 Okay, good morning. I would like to start by thanking the organizers for the invitation to come to this workshop What I'm going to discuss with you is experiments with a system Which is certainly very small compared to the size of systems that are being discussed at this conference I'm going to talk about experiment with trapped ions with strings of trapped ions and the system is not only small It also has very few constituents So I will present experiments with not more than 20 ions and what I hope to show is that these ions constitute a quantum system, which is extremely well controlled and which has long range of actions, of course and And that we can control the system in such a way that we can for example make ions behave as if they were interacting Acting systems of spins. And so I will spend some time on explaining how we do this and this will compliment What you have heard already from Igor on Monday on these trapped ion systems And then I will go on and present you some experiments where we can investigate then the some non-equilibrium Quantum dynamics in these systems particularly we can look at how quantum correlations spread in these systems How we can understand this spreading in terms of spectroscopy of Yeah, all the particles that we have in the system and then finally how to how we can maybe Characterize quantum states that we create in the system by doing measurements in order to characterize the performance of these Yeah, well-controlled quantum systems now This is work, which is which is done jointly in in sprux So these are the people involved in the experiment together with our theory colleagues Philip Hawke and Peter Soler in in sprux But it involves also collaboration with people from the group led by Martin Plinio and in ohm when it comes to the characterization of quantum states that I want to discuss in the Later part of my talk and also we had input from entry daily from the University of Strathclyde Now the experiments that I want to present belong into the domain of what is called Yeah quantum simulation that is the hope of having a well-controlled quantum system and artificially engineered quantum system that We can control to the degree such that we might be able to learn something about the physics of quantum Many body systems which are hard to simulate on a computer and there are a couple of such systems Which which are quite different. So there are for example experiments with also called atoms that trapped iron experiments But then they're also solid state systems like for example quantum dot arrays or systems of superconducting qubits each of these systems Has some Interactions that can be some yeah that are natural to the system and therefore they might be used to investigate Different different physical models what I want to focus on in my talk are trapped irons which are well suited for among others simulating easing models and so What I want to show you is that we can realize a Hamiltonian to Hamiltonian of a long range Transverse easing model where we have a spin-spin interaction term Together with a with a transfer magnet field that competes with these spins interactions Which these spins interaction have a long range and as described by a power law with a certain experiment alpha There are a couple of yeah different experimental systems Which can be used to simulate such easing models for example? Yeah, we heard about Rittberg atoms yesterday the Rittberg atoms in optical lattices are one candidate They are also yeah, so neutral atoms and molecules and optical lattices or for all the laser dressed trapped iron crystals that I want to discuss today Now there are Yeah, this is a slide just to illustrate what yeah, yeah, what we are doing and what has been done so these are the energy levels of a of a six six spin system and Where you see the energy as a function of the strength of the transverse field These yeah, the the idea of using trapped ions for simulating Transverse easing models has been yeah has been around for a couple of years and many experiments that have been done initially in the group led by Chris Monroe at the University of Maryland and so many of these experiments have focused on the properties of the ground state where people started preparing maybe they the system in a State which was easy to prepare and then adiabatically changed the parameters to go in a regime where the States start to become interesting what I want to discuss with you today is the is physics of the of none of the of excited states here and in a regime where the Transverse field here is considerably bigger than the spin spin coupling And so in that case what you can see here is that the energy levels here fall into yeah fall into groups Depending on whether you have either no spin excitation or one spin excitations two spin excitations and so on and then the The degeneracy of these levels here is then lifted by the the spin-spin interaction term that gives rise then to delocalized yet to delocalization of these these excitations and So what I want to show you is that Yeah, we can do experiments in this this first subspace where we have just one Excitation in our system. This is nice because we can then we can easily understand the physics of what is going on there And we can just look at what I want to show you is that we can look at the dynamics of Yeah, correlations in our system and spectroscopically investigate these low line excitations And then I also want to show you experiments in a regime where the complexity of the Hilbert space is much much bigger where we have half of the irons in the excited state and half of the irons in the in the lower state and in these systems we have been looking into methods of To to characterize the quantum states that are produced by the ising dynamics now Yeah, clearly the when you have a system of trapped irons the long range of action that we have at our disposition is a Coulomb interaction and Already mentioned that this Coulomb interaction gives rise when you when you when you put irons into an external potential and Cool these irons to to low temperatures gives rise to yeah crystalline structures So these weakening crystals where the irons repel each other due to the Coulomb interaction and then find equilibrium positions That is that are given by a by an equilibrium of forces of the Mutual repulsion of the particles and the effect of the confining potential now We make use of this Coulomb action for our experiments because what we what we want to study are not the Positions of the of these these charges that we have in our job But we are rather interested in quantum states that we can encode in the internal levels of our internal electronic levels of our irons and Therefore what we are looking for are now interactions that somehow couple the internal state of Yeah, a couple couple the states of one iron to a state of another iron that is far away And this is something we get we can do by making use of the Coulomb interaction together with laser Lights that that dresses this iron crystal and so the the basic idea behind this approach is is quite easy to understand And this involves a transfer of momentum from the light field to the to the irons So when when you think about for example for a free at for you think about a free atom an atom that absorbs of the photon Yeah obtains is obtains a momentum kick because it's just it's it carries the momentum of the photon However, if the atom is initially in the excited state Then you have the opposite process and you have a stimulated emission process And this process then after the atom gets de-excited You have two photons propagating in the same direction and therefore the atom obtains a momentum kick into the opposite direction So this creates spin dependent forces So the forces that depend on the internal state of the particle and now thanks to the Coulomb interaction if one iron in this way Obtains a momentum kick then this momentum kick is somehow transferred to all of the other irons And therefore if there's a second absorption event on another iron then somehow these these phonons that arise By the by the Coulomb coupling of the irons can then give rise to an effective spin-spin interaction that involves only the internal degrees of freedom of the atom I'll come to that come come back to that later in a moment now I told you we what we need are and trapped irons But also we need lasers to manipulate these irons and for this reason when you when you look into the lab There's what you can see here is the iron type apparatus in a vacuum system here But this takes only the smaller part is only the smaller part of our experimental setup The bigger part is occupied by lasers that we use for different purposes So what we need are lasers in order to create irons in the first place in the trap Then we make use of irons for cooling the particles to low temperatures and we cool them such that we have to Quantize the motion of the potential Have to quantize the motion of the irons in this external potential provided by the iron trap then and we also Make use of dissipative dissipative processes to initialize the irons into a pure quantum state and a pure electronic quantum state Then we make use of irons for here to create an excitation of the particles So this is what is needed when we want to simulate these spin-spin interactions and finally the laser the lasers are important for Detection measurements and so this is when we when we want to measure quantum measurements on the on the iron state Then once again lasers are involved now Whatever the time scales so it takes about a few minutes in general to load an iron string into this Such a trap what is nice is that these irons have provide deep potentials And therefore we can store the irons for a long time meaning that we can experiment typically With an iron string that we have loaded for a day without losing irons or having any other problems with it And this time scale is long compared to the time scale of an individual experiment So by an individual experiment I mean that what we want to do is we want to initialize the iron which involves laser cooling We want to conveniently manipulate them and in the end carry out a quantum measurement This process takes about 20 milliseconds And so if you if you compare these two time scales then you can easily conclude that on one day We can carry out something like a hundred thousand to a million quantum measurements now This sounds like like a lot But you also have to keep in mind that when we do an measurement a quantum measurement on our iron string We get only very little information because a typical quantum measurement for us means that we want to figure out Whether the iron is in its electronic round state or whether it is in some excited metastable state So per iron we get one bit of information and if you want to measure for example for an estimate and Observable it means that we have to do an experiment of this kind over and over again to get an estimation of the value of This expectation value So I told you about time scales Something yeah, it's also interesting to think about length scales So so you have seen that this is a fluorescence picture of an of an iron string that we typically confine in traps Which have oscillation frequencies of in the megahertz range now these oscillation frequencies set the equilibrium distance between the irons and This means that the the biggest length scale our Experiment is the distance between a pair of irons which is only on the order of a few micrometers Now this is nice because it means that this is much bigger than the wave lengths of of lasers that we use for manipulating the irons This is nice to two reasons first of all it enables us to Spatially resolve the fluorescence of these irons when we image them on a ccd camera So we can really look into we can really see the individual components of our quantum system And also it enables us to carry out Experiments where we manipulate the quantum state of one eye and using a strongly focused laser that does not interact with Lays with the irons which are sitting next to the one that we want to manipulate So this is something we can do with yeah with with strongly focused laser light the yeah the bad side of these comparatively large distances are that the The iron-iron distance is much much bigger than the natural length scale of our system Which is a bore radius so there are no naturally occurring spin-spin interactions for example in our system on Experimentally relevant timescale so if we want to to turn this iron crystal here into a quantum many body system We have to do it ourselves by the techniques that that's yeah Eagle already mentioned on on Monday But before coming to that problem, let me tell you about the irons that we use we Experiment with with calcium singly charged calcium 40 irons Which is an iron that is convenient for our purpose and here I would just want you to to look at the most relevant energy levels that we have so so a singly charged Alkali earth Atom here has one valence electron And so it has a level structure similar to what you know about hydrogen So we have an s one half ground state and this s one half ground state can be Yeah is Can be coupled to a state which you do not find in hydrogen and this is a meta stable state a D5 half state Which can decay back only on on an electric portable transition and for this reason it has a long lifetime long on on therefore For atomic physics experiments of about a second and this lifetime is long compared to the duration of an individual experiment Therefore we we can forget it more or less about the the finite lifetime of the state now these two levels here are What we yeah form what we call our our spin one-half system Or maybe I should call this a pseudo spin one-half system because it's not has nothing to do with the electron spin of our our iron and so we have a two-level system that we can manipulate now by a narrowband laser that couples these two transitions here and we can then for example also measure the system by coupling the The the S one half state to this short-lived P one half state that you have here using a second laser And if the iron gets excited to this upper level here Then the upper level can decay back only to the S one half state not to the other state And therefore if the iron is in the S one half state We can scatter millions of photons within one second and therefore we can when we observe the the fluorescence The image the fluorescence onto a camera We see that we have a bright iron in our trap if on the hand the iron is in the in this metastable state It does not couple to the laser and therefore will not scatter any any photons and therefore appear as a dark spot on the on the camera And so in this way we have a quantum state measurement capability that allows us to to detect models with unit efficiency Whether the iron is in in one state or in the other state if we are in a superposition state of the two Then this is a protective measurement which protects the iron on either the ground state or to the metastable state So if you think about this as a pseudo spin system here Then we it means that we identify the S one half state with the spin down state They the D five half state with the spin upstate and then that language then what you carry out here is a spin projection Measurement along the Z axis of the of our our sphere Now we trapped irons in a in the linear trap, which I don't want to describe here in detail What is important is that this linear trap provides a harmonic trapping Potential that we make very unisotropic in such a way that there's a weak axis here So along which the irons will align which has a Nostalation frequency which is about 15 to 20 times weaker than the these strong Directions which confine the irons in the redirection and in these unisotropic potentials the Equilibrium positions of a small number of irons are then linear strings and this is convenient for us because it means that we can then Address irons in a way as I told you using focused laser beams and And so we get them then pictures like this We can spatially resolve the fluorescence of the irons using yeah measurements where we detect the fluorescence for a few milliseconds and that enables us to now to Carry out quantum measurements on all of these irons for example what we can now do is so this would be then the state We're all the irons on the spin down state We can then for example use a strongly focused laser beam to flip the iron Here in the middle the spin of the in the middle in which case then There there seems to be an iron missing it, but it's just an iron that does not emit fluorescence here And in this way we can then yeah individually measure all the irons in our string This strongly focused laser beam can also be quickly moved from one iron to a different location And and this is useful in the context of what I'm Presenting to you today for two purposes first of all we can then for example Engineer certain states initial states where certain irons are flipped But we can also make use of this for measuring other quantities than the ones we can measure by a fluorescence measurement only So I told you this fluorescence measurement corresponds to spin protection measurement along the Z axis Now when we have this strongly focused laser beam We can make use of this for coherently rotating our spin on the block sphere And this means that for example when you have when we do not do these measurements these rotations And we have measurements then we can identify for example these pictures here with with the the quantum states that are sketched here Now what we can do is to have a coherent state rotation prior to the measurement And this allows us to measure other spin projections as well So for example if you if you want to measure the z direction I told you we have to project under the eigen states of this operator if we want to measure the for example the sigma x Operator and you want to project under the x axis We have some to find a way of projecting onto onto the the Superbitions of the ground plus excited state and ground minus excited state which is which can be done by Coherently rotating these eigen states using a laser pulse prior to the measurement And in this way we can then carry out by the so this combination of a spin rotation followed by a fluorescence measurement Corresponds and for example to a measurement of sigma x and in that case So so for example this picture here would then correspond to a measurement of Population in this state here and in this way we can then have then the possibility of carrying out arbitrary Spin correlation measurements that we entered in now How can we make the the spins interact with each other? This is the the the big challenge in in these experiments So the pool of interaction gives rise to this equilibrium positions But it also gives rise to the normal to collective Vibrational modes of the ions around these these equilibrium positions so this is the equivalent of phonons and in a solid state and We can now make you so yeah We can now make use of these collective excitations So this is an example of the center of mass mode of motion Which is not and this is not a not a small excitation that we have here But rather a very big one that in which the ions ring was Externally excited by by some electric field that at that was resonant with the ion motion and another example here Is this this freezing mode in the longitudinal erection now? We can make use of these modes in order to engineer our spin-spin interaction in the following way So this is what I have shown you here And I want to tell you now how we can can achieve such an effective spin-spin interaction so There are if you have n ions in your in a linear string You have n vibrational modes that describe the collective motion of the ion along the axis of the string And then there are a set of of n modes that describe the motion that in one transistor action another set of motions in the other transistor action and When the potential is very unisotropic it turns out that these these Collective modes that describe the ion motion the transistor action tends to bunch together in frequency space quite quite yeah, quite well, and so what you can see here is is a spectrum as we would measure it when we excite the ion on the On this internal transition between the electronic state as catch to you So we can do if we tune the laser to to a frequency such that it is resonant with the atomic transition Then we can excite the ion and it turns out that we can also excite the ion if we tune the laser to a frequency Which is equal to the atomic transitions frequency plus or minus the frequency at which these collective modes of motion occur and for the these chances modes of motion you have here then one set of modes and What we can now do and so sorry, this is this is one example where we have an 18 ion crystal You can see here that here the if you if you count you can see that there are 36 vibrational modes of that describe the transfer of ion motion and On that are that can be yeah can be seen in in laser spectra And that's a second set on on the lower frequency side of this what we call carrier transition now each of these Resonances here corresponds now to a process where upon excitation of the ion from the lower state to the upper state We also increase the vibrational quantum number in this particular mode by one unit by one quantum of motion Or where we decrease it by one quantum of motion now using laser cooling techniques We can cool all these modes to the vibrational ground state And in that case what you see here is that the spectrum gets modified because now if we are in the motion The ground state then there's no no if any equals zero There's no corresponding state to which we could excite the the ion on these red side been transitions Whereas on the blue side then this is still possible So what we do now is we make use of a laser that is by traumatic so that carries two frequencies And that is used to off resonantly excite these vibrational modes that you see here In such a way that the sum of the two laser frequencies equals twice the atomic transition frequency And what happens to the following so imagine that you have that you just pick two irons here Which I assume to be both in the spin downstate and where the where one of these vibrational modes is Prepared in the in the ground state of motion now if if one of these irons now absorbs a photon from this Blue laser here then its spin gets flipped and the vibrational quantum number increases by one unit This is what you see here now if in if the other ion now absorbs also a photon But this time from this this red laser here Then it's it's been gets flipped as well and the vibrational quantum number decreases again back to zero And there's a similar process and so you so you see here what we have done is now We have we have flipped the the two spins to the up upstate and the vibrational state is again in the ground state There's a similar process that involves the spin downspin upstate and that couples it to spin up spin downstate And it turns out that both of these processes have the same strengths So if you look at this now this looks as an effective spin-spin interaction as you find it in the in the easy model Now this is what what I get for a pair of irons the question is now what I what I haven't shown you here Is what are the coupling strengths and for this we have to look into this Vibrational mode structure and see what happens when we have a coupling that is mediated off resonantly by all of these modes here so the idea is to have a now a So a process where we have a to photon to iron process when the irons Change its own internal state mediated by these vibrational modes of motion of the iron crystal Now so this is yeah slightly busy slide, but don't don't worry. So what I want to show you here is Let's maybe first concentrate on what is plotted here. So What I show you here is the How the coupling gets mediated by one of the modes and this is the highest frequency mode the center of mass mode But all the irons move in sync with the same amplitude So this this movement is sketched here by by the blue bars that tell you by how much the iron move and whether they move in face or out of Face and as all the irons move by the same amount it turns out that the coupling strengths that we get Is the same for all irons because what matters here in this in this formula So here are the these are the the the spin spin coupling constant and and I told you in second or the perturbations here We can calculate now the strengths of these spin spin coupling constant And it turns out what we have here is a the sum over all vibrational modes And what enters into the sum is now the product of these factors be I am DJM where I Indicates which iron I'm looking at and M which mode I look at and these factors are then the the amount by with which a certain iron participates in a certain collective mode of motion and Then and so so have these these two amplitudes here And then this gets divided by by a term which more or less is the detuning of the laser from the From this resonance frequency that that we're looking at So if we have this center of mass mode of motion where all the irons move in sync It means that this all these coupling constants here are the same and therefore the spin spin coupling is a Yeah, it's a mean field coupling where each iron couples to each other iron with the same strengths And this is what you can see here and this is big matrix here also in the small one So here the color indicates the strengths and also whether the coupling is ferromagnetic or anti ferromagnetic And in this case we have an anti ferromagnetic coupling which is homogeneous When the coupling is mediated only by this center of mass mode now What happens if we if we pick then this the next the next mode which is the tilt mode where the iron crystal moves like this In that case we have now here a coupling which becomes in homogeneous And it turns out that you see here that ions which are close together They they move in in sync irons that are at opposite end of the string they move out of phase and therefore we have now a coupling that That is that yeah that reinforces the coupling of the center of mass mode if the irons are close together And that is opposite to the coupling that is mediated by the center of mass mode if the irons aren't opposite ends of the string So here you see that and for for close irons you have anti ferromagnetic coupling for irons that are far apart You have ferromagnetic couplings if you add up these two you get what you see here in this in this plot here now You can add more and more modes. So there's a next mode and then in this 11 irons I think so if you add up all of those then what you end up with here is a spin spin coupling matrix which Where you see that now the the coupling strength decreases as a function of distance So it falls off we have now a coupling of finite range and where the coupling is more or less how much in this Along the the direction of the string. So we have more or less the same Coupling for when it comes to nearest neighbor coupling or next to nearest neighbor coupling So because the colors here are the same and by by playing now on the one hand side with the Anisotropy of the potential and that I hasn't on the other hand with the laser detuning We have now some control over the range of the interaction that we can engineer Now what does it look like these are experiments with 15 irons where we looked at at quantum quenches So what we did is and so to hear you Let's just look at the the plot on the left hand side and so time goes upwards What you measure is the magnetization of the irons where the color encodes whether the iron is in spin down or spin up what you can see here as we flip the center iron and Then we have the spin spin coupling which gives rise to a hopping of excitations for this reason this excitation starts to spread out As a function of time and then there are some difference patterns and develop We can now change the range of the interaction and so this was with a with a so alpha here is the Exponent of this power law and so we can go from from yeah long range to maybe even longer range And and what you see here is that I don't know how well this can be seen the there are parts of the excitation which starts to spread out faster and faster but on the other hand the the excitation also stays the main part of the excitation also stays localized for a longer time and This can be so this behavior now can be understood in terms of Yeah, of course the particles present in the system and yeah since this is a workshop about long range of the action I think in principle I should should spend more time on talking about this because this is what by the Long-range character of the interaction comes in but on the other hand I have to hope that Michael first like who was involved in experiment at jqi Maryland which I published in Companion paper and both have some time to talk about this and and so I rely on him to say more about this the spreading of these excitations Now we can do more than just look at the magnetization We can also look at the spreading of quantum correlations so so because from from measurements here It is in principle not clear whether this yeah These are really quantum correlations that we that we see here propagating and for this we have done experiment in a smaller system With only seven irons, but the same type of system But now what we do instead of just measuring the spin projections along the z direction What we can do is now we can pick for example a pair of irons Let's say me the iron which is to the left to the right of this initially flipped iron and then Evolve the system for a certain amount of time and then carry out measurements And can measure all kind of spin correlations between these irons and when we have these measurements We're able to reconstruct the density matrix of the this reduced So the reduced to particle density matrix that you can see here for example for in a time of nine milliseconds This is a real part of the density matrix and you can see here But what is what what this tells you is so there are Elements here which tell you that either the excitation is residing in one iron or in the other iron or In the rest of the system that is not the two irons We're looking at but in addition to that you see that we have also off diagonal elements Which tells you that we have here a coherence of a position between the iron being on the left side and on the right side and From this measure density matrix you can then measure how much entanglement we have in the system This is just can be described by the concurrence and When you do this now for different irons and for different amounts of times you get the curves You can see here which shows you that so this is this this amount of entanglement that we have between the Spares of irons you see that the neighboring irons get entangled then get disentangled again And then the entanglement moves to the next to nearest neighbors two and six and then to the to the irons Which are at the edges of our interest Now this spreading of correlations can be understood in terms of quasi particles propagating in the system and The question is whether we can whether we can do spectroscopy of of these quasi particles Which are which are spin waves in our system now For this we can carry out a Ramsey experiment and many body Ramsey experiment in the Ramsey experiment With a single particle what you do is you have a pi over two pulse that is you you flip your spin then you switch on your interaction if Now the energy of your of the two levels does not coincide with the frequency or with the energy of Off your of your Coupling the two levels then that you will see in that the block vector will start to rotate And this is something you can detect using a second pile of a two pulse By carrying out the spin protection in the equatorial plane now when you have when we when you do this now with With a string of irons where we flip one of the irons and switch on then this easy interaction It means that now this localized Excitation has to be written in terms of the delocalized Excitations of which are the eigenstates of our system here And so now we have that means that we have here a superposition of eigenstates and Since these have different energies This is at the basis of what makes these excitations hop here so that when we then carry out the second pi over two pulse here we measure We can measure mechanization that function that changes as a function of the The time during which we have this easy interaction switched on and in this way we get then a signal which tells us about the Gives us information about the the energy separation between these energy levels that are occupied here from the ground state Now this is a bit difficult if we populate more than one of these levels here because at the moment our Resolution is not not good enough to resolve all of them But what we can do is we can try to create a superposition of the irons in the ground state and in one of these These was a part of insertation states and when we do this and measure that now these these Mechanizations here along different axis and combine the cycles in a suitable way Then we get here a signal that you can fully transform and from the full transformation we see that in this case this state here is shifted downwards by the spin-spin interaction and By it by a certain amount if we do the same with some other state We see it in upward shift this talc and from this we can infer that we have indeed anti-ferromagnetic Interactions and in principle it would be nice if you could carry out a spectroscopy of all these levels in this way now the Resolution in our system at the moment is is limited by the clearance time that we have for this reason We turned to a slightly different technique That is we thought it would be nice if you couldn't measure the energy level differences between two such states here where we have for technically reasons much longer coherence times and For this we then create a state which is roughly a Superposition of the ground state two of these states here and then some higher Higher excitation that we don't have to care about When the time above our system now under the easing interaction and these two states have different energies and therefore we get they will acquire different phase factors here and Then in the end we can carry out a protective measurement here of the this and measure measure the spin protection in the Z direction this time and then we get pictures like this now the the easing interaction Conserves in the limit in which we are the number of excitations in our system and therefore what we can do is we can now now Post-select only those measurements where we have the right number of excitations present in this way We can more or less get rid of population which is here and in these higher excited states And this is a good and in this way we have a signal which is which we would have obtained Had we initially prepared the irons and position of these two states here now? Doing this measurement here We get experimental data which now shows oscillation at a single frequency which match quite well the the Simulations to carry out when you do a Fourier transform of this data here and do this for different Superposition of states then we find indeed that we have here that we have that we can measure the energy separation between these different points across the particle state in this way we construct the Yeah, the the dispersion relation that we have I don't have time to talk about this But we can also in this way look at interactions between cause the particles and show that there are that these cause the particles do Interact in our in our system Now coming to the last part here I want to show you that we can also do experiments in the domain where the system is much more complex Because when you have a single excitation you have even if you have n irons You're the the relevant state space is only n-dimensional So there's still a very very simple quantum system when you have n irons and half of them are in the excited state in the Subspace the subspace grows exponentially with the size of the system and therefore it's much harder to yeah to analyze What is going on in such a subspace and so what we have done is to do similar experiments Starting by in a nail-like state where every second iron was flipped and then we switched on this interaction And then you see that yeah, some some hopping of excitations And at some point then it becomes hard to to to say what what is going on and What we would like to do is to understand in a bit better which kind of quantum states that we Which yeah, which kind of concepts are produced under this dynamic So we can of course do is we can do what I already showed you we can look at that measurements on Subsystems that is we can carry out quantum tomography for example of pairs of neighboring ions when you do this We measure all kinds of correlation functions. We reconstruct density made reduced density matrices And that tells us for example that now after a certain amount of interaction time all the ions become entangled and then become disentangled again But it does not tell us anything about the global state of our quantum system Now the global state of our quantum system can in principle be reconstructed using their quantum state of tomography methods, however The the number of measurements that would be required to measure the density matrix of such a 20 iron system is Is huge and it's clearly out of question to carry out such experiments This has been done this something we have done in the past with up to eight particles But even with eight particles you need something like close to ten thousand different measurement bases And so it involves a lot of measurements and since the number of measurements grows exponentially cannot certainly do this with With more than let's say eight or maybe ten particles so What we have done instead was to do a kind of quantum state tomography. This was in collaboration with our collaborators from Ulm where we Where we say now We have a state which is special in the way that initially the ions are In a product state and then we switch on the interactions We have long-range in the action But nevertheless ions tend to to couple much more strongly to talk much more strongly to the neighbors than to Einstein far apart and Based on this assumption. There's reasonable to assume that now the that now these these entangling elections produced some Yeah, produce correlations, but these correlations should be mostly local and under these assumptions There is a hope that we can make use of of Parameterizations of the quantum states which come in the name of matrix product states which might efficiently capture these correlations and which are Yeah, which are Much more compact way of describing the quantum states and then by by the full quantum state And so the the idea is that what we can try to do is when we evolve our system under the easing interaction that we now carry out measurements of quantum correlations for example of Triplets of spins which might capture the correlation that we have in our system and then based on these measured correlations Now we try to to find the matrix product state which best represents our system This is certainly an approximation because here we represent our system by a pure quantum state We know that our system is not an appeal state every experimental system is in the mixed state and But nevertheless it might be an approach to to learn something about the state. So so what we do is we have These quantum correlations here We have our first data set that we use for for reconstructing such a matrix product state And then in a second step, we have additional measurements that we can then Use in order to to here to to bound the To learn something about the state that we have in our in our lab. I think Given the time I will not say much more about this So so what you can see here are now measurements where we evolve the system This in this case eight irons for variable amounts of times and the the upper case here is so this is a small system But we can easily calculate the quantum state that we expect to find and then we can compare now our Reconstructed quantum state here with the exact quantum state and the we have taken numerically that this now this Reconstructed matrix product state should be a very good representation of our state So so if there's a deviation from if these two do not match perfectly This is not because we have limited ourselves to a reduced set of states But rather because we have some experimental imperfections which give rise to to a reduction of this fidelity that you can see here now but on the other hand if we if we have a curve like this then we We reconstruct the state and we can say how close it is to the state that we expect it to be but We do not yeah It's not clear what we learn about the state that we have experimentally produced because we know that this state is not what we have in our experiment and So for this reason it's also interesting to ask is it possible given such a reconstructed state that by doing additional Measurements that we can now put bounds on how close our our experiment is to this reconstructed state Which in principle which could be also useful even that maybe it and maybe we have the wrong Wrong theory assumptions, so maybe we we believe that we have we should have this state But in fact we have some control parameters which are different and therefore we it's slightly different but what you can see here are now now bounds that we find and It turns out that when we when you look only at it. Yeah, it's it. Yeah, these are bounds which are based on looking at correlations of one two or three spins So it needs at least three spins in order to to get a bound But under the hand these bounds tend to become quite quite weak after short interaction times, which is a bit disappointing now We have I don't have the time to talk about this I believe But yeah, if you're interested in this I would suggest that you come to me and I show you then the data We have here, so this gets worse if we have 14 irons now We can do something In addition to that it is we can try to carry out a direct fidelity estimate measurement of between our state in Our experiment and these reconstructed state This is also a task which becomes exponentially difficult when you have large systems But there are techniques of doing this and we have done this with 14 irons You see that that this gives now values, which I'm much closer to what we what we hope to find from this this simple Matrix producing reconstruction process. Yeah, so this is what I what I hope to show you I skip over this so we have we measure complex Yeah, long poly strings and make use of this to reconstruct this this state And so to to summarize I hope you have seen that that irons are indeed a well-controlled quantum system that can be used for realizing these investigating these long-range easing models and that we can carry out measurements along the lines I have have told you so it will certainly be a challenge to go to a larger number of irons because so far we are in a regime where we Can still understand everything from our Can calculate everything in principle if you could go to 50 irons and this would the situation would change then it would not be possible to do exact simulations of the the the states that we should produce also Creating higher coupling strengths and lowering the dequears in our system is is a concern we have and Then what is also interesting is to see which kinds of interactions Which are not only long-range easing interactions can reduce in our system. Thank you very much