 What we were starting with is we looked in the first lecture at what does it mean to talk about a single cavity mode? What does it mean to talk about a single mechanical mode? We understand the logic behind that that we always are dealing with in principle structures exhibiting many of these resonances and we deliberately assuming a high finesse for the cavity, assuming a high Q for the mechanical oscillator, we deliberately constrain ourselves to talk about single modes in both systems. We looked at the basic equations of motion for these systems including the open system dynamics, damping as well as the necessary fluctuating forces acting on each of these degrees of freedom. In the second lecture, we then coupled those two things together. We saw that the fundamental coupling in typical scenarios is nonlinear meaning the Hamiltonian is cubic. There is a quadratic term in the equations of motion and we linearized this nonlinear dynamics around a certain mean field which we induce by driving the cavity and this linearization led us to these final equations of motion which I wrote here already in a rotating frame at the detuning of the laser from the cavity resonance and in the rotating frame with the mechanical frequency omega m. So now we in principle are now boiled everything down to two harmonic oscillators. They are coupled to each other in a way which you can look at as a spring so you can say wow this is a boring system. But again let me emphasize this is a degree of freedom of a macroscopic piece of a solid of one of the resonances maybe the center of mass motion of a levitated sphere or of a gram scale mirror in a gravitational wave detector or it could also be an atom trapped inside a cavity. So this as describes our macroscopic system and it is coupled to a mode of the electromagnetic field in a way which we can tune remember that this linearized coupling g is some small fundamental coupling g0 times the mean number on the mean amplitude inside the cavity so alpha squared is the number of photons in the cavity. So we can increase this almost arbitrarily. The cavity is driven by vacuum fluctuations of the electromagnetic field and this is a channel which is firstly this is an interesting channel because it is essentially a temperature zero in the optical domain and the other interesting aspect is that this is a channel which we have access to. We can measure it there is a output to this cavity and this output we can measure. This is very different from the typical field we are talking about here. This is a thermal field these are whatever degrees of freedom there are in this solid body. I mean this is a field we don't have access to. This is just driving your mechanical oscillator with noise but what goes out from the mechanical the information which goes out from our mechanical degree of freedom into this bath of harmonic oscillators or whatever the environment is is inaccessible to us we cannot measure that. This is a channel which we can control and measure and that makes this system interesting now this is a macroscopic degree of freedom and that is a quantum channel which we can manipulate we can measure and if we increase this coupling g here sufficiently as we will see in today's lecture then this mechanical oscillator starts to be coupled more strongly to this well controlled zero temperature field of the electromagnetic mode then to its thermal bath and then we can do things like cooling this to the ground state we can measure it to a precision we can measure the mechanical oscillator via performing measurement on light to a degree where we see the measurement back action in the mechanical oscillator reaching maybe the standard quantum limit we can tune this interaction also in a way where we can create very controlled entangled states between the mechanical oscillator and light which is leaving the cavity I'm not talking about the intra cavity light leaving the cavity in this well controlled bath. So this is all hidden in this innocent looking linear equations of motion for these two oscillators and in order to see all this it's most convenient to make yet another approximate approximation and go to what I call here the weak coupling optomechanics so we assume that this optomechanics coupling is a lot smaller than the cavity line widths and this is a regime where the cavity mode here is a fast degree of freedom which is heavily damped out and if you have a heavily damped oscillator subject to a certain force which is here determined by the mechanical oscillator then the cavity field will adiabatically follow this force no it's like taking a swing and moving it around it will immediately respond to a slowly changing force and this is what is contained in the adiabatic solution which we derived yesterday so there the cavity field at time t is identified as something which is in some sense proportional to the state of the mechanical oscillator to this amplitude of the mechanical oscillator b and b dagger at time t so it's just directly proportional to the applied force this is the meaning of this adiabatic solution on top of this there will be noise yeah vacuum noise of the electromagnetic field now we take this adiabatic solution and insert it into the equation of motion for the mechanical oscillator which I wrote down here and you will see that when we insert a here and a dagger of course there then there will be terms which will be proportional to b if we insert a dagger here then this b dagger will turn into an b so there will be another term which is proportional to b and whatever is proportional to b we can move here and whatever there will be a complex number multiplying this b the real part of which we can interpret as a shift of the damping the imaginary part of this complex number we will interpret as a shift of the frequency of course when we insert a and a dagger that will be also b duggers and showing up here but they will be associated with an oscillation frequency which goes twice omega m you can see this here so there is an omega m in the exponent here and there and exactly for the b duggers this adds up and these are terms which I dump here and I neglect in a rotating wave approximation so these are fast oscillating terms which will average out in the dynamics but they're included already here the shift in the frequency and the shift in the damping and these are related now to these coefficients eta which I introduced already the other day so this is just a summary what what I already taught you yesterday the frequency shift is associated as I emphasized to the imaginary part of a certain complex number and the optical shift of the damping is associated to the real part of this complex number and this optical damping we can obviously decompose into two contribution which I call gamma minus minus gamma plus and they show up also in the noise and this is now radiation pressure fluctuations driving the mechanical oscillator and I decompose that there is a certain technical step involved here which I jump over to save time but I promise you that this a in will be decomposed into two con contribution a in minus and a in plus which are independent white noise processes both physically associated to radiation pressure and there is this a dagger notably and this comes from the fact that I have to plug in here a dagger from the cavity and a dagger from the cavity will have an a dagger from the driving field okay and this will play an important role in in what comes now so we can look at the dependence of the optical induced damping and the optical induced frequency shift versus the de-tuning so let's keep all other constants fixed and vary the de-tuning and I plot here the de-tuning in scales of the mechanical frequency this is in the regime where we are so called sideband resolved that means the ratio of kappa over omega is small and then I also scale the the optical damping to four cheese squared over kappa and then you will see that there are two peaks on the red sideband we have a positive peak at this lower sideband where the optical induced damping takes this value of four cheese squared over kappa and the line widths shift here in terms of cheese squared over kappa which is a unit which supposedly is much smaller than omega m otherwise we would be in trouble with stability this also takes on large values around the sideband but this we should compare relative to the scale of omega m so in relative terms this is typically on the order of maybe tens of percent of the mechanical line widths for for high frequency oscillators it could be a much more important effect for low frequency oscillators like if we have a pendulum on the Hertz level then this could be a really a dramatic effect making turning a low frequency oscillator into a high frequency oscillator or making it even unstable giving it the negative frequency but for high frequency mechanical oscillators this typically is not such an important effect various the shift of the line widths here gamma optically is a huge huge effect so the intrinsic line widths of the mechanical oscillators with high q values we're talking about here think of the example of the megahertz oscillator with a q of 10 to the 6 or even higher 10 to the 7 10 to the 8 we are in the Hertz or subhertz regime for the intrinsic line widths and now we are giving this thing an optical induced line widths on the order of four cheese squared over kappa which can easily be several hundred kilohertz or even yeah so can easily easily be in the in the kilohertz regime for for this scale that was talking about so this can be an appreciable fraction of the mechanical line itself itself yeah if we are on the on the blue side on the red sideband on the blue side when we see the opposite effect we see that the optical induced damping is actually negative and this is the explanation for the instability we have seen well i have seen you don't see this here but i promised you that now this plot looks like they're unstable here and here and the instability on this side of the tuning versus g so this is this plot here this instability is due to the fact that on the on the blue side when the optical induced damping is negative and the whole system gets unstable but let's proceed slowly and treat now each regime one after the other and we start with the red detuning we have any questions so far so this would now be subject subsection f the red detuning where we are talking about cooling state swap so one thing we can ask for in the regime where delta is smaller than zero and in the following i also want to focus actually on the regime where in particular delta is tuned to the lower mechanical sideband and we are sideband resolved this is the most interesting regime so let's look at this right away we can look at the steady state of the mechanical system so what we do is we take this equation of motion here and we solve it and calculate the average of b dagger b in the long time limit and i will not go into detail of this calculation because we already did essentially the same calculation i remind you that in the first lecture i where we didn't have these terms here this is radiation pressure noise i integrated this equation of motion for a free mechanical oscillator subject to thermal fluctuations so look up how this calculation worked and we use the solution of this equation to calculate the average number of phonons in the long time limit what we have to do now is essentially the same thing but the system is now subject to three forces okay but you can do this on your own important is that when we now look at normal ordered quantities like b dagger b all these noise contributions here so the first one will be also evaluated as a or give rise to a normal order contribution b in dagger b in which has a certain level of thermal noise this is vacuum noise so a dagger in a in for this noise for noise process does not contribute here it's zero it's vacuum but this one in the normally ordered expression will give rise to an anti-normally ordered expression a dagger in a in and this will be non-zero even if we are a temperature zero okay so technically the step for the solution to this quantity are the steps are the same as what you have seen you just have to pay attention to what uh processes contributing with contributing in which sense and I directly write the result here so this is the the term proportional coming from b dagger in b in and this is the term coming from the a in plus a in plus dagger so it's a the weighted sum of two occupation numbers n bar for the thermal background and one from the one unit of of radiation pressure fluctuations and then we have these prefactors here which we already know and now looking at at this table here in the regime we are focusing on we see that gamma minus here is essentially equal to gamma an optical and this is four g squared over kappa so we are sitting on this on this left peak here and this if we realize that the intrinsic line with this really a small scale here will be much much larger than the the natural line with gamma m so gamma optical will be the by far the dominating thing as compared to gamma this will make this a small term overall various gamma plus turns out to be four g squared over kappa kappa over two omega m four omega m squared and if you take so these are easy things you can estimate from the expressions I gave to you before then what we see is that the effective occupation number when driving on the on the red sideband would be gamma m n bar over four g squared so essentially one over this optical damping which is four g squared kappa so we have here four g squared kappa and then the numerator we also have gamma m n bar plus here we have the ratio of gamma plus to gamma optical gamma plus is four g squared over kappa so that's essentially the optical damping times this per assumption small term kappa over four omega m squared we can rewrite this quantity on the first term we can identify with the thermal cooperativity so we see that the remaining contribution of the thermal bath is suppressed if the cooperativity is much larger than one so when you crank up the coupling large enough such that the cooperativity gets larger than one the contribution of the thermal baths will be essentially suppressed and we are left with this second term here which is the due to radiation pressure noise so these are vacuum fluctuations of light driving the mechanical oscillator whereas these are thermal fluctuations due to the phonon baths of the mechanical oscillator suppressed by the cooling so let me point out that the meaning of the large cooperativity here is just that the optical damping which is four g squared over kappa is larger than gamma n bar which is the thermal decoherence rate so we cool the system through our optical channel faster than the system the mechanical oscillator is heated from its thermal environment so saying the cooperativity is larger than one is the same statement if we are in the red sideband if we are sideband resolved so all these assumptions of course go now into into this formula so there are many ways of explaining the mechanism of the sideband cooling and and I will not delve too much into it what happens when we drive the optical mechanical system with the laser is we create sideband photons now we have we have these fundamental processes where we take out or add a phonon to the mechanical oscillator if we add a phonon to the mechanical oscillator this energy is coming from the laser and the scattered photon will go to a lower sideband so we remove a little bit of energy and this adding a phonon is a heating process on the other hand if we take out a phonon we go to the upper sideband with respect to the laser and this will be the cooling process and when we detune by the mechanical line width from the cavity then we resonantly enhance this process of scattering to the upper sideband and suppress the scattering to the to the lower sideband and this on the one hand cools but there is necessarily this remaining heating and this is essentially proportional to this rate gamma plus here and this is what we see in our balance of the of the occupation number so this is all I want to say about the cooling maybe one one last remark we could now say wow this is wonderful we can prepare the mechanical oscillator in its ground state and have a very coherent system because apparently it's cold but there is a catch for the hot mechanical oscillator the thermal decoherence rate just to point that out is again gamma m times n bar now does this decoherence rate improve when we do sideband cooling using the laser now we could hope that we diminish this make it as small as possible maybe one so do we gain in the thermal decoherence rate when doing sideband cooling unfortunately no so the cooling works or is based by broadening the mechanical resonance so from gamma m we go to an effective gamma which is essentially given by the optical cooling which is much much larger than gamma m various the effective occupation number so the n bar the relevant number of occupation goes to an effective occupation number as we have seen so let's forget about the small heating term which is essentially 1 over c which no it's just gamma n bar over the optical cooling as i explained and now if we ask the question how much the thermal decoherence rate change so we take the effective line width times the effective occupation number well the gamma optical drops out and we are just left with the original decoherence rate yeah so while the effective occupation number of our effective bath is small our mechanical line broadened so overall we don't gain in this metric still i think it's a it's a very it was a very fantastic achievement in the experiments to go to the to the ground state and cool these mechanical systems to their quantum mechanical ground state but having achieved the regime of strong cooperativity which is witnessed by cooling to the ground state opens up other interesting possibilities and instead of going to the stationary state or looking at the stationary state we could also look at the time dependent dynamics so let's take our effective equation of motion for the mechanical oscillator and strip off all small terms so we are still looking at the lower sideband delta being equal to minus omega m and a resolved sideband regime where omega m is larger than kappa and in particular also on the regime where the cooperativity is larger than one so what i did is i go to the rotating frame where i take off the fast oscillation so in principle this is again this tilde operator i'm switching back and forth between these two representation but when i'm not writing the frequency of the mechanical oscillator there then it means i'm in the rotating frame with omega m or the shifted omega m and then there is the radiation pressure noise and here i already take out the relevant term where we have this large the spirit of the large optical rate here for g squared over kappa plus there will be thermal noise which i don't write here explicitly so i'm just asking what is the effect of this dynamics taking in the rotating frame with the mechanical frequency let's solve this and go to a time capital t so this would be gamma optical times t half capital t half b zero minus i gamma optical and then the integral zero capital t dt prime a in of t prime let's call define this quantity at time t capital t which is under our control so think of a pulse of light which we shoot in the cavity at the with the central frequency at the lower sideband and this pulse has a duration capital t and after time capital t the interaction is switched off meaning the optimal mechanical coupling g goes to zero so light leaves the cavity and there is no interaction anymore so then b out would be the mechanical oscillator at the end of the driving pulse and then what we see is that whatever was in the mechanical oscillator at time zero which we could call b in will be exponentially suppressed e to the minus gamma times capital t over two so if gamma is large or gamma times t is large then what was in the what the state in the mechanical oscillator will be damped out exponentially but we will get a contribution of radiation pressure noise of some quantity related to the incoming light field so how do how shall we interpret this this integral here so it's important now to get a feeling for what this a in means we are looking at the optimal mechanical system and for simplicity we will always think of this single mode cavity single sided cavity where we have a particular field of light talking to the cavity modes but this is a continuum of modes so let's cut this into pieces and we can think of this as little pulses and one of these pulses after the other will interact with this cavity and this a in is something like a creation or annihilation operator here for a pulse at time t yeah this is how you should how the feeling you should have for this process a in of t so these are modes temporarily localized modes which are described by this annihilation operator now when we have a integral over that as we we have here oh i'm sorry i forgot an important weight here so this is e to the gamma opt t prime half a in of t prime so this is important of course when we have such a linear combination what we do is we pick out a particular mode of light which is coming in here starting at zero and going to time t so this is the the piece which is arriving first and this is the piece which is arriving last but which is getting the most weight to do due to this pre-factor all right and we can do the following we can define an operator a in which is i write here a certain normalization an integral from zero to t dt prime e to the gamma opt t half a in t and this is one mode of the electromagnetic field it's not localized in space anymore in time or space anymore it's an extended pulse of light described by this operator this operator will have an adjoint and we can check the commutator of this operator so this would be one over set squared zero to t dt zero to t dt prime e to the gamma optical t plus t prime half and then we have to take the commutator of a in of t and a in dagger of t prime so i just take the joint here dagger means a dagger here and then i plug everything in this is white noise free field delta minus delta t minus t prime so this is one over set squared zero to t capital t dt e to the gamma optical t times t and we can solve the integral set squared one over gamma optical e to the gamma optical times t minus one and now we want that this is a valid mode of the electromagnetic field whose commutator should be suggestion three no it's a stupid question okay it should be one and we can assure that by by choosing the the normalization properly and we can read off that set has to be the square root of e gamma opt t minus one over gamma opt then this ratio will indeed be one and this will be a valid mode of the electromagnetic field associated to a propagating pulse of light so in principle we could with a magic machine produce a funny state in that pulse of light a foxtate or something more complicated that's possible yeah one can do that so defining this mode of light we see this is nothing else as what we have here this is what our mechanical oscillator is talking to up to the normalization and the normalization we now have to put in so i continue this equation here so this is gamma opt path b in and now this we could also absorb this minus i here this makes our formulas a little bit prettier which the minus i which we have here into the definition of our light mode and now we come with the we have to normalize this so we have to compensate by the set the square root of the gamma optical drops out and then we have the square root of e to the gamma opt t minus one but we have to multiply overall by minus e to the minus gamma opt t so what is actually here is a square root one minus e to the minus gamma opt t times a okay this is the evolution of the mechanical oscillator after the time capital t its initial state is damped out with this vector this can be really small and then if this is small then this square root here is almost one and what we see is that essentially b out is a valid for gamma optical times capital t is moderately larger than one moderately larger because we are we have this thing in the exponent yeah so what does an equation like this mean operator b out is a in i'm already saying that this is a state swap but this is maybe not obvious to you and we can convince ourselves of that so we're always working in the heisenberg picture where we track where we track the evolution of operators but what what do these mean things mean in the in the heisenberg in the schrodinger picture so let's assume we do what i said we have this pulse of light which is traveling towards the optomechanical system which is described by a in and this is described by b in and after the evolution for a time t our mechanical oscillator is described by b out i will talk about the light which is coming out in a second but let's first think about this so let's say this pulse is prepared in a particular state psi and the mechanical oscillator is maybe initialized in the ground state so this is light and this is mechanics we can write this as some superposition of fox states which would be a in dagger to the power n acting on zero for that so this would be the state psi expanded in the fox states right now our evolution turns a in into b out zero light zero mechanics now this b dagger out to the power n is exactly the state which we sent in in light now swap to the mechanical oscillator so i'm cheating here a little bit because we i didn't show you so far what happens to the to the state of light this could be something something more complicated maybe some state five or so but in fact it's zero and i will show this in a second so this is why i'm saying this is a state swap yeah so after time t the quantum state goes over from the state of light to the mechanical oscillator and it is a quantum state of one particular temporal mode which is selected by the dynamics of the optomechanical interaction now i deliberately left out thermal noise here and maybe also anti-stokes scattering which is so that the process proportional to this a dagger in and now we should estimate when such a clean dynamics actually would arise so when thermal noise is negligible so after all what i was asking for this swap to be to be good is that the optical rate here which again is 4g squared over kappa that the optical rate times the pulse length capital t which we are looking which we are assuming here is moderately larger than one so we could say although just let's take a very long pulse okay then gamma optically can be as small as as as it may be when we can still make this large but of course at some point the thermal noise will kick in and and if we use two long pulses in addition to to this clean dynamics there will be the decoherence setting in and this be out will be disturbed by by the thermal occupation by the thermal noise acting on the mechanical oscillator so t is constrained and we should choose a time t which is smaller than the inverse of the thermo the coherence rate okay so on the one hand we want that one over gamma optical is smaller moderately smaller than t which turns that which which assures that our interaction is sufficiently strong this is equivalent to this assumption but on the other hand we really want that this time t is smaller than the inverse of the thermal decoherence rate which is the coherence time of our system after this time t whatever we swapped to the to the mechanical oscillator will be gone will be erased by the thermal background and if we look at the outermost inequality here is what we see here what we need is the gamma optical over gamma thermal should be much larger than one and this is just the cooperativity so having a large cooperativity means we have enough time to send in a pulse which is much smaller than the decoherence time such that we perform the swap yeah so having the large cooperativity opens up this margin to do coherent quantum dynamics before things are decohered this is the one another meaning of the large cooperativity right so we can do the swap and now i want to go back to the question to what happens actually to the state of light so is something coming out of the mechanical of this system after this time t and maybe does this pulse which comes out have anything to do with b in the question is of course yes this is really in in the right limit a unitary process where the two systems swapped their states so there could be some more interesting state phi in the mechanical oscillator which would be swapped to a particular mode of the light field and i just want to indicate briefly how one can find this construct this light fields so what i didn't talk about so far when talking about optical cavities is how the outgoing field can be calculated from the incoming field and a they have no chance to to derive this formalism of course here properly but what i can do is motivate what is called the input output relation for cavity purity it's this so i again think of these fields as being chopped into pieces and there is a continuous flow of these pulses to the to the cavity some fraction will enter the cavity some fraction will be directly reflected of this mirror and there will be a continuous flow of of pulses out of that cavity and this input output relation here is a boundary condition tying all of these things together this boundary condition tells you is a recipe it tells you take a in of t that is the pulse arriving at time t at this mirror parts of it is directly reflected and this is described by this thing a in t and then minus square root of kappa the cavity line with a of t that is the state of the cavity at the same time t which happens to leak through this mirror and the interference of those two fields gives you the field a out of t leaving the cavity this is how you should read this formula it looks funny because you have this square root of the of the cavity decay rate here and you could say well what about units it is important that this is a dimensionless quantity yeah so a of t a dagger of t is one mode so it has commutator one various these two are half dimensions these are fields so for example a in of t as I wrote down several times now a dagger t prime is delta t minus t prime and that means that the physical dimension of a for process a in of t is well the delta function has a dimension seconds so that is second square root of second per second per second another way of seeing the meaning of this the physical meaning of these field operators is that this for example a out but the same applies for a in a dagger t at would be proportional to the photon flux operator or is the photon flux operator so it counts how many photons per seconds are flowing to a particular at a particular position here so that means a has to be of the dimension one over root second and then this formula here makes sense from a dimensional aspect now we are using it this to calculate the output of our optomechanical system so we can take the input output relation here we plug in the adiabatic solution for the cavity field and then we insert the solution for b of t now you remember that the adiabatic solution was somehow directly proportional to b of t with a certain proportionality factor which we determined then we have in the input output relation that there is the incoming light and b which also has some contribution which is proportional to b then b we solved so that this would be has some contribution of b0 plus the process a in of t which we already have here then you take the pixel right mode function which you can guess by looking at the equations this is just the recipe i'm not going through all these steps no but it really it's a systematic thing of going through that i'm plugging in the solutions for a and the ones for b which we derived and then you look at the equation and you ask yourself what is the right mode functional mode function i should pick out and you can recognize it from the from the formulas and if you in the same way as we did it for the for the incoming mode function and you will find a proper outgoing mode function which evolves like e to the minus gamma optical t capital t half a in plus square root 1 e to the minus gamma optical t b so we will find the same input output relation describing the evolution of the incoming mode to the outgoing mode as we found for the mechanical oscillator before that we had here b out and b in and on this side a in incoming light now you see that in the regime we were requiring gamma optical times t is larger than one this is essentially e in and we do indeed swap the states yeah you follow me any question so this is possible and and yeah it has been done it's a routine more or less routine protocol now in experiments and any sideband cooling experiment is actually nothing else than if you like a continuous state swap yeah because whether you measure or prepare these these initial states i mean any if you come in with with vacuum then any initial light mode here will be prepared in vacuum and will be swapped to the mechanical oscillator in a time continuous fashion you don't have to bother about pulse shaping yes you cannot yeah so this is the power of the cavity input output relation that works also if a of t fulfills a nonlinear equation in our case we have a linear one but you can use this also if you have a single atom in the cavity and or if you have an optimal mechanical system which would happen to be very strongly nonlinear so this holds generally yes um so the first thing which would matter when you're not in the resource sideband regime is that the anti-stokes process so the one where you heat instead of of cool will start to compete and then it's clear that uh for example in the in the state swap dynamics you will not only swap vacuum in but you will also add a certain level of heating uh due to the the stokes scattering so but it's a linear dynamics you you can you can solve this still and and work out the input output relations so we did that um perpetually but in principle you can solve it analytically we'll get very complicated and painful but in principle one can solve it for the general for a general linear dynamics and i'm just you know trying to convey the the most simple corners of this big parameter space here so the next thing we can look at is when we are detuned to the other side and here we will have to face the problem of instability but we will also find the optimal mechanical entanglement so let's think about the regime where we are blue detuned and for simplicity let's go to the most extreme point where we are sitting on the upper sideband and of course obviously uh we are interested in the regime of a large cooperativity in the end so in this case the effective mechanical damping which is gamma m plus gamma opt will be negative uh and this is the reason reason for the instability this is again the the peak here so asking for the stationary state now doesn't make any sense it doesn't exist or to be more precise uh the stationary state is not described properly by the linearized model the linearized model is fine as long as fluctuation stays small and fluctuations are too large we shouldn't use it and this instability tells us fluctuations will get large so we should go one step back and look at the non-linear model and then you would see that actually a stationary state exists also on the blue side and is described by classical non-linear dynamics and you can do quantum theory around that and so on but what happens in principle on this side is the system rings up and goes to uh uh stable limit cycles so you get a larger amplitude and then the system just starts oscillating I'm not covering uh this non-linear dynamics here so I want to restrict on the linearized model it's interesting enough so stationary state is not an option but we can look at the time-dependent scenario so the time-dependent solution so now the equation we have to solve is this one I introduced here gamma as the absolute uh value of gamma optical gamma optical will be negative in this case will be negative in this case so this will be again 4g squared over gamma the optical uh gamma is negative so I take the absolute value so we have gamma half times p I write here explicitly minus gamma half minus so overall we have an anti-damping and we can solve this equation again up to a time t now it would be gamma half gamma capital t half b zero minus and we could do the same calculation as before we have a particular integral over the incoming light field now with the exponential profile which is flipped before we had an exponential rising profile now we would have an exponentially uh decaying profile and we have to pick out the right temporal mode and we have to find the right normalization and plug everything in here I leave this to you as an exercise what comes out of this exercise is the input output relation square root of e to the gamma t minus one times a particular temporal mode a in dagger so let's call this again b out after time t and b in referring to the state at time zero and I promise you if you use the input output relation you will find a temporal mode which has a very symmetric shape for light so there will be an incoming light field whose fluctuations will be amplified e to the gamma t half and this is a out and there will be also an contribution of b in dagger in this case so this is the input output relation describing the dynamics on the blue side band so what does this mean this is the heisenberg picture representation of a two modes creased state of an entangled state in order to see this uh let's define an operator s which is a b minus a dagger v dagger and see how s this is a unitary operator transforms the operator a so this is now disconnected from from optomechanics this is just a little bit of operator operator algebra which I want to show you you can work this out this is another exercise and let's put here a parameter r what you find here is cosine hyperbolic r a minus sine hyperbolic r b and you can also calculate how b transforms under this unitary and you will find b dagger and a dagger importantly so this unitary mixes a's with b daggers and b's with a daggers so it's formally similar to what we found from our time dependent dynamics and now you can ask we can also now compare these things and read off what they are would be in our case the equivalent r would be something like arcus cosine hyperbolic b to the gamma t half and now we can link this parameter r to our physical parameters gamma t if you like now having this unitary here which is apparently equivalent to the overall dynamics which we generate on the upper side band you can also ask what does this thing generate in the Schrodinger picture to know to get an idea of what the states is state is corresponding to these input output relations so in Schrodinger picture we can take s and ask what happens when we apply it to a state where the mechanic loss to later is initiated in the ground state and light comes in in vacuum and you go through the algebra and what you find is this state so you go to a superposition of fox states where there is an equal number of fox states populated in both systems m and l this is not surprising because we see that somehow this s consists of products of this creation operators so whenever this product meets zero zero we will create a pair of fox states now in the in the limit where r gets large say that which is nothing else than in the limit of gamma t getting large this tangent hyperbolic r goes to one and we approximate this state which looks like a maximally entire state I mean this is nothing we will ever achieve because it's a state of infinite energy okay but we approximate this by making these weights here all more and more equal once we increase gamma t we have to do that again within a time t which is small as compared to the thermal coherence rate and we can do that there is time to do that if the thermal cooperativity is significantly larger sufficiently larger than one this is the same logic as in the case of the state swap okay any question if not I would like to flip gears a little bit and use the remaining time to treat a third case so we were now talking about cooling on the lower sideband entanglement instability on the upper sideband and what remains is subsection h resonant and this would be the standard regime of optimal mechanical force or position sending the standard quantum limit which comes in here so I'm not not sure I managed to to tell you all I I wanted to say but let's focus again at the regime where we are in the weak coupling regime now delta is zero and we find that here the shift of the mechanical frequency as well as the shift of the damping all are zero so we are not dealing with these problems and when we look at the equation of motion we just have the three oscillators subject to its thermal bath and then the light the only effect of light is an additional force which looks like a in plus a in dagger so you remember in the beginning of the lecture I introduced the quadratures for the light field and now we see what acts on the mechanical oscillator is a particular quadrature it's what we can call no sorry what we can call the incoming amplitude fluctuation so this would be one over spirit of two a in plus a in dagger so these are amplitude fluctuations which represent an additional source of noise for the mechanical oscillator when we are driving the optimal mechanical system on resonance we can also ask what happens to the light field here or maybe before that let's also to quadratures for the mechanical oscillator so if we rewrite the equation for b here in terms of x and p's it would be minus omega m times p m gamma m half x m plus square root of gamma m x in of t for make the mechanical system and let's put here an l in order to make sure we are distinguishing here the quadratures for the mechanics and for light so this is quantum fluctuations of light this is thermal fluctuations for the mechanical oscillator and p m dot would be minus omega m x m minus gamma m of p m plus square root of gamma m p in m of t and then p because of this i here it is p which is driven by the amplitude fluctuations of light so that's the fluctuation fluctuating radiation pressure force for the mechanical system and we can now take the adiabatic solution for our cavity field plus input output relation to derive that the outgoing amplitude fluctuation in this case is conserved so it's just the incoming amplitude fluctuations and the outgoing phase quadrature is the incoming phase quadrature plus square root of gamma x m of t so this is really directly what you get from the equations which we derived yesterday you just have to decompose it into amplitude and phase quadratures and specialize for the regime where we are on resonance right now what we see is that the phase fluctuation or the phase quadrature here gets an information imprinted on the position of the oscillator and this is the idea of the basic basis for using an optomechanical system as a position sensor so you just need to measure the outgoing phase quadrature and knowing your system dynamics meaning in particular gamma which is 4g squared over kappa we directly get an estimate for the position of the oscillator if there is some force acting on the oscillator this force will imprint its trace on the position of course and knowing the dynamics of the mechanical oscillator its response function we can through the measurement of the phase quadrature also infer the force so that's the idea of the force sense we have the safeties and over so imagine now we want to measure the position of the oscillator so what we would do is we integrate our it would measure the phase quadrature take the optomechanical system we have the mechanical oscillator we get the phase quadrature pl out of t we set up a homodyne detector not going into detail here with the right local oscillator phase i promise you we can measure with high efficiency the outgoing phase quadrature and we would integrate it for some time t so let's look at what we would measure here we would measure p out l of t we integrate it over a time t i always want to work with normalized operators so i normalize to the square root of t and then what i have here is something like again canonical operator p which has standard commutation relations so i can construct an x out of that this is via normalized with t and then we would see that this is pl in plus square root gamma times t x m of t and let's say x m of t for the moment is constant so t capital t is small enough that we don't see a change in t so t is small meaning x m of t is constant think of a very short pulse knowing this measurement what how would we infer the position of the oscillator well we would divide by square root of gamma t this is what we know and use this as our estimate for the position so the estimated position would be the scaled measurement result so this is one over so this would be the real position essentially plus one over gamma t square root p l in so it can measure this quantity and this quantity essentially corresponds to the operator we are trying to measure the mechanical position plus something which is connected to the short noise of our measurement this is the noise of light so this is added noise in our measurement and when we want to evaluate the precision of our measurement we can just directly look at the variance the deviation of this noise the variance of added noise is a measure for the measurement sensitivity and it would be the variance of this normalized operator which is one half over so let's write it down p l in squared the variance of that this is vacuum noise over gamma t so this is one over two gamma t and that means our sensitivity small a small number here is high sensitivity so this is small the sensitivity is good we can increase our sensitivity or decrease the the amount of added noise if we use a large readout rate so if we use a large rate 4g squared over kappa and if we integrate for a long enough time of course now there is a catch because we assume that t is small and xm is constant and this is exactly the origin of the standard quantum limit in this case so remember that the equation of motion for the mechanical oscillator look back I don't write here the full equation anymore but xm dot in the oscillator of course it couples to the to the momentum but it also if we would look at the free mass we would have the the momentum here of course being translating into a new position and pm dot being driven by noise of light so these are the equations I was writing down before so when we start to measure longer and longer what we would see is at some point and this higher rate gamma what we would see is that we would disturb our mechanical oscillator at the same rate and at later times of our measurement we would see the noise of light the amplitude fluctuations fed through to the momentum of the oscillator due to the radiation pressure fluctuations fed through to the position oscillator in our measurement record and this will ultimately give rise to a second term here of added noise which will limit the sensitivity which will also scale with a readout rate gamma so here the short noise is suppressed with the readout rate but the back correction of our measurement is growing with gamma and there is a trade-off and this trade-off is the standard quantum of continuous position sensors so what to do now I think the right way of analyzing these things is now to analyze the the dynamics in Fourier space where you really see the trade-off and maybe I just write down the result with this explanation you at least get a feeling for for this so now we would analyze everything in Fourier space and use this to measure the position or maybe infer a force acting on the oscillator and then we would evaluate the same sort of sensitivity which I wrote here and this sensitivity would look like 1 over gamma this is the same term as we have here and now when we are talking about the sensitivity for a force fluctuation this is the sensitivity for position measurement here for very short times and we when we do that for a force sensor then we get here also the susceptibility of the mechanical system in the denominator plus gamma this is the contribution of short noise chi m of omega and now this is all in frequency space and that's the sensitivity for a force at a particular frequency omega the susceptibility would be 1 over omega m minus omega plus minus i gamma m and this would be the back section noise again this is coming from the amplitude fluctuations here which drive the mechanical oscillator harder and harder the more power I use to measure the position or infer this force and now you see this trade-off we have a term scaling like 1 over gamma and one term scaling like gamma in principle there is also thermal background for this sensor but let's let's leave that out due to this trade-off we can find a bound no there is a clear minimum somewhere for this gamma and this minimum turns out to be 2 over chi m of omega this is now in dimensionless units we could multiply this now with a proper zero point fluctuations and then turn this give this s meaning in or the proper units where we have a variance of what we measure that would be a force per bandwidth because we are treating things in the Fourier domain now per second so the per second is the same thing as we have here this is this 1 over t and then the sensitivity means that within a certain averaging time we can reach a certain precision so this is how you should read such a statement about the sensitivity being given by essentially here the the susceptibility now the standard quantum limit and this is this thing on the right hand side four four sensors as this shape of the inverted susceptibility so here we are essentially constant below resonance frequency given by the mechanical frequency then this goes down to gamma m the line width and then for high frequency the standard quantum limit scales like the frequency so these are the the limiting values of the the inverse susceptibility all right so this essentially realizes the Heisenberg microscope I was I was telling that in the beginning and gravitational wave detectors are supposed to reach the standard quantum limit and they are typically working far above the resonance frequencies which is in their case like several hertz because they use this low frequency pendula they did not reach that because so far they are limited by what is also there the thermal noise and this is just the last line I'm writing and then I'm done we can also ask the question when is the back action noise larger than the thermal noise the thermal noise will have a level of gamma n bar in the sensitivity you know so we can also write this here gamma n bar so when is the thermal noise smaller than the back action noise in our position or force sensor well this is the case when the cooperativity gets larger than one yeah working in the cooperativity larger than one means we're running our Heisenberg microscope in a fashion where we are disturbing the system stronger than so we cannot localize our electron or our mirror because we are measuring too strong yeah so what we are seeing is essentially only the back action of our measurement it's a very remarkable regime once more and the challenge for the people reaching just going to the real standard quantum limit would be to have this smaller than than this value and this has so far not been achieved okay so this brings me to the end of my lecture I started this with talking about Einstein's reasoning of why the why photons are particles which was based on in the Duncan experiment using optical cooling optical cooling is a reality we can do that very efficiently even to the ground state I was mentioning the standard quantum limit we reproduced that in our description with weak optomechanics and I was mentioning the proposal of Roger Penrose for creating crazy entangled states of light and mechanics in a different way this can also be done in weak optomechanics so in reality like all of these early ideas in maybe some other form have been realized in this field and I hope you now have the necessary understanding to really go into the literature and reproduce the description of these things thank you very much