it contains domain walls, some due to imperfect state detection and typically our domain length at the moment is seven to eight atoms in the system. This is all great. What we did next came to us as a surprise. So what we did is the following. We prepared adiabatically as well as we can this anti-ferromagnetic state and then we quickly quench the system by jumping onto resonance, jumping the detuning very quickly onto resonance. And what we expected in the system is that all hell breaks loose, right? You have all these multitude of states so we wanted to look for thermalization. We're curious how fast does the system thermalize. However, what we saw when we did this sudden quench so we adiabatically ramped to the anti-ferromagnet here shown for a system of 11 particles and then we suddenly quenched onto resonance and we expected just a decay. But what we instead saw are these long-lived oscillations between this anti-ferromagnetic state and shifted an anti-ferromagnetic state by one. This surprised us because here there are many, many states available to the system. But the system starts to oscillate between one anti-ferromagnetic state and the other. It's not as easy as you think because as I told you before, neighboring Rydberg atoms cannot be excited. So to go from one anti-ferromagnetic crystal to the shifted one, you have to first bring the atoms back to the ground state and then excite the other crystal, but somehow in the process you have to retain memory that you had this other pattern before. So, but nonetheless, we think it's not mean field. As far as we understand, it cannot be described by mean field, right? Sorry, field that couples the atoms. So it could be, so beyond the next neighbor, there's a little bit of second next neighbor interactions and maybe that's how the system retains its memory, right? The other way how the system is written because of the oscillation, that's possible. The other way to describe is that you're going off resonantly through a state that has Rydberg excitations in it, but you're not really populating it, right? But that's just how the system behaves or retains memory of how to oscillate. It's a very good question, it's still open. The simple model that we can do is you can imagine that you have like these spin singlets where basic one atom is in the ground state and the other in the Rydberg state and then they effectively interact with one another. If you make that model, you would predict oscillations, but your predicted oscillation frequency would be root two of the Rabi frequency. That's however not what we expect. We observe a factor of 1.60 times the Rabi frequency instead of root two. So there's some physics correct but this is not the correct model. These scientists here led by Zoran Papich, they have a model for this. They call it quantum scars. In analogy to, in certain quantum chaotic systems, there are certain trajectories that are somehow more often occupied than others. And so they apply this concept to the many body system and they predict correctly in this model the frequency. So basically the idea is roughly speaking that out of this forest of quantum states that are available to the system, some quantum states are preferred or bunch up at certain energies which leads to these oscillations. And we can see here that the oscillation is remarkably similar. One of them is for just nine atoms, the red dots, and the other one is for 51 atoms. And basically you see the same vibration frequency, the same behavior, it's dominated by the bulk. And this is by a fit by this frequency as derived from this paper. So this is all about quantum simulation. We hope in the future to try adiabatic quantum computation simply by following the ground state in a more complicated system that we don't know what the ground state should be, maybe very briefly for the end to just tell you about two atom quantum gates. We have done initial experiments on that. So here we just take pairs of atoms and we look at them. So we can do single qubit Rabi flopping. This is the quality of the Rabi flopping that we had before. After we improved our laser line width, we can see very, very long lived tens of microseconds, many, many oscillations of a single qubit. So basically it turns out that laser line width matters. So we bought a very high finesse cavity and we filtered the laser beam through this cavity and locked it very tightly to this cavity. So instead of a few 10 kilohertz, the laser line width is now below, it's just maybe a few kilohertz. And that dominated partly. The remaining effect we think is the Doppler effect. The atoms are cold at a few 10 microkevin but not absolutely cold. And therefore as the atoms are moving, there's a small phase in the interact, the interaction strength is changing and there's also small phase effects, Doppler broadening, which limits this. Do you try to actually cool them further, like I don't know, big, why is this doing? Sure. I mean, we have done before Rahman's side, then cooling all the way to the quantum ground state and then that way you should remove all Doppler effects. We think we can do that just adding another layer of complexity but we'll do that in the future. So the moment we work with tens of microkevin but yes, we can cool them to the ground state. At least we have done before for individual traps and atoms so that should be possible. Here's showing again the same thing now, the decay when we basically use their magnetic fields and other effects, essentially it's a spin echo technique and if you do spin echo, then we can have very, very clean oscillations persisting for a long time for a single qubit. This one shows the two qubit quantum gates. So now this is oscillations with two atoms close together. This is the population of the ground state, the atoms start in the ground state, then one atom is in the ground state or this is the probability of having both atoms in the ground state, it starts at one and then goes to zero. This is the probability of having one of the atoms in the excited state and this is the probability of having both atoms in the excited state. If they weren't interact, the oscillation would go to here but because of the Rittberg blockade, you can see that we have essentially never two atoms in the Rittberg state and this allows the quantum gate. So basically we have also very clean oscillations for two qubit quantum gates in this system. And this shows just kind of a density matrix representation of the quantum gate. When we stop at the correct time to make a belt state, we can make this belt state of about 97% fidelity. And I should say we haven't really tried very hard for this. We have just improved the laser line so far, none of the ground state cooling and the other tricks that we think we can do. So we are pretty hopeful that one can go well beyond 99% in the quantum gate fidelity. So with that, this is all the results that I wanted to show to you. What is the future? We think that within the next one to two years, we can go somewhere between 100 and maybe 500 qubits. Beyond 1000 qubits, we are simply limited by the laser power. So we have a few watt laser for trapping the atoms. Then we need to talk about two or three or four lasers just because we are making so many traps. But I think up to a few hundred qubits, we should be fine with just kind of extending this technique a little bit. We can also go to a 2D arrangement. We are preparing that for now. So instead of having one delinear change, we can have two delinear chains. I would like to point out, if we can push the correlation length a little bit longer from seven to eight, what we have now to about 10. And then we go to a 10 by 10 qubit, 100 qubit system, we should be able to prepare perfect quantum state of 100 qubits where everything is quantum coherent. And what we are thinking about very hard now, and this would be an excellent place to solicit input from the theoretical communities, we would like to understand which, ideally NP-hard or NP-complete problems map onto this quantum simulator. Can we simulate them in a reasonable way? And one system that Michel Lucan and his theory poster has looked at is the so-called MIS problem. It's a maximally independent set problem. It's a problem where essentially you have vertices, you define connections between them, and you ask yourself, what's the maximum number of vertices that you can have that are not connected to any other vertex? And it turns out that's an NP-hard problem. And that, as you can see, maps quite naturally onto a Rydberg system because not exciting to connected vertices is like not exciting to Rydberg atoms that are closer together than a certain distance. But we are looking actively into other problems that map well onto this quantum system. And with this, I would like to thank you for your attention. Yeah, Kolokyuk, question about? Yeah, I'm interested in the experimental part. How do you split the laser into those many tracks? Do you use adaptive optics or something like that? No, we use a so-called acoustic deflector. It's similar to an acoustic modulator. So basically, you send a laser beam in here, you send various frequency sound waves into it, and each sound wave frequency corresponds to one of those traps. So that's how we do it. And with this, we can make about 100, 150 traps maybe. And then by controlling these radio frequencies that go into this crystal, we can turn off traps or by changing the frequency, we can move traps. Basically, the deflection angle is given by the frequency or the modulation frequency of the laser. So if you make the modulation frequency smaller, the beam moves in. And that's how we move the beams in real time. But certainly, there are other possibilities, micro mirrors, SLMs, and so on to do this. We like this approach because it's fast. So the other ways, adaptive optics and so on are relatively slow in changing things. With this, we can change things on the few microsecond timescale. But very good question. So that's how we make the traps. So you make also different patterns so that you are strict to that configuration. So if you do this acoustic deflector, then you can make an arbitrary pattern of traps in one line. The only limitations are your resolution. So you can make maybe 150 traps. And if you bring two traps closer together than one to two microns, then the interference effect starts heating the atoms. So we can't go closer than maybe two to three microns. If you combine two such systems, one in this plane and the other one in the other plane, then you can make in principle arbitrary 2D patterns. And the group of Antoine Brouvet has a beautiful experience with 2D systems as well. Can you realize frustrated lattices? I mean, just say triangular and deferral magnets. Definitely. I mean, because you can arrange these patterns any way you want, essentially. We think we can do triangular lattices, hexagonal lattices. And that would be a very interesting future direction to look at the frustration in these systems. Single photons or the laser source? In the first part of the? No, no, no. Here, when you're trying to just construct that traps, are you using? No, so these are just laser beams. And they are just coherent laser beams. And the photon statistics for this part are completely irrelevant. These are very strong laser beams. And we're not manipulating the light in any way. So in the second part of the talk, we're just using the light to manipulate the atoms and do the Rabi flopping. But it has nothing to do with the statistics of the photons. You can think here completely of the laser beams as classical beams. Could you perhaps compare this design with other designs of quantum emulators as a perspective for the future? Yes, so essentially, in my mind, there are three competing systems. The longest one is the ion traps, obviously, where quantum gates have been demonstrated in the 1990s between two ions, the superconducting qubits. And this one is a newcomer. There's also photonic quantum computation. In my view, it has no future to be scalable up because of photon losses. I don't see how it can be scaled. The superconducting qubits are very nice because they are the fastest, although this also, in principle, has potential to go to gigahertz frequencies. The ion traps are leading in terms of the quality of the quantum gates. They can make almost 99.99% quantum gates in this system. But they have also the longest history. What I like about this system is it's very easy for us to scale up to a large number of atoms because these are neutral atoms that are essentially not interacting with one another when we trap them. That's not the same in ion traps, where if you load bigger and bigger traps, you have more and more collective modes and problems. Similarly, in superconducting qubits, you have problems even with just classical wiring if you have many, many traps on a chip and crosstalk. So in terms of competitivity, I think we will be the first one to go to more than 100 qubits. I'm pretty sure in a controlled system. We have to see how well the system scales in terms of fidelity. So I would say at the moment, each of these systems has their own advantages, but we are hoping to beat Google to it. So I have a follow-up-provoked question. I mean, if you had to build this experiment now again, instead of rubidium, would you use strontium or yterbium so that you can maybe trap the Rydberg? So avoid losses and at the same time use single photon to excite the Rydberg? Very good question. So today, no. In five years, I think yes. So the complexity added by trapping strontium or yterbium, where you need two photon transitions and other lasers, at the moment, I think will be too costly. Because we are not at the limit where coherence time limits us. But in five years, I think we will switch most likely to a different atom when we understand better what is going on. At the moment, the goal is simply just to give you an idea, nobody can simulate anything, kind of brute force, more than 20 or 25 qubits. So 51 qubits is far beyond what can be simulated. And even these oscillations that we see, we can make models and see if they agree, but we can't really understand the system by fully simulating it. So what interests us is how things scale. That's the name of the game. You try to make the ground statement, you need to show how it will scale to higher and higher number. And for that, you need at least a decade of numbers. So we think that going from 20 to 200 will teach us something about the scaling and whether quantum computing works or not and how to overcome certain limits. But after that, I think it's likely we will switch to a different neutral atom. I agree with you, the earth alkali atoms are probably much better. Right now we can, if you want, we can maybe do 50 within the 10 microseconds or so. We can maybe do 50 gates or something like that. But at some point, this will be a limitation. There are certainly much better atoms for that. It's a very good question. Would you also engineer some topologically protected states We are dreaming about it and thinking about it, how to do that. It should be possible in principle because you have arbitrary arrangements. We are not understanding at the moment what advantages it would offer and how to do it. We don't want at the moment to make a fully digital universal quantum computer because we think that overhead in terms of error correction and number of physical qubits that you need is too large. We would like to explore the route to go either via adiabatic quantum computation or some other approximate method. But it's a very interesting area, I think, to think about topological state and otherwise protected states. Back to the early part of the talk. Two photons bind together through this mechanism. What would happen? Could you do it with three photons? Could you three, four? What would happen? So we did three photons. I didn't have much time to show. We see that three photons are more strongly bound than two photons and they are almost four times more strongly bound. And you get that just from having two delta function potentials. And then we teach that, right? You get a decay length that is twice as short or a binding energy that is four times as large. So we have done three photons. You can ask yourself when you have four or five photons you might get into the regime where now you have multiple bound states and things get complicated. So I think to four or five photons you can do it after that because of finite detection efficiency is just hard to detect the state. And to be sure that you had this, I mean, even now when we detect two photons sometimes we have three photons stated. We mistake for a two photon state because we didn't detect the third photon because of finite quantum efficiency. So you always have to keep the rates low not to confuse by other states. So up to four or five you can see things. Another way to think about these is as quantum solitons. You can think of these as solitons but they are quantum mechanical in the sense that the size of the soliton, the bound state depends on the photon number which is not what we have for classical solitons, right? The small change in photon number or intensity doesn't matter. Here you have a quantum soliton where each of these states. So up to a few it will work. We are now at the moment very excited to see whether we can have a crystal of light. Can you make a system where the photons crystallize? And you know, in the repulsive case. In the repulsive case. More questions or comments? I think we're basically through so we thank Professor Wurlitz again.