 OK, so good morning. Welcome to the second lecture on my little introduction to optomechanics. Yesterday, I mainly talked about optical resonators and mechanical resonances in solids as separated objects. And I introduced to you essentially the description of this open quantum dynamics, where we include damping. And we have these additional forces, thermal forces acting on the mechanical oscillators at the typical frequencies in the megahertz or low gigahertz regime. And for the optical resonators, it's mainly vacuum fluctuations of the electromagnetic field at the high optical frequencies. If we are talking about microwave domain, as in John Truffle's work, then we should also take into account maybe some thermal occupation at the relevant gigahertz frequencies of these circuits. Now today, I'm finally going to talk about the coupling of these two systems. And this will be Part C, the Optomechanical Interaction. And thanks to the other speakers, I can be very quick on that, I guess. So in principle, the main mechanism is that the cavity frequency depends on the position of the oscillator, as we heard in Yaroslav's talks already. And I will give two examples how this is achieved. But let's start by just assuming this is the case, that Hamiltonian for the cavity still this, and this also has to be understood in some sort of Born-Oppenheimer approximation. We assume that this oscillation of the mechanical oscillator is slow on a relevant time scale of the cavity. If you think of the Fabry-Barreau resonator, it would be the roundtrip time of the photo. So the motion of the mechanical oscillator should be adiabatic on the frequency of the cavity. And then we can simply assume that there is this parametric dependence of the cavity frequency on the mechanical oscillator. And we can perform a Taylor expansion of that. So we assume that xm has some equilibrium position and we expand about this. And then we have here a term which depends on the derivative with respect to this position. And we have a dagger axm. So this is yet the real position of the oscillator. That means I'm talking here about what I introduced yesterday as the capital letter x. And we can remember now that we introduced the dimensionless coordinate, which is the real coordinate scaled to the zero point fluctuation. And then what we have here, you can write a dagger a times the dimensionless coordinate xm. And I introduced the single photon coupling strength, which is the derivative of the cavity frequency with respect to the amplitude of the oscillator times the zero point fluctuation. So that is a rate g0. We like to express things in rates in order to make them comparable. And this has the dimension of a rate. We have a rate per length times length. And that characterizes the strength of our optomechanical interaction. And actually it is common to introduce this with a minus here. So we absorb a minus on this side. So if we look at what is achievable in experiments, I again take this catapult from the review of modern physics. I cited yesterday. And here you have a bunch of experiments of indifferent regimes, microwave, photonic crystals, levitated nano-objects, micro-resonators, mirrors up to onto cold atoms, where on the y-axis here we have the g0. And in this plot, this is compared to the cavity decay rate kappa, which is typically again in the range of hundreds of kilohertz up to 2 gigahertz. And the single photon coupling g0 is more in the range of hertz up to tens of kilohertz. So if you take the ratio of these two scales, then you see that at least four years ago experiments were mainly in the regime below 10 to the minus 2. There were some exceptions here, points 28 and 29. And these are all cold atoms. So for cold atoms, we can really have a ratio of g0 over kappa. That means that a displacement on the scale of a zero point fluctuation shifts the cavity frequency by a full line. This is what g0 tells us. How much does the cavity frequency shift if the oscillator is displaced by one zero point fluctuation for atoms? This can be a huge effect on the scale of the line width of the cavity for normal systems. Four years ago, this was on the order of 10 to the minus 3 or so. It improved in recent years, but still the strong coupling effect, like in cold atoms, trapped inside cavities as Oriole explained it in the previous lecture, has not been achieved in these micromechanical systems. OK, so I want to talk briefly about two examples of how this g0 is calculated from first principles. The most trivial or easy case is, of course, the one where we have the trapezoidal resonator. And we already heard yesterday that the cavity frequency here is a multiple of the free spectral range. And the free spectral range scales like the inverse of the length of this trapezoidal resonator. And now if this is a movable mirror, then the length itself depends on the position. So we take your coordinate system, where a positive displacement causes a longer cavity. And the longer cavity will have a lower frequency. And then you can quickly calculate that omega c of xm will be omega c. And in first order Taylor expansion, it will be minus omega c. And then this 1 over l times xm. So the g0, in this simple case, is just the optical frequency and then the ratio of zero point fluctuation over the length. Here you could say, well, I'm happy because omega c is a really large frequency. But the zero point fluctuations on the order of femtometer or so for typical micro-mechanical experiments in the length on the order of several wavelengths make this g0 still comparatively small scale on the order of 100 hertz or maybe a kilohertz or so. And typical micro-mechanical experiments with movable end mirrors. It can be larger. And the experiments achieving largest coupling in the case of optical fields. Microwave systems, electromechanical systems are different games. So they are in a better shape in terms of g0. But for optical fields, the systems achieving highest couplings are optomechanical crystals. And these are structures like these ones. Yaroslav introduced these systems also in his talk yesterday. So the idea, as he pointed out already, is to take a nano-object and structure it in a periodic array. So you take a dielectric material and you cut holes in order to make some periodic array. And in this way, as you know, one can shape the bands for optical fields. So this is shown here. I don't know. I don't go into detail here. But it's a periodic array. You should expect bands. This is what you get. If you change the periodicity, you can even achieve localized fields. So you can build cavities out of these structures. And because also phonons are propagating here, the same periodicity will shape the phonons, phonon bands of these structures. And by the same tuning of the periodicity, you can also localize phonon fields. So without going into details on how these phonon and photon modes look like, which typically requires some finite element analysis, which I don't want to do here, we can derive a formula for how these two degrees of freedom, phonons and photons, in such a dielectric structured system, coupled to each other. And there is a formula which is used in this community and that has been derived several years ago in a way I don't really understand. So we sat down and re-derived it. And I want to show this little calculation to you. So what we assume is a dielectric material. This would be, for example, this bridge here. And I assume that this is lossless and non-dispersive as a start. And this material has a permittivity epsilon of r, which is distributed in space and takes on two values. So here we would have vacuum in one part and the permittivity of the material in the other part. So when r is inside this material, it would be epsilon 1. When r is outside this material, say in vacuum, it would be epsilon 2. And I would like to write this piecewise constant function here over space in the following form. I write it as epsilon 2 plus epsilon 2 minus epsilon 1 times theta of r, where theta of r is a step function. It's 1 inside the medium and it's 0 outside this medium. OK, now when we are talking about phonons, we have to think about how the displacement of this material, how the phonons change essentially the permittivity I'm writing here and how that, in turn, changes the energy of the electromagnetic field. So what we need is the energy of the electromagnetic field in its dependence on the permittivity. And then we think about how the permittivity depends on the phonons. So let's write down the Hamiltonian for an electromagnetic field inside a lossless, non-dispersive dielectric. So we have the integral over space, which is the integral is over the energy density of the electromagnetic field. So there is 1 over the permittivity times the electric displacement field plus the contribution from the magnetic part, which does not depend on the permittivity. Displacement field D is epsilon of r times e. How now does the mechanical displacement field of these structures affect the permittivity? And there are two mechanisms how this is affected. So you remember the mechanical displacement field I introduced yesterday that's essentially due to the amplitudes of all normal modes of this structure. Maybe ultimately, we will be interested in only one. Then this displacement field will be maybe dominantly given by the amplitude and the mode function by this particular phonon mode we're interested in. But in principle, there are many, many of them involving all those bands available in the material. So let's talk about the fundamental thing, the mechanical displacement field. And this affects the permittivity via two channels. The first one is called electrostriction. And that is the change of epsilon 1 with the displacement field. So you can imagine that this mechanical displacement field will induce a variation of the density of this material. So the material due to the oscillations will get more dense in one place and maybe less dense in another place. And due to these density fluctuations, the index of refraction will change. And that will change epsilon 1, of course. This is electrostriction. It's a bulk effect. This is going on inside the material. And I'm not going to talk about it. One can treat it in a very similar way as I will do for the second process. And this second process is the radiation. And this is a change of the surface with u of r. So we neglect maybe a change of the index of refraction of epsilon 1 inside the material. But we're interested in the change of the surface of this object. How does the oscillation displace the boundary between the volumina we're looking at here? This will be, well, I'm talking about a surface here, a surface effect. And then it's not surprising that for small systems approaching the wavelength, this effect of radiation pressure is dominant. So let's treat this one. If you know how to treat this one, then you can figure out how to treat the electrostriction. Also, that's not very difficult. Yes? Pardon? A small system approaching the wavelength, essentially, roughly. In principle, I mean, there is no sharp transition. One can calculate both effects. And as you reduce the dimension of these systems, you will just see that there is at some point a crossover. And there, this crossover exactly really depends on the geometry and so on. I cannot make a general, give a general ruling. OK, so what we are trying to calculate now is the variation of the permittivity, depending on the shift of the boundary in this volumina, depending on the shift of this theta function. And when we know how the theta function changes with the displacement field, we simply plug it in here and calculate this integral. So that could be our first guess how to proceed. So let's calculate the first order correction of epsilon of r with u of r. So we want to calculate the variation of epsilon, which I call here delta epsilon, with this change of the surface. So we essentially have to look at the variation of this theta here. And this would be the variation of epsilon 2. The constant disappears. So we have epsilon 2 minus epsilon 1. Yes? Of course, it's epsilon 1 minus epsilon 2. Thank you. Yes. So it's epsilon 1 minus epsilon 2. And then the variation of this step function here with the displacement would be the directional gradient of this step function with respect to the displacement. OK, maybe I'll write this down in better handwriting. So we take the directional derivative of this step function with respect to the mechanical displacement field. Now from the mathematical point of view, what I'm writing here is a horrible object. We have something like a 3D step function. And now we are taking the gradient of it in the direction of u of r. So this makes sense only in a distribution, but we have to plug it in this integral. So we know how to deal with it. In 1D, you know that the step function, if you are on the real line only, would be 0 and 1. So let's say if r is larger than 0 or if it's 1, if it's smaller than 0, it's 0. And then the derivative of the step function would be a delta function. Now in 3D, we still have to expect that this is the case here. And I know I will show how to evaluate this volume integral here. Essentially, this gradient of theta will turn the volume integral in a surface integral. And this volume element will turn into the surface element pointing in the normal direction. And then we have to take this color product with this displacement field here. So in principle, we just have to plug this in here. Now, of course, we have to be careful, because in principle, we have here 1 over the permittivity. And what we calculated is the variation of the permittivity. Now taking the inverse of that, we should look at this as a delta function. That's not a good idea. So we should actually directly calculate the variation of 1 over the permittivity. And this can be done. So I'll keep the energy here. So 1 over the permittivity, we can express as 1 over epsilon 2 plus 1 over epsilon 1 minus 1 over epsilon 2 times our 3D step function. And then the variation of this would be 1 over epsilon 1, 1 over epsilon 2. And then the same thing, u of r gradient in the direction of theta in the direction u. You understand why we have to do that? And this is a step function when we want to take the derivative. We get this delta function. This is a sound way of writing this down, taking the inverse directly of this is not a sound way. Now we could plug this in the expression for our field, for our energy of the electromagnetic field and expect that now we get the variation of h from calculating this integral. So I simply plug in the variation here. So that would be 1 over epsilon 1, 1 over epsilon 2 u r direction of the derivative of theta. And then there is the displacement field squared. And now the volume integral would be evaluated as I announced previously into a surface integral, which I write here as dv times u of r. So that is the projection of the displacement on the surface into the direction of the surface element times 1 over epsilon 1 minus 1 over epsilon 2 d r squared. So this is what we would get, but actually it's wrong. Because the problem here is that the electric displacement field has a step. It's a discontinuous field on the surface, at least some components of the dielectric field. So when evaluating this here, we are evaluating a delta function. And the integrand here is not continuous at the surface. So this is actually not the correct answer. We have to be a little bit more careful. So problem d parallel is discontinuous on surface. So you know that field amplitudes can have discontinuities if we have steps in the index of refraction. And for t parallel, this happens to be the case. Now what we have to do to resolve this problem is we can rewrite our Hamiltonian at least close to the surface. So maybe this integral, we can restrict already to a small shell around the surface. And in this small shell, we decompose the energy density in terms of continuous quantities. And this is the parallel component of e and the normal component of d. And now the variation would be just the variation of these permittivities. And that finally leads to the correct answer. So we get, again, this contribution here. The displacement field projected into the normal of the surface element. And then the variation of the permittivity we already calculated. That's epsilon 1 minus epsilon 2 e normal, a parallel. And then there is a minus 1 over epsilon 1 over epsilon 2 d normal. And this is the final formula. You wouldn't guess it, no? But it is used by Oscar Painter and essentially everybody producing these optomechanical crystals to evaluate the G-NOTs. Now imagine that this U involves only one particular relevant mode. Then this will be essentially some mode function um of r times the displacement, the amplitude. And this field amplitude squared here in normal order. If there is only one relevant mode here, then this will be a dagger a, maybe times some field mode, parallel squared, and the same for d. So you see that structurally what we get out of this formula is, again, something like G-NOT xm a dagger a. So what is done is you perform finite element analysis for these structures, identify the relevant modes, and then you plug the mode functions in here, evaluate these integrals, and identify your G-NOT. Apart from the coupling of this particular single mode of the cavity to the single relevant mode of the mechanical oscillator, you can also evaluate this for a multi-mode system, and then you get all kinds of cross-coupling between photon modes and maybe different mechanical modes and so on and so forth. This is all covered by this formula. You can apply it also to extended systems and get preoline optomechanics out of that. But we continue with single mode or two mode optomechanics, having one mechanical mode and one optical mode. But this should just give you an impression of how G-NOTs are calculated in these systems from scratch. Any question? Could we proceed and believe that our fundamental interaction is of this form G-NOT xm a dagger a, and look at equations of motion? So now I take what I derived yesterday. I have my cavity mode of frequency omega c, power decay rate kappa, so the amplitude decay is at minus kappa half. Then on top of that, there will be these vacuum fluctuations driving the cavity. I'm just writing down what I derived yesterday. And here we have i omega m plus gamma m half frequency of the oscillator, damping of the oscillator plus its driving field. And now we add the optomechanical interaction. So I set h bar to 0 here. So what we evaluate is G-NOT a dagger a from b plus b dagger a plus i G-NOT a dagger a, b plus b dagger b. Evaluate the commutator. We get G-NOT b plus b dagger a. And here we get G-NOT a dagger a. These are equations Jaroslav was writing down also the other day. How is this optomechanical interaction interpreted in that language? You can take this term here and actually absorb it here. We pull out the a, and then we see that due to this i G-NOT, the whole thing looks as a correction to the frequency. Well, this is the way we derived it. On the side of the mechanical oscillator, we see that the mechanical oscillator is subject to a force, which is proportional to the photon number a dagger a. So the photons being reflected off this mirror exert a push, give momentum transfer, and also in these much more complicated optomechanical crystals, the photons will displace the mechanical oscillator and will give a certain amplitude to its motion. Now, we could go on and analyze this system and see what happens. Let me remind you at this stage that of the properties of these driving forces, so we assume that this is vacuum fluctuations. So a in dagger have a two-time correlation proportional to the Dirac delta, and b in of t, b in of t prime dagger would be n bar plus 1 delta t minus t prime. So this is a thermal field. This is a vacuum field, which is the relevant physics for optomechanics in the optical domain and in the microwave domain. Again, we could also include a thermal occupation here for the light field. Both fields are otherwise free bosonic fields. So they have this commutator. So I'm writing this down just for completeness. We have seen this already yesterday. All other correlators like a squared or b squared or so are 0. b dagger b is proportional to n bar times the delta function, which is implied by this commutator. So what happens now in this system due to the optomechanical coupling? The answer is nothing because the cavity is empty. We wouldn't see anything because we don't get photons out of this cavity. What we have to do to see some physics at all is we have to drive the system. And this still has to be put into the equations. So now we want to look at the optomechanical system. So far, we assumed vacuum fluctuations driving this optomechanical system. And now we will put a C number component to this driving field, which is our laser. And this laser has a particular frequency. So the C number component is oscillating at the laser frequency. So this is essentially giving an average value to this driving field A in. And the average value I pull out and treat it separately. Separately, such that this A in stays a vacuum field. So what we do is we replace A in by A in plus alpha in e to the minus i omega LT. And alpha in is just a real number. It's not an operator anymore. And it is connected to the drive field amplitude. It is given by the square root of the power over h bar omega L. So it's essentially the square root of the number of photons impinging on this cavity per second. It's proportional to the square root of the photon flux. The more power we inject, the larger will be this alpha. Now this alpha has a frequency omega L, the laser frequency. And it will be convenient to take out this fast frequency by moving to a rotating frame, laser frequency omega L. That means we define a tilde operators for the cavity where we take out this large frequency omega L. And when we look at the time derivative, we would have i omega L a tilde plus e to the i omega LT dot. And the a dot we have here, we plug it in there. And what we find is that a tilde dot is i delta not minus kappa half a tilde plus square root of kappa A in plus i g not B plus B dagger A tilde. What happened is that the frequency of the optical cavity essentially shifted to the difference between the dry frequency omega L and omega C. So it's common to define the detuning as the difference of omega L minus omega C, the laser frequency minus the cavity frequency. So we have the cavity resonance here. Now we can choose the laser drive and the detuning here is the difference between those two with the convention that below resonance, the detuning is smaller than 0. And above resonance, the detuning is larger than 0. A remark here, the definition of the detuning is different in different communities. In atomic physics, in quantum optics, the logic is my cavity has a particular fixed frequency. And then as an experimenter, I can choose my laser and tune it with respect to the cavity. So it makes sense to define the detuning in this way such that the detuning is negative when I'm below the reference, which is the cavity frequency. And it is positive when I'm above. There are communities where this is used differently and where the laser is the reference. And maybe a cavity is tuned with respect to the laser. For example, in gravitational wave detectors community, the detuning would be defined exactly the opposite way. And the reason is that they use like megawatts lasers whose frequency is written in stone. I mean, they will not tune their megawatts lasers. This is given by the laser system. And then the cavities which are involved in the gravitational wave detector are tuned with respect to the laser frequency. So the convention is exactly in this opposite way. But we in quantum optics like to do it this way. So I immediately will drop the tilde. And remember that the A's I'm writing are now defined with respect to the, are in the rotating frame of the laser field. So the equations of motion which I wrote here. And of course, there will be now this very important term. I missed to write down square root of kappa alpha in which is the laser drive. Now this is a C number or real number in our convention. And we expect, of course, that this inhomogeneous term here in this differential equation will cause a certain C number component also for A. Or physically, we expect that some field will build up inside the cavity. And now we want to treat this mean field of the cavity. And because the cavity exerts a push on the mechanical oscillator, there will be also a mean displacement of the mechanical oscillator. So we want to treat the C number components, the mean fields of the cavity and the oscillator, which are now induced by the laser drive. So we expect mean components of A and B. So we define, and this is something also Yaroslav started to do yesterday, wrote down yesterday. We define fluctuations around this mean field. And the mean field I call alpha. And I define a fluctuation for B. And this mean field for B I call beta. Or in other words, I replace A by alpha plus a fluctuation. And I replace B by beta plus a fluctuation. I write here once, had to make it clear that only the fluctuations are now quantum mechanical degrees of freedom. And with these answers, which I can always do, I mean there is no approximation whatsoever made at this point, I go into my equations of motion and rewrite those. Now there will be a bunch of terms. I look at the variation in time of the fluctuation. I assume I managed to find time independent means, so time independent alpha. So the only time independence here is in the fluctuations. This is, of course, an assumption which I will have to justify later on. So when we insert the A dot from this side and replace everywhere A by alpha plus delta A, then we will get three sorts of terms. We will get terms which are only real numbers or C number components. We will get terms which are linear in fluctuations. And we will get terms which are quadratic fluctuations. And I'm already sorting all these contributions for you. So this is the C number contribution. Then there will be a linear contribution. And there will be the quadratic contribution. So zeroes order in fluctuations, first order in fluctuations, second order in fluctuations. This is for the cavity. We can do the same thing for the mechanical oscillator where we have the change of the fluctuation B dot. Just plug this in the equations of motion. I was writing down beta plus i g0 alpha squared minus i omega gm half delta beta plus i g0 alpha delta A dagger plus alpha star delta A plus squared of gamma B in plus i g0 delta A dagger delta A. Again, zeroes order in fluctuation, linear in fluctuations, second order in fluctuation. Now our main idea of introducing these means here, taking out the mean fields, is to deal with the driving field we have here due to our laser. And by a proper choice of alpha and beta, we could try to cancel the whole C number component which we have here. And that would remove the driving field from the game. So let's try to solve these equations by setting the first line here to zero. So what we do is we set the C number components to zero. And we get the equation i delta kappa half alpha plus i g0 beta plus beta star alpha plus period of kappa alpha in is zero. And i omega m plus gamma m half beta plus i g0 alpha squared is also zero. Two non-linear equations for alpha and beta. We can, for example, easily eliminate beta from the first equation by just solving here for beta in terms of alpha. You see that beta would be proportional to alpha squared. If we set it in here, we get a cubic term in alpha. So the whole thing is equivalent to a cubic equation in alpha. Or equivalently, we can also eliminate alpha and plug in here and get a cubic equation for beta. Now a cubic equation in principle has three solutions. What does that mean? This means that the system might exhibit by stability. So there will be more than one state, stable or unstable, for this means. So cubic equation can have three solutions. And I will not go into detail. You can analyze this further. It's not hard. I mean, check the conditions when a cubic equation has three solutions. This occurs for delta smaller than the square root of 3 over 4 minus square root of 3 over 4 kappa on the red side. There can be a regime where for certain drive strengths, the optimal mechanical system becomes by stable. The unique solution exists for delta larger than minus 3 over 4 kappa. So now we can look, for example, at the solutions of the cubic equation for beta versus the drive strength. And then for deltas, which are exactly got it wrong here, so this should be flipped. For deltas which are sufficiently retuned, there could be several solutions for a given drive field for three possible mechanical displacements. And the further analysis shows that this will be an unstable point and this will be a stable point. These are things you know from stability or bisability analysis in nonlinear systems, I suppose. For deltas which are sufficiently blue, larger than minus square root of 3 over 4 kappa, we always have a unique solution. And this is what we are going to assume in the following. So we assume now we are in a stable point here. Assume we have a unique solution. Yes. Because my goal is to derive an equation for the fluctuations of this system, which I erased already. Well, it's still here for the mechanical oscillator. I want to know how the fluctuations evolve around the mean fields. So when I start to drive the system, I inject the laser field, there will be an effect on the mean, the average dynamic of the system. And this average dynamic is described by these nonlinear equations. These equations are interesting enough. So there is a lot of interesting classical nonlinear physics in there. But let's assume we find a mean field for the cavity and the mean displacement for the mechanical oscillator. And then we are interested about the quantum fluctuations about this mean field. So firstly, we assume a unique solution. And we neglect the quadratic contribution in equations of motion. So let's write the resulting equations of motion down once more in order to the speed dagger. And now I drop the delta. So I replace the fluctuations again by the a's in order to save some symbols. And we simply keep in mind that we are talking about fluctuations about the mean field. Plus square root of kappa a in and b dot would be i omega m minus i omega m plus gamma half b plus i g a plus a dagger plus square root of gamma m b in. And the g I'm defining here is g naught times alpha. So look at here. We have alpha and alpha star. And let's assume that alpha is real. A phase we can absorb in a suitable way. So what we see is that g naught is multiplied by this mean field. Yaroslav was pointing this out the other day already. And the same thing happens in the equation of motion for a, as you can check. So the interaction is now given by g, which is multiplied by alpha. And note that alpha squared is proportional to the number of photons in the cavity. So this is typically a huge number. And it makes this part in the interaction, which we have here, much, much larger than this part, which scales like g naught. So for alpha squared much larger than 1, and values here are typically on the order of 10 to the 6 or so, depending on your drive, the g will be much larger than g naught, which is the justification for dropping the quadratic contributions in the equations of motion. So these terms are there, but they are comparatively small. And we will not be able to see them. If we could see them, that would be a big thing. So that is still something we are looking for in Optomechanics. I saw a poster yesterday where this contribution of the nonlinear terms and the widening of the optic cavity was treated and observed, which is ultimately due to the g naught contribution here. So seeing a unique sign of this g naught dynamics would be very interesting. But even this dynamics described by g is very useful and interesting, as I will show in the following. So these equations of motion result from an effective Hamiltonian is composed of the energies of the two systems. Now with the effective energy of the cavity in the frame rotating at the laser given by minus delta a dagger a plus g a dagger a plus a dagger b plus b dagger. So when you read papers in Optomechanics and you see in equation one, this is the Hamiltonian for Optomechanics, then the whole story of what I told you before is behind it. And then what you see in equation two after this Hamiltonian are these equations of motion where this part and this part follow from the Hamiltonian. But the damping and the noise, of course, doesn't in order to see this one has to treat a larger system and reduce it, blah, blah, blah. So you should now I think you are able to understand equation one and equation two and you know what is behind. You don't need to go this long path every time, but now it is important to go this path once at least. It is also important to remember that these a's and b's describe a fluctuation around a mean field. Another important thing to note with this Hamiltonian and these equations of motion is that now the Hamiltonian is quadratic in creation and annihilation operators, where it used to be cubic initially. What we did is we linearized the model going to a quadratic Hamiltonian and to linear equations of motion. Now you immediately can say, well, linear dynamics, OK, this is classical. And you're right. By Ehrenfest's theorem, we know that averages of these quantum equations of motion behave in the same way as a classical system. And a classical physicist, whatever comes out of these equations would not be surprised to see because this looks very much classical. But because we believe in quantum mechanics, we know that these are vacuum fluctuations here. And now we can track what happens in these dynamics with a macroscopic system, which we have here, driven in some way by vacuum fluctuations of the electromagnetic field. And this is the fun about optomechanics. So we have this handle. We can switch on this interaction. It's linear, OK? But we have this very clean vacuum fluctuations of the electromagnetic field, zero temperature, essentially, driving the mechanical system, macroscopic system. What comes out of this? Before we look into the dynamics of these equations of motion, we should, however, check whether the model makes sense at all because the fluctuations we are treating here might actually be unstable. It might be that a small initial fluctuation, maybe due to vacuum fluctuations or to some thermal fluctuations, will get amplified and explode. And we will not be able to describe a stable linear dynamics with this model. So we should think about the stability of our dynamics. And I will not go in much detail here. You can work this out easily on your own. Let's define a vector a, a vector of operators with this composed of a, a dagger, p, and p dagger. And rewrite our equations of motion as a dot is m times a. It's a linear system. So we can write it in this form and read off from these equations of motion here a matrix m. Plus there will be some forces a in. So this will be a vector of a ins and b ins and so on. And condition for stability, so this thing is stable if, and you understand this intuitively, the real part of the eigenvalues of m are smaller than 0. If the real part of the eigenvalues of m are smaller than 0, then these dynamics will be some sort of damping dynamics. If there will be at least one eigenvalue which has a positive real part, this will just blow up. This will amplify the initial, the external fluctuations and the dynamics will not be well defined for t getting large. And we should assume that if the amplitudes get larger and larger, our linear description will break down at some point. And we should go back to the non-linear dynamics of the system and maybe include things which are not at all in our model, like mechanical properties of the structure, et cetera, et cetera. Now for optomechanics, you can just read off this matrix m from the linear equations of motion, calculate eigenvalues, take the real part, and plot it in now here a plane where I plot the detuning on one axis. And the coupling strength, which note is proportional to the amplitude alpha. So let's write this down once more. g is g0 times alpha. And alpha was the square root of the photon flux. So when I plot something with respect to g, that means plotting against power. There is a bistable corner. Oh, aha. And you don't see the colors. Funny. So this plot in reality looks like this. So here is delta. Here is g. There is a bistable lobe for small enough delta. And then there is an unstable load for large g. And a big unstable area for blue detuning. So this is delta larger than 0. And this is delta smaller than 0. And this is a bistable, which g. So we are unstable almost everywhere on the blue side. We get unstable for large driving fields, large couplings here. And I plot this here in omega m in terms of the mechanical resonance frequency. And this is roughly about 1 or 1.1 or so. So when the coupling gets larger than the mechanical frequency, we are in trouble also on the red side. Let's still leave room for physics. Let's start first on the red side for sufficiently small g's. And this is what I'm going to treat now. I'm going to treat first the regime of weak coupling. So that would be this area here. Weak coupling is, I assume, that g is smaller than kappa. Maybe much smaller, but decently smaller is typically fine. So a factor of 3 is typically sufficient to make the formulas which I'm going to derive now apply applicable. So our aim now is to derive an effective equation of motion for the mechanical oscillator. In this regime of g much smaller than kappa without making an assumption on how omega m over delta compared to kappa with respect to kappa. So we want to treat the regime where kappa is maybe larger than omega m or delta or smaller. We just want to use this as a small parameter and get an effective equation of motion to first order in g over kappa in this regime. So what we do now is we move to a rotating frame with respect to the mechanical frequency and the detuning. So for example, for the cavity, we introduce again tilde operators now oscillating at the detuning. And for the b, we introduce tilde operators oscillating at the mechanical frequency. And for the tilde operators, the equation of motion for the cavity, for example, looks like this. So this is still exact. And now we integrate, we get a formal solution, which is e to the minus kappa of tau e to the i delta plus omega m t minus tau b t minus tau plus e to the minus i delta plus omega m t minus tau b dagger t minus tau plus square root of kappa plus the vacuum fluctuations from outside. So there will be also terms which depend on a naught, but they will damp out this kappa. And we're interested in times which are larger than kappa. And now you see that the field of the cavity will get some contribution from the mechanical oscillator proportional to g. Then there will be some response function, time integral. And we have this convolution here with the operators from the mechanical system. But now these b tilde's, they evolve because we are working in the rotating frame. They evolve at slow rate. So this is essentially b tilde of t plus something on the order itself scaling like g. And in order to get a equation for a tilde at t, which is correct in first order of g, we can neglect this thing and replace b tilde of t minus tau by b tilde. And the same we can do here. So this is assuming that the cavity decays fast on the scale of kappa. And b tilde is slowly evolving slowly on the same scale evolving at the order at the scale of g. So g over kappa is our small parameter here. This will become more evident once we take the integrals. So a tilde of t will be i g. And now I define a symbol eta plus b tilde e to the minus i delta plus omega m t plus eta minus b tilde dagger e to the minus i delta minus omega m t plus 2 over square root of kappa a in. And I call this a in tilde. I'll explain it in a second. And this coefficients eta plus minus are 1 over kappa half minus delta minus i delta plus minus omega m. Sorry, it's defined the other way around. Minus plus. So it's exactly the flip sign here. So these coefficients eta here, they come from taking this integral with respect to tau. And the integrand here is e to the minus kappa half. And then on this side, minus i delta plus omega m. When we integrate with respect to tau, we get this complex number in the denominator. And we get once kappa half minus i delta plus omega from here. And then this is, of course, delta minus omega. Sorry for this confusion here. So this is important. This is very important. This thing here, I'm a bit sloppy due to time running out. This is a filtered noise process. So whatever happens from here to here is hidden in the tilde. But it's still essentially a wide noise. And this solution, which is called the adiabatic solution for the cavity field, this solution we can plug in the equations of motion for the mechanical oscillator. The procedure we're applying here is adiabatic elimination of a fast variable. In this case, the fast variable is the cavity. It's fast because it's decaying as kappa, which we assumed is the largest rate in the problem if we are working in the rotating frame here. So we didn't assume anything about the size of delta and omega m. Adiabatic elimination, we will have maybe seen in the literature. And typically, it's also something which is confusing for beginning students. It should serve here as a hint of how to deal with it. So this solution, we can plug into the equation of motion for p tilde. And I have to stop at 12.30, no? So I will now jump over a few lines and give you the result of this plugging in of our adiabatic solution. You can check that finally you arrive at an equation of motion which you can rearrange in the following way. Delta omega m is a shift of the mechanical frequency, which is g squared and then the imaginary part of eta minus minus eta plus star. Eta are the coefficients I introduced before. Gamma opt is an optically-incuse shift of the mechanical damping. It's defined as 2g squared and the real part of the same quantity. And this we can write as the difference of two rates, gamma minus and gamma plus, with obvious definitions. And these gamma minus and gamma plus rates show up here in the noise, which is radiation pressure noise on the mechanical oscillator. And these two noise processes here refer to vacuum fluctuations at side bands. So a in plus minus of t, a in plus minus of t prime are white noise processes. And they still correspond to vacuum noise. So this is the final result of the final equation for weak coupling optomechanics. That's the equation of motion of the mechanical oscillator dressed by the cavity field for arbitrary detunings, arbitrary mechanical frequencies in the regime where kappa is larger than g. And it still contains a lot of interesting physics, which I'm afraid I have to teach you tomorrow. It contains, in particular, sideband cooling effects. It contains coherent state swaps. It contains coherent entanglement. It contains the standard quantum limit. So there is a lot of physics inside. And this is finally the subject of tomorrow's lecture. Thank you very much. Questions?