 Lecture is ready. Great. So then the next lecture will be given by John Teufel from NIST. And he's going to introduce you to microwave optomechanics in the next 90 minutes and then later on during the week. Can everyone hear me? Do I have the microphone close to somewhere? Good. I'll start thanking the organizers for the opportunity to speak, like other people. Just to introduce myself, my name's John Teufel. As Ava said, I am an experimentalist from NIST Boulder in Colorado. And the circuits I know and love are represented in this artist's conception that look like this. We describe them as microwave cavity optomechanical circuits. I'm going to use today's lecture to introduce you a little bit of the history, the nomenclature, and a bit of the mapping back and forth between microwaves and photons, which you might be more familiar with. So as I said, as I go through some historical concepts, some of the things I'm going to be highlighting are what are a bit of the figures of merit. Some of the themes I want you to look for, not just in my lecture, but other people's lectures, which of the physics that people are presenting would really change if our world was more classical instead of quantum? What would happen if H bar were different or zero? Would that change the experiment or not? Do you need it in the derivation of some of the nonlinear physics? Some of the other things you should be looking at is when people show you experimental devices, asking questions like, well, why did they choose that frequency? Why did they choose that size? Why not something else? The answer almost certainly is a compromise of 100 different things, but understanding which trade-offs go as you change why you try to change just one parameter is one of the key questions and a way of also looking at a variety of different experiments. So I'll start by telling people things they know. One of the things I really like, this Nobel Prize in 2012. This is not just because I worked down the hall from Dave Weinland, but the topic of this Nobel Prize in physics you can read from the top. But from my perspective, it's two things. It's one, using light to study quantum motion. Dave Weinland is one of the seminal players in trapped ions, where the mechanical degree of freedom is the motion of just a trapped ion, kind of the lowest mass mechanical system you can think of. And trapped ions really represent the state of the art as terms of quantum control and read out of a mechanical system. Nowadays, we're very loose in what we call, we usually, when we say mechanical, we mean massive, bigger than that. And all the things that we show that are nanofabricated are hundreds of billions times larger, and a lot of what we do is try to just exploit the same techniques that have been pioneered in these types of systems. The other half of it is Serge Hirosh, which wrote the textbook on quantum optics, really defined a lot of things. But what I love about all the quantum optics is his optics were microwave photons. And this is something that people tend to forget. It was very quantum, because they worked at low temperatures and they had atoms in good two-level systems to resolve the quantized nature, but all of the photons were done with microwave signals. And that's one of the themes that I'll have. When we look at Maxwell's equations, or even H bar omegas, there's no characteristic frequency scale in there. So all you have to ask are, what are the effective frequency and temperature of your bosons? So again, if you just have a harmonic oscillator, which is the only thing I know and love, that's what all of our cavities are. It's what all of our mechanical systems are. You can think about its population just by some bosonstein factor. These are the things that people want to get as far close to zero as humanly possible. This is the idea of preparing things in really pure states of motion. That's the realm where you really talk about quantum optics. So in mechanical systems, there's been quite a bit of a race just saying, how cold can you get? What if you put it in these kind of normalized dimensions of this occupancy factor, this N? How far can you get below one? In other words, how pure is your system? Now, I will go back and forth talking about mechanics and electromagnetic waves in the same language. Some of the pros and some of the cons come just in the different scales that you have for these things. So for example, if you were just talking about taking a 10 megahertz oscillator, what that would look like, there's a beautiful picture I found from CERN in the 1960s of about a 19 megahertz cavity resonator. So you can see it's kind of got a pillbox shape, it's a cylindrical cavity, and the scale is about the size of a person. You're talking about a meter length scale, and that is just set by the wavelength of 10 megahertz, which is meters, and the speed of light. This, for example, is one of our devices, a mechanical oscillator, vibrating at about 10 megahertz. Again, the two boundary conditions are kind of these edges over which it's suspended. It vibrates like a circular membrane. The characteristic length scale, it's about a lambda over two resonance there, but you can see that wavelength is about 10 microns or so. And that's just telling you the difference in these scales, and that's something you'll see throughout, whether you're talking about photonic crystals or what have you, this various kind of broad difference between the mechanical degree of freedom and an electromagnetic. Now, as I said, we know very well quantum optics. What we'd love to be doing is quantum acoustics, really taking all the techniques that have been developed to measure quadratures and fields in the light field and transfer all of that information and knowledge to mechanical systems. Part of the problem with mechanical systems is that they naturally live at lower frequencies. You can make mechanics that vibrate at very high frequencies, they just become increasingly small and hard to couple to. Also, conversely, hard to listen to. Things that vibrate at high frequencies vibrate with very small amplitude, it just puts all the burden back on the readout. So for example, the reason we say quantum optics is because at room temperature, taking normal laser frequencies, you're already at an occupancy much, much less than one. You never have to think about the thermalness of the photons. As a microwave person, I really like doing quantum optics with microwave photons, but that requires us to go to low temperature. At room temperature, the microwave photons has hundreds or thousands of quanta in the field. So you can still do quantum optics, but all of your quantum optics would be with displaced thermal states, displaced thermal states in the language of quantum optics. Luckily, you can buy commercial cryostats, cool down to 10s of Mila Kelvin, and now you're deep in this kind of pure regime. The mechanical degrees of freedom, the ones I will talk about tend to be in the giga, sorry, in the megahertz regime, although as we've seen, they can go all the way from hertz up to gigahertz on typical cavity optomechanical systems. And there, if you just wanted to brute force, cool them to the ground state, you would need to achieve kind of 10s of micro Kelvin, which gets unfeasible for these types of systems. So this is the kind of scale that you should have in mind. Again, when I say quantum optics at microwave fields, I really mean we're already working at moderately low temperatures. We can use this as our kind of pure resource to couple the mechanics too, so the mechanics can inherit its quantum properties to the quantum resource of the microwave fields. Even though the mechanics starts at the same kind of 10s of Mila Kelvin temperature, you can do things like sideband cooling where you couple strongly to something very cold and you can initialize a mechanical state in a very low occupancy state. So here's my cartoon introduction to cavity optomechanics. I think of it from a measurement perspective. Let's say I have an object. I've drawn it as a mass on a spring, something that can move around it. I wanna know where it is. How do you know where it is? Well, you bounce something off of it. You have to interact with it. The cleanest thing you might think of is just bounce a laser pointer or laser, whatever you have off of this. If this boundary condition moves, it slightly changes the path length that you have here. And in the output spectrum, if you looked at the phase as a function of the position, the boundary condition move, you've just changed the standing wave that goes through here, the traveling wave. Now if this boundary condition is moving harmonically like a mass on a spring, and you look in the output spectrum in frequency, you'll see this encoded as phase modulation sidebands. Now this is very intentionally a very classical picture because I want you to know this exact same E and M equations that you have here. A lot of what is beautiful and interesting about cavity-optim mechanics or even interacting light with motion is the fact that the coupling is parametric. Now as I said, we only deal with harmonic oscillators. You have a beautifully linear harmonic oscillator of the cavity, a beautifully linear harmonic oscillator of the mechanical mode. So how do you really couple them? You need something nonlinear to mix these frequencies. And this kind of parametric coupling where you move a boundary condition provides exactly that. Now actually as I've drawn it in the cartoon right now, it's not even cavity-optim mechanics. It's really what I would call optim mechanics. You're just bouncing light straight off a mechanical boundary condition. Of course we know if you create another boundary condition, you can create a high-Q cavity, make that same photon interact with the mechanics many many times. In a transduction kind of picture, what a cavity does is it steepens the transduction between a given motion and the phase. You now get a sharp resonance here. This steep part of the resonance is just given by the Q of the cavity. The one over the kappa essentially. And when you do that, that increase the size of these photons. For the same mechanical displacement, you get a much bigger signal in your output field. Conversely, we also know because the photons are interacting more with the mechanical degree of freedom, that is also emphasized things like the back action and the ability to impart forces back on the mechanics. This Fabry-Perot system with a mechanically compliant mirror, that's kind of the canonical cavity-optim mechanical system and with every other incarnation that people invent, they kind of map it onto this general picture because it captures all of the relevant physics that you have. This same exact picture would apply for a microwave resonance. Now, you could have a microwave Fabry-Perot cavity, that's precisely what Sarah Charros does to make these ultra-high finesse cavities. But here, what we also have access to in the microwave regime is the existence of lump circuits. By lump circuits, I mean you can have circuit elements that are much much smaller than a wavelength. And the ability to concentrate fields with very small mode volumes is one of the keys to enabling large coupling in these types of systems. So what I've drawn here is an LC-resonant circuit. The L and the C determine the microwave frequency. I will always refer to throughout this talk as the cavity, just to stay with the language of cavity-optim mechanics. And if the capacitor vibrates up and down, it tunes the frequency and gives the exact same interaction Hamiltonian. Now, this sideband picture as I drew it was just the fact of the cavity's only role was to enhance the interaction. But of course we know that cavity can also serve another role. So this cartoon picture again in the spectral domain, what's shown in purple is kind of the susceptibility of the cavity mode. You can also think of it in the scattering picture as the effective density of states for these processes. The ability to achieve very narrow cavities where the line width of the cavity is much much less than the mechanical frequency now gives you the ability to address individual mechanical sidebands. You no longer have to get pure frequency modulation, which is by definition equal stokes and anti-stokes. But you can do things like apply your pump detuned so that one is resonantly enhanced and one is non-resonantly suppressed. And this is the whole heart of things that are referred to as resolved sideband cooling. Again, stealing exactly these ideas from the Trapt ion community, augmenting them to think about just linear harmonic oscillators. And it's this same kind of resolved sideband regime that is a bit of what people got very good at achieving in the last decade, experimentally being able to do these things and really tune these parameters to what they want is a bit of the experimental breakthroughs. So just for context, as I referred to in the Trapt ion regime, if you look to the late 80s, it's then that people were really starting to talk about laser cooling, mechanical motion to the ground state. They really look at these stokes and anti-stokes sidebands. Again, these two sidebands, you should think of as non-linear process where a photon comes in at a pump frequency and gets scattered to a sideband frequency. It really is absorbed or giving off energy from the pump field and that's the thing that can lead to heating or cooling. Measuring the asymmetry of these two rates is a primary way of doing thermometry that in the past three, four, maybe five years, people have also been doing cavity-optim mechanics. Again, I show this just to show kind of where the bar is and how many decades behind Trapt ion community we are. So the interaction Hamiltonian, I don't think anybody has actually flashed up yet. I will write it like this. I've already jumped into the dagger notation of creation and annihilation operators. I think we heard a little bit in Clemens' derivation this morning of how you actually get to the B's and B daggers and A's and A daggers from X and P. I think in the following lecture, we'll also hear more about specifically how to think about quantizing voltages and currents formally. But I want to point out here is all we're saying with this whole interaction Hamiltonian is you have a displacement X, which we've written in a fancy operator notation that is coupled via some force to a cavity resonant frequency. So in other words, if the mechanics moves, I have some transduction factor given by these two constants in the center will shift the cavity frequency that's characterized by the A dagger A of the cavity mode. The whole game that you have in all these cavity-optima mechanical systems and many, many more is trying to optimize and strengthen this coupling. I would say cavity-optima mechanics as a field has really been starved for coupling and a lot of the revolutions experimentally have been jumps in this G knot, this kind of combined parameter that quantifies how much zero point motion you have and how much zero point motion gets transduced into the cavity frequency. So in other words, it is this product that's literally what we think about as a G knot, a vacuum coupling rate. And despite all the advances we have in the field, this is still something that's depressantly small. It tends to be one of the smallest timescales in the whole system. And so much of what I will talk about when we hear other people talk about is taking this vacuum coupling rate as big as we can and then parametrically pumping it to effectively increase it even further. As I'm sure you've seen, I'll just highlight kind of a zoo of different types of technology. Many of the ones at the top really take the form of a Fabry-Perot cavity with a mechanically compliant mirror. You can see some of these early things really putting like multi-layer, high-reflectivity brag mirrors on top of things like an AFM cantilever. And those really realize the interaction Hamiltonian. I would say there's also these more microscopic things where you have microtoroids or ring-redness in itters or zipper cavities. Now, without doing any math or thinking too hard, generally if you want to know which system has the biggest vacuum coupling rate, you should say whose cavity fields overlap the most with the mechanical motion. And in general, with the Fabry-Perot, you can imagine your electric fields are living largely away from the mechanical degree of freedom. But in these kind of micro-fabricated things, you can see the mechanical degree of freedom and the cavity in the same picture. And that's already hinting at the fact that there's probably going to be quite a large overlap. If I somehow deform the shape of this or this or this, that is a mechanical degree of freedom interacting with an electromagnetic mode. If we go to what I would call microwave cavity optomechanics, again what I mean is the same interaction Hamiltonian and think a little bit about the history. I would say the history actually started predominantly in the context of gravitational wave detection. People were thinking about these things long before actually they had lasers. They were starting off with microwaves. Some of the seminal work you think about is Brighinsky and Moscow State University. There's an experiment here. I'll explain a little bit more in detail. In the 70s all the way through the 2000s, there were a number of groups really going after these massive systems read out by microwave readouts. A lot of work out of Western Australia. I mean, there were LIGO collaborations in Japan, in Maryland, in Italy as well. And the idea was the same. Have some super massive chunk of metal and hope a gravitational wave would come in and disturb it and try to read it out in the lowest noise ways humanly possible. In the kind of late 2000s people were really trying to take these same ideas and apply kind of nanotechnology, nanolithography to these same types of ideas. Highlight some of the work I was doing at JILLA at the time. But a number of groups were doing these same types of things where essentially we were fabricating little nanowires, letting those wires vibrate near another piece of a metal. And that's the changing capacitor that would give you the cavity optomechanical interaction Hamiltonian. So again, how do I think about the G-naught in comparing the optical or the microwave systems? Well, this vacuum coupling rate for the canonical Fabry-Pro, it's a calculation we could write out on the board but it almost takes more time to write it on the board than to just write down the answer. Essentially the frequency of a cavity is just given by the speed of light and the length of the cavity. And you pick up given mode, if it were to move around and you were asked how much the cavity frequency would shift, you just ask how much did the mechanics displace compared to the length of the cavity? So here's another way to see if I want a strong interaction, what I wanna do to optimize this problem is make the lowest frequency, the smallest length cavity humanly possible that gives the smallest mode volume and it increases this ratio of the zero point motion to the length. Now unfortunately there's a problem in this at least cartoon picture, the lowest cavity mode I can have is when I get a half wavelength resonance there. So there really becomes a hard limit of how small L can be. Now the wavelength of light, you know we're talking about hundreds of nanometers, the zero point motion, it depends on what you have but for most of these micro fabricated things, we're talking about femtometer scale. So we're in the safe regime where we can think about small perturbations, we're safely in the lamb-dicky limit in the jargon of trapped ions, that's not what we have to worry about. But all of the game has been just trying to really push these effective lengths as small as you can go. If you do the same kind of calculation for an LC resonator, this is a figure out of Brighinsky's book which I didn't know about until long after I started doing these things. If now the character's lake scale is just the separation between the two plates of the capacitor and I imagine one plate moving up and down, I can see this very similar form for the G-naught. Out front, the characteristic frequency scale is always that of the cavity which at first blush seems like a really bad thing. We've just gone from a few hundred terahertz to 10 gigahertz, we took a hit in a factor of 10,000 in this pre-factor. The zero point motion, we have some hopes of making smaller just by engineering very small systems but the crucial part here is the characteristic length scale is not given by the wavelength of the light which is good because the microwave wavelength of the light would be very, very large, centimeters or so. Here it can be very sub-wavelength and it can be the gap between these two plates and this is something we engineer now to be tens of nanometers and that's what wins back many of these factors and really gives us G-naughts that let us compete and do interesting science in the microwave regime. The last thing I'll write here is as I drew it in the cartoon, the only capacitor in the problem was that that's mechanically compliant. Now, if anybody's ever really tried to design circuits, you would understand there's parasitics that arise there and in fact, if you have other parasitic capacitance like in your inductor, for example, that dilutes your participation ratio and that dilutes your G-naught by a little bit. So there's this last factor that's the ratio of the mechanically compliant capacitance to the total capacitance that you have and if you're engineering right or doing your job right, this is something you get very close to one and leave you close to the approximation that I showed you earlier. You'll also note that this little factor of two, this factor of two is nothing mysterious or surprising, this just comes from the fact that the frequency of an LC oscillator is one over the square root, so this one half here is just that square root in this kind of linear expansion. So again, if we were to talk about some history about what we have, I found this paper, this was a review article with 200 references talking about how there's all sorts of parametric coupling in the world, mechanical, optical, what have you. What I like is he kind of refers to here early work of Faraday, Melda, and Raleigh, all in the 1800s considering mechanical systems. And over here, I'll blow up one of the quotes that I actually really like that's again really emphasizing this idea of you need something non-linear to do something interesting. So perhaps the best-known example is a form of Melda's experiment in which a fine string is maintained in transverse vibration connect by connecting its extremities with a vibrating prong of a massive tuning fork. The direction of motion of the point of attachment being parallel to the length of the string. This is really crucial. Jack Harris will talk later this week, one of the quotes he likes to say is the way you get non-linear coupling is you can't have two guitar strings, they don't talk to each other well, but if you couple a guitar string to the tuning at the end of it, that gives you the parametric coupling where the motion of one tunes the frequency of the other. And that's the thing where they were surprised to give you this non-linear interaction. You drive at one frequency and you see interactions at multiples or sidebands of the other frequency. Another review in the journal Bell Labs, this is from the 1920s, a guy named Hartley was a predecessor of Manly and Rowe. Bell Labs was actually where they would then shallow in towns would be working on the laser in the coming years, but before that they were really starting to think about non-linear optics in the way of how do you create frequencies at multiples or sidebands of what you're doing. And in this article he has this striking little quote here. It says there's a resemblance between the phenomenon just described and the Raman effect in atomic physics. The circuit they were doing is they had LC circuits in here and what you can see just off the screen is they stuck a little tuning fork in there. So this tuning fork's vibrating around, modulating the little bit of capacitance. And what they were amazed by why they were interested in this is again that same kind of non-linear coupling. Now the ideas of the analog to the Raman effect, again quantum theory itself was not even that old. People were just starting to think about photons as particles. The fact that you could get this kind of inelastic scattering with these deep analogies to atomic physics and circuit theory, well that I'm very excited, that's the heart of what I do today at most of my research. Just to put this in context of Bell Labs on the next page of the journal, there was an article because they had recently invented the telephone and decided it would be useful to have a ringer that tell you when you have a phone call. So while they were thinking about these atomic analogies, they were also thinking about maybe telephone should be able to ring. Going again through history, if we zoom into Brighinsky's work, there's a ton of work in journals throughout. Brighinsky did a fair amount of work on both the theoretical and experimental side. One of the papers where they did, the apparatus looks like this. It's a little hard to see. What he did is he took a rectangular wave guide, that's this long length here. This is a rectangular wave guide that's designed to work in kind of the gigahertz band, a rectangular wave guide of about this cross section or so. On one end instead of shorting it, he dangled a piece of aluminum foil and that's his boundary condition. He suspended that piece of foil by very fine wires and as it's free to vibrate around, he said, what's happening now as I energize this rectangular wave guide with as much microwave power as humanly possible? Now what's interesting is he didn't actually use the microwaves to read it out. He bounced a wave off of this side and looked at the interference to really know what the motion was doing. But just to put this in scale, he had a wave guide resonating at about seven and a half gigahertz because wave guides can be very good. Actually had a queue of that electromagnetic resonance of about 20,000 even at room temperature. This little piece of aluminum foil vibrated at a three hertz resonant frequency and he used very fine wires. He was also getting queues of about 100,000. If we were to put this in the language that I just described, you can calculate his d omega dx in a zero point motion. You could calculate his zero point, his vacuum coupling rate. It would be about a microhertz. But as we know, if you have a very small coupling rate, the first thing you should do is pump the hell out of it and see what you can do. That's precisely what he did. He put tens of watts of power in this and what he did was he measured the ring down time of this little pendulum, whether or not this microwaves was applied red detuned or blue detuned. And he saw in one case, it was a little bit, no, over an hour. In the other case, it changed by about 10% or so. But this was sort of the first microwave experiment I can see that really showed kind of the damping that's necessary for sideband cooling. This is radiation pressure damping, really showing that you can change both the ring down time and the temperature of the mechanical mode. Again, to zoom in on previous work I did as a former postdoc, the ideas were what if you took, instead of a thing of aluminum foil, a kind of nano wire. So what's shown in red is one electrode with a little wire, tens of microns in length, trying to be as close to this other blue piece of metal as humanly possible. This is at one end of a coplanar waveguide resonance that meanders around and goes off here, but it's essentially lambda over four long. So that gives you a cavity resonance. And now you can see as this wire would vibrate in the plane, it's gonna change ever so slightly the capacitance just between this electrode and this electrode, which will tweak the frequencies that you have here. Kind of surprisingly, even though we're using nanolithography and superconductors, we're talking about very similar parameters to Brighinsky's cavity. Frequencies of around five gigahertz or so, quality factors here over a couple of about 10,000. What really helps a lot is the fact that now the mechanics because it's so low mass is very high frequency. Whether high frequency is a good or bad thing can depend on your perspective and what you're trying to do, but this at least allows a lot of things you would want, things like resolve sideband cooling, where you want this frequency to be much, much higher than your cavity lightning. And again, if you take this product of transduction in zero point motion, you get G knots of order one hertz or so. Other incarnations, we then took similar wires and really pushed their length, pushed the gaps down, going more into a lumped element regime instead of having a lambda over four resonator, think of it more like a meander or inductor and push that as much as we could. All of these things, what they really, really suffered for was not just the small G knot because the capacitance and the gap was not very big, but it was really this participation ratio that I alluded to earlier. The fact that this wire could really only become like a femtofarad capacitor, which is quite negligible compared to the tens or hundreds of femtofarads you get for the rest of your oscillator. And this was kind of the same case with a lot of these same experiments. As you push this participation ratio better and better, you might ask, how does that affect other things in the system, the mechanical quality factor, the cavity quality factor, and all of the frequencies of the system. So the circuits we started developing NIST, we're really trying to trade every parameter we could for coupling, really trying to maximize this participation ratio. So this cartoon that I showed you earlier, take a spiral inductor to really get the largest inductance per unit area and couple it to really a parallel plate capacitor. This parallel plate can get orders of magnitude more capacitance per unit motion in these little wires, just mostly because it's large enough to couple two. And that's the same type of trade-off I talked to you earlier about. If you make these very nano things that can be good for what you wanna do, but you still have the burden of having the couple to it. So in a lot of times the balance is being big enough to actually couple two strongly enough, but small enough to be nano, to have large zero point motion in the frequencies that you want. The whole idea is we're gonna measure this motion moving up and down. This is certainly not to scale, to put it in scale. We have gaps of maybe 50 nanometers and we're gonna be looking at vibrations of the femtometer level. So we're still talking about this part in 10 to the seven, part in 10 to the eight, kind of fractional vibration here, but that's something we can read out these very small phase shifts of the cavity resonant frequency in microwaves as well as in optics. I should say this kind of grew out of technology they were developing at this before I got there. Part of the idea is they were just trying to make these suspended vacuum gap structures, not study nanomechanics, but just to make the highest Q capacity as humanly possible. If you have to store your electric field in something, well, vacuum has about the highest Q we know. So instead of storing it in some dielectric, the idea was to store the fields in these gaps. It's these same kind of fabrication procedures where you can take one piece of metal, a sacrificial layer, another piece of metal on top, and then go back in and etch through these tunnels and holes to get rid of everything in between. That makes these kind of magic mechanical sandwiches. So here's what a circuit actually looks like. We do way further level lithography. This is kind of standard optical lithography. We dice it up into little chips here. In this picture, what's shown in black is actually the metal. It's aluminum in our case. Aluminum is great for a couple reasons. Wine, if you're at low temperatures, it's superconducts. But then two, it's also nice and light, much lighter than gold or lead or other things you would want, and that gives you very good mechanical properties. Now, each one of these is the kind of bond pads we use to connect to our coaxial cables, and there's a small little transmission line that runs from one pad to the other, and each one of these little squares is a resonator. We can package these little chips and sample holders, as I said, that let us get off the chip into a nice coaxial mode, something that we can mount in a cryostat and really have good access to. So this is kind of the zoom in of the chip and the idea of the measurement. On one side of the feed line, you send microwaves down this little transmission line coming out. When the microwaves interact with the frequency of one of these resonators, you can measure your cavity and everything about your mechanics. And I want to emphasize that we're doing everything what I would call spectroscopically. We're never actually driving or listening at the mechanical frequency. We're always driving and listening at the microwaves and inferring what's going on at the mechanics. This is a further zoom in, showing you what some of these look like. We have many of these little LC resonance circuits. We designed them each to have a different resonant frequency. You can kind of see there's a slightly different sized drum that lets you address this one at six gigahertz, this one at seven gigahertz and so on. And it's kind of multiplexing that comes for free in your circuit. Going back to kind of the false color SEM, you can see the same way we fabricate these vacuum gap drums. This edge here is what you have that's your suspended perimeter. That's also what lets you make nice little crossovers to make a spiral inductor. And again, the spiral inductor really gives you much more inductance than a meander because all of the flux in each turn of the coil really adds up. What you can also see here is we have a plate separation that's about 50 nanometers. What's nice is we don't have to do the fanciest lithography. The length scale that we care about is just a thickness of a layer which is much easier than making drums that are kind of nanometers in diameter. So this is all done with standard optical lithography. If we were to calculate this in terms of mode volume, you get a mode volume that's kind of ridiculous in the terms of lambda cubed. And that's because this gap separation is so sub-wavelength. So in a mode volume argument we're at something like 10 to the minus 13 lambda cubed. But again, that's mostly a statement of how big lambda is of a microwave resonance. If you were to calculate the electric field, you get, again, this 15 nanometer gap gives you very large kind of electric fields for a given photon in the cavity. And by photon, I mean just energizing this with an LC current that would give you, on average, one coherent excitation. The mass of these systems, the scale we're at is about 50 picograms. That's kind of a nice ballpark or anything that's nanolithography around here. If you really go to much smaller things like the zipper cavities that can win factors of 100 or so smaller, if you take silicon nitride membranes that are bigger, you can go factors of 10 the other way. The point is there's all in a completely different class than things like trapped ions that people were thinking about on one hand or kind of hand-machined resonators like people were thinking about, like LIGO in the early 90s. The frequencies we chose are about 10 megahertz and that's convenient frequency allowing us to get this resolved sideband regime. And again, the zero point motion here is about four femtometers. So the ideas and the measurement on the experimental side is just as I described. We're gonna put this in a commercial dilution cryostat that's gonna cool us to a temperature of 40 millikelvin. That's both the electromagnetic mode and the mechanical mode. We're basically going to send microwaves in and listen on the way out. Now on the way in, we have to heavily attenuate and filter in order to make sure the room temperature, Johnson noise, does not really reach the sample. Cause again, we don't have pure coherent states of the microwave at room temperature. It's only at the base temperature of the cryostat that I can really think about a pure coherent state of the microwaves. Then on the output side, we're gonna do our best to amplify with the lowest noise amplifiers we know. So the first thing you can do is you can do a network analyzer measurement, a vector response of what comes out to what comes in. You can do this in magnitude and phase. These are the types of systems where if you zoom in near one of these resonances, you basically see a notch on your feed line response. So a way for many of these resonant frequencies, everything that would go in would come out. That would be zero dB on this scale. But near one of these resonances, it actually reflects most of the energy back towards your source and you see this beautiful Lorenzi and dip. You can already read off kind of the frequency in the capo, two of your parameters for your Hamiltonian. This one is about seven and a half gigahertz and the line length here is overcoupled to be about 200 kilohertz. Again, we wanna compare that 200 kilohertz to the kind of 10 megahertz resonant frequencies we have which can safely say being in the all upside band regime is not gonna be the bottleneck for these types of systems. And that's nothing special about these drum devices. I would say for the microwave systems in general, that's not the hard part. So now what if you wanna know your mechanical parameters? Well, for example, what you could do is you could just apply a drive somewhere near this cavity resonant frequency and listen on a spectrum analyzer. Not at the frequency you were pumping it but at a mechanical side band away. And so if you do that, if you pump and listen 10.689 megahertz away, you can see there's a little noise peak in the output spectrum. And again, this is noise coming from this mechanical resonator still being far from its ground state. Even though it's in a dilution refrigerator cryostat, it's still vibrating with hundreds of quanta of motion. And resolving this thermal motion is what's coming out here that already now has told you what your mechanical frequency and mechanical line width are. For these sorts of systems, we deal with line widths of tens of hertz or so. We're talking at a queues right around a million. Again, part of that is because these drums are operated at low temperatures. The queues tend to get better and better at low temperatures. Another thing that we benefit from is the fact that when we cool them down, thermal contraction actually pulls these drums tight. And storing energy and tension instead of the Young's modulus of the material itself helps you really get the highest queue resonators. That's something that's known to be kind of a way to get higher and higher queues. Maybe I should pause, see if there's any questions as I've been rattling away. Are there any things I can explain further or keep going? Yes, please. I didn't explain it. Absolutely. Here's the argument as I understand it and I will literally wave my hands as I'm saying it. But essentially, if you can apply tension, a tension stores more and more energy in the tension. And as far as people can tell, the kind of imaginary part of the tension, the loss of the tension is negligible compared to the loss of the material. So when you take a mechanical oscillator that would have been very floppy, like silicon nitride, and you put it under natural tension, what people see experimentally is you increase the frequency without touching the line width. So the dissipation is what it was, but the frequency got much bigger. And now when you look at the normalized quantity, the queue, it looks like tension has just won in the queue. And this same argument, I would say applies qualitatively to most of these systems. Things like nanotubes or graphene nitride membranes, I think are the clearest example. Roughly speaking, adding tension didn't change the line width. It kind of left it untouched, just saying it's not contributing at that level. And so that's why people love the silicon nitride membranes where you can really get a giga Pascal and take something that would have been at 100 kilohertz all the way up to 10 megahertz or so. Are there other questions while I'm pause? Yes. Yes and no. So if you use console or some software, you can see what the effects are. It can be a little bit tricky. Roughly speaking, if you did a back of the envelope calculation knowing the thermal contraction coefficient of aluminum and the thermal contraction coefficient of the substrate, which is sapphire, you can guess at what the tension you would have and that's very close to what we find when it's cold. I know, I mean, it's large factors, factors of two to 10 or so, I mean, could really be. I don't know off the top of my head if there were no tension where this would vibrate at but I think it's less than a megahertz around a megahertz or so. So it's very strong in the tension limit, which is why we really think of it as a membrane and not like a deformed solid. Yeah. If the tension goes up, is it a two, or no? Let's see. If you try to estimate how much a chasmier force is contributing, it's a little tricky because the theorists can't quite agree on what the right infinite boundary conditions are but you would estimate that it's probably pulling you in by an extra 10% or so. Part of the problem is every force is attractive at this scale whether it's something sexy like chasmier or van der Waals or just patched potentials. In Conrad Lander's group, they have drums where they put a DC electrode so they can actually apply a voltage and pull it in and you can measure the mechanical kind of separation as a function of voltage and what they find is you really need a strong correction term, which could be chasmier, it could also, like I said, be explained by other type patched potentials but there are very attractive forces and part of the game we have is we're getting very close to a cliff. If you get much closer, it will just collapse completely because all of these kind of forces scale very strongly in that gap separation. I don't know, in the last part of your question you mentioned about how the chasmier might affect the Q. That's, I would tend to think of it as another thing that gives you a little bit of attention in a nice, lossless way is how I would want to think about it. So I think it would be a good thing. If you just wanted to apply attention in that way, like I said, you could apply a DC voltage itself to pull it down, like we heard about with the nanotubes and things. Yes? Yes. So the fact that we can couple to them says that they radiate and in fact the kind of LC as I drew it as this lump circuit actually radiates a lot. There's nothing to combine the fields. So if you really think about the magnetic fields, they're going very, very far out. In principle, they're combined by the sample box that we put around them. The way I usually think about it is ask what the Q of the LC resonator would be from just radiation damping, talking to its environment. I forget what the numbers are, but they are very, very, very negligible. So you have to be a little careful about your environment and what you put around it, but we can really, really over a couple of these things nicely. So radiation is completely negligible compared to the internal dissipation and especially negligible compared to our intentional coupling. Likewise, you could also ask the same thing about the drums and their ability to radiate phonons into the substrate. That actually you get a bit for free just because there's an acoustic mismatch between this vibrating drum mode and actually launching sound waves into the sapphire. There's things like quits and resistance that really if you start to get very, very high cues, you could start to worry about these things. I think we're not in that limit, but we'd love to be. So I mean, it's not a crazy question to think about. Anything else while we pause. All right, I'll keep going. Feel free to stop me or wave if I hit something. So again, getting back to the bare interaction Hamiltonian. As I said, we have this beautiful nonlinear Hamiltonian coupling the number a dagger a in the cavity to a displacement or a quadratree of the mechanical field. If this vacuum coupling were strong enough, we could see lots of beautiful nonlinear effects. But really you need this vacuum coupling to be strong compared to the dissipations in the system. Being large compared to the dissipation of the mechanical system, that's actually not so hard. I would say many kind of experiments are almost in that regime. Unfortunately, nothing special really happens if you get G not just bigger than gamma. What you really want is G not bigger than gamma and kappa and getting G not of order kappa bigger than kappa is something that's very, very hard to do. In most kind of micro fabricated experiments, that ratio is very small. And things where you use atomic clouds of atoms to realize kind of a pseudo cavity optimal mechanical system. That's where you can start to get that of relevant order one. The thing we mostly do is drive the cavity mode with a large coherent state. In this regime, you can think about the coherent state as some large drive alpha plus some small fluctuations on top of it. And if you now linearize your interaction around this, which is the first step to every experimental paper you will ever see, what you get is this linearized coupling. Now, unfortunately, it's no longer not linear. It's not a dagger A. It looks like a quadrature of the cavity coupled to a quadrature of the mechanics. But the crucial part is this G out front really is now a parametrically enhanced G. You can think about this vacuum coupling rate really increased by the square root of the number of photons in the cavity. In other words, just the length of the vector in this kind of ball and stick picture of quantum optics. And this is the thing, because we have beautifully clean linear harmonic oscillators, we're allowed to drive them very, very hard. And one of the fears of merit for cavity optimal mechanical system that people don't talk about as much as how hard can you drive it before something else goes wrong? Every cavity or every mechanics will go nonlinear at some level. And so a lot of the best systems start off with a moderately big G naught and then pump very hard with millions or even billion photons to now be able to parametrically enhance by orders of 10 to the five or so. And this kind of single photon cooperativity getting increased by a square root of a number of photons. This is very analogous to other kind of cavity QED systems where you can think about N atoms interacting with the same cavity. You get a kind of incoherent adding of all your individual particles. And here it's a parametrically enhancement from the light. So now with this just linearized Hamiltonian you can start to ask, what can you do? I mean, some of the first things you can think of doing are you can drive red detuned. This is the kind of sideband cooling probably the easiest picture to think about. In the language of quantum optics what we're isolating is the beam splitter Hamiltonian. I create a photon by annihilating a phonon and vice versa. And driving this red detuned really gives you the effect in this parametric rotating wave of looking like your mechanical harmonic oscillator is brought into resonance with your cavity harmonic oscillator. And if that coupling is strong enough they will coherently exchange information back and forth. Hence the beam splitter part of the Hamiltonian. If you were to drive red detuned, instead of red detuned, drive blue detuned we would normally think about this as heating the system. Now, heating is fine qualitatively but you should think of it more like a coherent amplification. Yes, when you pump blue detuned you start to narrow the line width of your mechanical mode. It becomes higher and higher Q but it gets hotter and hotter. And again, hotter is not quite the same as becoming more and more energetic. In the quantum optics line which what you have is a dual mode squeezing Hamiltonian. This is something that cracks a pump photon and puts one quanta in the mechanics and one quanta in the cavity. And it's something that's beautifully coherent. You can use to do lots of interesting things. Things like entanglement or amplification. But if I were to just trace over the cavity and look at the mechanics I would see the mechanics looks hot. It's in a state that has a lot of energy uncorrelated with anything because I traced over the thing it was highly correlated with. The last thing I'll talk about is you can imagine ways to use both. One way to use both would be simultaneously pump red and blue detuned. This realizes kind of a canonical Q&D measurement. This is something that Brighinsky and colleagues were seminal and thinking about back in the 1980s. The other way in which you get both parts of this Hamiltonian is actually the simplest way when you drive exactly on resonance. The stoke scattering is just as relevant as the anti-stoke scattering because you're pumping exactly in the middle there's no damping or anti-damping but there's still strong interactions. So those are kind of two cases where you can think about using both. Today's lecture I'm gonna be concentrating mostly on this kind of beam splitter interaction. So, as I mentioned, one of the first measurements you can do when you get an experiment cold is first you characterize the cavity that you have and you do that just by sending in a coherent signal somewhere near the cavity resonance and you look at the amplitude and phase of what's transmitted or reflected. But now we can start to do something else. If we were to pump red detuned with a strong pump as I've drawn here there's an upper sideband on the mechanics that's nearly on resonance with the cavity. And we can ask what is the mechanical frequency and the mechanical line width as I would move the detuning of this pump around. These are what I measure here. On this y-axis is the mechanical frequency and this is the mechanical damping rate. The x-axis is the detuning between our pump frequency and where I'm measuring this mechanical sideband. We'll start here first with the damping. You can see when I get to a detuning of exactly the mechanical resonance frequency, the jargon of pumping perfectly the red sideband, what I see is I get beautiful damping, strongly enhanced. And as I move away from resonance I get less and less damping. If I asked what's the line width of this kind of damping I get, this is given precisely by the density of states of the cavity. This is really the cavity susceptibility being imparted in the back action forces. Now from a Kramer's-Kronig type of relationship you can imagine if there's a real part to this damping there's an imaginary part. And maybe not surprisingly, as you've seen in many other forms of physics, when you get one that acts like a Laurentian the other looks like a derivative of a Laurentian. And this is just the other side of the coin. This is just the mechanical spring. In the resolved sideband regime is where this looks beautifully like a Laurentian and derivative of Laurentian. Again, exactly at the mechanical sideband the optical spring actually goes right through zero. So you get pure damping. And that damping can be very, very strong compared to the intrinsic damping that you have here. Mapping out these kind of curves just in what the mechanical frequency in Q that already is a pretty good way to characterize what your system is. You can fit these to canonical forms for the optical spring and the damping and you can pull out what your parametric G is in these cases. And these kind of fits to dynamic back action are the things that can tell you what your transduction rates. For these sorts of drums, we now have transduction rates of about 50,000 kilohertz per nanometer. I don't know that that unit should mean anything to anyone. Again, to remind you, the vacuum motion of the drum is about femtometers. So we're talking about shifts in the cavity frequency of a few hundred hertz from a few femtometers of motion. So now, one of the things to keep straight is all of the frequency scales in the problem. As you've already learned, I apparently like to think in the frequency domain, you can characterize all these things in the time domain as well. But what I like to think about cavity optomechanics is a balance in competition of rates. When we say cavity optomechanics, we usually mean mechanics coupled to a cavity mode. We also implicitly assume that cavity is the high frequency one with its larger line width. So very typically, the hierarchy of scales is something like this. You have a cavity frequency that's very high. You then have a line width of the cavity and a mechanical frequency that are somewhere comparable. Depending on the experiment of the regime, these might flip flop around. And then far, far down, you have your mechanical dissipation all the way down here. And that's the typical hierarchy that you want. When I say resolve sideband regime or good cavity limit, that's specifically referring to ratio of how it compares to omega. And often in that regime, you want the mechanical frequency to be bigger than capital. Now, as I said, the coupling between these two things is something that we can tune in situ just by our parametric pumping. We start off with a vacuum coupling rate in our case of a few hundred hertz and we can drive really hard, drive that up to several megahertz. And that's what lets us go all the way across this regime. The regime becomes interesting if your coupling starts to get bigger than any of your dissipations in the problem. Now that's a strong coupling regime where you can really think about coherent swap operations where these two modes can interact with each other faster than it can be dissipated out of the dissipation of either mode. There's one last kind of rate in this problem that's the thermal decoherence rate. Another way of thinking about that, every other thing I've shown here is kind of a driven response. The thermal decoherence rate would be asking how fast do the environment want to decoher or heat up the sample? If by some magic I prepared it in its ground state, how long before the first phonon comes into the system and kind of increases the entropy of the system? And that thermal decoherence rate is something you can't measure from the driven response that I showed you before. So if you're doing it right, what you'd want to achieve is a thermal decoherence rate, which classically you should just think about as a heating rate. How fast does this thing heat back up to its environmental temperature that's kind of high? Kappa is the line width of the cavity. That's what's going to serve as your cooling rate. The way you're going to sideband cool is you're going to scatter away mechanical quanta to the light field. The light field leaves the cavity at a rate kappa. So that's the thing that's your kind of maximum cooling rate. G is your coupling rate. That's your ability to get information from the mechanics to the cavity and the cavity back. So again, if you're allowed to make G big, that's good. And then lastly, you have the mechanical frequency itself. I wrote this last inequality because you have to be a little careful. If you start to swap things faster than your mechanical frequency, that really achieves kind of an ultra-strong coupling regime. I'll talk a little bit more about that, but you imagine when I have two harmonic oscillators and I couple them strongly, I get normal mode splitting. If the G becomes bigger than the frequency of one of the modes, I've actually normal split that mode all the way through DC, which is another way of saying every one of your rotating wave approximations you made has started to break down, so you should be more careful in the way you're talking about it. I apparently put this slide in twice for no reason, so I'll skip past it. So that was the effects of driving and putting photons affecting the mechanics, but now we can ask, what's the effect of the mechanics on the cavity? So now we're gonna do a very similar thing. We have our cavity density of state shown in green. We're pumping nearly optimally red detune so the mechanical sideband is nearly on resonance. And now what we're gonna do is send a probe toned through to probe what the cavity looks like in the presence of this interaction. So if my pump is turned all the way off and I send a probe toned through, I just measure this Lorentzian dip that I showed you before. This is just the bare susceptibility of the cavity and it's bare cavity Lorentzian. But now as I turn up the amplitude of this pump, if you can squint and maybe change the lights, you can see there's a very sharp interference here that shows up in the center. And this is really the mechanical sideband being re-encoded in the cavity. Essentially, as I say over and over, we have two coupled harmonic oscillators and we have their two susceptibilities now interfering with each other. If I increase the amplitude of this pump, I now increase the size of this interference and you can make it stronger and stronger. Now this type of interference is very analogous. I mean, as sure as you've seen from the literature, the analog here would be electromagnetically induced transparency, something that's well known in optical physics, how you can make a parametric interaction from a lambda resonance really show up as an interference spectrum. Some things that are crucial is this interference is really coherent and what's nice about it is it's really in the cavity susceptibility is inheriting things from the mechanics. One of the things it's inheriting is this very narrow feature. Narrow features spectrally are also long lived. So this is a way to achieve very long delay. You put a photon in the cavity, it rings around in the cavity, it then couples to the mechanics, rings forever in the mechanics and comes back out. So EIT is a well-known way of storing and slowing light. In cavity optic mechanics, we get to kind of play with the same type of physics. So just from a practical sense, you can already see this kind of EIT or to be as Kippenberg likes to call it omit, Optomechanically induced transparency. It gives you a naturally tunable kind of narrow feature that you can move around. And you can do things like this. I mean, this not only could be useful for filtering, but one of the ways you can really think about it is now getting photon storage of order the mechanical lifetime, which is much, much longer than that of the cavity. So I'm gonna try to walk through a little more carefully in Optomechanically induced transparency. What's interfering with what? I've tried to tell you it's coherent, but why is it coherent? What I wanna emphasize is I don't have to go to the A's and A daggers to explain the coherence. I'll explain it more in terms of a mixing experiment. So let's say I have a cavity mode with its density of states like I keep drawing here, and I pump it somewhere red detuned. Now there's the mechanical sideband at the upper sideband and the lower sideband. And right now I'm not pumping the mechanics at all. So the mechanics is just vibrating away with its thermal motion. And there's a little bit of a signature off resonance here, but you can see it's not nearly as dominant as this one that's on resonance with the cavity. Now in Optomechanically induced transparency, there's a pump and a probe. So this is the pump that's mediating the interaction between my two harmonic oscillators, and now I'm gonna come through with another probe just to see what's going on. Before I turn on the probe, if I turn up the pump, I do all the things I just described about red detuned pumping. As I pump harder, I start to damp the mechanics so it gets wider and wider, and it also gets colder and colder. Now if I were to come through with a probe, and that probe would hit this mechanical density of states, if I look at my light field, my probe and my pump are exactly a mechanical frequency away. That means I get a beat force on the mechanics at that different frequency. Because as we've seen, the interaction in Hamiltonian is just kind of the energy in the cavity is the thing that pushes back on the mechanics. So when my probe and my pump are a mechanical frequency away, I now drive the mechanics with exactly this nice coherent state. So if I were to listen just at my mechanics while I was doing this, I would see somebody just drove it with a huge sine wave. But if the mechanics has a huge side wave in it, that gets re-encoded by the pump at the upper and lower side band. So you can see how this pump now got mixed. Let's try this again. The probe I applied beats with the pump, drives the mechanics. The driven mechanics is re-encoded by the pump up to the same frequency you probed at, and this idler frequency in this type of language. Because that whole process is coherently mixed, given by the phase of the pump, this probe re-interferes with itself. That interference is precisely what we mean when we see optim mechanically induced transparency. And that's why we see a coherent interference in amplitude and phase in the cavity spectrum or in the language of cavity optim mechanics in the dressed susceptibility of the cavity. Actually, if I were to look at driving over here at what comes out over here, there is kind of a frequency converting, also coherent interference that happens at that side band as well. So again, just to say it again in words, optim mechanically induced transparency is the mechanically mediated interference between the probe and the driven side band of the pump. And what's nice is you can see it no matter how cold or hot the mechanics is, because it's a driven response. You're driving the hell out of it with the beat force between your pump and your probe. So even if your mechanics is beautifully in its ground state, if you try to just listen to the noise spectrum, that's a very hard measurement with low signal to noise. This gives you a high signal to noise. And I think of it because it's a driven response. So now everything I've described has been in the so-called weak coupling regime. Been thinking about coupling the mechanics and the cavity at rates which are not faster than that of the cavity. So again, if I were looking at this dressed susceptibility, I turn on my pump and I see a little bit of an optim mechanically induced transparency. I turn up that pump more and more. You can see the mechanical interference here is getting wider and wider because I'm pumping more and more red detuned. But I've also saturated the interference up here. Up here, a photon I apply goes all the way to the mechanics and comes coherently back out and nothing is getting reflected. What happens as I turn up the pump even more and more? Well, I know the mechanics has to get wider and wider, which means this interference in the center has to get wider. But now what you start to see is when the mechanics gets wider than the cavity, this now just separates into the well-mown normal modes. And again, this is the same thing you would see if you took two electromagnetic cavities and just coupled them resonantly, tuned them to be on resonance with each other, whether it's two Fabry-Perot's or two LC circuits, the eigenmodes of the system are no longer that of the bare ones, they're split into normal modes. And in fact, if you're allowed to pump harder, you do. So when you do, you can split by much, much more. And in fact, you can split so much more you're off the screen here. So if you were to zoom out and think about what is the coupling you can get, in this experiment we were achieving coherent coupling rates of around a megahertz, which is much, much lighter than the dissipation in the system. If I were to think about what's the dissipation of these two eigenmodes, essentially it's all the intrinsic dissipation I had in the first place, which is only gamma and kappa. Gamma is completely negligible, being the mechanical dissipation, so the line with here is essentially half the kappa that you have. And this is the so-called strong coupling regime. Just to be very clear, this is a driven strong coupling regime to be contrasted with a kind of a vacuum strong coupling regime. That's something that is also very exciting for another class of physics, something we'd love to be able to do. But this driven strong coupling regime is at least the thing that can let you think about coherently interacting, really performing true swap operations, taking whatever state you had in the cavity, swapping it perfectly to the mechanics, or vice versa. One of the figures of merit that we love in cavity optomechanics is cooperativity. And again, that's just normalizing this coupling rate by the dissipations in the problem. This is a very analogous to again cavity QED. And to put it in this language, our kind of driven cooperativity can now be of order 10 to the five. Just for scale, cooperativity is definition of what damps you by a factor of two. When the damping from the light is equal to the damping of the intrinsic dissipation. So cooperativity of one, is it what you need to cool by a factor of two, pardon me, if you're pumping the red side band, or what you need to parametrically oscillate if you're pumping the blue side band. The ability to pump by cooperativity of 10 to the five means you can do much, much more. So this animation over here is showing you what data actually looks like in our system. This is just sweeping the pump power from low power to high power, where you can really see this evolution from what I would call an omit or a interference into normal mode splitting. I wanna emphasize these are really just two sides of the same coin. This is just two coupled harmonic oscillators, one with a very asymmetric dissipation and one with a very strong coupling. This is the plot that shows you kind of the evolution that you have here. We have a nice parametric tuning knob that's just the number of drive photons. We're varying it from much less than one to about a million photons in the cavity. And we can see this coupling rate goes exactly as the square root of number of photons as we would expect. This is just the thing that's telling us we really do have a clean interaction Hamiltonian. And that's the other point I really wanna emphasize about cavity optomechanics. It's well described by two harmonic oscillators and a coupling between them. I've done a lot of measurements in my history using exotic kind of mesoscopic amplifiers, things like squids or single electron transistors. And part of the problem there is the extra degrees of freedom you have in these mesoscopic systems. I think a lot of the power of cavity optomechanics is the fact that it's clean and clean in the most sense so that if you really measure this, you know that's exactly the interactions you have in nothing else. If your experiment is missing the quantum limit or missing a quantum interact for some reason, there really aren't that many suspects because there's only very few things that you're actually addressing here. So that was varying the pump amplitude. What you can also do is you can look at this same kind of plot. Again, in the y-axis, this is one slice would be that Lorentzian dip that I was showing you before. So this is just the cavity width capa here that you're seeing. As you would tune the frequency of the pump to bring this mechanical sideband into and out of resonance. For this very weak kind of pumping strength, you can just see there's some action going on and it scales linearly. So you might think it's a sideband that's well sited. If you turn up the pump strength, now you can really see the fact that the coupling really goes through the density of state to the cavity. When the mechanical sideband is far off resonance, it's very narrow, more like the intrinsic line width. And as I bring it into resonance, I now think of this as normal modes, each inheriting half the line width of the cavity. And then as I go back out of resonance, you can see they become more and more uncoupled. Again, this is data. If we pump it as hard as we possibly could, we can also get kind of deep in the strong coupling regime. Just showing what's possible in many of the state of the art systems. I think this was a little bit unique at the time. This was six or seven years ago now. Just to show again what these line cuts look like as you would sweep the mechanical sideband through the cavity resonance, you can really see the interference you have. And this is exactly what you would get from diagonalizing a two by two matrix of just two coupled harmonic oscillators. And you can see the interference really gives you this fanoness when it's off resonance. And that's showing you it's coherent. I've been showing you just the magnitude of the transmission. We can measure the magnitude in the phase. These vector measurements are very easy to do with the network analyzer and the microwave regime. And again, you can see you both have the, exactly where you have the magnitude dip. You have the canonical phase response you get with the Lorentzian. You get a narrow one for that of the mechanics and a wider one for that of the cavity. And you get that vector interference whether or not the sideband is on or off resonance. So, but as I've been saying, all of this driven response and strong coupling regime, that's nice. This data would look the same if everything were at room temperature or in a very, very classical thermal state. This doesn't have to be cold to do these sorts of things. So the next questions we wanted to ask is what is the temperature of the mechanical mode or the temperature of the light mode for that matter? Is it quantum? So this is mostly linear. Couple harmonic oscillators and this is the driven response. Now what we're going to have to do is look at the noise spectrum to really know what the fluctuations are, what the thermal energy is in the system. And that's precisely quantified by this thermal decoherence rate. So what is a thermal decoherence rate? Again, the way I think about it phenomenologically is if I prepared it in the ground state how long before the first phonon came into the system? If we want to do the math, it's essentially the product of the intrinsic mechanical damping rate and the equilibrium thermal occupancy. Another way to say it is I don't really care at the beginning whether I couple very slowly to a hot bath or I couple rapidly to a cold bath. Both kind of have this same thermal decoherence rate. And when you do the math, what you find is actually the mechanical frequency drops out. In this thermal decoherence rate, you can only make better by either making a colder starting temperature or a higher mechanical cube. That's all you have. And when you turn on sideband cooling or even blue sideband heating that doesn't change this intrinsic thermal decoherence rate. It just changes the equilibrium values of the system. Again, if we want to be able to cool to the ground state because Kappa is going to be our cooling rate, we really need this thermal decoherence rate to be small enough in the first place. Because once I get into the strong coupling regime, there's no more damping to be had. I don't get more dissipation to the mechanics. I just shift its frequency. So I'm stuck at whatever temperature I get whenever I hit the normal mode splitting. So again, to think about sideband cooling, I like to think about a detailed balance. Kind of just this bubble picture. I have some mechanical mode at frequency omega. Now every mechanical mode has some Q, which means it's coupled to some environment. This is just the fluctuation dissipation theorem that says whatever is giving you your loss is the thing from which you inherit your temperature. Even if I don't know what the microscopic dissipation mechanism is of the mechanics, I can measure that it's in resonant, it's in equilibrium with the thermal environment given by that at the dilution refrigerator. And that gives me an effective occupancy of my bath of KT over H bar omega m. Now what we're going to do is we're gonna couple this mechanical harmonic oscillator to something else. It's a cavity mode. The cavity mode is at the same temperature, but crucially the same temperature for the cavity gives you a very different occupancy. And for these kind of frequency translating parametric operations, the way to keep score is always in terms of quanta. That's the thing that's fundamentally conserved. So even though both are at the same kind of 50 millicolumn temperature, this is effectively in its ground state with an occupancy much, much less than one. And this is very thermal at an occupancy of about 100. Now we start to turn on some coupling. Well before we turn on any coupling, there is some vacuum coupling here. But this vacuum coupling is completely negligible compared to this thermal decoherence rate, compared to this kappa. So there's really no signatures that we can see just with the static coupling that we have. But now we're gonna drive the red sideband, as I said before, we're of course gonna parametrically enhance this G naught. And now the picture I have is the mechanical mode feels a coupling to the cavity. As soon as I couple anything to the cavity, it immediately leaks out the cavity. That is the weak coupling regime by definition. So I start to swap something from the mechanic to the cavity. The cavity is immediately cooled out its bath, and this is the canonical sideband cooling. Tracing over the cavity and just looking at the mechanics, the mechanics basically sees a heating rate coming from the environment. This N thermal that I was telling you that's the intrinsic line width multiplied by the equilibrium occupancy. And now it sees an effective damping, this 4G squared over kappa. This is additional damping. And crucially, this damping looks purely cold. So there's no heating coming back in the resolved sideband regime. So now I can do my detailed balance. I have a mechanical mode with a heating rate this and a cooling rate this and say, what equilibrium temperature do I get? And basically what you get is the ratio of these two, which is just kind of diluting the intrinsic occupancy by the ratio of the intrinsic line width to the damped line width. And so I get an N final that's something essentially like this. And this is correct just to be clear in the quantum regime. This is correct as I get much less than one. There's nothing special that happens there at N equals one or not. Now if I were to continue pumping my G so that I go into the strong coupling regime, now it's this regime where I should really think about mechanical quanta go to the cavity, but before they can leak out of the cavity, they actually get coherently swapped back to the mechanics. This kind of swapping is precisely the thing that gave me my normal mode splitting in the driven response. And as you can see in this picture, if I have too many mechanical heat quanta coming in, swapping them around faster and faster doesn't help me get colder. The only way I get colder is to have dissipation. This is a well-known game in atomic physics. When people are doing things like sideband cooling, in atomic physics you can actually run into resonances that are too high of Q. They don't have enough dissipation. So you have to damp them to make sure you can lose your energy. Because losing the energy is the only way you can really have this cold bat to dissipate. So again, in this strong coupling regime you can really think about Robbie swaps between them. Just to be clear, in this regime, I could achieve my cooling, instead of just turning on a pump and waiting for the steady state, I could actually think about doing something like a pipe pulse. I could take all 100 quanta from the mechanics, do a pipe pulse, a turn on this swap for exactly the right amount of time. So 100 quanta went to the cavity, the zero quanta in the cavity went to the mechanics, and I'm done. That's all I had to do. Unlike the Jane's Cummings Hamiltonian, where I used two level systems to cup to the cavity, this doesn't have to get the mechanical occupancy out one phone on at a time. There's no two level system here to be the bottleneck. And so one of the benefits of these parametrically coupled linear harmonic oscillators is the large Hilbert space of the two harmonic oscillators. This same swap that I described would describe whether I was cooling a thermal state, swapping a thermal state, a foc state, a cat state, whatever I have. And it's this kind of coherent regime that's a lot of why we're excited. In superconducting circuits, in microwave circuits, people have gotten very good about making these sexy states of microwave fields. Things like John Martinez or Rob Schollkopf's lab make cat states or arbitrary quantum states of the cavity mode. That same power, if you can make that mode and turn on this interaction, you can then coherently swap it into the mechanics or reverse swap it back out in order to do the mapping. And that's part of the power, again, of taking a mechanical system that we're kind of not nearly as sophisticated in our measurement and handling and map it onto a system that we know and love very well. The same thing, of course, with the optical photon. Again, just to think about it, we wanna engineer the system so the thermal decoherence rate is small, cap is bigger. This G less than the omega lets us still give us the rotating wave approximation and this is the regime that lets you enforce coherent swaps. Really talk about a quantum coherent regime, again, as the Kiffenberg lab likes to call it. So now on this same plot of all our rates, we know our coupling rate, we wanna know where our thermal decoherence rate to really know if we have this hierarchy. So now what we're gonna do is these same type of measurements I showed you before. We're just gonna turn on our pump to mediate the interaction and the only thing we're gonna do is listen to the noise spectrum of the mechanical sideband. The area of this sideband is the thing that's proportional to the mean square motion. If you calibrate all your transduction factors right, the area of that little Lorentzian tells you what your occupancy is. Just to be clear, we're trying to look at very few mechanical quanta being transduced to very few microwave quanta. So this puts all of the measurement problem on having the best microwave measurements you can. The things we use tend to be semiconductor amplifiers at four Kelvin as our preamplifier and then we make our own quantum limited parametric amplifiers out of superconductors and jostles and circuits. I won't go into detail about this today. If people want to talk about it, you can happily be convinced to talk about parametric amplifiers as much as you'd like. The way you should think about them, the parametric amplifiers are very much like an optical parametric amplifier. It's a nonlinear medium that when you can pump, you can get gain out of. The gain is very clean and nearly quantum limited. So it's our analog of doing nearly shot noise limited homodyne detection. Again, I won't go into detail, but the way we make nonlinear media in the microwave regime is to use jostle and junctions and squids. That provides the ideal nonlinearity. And then you can make these nonlinear cavities that allow you to make the analog of an optical parametric amplifier. I'm gonna skip the thing that shows you how quantum limited it is. It works pretty well. And the whole idea, if you wanna cool to the ground state and verify it, this whole system essentially looks like this in cartoon form. You're gonna pump at the right frequency, make sure your pump is purely clean, really a coherent state, coupled to a mechanical degree of freedom in the resolve sideband regime, and measure what comes out with the lowest noise amplifiers possible. This, for example, is a snapshot of what one of these dilution refrigerator cryostats looked like. This particular one, this is a vacuum can, kind of as tall as a person. They don't have to be that big. We tend to make them that big because that's the scale to put all of our microwave components that we need in there. They could be much, much smaller. Making them much bigger, you can also do, it just costs more money. There's already plenty of room in there to cool, to 15 millikelvin, whatever you want. Crucially, the 15 millikelvin, it's far below the transitioning temperature of aluminum, which is about one kelvin, and more crucially, it's below the kind of h-bar omega of our seven gigahertz resonance. This ratio is enough so the thermal occupancy of the cavity mode should be part and 10 to the four, at those temperatures. That's the detailed circuit diagram just to scare everyone to show that we do do some science in there. Again, some of the technical details we do are things like really filter the cavity to avoid any excess noise. We also do lots of kind of thermal distribution so that we attenuate hot photons and re-radiate them cold. Lots of isolators on the output to make sure noise doesn't come back in our measurement and drive. Because as I explained, if your cavity-coherent state ends up in a displaced thermal state for any reason, that gets directly written onto the mechanics, and your mechanics would end up in a thermal state. So if there's any noise on your drive or near your drive, those noise photons beat with your pump and make your mechanics hot and shoot you in the foot, kind of go against everything you're trying to do with sideband cooling. Just as an advertisement for what things look like with these parametric amplifiers, I would say that's been one of the weak points of microwave measurements because we don't have the good photodiodes you have naturally at room temperature for optical photons. Of course, we're trying to measure photons that are 10,000 times smaller, so the problem is harder. Without a parametric amplifier, if we just measure our white noise floor, it looks like that. And there is a little mechanical sideband buried in there that you could average away and see. But if we turn on the parametric amplifier and refer back to the input of the gain, we see our noise floor dropped by a factor of 30. And now in the same integration time, you can clearly resolve the mechanical sideband. Again, in terms of practical limits like integration time, that really revolutionizes what it takes to do the measurement. Just to remind you, our thermometer now is just going to be measuring the area of this Lorentzian and of the deviation from this white noise background. Everything there we know is mechanical energy and that's how we're gonna do our thermometer. The calibration that we do in the first place, I apologize for the equation, is essentially just using aquapartition. The first thing we do is we don't do any sideband cooling. We just probe very weakly and ask what the area of the kind of curve and we vary the temperature of the environment just by varying the temperature of the cryostat. If all is right with the world, we have aquapartition, meaning half kT of energy per degree of freedom. So this mean square motion should just scale linearly with temperature. This linear relation is the thing that serves as our primary calibration here. This is what shows us that when we say our fridge is at 20 mili Kelvin, how do we know the drum is at 20 mili Kelvin? Well, it's because we see this agreement everywhere throughout and we see it doesn't deviate, although in principle it could. And so we're starting with a quanta of about 30, 20 mili Kelvin and a 10 megahertz mechanical oscillator. And that thermal decoherence right now with the Q of the mechanics is talking about one kilohertz. So that means you have about a millisecond before these mechanical quanta start to come back into your system. We need a cooling rate much bigger than this kilohertz to cool to the ground state. As I said before, we engineered kind of a line width of about a hundred kilohertz to naturally be deeply in this regime. So again, these are the mechanical noise spectra. Just as I turn up the strength of the pump, as I turn up the pump, you can see three things happen. One, the noise floor goes down. That's not because I waterfall the plot. That's because here I'm plotting in mechanical units in meters squared per hertz. And in an information theory kind of perspective, I'm pumping harder and harder. I'm acquiring more information about the mechanics when I pump harder that gives me a lower noise. This is just beating down the imprecision of my measurement by driving the interferometer harder and harder. The next thing you see is these mechanical line widths are getting wider and wider. That says I'm doing this damping that I described earlier. And the area of these curves is getting lower and lower. That's hard to see because I plot it on a logarithmic scale, but trust me, on a log scale, each one of these is going down in temperature by as much as it's line width. Just to talk about the displacement sensitivity, here this is just plotting what is this kind of white noise level? How good of an inference are you making about the mechanics as you turn up the strength of your measurement, meaning the strength of your pump power? Just to give you a sense of scale, we're talking about displacement sensitivities much less than a femtometer per root hertz. And that kind of femtometer scale is crucial because that's kind of the zero point motion of the scale. Driving our hardest in some of our best circuits, we get things of about 10 atom meters per root hertz. We're not quite doing as well as LIGO, but I'm okay with that. Their budget is a little bigger than mine. The other things you can see as you drive harder and harder and turn up the strength is you can measure all the different rates in the problem. The red is the mechanical line width. When I pump very weakly, it's just the intrinsic line width of 20 hertz or so. As I pump harder and harder, it just increases linearly. The blue is the cavity line width, which in principle shouldn't be changing at all. It's changing a little bit here for technical reasons. And the green is the coupling rate. And you can see that we can vary the coupling linearly in the number of photons, the cavity. If you were to think about the relative displacement sensitivity, one of the things you wanna talk about is just not how many femtometers per root hertz, but how visible would a single phonon be? In this resolved sideband regime, as you pump harder and harder, your noise floor goes down, but your mechanics gets lower and lower Q. It will naturally plateau to something. That plateau is a direct measure of your quantum efficiency. And in these kind of experiments, we were achieving kind of this efficiency of about five H bar. The dominant loss was just the fact that our microwave measurements were not quite perfect. They were missing the quantum limit for measuring the light by a small factor, which means we missed this kind of quantum limit for the mechanics. But that type of scaling is some of the south calibrating things that come out in these very cold experiments. So with the last 30 seconds, I will just show you what the data looks like as you cool to the ground state. You start off with about 30 quanta. You pump harder, harder and harder, cooled to just less than one quanta by damping very wide. You'll notice I'm very zoomed in there to the mechanical sideband. If I take this same blue curve and just zoom out on this y-axis, instead of being on kind of the one kilohertz scale to maybe the four kilohertz scale, if we were to zoom out, what you could see is your little mechanical sideband is riding on top of a broader Lorentzian. This is again just showing you the two Lorentzians in the problem, that of the mechanics and that of the cavity. The cavity shouldn't have any occupancy, but when we stare really hard, we can see a little bit of occupancy. And now as we pump harder, what we can start to see is we evolve fully into these normal mode splitting that I showed you before. There's a little bit of thermal occupancy equally distributed between the cavity and the mechanical mode, all happening at a kind of occupancy less than one. And this is the demonstration of that quantum coherent regime of simultaneously being strong coupling and fully cold, a total entropy of the system less than one quanta. These are what the data look like. I think I'm out of time, so I'm gonna stop there for today's lecture. Please find me throughout the days, throughout the weeks for other questions. And if there's other things you'd like to know more about, I'd be happy to incorporate those into my later lectures. Thank you.