 A sketch of how you can derive the amplitude frequency relation for the Diffing oscillator, because I will need this relation for what I'm going to explain next. Now, OK, that's the slide we have seen in the very first. That's my very first slide of the first lecture. And that's a response of the linear resonator, which is Lorentz and concentrated around the eigenfrequency of the oscillator depending on how you drive it, with what frequency you drive it. Now, that's the Diffing oscillator. And now I put it, I mean I could have put just the final result. But I thought, OK, Mark has derived it, spent some time. You could have got an impression that it's really difficult. I hope you didn't, but you could. And this is something you actually, if you are working with something nonlinear, you come all the time against very similar questions. I mean, nothing, OK, now you have seen it. You can go to Mark's lectures. You can go to literature. You can find the result. But then next time, instead of nonlinearity power 3, you have also nonlinearity power 5. What do you do? And then you have nonlinear dumping. So you have dumping plus x squared. What do you do then? And then you have a parametric excitation. What do you do with the parametric excitation? I mean, there is just this framework that would help you to do all those things once you need them. And it's actually easy. The only thing is, indeed, you have to be careful and you have to understand what you are doing. And what I'm now summarizing is the last half an hour of Mark's lecture. So I'm not going to show anything different. Right, so we start with this equation. First thing we do, we go to rotating frame. So instead of this, I'm using slightly different rotations, but I hope it's OK. Instead of coordinate and velocity or coordinate and momentum, we define these new things duently. Now we are driving the oscillator with the frequency omega. So we expect that the oscillator responds at the frequency omega, or maybe close to omega. And because of that, that's exactly rotating frame. It means that those u and v are time independent or very weakly time dependent. That's a scale separation. Right, now if we do that, we get those equations for u and v. So u dot and v dot are equal to that. And that's exact. That's an exact transformation. I didn't do any approximations. Now we need to do approximations because we cannot solve this exactly. And if we start looking at that, we see different combinations of the sines and cosines. Like we see, for instance, sine times cosine. And we know that's sine of a double of 2 omega t. And we see here cosine times cosine. We know that's 1 half plus cosine squared over 2. And we see here some crazy things from the third power. And most of those things oscillate in time. And they oscillate in time with a double frequency, with a triple frequency, with a quadruple frequency. And there are a few things which don't oscillate. Like this cosine squared, there is a component which doesn't oscillate. And then we do this. Let me go to the next slide because I have it copied on the same as we have seen. Then we do the rotating wave approximation. So technically speaking, the average, the whole thing, over time. And if the average is over time, all those fast oscillations disappear. And we only have terms which oscillate very slowly or don't oscillate. And then we just look what are those few terms which don't oscillate. And then as I said, we have this cosine squared, which is, on average, 1 half. And we have cosine to the full, which is, on average, 3 eighths. But then we have other things like sine omega t cosine cube. Omega t, which is, on average, what? Zero. And then if we do all that, we get a much simpler equation, which still have u dot and v dot on the left-hand side and something on the right-hand side, but no time. So all these cosines and sines are gone. And then I say, OK, fine. Maybe in the first instance, we don't care about the time evolution. Let's just take them time-independent. And then we just say, OK, this is zero and this is zero. And then we have just two algebraic equations, which are third-power equations in u and v. And then we can plot it. We can define the amplitude, which is a square root of u squared plus v squared. And for this amplitude, we get this nice equation, which is an equation with respect to r squared. That's a cubic equation with respect to r squared. And this cubic equation Mark analyzed it yesterday quite spending one some time. So sometimes it has one solution, and sometimes it has three solutions, one real solution and three solutions. And sometimes, meaning depending on the force and the frequency. And this, he didn't have this plot. But I needed this plot in this form, but he had something equivalent. He plotted it differently. So what is plotted here is, well, essentially, the amplitude of the oscillator versus the frequency. And the frequency here is normalized in such a way that one means we are exactly driving at the resonance. So that's actually omega minus omega not in my annotations. And this is omega be exactly equal to omega not. And then what you see is that if this non-linearity, I'm sorry, this is an image from somewhere else, so it doesn't have my annotations. But so this beta is my alpha. You see that, well, first of all, depending on signs, it does either on the sign of alpha. It either goes that way or that way. And then you have this kind of big shape of the response. So instead of a Lorenzen, which goes nicely like that, you have this response, which goes like this, and then goes back, and then goes like that. And you can also show that this part, when it goes back, is unstable. It's actually, well, you can show it formally, but it's also, you can easily understand it by kind of thinking that this amplitude, I mean, every solution is a minimum of energy. It's not trivial to understand what this energy is, but every solution is some minimum. And if you have a minimum here and a minimum here, that must be a maximum in between. And if you have a maximum in between, it's unstable. So if you have two stable minima, and the maximum is an unstable solution because the system can go that way or that way. And so in practice, if you drive a system, by, for instance, increasing the frequency, you will have a hysteresis. So you will follow this branch, and then at this point it becomes unstable. And so you will just go down and follow that branch. And if you drive it by decreasing the frequency, you go all the way up to here. Then here it becomes unstable. Then you jump to the stable branch and go down. So this hysteresis would be a kind of signature of this definite behavior. Okay, I think that's what I need. And I will now go to this Optomechanical Induced Transparency. And that's a slide which I already have, so I will not explain it again as much and also John explained it very nicely. Just to repeat that we have, this is a setup for an experiment in which we drive a cavity by a pump laser at the frequency which is cavity minus mechanical. And we have a probe laser which probes the transparency of the cavity close to its resonance. And I spent some time explaining it. John spent some time explaining it. We have seen the results. We know that there is this Optomechanical Induced Transparency peak which develops, which is very sharp and very hard. Now, that's all fine. Imagine you drive it harder and harder and harder. What happens? I mean at some point your cavity is linear, so nothing happens to the cavity. But at some point, because you drive it, you also excite the resonator. And at some point, the resonator starts moving with such a big amplitude that it becomes non-linear. And in this case, you can basically assume that this is just not a linear resonator anymore, not a linear oscillator. That's a nothing oscillator. And you can ask what is the result of the, what is the effect of the nothing oscillator on the Optomechanical Induced Transparency. And the first actually had an experimental answer that the experiment by the group of probes given back. So that's exactly what they have done. They were driving the cavity harder and harder and they were looking, that's just a theoretical picture. So that's this history that I explained to you. But this is the experiment. They were just looking at the response. And what they're plotting, remember the Optomechanical Induced Transparency is when you have transmission versus frequency. And you have a cavity resonance. And on top of this cavity resonance, you have this peak. And what they are plotting is actually this part. So in that picture, this is almost flat. So that part would look like this. And eventually it would go down, but they don't plot that because it's not interesting. So this is the cavity and this peak is Optomechanical Induced Transparency. And you see if you increase the drive and you increase the drive from this curve to that curve, this one is pretty much Lorenzen, as we should expect. And this becomes, that clearly shows history, this is very similar to the theoretical curve for the Duffing oscillator. I believe they also actually could fit it with the Duffing oscillator formulas. Okay, so that's a very clear message, right? So when they even have it written explicitly in the paper, that actually the shape of the Optomechanical Induced Transparency peak repeats the response of the mechanical resonator. But if this is the case, that would be great. If you can take, you can classically bring the resonator to any kind of crazy motion and you would be able to use Optomechanical Induced Transparency to basically image this motion. That that's much easier than to observe the resonator directly. Because it oscillates somewhere, you need something to observe it directly. It's not easy, I mean that requires some additional things in the setup and here you have it just easily, okay? Now those are the Delta experiments from Gary Steele group, which I'm also involved. That's exactly the same thing with the exception which I already mentioned. They have a single port cavity, so they don't have transmission, they only have reflection. So their cavity is the Optomechanical Induced Transparency is actually Optomechanical Induced Reflection and that's not a resonance upwards, it's a resonance downwards and the deep here. But other than that, it's the same. So we could just take this one, take a mirror image and get the result. And you see what, this is what they observe experimentally and that's not at all a Duffing response. That's something which is clearly different. I mean, let me maybe sketch it. Duffing would be that and then it goes down, otherwise it would just go like this. So Duffing would be that and what they see is this, which is not Duffing. Okay, now that's why they came to us and we made a theory. And the theory I could have put it here because it's actually based on input-output relations and Clemens even reiterated, not probably having that in mind, but reiterated that input-output relations are very general. Whether you have a linear system, whether you have a non-linear system, whatever system you have, you can dump everything in and if you dump it in and if you are able to solve it, you will always get the result. And that's exactly what we have done. We have just in this input-output relations, we just put the Duffing force acting on the resonator and then we have solved it. We couldn't solve it exactly because it's non-linear, but we could solve it in the same sense. I have just shown you how you solve the Duffing oscillator. If you don't care about the total response, but you only care the response of the frequency you drive, you can do this, you can go to the rotating frame, you can do an average, you can go to rotating wave approximation and that's exactly what we have done, but not on the level of the Duffing equation, but on the level of the input-output relations. That's a bit too cumbersome, so I'm not going to show it here, but we learned one thing which kind of helped us to explain why we have this response. We learned one thing and the thing is that this is the response for the amplitude of the oscillator, but the oscillator doesn't just have an amplitude, it also has a phase and the phase is also doing some non-trivial things. Like for instance, we know, I don't have a picture here, but we know that for a linear oscillator, if you go through the resonance, the phase goes, the phase jumps by pi and he does something even more crazy. And if you take this phase into account, you come to, you can theoretically reproduce those results. Let's see whether I have it on the, yes, this is our theoretical course. Okay, now that's great, but then of course we are in the situation and we have two experiments which explicitly contradict to each other, right? We have that by Tobias Kippenberg and we have that by Gary Steele and Libertin has the first answer to measure that. And then what you do? Well, you start looking whether the experiments are actually done, have been done under the same conditions. Well, I mean, they will not have done under the same conditions because I didn't mention that, but this is an optical cavity and this is a microwave cavity, but still, well, I mean, it's still a matter of parameters. So I mean, if you have a correct expression, you have different frequencies, whatever, I mean, that's fine. That should not be a reason why the results are different. If you do everything properly and if you should get the correct results. Now it turns out that the difference is actually this coupling parameter. Remember, I spent some time on my last lecture and I think other, you can see it also in other lectures. So there is, there are losses and there are losses in the cavity and there are losses in the external circuit, which in this case is a wave current. And to have this parameter, I believe I had, I call it either C last time, nevermind, which is this ratio, which is always between zero and one. That's called coupling. It's not the same as optomechanical coupling. It's a coupling to the external circuit. It's something else that has nothing to do with optomechanical coupling. And if this is exactly equal one-half, it's called optimally coupled, if it's below one-half, it's under-coupled and if it's above one-half, it's over-coupled. I mentioned this last time, but now I'm repeating it. And the delt cavity is over-coupled. The Munich cavity was under-coupled. And then we have actually repeated our theory also for under-coupled cavities because in the theory, I mean what we do, we just do all the calculations and then in the end, we go to the experimentalists and take the parameters they used and try to fit. And of course, we're using the delt parameters. Now we have also repeated the calculation. I have taken the Munich parameters and that's what we get. That's the theoretical results. Let's look only on the left panel. So if you have an over-coupled cavity, that's those are the pictures for different drive and that's reproduce the, I know, I lost it somewhere. Let's reproduce this one. So that's the delt experiment. Now, theoretically, they haven't seen it in the experiment but theoretically if you drive harder, at some point it flips. So instead of being, in this case, optomechanically induced reflection, it becomes optomechanically induced absorption and then it becomes ducking shaped. And if you start from an over, under-coupled cavity, it's from the very beginning ducking shaped and that the Munich results. And then we have also done, that has not been done experimentally but we have also done the calculation for the case you drive it at the blue side bend and we see that the opposite. So for an over-coupled cavity, it would be from the very beginning ducking shaped and slanted on the left. For the under-coupled, it would be first this funny shape like that and if you drive it harder, it flips and becomes ducking. So it's just total opposite to the over-coupled. Okay, now why I'm talking about that, right? It looks like some minor detail you know, coupling to an external circuit. Why the hell should anybody ever care about coupling to an external circuit even if it flips the shape? Well, because there is some physics here and the physics is nice. And let me explain the physics. Now let's plot first before any optomechanically induced transparency. Let me plot the minimum transmission or minimum reflection as a function of this parameter. Again, the plot is taken from somewhere. So this copper internal is what I call here what I call here copper, nevermind. And this minimum transmission or minimum reflection depends on this copper and it becomes zero when the two losses are the same. When this copper external equals to copper internal. I think for experimentalists it's probably easy because that's impedance matching. For theorists, I don't have a very good argument but for instance you can think about a transmission through a Fabry-Perot cavity and then a Fabry-Perot cavity if both transmissions of both mirrors are the same you get the maximum, you get one. So something always happens if you have these two elements something always happens if the elements are the same. And that's kind of the same thing. I mean, I can formally derive it. I don't have a good hand-waving explanation why it should go to zero but it definitely has a minimum and that just you can derive it, you can try to argue it but that's a fact. Now, remember I showed this formula last time and actually Clemens was talking a lot about that and then derived it. Now the line width of the cavity which are the losses in the cavity is renormalized by the optomechanical coupling and that goes to plus and minus and I think plus is for red detuned drive and minus is for blue detuned drive and this correction is related to cooperativity. And remember last time I used that expression saying, okay, now we have to compare this correction to couple itself and if this becomes too big, then and this means cooperativity becomes one because it's exactly equal to couple when the cooperativity is one. So if cooperativity is one, then as the blue detuned drive, we get copper which is zero, renormalized copper which is zero and that's instability and Clemens spent quite some time explaining what do you have in this instability regime and what happens. Now, okay, so if we change this copper, then it means that for the, our system for the red detuned drive always moves to the right and the harder we drive, the further it moves to the right. Now, if you write it as the blue detuned, it moves to the left. Now, under coupled cavities, sorry, I'm probably screwing up things. I would need to check, I could have seen that that's actually, it's probably external losses. I would need to check, but anyway, so the corrections of errors is correct. So if they are under coupled, we should be here and then if they are under coupled and red detuned, we're only moving that side and nothing happens. I mean, quantitatively we change things but qualitatively we don't change anything. Now, if they are under coupled and blue detuned, then we are moving that side and here horrible things happen. So we first pass through the zero, we go on the other side, so here the slope is different, physics is different and that's where we actually switch from that shape to that shape. And then if you drive even harder, which not has been shown in that experiment but we know what should happen, then we heat up the instability and then we have amplification. Now, if you start and that's exactly what our theory gives and that's also why for an under coupled cavity and in the unit experiment, they only see one shape and this is a different shape. Now, if you start from over couples, if you blue detuned and drive it, then it just goes at some point amplification but nothing happens in between. But if you red detuned drive, it goes down, goes here, so here we switch from one shape to another one and flip the, flip the, flip the peak. And then it goes here. And that explains why actually those two have very different physics and that results in this case and all these strange shapes. Good. Okay, if you got lost, now you get a second chance because now I will be talking about non-linear cavities. And as I mentioned, an optical cavity, we just, I think, safely say it's very difficult to make non-linear. So I will be talking specifically about microwave cavities and I mentioned last time that microwave cavities are typically superconducting. For some of you know very well what superconductivity is but for others it's just a buzzword. So I will spend a little bit of time explaining what superconductivity is and what the Josephson effect is. If you know that, just skip the next three slides. If you don't know, listen, that would be helpful. I will not explain all things. That's mostly the textbook material. I will put on slides more than I want to explain and slides will be online in an hour. So you can check it as a first reference and if you need to know more, you really need to go to solid state courses and read. So that's very brief crash course in superconductivity and Josephson effect. Now superconductivity is a state of matter different from a metal. So if you take a metal, some metals at very low temperature lose electric resistance and they really lose electric resistance. It not just becomes so small that you cannot measure it, it really becomes zero. And that's a phase transition into a superconducting state. So this stated at zero temperature, very low temperatures is superconducting. And temperatures are really low. I mean for aluminum it would be 1.2 Kelvin I think and for niobium it would be nine Kelvin and those are I think for molybdenum, rhenium, those are materials you typically work with in this business. That would be like, I don't know, also like about 10 Kelvin. There are superconductors which have much higher, much higher transition temperatures, but they are not interesting from the point of view of those applications. Now that has been very well understood they have been noble prices given out, I will not go into details. What is important are two things that first of all a mechanism of the superconductivity is that in a superconductor you don't have three electrons and the electrons are bound in pairs. They're called Cooper pairs and each pair contains two electrons. And so the charge of a pair is double electron charge. And another one is that I said so that's a new kind of state, that's a coherent state. Not coherent state in a sense of that or not immediately, but that's a macroscopic state where you have the whole thing is coherent. And because it's coherent it is described by a single phase. So you can introduce something which is called a nodoparameter or you can also call it a gap. And what is important for me is that this nodoparameter gap has a phase phi. So if you have a piece of superconductor this phase is very well defined. It also has an amplitude which tells you how close the system is at the transition and the transition disappears and then in the noble metal this gap is in. Okay, that I said it's a crash course in superconductivity. Now let's go to Josephson junction. Josephson junction is if you have two superconductors which are put next to each other and they are separated by some barrier. In the simplest case by an insulated barrier. Technically you basically take a piece of aluminum and you oxidize it and then you put another piece of aluminum and that's your junction. Now those two superconductors could have different phases. And I call them here minus phi over two and phi over two. So what we are going to be operating is the phase difference. That one minus that one and that would be phi. Now that was an insight of Josephson for which he got a noble price and then that's called Josephson effect is that actually if those phases are different then there is a current going through the junction. And this current is going through the junction in the absence of any voltage so that dissipation less current. So you have current but you don't have ohmic losses. Which is maybe not so much surprising because also in superconductor you could have currents but you don't have ohmic losses because you don't have resistance but here you do have resistance in the interface but still you could have current without losses. And now you can ask okay what this current should be how it should depend on the phase. And your first idea that in any case it should be periodic because if you rotate a phase by two pi nothing changes. And so in any case whatever you do this current must be periodic function of the phase. Now what it exactly is depends a little bit but for that you can convince yourself that you can write the energy of the junction and the energy would be cosine of the phase difference. And the energy has a minimum at phi equals zero if I've written it like that and that means the junction wants to be at zero phase difference and that's an equilibrium. And if you take it out of equilibrium by putting different phases it wants to go back to equilibrium and then you will have current flowing through the junction. And now I didn't mention you can you come to that expression by saying okay maybe we have that's a bit hand waving I mean you can do it much more precisely for which I have no time but you can say well you have cooper pairs and maybe a cooper pair can just go maybe you can have a cooper pair which penetrates from one superconductor to another superconductor. And then if you start reasoning in this way then you come to the conclusion that that should be the overlap of those cooper pairs and the overlap would be proportional to the cosine. Now I will show you what the current is on the next slide but there is one more important thing which I need. I need this relation. So and this relation tells you that actually those phases are related to voltage. And remember I just said that if you have if you have different phases then the junction goes out of equilibrium and then we have current. But voltage is also a way of putting the junction out of equilibrium. And then this shows and there is this gauge argument which I will not go through that the gauge invariance argument which tells you that actually the time derivative of this phase is proportional to voltage. So those two things putting phase difference and putting voltage across the junction are almost the same. And this is called the Josephson relation. You can also say okay let's just take junction and applied voltage. Normally if you have a normal junction that would mean well there is a current which is proportional to the voltage. But if it's not current you will just rotate the phase difference. Current is already there but you will be just rotating the phase difference. Right now once you have the energy which is phase dependent you can also use this gauge invariance argument and you can derive the current which in this case would be proportional to the sign of the phase. And that's it. So we have the junction. We have the phase. Current is proportional to the phase. So I see the constant which is related to the properties of the junction. It's actually proportional to the transparency of the junction absolute value squared. And the phase, if the phase is time dependent you also generate voltage. Now okay that's the simplest picture you have. Now you can start adding more design. Now I don't have it on my slides and I don't have too much time but let me still try to spend a little bit of time because I think that's important that you have seen it once. Now what we have derived is just basically a circuit element, right? So that the new circuit element Josephson junction and we know that this current is just proportional to the sign of the phase. Now in practice you have other stuff going on. For instance, one thing you also have electrical resistance. You cannot easily add it to the Hamiltonian as we know but it is there. You also have some charge accumulation. And this charge accumulation is described by a capacitor. There is somewhere some capacitance around. And what people commonly do, they use the mod, so-called model of resistively. I will write it first, okay. Resistively and capacitively shunt of junction and which is abbreviates as CSG resistively, capacitively shunt of junction sometimes in the literature you see just RSJ, whatever. That's all about the same thing. People will say, okay, let's just forget about physics. Let's stupidly describe it as a circuit element. Okay, so we have, well, junction are just three things in parallel because they all happen at the same time. So they should be in parallel. You have this Josephson junction proper. You have resistance and you have capacitance. And let's see what the junction is doing. And let's put some current and the current must generate some phase. Let's see what phase does it generate. So the current equals, first of all, because that's a circuit, it equals to the sum of the three. So it equals to Josephson current. I will explain in a moment what I mean. The current through the resistance and the current through the capacitor. So Josephson current, we know it's just IC times sine J, sine pi, sorry. We have capacitance, resistance, that would be just voltage, whatever it means, divided by the resistance. And we have capacitance, which is CV dot, CV dot, right? And we know that the is proportional, the coefficient is over there, is proportional to phi dot. So in fact, this term is phi dot and this term is phi double dot with some coefficients, which I can derive if I take that integral. Okay, that actually means a lot of things. So we have that equation, right? So we have phi double dot times something plus phi dot times something. Equals to I sine phi minus this current. Now, basically we are, well, some of us are theorists and one of the kind of main part of the job description, you should be able, if you see an equation, you have in some new, in some setting, which is new for you, you should be able to figure out what does it look like. Do you know any other systems which are described by the same equation? Okay, and we do know. For instance, let's just to make it kind of more familiar, let's write an equation where this phi would be x, right? It would be x double dot times something, every dot times something, and some function, which I will call f of x, which in this case, well, that equals zero, and this f is this I sine phi minus I. All right, and if you look at that equation, I mean that you would probably be all able to recognize it, right? That's an equation of motion for a particle. That's a mass term, that's a friction term, and that's the force which acts on the particle, which is the derivative of the potential energy, and what the energy would be to give this force. The force is Ic sine phi minus I. The potential would be Ic, indeed, for sine phi, with the minus, because it's, no, okay, let's write it like that. Minus I phi, right? This gives me constant, and this gives me sine, and I should write this force as minus d, over df, ddf, so that there should be minus, which, okay? And now if I plot it, how it looks like depends on the relation between I and Ic, because I have in the energy, I have Ic and I, external current and the Jordanian current, and if I is less than Ic, I have that, I have that, and if I is bigger than Ic, I have something like that, right? And that means what does, now we understand what the particle is doing, right? Here, just moving down. It goes to the minimum, the potential, potential doesn't have minimum, just going down. Here, it's sitting here and oscillates, maybe in a minimum, but to go here, maybe it should really go through that maximum. Now, what does it mean? Okay, that's phi, that means that in this case, phi dot, which is voltage, is not zero, so the system generates finite voltage, and here, maybe at any moment voltage is not zero, but if we average it out, it's zero. So there is no voltage in this case, in this case, and voltage in this case. If you want to generate voltage here, you need to go through the maximum. And that's not easy, I mean, you need to supply energy to the system, for instance, at finite temperature, whatever, but that becomes, becomes. And another thing is that actually what it is doing in the maximum, in the minimum, is determined by the mass term. So this capacitance has a role of a mass of a particle, and you can actually, if you write the energy, you choose all correct, you add all correct coefficients, and you see that you have an energy of the Josephson junction, which is proportional to phi dot squared with this coefficient, which is proportional to the capacitance, and this is like a mass term, that's like inertia term. All right, now I, okay, I have to speed up a little bit. So now one thing which is important is that, remember I mentioned that we will, we will be talking about inductive coupling, and we will in a couple of slides, but let me first show that if we have this expression, we can actually easily derive the inductance of the Josephson junction, right? Because I just tried those two equations, that's non-linear, so that's not useful. If you want to define inductance, we need to go to the linear regime. Linear regime, meaning we replace, for the time being, sine with phi. So we have this current, we take the derivative of this expression, and on the left I have i dot, on the right I have phi dot, which I convert into voltage, and we get a relation between i dot and voltage. And we should, I have it on my slides. So the general relation is that i dot is inductance times voltage, and we just read out inductance, and our inductance is just that. So the Josephson junction has an inductance. That's useful because they're going to give. Now that's a squeak. That's a system of two Josephson junctions. We just make a loop. And that's for simplicity, take two identical Josephson junctions. You can also do it if they're different, but for simplicity I'll keep them identical. Now that's called DC-squit actually. Squit is for superconducting quantum interference device. There is also something else, which is called RF-squit, and that's a different thing. I'm not going to talk about that. So for me, squit is DC-squit. Now let's just take these two junctions and see what would be the total current through the squit, so from here to here. And that's obviously the current through the left branch and through the right branch. So we just write it like that. Now what are those phases? Okay, one thing which we know for sure that if you walk around, then we get some phases in the superconductors, and here some phases in the superconductors, and also if you put magnetic field through the squit, magnetic flux, we should also get phase, which originates from these flux. And we know that the total phase increment should be zero or maybe two pi times an integer because we should come to the same physical state of the system. Let's say zero, so it's irrelevant whether it's which integer they take. And now if we just calculate everything, so if we go through the junction, we get a phase difference. If we go through that junction, this way we get minus pi two because we're going in the opposite direction. And magnetic, the total magnetic phase we get would be actually equal to two pi magnetic flux divided by phi naught. And this phi naught I've already had it on my first lecture. This phi naught is a combination of fundamental constants and that's different from this phi naught which I have on the first lecture by a factor of two here because here e now is not a charge of an electron or to charge, so this e is a charge of an electron, but what we should have in the flux quantum is a charge of the Cooper pair, which is doubled. And that's why this superconducting flux quantum is twice as small as the normal flux. Okay, now if you plug it in, that's the expression which we get for the current. So this phi is phi one plus phi two over two, so I should have it on the slide. So it acts like a Josephson junction, so also it's proportional to the sine of the space times some amplitude. And this amplitude is two times the critical current of a junction times cosine of this pi times pi divided by pi naught. And that's actually why those quits were originally developed for precise measurements of magnetic field because you can measure magnetic flux with the precision of phi naught, which is a very small number, but flux is field times area. So if you take a very big loop, the only thing you have to care about that the whole thing is coherent, you don't lose coherence when you go around the loop. If you can make it meter by meter and it would be stick coherent, it's still fine. And then you get very precise measurement of magnetic field and that's why it was designed. It's actually used, it's still used for that. I mean, there are the scanning quits, you can measure magnetic fields locally very precisely, whatever. Okay, now we are ready for inductive coupling because now we have everything. So imagine I have this quid loop and I have flux and the flux is not perpendicular to the area but has a little bit of an angle. And I have part of this loop which is mechanically compliant so it can oscillate. And if it oscillates, it actually changes the area which the magnetic field sees to form the flux. So the flux would be dependent on the position of this mechanical part. And then if it's dependent on the position of the mechanical flux, but sorry, I have inductance which is remember it's proportional to the critical current and now it's proportional to this five which depends on X. And this is inductive coupling. So if I change the position of the resonator, I change the coupling and then just, yeah. So I change the flux and it couples to the, to the critical current. That's an idea which was proposed in those two papers by Joan Miesel and by Gell-Bupps and Miles Ben-Pau. Miles will be next week here. That's an experiment which shows that that's from Del from the group of Harry-Hundersand. That's actually this quid loop. Here maybe if you have sharp eyes you see a Josephson junction. Here is another one. So this white line. The whole thing is suspended. I mean, they started with something a bit less fancy and that was the most advanced device they had. And this is, well, actually that's on a different device but the same idea, different paper. What they did, they could drive the resonator mechanically. So they had a piezo substrate and they could drive it with some frequency and then they measured the voltage across this quid. And then they see in the voltage they have a peak and the peak as a function of frequency and then identified the position of the peak with the frequency of the resonator which is two megahertz and perfectly makes sense. And also they could get a quality factor which was in that case 18,000 and that's also they could also measure the phase which is doing exactly what the phase of a driven resonator should be. Now I probably, let me skip that, that's nice stuff but I will spend more time explaining that. And I have like three minutes. Let me give you a little bit of perspective. I mean, all these things I showed you for the squid or things I didn't show you for the squid, they are all DC. Well, I know DC, we have seen it's two megahertz but a typical frequency of a squid, well, squid is now our cavity. It's some fancy cavity but it's a cavity. We can operate it in a linear regime and we know that the frequency of the squid would be in the bigger range. So the mechanical resonator is much, much slower than the squid. And we operate our cavity very far from the cavity resonance. So we operate it with the frequency of the mechanical resonator. And you see things I used, I mean, why I had to, why I had to spend 20 minutes and use the whole boat is because that's not actually optomechanics. It's completely different physics. It's very nice physics, maybe even nicer than optomechanics but it's not optomechanics. Now let's see what happens if we add Josephson junctions to the cavity for simplicity if we just take a squid, whatever. What can we expect? Nobody ever, as far as I know, measured that at the frequencies of the cavity. So that this non-linear optomechanics in terms of the cavity with Josephson junction just experimentally doesn't exist. Let's, in the last couple of slides let's try to understand what we can expect. And I only have time for very simple things. I don't even have time for simple things So let's first of all quantize. So we know what the energy is, we know how we quantize it. Right, and one thing is we have this Hamiltonian for the cavity, which is usual Hamiltonian but we also add this Josephson energy and the Josephson energy is difficult to quantize. We can do it but let's expand it and let's treat it at the level of the of the Duffing Oscillator and quantize and Mark mentioned that what we get is called the quantum mechanics care Hamiltonian. So this care Hamiltonian is that that basically we know that a dagger a is number of photons so that essentially number of photons squared up to some commutation relations. So we know everything about the cavity. Now interaction, you cannot quantize. Because interaction you have cosine and if you start expanding things it doesn't work. So you cannot quantize it generally but you can't quantize it in two limiting cases. You can't quantize it for the case we just treated. So when the cavity is very slow and then what you get is a beam splitter interaction and you get the non-linear term which is we call cross care interaction. So that's number of photons times number of photons. Now the funny things here is first of all beam splitter you usually get by linearizing radiation pressure. Here it's not, you just get it properly from the Hamiltonian. Another thing it's actually disappears if your squint is symmetric if the two junctions are the same. So you have to do something more complicated you have to take different junctions. And that's actually part of the reasons why these physics are explained to you is different from cavities. It's something else, I mean it's not a cavity. Now if you go to the cavity frequencies you can also quantize it and then you properly get radiation pressure with the constant coupling which depends on the parameters of the squint and actually by all theoretical calculations you can go to very strong coupling. And again you have this cross care which is the simplest non-linear interaction. So at this regime it should operate as a non-linear cavity. You don't have non-linear interaction and you have here in the non-linearity of the cavity and actually all those parameters are if you're thinking about the flux cavity and the squint cavity all those parameters should be tunable by the magnetic flux which is additionally. I have one slide. I apologize again for my drawing that I had to do yesterday night and okay. So what should we expect if we just drive the cavity? I mean in the first instance okay if the cavity is linear we just get a Lorentzian. Now this is a non-linear cavity. We should get something which responds as a duffing oscillator which is supposed to be staying in the middle. So that part is unstable and this is what we get. Now we know that we should also get a mechanical sub-bends in a linear oscillator if it's coupled to a mechanical resonator. How do the mechanical sub-bends look like? Naively you could say maybe each of them is also a duffing response. No, nothing like that because duffing response is only in the limited range of frequencies. If you go very far here it's not duffing. It's a tail from the Lorentzian peak but it's the same. Like if you go very far from the resonance you cannot say whether it's Lorentzian or whether it's duffing. And those things, well I mean if you have a good cavity if it's very well resolved those things are sitting at the tails and then they should be Lorentzian. And maybe then if you increase the drive maybe that one would actually merge with those side bends and then you will have something completely different and that would be different physics which I don't know anything about. And the last thing and that's really my last two slides and I will not go through the details of the slide because I have no time. There was an absolutely brilliant paper 1986 by Jochen Stoller. You know 1986 that was way before any, well I mean that was already, Optomechanics already existed but people were not thinking about quantum state transfer and they were not thinking about quantum state transfer they were thinking about something completely different. They were thinking about some non-linear medium and propagation of waves and what happens, well I mean that was at Bell Labs and you know they're interested in whatever cables and all this kind of stuff. So doesn't matter. The problem they solved is they have taken that Hamiltonian so that's a usual cavity. There is no mechanical resonator only cavity and they didn't even call it cavity and that generalized care. So you have this power P which is an integer and for P equal two it would be care. And now you can say okay fine let's start with the Cochirin state and let's see what this state is doing as a result of evolution with this Hamiltonian. And you can solve it with the exaction. And this is the result. So this function is a function of time. The only thing that changes you can see is the face which is IK number of photons which is in the Fox state to the power P times time. And P is integer and that means that this function is periodic. So if you take time and time plus two pi over K that the same. So two pi over K is a period of the oscillations of that. And now if you look what happens after a quarter of a period you just take this time which is pi over two K you get a K state. So you get this combination which, well you get something which you can rewrite as that. So you have a Cochirin state with alpha and another Cochirin state with minus alpha with some phase factors and this plus. So it means that specifically for even P so that already for K that's already for K if it's odd P there is something else. So it means if you have a K interaction in a cavity you just take a cavity, wait for quarter of a period and instead of a swap you create a K state and that should be doable with the Josephson junction and that has not been done. And then you just take the same cavity and transfer to mechanical result which is conceptually kind of easier than first preparing cavity in a crazy state and then trying to transfer this crazy state. That's as I said, so that's my last slide. I'm done with that. As I reiterate that none of this last part has been ever experimentally done. Maybe one of you will do it at some point. That's something which I believe will be kind of expanding. Right, thank you. Yeah, yeah, but not transfer to the mechanical resonator. Sorry, I probably didn't formulate it correctly. I actually mentioned that in the very beginning, but yes. Sorry. But there may be a few teams where only one person or maybe two of them are there next week and those teams will support that and I think that's a good point. Sure. So I don't know if this is kind of weird. Well actually, the simpler question first, that differential equation for five, that looks just like a damn pendulum, right? It is. Okay. So I know that's a hard differential equation as well, but since we've been thinking about damn pendulum for a long time, we kind of know everything we need to know about these. Yeah, that equation has been solved. Okay. You can. Interference with size five in it. Yeah. So you can, first of all, let me switch it off. Overdump function? Because then it becomes first order and basically, I even wanted to solve it here, but I didn't have time to do it. That's all known. Then if it's underdumped, you can solve it, but then it becomes treated and then there are some special functions involved. It's also in the literature, but it's more complicated than that. Now, if you really want to do both at the same time, I believe it still can be solved, but then you really need to give in the literature. It might be in the book of Steve's novels, maybe not in the novel, but definitely in the novel book, numerically. Actually, if it's already, it's underdumped, it starts getting complicated for a bit, it starts getting complicated. But that should be all in the literature. Okay. Do you know what's underdumped? Oh, so I don't actually do this stuff, but it is super interesting. Yeah. So I just kind of want to, I recognize that equation from trying to solve, like in our classical mechanics course in college, I prefer we first spoke about the great propensalum and then we linearized sine five because it's hard to do when you're in college, I guess. But, okay. I mean, maybe one more thing about this. So in this description of five in this potential world, this fluctuations, can it handle through these things or is that like a weird thing to have? No, no, that's a good thing for us. That's a classical description. Okay. But then you can eventually quantize it, then that's actually why I needed this kinetic energy. Because without kinetic energy, it doesn't make sense. Okay. Yeah. If you quantize it, it's a little bit like, like in this kind of, it's a little bit like a little bit of a harmonic oscillator. And then indeed you actually provide and it will be some momentum in the field that's called microscopic momentum tunneling, and then you can indeed be able to generate more pressure. But that's a little bit of a particular thing. I think for junctions, it's pretty well known, but I can imagine there are, and then you're gonna go to more complexity. I think, I mean, I'll show you around, see if we should move on with this. I will put it down here. Okay. I'll do the, I'll put it down here. We could go and bring it to you by the way we're on it. I'll put it down here. Sure. I'll put it down here. I'll be messing with you, don't put it on the screen, right? It's with consistent two junctions. Yes. Yes. Let me see if you want me to do that. I'll give you a score here as well. I'll put it down here. Yeah, let's do it. We hope you can, I think we'll do it. But here, I mean, it's different. Yeah, that's probably correct, but we're going to drown it. I mean, we really want to put it down here. Yeah, put it down here. But the peak normally we call the, it's a big difference between the person here and the peak. Yeah, well, usually what you're doing, okay, my question, usually just like that, like this, and this is the contact. Now I want to do that. I'll give you a good start. That's my question. Yes. And here, the whole thing is fake. That's probably fake. Without the block, the whole thing would have the same things. And the whole other thing would also have the same things. Mm-hmm. But with the flux, with the flux, it'd be a problem. No, you know, this is like in a year here, don't you think? Yeah. Yeah, which is typical, which means also when you're going to go around, you have to pick up the face, which is... So, the next thing is actually, when you actually speak, don't you just have to say, you know, the next thing is actually the face, you can put it back here. If you put it back here, right? I mean, they have a bit of a synonym here. It's true. Or when you have to make them control the face, or control the face. How do you make it? There are only two things. I mean, one thing you can do is, even get what, that's why they do squeak, because then you control by flux. You can also, in principle, control by voltage. I mean, you can apply, for instance, AC voltage, and then the face could be voltage, but the voltage just makes them have that kind of difference. Yeah, you cannot put such a voltage that the face would be constant. That's not possible, because voltage is pi dot. If pi dot is constant, pi dot is equal. But you, in principle, you can still give a pulse and then the face will kind of pulse. But normally, for example, I don't know if you recommend me to separate the two sides of the jumps, but normally, it sounds a bit different with the other. I mean, then you do something with the junction. For instance, you pass current, and then you have all the story, and then the face is determined by the current. Or you apply voltage, and then the face is of the voltage, or you couple it to something which has happened, it's different, different state, and then you, for example, take different state. Okay, one last question about humans, when you came with drugs, this non-linear reason, you knew high-duty in this region, so you think it's a collection. So normally, it would be that one job, high-dependence, high-dependence for the whole state, if I respond to it. So basically, Yeah, you kind of talk, it's a non-linear equation, you kind of talk with the detector. No, no. It's written to this, that for me, we cannot talk, and that's the problem. No, no. You can actually, you can start asking, I mean, for instance, you can say, good, you can thrown away some films, what is the role of those films? And maybe you can take them into account, as participation, and see what they do. That's a reasonable question. But you cannot do it exactly, I'm not sure you can do it in America, but analytically, it's just all over, like that. That's what we do, okay. But it could be the equipment for transportation, because after you do this, you're just some, I can only do some, yeah, after you've done that, it's all. But you can still say, okay, maybe you are not happy with the rotation. Maybe you want to take those fast, which are taking something to account. Probably not important in this process, but it's, for instance, if you have like open mechanics, they also, you also have something to do, or if you did, you can also, for some point, give you the right-hand data approximation, and then you throw away some terms, which could be in some situation, but that's okay. So that's the most effective, that's how you make it, or just pay and let them do it. What do you mean by, how do you keep? Are we talking about interaction, or are we talking about the very last part? I mentioned it on two, I mentioned lots of say transfer in two contexts. First in the very beginning, then I was talking about non-linear interaction, and then on the very end, on my last slide, then I had this character, which one I asked about, or were you talking about both? How is it? Okay. Okay, now efficiency, I mean, ideally, in theory, of course, you have found it, I think your story is 100% for the original problem. With the non-linear interaction, I draw some estimates, which I don't have all the top of my head, but you have very high probability. But then, of course, once you go to the experiment, then you have all kinds of pictures. Yeah, yeah. But in the last two slides, I even didn't take the 100% high high. This is your story, without mechanical reasoning, for example. It's already, if you have a cavity with the additional monomeric of the cavity, and then you just let it evolve, you already create this aspect. And then the idea is that if on top of that, which I didn't show, you had it in the same place. So when you do that, it's great as a non-linear problem. Mm-hmm. It's more near the course of the experiment. No, no, no, it's near the course, in the first instance, the course of time point. Oh, because of the, oh, like that. Because of the non-linear theory. You can also, of course, in the tradition, you can also take the mechanical process. First thing, that's the course of time point. That's also in the unknown, maybe the, I mean, it's still there for a while. No, no, but the time fly also gives you a non-linearity of the cavity, because we know that. And then you also have a linear coupling, but the biggest effect is the coupling. They need to be resolved upon the function of the linear coupling. But the cavity can become more than in the weekend. And in this field, then we, okay, I'm sorry. I'm sorry. Well, well. It's super. Yeah, it's awesome. Exactly. It's still tunable, so you could have some situations when you can tune linear interaction to the limit. And then it will be only non-linear, but it's still there for a long time. Well, I mean, you know, yeah, okay. Linear is actually common. That's just my numbers. But non-linear, I, all numbers I have ever seen, but if it's weak, and it's lonely, I think it is. I think it is. Yes. I've been talking about this radiation pressure, which is non-linear. Yes. Okay. So really, it's very, it's very, it's very, it's very, it's very, it's very, it's very,