 All right, thanks Andrea for the introduction. I noticed that there was a slight additivity problem in the times, because it seems that there are two talks within the next hour, and one is 60 minutes long, and one is 20 minutes long, but I can talk for 60 minutes. OK, I wasn't sure. OK, if you get bored, you just boom me off the stage. So this second talk is dedicated actually to the Rydberg gases that I was kind of introducing yesterday, and that way elaborated on today by Robert Löw. So what I'm going to talk about is many body physics today, and in particular the out of equilibrium dynamics of long range interacting Rydberg gases. So just like yesterday, this is the slide, local hero Robin Hood. There's a bunch of names here highlighted in red, as I hope I haven't forgotten anyone. Those are all the people that in one way or another have contributed to what I'm going to talk about in the next, yeah, 50 minutes or so. Right, so this is the outline of the talk. As I said, focus today will be on many body physics, and what I want to advertise in the first part of the talk is something we have been pursuing, and I'm not going to know for quite a while, that relates to soft matter and quotes type of physics that you can conduct with excited atoms. I want to convince you at the beginning, so that dynamics is more than static. So this allows me to make a point in the sense that I hopefully will be able to convey that this somehow makes sense to think of soft matter like problems within the context of Rydberg gases. I will talk a bit again about atomic physics, just basically doing a recap of what Robert Löw has been talking about earlier today. And I will talk about noisy systems, and in particular how noise effectuates particular kind of many body dynamics that can be described by so-called kinetic constraints. I will talk about this in detail what that means. And then I focus on actual dynamics of the system. I will explore how a gas of driven Rydberg systems that is excited on resonance actually evolves, and we will find some interesting behavior, at least I believe so, that this is also interesting to you. Namely, we will find that there is a self-similar evolution of the system. And this, in fact, the evolution can be described by a so-called scaling solution. And then in the third part of the talk, let's see how far we get, I'll discuss a non-equilibrium phase transition that you potentially can also study with the help of excited atoms. I will also show you some experimental results, and I will particularly highlight that these Rydberg atoms allow you also, in principle at least, to explore really the competition between noise and quantum effects in these systems. And last but not least, I want to connect to a rework problem where we apply the methodology that is being developed here in the context of Rydberg gases to something that is actually relevant, for instance, in medical imaging, namely nuclear magnetic resonance imaging, and I will show you how one can actually exploit and understand non-equilibrium methods that go under the name dynamical nuclear polarization through this framework that is being established here in the context of Rydberg atoms. All right, so first punchline is dynamics is more than static. So what do I mean by that? So let's make it simple. Let's just consider a system formed by our beloved spin one-half particles, and just make it very easy. Let's suppose we have a classical evolution in the sense that transitions between those two states are described by rates. One rate is for going up, one rate is for going down, and the system is even simple, although it's many bodies non-interacting. That means spins flip independently so you wouldn't expect anything fancy to happen. So just keep this in mind for a second, and then I want to contrast this now with a so-called kinetically constrained system. That means spins can only flip from up to down provided that, for instance, the neighbor, the left neighbor, is in the excited state. So now you will think clearly, these systems are completely different. One is non-interacting, and one is interacting. And of course now, if you look at an ensemble of those spins, you find here black, this axis is space, so this is just a line, and this is time, and you see spin-ups are represented by a dark dot, spin-downs are represented by a white dot, and you see indeed these systems behave differently, so far so good. You see here, as you expected, and all the spins flip independently, so there's just a bunch of noise, and you see the system relaxes very quickly, whereas in this case, all to the kinetic constraint that allows spin flips only when the configuration of the neighborhood is in a specific state, you see a highly intricate dynamics, which is clearly correlated, and the funny thing now is that if you analyze those models and you look at the stationary state, the equilibrium state of those problems, they are in fact identical. So if you go to long enough times and you take a slice here, and you take a slice there, you would think, okay, they come from the same non-interacting problem, and this is what makes the point. The dynamics is clearly more than statics, and although the stationary state of these systems is trivial, you find that in fact, in this kinetically constrained system, you have a highly intricate dynamics with which the system approaches the stationary state, and those models are used, for instance, in the context of glass physics, because they are idealized representations of substances that undergo slow relaxation. All right, so now how does this link to Rydberg atoms? So the question is now, look, I have represented, I have shown you an example of this kinetic constraint, this looks a bit artificial. The question is now, can these kinetic constraints actually emerge rather naturally in one or another system? Of course, well, I wouldn't be standing here and talking about this, if this wasn't the case, for instance, for these cold gases that are made up of Rydberg-excited atoms. So what I will show you in this talk is in fact that these systems allow you to study features like slow and glassy relaxation, meaning, okay, emergence of new time scales, observation of transient phenomena, and meta-stability, for instance, and will also allow the realization of systems with non-equilibrium stationary states. They allow you to study interesting things like collective dynamical features, such as nucleation, aggregation, and growth, and an interesting aspect of this, I think, is that it allows the controlled inclusion of quantum effects. On top of this classical noise that gives rise to the spin flips, you can also include coherent fields that make the flips undergo coherent evolution, for instance, between the up and the down state. And this degree of quantumness is in fact controllable. But also what you can do is you can use those Rydberg states to kind of engineer exotic interactions. I don't really want to discuss about this. I'll give you a teaser at the very end of the talk. Let's see how far we get. All right, atoms and Rydberg states, this is the overview again. So you look at our periodic table. Atoms we are going to talk about are located here in the first column of the periodic table. They all have a very simple level structure. Robert talked about this earlier, so you can understand the levels, basically within the same framework that you're also using to understand the spectrum of a hydrogen atom. Basically, the atoms are labeled by quantum number N, which is the principle of quantum number. In the case of hydrogen, they form highly degenerate multiplets. This is not the case for these heavier atoms like rubidium or lithium or potassium. We see that these famous Rydberg formula is actually modified through the inclusion of a quantum defect, which kicks in for low angular momentum states, and this means in practice that states, like for instance, an electronic S state, which zero orbital angular momentum is split away from this highly degenerate manifold of states, which is good because you now can spectroscopically actually address them, which would be a bit more difficult if they were hiding within this manifold and mixing, for instance, under the influence of fields with the rest of the states. So just to show you, okay, how a Rydberg atom looks like, well, this is the radial wave function for the rubidium 39 S state, and you see there's a large concentration of the electronic density far away from the origin where the nucleus sits. And therefore you can think of those atoms as forming some kind of dipolar structure with a positively charged core and a negatively charged valence electron, and the distance here can be on the order of one micrometer, as Robert also said. So in a nutshell, hydrogen-like, simple level structure, long lifetime, this is important to do coherent physics, and there's a large displacement between the charges, which gives rise to strong interactions as we discussed today and for the case of S states. It's just like for the polar molecules that I have been discussing yesterday, you end up due to the fact that those states are really spherically symmetric with the van der Waals interaction, typically which scales as the six power of the interparticle distance. Just this is exactly also the same for ground state atoms as Robert emphasized today, but the interesting thing is that this interaction coefficient scales with the 11th power of the principal quantum number. So in this, therefore it becomes extremely large as compared to ground state atoms. So typical interaction strengths are 10 orders of magnitude larger than what you would expect typically for atoms in the electronic ground state. All right, so now what we want to do in order to do some theoretical many body physics, we want to simplify the description of the system. Clearly you don't want to carry with you all this complication of this level structure, because still though it's simple, there are too many levels to do many body physics in a useful fashion. So what we do is we just focus on the essentials. Okay, we know that we typically find the atoms in the ground state. This is of course also not a single state, but let's for the sake of simplicity assume that and we couple this state with an excitation laser via intermediate states to this NS state. And in practice you ignore also things like the fact that you need two lasers for that. You just say, okay, this becomes a single laser. We isolate those two near resonant levels and end up with our fictitious spin one-half particle just along the lines as I was showing you yesterday. So now the excitation laser is typically parameterized by a Rabi frequency, which corresponds to the electric heat strength of the laser and the detuning, which is basically the difference between the laser frequency and the frequency corresponding to the atomic transition. And now if you wanted to write down a Hamiltonian for the single atom excitation you would end up with this expression and okay, this is also what I showed you yesterday. You have the sigma x and this n, which is the projector on the excited state. So this is the single body physics. And now we want to do many body. So what do we do? So the convenient way of formulating the many body dynamics here is in the framework of a quantum master equation I also showed you examples of that yesterday in the context of photon mediated dipole-dipole interaction. So the master equation looks like this. So we have here a coherent part that describes the evolution of the density matrix of the atomic ensemble. And you see this is governed by the Hamiltonian, by the quantum Hamiltonian. Here you recognize the bits that I just showed you before. So the Rabi frequency of the laser, this is the detuning, the frequency mismatch and you just promote this to the many body level by summing over k, yeah, k is the atomic label. And then of course here you have the interaction potential which is a density-density interaction. I showed you also examples of that yesterday which is parametrized through these coefficients here which obey this Van der Waals one over r to the six power long. So this is the coherent part. So now the dissipative part is described by this structure. It has this Linn platform and it's actually diagonal. So I mean there are no collective effects. For reasons one can motivate and explain. So actually this is really well tested. So there's a good, very good agreement between the theoretical description given by this equation and actual experiments. So and the two processes that we want to consider here, the ones that are dominant I would say in the context of these Rydberg gases are given by either defacing. Yeah, I focus on this big time in the next few slides. So I don't want to discuss this now. And decay, yeah, which for simplicity we say is happening from the Rydberg state directly to the electronic rounds. This is of course an approximation in practice. I mean there are many states in between but for the sake of extracting simple physics this is actually a fairly good description. All right, so those are the two processes that we for instance are able to capture within this formalism. And what I want to do is I want to focus on this defacing in the following. Which quickly removes any sort of quantum coherence. In a second I show to you in more detail what I mean between the two states of this atom we are considering. Yes, exactly, yeah, or for instance a laser that has phase noise. So as I showed you, we can describe the evolution of our many body system in terms of a quantum master equation which is notoriously difficult to solve. We can nicely formulate it and it's a very transparent representation of the physics there, but in the end you have to solve it. Which is almost always difficult if you want to do this for many bodies. And what you want to do is you want to assume and exploit in the following separation of timescales where we just decompose the evolution of our master equation into a fast part and into a slow part. So the fast part we say is just given by this diagonal bits of the interaction energy. So it's the de-tuning and the interaction energy which is a density-density interaction is sort of say a classical interaction energy. Together with the defacing noise. And defacing noise, what does it do? I said before, if let's say you have a quantum superposition of the down state and the up state, it will very quickly destroy this quantum superposition and convert it into a mixture of atoms pointing up plus atoms pointing down, a classical mixture and the typical time at which this is taking place is one over gamma, one over this defacing rate. And in the end, once this defacing noise has acted only classical configurations remain. So now, this is the fast dynamics and clearly, okay, this is all diagonal. So something needs to happen on top of this and you might have guessed it already. So we take the laser because the laser is now driving dynamics here. The laser wants to flip atoms from up to down and from down to up. And we assume that this is the smallest scale in the problem for the moment. So now what we can do is we can adiabatically eliminate the fast dynamics from the problem. So what does it mean? So we now look in Liouville space in which this state vector lives of our system. So we focus here on this corner where we are only having classical configurations because those are the configurations that will very quickly emerge due to this defacing which converts all quantum superpositions to classical states. So then we have some states with large rates that decay quickly and that rotate quickly because they are strongly interacting and they have a large detuning. And then those classical configurations are connected to those states via the laser. So on this we consider weak and you see already the generic structure that pops up when you try to do some kind of second order perturbation theory. And this is what you do now. And once you do this second order perturbation theory which is not difficult, you end up with an effective equation of motion that looks like this. Okay, so let's analyze this. So if you now look at this part here, you find it just describes an ensemble of spins that flip up and down with the same rate. Here you see this immediately here. So if rho is for instances down, then the sigma x promotes the component of the density matrix to up and vice versa. So this bit effectuates just spin flips where the up rate is the same as the down rate. And the interesting thing is also that this bit which determines the dynamics of the system also tells you what the stationary state of the system will be within the approximations we have been employing. And you find that the stationary state is in fact trivial because by trivial I mean that it's proportional to the identity. All configurations appear with the same rate. You see if you replace rho by one, then you can see that sigma x times sigma x is one because it's a poly matrix. Mine is one is zero. So the stationary state of this dynamics is completely trivial. But of course I wouldn't tell you this if there wasn't a caveat here. And of course this kicks in because of this gamma of k which is now not a constant but instead an operator. And I will show you in detail in the second what a shape this assumes. But this operator is exactly this kind of kinetic constraint that I was advertising before. Because it somehow relates to the configuration of the surroundings of a given spin. And depending on the state of the configuration it will either favor or disfavor. It's state change of the spin we are considering. All right? Okay, so now let's analyze the specific structure of these gammas, of these kinetic constraints that we can actually expect. So and we now distinguish between two cases. One case is resonant excitation and one case is off-resonant excitation. When I say resonant excitation and I mean that this delta, this detuning is actually put to zero. And if we put this delta to zero and we just consider a very simple situation where we have only one excited atom at the origin which we assume to be fixed, then we can now ask, okay, it's just coming out of the second-order perturbation calculation. What is actually the rate for another spin here and its vicinity to flip? And you see, this is the rate. And as a parameter enters this kind of capital R which is sort of say the blockade length or blockade radius, also very much related to what Robert said, but now in the incoherent setting. So this sets a length scale which is determined by the interaction strength. And you see now all spins for which small r, so for which the displacement of the excited atom is larger than this big r, they will flip fast because the rate is one. And all spins that are close to this excited atom, they will undergo a very slow evolution. So they are more or less stuck or blockaded. So this is how this constraint works. So it makes dynamics far away from an excited atom fast and close to it slow. So then we can have a different situation where we start with an off-resonant excitation with a detuning that should be delta. So this small delta here. And we'll see actually there's only one slight modification. So in this bracket, there's now this small laser detuning coming in. And you see now what can happen is if this detuning is negative, there's a certain distance at which this bracket becomes zero and the rate becomes one, so it becomes large. So that means there is some kind of specific radius, so-called facilitation radius which is given by the detuning and the properly scaled unit. So that's why it's lower case delta and not the capital delta at which you favor frequent spin flips of an atom whereas everywhere else, the dynamics is slow. So I mean, we have studied this theoretically together with Robert, but please forgive me, the PISA group is having the nicer data, I think. So that's why I'm showing the data from the PISA group in the next slide. So just to say for comparison with the predictions that are coming out of this many body dynamics. Okay, so let's look at this curve. In fact, three curves, let's see what's plotted. So here is time. And here's the mean number of excitations. So we start in a state where all atoms are down so there are no excitations. And now let's look at the case where we have the first situation, this one here, where we are resonant with the laser with regards to the atomic transition. Then you see very quickly the rise of the excitation number and the levels of at some point. So this is the resonant case. Well, and in contrast to that, you can now look at the facilitated case. You see, it picks up much more slowly because there's only, at the beginning, there's no excitation happening. There's no excitation present in the system. So it means it takes a long while because the rate is slow. There's no facilitation possible since there's no excitation. So the first excitation requires some time to be created. But once this created, it's created so this picks up very quickly and further excitations are created. No, but you can estimate. You can more or less calculate it. I'll show you a theory curve in a second. So I will comment on that in a second. So then, alternatively, what you can do is you can now actually change the value of the tuning. So you can, instead of exciting the atoms with a negative detuning, which means it's the laser's off-resonant towards the blue, you can also go off-resonantly towards the red, which for short times shouldn't make a difference because there's anyway no facilitation at the very beginning because you need one excitation to set off this avalanche. So at very small times, the two situations should coincide which indeed they do. But you see in the green thing, unlike in the blue curve, so this off-resonant case doesn't show this kind of avalanche effect in the end so it just levits off at very low excitation numbers. So now you can do the theory and on purpose I don't show any explicit numbers here so there has to be some rescaling done here. Robert is smiling at me already because there are some intrinsic uncertainties in the experiment, like what's the actual atom number? So what's the actual Rabi frequency? There are also inhomogeneities, yeah. But what I want to stress is that, okay, the ballpark figures are okay, but what I want to stress is I now take an experiment, a numerical run for fixed parameters and now calculate those three curves with exactly the same parameters. So relative to each other, they are fixed. So there's no tweaking of these curves relative to each other. And now I can superimpose this and you see it more or less works. At least, okay, we understand to a relatively good degree what is happening in the experiment. Okay, so let's now move a bit beyond the experiment. And let's have a closer look at the resonant dynamics because in the experiment so far we have only measured the density. I want to motivate that it's maybe a bit more interesting to look actually at the system in a somewhat higher resolution where we really see the configurations evolving and we can also look at the correlations. So, and in fact, in theory, we can also tune now at will, so to say the parameters of the system, for instance, this blockade length that I discussed before. So, and what I'm showing now in this plot here on a log-log scale is the evolution of the density of excited atoms starting again from a state where there is no excitation for different values of the interaction. So the black curve here, so this is the black curve, is more or less the non-interacting case. So nothing much is happening, blockade parameter is one, whereas now the other curves show an increasing, the situation for an increasing interaction strength up to r equals eight. And what you see now is that when one atom is actually blocking eight atoms, you see a very striking power law dependence here in the growth of the density, which is actually interesting, I find. So initially we are uncorrelated, but at some point you have a correlated dynamics which gives rise to this power law behavior and the power you can also estimate in one D, it's one-thirteenth. And okay, this is increasing interaction strength. You see it becomes more and more pronounced, the larger the interaction becomes. Okay, so now let's look at the system, let's look at the microscopic configurations of the system. And now, okay, what you would expect now is what you also see. You see, expect that the density of the excitation increases as time passes. So low density, intermediate density, higher density. But now what you can do is you can actually rescale the magnification factor, so the zoom and factor that you use to observe these patches. And if you do it right, yeah, so then you can scale your magnification factor as a function of time such that the system actually is not evolving, that it is getting stuck in this kind of configuration. And this hints towards a similar evolution of the problem. So where time and space, so to say, don't evolve independently, but you have one variable combined of the two that is actually describing the evolution of the system. And we can now look at this in a bit more detail because one can actually extract this more or less analytically. So why do we observe the scale invariant relaxation? So first approximation or first consideration that leads to an approximation eventually is that the rate function that I show here again is actually highly peaked, so what does it mean? So if I have an excitation here and I have an excitation here, so the next excitation is most likely to happen in the middle of two already excited atoms, just because there the interaction strength is weakest. So, and then what I also said before is that you have an excitation and a de-excitation at the same rate. But now we start actually from a situation where all atoms are down. So at the beginning, the excitation, the creation of root work atoms will be far more important than the de-excitation of already existing root work atoms because they are far less root work atoms than actually atoms in the ground state. So you can actually remove this kind of de-excitation and everything becomes sort of deposition dynamics where you just deposit excitations within your system and never remove any. So, and you just ask, okay, is this approximation a good one? Well, okay, this is no rigorous answer but you can just run the simulation with and without this de-excitation process and you'll see then for this regime where you observe the power law behavior, both dynamics coincide so that makes you confident that you can actually neglect this removal process. So, and then with these considerations, you find that the dynamics of the system more or less proceed like this. So you have a certain density of excitations. So by the nature of these rate functions, they are more or less evenly spaced. And then it's most likely to create the next excitations right in between already existing ones and the next ones as well, right in between the already existing ones. So like this, the system you see already, a glimpse of it. The other system looks more or less the same at all times if you were just to zoom in further and further to compensate for this increase of density. All right, so let's formalize this a bit more. So this brings us to an analytical solution of this problem. So what we do is we take this rate function and we assume already that there are a number of atoms inside the system so that there's interaction. So because if you look at the two dilute situation where there are only few excitations, there's anyway no interaction. So in order for this correlated dynamics to kick in, there need to be interactions. And if this is the case, you can actually neglect this one here in the denominator of this function and you can approximate the rate function like this. And what you also assume now is that the distances are quantized. So that always powers of one, that they are always given by powers of one half because the motivation is that excitations appear predominantly in the middle of already excited atoms. So, and now think in terms of particle distance. So we start with distance one. So then you place one excitation in the middle. So this is now two distances of one half. Place another particle, this is two distance of one quarter and one distance of one half. And now what you can do is you can write down a rate equation for the distribution of these interparticle distances. It's just simple observing what this dynamics is and then reformulating it in this manner. And then you can go to a continuum limit by just expressing this in terms of derivatives here doing some Taylor expansion and you get a function like this. It's not really important what it looks like but this is an equation of motion for this distribution function of interparticle distances. We now went from a discrete variable N to a continuous variable X. And this we can actually, ah yeah, this is what I wanted to say. You find this funny power here, 13. So this is of course related to the power of the interaction. So whatever power alpha of the interaction appears in our system here at six, will manifest itself in this exponent. So now you can analyze this equation and you can find that actually it is solved by a function like this. So you see X and T space and time don't go independently but they enter as this kind of product. And this is exactly so to say this scale and variance that we have been observing in the numerics. So now if you take your numerics and you plot the distribution function as a function of time then you find for short times it's of course peaked at large interparticle distances and it's time evolves. It goes to smaller interparticle distances but now if you apply this rescaling you find that all these curves here fall nicely on top of one another. So this is the numerical data with the scaling and this confirms nicely that we indeed have to sell similar evolution. So the conclusion there from this is that even in this very simple setting I guess I don't know what you can object here but this is probably the most simple thing you can do in an experiment. Turn on the laser, a bad laser on resonance and see what happens here. Even in this surprisingly basic setup you find this highly correlated dynamics. Okay at short times of course excitations are created at random but then for longer times when the interparticle distances becomes comparable to this blockade radius you enter a regime in which you proceed with this kind of self-similar evolution towards stationarity. All right so and this also works in higher dimension I just want to show you this plot. And you see also in what sense this works it's only approximately true. So this is a real simulation but you see this kind of I'll leave it on for a moment. So we see the system with this kind of checkerboard pattern and you see it's not proceeding in a way that this checkerboard was preserved for all times but you see here these regions like here or here where you see okay this is more or less looking like the situation we started from. You again have this checkerboard pattern is a bit distorted but you see then this is serving as initial condition for the next time period. So you just zoom in and the system looks the same again and you wait again for a moment, zoom in and the system will look the same again so in 2D this works quite nicely but it's not to be understood in an exact sense it's more like you see it becomes you see also this square structures okay it becomes increasingly difficult to observe but okay if you do the analysis of the correlation functions you see really this self similarity persisting. All right, are there experimental signatures? I mean this is a bit challenging and I just go back to this PISA experiment and theoretically this curve should correspond to the situation I was just considering and then you can now superimpose this with the numerics and you see okay this black is curved, the dashed curve is the T to the one fifth power. It looks more or less okay but this is also a Lin Lin scale so I mean it's not really convincing so one would have to do a very carefully designed experiment I think too in order to observe this algebraic growth of the power so one fifth is what one would expect in three dimensions, okay so I still have time okay so this was everything I had to say for the moment concerning this resonant excitation but now let's look at the case in which we excite atoms of resonantly and before I come back to the Rydberg atoms I just want to have a small detail the detour and I want to introduce you to the contact process which is somewhat going on the different names game of life like it's in essence it's a cellular automaton which obeys a number of rules and I come to the rules and the second the important thing is that you have an order parameter that you think describes the physics of the system which is just given by the density of active site so each site can be either alive or dead or up or down or Rydberg or ground state use the notion that you like best and now let's consider the rules with which this process is taken place so one rule is just death or decay so you can remove an excitation from a site and then it's gone and another rule is branching so one excitation can spawn another excitation on an adjacent lattice site and what you find is when you look at this dynamics is that there's actually a unique absorbing state that means the state in which everything is dead will never come back to the living so because you need this facilitation and this branching in order to create density so once everything is decayed you will get stuck in this absorbing state so but now what you can do is you can of course let the decay fight against the branching and this is what's plotted here so branching, relative strength of the branching is increased into this direction and then you find if you exceed a critical value then you go from this inactive phase which is formed by this absorbing state which is non-fluxating to an active phase and what delimits those two phases is a second order phase transition which belongs to the directed percolation universality class so why is this interesting? Well, it's allegedly the simplest non-equilibrium universality class and although it's so simple we have hardly any experimental realization of this so there's one experiment in 2D conducted in a system that comprises liquid crystals don't ask me exactly about the details and then there's another nice work that appeared this year actually where I think they also looked at the evolution of the fluid through some porous medium so both of these systems realize in one way or another an instance of this directed percolation universality class so far in 3D there doesn't seem to be an implementation so now you can ask naturally how do we do with the Rootburg atoms is this feasible with these Rootburg atoms because clearly we have the branching so I advertised before this facilitated excitation this we have, decay we have as well, no problem and the problem here is that a unique absorbing state does not exist because I showed you already in these experimental curves before that even if you start in a situation where all atoms are de-excited at some point you will excite one due to off-resonant excitation processes exactly but you'll do it eventually and you cannot really inhibit this for real so this unique absorbing state does not exist fair enough and interactions also of course extend beyond nearest neighbor which is also an issue and still you can see okay what can we do and on top of this if in any real experiment you're of course not in the thermodynamic limit but you have finite sizes you have tens of perhaps hundreds of atoms and you are thinking of lattice experiments alright so let's now look at the small system which implements a clean version of this contact process so that means okay we now look at the density and see how it evolves as a function of time for different values of this lambda of this branching ratio you see when the branching ratio is below the critical branching ratio then these curves pretty much approach the absorbing state so density decays to zero and if you are above this critical lambda above the second-order transition you find okay those states those initial conditions bring you to a state with finite density and in between there is a critical curve which exhibits at least over some parameter regime over some time domain in algebraic scaling so this is the critical curve and okay you can also collapse the data if you rescale these curves accordingly until they collapse on two master curves so this is the dynamics so this is the evolution of the density and time and what you can also do is you can look at the statics and of course now that you are having a finite size system 50 sides you expect of course no sharp transition but a smoothed out transition but then if you pinpoint this transition point more or less precisely you are also able to at least superimpose this expected power-low curve and you see it's not too far off I just want to emphasize this is no fitted data so it is just taking the known value from the literature and superimposing it here on the numerics alright so now you can ask what happened if you were to use Rydberg atoms for this and you see now the behavior is pretty much similar but you see this bending here so you see the bending here and where you had in the clean system this algebraic decay for a long time at some point this algebraic decay in the Rydberg system is leveling off here and this is due to the fact that we have these off-resonant processes which prevent this unique absorbing state from existing and at the end of the day they will change the physics if you look at the system only for long enough time scales so looking at the Rydberg gas in the stationary state doesn't really make a lot of sense and well for several reasons at some point your experiment also stops working so but what you will see is that there's a manifestation of this closeness to the directed percolation transition here in this intermediate transient regime and you see here over roughly an order of magnitude at least a scaling and a behavior that is reminiscent of an algebraic decay so you could be hoping that at least in an experiment you would be able to see something like this so and this is of course conversely the curves here so it doesn't look better than in the ideal situation still it's smoothed out and it's difficult to fit the data so now what you can do is you can do an experiment I just want to emphasize that this is extremely preliminary data from the PISA group so what they do is they look at the quasi one-dimensional cloud it's a similar experiment to the one I showed before they excite atoms in here off-resonantly and then they count in the end after applying an electric field pulse that converts the Rootberg atoms into ions the ions that are being created so those are proportional to the number of Rootberg atoms that you had before in the cloud so now if you change the Rabi frequency here so the strength of the laser driving which is so to say proportional to the branching so think of this as the branching strength and you see there's some kind of phase transition like smoothed out phase transition like behavior and in the fluctuations you also see there's some kind of peak structure and if you wanted to and okay this is really preliminary you could also recognize some scaling within the exponent that is not too far off the one you would expect for one-dimensional directed percolation but okay this could well be a coincidence but okay it's at least an interesting direction to pursue and okay we will definitely hopefully firm this up and then come back to you with a stronger conclusion alright so now you can ask okay what's the point so why would I yes you would need to increase the detuning it's a bit of a technical bit yeah but also not excessively long I mean in the end look I mean those were orders of magnitude on the time axis at the end your experiment I mean you know better than me but you are nodding your experiment does something I mean it doesn't run forever not over six seven orders of magnitude I'm certainly not able to comment on this we have in the in the PISA experiments the absolute time scale is on the order of a millisecond so far we had this effective classical model so where we had coherent excitation and interaction fighting with decay and defacing which eventually turned everything into some class effectively classical model although of course the physics the underlying physics is quantum but now the question we can ask is okay what happens actually if we get rid of this defacing yeah if we really have quantum coherent excitation fighting against decay and this is the strength of this root work system yeah it's not so to say so much that you can reproduce another instance of directed percolation it's really a platform which allows you to he agrees with me yeah which allows you to study the influence of quantum effects in such systems and make them let them compete also with classical rates okay let's remove this defacing and let's go to quantum description and the simplest version that you can write down and you can approximately realize this up to some corrections with root book atoms is this kind of quantum kinetic constraint you see this is just a coherent evolution yeah so this is a spin flip from up to down coherently which only takes place if either the left neighbor or the right neighbor of your Kate atom is excited so this is kind of a quantum branching and this is now going in both ways alright so I don't want to talk too much about it just motivate I mean what you what you might be able to find you can do some mean field treatment of the system this is the first thing you can do and then you find that instead of a second order transition that you encountered in this classical problem where you had classical branching instead of this quantum branching you find the first order transition you still have an absorbing state but you see if you start now you increase your branching your system your density jumps up to this state so it's the first order transition you can also do some numerics and you see indeed bimodal distribution which is a hallmark of a first order transition there's a much much much much much more thorough analysis of this given in the poster by Matteo Makuzzi that's hanging over there I invite you to talk to him about this and I just show you this kind of phase diagram that you can now plot when you let this quantum branching fight against the classical branching and you find new structures you find okay this second order transition line so here's the classical DP so this is the second order direction transition so this extends into the quantum regime and then at some point it becomes the first order transition and there's some other critical point in between Matteo is the expert on that he will guide you through this if you like what I want you to know about is actually this yeah in the last five six minutes it's relating to actual efforts that might be relevant ultimately maybe in the domain of magnetic resonance imaging and they relate to so-called hyperpolarization methods with which you can polarize ensembles of nuclear spins so this is important in the context of magnetic resonance imaging because the sensitivity of your signal and that you extract is actually dependent on how well you can polarize your nuclei of the tissue or the material that you want to scan and for this it's important to have an imbalance and the probability of those nuclei pointing up versus pointing down and you now find that in order to create an imbalance of course in thermodynamic equilibrium you have to lift the degeneracy of those two spin states and for this you apply a magnetic field on the order of maybe 10 Tesla or so and then if you look at this at room temperature you find that this imbalance is one part per million so it's not impressive so you can go better than this you go to 0.01 Kelvin and a magnetic field that is twice as strong and then you get a tenth so clearly this is brute force you really have to invest a lot but this is for thermal equilibrium now what you might be thinking of why bother with thermal equilibrium let's go out of equilibrium and this is actually being used known already I think for tens of years 50 years or so that once you drive a system of these nuclei out of equilibrium you can actually achieve polarizations still at relatively low temperature of a Kelvin of 0.6, 0.8 at moderate magnetic fields and how does this work so you have your ensemble of nuclear spins so this is the simplest manifestation of what one can do so we have an ensemble of nuclear spins and now in this ensemble there is immersed a single electron and what you can do is now you can drive this electron with the microwave field and saturate this transition and you can transfer the polarization from the electron to the nuclear spin so at first this transfer of polarization is taking place via spin-spin interactions exchange interactions so from the electron to the nuclei and then later within the bulk of the nuclei through spin-spin interaction between the nuclei this takes a while so I mean this is a simulation and you see this is on the order of minutes and it can be even hours so you see the electron is very quickly polarized with this method but then the spins take a while before they pick up the polarization and actually assume relatively high polarization values so now what are the challenges here from theory there are lots of theory on this but it's mostly I would say I don't want to shout too loudly because I'm by no means an expert on this but it's kind of phenomenological and it's only looking at very small systems for which you can then come up with exact solutions like 10 spins for instance but it doesn't really allow you to study systematically for instance this diffusion of polarization from the electrons through the nuclear bulk because for this you need large ensembles of thousands of spins to really confidently be able to establish this so understanding the role of diffusion is one challenge for theory then of course the simulation of large scale systems and in the end once you have achieved this you also want to optimize your systems for instance you want to find strategies for achieving even higher polarizations ultimately in the stationary state so and well at least our contribution to the solution is to adapt this methodology that we have been developing in this context of the Rootberg gas so you can then by the same strategy with which I derived these classical equations you can also understand the dynamics of this coupled electron spin nuclear spin system within the framework of rate equations that you can derive and you can in the end you get diffusion which features kinetic constraint and not just single spin flips that are governed by kinetic constraint you really get two particle processes which are factored by these constraints and then you find that it actually describes the physics of the system fairly well at least in regimes where you can establish this connection where you can really compare the exact calculation to the effective calculation and you see all these colorful curves are superimposed by black curves one is showing the effective theory this classical theory which allows it to simulate large spin systems compared to the exact theory so and also what I want to emphasize is that this is actually I would say it's really surprising that it works so well because you have a huge separation of time scales here we look here at 10 to the 3 seconds and the microscopic dynamics is on the order of gigahertz so it's 10 to the minus 9 seconds so there's a huge mismatch of energy scales and yet you are able to describe by modeling this microscopic dynamics really the long time behavior of those systems this allows you to simulate large systems for a single spin you really see how this polarization develops I mean you can include effects like dipole-dipole patterns you see it's not isotropic the interaction between the electronic spin and the nuclear spin this you can include and now what you can also do is look you can drive this a bit further because this is the simplest manifestation of this physics where you have just a single electron you can also have two electrons which then give rise to other processes spin flips where you have a collective flipping of the two electrons that take with them another nuclear spin you see okay this flips from down up down to up down up and you see you have generated polarized nucleus you can understand this you can optimize so to say the rate for those three body processes you see here two electrons that depends on the distance this is a measure of the distance between the electrons and you see you find in certain regions where these three processes are more important than the two body processes and we can really use this as a means to optimize the physics and the polarization build up okay so this brings me to the summary I hope I could convince you that Rootberg atoms provide a rich playground to address problems that might be of relevance for instance in soft metaphysics or at least resemble situations that you might encounter in soft metaphysics so the name are kinetic constraints which make the relaxation very intricate but might still favor trivial stationary state I showed you an example of this is the self-similar relaxation of the Rootberg gas I talked about this non-equilibrium phase transition in which you can now in principle also introduce quantum effects and you can study what they do to the phase diagram and then in the end I showed you how this methodology can be applied to some real-world example for in the domain of nuclear magnetic resonance and okay you can also drive whether you can go to lettuces you can go to multi-species system so there's lots of things that can be explored and that I can back the experimentalists with so I hope you enjoyed this and thank you very much for your attention