 This is what he's going to lecture about. Thank you so much, please go ahead. Yeah, thank you for the invitation. And hopefully, I can give you some introduction, some useful introduction to trapped ions, atomic trapped ions, when they are used for quantum information science. So quantum information science is a very wide field. It ranges from quantum simulations to quantum computation, quantum sensing, and fundamental investigations into entanglement, for instance, and stuff like that. And for all these purposes, atomic trapped ions have been used. And certainly for quantum computation, they have been the gold standard for a long time, and they probably still are. So it's useful to learn something about atomic trapped ions, even if you do not work yourself with them. OK, now let's see if this turning it on helps. Yep. OK, so the program for today is, first, I'll show you where we want to go eventually. What do we want to understand? Some recent up-to-date experiments using trapped ions. And you will not understand what I'm saying here, but the goal is that after the lecture series, you do understand what I show you now. The first few slides are just there to impress you about the state of the art of quantum optics experiments with trapped ions. And then I will start with the proper talk. So I'll give you introductory material about atoms and ions and stuff like that. And I'll show you how we trap ions, how they can create qubits, and how they can mate to interact these qubits. OK, so in this prelude, like I said, I just show you some rather spectacular recent results using trapped ions. So this is a result from the INSPRO group where they implemented the Schor algorithm on a five qubit trapped ion quantum computer to compute. So when you do the Schor algorithm to factorize numbers, the basic task for the quantum computer is basically to calculate a period of a function. So that's the basic task. And that's what you see here in these. So they factorized the number 15. And yes, I am not going to go into details now. As I said, they factorized 15. And then they have to find a certain period of a function. And for different cases of this algorithm, you find this period 2, or you find the period 4. And when you repeat this algorithm a few times, you can, with 99% fidelity confidence, you can make a statement about the factors of 15. So you can do this with 100% fidelity, but a quantum computer. I mean, that's the beginning of the quantum computing age, maybe not the beginning. It's the beginning was much earlier, of course. So this is another example, a very nice experiment from the group of Chris Monroe in Maryland where they used trapped ions to simulate the, not to simulate actually to observe the propagation of entanglement along a chain of ions. So you have a chain of ions, a spin chain. And then you do something to one ion. And then the entanglement propagates between those. So what you see here is basically the separation between the ions and then the time it takes for correlations to spread. And then this alpha here, that's the exponent with which the interaction scales. So they can vary this exponent and then look at how entanglement spreads through a spin chain. OK, like I said, that's only a teaser. I'm not going to go into details now. I just want to show you where we are going to go. And this is one recent result from our lab in Ziegen where we use a particular method of manipulating ions. Magic sounds good, doesn't it? So magic stands for magnetic gradient-induced coupling. So that's a different way of coupling spins in an ion trap quantum compute. And one of the interesting things here is that this was done with long-range interaction, a coherent QFT quantum Fourier transform, where the whole quantum algorithm takes just the time you usually need for one two-qubit CNOT gate. So that's one of the features of this experiment besides the fact that it's using this magic stuff. OK, I'll explain this to you later on also. OK, so that's just some state of the art experiments more or less randomly picked from current literature, but rather impressive results I find. OK, and now let's go back 2,000 years and think about the notion of single atoms for a while. So just to put things in a historical perspective. So you know there was for a long time this discussion between people who thought atoms exist, so the smallest thing you can have, there is some indivisible atom when you chop up this desk here into smaller and smaller pieces, you end up with the smallest pieces and then you cannot chop it up anymore. So that's a view many ancient Greeks had, and but there was the opposing view where people thought space is uniformly filled with metal. So if you chop up this desk, it will always give you smaller pieces of the desk, but there's never a smallest thing. So like I said, at about this time there were very smart people adhering to either of these views. And then in a very simplified picture, 2,000 years nothing happened, and so we make a large jump in time. And then you see all these familiar names from physics basically. And there were two different views into the 1900s, certainly. But one thing changed decidedly. So these people were not only thinking about what metal is constructed of, but they did experiments. So you certainly know about these experiments of these people on thermodynamics and chemistry. So they worked with the assumption or they came to the conclusion that metal is made up of atoms, but there were also very prominent and smart people still adhering to this view here. And so Leibniz is in gray and Planck was in gray because they changed their mind later on. But there were people who did not change their mind. And one of them was Ernst Mach, a very prominent physicist. And he was a real skeptic. And he said, atoms, that's just a thing of your thoughts. You just think atoms exist, but you can explain everything also in a different way. And so he said, basically his argument condensed was, have you ever seen a single atom? Have you seen one? And people could only answer, of course, no, we have not. So still after 2,000 years, this was going on, this discussion. And for those of you who speak Austrian, that's the original quote. OK. And then Alvin Schrödinger, one of the founding fathers of quantum mechanics, even in the 1950s, he doubted that it will ever be possible to see individual atoms. So there are these two quotes from this article here. And the emphasis is by Schrödinger himself. So he said, we never experiment with just one electron or atom. So he also thought, atoms are a convenient thing and electrons to think about. But we can never do experiments about it. And then he quoted, he liked Zeus. You all know Schrödinger cats and stuff like that. So here he's talking about certain dinosaurs. So he just made this very statement to make clear that he did not believe in observing individual atoms or ions or anything. OK. And so history decidedly changed when this person, Werner Neuhauser, observed for the first time in mankind's history a single atom with his own eyes. And you can see a picture of this single atom here. So that's right here, this little spot here. Oops. And it's gone. No. OK, it's back. OK, so what you see is scattered laser light by a single atom. And so this is all garbage, basically. This is scattered laser light of some experimental apparatus. The interesting thing is this little faint spot. And so that's an original entry from the lab notebook of Werner Neuhauser, where he observed this for the first time. So this was work in the lab of Peter Toschek and initiated by Hans Dehmich, basically. And it was published in fifth ref A. This was the very first time. I mean, I'll show you in a minute all samples have been observed. But you never could see a single individual. Of course, later, then you also have these scanning false microscopes and stuff where you, but this is really seeing, because we see it with our eyes. We can, it's conceivable. I don't know of any realization, but that's conceivable. Yeah, so a little bit of sociology and science. So this was submitted to physical review letters at the time, the only, no, not the only, but one of the most respected physics journals, as you know, of course. And it was rejected because the referee said a single atom, so who cares? So later, Hans Dehmich received the Nobel Prize for this and also Wolfgang Paul, who I'll come back to in a minute. OK, so this was definitely changed history. And so this was a single atom, but actually a little bit before, earlier clouds of atoms have been observed by Peter Toschek and Hans Dehmich and Werner Neuhauser, as I showed. And at the same time, so look at the received date here, basically at the same time, one day later, the group from around Dave Weynand, who recently received the Nobel Prize for his pioneering work, also observed these trapped ions. So, and Dave Weynand, as you all know, did spectacular work at the time and ever since. And just to show you the ultimate control what we have today over individual atoms, I want to point out recent experiments by the Weynand group, where they use individual ions as a frequency standard. So when you trap your ions, you get rid of Doppler shifts because your ion is basically at rest, so no more moving around in Doppler shifts. If you laser cool them, time dilation basically goes to zero, very close to zero, very small. And you trap them at ultra high vacuum, for instance, typically, and you laser cool them. And so you don't have any perturbations. And so this is a table from a paper from the Weynand group where they have an error budget. And I don't want to go through this error budget. I just want to draw your attention to the uncertainties that we have here. So this is a rather unbelievably uncertainty in the 10 to the minus 18 range. So that's how well these individual atoms are under control. So that's OK. And of course, all this work relied on the seminal work of Wolfgang Paul, who invented these Paul traps that were used for the experiments that I showed you. OK, so that's now today. Now we are at today's state of the art, basically. So individual trapped ions, they are, like I said, they are very well localized in space. So they don't move around anymore. You can, typical localization is a few nanometers. You can laser cool them. So they are at milli micro-kelding temperatures typically. And you can prepare them individually. You can let them interact in a deterministic fashion. And you can store them for a long time. So what is long? Long can be anything between minutes and years. So there have been incidences reported where a single ion has been stored at least over many months, possibly over more than a year. So you leave the lab and you come back day after day. And it's still there, your single atom. So months, I know of single ions. When you have more ions, it reduces, typically, your storage time. But still, it can be days. Yeah, right, of course. So that's an interesting question. When you have a single particle or a few particles and you talk about a temperature. So what is really meant here is the root mean square of the velocity and then equal to kT, Boltzmann's constant times T. And then you say that's the temperature. Oh, yes, but the trap is always on. So you keep it actively trapped. And the trapping forces are much larger than any gravitational forces. OK, and yes, today, these trapped ions, as I already mentioned, they are used for some of the most accurate clocks that we have around. And using this high precision, we can now look to experiments and see if fundamental constants, like the fine structure constant, is this really a constant or is it changing in time? So instead of waiting billions of years and observing distant stars, you can go to the lab and wait just for an hour or a day or so, or maybe a year. And then because you have this high accuracy here in precision, you can then look for these fundamental things. You can, there are experiments going on making anti-hydrogen. You can, of course, do molecular spectroscopy. You can investigate chemical reactions between individual atoms, molecules. But what we are interested in here is quantum information science, as I already said. So that's the focus of this presentation here. And quantum information science to make this a bit more precise, like I said, it's fundamental questions of quantum physics that we're interested in. We are interested in ways to universal quantum computation. We're interested in quantum simulations and precision measurements. So that's what trapped ions are currently being used for. And the goal of this talk is to give you a basic physical picture of the necessary ingredients to do such experiments. OK, so that's now the introductory part. And now we'll start with the real stuff. And my emphasis will be in this first presentation, in any case, to not use any or hardly any formulas, but to give you a physical picture of what's happening in and with trapped ions. So that's the goal of this first lecture to give you a good, as Martin already said, intuition about how these things work. And then later on, we go into more detail and try to better understand what's going on. OK, so that's a generic Paul Trap who has seen a Paul Trap before, who has not seen a Paul Trap before, who is undecided. OK, so there was only one person who admitted to have seen a Paul Trap before. So I'll say a few more words about this. So this is just four electrodes. Let's consider these four electrodes, just four rods made out of metal, typically. And now we cut them in this direction. So we look along this direction. So we see these four rods cut. And now we have a positively charged ion sitting here. And we apply voltages to these four electrodes such that these two electrodes are positive and these two electrodes are negative. And now you see the field lines here. So you have a quadrupole potential. And of course, your positively charged ion wants to move away from the positive electrode towards the negative electrode. And if we look at the coordinate system, so this is the x and y direction, this is the potential and ion sees under these circumstances. If you have a fixed potential fixed in time, then if you go along the y direction, you see a binding potential. So that's the y direction. The positive charge wants to stay away from these electrodes. And in the x direction, you see an anti-binding potential. So in the x direction, the ion wants to move towards this electrode. So this does not trap, obviously. So this is not a trap. And actually, you can show in very simple terms using Ernst's theorem that you cannot make any static potential such that a single charge can be confined. But what Wolfgang Paul realized is that you can make a dynamic potential. And that's what I'll show you now. So now we apply these voltages alternately, positive here, then positive here, and so on. So that's the potential that you get then. And then if you do this fast enough, your ion just sits in the middle and doesn't know where to go anymore. And you can show that on time average, you get actually a harmonic potential here. So you have now trapped your ion in an effective harmonic potential. And so we have talked about the x-y direction. But now you also have the z direction. And you simply apply a static voltage along the z direction. And then you can confine one ion or more ions along this axis here. If you have more ions, they keep each other at a distance because of the Coulomb repouch. OK, so people from the first row were asking questions. But everybody is, of course, invited to interrupt me at any time and ask questions. OK, so now we can trap our ions. So this was a very generic trap. A trap that is not used in this way, or it is still used for several purposes, for mass spectrometers and stuff like that. Such an arrangement is actually used. But for most modern experiments, the traps look very different. You wouldn't recognize them at first sight. But the basic principle is still what I explained to you. OK, so you have your ions. But nowadays, of course, they move around your trap at high temperatures, typically. And then you need to cool them down. So you use laser light to cool them. You use a fast dipole transition on your scatter laser light and cool your ion down. And then they form such a crystal here. And at the same time, you can look, you can detect the scattered light and see the ion. So that's a picture taken of three ions, three individual ions. OK, a Doppler cooling. Who knows Doppler cooling? Who does not know Doppler cooling? OK, so it's 50-50. So I'm not sure how fundamental I should be here. But this will take two minutes anyway, only so. So let's look at a two-level atom. You have some laser light. And the two-level atom is moving at velocity v. And then you have, because it's moving, you have some Doppler shift. And so when your laser light is detuned by about Doppler shift, the atom sees in its rest frame actually resonant excitation. So it's very efficient to excite a moving atom. If the atom would be at rest, then you would be red detuned here. And it would not be as efficient as if the atom is moving towards your laser light. So you use the Doppler shift to make this resonant. OK, so then this photon comes along. It carries some linear momentum. And then this is absorbed by the atom. So the atom gets a momentum kick and slows down. So it changes its velocity by this momentum divided by the mass of the atom. And then at some later time, it emits radiation in an arbitrary direction. So you always get kicks in one direction, but you emit in arbitrary 3D directions. So your recoil is in an arbitrary direction, which means on average, you cool down, you slow down this one-dimensional motion here. OK, and if you do this often enough, so one photon momentum is by far not enough. But then you do this often enough. So n times, you absorb this linear momentum h bar k. And then you change the momentum of your atom by this. But then the atom emits, like I said, randomly. And you have to, so it diffuses in phase space. Not only in phase space, it's like a Brownian motion. It just moves randomly around. So you cannot cool it to zero temperature using this method. But to a final temperature that's determined by the spontaneous decay rate that you have here. So this determines the width of your absorption line. And you can show that the final temperature is given basically by this expression here. OK, so that's basically Doppler cooling in a trap. You don't need to cool from all three dimensions. You don't need six laser beams. You just need one laser beam that has a projection onto all three spatial axes. So one laser beam is enough in a trap to cool. And to come to give you some numbers here, if your trap frequency is about 1 megahertz and your transition frequency here is about 20 megahertz, then you end up at a mean occupation of your vibrational states around 10. So this is already pretty close to the ground state. Depends how you define close. But OK, so and this is now an example for one of the traps that we use in our lab now. So they are microstructured. So as a size reference, we have the Italian one cent coin here. So this is a microstructured trap, like I said. And also the vacuum recipient is very small. So the whole thing is about the size of my fist. And what I'll show you now is a live movie of filling this trap. If I manage for some reason this cursor disappeared. Why does this work now? It's a nice little movie just to show you what we see in the lab basically every day. Let's see if this works. Pretends to play a movie, but it doesn't. Oh, there it is. OK, so this is now a live view into this trap. And you see how you can deterministically trap a chain of ions, so first one, then two. So this is up to 10 ions. So this is stand out in many labs now, not with this trap that I showed you, but with other traps. OK, and anything? Nope, strange. And this is just an example, maybe this time. This is again trapping ions. Now they are green. They come in all colors. And just to show you that they also form very interesting spatial structure. So if you trap more and more, you have seen this transition from a linear chain to a zigzag chain. And here you see a three-dimensional crystal actually, so there's also very interesting physics hidden in these structures. OK, so that's just the movies. Now we want to come to quantum information science. And so we have our ions trapped. And we can cool them. We can detect them. And now we want to have qubits. So we choose two internal states of each ion as a qubit. So we want a long-lived state. For instance, a meta-stable state that doesn't decay spontaneously using this as a qubit would not be a good idea, because it would disappear within nanoseconds. So you choose a long-lived state and some other long-lived states as a qubit. And then you can state-selectively detect them by scattering light. For instance, you scatter light of this state. So if you shine in light, you see something. If you shine in light here, you don't see anything. So by the state-selective scattering of resonance fluorescence, you actually do a projective measurement of each individual qubit. So that's OK. And to show you a bit more how this works, so this is a concrete example. Those are now barium ions. And to make them fluoresce, you need this green-blue laser here on this fast dipole transition. But then this state might also decay into this meta-stable state. So meta-stable, because a transition from a D state to a S state is not a dipole transition. So it's forbidden. So this state has a long lifetime. And whenever your atom decays into this state, you have to re-pump it. You have to bring it back into the fluorescing cycle. So if you apply these two laser beams, then you see fluorescence. And that's what these people observed here. But then they all of a sudden see these instances where the fluorescence essentially drops to zero. So this was the famous observation of quantum jumps. So this was simultaneously observed in different labs, one of them in Hamburg by the same people who trapped this single ion, which also was a barium ion, by the way. OK, so what happens when you're here? There are certain processes that bring your ion into this meta-stable state. And if it's in this state, you don't see any fluorescence. So you really see these quantum jumps. The ion is jumping between two possible states that you can choose as qubit states. So this can be your state zero. This can be your state one. And if you're in state one, in this case, you don't see any light. And if you're in state zero, you see light scattering. So that's the basics. But this is a very important aspect of quantum information science. How well can you detect your individual qubit? So I'll say a few more words about this. So when you're in the bright state, as you've seen already, you don't observe a fixed number of photons. You observe typically a Poissonian distribution of photon counts if you wait for prescribed time. And then if you're in the zero state that does not scatter light, you typically still see some light, some background that you cannot avoid. And then you need to choose a threshold where you say, this one is in zero and this one is in one. And of course, if you choose this threshold, you notice that you always have some wrong assignments. So this Poissonian distribution extends below the threshold. And this one above the threshold, so your detection efficiency is limited by this overlap, typically by this overlap of the Poissonian distributions. But fortunately, this limit can be a very high limit. So detection efficiencies of better than 99.9% have been achieved. So this works quite well, but you have to be careful. And now we have our ions in the trap. We can cool them, we can detect them, we have qubits. And now we want to talk to them individually. That's one of the prerequisites for quantum computation. You need to be able to talk to each ion individually. And if you look at the typical distance of ions, that's a few micrometers. So you have to focus your radiation that you use to drive the qubit to a spot size that is much smaller than the distance between the ions. And because of the diffraction limit, you need a wavelength that is much smaller than this distance between the ions. So that's one of the reasons why people use laser light in the optical range for doing such experiments. OK, and so now let's assume we are able to address individual ions. And now we want to do single qubit rotations. We want to be able to prepare qubits in an arbitrary state. So what you need, you have your two levels, your two qubit levels, and you have your electromagnetic radiation at frequency omega L. And you have your transition. So in general, this is slightly detuned. And you have your splitting of your qubit at frequency omega. And then you have some coupling between them. So you have some dipole moment or quadrupole moment times your field. And this divided by h bar gives you your Rabi frequency omega. That's a measure for the coupling strength. And this thing d here, that can be an electric quadrupole moment, typically, or a magnetic dipole moment between hyperfine states. And correspondingly, this f can be an electric field or a magnetic field strength. And then if you go through the mathematics, you find this Hamiltonian here in the resonant case. So when you tune your radiation to resonance, so this is the Rabi frequency that I just indicated here. And in the rotating frame, so you do some transformations and go to the rotating frame of your radiation, then you find this Hamiltonian here. So sigma plus is just the Pauli matrix, the raising operator, and sigma minus is the lowering operator. And phi is the phase of your radiation. So you apply a field f that, of course, has some polarization. Then it has some amplitude. Then it's typically a harmonic perturbation. And then you have a phase phi. So in what theorists often do, when they do all kinds of calculations, they never care about this phase phi here. But that's very important for experimentalists, because you don't pay attention. You never get any useful result from your experiment. So when you look at theory papers, people often say, OK, so now I said phi equal to 0. And I forget about it, because it's arbitrary in the beginning. And then I forget about it. And then they do complicated calculations. And in the end, there's no more phi. But for the experimentalists, it's very important to know where phi appears. And because, yeah, otherwise, your coherence is gone when you don't keep track of the phase of this phase. OK, so this is then the Hamiltonian that allows you to do basically arbitrary rotations of your qubit. And so that's the time evolution operator. And so I'm not going into the mathematics, OK? But it's just a 2 by 2 rotation matrix then. So if you have a spinor notation for your two states, if you write them just as 1, let's say, corresponds to 1, 0. And 0 corresponds to 0, 1. And then your Hamiltonian, your time evolution Hamiltonian, is just a 2 by 2 matrix acting on these spinors here. So it's just a rotation of your spinor. And you probably all know this. So you can represent your qubit on a Bloch sphere. And then you can do rotations on your Bloch sphere using this Hamiltonian. So you can prepare arbitrary states. OK, so you need an electric field that keeps its face stable for a long time, or a magnetic field that keeps its face stable. And you typically have an electric quadrupole moment here, or a magnetic dipole moment. So that's typical cases. OK, yeah. And just to show you how well this works, some recent, or maybe not so recent, results from single qubit operations in trapped ions. So that's, again, a more or less random selection of some outstanding results. So the single shot read out fidelity. So single shot means you do the measurement once, and you know exactly what it is. So it's not, you don't have to repeat it. It typically can reach this range here, 99.9%. The preparation of single qubits, there's even one nine more, or sometimes even two nines, depends. And the coherence time can be very long. So you can keep the coherence of your qubits even for 10 seconds, or 50 seconds, or minutes. So this is, yeah. OK, so this works very well. And now we have, according to my watch, three minutes left. Is this true? Six, seven minutes. OK, so yeah, I showed you the basic physics principles without going into computational details or experimental details yet. This will come later. I just want to give you an overview of some important things. And the second important thing is, of course, individual qubit rotations are nice to have, but you have to let them interact your qubits. Otherwise, you will not have a quantum simulation or entanglement or anything interesting. So the question is, how do they interact? And now, let's see, they have electric fields, like electric dipole moments, magnetic dipole moments. And yeah, let's look at the magnetic dipole moment, for instance. So you have your magnetic dipole moment of one ion. And then this influences the field at the position of the second ion, the magnetic field. And so you could think, for instance, that you have a magnetic dipole interaction between these ions and entangle them in that way. But that does not work simply because they are too far apart. If you do a calculation for typical atomic dipoles and typical distances, you end up with an interaction strength in the range of millihertz. So that's not really useful. But in recent experiments, people are preparing Rittberg states of atoms that can have a huge dipole moment, three orders of magnitude more typically than in the ground state. And then you can use such things. But so far, this has not really been used for quantum computation or simulation or anything. OK, then you can think about the exchange interaction. So you have two particles, but to calculate the exchange integral and for the exchange integral to be appreciably different from 0, you need overlap between your spatial wave functions. So and if you do a calculation here, you find that's essentially 0. So there's no exchange interaction either. So you can forget about this also. So they basically sit in their trap and they don't feel each other at all. So that's bad news. But of course, there's a remedy for that. And this was realized by Ignacio Zirac and Peter Zoller. How you can use the common vibrational motion of your ions as a quantum bus to mediate an interaction. And that's what I want to show you again in pictures. So you have your radiation that is absorbed by one of your qubits. And then you also impart linear momentum again to your ion. And because they are all coupled by a Coulomb repulsion, so you give one ion a kick and all the others also move. And now if you do this in the proper way, you can do a conditional quantum gate. Because this ion will now know if something has happened to the first ion or not. Because it's moving or not moving. So this is the basic physics. And the strength of the coupling is measured typically by this so-called lamp-dickey parameter, which is essentially the linear momentum of the photon that is absorbed by the ion in the natural units of your harmonic oscillator potentially. So p0 is the extension of the ground state wave function in momentum space of your harmonic oscillator. OK, so eta needs to be large enough in order to enable this transmission of, in the end, quantum information between different ions. And if you do calculations, then you also find h bar k, in order for this to be large enough, you need an optical photon to give it enough momentum that the ion starts to move. OK, and yes, I think so in the next part is now a more detailed explanation of how a conditional quantum gate works with trapped ions according to the recipe of Sirak and Zoller. And then, so the next slides are also nice picture slides, three. But then it will become a bit more involved and I use the blackboard tomorrow to give you some more details about how things work. And then at some point, we come back to the picture book. OK, so hopefully you got some. Actually, I'll show you what your summary. OK, so this I'll show you tomorrow how this works and summary for today. Yeah, that's the summary for today. So we talked about the history of thousands of years discussion about single atoms. I introduced to you the research fields of trapped ions. I showed you how the Paul trap works and I showed you the physics principles of these things. This one not really yet. OK, our chairman wants to keep you here. So I'll do that. Yes, so that would be in that's what we had. OK, so now let's look at the C0 quantum gate. So a controlled not gate, which means one ion change flips its state depending on the state of the other ion. And this is the truth table here. So we're looking at I and A and I and B. So if I and A is in state zero, then the truth table of the C0 gate says nothing happens to I and B. And I and A stays in zero anyway. So that's what truth table says. And now let's prepare both ions in their ground state zero. And now we simply try to excite this ion. But we're not on the resonance of the internal transition. We are missing a bit of energy. And this bit of energy is exactly what we will have later from the vibrational motion here. So this truth table is very easy. You try to excite I and B when I and A is in zero and you're not successful, so everything stays as it was. So that's the simple part. But now I and A is in state one. And now you want to flip I and B, the state of I and B. And this works by having laser pulse on I and A and de-exciting I and A and writing this excitation into your vibrational motion. So now you have one quantum of vibrational motion excited. And I and B will notice, of course, so that's now our new level structure here. The whole string is moving. And I and B is also moving. So the whole thing is in a vibrational, excited state. And now you do the same thing that you did before. You apply a pulse to I and B. And now this missing gap here is bridged. So you are successful in exciting I and B. And that's what is mirrored in this truth table. So if I and A was in state one, and I and B was in state zero, then I and B, A is still in state one, so nothing happens to A. But the state of B is flipped. And exactly the same thing happens. If I and B was in state one, you're successful in de-exciting it into state zero. So you see, by using the vibrational motion, you can realize the truth table of a controlled not gate. So this is the basic principle in the first proposal for conditional quantum gates with trapped ions. OK, so now I'm done. OK, so the final. Thank you very much. Thank you.