 Oh, I should put a microphone, I'm sorry. Now it is done, I don't know, because I have that. Oh, okay, great. I still need to put it somewhere. All right, so let me first take a couple of minutes to explain what I'm going to do. So I will actually take an approach almost opposite to that of Clemens. I will try to be broad. I will try to cover many things. If I have to be broad, I will necessarily be shallow. So I will not have really time to go to long derivations. I think on a couple of occasions I will try to sketch the ideas of the derivations, but I will not have time to complete them. I will mainly be staying with the slides, only occasionally going to the blackboard. And my idea is that, I mean, if you see something that you think is cool, that's in the literature. Just go find the literature. I will help you by providing some references. Everything is available, just read it and see that it's indeed cool. All right, I will have some overlap with other lecturers. We didn't coordinate, really. So I will have, definitely I will have overlap with Clemens. I will have overlap with John Tuffel, which I will only figure out after his first lecture, and then my second lecture is immediately after him. So I will not even have time to adjust. I will have overlap with Mark Dickman and probably less with other lecturers, which is also fine because if you see the same thing explained by different people, it usually helps. So I will try to explain things, and as I said, I will show you probably a lot of things. But my lectures are not about priorities. I mean, there will be experts in the audience for some of the topics. So if you see, well, I mean, I will probably be heavily skewed towards dealt results just because it's easier for me just to take pictures from my colleagues than to cut them out from the papers. So I mean, if you see that the time I'm doing something and something needs to be added, please add it, just feel, feel, feel free to interrupt me, feel free to ask questions, and of course I will be also available after the lecture and during the workshop hours. Now I will give three lectures. One today will be about nanomechanics. One more will be also today, the last one, that will be about microwave optomechanics. And the main idea of that lecture will be to show you that you can do the same things with microwaves, which you can do with optics, with real optomechanics. And then I will have one more lecture on Thursday, which will also be on microwave optomechanics, but there I will concentrate on the differences, on non-linear effects, and that will be close to my own research. So there will be something else. All right, now let me go through that one. So I have like four themes. I will first go and explain what are mechanisms of coupling of mechanical resonators to everything else. Then I will go to something which I write here, detection. And what I mean by this detection is that often people don't particularly care about mechanical resonators, but they want to measure something. They want to measure some effect, and it's just easier to use mechanical resonators as a tool to measure this effect. I will give some examples, but let me give one now. I see Robin Doleman in the audience, and he is measuring several properties of graphene, which he has a poster and he will be, I hope, happy to explain all of you what he is exactly doing. I'm not going to talk about that. But his focus is that that's what he wants to measure, but the tools he is using are mechanical. And I will show some examples of the experiments with that idea. You don't learn so much about mechanical properties, but you use them as a tool to measure something you're interested in. Then I will spend some time explaining doubly clamped beams, and hopefully we will learn something about some things which are related to that. And then in the very end I will have, I will talk about graphene and 2D materials, also concentrating, of course, on what is related to mechanical properties. Right. So we are at International Center for Theoretical Physics. I am a theorist. So let's start with an equation. And that's an equation which we have seen already today. You have actually seen it many times before. If everything is fine, you should have seen it at high school. And definitely, if not, definitely at the first year of the university. And that's something which we all very well know. Right. That's an equation for a single mode mechanical oscillator. So we have inertia. We have friction. I've written it so that this friction coefficient is the frequency divided by the quality factor. And this is just the return force of the pendulum. So that's a linear oscillator. So this return force is linear. And that's a driven harmonic oscillator. I've written here. The mass goes to the denominator. Otherwise, it would be in all these coefficients here. And here I've written the harmonic force which drives this oscillator. And again, that's something which we're very well known. We know how to solve that equation. And we know what happens. Right. We know that the solution would be the motion of this oscillator with the frequency which is equal to the frequency of the driving force. And the amplitude, there could be some phase involved. And the amplitude is strongly dependent on the frequency with which you drive the oscillator. Right. So it depends something like that. That's supposed to be a Lorentzian. I excuse for my drawing skills. And actually, I will have, you know, you will have more examples later on in this and other lectures that I cannot really draw. So I just apologize now for all of those examples. So that's supposed to be a Lorentzian. And then we know that the highest amplitude is actually if you drive it at resonance. So if this frequency equals to the eigenfrequency of the oscillator. And we also know that the amplitude and the width of this Lorentzian are determined by the quality factor. Right. That's something which I hope we kind of commonly share this knowledge. So I don't need to go into details and explain. Now, what is nano-mechanics and optomechanics as a part of nano-mechanics about? Well, from the theoretical framework. It's about solving similar equations but more complicated. Right. Now, if you only care about properties of this oscillator. I'm sorry that should not be harmonic. It should be more advanced oscillator, mechanical oscillator. It's not harmonic what it's been. Now, you start adding some details and details. Well, I mean, you add some force which is non-linear. That's why it's not harmonic. Your quality factor may start depending on the position. Your driving force may depend on the position. And then you start solving those complicated equations. And that's actually a large part of what many of us are doing. I just know that there is something going on. I mean, you add parametric driving here. You add nothing terms here. And you see what happens and what are the properties of this resonant. I think actually I anticipate that what Mark Dickman will be talking about is a large part is about that. It's about how can you solve very complicated situations which arise when you start adding these crazy things to the equation. Now, okay, as I said, that's a theoretical approach. Now, of course, you also have another part which is maybe more interesting is how do you arrive to that equation? How do you know which equation you need to solve? Because solving equations is, well, mathematics. I mean, you still need to make some approximations. And then because we are making those approximations, mathematics becomes physics. But it's much more physics in how you actually write and derive those equations. So you basically need to start with a real system which you're interested in and see what this system is doing. And what do you need to add to describe the system and what equation would be describing it? And that's, for me, it is much more fun than just solving the equation which you get from somewhere. And we will be also talking about that. Now, it's usually, it's not enough just to look at the oscillator. Usually things are much more complicated. Usually you don't just have an oscillator. It's coupled to something. And this something also has its own dynamics. And this something here in this equation I added y here. And this y also has this equation of motion. It has its own evolution which you also need to consider. And then they are coupled. And then things really become complicated. And we have just heard in the lecture of Clemens an example of those. So we had an equation for the oscillator, which in his case was a simple harmonic oscillator. But also this oscillator is coupled to the field and the field has its own dynamics. And we had just half an hour ago equations for this dynamics on the blackboard and coupling we didn't even consider yet. So that we'll be coming on the next lecture. So in this case you need to solve simultaneously an equation for the mechanical oscillator and the equation for the fields. And they are coupled and we will see they are coupled actually non-linearly. And the field is quantum and the oscillator is sometimes quantum. So things can, the degree of complexity can actually increase infinitely. And it's also nice. Now, okay, as I said that theoretical framework, let's see it at more specific examples. And in real life, I mean of course mechanical resonators could be coupled to anything. In real life, at least in real life, which we kind of all are part of, they are typically coupled to five different things. So they're coupled to charge and the main mechanism is capacitive coupling. Now I'm just giving names. I will go immediately explaining what it means. They are coupled to electromagnetic radiation and the main mechanism is radiation pressure coupling. They are coupled to spin and I have actually no idea whether there is any name for this mechanism. I put it blank. They are coupled to flux or magnetic flux and that's inductive coupling. But actually, well, I mean from our everyday experience, we know that the easiest thing you can couple a mechanical resonator is another mechanical resonator. We have a lot of things which are described like coupled pendulum. Now it actually turns out that if we go to the nano world, to nano mechanics, things become more complicated and it's more difficult to couple them. And I will not even, I will now have four slides, but eight slides about that. I don't have slides explaining coupling between mechanical resonators, but I will later on, I will show you an experiment where this coupling is important. And then we will see what is interesting about it. Okay, now let's, oops, something has gone wrong. I think it looks like my battery is dying, so I will probably use that one if it, oops, now. Okay, that's what I want to know. Yeah, I think I have that one thing. So, well, for every mechanism I'm going to discuss, I will give just one theoretical picture and hopefully an explanation. And then I will show you an actual experiment which uses that. Without, as I said, without going into much details. And I think this one should be fine. If they're compatible, which I'm not even sure. Nothing else to be, seems to be fine. Yeah, great, thanks. Right, so let's start from the explanation. So, well, it's capacitive because we are talking about a capacitor. Right, so the capacitor is something where you have charge and you apply voltage and charge is proportional to the voltage. I see that I have some formulas missing on my slide. So you have q equals to Cv. C is capacitance and you can write the energy, which is the simplest case would be q squared over 2C. Now imagine that one of the plates is mechanically moving. And then if one of the plates is mechanically moving, then this capacitance would be a function of the position of the plate. Now it's actually much worse. Because imagine it's, well, we don't have to assume it's moving, but let's say one of the plates is somehow deformed. Now like on my picture, that's supposed, it doesn't really matter what you do, but here we have three electrodes. So we have left and right and you have one plate which is sitting between them and another plate is a gate. And usually you just apply voltage to the gate and see what happens with the transport from left to right. It's not really important for my story. Right now it will be important later. But what is important is that if this plate has a non-trivial shape, then your capacitance depends on this shape. And not just on one coordinate, but on the deviation from the horizontal at every point. And that's a big deal, right? The displacement becomes a field and then you need to treat this field and whatever. Now what people usually do, that's already, Clemens already mentioned, people usually assume that it's not just an arbitrary displacement, but you go to the normal mode, you go to the description of the mechanical oscillator in terms of normal modes and you just select one mode. Which in this case, I have selected, I mean those modes are, let's say if these two ends are fixed, then this mode is just a sign because sign is the only function which is harmonic and which is fixed at both ends. So different modes are just different signs and this is the first mode and the second mode would be doing something like that and maybe I should draw it. The first mode is this, the second mode would be that, the third mode would be this. But every mode is only characterized by one number and this number is essentially the amplitude of the displacement. And if it's just one number, you can say, okay, we're just in one mode, so we have this capacitance which depends on only this one number and for this number has its own dynamics, that's a harmonic oscillator. We can write this equation for harmonic oscillator, for the z, which, yeah. And, okay, fine. Now what we have is the energy of the capacitor which now depends on the position of the harmonic oscillator and that's coupling. Because by definition coupling is if you have two systems, in our case it's capacitor and mechanical resonator and that's a little bit semantic because if you look at the picture, right, then the mechanical resonator is a part of the capacitor but still one of them is electric and one of them is mechanical and that's why we can separate them. We know that there is energy of electric system and energy of mechanical system and if the total energy is the sum of the two, then they are non-interacting, right, does everybody follow? But now if the total energy is not just the sum of the two, then they are interacting and we can write the interaction explicitly and I will have it later on on the slides but we can actually take this formula, so this q squared over 2c times that and we can say, okay, maybe the z is very small and typically it's very small, it's probably much less than the distance between the plates and then the change of the capacitance is very small and we can expand it, right? And then we can say c is some c0, whatever it means, which would be at z equals 0 plus dc over dz times that, right? And then we can put it over there and then this q squared over 2c of z would be q squared over 2c0 minus, let's see if I can write it immediately correctly, 1 over 2c squared c0 squared dc dz times z times q squared. Now, the important thing here is that that's the interaction. So for this capacitive coupling, the interaction is just a product of mechanical coordinate times charge squared or if you want voltage squared. And I will explore that a little bit further in the lecture today in the evening. Now, I spent some time on that. Let me mention one thing which I will probably be mentioning by every coupling mechanism. Now, I said, okay, if we have interaction of mechanical resonators and charge, it's typically capacitive, but it's not always capacitive. You could also imagine other mechanisms which couple mechanical resonator and charge. And I just give here one example, for instance, piezoelectric coupling. I mean a piezoelectric is a material where you can induce strain by adding by by applying voltage. And now if you induce strain, it means you also have mechanical displacements inside the material. You have some rearrangement of some parts of the material which cause strain. And if you can use it for any nano-mechanical experiment, it's not easy because those displacements are happening in some bulk system. But if you can use them, then you will have piezoelectric coupling of charge and mechanical resonator. That's not the mainstream, but people are using that. I have chosen a random picture which illustrates this experimental picture which illustrates the capacitive coupling. I will have more pictures later on because that happens pretty often. That's an experiment of Mark Bockret who was at the time of UC Riverside. And that's about their group was studying graphene. And so what they do, they put a graphene plate over a hole. So they have holes. And beyond the hole, there is a back gate. And this back gate is exactly what I put here. So you just put a voltage by voltage, you can deform the graphene and they could study it. And that's exactly their capacity. Coupling, they could also put some time-dependent signal, but I will not go into the details. Right. Next one is interaction with light. I will be very brief because Clemens will spend some time on that in his next lecture. So that's an optomechanical cavity. We have seen that there is, if we just take a single mode of light, the cavity frequency, which is well-defined. And now, if, well, the frequency, the cavity is two mirrors, and if one mirror is moving, then the cavity is modulated because the frequency is determined by the distance between the mirrors. So you can say, okay, if there is a mechanical motion of one of the mirrors, then the cavity frequency depends on the position. And similarly to that story, I can actually expand it. This displacement is very small. I can always expand it. And if I expand it, I take this Hamiltonian, which we had and Clemens had it already. So h bar times the cavity frequency times the number of photons. And from this expansion, I will get the radiation pressure term, which is proportional to the number of photons times the displacement. And that's radiation pressure coupling. Now, what I want to mention is that actually again, that's clearly the mainstream mechanism of interaction of light and mechanical resonator. But that's not the only mechanism. You can, for instance, think about optical phonons in solids interacting with light. And they have completely different mechanism because that's dipole moment which they create interacts with electric field. That's not, people are not using that. It's not convenient. It's difficult to make cavities, but it's something which in principle provides you with the alternative mechanism. And experiments here, that one I've picked from my colleagues in DEL, the Simon Roblachas group. Now I will maybe spend, Simon will be here next week, possibly talking about this experiment. I'm not sure. Maybe I will spend half a minute explaining what you see. Because, well, cavities to mirrors and you expect to see two mirrors. But here there are no mirrors. Here there are those strange things which look like black and white and which contain some strange inclusions. Right, those things are actually parts of photonic crystal. So those are basically you take something and you drill holes. Now why drilling holes is a mirror? I mean that's something which I maybe spent a bit of time. I mean one thing which most of us know from basic solid state physics or you can also know it from basic optics that if you have a lattice, a perfect lattice and you send a wave into this perfect lattice, there is no scattering. That's something which is normally taught in solid state physics. Not necessarily, it's not necessarily emphasized, but it's kind of all but implicit. In solid state physics you need impurities to have scattering. If you have no impurities, you don't have scattering. Everything goes through. So if you just take this thing and drill holes which are all the same at the same distance, scatter light and it will also never scatter sound. You can send light along the crystal. You can send sound waves along the crystal. If the crystal is perfect, they will never be scattered. So if you want to make scattering, you need to take those holes different and then they can actually do it in some clever way so that these different holes act a little bit like those two mirrors. They act like a cavity and this cavity is also at the same time cavity for light and cavity for sound, for phonons. And in this way they actually make this the enhanced interaction between light, light and phonons and I will not go into any other details even though I will show you the result of this experiment later. Now coupling to spin there are different mechanisms. The mainstream mechanism is here so that that's actually the cartoon taken from the experiment of den Ruger at IBM 13 years ago. The idea is that if you just take two magnets they exert mechanical force on each other. So what you can do, you can just take a cantilever, you can put one magnet to the cantilever and you can use this cantilever for sensing of a different magnet somewhere around. So if there is a magnet somewhere in the substance some magnetic moment you can use this cantilever with a magnetic tip to sense this magnet and actually it was demonstrated in that very experiment then you can actually in this way you can see a single electron spin. The sensitivity is so high you can see just a single spin. And that was actually later on that as I said it's quite some time ago it was later on developed in the separate field which is called magnetic resonance force microscope. Now the next picture would be a picture of an experiment that it's notoriously difficult to find a good picture of a spin detection experiment probably because they're very boring. Because well basically you need a cantilever with something in the end that's one of the best pictures I have so that's from the group of Alephia Axis in Grenoble. They used this tip which in the end you see this shining point which is increased here doesn't help much but okay that's the same and that's an envy center. Now I guess probably some of you must have thought about envy centers that if you take diamond then sometimes there are natural defects so you can also use those defects if you want where instead of a carbon atom you have well diamond is pure carbon but you can sometimes replace carbon with a nitrogen and the next to the nitrogen there is a vacancy and for whatever reason those complexes so those defects they have spin one itself and they are they use for very good they use a very good center so this something with an envy center is becoming a mainstream and there will be more and more and more groups coming. Now I will not be talking about spin anymore in my lectures and I don't expect to be honest other lectures to talk about that so for spin I will give a credit and I will add one more slide and by the way I think tomorrow at 2 we will have a symposium or whatever it's called by Robert Schacht and Sanyat Golarelik in the up there in the main building of ICTP talking about nano-methanical spin tronics so that's related to the things I mentioned. Okay so I prepared another one again there is no good picture of the experimental device but there is a good picture of explaining what they are doing that the experiment of the group of Patek Malikinsk in Basel they also use envy centers and envy centers have another property the energy structure of an envy center is strongly sensitive to the strain so if you deform the diamond that would change the electronic structure and actually what they have done they managed to put an envy center in some other substrate they put it on a company driven by the other substrate and because of that they were able to drive the envy center itself and this is the coherent drive of an envy center which is oscillating between the probability of occupation of one of these things. Right, now inductive coupling I will actually spend quite some time on my last lecture on Thursday so I will skip discussion of it now let me just mention that inductive coupling is about inductance inductance is the proportionality coefficient between flux and current and if you take a time derivative of that time derivative of flux is minus voltage so you will get voltage is inductance times I dot and then you can also write the energy which is inductance times current squared over 2 and in some situations which I explained because the inductance could be position dependent and then you can similarly to capacitive coupling you also get inductive coupling would be also not surprisingly in the same way x times I squared. Now as I said let me go to to those things let me give you some example where people could use mechanical resonators which were difficult to measure before that and let me start with the example of the experiment of Jack Harris I will show the experiment on the next well in two slides I think but let me first explain what the experiment is about the experiment is about measuring persistent currents and persistent currents is a big deal so it was a big deal in the 90s so the idea is imagine you have a metallic ring and there is some flux going through the ring so perpendicular now if you have flux you will induce current in the ring even if there are no contacts usually we think ok if you have some conductor then if the conductor is equilibrium there is no current but if you put voltage then there is current and here there is no voltage but there is flux which actually still induces current and to understand that let's very briefly go through the sketch so you just take an electron which is sitting in the ring and the Hamiltonian of this electron is just p squared over 2m but since there is magnetic field this p is not p but the so-called long momentum so you need to add the vector potential now if that general now since they are sitting on the ring can only move in one direction then the only relevant variable the only relevant coordinate is the phase along this ring I can just write everything project on a phase so then I will get the derivative of this phase which is just momentum and I get flux divided by flux quantum which I for whatever reason didn't write here the flux quantum is just this combination of fundamental constants so this is speed of light and this is charge of element and now I can say fine we have this Hamiltonian obviously the motion has to be periodic so the because it's periodic the wave functions can only be like that so it would be e to the e times some integer times this phi and we can ask okay fine what are the energies of such a Hamiltonian with such functions that's trivial to calculate same way you are calculating whatever levels of a single of one dimensional infinite square well you can just do it and then this is the result you get so it depends on flux it depends on this m you can say that it is given so at every given n as a function of flux the energy is a parabola which is drawn here so that's for different values those parabolas are for different values of the flux and then for every for instance for every given flux the ground state is whichever parabola is the lowest so the system adjusts its number its number n to the flux so you just take the n which is closest to phi over phi naught and then you are in the ground state okay but that's not so much important what is important is actually that the energy depends on the flux the energy of the ground state in particular of any state depends on the magnetic field and if the energy depends on the magnetic field then there is a current and then we get current in the ground state and okay that was well I mean this you can also go on and estimate the amplitude of the current how strong would it be and unsurprisingly it's actually not strong it's very weak it's very difficult to measure and then if you start going into details that's actually why it was a big deal you can for instance easily do it for just a single ring but for a single ring it's difficult to measure because the signal is too weak you can also go to disordered rings and look at ensembles of rings you can just take many rings and see what would be the average current and then there are many interesting things for instance there is a difference between if you calculate it in canonical and grand canonical ensemble and that's actually why it became one of the main most important things in metascopic physics which I will skip for the time being what is important and what actually that the quote from this paper of Jack Harris's group is that there were a lot of theoretical results there were some experimental results and they didn't match very well so there was some controversy which at some point people gave up stopped measuring and that remained unresolved for whatever, 15 years or something and now this paper came so it was 2009 what they did they put those rings which are here on a cantilever actually that's a sketch the real structure is here there are just rings and rings and rings and rings around that's many of them because you want to enhance the signal now what happens is that you put them in the magnetic field and magnetic field produces this persistent current and the persistent current actually produces torque and shifts the frequency of the cantilever and you can measure this frequency shift and actually the frequency is the quantity you can measure with the highest precision of everything of all measurements which are around the most precise measurement is the measurement of frequency and I measured the frequency and then everything was fine with the signal there is maybe one more thing I want to emphasize before I go to the next experiment I think most of you are serious but there are some experimentalists around just look at the list of authors so they have one, two, three experiment... three theorists on the paper so Leo Glassman is a senior person who just moved to Yale I believe at that point Felix von Oppen is a senior person, he's a collaborator and Arangina Sir I think he was a postdoc at the time so you basically need three theorists on experimental paper to make it into sense think about it that's a different story that's an experiment of Eva Vaik when she was still in Munich 2013 she will be coming maybe even during my lecture here today what they looked at they looked at two coupled oscillators so I said okay we can easily couple oscillators well maybe not easily and this is one example now there are oscillators if we think about couple oscillators we think about couple pendulars we have like two pendulars and then try them maybe and see what they're doing now in their case it's not at all like that in their case they have this some complicated structure and when they talk about coupled oscillators they talk about actually two different modes two different normal modes and one of the modes is in plane motion and another mode is out of plane motion so those are just two different modes which are coupled because of some properties of this material and now what they were interested in they were interested in coherent manipulation of those modes so they have for instance they have a control parameter which is gate voltage so it's again about capacitive coupling and with the gate voltage they could drive to start with those two modes and they see that at some point they become very close to each other and if they become very close to each other well I mean you can populate the lowest mode and the highest mode and see what it does so you can just look at this mode and the power is the amplitude of the motion essentially you can look at that mode and see what this mode does and you can actually see that if you drive it in a smart way then the power goes from one mode to another mode and oscillates between the modes which is very similar to what people do in qubits you have two states you populate one state and then you have Rabi oscillation between the states and actually they made this analogy even further they looked at the block sphere because the qubit is essentially spin one half and for spin one half you can put all the states of the spin one half on this block sphere and see what it's doing I will not go into detail in the next segment which shows that you can do a lot coherently with mechanical systems now to finish that part let me talk about mass detection the idea of mass detection is actually very easy and that was developed for many years in the lab of Michael Rookison Caltech the idea is that you take some mechanical element it could be double-double-dim it could be a cantilever I will show it and you just put it into some medium and then there are some molecules coming and then molecules stick to this beam and if the molecules stick then the mass increases and if the mass increases the frequency goes down and as I said measurement is the most sensitive measurement you have around so you can just by measuring frequency you can measure mass I can say okay well that's a massive system right it's nano but it's still a big big piece of some material I actually don't know which material it contains zillions of atoms what if you just add one molecule why should it change anything well okay depends of course what is your sensitivity and what molecule you are adding so this is the experiment of 2006 and they had they had a sensitivity of 100 zeptograms which is what is it 10 to the minus what is it zeptogram 21 right yeah 10 to the minus 21 just to give a perspective for the LIGO measurements which Clemens showed the expected sensitivity was zipped to meters so 10 to the 21 meters and they didn't reach it because they were lucky with this with this supernova explosion so they could go a little bit higher right now zeptograms have been achieved 10 years ago what they do that basically time it means they just open this shutter and they start adding molecules and they just see what happens and the frequency goes down and actually it goes down in almost in steps and each step corresponds to this zeptogram sensitivity now they actually the same group that 2012 experiment they have done it with a different device they claim they achieved a single molecule sensitivity now of course it's not a hydrogen molecule it's something like a protein so which is still pretty heavy now that was roughly same time the experiment of the group of Adrian Bartelt in IKFOR and they claimed zeptogram sensitivity zeptor is 24 I think but they used carbon nanotubes and that has certain limitations but that's how you can just I mean that and I think it's still improving so that's the way how you can you can do things and now as I said so in those experiments you particularly don't care about mechanics you just use mechanics as a tool to do something I mean here you can measure frequency very well you just see what happens what happens with the frequency and you just use something which is completely mechanics unrelated you can use it as a sensor or whatever concentration of some substance which whatever right now these two doubly clamped beam part I will organize it in this way so that's one particular experiment by the group of which I must admit I learned earlier this year about the existence of this experiment from Fabio and he is actually the person who probably have done the most in this direction but I will only go through very simple things so let me first explain what the experiment is what they actually measure what they see and then we'll spend some time understanding what they have seen right so what they use either and by the way there are also two highly qualified theorists on this paper what they use is a carbon nanotube which you can hardly see here they helped us but I'm not sure I mean I can see it but probably probably those of you who are sitting on the back cannot this carbon nanotube is suspended between source and drain which means you send current from source to drain through this carbon nanotube now carbon nanotubes could be metallic or insulating and in this case it's relevant for the reasons I will explain in a moment now on top of that they have gates this one, two, three, four, five those are gates they are lying below and what you can do you can selectively put voltage on one of the gates you can also put on several but that would make things more complicated so let's say one and by putting voltage on one of the gates you can actually deform this nanotube so you can for instance put this capacitive coupling which I outlined so for instance if you put voltage on the gate one then this part of the nanotube gets attracted to gate one so you will get something which is more straight and more deformed and for instance if you put it on gate three which is in the middle then it's something which would have something like this picture which I showed you before which has closest parties to the gate three okay now on that they measure current as a function of this gate voltage well this is shown it is shown for two different gates this is gate three, this is gate four but they have it for all gates oh yeah here I should never mind and what they see there are two important developments one development is that the frequency goes up with the gate voltage and second is that on top of this monotonic dependence you also see this strange minima so sometimes it stops going up goes down then goes up again but on average it goes up and these are two things which we need to understand that the picture which I already shown you and actually it's that the picture which specifically I've taken from one of my very old presentations because we were looking at carbon nanotubes so now we can think that this line is a carbon nanotube and if this carbon nanotube is deformed it becomes like that and that modifies the capacitance also I should mention that that's of course not the first experiment with carbon nanotubes around that's another one from Delft by Gary Stillman he suspended the carbon nanotube between those two electrodes and the gate is just underneath so it's a very similar idea but he just had one gate and now here I put some scales and I screwed something up but anyway so the size you see it's 400 nanometers so the length of this carbon nanotube is 500 nanometers and the frequencies in his case was 140 megaphones I think in Shahalilani experiment the frequencies around well it starts at 200 and it starts at 80 it depends exactly what you are doing with it nanotube but in any case it's in the range of 100 megaphones okay let's first understand why the frequency goes up well very roughly speaking and kind of an intuitive level if you apply voltage to the gate you just pull the nanotube right that's what capacitive coupling is about that's still on the blackboard behind me if you well here could be let me just add it so we have it that's proportional to the displacement times voltage squared so that basically means if you apply voltage you pull it mechanically right you have a mechanical force on the nanotube you just pull it now from everyday life you know if you pull something if you just pull it like that or like that if I play a guitar I increase the frequency and that's exactly why the frequency is increased there is nothing special about nano except for that's probably easier to pull it by gate than by pull it by whatever by hand but there is nothing really special here so that just the frequency goes up because you pull it by the gate and because it's v squared it doesn't matter whether you put positive voltage or negative voltage whatever you do you pull here it's actually much it's a bit more complicated but let's go through the detail so again this is this is an equation which I had on my field slide so K there is a nanotube in the first approximation is a harmonic oscillator so it has inertia it has friction it has this return force now we don't drive it by itself we don't apply any harmonic force which I had in my equation but what happens we well we pull it by the gate and there is this force coming from the gate and we just need to calculate this force it's not trivial I'm not going to do it it's some non-trivial electrodynamics I will just show you the result the result is essentially that so this is the energy and the force is the derivative of the energy of the coordinate right so the force which we get here would be proportional I don't actually know where how one half is coming probably because it just divided by two anyway so there would be derivative of the capacitance and here that would be normally just voltage squared but also because we have a gate it's not voltage squared it's voltage minus gate voltage squared and the voltage on the nanotube itself that's something which you would need to calculate you just solve this simple electrostatic problem like you have this nanotube the capacitance to the gate capacitance to the source capacitance to the drain you just write all those whatever energy differences and then this is what you come now what is important that this first of all that this force is still dependent on the coordinate I mean you can of course approximate it by this and this by linear relations but then we don't get any interesting effects but if you go beyond linear relations if you take capacitance which is more than eggplant which is x squared and if you take that which is x squared then we still have force which is dependent on x and now what we do we can first of all expand this force there will be terms which are proportional to the x to the displacement terms proportional to x squared and you see here if you have on the right hand side we have terms proportional to x they would renormalize the frequency of our resonator right because we just add that and that and then we will get a new frequency squared and that's exactly why the frequency is changed on the level of equations and now that also if we take these terms into account then our resonator which is by itself was a harmonic oscillator it becomes a non-linear oscillator and it could show all kinds of things which non-linear oscillators show and I will not be talking about that now but I will return to that in my lecture 3 when I will be talking about non-linear so that so far we understood why it goes up we still don't understand this minima the minima are not or at least not trivially in this expression we need to understand something more to understand the minima and there's something more it's called Coulomb blockade and Coulomb blockade is let me first start with the intuitive explanation what is it so imagine you have some very small object and this object has which is a part of electric circuit and this object has very small capacitance now if the capacitance is very small then this energy q squared over 2C is very large now imagine I want to transport charge through this object now I just put it here so I put a source and I put a drain and I put voltage between source and drain and now to transport charge I need first to put electron into that object and then take it from the object but if I want to add it to the object I have to pay a lot of energy because this electrostatic energy is very large and this energy must be coming from somewhere so it's either should be provided by temperature temperature should be high enough so temperature should be KBT should be higher than that or I usually because if you want to see something interesting you want KBT less than that otherwise if I want to overcome that I need to apply voltage which is very strong which is very high electrostatic energy and if my voltage is not high enough to overcome this electrostatic energy then there is just no transport and that's called Coulomb look-it people started to study it actually in quantum dots but right now for instance it's more much more common to have experiments on single molecules and whatever which can also surface conductors now let's go into some details which again I will not provide too much if you want to know more go and read it in the literature answer okay yeah basically I didn't say that but I will say that now the reason why we have Coulomb look-it is that because electron charging quantized right you cannot have half electron charge you can only have 0, 1 or 2 or 70 but you cannot have 1 half or 3 quarters and that's exactly why you need if you put an electron you put the whole electron that's why you have E squared over 2C and you cannot have less than E squared over 2C because that would mean you will have less electron charge than less charge than the electron charge now I can say fine let's say we have some state where we have N electrons here and some state with N plus 1 we could have for instance N equal 0 and N plus 1 equal 1 doesn't matter and now if those energies of the states of N and N plus 1 are drawn like that then we should have no transported zero temperature because if you want to have transport we must have an electron from here from the left which is somewhere in occupied states going here so electrons from occupied states can only go to this level but not to that level because they should do it at the same energy but then the electron also should be able to go to the right and if it wants to go to the right we should have an empty state to the right at this energy so we would only have transport if we have a state here but we don't have a state here we have one state which is again occupied levels left and right and one state which is again empty levels left and right so if we have in this situation we are in the Coulomb location we can formalize that by just calculating those are the conditions we have one level should be bigger than both voltages left and right and here I put the right voltage zero it doesn't matter and another one should be below those two and of course there are four conditions and one is redundant because I assume that this one is bigger than that one but it doesn't have to be like that it could be a cent bigger than a cent plus one and you can actually calculate those energies and those energies what is I'm not going to do that but what is important is those energies are always linear functions of the voltages on the left and on the right on the gate and so if you just start plotting that usually people plot it like this so you have the gate voltage here and you have bias which is a difference here left minus right then there are these structures which are called column diamonds and within every diamond so like here the number of electrons is fixed like for instance in this one the number of electrons is zero this one is one and this one is minus one here it's two and so on and if the number of electrons is fixed it means you have column blockade because if you want to have transport you need to be able to change the number of electrons you need to add one electron here and then from zero it would become one and then it would you would remove it to the right and then it would become zero again and that's not possible if you are sitting inside the diamond but that's possible outside the diamond here and here and you can actually take this into account because that's all in this formula you can take this into account and you can calculate the force and the frequency shift which I'm not going to do and in the frequency shift you have a term which is actually proportional it goes with minus and it's proportional to the derivative of the average number of electrons over the gate voltage and now I apologize for this drawing I mean it for whatever reason every time I put a background it's not my power point the background is a piece I don't know why so we have to look at it like that you see this picture shows this average number of electrons and that's an exact derivative of that picture because for instance if you take bias equals zero and you go along this line then you see that it's zero and then it's suddenly one and then it's suddenly two and so on so it just can be only an integer number because it changes in steps and now if you take a finite bias then those steps disappear and then become kind of a smooth connection between the plateaus but in any case you have plateaus which well this is another example the two plateaus are joined smoothly and if you are somewhere between the plateaus it means okay and now we know that if you just take this and take a derivative then we exactly get those minima which they see and they were able actually to fit them to the theoretical right and this is the similar experiment in the group of Gary Steele where they had also carbon nanotubes and that's also they could fit it to our experimental the experimental theoretical coefficient you see here when you add low bias you have one minimum and then when you have high bias you have two minima because then you are crossing this point now there is one more thing and that's something I would like to spend some time that you renormalize because of this column blockade you renormalize the frequency but you also renormalize the quality factor and frequency I kind of could hand waving explain quality factor I cannot well I mean you can of course say that I mean always if you couple something to something else if you couple mechanical resonator to something else then mechanical resonator loses energy and if there is coupling to some other system it can also lose energy via decaying through this other system so the the damping in this hand waving explanation should always go up which sometimes it fails and the damping goes down but and this is something which is very well known for quantum optics and it's called the Percelhex and I should have mentioned the shift of the frequency is called optical spring effect but to take it to the quantitative level it's actually difficult I will give you the formula which has been derived by by in those two papers which Fabio is in the audience and other people are not and I will on the next slide will sketch how this derivation what step this derivation goes through that will be probably hard so if you cannot follow just don't follow I will come back to some more easier things but before I go through that let me show this is the same experimental paper by Gary Steele's group they could also they could also fit the quality factor and they fit it with this expression and which against here you can you see that there is the derivative of this average number of electrons and the feet again then sometimes they have one and sometimes they have two minima for the same reason as the frequency and the agreement is not fantastic but at least you see the same tendencies now let me sketch the steps as I said that's probably difficult to follow if you find it difficult just don't follow and wait until the next slide right so the tool to describe this column blockate which I so far avoided to explain is the master equation and the master equation operates with this object which is a probability to find exactly n electrons and in usual column blockate without any mechanics this probability depends on the number of electrons and it may depend on time and you write an equation for its evolution in time and this is the master equation now if we also have mechanical resonator then this probability also starts depending on position and velocity of this resonator or if you want position and momentum it doesn't matter and you have to write the equation for the whole thing and this becomes I'm not even sure that if I do that the master equation is a good name first and probably it's already the Boltzmann equation but that's semantics and let me because there is one thing which I don't have on the slides so if you don't have mechanical resonator the equation is well then it's just total derivative dpn over dt is whatever I call the collision integral and I will specify right now what is it now let's imagine for instance two states like zero and one right then I can write dp0 over dt equals to what why should I have this probability to have zero electrons in my system why could it change at all well it's obviously the only reason why it could change is because electrons there are not zero electrons anymore but there is one electron it could be also minus one if I take this into account or maybe there is even two but the electrons go in this language they go from the state zero to other states how do we describe it if for the case when there is only zero and one nothing else then I say well there is some rate to go from zero to one I write here p0 and I put here minus this rate is probability to go from zero to one to unit time so it makes sure that I have the correct dimensions now it should be proportional to p0 because the more electrons I have the faster they disappear and it's minus because if they disappear my probability goes down right now there is another process I can only go to one because I've only chosen that there are two states if I choose more states they will go to all different states but it would be still proportional to p0 and go with minus right now there is an opposite process electrons can go from elsewhere to zero in this case they can only go from one to zero so I will have another term in this equation which is now proportional to p1 and with plus because if they come to zero then it becomes more the probability to get into zero gets higher so that's usual master equation I have written it for the case when there are only two states but I can write it for more complicated cases and that's basically what people do then you can solve it you also need another one for p1 which is written in a similar way you also make sure that the sum of all probabilities is one which for two states make problem very easy because p1 plus p0 is one you can only look at one of them I will not go into that there is one more step which is needed we still don't know what those rates physically are and for those rates we need to look at our actual system which is we have tunneling here we don't have any tunneling here it's purely capacitive connection so we have tunneling from the left to the tube, from the right to the tube from the tube to the left and to the right and then those will make up those four rates those four will make those two rates now in this case because it also depends on the position of the resonator I have to add additional terms so I keep this one there my rates now may depend on the position but other than that it's okay and what I should take into account that this total derivative now becomes partial derivative plus dpn over dxx dot plus dpn over dvv dot and now I know that x dot is v and that's here and I know that v dot is just proportional to the external force and this force is the return force of the oscillator, it's friction and there is also this force which comes from electrostatics which is proportional to n so for 0 there will be no force for 1 it will be false it's a stochastic force because every time you tunnel the electron sits there and generates a force and so it's there when the electron is there and the tunneling process is stochastic so this is actually a stochastic equation and then there is some way of dealing with stochastic equations so this is some of those terms are kind of random functions you can say this kind of noise you can take into account that I'm not going to go into any details but you can rewrite this equation as the Fokker-Planck equation and in this way you see that those rates, the role of those rates here they renormalize the quality factor and they also create diffusion in the velocity space that's all the complicated but at least that provides an expression for this quality factor which I put on the previous slide now we have 10 minutes and I will now go to something completely different so that if you got completely lost in that part you have another chance to understand what I'm going to talk about I will talk about graphene first and as I said graphene is a two-dimensional layer of carbon which a lot of people are using because it has excellent mechanical properties well it actually has a lot of excellent properties we know that there was a Nobel prize given out for excellent electric and usually electric properties it also has excellent mechanical properties and that's what makes it attractive and it's used as a resonator in a lot of experiments which many of those don't at all care what properties of graphene is and I will not be talking about those I will actually show you a couple later, not in this lecture so for the time being it's enough to know that that's an excellent membrane which you have a lot of control over but what I want to to show you are things which where you actually engage with the properties of graphene itself so what interesting could be done what do you learn about about graphene if you look at mechanical vibrations and now that would be pure theory and I don't know any experiments on that but the theory is nice and let me spend 5 minutes going through it so the idea is due to first Suzuru and Ando before graphene actually I think they they were doing it in carbon nanotubes and then by Diney, Katnalsen and Bob Mediana was that actually if you deform graphene you have to create local deformation and this deformation electrons in graphene see as fictitious magnetic field actually as fictitious vector potential and there are some there are some formulas you can if you know what those deformations are they are usually described as components of this deformation tens or stress tens or whatever then you know what you should what potential components you generate and then you have those components and then you have Dirac equation you just add those vector potential as gauge fields in the Dirac equation because electrons in graphene are described by Dirac equation and then the first simple thing you can do is just see what happens and then for instance you can have some fun by taking some shapes or deformation and that's what we have done in this paper you can have some fun looking at what actually what potentials those deformations would produce and what effect would it have on the properties of graphene because you have some funny time position dependent magnetic fields it's not something you usually do in solids you usually have in solids okay so I that's fun but that's not what I wanted to talk about what I wanted to talk is is that that's another paper so it's Fogler, Pinae, and Kant-Nelson and so what they have done said okay well let's just take a graphene membrane suspended over the gate well they didn't even have a gate so just suspended and see so that deforms it let's calculate let's take the deformation calculate calculate those vector potentials and plug it in the equation and see what they do now you know this is the thermal surface of graphene so the spectrum of graphene I don't do I have it on the picture? the spectrum of graphene has so-called Dirac points and at every Dirac point you have a linear dispersion so if everything is exactly compensated you don't have electrons in graphene because you are exactly at the point that it's compensated now if you add a little bit of electrons then the electrons are staying in the case space in the reciprocal space close to those Dirac points and your thermal surface is just that the electrons are leaving here and they are nowhere else there are no electrons here and no electrons here now what they said is this vector gauge fields and those gauge fields are actually shifting the momentum they would be the role of those fields they would be shifting the position of those Dirac points and if they shift the position of the Dirac points this shift is different depending on where you are I mean in this point it would be the strongest probably and in this point it would be the weakest but in any case if the electron has to go from here to here to here all the way to get on the other side it will have to go from let me again draw it at one point for instance it would be sitting here but at the other point the position of this cone shifted and maybe if the formation is strong at some point it shifted so much it doesn't overlap with the original one anymore they should be same size and then they argued that if the shift is such that there are two points where they don't overlap so like at one point in space here for instance the electron spectrum is that and at another point in space the electron spectrum is that then there will be no transport at all because there is no k at which electrons can actually go through the whole material they will have always see a point where they cannot go anymore because there are no states and then they said okay there should be metal and solar transition so if this is correct that has never experimentally been observed as far as I know and there are some good reasons for that but if it's corrected it would be very difficult to observe in any case but if this is correct you have metal and solar transition induced by mechanical deformation which is great now what is actually part of the gate is what sorry is what my student has done some time ago we put a gate and if you have a gate then you have another effect and this other effect is similar to what we have discussed before okay for instance there will be there will be more electrons at this point than at this point because the potential is lower so you don't just shift that you also increase the radius if you move along the membrane and whether you will actually have and where it shifted most the radius is actually the biggest and then you need to look at the details whether you will still go through the metal and solar transition or not and our conclusion was that actually it depends on you must have some residual stress so this graph in sheet should be stressed from the very beginning if you have some and here is the gate voltage and I think our conclusion was that in this area if you have very large stress and very small gate voltage you still stay with the physics of metal and solar transition and if this residual stress is insignificant then you will discharge the distribution below the steel transition so I have two slides more in like two minutes so I will just finish on time now again most of you have probably heard that next to graphene there are other two dimensional materials they are called van der Waals materials which have similar properties they can also be produced in monolays with one important exception they are all semi conductors graphene is technically speaking also a semi conductor but there is a zero band gap semi conductor so for all practical purposes it's a metal but those things are real semi conductors and that's a picture which I borrowed again from the Delta experiment by Andreas Castellanos who was at the time also postdoc that's that's a molybdenum diseline it's one of this family of these materials which is a semi conductor I think that's a monolayer and that's a real microscope picture and you see that actually they cannot really fabricate really flat membranes what they fabricate has these wrinkles and these wrinkles I mean they probably look a bit saturated but in these wrinkles there is a lot of stress concentrated and so you can actually to start with those materials you can see what happens if there is big stress I mean if you have stress or strain which is of the order of several percent if you think about that that's a lot I mean if you just take a crystal just a usual crystal regular whatever 3D crystal and if you would be able to induce several percent of strain this crystal would just disappear it would have all kind of problems it would not survive now of course in practice it just disappears much before because of other problems that's a different thing so you have some areas where you have a lot of strain and you can actually see what happens and the strain is so strong here electronic structure in significant way I mean I have given here an example when the strain affects electronic structure but here it was because there is this Dirac point and this structure is very sensitive here there is no Dirac point it's just a real semiconductor with a large gap it's like electron volt gap and then this is a picture for instance they know how much this this strain this strain is on this side how much this strain affects the gap and you see here they have a lot of a lot of points and it's difficult to conclude but in any case it's something between between 1 and 3 percent the gap change and on the left I sketch very briefly the way they actually know it they know it from a combination so an exciton is a complex of an electron and a hole so you basically excited by shining light so if you shine light if you eliminate something then you give energy and this energy goes to electrons so that create one electron and one hole so one electron which is sitting in the valence band in the sorry electron sitting in the conduction band hole sitting in the valence band and this this energy should match the size of the gap to make it efficient and now if you just go to a different point at this point the gap would be different and the energy of the exciton would be different it's of course very difficult because you need to conserve energy so the energy must be going somewhere it should probably go to photons or whatever but if they do it sorry to photons but if the exciton can move here and can here they can recombinate so the electron hole goes together and excites a photon this photon would have a different frequency and by measuring this different frequency and there is some good argument why it should happen exactly at that point it's not here which I will skip then this would be different frequency and from this frequency you can measure this change of the gap right sorry I'm over time that's my last slide anyway so I'll stop here