So it's Friday. We made it to the homestretch. I'm going to give my third lecture continuing on the same lines that we have beautifully building off of, hopefully, what all the other lectures have laid the groundwork for so that I can show you some of the experiments and some of the pictures that go along with the concepts and ideas that you've been seeing. I didn't mention it as much in my previous lectures. I do want to point out all of this work is in the Advanced Microwave Photonics Group at NIST. That's myself and two other PIs, Ray Simmons and Joe Amantado. Today's lecture in particular will also use a lot of slides and experiments done by two former post-docs, Jeremy Clark and Florent LeCocque. So the title of this slide is Quantum Light for Quantum Mechanics. I think I'm going to talk about a number of different things to tie up loose ends of other things I didn't get to. But one of the themes will be how we can use systems that we can engineer non-classical states or interesting states, namely the light field, and use that to influence the mechanics and do interesting things from the perspective of mechanical measurement. So I think right where I left off last time, I was explaining a series of measurements, things about the standard quantum limit, things where we were driving on in resonance. And when we drive on resonance, we no longer have this want or need to be in the good cavity limit. And as I explained a little bit, actually being too far in the good cavity limit can actually harm you a little bit. It can make your mechanical sidebands be outside the density of states of the cavity or in filtering language, just low-pass filtered by the time response of the cavity. Whereas there's a number of good physics that we absolutely do want the good cavity limit, things like sideband cooling or normal mode splitting. Any time we want to isolate one interaction like the dual mode squeezing Hamiltonian or the beams flitter interaction. So in general, from an experimental point of view, part of what we want to think about is, where's the balance? How do we get the best of both worlds? These are part of the reasons why we're excited about these engineered quantum systems. If you want more degrees of freedom, you can build them in. And so what you can naturally move to, if your system doesn't already give it to you for free, is now making multi-mode circuits. You can have multiple mechanics coupled to the same cavity or multi-cavities coupled to the same mode, the same mechanical mode. So just to highlight a little bit, give you an example of one type of an experiment. Here we have a single drum. In this red bubble is our mechanical mode of interest. And we're going to couple it to two different LC resonators. One over here for a cavity mode and one over here. The idea could be, for example, make one be a very low-Q cavity, which is a nice fast cavity that's good for readout. Make the other one be a high-Q cavity, deep in the resolve side-band regime. And in that way, you can use one for your state preparation to get your interactions, the other nicely for your readout. These types of ideas are very much something you'll see in superconducting qubits or other forms of quantum information, where you just build in whatever functionality you need if you can nicely engineer extra modes. So to zoom in a little on this circuit, just to show you what's going on, here you can see a drum. It's coupled to this inductor. Don't be fooled because it looks like two inductors. This is actually just a single radiometric inductor. And part of this game, part of the trick here, is we want to get strong coupling of this resonant mode to this feedline coupling. And we want weak coupling here. So what we can do is use a little bit of symmetry. Our drive and readout port, again, what's shown in silver is the superconducting metal. It's the aluminum. I like to think about this center and the two conductors on the outside as a co-planar wave guide, this two-dimensional coaxial mode, that we send our microwave drive down. You can see that current splits symmetrically, goes out here and around and out here and around. And those currents, if I did my winding right of these coils, will couple very strongly and constructively into those two coils to give large coupling to an LC cavity mode that looks something like this. A bottom plate of a capacitor down here, resonating with this L, we designed it to have its own resonant frequency and designed to have relatively large kappa. And again, large compared to what? Well, usually we're thinking of the mechanical frequency around 10 megahertz. So this is kind of in this right in between or optimal in between good and bad cavity limit. Then the other half, you have another cavity. You give him a different frequency so you can address him individually. And you can choose his kappa to be just what you need. Here we chose right around two megahertz. And then what you need is a drum mode that couples to both of those. If we zoom in a little bit on what the drum is, you can see we've deviated from the perfect circle, made it slightly more ellipticals, not the word, maybe stadium-shaped. But more importantly, what we've done is we've put a split capacitor in the bottom. You can very faintly see the kind of imprints of the bottom plate of the capacitor that's on the top piece of the metal, making these little half circles. If I highlight them in color, you can see you have one plate of the capacitor that couples out to that mode, the other plate couples this way. And again, these are two of your two different readout modes. Now you can ask what mode of the mechanics of this couple to. And basically, you can almost look at it and decide by symmetry. You can see the fundamental mode does go up and down with both of these. So it will have some coupling constant g. And in fact, it will also couple quite nicely to this symmetric mode, this first harmonic mode that would go up and down. There is a sign change in the coupling in the g, but that's not something we would usually care about. It's something you might care about if you really started to interfere these things, but not something I'll really talk about here. In general, I mean, I want to point out, usually the problem is our resonators have plenty of modes, whether we want them or not. We spend a lot of our time trying to break degeneracy or make sure we're not trying to isolate some mode in the forest or a mess of other modes. In these kind of lump circuit elements, it's nice because you can really define clear modes, and the higher order modes are well out of your way. You have plenty of room to work spectrally. Did you have a question? Yes, directly, because of the two problems. That's right. So same as my answer yesterday, any time you see two things close to each other, you should assume they couple until told otherwise. To be fully honest, there is capacitive coupling between the two. There's also inductive coupling, because the two inductors will talk magnetically. In practice, it's completely negligible, because their eigen frequencies are so far away. So any slight renormalization you have, whether you're talking about some slightly changed normal mode, doesn't make a big difference. If we tried to really bring them in resonance, then we could really see how coupled they are, and that could have real consequences. I mean, people engineer this on purpose to make light and dark modes, things that strongly couple or don't. And just as a springboard off that question, if you were really worried about this capacitive coupling and you were a very good microwave engineer, you could exactly cancel it with inductive coupling so that you null it out and you get back to your bare basis. But that involves both a bit of engineering and a bit of really good fabrication. These are some of the tricks we play in other circuits, but here we didn't have to worry about it. So the demo kind of experiment I'm going to show you what we do with this is just the idea of Raman thermometry. This is sometimes referred to as sideband asymmetry. This is something that's been measured by a number of groups these days. But it's just this idea that if I pump a cavity exactly on resonance, of course, my mechanical response is at the upper and lower sideband. In the last lecture, I went ahead and described everything as if I'm going to immediately homodyne and just interfere these two peaks together and look solely at the phase quadrature that contains all the mechanical information. I think what you might know is that if you do look individually at the upper and lower sideband, there is information in them. And namely, the peak heights of these two things, even though they're coupled to a drum or a mechanical mode of the same occupancy or temperature, one is proportional to that occupancy, the other is proportional to that occupancy C plus 1. This plus 1 people get very excited about because they say ha-za, quantum mechanics, which is great. People can give whole lectures about the meaning of the one. What I will certainly say is the one is a signature of the vacuum fluctuations. Part of the debate is whose vacuum fluctuations? Is the vacuum fluctuations the light or of the mechanics? People also debate whether you could explain this in some sort of semi-classical formalism. The answer is maybe. But what I will say is if you can see these clean signatures, one, it gives you another way to kind of reinforce the thermometry that we already think we know. Like I said before, we calibrate just looking at our mechanical motion at a known temperature and extrapolate to low temperature where we don't know. So this can be kind of a double check on our thermometry. And if we really do indeed understand everything about our system, all of the noises that we thought were pure vacuum, are they really pure vacuum? Are they not? And again, I just want to reinforce, this is real primary thermometry. If I measure some power spectral density in whatever units, it doesn't really matter because all I'm gonna measure is the ratio of this to this and that ratio will directly relate to the temperature of the mechanical mode. So now that we have this extra degrees of freedom, we have two cavities to pump. So we can do what we want. We're gonna take our upper cavity, which is in the good cavity limit. That's gonna be our cooling cavity. So we're gonna apply a drive, red detune so the upper sideband's on resonance and this lower sideband, that stokes process is pretty negligible here so we don't have to worry about it. Then with our readout cavity that's nice and wide, designed to have a line to match roughly these mechanical sidebands, we're gonna drive exactly on resonance and instead of homodining, we're just gonna look at the heterodyne spectrum. We're gonna look at the absolute power of the upper and lower sideband. I think the plot I have just jumping into the data shows way too much information all at once. These kind of family of curves, if we go slowly, what you can see on the left and the right are the upper and lower sidebands respectively. The x-axis is in frequency away from that center pump. Here for this mode, we have a mechanical frequency that's about 5.7 megahertz, which is why you see them at plus and minus 5.7 megahertz. And the family of curves are for different cooling powers that you apply to the second cavity. Now what's nice about this, in most of the experiments we do, we designed them to be very simple, with a single cavity, a single readout. And that readout is the same thing that's doing the preparation and the preparation is doing the readout and it's convolved for better or for worse. There's reasons why that's really good for calibration, but there's reason why that's annoying. So what's nice about this is the readout stays completely fixed. We don't change anything about this with this whole family of curves. The only thing we do is we change the strength of the cooling pump. As we do that, we start to damp from the red to the orange to the green and you see both of these going down. Both of them show kind of the nice optical spring and optical damp features. You can see how they nicely shift together in frequency of order of the line width. That's mostly a plausibility argument that it's the nice optical spring that's doing it for you there. The other thing you'll notice is when we cool very weakly, that's when our mechanical peaks are very narrow, IQ and very hot, hot being around 100 quanta or so, the temperatures we work and we can't really resolve much asymmetry at this level. As we go down and as we go down, hopefully if we're doing our job right, we should be able to see the difference in these peak heights. So what I'll do now is plot the area of these two curves, the area on the lower sideband and the area of the upper sideband as a function of this cooling power. And now the natural units of the cooling power are just in the mechanical damping rate. So this is, roughly speaking, just the optical damping rate. This is the cooling that you have. If all is right with the world and you're in the good cavity limit and all your extraneous noise sources are small, you should exactly trade off damping for temperature. But now when you, yeah, Jack, absolutely. That's something I would say you can go back and check and see if it fits to a single temperature. It's more that the data tells us that other than that we assume it a priori, but I agree that you have to be careful, especially as you start at lower and lower temperatures with these nanomechanical devices of some sort. We work very hard to isolate them, make them high Q, which means they aren't cooled very well to their environment. So if you blast them with power, you should always be suspicious that maybe they're actually decoupling from their environment. Perhaps it's heating or fluctuating, but those are things that you have to be aware of. The fact that these kind of fall on this line is the thing that is evocative, that it's described by a single thermal decoherence rate, a single environmental temperature and an intrinsic mechanical damping rate. And if you look at the area of the upper and lower side band and you were just to naively infer what the occupancy would be, you can see that they really do scale. One keeps going down and down and down without asymptote. The other is clearly starting to trickle off. We can calibrate this y-axis in two ways. As I said before, we can do it in our usual kind of calibration way. We can carefully measure our G-naughts and our detuning and our echo partition theorem in order to know what the area is on our spectrum analyzer. But then two, we can cross check to see how well that agrees with the asymmetry you would infer with no a priori knowledge. And again, the basic assumption here is that this would be n and this would be n plus one. This data here, I don't have the full calibration between the two. I just wanna show you a flavor of what this sort of data looks like and mostly points you to a number of other experiments that have measured similar things. For example, in the painter experiments, instead of driving exactly on resonance and looking at the upper and lower side band, they compared pumping blue detuned to measure the lower side band to pumping red detuned and measuring the upper side band. These are all different ways to sort of infer the same information. And this is now kind of a state-of-the-art technique that I think more and more groups are appreciating the power of, but also, I mean, more importantly, from my perspective, it's a way to debug to know if any of the things you were assuming naively really do make sense. If you have excess noise, for example, of your cavity, of your drive, of your laser, those are things that really show up here or would kind of become much more obvious. And you would get very unphysical parameters here. So I think that's all I'm gonna say about this. This was partly just to talk about heterodyne spectroscopy and the types of measurements that we can do. So now, I think I do have an actual spectral density. So just to show you, I mean, we're cooling to around 0.3 quanta that's usually about the same level where it's kind of as hard as we can drive and things are still well behaved. What's nice is this corresponds to temperatures of about 200 microkelvin and the fact that you can measure this small asymmetry over this factor of 100 says you're doing primary thermometer in this range. And there's only a handful of thermometers that really work well in this range. This could be one. And at the otherness, the one in Gaithersburg, people like Tom Purdy who worked on some of these things in Cindy Regal's lab has been doing much more recent experiments really trying to push on the absolute kind of precision and accuracy of this as a thermometer, not just at cryogenics, but even at room temperature. Just to show you a little bit about what the actual data looks like of these curves. For the coldest curves, you're looking at two sidebands that look something like this. Some things you should always note is the y-axis here. The units aren't so important, but what you notice is zero would be all the way down here. And this is another way of saying that our quantum efficiency is, I think the polite word is modest. It could be better. Even if you have a poor measurement efficiency, a large effective noise, in principle, if everything is clean and stable, you can average as long as you'd like to resolve these small features. And that's what we've done here. You can see kind of the statistics and the noise that you have. The other thing I want to point out, this much as the thermal motion, there's about that same fraction of thermal motion over here, but the whole rest of this area, if you asked what you were looking at, I think it's fair to say that is literally seeing vacuum fluctuations on your spectrum analyzer. This kind of dual mode squeezing process that gives you your stoked sideband is in fact amplifying the quantum vacuum. And it's something that is now at real, physical size voltages that you can see on your spectrum analyzer oscilloscope. So again, we're around this 200 micro-Kelvin range. That's mostly set by the frequencies we chose, the five megahertz. That's basically the H bar omega over a half is right in this range. If your goal in life was really to get better energy sensitivity to quote lower temperatures physically, I think what you would want to do is use lower mechanical frequencies. So moving on, some of the things that we've been hinting about over and over are non-linearities, how beautiful they are, how hard they are to get. Here's a picture of me when I was a kid and my advisor was showing me how to use a screwdriver or something. But one of the things they'd love to say was that the Jocin injection is the only non-linear, non-dissipative circuit element. And there was always triple underline on the Englely. And this is a bit what we heard in the previous lectures. The idea is a Jocin junction, the Jocin SIN effect. At DC is just a short. You get a super current across these two metal electrodes that aren't in contact. So there's a tunneling current that passes through it without resistance. If we were to think about what that looks like at AC frequencies, well things that are short circuits to first order, you might suspect they look like inductors. And that's in fact the expansion we saw the other day. If you expand what this circuit element of a superconducting Jocin junction looks like, it does look like an inductor. And it's an inductor whose inductance depends on the current that's flowing through it. And it's in that way that we talk about Jocin injunctions all the time or junctions resonating with capacitors as Kerr media. So that's an impedance that depends on the current that's flowing through it or alternatively the number of photons you put in this LC resonant circuit. But we've heard over and over about another circuit that does kind of the electromagnetic dual of this. And so I would say another nonlinear dissipative circuit element is a mechanically compliant capacitor. Just like current changes the value of the inductance, if we put power in this LC, you can think of it physically as putting a voltage, the voltage pulls your plates together, changes your capacitance. And it gives you the same kind of circuit relations. Just to zoom in on it a little better what we're actually talking about when we talk about some of these things. Here's a beautiful photo of a tunnel junction close up in an SEM image. This is a double angle of aberration aluminum. You put down one layer of aluminum. You let it oxidize for about a nanometer or so. That's the size of the tunnel barrier. And then you put another layer of aluminum on top. So you get this super current that tunnels really from this electrode over here across I traced in red where that tunnel barrier might be. And that's the super current. That's this magic nonlinear element that's enabled all of superconducting quantum computing that makes our parametric amplifiers that does a lot of the workhorse. Defines the Josephson voltage standard. For example, at NIST does a lot of the detectors for radio astronomy. And the idea is even without anything about superconductivity, without things about Josephson relations or Bougalabab transformations, a mechanically compliant capacitor really does give you the similar nonlinear physics. Here we absolutely do use superconductivity, but I just wanna emphasize that's only just to minimize the loss to get rid of loss more and more. In principle, this effect could work with normal metals, could work with dielectrics as we've seen. This is just the radiation pressure in Hamiltonian which doesn't have to be written in this circuit form. Just to reiterate, even though we refer to these things as cur media, how non-quantum this has to be, I just wanna do the back of the envelope derivation for this nonlinearity. So as I described in words, you can think of the energy in this capacitor as one half CV squared. The force is just the derivative with respect to X. X is the distance between the plates of the capacitor, but we know they're free to move. They're free to move if a force applies by the spring constant, just the mass times omega squared. And then the last thing we need to know is that the capacitance depends on this displacement as well. If you put all these things together, they give you the radiation pressure physics. What we like to do is write it in fancy notation. You can write it as a cur medium. No one's going to stop you. You have a cavity frequency that depends on the number of photons in the cavity. And the pre-factor that kind of cur constant out front involves G not in the mechanical frequency. So this is one of the scales you'll see people talk about in reviews and things. If you could make this factor really be large compared to Kappa, well that's an interesting regime because now a single photon would move you more than a line width. And those are kind of the requirements you need to really talk about isolating individual quantum states just like Jack was describing. This is really something that could be considered a qubit if that were big enough. If I had to really defend my advisor now in saying that the Jocelyn injunction is the only non-linear, non-dissiputive element, what we've also heard is what is not the hardest thing in the world is getting non-linearity. What is very hard is getting non-linearity without loss. Or it's this ratio of non-linearity to dissipation that's kind of the precious resource. I think the quote I like is describing a system as non-linear is like describing an animal as a non-zebra. It's true, it's just not very helpful and everything's non-linear if you drive it hard enough. In fact, it gets hard to get things that are linear up to arbitrarily high energies. But again, this idea of getting non-linearity at this few photon level, the few quanta level, while still having the loss be very low, that's what we like. So the Jocelyn injunctions, you can really tune this non-linearity to be really as big as you want. You can take it from a mild non-linearity which is what we use for our parametric amplifiers all the way to non-linear at the single photon level. That is the qubits. Optomechanics, well, that's not necessarily what they're good for. Things we're excited about are one, studying massive objects, studying some of these low frequency modes. One of the things we're actually most excited about is this non-linearity of radiation pressure really works all the way from DC to daylight. There's no kind of frequency scale in it. So it works for kind of kilohertz circuits, kilohertz mechanics, or gigahertz circuits, or gigahertz mechanics, or terahertz. And I really wanna contrast that to Joseph's injunctions. We think of them in the microwave as good inductors, but that doesn't work to arbitrarily high frequency. For example, if you shine light on a superconductor that has enough energy in a photon to break a cooper pair, it doesn't look very super anymore. In fact, from the perspective of the light, it looks almost like the normal metal. For things like aluminum, that frequency is about 80 gigahertz, and that's part of the reason why we love the Joseph's injunction dearly, but it's not gonna solve by itself things like converting microwaves to optical. But there is some hope that a non-linearity like this could actually do that, where you could couple both gigahertz, megahertz, and terahertz modes all to the same thing. All right, I think that's most of my speech there. Again, just as an advertisement, that people didn't already know, it's with these Joseph's injunctions, kind of the quantum information in the superconducting cubic community, where people have made these very, very non-classical states. Here are some sexy Wigner tomograms from UCSB. This is Andrew Cleveland, John Martinez. I'll highlight also some work from the Sholkoff lab, making these cat states, making these very non-classical states. The ability to do these sort of yurky-stolar interactions like we heard, that's the thing that we know they had both non-linearity and low dissipation in order to achieve these things. That's what we'd love to achieve directly in the mechanics. And if we can't do it directly, well, maybe we can just take these resources and figure out a way to incorporate them into the electromechanical circuits. So now I'm gonna pull back the bar now that I've showed you the most negative Wigner functions humanly possible, very complex interference fringes. And I'm gonna go to what I would call moderately quantum states. Most of this talk for the rest of the lecture, I'm gonna describe in detail squeeze states. If you ask an average person what makes a quantum state or where the line is, I think most people will agree a thermal state is not very quantum. Coherent states, they're pure, they're nice, but they're also very much in the correspondence limit. As we heard, even cooling to the ground state, we were very excited, but it is the world's most boring state. It is both a special case of a thermal state, a coherent state, special states that it was squeeze state, I guess too. So squeeze states are right in this line in the Venn diagram of kind of quantum states. It's the one that's Gaussian but still qualifies as quantum. And it depends a little bit on who's talking or which metric you're using. I'll give you a little bit of history of squeeze states. In context for them, then I'm gonna talk a little bit more about how they're used in optomechanics, both on the opto side and the mechanical side. So just to be fully clear, I think Clemens talked a little bit about squeeze states, writing down the operator that would generate them. The mental picture you should have, picture all of these states in phase space because that's now how we've learned to think. If you just have a vacuum state, the area of this curve is just given by H bar. That's how you know you have a pure state. The squeezing operation is the thing that preserves the area and really compresses one quadrature while expanding the other. So there's no violation of uncertainty principles or things here, but it is a very nice and powerful state. And in fact, if what you want is low noise in any one quadrature in one direction, that's precisely what squeeze states are good for. Here's a nice picture of a Wigner Tomogram with the projections on the other axis. This is from a review paper from Roman Schnabel, one of the people really in charge of doing the best squeezing in the world and in fact providing the squeeze light for things like LIGO. What you'll notice about the Wigner function is it still is completely Gaussian.  What's special about it is that the Gaussian can be narrower than the vacuum wave packet. That's what we think is special about a vacuum squeeze state. The other thing you will notice is it's everywhere positive. The squeeze state does not have a negative Wigner function. So it doesn't meet the criteria that some people have that you have to have a negative Wigner function in order to have a non-classical state. That's part of the reason why it's kind of right on that hairy edge. Most other non-classical states you can think of do have a non-negative Wigner function. The other thing I'll say, I don't think I have a diagram of it, if you really expand a squeeze state in the number basis, there's something very special that happens. And that's that all the odd elements in the number basis really disappear. And that's a consequence of the fact that a squeeze state is really all about correlations. You're using quantum correlations to in fact get things to have line width less than the vacuum line width. And you should always, when you hear squeezing, think about pairs. So in a single mode squeeze state, you're always, for example, down converting a pump photon into two correlated photons. So you get things in the n equals two, n equals four, et cetera, et cetera, but only in even pairs. This also highlights why they're very fragile. If you lose one half of those pairs, you're gonna lose quite a bit of the coherence that's in here. So, I mean, this is one of the themes that we'll see over and over. These very quantum states are highly useful. They're really powerful, but usually they're equally fragile to all sorts of losses. And that's part of why just being able to make them and characterize them helps also characterize your ability to measure and detect. So again, if we were just to look at what you would see if you were looking in kind of the classics quantum optics experiment, you generate a squeeze state. A squeeze state has some angle around which you've squeezed. You've reduced the fluctuations in one quadrant at the expense of the other. Let's say you put it on a perfect homodyne detector and you tune the phase of your local oscillator around till it matches this phase here, the angle, pardon me, of your squeeze state. If you were to look at the power spectral density of just the noise on your spectrum analyzer, what you would see is it would make this McDonald's plot. Maybe we only call it that in the US. The idea is in the anti-squeeze quadrature, it went up here by 10 dB. And if it's a perfectly pure squeeze state with no entry V, no loss, you get 10 dB of squeezing there. That's also highlighting that the squeezing operation is unitary and reversible. So it's completely pure. It's a nice thing you can do. What on the game side of this, this is exactly when we look at these things to operate parametric amplifiers. This is what we use to amplify vacuum fluctuations. So having the anti-squeezing is not always necessarily a bad thing. And part of the time what you want is the gain. And part of the time what you want is really, attenuation is the wrong word. You should think about coherently de-amplifying, not really attenuating the noise. Because if you attenuate noise, it has to radiate with its own vacuum noise. And in this notation, as I said, just like a coherent state or a vacuum state, if you look at the mean square fluctuations in X and Y and their product, they really are conserved in these dimensionless units of one over 16 is one of the conventions in quantum optics. If you did have loss, the way you can model loss is as if somebody stuck a beam splitter in between your beautiful squeeze state and your detector. The thing you should always remember about a beam splitter, not only does it route some of your delicious signal away from you not to measure, it has to let in vacuum noise in that other port. So it's doing two things that really hurt you. It's getting rid of your beautiful correlated photons and it's adding in incoherent vacuum noise. And so if you have a detection efficiency that's less than unity, now instead of measuring this dotted line, you'll measure like this green curve. And what you see is your gain went down by your efficiency, but actually your squeezing went away much quicker. And that's essentially because you're dealing with a state that's below vacuum noise. So when you add in the vacuum noise, that's a very big correction. And this is one of the other reasons why it's very, very hard to measure deep levels of squeezing. Any small bit of loss gives you a big hit in the kind of raw squeezing you'll measure at the output. Apparently decided to go back to my beginning of my slides somehow. Are there questions? Well, I'm thumbing through my slides. Yes. Should we go to the actual floor? Oh? No, it's okay. It's leaving everything up to go down because the vacuum noise from that actual floor in the frame of the vacuum noise associated with the squeezing. And I wouldn't- Is that good? Let's add in the vacuum. Yes. I can write it on the blackboard. I wouldn't even say interfering. I would just say adding in quadrature, adding in an incoherent way. If I have some squeeze noise and I put it into a beam splitter with some loss and I wanna figure out what I get, I take the squeeze noise and I reduce it by the eta, the efficiency that I have. But every beam splitter, and this is just like when we say the words beam splitter, Hamiltonian, it's always sending the power somewhere. A beam splitter itself doesn't actually lose energy. It's just splitting it up and dividing it. So if eta came from here, there has to be a one minus eta term. And what comes in that port if it was a cold vacuum coming out of that port would just be zero point or vacuum fluctuations. And this equation is nothing more than what I'm plotting here. If you can imagine if this quadrature is the anti-squeeze quadrature so that it's huge, this term is pretty negligible. But if this term is the really squeezed quadrature, then this becomes quite dominant. And so for example, if you have a 50-50 beam splitter, you can put an infinite squeeze state into it and you're only gonna get 3 dB of squeezing out. Are there other questions? Another way to think about it, if you have loss, it's a little bit counterintuitive. You started with a squeeze state that was purely, completely pure. You had a beam splitter which added in vacuum noise and vacuum noise is also completely pure. But now if I look at the squeeze state that came out and I look at the product of X and Y, it's now, it's gone up. It looks like there's entropy in the system. It looks like there's impurity in the system. And in fact, there is oftentimes in squeezing, it's quantified almost like an effective thermal state. But I just wanna reiterate, you get this kind of effective thermal state even if everything is perfectly cold. And it's just saying that again, if you think in the Fox basis, you've thrown away some of your correlated photon pairs, they're no longer have a correlated partner.  And so the product of these two things no longer has to be just H bar. It can be much, much bigger than that. The... I guess Donald, you're mixing into the... So for, I mean, I don't know if you can read it out. It looks like we went from 10 dB to maybe six or seven dB here and we went from 10 dB to like three dB here. And that's exactly the point I'm trying to make. The fact that the squeezed is much closer to the unity line than the anti-squeezed is precisely because of this relation. And if we wanted to work it out, we could do it for the squeezed and anti-squeezed and look at this product. And it really gets bigger and bigger. Interestingly enough, actually 50% loss is maximally bad. And one way to think about it is if you have 100% efficiency, well, you have a pure state. If you have 0% efficiency, you have a pure state because it's all just vacuum. It's not squeezed, but it's pure. And so 50-50 actually tends to be worse from an entropy perspective. The other thing I'll say is a lot of the things that wanna take advantage of squeezing oftentimes really care about this raw squeezing that you have there, what you deliver right at the device of interest or the quantity that measures this. Anything else? All right, so I'm gonna move kind of quickly, but I wanted to give a bit of a history because I found myself yesterday over lunch that you've been not quite sure of my timeline of who did what when. Squeeze states were something that were talked about theoretically for a long time. Certainly in the 60s, there were review articles about such things by people like Glauber. In the 80s is when things really picked up experimentally. I also learned reading a review that I think it was caves that came up with the name squeezing also sometime in the 80s. So people knew these states, loved these states, but they didn't call them squeezing. The first experimental measurement, I think most people agree, was in 1985, it's at Bell Labs, it's Slusser, Yerkees on there, some other people, but really these four groups are there because they were all racing to do very similar things. I think they were at Japan, maybe IBM and University of Texas. You'll see some names you might recommend that is like Yamamoto or Kimball in here, Walls, and all of this race was just partly in the generation of the squeezing and how good it can be, but also in the detection. And right within these kind of 15 months or so, there were four demonstrations of squeezing below vacuum. And just to be clear about nomenclature, in general, you can have thermal squeeze states. It's easy to take a thermal distribution and make it look asymmetric. That's not nearly as special because if something never goes below vacuum in one quadrature, I could say just start with a hot state and make it even hotter in one quadrature. And so it's asymmetric, it's squeezed, but it's much less interesting. So part of the game, part of the threshold was really get one quadrature below vacuum. That was with light. What about the light that I like, the microwaves? Actually, also at Bell Labs, not too much later, 1988, people like Yerke and coworkers used arrays of Josephson's injunctions to make the first parametric amplifiers. They demonstrated thermal squeezing of four Kelvin noise and then one year later, two years later, 1990, they measured the first vacuum squeezing. Again, this was kind of really showing the power of Josephson's injunction technology all the way back then. A lot of it, there weren't a lot of driving applications for it. Much of this really kind of just sat there in the literature people quoted and knew about, but really didn't get revived until the mid 2000s. And most of that impetus was within the context of quantum information. What about the mechanics? For example, in 1991, there's a very nice paper of Dan Brugar, who you might know from MRFM. He did a thermal squeezing of a cantilever. So they just drove a cantilever. And in general, if you want squeezing, one way to think about it, if you had some way to take a parabolic potential, the restoring force of a mechanical oscillator, and modulate it at 2 omega, the canonical picture that people like to say it's like a kid on a swing, this is something that if you pump a parameter like the spring constant at 2 omega, you can really coherently add energy to one quadrature that's in phase with it and de-amplify the other quadrature. And so for example, they showed distributions that looked very much like this. But just to highlight, this was very much just thermo-mechanical squeezing, not at all below vacuum at the time. I also always like to keep track of trac-dions, certainly a mechanical system, even if it's a mechanical system with the massive only one atom. I think the first measurements people agree on were at Ness Boulder. They used their Jane's Cumming Hamiltonian to actually measure it in the energy eigenbasis. They don't show the number distribution for some reason. They show some curve from which they infer the number distribution, from which they infer the squeeze date. But I think they agree this was one of the first squeeze dates in trapped ions. Just to jump ahead of what state of the art looks like, this is a paper from last year, two years ago. Jonathan Holmes group at ETH also using trapped ions. Here instead of driving the potential at 2 omega, they use some of the ideas of reservoir engineering. And here they actually show the number distribution of the motion of a trapped ion for a coherent state and a squeeze date. And again, the way they infer it is from the detailed fingerprint in these vacuum Robbie oscillations in letting a two-level system involve with a cavity. Just to look, again, the number distribution of the coherent state, it looks pretty boring, kind of like you'd expect. It's got a center and a mean related to each other. The number state, it's almost hard to see because their data is too good, but all of the odd things are missing. What they have is they have the 2 and the 4 and the 6 and so on. And from that, they infer a squeezing of something like 12 or 13 dB. What they also went on to do here was also verify the phase, which isn't something you necessarily get from the energy distribution alone. To highlight some other work in the microwave, again, in the mid to late 2000s, a number of groups were making parametric amplifiers. This is work from the Lanark lab at Gila. There was also lots of work at, say, ETH and Andreas Wallroff on a number of other ones, I'll forget. This is just a good example of you take one JPA, essentially, as your squeezer. You couple it as low a loss as you can to another JPA as your beautiful parametric amplifier. You rotate your phases around. You do some inverse radonning. And you can actually determine what the measured beginner function is. The reason I picked this work is this was one of the works that goes out of the way to really plot for you the total raw squeezing you see at the output without any assumptions about your measurement efficiency. As I said before, if we really calibrate our detection very well, we can refer what our squeeze date would have been before it hit any loss. But kind of the most brutal and most useful figure of merit is high fidelity is the squeeze date when it actually gets to the end of your detection. And here I think this was squeezing of about 2 dB. Maybe city of the art these days is 3 or 4 dB. And most of it is not the fault of the squeezer or the amplifier.  It's the isolators and directional couplers, the things we leave out of the circuit diagram. That lets our signals flow in one direction and not back act in the other way. I think this is the going world record for light. Again, this is Roman Schnabel's group. Every couple of years, he publishes a new record because they get better and better. The measurements are really phenomenal. This actually shows squeezing as a function of frequency. It's really right on the 15 dB line. If you want to, you can back out the impurity has by looking how this 15 dB compares to the anti-squeezing. Again, it's 18 to 20 dB or so. This is 15 dB, again, without subtracting any losses of the measurement, without doing any corrections. This is really getting everything right. And this is hopefully the 15 dB of squeezing that something like LIGO could actually take advantage of. And here's where, again, I'll just reiterate. I would say optical technology is just much more mature. At these specific wavelengths, the ability to get every component to be very, very low loss right up to the detector all the way from the squeezing source. Again, 15 dB is phenomenal to me. So now in the context of the nanomechanical systems, I think I'll speak for myself. A lot of people in the field really looked at this PRA by some names you might remember for a way to use reservoir engineering in Optomechanics to initialize directly a squeezed date of the mechanics. I want to point out there are analogs of this that were done for the ideas of Trapdions, Sirak and Zoller. But again, Trapdions are analogous, but there's a Jane Cummings Hamiltonian there, not the Optomechanical Hamiltonian. The analogs, some things you can take, some things you can't. The basic idea, I won't go into too much detail for, but the idea is that you pump an Optomechanical system in the good cavity limit at the red sideband and the blue sideband simultaneously. Depending on who you ask, they could give you different intuitive ways to explain why this should give you squeezing. I'm not going to give you a full derivation. I'll refer you to some of these very nice papers like this one that do explain it in detail. What I will say is if you applied the red and the blue pumps of exactly the same strength, that's almost as if you drove on resonance and just 100% amplitude modulated your signal. In other words, blinked on and on your signal, stroboscopically, at a period that's commensurate with the mechanics. This is a way you can imagine if the drum is vibrating away and you want to make sure that you really measure amplitude or phase, but don't acquire simultaneous quadrature information so that you're not subject to the Heisenberg uncertainty principle, you can imagine what you would do is just make sure you closed your eyes every half period so that you, nor anyone else in the universe, really knew what it was doing at that point. Those are the ideas that are called back action evasion measurements. They're things that people like Brighinsky have been talking about for a long time. They're very analogous. The idea of the squeezing here is actually to slightly deform from that picture. So again, the fact that you're pumping red and blue detune means you have two pumps that are separated by a frequency of 2 omega. That again gives you a flavor of why you could talk to quadratures because you're doing something kind of periodically with the mechanical motion. But now what you do is you actually pump a little bit harder on the red than you do the blue. Now there's some dynamical back action. There's some damping and there's some cooling. So that means you're not going to leave the mechanics in the same state it was. You have to damp it. And in the language of reservoir engineering, when the red and blue are on, instead of cooling to the ground state, what you would do is cool to a squeeze state of motion. That would really be the ground state of this interaction. It looks as if you're cooling to a squeezed reservoir. So again, this paper came out in 2013. There were three groups that really did the almost the same exact experiment at the same time. Our group was one of them. Key Swabs was the first one to really measure it. Mika Silimpah was just at the same time as well. Ironically, all three of these were really with these drum circuits like I've been showing. There's no reason you couldn't, in principle, do it in optics. I have a feeling the electromechanical circuits and the drum circuits are maybe a little bit more nice for having multiple tones, having them stay nicely phase locked and coherent and stable over a long amount of time. We can ask some of our optics colleagues why they would choose to tackle or not tackle these problems. So just quickly I'll walk through what these measurements look like. What we wanna do is ideally have one cavity for state preparation and one cavity for measurement. And what we're gonna think about is we have a mechanical mode all the way down here. We're gonna apply a pair of pumps all the way up here. And these two pumps, it only realizes this perfect squeezing or perfect back action evasion if the separation is really exactly two omega. If you wanna calibrate before you do that measurement, you can just detune them epsilon away and you kind of break this perfect timing of this whole thing. It gives you a way to compare kind of the back action evasion to the non back action evasion. So for example, as I said, if the pumps are perfectly balanced, this gives you a Q and D measurement of the mechanical quadratures. And if I slightly imbalance one, then it gives me a squeezing operation. So in our experiment, it looks something like this. We send our drives in, couple to a circuit, look at what comes out. As I said before, we're gonna engineer a circuit where one mechanical mode is coupled to two cavities. These two cavities will just give us more room in the spectrum, one to do our measurement, one to do our state preparation. And that means we have to actually drive it with four pumps. We're gonna drive one pair of tones on one cavity that's our perfect back action evasion, one pair of tones on the other that's gonna be our squeezing operation. The circuit looks something like this. This is a little coupling feed line. This is now a drum circuit similar to what I showed you before with the two plates. The two plates coupled to two separate inductors. Here the two inductors give it the two cavity resonant frequencies. Here is absolutely a case where taking into account the coupling, both capacitive coupling and mutual inductance coupling is part of what's setting these eigen frequencies. But in those eigen frequencies, we can just think of them as normal optomechanical systems in that basis. Just to zoom in a little bit more, mostly because I like the picture because it came out really nice. You can really start to see inside of these holes, inside these tunnels to really see the top plate is suspended. And again, that plate separation that we have at room temperature, it's something more than 100 nanometers. Part of the magic is the fact that when it cools down in the cryostat, thermal contraction brings it in so that it's only maybe 35, 40 nanometers separated that gives us the G-naughts that we have. The mode we're gonna look at is this kind of even odd mode. Not sure of my nomenclature for my circular drums. I should bone up on them. It's this mode. So here's what the data looks like. Again, the way we're gonna start, with our state preparation cavity, we're gonna start just by cooling it. First, let's see if we can measure something that's cold. And we'll see if we can measure something cold with our back action evasion measurement. Now what we did in the beginning, our cooling cavity, even though we're just thinking of a state preparation, we can always read out its signal so we can measure his power spectral density. It already tells us what we think the answer should be. Then with our other two tones and our other cavity, if they're not detuned properly, we'll actually see the upper sideband of the lower pump and the lower sideband of the upper pump separately. And this is another way to think of this heterodyne thermometry that I showed you before. They're measuring the same mechanical mode, but one has an n population, the other has an n plus one. Now the magic, when you do everything right, I guess before I do the magic, what you wanna see is if you turn up your measurement strength higher and higher, you can see if in fact you do heat the mechanical mode. By heat, we mean back act on with the quantum shot noise of the microwave light. And what we see is when we increase all both of these pump strength together, the mechanics gets hotter and hotter. It started somewhere at point two or so quanta, heating up to two or three quanta. And we see that signature, both in the measurement cavity and in the cooling cavity, which is also measuring. Now what we can do is take these two peaks, move our pumps in so they interfere on top of each other. And here I really will use the word interfere because you notice I took a noise peak this tall and a noise peak this tall, if I just would incoherently add them, when I put them on top of each other, it should have been taller than either of them and it actually went down. This interference is the thing that tells us we're looking at its single mechanical quadrature. And now when we have this perfect interference condition, we can ask what happens as we change our pump strength. And so when we change our pump strength, that's these blue dots here. What we see is the idea of a Q and D measurement. What we see is we see no back action in our measurement. To be fully rigorous, what happens is I'm still back acting. The beauty and the symmetry of the coupled modes here is that the back action is exactly in the quadrature that I'm not looking at. So it doesn't couple into this measurement. And the way you can see that is if I just look at this cooling tone as a readout, he still sees all the back action because he's not phase locked to the same kind of stroboscopic measurement. So for example, when we tune this perfect back action evasion measurement from the perspective of that measurement, we see no back action. From the perspective of anybody else in the world, they say, hey, somebody heated your drum. So that is the Q and D measurement. Now we can start to use that to do tomography of our squeeze date. You can take your two tones. You can rotate the phase of one pump with respect to the other that will rotate around the quadrature you're looking at. And hopefully you can see you're squeezing. And that's what I show you here. These three curves from this paper here, the middle one is our best cooling of just a vacuum state. And you can see we put this in units of vacuum. So the peak comes just above the vacuum. That's because there's a little bit of thermal access population. Then the other two color curves are gonna be when we apply our squeezing to the other cavity. These asymmetric two tones, exactly in phase, you can see that one goes up and one goes down. These are exactly the hints we're looking for. In the end, if we plot our two quadratures as a function of our squeezing parameter, we see something like this. Again, I'll use the word modest. We were super excited. It does, in fact, go below vacuum. The fluctuations in that quadrature were below vacuum and measured to be. There's still open questions of why it wasn't better or what inefficiencies there are. But this kind of state of the art of showing squeezing right around a DB or so is about what the three experiments are. I believe Keith Schwab experiments now have actually shown slightly more than three DB, but that's kind of the state of the art. The thing I would say really remains is here, this squeezing is really inferred squeezing. What I mean by that is we're subtracting our noise background. We're calibrating the hell out of it so that we know just what our mechanical mode noise is, so we know what to subtract. What still remains to be done is to observe pure squeezing or observe total noise that's really below the SQL.  Nothing fundamental.  So I'd say that's part of the motivation of things still to be done and that's a tool that would be used for tomography of any quantum state you made. Didn't have to be squeezing. It's exactly this back action evasion measurement. I could rotate around, inverse radon transform and show your Vigner function. So if somebody told me they could make a cat tomorrow, the problem still remains to be seen. You have to be able to verify that cat and really what you wanna see is that raw Vigner negativity. All right, so now I'm gonna switch gears again. Instead of talking about squeezing of the mechanics, now we're gonna think about the question of what if we squeeze the light that we're illuminating an optomechanical cavity with? Now most of this story, I've actually already told you yesterday, trying to be careful in how I planted the seeds and talking about illuminating an optomechanical cavity with a pure coherent state of light. And now what we're gonna do is think about instead of illuminating with a pure coherent state, what if it's some squeeze state? In the language of quantum optics, not just a squeeze state, but a displaced squeeze state, meaning far away from the origin.  Just to make clear all the assumptions, standard optomechanical Hamiltonian, we're gonna learn in the linearized regime, we're gonna work with large displacements here with our parametrically enhanced G. We're gonna be thinking about this in the weak coupling regime, G less than kappa, that's just for convenience and clarity. We're also gonna be in the deeply overcoupled regime where we can ignore internal cavity loss because almost all of our loss of our cavity is back out of our measurement. Over and over, I'm gonna kind of quantify things in terms of cooperativity. This is just the good dimensionless parameter to quantify the measurement strength or the coupling strength over the dissipations in the circuit. So the first thing we're gonna consider is this problem of reaching the SQL. The idea if I drive an optomechanical circuit on resonance and I wanna measure the standard quantum limit of mechanical fluctuations. What is squeezing do for you? Can squeeze light help? Does it hurt? Does it do nothing? So just motivate the cartoon picture first. Normally what I would say is we have some coherent state that sticking out like this interacts with the optomechanical circuit that puts phase modulation on it. You see this little noise ball moving back and forth. Kind of the ratio of that signal to the length, this noise ball to the length of the aero, that's like the sensitivity of your interferometer as a phase meter. If you want better phase noise, the easiest thing to do is just turn up the power of your laser. That takes the same noise ball, it divides it by a longer lever arm. That's the easiest way. But in many systems, you can't just arbitrarily turn up the power. And so what you can ask is what if in chat of changing the length of the vector, what if you talked about possibly squeezing? Now, since our signal is a phase modulation, the most logical thing to think about would be to do phase squeezing. In other words, squeeze along the axis so that now when you move side to side, you get bigger separations between these noise balls. But as we see from this picture, we know that when we squeeze that quadrature, we anti-squeeze the other. We know the amplitude fluctuation, the light have consequences. So part of the question is, where does it come out in the wash? Does the amplitude noise hurt? Help? Are you immune to it? So here's the same picture. Again, first start retelling the same story. An optomechanical cavity. Again, I draw everything in the optical analog because I try really hard to work on my ability to draw beam splitters and things. Our experiments are gonna, of course, be microwaved, but I find more people can touch base with this. If I just look on my spectrum analyzer, what comes out in my homodyne signal of the phase quadrature as I drive exactly on resonance, I see this purple background that's just the shot noise of my interferometer and this mechanical peak on top of it. Down here, I have all the different components that I will have, the shot noise of the light that gives you your imprecision, the mechanical zero point motion and the radiation pressure heating. And what I'm gonna play is a little movie where you increase the cooperativity. And in the raw units coming on your spectrum analyzer, all you see is this peak gets taller and taller and actually as you get high cooperativity, it gets taller much faster than you expect. It gets taller for two reasons. You're doing a more sensitive measurement and the mechanics is getting hotter. The way we usually think about it is to take this noise divided by the strength of our measurement to really work in phonon units, to refer this back to the input of our measurement. Now in this way, if we start at low cooperativity with a really weak coherent state probe, it's like we have a very noisy interferometer. So our shot noise level is up there. Now as I turn up the cooperativity, what I see is we beat down this noise floor. It's just doing better and better phase measurement. At some point, the mechanics gets a very good signal to noise. I've frozen the movie right here. This is the optimal and again, what we're trying to optimize is the total noise on resonance. So you notice from this point, it's always been going down and down and down. But if I continue to play the movie, what I start to see is now even as I push this down, the back action means the total noise goes up. This is another way just to show you the curve that I've shown now five or six times. The idea is you have imprecision trading off with back action. If your mechanics is already perfectly cold, the only thing you measure at the crossover is the sum of these two added noise and the vacuum fluctuations of the mechanics. Now what happens as we squeeze? This was for a perfect coherent state. That's my disclaimer about the standard quantum limit, but I think I've already given that speech, so I won't repeat that one. If I now inject phase squeezing, we know that should push down my sensitivity, that should push down my imprecision noise. But we also know it should increase my back action. And in fact, the sum of those two things, well, it's kind of depressing. All it really did was move the curve over. The imprecision went down, the back action went up. The minimum I reach is still the standard quantum limit. And this is part of the reason why I want to reinforce. The standard quantum limit is a real limit in this sense. I'm measuring both mechanical quadratures with the minimum added noise allowed by quantum mechanics. So if my squeezed state allowed me to do something better than that, I better be breaking one of my assumptions. Most likely, somehow not measuring both quadratures. But here it really is. Now, for example, if I were to, instead of phase squeeze, amplitude squeeze, the same logic applies. It actually moves over here. Now, lots of people have used squeezed states to look at optimal mechanical systems or analog interferometers. What it was kind of knew here is being able to actually resolve the back action of the squeezed state and see the full consequences.  It doesn't really do anything good for you. One of the things I want to point out is the effects of loss. So if your measurement isn't perfect, two reasons it could be imperfect. One, you have too much back action. You have a noisy laser. Or in our case, our main imperfection, as I keep saying over and over, is our detection efficiency. In the presence of poor detection efficiency, well that tipped this imprecision line and moved it up by a factor of one over your detection efficiency. In our experiments, our effective detection efficiency from wall to wall photon in the cavity to our real oscilloscope is about 3%. And so our curve looks like this and our minimum misses the standard quantum limit. But now, if I do the same argument of what would squeezing do? Now, since we have loss anyway, squeezing will hurt in one quadrature and help in another quadrature, but as we saw before, not symmetrically.  There's a new trade-off that you have there, but for example, when you do a little bit of squeezing, it doesn't hurt much on this side because we were doing a relatively poor measurement, but it helps a lot over here because the mechanics knows about the real squeezing in that quadrature. So if you squeeze a little bit harder and harder, you can see you eventually do pay a penalty over here, but the idea is on one side, you're winning more than you're losing on the other side.  So even in the presence of poor detection efficiency, with arbitrarily large squeezing, you can get arbitrarily close to the standard quantum limit. And as a standard quantum limit, it's not something that people are ever going to hit. People are always going to talk about how epsilon away they are, depending on how they want to keep score. I will just advertise squeezing as a thing for whatever is limiting you, could always let you do a little bit better if you have access to pure squeezing. So let's look at the data. Again, this is the optical picture. We're going to take a squeeze state, we're going to combine it on a 99 to one beam splitter, or as I like to call the microwave a directional coupler. And this is what's going to allow us to make a very pure displaced squeeze state because we're going to add a huge coherent state with the squeeze state, bounce it off of our optimal mechanical system, and look at what comes back. This is what the circuit looks like in the microwave circuit diagram. This is our usual optimal mechanical circuit. This is our amplifier and homodyne readout. You can see the isolators and circulators that have drawn there. The directional coupler I put in for actually displacing the squeeze state. And this magic pink box is the Joseph and Parametric amplifier that generates the squeezing. You can see in the circuit language, all it is is a capacitor resonating with an inductor. The inductor is two Josephson injunctions in parallel, which you might know as a squid. The fact that it's a squid means if you apply some flux through another lead, you can tune its critical current, which means you can tune its inductance. So now this is a fancy way. I've made a tunable oscillator where I have the ability to modulate its resonant frequency at two omega. And that's precisely what we're gonna do. We're gonna pump the flux at two omega to modulate this guy's resonant frequency at twice the frequency we wanna squeeze. Again, the fad pictures look like this. I'll skip over the drum. I'll go to the Josephson parametric amplifier itself. This doesn't have the false coloring in it. These are something we make at NIST. This is actually made out of niobium, another one of our favorite superconductors. This big plate that you see here is just a parallel plate capacitor. Now it's a big sandwich with a real dielectric, not a vacuum gap thing. This is the action end where there's a little squid. We zoom in and zoom in.  You have to put about a gauss of magnetic field to tune this by a flux quantum. The junctions are these areas that you see here and here, and that's what gives you the squid circuit. And this line is just the line we run current to modulate the flux to pump the parametric amplifier. And then the idea is, go to this. The idea is that all of your signal goes out very nicely. All of your signal goes out this port. There's three leads that lead to a coplanar wave guide that get us to our SMA cables and such. So here's what the data look like. Again, we're gonna start pretty cold. We're gonna pre-cool to about 10 quanta. We're gonna measure our imprecision and back action and see this very nice trade-off, just illuminating it with a coherent state. Now what we can do is instead of using a coherent state, we can use a phase-squeeze state. And we see exactly the effects I described before. What we see is the increased amplitude noise gave us a heck of a lot more back action. But the fact that we have modest detection efficiency means we played a very small penalty in our imprecision. And in fact, that's what this inset is. It's just taking this line, taking out the one overpower dependence so it's a flat line and showing you there is a very small but measurable difference. The fact that this side went up, this side did in fact go down. If we squeeze the other way, we see the converse. We can see that our imprecision got worse, but our back action got much better. And again, part of the claim of the paper is the global minimum is slightly better than we could have done without it. Most of this was not claiming some record number for this sensitivity, but more demonstrating the physics that go hand-in-hand of using squeeze dates in an interferometer. Just to show you what the mechanical sidebands look like, this is in the homodyne spectrum. So what's shown in green is if you just have an unsqueezed drive, a nice coherent state. And then what is in red is the data for the squeeze state and the black is the theory. Now everything I've been showing you has been perfectly aligned to either the phase or amplitude quadrature. I'm gonna play you a little movie where we rotate all the way around in between phase and amplitude. And now what you see at the two special points of the top and the bottom, those are the beautiful perfect Laurentians I was showing you that just look like your peak height went up and down. But what you'll notice is that at the in between phases, you see the wings go up and down. This is the thing that's actually showing you there is interesting interferences going on. These kind of going below on the wings are precisely the reason that LIGO is interested in their squeezing. They don't ever wanna measure anything at their mechanical resonant frequency. They're far away from resonance. And what they know is if they illuminate with squeeze light, they can get this line to go below. And that goes below a different type of SQL that they define over there. The SQL there is just if you have only fundamental noise sources of radiation pressure shot noise, you ask, can you do better at one of these Fourier frequencies in the total noise you measure? This is what the McDonald's plot looks like for our data. Again, we can use just our usual homodyne detection without the mechanics to measure the fluctuations go up and go down. We do measure raw squeezing at the output, but it's very little. Again, because of this 3% detection efficiency. But the thing we saw earlier is actually what's much more sensitive to the squeeze date is the mechanics when you're back acting. So instead of not pumping the mechanics and just looking at our inefficient receiver, why not use the mechanics to measure? Now, I've already shown you just from those back action curves you would see a big signal. I'll make a slightly higher order point. And that is that the Bayer interaction Hamiltonian really does look like this. What you should notice about this interaction Hamiltonian is that it is a dagger a that commutes with the Bayer Hamiltonian. This b dagger plus b does not. What this means, for example, is that the optimal mechanical Hamiltonian commutes with amplitude fluctuations of the light. And in fact, driving exactly on resonance is the thing that not only does a good job of measuring the amplitude fluctuations, but actually goes beyond that, measures them and does not disturb them. So amplitude fluctuations shine on your optimal mechanical circuit, drive the mechanics, get re-encoded in the phase, and come out in the phase amplified. But the amplitude fluctuations just go along for the ride. They get perfectly reflected, ideally undisturbed. What's nice about this measurement is usually the way we think about hiding one quadrature for the other. In parametric amplifiers, you amplify one and you squeeze the other. That ensures you only get information about one variable at a time. Here, the way that Hamiltonian makes sure you only get information about one variable is it encodes the amplitude quadrature into the phase quadrature with gain, which buries the original phase quadrature, which means in the end you only have information about the amplitude. So again, this is what the power spectral density looks like. If we look in the phase in the amplitude quadrature, this is just a way of saying we have an enormous signal in the phase, but ideally in the amplitude, there's just nothing going on. And in fact, if you look at the ratio of these areas, we can put bounds on how Q and D we are just by saying that we don't see anything at the more than 30 dB level. And in fact, if we quantify how good we think our mechanical detector is of these squeeze light fluctuations, well, again, we have all the loss of the squeezing getting up to the optimal mechanical circuit, which we can know very well. And now if we just ask, what's the inefficiency that we can blame on the optimal mechanical circuit itself? In the high cooperativity limit, we have something that's like 94% efficient. That's great and exciting. But again, it only gets really, really exciting if you can get all your efficiencies up to that end so that you can really get wall-to-wall efficiencies on that level. That's the things you need to do to show these 10 or 15 dB of squeezing that's the state of the art. All right. So for the last couple minutes, what I'm going to talk is one last thing we do. All of this work in talking about interferometry and improving signal noise led us to another question. How does squeezing affect sideband cooling? And I'm going to go kind of quickly through it, mostly because I've already talked in detail about how to think about sideband cooling. And I'm going to take that same picture and to form it into the way we think about it. So again, just to recapitulate the way I describe sideband cooling, we're going to pump red detuned, optimally red detuned so that we get preferential anti-stoke scattering over the stoke scattering. We can describe those scattering rates from a Fermi's golden rule calculation just by these gammas that involve your parametric g and where you fall with respect to this cavity, Lorentzian. I already described how these are kind of the rates per phonon. One gets weighted by the final occupancy of the mechanics.  Now we know one of the definitions of losing your cooling power is when you do just as much scattering up as you do down. So when you really reach the limits of sideband cooling, meaning the limits by this residual stoke scattering, what happens is these two sidebands at the output of your spectrum just equalize. And that's just saying, even if you turn up your power more, you're going to cool faster and heat faster. And in that, you're going to stay the same temperature. That effective temperature occupancy you plateau to is something that only depends on your sideband resolution. That's this nm naught in this notation. And so again, the signature you have if you reach the limits of sideband resolution, just from the quantum noise of the coherent state, is that your sidebands will be equal. And this final occupancy, like I said, it's just a ratio of the stoke scattering to the total damping of the system. And ideally, this is the thing that we say in the good cavity limit. We can make it arbitrarily small. If you want to get colder, we'll make a narrower cavity. The question is, can you get colder without making a narrower cavity? Can you do better with squeezing? Spoiler alert, maybe. So if we just tried to guess some answers to think about how it should be, how might squeeze light affect the sideband cooling? One thing you might say is it shouldn't help at all. Why would putting real photons into the cavity field help? The coherent state that we were starting with was perfectly pure. There was no entropy there. There's no reason to expect a priori that putting in this thing that has more energy than the vacuum state really should help. Luckily, that's not right. The next naive guess you might have is something like we would normally say. Well, maybe it's just proportional to the raw squeezing. Maybe if I squeeze by 10 dB, I get 10 dB colder. That's also not the right picture, which was a little bit surprising to us as well. The correct answer, and I'll say it in this almost rigorous way, two rigorous way, for any red detuning, there's a unique squeeze state of light that yields a zero temperature bath for the mechanical mode. And what that means is there's one and only one Gaussian state. That Gaussian state's gonna be a squeeze state of a certain amount of squeezing and a certain angle that will perfectly null all of these kind of what look like effective temperature. And this is what I'm gonna try to spend the next couple minutes just to explain. So again, we have this system bath picture. I have mechanics always coupled to an environment through an intrinsic damping rate. We're gonna turn on our cooling tone that's gonna damp it with gamma optical, but we know we should think about this not as a zero temperature bath, but an effective occupancy that we just arrived. The detailed balance gives us final occupancy that looks something like this. Now where we're gonna work, where state of the art experiments work, is where optical damping is much, much bigger than even our thermal decoherence rate. This term in green is 30 times bigger than, for example, this. That's not where I wanted to skip. Give me a second. Sorry, I don't know why my computer decided to freeze. So again, we're gonna be in this limit where essentially these original environmental temperatures become negligible. You can see the optical dampings in the numerator and the denominator. And so our final occupancy is basically the effective temperature of the bath. And that's the reason we refer to this as the effective bath temperature. It's the temperature the mechanics would have if it's the only thing in the world it was coupled to. Just to motivate, this is experimentally relevant. I'll point out this work by Cindy Regal's lab. This was just last year or so. They were showing they could really reach these limits of sideband cooling. They really hit this plateau where their upper and lower sideband, their cooling and heating rates were equal at about two quanta or so. This is the point where you either have to design another cavity, change your mechanical frequency, or possibly use squeeze-late. So again, now the question is gonna be how do we think about this bath temperature if instead of a coherent state we have some squeeze-date? So this is the plot that's gonna plot this effective temperature of the bath in occupancy units. The color scale is everything that's this maroon red is above one quanta of occupancy. We're gonna be interested in all these other colors. Those are the colder. This is a long plot of the detuning. This would be when you're optimally red-detuned. Going this way would be more and more red-detuned away from your cavity. This would be logarithmically closer to being on resonance. This is your sideband resolution in the x-axis.  This is just plotting the quantum theory of a coherent state. This is nothing new or exciting.  And if your sideband resolution is moderate you can get moderately cold. Now I'm gonna play the movie of what happens when you squeeze light that's tuned to exactly the right angle at each point. What you see is a lot going on. The things you should be excited about is that there are these deep blue regions. The deep blue regions are very, very cold, kind of arbitrarily cold. And what you also see are these deep blue regions now start to protrude further and further into the bad cavity limit. What's not so intuitive or obvious is the fact that it looks like there's multiple Ottoman cooling points that evolved out of this. And that's part of the things that puzzled us in the beginning. But the whole idea here is if you have access to tend to be of squeezing you can be fairly in the bad cavity limit and still get very, very cold. You can use squeezing as your resource that really gets you arbitrarily cold if you have very pure squeezing. So the experiment we're gonna do we are operating at a sideband resolution of about two and a half. We're gonna be operating some point about here. So now I'm just gonna show you slices across. These are the slices of the effective occupancy of the bath as a function of your detuning. So the black curve is just the usual coherent state where there's just one minimum. That's what we're used to thinking about. And this minimum, the value it has is set by the sideband resolution. So here, even though we're barely sideband resolved we can cool to about 0.3, 0.4 quanta. If we have a little bit of squeezing you can see it gets deeper. And if we have more squeezing for example this four and a half dB you can see that's the perfect amount of squeezing that actually makes the effective temperature of this bath go to zero. It really gets coherently knold and arbitrarily low. But then as we go beyond it in squeezing now there's still two points where we can get very, very cold but they're actually at very different detunings. I would now have to change the detuning of my pump symmetrically either closer or further away from the cavity. Again, this whole idea is that this is giving you what this effective bath temperature is. And if this seems confusing I think I'm with you. We've developed quite a bit of intuition. I mean the equations are just coupled equations of motion. They're not that hard to solve but trying to understand what they mean. The understanding we have is as follows. This NM knot, this residual heating is from this stoke scattering process. If I were looking in a heterodyne spectrum as what comes out when I illuminate with squeezed light again just to be very clear I haven't really told you what the reference frame of the squeezed light is. The reference frame of the squeezed light is always that of the pump. That's the one that's gonna give us correlations above and below and that's the thing that the mechanics really feels. So that's the reference frame of the squeezing. And if we're looking in a heterodyne spectrum and we illuminate with squeezed light so first with a coherent state we see something like this. Now as we squeeze, well we see what looks like excess noise and this is just seeing our squeezed ellipse rotating around because we're not in any reference frame with it so it looks like it has more noise. But you notice what happened when this noise floor went up these sidebands started to disappear and in fact if I turn up the squeezing a little more there's a point where they perfectly disappear. And that point not so coincidentally is exactly the point of zero cooling where there's really no signature of the scattering anywhere in the spectrum. And now if I go beyond that to more squeezing well now I start to do a thing that's familiar to some of you from a squashing perspective or if I try to do active feedback cooling actually I dig little holes where my mechanics is. It looks very interesting in the spectrum but from the perspective of the mechanics it's actually getting hotter here and that's what you're seeing the temperature actually went up if you stayed at the center of de Hoonig. So I'll just try to say it one more time in words what is the optimal squeeze state for cooling? The way I now understand it is the injected squeezing is exactly the correct amplitude in phase to cancel out all stokes scattering. And again that's getting the squeezing parameter kind of the R parameter right and the angle right. Coincidentally if you go to the optimal detuning for cooling the thing that's a mechanical frequency away in the good cavity limit or a half kappa away in the bad cavity limit the phase you want is exactly 45 degrees. It's exactly in between amplitude and phase squeezing. And this critical squeezing parameter is just related to only your sideband resolution and nothing else. So I'll just show you the data with the last 30 seconds of the talk. Again our circuit looks something like this. We take a squeeze state, we bounce it off a circulator off our optimal mechanical circuit and amplify what comes out. We're gonna do heterodyne thermometry in this setup. This is if we do our best sideband cooling with our highest power coherence state. You can see in the raw output noise spectrum these two sidebands are almost equalized. That's what's telling us we've almost reached the limits of sideband cooling. They're never nearly equal as expected. Now we're gonna turn up the squeezing and what we see is the size of these peaks goes down.  These are exactly the signatures we'd expect. The other thing we can see is if we don't go to the optimal phase but we actually rotate the phase around, you can get these interferences in phano shapes like you'd expect. And it's only for pie away from the optimal theta that it makes perfect Lorenzians up and down. So the last thing I'll show, this is the final cooling we infer.  The red and the blue are the temperature of the drum we infer with the squeezed state cooling from the upper and lower sidebands. And we go below the resolved sideband limit by about a factor of two. That's what the spectra look like. I guess I do have one more slide just to summarize some of the quote quantum states we've made so far in the drums. We have a ground state, which I've now told you is the most boring, trivial state, entangled states. These I didn't talk at all about but I can refer you to this paper. This is using dual mode squeezing to generate unambiguous entanglement between mechanical motion and the cavity. Squeeze states I described in some detail, our group as well as a few other groups. I'll also highlight a recent paper that just came out in Nature of Physics this week. This is in Conrad Lander's group. I helped a little bit with, it's using a very similar drum. And they're actually illuminating it now, not with squeeze states, but with a real qubit, circuit QED. This is a 3D transmon. And so they can illuminate it with real Fox states. Here they're showing kind of their inference from their measurement of superposition states of the microwave photon field. They swapped the mechanics and back out. I will stop short and say, I think they genuinely are making Fox states of the mechanics. This is not yet showing negative Wigner tomography. These are kind of subtracted histograms that show you the same type of physics that's there, but it's still on top of a big noise background that a very skeptical person could say, how do I know that's just the fault of your measurement? That could be the fault of your state. So I think that's still the bar that you have. With that, I will stop there. I do wanna highlight there's a Gordon Research Conference next March, February actually, that Marcus Aspelmeyer and myself are planning. And if you guys are interested, you can apply online. Thank you.