 if my microphone is placed moderately OK. I'm going to continue as soon as I find my laser pointer. Continue a bit where I left off in the previous lecture, but I think also very nicely from things we heard this morning, especially Clemens' nice derivation of input output theory and thinking about measurements and cooling. I'm going to reiterate much of that same story. Again, give slightly my own language, show some more experimental pictures to go with it and some more data of what some of the experiments look like. As always, jump in, flag me down if you have questions. So in general, part of my goals today, last time I talked mostly about driving the red sideband and the things you could do there. Things like swapping or cooling or damping or normal mode splitting. Much of today's lecture is going to be doing in many ways the simplest possible thing, driving exactly on resonance and talking about the similarities and differences there, what you can achieve, how you think about various standard quantum limits, and also how you think about in-cavity optomechanics, we constantly have two subsystems, the cavity mode and the mechanical mode. And often a lot of the confusion comes from the fact that you can ask many different questions. You can constantly flip around who's being measured, who's the probe, who's the meter. So I'll talk about using light to measure the mechanics and then kind of turning the same experiment on its ear to think about the mechanics measuring the light. So I'll reiterate kind of the same Heisenberg microscope type picture. This is the canonical kind of measurement picture. You have some system of interest. You want to acquire information about it. Therefore, you have to interact with it. I like to think about interactions in this scattering type picture. You scatter something off of it that acquires information and necessarily disturbs it. And that is the canonical Heisenberg microscope. And cavity optomechanics is really kind of cleanly realizing this, where the system of interest is our beautifully nanofabricated mass, the thing we want. And the thing we're going to shine off of it are photons of some frequency, be it microwave or optical. We'll put cavities around things to enhance some of these effects to really get these things up. But most of what we can think about is measurement rates are really deeply related to these same kind of rates we see over and over, the things that are quantified by our cooperativities and everything that we've been talking about. So in these nice, clean measurements, we can think very nicely about the photon field as the only thing that interacts. It's our beautiful, delicate probe that stands between our classical world and this delicate mechanical object. And ideally, if we're doing everything right, the only noise source should be the fundamental quantum noise of the light interacting with the system of interest. So if you want to do things right, you take your nice, clean probe. You make sure it interacts cleanly, only in the way that you want. And hopefully then just driving the photon field will be able to turn up cleanly your measurement strength to tune it and optimize to whatever your goals may be. And as we'll see throughout the talk, there's some regions where you might want a weak probe. You might want to really impedance match your probe. In other words, tune your measurement rate to match other rates in the problem or many other systems you actually want to be in the very overwhelmed regime. Measure very, very strongly. So this is the cartoon picture most people throw up when they talk about the SQL or standard quantum limit. Again, I'll throw the caveat, the standard quantum limit can mean a lot of things to a lot of different people. So I'll try to be very specific what I'm referring to here. In the cavity-optim mechanics community, when we draw this kind of plot, the x-axis is something about our measurement strength. And that is usually just our power of our coherent state that we're using to probe the system. On the y-axis is the total sensitivity of the thing you're measuring. And here, the thing we're measuring is the mechanical degree of freedom. Now, whether you call it the intra-cavity mode operators or the force operators dig on the mechanical mode, that can depend on your point of view. But ideally, from the output of your measurement, all the information you get, every source of noise, you trace out this kind of curve. And what's implicit here on the y-axis is we put this in some sort of mechanical units. Or we've already kind of referred this transduction back to the input of the sensitivity of our measurement. If we probe very, very weakly, as I said, you acquire very little information. Most of what you see at the output is uncorrelated noise that has nothing to do with the system you're measuring. It's just the noise of the light. As you turn up the strength of your coherent state, you can measure its fluctuations better and better. You can beat down this imprecision. But eventually, you start to disturb the thing of interest. And that's this trade-off. The one goes up and the one goes down in a very evocative Heisenberg way. And somewhere here in the middle, there's a balance where you can think of your measurement tool as adding the minimum amount of noise. So to me, in this context, when I say standard quantum limit, I mean measuring the mechanical variables and thinking about your cavity probe as your ideal measurement, which if it's acquiring simultaneous information about the two quadratures of the mechanics, the sine and the cosine, the phase and the amplitude, position and momentum, however you want to say it, then Cave's theorem tells us that we have to fundamentally add noise. If we, what we've done, whether we know it or not, is made an ideal phase insensitive amplifier. So our total measurement has to at least double the vacuum noise. That's the simplest way I can think about stating Cave's theorem without going into factors of 2 and nomenclature. And so if we're doing it right, right at this perfect balance, we'll have half of our added noise of our measurement from the imprecision, half of our added noise of our measurement from the back action, and together, they will equal the vacuum noise. And that's, I say standard quantum limit, really meaning that's the limit, that's the best I can do if I'm going to acquire simultaneous information about the two canonically conjugate variables of the mechanical degrees of freedom. As we've heard throughout, LIGO uses slightly different definitions depending on different contexts, depending on what you're measuring. But that's what I will mean when I say this throughout. So again, just to get back to notation and how we relate things in the lab, we will often throw things on a spectrum analyzer, which means we'll be looking at the power spectral density. This is just a way of characterizing the noise in the frequency domain by partitioning it up into every little frequency bin so we can really see where all the action is. And again, the definition of the spectral density of any quantity, it's the thing if I integrate over all frequencies, I have to get back to the total mean square fluctuations. So often when we're talking about sensitivity in mechanical units on the y-axis, we will put a displacement sensitivity in meters squared per hertz. Oftentimes, we might normalize it or put it in force units or something proportional to this. And what's most important here is not necessarily our definitions, but again, just to keep a track of what's the thing of interest we're trying to measure. So for example, often what we're trying to measure is a little mechanical Lorentzian. This is the thing that tells us about how the drum or what have you is vibrating around. And if we look at the total noise on resonance, we can see kind of four contributions that I'll go through here. So this white noise background, that's just the detector noise of our interferometer. If I'm thinking of a Fabry-Perot, these are just the phase fluctuations in the light, the normal shot noise of the light. The thing that if I turn up the coherent state more, I can beat down that phase noise. Has nothing to do with the thing I'm trying to infer, nothing to do with the mechanics. But if I were just looking blindly at the total noise on resonance, I can't necessarily tell the difference. One of the beauties of the power spectral density is we can look nicely off resonance. Look over here where there is no mechanical motion and really see that this is our white noise floor. And that's the clean way we can always separate out this imprecision. The next thing we look at is what is the peak height of this Lorentzian on top of that white noise background? Now, people will go back and forth between talking about the height of this or the area. I will use those somewhat synonymously because you can always go back and forth just by multiplying by the effective bandwidth of a Lorentzian, something that is just a mechanical line width, or I think vigorously it's like a gem over four. And so often what's useful in the lab in thinking about this total height is to know what you're measuring there. This is talking about the real motion of the mechanics. So that's any thermal motion it possesses, any driven motion from any external forces. It also includes any back action of your measurement, anything you've done to disturb it. And as I said before, because we're measuring both the sine and cosine components of the mechanical quadratures, we also measure the mechanical zero point motion. But just to be clear whether or not you measure the zero point motion, that can be tricky depending on exactly how you're pumping or probing what you're listening to. What I have in mind here is again driving a cavity optomechanical system exactly on resonance. So there's stokes and anti-stokes side bands. I'm going to put those into a homodyne detector where they coherently interfere with each other. And I will look at that linear combination that is the phase quadrature. In that phase quadrature, I will see all of these things. Because spectral densities are ugly both to write and to think about what we often do in papers and in the lab is just put these in terms of quantum units, put these in kind of dimensionless units. And you can do that simply as follows. The total noise of the spectral density on resonance has these four components, your imprecision, back action, thermal motion, and your zero point motion. Now these spectral densities, if you pull out all of your units, for example, just saying if you knew this Lorentzian corresponded to a drum vibrating with 100 quanta, you can ask how that 100 quanta, what the pre-factor would be between going from that quanta, the amount of energy in the mechanical oscillator back to the spectral density. Just to motivate the units here, what you see, well, the h bar shouldn't be so surprising. The h bar over m omega, that's something like a zero point motion or a zero point motion squared. And then the gamma m is something like an effective bandwidth. So without even thinking too hard, the natural pre-factor out front here is something like a zero point motion squared divided by the bandwidth of the mechanics being it's just natural line width. Now what's nice about these units is we put them already in terms of quanta. So for example, the zero point here, we already know that's going to be the half quantum that we define from quantum mechanics. And then what we can ask is how big or small can the rest of these terms be? So for example, I won't go into the details but you can read in any of the reviews, the review of modern physics, what we know is the imprecision and back action. They trade off with your measurement power. If one goes up, the other goes down. And there's going to be some minimum where they equal each other. In these units, the best you could possibly do is choose the power where the imprecision comes down to a quarter quanta. The back action rises to a quarter quanta. Hopefully you're working at zero temperature because you have some magical fridge or you found some way to pre-cool and then you measure the zero point motion. This would be the fundamental idea of the best you could possibly measure, a perfectly cold mechanical oscillator. And this is what it's often referred to as the standard quantum limit of measurement. This one half from the vacuum noise and another one half is the added noise of the measurement, exactly doubling it, giving you one full quanta or a standard quantum limit that goes here. So now what I'm going to do is try to touch base back with this picture of trade off of imprecision and back action and measurement strengths and see if we can connect back to some of the same rates we talked about in the last lecture. The rates of heating and cooling just from resolved sideband cooling from the stokes and anti-stokes scattering. And what was a little surprising to me even after being in the field for quite a bit, I always thought of these as totally separate. I would have my own way of deriving it in each case but really the coupled equations of motion fully contained the dynamical back action and the dynamical back action perfectly describes the cases of whether your pump is detuned to the red, detuned to the right or exactly on resonance in this kind of canonical measurement picture. So again, just to get the notation, right? I apologize, I think I have switched the plus and minus definition from the lecture you had this morning. I had good reasons for doing this at some point. I forget what it is. But just to be clear, I'll always refer to the upper sideband as the anti-stokes. It's the thing that does the cooling. It extracts energy from the mechanics to up convert a pump photon. The lower sideband is always going to be the stokes sideband and it does the reverse. It always adds energy to the mechanics and that gives you effectively heating. Of course, we're always going to put heating in quotes and we're going to know this is actually a quantum optical dual mode squeezing operation which is providing quanta to both the cavity and the mechanics simultaneously. So now if you were to calculate the scattering rates that you have here and I have very much the notation of the review of modern physics in mind or even Florian's PRL from back in 2007 you can think about this as just a fundamental scattering rate. You have the intrinsic number fluctuations in the cavity multiplied by the pre-factor that's the vacuum fluctuations here. And you see this very familiar looking form. This gamma is exactly the things we use to calculate the optical damping rates or anti-damping rates depending on whether you're red or blue detuned. It looks a little bit more complicated here and that's just because I haven't made any rotating wave approximation or assumed where I'm pumping. But just to reiterate, if I were pumping optimally red detuned that would be a delta of minus omega m. This term would cancel and I'd be left with the four G squared over kappa the thing we keep seeing over and over as the ideal damping rate after you've made a rotating wave approximation and ignored this other process. But just to be clear, in general we will always keep track of both first order sidebands. So we don't have to make a cavity rotating wave approximation. We don't have to assume our priority if we're in the good or bad cavity limit. And in fact, if we're gonna be driving on resins it doesn't matter if we're in the good or bad cavity limit. We have to keep track of both processes because they are equal. The filter of the cavity doesn't do us any good to throw one away. The last thing I wanna say here these gammas really are the damping rate. We would really, if we were pumping just as I've shown here damp the mechanics from its intrinsic line width to an effective line width of four G squared over kappa. But these are also literally the scattering rates. The rate at which you scatter photons out of your pump scattering off the phonons. So they are the scattering rate per phonon. That way they really are telling you how much light is getting scattered out of your pump to measure on your spectrum analyzer. It's also telling you how hard you're measuring the system because it's telling you how many of your pump photons are actually interacting with the phononic system of interest. So that's the other half of the problem why we really like to use these rate equations and to think about it in every sense of the word. So again, using this notation, as I said here, you have some optical damping. In principle, it's the difference between the stokes damping and the anti-stokes damping. And these two are defined with the plus and minus here just depending on where you are falling with respect to the Lorentzian of the cavity. Whether or not this sideband process is on or off resonant. Usually what we would say is in the good cavity limit, one of these is your 4g squared over kappa, and the other one is really, really negligible and you entirely ignore it. But that ability to ignore that other sideband is this thing that gives you a lot of fundamental physics, and that's a lot of what we're going to talk about. So for example, the other quantity you can think about other than just this optical damping rate is also this ratio of the damping rates. This gives you an effective temperature of your measurement. Now, this is really a weird concept. What I want to emphasize is you assume you have a cavity bath that's perfectly cold. It's at zero temperature. It has no entropy. It has nothing. But when you couple it to the mechanics in this parametric way with a strong pump, unless you can make 100% perfect rotating wave approximation, you will always have some stokes process. And that stokes process will always look like a little bit of extra energy. And in the way we normally do, we're kind of tracing over that energy. And it looks like extra noise or extra heating. This is precisely what people are referring to when they say the quantum limits of resolved sideband cooling. This is just from the fundamental noise that's there in a pure coherent state. But the fact that there is a little bit of stokes process, stokes scattering, if you balance that scattering rate, this kind of heating rate that you have to the total damping of the system, you get an effective temperature. And this is the thing that we'd love to say, again, if I were exactly red detuned and in the good cavity limit, this number can be far less than 1. And we're all happy. We can get as cold as we want. And if we need to get colder, well, we just need to make a narrower cavity or a larger mechanical frequency. But over and over, I just want to emphasize these two numbers kind of the optical damping and this effective optical temperature, both of which depend on where you're pumping because there's an explicit detuning in them, fully characterize the measurement and the interaction between these systems. So let's walk through it. Again, the cartoon picture I have in mind over and over, I have a mechanical system of interest. It's always coupled to its intrinsic environment through its intrinsic damping rate. That's what I ever gave it, its intrinsic Q. And the thing that's given it its temperature in the first place, if it didn't have some intrinsic Q, putting it in a priostat wouldn't do any good. Because if it doesn't have a Q, it can't couple to that environment. So this is just how I would normally describe a mechanical element coupled to some thermal environment, giving an equilibrium temperature of nm. Then we turn on our light. We pump somewhere. Depending on where we pump, we turn on some coupling to another bath. And that bath has an effective temperature. And again, I always talk about temperature in terms of occupancies, just to be clear. And that effective occupancy could be coming because this other bath is hot. I mean, if your cavity isn't fully cold, you can't get your mechanics any colder than that. But even if that cavity were perfectly cold, that residual stoke scattering provides an effective temperature. So again, this is the environment. And this is what I would call the measurement. Now, normally when we think about sideband cooling, we don't think of it necessarily as a measurement process. But over and over, I mean, what we see is that same pump we applied to cool the mechanics is the same thing that's giving us something to read out. So I really will use these kind of state preparation or measurement in the same language because you're doing both at the same time. And then in the steady state, you can do a simple detailed balance. You just take the temperature of each bath and you take a weighted sum, weighted by how strongly you're coupled to that bath. So you'll notice the denominator is the total damping the mechanic sees. It's intrinsic plus whatever anti-stokes cooling there is minus any stoke scattering that would anti-damp the system. So I get my total line width down here. Usually when we say sideband cooling in the resolved sideband regime, we're just looking at this first term here. We have our initial heating rate, which we've heard over and over, is the thing that's the environment throwing so many quanta per second at us. And then when we couple to another cold bath, we just dilute that thermal energy. And on average, the mechanics got much colder, colder by exactly the factor at which I over-damped it. What we need to keep track of, if we're going to be very rigorous, is this other term over here. Any effective temperature of that bath you're coupling to, if for any reason it's not perfectly cold, then even when I damp the hell out of it, I will still end up asymptoting to some other temperature. And that's this effective temperature of the bath. And that's kind of how we define this thing. It's the temperature the mechanics would have if somehow you turned off its intrinsic coupling to the environment if that were the only bath it were coupled to. And so this term over here, you can think about it in many ways. You can think of this as the limits of sideband cooling. You can also think of it as your measurement back action. Or you can also just directly think of it as the manifestation of the radiation pressure shot noise. So these are the terms we're going to look at. And for me, what's interesting is, again, look at it in the two special cases, where I'm optimally ready to tune in trying to cool, or where I'm zero detuned pumping on resonance and trying to measure. So let's set up a little chart. I think this is heavy as I will ever get with the equations. So if we look at this effective bath temperature, again, it's got this maybe not too complicated looking formula. What you notice is there is a detuning in the denominator. So for example, the usual case where we evaluate this, we say let's pump optimally ready tuned by a mechanical frequency. And this is the thing that goes to a cap over 4 omega m squared. This is the thing that in the resolve sideband limit is much less than 1. And if we ran into that, if you were cooling to that limit and hitting that limit, I would say you're hitting the fundamental back action limit of cooling. It's very analogous to the Doppler cooling limit in cold atoms. But again, if you look at what happens when delta goes to 0, well, delta goes to 0. This quantity blows up. It goes to infinity. Seems like a problem. Luckily, if we look at the other half of the problem, this kind of optical damping, these are the two terms. The damping from the stokes and the damping from the anti stokes should have pointed to them in the other order. And if you ask what is that when I'm optimally detuned, well, primarily it's this 4g squared over kappa. We see over and over. And then there's this small correction term. This is the correction term that goes to 1 in the good cavity limit. But of course, nobody's infinitely in the good cavity limit if you keep track of it. It's this. You don't get quite as much damping as you thought you would have because there's a little bit of stokes scattering, no matter how off-resident it is. Again, if you evaluate this exactly on resonance, well, that's by definition when these two damping rates are exactly the same. I've perfectly balanced my stokes and anti-stokes scattering. So there's no net damping from the system. I've left the mechanics back with its bare mechanical susceptibility. Even though I've dressed the interaction with this enormous pump, if I keep exactly on resonance, how exactly I'll go into it in a little more detail. If I'm precisely on resonance, the mechanics has its intrinsic line width with no excess damping. Now, as I said on the last slide, what you care is not about this or this. You care about the product. That is your back action. That is your measurement rate. And even though we have infinities and zeros over here, this kind of product that you have is always well-behaved and finite. So again, if I look at it when I'm trying to sideband cool, well, this is just reminding me of what my stokes scattering rate is. That's all this is, is a fancy way of writing what that stokes scattering is. And that's this kind of dual mode squeezing operation that's going on whether I want to or not because I have no way to shut it off. Now, if I drive exactly on resonance, well, it has this form. It's maybe a little more complicated than we're used to thinking about. In the denominator, there's this kappa over 4 omega m squared. Usually, people write this very much in the bad cavity limit because when you're driving on resonance, you don't want to filter any of the sidebands. And this is the thing that looks just like a 4G squared over kappa. This last little plus 4 omega m squared, that's keeping track of the fact that you're trying to scatter to sidebands that are slightly off resonant with the cavity because you're pumping on the center so these two sidebands are happening at the wings, and they will get a little bit of filtering. This is the quantity that, whether we know it or not, everyone's interested when they say they want to measure the radiation pressure shot noise or see the fundamental back action. What's interesting to note here, if what you are starved for is the coupling rate G, if that's what God gives you in your system and you can't make it any bigger, and all you can optimize is the rest of your parameters, if you want to see this back action, if you want to measure as strongly as possible, that leads you to work not in the good cavity limit or the bad cavity limit, but kind of optimally in between exactly where your sidebands are following at the half line width of your cavity. And again, this is just the balance between, if you go to a super low Q cavity, you lose any of the Q enhancement of the interaction of your cavity, but if you go very, very far to a very narrow cavity, you end up filtering those same interactions and just reducing the density of states for those scattering processes. So again, here's the picture that we'll have. We have some nice Lorenzian. Hopefully we're gonna tune it to have an optimal line width. We're gonna pump exactly on resonance. If you look at the spectrum coming directly out of the cavity, there's gonna be a stokes and anti-stokes sideband. That would be if you did some sort of direct measurement of the light or a heterodyne measurement. Oftentimes what we'll do is homodyne it with a local oscillator at the center, in which case we interfere the upper and lower sidebands. And we can now, instead of talking about this in the basis of upper and lower sidebands, we can talk about it in amplitude and phase quadratures. And again, what we're gonna do is engineer it to be right in between this good and bad cavity limit, kind of this medium cavity limit, which coincidentally is when where many of the experiments are for free. This kind of regime of having about this is where many people naturally are, even if they could engineer it to be differently. So if our goal in life is to say we wanna try to measure at the standard quantum limit, let's think about the things that are gonna make us miss this quantum limit. That's the platonic ideal of the best our measurement could ever do. So for example, if your cavity has any loss that's not directly out to the port you're gonna measure, not the light that's gonna go to your homodyne measurement, that's an infidelity. That's lost information. That's information that somebody else could have and you don't. So that means you would necessarily miss the quantum limit. Likewise, if your homodyne detection is not 100% efficient, you miss in the same way. Ideally, you should measure only the phase quadrature of the light and nothing else. As maybe we've seen this morning and I'll talk a little bit more about when you drive exactly on resonance, all of the mechanical information is pure phase modulation. And that comes from the dispersive interaction that you have. In a nicely over-coupled cavity, the mechanical motion just phase modulates the cavity. And so if you were to measure both quadratures of the light, well, then you're adding noise you didn't have to and you would miss the quantum limit. So ideally you wanna measure only the phase quadrature of the light and not the amplitude quadrature at the same time. Another way to say that is here you should do a homodyne measurement, not do a heterodyne measurement if that's what your goal is. And the last one is that you need to be able to adjust your power so that your measurement strength can be strong enough to reach or overwhelm your thermal motion. Yeah. So on the second point. Yes, but more than that, I would say if I'm a photon in the cavity, I'd better make it all the way through that detection with 100% efficiency. So that's really wall-to-wall detection. From every photon that interacted with the mechanics in the cavity, should make it out of the cavity being 100% over-coupled all the way to your detector and then detected with 100% efficiency. So if you didn't fully orthogonalize or fully darken your dark port, that would look like an inefficiency. But even if you did that right, you still ask, oh, did I lose something coming out of my cryostat or launching into my fiber? Other questions while I'm paused. I think I've done with equations for now so we can go to some pretty pictures. Again, the system we're gonna talk about because it's the only thing I talk about are these sort of electromechanical circuits. Just to flash some fab pictures again. This is done with standard optical lithography. There's a picture of me holding a kind of three inch wafer. The black that you see there is the metal. The substrate is on pure sapphire, which is nice and clear. It makes a nice, good, low-loss dielectric in the microwave. And it's also fully compatible with the etch recipes we use to release these drums so that we make sure we don't etch away the substrate when we're trying to etch away the intermediate layer between the two pieces of metal. And because it's optical lithography, you can do this all at once. Dice it up into little chips. Each of the chip has lots of little multiplex experiments on it. The drums in this generation were moderately redesigned, but essentially it's the same. You have a spiral inductor of maybe 10 to 20 turns, giving you an inductance of 10 to 20 nanohenrys. If people wanna think about the impedance, it makes a fairly high impedance resonator. You can actually get something close to a kilo of characteristic impedance, which is kind of hard to do without lump circuit elements. So that's one of the nice parts about using these inductors. You can get a relatively high impedance compared to, say, the impedance of free space or that of a standard coplanar waveguide. The drums are suspended over a 20 micron diameter. You can see here we put these little holes throughout it. That's just to make sure we can nicely get in there to etch underneath it. But essentially these things are comparable the same. We've played separations of around 40 nanometers, sometimes a little bit less if we're lucky. We have electromagnetic cavity frequencies in the gigahertz regime. Again, around six gigahertz or so. Most of our infrastructure tends to be kind of four to eight gigahertz. There's nothing particularly special here. And the mechanical frequency then lies around 10 megahertz. That guy? That's the one place where we have to connect the bottom layer of metal to the top layer of metal. So it's just a via. And we probably went overkill of making it way larger area than it needs to be. But just to ensure we have really, really good contact between the bottom layer of metal and the top layer so that there's not any resistance to that current. So we take our chips. We mount them in some sort of sample holder. I think I showed you another version the other day. If you squint or zoom in, you can see there are actual little wire bonds that connect the electrodes on the chip to the electrodes off the chip. Those wire bonds also are aluminum and superconducting. Although I don't think it would matter if you used a good clean metal at that point. And for example, one of these coaxial modes that you see here comes down to like these three probes of the launch. These three probes that you have here, you can think of like a coplanar waveguide. Coplanar waveguide, if you're not familiar, is just the two-dimensional version of a coax cable. You have a center conductor and then kind of a ground on each side. It's a nice, well-characterized transmission line that we take all the way up to the chip. And then we terminate this transmission line doing something that looks pretty stupid. We just short both ends. This kind of short condition is the thing that gives us basically like a little inductive antenna. So these shorts here will carry some current. That current will flood red flux through each of these little resonators and inductively couple to these six little resonators that you see here. Again, we made six just because it kind of, you can multiplex for free. You can address this one at six gigahertz, seven gigahertz, eight gigahertz, put them around where you want. You can also almost read off here, these two resonators that are inside these loops are gonna be strongly coupled. So they'll have a nice big over-coupled kappa. These ones here will have a moderate kappa. And then these ones over here are very weakly coupled, have a very small kappa. And that's the type of thing we can do to really work with devices that are critically coupled or over-coupled or what have you. The honest answer is yes, always. And then the better answer is at what level? Because they're separated by hundreds or thousands of line widths, the frequency and queue of this resonator has almost nothing to do with any of the other five. You can, if that were a problem, you can engineer it. But in principle, when you see things like that, you should always worry about it and make sure that that's not a problem. If they were all at the same frequency, I would think it would be more of a problem. Each one of these inductors really does have kind of a far magnetic field. There's no ground plane around it to kind of isolate your fields or really kill any crosstalk. So it is something you could in principle worry about. So again, our measurements are gonna be our microwave analog of a single port of fabric pero cavity. We're gonna send a microwave signal down exactly on resonance. We're gonna bounce off this transmission that couples to these six resonators, take what comes back, amplify in the lowest noise way we know possible. We'll then mix that with a copy of our drive in order to do a homodyne measurement. And we're gonna look at the phase quadrature on the spectrum analyzer. And now we're looking out on the spectrum analyzer because we've done this homodyne measurement is now directly at the mechanical frequency. So in other words, we drive with six gigahertz. The six gigahertz comes out with 10 megahertz sidebands. We mix that with six gigahertz and we just look at this 10 megahertz phase modulation sideband. There's a gory ugly picture of a dilution refrigerator mounted with a lot of stuff. I think most of this is actually a separate qubit measurement that was not these measurements. This optomechanical device is this holder with one single coax cable. And most of what you see here are filters, directional couplers, circulator and the amplifiers up off the screen. I wanna emphasize, this is kind of standard commercial technology to get in these tens of millikelvin range. Depending on how new or well-tuned your fridge is, maybe it's 10 millikelvin, maybe it's 50 millikelvin. But these are all at temperatures that are far below the superconducting transition temperature of aluminum and far below the H bar omega of the cavity. So in other words, with the six gigahertz cavity mode at these temperatures, the thermal population, I think if you did the math, should be like a part in 10 to the five. You're really in the fast part of the V limit. So it's really dropping exponentially. And that's why it doesn't make a big difference from the cavity perspective exactly which temperature you're at. Of course, we always want the mechanics to start at the coldest temperature possible so the colder, the better. Let's see, so what we're now gonna do is zoom in on one of these resonances. We're gonna choose one that's nicely over-coupled. Now, if you look at an over-coupled resonance, usually I would show you the magnitude of the transmission or the reflection. But here we have a single port cavity that's perfectly over-coupled. And what that means is there's very little signal in the amplitude. Whether you drive on or off resonance, the microwave signal is always fully reflected. And so your cavity resonance only shows up as a phase shift as a dispersive interaction. So instead of plotting the magnitude, you can plot other things. One of the things I like to plot are the reflection in the complex plane, the kind of real and imaginary parts or in the microwave, these are often called the in-phase and quadrature components. I don't know whether you've ever thought about what a Laurentian looks like in a complex plane. It just makes a Laurentian circle. And ideally, if you're fully over-coupled, it should be a circle with a unit radius. That's another way of saying if you looked at the magnitude squared of what comes out, everything you send in comes back. Everything is fully reflected. But you can tell there's a nice resonance there because you get this strong phase shift, this rapid phase shift. Don't think I have a slide on it. This might be worth writing on the board. I apologize in advance for my handwriting. If you want to think about any resonator in this complex plane, there's many ways to think about it or drive it, but you can actually use the simple input-output theory we were talking about earlier. And it's the easiest case. Just do the input-output theory for just a cavity, not coupled to anything. So like we saw over and over again this morning, you have some a-dot. I will get my signs wrong and see if I have. I thought I had a cheat sheet. Apparently I lost it. Yeah, so you'll have to watch me guess in real time. Again, I will get the factors wrong. You have your usual differential equation of just of a cavity. This is straight out of the review of modern physics. What's important, this is your detuning from the resonance. That's the total line width of your cavity. And then this is the coupling to the port that you're going to excite wherever you're putting the energy into. I wasn't bad for my first guess. Yeah, I agree. So if you take this and combine it with one other constituent equation, like how a-out relates to a-in minus the same square root of capital X, a. You can basically solve this in the steady state. Steady state means you set this to zero and now you plug in for a and you can get a-out over a-in. And these are what I would call the normalized reflection parameters. Or in the microwave, these are the s-parameters or scattering parameters. And so you can get a-out over a-in. Here again, I'll let people correct me on the fly if I'm a bad guesser. So again, this is a purely vector measurement. You put something in and look at what comes out. This is classical or quantum. Doesn't have to do with the operator notation at this point. Kappa naught would be the internal loss, the loss of the resonator that doesn't have to do with the port you're exciting at. Kappa X is what you have here. You can see if you were 100% over coupled, the Kappa naught would go to zero. And what you would get is minus Kappa X in the numerator and plus Kappa X in the denominator. And so basically what you can see is if you were exactly on resonance, you get reflected with a unit and a minus sign. The minus Kappa X over Kappa X. And if you were far off resonance, so this dominated, you would reflect just with no minus sign on it. This is what I'm plotting over here when I plot the measured unit circle. This is just measuring what you have in the real and imaginary parts of the complex S-parameters. So measuring the radius of this, how close it is to unity is telling you exactly how negligible this Kappa naught is in the total Kappa. And we can put bounds on it here. We're over-coupled by something like a factor of 200. The other nice things we could look at, you could choose to look at the phase, taking the phase, kind of the arctangent of the imaginary over the real. That's the thing that gives you the thing where you see it over and over a nice curve that looks like this, that's described by an arctangent function that goes from plus pi to minus pi. The width of the transition tells you the Q of your resonance. Even nicer than the phase, you can plot the derivative of the phase, what's known as the group delay. And that's what the plotted over here. And that gives you back a nice Lorentzian form that shows you a nice Lorentzian peak that you can just read off your line with from it directly. So again, I'm going to do it a little in detail. This is what we would look at in the nicely over-coupled resonances that we want to not have any lost information. I'll also come back to these resonance circles a little bit later when I'm talking about going back to some of the EIT interferences. So what's nice is fairly straightforward here and the microwave is the ability to over-couple, to get all of our photons out of the cavity nicely. This is something that's not so trivial. For example, in the zipper cavities, they spent years engineering nicely how to go from a tapered fiber that couples light out both ways and kind of is fundamentally 50% inefficient to having ways to have a fiber just nicely reflect off of the zipper cavity. And that's what you'll see in all of the latest experiments from the Painter Group or the Delft Group. They're now moved to this kind of single port geometry in order to not be quantum inefficient. So again, you can see it here on the device, a little bit more of a zoom in. We have one of these. We have our coplanar waveguide. We have our wire bonds there. This is what we're measuring. And again, this being over-coupled by a factor of 200 is what tells us we kind of have 99 point something efficiency at least with respect to photons in the cavity to photons outside the cavity. If everything else were perfect, I wish this were the bottleneck. In principle, we're just excited we can eliminate that as a problem here. So now, as I said, what we're going to do is we're gonna measure the homodyne quadrature, stick it on a spectrum analyzer and look at what's coming out at a mechanical sideband frequency. Here, nine point something megahertz away. I've gone ahead and taken the spectral density that we measure directly coming out of the cavity and divided it by the pump to refer it back to the input of the measurement to put it in these mechanical quanta units. Pardon me, I described a little bit the other day. There are many ways to do this calibration. Often what we can do is take the temperature of the cryostat or the environment to some well-known temperature where we know for a fact this Lorentzian corresponds to a peak of exactly 230 something quanta and use that to calibrate our transduction, use that one factor and then change the temperature of our fridge and just ask the question from this Lorentzian what's the temperature I infer of my mechanics? So again, this is the drum vibrating here at 40 millikelvin or so for some moderate probe strength. What we can now do is turn up the strength of the measurement. And as we turn up the strength of the measurement, we're gonna look at a couple of things. We're gonna be watching what the white noise background does and what this thermal motion does. And already, yeah, that's right. So there's a bit of an arbitrary choice there. And this is the same question as asking what's the effective mass or what's the coordinate system that you're using? Here we've kind of chosen as a convention to use the total mass of the system so that our displacement coordinate is kind of the effective coordinate weighted over the mode shape. Some people might choose a different one. Oftentimes the time you would definitely wanna choose a different one is if you were actually trying to measure the displacement at the center of the drum or a little bit off center of the drum, then you might wanna normalize to that specific coordinate. I would say usually this is what we do here. When we compare to quantum limits, it actually completely falls out and doesn't matter what the choice is. It only matters when we quote an absolute number for this meter as per hertz. But in principle, depending on exactly where on the drum or if we were using the weighted average, that could change by square roots of two, something of integrating over that vessel function mode shape. Anything else? Already just looking at this data, not knowing anything about this y-axis, if I told you that this Laurentian was 100 quanta, you can already read off how well we're doing, whether we're hitting the standard quantum limit or surpassing it just by asking what's the signal to noise of this peak to this background? And it'll become more apparent as we change the measurement strength, how you move these two things around. But essentially that's part of what you're looking at here. And the whole name of the game is just to optimize the measurement of this mechanical signal. So this is for a very weak measurement. You can see this Laurentian is barely showing up in the noise of our interferometer. This is just the white noise of the coherent state of us trying to detect the phase quadrature as well as we can. Now as we turn up the power, well, our noise floor went down. Again, I always wanna be careful because people will plot different things. If the reason it went down mostly is because I'm putting it back in mechanical units, which means I'm dividing by the pump. So if I drive harder, my phase sensitivity improved and that went down. But if I look at the absolute power on my spectrum analyzer, it wouldn't move. So again, this is looking at these noise spectral densities in the mechanical units. If I turn up the pump harder and harder, you can see the noise floor going down and down. But if you look closely at this mechanical peak height, it's pretty much staying the same. It's not really doing much exciting. But now we know if we push this noise floor down too much, we have to start pushing back on the mechanics. And at some point this Laurentian has to get taller and taller. And as we push down, that's then what you can start to see. And in these kind of circuits, again, we have access to very high cooperativities. We can strongly go into this back action limit where we overwhelm the thermal motion. We really dominate with our back action and push our imprecision very, very low. A couple more clicks. So here's what the data looks like if you were to plot them to make that canonical SQL X-Men plot that you have. On the X-axis is again our measurement strength. I'll point out here, when we say we're measuring at high power, we're talking about 10 nanowatts. And this is one of the other advantages with microwave photons. On one hand, microwave photons are very wimpy. On the other hand, they're very wimpy. So I can measure with millions and billions of photons and that's still only 10 nanowatts of power. All fully compatible with ultra-chrysogenic measurements. And so that's one of the things of why the microwaves tend to be good is we get to measure at these temperatures almost for free without a lot of thinking about. Whereas in the optical, this kind of power you would get for even exciting with just a few photons. The Y-axis is the mechanical noise and I put it in the quantum units that I described before. The blue dots are just plotting what that white noise floor is doing. You can see it goes beautifully down as this one over power. And that's just saying that there's no excess noise that creeps in here. As we turn up and up our power, everything stays very clean. And that's just saying we're measuring our interferometer nicely with the vacuum noise. The green points are the height of the Laurentian. So at low power, the height of the Laurentian is dominated by the thermal plus the zero point motion. And the back action would be this white dotted line. It's completely negligible down here. Most of this thermal motion is exactly what it has from being in equilibrium with the 40 millicelvin bath, about 100 quanta. And as we get to a certain power, we can see we start to heat. And in fact, we go up and up and up. And what's very nice is we measure this imprecision from the imprecision. If you know your quantum efficiency of your measurement, which we can also calibrate, you can predict with no parameters what this back action should be. And that's what this line is that we plot. And so what you can see is it nicely confirms that this back action is not stupid heating. This isn't just because we were dumping a lot of power and cooking our food with our microwaves. We're actually seeing the photon shot noise impinging back on the mechanics. And this is the trade-off that people had talked about as the Gadanke measurement for years and years. And then just recently, within the past five years or so, many number of groups have the parameters to measure these. I'll refer you to the Stamper-Kern group where the Cindy Regal group also have beautiful measurements. The Kippenberg group now has also some of in their feedback cooling measurements, some of the cleanest data that looked like this. Again, just showing this is a clean interaction Hamiltonian and you can really nicely achieve a beautiful measurement. Yeah. So yeah, I'm gonna try to zoom back to it, but it will take a couple clicks. Just because this peak is smaller than the background doesn't mean I can't average and see it. So in principle, if I had patients, even if the background were 10 times higher, I could average until I can resolve a little 1% bump on there. And you can measure it arbitrarily low. It's, like I said, it's more about stability and patience in averaging. Technically, what we do here is keep the same resolution bandwidth. If I change and change the resolution bandwidth, I would at some point not be able to resolve this skinny peak. So then the only thing left is integration time or number of averages. So you just repeat it that many more times. This is noise, so you can beat down the noise of the noise and just lower the uncertainty there. And so yes, when you look at other people's papers and you ask why they didn't plot those points. On one hand, there's not much action going on here. It gives you confidence. Usually if you can measure it, you do. If we could zoom in and look at the error bars on each of these, the error bars start to get larger and larger over here. It's just a smaller signal. So in these devices, we were very excited because we're able to go to very strong measurement strengths, something like 200 times the thermal noise. And this is really in the kind of quantum cooperativity that Clemens was talking about today. This is a measurement of 4g squared over kappa over the thermal decoherence rate. That's what's telling you how much you overwhelm this intrinsic heating rate that you have here. On the other side, what you should say, you can also ask, well, what's the imprecision of your interferometer? And here, we're getting down to something. In these units, it's 10 to the minus 4 quanta. You're 1,000 times below the imprecision at the SQL. But again, you shouldn't be impressed by any one of these numbers. This can be as low as possible. That's not violating quantum mechanics in any way, shape, or form. If you were to put this back into kind of the meters squared per root hertz, we're talking about 10 atom meters per root hertz, 10 to the minus 17. Again, it's not LIGO, but you're talking about things that are way, way less than a nuclear distance. I mean, these are length scales that are becoming ridiculously small. And that's just kind of a measure of how strongly you're probing this interferometer. There's the numbers. Again, the thing that quantum mechanics requires is the product of this imprecision in back action is limited to the H bar. And again, our back action is nearly ideal. The main non-ideality in these measurements is that these blue dots don't fall on this dotted line. And that's because in these measurements, we have a relatively poor quantum efficiency of our measurement. We're using standard hemp amplifiers that add noise that are something like 20 times the quantum limit. That gives us an effective measurement efficiency of a few percent. And that's why this line is not here. That's why we miss exactly this factor. This is something that if we use the most state-of-the-art Joseph and parametric amplifiers and the most low-loss circulators, now you can really push this bound all the way down to that line. How close depends on the experimental details. For this proof of principle, we were just showing the clean trade-off between imprecision and back action. But that's the main thing. That's the main inefficiency in our measurements. It's just our sheer detection efficiency of our microwaves. And most of it is not actual loss at the cavity. Some of it is lost before we get to our amplifier, and then some of it is added noise out of the amplifier itself. So the question I normally get at this point is that's really cute that you could pump all the way over here, but why would you want to? The SQL is all the way back here. Isn't this where you've done the best mechanical measurement? And the thing I will say is absolutely, if that's your goal, is to make the best inference about the mechanical mode, you don't want to be operating here. Where you're swamping it with radiation pressure shot noise, you have phenomenal imprecision to read out that radiation pressure shot noise, but you do a horrible job of making an inference of what the original thermal bath is. So yes, if the mechanics is the thing of interest, what you want to do is you want to operate at this balance power over here, exactly where imprecision and back action for your measurement are balanced. That's where you add the lowest amount of noise, and that's where you would have the highest signal to noise between just the thermal component and the background. And that's what I would say in kind of a scattering language. That's where I've impedance matched my measurement rate to the thermal decoherence rate of the mechanics. I'm measuring it just as strong as the environment is. And in other words, the mechanics and the cavity optimal mechanical system is now one effective transducer that takes beautiful thermal fluctuations from the environment of the mechanics and transduces them up to my measurement apparatus to measure with very good signal to noise. So if now we said, why would you want to go up to this strong measurement power, here's where I'm going to ask the question of who is measuring whom. You can ask different questions about what's going on here. From the perspective of the mechanics, it looks really fricking hot. It's 200 times bigger fluctuations that we had. But again, in these clean systems, anytime I tell you this distribution looks very thermal, what we know is it's highly entangled with some other degree of freedom that we are tracing over. And if we really kept track of whose fluctuations they are, we could ask, what is this useful for? So again, this is what the power spectral density looks at, this highest measurement power. The signal to noise is phenomenal. I mean, it really blows up because your signal, this peak, gets higher and higher, and your background goes down and down and down. The area under this curve of this very narrow Lorentzian is precisely what we're using to determine these occupancy numbers right here. Now what I want to emphasize, again, this beautiful Lorentzian we're seeing here has almost nothing to do with the thermal environment of the mechanics. If the mechanics were 10 times colder or 10 times hotter, these data would look almost the same. And that's just another way of saying the thermal environment is negligible compared to all the VAC action. So this thing that we call VAC action and that has all this bad connotation, as I keep beating over and over, what is it really? Well, this is the quantum noise of the light, the quantum noise of the probe that we're using to probe the mechanics. Very specifically, it's the amplitude fluctuations of the light that drive the mechanics. The mechanics is then hot and driven, which gets re-encoded on the phase quadrature fluctuations of the light. So if you want to think about what this signal is, what this signal is is a beautiful amplified measurement of the amplitude fluctuations of the light that got re-encoded through this optimal mechanical black box into the phase fluctuations. So it's at these very high measurement powers that what I would say you're doing a very good job of doing is using this optimal mechanical transducer to get a very sensitive metric of the light. And in other words, what you're looking at here is mostly amplified vacuum fluctuations of the light. Very specifically, it's the amplitude fluctuations of the light, a mechanical frequency away from your pump. They beat together to give you a driving force. The mechanics responds very sensitively to that, and they get re-encoded. So again, over here is where I would say your impedance and batch to the mechanical environment, where you do the best job of having the light measure the motion. Over here is where you're very, very, you're measuring way too strongly. Your measurement is way over coupled from the perspective of the mechanics. And in fact, now if you ignore the mechanics, this is where you do a good job of using the light to measure the light through the mechanics. So this is the story I'm gonna go in a little bit more, talking about these same optimal mechanical interactions, the same driving, the same measurements, but now as the perspective of how well is it amplifying or measuring the light? What is it actually doing? So again, just cartoon form we're driving on resonance. If we were to look at the output spectrum coming directly out of our cavity, well, I've cut out the pump. There would be an enormous delta function pump in the middle, but there's an upper and lower sideband that are very, very big, and they're very, very symmetric with each other. This is how we know their good phase modulation. And now I could ask the question, what if I put a little test signal in? What if I didn't let my light field just be the vacuum fluctuations? What if I put some probe in there? What happens to this probe as it interacts through the mechanics and comes back out? What I'll show you is that essentially what you will do is you will get gain right around these two mechanical sidebands. I just got done telling you that this large noise power you see in the upper and lower sideband is just amplified vacuum fluctuations of the light. If you put some signal in there, it too will be amplified. So again, just to touch base with the Optomechanical Induced Transparency I described the other day. When I showed you these kind of cartoons, this was working in the good cavity limit where I was probing optimally red detuned and asking when I send in a probe what happens when it comes back? Now, as Clemens described very nicely today, if you're pumping red detuned and you're in the good cavity limit, that is a pure beam splitter Hamiltonian which is beautifully unitary. It can swap information from the cavity of the mechanics and back, but there's never any gain, never any energy that came from anywhere. It's in one subsystem or the other. And that's why no matter how hard we pump here the only thing we can ever do is get back to unity reflection. And this nice interference goes through and we get these interferences. Now, the soon as we stop making this beautiful rotating wave approximation and we allow there to be some scope scattering, now we have this non-conservative process. We have this a dagger a times b dagger, a dagger times b dagger, which is the dual mode squeezing Hamiltonian. The thing that cracks pump photons and creates correlated pairs and the total energy of the system goes up. And so now we can think about, again, in this perspective of optim mechanically induced transparency, what's happening? So I'm gonna reiterate the same story I told you before. Now I'm just gonna run through this same cartoon, but thinking about pumping exactly on resonance. So we have our mechanics vibrating away with its hundred quanta down to 10 megahertz. We pump the cavity exactly on resonance so we see its little mechanical side bands, whatever they are. If we were to take a test signal, a probe and put it in here, again, it beats with the pump. When it beats with the pump, it drives with mechanics. When the mechanics is driven, it gets re-encoded as side bands, both upper and lower. So you get a signal that comes out back where you put it and a signal that comes out here on the other. And you can see this directly on a spectrum analyzer. If you put a tone in here, you can see it comes out nicely here. It comes out with some gain over here. If we had the ability to listen at the mechanics, we could also see it down here. Normal in these experiments, we don't bother doing anything down at the megahertz. We do all our inferences from the light. In the language of amplifiers, what you can think about, this probe is your signal. If a signal is gonna come out with gain, it's gotta be getting that gain from some idler. And here you can think about what are the idlers in the problem. And those are exactly the mechanical mode and this other side band of the light. So there's optimal mechanical Hamiltonian. The fact that the radiation pressure interaction is nonlinear, all we've done is made this really complicated doubly resonant mixer. We're depending on how we pump this mixer, we can kind of really specifically address which tones we want and get exactly the processes we want. So over here, where we put our signal and the signal comes back taller in the language of parametric amplifiers, that would be our direct gain. The fact that the signal comes out with a copy over here also amplified. That's our indirect gain or idler gain. So part of the reason I went through the story of explaining what these resonance circles are, for me what's very interesting is to think about how the cavity susceptibility, I'm just drawing that same cavity resonance circle here, what that's gonna look like in the presence of these interactions. In other words, once we pump it, we're gonna see this cavity laryncean interfere with mechanical larynceans. So just to be clear, what's plotted in green is just what I showed you before. It's just a unit circle of our nice over-coupled cavity. The line width of this is kappa. It's how fast I go around this resonance circle with frequency. And if I just measured the reflection in dB, for an ideal over-coupled cavity, it's just flat. There's no signature whatsoever. So now this exact argument works for circuits or the canonical Fabry-Perot single coupled cavity, single port cavity. If I turn on an optimal mechanical pump, and I've made a couple of assumptions, these are theory I'm just plotting here. So I'm gonna pump exactly on resonance. I'm gonna choose right in between the good cavity and bad cavity limit. I'm gonna put the mechanical sidebands right at the half width of the cavity. And I'm gonna start off at some lowish cooperativity. And again, cooperativity is just normalizing my measurement strength by the intrinsic line width of the mechanics. It's this usual 4G squared over kappa. So what happens is if you took a very weak probe, a very weak pump of just cooperativity of a quarter, what you see as you would sweep through frequency at the lower sideband, there's a little gain peak. And at the upper sideband, there's this dip. And this is what we've seen in a couple of the lectures. The fact that it changes sign, whether it's red or blue. What's more informative about whether these are peaks or dips to me is just thinking about what these little resonance circles do. They're circles because that's what the complex susceptibility of a Lorentian is. It's a circle in this complex plane. So as I would sweep from low frequency, I would go like this, I would whip around this little circle, go around, come to here, whip around this circle and go around. Now as I turn up the cooperativity of the pump, what I'll start to see is stronger and stronger interference. So if we go to cooperativity of a half or one, and I'll go back and forth a couple of times, in the polar plot, it's nice. All I'm doing is increasing the diameter of that little interference circle. Over here in the magnitude, I can see I get a perfect dip that goes all the way down when I hit cooperativity of one. I've done something very special. And I get exactly, I believe it's 60B of gain here. That's a nice round number that I will always get wrong. I think it's some coherently adding factor of two in amplitude, which gives me 60B. And so what you can see, this blue dip goes all the way down because I've made this little blue interference that goes exactly back to the origin. The origin is where my pump doesn't come back with any reflected power. If I think about this again from an impedance matching perspective, what did I do? I sent my probe in, it got transduced to mechanics and perfectly transduced out the mechanical environment, never to be seen again. And that's the sense in which it's impedance match at exactly this frequency. What's more interesting now, since I'm driving exactly on resonance, I can keep pumping harder and harder. So if I go to cooperativity of two, well, my upper side band circle retraces my unit circle. I lose all signature over here and I get a bigger gain peak here. And if I keep going and keep going, those circles can get arbitrarily large. These are the peaks I would see in magnitude. And this is really what I mean when I say an optomechanical circuit acts as a reflection amplifier. A signal coming in near here goes into the mechanics, comes back out with gain. At cooperativity of 10, you can already see where at 20 dB of gain on both the upper and lower side band. And that's the large gain limit. That's back in the limit of just reiterating this language. Of large cooperativity is now doing a very good job of using the light to measure the light. So these are what the magnitude of the direct gain would be at the upper and lower side band. As I just showed you, as a function of cooperativity or pump power, the upper side band goes through this perfect null and then gets larger and larger. Well, the lower side band just gets larger and larger. That's the theoretical plot of what you should get just from solving the equations of motion from the input output theory. Pardon me, keeping track of the stokes and the anti-stoke process. Again, because we're driving exactly on resonance, there's no damping or cooling of the mechanics, kind of the bandwidth is unchanged. So that's what the theory should be. Again, we have a device we can measure. We can put measurement dots onto these and you can really measure exactly these types of interferences. It's doing exactly what it should do. The part that becomes ridiculous for me as an experimentalist is you can get things like 80 dB of gain. Again, you're amplifying by a factor of 10 to the eight. This is a reiteration of just how beautifully linear the Cavian mechanical modes are. We took the 100 quanta of thermal motion in there and we replaced it with now 100 million quanta because we amplified the vacuum quanta up and it's still all well-described by these nice clean Laurentians. So depending on your perspective of whether you like things to be linear or not, this can be a very good thing. What's interesting now is I told you I was driving exactly on resonance. How on resonance was it? Well, I had to drive on resonance by something like less than a per per million because if I was a little bit red detuned or a little bit blue detuned, I would take my 10 Hertz line width and I would turn it into 20 Hertz or I'd turn it into zero, which means parametric oscillations. So part of the hard part is it becomes very demanding of this exactly how perfectly detuned you are. On the other hand, you can use that as a feature. So this is plotting just the gain of one of those curves. It's now going into logarithmic frequency on the x-axis away from the center of the gain. So too many animated slides. Again, if I think of this as a nice little Lorentzian gain peak, I'm now going into a reference around that center and saying what's the gain, some small Fourier frequency away. And I'm plotting it in this kind of boat plot eight where I go from one Hertz to 100 kilohertz. So this is when I'm driving exactly on resonance. I can read off the 10 or 20 Hertz of intrinsic bandwidth and the 80 dB of gain. But now if I just red detuned by epsilon, well, I can damp this hot mechanics a little bit. I can reduce my gain, my laser pointer died. I think that means I've been talking too much. If you red detune a little bit, you can tune down the bandwidth and the gain of your system if that's what you wanted to do. And this is nothing more than just detuning a little to the red and getting large amounts of damping. No, I can't use this to switch. All right, with the last couple of minutes, I'm just gonna sum up. When I talk about this amplification, part of the reason we get excited is because this should be a very clean parametric amplifier. The hard part with amplification in general is not getting some gain, but it's getting quantum limited gain. Gain with exactly the added noise that quantum mechanics requires. No more, no less. You got less problems, so you should talk to me. And what I wanna think about this now is again, in terms of like a cave's limit of an amplifier, how does this compare as a phase insensitive amplifier for the light? To me what I would say is if you have a linear phase preserving amplification, it amplifies the sine and cosine quadratures in this ball and stick picture that we'd like to draw in quantum optics. The stick is the coherent state displaced from the origin. And if you're doing everything right, the ball here is just the vacuum noise. Now if you amplify it with a linear amplifier, you make your stick longer. That's what means you have a good amplifier. You have to amplify the noise that you had because that was there. And if you amplify both quadratures like I've drawn, well, you actually have to not just amplify the noise, but also double it. And that's what I would call the quantum limit. And these are the same kind of relations you can derive. Things like caves did essentially just by preserving commutation relations on the input operators and the output operators. You can see that there had to have been added noise. And I think of that as a nice reinforcement of the Heisenberg uncertainty principle. So now how do I think about the added noise in this mechanics? So I'll go back to the same cartoon picture. We're pumping on resonance. The mechanics is kind of hot. It's got 100 quanta far from its quantum ground state. It's got these little mechanical side bands over here that are just thermally noise limited. But now if we pump harder and harder, well, we know once we pump harder and harder, we backact on the mechanics. The mechanics looks even hotter, again, hotter with the air quotes on it. So these noise got bigger and bigger. But what we know is all of this excess noise had nothing to do with the original mechanical environment. And so the statement that was surprising to me, even after thinking about this for quite some time, even though the mechanics started hot with 100 quanta, and the only thing we did was measure hard and backact on it, the noise we get at the output is purely the quantum noise and the light. And as long as you overwhelm the intrinsic thermal motion, you can hit the quantum limit of amplification, the quantum limit of adding the noise, only the noise that you have to as required by quantum mechanics. And if you had asked me just over a beer or thinking about it, I would have said, yeah, you could probably get quantum limited amplification if you pre-cool the mechanics to the ground state, because then it can be a nice cold idler. What's interesting to me here is you don't even have to pre-cool it. As long as you overwhelm it, it just is an incoherent noise source that becomes arbitrarily negligible and adds a negligible amount to your total noise. So we spend a lot of time exactly calibrating the gain and noise that you get from this very narrow Lorentzian around one of these mechanical sidebands. So again, this is at a cavity frequency of 6. something gigahertz. I'm pumping a mechanical frequency away. I'm looking at one of the sidebands, the upper or the lower. I can see I get large amounts of gain just controlled by my cooperativity. And if I look at the added noise referred back to the input, right at the center of this gain peak, I really almost hit this line of the quantum limited amplification. As I've been saying over and over, in the microwave we're a little bit starved for perfect amplifiers. The fact that the mechanics might be able to serve niches and do very quantum limited amplification is something that could be useful. The full confession would be, we have Josephs in parametric amplifiers. They can also be very good. They can also be much broader band than this. This is the world's most narrow band amplifier. It can amplify any signal within 10 hertz of where that is. I can think of reasons why that might actually be a feature. If what you want is to amplify a signal that's small and not something nearby. But the proof of concept here, what's really interesting about the mechanics. One, that can be clean and quantum limited. And the two, it's fully tunable with my pumps in terms of gain. And the other thing is really the dynamic range, which for many applications people really care about. This idea that you can get 20, not 20 dB of gain or 25 dB, but 80 dB of gain is something that really makes it a nicely linear amplifier in the sense of the word that we'd like. So I think that's all I have. Let me just check. Yep. And tomorrow's lecture, I'll talk with, I think it's actually Friday. I'll talk a little bit more about if you want sometimes to be in the good cavity limit, sometimes in the bad cavity limit, how you can engineer devices to get the best of both worlds. I'll also describe what you can do with squeeze states. That's something we have access to and we've done quite a bit of research, both in ways to squeeze the mechanics or squeeze the light and think about the implications they're in, what it does for you and what it doesn't do for you. But I think with that, I'll just stop there for today. Thanks.