 So we were talking about modifying an ion trap by adding a magnetic field gradient. And I showed you that you end up with this magnetic gradient induced coupling. And we quantitatively looked at this. And now I'd like to go back to a view from the lab. So away from the formula and mathematics and back to the lab, what does this really mean for the work in the lab? And just recall, so all these experiments, or many of these very nice experiments, they have been done using laser light. So if you have two hyperfine levels, you drive them using a Raman transition instead of driving them directly using radiofrequency. And this involves the use of lots of optical elements, so polarizing beams, splitters, acoustic-optic modulators, electro-optic modulators, of course, the lasers themselves, frequency doubling, and all kinds of stuff. So you saw this optical table. There's lots of stuff. And then you use this laser light to drive this qubit. But now, if you take a second careful look at this experimental table, you will realize that almost all of these elements that actively do something, they are controlled by radiofrequency sources. And these radiofrequency sources typically act exactly as they can act, but also at other frequencies, at this transition frequency. So now using this magic thing, you can get rid of this complete laser stuff and drive your qubits directly using your radiofrequency sources. So that's a benefit, a technical benefit from when you look at the lab, what you have to do in the lab, or when you think about scaling up your quantum computer. And so if you summarize the advantages that this magic scheme has, so using radiofrequency waves instead of laser light, you can get very stable radiofrequency sources off the shelf so that you require them anyway for most of these lasers. Phase and intensity stability are much less of an issue. Beam quality, pointing stability, diffraction. In most cases, you don't have to care about them. You get rid of all the spontaneous scattering that can happen when you drive such transitions. You can address much better, as I will show you then with laser light, even though you have long wavelength radiation, and you're much less sensitive to thermal excitation of your vibrational motion. So to illustrate this concept, I brought my six-year-old mobile phone. And if you chop this up, this mobile phone, and it's about time to do this with this old thing, then you will find in here, you will find radiofrequency sources in the microwave range. You will find amplifiers. You will find phase locks. You will find antennas. So everything is in here as opposed to these huge optical tables. So I'm exaggerating a little because this does not work at the right frequencies that we use for our qubits. But I mean, that's just a matter of technical developments. Mobile phones are a huge market, and so companies are very eager to make something good and reliable. And doing this at a different frequency wouldn't be such a fundamental problem. OK, so that's the vision, basically. And now let's look at this. Let's look at a real experiment now. And I'll come back to this picture. You've seen this before. So when we drive our ions, we have carrier transitions that only change the qubit. We have sideband transitions that modify also the internal motion, the external motion. So we have this red and blue sideband. So the smoking gun for coupling internal and external states, that's exactly these emotional sidebands in an excitation spectrum of your ion. And that's what I want to show you now. The first observation of these sidebands. So our favorite guinea pig is the uterbium ion. And in this particular experiment, we used Caiman levels of the D3 half state as some kind of qubit. So you can manipulate them using radio frequencies. There are four states in a magnetic field. They split up. So you have four different energy levels. And then you have these upper states here. They split up into two energy levels. And now what we do in the experiment, we pump population into this metastable level by just exciting this transition here. And after some time, the ion will end up here. And then we re-pump it using a second laser, bringing it back into this fluorescing cycle. But as you see in this picture here, the second laser has bipolarization. So it only empties these two states. And it does not empty these outer two states. So if you turn on your two light fields for a certain time, then you optically pump all your population gets trapped in these two outer states here. So the ion gets dark. And now you turn on your radio frequency field, resonant with these transitions, and bring back the ion into the fluorescing cycle. So this is a so-called optical radio frequency double resonance spectroscopy. So what you tune in the end is your radio frequency. Your lasers remain fixed. And then you look for excitation depending on this frequency. And that's what you see here. So here you see this Seyman excitation here in this D3 half state. How this thing gave up, OK. We have batteries, maybe. Our second one of these things. A pointer. Too bad. Technical assistant is missing. And I didn't bring my spare batteries too bad. Shake the batteries at house, OK. Maybe I can throw them at the floor. OK, so maybe I'll try to use a pointer here. Yes. Oh, you do have batteries. No, it's these small ones, unfortunately. Thanks for your effort. There's always a point where they give up. OK, my arm is too short. OK, I'll use the mouse pad in this little arrow. Oh, that's difficult to position accurately. OK, so what you see here is this resonance spectrum when you excite this radio frequency transition. And now this is a symmetric transition. So we only look, we folded it onto one part, onto the left side, basically, to slightly increase the signal to noise ratio. And you see there's no magnetic field gradient applied here. So that's just the usual optical radio frequency double resonance. And now we turn on this magnetic field gradient. And then you see this little bump appearing in the spectrum at exactly the frequency of the ion oscillating in the trap. So this is a small but unambiguous emotional sideband that shows you that you actually couple internal and emotional states. So this was the first demonstration. And soon after, this got much better. So this is now a sideband spectrum of a single trapped ion using a microwave transition in the ground state of this ion. So you see in the middle the carrier, and then you see these two emotional sidebands that signify emotional excitation. Oh, thank you very much. And do you also have a receiver for that so that I could use it? No? I take out the batteries and use it with mine. Oh, there's yet another one. OK, blue cheese. Was very short. Damn it. So maybe it's the device itself that this is another one. Sorry for this interruption. OK, great. Thank you very much. OK, now we are already, oops, what is this now? Yeah. OK, so here you see a sideband spectrum where very clearly and completely unambiguously can identify these emotional sidebands. So you couple internal and emotional states by having this microwave radiation. OK, now let's look at, so this was done in our lab. Other people also do very nice work on microwaves or radiofrequency waves with trapped ions. So this is work from the group of Dave Weinand, where they implemented single qubit gates using microwaves. And so they implemented microwave gates with an error of the order of 10 to the minus 5. So that's much better than any laser implemented gates. Or this is an example from the Oxford group with a particular trap, where they implemented even better gates with even smaller gate errors. So these microwave gates, they work very well. Those are single qubit gates. And I'll come to conditional gates in a little while. OK, now let me tell you about how you work in the lab with these magic traps now. So we use, like I said, radiofrequency for everything coherent. We still need laser light to read out the ions and to cool them. This is very undemanding laser light. And what I want to show you first is that the individual addressing works very well, even though the microwaves have centimeters of wavelength. You can resolve ions that are just a few micrometers apart by having this gradient. So this is a quantum byte, so eight bits in the trap. And then there's a magnetic gradient of 19 Tesla per meter applied here. And if you choose your hyperfine states of the ground state of the terbium, you have such a structure here. So the lower state is basically unaffected by the magnetic field. The upper state changes its energy as a function of position now. So by dialing in the right frequency, you can talk to only one of these ions. And so this is one result here. So here in this case, we do Rabi oscillations. So we drive the spin of ion one continuously. And we check whether there is anything happening to ion two. And you see that there's basically nothing happening to ion two. But now the question is, what is nothing? How do you quantify nothing? And for this purpose, we do a so-called randomized benchmarking method where we apply lots of gates, actually thousands of gates, and then look at the accumulated error after thousands of gates. And that's what is shown here. So in this case, so what is plotted here is the state fidelity of ion four when we drive ion five. So now we do gates on ion five, and we check what happens to ion four to the neighboring ion. And then we do, in this case, up to 1,250 gates on ion five. And we look at the fidelity of the state of ion four that we prepared before. And then you see the fidelity goes down as a function of the number of gates. And then you can extract an average gate error of your neighboring ion. And this is the average gate error. And that's the worst case that we measured, actually. If you look at the error of ion one here, then this is already quite a bit smaller. OK, and then just I don't want you to read all these numbers, obviously. I just want to emphasize here that this matrix has been completely measured. So you find quite a bit of addressing and crosstalk measurements in the literature. But you have to be a bit critical and look at what has really been measured and what has been deduced from certain measurements. So this is a complete crosstalk measurement. So every number is a measured number. So we measure ion one and check what happens to ion two, three, four, five, and so on. And so we get the whole crosstalk matrix. And this contains all the errors that happen. So there is no assumption in these numbers. That's what the lab spits out, basically. And an important point, this is well below an accepted error correction threshold of 10 to the minus 4 bar. OK, so this is the individual addressing. But now we come to conditional quantum gates. So we need to be able to not only talk to individual qubits, but also to have them interact in a prescribed way. And that's what I want to show you now. So that's under the heading of spin-spin coupling. And actually, this magnetic field gradient allows you to individually address your ions. But it also allows you to have a spin-spin coupling. And I haven't mentioned this before. So that's the coupling of i and i to i and j with some coupling constant. And I'd like to give you a physical picture of where this spin-spin coupling comes from, and which is then useful for quantum simulations, for quantum gates, and so on. OK, so this is our harmonic trapping potential. Then we have our ions. They have a spin 1 half. So it can point up or down, basically. That's indicated by these red arrows. And then we have a cooler interaction, which is here, indicated by a spring. And now we look at the energy of a spin as a function of position here. So this is the energy of this ion when it's been down. This is the energy of this ion when it's been up. So there's no gradient so far. Now we apply this magnetic gradient. So we see this energy level is shifted as a function of position. So the spin up states now all have different energies. And now look what happens when I excite this ion here. It slides down this potential hill and seeks a new equilibrium position. And because of the Coulomb repulsion, all the other ions will notice that something that this one has flipped its spin. And they will also move their position in this magnetic gradient. And therefore, they will change their energy depending on the spin state of ion 1. So you flip one spin and all other ions change their internal energy. And that's exactly what this mathematical expression means. So the energy of the whole system depends on the relative orientation of each pair of spins. So that's what you get when you go through the mathematics. And I think I'll jump a bit ahead and show you just a rough outline of how you calculate the spin-spin coupling. So let's look at this interaction Hamiltonian. So it contains n ions. And each ion is characterized by a certain transition frequency. And so that's just the Pauli set matrix. And then you have this term, which now contains this gradient. So that's the derivative of your transition frequency with respect to space, because you have this gradient. And qn is the deviation of the nth ion from its equilibrium position. So you see that qn times this derivative gives you a frequency again. So that's an energy. So you have a positioned dependent frequency. And you can write this in a slightly different way. You can put all your constants together in a false term. You have some false, which is dependent on the position of all ions. So false times distance gives you an energy. And so yeah. And now you also look at the trapping potential. And so we assume that the total potential is a harmonic potential. So it only depends linearly on each coordinate. So you have this quadratic term here in your potential quadratic in the coordinates. So the false is linear and the potential is quadratic. And what you find is when you go through the mathematics of this thing here, you find n-uncoupled harmonic oscillators. OK, so we looked at the internal states now. And here we look at the external energy. And now we have to bring everything together, of course. And so you do some suitable transformations and everything. And then that's the Hamiltonian you end up with. So you have your internal energy terms. You have the external emotional states. And then you have the spin-spin coupling term here. And let's now focus on these spin-coupling constants. How strong is this coupling? So these Jij, so coupling between i and i and i and j, is given by this expression here. And these epsilon inij, they are essentially this generalized lambda-dickey parameter that we had before. Except there's this additional factor S-i-n here. What does this mean? This means how well does i and i couple to emotional mode n? If you have just two ions, they couple equally. So then this is just the same for both ions. OK, so what you have here is basically your generalized lambda-dickey parameter. And that's the matrix element that tells you how well i and i couple to mode n. And then your j-coupling is a sum over all these coupling constants. So you sum over all vibrational modes. And you have an additional factor here. And that's what you get in the end. So this is the mathematics and the physical picture I showed you before. And yeah. OK, so now that's the system we work with, a string of trapped ions gradient. This gradient allows us to address them individually. And it gives us the spin-spin coupling. OK, now the next step is going to the lab again and measuring the spin-spin coupling. And in order to measure this, we use techniques that are well established. And one of them is the so-called REMSI type measurement. And that's something everybody in quantum optics or quantum information should know at some point. So who has heard about REMSI measurements before? Who has not heard about REMSI measurements? OK, so I'll say a few words about these. So that's an important concept that you will encounter in many different contexts. And so therefore I will shorten the presentation of the real experimental results in favor of presenting these concepts. And then I'll give you just some references where you can look at the real experimental results. OK, let's look at these REMSI type measurements. How does this, with a REMSI measurement, you can measure the phase, for instance, of a qubit. So far, I always talked about state selective detection, where you project on one of the states. But a qubit can be anywhere on the block sphere, of course. So you also want to be able to measure the phase of your qubit. So and just to not lose most of you, let me remind you, you have your two qubit states. And you can prepare them in any state, in any superposition state, for instance, can be parametrized in this way. So you have two angles, theta and phi. And you can visualize these two angles on the surface of a sphere. So any point on the surface of a sphere represents a state, a superposition state, here. So this REMSI measurement proceeds as follows. Let's say you have your single qubit in the spin-down state. Then you apply a so-called pi over 2 pulse. So you rotate your spin by an angle of pi over 2 around this red axis here. And then you just let it sit there. And then after a certain time, you apply a second pi over 2 pulse. And you rotate this all the way up. So what we did here, our second pi over 2 pulse had a relative phase of 0 relative to this first pi over 2 pulse. And so what we did effectively is we rotated pi over 2, we wait, and we rotated again pi over 2. And then we do a state measurement for this phase phi equal to 0. And we measure basically the spin-up state. OK, but now we do the same thing again up to here. But now our second pi over 2 pulse has a rotated phase relative to the first one. So this was a rotation of around this axis. And now we rotate around this axis. And as you immediately see, a rotation around this axis doesn't do anything to this state. So your superposition state remains what it was. And then if you measure the excitation probability, you find in an ideal case just 1 half for spin-up and spin-down. So that's what you measure as an excitation probability. And now you can have a phase pi here for your second pulse. So you change the direction of your rotation axis by changing the phase of your radiation pulse. And then you start spin-down, you rotate it up. And then in the last step, you rotate it down again. And then you get the state 0, basically, essentially. So in an ideal experiment, this goes all the way to 0. But this might be a 0. So you see by changing the phase of this second pi over 2 pulse, you get these different excitation probabilities. And now you can have any intermediate position, of course, for your phase. And then you get these typical Remse interference fringes that indicate that you have a coherent system. So remember these different excitation probabilities, they rely on the fact that you have control over the relative phase between your qubit and your driving field. And if this phase is fluctuating, then you just average out these probabilities. And you don't get fringes, you just get a constant probability. So the fact that you see such interference fringes is proof of the existence of a coherent superposition and how well it's preserved. So yeah. OK. And now we use this technique to measure this spin-spin coupling that I just introduced. Sorry. So in these measurements that I show you now, we do basically a Remse measurement on one qubit. But now the phase of this qubit relative to our driving field is controlled by a second qubit. And this gives you, and this is due to this J coupling. OK, let's look at this. So we have our two qubits, the Bloch sphere of two qubits. So both qubits start in the spin upstate. And now we do this pi over 2 rotation on our target qubit. So basically, like I said, we do a Remse experiment on this qubit here. But the phase is now controlled by the state of the second qubit. OK, so and the phase comes from the time evolution under this spin-spin coupling Hamiltonian. So this is just a time evolution operator with the spin-spin coupling and the time during which this spin-spin coupling is active. And then we can adjust this value such that it gives you this exponent here. We can adjust tau. When J is given, we adjust tau such that we get pi over 2 here. So there's a spin-spin interaction. And this is for spin up. We get a positive sign here. We get a conditional phase shift of this second qubit. And now we apply the second pi over 2 pulse and we flip the spin up. That's exactly what you do for a C0 gate. So you start with spin up, spin up, and you end up with spin up, spin up. So control and target. Remember, I introduced to you the truth table of the C0 gate earlier. OK, and now if the first qubit is in the spin down state, you have a minus sign here up here. And so you get a negative phase here. And if you then apply your second pi over 2 pulse, your target qubit that was initially in the spin up is now in the spin down. So that's exactly what a C0 gate is. And so those were the two extremes, so to say. So you let the phase of your target qubit evolve until the second pulse brings it up or down. That's a C0 gate. But you can let the phase evolve for any time. And then you just get your ramsey fringes again. And I think you can see on the next slide. OK, so that's exactly now a real experimental result. So you see the ramsey fringe of a single ion. And then we put in a second ion into the trap. And we have the spin-spin coupling. And then depending on the state of this control ion, you see a shift of these ramsey fringes. So you see that the control ion shifts the phase of the target ion. And so in this way, you can then measure the spin-spin coupling. So you know what time you applied for, during which time you applied this coupling. And you can read off the phase shift from this graph here. And then you get your j-coupling. And yes, you can vary the j-coupling. So this is proportional to the 1 over the square of the trap frequency. And that's what you see here. So you vary the trap frequency. And you change this j-coupling with the gradient held constant. OK, now I think I'll go ahead a little. So that's how much time there are left, 12 minutes according to the minutes. OK, let me see what we can do in 15 minutes. So I don't want to go into the details of these experiments now. I jump to this slide here. Yeah. OK, so what I just skipped is the measurement of all these things here. So you can show that you can have the spin-spin coupling. That's what I showed you, the spin-spin coupling. I showed you that you can adjust the magnitude by changing your trap. And what I did not show you, but what you can read in these articles, is how you that you can have simultaneous couplings between all ions. So if you have a string of ions, each ion talks to each other ion. Of course, you have to show this experimentally. That's in this paper here. You can, during any time of your evolution, you can turn on and off your coupling by simple microwave pulses, by preparing your ion in a magnetically insensitive state, turns off the coupling. So a simple microwave pulse turns your coupling on or off. And by choosing different hyperfine states, you can also change the sign of your coupling any time during the quantum simulation or on algorithm. So this is a, you have just a stationary string of ions. And then all you do is, for complete control, you apply microwave pulses. So that's the whole story. So that's shown in these papers here. OK, and actually, yeah, so, and then there's this quantum Fourier transform, also in this paper, which I will skip. You can read about this if you're interested. And I just want to emphasize one fact that I mentioned earlier. So this uses this long-range coupling. So these, so that's the three qubits. That's the timeline of these three qubits. And this signifies interactions between all qubits, simultaneous interactions. And you can show that this QFT can be done in a time that we usually would need for a single two-qubit CNOT gate. So this is a substantial speedup of your quantum algorithm using this long-range interactions. Yes, OK, so that's what I want to say. The rest you're welcome to read about. It's relevant for you. Sorry for going through these slides. But yeah, I want to say a few more conceptual things. Yeah, so now what I did not mention so far, one could, so what is the cost you pay? Life gets much simpler in the lab. And this works very well. Everything, so the question is, what is the drawback? What is the thing that does justice to the philosophical statement? There is no such thing as a free lunch. So the thing is that you have magnetic fields, sensitive states, and they are sensitive not only to the fields that you apply purposefully in the lab, but to any garbage that flies around. So that limits the coherence time of your qubits, but there's a way out. So as has been shown in this paper here, using so-called dress states, you can make very stable qubits even in the presence of a very rough environment, very fluctuating environment. And so yeah, this has been first shown here. And at the University of Sussex in the group of Winnie Hensinger, they did some great experiments on using these dress states and showed high fidelity to qubit gates using these dress states, so almost 99% fidelity. So this is the magnetic field sensitivity. It's not really a drawback once you find a cure for it. So that's what happened here. And then there's a different variant of applying a magnetic field gradient, so-called dynamic magnetic field gradient. So what we saw so far is our B field depends on the position of the ion. But now you can also add a time-dependent term. And with this omega close to actually driving the spin flip resonance. And this has been done, has been pioneered by Christian Ostberghaus in Boulder with the Boulder Group, theoretically and experimentally. There are great experiments in Oxford, also with high fidelity to qubit gates. And Christian Ostberghaus' group in Hanover works on this. If you're interested in the connection between dynamic and static field gradients, there's a recent paper that we published in New Journal of Physics that makes the connection between the physics of a dynamic gradient and a static gradient. Let me see what we want to do next. So according to my watch, six minutes. Is this correct? OK. So what else do we have in store here? So I have some slides on explaining this thing. But I will skip this also and read about it and ask me questions if you like. I want to go now completely to a different thing again. So what I showed you so far is spin-spin coupling using this field gradient. But now people also do great experiments on having spin-spin coupling induced by an optical dipole force instead of this magnetic gradient force. And this has been described first in this paper here. And once you know how this magnetic gradient induced coupling works, actually it's quite simple to appreciate how this works. So in this case, you have an optical, let's say, a standing wave for simplicity now. So you have an optical standing wave. And you have your ions sitting where you have the strongest gradient of your wave, basically. And if you now flip the spin of one ion, you again have a displacement that's exactly the displacement that we had before. So you have a force in your harmonic oscillator potential. And then you can calculate a coupling constant. So this is a classical estimation here. So your coupling constant, your energy is just this force times by how much you displace your ion. And so you end up with exactly the same expression that I skipped in my presentation for the magnetic gradient. Magnetic gradient case, this is the optical gradient case. And if you have a standing wave, then this is a static force. But in nowadays experiments, people mostly use two optical waves with a relative detuning. So you have a beat node between your two waves. And then you get a modulated force on your ions. So remember, when we looked at the spin-spin coupling of a gradient, we had a force times the displacement times sigma set. And now we have, in addition, this modulated term here. And if you go through the mathematics, the slide I showed you before, now it's, you remember maybe this term here. So what's new now is this modulated term. That's all the rest is exactly the same. But now you have this modulation here. And then you can show that you again get a spin-spin coupling. But now the spin-spin coupling can also be tuned by changing your modulation frequency. So that's a knob you have in this optical case. And that's what has been done in recent experiments. So the first experiment was in the group of Tobias Schetz. Where is it here? So where they looked at the transverse ising model, basically, of just two spins. So they used this. They had an effective transverse field, the sigma set coupling here between two ions. And then they looked at, in quotes, phase transition. I mean, two particles and phase transition is a bit questionable. But this was a very nice and pioneering experiment that showed the principle of doing such a thing. And yeah, you can read about this here. And there's another very beautiful demonstration of the spin-spin coupling in a panning trap this time. So where you have a large array of ions in a plane. And this plane is actually rotating. And then you can measure interesting properties. So there's no time to explain this. But you can read about this here. And this is now a bichromatic force that I explained to you earlier. This has been pioneered by the group of Chris Monroe in Maryland. And yeah, there's no time left to really explain this. But that's the reference. And that's a very nice recent experiment they did on the propagation of entanglement in a spin chain using these optically induced spin-spin couplings. So again, I recommend to read this if this is relevant for you. And now I finish by summarizing a few things. So trapped atomic ions in the very beginning, I said trapped atomic ions have been and probably still are the gold standard in quantum computation, quantum simulation. And I think that's well justified to say that. So all ingredients for scalable quantum computation have been shown, high fidelity gates with many nines, single qubit gates, multiple qubit gates. And small quantum computers have been demonstrated. I showed you some examples. Quantum simulations, very nice examples, have been demonstrated. The future of scalability probably lies in having two-dimensional traps, arrays of two-dimensional traps where you can have lots of ions. And there are concrete proposals, concrete realistic proposals for building a scalable quantum computer. So one of them, so that's an example for such a 2D trap. And one of the proposals came from the Weyland group to make a large-scale quantum computer. Another proposal is based on photonic interconnects. That's what the Maryland group vigorously works on. And then there's a recent new proposal based on magic on this microwave-driven trapped ions. So that's actually, the paper's title is a blueprint for a quantum computer. And blueprint means you can give it to an engineer and he builds a quantum computer for you. So that's the spirit of this paper. It remains to be seen whether the engineers already have enough quantum physics knowledge to realize this. But all the possible issues that we think of today have been addressed in this paper. So this is really from today's point of view a realistic scheme for building a large-scale quantum computer. OK, so that's where I want to stop. And thank you very much for your attention. Thank you.