 Before we continue, there have been a couple of questions about zero point quantum fluctuations, so why I'm not including the one-half in the harmonic oscillators. I forgot to mention, of course, one should do that if objects are very close to each other, such as so that Casimir forces or other type of dispersion forces might be relevant, but here all the distance will be very large, so there will be no Casimir forces whatsoever that are relevant, so we can then just shift the Hamiltonian, the zero of energies in the Hamiltonian sides that we forget about these one-halves, okay. Right, so now let's discuss finally the interaction, the optomechanical interaction, so we have defined the two normal modes, one for the mechanical mode, one for the electromagnetic degree of freedom, now it comes the interaction, and again with our plot, we have this mirror, then this movable one, then there was this motion here which we call x, and then there is this cavity mode here, omega c, and the mechanical mode, omega m, and then there is now the length of the cavity L, that precisely because one end of the cavity can move, now depends on x in that setting, okay, depends on the position of the mechanical oscillator, so since the position, since the length of the cavity depends on the position, then this means that the resonance frequency of the resonator depends on the position of the mirror, and typically since these fluctuations will, or this motion will be very small in amplitude, you can tailor expand this function, and then you get the zero order term, plus a term that is linear in x, plus there will be terms of order squared or more, that to start we will neglect, okay, and this is of course this, there are very different ways to obtain the same result with different level of rigor, this is not the most rigorous way, but it gives a very nice physical intuition, so that's the mechanism, the resonance frequency of the resonator depends on the position of the mechanical degree of freedom, and by linearly expanding you get this dependence here, which is linear in x, so then basically the interaction is constructed, or the Hamiltonian then is always constructed in this way, now you say, oh the resonance frequency of my oscillator depends on the position of the mechanical degree of freedom, and in the sense by just doing that, that's a very fast way to derive this optomechanical interaction, okay, so that's again, you see suddenly these are, before I didn't put this dependence, so these were two independent not not coupled harmonic oscillators, now we're making that the resonance frequency of the cavity mode depends on x, suddenly you see there is an interaction here, okay, and then what you do is, since you know these fluctuations will be small, of the order of the zero point motion, you can expand these dependence here to first order, and then you just get this term, you get, now you get, you do a Taylor expansion, then you have this term, which is the resonance frequency of the resonator when the mechanical mode is at equilibrium, at zero, and the first term in the Taylor expansion give me this term that couples x, the position of the mechanical mode with a dagger a, the number of photons in the cavity, okay, and then I just redefine things in the following way, and now I now do the following, I now write x in terms of b plus b dagger, recall, and I, the constant in front I define minus, for convention, h, j zero, and this is the famous optomechanical Hamiltonian, where j zero is a frequency that has been defined as, this is the so-called single photon optomechanical coupling, so this is then the fundamental optomechanical Hamiltonian that models these dynamics of two mechanic, two modes, the electromagnetic field mode, and the mechanical mode that are coupled in this way, in this term, okay, and this Hamiltonian even though looks kind of very, it's very, you know, you can write very, you know, one line is actually not so simple, it's kind of a complicated Hamiltonian, mainly because the Hamiltonian is what is called non-quadratic, okay, so non-quadratic Hamiltonians are Hamiltonians for which there are terms that contain more than two creation and elation operators, okay, these are non-quadratic Hamiltonians, and you see this term here is non-quadratic, it contains three creation and elation operators, and Hamiltonians that are non-quadratic are also called non-Gaussian, why? Because if you would start from a Gaussian state of both the mechanical mode and the harmonic oscillator in the presence of that Hamiltonian by evolution under this Hamiltonian, you could create non-Gaussian states, the states which would have a negative being a function, okay, so this is actually great news because this Hamiltonian is kind of non-Gaussian and hence could lead to non-Gaussian physics, but there is a drawback, that the G-node, what is this G-node, what is the coupling strength, the coupling strength, what it is, physically you can also understand what it is, it's this number is telling me how much my resonance frequency changes if I move the mechanical mode by a zero point fluctuation amplitude, and I said before that the larger the mass, the smaller the zero point motion is, hence you can already expect that if you basically move very slow, very little the mechanical mode, you will not change so much the frequency of the cavity and that's the case, okay, so the larger the mass, the smaller G-node is, and hence even though this Hamiltonian is non-Gaussian and very interesting, it is very hard that this coupling is strong enough to be useful, okay, and today in Optomechanics, many experimentalists try to devise clever experimental settings where this G-node is enhanced as large as possible, but it's hard, okay, so just stay with that in this message, okay, so again I say, recall that this G-node then has two interpretations, first G-node tells me, says how much, how much the frequency, the resonance frequency, the resonance frequency of the cavity mode shifts, shifts if the mechanical mode, if the mechanical mode is displaced by x zero, as you can already guess, if you have a resonator you might measure this resonance frequency with a lot of precision, by maybe shining light and see how much light enters into the cavity or not and so on, now of course if your resolution would be such that you are able to see these tiny changes in resonance frequency by the motion of x zero then you would be able to detect the motion of fluctuations even at these small scales and this sometimes can be done, okay, this is one interpretation, then another one from this Hamiltonian, another interpretation from this Hamiltonian, there are two physical things that this Hamiltonian tells me, one, the one I said, the resonance frequency of the cavity shifted by the motion and the second one is that now you can also see that as you remember a force can define a force as the derivative of the Hamiltonian with respect to x, okay and if you take the mean value of this force, the force that the mechanical mode feels is, as mean value is from this Hamiltonian is given by that because I just take the derivative of with respect to x in this Hamiltonian and I have a force and you see these forces, the force, the mechanical mode, the mechanical mirror feels due to the presence of some photons in the cavity, okay, so this is the so-called radiation pressure, this is the, you know, light exerts pressure, makes force into mechanical objects because the scattering of a photon from a surface imparts momentum, that's why the comets, the tail of the comets always point away from the sun because the sun is making radiation pressure into the comet from the electromagnetic field radiation, that's the same type of physics, okay, so this is then, this is the so-called radiation pressure, then from here you can also understand a bit, give another meaning to J0, so you see if you have one photon in the cavity J0 divided by x0 is the force this photon makes into the mechanical mode, so if J0 would be larger for a single photon mode, for a single photon you would make more force into the object, if you make more force to the object you could actually move it away from its equilibrium point, namely you could excite phonons, okay, so you see that's why these terms here, B and Vdaga, they can create phonons and the rate at which they create phonons depends on A-A, so the more photons you have in the cavity the more phonons, the more displacement you generate, okay, so this is also encoded in this Hamiltonian, so in particular now you could ask how much displaced is the mechanical oscillator in the presence of some electromagnetic field state in the cavity, so what you can see is if you take this Hamiltonian now I write the mechanical mode again in its p-square plus x-square form and I add this term, I'm writing this Hamiltonian, now I write like this part I write like p-square plus x-square and then this interaction term I write like that, okay, I put x and divide x0, so it's the same, so this Hamiltonian then actually you can also understand it as this, as having now your mechanical mode being displaced from its equilibrium point by a length scale xd, so displaced, okay, so you put some, there is some light now in the cavity and this the mechanical mode is displaced from the equilibrium, xd, where xd right here is nothing else but xj0 x0 m over m, so I write the final form and I think this is even better, it gives a better expression because it tells me if I have one photon in the cavity I displace the equilibrium point of the mechanical mirror by a distance given by x0, the zero point motion times g0 over omega m, but that is the mechanical frequency and this actually tells me something, it tells me oh if g0, if the mechanical single photon coupling rate will be larger than omega m a single photon would displace the mechanical mode more than the zero point motion, which means that if you would be in the ground state where you have an harmonic, a nice wave packet, a single photon would displace the ground state wave function to a distance such that the new state is almost orthonormal to the other one, okay, and this would only happen if g0 is larger than omega m, okay, but of course that's hard, g0 is typically way smaller than omega m, but as I said before some experiments try to achieve this regime where g0 is comparable to omega m and so on, okay, so this is all these things you can extract from this Hamiltonian, okay, but again all this discussion has to do with the Hamiltonian which describes just the coherent dynamics, okay, and then I already said, okay, if g0 is comparable to omega m, that's interesting, but of course as always in quantum optics what is important is how large interaction is compared to the decoherence rate, to the dissipation rates of the two modes, this mechanical mode and this optical or electromagnetic field mode they are not decoupled from the environment, they couple to the environment so they can decay at some time scale, if you put a photon into the cavity it will not be there forever, if you put a, if you excite a phonon into the mechanical mode it will not be there forever, it will decay and what matters is then how these decay rates compare to the interaction strength, okay, so that's why now we right away move to the open optomechanics, okay, so the idea is how large is g0, in particular here what is relevant is not the absolute value but how does it compare with decoherence rates, two questions about that, this is the fundamental Hamiltonian, it looks very simple but it allows you in many situations to model very well the physics of a resonator mode coupled to a mechanical mode, in many, many different experimental scenarios, the microwave regime, in the optical regime and so on, okay, it's almost always boils down to this, good, now as I said before the system is open, so this degrees of freedom couple also to the environment and hence one needs to describe optomechanics as an open quantum system, so the idea is both modes, both modes are coupled to a bath and recall that a bath we typically use in physics as a way to say other degrees of freedom we don't control or we don't, we cannot, yeah, we cannot measure and so on, okay, so in our picture of the optomechanical system, okay, now what happens is that the optical mode can decay or can couple to the bath with a rate kappa, I will use kappa and the mechanical mode can also decay with a rate I will use gamma, okay, and this means optical mode couples to other degrees of freedom, for instance it could couple because the mirror is not super perfectly smooth there are some defects then a light a photon that is scattered is reflecting from the mirror suddenly reflects by the impurity away from the cavity and then instead of coming back to the cavity it goes away from the cavity so you lost it so you couple to other electromagnetic field degrees of freedom or the mirror is not perfect so it absorbs from time to time the photon and then the electromagnetic field couples to the charges in the surface that couple to other degrees of freedom and dissipates to the rest of the of the wall the mechanical mode the same these vibrations suddenly you know my excite other vibrational modes in this structure so if you have a mirror you are looking at the center of mass mode but maybe from time to time you also excite kind of internal vibrations or this mode then couples to the rest of the other acoustic modes in the material and so on so it can happen of course okay so let us focus then first on how to describe a lossy and mechanical mode so and the idea is it will be should sufficient to just treat it as a mechanical mode coupled to a thermal bath a thermal bath of temperature T with damping rate and this will come in a second how I define these things dumping rate gamma so the idea is now you have this harmonic oscillator which has is defined by a frequency h bar omega m and this harmonic oscillator is not isolated in the wall it couples to many other degrees of freedom that they have some temperature because the the the biome is in equilibrium at some temperature T and very generally I just say okay it couples to a bath of temperature T okay and the coupling rate it will be related to some so-called dumping rate that will come in a second gamma okay so then there these two new parameters appear the temperature of the bath T and the dumping rate gamma okay and in nature you can never be completely isolated you can never make dumping equal to zero it will it's amazing how well you can isolate these systems but not perfectly fundamentally you will always isolate for instance if the object has some temperature it means it's it has some temperature this massive massive object is emitting electromagnetic radiation through black-body radiation because it has some temperature emitter radiation every time I am in radiation I send momentum away from the object so I recall and since this happens stochastically I'm recalling in all directions so I also heat up the center of mass motion so there is always fundamentally a way to couple to the bath okay and we will parametrize it in this way in quantum optics what you can do maybe you will discuss that already in the lectures on open quantum systems you could have a mechanical mode so you have an harmonic oscillator and the standard model is you assume it is coupled to an infinite bath of harmonic oscillators you trace them out and you derive an effective master equation for instance like a so-called quantum Brownian master equation all the all the cal data legged model and so on so I will not discuss this how it's done I will just tell you the result so typically if you in the presence of an harmonic oscillator coupled to such a thermal bath and the usual conditions that the time evolution of the of the density matrix can be written as this it contains first the Schrodinger equation part of it the coherent evolution this just that's the evolution due to the Hamiltonian plus some terms that account for the dissipation I write like that okay so this is the coherent evolution and these terms will describe the dissipative evolution okay if I have a single harmonic oscillation again this Hamiltonian is just h bar omega m b data b if there will be no dissipation my the state of the mechanical mode would only evolve through the Schrodinger equation by at least Hamiltonian but now there is dissipation and then there is additional term here and this additional term can always be written in the following way so it contains in general two contributions when you couple to a thermal bath it contains a contribution of that type okay so it contains this term gamma n bar n plus one and then these operators b row bidaga minus one half b bidaga sorry bidaga b and this is an anti commutator so this is this has the form of what is called a limb blood form which guarantees that the evolution of the density matrix in the presence of this dynamical equation will send a density matrix to another density matrix so it will keep the trace and will keep the positivity of the density matrix so it then it has this form and it has this coefficient gamma n bar plus one and recall that this limb blood form always has the same structure it's always has some operator row it's daga minus one half of then you put the second operator in the first place and the first operator in the second okay it has this form and you should remember that and if you remember that then this the first operator on the left of the road tells you what the process is describing so this is telling me that this process is describing the fact that the number of excitations lowers so this is the term that describes the decay of energy from the mechanical mode into the bath okay and this process has a rate gamma the damping rate then n bar plus one where n by n by I define a second but is the phone is the mean number both answer mean number occupation for the frequency omega m and temperature t the second term is the term in which you can put energy from the bath into the oscillator and this goes with the rate like that and then they have the bidaga here and then these limb blood terms there are a bit of an ambiguity sometimes we write it like that sometimes you will put you will divide by two here and multiply by two so that you have a two here and a one and then you would have here two times gamma and two times gamma some people will define it as gamma so then there might be a factor of two always in the gammas that one has to be very very careful okay there is no clear agreement on how to do that but there are all always these two terms this term reduces the energy of the oscillator and this term increases the energy of the oscillator okay so this term describes this arrow going up the first term describes arrow going down and the point is that this n bar we defined before but I repeat is like that one over h bar omega m divided kbt where t is the temperature of the bath and then from this master equation okay you can now check in a second the following so from this master equation one could now calculate what is the time derivative of the number of excitations in the mechanical oscillator as a function of time due to this master equation technically this is nothing else but calculating the trace of rho square dot so rho dot times b dagger b okay and then you substitute rho dot by the master equation there and you do these traces okay which using commutation rules you can always end up expressing as mean values of b dagger b and then what you would get is actually the following you will get this equation if you do it as an exercise the time derivative of the number of phonons is basically given by that which again this is the n bar over omega m which this can then be solved very easily and then you have that the number of phonons as a function of time due to that master equation is basically given by that this is a very nice very nice equation this is the number of phonons I had initially okay first of all note that the t equal to 0 basically this n cancels with this one so I have I have this then at t going to infinity since gamma is positive I get that that's my final state it of course it gets it equilibrates with the bath but now depending on whether my initial number of phonons is larger than the mean number of phonons in the battle lower I either decay or I hit okay so there are always these two processes so for sure as a function of time so for way larger than one I end up with m bar so here there is the m bar and now there are two processes either if I start it started above m bar I just decay and if I start below I just hit up but I always end up okay so that's what you would expect from an harmonic oscillator couple to a bath and this dynamics go as the damping rate that's why we call gamma the damping rate because that's the rate at which energy decays or increases okay this comes from the master equation very nicely now here it comes finally something relevant which is not see that what is very important in this coupling to the bath is what is the ratio of h bar omega m divided kbt again that's the frequency of the oscillator and that's the temperature of your bath that's the temperature of your experiment then it is very useful enough in quantum optics to recall the ratio of h bar divided kb anyone so what is the value of h bar in as a unit what is the power 10 to the minus 34 and kb 20 23 so the ratio which is the power and 11 that's what you need to recall 10 to 11 or 10 to the minus 11 if it's like that because then it tells you the following if you have here a 10 to the minus 11 if I have one Kelvin okay then for omega m's larger than than 10 to the 11 this will be way larger than one okay and you see since is a 10 to the 11 it's a funny power because microwaves are 10 to the 10 and optical frequency at 10 to the 15 so here now there is a very important important thing which is that optical frequencies even at room temperature they will have a pre-factor this number will be way way way larger than one whereas microwave frequencies at optical at room temperatures this is smaller than one way smaller than one okay so why is it relevant well I just write it again so I can write this as h bar kb 2 pi over some natural frequency and this pre-factor is 5 minus 11 and this is f over t well first of all so this one okay I went a bit too fast so this ratio is correct but now we are talking about the mechanical mode and the mechanical mode I said before that typical frequencies for the mechanical mode out of the order of 10 to the 6 maybe even 10 to the 9 Hertz so this means that a mechanical oscillator at room temperature is of course not in the ground state as you would expect even yeah even at its highest frequencies but you might know that today quantum optics experiments can also be done in so what is the range of frequencies of temperatures okay so recall so basically if we put temperature there is here at 300 there is room temperature experiments okay these are experiments where you go to your colleagues and you enter into an optical lap and of course everything is a room temperature then you could use kind of cryostats which are cheap which this will be helium type of refrigerators where you are around 4 Kelvin and then you can go to the most expensive refrigerators which are called dilution refrigerators which in principle allow you to even go to 10 milli-kelvins okay so 10 to the minus 2 this is temperature in Kelvin and this is important to so experiments are either done at room temperature at around 4 Kelvin's or at 10 to the minus 2 or so okay 100 milli-kelms then you might be surprised that the first experiment that showed a mechanical oscillator in the ground state was done by just taking a very nice high oscillator frequency 10 to the 9 10 to the 10 and putting it in a dilution refrigerator and then it was automatically cooled to the ground state and they observed that that was done in the group of Andrew Cleland in Santa Barbara and in 2010 you can check that and moreover they coupled this mechanical mode to a qubit and they saw some nice ravi oscillations okay if you want then to cool a mechanical mode of lower frequency or you want to cool a mechanical mode at room temperature you need to do something else you have to remove this thermal excitation somehow and this will of course be done by use the coupling with the lasers and this will be the discussion this afternoon how to do laser cooling okay but please keep in mind these these numbers here which are relevant okay last comment about the mechanical mode is that in optomechanics to give a strength of how well the mechanical mode is isolated we talk about mechanical quality factors mechanical quality factor which is defined as omega m over gamma this is a dimension less parameter and recall that the mechanical quality factor you can already see from this equation it tells you basically how much energy can you put or remove from the system in a in a oscillating period because if in a period gamma times period would be of the order of q and how much energy changes in in in such a time is given by q okay anyways that's a definition and for a given frequency tells me what is a damping rate and ideally for optomechanics you want mechanical you want to devise mechanical modes which have q's as large as possible okay if you if q is not large it will be hard for you to do interesting stuff in optomechanics and these q's now are amazing so there are systems where these q's typically range from 10 to the 5 okay up to crazy systems where q's are of the other even 10 to the 12 okay which if you put numbers you would see that the phonon so a vibration 10 to the 12 there were some nice experiments in Caltech in the group of Oscar painter with very tiny structure where you have phonons that vibrate so in a such isolated way that this vibration if you put it in a length it would vibrate over kilometers even distances longer than what the photon survives in an optical fiber really amazing so yeah so you can now there are you check quality factors in optomechanics you will get these nice plots where they tell you different experiments of what is the q they have good okay so that's regarding the mechanical mode so that's how we will model a mechanical mode coupled to the environment by this dissipator then what about now the cavity mode which will be also open the important thing is that typically the cavity is open also on purpose because you make one of these mirrors not really perfect you want these mirrors to be sorry a bit transmittive so that they can they can transmit radiation because then this is used to actually put light in it because you drive here and then some of the light enters and also to read out the state inside the cavity and I put always these drawings but the this concept applies also to microwave cavities okay so on the coupling if you want here is given by the rate kappa she's called photon decay rate or so on many names but it's kappa the important thing as opposed to the mechanical mode is that for sure we will always be in a scenarios where then the phone on number the photon number occupation at the resonance frequency will be basically zero so the electromagnetic field mode in my experiment is always in the ground state this will be always a condition if it is not then you will not be able to do it will be everything more complicated is it easy or hard to achieve that condition well you see now from here so if my optical if my mode is optical I have a frequency of 10 to the 15 here so even at room temperature this is way fulfilled okay so this is always nice to recall that optical modes when when you switch off so if you put all this room in the dark so you really switch off all the light even though we are at room temperature actually optical modes are empty they are in the ground state in the quantum ground state so that's why we don't see light whereas if we would go if our eyes would be sensitive to microwave radiation even though we would be in the dark there will be you would see light in that sense okay so at room temperature but then this then means optical experiments require room temperature whereas microwave experiments microwave experiments require these temperatures the illusion refrigeration because if f is of the order of 10 to the 10 you need these of the order of 10 to the minus 10 to the minus 2 to make this very large so that the n-bar is smaller than 1 so that's very important require dilution and actually that's why all the experiments that have to do with microwave optics such as super conducting qubits microwave cavities or microwave optomechanics and so take place always inside these amazing dilution refrigerators that they cool down the environment to hundreds or less milli kelvins okay which is no problem they are very expensive but if you have the chance to have one in your laboratory then you can run experiments inside and the electromagnetic field radiation will be in vacuum what is amazing and we are not I think we are so used to it that we never think about it is that why all these quantum optics experiments are done in laboratories that you can visit and see there the experiments going on is because at room temperature the electromagnetic field modes are in vacuum are in the quantum regime okay so that's why you can open which is very very nice of course the other modes such as the mechanical motor room temperature will not be in the quantum regime but the sense of why we can use lasers to cool objects to the quantum regime is precisely this one as I will discuss later which is that electromagnetic field modes at room temperature are in the quantum regime are in the vacuum that's why you can actually cool things also room temperature to the quantum regime okay because we will discuss a bit more but cooling has to do with exchange of entropy not so much about exchange of temperature okay and in the language electromagnetic field modes at room temperature have very little entropy they have basically zero entropy they are in the vacuum okay so all of this comes from here that's why it's so important all right so this means that the cavity mode has zero n bar so the dissipator will be the same but n bar is zero and if n bar is zero this term disappears and only this term survives which makes sense because this term is the term in which you provide energy into the bath which you can do but of course this term should be zero because you cannot extract energy from a bath that is effectively at zero temperature okay it makes total sense so that's why one if one understands this you don't need to memorize this a structure it has to be like that the term that goes with be that I should be proportional to m bar the terms that is with be should be n bar plus one okay so hence the cavity mode when it is open it will always be described as the Hamiltonian part as before plus now only one term here kappa which describes the decay where this again is all right then in the open cavity there's also some very important aspect which is that this cavities this electromagnetic field resonators can be driven the fact that the mirror is open it also allows me to put energy inside the cavity and in the Hamiltonian this can be described by adding the following term to the Hamiltonian so driving the driving in the cavity the driving in the cavity now can be described as that you cannot a new term which are right like that okay this new term is a mission as it should be and it contains two terms there is this driving strength which basically is proportional to the power at which you are driving the cavity and the frequency here is the laser frequency or the driving frequency this term I put now by hand but can be derived very rigorously for those experts what happens is that this mirror by not being perfectly reflective it basically induces a tunneling between photons inside and photons outside okay and this tunneling term will be of the form a data a for the two modes and now one mode since it's driven it's basically classical so you can displace by a coherent state and then the operator disappears you only get a C number and typically describe this also in the interaction picture with that mode so that's why it's oscillating okay so that was it appears a time-dependent term we don't have time to justify this but just believe me that's the case all right so then that means that now finally I can describe the most general or or at least the standard setting in optomechanics okay which is you have the optomechanical system which typically in generally can be driven the resonator mode is driven and of course the modes decay so hence the total dynamics of the system can be described now in this beautiful and complete way so basically what we have is the total dynamics of the system which now this is the density matrix of the joint mechanical mode and electromagnetic normal mode is then described by by the following it has the Hamiltonian part plus the dissipative terms and the Hamiltonian parties okay the Hamiltonian part is just the optomechanical term plus the term describing the driving of the electromagnetic resonator which you can always put to zero but in principle could be there at a given laser frequency and driving strength plus the dissipators and the dissipators now are three terms is the dissipator of the cavity which only decays the decay of the mechanical mode plus the fact that you can put energy into the mechanical mode from the bath so this is a very nice model described such a general physical setting of a mechanical mode coupled to an optomechanical mode so again see what are the parameters that are relevant the frequency of the cavity mode the frequency of the mechanical mode the single photon optomechanical coupling the driving strength of your resonator the laser at the frequency of the driving the decay rate of the photo of the cavity mode the damping rate of the mechanical mode and basically the temperature of your bath okay given all these physical parameters now you can explore a lot of different physics and that's what people doing optomechanics but the beautiful thing is that this model theoretical model allows you to describe a huge variety of physical systems in the microwave in the optical all type of different mechanical modes in a sense all boils out boils down to this description okay so then as a quantum if you are a quantum optics theories as soon as you are given that you already know you can study a lot of things okay so what we will do after after lunch is basically to study a particular regime which is a regime in which I assume I really drive I'm really driving the cavity okay quite a lot so that I put many photons into the cavity mode and then I want to look at the description of the fluctuations above this mean field where I have a lot of cavity photons by doing that we will do a trick where we will see that this Hamiltonian effectively linearizes namely becomes quadratic and by becoming quadratic everything even simplifies more you can solve the end of the dynamics exactly and then we will focus how this linearized Hamiltonian in the presence of driving can be used to cool the mechanical mode to the quantum ground state okay that's the idea but but from here you could study many other regimes now you could do research oh what happens when G note is larger than gamma or G note is larger than kappa gamma and you know depending on the hierarchy of the physical parameters involved in this there are a lot of different type of physics that we could study of course should we have more more hours but my idea is after lunch to focus on the perhaps most important application namely that you can cool this mechanical mode to the ground state using these interactions so do you do you have questions you ask me some very good questions in the break maybe you want to ask them now so that we can discuss with all of you at the same time yes yeah good very good question because