 Actually, a separate activity which happens to take place in parallel with our school, and there will be two special lectures on a nanomechanics-related topic today at the other building of ICTP. It's called the CMSP special lecture series, Nanomechanics meets Spintronics, and there will be talks by Robert Schecter and Leonid Gorelik this afternoon during our mini-project time. So in case you're interested in the topic and you agree with your fellow team members that you might want to go to these talks and catch up on your mini-project during some of the other mini-project time, please feel free to attend. This is an extra opportunity if you're interested. Besides that, it's already posted outside, hmm, it's not working. We will have a conference photograph taken tomorrow unless the weather is too terrible, so we'll announce that again, but keep in mind tomorrow in the coffee break there will be a conference picture. And for those of you who will stay for the second week in the workshop, I would just like to give you a very short peek at what's going on there. We're going to have an excursion on Tuesday to visit these beautiful caves, or I mean, you can go to the cave and take a hike around if you don't want to go inside. There will be more information next week, but so that's coming up on Tuesday. And on Wednesday, we're going to have yet again another event, an ICTP colloquium given by Florian Makwart. There will be more information next week. So back to this week, we were briefly outlining the lunch with the experts yesterday, and we decided to write it up in the form of sort of one compact slide for everybody to catch up on that. As said yesterday, it's an opportunity to interact with the speakers, with the organizers in an informal way over lunch. The participants which are supported by the ICTP, so who had their local expenses or their travel supported, are obliged to attend these lunches. If there's a few empty seats at the table, the other participants are, of course, invited as well, but the supported participants have preference, and they have to be there. This means when we're done with the morning session, please proceed to lunch immediately such that you won't spend half of the time in a queue. And then once you have your food, find the table with the sign lunch with the experts and join us. We're looking forward to talking to you. And finally, the mini projects. First question, have you found your team? And I think Florian will take over with the list in a minute. If you have not found your team and you don't know what to do, please contact us and we're happy to help you. So, I mean, as you saw in the schedule, there's plenty of time to work on the mini projects. We'll be around if there's questions or need of discussion, but so this is essentially your time to do this together. The results of the mini projects will be presented mostly during the second week, during the workshop. There are some teams we learned which are essentially gone next week. So if your team will be mostly home next week, please inform us, let us know, and then we will assign your presentation at the end of this week. These presentations are intended to take about five minutes, so I would say roughly three slides to tell us what you did and what you came up with. Okay, and so this is the team composition as far as we can tell. A few people have not arrived yet or have not signed the list. These are highlighted in red. A few didn't specify their team number on the piece of paper, which was sort of passing through yesterday. You may want to fix that. And a few people have canceled, and so these are the names that are crossed out. So if you were missing these people, so you can stop worrying, these aren't actually here. I think now I quickly end on. Okay, just everyone, this is the list again that we passed around yesterday. If you have not yet signed, please sign there. Even if you have signed but didn't insert your team number, please insert it. If you don't at all know your team number, just contact us and we can work it out and maybe assign you to some team or show you the list of the topics again and you can be assigned to a team. Thank you, so this list is important. And the first lecture of today is Oriol, Romero Isar, who will actually give a blackboard lecture. And this has to do with a very interesting topic of levitated optomechanics, which is also connected to the foundations of quantum physics. And he gives great lectures, so we're looking forward to his lecture. So good morning everybody. So since we have the blackboard with our microphone, maybe the students in the last room, we have three lectures, two today and one tomorrow. Usually we have some things related to the levitated optomechanics. In particular, I plan to do the following. The first lecture is going to be an introduction on the levitated quantum optomechanics, the similarities and differences between the electric nanospheres and the digital atoms. Lecture two will be the sources of the convenience and heating in this scenario, in this levitated nanosphere. The lectures will be devoted then to study the... Sorry, so can you use the microphone please? Okay, for the recording. Yeah, it's okay? Okay, thank you. Okay, perfect, thanks. And the third one will be on studying just wave packet dynamics, as we study in the first courses of quantum mechanics, but now looking at the interplay between the coherent dynamics and the dynamics in the presence of the coherence. And this is of course related to the fact that wave packet dynamics is a way to expand the wave function of the center of mass of this nanomechanical system and aims at the possibility of preparing really large quantum superpositions for these massive objects, okay? So wave packet dynamics, coherence versus decoherence. And then today with the first one, okay? And please let's make it very informal. So any question, anything you don't understand or anything you're very interested in and you would like to hear more, please let me know. We can have these lectures as an informal discussion among all of us. So please feel completely free to interrupt and ask whatever you want. Okay, so let me first try to motivate you why some people think levitated nanospheres are interesting for optomechanics and nanomechanical and nanomechanics in general. So the idea of what we want to do here is now as you have seen already in some of the lectures, most of the nanomechanical systems are always clamped to some substrate. And here now the scenario is completely different. We want to consider this nanomechanical system to be unclamped, to be really levitating, ideally in high vacuum, okay? And the easiest shape to think about it is just to have a sphere, that is just in high vacuum. So why is this interesting, this scenario? Well, we will see that in the field of cavity optomechanics, this is interesting because first by the very same scenario, as I said before, there will be no clamping losses. The degree of freedom that we want to bring and control in the quantum regime, namely the center of mass of this sphere, will be in principle unclamped from the mechanical path of other modes, because it is levitating. One good thing, well, you want to bring and control to the quantum regime the center of mass of a big object. So still the center of mass could be coupled to all the internal vibrations inside the sphere. But something that will be crucial is that this sphere is so small, that these internal vibrations have very, very high frequencies, okay? Actually, they have frequencies of the order of the speed of sound divided the size of the object. And for objects of the size of 100 nanometers, these frequencies are rapidly to the regime of 10 to the 10, 10 to the 11 hertz. This means that the center of mass, which will be typically a megahertz, is completely or resonantly coupled to the internal vibrations. So effectively, this is completely decoupled, okay? And this reason is also what also brings another feature, which is that this center of mass degree of freedom is decoupled to internal defects, okay? So which in other nanomechanical systems, this is a problem. The couple from internal defects. Because the coupling to internal defects is typically mediated by a strain. And strain means that it is mediated by phonons. And since the center of mass is decoupled to phonons, then there is effectively no coupling between the center of mass and this potential internal defects that could create decoherence, okay? And these things, for instance, have already been demonstrated experimentally, as I will comment later. And last but not least, another nice property in nanomechanical systems for the field of nanomechanics is that now the frequency of the mechanical mode, the center of mass, is a variable that can be changed. Because as we will see, this mechanical frequency depends on the intensity, for instance, of the lasers that you use to trap the particles. So and since you can just reduce the intensity, you can change the mechanical frequency. So it's a variable that you can tune. So mechanical frequency can be modulated in cavity or in optomechanics. These two properties of here automatically imply that in principle, it should be possible to cool to the ground state, the center of mass of a levitated nanosphere from room temperature, okay? So in principle, ground state cooling should be possible starting from room temperature, okay? And indeed, this has not really been demonstrated experimentally, but there are experiments in which an optically levitated nanosphere is trapped in high vacuum, and they apply feedback cooling, and they have already demonstrated cooling to a few, few phonons, to 10, 10 phonons between 10 and 20 phonons, okay? And experiments in the group of Lukas Navalny in ATH Suric and Romanke Dan in Tikfoil, Barcelona, okay? Good, very good. So then another area where we believe the electron nanospheres are also interesting is in the area of matter wave interferometry. Most likely, all of you have heard about these beautiful experiments done in Vienna about showing matter wave interference of very large molecules. Like for instance, the fuller end molecules containing 60 carbon atoms in the late 90s and even larger molecules more recently, okay? So the idea here is always to start with a massive object whose position is somehow sufficiently cooled such that it behaves according to the quantum, the laws of quantum mechanics. Then you expand this wave packet and then maybe you put a double slit and then you prepare a square position state and then you see interference, okay? Of a massive object. And one goal is to actually do these experiments with yet more and more massive objects, okay? And using levitated nanospheres, one feature that you have is that actually you can prepare very nicely the initial state of the matter wave interferometry, namely you can prepare a significantly large object whose center of mass is already in the ground state by applying the techniques of cavity optomechanics, okay? So basically you are able to prepare initial pure states very nicely in a very controlled way because in a very controlled way because in matter wave interferometry experiments, typically what they do, they have an oven and they just emit molecules and they try to filter such that they really get very cold in one axis. This is done nicely, but to go to even larger masses you need further control. So, and this is for sure then a very nice possibility just to trap these nanospheres, cool the center of mass close to the quantum regime and then switch off the trap and let the particle fall to a matter wave interferometry. And actually in the group of Marcos and in the University of Vienna, they are actually very interested in this direction. The second one is that you could also think to use then what we know and what we learn from optomechanics, namely to control these degrees of this center of mass degree of freedom, to actually manipulate the wave, to maybe be able to expand the wave faster, to break it into two states, into a superposition state and so on. So we can use actually the proper the tools of nanomechanics to control the matter waves, okay? Where these nanospheres might be interesting is for sensing, okay? And this is of course an area where basically almost all nanomechanical systems are very interesting because they are massive and they can be used for sensing. And once they bring them to the quantum regime, they are very fragile to the environment so the environment can already imprint a feature that can be read out from the nanomechanical system. What is a bit peculiar for the levitating scenario is that or a nice property that all nanomechanical systems have is that the object by definition, since it's a solid, has a high mass density, okay? And why is this interesting? For instance, a very fundamental question is to measure gravity at very short distances because there are many theories that predict that the Newton's law might be not correct at very short distances. So at very short distances, the gravity could be much, much stronger than what is given by the Newton's law, okay? And actually it's very interesting to know that for instance, at the range of few micrometers or even below one micrometer, actually gravity has almost not been measured. So we don't know whether at these distances gravity is described by the Newton's law or it's much stronger, okay? So to be able to measure that, then you would like to put a sufficiently massive object close to another one. And in this sense, one strategy was always to say, okay, I have a surface and then I put a gas of atoms, okay? Which are nicely isolated from the environment. I place it close to the surface and I try to measure the interaction between the surface and the cloud of atoms. Of course, it's much more interesting if we can increase the mass density such that all this mass that is distributed over a large cloud now is placed onto a very, very tiny volume of a nanosphere, okay? The same mass is now concentrated on a very small volume of 100 nanometers. Typically a cloud of atoms occupies a size of few micrometers or even more, okay? So now you have a very high mass density. So this is like a point source of gravitational field, which now you can place close to another source and try to measure gravity at short distances, okay? And of course here, the levitation would help in the sense that the system is better isolated from the environment, but not necessarily, okay? Another interesting aspect of this is that having such a point source to be that small, it's beneficial because you could consider that the mass density is homogeneous. And this is interesting because of the following. So again, related to gravity, you might know that the worst, so the big G and a constant in nature, a very important one, is actually quite badly measured, okay? This is the worst constant of nature measured because it has been measured to four digits, which is maybe not that bad, but the problem is that this measurement doesn't agree between different experiments, okay? Because there are systematic errors that cannot be controlled. The measurements are sensitive, but not accurate, okay? And this is actually a very important open question in the film. And one of the reasons why a very strong systematic that they have is that to measure big G, they need to use very large masses, okay? And then these large masses, so and in these large masses, to measure big G very accurately, you need to know where the center of mass of this big mass is, okay? And this is very difficult to know because there might be some systematic errors like, for instance, in homogeneities in the mass density that already gives you some uncertainty where the center of mass of this big mass is. So in this sense, having a point source, a very, very small source of gravity like a nanosphere, it actually might be beneficial in terms of locating where the center of mass of this source of gravity is, okay? And for instance, this is a topic that the group of Marcus Aspelmeyer also in Vienna is very interested in, okay? And then, yeah, and these things that I explained here for masses related to the mass, that would be if you want to use the spheres as a source of gravitational fields, but other type of spheres might have, might be sources of other fields like as sources of electrostatic fields if they contain charges, or sources of magnetic field if they are magnetized. And then the same things would apply there, okay? The same thing would be beneficial in this scenario too. So if you have, for instance, the particle to be charged, you could use it as a very, very good electrostatic field sensor, and there have been recent experiments in the group of Lucas, no one need, or if the particle is magnetized, maybe you can also use it as a very nice magnetic sensor which is very small and very sensitive, okay? And if you want to also extrapolate in the long term, if we were able at some point to prepare really large superposition states of these nanospheres, imagine you are able to prepare a state in which the nanosphere is in two different positions, okay, left and right, where these two positions are separated by a distance larger than the size of the particle, in quantum optics you could consider that this is kind of a noon state for measuring forces that are proportional to the mass such as gravity, okay? This would be like a noon state, a noon state. Maybe you remember in quantum optics, a very sensitive measurement is to in an interferometer to prepare a state where all the photons are either on one arm of the interferometer and on the other one in a superposition fashion, okay? And these measurements are very sensitive, and these states are very sensitive if in one arm of the interferometer there is a phase imprinted, okay? So if we would do the same, imagine you have a source of gravity and now you prepare a superposition state where I have two different positions, one very close to the gravitational source and one very far, and you are able then to prepare the superposition state and measure the interference pattern downstream, this would be a very, very sensitive measurement to measure gravity, provided of course these states are free from decoherence or decoherence doesn't scale also with the superposition size and so on, so on, always has to be a bit careful, but in principle that would also be possible. Good, then another area where nanospheres might be interesting is in the field of nanophysics. So so far I've been telling you that one possibility is to have this object at the nanoscale really nicely isolated from the environment and for these applications that I commented before we are interested just in one degree of freedom for instance the center of mass, okay? The position of the center of mass and how it behaves and so on. But now you can also think, okay, I have a piece of matter which has a size of maybe 100 nanometers and now, and this piece of matter is really well isolated from the environment, so now what I can do is I can look inside what happens inside this object, okay, and what are the physics that happen inside this object. In a very peculiar scenario, which is that this piece of matter is actually isolated from the environment, it's not on a substrate, okay? And this, of course, it's a very interesting scenario, in particular when the object is very rich. So for instance something we are very interested in is to consider this object to be a nano magnet, a single domain magnetic nanoparticle. Then suddenly you have a magnetic domain that is isolated from the environment and now you can look very nicely at what happens inside. What are the physics of nano magnetism inside the object? How this physics coupled to the center of mass, to the rotation and so on, okay? Yeah, you could also think about, for instance, already in the experiments of Lucas Navonia, Comencridan, they have an optically levitated nanosphere as I will describe later and they have the following scenario. This object is in vacuum and is inside a lab that has a temperature, it has a room temperature, okay? The center of mass, we are able to cool it to a few phonons. So the center of mass is, let's say, at the Mealy-Kelvin regime and the bulk temperature, if you look at the temperature of all the other internal vibrational modes, it is super hot. The bulk is, actually they have measured to be above 1,000 Kelvins, okay? So, and this difference in temperatures, center of mass being much, much cooler than the bulk, is a consequence of the fact that I told you that the center of mass is decoupled from internal vibrations, this has been measured. But this scenario also opens interesting questions. You have now a system completely out of equilibrium, of thermal equilibrium, so in vacuum and now you could study things as, for instance, how the system thermalizes, how it exchanges heat with, for instance, another particle that is in the vicinity of this one, and so on. So you could also look at this type of physics, what is called radiative heat transfer, radiative heat transfer with sources of heat which are finite, so its temperature can also change as a function of time, it's out of equilibrium and so on, okay? This is, I think, also a very interesting direction. And the last point I would like to comment, as Florian was motivating in the very beginning, is that of course these scenarios might also be interesting for the quantum foundations, or for the foundations of quantum mechanics, and the reason is the following. So imagine I consider a rigid body, so I consider a solid, and this solid has some mass density, rho, and some radius r. So the mass of this object is basically scales with the mass density and the volume, and now imagine I'm able to actually prepare a superposition state of the center of mass, such that this solid is now preparing a superposition state where its center of mass is the localized over a distance b, okay? Then now you could think about the following plot. You would like to be able to prepare superposition states of objects of radius r, and in a superposition state where the center of mass is the localized over a distance d, okay? And this then forms a perimeter space. So far, experiments that have been done either use small masses and very large superposition states, for instance, something like that. Imagine this distance is here, I put the half, okay? And this, for example of that, there have been this amazing experiment at Stanford where a single atom is preparing a superposition state of half a meter, okay? Where an atom is the localized over half a meter, okay? So there are, this parameter regime is very nicely explored. Then the field of nanomechanics aims at exploring this parameter regime here, okay? Where I have very, very large massive objects of billions and billions of atoms, but the delocalization of the center of mass has a length given by the zero point motion, which as you know, then d would be at the order of the zero point motion. Once you put numbers into that, this delocalization distance is typically smaller than the size of a single atom, easily at the 10 to the minus 12 meters and so on. So you have a very, very large massive object whose center of mass is delocalized over very tiny distances, the smaller than the size of an atom, okay? So if you want here, there are then the matter wave interferometer experiments with atoms and small mass molecules, and here we have the nanomechanical systems, okay? Then one could ask, okay, so this we have tested quantum mechanics in this area, but what happens here, okay? What happens, is this allowed by quantum mechanics? Of course, yes, in principle, why should not be allowed? But there are some people that from many, many decades have already or have conjectured that perhaps quantum mechanics breaks down once you have a very large mass that is delocalized over a very large distance, okay? And the typical name that always appears is Roger Penrose, but there are many other people actually here in Trieste, like Professor Girardi, who already in the 80s also proposed models beyond quantum mechanics that would predict that quantum mechanics is perfectly okay in this regime, but it's completely different in this regime, okay? And these are conjectures that cannot be derived from first principles, but give a very clear prediction. It tell you, okay, if I prepare a superposition state here, even in the absence of the coherence, this guy will not exist, or you will not be able to prepare it. It will have a very short lifetime because of some mechanism that we don't know, but they give a prediction. So therefore, many people find very interesting to actually do experiments in that regime, to explore quantum mechanics, whether it works, and to falsify these conjectures, okay? And in this sense, elevating particles is maybe a good scenario to explore this regime because I can be here, I can have very large masses that I can cool to the ground state and prepare already a quantum state, but then by the fact that I can switch off the trap and I can let the wave packet expand, I can go from this to this, okay, in that direction. And this is a unique feature that these levitated particles have because I can switch off the mechanical oscillator. If I have a clamped oscillator, I cool to the ground state, I have a very nice super, a very nice quantum state, but then if I want to have this motion delocalized over very large distances, it's gonna be difficult because it's clamped and this creates the coherence. Allowing this particle to fall in vacuum, in the absence of fields and so, perhaps I'll help you to do that, okay? Good, that was a long introduction motivation. I hope you feel a bit motivated now to hear the rest of the lectures. Do you have questions about that? Yeah, it's a different approaches. So, of course there are many people that thought for many years with this clamped scenario how to prepare superposition states first, which is not easy because you need interactions which are nonlinear, you need nonlinearities, sources of non-Gaussian states to prepare superpositions and then you could think about expand, but it's gonna be difficult because you have, because the system is clamped, there is always the coupling to the thermal back and this is a limiting factor. The main advantage or the main difference in the levitating scenario is that you don't have this source of decoctinus. There might be other problems, but at least this one you don't have. Yeah, it's difficult to say. Good, okay, so now let us concentrate on the optical scenario of levitated particles. So we will consider particles made of glass, made of silica, which are optically trapped using lasers and placed inside an optical cavity, okay, to do cavity quantum optomechanics. Other scenarios could also be considered such as magnetic levitation of either superconducting microspheres or magnets and then typically you couple them to either spin qubits or microwave cavities that for instance have a frequency that depends on the flux that is applied to this cavity. This could also be explained, but today we will focus on the optical scenario. And the first thing to do is actually to discuss about what is called the optical dipole force. So as I said before, we will consider an object that it's polarizable, okay, it can be electrically polarized and then we will consider this object to be in the presence of an external electromagnetic field, typically a focused laser beam, okay. And we will describe the forces that this beam exert into this polarizable object and we will see in a second that this for instance creates a trap. We'll create an harmonic trap, typically in the three dimensions, okay. So important thing is this guy is a polarizable object. So, and then, so the idea is we have this polarizable object in the presence of an electromagnetic field and we will define the following. So consider we define the time average electromagnetic field, okay. So I take the electromagnetic field which is a field that depends on position and time. I take its modulus square and I take the time average, okay. So this is a time average quantity, okay. This is time average. Then if this, if such an electromagnetic field interacts with a small polarizable object, then the relevant wavelength of this electromagnetic field is much larger than the polarizable object, then it is known that this object will feel a force, okay. And the force typically has the same, the following form, okay. So the force is basically proportional to the gradient of E square. And of course E square has to do with the intensity of this, let's call it laser beam from now on, okay. And the constant of proportionality is gonna be super important, alpha. And this is the real part of the polarizability of the polarizable object, okay. So alpha is the real part of the polarizability, okay. And this parameter, it's gonna be a crucial parameter in cavity optomechanics for this polarizable objects, okay. And this again assumes that the wavelength of this electron, the relevant wavelength of this electromagnetic field is actually larger than the size of the polarizable object, okay. So you have a sphere, then that's much larger. I'm defining now here just the relevant part. So the relevant, the case in which, for instance, the particle is completely reflective. It doesn't absorb light. So then there is still a force that is related just to the real part of the polarizability and this is the optical dipole force. The imaginary part of the polarizability enters into what is called the gradient force, but we will not assume that. So imagine that the field has a phase that does not depend on the position. Then the only force that exists is the optical dipole force. So this is the optical dipole force. And now what I will do is to characterize. So well, first of all, let us discuss now what is the value of this polarizability. For a dielectric sphere, for a dielectric sphere of radius R, which is much smaller than the wavelength, well, and for low intensities, sufficiently low intensities, sites that the polarization is linear with electric field, then it can be shown that the polarizability, in this case, is given by the following. It's given by three times epsilon zero, so the vacuum permeability, the volume of the particle, the refraction index squared minus one divided refraction index squared plus one. And of course, the refraction index is evaluated at the frequency of the, in that case, the laser, okay? And these edges size you can see that's in Jackson. This is section, you can see it in section 4.4 of Jackson. Can we say from here already, so that for a dielectric nanosphere, the polarizability, there are two important features I want to tell you. First, the polarizability of the dielectric nanosphere is always positive. It's a positive constant. And this positive has already a very important consequence, because if I look at the force, it tells me that this dielectric nanosphere will fill a force that is pointing to the direction where the electromagnetic, the intensity of the field, there grows, okay? So the particle wants to go to the maximum of the intensity, okay? And therefore, if you go and Google now optical tweezers or you go to YouTube, you will see all these videos where they focus a light beam and then they see that the dielectric objects go to the maximum of the laser, they go to the hot spot, okay? That's because the polarizability is positive. Second feature, which is also very, very important for nanomechanics is that the polarizability scales with the volume of the particle. Okay. This is gonna be very, very important later as I will comment further, okay? Good. Now, let us try to do the same, but not for the dielectric nanosphere, but just for a two-level atom, okay? So a neutral atom, you might know, it can be trapped also with light, can be manipulated with light. So, or if you want, a neutral atom is also an object, a polarizable object. It's an object that can be polarized, electric sphere. And now let us discuss for the atom case, okay? So, for a two-level, for a two-level atom, okay? The simplest model. So basically, what we assume is we have an atom which is neutral and has some electronic structure and from this electronic structure, there are two eigenstates which are nicely separated from the others, okay? And these two states, I call them ground and excited, as always, and this forms an atom. And this has a frequency which is given by omega zero, this transition frequency. And then I take this atom and then now I send light, okay, that has a frequency omega, like the one before. So I want to place now this atom inside the optical piece, okay? Then there are some important parameters that enter always when you want to describe the interaction of a two-level system with a laser beam of frequency omega. The first important parameter is what is called the spontaneous emission rate. So as you all know, in the absence of laser, in the absence of external fields, if you have an atom that is excited, at some point will decay to the ground state and there is a rate for which that happens and this is a rate you should always remember, okay? At least the scaling because it's very important. It's called the spontaneous emission rate. And what you should know is that this spontaneous emission rate at least scales with the dipole moment of the transition and the transition frequency to a power of three. And then there are some prefactors below to match the units, epsilon pi h bar c square, c to the power of q, but the defendant is very important. D here, okay, here actually just to be precise, I'm assuming that the transition between the ground and the excited state is isotropic, so, okay? Epsilon here is the polarization of any mode that the atom is interacting with and recall that this is the dipole moment operator, which is nothing else but the sum of all the charges inside the atom times the position of this charge from some reference frame, okay? That's a dipole moment operator. And this term basically tells me how well these two states are coupled by an electric dipole transition, okay? Good. Then, and then recall that for instance, this guy tells me that the probability to be in the excited state, so the time evolution of the probability to be in the excited state, I call it like that, so the population to be in the excited state decays exponentially with this rate, this is if you recall the optical block equations for a two level system, then you always have this term here, okay? The population in the excited state decays in this way. Good. Then, there is also another parameter, which is called the defacing rate, which is gamma, and this is related to the spontaneous emission rate, but I have an additional contribution, okay? Which I call gamma C, and this is just the rate for which the off diagonal terms of the density matrix decay exponentially, okay? So basically now the off diagonal terms will decay with this rate, okay? So in the absence of any additional defacing, at least they decay with the spontaneous emission divided by two, but they can decay faster if there is some spontaneous, some additional defacing. That's called, for instance, when there is an additional defacing, that's called in quantum optics, in homogeneous broadening, yeah? That comes from when you derive the optical block equations, and this is just related basically to conserve also that the matrix is positive, so the density matrix has to be positive, and then there are so constraints. So it cannot happen that these things decay separately, okay? So if you want to review that, there is a very nice source which is available online that I strongly recommend. There is the professor Daniel Steck from the University of Oregon. He has some available PDF notes on quantum optics, which have like 1,000 pages in PDF that he let it. He's an experimentalist that while he was learning quantum optics, he just typed everything, and this is really pedagogical with a lot of useful information. I strongly recommend that it's available online. So Daniel Steck, and the notes, just you should find it easily on internet, but it's called quantum and atom optics, the notes. And you can review all of these things, okay? Then there is the defacing rate, which is related to the spontaneous emission rate. As I said, this is a very important parameter to remember. Then there is a parameter that tells me how well these two level system couples to, for instance, a laser beam omega, and this is called the Rabi frequency, the Rabi oscillation frequency, and in the way I define things, I define it like that. So this is just related to the dipole moment times the time average field divided H1. The more important thing, this is just dipole moment times the intensity of the field, E0. Just for the definition, let me define this guy here that I defined before as E0 squared. E0 squared, that's the time average field. And I don't put the arrow to indicate that was a vector, okay? That's just the definition, E0 squared, and this is the E0 that appears here. Okay, so this tells me how strong is the coupling between the two level system and this external field. Good? Okay, then with these parameters, I can redefine another parameter that is gonna be also, that is as important as the spontaneous emission rate. That is called the saturation parameter. So the saturation parameter is a dimensionless parameter that is defined as the Rabi frequency. This parameter is dimensionless and depends on the Rabi frequency, the spontaneous emission rate and the detuning. And the detuning means that the difference between the frequency of the laser and the transition frequency, okay? And this is a dimensionless parameter. It doesn't have units. And for instance, and in quantum optics with this parameter and the spontaneous emission rate, you basically almost can describe everything. So for instance, I ask you a question. So imagine I have this two level system, okay? And now I shine light at frequency omega. And therefore the atom with some probability gets excited. And so in the steady state, what is the probability to find the atom excited? Do you remember? Can you make sure that in the steady state this is 100% excited or not? What is the limit? Okay, but more precisely, the limit in the steady state is actually this one half s divided s plus one, whereas it's the saturation parameter. That's why it's called saturation parameter. Okay, so only in the limit where the saturation goes to infinity, you get the asymptote. But if the saturation parameter is smaller than one, actually the probability is even smaller than one. And this makes sense. You see the saturation parameter is proportional to the Rabi frequency that I said before is proportional to the intensity of the field. So if I increase the intensity a lot, a lot, a lot, a lot, in principle, I would saturate the transition. And I would do that more effectively if I'm on resonance. For instance, if the detuning is not very large because if the detuning is very large, then of course for the same intensity I also decrease the saturation. So I increase the probability to be in the excited state. But this is just an example. Good. So why I define it all of this is because assuming, now under the assumption, assuming that the detuning is much smaller than the transition frequency so that the detuning is not completely crazy so that the frequency of the laser is close to the resonance, that it's not very, very far from the resonance. So that in quantum optics, this means you can do the rotating wave approximation. You can also do what is called the Born-Oppenheimer approximation, which means that the motion of the atom inside the presence of the electromagnetic field is actually sufficiently small such that the internal structure is always in the steady state. So the time that I need for the two levels to go to the steady state is much, much faster than the scale at which the atom moves. That's the Born-Oppenheimer approximation. If these two conditions are fulfilled, then this object, this atom, fills a force, that we call it optical dipole force that can be written in the same way as before. So namely, it can be written again as a force. It is of this form. It's the gradient of E0. It has exactly the same form as the other case for the electric nanosphere. But of course now we have to tell what is the polarizability of these two level atoms. And the polarizability in the most general form as I've defined things. Can you read here when I write here on the last row? See? And because that's an important one, so I'll write it on the other side. So in that case, the polarizability of these two level atoms can be written like that minus two times the detuning. Let me write it. That's the most general form. It depends only on the parameters I've defined, the detuning, the dipole moment, the spontaneous emission, the saturation parameter, and the total defensing. And for instance, let's make an assumption, assume far off resonance and sufficiently and not extremely strong driving, so not very, very strong fields, such that the saturation parameter is actually much smaller than 1. So if the detuning is much larger than the defacing and this is not extremely strong, once you put numbers, you always see that the saturation parameter is much smaller than 1. So let's assume that as much smaller than 1. And also, let us assume, so if this is the case, the saturation parameter then is just given by this ratio. So traby frequency squared divided the tuning squared plus a pre-factor that is the defacing over the spontaneous emission rate. That's for large detuning. And now let's also assume that you have a clean two-level system. Namely, there is no additional defacing than the one produced by the spontaneous emission. So let's assume that gamma c is equal to 0, so there is no inhomogeneous broadening, namely that the total defacing is just a spontaneous emission divided by 2. And then that's also the cleanest or the simplified expression of the saturation parameter, which is also worth remembering, just traby frequency divided the tuning squared with a pre-factor. If this is the case, s much smaller than 1, so this guy disappears, s is much smaller than 1. And as I write like that, I can also simplify the polarizability to be minus detuning d squared. And this guy just simplifies to, which is also a very nice expression. So it's good to remember that the polarizability of a two-level atom, far of resonance, is just d squared divided h bar detuning. The dipole moment squared divided h bar detuning. So some comments. First of all, the polarizability now, it's not always positive. It can be either positive or negative, and this depends on the sign of the detuning. Whereas for the electric nanosphere, the polarizability was always positive. And as I said, this means that now, atoms can feel attracted to the maximum of intensity or to the minimum of intensity. And this depends on the size on the sign of the detuning. So if you are a red detune, so if the detuning is smaller, namely the frequency of the photons that come to the two-level atom has a frequency that is below the transition frequency. They are low frequency compared to the transition level. The detuning is negative. This means the atom feels attracted to the maximum of intensity. So this is called red detune. If the detuning is positive, namely, and of course, this depends, don't remember positive or negative, because this depends on your definition of the detuning. So what is it good to remember is that if the photons of the laser have a frequency below the transition frequency, then the two-level atom wants to go to the maximum. And in the other case, they want to go away from the maximum of intensity. So if you are blue detune, then the atom actually is repelled from the maximum of intensity. This is blue detune. That's a very important difference as compared to the nanosphere. Interestingly, I can rewrite this polarizability now in another way. So you see this is proportional to d squared. And I told you before that the d squared is basically proportional to the spontaneous emission rate. I told you, remember that the spontaneous emission rate scales with d squared and the transition frequency to a power of 3. The transition frequency I can write now as a wavelength. The wavelength associated to the transition frequency. Let's do that. So you see, if I write now d squared, I remember that this squared can be written like that. I can write d squared using the spontaneous emission rate in this way. And now let me define this guy as 2 pi over the wavelength of the transition frequency. And this guy can be written as proportional to the spontaneous emission rate. I just rewrite it like that so that the polarizability of the two-level atom for the far of resonance case then can be written like this. I mean, I just rewrote this guy. Instead of writing in terms of d squared, I write it in terms of the spontaneous emission rate. And I write the transition frequency of the two-level atom in terms of the wavelength. And then I get this expression. And why is this expression interesting? Because now, compare this expression with the polarizability of a dielectric nanosphere. I said before that for a sphere, I have 3 epsilon 0, the volume, and refraction index minus 1, refraction index plus 2. So the unit should coincide. I have an epsilon 0. I have an epsilon 0. And then I need a volume scale. Here is the volume of the sphere. Whereas here is the optical wavelength to a power of 3. That's a volume. So the units are correct. And it's actually pretty similar. Indeed, note that based on this analogy, you could say, and it's actually very, very fair to say, that a single two-level atom with a transition that is close to the frequency of the laser behaves as an electric sphere of size lambda 0. So even if you have a single atom, from it gets polarized so much or similar to a sphere of size of radius lambda 0. And recall that lambda 0 can be an optical frequency. So these two-level, therefore, this would be at the scale of a micrometer. So this means that being close to a transition frequency in the two-level system boosts the polarization of a single atom to be similar to a sphere of almost a micrometer that contains billions of atoms. Indeed, it is very illustrative to compare the ratio between the two polarizabilities. Let's do that. So let's compare the polarizability of the two-level system and the sphere. So if I take the ratio, I exactly get the following. And now, for instance, let's put some numbers to get the scale of this ratio. For instance, assume a sphere with a refraction index of, let's say, 2 of the order of 2, like silica. Consider that the transition frequency of the laser is the tune from the transition frequency by an amount of 100 times the spontaneous emission rate, which is quite OK. And consider the transition frequency wavelength to be 10 times the radius of a nanosphere. Say, if I have a nanosphere of 100 nanometers, I automatically get here around half a micrometer. Or if I have a sphere of, yeah, 15 nanometers radius, then this would be 1 micron. Is it reasonable? Then if I plug these numbers into that expression, I automatically get that the polarizability of a two-level system compared to the polarizability of such a sphere is 20%. This is a very large number. If you compare, if you recall that this nanosphere of 15 nanometers radius will contain of the order of 10 to the 7, 10 to the 8 atoms. OK. So what does this mean? That in glass, since you are very, and if you shine glass with optical frequency, you are very far from any resonance. So all the atoms inside the glass get very, very little polarized as compared to a single one that would be somehow close to the transition frequency. So the polarizability per atom in glass is much, much smaller than the polarizability of an atom that has a transition frequency close to the frequency of the laser. That somehow makes sense. But still, the number is very, very amazing. And that's the reason why actually, for instance, Clemens Hamara some years ago had a very nice proposal where he showed that actually a membrane can be coupled. So if you place in a cavity a membrane and a single atom, the atom and the membrane can be strongly coupled via the cavity. Because actually, both objects are almost equally polarizable. Or that's also the reason why Philip Troiland in Vazel is doing this beautiful experiment where also couples called atoms to nanomechanical systems. Because the polarizability of these tiny atoms is actually not that small. That's the opposite. I don't know. That's OK. Maybe if I make a typo. So I made a typo. And so the polarizability of the nanosphere is what I wrote here. And again, recall that that's a refraction index at the frequency of the laser. The refraction index, of course, changes. This is a function of the frequency. So that's a refraction index at the frequency of the incoming beam. Good. Questions about that? Now we will proceed. And I will now describe the optomechanics of this object, of this polarizable object. And you will see that basically what enters always is the polarizability parameter. And therefore, the discussion, you can apply it to a single atom or to a nanosphere. You just need to change the value of the polarizability. Before we do that, just one comment. So note that I said that the optical dipole force can be written like this, the gradient of E0 squared. And then provided the polarizability does not depend on the intensity of the laser, which is the case for a dielectric nanosphere at sufficiently low intensity, such that the polarization is proportional to the electric field. So it's linear with electric field. And the polarizability is independent on the electric field. Or for an atom that is sufficiently far off resonance, such that the saturation parameter is smaller than 1, because otherwise, note that if the saturation is not much smaller than 1, this guy depends on the Rabi coupling. And the saturation and the saturation also depends on the Rabi coupling. And the Rabi coupling depends on the intensity of the laser. So actually, the polarizability of a two-level atom is only independent from the field either when it is of resonance and sufficiently low intensity. So otherwise, this will depend on the intensity of the field. But in that scenario, no. That's important. So when this is the case, of course, then I can write an energy associated to this force. Then I can easily integrate this force and get an interaction energy, which has the form, the interaction energy. I can easily, so for alpha independent of the electric field, so for a linear polarizable object, then I can easily integrate and get this guy. And this is then just the form of the interaction energy between the field and the polarizable object. Also very important. Let's move on. So now we talk about, after having discussed about the optical dipole force, now we can finally start discussing about optomechanics. And in particular, we will talk about what is called dispersive cavity quantum optomechanics. So the scenario we have is the following. We assume there is a hypheness optical cavity. And then we place this polarizable object inside the cavity. And this object could be either an electric nanosphere or a neutral atom, far of resonance. Then this cavity, we will assume, has a very well-defined single mode in the cavity that we write like this. And this is going to be what I call, and this mode has a frequency omega c from the cavity. Then I assume this cavity is driven externally from some beam that has a frequency omega. Then I also assume that this particle is placed inside the cavity because there is some optical tweezer that is trapping the particle. So this is this external one. And this will create a potential into the particle of frequency omega t, mechanical frequency omega t that we'll show in a second. And then what can happen is that we will see the motion of this particle then can have some decoherence, some source of decoherence with a rate that I will call gamma m. So the motion can decoherence. And also the photons can be lost from the cavity with a rate given by kappa through the mirror because the mirror is not perfect. So you can lose the photons because the mirror is not perfect, so there is some decay rate. And actually, this picture is the one that is happening now in some labs. As I said before, in, for instance, many labs. They trap an electron in a sphere with some external optical tweezer. They place the sphere inside a hyphenous optical cavity. They drive the cavity. And then they look at the physics of one degree of freedom. For instance, the motion of this polarizable object along the cavity axis. OK? Good. So how do we describe the optics on the quantum optics here? Well, first of all, from the quantum optics point of view, yeah. So from here now there are the electromagnetic field. Now you could describe in different modes. There would be the modes. For instance, imagine there is only a single mode. There would be a single mode inside the cavity. Then there are some peculiar modes, which are the modes that you use to drive the cavity and the mode that the photon occupies if it leaks the cavity. And then there are the free electromagnetic field modes, all the other ones. OK? So in quantum optics, in cavity query, the typical thing you do, the first thing you do is to trace out the input or output mode of the cavity. OK? And then typically effective dynamics you already get can be described by a master equation, which is nothing else but just the Hamiltonian dynamics plus a term of this form. If already, and this is just to tell you, that would be our starting point. So here I just have traced out the output mode. OK? So therefore, the photons inside the cavity, which are described by the creation and elation operator A. OK? A dagger creates a photon inside the cavity at this single cavity mode. And A annihilates that photon. Then once I trace out these output modes, then I already have a decay channel for the photons inside the cavity. OK? For those that are not familiar with master equations, it's not so important. So basically, just recall, this is the Schrodinger equation. Just the Schrodinger equation written for density matrices. And whenever I have a guile of this form, OK, which always have this structure, sum operator rho, the dagger of such an operator, that's typically a decay process. And a very nice memotectic strategy is to recall that the operator on the left of the density matrix is telling you the physical process that is happening. So the operator on the left is an A. This means that this guy is describing me the fact that I lose photons from the cavity. OK? So if I would have this guy written with an A dagger here, this would mean that the process that is putting photons inside the cavity. That's always very good to remember. More technically, that's the jam operator that described the decoherence process or the nice process. OK? So in this case, this guy just tells me that I'm losing photons from the cavity with a rate given by the photon decay rate, which has to do with the coupling between the internal mode and the output modes if the mirror is not perfect. OK? Good. And now this Hamiltonian should describe all the other degrees of freedom in this situation, which is the degrees of freedom inside the cavity, the cavity mode, the free electromagnetic field modes, and the center of mass mechanical mode. So let's write that. Then this Hamiltonian can actually be written in this form. It will contain the following path. It will contain the term that describes the dynamics of the cavity mode, which is really simple. It does an harmonic oscillator with frequency omega c. It will give me the motion of the kinetic energy of the particle, p squared divided 2 times m. It should have a term that describes the dynamics of all the free electromagnetic modes, which I will call H3, and I will define in a second. And then the crucial one, there should be the Hamiltonian that describes the interaction between the particle and the cavity mode and all the other electromagnetic field modes. These I call H interaction. OK? So please interrupt me if there are things that are not defined or that you are not familiar with or you don't understand. So the free one, it's also kind of simple. This should be just the term that describes the dynamics of all the free electromagnetic field modes. And this is something that we know how to do. So we quantize the electromagnetic field in free space, and then we just have a set of harmonic oscillators, which are defined by the following quantum numbers, a vector k that tells me the direction of the photon and a vector epsilon that tells me the polarization of such a photon. And then I have to sum over all possible k's and all possible epsilons. So I recall that the epsilons have to be transverse to k, because the polarization is always transverse to k. And there are two possible values, one or the other one. So then I have an integral over all k's, which is all directions and modulus of the k vector of a photon, and a sum over epsilons. The two possible polarization states that are perpendicular to a given k. And then I just sum by these terms. And this term here tells me I create a photon that has momentum k and polarization epsilon. So I have to sum over all possible states. From the technical point of view, you could say, well, but somehow these ones and these one and the one you trace out, they are not completely orthogonal. And that's actually true. But from the technical point of view, this mode has just one mode over an infinite. And this one is also almost one mode over an infinite amount. So I can here just double count them. And this is not a problem. Not to get complicated about this, but this just would be all the other modes that are not included either in the cavity or this particular mode that has a given koo, which is going in that direction, and a given polarization. This would be one mode from all this infinite set here. Just for those who are more theoretical. Good. So then this free is just given by that. And then the crucial one is the interaction. And the interaction, this could be justified more rigorously than what I will do. But from the first section, we arrived at this expression, which is telling me that if I have a polarizable object interacting with an electromagnetic field, the interaction energy is just given by the total field at the position of the particle squared. So I can actually already, I hope you feel that this should be related to that. And more rigorously, you can show that it's like that. So the interaction term is described by the following. So the interaction term. Now I promote this interaction energy to a Hamiltonian. Let me write this. It's minus alpha 4. And now here will be the field, which can be an operator, evaluated at the position of the particle, which is also an operator because I want to treat in the quantum regime. Of course, sorry. Thank you. This one? This one? This one? This one? OK, this has to go with that. So this is kind of the form that this term has to have, such that if you have a density matrix that defines a state, a physical state, when it evolves in time, you want that this state goes to another physical state. More rigorously, you have to be sure this density matrix has two properties that are very important. So it has to have that the density matrix operator is positive so that its eigenvalues are positive. So this means probabilities are always positive. And you also need that the trace of rho, that the probability sum to 1. So it's 1. So if you would not include that term, it might be that starting from a state that fulfills these two conditions, you end up into a state that does not fulfill these conditions. It's just a mathematical restriction. So you need this term. This is called the Limbland form. And this is an anticomutator. Good. So now what we do is that here, we now put the total electromagnetic field that this particle fills inside the cavity. And this has different contributions. The total field, it's going to be the field generated by the cavity mode. The field generated by the external classical field that I put a laser with a lot of photons. And this is what I call the classical. All right, let me just follow my notes. I use the same notation. Yeah, the classical that I don't put ahead because you will see in a second it's going to be just a classical field. But it's evaluated at the position of the particle, which I described quantum mechanically. And then there is the field that would be generated by the free modes. They are in vacuum. So the mean value is 0, but its fluctuations are non-zero. So that would be the free. So that's the three contributions that I have here. The cavity field, the external field, and the free electromagnetic field. So these guys can be written in the following way. So the cavity field. The cavity field. And let's assume now the particle only moves along one axis, x, the axis of the cavity. This is then when the typical, if you can quantize the electromagnetic field inside two mirrors, and there is a pre-factor. This would be the mode form. So usually you have a sinus. There is a pre-factor here that tells me the strength of the field of the cavity, basically. And it goes with omega c, the frequency of the cavity, and the cavity volume. I write the fields and I stop there. And this is what is called basically the volume of the cavity, the cavity volume. And then here I determine that basically creates or annihilates a photon inside the cavity. And of course this sinus is evaluated at the position of the particle, which is a quantum degree of freedom. That's the position where the polarizable object is. And you already see this creates a coupling between photons and mechanics. Then there is the free, which just have a sum over all the free electromagnetic modes. There is this guy here. There is a pre-factor. So again, I sum over all possible external modes, over all k's and all epsilons. But the important thing is this guy. So this guy, what it does is you see, this is the process in which basically you can think the particle would scatter or would absorb a photon of momentum k and pre-position epsilon. And because of this scattering event, the particle fills a kick. Recall that this in quantum mechanics is a momentum kick, an exponential of an operator that is i times x is a momentum kick. So it changes the momentum of the particle. That's a recoil kick that can have one direction on the other, depending on whether I have an a or an a data. So this will create a coupling between the motion, the mechanical motion, and the free electromagnetic modes. And the last term, it's going to be the classical one, which I will come in a second. I will do that in the next lecture. But the classical one is just an external field that describes the tweezers. Good. So in the next lecture, we will see that now. Actually, how these three terms, once you plug them here, actually give you the auto-mechanical Hamiltonian that you are familiar with. Plus, it describes also the interaction between the cavity mode and the mechanical motion, the optical mechanical, and the coupling between the motion and external photons, which is what described recoil hitting, a source of hitting in the motion, and also a new channel for losing photons from the cavity. Because now, a photon from the cavity can be scattered off from the sphere outside the cavity. And we will discuss the rates and so on. Please, we will do in the next lecture. Thank you very much. I know if there are any questions.