 Okay, that's much better. So my lecture will be mainly a theoretical lecture. There will be some formulas, and I decided to have it really as a lecture. That means I am going to mainly work on the blackboard. Now there is this pillar here. So if you want to move in front to have a better view on the blackboard, then please do it now. There are many seats available in the front. Use your chance. And important, so if you don't see the blackboard now, then it's maybe better to change your seat now, because you will miss half of what I'm going to write. I will show some graphs and so on, which I don't have to draw then on the blackboard. Another important aspect of having a lecture is also to have questions. So I really ask you and invite you to ask questions, stop me, interrupt me and ask if I'm too fast or also too slow if you want to hear more or less. And in order to improve this, I bought some candies on the airport. So whoever asks a question will be honored, rewarded with a candy. Let's start. And the first few things I'm going to tell you I actually have on slides. This should be more or less a motivation for the field of the mechanics and a little bit of historical introduction. So if you look back in history and dig deep in the journals and search for neurons, micromechanical systems coupled to light, then we arrive as usually at Einstein. So here is a paper by Einstein, it's in German, from 1909 here and it was published in the Physikalische Zeitschrift at that time. So this paper summarizes a talk he was giving at the meeting of natural scientists and medics in Salzburg and it's about the development of our conception regarding the nature and constitution of radiation. So he was interested in finding out what is the fundamental theory of light. Four years earlier he published his paper on the photo effect where he postulated that light is somehow composed of particles and the momentum or the energy of this particle is connected to its frequency via h bar omega. And now he elaborated on this and wanted to know more. So what did people know about what we know now as the fundamental theory of radiation, quantum theory, quantum electrodynamics, what was known at that time was only the black body radiation law by Planck. So Einstein wanted to build on this foundation and know more about the nature of radiation. So he said assuming Planck's formula to be correct, what can we deduce about the constitution of radiation? And what he did is a Gedanken experiment as usually. So he imagined a box of temperature T filled with a gas of photons as we call it now with radiation and this radiation is a thermal equilibrium with temperature T so it's described by Planck's law. So this is black body radiation at temperature T. Now he imagines perfectly reflecting mirror being hung there which is free to move, maybe oscillating. So the box shifted and in addition to that gas of photons, there should be also an ideal gas which keeps the mirror in thermal equilibrium at temperature T. Now, I'm sorry, the box somehow shifted but Einstein then looked at was the, redraw that in order to have the arrows right, but he looked at was the reflection of photons of that mirror. So this mirror at some instant of time will be moving with velocity V in one direction and then there will be photons impinging on that mirror and being reflected of that and due to the Doppler shift, the reflected photons will have a different wave number K prime which is given by this formula. So this proportion and this factor here proportional to the velocity is simply due to that Doppler shift. Now, the photons being reflected of the mirror will transfer a momentum to this mirror which is the difference of the two wave vectors here, wave numbers of opposite sign and this momentum transfer is a radiation pressure force. So what we get is a radiation pressure force which has a component which is proportional to the velocity of the mirror. Now if you have a force which is proportional to the velocity then this acts as a friction. So the momentum of the mirror will be damped out and this friction coefficient you can work out if you put in all the factors of proportionality here, the stamping rate of this friction force is connected to the power of the photons of wave vector K divided by the restness of this mirror. So what Einstein found here is the Doppler cooling of the mirror via radiation pressure. Now he says if this is true, there are photons of wave vector K, all of them will contribute a radiation pressure force here. If this is true, the mirror will be damped and the thermal equilibrium, there is no thermal equilibrium so we will just suck out energy from the ideal gas heating the mirror and transfer it to the gas of photons. So how is thermal equilibrium assured in this case? So if both gases, the photon gas or the electromagnetic field is a temperature T described by Planck's law and if the ideal gas is a temperature T, nothing should happen. So what is wrong here? So he went on and argued, well, the radiation pressure here is applying maybe some damping but at the same time there will be also some fluctuations in this force. So this radiation pressure force also has a mean component here connected to the power and this power will be a fluctuating thing according to Planck's radiation law. So if there is a force which applies damping and there is a fluctuating part of that force, this fluctuation will cause heating. So one can calculate from this argument the average squared momentum which is transferred due to the radiation pressure fluctuations and for this he used Planck's law. And what he found is this, I don't go through the details here, this should just serve as a motivation. There is one part which looks like, which it scales with the density of the energy density of the electromagnetic field and this looks very similar to what you would expect from the momentum transferred from the ideal gas composed of particles. So this looks like particles moving at energy h bar nu and then there is a second term which scales like the density squared which you would expect from a classical theory of electromagnetism where you have interfering waves creating fluctuations in the radiation pressure force. So he pointed out that in addition to this thing which you would expect from a classical theory there is this other part coming out which looks like coming from localized particles. And he also points out that at low radiation, at low energy densities, when rho is small then this first part actually is the dominating one because this scales like flow squared. So Einstein in the year 2009 performed this as a gedunking experiment with this main intention being to figure out what is the true fundamental nature of the electromagnetic field. Now 100 years later we can say this can be done, not in that way but we can cool mirrors through the radiation, we can see the limiting, the limit of this cooling mechanism being given by quantum fluctuations of the radiation field. So these are things which have been achieved in the last years in the fields of optomechanics. That is a very early realization in theory at least of optomechanics. Let's make a big jump and go to the year 1980. I flash here another paper by Carlton Caves on the radiation pressure fluctuations in interferometers and I show you this paper here also because it has a very nice abstract. If you ever write a paper, this is how you should phrase an abstract. He says the interferometers now being developed to detect gravitational waves work by measuring small changes in the positions of free masses. There has been a controversy where the first sentence puts the context, the second sentence puts the problem. There has been a controversy where the quantum mechanical radiation pressure fluctuations disturbed this measurement and the third sentence gives the solution. These latter results, the controversy, they do. So what he was looking at is a gravitational wave detector. That's a big Michelson interferometer, light is sent along two orthogonal arms and the main aim is to measure the difference of the length of those arms to a mind-blowing precision, precise enough that if it happens that the gravitational wave runs through this detector, the space-time distortion changes the length to a degree which is measurable in this interferometer. That's the idea of a gravitational wave detector. Now why do radiation pressure fluctuations come in here? You all know that in an interferometer, the precision of the measurement of the relative length of these two arms here, the precision, grows with the power you inject into this interferometer. What you measure is a phase change. If the two arms are slightly unbalanced, you will see a phase change, the two beams get a relative phase change and you will see light coming out in this detector. Now the phase of the electromagnetic field you're measuring here has a certain quantum uncertainty and the phase change scales with the power you inject. So in principle, you can just put in more and more and more power, increase your signal relative to your measurement uncertainty which is given by the quantum fluctuations in the phase of the field here. So that sounds like a free lunch. You just have to use a lot of power in order to increase the measurement sensitivity relative to our measurement noise. Now there is a catch and this is pointed out by Carlton Caves and actually this paper maybe marks the end of a longer controversy which, as he pointed out, you can read in the literature. So this question whether there are quantum fluctuations disturbing this logic goes back to, started with Braginzky and then many people like Bill Unru and so on contribute to do that. Now Carlton Caves here gives I think a decisive answer to it. So what is the catch? The catch is if you increase the power of this laser driving the interferometer, at some point what happens is that the radiation pressure acting on those mirrors very much in the spirit of what Einstein argued, the radiation pressure fluctuations grow with power. The more power you use, the higher will be the radiation pressure fluctuations. So you will actually change this length you're trying to measure by your measurement apparatus. It's the measurement of back action which at some point limits your measurement sensitivity. So what happens is if you increase power you first increase your sensitivity but then there will be an optimal spot after which the measurement back action will spoil the sensitivity again. And this is what is called the standard quantum limit in gravitational wave detectors. And it is a realization of Heisenberg's microscope. You know this gedunking experiment of Heisenberg, he imagines an electron which is measured under in a microscope. You shine light on the electron, you collect the scattered light and you try to localize this electron in space. You want to see it in the microscope. So the spatial resolution of your microscope is of course connected to the wavelength you're using but it is also proportional or inverse proportional to the angle of the collection angle of your microscope. So the larger your collection angle is, the better is your resolution. But there is also a catch. The photons which are scattered off the electron will impart a momentum to the electron and this momentum transfer will have just the comp and recoil simply. The momentum transfer will have an uncertainty which scales also to the opening angle. The wider is your opening angle, your collection angle the less you know about the momentum transfer onto this electron. And you see there is a trade-off here and you will never go below Heisenberg's uncertainty. So the measurement-back action limits the sensitivity of this microscope and this is exactly the same thing as the standard quantum limit in the gravitational wave detector. By the way, the standard quantum limit in the gravitational wave detector has not been achieved. The standard quantum limit has not been achieved in any experiment also not in any other realization of the mechanics. What has been achieved in experiment is to see the measurement-back action on top of a thermal background. Reaching the standard quantum limit means reaching the sensitivity where the measurement-back action and the measurement noise, the short noise of light, contributes on equal parts and this is so far still elusive. I think this is one of the big goals in the field to see this. And gravitational wave detectors are designed to actually reach ultimately this sensitivity but they are not there yet. Now that's a second highlight of optomechanics. I want to point out here in this little historical review. And then as a third one, I would like to flash this paper by Roger Penrose and Dick Barmeister, published in the year 2003. So almost 15 years ago now. In retrospect, I think probably the authors themselves would not write this paper in the same way but I think it was a really visionary paper at that time and it triggered a lot of subsequent development, technical development and ultimately in the second order effect triggered a lot of other maybe even more interesting experiments than what they suggested. But what they were thinking about is can we perform an experiment of this sort? Can we set up, again, a Michelson interferometer and in one arm we put a little moving mirror? Now we take a single photon injected into this Michelson interferometer so the photon goes into a superposition between being in arm A and arm B. Now here it's just an empty cavity so the photon will run in and run out and if the photon is in arm A here it will enter this cavity and it will impart a momentum transfer to this mirror. Now this momentum transfer will displace the mirror, the photon will come back. So what we created is actually a superposition of the photon being in arm B and the mirror maybe in its ground state plus the photon in arm A and the mirror in some displaced state, say a coherent state due to the momentum transfer it received from the photon. So this creates an entangled state between the two systems and the mirror they argued is in macroscopic, relatively macroscopic thing. We know we can create superpositions of two photons but can we also create a superposition of a massive mechanical oscillator like this mirror? So this is motivated of course by the Gedanken experiment of Schrödinger with his cat where the idea is that you have a radioactive, at that time a radioactive particle decaying or not decaying so this is ultimately also a coherent process where the atom can be decayed or it is not decayed, we don't know. So the weights of these components will shift with time in the known manner but it goes to a superposition and now we somehow managed to create a mechanism which kills a cat if the atom decays or not. So we would go to the superposition of the atom decayed or the atom still not decayed so no decay and the cat alive plus the atom decayed and the cat dead. So the idea was now can we create a Schrödinger cat with this mechanical oscillator? And they went through the numbers and worked out the conditions for this to work and made some bold claims that this is going to work until Baumeister is sort of still pursuing this idea. Again I think it wouldn't be written in the same way now 15 years later but the main ideas of using light to create this sort of entanglement between light and atoms this works not in this way but we will see in different ways you can actually do this. You can create these sort of superpositions even with macroscopic objects. So this is the last historical flash I would like to show you to motivate this field of optomechanics and after these lectures you will hear you will be able to understand all of these three aspects I mentioned the cooling plus its quantum limitations the standard quantum limit of measurement and what it means regarding quantum effects in optomechanics and ultimately also something like optomechanical entanglement entanglement state between mechanical oscillators and light so that's a bit the outline of the lecture and that I would like to start as promised well it's really that no the lecture on the blackboard here so this is supposed to be an introduction to optomechanics please if you cannot read it too small size okay or too small you cannot read it oh wow try to improve my handwriting well you see you also hear my words okay so this is an introduction to optomechanics but what is the problem it's my handwriting it's better than before no okay so we should meet halfway somehow so the first part is on mechanical oscillators and what we will mostly be dealing with in this course is the deformation solids so imagine you have some piece of solid and mechanical oscillations will deform the solid in some way now the this deformations will be described by a displacement field which is a vector field so assigns to each point in the solid a displacement how much this particular infinitesimal volume is displaced by the vibrations in this in the solid from its equilibrium position there will be lectures on nano mechanics and I guess you will hear much more about that but what I can promise you is that it is possible to decompose this displacement field here into a sum over normal modes so these solids will have particular eigen modes of oscillation and these eigen modes have a particular mode function u m of r and x m of t is the amplitude of eigen mode n so the time dependence goes into the amplitude the spatial dependence is in the eigen mode u m of r for small displacements these amplitudes here will just fulfill the equations of a harmonic oscillator which is driven by whatever external force there is what goes into this equation of motion is also a parameter which is called the the effective mass of the eigen mode u m so omega m is the resonance frequency obviously is the damping rate m effective is the effective mass of eigen mode u m of r external or whatever external forces are acting on this object so one important number dimensionless number which we can assign to any particular eigen mode mechanical eigen mode of such a body is the quality factor which is the ratio of its resonance frequency omega m to its damping rate gamma m and we will be mainly dealing with high quality mechanical oscillators so a high quality means we are in the underdamped regime of of these modes so we can see many many oscillations of this mechanical oscillator within the damping the decay time the ring down time of of this eigen mode so if we are in the high Q limit maybe you know 10 to the 8 or so can be achieved so we make 10 to the 8 oscillations which before an amplitude dies out of these mechanical oscillators you might say well this is such a good oscillator let's forget about the the damping here but actually this is of course not allowed especially when we are interested in thinking about quantum mechanical effects performed with these sort of oscillators along with the damping which we have here by the fluctuation dissipation theorem there has to be also some fluctuation in the force which is acting on this mechanical oscillator ultimately from a microscopic point of view you can think of this damping as an effect of this eigen mode being not a true eigen mode of this system but this eigen mode is a pretty good resonance but it is still coupled to some environment maybe this body is you know some ported to some frame and this frame also has eigen modes and phonons and they you know talk to this body via the support so these eigen modes are damped exactly due to this coupling to the environment and then through the same coupling you will also have some random force acting on this mechanical oscillators and in particular in this external force there are always thermal or at least vacuum fluctuations which I call f t of t and in the following we will assume that this is actually that the dominant part of our noise so what about numbers so let me summarize this again we can think of a structure like like this drum mode here which was introduced by Jack Harris and this drum mode you can imagine have these sort of eigen modes so if we perform the analysis and really figure out the eigen modes of this system here we will find the combo frequencies associated with particular eigen modes and we have this displacement field now this is one example the sub suspended membrane here but there are many more examples like mirrors which are hung or suspended micro mirrors which are more in the spirit of what I have sketched on the board here we can have micro toroids with particular mechanical eigen modes we will or this sort of nano disk nano disk coupled to waveguides John will tell us much more about drums coupled to micro with circuits here you can have photonic crystals I will mention later on in my lecture you can even go to cold atoms being trapped in optical fields and talk about their mechanical oscillations and you see that these realizations of mechanical systems spend a huge range of effective masses going from the rum or even kilograms scale down to the scale of atoms on the order of sector grams so this should give you an impression of what what range of masses we can cover with this logic here is a scatter plot which I take from the review of modern physics by Marcus Aspelmaier Tobias Kittenberg and Florian Markwood from 2014 where we look at the mechanical quality factor and the mechanical frequency and you see that we have typically mechanical frequencies on the order of megahertz or tens of megahertz going up to the gigahertz regime and mechanical quality factors say below 10 to the 8 so this is from the year 2014 and since then this a new scatter block will probably have moved up quite a bit and there are experiments now demonstrating quality factors above 10 to the 8 so this is yes so a candidate for a candy one can come up with a microscopic derivation of this line width and also of course then for the fluctuations due to a remaining coupling of your eigen modes to the environment so you can do a microscopic theory this is not what I'm going to do here I will use this description more on a phenomenological level and this is the line which which is ultimately method measured and then along and this could be due to several processes I mean this could be due to the support of as I sketched here of this system but this could also be due to some sort of impurity of the material itself maybe there are two level fluctuators some electronic degrees of freedom which couples do these vibrations and also contribute to the line width so what one can come up with microscopic models giving an answer to what what the expected line width is and of course if you are clever and you know take into account maybe parts of the preparations outside then you can in your description improve the the eigen modes you're working with is a certain degree up to you I mean the cut between what is your system and what is the environment is up to you but here we will dump everything which we cannot control into into this damping rate and the fluctuations which are driving this. All right now the next step this is still valid also on a classical level here and now we will quickly move to the quantum description of the same system so from in the following we restrict to one resonance so there will be many there will be a comb of them in general for some structures there will be actually only one that really depends on the realization in our toy model we are going to develop here we are mainly now focusing on one particular mechanical mode and then you can look at the index M I was using as just indicating this is the mechanical part of what I'm going to talk about. In a quantum treatment we just impose on the amplitudes of all of these eigen modes a canonical commutator X M and P M the canonical momentum commutes to I H bar and there is an intrinsic length scale in a harmonic oscillator which is the zero point fluctuation it's H bar over 2 M effective omega M as you know from your basic course in quantum mechanics so this is the variance of your position coordinate in the quantum mechanical ground state and with this length scale we can introduce dimensionless parades and these I keynote by small so this should be a capital letter here and in them I will quickly now switch to this lower case letters and only use them essentially in the following so there is no danger of confusion but on this particular line so lower case capital and there is a factor of 2 so it's the position coordinates k to square root of 2 zero point fluctuations and a dimensionless momentum variable which is P M of 2 X zero point fluctuation over H bar times P M so the momentum is of course scaled to the characteristic momentum length scale here which is connected to the zero point fluctuation via H bar and these dimensionless operators fulfill that they commute to a set of I H bar so we go to dimensionless coordinates which is always convenient and you can introduce creation and annihilation operators that's one over square root of 2 X M plus I P M and the dagger the joint creation and annihilation operators of a harmonic oscillator the canonical commutator implies that they commute to one as you all know we will have an harmonic oscillator Hamiltonian which in the dimensionful variables is the kinetic energy and the potential energy of the harmonic oscillator we can represent it in terms of the dimensionless quantities then this is you can do this easily on your own this is just H bar Omega H bar Omega M half X M squared plus P M squared and we can convert this also in the language of creation and annihilation operators and this then this will be H bar Omega M V dagger B plus one half so this is a simple consequence of these definitions and things you are aware of from from your courses in quantum mechanics now the equations of motion which are implied by by this Hamiltonian so maybe to write here so we are working here in the Heisenberg picture so we track the time dependence of quantities like position and momentum would be I over H bar and then the commutator of H with the position now we can for example take this form of our Hamiltonian so there will be the part proportional to the potential energy X M squared which does not contribute in the in the commutator and then we have I Omega M half commutator of P M squared and X M and this is just Omega M X M at P M sorry P M and in the same way we derive P M dot minus Omega M X M in the dimension less representation and now you can say wow but this hammer this equations of motion do not correspond to what I have written down before where we also had damping and and maybe some fluctuating force associated with this damping of course not because this Hamiltonian describes only our eigen mode in order to also see the damping or the fluctuations what you would have to do is to add to this Hamiltonian the Hamiltonian of the environment and the Hamiltonian of the coupling of the environment to our particular egg mode and then to a proper theory which would lead ultimately to a damping showing up in these Heisenberg equations of motion this is possible and I suppose we will hear more about that in the lecture on nanomechanics I'm not going to do that I'm giving you the result of this and this is just what we had in the other equation what you would find is something like this so this is now put in by hand so this would follow from a microscopic theory password here is the Kandaira legged model which is a model giving rise to this and for a particular concrete realization you in principle have to do that from scratch every time you have a new system there will be a new environment which gives rise to damping so a microscopic theory really has to cover each system specifically still there are certain standard models which are applicable in a huge range of physical scenarios and the Kandaira legged model is one of them again this would give rise depending on which model you use give rise to such a Brownian motion like model where you have a viscous force acting on the mechanical oscillator and certain fluctuating thermal force acting on it now this f t I'm writing here is the essentially the thermal force I'm introducing before but now we go to dimensionless units so we scale also this force here to something which is almost dimensionless so that is scaled to x set the f over h bar that's essentially the characteristic momentum scale and it is convenient to divide also by the square root of the damping and the dimension of this lower case f of t is spirit of Hertz one over a spirit of seconds for the reason which will become clear in a second so now we want to know say a little bit about the properties of this fluctuating force here and I'm giving here only the behavior of this force in the high temperature limit and high temperature means here that I assume that kbt the thermal energy is larger than the quantum of the harmonic oscillator h bar omega m so this is what I mean by high temperature here and also the high q limit so the high mechanical quality factor limit for these two limits it is justified to use zero mean white noise model for these forces at least to a zero approximation of the damping so I'm spending a little bit of time here to properly or sufficiently at least introduce the description of open quantum systems which is maybe something you have heard and maybe there are many of you who did not hear about that so I think it's important for this meeting this is why I'm spending this time this zero mean white noise model for the fluctuating forces assumes that zero mean that's not a particularly surprising thing and then there is a particular correlator in time so we can ask when I say this force is fluctuating we can ask how much it is fluctuating so how does the value of this force at time t compared to its value at some other time t prime I symmetrize that expression and take the average the average is respect is with respect to that or with respect to the state of the environment now this thing is fluctuating in time and if this is a stationary process we expect that this depends only on the difference of these two times and in the white noise limit these forces fluctuate so strongly that actually they are proportional to a direct delta so they have something to do with each other only at the same time and if they are separated by an epsilon they are already uncorrelated the values of this f are strongly correlated only at equal times tt prime and there they take on a value which is 2n bar plus 1 there in the high temperature limit we can simply set kbt over h bar omega m so I assume again that in the lectures on nanomechanics you will hear more details about the derivation of these things but for our purpose this is the minimal model we are going to use to describe the thermal fluctuations acting on the mechanical oscillator and by assumption this is much larger than one in the high temperature so the n bar is the average formal number in thermal equilibrium and how large is that let's look at a little table here I plot here this number n bar the average thermal phono numbers versus frequency for particular temperatures so at room temperature we see that the megahertz up to Tiggerhertz oscillators we are dealing with in optomechanics at room temperature have a huge thermal occupation hundreds of thousands several million if we would like to go to the ground state by just cooling the environment let's say here 200 Kelvin or we can maybe also go a little bit lower to 20 milli Kelvin or so you really have to go to high frequency oscillators in the Tiggerhertz tens of Tiggerhertz range to go below one which would be the ground state so John is I'm sure going to show us some examples of where you can actually do that high frequency systems at low temperatures which are in the ground suggest by passive cooling but for typical optomechanical experiments working with light we are dealing with huge thermal occupation numbers in thermal equilibrium so it's also interesting and useful to take these equations of motion and rewrite them in terms of the creation and annihilation operators so if these are one here then we can write one is equivalent so the figures I will upload on the web page so you will be able to to get them from there so B dot would be one of our square root of two X m dot plus I P m dot and we just use our equations of motion here to rewrite this as minus I omega m B m so this comes from the part which is actually covered by by the Hamiltonian so this will give rise to this minus I omega m B m and then on top of that there will be the part due to damping gamma m B plus B dagger plus I square root of gamma m F T T and now there is yet another model for describing this damping this is the Brownian motion model where we have the viscous force which takes out which which just which is proportional to the velocity and we have a force acting on the momentum as you would expect for from a mechanical oscillator but sometimes it's more convenient to use these equations here and then for the high frequency oscillators so for high quality factors it is justified to drop the B dagger here in a in a rotating wave approximation which you are maybe familiar with from courses in electrodynamics and when working in the creation and annihilation operator language it is common to describe the forces as be in and we can read off here that be in of T is I F T T in terms of these we have the correlator in time of these fluctuating forces being given by by this expression here this is the same as I have written before yeah so that if you imagine that we take out the fast oscillation here at frequency omega m yeah so we could go we could define something like a be tilde operator which is e to the i omega m be so take let's go to a reference frame rotating with the oscillator then this guy here would in this language of the tilde operators oscillate at twice the the oscillation frequency omega m this would be a slowly varying operator and this would be so we can write this once now we introduce this be tilde and rewrite the whole equation for the tilde operators and then here we would have a 2i omega m t and this is a fast oscillating thing oscillating much faster than oscillating very fast on a scale of gamma m and then we can drop it this is true for high quality oscillators where omega m is much much larger than gamma m okay then essentially we can symmetrize the damping so if you take this equation here for for b and reconvert it to position and momentum what you would see that the momentum would still be damped by gamma but now only with gamma m half plus there will be some noise acting on the momentum but also the position in this model of damping would be damped in a symmetric way so that's just saying if you have a Brownian motion damping model where the oscillator feels a frictious force and a fluctuating force acting both on the momentum if you are if you have a high q oscillator this thing is oscillating so fast that both canonical variables x and p feel the same damping on average averaged over a few cycles and both are subject to fluctuating forces okay this is the physics behind this this opportunity okay so that was a question actually there was a question before you want yes so I claimed always that that this n bar here is the occupation in thermal equilibrium as an example here we can we can check that briefly so let's let's solve this equation here the solution to the equation of motion which we have here is e to the minus i omega m plus gamma m half t b of 0 plus square root of gamma m integral 0 to t t t prime e to the i omega m plus gamma m half t prime b of t prime so b 0 is the initial condition the solution to the homogeneous equation and then this integral here over b in is the solution to the inhomogeneous part we can quickly convince ourselves that this is a solution if we take the derivative we will get down this prefactor i omega plus gamma m half in front of everything which is b that reproduces the first two parts the homogeneous solution or you can save the index b here and then the inner derivative here in the brackets evaluates the integrand at time t then the exponentials here cancel and we get g m gamma square root of gamma m times b in as the inhomogeneity so this is indeed the solution to the equation of motion here now the occupation number would be calculated from from this quantity b dagger b average and this average is now with respect to the environment the initial state of the system now let's use our solution so this is minus minus omega m plus gamma m half t times i omega m plus gamma m half t so we get this prefactor here once conjugated because we have an b dagger and then there will be b the average of b dagger 0 v of 0 so that is the average with respect to the initial state and then there will be two time integrals similar prefactors in here minus i omega m plus gamma m half t prime e to the i omega m plus gamma m half t two prime and then the average of b dagger in t prime b in of t two prime so that is the average with respect to the environment plus something which i don't even bother to write down which for example depends on the correlator between the initial state and the bath the fluctuating forces and this is zero no correlations at close so we are left with two terms now for the correlator of the fluctuating force we can use what i have written before this is n bar plus one half delta t minus t prime so that allows us to to take the the integrals so from from here we also see that these frequencies drop out and we get minus gamma m t and this is the initial occupation number average of your occupation number n not this is b dagger 0 b 0 and then we can take one time integral using this delta function i'm sorry there is a factor gamma m half gamma m missing from the square root of gamma m squared so this is gamma m n bar plus one half and what remains is an integral over time t t e to the gamma m t so for equal arguments with the time here the frequencies also drop out because they have different signs and the gamma gammas add up now what is left is to evaluate these integrals and what we find is that the occupation number at time t is e to the minus gamma m t initial occupation number plus take the integral e to the minus gamma m t n bar sorry this should be n bar yeah this is actually true for bb dagger here the one half and then b dagger b i'll check it later let me write this this out first so this is n bar which which survives here so what we see is that the occupation number at time t will be damped out so the initial occupation number will be damped out and it will be replaced in the long time limit with n bar the average occupation number as i i promised yes it should be n plus n plus one yes so thanks yes and there will be this commutator between the noises bt b dagger in t prime delta t minus t prime which ensures or which implies then that b dagger b is delta correlated with n bar okay thanks want a candy okay good so if we would have initially started with an arbitrary level of the fluctuating force then this calculation would ultimately tie this arbitrary level this number which we could have left open in the beginning would have tied it ultimately to the thermal to the occupation number in thermal equilibrium so far i was talking about mechanical oscillators for optomechanics we also have to talk about optical resonators before we can talk about the coupling among mechanical oscillators and optical resonators a toy model for a optical resonator is a fabri puro resonator so most optomechanical experiments actually employ something physically very different but let's still talk about a fabri puro resonators first so we have two etalons separated by length l i think i even have a figure here and if we treat solve maximals equations with these boundary conditions then we find in principle the same thing as we found for the mechanical oscillators we will find a set of eigen modes and a fabri puro resonator exhibits eigen modes at integer multiples of some fundamental frequency which is the free spectral range so we will have this comp of resonances all separated by a free spectral let's call the free spectral range and this is essentially connected to the round trip time of the field inside this fabri puro resonator c over 2l times 2 pi we are talking about angular frequencies apart from these in principle on paper ultimately sharp resonances in reality we will always have a certain width of these resonances and these are at least due to the finite reflectivity of the end mirrors which gives rise to a power decay so to fields leaking out of that cavity at rate which is connected to the width of these resonances an important parameter in the game of optical resonators is the so-called finesse which is the ratio of the free spectral range over kappa and you can imagine that optical resonators become interesting especially in the regime of a high finesse where these resonances are much sharper than their separation so that means f should be large and just as we did in the case of the mechanical oscillators we will now based on this assumption that there are resolved resonances we will pick out one of these resonances and forget about all of the others so we will focus on one resonance frequency let's call this here omega c where the c reminds us that this is a frequency of a cavity so now we are interested in deriving a quantum description of this cavity mode and again I will just summarize the result and it will look very similar to what we arrived at for the mechanical oscillators so we will describe this particular cavity mode in quantum electrodynamics by a harmonic oscillator described by creation and annihilation operators commuting to one the electric field would be given by some mode function for this particular cavity resonance we're looking at times that's for convenience put here a factor of one over square root of two a plus a dagger so this would be the observable corresponding to the electric field connected to the creation and annihilation operators of our cavity mode this suggests that we look at what is called the amplitude and phase quadratures of the cavity mode so maybe let me note here again that this is the amplitude a single amplitude and wave function the mode function a single photon so by solving Maxwell's equations we find these eigen modes and they now come out of Maxwell's equations and then quantization means we replace amplitudes by creation and annihilation operators with proper normalization so due to the fact that the electric field would be given by a plus a dagger essentially we can define the amplitude quadrature xc and the phase quadrature the conucrate variable pc and these are canonical variables as we had it for the mechanical oscillator can repeat the game we played for the mechanical degree of freedom so this particular cavity mode of frequency omega c would have this Hamiltonian here which implies the equation of motion a dot is minus i omega c a so I did down once i over h bar some Heisenberg's equations motion and this is treating an ideal cavity without losses but let's do that for a second and see what comes out so a of t would be e to the minus i omega c ta of zero and with that we could evaluate for example what the electric field time t would be so that would be this mode function a of r inside the cavity and then we have here a e to the minus i omega c t 1 over square root of 2 plus a dagger e to the i omega c t and we can decompose this into xc cosine omega c t plus sin plus pc sin omega c t so in an electric field we will have in a field mode of frequency omega c we will have a component oscillating like cosine omega t and a component oscillating like sine omega t trying to just convert that and even we will see that you will have the so-called amplitude and phase quadrature appearing here and this should explain the the notions the terminology of amplitude and phase quadrature so xc describes the component of the electric field oscillating in phase with cosine omega t so that should be our reference the amplitude and pc describes the field oscillating pi half out of phase that's the phase quadrature okay so this is where where this terminology comes from so xc again is the amplitude or describes the amplitude of field oscillating like the cosine and pc describes the amplitude of the field oscillating the pi half shifted field the sine component these fields are not independent they are canonical conjugate variables okay so they fulfill the canonical accommodator and the canonical accommodator implies in particular Heisenberg's uncertainty so this is why we cannot have a field with an infinitely well-defined amplitude quadrature and phase quadrature at the same time which will give rise for example to the standard quantum limit I mentioned before when we look at an optimal mechanical system as an force sensor or a sensor for gravitational waves good so the last thing I want to write down is just the equation of motion including noise so we already argued that these cavity resonances will not be infinitely sharp there will be coupling to the environment in particular to the lossy mirrors of these two etalons and there will be electromagnetic modes outside photons are able to leave the cavity and tunnel to the field modes outside on the other hand also at least vacuum fluctuations from the outside will drive the cavity mode through these open mirrors and this can be modeled again in principle on a microscopic level very much in the spirit of the caldera legged model or the rotating wave approximation of that and I just state the result so the equation of motion would be i omega c plus kappa half a so there is an overall minus that is the the damping the field amplitude so a is always associated an observable it's not an observable but it's an operator associated with the amplitude of the field as I have explained will decay at kappa half kappa is a power decay and associated with that decay there will be vacuum or maybe also thermal noise fluctuations driving the system so now we have to imagine that one of at least one of these mirrors is lossy and what comes from outside is a in of t and this is as before a zero mean field adopting again the white noise model the correlator in time is delta t minus t prime so now I'm writing here a a dagger the commutator of these fields is also delta correlated and this is now valid at temperature zero zero if there was a non-zero temperature we would have some occupation number here in the correlator so let's name these two and if you take equations one for the mechanical oscillator and two for the optical resonator then these two equations will be the basis of the formalism of optomechanics what remains of course to be derived and this will be the topic of next lecture will be the the optomechanical interaction so how are light and mechanical oscillators actually today I was mainly focusing on the proper description of the open system dynamics of each of those two parts because my experience is that this is maybe not so familiar to many students in the beginning of their PhD and this troublesome and I am of course fully aware that for each of these parts I'm writing here one could add you know a separate lecture on deriving all of these things so what I told you was as compromised between getting done in one and a half hours with this description and still being reasonable self-contained are there any further questions yeah yeah yeah okay yes yes okay so I will hopefully be able to talk about the standard quantum limit in the third lecture that is my plan and I think the ultimate answer will come then but as a warning the notion of the standard quantum limit is very much non-standard so it's used in different communities in different aspects so when you forget about the business or the issue of measurement back action in in due to radiation pressure in optical interferometry then what you will find is that the short noise of your measurement is a quantum limit there is our quantum fluctuations of light and they will limit the sensitivity but they will this sensitivity will scale like one over power you're using and then in one community dealing with optical interferometry this is used as this is a termed the standard the standard quantum limit the sql the one over n scaling with the number of particles in this case photos when checked here or the the community of frequency metrologists using you know clouds of atoms to measure time they also have quantum noise in this case it's fluctuations of their of their spin and this strength of these quantum fluctuations also scales like one over the number of involved particles or one of a square depends on whether you look at the square or the up and this is called the standard quantum limit in in those communities in the community of gravitational wave detectors this is called the measurement noise and what is called the standard quantum limit is the point where they see the measurement back action coming up it will become clearer later okay but there are different notions of standard quantum limit and when you read a paper saying we measure below the standard quantum limit then this is this first notion this is not what what the gravitational wave people would would call such as well you can get a candy sorry absolutely yeah but actually on a deeper level one actually has to go below the wide noise model because the the Brownian motion model in the wide noise limit creates a non completely positive map on a deeper level so it's not in principle physically allowed dynamics you have to use a non-wide force to to fix this and get a mathematically sound description but in the high temperature high q limit this works pretty well and allows to understand things properly in particular systems one should really sit down and look at what is my noise really is it white what is its color for example the people especially when you look at like anything which which is broadband in any sense for example again people in gravitational wave detectors want to see gravitational waves in the audio band over tens or hundreds of hertz or in the range of 100 hertz or so in current detectors and they really want to know exactly what is their noise because whatever signal is is coming in is you know swamped by all kinds of noise sources and they have a microscopic model for you know a soup of noise and they're all non-white so non-white the white noise is a zero is a fine zero order approximation to experiments and then one really should sit down and do things properly for case to case yes we should stop thank you very much