 My name is Uriel, and I will be teaching today on quantum optomechanics. Is it not working? I think I've activated the microphone. Is that OK? Or it's just now better? Bit louder? OK. So I plan to teach these four hours in blocks of 50 minutes. We will do 50 minutes and 10 minutes break, if that is OK with you. Please ask as many questions as possible, ideally during the lecture, so that we can make it more interactive. I think that would be a good idea. So let me, first of all, introduce myself. So give my email in case you want to contact me. That's at the University of Innsbruck in Austria. That's the email. Also, we have a research group that does theory on quantum optics and quantum nanophysics. And here is the web page. And also, quite recently, we have been a bit active in the social networks, especially promoted by our students. So we have a Twitter and Instagram account. So if you want to check, so that's just some information. Good. So then the lectures will be on quantum optomechanics. And before we start, I would like to have a bit of feedback to know what is your background. So first of all, from the students, could you please tell me how many of you are doing currently the math? Well, first, how many of you have studied physics? I assume, yeah, most of you. Good. How many of you are doing the masters right now? OK. How many of you are doing the PhD right now? OK. And how many of you are postdocs? And how many of you have never heard about optomechanics? How many of you you consider to be kind of expert in optomechanics or really know quite a lot already? Because maybe you are doing research on that. All right. And I think it's kind of OK. So the idea is I will do this for hours to be pretty starting from zero and very accessible for people that have never heard about optomechanics, which then perhaps has the disadvantage that for the experts, some of the parts might be too basic. But anyways, I always think it is good to review even things that you know to understand it better. So the plan of the lectures, basically, I've divided them in five parts. So the plan of the lectures, they have five sections. First, we will start with an introduction and motivation to the topic. Then section one will be on what I call closed optomechanics, namely to understand the dynamics of the system in the absence of dissipation and how to describe it. Then we will discuss what happens when the system interacts with the environment, namely it is open. So I call that open optomechanics. Then section three will be devoted to a very important regime in optomechanics, which is called the driven case, the driven scenario, which basically linearize the dynamics. And that is happening in most of the experiments today. So this is called the driven in brackets linear optomechanics. And we will conclude by applying this scenario of driven linear optomechanics to discuss the fact that you can actually cool to the emotional quantum ground state mechanical systems using electromagnetic field radiation. So we would conclude with that about ground state cooling. So our group is a theory research group, but we collaborate a lot with experimentalists. So I myself am not an experimentalist, but we are used to talk to them. So if also you have questions from the experimental point of view, also feel free to ask. And I will do my best to try to answer, OK? Good. So that's the plan of the talk. These four sections plus introduction. And then just for bibliography, for you to know that, I mean, this is a very active research field. So there is a lot of material, a lot of papers. But perhaps now the most famous kind of reference is a review modern of physics called cavity optomechanics. This is a review modern of physics article published in 2014. So six years ago. And since the field is very active, of course, there might be some things that here are already a bit outdated, but for getting access to the basics and the background, for sure that's a very good one. The authors are two leading experimentalists and one leading theorist. So Marcus Aspermeyer from Vienna, Tobias Kippenberg from EPFL in Switzerland. They are both experimentalists. And Florian Marquard from the Max Plan Institute of Light in Erlangen, who is a theorist. And recall, as always, this review modern of physics. These are reviews which, of course, ideally they should be also accessible for students. But at the same time, they should for scientists and researchers. So they are sometimes the level is a bit high. So if you have never heard about optomechanics and you start reading that review, that might be a bit challenging in the beginning. But hopefully after these five, four hours, this review will be even more useful. Good. So then let's start with the introduction and motivation to the topic. Good. So let's first of all think about the name of these lectures. This is called quantum optomechanics. Let's focus. I mean, quantum, we already know what it means. Probably this system will at some point behave according to quantum physics. But what does optomechanics mean? Basically, optomechanics contains, in a sense, two words, optomechanics. And here what is important is that what we refer from opto, which you could think has to do with light, we should be even a bit more general. More than light, we should think this refers to the electromagnetic field. Not only radiation at visible light, but it could also be radiation at other frequencies, such as microwave frequencies. So from a more general point of view, optomechanics refers to the dynamics that describe the coupling between the electromagnetic field, electromagnetic field coupled to mechanical motion. And of course, when we say mechanical motion, we always should understand this has to do with the motion of something that is massive. And here the mass will be something very relevant in this field. So at the end, we want to couple the electromagnetic field to some emotional degrees of freedom that are embedded in a system that has mass. And I can already tell you, of course, you could say, well, this is quantum optics in general. Quantum optics also describes the couplings of electromagnetic field with the motion of massive particles, namely electrons and protons. That's correct. But here, as you will see in a second, has to do more with the fact that this mechanical motion is related to the motion of a very big object, almost microscopic, almost, you can see with your bad eye. And that will be the main property. We want to couple the electromagnetic field with very massive objects. OK. And for instance, just to give you a bit of intuition of how you can do that and also to define two standard regimes in optomechanics, the so-called optical, so optomechanics in the optical domain. And a typical scenario would be the following. Imagine you have a very nice optical cavity with two very good mirrors. This one is completely fixed. But now this one can actually move because imagine it is actually connected by a spring to a big mass. So you have now two mirrors, but this one can vibrate. And then use some colors. And as you know, in cavities, if you solve the Maxwell equations in the presence of these boundaries, you find modes, namely vibrations or states of the electromagnetic field, which are very well confined within this cavity. This will be optical light, optical modes. And this is what I call the electromagnetic field. Some of these modes can be excited with photons and so on. And there is this mode, the optical field. And now you see, because this mirror can also move, that's another degree of freedom, which is mechanical. So now this mirror can vibrate. And this is the mechanical motion, the mechanical degree of freedom. And you can already guess from the physics point of view that in such a system, I will definitely have some coupling between the degrees of freedom of the electromagnetic field and the degrees of freedom of this mechanical motion. Because, for instance, you can think it in two ways. If you have some light inside this cavity, you know light exerts pressure. It carries momentum. So when photons reflect from this mirror, they impart a bit of linear momentum that will shift the position of the mirror. Now by changing the position of the mirror, you know that the resonance frequency of the cavity is also modified, because the resonance frequency of a cavity depends on its length. And now its length is actually changing. And by changing the length of the cavity, maybe the light you were sending to the cavity is not resonant anymore, it doesn't enter, and so on. So you see that there are some intrinsic dynamics here between the electromagnetic field, degrees of freedom, and the mechanical motion. And that's a bit, this is one particular setup. There are many in optomechanics, but the principle is always the same. You have some optical resonator whose resonance frequency can be modified by the motion of some mechanical degree of freedom. That's one example I will show you then, other examples. But this is important. This typical setting is in the optical scenario, where there are optical modes. There is the same concept applies in a completely different regime, not in the optical regime, but in the microwave regime. So recall, optical frequencies are 10 to the 15 hertz. Microwave frequencies are 10 to the 9, 10 to the 10 hertz. Another scenario is you might remember from our basic lectures of circuits that there is this circuit, the so-called LC circuit, where you have an inductance and a capacitor. And the dynamics of this LC circuit can be described by an harmonic oscillator also. Now imagine I do the following. In this capacitor, I have two plates. And now I do the same trick as before. I allow one plate to actually move because it might be on top of a spring. So for instance, I connect one of the plates. It's actually come by right because it's on top of a spring. And I connect the circuit like that. So this is now an LC circuit. It's an LC circuit where current, or if there is some current flowing, it generates some microwaves. So here you could see this is kind of a microwave cavity. And now the resonance of this microwave cavity depends on the position of this plate because if it changes, it changes the capacity. And this changes the frequency of the oscillator. And it's again the same concept I have now. Microwaves here, microwave electromagnetic field coupled to the motion. And this somehow defines two very different settings for optomechanics, either optical optomechanics or microwave optomechanics. Sometimes it has also other names. People use it, electromechanics and things like that. But this is important because these are completely two different regimes. Here the electromagnetic field has optical frequencies. Here the electromagnetic field has microwave frequencies. And during the lectures, we will see that this has important differences, not fundamentally, but in terms of experiments. Because experiments here look very different than experiments done in that setting. So at the end of the day, this is kind of just doing standard quantum optics. But now you couple electromagnetic fields to the motion of you already see from these figures of kind of big objects. And this is actually the distinctive or the main difference. So the main distinctive difference is precisely this one. That is that the mass m associated to mechanical motion is somehow microscopic. And here by microscopic, we mean that the mass, for instance, if I write the mass in units of a single atomic mass unit, so n times an atomic mass unit, which is somehow the mass of a single carbon atom, 10 to the minus 27 kilograms. For these objects, this n actually can be very large and depends a lot on the particular experiment. But current experiments go from masses that are relatively small, which means they contain of the order of 1 million of atoms, two really huge masses containing 10 to the 19 atoms and even more. In particular, perhaps the biggest masses in which you experience optomechanics are used in LIGO, in the interferometers for measure gravitational waves, where they have suspended masses with a mass of a kilogram. And they basically have to isolate them very well. And they are part of the interferometer. And the motion is affected by the reflection of light, in a sense. So these spans are completely large parameter regime in regards with masses. And that's what makes this feel different from quantum optics, in the sense that traditional quantum optics was dealing in the interaction of light with atoms that contain the mass of an atom, which would be of order 1. Here, now you deal with way bigger, bigger masses. So large these objects are that sometimes you can, of course, see with your bad eye. Good. And important is also to have this, OK, to know that there is basically a plethora of experimental configurations where optomechanics is implemented. If you search for optomechanics, you will immediately see these beautiful images of all type of different experimental scenarios where people talk about optomechanics, meaning that they essentially have that. So the principle, as I said, in all these scenarios is that the resonance frequency of an electromagnetic field resonator depends on the position of a given mechanical oscillator. So that's always the principle. The resonance frequency of an optical resonator, which is some space where single or few electromagnetic field modes are confined and resonate as a single harmonic oscillator, this frequency always depends on the mechanical position of some mechanical motion associated to some mass that is also described as an harmonic oscillator. So at the end of the day, as you will see in a second, it has to do with the dynamics of few coupled harmonic oscillators. Just for instance, to give you an example of how you can modify these ideas with experiments that already exist, another configuration similar to that one would be you take a cavity whose ends are fixed and you just put an electric object inside the cavity. So for instance, you put a cavity that is fixed and now you put some of the electric object in the form of a pendulum which is just a dielectric material that can oscillate inside the cavity. Since this is a dielectric material and you know that the resonance frequency of an electromagnetic field resonator depends on the electric properties in its interior, now you see this cavity is vacuum everywhere with exception of a space of volume where there is some dielectric. Now by this moving also changes the resonance frequency, and then you would couple the light to this motion. Here also another setting is now the plates of the capacitor remain fixed, but in the interior you also put the electric object which now is vibrating because any object vibrates due to its acoustic phonons. Now by these vibrations you also modify the capacitance of this capacitor and hence you modify the resonance frequency and then you couple this mode to the vibrations of some piece of material and things like that. This concept really appears in a multitude of different experimental situations. So what we will do in this lecture is to discuss the theory that actually then basically applies to any of these situations. Good, so questions up to here for this introduction. All right, so this is a bit to describe what optomechanics is. Now let me tell you why optomechanics is interesting. Why should you be interested in optomechanics? And mainly there are three clear applications of these systems which makes them very interesting for a broad community of scientists. The first one as you can already guess from my emphasis on the big mass has to do with a bit more of a fundamental character. Namely I call it macroscopic quantum superpositions. Of course here I described optomechanics, I didn't use the word quantum yet but you can already guess that we use the word quantum because we will use these interactions to actually bring this mechanical motion to the quantum regime. Namely we would like that these vibrations end up behaving according to quantum physics. And what quantum physics also allows is actually to take this mechanical mass and prepare it into a superposition state. Where this mass will be doing two different things at the same time, either being a one position and another or vibrating in one way or another one and so on. So really. And then of course there is a fundamental question which is to ask whether is there any fundamental limit? Is there any fundamental limit in the validity of the superposition principle? Or a similar question is can arbitrarily large masses be placed in a quantum superposition state? So these questions are basically motivated by pure intellectual curiosity. And for the, we as human beings, we strive always for discovering things and for exploring things that we don't know. So this should raise the curiosity of any scientist to think whether these questions have an answer. And of course the only thing we can do is just try to explore that. Just to do experiments, see if we observe the quantum superposition principle with yet bigger and bigger masses. If this is the case then we have an answer to this which is no. If this is not the case, of course that's always hard to answer this because it could be that you don't observe that because you are not doing the experiment correctly or you are not isolating well the system and so on. But at least if you observe superpositions with larger and larger objects, you give a no answer to these questions. And of course, I mean, compared to traditional quantum experiments, this is a big leap in mass. So for instance, large superpositions. So the largest mass that has been prepared in a microscopic superposition in the context of matter wave interferometry doing this double slit type of experiments. Today, the record was very recently achieved in Vienna by the group of Markuzarn and this is of the order of 10 to the four atomic mass units of masses that have 10 to the four. So this explores much larger regimes. So this is a bit of a fundamental question which is actually also very relevant here in Trieste because traditionally there was Professor Girardi who had major contributions regarding this question and today there are still research groups here working on this such as the group of Angelo Basi with Matteo Calezzo who is also here which from the theoretical point of view, try to derive alternative theories to quantum physics that are consistent with quantum physics at small scales but predict that quantum physics will break down as soon as we understand it as soon as the masses are sufficiently large. And these are theories that give very nice predictions. They predict you will not be able to prepare a quantum superposition of this mass at this scale, hence doing experiments and showing that you can would falsify such theories, okay? So this motivates very well these questions. Another actually super interesting related question is to think that these objects start to be so massive that they generate a gravitational field that could be measured. Then there is also fundamental question. If I have a mass that is so large that it generates a gravitational field that can be measured, what happens if this source of gravity is placed into a superposition state? What is the gravitational field that such a superposition state generates? Okay, so namely what is the space time metric generated by a mass in a quantum superposition state? And of course this question is relevant because as we know, unfortunately, we still don't have a unified theory of gravity and quantum physics. And this somehow my approach, this terrain where we don't have a very well defined theory from a very different regime, a very low energy kind of experiments where you have very low energy experiments but that nevertheless prepare a big mass into a large superposition state and then you look what is the space metric that such a superposition generates, okay? And there are many people that are very interested in that. And somehow also related to that question of the interplay between gravity and quantum physics there is also another yet very interesting question which is whether can we entangle the motion of two massive objects using the gravitational interaction? So the idea would be imagine you have one of these two big masses suspended, okay? They can move and now you want to entangle this motion of degrees of freedom using the gravitational interaction, okay? And many people believe that this is a very interesting question because if you were able to entangle the mechanical motion via the gravitational interaction that they have between them, this would indicate that gravity needs to be quantized, or otherwise if gravity could not be quantized it's completely a classical communication channel then there would be no way that you can entangle the motion between massive objects, okay? There are papers about that which I can refer to but I myself I'm not an expert on these topics, okay? So don't expect very qualified discussions on this, okay? But that just motivates and at least gives a very nice figure of merit because if you now try to actually see how large the mass should be so that this entanglement is a generation, this interaction is large enough so that you can entangle, of course you need big masses. If you put small masses that's impossible, okay? And again this hints onto exploring experimentally the interplay between quantum physics and gravity, okay? So this is this type of questions regarding more fundamental character of why optomechanics is interesting because it allows us to bring massive objects in the quantum machine, good. Another questions about that? Okay, so a second aspect somehow related has to do with quantum even though sometimes you might not even need quantum physics but at least to use these objects, these vibrations of massive objects as extremely good sensors. As you can already guess that if we are doing experiments where we want to bring this mechanical motion to the quantum regime, this will require these mechanical degrees of freedom to be extremely well isolated from the environment. If they are very well isolated from the environment this also means that as soon as the environment does something that you don't expect, you will see this effect onto the motion. So these objects will be extremely fragile to the environment which also means extremely sensitive, okay? And hence a side application of this and many people are interested in that is in using these optomechanical systems and as extremely good sensors, okay? So the idea is that this mechanical motion is ultra sensitive to environment, okay? Because a bit you see already from the physics they introduced that it is as soon as it displaces so and it can be measured very well because you already see if there is a bit of displacement under the mechanical motion this changes the resonance frequency of the resonator and of course using electromagnetic fields you can measure this with a lot of precision, okay? So and indeed this is a bit what is behind why these optomechanical systems are used in LIGO because the combination of this, of the fact that the mechanical motion is so sensitive to external perturbations and the use of optical or electromagnetic field interferometers leads to very good sensors and that's why they are used for instance at LIGO. Set in combination with optical interferometry for instance this can lead to something as sensitive as LIGO is and also from a more applied point of view maybe not so fundamental but still very interesting and applied point of view you can use these systems for very nice things. So imagine this typical optomechanical setting. Now I make the optomechanical setting like this. I put a cantilever, a trampoline, okay? And on top of the trampoline I put a mirror. Wait, let's do it like that. Okay, I put a trampoline and here below the trampoline there is a mirror that connects with another one that is fixed and then I have my nice resonator whose frequency depends on the position. Now imagine that now on top of this cantilever I put a very tiny mass, okay? Something is deposited on top. Say a virus or a molecule. I don't know how to plot it. Whatever, some extension. I put a molecule on top. Then the mass of course will displace a little bit the cantilever and this will change the resonance frequency of the resonator which then I can measure with a lot of precision. Okay, so for instance these systems can be used as a way to weight very tiny masses and actually the numbers are amazing of you know these are, you can weight the smallest masses with these type of systems. Okay, so for instance this allows you to make a very accurate weight of small objects. You could also use them for, also for accelerometers because if this is now imagine inside this concept is inside an object that is actually moving such as a car, imagine now when we accelerate this would also create displacements onto the cantilever and you will be able to measure that with a lot of sensitivity of whether it is rotating and so on. So this same concept applies for applications in accelerometry and inertial sensing in general. As well as also in terms of sensing can also be used for fundamental applications. In particular to measure so-called short distance, short distance forces. Okay, also with this concept now imagine I do the following thought experiment. Okay, I put my optomechanical device here which can be vibrating and then I approach this mass here very close to a surface, okay? Or put it, yeah put it like that. Very close to a surface, okay? And now imagine that behind the surface I either approach a mass and this surface imagine it's made of a perfect conductor so that all these Casimir forces are shielded. Then behind this wall I just approach a mass. I approach a mass and I move it, okay? Then of course when I approach a mass it generates a gravitational force to this cantilever so it will displace it. And if this mass is approached and removed periodically it will give a driving force, okay? And with that I should be able to measure gravity at short distances. Actually this is very interesting from the fundamental point of view because Newton's law at a scale below one millimeter has been measured really not very accurately, okay? So it is not falsified yet that Newton's law should is way, way stronger at shorter distances that at large distances. And this is very interesting because there are many models in high energy physics such as models in for instance in extra dimensions which has a prediction that gravity at short distances should be way, way stronger, okay? And this has not been falsified yet. And this type of experiments could do that, okay? All right, then the last point so therefore, yeah people in Optomechanics always say there are these three areas of motivation, microscopic quantum superpositions, the area of quantum or in general sensing and the third one has to do with quantum transaction or in general which is relevant for quantum information processing and the idea is the following. The idea is for what it's meant by quantum transaction is in many applications in the context of quantum technologies it might be relevant to be able to couple quantum systems of very different nature, okay? Because some systems are very good for doing something other systems are better for something else and you would like to use them both but at some point you need to couple them. How do you couple them, okay? So mechanical motion is a very nice degree of freedom to actually couple all type of different mechanical systems, a different quantum systems and I will give you some examples. So the idea is here to use the motion of some massive object to couple different quantum systems for quantum information processing, QIP. And I give you an example. Imagine you want to couple the spin degree of freedom with the motion of the degree of freedom of an ion, okay? The spin degree of freedom that you might have in a solid spin like an Mb centering diamond and you want to couple it to the charge to the motion of an ion. How would you do that? Okay, as a thought experiment, imagine you could use one of these nice cantilevers that is vibrating and on one part of the cantilever I put a magnetic tip that generates some magnetic field lines that will couple to a single spin that is on the surface of some solid. And on the other side of the cantilever, on the other wall I put some charges that creates some electric field that now couple to the motion of an ion. And that's the idea. So this now by moving, so you see as soon as the ion moves it makes an electrostatic force to the cantilever, the cantilever is displaced and by being displaced the magnetic field at the presence of the spin qubit is modified. And now in this way you just can couple the spin degrees of freedom to the motion of degrees of freedom of an ion. And this is achieved by using a mechanical system that needs to be in the quantum regime such that this coupling is coherent, okay? This is the idea. And the same concept applies to many other type of systems. And in particular one that is very important is to couple electromagnetic fields but of very different frequencies. In particular it would be super interesting to be able to couple coherently microwave fields to optical fields. Because optical fields are great to communicate over long distances because they can be put into fibers. And microwave fields inside dilution refrigerations are nice degrees of freedom to couple to superconducting qubits. So say you have now a very nice quantum computer inside a dilfridge made by superconducting qubits and now you want to, after some computation you want to send the quantum information to another computer which is in another city. You would like to do that by optical light of course to use the fibers. So how do you couple microwave fields to optical fields which have very different frequency and very different nature. So then again you could think about the following device that actually is implemented experimentally not in the form I draw it but the concept. So to couple microwave fields to optical conversion you could do the following. I will basically merge these two diagrams into one. Okay, this one and this one. So now imagine you do the following. I put one of my mechanical motions like that. But this mechanical motion, this mechanical object on this part will have a mirror and on this part will have a conducting plate. And then that's the idea. Here you connect to an LC circuit and this you connect to a nice optical cavity. So this is the thing that is moving. It's a bit big but you know what I mean. So it's a conducting plate and here is a mirror. And here then you have optical waves and here you have microwaves, okay. And this is again a way that by coupling microwaves to the mechanical motion of this system this mechanical motion then couples to optical light. Okay, and there are experiments where basically they implement this concept. And then they can send for instance one, they can transfer one photon in the microwave regime to a photon in the optical regime. And again they use this motion. Okay. Good, so this I think gives you a fair overview of the topic of quantum optomechanics and it's motivation. And so I don't know if you have questions about that because this would conclude the introduction of motivation. Are you motivated for optomechanics now? After this? Hopefully yes. So then the three next hours will be interesting. So now we will do then a 10 minutes break and then we start with the theory. Do you have questions? Good. Well, I'm still not sure.