 I can do single qubit gates, and we also looked at interacting ions. And now today, I want to start looking at the interaction between electromagnetic radiation and trapped ions or trapped atoms. Actually, what I'm going to do now, most of it is also applicable to any particle in a trap, any particle that has an internal structure in a trap. OK, but first I need some space. OK, so what we want to look at is at the interaction between trapped ions. And like I said, you can replace this with neutral atoms or something with electromagnetic radiation. So that's what we will do here on the blackboard. OK, so where do we start? Let's start with a picture. We have a very simplified view of our ion, of course. We only talk about qubits. So just two states that are relevant, 0 and 1. The energy separation is given by omega. And then we have the atom in a trap. So we have a harmonic oscillator coupled to the ion. So that has energy levels separated by frequency nu, angle of frequency nu. And then in addition, we have some radiation field. I'm talking about two stable levels of the ion that do not decay like metastable states or hyperfine states. So this omega could be in the optical regime. It could be in the radio frequency regime. So what I'm going to show you now is completely general. No particular assumption on omega and on the strapping potential. OK, so now we have some radiation coupling. You will see that this will couple the internal and the external dynamics. That's what we want to see in the end. And so we have radiation, near resonant radiation at a frequency omega l. And then we have a coupling between the atomic dipole or quadrupole or whatever we consider. So if you have a dipole, then your interaction Hamiltonian is simply hl. Then this is simply given by minus d dot e. So that can be the electric dipole moment of the atom coupling to the electric field, or it could be a magnetic moment of the atom coupling to this explicitly minus mu dot b. So that's a magnetic moment coupling to a magnetic field. OK, so but first let me specify this field here. So the field, I call it, so I call the field now f, and f can stand for either e or b. So f has, like I wrote down yesterday, already some unit vector that indicates the polarization, some amplitude, and then it's supposed to be a traveling wave, a linearly polarized traveling wave. So we have for the traveling wave, that's k times set. So the wave is traveling along the set direction. k is the wave number. So k is 2 pi over lambda. Lambda is the wavelength of your radiation. Omega is the frequency, and phi is the phase. So you just have a traveling wave. And then you can write your interaction Hamiltonian, this hl, as, so I'm just, you can write this as h bar omega Rabi frequency. That's what I introduced yesterday. So the Rabi frequency, h bar times the Rabi frequency is just, for instance, d dot e, or it's u dot mu dot b. So this gives you the energy. That's the interaction term. And then you have a term sigma x that couples your two states. So actually the internal Hamiltonian is given by 1 half h bar omega times sigma set. So that's the usual Pauli matrix that you know. So this is just a two level system that can be described like a spin. So you have spin up or spin down. You can imagine this as being spin up or spin down in a magnetic field, for instance. And this is your Hamiltonian. You all know Pauli matrices, right? Nobody shouts, yes? Could I assume? OK, so this is our designated space for the basic stuff here. Let's write them down. So that's just for our two, we describe as a spin 1 half or two level system. So we just need a 2 by 2 matrix. And that's just sigma set. And then we have sigma x which couples these two states. So if you have a sigma x term, you have a coupling between these two eigenstates. Or you can have a sigma y term that looks like this. That would be the identity. OK, so you know Pauli matrices. Very good. OK, so here we have the internal Hamiltonian. Here we have the external Hamiltonian. And so this is just a quantum mechanical harmonic oscillator. So we have h bar times nu times a dagger a plus 1 half. So a dagger is the creation operator and annihilation operator for the harmonic oscillator. So the annihilation operator creates one quantum of excitation and the creation operator creates one quantum of excitation and the annihilation operator gets rid of one quantum of excitation. So you probably all remember this from your quantum mechanics lecture. So if you should have any questions during the course of this fertile derivation, just interrupt me. OK, so that's internal and external Hamiltonian. And now we have this coupling between, so this is just the atom sitting in the trap and now we add radiation. And that's what I want to look at now. So I look at the coupling Hamiltonian that couples the radiation to the trapped ion. OK, so that's the strength of the coupling, the energy of the coupling. That's the sigma x term that couples these two terms. And then we have, of course, our wave here. OK, and we rewrite this in the following way. So sigma x I can write as sigma plus plus sigma minus. So those are the atomic raising and lowering operators. So sigma plus is just this matrix here. And sigma minus is this thing. And if you apply this, OK, so and our, you've seen this yesterday, so 0 in the bracket notation is in the spinon notation is this thing. And 1 in the bracket notation is 1, 0. So now you see what these raising and lowering operators do. So if you apply sigma plus to your spin down state, you get, just multiply this, you get spin up. And if you apply sigma minus to spin up, you get spin down. So you flip the spin of your two level system. So A and A dagger create and destroy excitation in the harmonic oscillator. Sigma minus and sigma plus create and destroy excitation in the atom. OK, and the cosine term, we write now in a slightly different form. So I'll simply write this down. So that's easy to spot what I did here. And I have to include a factor 1 half because I have now two cosine terms. And I have to get rid of one of them by this factor 1 half. OK, so that's just in a different form. And now I started with a classical wave here. And I'm going to continue with a classical wave. So this is a semi-classical approach. The radiation field is a classical field. The atom and the trap is quantum mechanical. OK, now let's look at this set. So set is the position of the atom. And if I have a quantized harmonic oscillator, then I need to have a position operator, so set. And the position operator, as you certainly recall from your quantum mechanics calls, can be written as a dagger plus a times some unit factor. And this delta set is just the extension of the ground state wave function of your harmonic oscillator. So what is the ground state wave function of the harmonic oscillator? What kind of function is that Gaussian? Exactly. So we have a Gaussian wave function here in the ground state. And the width of this Gaussian wave function, that's delta set. And you can write delta set as square roots of h bar divided by 2 mu. OK, so we introduce, instead of the classical position, we introduce the position of my atom in the trap in terms of an operator. And then remember yesterday, we started to look at the Lamb-Dickey parameter. So I introduce now this Lamb-Dickey parameter. So that's a very important quantity, eta. And this is defined as delta set times k. So you will see that's exactly what appears here. Delta set times k. And you can rewrite this. For instance, k is 2 pi over lambda. So that's just a wave number. So you can write this as 2 pi times delta set over lambda. So it measures this Lamb-Dickey parameter, measures how well is your atom localized relative to the wavelength of your radiation of the atom. And another physical interpretation I gave you yesterday, you can also write this Lamb-Dickey parameter in terms of the momentum. You can write this as h bar k divided by 2 times, now I call this delta p for consistency. Yesterday it was called p0. So this is just the extension of the ground state wave function in momentum space. So you can also look at it as the Lamb-Dickey parameter measures how much momentum you transfer to your atom. And there's yet another way to write it, which might be also illustrative. You can write it as h bar k squared divided by 2m. So what is this now? That's the recoil energy that an atom absorbs when a photon is absorbed. And this again in units of the energy of the harmonic oscillator. OK, so I'm writing this in all these different ways because the Lamb-Dickey parameter really has an important physical meaning. And in case you don't remember anything after lunch about this, you should remember the Lamb-Dickey parameter. The atom has an important individual. So I guess you're thinking of the collective. It's a very interesting and relevant question, of course. So I haven't said anything about this. So everything I'm doing here is valid for either a single atom or a collection of ions for a normal mode of a collection of ions. This harmonic oscillator, these modes, they could be the mode of a single atom or of several atoms. So I didn't specify this yet. Both are possible. OK, so now I'll write down. So we find hl 1 half h bar omega r times sigma plus plus sigma minus. So that's what we had so far. And then I simply rewrite this as e to the i eta a dagger plus a minus omega t plus plus the Hermitian conjugate. So just to make it a bit shorter. OK, so that's now the Hamiltonian that we have. And you can, of course, now do numerical simulations of this Hamiltonian. But there's a better way to approach it or an analytical way, which is often the better way. Let's put it this way. And in order to find an analytical solution, we go into the frame rotating with the frequency of my atom. So if you think about a spin, spin 1 half precessing in a magnetic field, going to the rotating frame means you go to the rest frame of the atom. So your new rotating frame is now rotating around the magnetic field like the spin. So the magnetic field gives you the lab frame. And then you go into this rotating frame that rotates with the frequency, the Larmor frequency of your atom. So if this were a magnetic moment, a spin, then you would simply sit on a carousel and rotate like this with the spin. And now we do the mathematical counterpart of what I just demonstrated you in a circus-like fashion. OK, so we want to calculate HL, a unitary transform of HL. And this is done in this way. So and H0 is the internal Hamiltonian plus the external Hamiltonian. So this is just sigma z plus. OK, and so you have to do some calculations. You notice from your quantum mechanics course you can use Baker-Hausdorff lemma, for instance, to calculate this. So I'm not going to go into the calculation now. So there is a bit of paperwork involved here. And then I'll give you the final result that you end up with. And so we have HL is given by this. And then we find in the rotating. Actually, this is not good to have this down here because this is very important. Rather write this up here somewhere. HL, so after some calculations you find this HL can be written in this way, e to the i omega minus omega l. So what appears here is the difference frequency between the atomic procession and your applied radiation field. That's an important point. Sigma plus, and then you have e to the i eta. And now your creation and annihilation operators become time dependent. So this is a times e to the i mu t plus a dagger plus a times to the minus i mu t plus the mission conjugate of this whole thing. OK, so now why does this help us? It doesn't help us yet, but you will see in a second that we're very close to an expression that, again, is amenable to a physical interpretation. So now let's look at this exponential function here. And we now have to look at a real physical situation to determine the value of this eta here. So how large is this lamdike parameter? And for typical traps, so remember the lamdike parameter involves the trap, basically. How stiff is your trap? How wide is the extension of your ground state wave function that's measured by delta z relative to the wavelength? So what you need is you need to know your trap frequency, basically, how stiff is the harmonic oscillator potential. And you need to know lambda of your radiation. And now if you look at typical situations like optical radiation in a trap of 1 megahertz, you find that eta is very small. So for typical eta, which means eta is much smaller than 1, you can do now an expansion of this exponential function here. So expansion in this parameter eta gives you this energy term. Of course, this remains unaffected. Then this first term also remains unaffected. And so then I expand this. So the first term is just the constant is 1. Then I have plus i times eta times a dagger of t plus a of t. And I omit the higher order terms now. And then there I have to have mission conjugate at some point. OK, so now we have expanded this. So nothing happened so far except some math with the physics background that eta is small. And now we look at the individual terms. And as I promised, we're very close to some very useful result, so I'll just write this in a different way. So OK, we have now e to the i omega minus omega l. So that's a difference frequency times sigma plus. So this is just this term here. And then I get a term e to the i omega minus omega l plus u because this is a dagger of t. That's this frequency here times, OK, I put the i eta in front of this term here times sigma plus. Here I have, what am I doing? Yes, this should be right. So that's the first term. That's the second one sigma plus times a dagger. We're getting close plus a term e to the i omega minus omega l plus nu times t. So that's this one sigma plus times a. And then these higher order terms that I neglect. And then the Hermitian conjugate. And then we close this bracket. So now why is this useful? Because we now can look at what happens when we tune our radiation to different frequencies. So what we do now in the lab is simply turning a knob on the frequency of our driving field. So we change this omega l. And now look what happens here. For instance, if I tune, so I give you an example now. So I'm looking now at these terms here. At these time-dependent terms. This is not really read. OK, so as an example, let's tune the radiation exactly to the resonance frequency of the atom. And then you see that this term vanishes. This term, this is zero. So this term rotates, evolves at a frequency nu. So nu is your trap frequency. At this frequency, you have a term that does something. And also this term here does something at your trap frequency. And this term is constant. So what you do now is a so-called rotating wave approximation. So we are in this rotating frame. And now we neglect terms that are fast compared to other terms. So they evolve very quickly. So if you look at typical time scales determined by this frequency here, then these terms average out. So they don't do anything. So what I am left with, if I tune the radiation exactly to resonance, I find this is 1 half h bar omega Rabi. And now this is just 1. And now I did a little mistake and nobody noticed it. I started with this phase phi here. And somewhere I lost it. It's still here. Here it was lost. Yeah, I think so here it's already lost. So I'll add it here again. Then I have it here. No, that's not times t. That doesn't evolve. Sorry, that's plus phi. t plus phi. That's this. And then I still have it here. OK, so this term is simply 1. And all that remains is sigma plus times e to the i phi. And these terms I simply neglect because they are fast rotating. And then I have the Hermitian conjugate, which is sigma minus e to the minus i phi. OK, and this now should look familiar to you because, well, this was on a slide yesterday. So we looked at single qubit operations. And what I showed you now, I treated the interaction between electromagnetic radiation and the trapped atom. And then if I tune the radiation properly, I do nothing with my motion. So there's no motion at all involved. I only rotate the qubit itself. So you can write this also to illustrate this. So sigma plus e to the i phi plus sigma minus e to the minus i phi. You can rewrite this S. So that's sigma plus plus sigma minus. So this gives you sigma x times cosine phi minus sigma y times sine phi. OK, so you can rewrite this term in terms of sigma x and sigma y. And now you see, when I write it in this way, I can rotate around the x-axis or the y-axis on my Bloch sphere. So remember, you can visualize these qubits as a point on a Bloch sphere. And the time evolution operator that you can derive from this Hamiltonian is simply a rotation of your state on this Bloch sphere. And you rotate around the x-axis or the y-axis. And you can choose this by choosing the phase of your radiation around which axis. So just to illustrate this, so if you start, let's say, with a spin up state, we start with a spin up state. Then we let the system evolve under this Hamiltonian and choose the phase phi equal to 0. Then we rotate around the x-axis, so around this axis here. So we rotate to a rotation in this direction. And then you can stop your interaction at any time and prepare a superposition state or flip the spin all the way. So that's exactly what Mauro demanded in his last slide. So remember this Hadamard gate, where you want to prepare a superposition of two spin states, basically. And that's exactly what you can do here. So you've rotated in this configuration space around the x-axis. If this were a real magnetic moment that I'm treating here, and this whole treatment also applies to real magnetic moments, then this whole dynamics is very classical-like with some exceptions. So you have a spin that's oriented in a magnetic field. And your magnetic moment precesses around the magnetic field direction. And the precession frequency, that's exactly this omega here. And then we apply an additional field and look at what does this additional field do in the rotating frame of my rotating spin. So I sit on the spin. And now I see an additional field. And this additional field flips the spin from spin up to spin down or back and forth. And the description here is for any two-level system. So it's also valid for magnetic moment, spin 1-1.5, but also for any other two-level system. But it's easier, of course, to visualize a real magnetic moment than to have this abstract configuration space. So I can relabel the axis. I don't want to do this now by having here a real space and a real magnetic moment. OK, so how much time do we have today? OK, so one of the important points is simply you have one knob in your lab that is responsible for your driving frequency of your field. And by turning this knob, you can create this Hamiltonian that makes rotations on the Bloch's field. And now, but of course, we have other possibilities as well. So we can tune now the radiation to a so-called emotional sideband. So I tune now the radiation, for instance, to such that omega l equals omega plus u. And then this first term will vanish. So now tune omega l such that omega minus omega l plus u is 0, which means on a frequency axis. So if I got your frequency axis of my light field or whatever it is, I have first I showed you the case. No, I just called it omega. I showed you this case here. And now let's use these colors that are not really colors. They all look the same for me, at least. So I tuned to this frequency. And so that was the first case. And now I tune to this frequency here. So that's this case. And all we have to do is look at this Hamiltonian what happens. So then this term, the time dependence, will disappear. And these other terms, they will be fast rotating again. So there will be some dynamics. Remember, if you want to have the time evolution operator, you take e to the i times this Hamiltonian, basically. And this Hamiltonian has in front this time scale here that's omega ravi frequency. And as long as omega ravi is much smaller than these terms, we have a slow evolution. And these fast rotating terms, they just average out. OK, so now we have tuned to this so-called blue sideband. So all terms disappear. And only this one remains. So we end up with our HL. So we have here, so then we have sigma plus a dagger. Oops. Yes, right. OK, plus sigma minus a. And then I can do the same thing. I can tune to the red sideband. So I choose omega minus omega l. There's a small mistake here. This is a dagger plus. This is minus. So this is supposed to be 0 now. And then I end up with the Hamiltonian HL. And now, again, everything else disappears. Only this term is 0. So I have a 1 here. This is fast rotating. This is fast rotating. And all that remains is this term, g's. OK, pluses in minus signs are, let's see. OK, so we have, yeah, that's correct so far. Yeah, yeah, it's fine. Sigma, yeah. So we have sigma plus a plus sigma minus a dagger. OK, and now, for some of you who have had some exposure to quantum optics might recognize this Hamiltonian, or does anybody recognize this Hamiltonian, the very famous Hamiltonian? It's called the Chains Cummings Hamiltonian. Yeah, exactly, yeah. And this was first introduced for describing the interaction of a quantized light field with a two level system. So here we have a completely different physical system. But you can, of course, if you quantize your light field, you treat also a two level system coupled to a harmonic oscillator. So in that case, the harmonic oscillator is a light field. And this Hamiltonian was derived in the case of two level system coupled to a light field, or some electromagnetic field. And it was the basis of many experiments in cavity QED, so like the experiments of Saar-Saharosh and other people in cavity QED. OK, so what I wanted to show you here is now by simply changing one knob in your lab, namely the frequency or wavelength of your exciting light. You can create very different Hamiltonians. So one knob is the frequency. The second knob is the phase. Don't forget about the phase. You have to control the phase of your field because this phase appears here again and determines what you really do. So frequency, phase, and what is the third parameter? You have a question? But I was asking you a question. So let me finish this sentence. So three parameters so you can change this Rabi frequency here by changing the strength of your intensity of your driving field, and therefore your E field or your B field. So that's the three knobs by which you control this Hamiltonian, frequency, phase, and amplitude. OK, now I'm ready for your question because I'm considering a quantized harmonic oscillator. OK, so what we do is we look at the quantized motion of the atom described by these operators here. And then the position operator expressed in terms of these harmonic oscillator ladder operators is just given by this expression times a number. And this number appears then here in this eta. Could you speak up a bit? So delta z is a fixed number. And the position operator is delta z times a dagger plus a. So that's the position operator of the position of the ion. Yes, I mean there's only one coordinate system. And we fix our coordinate system at the equilibrium position of the ion. So that's our z axis here. And we define this as the 0 of our z axis. So z always gives me a position. So here it's the position of the ion. It's just the position of the ion. Where does the ion sit relative to the exciting wave? Exactly, exactly, yes. Actually, that's a very important point because depending on the spatial phase of your field, you also get a phase factor over here. So depending on this where your ion exactly sits, it sees a different phase of your traveling wave. For a traveling wave, it doesn't matter. But if you had a standing wave, then it would matter. Yes? You get a linear coupling by expanding the x-axis. But usually in a system, you don't need an external, like the question is, is there any reason why that term is neglected here or coupling to the field? I feel this is much more stronger than that. There is nothing else. That's the whole point. You're probably thinking about solid state systems. And solid state systems, you always have phonons and couplings to everything, couples to everything. Here you have an isolated atom or some atoms sitting completely isolated from the rest of the world. So that's the complete truth. There's no approximation here except the expansion set I did here. But the Hamiltonian is completely, even if you have several ions, you can, as I mentioned before, you can treat the collective modes individually by these operators. So that's still the whole truth of what's happening in the trap. So there's, yeah, in a harmonic trap. If it becomes unharmonic, then life is different. But we're talking about harmonic traps here. Any other questions for today? Our chairman is busy. Yeah, there's a question. How would you do a sigma set? How do I do a sigma set? Yes, interesting question. So one thing is you can tune if you're not exactly on resonance, you get an additional sigma set term. And you can easily imagine where this comes from. If you're in the rotating frame and you're exactly on resonance, you exactly rotate with your spin. But if you're slightly detuned, then you see a small change in time between your reference frame and your applied field. And that gives you a sigma set term. So if you're slightly detuned, you can have a sigma set term. Or what you can do, you can apply a sequence of proper pulses with proper waiting times. I can show you a concrete way how to do this. But it's an important question, of course. I don't have any. OK, thank you for your attention.