 OK, it's a microphone. So then let's move on then to the theory. So as I said, let's start then with section 1, which is close, which I call closed optomechanics. And here by close, it has to do with the difference with open quantum systems. You had already lectures about open quantum systems, which means a system coupled to the environment. Here now I refer to the system that is isolated, still not coupled to the environment, just to describe the so-called coherent dynamics. So recall that symbolically, our optomechanical system, I will use this image again, this cavity with a movable mirror. And in quantum physics, whenever we want to describe the system, we need to identify well the degrees of freedom that the system has and the potential interaction. And here, basically for us now, this big massive system and big will be analyzed as basically the coupling between one degree of freedom associated to the cavity, which I call it cavity mode. And recall that in quantum optics, we use mode. The word mode, in a sense, can be translated to a single harmonic oscillator. Any mode is an harmonic oscillator, basically. So here, there will be one harmonic oscillator associated to the cavity. And cavity here, I mean the electromagnetic field resonator. And then one mechanical mode. The excitations of the cavity mode, we will call them photon. The excitations of the mechanical mode, we will call them phonon. Basically, there is a cavity mode, a mechanical mode. And then there is the interaction between the two. So that's the idea. You will see that now we will suddenly model this in a very rather looking simple way, but actually it contains and it is able to describe very well the physics of the system. So let us then focus first on the mechanical mode. This is nothing else but a quantum harmonic oscillator. Let us focus on the mechanical mode. So namely, this motion here of this mirror. So basically, you see already from the picture that here, what I'm interested is in the one-dimensional motion of this mirror along the axis x, which is vibrating. So hence, we can use the model of or the description of a one-dimensional quantum harmonic oscillator. And recall that the Hamiltonian of a quantum harmonic oscillator is this one. Now I write quantum mechanically already with operators p and x, which fulfill the following commutation rule. And this Hamiltonian introduces two parameters. One is the mass of the oscillator, which is relevant. And the other is the resonance frequency of the oscillator, omega m. And in the quantum harmonic oscillator, we can make a change of variables. We can write x and p in terms of creation and annihilation operators in the following form. And I know this might be very basic for some of you, but we need to review it because what appears here is actually super relevant for Optomechanics, which is this parameter here. Recall, this is just a change of variables where now these creation and annihilation operators fulfill the commutation rule bb dagger equal to 1. And I call b for the mechanical motion because I will use a for the cavity degree of freedom. And here, it has appeared a value here that is very, very important. This is a length scale. We call it in Optomechanics the zero point motion. And I write it here, the zero point motion, x0, which is the square root of h2 mass and omega m. This is the typical length scale that appears in the quantum harmonic oscillator, which is a theoretical physicist I think is very useful to remember because it's very good. Then, why is this important? Because now, as I said in the introduction, the main feature of Optomechanics is that we deal with very large masses, very large masses. Here will mean that this parameter will be way smaller than for a single atom. So let us put actually some numbers to that. So if we take, if we write the mass written as n times an atomic mass unit, recall that atomic mass unit is 10 to the minus 27 kilograms, and we consider for a reference a mechanical frequency that I didn't mention yet, but typical frequencies for Optomechanics are of the order of megahertz. So the mechanical motion is typically oscillating at megahertz frequencies. So it makes one million oscillations in a second. If we plug this number and this number here, we use the value of the h bar, we will get that this x0 is of the order of 71 divided by the square root of n, where n is a number of atomic mass units. The mass contains times 10 to the minus 9 meters. And this is actually, it's very good to think about these numbers so that you have a mental picture of the system. So this number tells us that if the mass would be comparable to a single atom, and you would take this atom, place it in harmonic potential of frequency of 1 megahertz, and for instance, you would be able to cool the motion to the ground state, the atom would fluctuate according to quantum fluctuations in a land scale given off around 70 nanometers, which is actually pretty impressive, because if you recall the size of an atom is 0.1 nanometer, more or less. So an atom in the quantum motion or ground state in a typical harmonic potential is delocalized due to quantum fluctuations over land scales of the order of 100 times its size. So it's really like a cloud, really. However, if the object is much more massive, for instance, I talked before about n's that could be from 10 to the 6 to 10 to the 19, you decrease this by a lot. In particular, if you take the typical masses that you have in optomechanics, say 10 to the 12, then you reduce this by six orders of magnitude. So you go to land scales of the order of 10 to the minus 12, 10 to the minus 13, even more. So basically, the quantum fluctuations of this big object is way smaller than the size of an atom, even comparable to the nuclei of an atom. This is important, because this land scale appears a lot. So this parameter is actually very relevant. It will appear in all the discussion. So as I said before, recall that if you would have such an harmonic potential and you would cool to the ground state, recall that the ground state of an harmonic potential is a Gaussian wave function. And this size here is comparable to the zero point motion. So that's the idea. The center of mass of this massive object, if you cool it to the ground state of this harmonic potential, will have a size given by the zero point motion. So then using this change of variables, you know well that then the Hamiltonian of the harmonic oscillator just can be written as this, up to constant shifts that I can always shift away. And recall that now in the quantum harmonic oscillator, there are different relevant families of states. Once are the so-called Fox states, which are the eigenstates of b dagger b, that in a sense counts how many excitations do I have. So the Fox states are defined as being the eigenstates of this operator, b dagger b, namely, the eigenstates of the Hamiltonian H. So these are energy eigenstates of the Hamiltonian. And once you implement this change of variables, there is this nice kind of algebra with these operators b. So the energy eigenstates are Fox states, eigenstates of b dagger b. And if you apply b dagger to n, you just increase n by 1 with a pre-factor square root of n plus 1. If you apply b, you decrease with a square root of n. And out of these Fox states, there is one that is particularly relevant, which is the ground state, which is defined as the state that is annihilated by b. This is the ground state. And in optomechanics, there have been already experiments done where the mechanical motion of these huge objects is cooled down so much that they reach the ground state, so that they really remove all the thermal excitation so that it's in a very nice ground state of an harmonic oscillator. And this is for an object so much as the ones I was saying before. And in the quantum ground state, what has that, as I said before, the fluctuations of x of the motion is precisely given by the zero point motion square. So that's why this length scale is important. So they have, I repeat that, they have done experiments where they slow down the motion of a big object that you can see almost with your bed eye, such that the fluctuations of the center of mass are of the order of x0, which is of the order of 10 to the minus 14 meters. So basically, this thing is only moving due to quantum fluctuations. So it's very impressive, of course. Sometimes we call these quantum fluctuations the fact that the center of mass is not in a given position in a deterministic way. We call that the center of mass is delocalized. It suffers quantum delocalization. So this is the delocalization distance, the localization distance of the mechanical mode. Quantum delocalization distance of the mechanical mode. And then you see something that is happening. Because of this scaling, you see that it goes like one over square root of the mass. It does imply that the more massive, the more massive the object, the more localized it is in the ground state. And this is an important, relevant comment that brings some point for discussion. Because as you saw in the motivation, I was saying, A, optimal mechanics is great. You could observe quantum physics with very macroscopic objects. And that's true. But at the same time, looking at this scaling, it also tells us that the more massive it is, the more tiny these quantum features will be. Because, for instance, the quantum fluctuations will go to really, really small values. So hence, if you want, one could think that this is another important relevant, not only if the system can be brought to the quantum regime, but also what is the size of this quantum effects, the size of this quantum, for instance, the quantum delocalization distance. And indeed, there is some research also on investigating how, what can you do once you are in this quantum regime where the fluctuations are small, what can you do to increase these fluctuations to even larger scales? So ideally, we would like to do experiments where we place one of these massive objects at two locations in space, which are really separate. Like one, the mass is in this room and in another room. This would be a very large localization distance over 20 meters. Once we are in the inside the harmonic oscillator, all the land scales are of that size. So when we talk about superpositions, they would have of that size. So we will not have time to enter, but it is a relevant question to think how to increase this land scale over microscopic scales, or whether this is actually relevant for addressing some of the fundamental questions I mentioned in motivation A and so on. That's something to take into account. Good. So this is regarding, you see, we have introduced now the Fox states. And in particular, one Fox state, the ground state, which is very relevant in the context of optomechanics. Another type of state that is also very important is the so-called thermal state. The thermal state of a quantum harmonic oscillator is defined as the density matrix rho, which you obtain by taking the exponential of minus beta times h bar, and dividing by the trace of such operator such that the trace of the density matrix is 1, where beta is as always 1 over kBt. So that's the definition of a thermal state. Basically, a thermal state depends on a Hamiltonian h and some parameter that we call temperature. This defines a thermal state. If you take the Hamiltonian to be this one, then OK, that's the thermal state of the quantum harmonic oscillator. It basically depends on two parameters, in temperature and in the frequency, omega m. And for instance, something relevant from this state is that if you calculate the mean number of excitations in the thermal state, which quantum mechanically we would calculate as the trace of rho times beta over b, then this is nothing else but the Bose-Einstein occupation number, famous occupation number, 1 divided exponential of beta h bar omega minus 1. And recall what this is telling us. So this is telling that if temperature is way, way larger, so if kBt is way larger than h bar omega m, then this exponent is huge. So the denominator is huge. Sorry, no, the opposite. So this exponent goes to basically 0. So e to the 0 is 1. So you have 1 minus 1, 0, 1 divided by 0, infinite. So if this is much for a very high temperature, n bar basically goes to infinity. So there is a lot of excitations on the oscillator. Whereas if kBt is smaller than h bar omega, then n bar is very close to 0. And this is actually very relevant to recall that. That if you have a quantum harmonic oscillator in a thermal state, if the temperature is smaller than omega m, the thermal state will be basically empty, will be in the ground state. Whereas if the thermal state has a temperature t larger than omega m, then it will be very occupied. This is kind of basic maybe, but actually in optomechanics, will be very relevant in a second. I will allude to this later in the lecture, so keep this in mind. Just comment. So then also something to recall from a thermal state. If you calculate what is the probability that if I take a thermal state and I measure how many excitations do I have, I obtain the value n. Such probability, which quantum mechanically you calculate by taking this bracket from the thermal state. This is just given by this expression. Don't confuse the n bars with the n. n bar is the thermal mean number occupation. n is the number associated to the probability you are asking for, to obtain n excitations. And also, you calculate what is the variance on the number of excitations for a thermal state, which is calculated like that. This goes like this n bar squared plus n bar. So for n bars larger than 1, it goes as n bar. So the fluctuations, so if you have a thermal state, you measure how many excitations we have. The fluctuations are of the order of the mean value for large n bar. For n bars smaller than 1, the fluctuations go as a square root of n, because then it dominates this part. Anyways, but perhaps the useful thing of a thermal state is that I can always write the thermal state in the fog basis. And in the fog basis, it has this form, which is very useful to remember. There is some pre-factor, but the important thing is that a thermal state in the fog basis, it is diagonal. A thermal state in the fog basis only has diagonal terms. There are no correlations between fog states. And why it is relevant? Because if I ask you now to calculate what is the mean value of the position for a quantum harmonic oscillator in a thermal state, you should be able to answer that immediately. So what is the mean value of x for a thermal state? It should be 0, because when you make the bracket, you will decrease with the b dagger's n. And when you do the trace, it gives a 0. Or for the quantum electromagnetic field, the mean value of the electric field in the thermal state is 0. Just the fluctuations are huge. Good. So this was to introduce the thermal state. Then another important state of the quantum harmonic oscillator is the so-called coherent state. The coherent state is defined as the eigenstate of the annihilation operator. And since these operators b are not a mission by definition, this alpha, which is the eigenvalue, can be complex. So alpha can, in general, be complex. So it depends on two real numbers, the real pattern and the imaginary pattern. And the coherent state, I recall, that can be written in the fog basis as that, is a linear combination in the fog basis. And you should, of course, appreciate the difference between this and this. This is a mixed state, which is the density matrix is diagonal in n, whereas the coherent state is a pure state that can be written as a linear combination of fog states. So this is a coherent superposition. This is a classical mixture. And from the coherent state, there are basically two things, or three things that are important. First is that the probability in a coherent state to find n excitations, which you would calculate quantum mechanically as this bracket, just give me this Poisson distribution. This is a Poisson distribution. And the mean value of n is just given by alpha square, the modulus alpha square. And these are important for a coherent state. This complex number alpha, the modulus square of this complex number alpha, tells you the mean number of excitations, so the mean value of B dagger B. Now, for a coherent state, the standard deviation of the number of operators goes as the square root of the mean value, so namely, basically, as the modulus of alpha. To compare the difference also between a thermal and a coherent state, the standard deviation for a coherent state goes like a square root of n, whereas the standard deviation for a thermal state for large n goes as n. So the thermal state has much larger fluctuations in the number of excitations. Then another property of coherent states is that they are not orthogonal. So namely, if you calculate the bracket between two coherent states of different alpha, this is not just a delta, but almost, because they decay exponentially with the distance between two points in the complex plane. But they are not orthogonal, but they are complete. So they complete the basis in the sense that if you integrate the ket bra over all complex plane and divide by pi, this gives you the identity. That's useful. So coherent states are not orthogonal, but they are complete. They can expand all the hub space of the harmonic oscillator. So I need to introduce this because I will use it a lot, and I wanted to make sure you all know that, and also to introduce some notation. So I go a bit fast. So that's a coherent state. Interam, if you have questions. Then from the coherent state, I can define another type of states, which are very fancy, which are the so-called Schrodinger cut states, which, depending on the context, we use this name cut state to define all type of different states in the context of quantum optics. Typically, for cut states, one understands a superposition of two coherent states with opposite phase. Namely, you could define a cut state defined as that. You take a coherent state alpha, and you make a superposition with a state minus alpha. And here you allow to include a phase with two particular relevant values, phi equal to 0, which means it's the plus superposition, and phi equal to pi, which is the minus superposition, what people call an even cut state or a odd cut state. And for instance, just to recall, if you have an even cut state, then this has a superposition of only even number of excitations. So it will only have a superposition of 0, plus 2, plus 4, plus 6, but not odds. So when you sum a coherent state with a plus, you only get odd number of excitations. That's why it's called an even cut state. When it is odd, then you do the same, but you have here 1, plus 3, and so on. So how many of you have heard about the Binger function? Did you have? Sorry? The last one? Yeah, so the pre-factor is the same, and then you have 1 plus 3 for the even. There is a typo? Yeah, it's also. Just wanted to recall that you have superposition of odd excitations or even. So very rapidly. So a tool that is very useful in the context of the quantum harmonic oscillator is the so-called Binger function, which is defined as follows. I just define it, and I discuss a couple of properties, which then allows me to define a concept that appears in optomechanics that is very relevant. So recall that the Binger function is a function that depends on two real numbers, x and p, associated with units of position and momentum. And I obtain it from a density matrix by doing the following strange operation. So that's the Binger function. It's defined like that, so definitions are never wrong. So this is just a definition. And just make sure we understand it. So here we have the density matrix. So for any density matrix, you can calculate the Binger function. And what you need to do is you take the density matrix, you do this bracket, where these states, these are eigenstates of position. These are the eigenstates of the position operator. You just take this bracket, then this will be a complex number that depends on x and y. And now you integrate over y, kind of doing a Fourier transform, and then you will get something that depends on x and p. That's it. Why this definition is useful is because it has the following. Well, first of all, it's a mapping. From any quantum state, you have a Binger function. And this Binger function, it has the following properties. First of all, the Binger function, it can be plotted. So the Binger function is real. So it means it can be plotted. Now in a nice three-dimensional plot, where I would put, for instance, or in a contour plot, where I would put x and p, and I will plot here with the color, with a bar, the w. Can be plotted, or in a 3D plot. This means that now any quantum state of the harmonic oscillator, you can have an image, can have a photo of the state, which is very nice. I'm very pedagogical. So I urge you to actually do that. To have a little code where for all these famous states, you just have an image of the Binger function. It's very useful. Then it has the following important properties. First, it's well normalized. So if you integrate over all phase space, so the space of position and momentum in physics, we call it phase space. So if you integrate over all positions and all momentum, the Binger function, defined as it is, it is normalized. It gives you a 1. Then as I said, it is real. And then there are two super nice properties of the Binger function, which is that if you integrate a margin, so if you take a marginal of the Binger function, namely, you integrate p, you get a function of x. And this function of x is actually something very physical. It's the probability distribution to find the harmonic oscillator in position x. That's a marginal. So basically, it gives me the probability distribution to find the state rho at position x. And if you take the other marginal, then you get the probability distribution for the state to have momentum p. And then this means that if you have this nice three-dimensional plot of the Binger function, whether here I have x, I have p, and I have w, and I have some nice three-dimensional function, if you now just project this function into this plane, then you would have some one-dimensional plot or in this plane. And this is just p of x. And this is p of p. So the projections onto the screen in the plane pw or xw just give you the probability distribution. So then the whole point is that you might be tempted to say, hey, I have some function that is real. If I integrate it, it gives me to 1. So is this not just a probability distribution? It is this function not telling me what is the probability to find the particle with position x and position p? Well, point is if this particle would be classical, yes. But since it is quantum, you know that in quantum mechanics because of the Heisenberg uncertainty principle, you cannot know x and p with certainty. So a manifestation of that is that this is not a probability distribution because there are some points where it can be negative. So very importantly, the Binger function w can be negative. Can be negative. So it is not a probability distribution. It's what is called a quasi-probability distribution. Yeah, you can calculate. So from the Binger function, since it contains all the same information as rho, any mean value of any observer can be calculated. So basically, if you calculate the mean value of x, you just integrate over phase space w times x. If you want the mean value of p, you multiply by p and you integrate and so on. It is? No, no, no. So that's the point. So if you want to now represent, so what I think if I understood your question correctly, you are kind of asking, how would you obtain the Binger function given the mean value of some observables? And that's, of course, a delicate question because you should obtain a lot. I will come in a second to that because I will use this question to motivate something. But now, first thing is that w can be negative. And it's very nice because it defines now you can make two classes of states of the harmonic oscillator. Harmonic oscillator states for which the Binger function is always positive. And quantum harmonic oscillator states for which the Binger function has points which is negative. And this defines two classes of states. And actually, you can now make a definition. You can say, hey, I define states which have a negative Binger function. And these states, if you want, in the quantum domain, they have some, they rank a bit higher than the others. Because not only because they really feature some quantum, very clear quantum features. Namely, you have now a phase space quasi-probability distribution with negative values. But it's also because it's actually not so easy to prepare them. So it just makes a definition. And that's why I wanted to introduce the Binger function because in optomechanics, it is relevant to make experiments where you will prepare the mechanical motion into a state whose Binger function is negative. And that's, if you do that, that's a very nice experiment. And only very recently, experimentalists are doing this type of experiments. But also, the Binger function also allows me to define another class or divide states into two classes which are better related to this division between negative and positive, which are the so-called Gaussian states. I want to define these Gaussian states because they are very important. So Gaussian states are defined as states whose Binger function has a Gaussian form in its more general form. So Gaussian states are defined as states whose Binger function can be written in this very general form. And I think that tomorrow you have lectures about continuous, so quantum information with continuous variables. And I'm sure this concept of Gaussian states will either be introduced or assumed that you know, so it's useful that you see now. So a Gaussian state is defined in this way, so it has a Binger function which can be written in its more general form along in this way. I introduce now everything. This r vector contains this coordinates x and p of my Binger function. Hence s is a matrix. And sorry, or I write it like that. And this s matrix is a 2 per 2 matrix that contains the following entries. The mean value of x squared, the mean value of p squared, and the mean value of x p plus px. Wait a second. Please, one second, Tony. So then you see this Binger function is just defined by that. So given the mean value of x, given the mean value of p, given the mean value of x squared, p squared, and this, these are five real numbers. Given these five real numbers, I construct this Binger function, which is Gaussian. And this is the more general form. No, no, no, this is the mean value of x squared. I think I should check, but so I remove the mean value here. So no, no, no, the x is a parameter. It's the parameter, the argument of your function. So the mean value only appears here. So basically the Binger function is centered in phase space in the mean value. But the important message is Gaussian states fulfill two things. First, they are positive. By definition, the Binger function is positive. And second, it only is defined is a quantum state that is defined only by five real numbers, the mean value of x, the mean value of p, which is kind of boring, because if it's not 0, you just receive the coordinate axis such that they are 0. So basically there are only three real numbers that are relevant to define a Gaussian state, which are basically the value of its second moments, x squared, p squared, and xp plus px. So it's very nice. It's a quantum state that only depends on three real numbers. Non-Gaussian states will depend on potentially an infinite amount of real numbers, all the higher moments. So Gaussian states are kind of very simple in that sense. Now from the states I've introduced before, I can tell you which ones are Gaussian and which ones are not. So for instance, is a thermostat Gaussian? What do you think? Yes, thermostat is Gaussian. It only depends on you can now calculate the three numbers and you have the Gaussian state. What about Fox states? Are they Gaussian or not? Who says it's Gaussian? OK, so you are neither perfectly correct nor completely wrong, because the ground state, which is a Fox state, is actually Gaussian. Whereas the Fox states with n different from 0 are non-Gaussian. And furthermore, basically non-Gaussian and negative being their function is almost a synonymous. There are very few states which are non-Gaussian and positive. So most of the non-Gaussian states are also negative. So Fox states with n equal to 1, 2, 3, 4 are negative being their function non-Gaussian states. But the ground state 0 is Gaussian. Recall also that the ground state and that next question, uncoherent states, which are pure quantum states, are they Gaussian or non-Gaussian? Huh? Gaussian. You could already guess they are Gaussian because they only depend on two real numbers. The real part of alpha and the real part of alpha. So they are not so complicated. They are Gaussian. Then we didn't talk here about that, but there are also squeeze states where you take a coherent state and you reduce one fluctuation, one argument really, really tiny. And people say, oh, these are very nice quantum states. Squeeze states are also Gaussian. And also recall that the ground state is very special because the ground state is basically everything. The ground state is a Fox state with n equal to 0, is a thermal state of 0 temperature, and it is a coherent state of alpha 0. So ground state is both a thermal state, a Fox state, and a coherent state. So it is Gaussian. Good. OK, so I said by worse, Gaussian states only depend on three relevant real numbers plus two that depend where they are centered. And examples of Gaussian states are thermal, coherent, and squeezed. Good. All right. This was just a fast introduction to the states of the harmonic quantum oscillator in the context of discussing the mechanical mode. Now let me start the discussion about the cavity mode. I will use all of this notation. So the cavity mode in general, what we mean here is that we always have an electromagnetic field resonator. Let's assume for the drawing point of view to consider an optical one, which is basically the electromagnetic field that you can have in between two mirrors or in a confined space, and this dimension is L. And I assume then from these mirrors, these optical resonators in this cavity scenario, we define the so-called cavity axis, which I used to be X here. And then the procedure of how you should do this rigorously and it's done, what one should do is to solve the Maxwell equations inside the Maxwell equations with these boundary conditions. Maxwell equations with boundary conditions but in this position of space, there are perfect mirrors, which means perfectly conducting objects, namely that the transverse electric field on the surface is 0. Then you solve Maxwell equations in the presence of these boundaries, and then one can show that the Maxwell equations in the presence of these boundaries can be written as the same form of an eigenvalue type of equation, namely that there will be equations or solutions that they will evolve in the Fourier space in a very trivial way. They will be eigenvalues. They will be eigenfunctions of these equations, which means they just evolve with a single harmonic oscillator just oscillating. And we call each of these solutions modes, electromagnetic field modes. When we do that, when we solve Maxwell equations in free space, plane waves are already modes because a plane wave just emolds with a frequency of omega t. As soon as you put some boundaries, plane waves are not longer modes of the electromagnetic field. There will be other type of waves that are modes. So if you do these exercises here, you could do that, find the modes, and then once you have the electromagnetic field modes, you can quantize them very easily using canonical quantization. And then you could find something such as that the electric field operator will be precise, the transverse component of the electromagnetic field operator as a function of x can always be written as a sum over all the possible modes that exist in such a resonator, which typically you can write in this form. There is I epsilon, which is the polarization. And there is plus emission conjugate. This means the electric field operator, I can always write as a sum of mode indices. Here in this case, the polarization, the two potential transverse polarizations, and n, which labels the level in your discrete spectrum that you have in the presence of boundary conditions. Then there is the annihilation operation for the mode n. Then some mode function, which for two mirrors is kind of a sinus, like the modes of a rope with fixed boundary conditions that you always have a sinus, and then basically labels a number of nodes that you have in this mode. And some pre-factor. And this pre-factor is always h bar omega n divided the volume of the cavity. So here, basically, I use all this notation. So basically, and that's why I say always that these modes are just harmonic oscillators, because now these modes n fulfill the accommodation rules of harmonic oscillators. This kn is the wave number associated to the mode n, which has a wave number n pi over l, where the minimum wave number is 1. So it's n equal to 1. And an omega n is just the frequency associated to the mode. And b is the cavity volume, which is basically of the order of l times some area, which is somehow the area associated to these mirrors or the area associated to this beam in cylindrical coordinates. So two comments about that. First of all, the smaller the l, the smaller the cavity, the larger this value will be, so which means that for the same kind of excitations, the larger the electromagnetic field will be. And this is important for some experiments if you want small cavities to make sure that these fluctuations have very strong magnetic electric fields that couple maybe then with other systems. And second, there are in principle different modes in this cavity, different harmonic oscillators. So I used two minutes, and then we stop. So then what is important is there are many modes. And these modes you see, because they are confined in a space, they have different frequencies. And the difference in frequencies between two consecutive modes can be calculated as follows. So the difference in frequency between two consecutive modes, n plus 1 and minus the frequency of mode n, is just given by pi c over l. And this is called the so-called free spectral range of the electromagnetic field resonator. So this means that if the cavity is sufficiently small, these modes are very well separated in energies. And if they are very well separated in frequencies, then as soon as you put another system that is somehow coupled to the resonator, what will typically happen is that this other system will only excite one of these modes, not the other ones. So if the free spectral range is larger than any other frequency in the system, then typically what will matter is one mode. So the system will, or in our case in optomechanics, the mechanical motion will only couple to one mode of all these n. So basically, this allows me to consider the cavity as made by a single mode. This is, if you want, called the single mode approximation or the single mode assumption, you should make sure your system incites that it fulfills that, that it only couples to a single mode, which is easy, not so hard to fulfill, to be honest. But a single mode approximation, and then basically the cavity field operator as a function of x then can be just approximated by some particular mode, which we will call the cavity mode, which will have the frequency that is the one coupling to the other degrees of freedom we are interested in. It might have some polarization, the one you're coupling to plus the harmonic conjugate. And hence, now, the dynamics of the electromagnetic field mode in the absence of an interaction is just given by a nice harmonic oscillator, which I use now the label A. So that's then the idea. In optomechanics, now I introduce the two modes. The mechanical mode, which was very easy to define, just the mechanical motion of something, which is an harmonic oscillator, which has a frequency omega m, which I set is typically at the scale of megahertz. And I use the symbol B. Then there is the optical mode, or the electromagnetic field mode, which is typically you solve Maxwell equations with some boundaries. Then you find solutions of the electromagnetic field, which are oscillating with a single frequency. These solutions are very separated in energies. So I can always focus on one of those, one mode, which will be the relevant one. And I call this mode the cavity mode. And it has associated a frequency omega c, which will be, if it's optical, it's 10 to the 15. If it's microwave, it's 10 to the 10. And it will be just a single oscillator with frequency A. Now, after the break, we will build the third relevant term, which is how these two modes couple. Questions about this? What tends to 0? Yeah, L tends to 0, which means that then the frequencies are very well separated. And then if L would be so small, so small, so small, the ground state mode would have an energy pi c over L. And if you would make LL very small, this frequency will be so large, so large, so large that it's not optical anymore. And if it's not optical anymore, physically, the mirrors will not be mirrors anymore. It will be transparent, and there will be no cavity. But perfect, but a mirror, so a mirror has a mirror. So the properties of a material, so the properties of how the electromagnetic field interacts with a material depend very much on the frequency of this field. So a mirror is a mirror for visible light, but it's not a mirror for x-rays. If we would have a mirror for x-rays, if you make it even smaller, then it will not be a mirror for gamma rays. The point is there are no mirrors for a very broad range of frequencies, unfortunately. We don't have any material that is a mirror for a very broad range of frequencies, yeah. Did I answer? The field? Not the field, the fluctuations of the field. But you see two, OK, I know where you are going, but so the point is this is the constant in front of A. Then you would need to be able to prepare the state of the electromagnetic field in a non-bacuum state. And of course, if the cavity is very small, you might need a lot of energy to excite such a photo. But yeah, in principle, but of course, you have to be careful, this limit theoretically, it's not consistent with the assumptions of your theory. The assumptions of your theory are sites that you don't, you are not able, so you don't consider very highly energetic electromagnetic fields because if they are very highly energetic, they would excite electrons to speeds comparable to the speed of light. And then you should consider charges relativistically, which we are not doing in quantum optics. So also to be consistent with your theory, you should always put a cut off to the energy of your electromagnetic field. OK, and I think there is now a 30 minutes break, and then we continue. OK, thank you.