 So the cafeteria to the terrace and there will be the welcome reception and after the welcome reception That's actually the poster session again down there. So seven o'clock welcome reception and the second thing is in the last lecture we asked through this list with your names and Ask you to sign and put your team number there now if you have already signed and put your team number there No problem. Everything is done, but we are still missing some names. So I will pass it through again We have already signed just pass it on And we want everyone to sign and want to put their team number there if you don't Remember your team number or you are not yet assigned to a team for the many projects Then please approach me because we want everyone to be part of a team Okay, so I will pass this through the rows and then enjoy the next lecture by Thank you I I the second lecture on microwave optomechanics and of course I knew I will have an overlap with john Which I thought it would be like one quarter now. It looks like more like three quarters On the other hand, it's it's not such a bad thing as I said also Because I will be explaining maybe mostly the same things But I will be explaining them differently. So if you didn't understand him You may still have a chance of understanding me But if you understood him then probably you will not understand me So just you can try and see Now so I will be talking about First microwave cavities what it means and how how they are quantized I will talk about optomechanical coupling Very briefly because Yeah, okay because Because we will we will hear that again from from clements And I don't want to duplicate very much I will talk about optomechanically induced transparency But show you something john didn't show you And and then I will in the end very briefly talk about quantum states I will be also talking about quantum states on my last lecture. So this will see we'll see how it is All right, so now We already all know that optical cavities is the best way to To think of optical cavities is is like a good example is a fabrizero cavity When we have just two mirrors and the light is trapped between two mirrors Uh and because it trapped, uh, there are just resonances which which correspond to certain frequencies Right and then we said good if maybe one of the mirrors is moving then it's an optomechanical cavity But that's with optical light or maybe far infrared Now if you want to do the same with microwaves How do we make cavities? Do we have microwave mirrors? Or what the hell do we have? And and the answer is okay. We don't we don't have mirrors strictly speaking or at least if we do they don't look like that But we have the simplest example of a cavity is just an LC circuit So that's an electric circuit. I mean microwave cavities are not cavities. They are not optical They're just electric. They're electric circuits And this electric circuit just has two elements. I mean it's a capacitor and an inductor I already apologize for my drawings, but no things become aggravated It was better than before before I export it as a pdf so for whatever reason some lines Shifted and it's likely to propagate to other slides But anyway, so that's supposed to be a capacitor and that's supposed to be an inductor And let's spend some time on this slide and go to some details to make sure that we all understand what's going on right, so if we have Capacitor well, it means we have sometimes we have well some way we have voltage like between Between between those two plates and we have the charge on the two plates and they are related and that's something I already discussed in my first lecture today. So the charge is proportional to the voltage and the coefficient is there Now if you have inductance Yeah, we can also rewrite it like that so we can take a time derivative of those time derivative of charge is current going through the capacitor And so then then we get an equivalent relation that the current equals to capacitance times the time derivative of the voltage All right now, we also have flux whatever it means And flux is related is proportional to the current and the proportionality coefficient is inductance Now that's probably less familiar, but it becomes more familiar if we can take a time derivative because the time derivative of flux is minus voltage and then voltage equals to it should be minus inductance times time derivative of the current And now instead of these strange things like instead of charges and fluxes We just have two equations which relate Current and voltage and we can easily transform them into one equation For instance if I take that one and take another derivative Right so on the left hand. No, let's let me take that one On the right hand side, I will get l times i double prime Which is here and on the left hand side, I will get v dot And v dot we know is i over c Right, so if I write i over c equals to l i double dot and I put c on the other side I get that equation And we can also write the same equation for voltage if you want And now if I get that equation that of course something which And again, that should be minus sign which propagated from here So that we know how to solve that equation Right, so that that's something which we learn in the first year at the university And we also know that this is the same equation as describes a harmonic oscillator Right, we can say now current is coordinate And if there is no friction the friction is not in this equation The the the second derivative of the coordinate equals to the first Equals to the coordinate times frequency squared Right, so our conclusion is first of all that this thing acts as a harmonic oscillator And then this harmonic oscillator is just a single mode oscillator With the frequency we can which we can read out from this equation And if we do it then we get that the frequency is just one over square root l times c Now, uh, there are two things which I I need to add One is as I mentioned we don't have dissipation in that equation Now clements already spent some time explaining this that actually adding dissipation is not a trivial thing We can just add it phenomenologically saying, okay, there is also resist around And if you have a resistor then we know current is proportional to voltage and in the end that would Make this equation more similar with to to the equation of a harmonic oscillator with the dumping term proportional to the resistance But if you want to quantize it and I will quantize this equation On a couple of like couple of slides later, then we have all kind of headaches. You can easily quantize Oscillator An oscillator where you have friction and and and you have to go to complicated model of quantum dissipation Which is called a relegator or whatever Feynman-Vernon or whatever you want And and that's not easy and that's not something I'm going to So we will for the time being we will disregard the The resistance at some point it will reappear and I will show Now second thing is in okay, so that that's far So far it's it's it's just It's just A theoretical analysis I mean and there is no reason why I should call this a microwave cavity, right This frequency can be anything depending on which capacitances and inductances you can fabricate And There are basically a broad range of what you can what you can make out of it. And that's not necessarily a microwave now Actually in in practice in in this optomechanical experiments people use frequent people use cavities with the Frequencies and the microwave range which is one to ten gigahertz or even more narrow actually I think John had seven point five Five and that's around what what you have like five to seven. I think five five to eight Now here there is one more thing which is all experimentalists No, we theorists usually usually don't care, but that's a good place to care about it This frequency is omega When you talk to experimentalists and ask what their frequencies is not omega. It's f Which is omega divided by two pi So we have to be careful every time we take actual numbers Because f is one to ten, but omega means two pi to ten times two pi Right, so six two whatever 60 Uh, and why why they use that? I'm not in a good position to explain But I was told that there are some practical reasons. There are some filters which only work on that frequency. I don't know Something which which for me is completely Not understandable, but they're just life. They just use it Now that's also actually practical because If you want to do any useful experiments, we don't want temperature to be very important If we don't want temperature to be very important means we actually have to compare Oh, yeah, I have it on the side. We actually have to compare the energy of the Of the microwaves one quantum of of the radiation, which is h bar omega or h times f if you want We should have to compare that with the thermal energy, which is kbt and for instance if we take If we take F, which is five gigahertz Then we get temperature, which is 30 milli Kelvin and you can Again in practical terms you can in a dilution freeze you can go down to 10 milli Kelvin. So you still have You still have that which is bigger than kbt without making too much effort And and that's useful for all the experiments We have seen again that for mechanics that that doesn't work and you need to cool it But but that's a separate story at least you don't need to cool that Now that was just an isolated cavity Now what people do with optical cavities they measure transmission and reflection Now with microwave cavities, you also want to measure transmission and reflection And okay, you just need to to do the same as as you do in optics. You need to add a wave gate And the wave gate is also some electric socket Could be a transmission line or could be whatever you just sent Send some whatever voltage along the line the voltage comes here something happens and gets out or it gets back and depending whether it gets out or back it's Depending on which side gets out at transmission or reflection Now I I I promised to mention at least that there are losses Which could be due to resistance And there are losses which happen outside the cavity and inside the cavity So inside the cavity is just resistance of the of the cavity outside is the resistance of the transmission line And that's something you you also can take both Some both. Okay. I must I must explain what is that actually kappa is the the line width So kappa has the dimension of of energy or frequency and that the line width of the cavity Uh, and you can assume that those losses are independent so the total line width would be the sum of internal and external losses and Actually, it will be important in some in some situations and I will show you in which situations it will be important Where losses actually happen? Right. Okay. Now now we have to quantize it and make sure that it's similar to To the cavity and again, we already know That it's a harmonic oscillator And we know how to quantize a harmonic oscillator Clemens spent quite some time today in the morning to show that So what what we are going to do is just to do exactly the same as what he has done All right. So first we write the energy classically And there is an energy associated with the capacitor cv squared over 2v is voltage And there is an energy associated with the inductance Li squared over 2i is current And now we know that this v and i are like Coordinate and momentum Of a harmonic oscillator and what which one is coordinate and which one is momentum. It doesn't really matter It's up to us to choose Uh, so I've chosen voltage to be coordinate and current To be the momentum, but but it doesn't matter as I said And so what we need to do now is to Write this current this voltage and current in terms of this creation and annihilation operators Or you can say differently you can introduce creation and annihilation operators in this in this way And so what you do, you know that somewhere should be some combination Should there should be a plus a dagger and somewhere else should be a minus a dagger And you want both operators to be emission And if they are emission Then this coefficient must be real and this coefficient must be purely imaginary Otherwise you have trouble And then what is exactly here is well you need to to derive And you derive it From from general consideration So first of all, you know the commutation relations That should be the commutation like that And second thing, you know Is that you should reproduce the classical equations of motion So if you take this now voltage as an operator and calculate d dot That should carry that that that should obey the Heisenberg equation And this is something which you can calculate if you know the commutation relations And okay, that's that's that's that's a bit tedious thing but but Clements has already done it. So I will not do it And I will show you the results So the result is basically here. So the coefficient should be like that If you want to reproduce the classical equations of motion Or sorry, I should probably Read what it should what it is. So in the end you should get On one hand is that but on the other hand, you know That you should get What is it? Is it l i dot You should get you should get this so you should get i divided by c and i is also an operator which you know Right. So first of all, you get two coefficients here, which are indeed real which which contain h bar Which is fine which which contain one of them contains Capacitance another one campaign contains inductance You can check I will not going to do it, but you can check that both equations of motion for voltage and current are the same as we expect And then you can also derive the Hamiltonian, which is not surprisingly just that and that's something which we perfectly expect for For for harmonic oscillator, and that's also clements derived in the morning for the for the cavity So so far we just have exactly the same quantization So in quantized form our microwave cavity We have exactly the same weight as an optical cavity, which is not really surprising because both of them are kind of cavities for electromagnetic radiation Right Okay, now that was so far. I didn't have any mechanics. It just just an electric circuit Now let's include mechanics How we include mechanics That's something which I discussed in the morning we have We could have capacitive coupling and we could have inductive coupling We could have capacitance, which is position dependent and we had some examples of that And we could have inductance, which is position dependent and I didn't have examples. I referred you to the lecture on Thursday So let me stick to the capacitive coupling the results that don't depend on what what you choose You can choose even both. It doesn't matter So let me assume that capacitance is position dependent Because for instance something is suspended over the gate. I mean some part of the circuit or whatever Some And I assume that inductance is not position dependent doesn't matter So if capacitance is position dependent, there's also something which we have done in the morning Is it okay? Well in the simplest situation this This deviation This Displacement this mechanical displacement x is small so we can just The Take a derivative of of capacitance and write something like this And if you write it like that that gives Rise to the energy of the cavity without any mechanical motion And that is proportional to x that gives such a term Which is the interaction term because it depends on the both the displacement and the Degree of freedom of the cavity, which is in this case voltage right Now Okay, that's classical now. We need to quantize that We know how to quantize voltage Voltage is proportional to a plus a dagger with some coefficient So from voltage squared we get that We know how to quantize the coordinate that that clements has done in the morning I don't have it on the slide, but I will write that x Is just the amplitude of zero point motion Times b plus b dagger and so a are creation annihilation operators for photons In the cavity and b are creation annihilation operators for for for mechanical motion for photons if you want And this is just the amplitude of zero point motion, which is What is it h? 2 m omega or something like that Now, okay, I get I get that term Which is a bit complicated, but we should understand That's actually those operators a and b are still time dependent And if they're time dependent Here we have four combinations Well five combinations So we have a a no actually more We have like a a b And a a dagger b dagger and all this kind of combinations And most of them are very quickly oscillating And if you look those which are not oscillating quickly That the only the only thing up to the commutation relations which survives and All others if I if I Take the if I if I go into Rotate into rotating wave approximation, which is exactly averaging out terms which rotate fast That's exactly what this rotating wave approximation is about Then all others I would disregard if they If they are fast Now again, there is a cavity coming with that That's in principle a perfectly correct thing to do but only If your coupling is not very strong And I will I will talk about that And and john talked about that today And I don't want to go into any details at this point But but the the important thing is that coupling is is very strong and I don't want to quantify it now then It's not sufficient to to only keep those terms you should also keep some other term well all other terms essentially because they would be Quickly oscillating and quickly oscillating meaning They are not resonant so they will Okay, but but but but but but but since they are multiplied with them with a big number They could still be important But for the time being I'm not just disregard them I get this coupling I call it radiation pressure Is it really radiation pressure? I mean is it is it something which Which which which we usually get In optomechanical cavities so let me spend several minutes And review optomechanical Coupling in in in optomechanical cavities So that's something which john has done And something which clements will do But still I need it now. So I will I will still still still spend this couple of minutes Right now we already know what is radiation pressure on the classical level. I showed you in the morning Now we need it on the quantum level and and radiation pressure is this term Which is coming from the position dependence of the cavity frequency So remember it's x times n number of photo of photons number of photons is here So it's a digger a and and and and this is This is uh b digger b b plus b digger is coming from the quantization of the of the coordinate So in the first instance that the same thing. I mean he is with minus here. I wrote it with plus doesn't matter It's a it's a matter of convention Now we can start thinking about Okay. Yeah, now now it's good good good point for me. So so it's it's the same thing. So we get the same radiation pressure now, uh, I I I I need to talk about the scales because that's important And again, I will have an overlap with with what was Before me and what is coming up to me, but still I will I will spend some time explaining exactly those points So you have several energy or frequency. Let's say frequency scales And okay, so you have the cavity frequency, which is the biggest You have the mechanical Line widths mechanical dissipation Which is the smallest unless you do something unless your own purpose make it make it make it big Which you usually don't want to then you have cavity line widths and you have mechanical frequency They are both in between And they could be of the same order and then Actually mechanical line widths could be bigger But usually we want to be in that regime Which is called result sideband regime And john already explained it in quite some detail what it means Uh, basically if we generate side bands of mechanical motion And the the distance between those side bands is mechanical frequency, but the widths of each of them is kappa. So if We are in this situation, then they are resolved. You can just see them separately If you are in the opposite situation, you can't resolve them. You just see them at one peak And and kind of drinks Okay now on top of that we need to add this coupling which also has dimensions of frequency And this coupling I I didn't actually uh emphasize that but this is something which is made of This the zero frequency amplitude of zero of zero point motion And of this derivative of the capacitance. So that's something which which doesn't Which we don't have very good feeling about. I mean it's some combination of some of some things which which are specific to the To the device we are talking about Okay now in the literature if this coupling Is below kappa It's called weak coupling regime. If it's above kappa, it's called strong coupling Now depending on where this kappa is you can still differentiate. You can call something ultra strong coupling That's that's not relevant for what I'm going to talk about. So for me, it's only relevant weak and strong Now What is important that actually strong coupling in this sense Has never been achieved. Well, I mean there are some caveats to that which I don't want to discuss There are some people who can reasonably claim that they achieve that but then it's not exactly the same So I will not go into that. I think as a as a kind of general statement. It's okay And it's very difficult to achieve It's called single photon strong coupling Now what you can do I can say okay, maybe we don't need that maybe What what we need Is we can We can put a lot of electrons or sorry a lot of photons into cavity And if you put a lot of photons into cavity Then the system is almost classical right because if you're in the regime of the number of photons much bigger than one It's almost classical and we can linearize it And say, okay This a Is approximately Square root of number of photons. I will explain in a second why it's approximately not exact Now then okay, so let let let me call it the average The average let me just call it like that. It's still approximate, but but I will come to that Now I can say good fine. Why don't we just try that a is Some average plus delta a And we just take this expression And and and write it as a plus delta a and a sorry a dagger plus delta a degant a plus delta a And then this equals to a dagger a And then we have a dagger delta a and then we have a Sorry a dagger delta a and then we have l delta a dagger And then we have delta a delta a dagger delta a I can say okay, this is small right the cavity is almost classical. So that's almost that So this should be small. Let's throw it away This is there that the biggest term, but that's not interesting because it doesn't know anything about the fluctuations So we are only left with those terms, which are responsible for the for the interaction And those terms exactly is what I what I have written here Well, not exactly, but but I will I will come to the next slide and explain why it's not exactly so And and then you have this new coupling which is which is well this g naught times the square root of the number of photons in the cavity and the number of photons in the cavity is actually very big So what you do you enhance a lot the coupling And so this new coupling you call multi photons strong coupling Because there are many photons And this one actually can achieve strong coupling And it has achieved strong coupling and and john was talking about that for half an hour Right now he also had this slide Well equivalent of this slide, but let me still Still go through that because I want to understand that we are all on the same kind of page Now look if I If I if I If I look here And now think about What terms are resonant if I want to make to to to make rotating wave approximation Which terms should I keep? A kind of naively I come immediately to a conclusion that I should not keep anything right because Again, they are time dependent, but a dagger Oscillates with the frequency of the cavity And b oscillates with the frequency of mechanical resonator And here you take a difference, but this difference is cavity minus mechanical. It's still a very big frequency So it looks like they actually oscillate very quickly and they are not resonant at all And then why didn't we just keep out other terms, which are also not resonant and If this is corrected kind of invites a lot of questions Now in fact what happens is that actually our uh, what what what I call Average A Is also time dependent So its amplitude is square root of the number of photons But it has a phase Which is oscillating And if you oscillate it it oscillates with a driving frequency plus or minus I never remember Plus This is a correct statement And what I have over there is kind of short way of writing this correct statement Of course the number of photons doesn't oscillate Right, so but but it has this a has a phase which results Results in like like in this g we have this time constant All right, so then we have if we drive the cavity here We have the driving frequency and here we have the cavity frequency and here we have the mechanical Frequency and now we need to take and to to add them up and see which one is resonant And that's what john has done That depends on how you drive it It's not coming by itself If you drive it for instance at the red at the red sideband So if you drive it at the cavity frequency minus mechanical Then only those two terms are left So you have This a dagger gets gets with a minus cavity and becomes with plus mechanical and they all free cancel And and that's a complex conjugate a Hermit Hermitian conjugated term Now and and and and and then you are left with interaction which only has those two terms And that's beam splitter interaction You can also drive at blue sideband And if you drive at blue sideband then those two terms will will be Will will will be resonant and then generate squeezing And then those terms are resonant and those terms are not resonant And again if you drive at some other frequencies you want to keep all the terms which you have and Because they can be all important in particular if you drive at resonance then all terms could be important Okay, I'm I'm now going to the experiments And my as I said my general idea for for for this lecture Is to convince you or or kind of reinforce if you're already convinced that Microwave cavity can do the same things as optical cavities. They are on some level they are just Very similar only parameters are different And To kind of help you I put here two pictures Four pictures of cavities and two of them are optical and two of them are microwave And I'm pretty sure that if you don't if you are not in the field if you are not doing experiments with optical or microwave optomechanics And if you don't know the groups which I cite you will probably not be able to tell which ones are microwave And which ones are visible light Now in fact those two are visible light and those two are microwave So those this is a photonic crystal structure from from oscar panthers group in kaltecht and those are How they are called toroidal resonators of tabis kippenberg ATPFL um They are both optical cavities and here the the photos are trapped like that and here they are trapped In this constriction as I already explained in the morning On a on a different example And we have actually seen both of those cavities in other lectures and I will cite them later today When I because they're both experiments were actually done to demonstrate the quantum nature of mechanical resonator And and and those two I I borrowed from from My colleagues and dealt from gay steels group. I'm actually on both of the papers. So borrowed is not probably a character This is a a cavity with a graph in membrane And they have like this is them. This is the Feed line so that that's that the waveguide and that that's a gate And where is the cavity? I don't well it says here, but I don't actually know what what is exactly the cavity And that's from the same group but completely different cavity. So that's originally Yale design From rock shell cop group. So this 3g cavities Which is really like 3g books out of aluminum And then you have a membrane in the middle of this cavity, which which is a oscillating oscillating element And actually this In in this publication they they set the world record for the temperature which is 48 My credit for for the for the mechanics All right, uh, now let let me um Okay, I should go through this slide So that that kind of sums up What I was talking about for for 50 minutes 40 minutes Uh, what what what what what what can be? I mean what what what what what are microwave cavities? What what can we do with them? Why are they useful? First of all, I hope I convinced you That we can at least do the same things as we can do with visible light So whatever experiments we can We can do with optical cavities. We can also do with Microwave cavities having in mind that frequencies are different and quality factors are different, but but So the different regime, but but that's same physics Uh, now there are two other things um One thing I will Not explore in my lectures. I will mention briefly today in the end of the lecture uh, that The the cavity frequency for microwaves could be comparable to mechanical frequency Um, well for optical cavities, that's not possible Because if you are In invisible or far infrared regime you are using hundreds of terahertz radiation There are no mechanical resonators With hundred of terahertz frequency as far as I know So the the highest frequency I've ever thought about Which was useful in context of Optomechanics was 20 gigahertz And people usually don't even use that Uh, there are some experiments and I will show you one of them Where you can use Low gigahertz resonators um Usually people don't do that they they go to megahertz or kilohertz So usually in in in all microwave Definitely no optical experiments The frequency of the cavity is much bigger than the frequency of the mechanical resonator But in principle if you have a microwave cavity You can make them resonant You can if you take a particularly stiff mechanical resonator with With the frequency of several gigahertz you actually can Make it in resonance with the cavity which would change all the things I was talking about Right because all this analysis Whatever at sideband blue sideband It assumes that the mechanical frequency is much less than the frequency of the cavity If it's not the case then this analysis is actually not good. You have to to look at things completely different That's not done very much not not Not theoretical and not experimentally, but but that's a good direction to to think about and Whether you can get any advantage from from this Uh now another thing which I will not be talking today, but I will be talking on on on Thursday Is that you can make Microwave cavities non-linear I didn't yet mention that All whatever not all but but most Microwave cavities which are used as superconducting You don't have to know what the superconductivity is. I will If nobody does it before me, I'll explain it on my left hand Thursday Um, but Superconducting junctions which which are used in those cavity They have some additional properties and one of these properties is It's Josephson effect, which again, I don't want to explain right now what it is But that's some non-linear effect In in current and because it's non-linear then your cavity becomes non-linear All right, and an optical cavity is very difficult to make non-linear You need to put some whatever non-linear media inside And that's not not trivial But you can relatively easily can make using this Josephson effect. You can make A microwave cavity non-linear and that's something I'm going to talk about On on on Thursday And so for the time being I'll stick to that And I will start with With that That's something which I already actually mentioned in the morning in slightly different terms But it's useful to see it again Now for instance what we can do We can say okay, we don't care about the cavity We only want to know what are the properties of mechanical resonator Then theoretically you would need to well experimentally you would need to measure resonator or measure it by means different differently from the cavity Which is not easy but possible Theoretically you would need to eliminate the cavity. So you would need to solve Let's say equations of motion for the cavity. They would depend on the position of the resonator You would need to to find the number of photons and plug it in in the Hamiltonian For the for the number of photons in the cavity which then becomes the full Hamiltonian for the mechanical resonator Now if you do that Then you find that there are some things which happen And they they they even happen if you don't drive anything so They they have so You you you you you if you don't drive the resonator you just Everything is coming from the cavity So one thing is frequency is renormalized I showed you it on example Without a cavity, but it doesn't matter that's In this business, it's called optical spring they said Now dumping coefficient is renormalized. I also spent some time explaining that And there is a very easy physical intuition. Why? Because you now have an additional channel of dissipation of energy Which which which which effect dumping Now this this Force can become non-linear. So the resonator can become non-linear if you do it properly That people didn't explore but but but it would happen And another thing which I didn't discuss But it's also trivial that the equilibrium position of the resonator could be shifted and that happens because remember We had this equation of motion of the resonator. Let's say m x double dot plus m omega naught over q x dot plus m omega naught square at x equals to 2f of x and if you expand f of x you renormalize the frequency for instance But you could also have a constant term You could also have a term which is just Let's say plus f naught and if you have this f naught The only effect of f naught is to put it here And renom and and renormalize the the position of the the equilibrium position And now while you can also look at the cavity and so the same thing happened with the cavity the frequency is shifted and The the damping is renormalized and I will actually need this formula for the later So I will just bring it without any attempt to derive it. It's not trivial to derive so the total damping of the cavity Is renormalized so it goes either up or down depending on whether you drive it at the red side bent or blue side bent and the The the the the shift of the damping is just four times this Coupling square divided by by gamma and now if you look at it carefully You see that we are comparing kappa with four g square two of a gamma meaning We are comparing four g square two of a gamma kappa with one Which with jon called Coaterativity and that means that some horrible things happen When cooperativity becomes one All right, at least if you drive it at the blue side bend then at some point this thing gets renormalized and gets negative and if you have damping which gets negative it means you have instability That if you have this Oscillator with negative damping negative dumping meaning negative quality factor and you drive it Uh, the role of this Tom is usually to damp the oscillations out But if instead it's positive it amplifies the oscillations and instead of damping they just the amplitude goes up this time And that's an instability and then that's something which You you you need to do something about that. I will probably will have an example on on my last in my last lecture Um now again, I already showed you this this this cavity Just just to emphasize that I will be showing the experiments coming coming out of that um, and I will spend some time again on Optomechanical induced transparency, but I will not attempt to To to give the same explanation as jon gave so to explain. What is exactly? What is exactly? Interfering with what I he was he was clear enough and I will instead I will I will show you the results Right, so the experiment is well, we have a cavity resonance. So this is uh This shows basically the line width so that the cavity has a function of of frequency Now we have a drive and we drive it At the red side band so at the frequency which is cavity minus mechanical And we can strongly drive it and we also have a probe laser which collects which measures the the transmission close to the cavity resonance So we drive it with one laser here and we probe the transmission Okay Now you have seen the result and I will also show you the result But before I show you the result, let me on two slides torture you with some uh formulas Which which I will not explain comprehensively I think clements is planning to do it at some point But at least I will give you an impression how these results are derived I will not go through the whole derivation, but I will show you the result today Now what technique which you used to derive this result are so-called input output relations And those input output relations are just fancy words for the equations of motion For the cavity and for the mechanical resonator And here there are that's a lot of symbols which which can look completely kind of unfamiliar But if you look carefully That's not such such a difficulty I mean if you look at first equation Let's first ignore those two terms We only have that that and that and that you can actually recognize You can actually recognize as an equation of motion for the operator a I because you have da Over dt similarly to that you have i over h bar a And then in h you have cavity You have mechanical resonator cavity is a dagger a And mechanical resonator is b dagger or let's say x squared for the for my purpose and Interaction Which is a dagger a x And then if you start thinking about that, okay So you have this term would give you the first term and this delta is the detuning. So that's a difference I thought I have it somewhere No, I don't have it. So that that's a difference on on On the driving minus Minus cavity, which is typically minus mechanical frequency Now you have mechanical resonance or x squared commutes with a no problem Doesn't produce anything and in the interaction if you calculate this commutator you get a times x And and that's what you get from from from uh from the Hamiltonian Well, there are some kind of couple of technical steps hidden Like you need to go to the rotating frame and I will I will not do that but That's my my kind of purpose is to convince you that it's not black magic Or not exactly black magic. Uh now There is well, there is a little bit of black magic actually because There is one thing which is here is the cavity line width And as I mentioned before that's not something we can derive Or at least cannot easily derive We can only put it by hands And I basically put it by hands Uh another thing Are those terms And they correspond to the input so what you send to the cavity Uh and you also have quantum noise I will not be Explained in any reasonable way in two minutes why they are written like that Why here you have this external? Well, okay, you can think maybe Input should only depend on the external losses, but why should be the square root? I don't I cannot explain easily and I'm not going to do that But at least we should understand that it's not kind of strange that we have input and we have quantum noise Right and then that's all in the literature. So every time you need to do this calculation I'm going to show all of you can do this calculation if you really, you know press to the wall and and Set to do it by your supervisor Uh this literature can go to the literature and you have enough knowledge to To understand what they've done And and to reproduce Now those two are equations of motion for the mechanical resonator So we have x dot which is p over m which is not very much surprising I guess And we have p dot and then p dot you have actual force acting on the oscillator So you have return force of the oscillator You have force coming from from the uh from the Radiation pressure which would be proportional to to a dagger a You add by hands mechanical damping And if you want you can also can add thermal fluctuations and Here we assume that that thermal fluctuations are irrelevant for Focus for the cavity, but if not we can also add them in the same way Okay, so you you get a set of nice non-linear equations Which are all time dependent and and you're completely lost and then Then then what you do is You try to figure out what the solutions could be for instance if you draw it with some frequency Then then in this rotating uh in this rotating frame the frequency of the drive is that so the drive minus probe And And then your solution will also have that frequency And you linearize and this a plus and then minus And you calculate And then you have linear equations Which are all at the same frequency and you can solve them and in the end there are some Recipes how you calculate the transmission through the cavity, which I will not go through that that's all written for instance in that paper In the supplementary So my message is not that that you can instantly learn how to do it, but but that it's Nothing really complicated. That's all doable all described in the literature and can be easily adopted to your situation If you need to modify it in some way Okay, here are the results I will have experimental pictures on the next slide and john shown have john Has shown you similar pictures, but let me sketch what you get So as I said for transmission as a function of frequency Now without any optimal mechanical induced transparency, we would just have a cavity resonance Now with optimum and and and and this is something so the line which here is the line width of the cavity Right now if we include this optimal mechanical induced transparency, we have additional peak Which is exactly at the resonance And which is very narrow so the Okay, very narrow and very high And now here I quantified There is a full expression which I could put on the on the slide, but it's not particularly interesting so I will rather rather show The the the the line width and the and the height So the line width is just that I think I probably forgot I forgot for here. I'm sorry for that And if you write it in terms of this co-operativity, which john already introduced Then it just mechanical Mechanical line widths one plus c So if you have weak coupling with mechanical line widths, if you have strong coupling that's mechanical line width times co-operativity So that it's why it's much more narrow because mechanical Line width is much less than the cavity line widths Right now you have also the the the height of the peak And that's the transmission and now I promised you that This this losses will play a role This e to c I have it on a different slide That's a ratio between External losses and total losses so external and External plus cavity now in principle you can yeah usually If this e to c is bigger than one half The cavity is called over coupled If it's below one half it's under coupled and if it's exactly one half it's optimally coupled and and there is Sometimes there is a good reason to to have an experiment at exactly one half So for instance, yeah, if if if the cavity is optimally coupled So if this thing is one so two e to c is one Then we just have one minus one divided by one plus c which is c divided by one plus c and the whole thing is squared And so if you are if you are at strong coupling you are actually are That's almost one if c is much bigger than one then it's one So you can go up to transmission one If you are not a strong coupling then then it's below but okay It's still it's still something which can be considerably bigger than than the cavity itself And another thing which I I will mention is that actually you can use this Optomechanical induced transparency to calibrate c so it's probably one of the easiest of the easiest ways to See how strong is the coupling? How strong is the optimal mechanical right now? Um This is not the original experiment the original experiment is that paper But that's the experiments from again from delft From this cavity. I showed you that one And okay, so they see this and I should explain why they see that because they have a cavity Which is a single port So from a single port cavity you cannot measure transmission because it cannot transmit anywhere. It can only reflect And indeed what they measure is reflection So for reflection You don't expect a peak like that For reflection you actually expect a deep at the cavity resonance And that's that's what they see we should look at the lower curve and then the deep becomes even deeper if Uh Because of the optomechanical induced reflection in this kind And it can ideally go down to zero. So he doesn't but but That depends on how you couple what what are the losses in your cavity? What is this parameter? It is c and and whatever Now that's another one, which I didn't mention before but you can also Drive it at the blue side bent And you also get optomechanical induced transparency But you see in this case it has an opposite sign So instead of suppressing the reflection it enhances Yes, it enhances the reflection So you don't have a reflection you have something else and since they don't have transmission if it's not reflection it's absorption Rather the only way you can get rid of the of the radiation. And so they call that optomechanical induced absorption and and this is optomechanical induced reflection Now before I I I go to my last topic. I'm actually I have a lot of time, which I'm probably not going to use Let me show you something else concerning this optomechanical induced transparency And that something else is is related to parametric driving Now parametric driving is is a fundamental concept Which I think all the speakers today already mentioned and I expect actually mark dickman to to spend some time on that So I will not spend too much time on this But I still introduce you what is that and and what are the results which we can get So parametric drive parametrically driven oscillator is that oscillator So you see that the equation which is different from from the one which I had here and erased by two by two kind of features One feature there is no external force, which is f cosine of omega t Instead of that you have something here You have renormalization of the frequency. So you have correction and this correction has a double frequency You can say, okay, that's still an external force. So it's don't externally driven But this external force is proportional to x It's not what we usually call it's not it's not a driven oscillator. It's something else So that's on the level of equation Now that's actually something which we know very well Because like the the simplest example of this Of this parametric oscillator is a swing And you have ever been on a swing, you know that you Well, there is no force except for what you do with the swing And you have actually To force it twice during the period Which means you are operating it with a double frequency And that's why why why it swings All right now that there is some okay, that that's something which you can solve and that has been solved Whatever 200 years ago And and and here I summarize the results Now if Okay, so what I plot here is Is not an amplitude That's That that's that's that's a a plane in the parameter scale. So we have two non trivial parameters One parameter is the amplitude of the parametric drive, which is omega p squared And another parameter is Is Detuning so how close you are driving it to the resonance And then if you don't have dissipation so if q is infinite Then there are these two straight lines And above the straight line the system is unstable meaning you can drive it to infinite amplitude Unless something happens typically something happens. I mean it's non-linear and then amplitude is stabilized or there's something else But if we are talking about that equation that equation Is in this equation the amplitude is driven to to infinity And here it's not driven and if you include dissipation then this this this This this this line actually becomes becomes curved like that and it goes a little bit up. So you still Here it's still still straight line, but you have some Some parameter space where you can still parametrically drive it and not Go to infinite amplitude Okay, that's something very general actually has nothing in principle Nothing specific with optomechanics Let me show you that that I believe that the only Non-published data i'm showing In all my three lectures That's again That's happening in gary steels group and daniel botner will be here next week and I expect him to Either present it as a poster or Or basically be around it if you want to talk about About the details of the experiment. So it's He is the right person to talk Now the idea is okay, so let's imagine the same Optomechanical induced transparency experiment for some additional feature And this additional feature would be let's also drive on top of this laser of this drive laser and the probe laser Let's also mechanically drive the resonator at a double frequency So that's possible. I mean you can put it on the piezo substrate for instance, whatever There are many ways you could drive a resonator. That's a much lower of course much lower frequency Than than than the cavity Now we know from From here that if we drive it Parametrically if it would be just a resonator it would eventually become unstable Now it's not just a resonator. It's a resonator coupled to the cavity We have this non-linear coupling. We have everything What would it do and and can we actually see what it does? Now the answer is which I'm not going to to derive I would be happy to talk if somebody is interested, but I I didn't put it on my on my slides The answer is The the the this this parametric driving of the resonator would result in enhanced Optomechanical induced transparency. So in this experiment you can get a peak of the transmission above one So you will get transmission more than one Which is not very much surprising right because it just means that we Somewhere generate more photons in the system by By when when when when the light transmits through the system. So there is nothing I mean, it's it's unusual, but there are no physical laws which prohibit that Transmission of light can be above one and the mechanism why we generate those photons is exactly because we have this This oscillator which is swung swung to the amplitude Which which which is very much Okay Now I have like four slides left and I will probably finish ahead of time Which is probably not bad Given that the last lecture of the day Let me talk very briefly about quantum effects Now what what would be cool? If okay our cavities both microwave and optical are operating in the quantum region So we have full quantum control Of of those cavities It would be it would be great If you would also have a full quantum control of mechanical motion Well, because then we could for instance We can use this mechanical resonator as a transducer Of course, we have seen mechanical resonators are coupled to everything And this everything can be made quantum Some More easily and others are less easily But but they can be made quantum And we could generate quantum states. I don't know of Flight of spin and maybe we can use Mechanical resonator to transfer quantum states of light to spin Which are by the way on the same frequency Maybe we can store that quantum states Because we know there is We have seen that the line widths of the cavity is much bigger than the line widths of the mechanical resonator Which means that the cavity states coherent much shorter than the mechanical resonator could stay coherent And this difference is several orders of magnitude and can be improved So maybe we could just Generate quantum states of the cavity And then upload them to mechanical resonator and then read them out On the scales for which cavity would not be able to keep them And if you're kind of Really thinking about you know Technologies, maybe we can use it to build a new quantum internet or something like that Now I will actually not be talking about quantum states. I will talk about them a little bit on my lecture on Thursday But kind of the first Point is not really to talk about quantum states, but about can we at all make mechanical resonator quantum And here there are two obvious problems And one obvious problem is that if you want the mechanical resonator to operate in the quantum regime We need to make sure that the frequency of the mode is much bigger than kb2 And we have seen that for mechanical resonators unless you go To really very high frequencies to two gigahertz If you take usual mechanical resonators, that's not really possible. So there are two ways out. Okay. I should give numbers I I've we have already seen them, but let me repeat. So for instance, if you take one kelvin The frequency omega not f corresponding to this one kelvin would be would be 100 gigahertz And we just don't have mechanical resonators at this frequency So you even need to to go to really high frequency. You can maybe take 20 or or 10 And hope that it's still okay if you go to to low millikelvin And and that's one direction and another direction is to cool this mechanical mode I will not be talking about cooling. There will be enough people on this school talking about cooling Uh, I don't think I can add anything but but that's they should always keep that this is an option So there are only two options. Either you work at high frequencies mechanical frequencies or you cool down the system If you don't do any of that then then you're not quantum Now once you solve that there is another problem That you actually need to make sure it's quantum That and it's not as trivial as you think Because you need to identify some features Which are quantum You need some what what people call So you you need something which you would look at and you will be sure That this is a feature of the quantum quantum motion of the mechanical resonator And that's actually not trivial because if you have a harmonic oscillator Classical harmonic oscillator and quantum harmonic oscillator almost similar I mean, of course you can quantize it But once you start calculating things like average displacement or Average fluctuations of the displacement, you will see that they are the same in quantum and classical case So we need here to first of all, you need A quantum detector. So you need a system which would be able to measure quantum properties And for instance, we were we were talking about Uh in in my first lecture I spent some time on On capacitive coupling and and on charge detectors. I didn't call them charge detectors, but But but you have you can read out the position with charge And and we were talking about that but they're classical So even if the oscillators quantum and you are using exactly those methods You will not notice that the oscillators quantum So you need something else And and you need to decide what you want to measure which are the signatures of mechanical motion Now today, I will stick to the experiments And and there are not so many things which you can measure Um and and the first experiment, uh, that's by by Andrew Killent and John Martinez groups So that's 2010 what they have done. They actually didn't use cavities They used they here they sold it in in terms of of of of taking a high frequency high frequency resonator So they had a specially fabricated resonator with six gigahertz frequency And they uh, they still needed to solve the second problem and they coupled it directly to a qubit Now again, I mentioned qubits today and I'm not going to explain what their qubit was But but the qubit is a two level system over which you have full degree of control So you can uh just a second you can basically If if this qubit coupled to another quantum system, you can transfer this quantum state to the qubit and you can measure this quantum state You can get all the information about it. Yes, please Is the question is why all the qubits have this frequency? I I believe I that's that's exactly that's exactly the same reason why the the superconducting microwave cavities have Frequencies around six seven gigahertz and I believe that's a technical reason But I there must be experts in the audience who know better than I know What I was told it's something genuine has to do with the external circuit Can anybody John can you confirm that? I mean, I would say below like three gigahertz, you know, you want to be far from the gap between the gigahertz There's a lot of little things that no real reason why five gigahertz or 15 gigahertz they could be I think it's more than minutes Good and so what they what they did they did couple it to to couple it to To the qubit and they measured the state of the qubit and got information about the state of the Mechanical resonator and they actually claimed that they could They could manipulate like they could address the too lower state of the mechanical resonator zero and one and coherently manipulate them Now this expert actually nobody ever Have repeated that experiment or have done in this direction And I think that's because they they encountered a number of problems And in particular one problem was that they The the the lifetime of this quantized states of mechanical oscillator Was very low and that's unexpected because as I argued Actually, the mechanical line width is very high and you expect this time to be mechanical line width But for whatever reason it was it was it was much much the time was much lower than Than than than the line widths and because of that they actually don't have very good pictures. So I Filled my slide with pictures which are kind of supposed to to reinforce what I'm saying, but I I'm not actually going to show you what they have actually right now this picture we have seen That's the experiment of of john measured by him And That was the first experiment which actually measured which addressed the occupation So you can say okay fine. Let's If it's too difficult to manipulate to manipulate the levels, it's of course great if you could manipulate the levels But it's difficult. So instead of that Let's let's start with some simple things Let's for instance Look, what is the average phonon or mechanical resonator occupation? And if it happens to be below one, we just call it quantum limit Well, I mean that's you can you can you can debate whether that's a good definition or not, but that's a reasonable definition Right and there have been at least three experiments Showing that that you can bring there if you can see below one and this is one And and then if you are below one you are at the quantum state Now in his design Actually, the cavity is still seven gigahertz, but the mechanical resonator has a low frequency So you have to cool it down a lot And that's the first of I I said there are three that the first of the three and the two I don't have pictures of what they measured but they measured the same thing And I have already shown you the experiment themselves Which are those two experiments by oscar pentas group and and by by That'd be skipping this group Now I will have two slides left And that's again something which which which which my colleagues in delft did so that simon drop lacher's group simon will be here next week And I is you he will be talking about that though. I'm not sure Uh, you can also I mean there are a lot of other things you can measure Uh to convince yourself that your select is quantum Now one thing is is the is is the occupation of the of the states, but you can also For instance, you can create entanglement and there are entanglement measures And then if you are measured something And and you got the result which is compatible with the entanglement you are sure your system is quantum I don't know belt test or something Uh, now what they have done they have actually measured the correlation function The correlation function is defined as that so you take that the correlation function of Photons of phonons, sorry So be a phonon so mechanical resonator You take two creation operators here 20 relation operators here at different times So they are delayed That's car is you can measure it directly. Okay. No just a second you you normalize it by that And this is something which is always below zero and two And there is a statement which is known from quantum optics. I'm not going to derive it But if you take that at time zero so time delay zero so all those Things are at the same time Then if this correlation function time zero happens to be below one then you are surely in the quantum state in a non classical state And they have measured that the the problem is it's of course very difficult to measure It can be easily measured in optics There are interferometry experiments which can measure this correlation function for photons And that's something which exists already for like 30 years or 40 years People can measure it very reliably with with the interferometry experiment With which two detectors and they they they look at two detectors and then they see if there any correlation Now the the added value of this experiment they actually find a way of translating Photons Phonons into photons So they have some protocol which is too complicated to be explained here This is by the way the cavity I explained in the morning. I've taken it from that slide And they found some way of addressing Phonons by looking at photons And they got a clear message. So they I should also add I don't think whether I have it. No, I don't have it on the slide But they used resonators with very high frequency. So they had was it six gigahertz So they had the resonators which they don't don't need to cool down And uh, that's uh, that's a bit difficult to see from the figures But for instance here, it shows this correlation function at zero as a function of the initial number of Of Phonons and that actually Uh, it shows that that there is a reasonable range when you are clearly below one meaning you are clearly the mechanical resonator is clearly in the context um, okay, so There could be still some things here to do That could be as I said, it could be cool if somebody could generate entanglement But probably the next major step would be to address quantum states to really create a given quantum state In the resonator in a mechanical resonator and be able to read it out And that's something which I'm not going to address today, but I will try to address it in my last lecture Uh on on Thursday and I think for the time being I'll stop