 Okay, so is the microphone working? John Luigi, do you hear me? Good, thank you. Okay, so good afternoon, everybody. Well, I'm sorry for this little delay because of the computer. So first of all, I would like to thank the organizers for the invitation. It's really a pleasure and a great honor to be here on such an occasion in such company. So I will tell you about a joint work with my experimental colleagues from Grenoble. So the inspiration comes from this paper by Boris and some other famous physicists which are also around. So you see, it's about electron relaxation in a quantum dot with discrete spectrum. So I got involved in discussion with my colleagues because they were interested in this electron-electron relaxation, and so my interest was maybe to have a grip on this physics in some experiment. So, but eventually our discussions, they drifted away from this, so the discussion evolved and in the end, nothing was left from this. And instead we ran into something else, and I will be telling you about something else. So I will be discussing mostly theory, but there is some experiment, and if time permits, I will show you the experiment. Okay, so let's now go slowly one by one. What is lambda-uzinir? Of course, everybody knows here what is lambda-uzinir, but I want to focus your intention on some specific aspects. So we have two level system, one level is standing, the other one is moving at constant speed, and there is coupling between the two. Originally, my particle at time minus infinity, my particle was on this low level, so here there is phase which I don't care about. So the particle sits on this level, the level is going up and it crosses this other level, and I'm interested in the probability at infinity whether the particle will stay here or will be transferred to this one. So there is one dimensionless parameter in the problem, it's v square over v, and you have clearly two cases. One is fast, when the particle is moving fast, then it just passes through this crossing point very quickly, and with probability almost one, it stays here, and only weak perturbative correction is here on this second level. The other limit is when you move slowly, when the level is moved slowly, and then it's better to find instantaneous eigenstates of the system, and with most probability, the particle will stay in the instantaneous ground state, and you have an exponentially small probability to be transferred to this higher excited state. So if you do this reasoning, you will never guess why this should be, the coefficient here is equal to the coefficient here. So this is actually the exact result, right? You know that if you solve the problem exactly, you get this exponential and this is it, but matching the perturbative part and the semi-classical part is not obvious why it should be the same. Okay, so this is, so here if we take now instead of linear dependence, if we move our level by parabola, so it goes from minus infinity, crosses, and then goes back, here you have actually two paths, so each crossing is like a beam splitter, so you have some phase accumulated along each path and you may have interference. Okay, so this is for Landau-Zener and that we will discuss, we will come to this later. Now let us discuss the turnstile. So what is quantum Sienis turnstile? So Sienis stands for superconductor, insulator, normal insulator superconductor. So originally this device was proposed by Pekola, but what I will discuss, what I will show is Grenoble version, which is a little bit simple. So we have two superconducting electrodes and one small particle between them, so it's a small metallic nanoparticle, tunnel contacts and this particle is so small that level spacing is large and coulomb energy is large. So you can think of it as having just one level which can be occupied either by zero or one electron, you forget about double occupancy and you can control the position of this level by great voltage. Okay, so then what do you do? So you apply a bias to these electrodes, so because of the bias, the levels here are aligned, are shifted, so if you move your nanoparticle level sufficiently low that it can pick up an electron from this, from these field states, but not from here, so you get an electron on this nanoparticle, then you apply great voltage, you move it up, again not too high, then the electron can jump into the right electrode, then you go again down and then you repeat the story. And this way you hope to transfer one electron per cycle from the left electrode to the right electrode. And so this is important for meteorological applications, so if you have, you can make a current standard of this. So frequency usually can be controlled very precisely and you can hopefully use this device to make a current standard based on the frequency. So this is why people are interested in this device. Okay, so yeah, here the original proposal from Pecola group was with big dot, so they had many levels. And if you have many levels, then you have to worry about, so it's a non-equilibrium system, you have to worry about the distribution function, you have to worry about relaxation and this is why this classical pioneering work is become important. But in my Grenoble colleagues, they just bypassed all this by taking very small nanoparticle and then you don't have distribution function, you just have one level. Okay, and you hope that you will be happier, but it's not that simple as it turns out. So let us focus on this step when you have an electron on this level and you want to jump on one of the electrons. So let us forget all the rest, let us consider this just single particle problem where you have one field level, you have empty quasi-particle states in the superconductor so you want to decay into this continuum. Well, okay, you can calculate the rate. So you have some tunneling coupling, so you have some finite rate and then you of course want your jump occur with probability one, right? If you don't let your electron, if you don't give it enough time to jump, if your time is not as short compared to the rate, then you get errors in your turnstile operation, so no metrology. So you want to get rid of errors and then you think that maybe you have to wait, you have to do this process the slower the better, but this is not true. And it's not true because of quite peculiar phenomenon. So if we write now this golden rule, the rate for the transfer, so you have to put here density of states in the superconductor and you get this divergence, this square root singularity. So golden rule is not sufficient and then we have to do better. So now you already know what the result of this is because Leonid Glasman explained yesterday that you have Shiba states. So if you solve the problem exactly, here it's a single particle problem. It's just discrete level coupled to continuum with this funny structure. So you can solve this exactly. So I'm interested of course to be near the singularity. So here I have self energy for this discrete level. So if I'm near the singularity, suppose my delta is large and my level is close to here, then I take the leading part, the leading divergent part, I count my energies from the singularity and here is my inverse Green's function on the discrete level. And it turns out that this Green's function has a real pole at negative energy, no matter. So you can take your discrete level and pull it arbitrarily high in the continuum. Nevertheless, the two lines always cross because your real part of self energy is divergent. So you always have this bound state and this is the same thing as Shiba state. Here I have no spin. I have no superconductivity because superconductivity, I forgot about it. I just have my singularity. I only need the singularity. And so singularity means divergence in real part of self energy and this is level repulsion. So basically this bound state, it's due to level repulsion between my discrete level and the singularity in the dose. Okay, so what does it mean for our term style? For our term style, it means a horrible thing. If now I move my level by great voltage slowly, then the electron will always stay in this bound state below the continuum and it will go back down without even noticing the continuum. So if I do everything slowly and adiabatically, the probability of error is one. So to estimate the scale of the disaster, we have, okay, so here are properties of this bound state. Let's keep. So I have to face the dynamical problem. So if I want to see somehow how this, whether my device will work or not, I have to solve this dynamical problem time dependent and this is basically, so you can write time dependent Schrodinger equation for your wave function which has component on the discrete level and component in the continuum. You eliminate the continuum and so you have this differential equation for the component on the discrete level and you have memory kernel which is nothing but the Fourier transform of self energy. Okay, so it's the same G minus one. So it's this written in the time domain. Okay, so we have this, yeah, so and we are interested to find the error probability, we should do what? We say that at minus infinity, our level was far below. We forget about delta, delta is very large. So our level was far below and the electron was staying on this level and then we do our half cycle and we want to know what is the probability to stay on this level. So this would give us the error probability. Okay, so you can solve this numerically of course. So the kernel is not very nice. You see a singular long range. So the numerical solution is doable but consuming and it's not very useful. What I find more useful is that this problem can be solved analytically for one special case, for the quadratic dependence. So it's not just academic. If your, I mean if your gate voltage profile is sinusoidal then you can approximate the top of the sinusoid by parabola and this parabola is nice because if you now go back to energy domain as was noticed long ago by Dimkov and Asherov for the linear dependence, here we have quadratic but still you have, your time becomes derivative and then you have Schrodinger equation. So this is just not in coordinate but in energy. So it's just very usual, most usual Schrodinger equation with only thing that here, this of energy is an analytic function and so it can become complex. It can become imaginary. So you have to continue it through the upper half plane. So for positive energies it is complex. Which corresponds to the decay rate of course. So nevertheless with this Schrodinger equation you can deal by WKB in the region of internal. In the region where we are interested. So you can formulate this transition probability as scattering problem for this Schrodinger equation and I will tell you the results. So for this parabolic problem we have two dimensionless parameters. So first of all we have natural energy scale which is determined by the singularity and by the coupling and we have two parameters in the parabola. So H is how far your level goes into the continuum and W is how wide your parabola is. So you have this two dimensionless parameters and we have three regions it turns out. So of course when everything is of the order of one the error probability is of the order of one. So we have a simple regime where either your level is does not arrive to continuum or it arrives and goes back so fast that you do not have time to eject. So then your error probability is close to one. Then there is indeed this adiabatic regime where you almost surely stay in this Schieber state and indeed the correction is exponential. And there is the regime which I call Markovian because here your error probability is basically determined by the instantaneous golden rule as if you have at each instant of time you can apply golden rule independently. Okay, so these are the three and if you want your turnstile to work of course you have to be in this Markovian regime because in the two others you just always at fault. So now what I want to do is to study a little bit the crossover between these two. So this Markovian result is not the end of the story I can calculate a correction. So this is the correction. So here you see this is the expression in terms of the parameters of my time dependence. So this is the decay due to Fermi golden rule. This is the phase which is accumulated. So this is the old part and this is the correction. And here surprisingly again we have the same combination that appears in the exponential in the adiabatic. And I don't know if this is just a coincidence or there is something deeper behind this. And so you'll also see that you have interference. So this result you can interpret as interference between two parts. So you have your continuum here and you can either go through the Shiba state or you can go through the decaying state and this thing tells you that you have interference between the two. So and the answer looks like you have, so here when you cross this square root singularity you can either go to this state or you can break through the singularity and then you decay. And this is similar to Landau-Ziener but here we don't have a discrete level. So we have something Landau-Ziener like between a discrete level and the singularity in the dose. Okay, so this is the theoretical part and now I want to discuss the experiment. So this is how the device looks like. These are aluminum electrodes and somewhere in between there is a tiny, tiny gold nanoparticle. Probably you don't even see it. So the parameters are such. I already told you that they respect this limit that we really have just one level. We can forget about the double occupancy. So what is important is that of course this square root singularity which was crucial for me in any real device will of course be smeared on some scale. So the good thing about aluminum is that this smearing is extremely weak. So it's characterized by the so-called Dyne's parameter which is this energy scale of smearing in the units of delta. So this is 10 to minus four, 10 to minus five. It's known that it comes mostly from the external noise from the circuit because people, so the same Pecola and his group, they managed to suppress this to 10 to minus seven by carefully shielding their electronics circuits, et cetera. So parameters are very good. We have plenty of room for this effect actually where it can become relevant. Okay, so and now the experimental results. So here I have to explain, spend some time explaining what we see here. So this is color map. What is plotted is differential conductance. So Di over DV. So where this is black, it's constant current and where you have something colorful, it means that something is happening. So what is on the axis? So on the axis, here we have AG is the amplitude of the oscillation of the level due to gate voltage. Here we have the bias between the two electrons. So this is a map of our turnstile operation. So here they apply the profile of the gate voltage which they call square wave, which means that you have something like this, like this, like this. Of course in reality it's always smooth, but nevertheless, so here in this region, the current is practically zero. Why? Because our amplitude is not sufficient to reach the, is not sufficient to reach this, it's not sufficient to reach the quasi-particle state. It's just always staying in the gap. Then, so here when your amplitude is large enough, you hit this square root singularity and your current rises, so you have a peak in DIDV, and here you have another plateau which corresponds to current equal to one in the unit frequency, electron per cycle. And then if you increase your amplitude further, then at some point you hit the next singularity of the electrode which was the original one, then your electron can go into the wrong electrode, and then you get a drop in the current. So this is the onset of the operation, and this is onset of back tunneling which destroys your turnstile. So this is for square wave, then they applied sine wave, sinusoidal. And you see that when you lower the frequency, your turnstile operation plateau shrinks. So you have this gap which is opened, and it looks like indeed this line is pushed away from the singularity, and eventually at such low frequencies, you have almost no operations. So now, we of course want to compare this experimental result to the theory. So here the good thing is that we know almost everything about the device. We don't need fitting parameters, we know the coupling, the gap. What we don't know is how to calibrate the amplitude of oscillations because these people in experiment they have the knob in the microwave generator, then it applies gate voltage and how it translates in the actual energy of the discrete level that we don't really know. We can only guess that indeed this corresponds to onset of the operation and this corresponds to the back tunneling. So we can do this for calibration which is not very precise and it depends on frequency also because all this transmission of electronics is frequency dependent. Okay, so what we can compare, one thing that we can compare is the gap with, ah, sorry, yeah, how do I do? Yeah, assumptions behind the theory. So on each half cycle we have this probability of error which you can get either from the parabolic solution or from the numerical solution for a sinusoidal wave and the difference is not very large. So this analytical solution works really well. And then I assume that the different tunneling events are incoherent. So then when I have these probabilities, I plug them into rate equations and calculate the current from the rate equation. So here we have the gap width as a function of frequency. So you see the experimental theoretical, the agreement is by order of magnitude is okay, quantitatively not very good. Another thing that we can compare is the vertical section. So here if we take a point where this peak crosses the bias say equal to delta and make a vertical section, here the voltage we know what the voltage is so we can really make a section and compare the curves. So here and here you see that this is the theory. Numerics or analytical result, they really have almost no difference and the experimental although the orders of magnitude here are maybe similar, there is something going on here on the high voltage side which does not correspond at all. So if we take this left half width and compare with experiment, again we get something of the right order of magnitude. But here on the high voltage side, agreement is not good. So for the moment I don't have a better theory for this. So I can say that this region, so this region, it corresponds already to being, when the level is quite deep in the continuum, so it's quite far from singularity. And then the theory is not supposed to be applicable quantitatively and there are also other things that one should worry about in this region. So if the nanoparticle level is deep in the continuum it means that this Shiba state is very shallow and here it becomes so shallow that you should really worry about the smearing of the singularity and clearly when you smear the singularity it helps you to eject the electron. So here probably smearing of the singularity matters then it's quite possible that neglecting the, so when you have, when the excursion is of the order of delta beyond the threshold, then you have to really take care about the lower, about the superconducting gap and include all this properly. So at the moment we are at this level we see that indeed there is this effect which is observed experimentally but its theoretical description still has to be improved and these are my conclusions and thank you for your attention.