 Can you hear me? So I think we can start. Hi, my name is Triampo Durario. I'm one of the co-organizers from Ulm, and I'll chair the first session. So the first talk will be from Carles Altimiras, from SACLE. And it's about the experimental test of the Kubo relation in a non-linear conductor driven out of equilibrium. So please, Carles. OK, so I would like to thank the organizers for giving me the opportunity to present this work. So this work has been carried in SACLE in the collaboration amongst the nanoelectronics and quantumics group. And all the measurements that I will present were performed by Zuber-Iftica and Jonas Müller. So the story I would like to talk to you about today is about fluctuations and dissipations. And this is a long story in electrical circuits, because we know from the Nyquist formulation of Johnson noise that you can essentially describe the fluctuations of a dissipative system by considering that a resistor is just a transmission line carrying the information to infinity, losing it. We know from the 50s also that the zero point fluctuations of quantum systems give a noise floor, providing an asymmetry between positive and negative frequencies of the spectrum density of fluctuations. And this was understood by Kubo. This noise asymmetry is generically linked to the linear response of your system to an external linear coupling. So all these ideas were somehow put together by the Kaldirelle description of dissipation in viscose or dissipation, that it's indeed this coupling to bosonic bath. And with that, well, there are many phenomena that can be predicted in elastic tunneling, dissipative quantum phase transitions resulting from this coupling and so on. So I would like to stress that even though people use a Kubo formula by computing these current fluctuations in a perturbative way, this structure holds far from equilibrium. So this was already a knowledge in the 80s where rather than doing a perturbative response, they were using a variational computation of the perturbation. And the idea is that the linear response is still described by this structure, the asymmetric part of the fluctuation spectrum. But this fluctuation spectrum must be computed in the very out of equilibrium situation you might end up. And this was generalized 10 years ago to arbitrary time-dependent states in the sense that you have an arbitrary time-dependent classical drive. So why is that general this structure? Well, if you look to a quantum system classically driven by a parametric drive which is coupled to just a linearly coupled to one observable of the system, just a simple calculation of the power dissipated into the system tells you that indeed the power dissipated in the system is provided by the real part of the admittance as computed by the Kubo formula. So it is just the dual effect. Of course, when your system A is driven by a quantum agent, well, you cannot conclude things that easily. So you need to simplify your coupling to understand what's going on. Usually, we can take a simple system, just a quantum conductor coupled to a resonant mode with his own losses. And when you do some approximations that the energy stored into the resonator has no memory, so it's leaking faster than it is injected, which in practice corresponds to a large impedance mismatch between the current conductor and its electromagnetic environment, you'll find that the power which is being dissipated by the quantum conductor takes this very simple form which is a difference between negative frequency current fluctuations weighted by one pre-factor and with an opposite side, the positive frequency current fluctuations. So with this convention, we find that there are indeed three terms, one term which doesn't care about the state of your electromagnetic environment, and this is just the spontaneous emission of the quantum conductor into its electromagnetic environment. As you put some energy into the environment, for example, with your input port, you get some stimulated emission from your conductor, but now since there is some energy to take, the quantum conductor can take it, and this is the stimulated absorption. So we find the same things as in quantum optics. So coming back to Cuba formula, you can of course rearrange these terms and to just express things proportional to the energy which is stored in the environment. And when you look at that, you see that indeed the rate at which you dissipate this energy into the conductor is given by these asymmetric current fluctuations, which is just the real part of the admittance by Cuba formula. Okay, so can we measure separately these fluctuations? Can we measure these elements? Well, historically people were looking at what you can do into the fields propagating into the lines. So if you take the classical Johnson setup, which is you have some fluctuating field, you narrow-basp filter it, and you look at how much power you get into this band by some rectifier. It can be a diode, it can be a light bulb. And if you do the quantum theory of it, as in input-output theory, you get that the rectified signal is indeed the symmetric combination of the voltage fluctuations carried by the transmission line. So indeed this is an observable, so the observable must be a real valued object. So it is this, it has always this shape. I think that's why many people have believed for many times that one can only measure symmetric noise. But then in 2000, it was realized that if, rather than using a classical diode, you use something like a quantum diode, so for example, a nonlinear tunneling device, then you can imprint the spectrum of fluctuations of the current conductor into the IV properties of the tunnel device. So it works with any nonlinear tunneling element. It can be SIS tunnel junctions, quantum dots, double quantum dots. So the idea is that you have a sort of current fluctuations, which thanks to some coupling circuit, imposes some fluctuations of flux at the input of the nonlinear tunneling device. And to give you an idea, this is taken from the group in Orsay. So how an SIS junction works as a photon assisted transport detector is it explains the fact that its energy spectrum is bounded by below, so you have a full range of operation where the device is only sensitive to the absorption of photons and this absorption of photons gives rise to a rectification of the IV curve, which is only proportional to the emission noise which is sent to the device. When the bias is larger than twice the gap, the device is sensitive to both. So the photon assisted transport will be a combination of the emission noise and absorption noise which is imposed at its input. Still one can use one part of the photon assisted transport current to characterize just the emission noise of the conductor and hope to be able to subtract it from the part when you have both emission and absorption. This gives a well-defined scheme to measure separately the emission and the absorption noise and indeed it was used in 2000 and tell to measure separately the emission and the absorption noise of the thermal noise of a single mode resonator. So the single mode resonator had its temperature being changed and as the temperature is changed one could see the two parts of the Kallon-Welton noise and the difference between the absorption and the emission noise was indeed checked to match the real part of the impedance of the source of thermal noise. The problem with this scheme is that it has never been possible to extract the absorption noise of an active conductor and the reason for that is that essentially the schemes that were used were completely symmetric. So if you want to measure the absorption noise of your source your detector must be polarized in a regime where he's also sending some emission noise into the conductor and if you're coupling circuit is symmetric you will have photon acid that transform from source to the detector but you will have also photon acid that transport from your detector to your source. So this becomes strongly non-linear and they did not manage to get quantitatively the absorption noise of the source because essentially both devices became a source. So I would like to come back to the power spectral density setup and I would like to insist in something is that indeed when you do a local measurement of your field with a quadratic detector of course is the symmetries voltage fluctuations but when you just play with the algebra you see it's just the occupation plus a zero point motion of the detection line. So whatever you're doing at its input the zero point motion of the line is just the zero point motion of the line. The question is how is the power transferred by this line depending of its input conditions. So this is what we were seeing before in this 3D coupling between the quantum conductor and the electromagnetic environment and the idea is very simple when there is a strong impedance mismatch between the quantum conductor and the electromagnetic environment the quantum conductor is imposing its current into the detection scheme and then the power that you detect at the output is just the weighted sum of the current fluctuations of the quantum conductor weighted by the voltage fluctuations of the electromagnetic environment. So when the electromagnetic environment is sitting with zero population close to vacuum when you measure the emission noise into the differential noise how much power is the quantum conductor depositing into the quadratic detector when you turn it on then it will be directly proportional to the emission noise only of the quantum conductor. But now if you do this very same measurement with some population into the electromagnetic environment you will have some contribution of the emission some contribution on the absorption and if you have calibrated this population you can extract the emission contribution to deduce the absorption contribution. So this is again a well-defined scheme to access both. So here is the experimental implementation of this idea we chose an SIS tunnel junction as a quantum conductor it's seeing a circuit via a bias T the bias T enables us to voltage bias the SIS junction and the capacitive port of this bias T is sending the electromagnetic radiation to the radio frequency part of the circuit. So the SIS junction only sees the environment via cavity filter sitting at 6.8 gighertz this is roughly 10 times the excitation quanta at 6 gighertz it's roughly 10 times the thermal energy so we are sitting very close to vacuum. Then the sample is protected from the detection line by a double circulation it's amplified, routed to room temperature where we just perform a power measurement with the diode. This is very standard the main difference with other setups is that the circulator sitting closer to the junction has a 50 ohm match normal tunnel junction which is itself a voltage bias via this inductive port. So the idea is that this junction can also dissipate the noise power coming back from the amplifier but at the same time it can be used in order to impose some population into the modes hitting the SIS junction via the cavity filter. So this is our experimental knob to control the number of photons which are seen by the SIS tunnel junction. Okay. So this is the DC characterization of the tunnel junction so we just voltage bias it and we measure it's current. So indeed it behaves like an SIS tunnel junction I would like to stress it's been designed in a squid geometry so that we can cancel the Josephson contribution. Also for specialists we needed to apply some not negligible magnetic field in order to avoid the back bending of the structure. So this is why the coherent peaks are somehow broadened but still it's quite sharply defined. We see that there is a DC transport gap of roughly 400 microelectron volts which is indeed quite standard value for thin film aluminum junctions. So this is roughly 15 nanometers and 13 nanometers thickness. And the important thing is that the DC transport gap that we get from this DC characterization is roughly 400 microelectron volts. So this is the DC characteristic. Let's look at the power the SIS junction is depositing into the detection circuit when we apply a voltage on it. So this is what we measure. We just make the difference between the power which is present into the line without applying a bias into the junction and the power that we measure when we have applied the bias into the junction. So it looks very similar to the IV characteristic of the SIS junction. This is expected from the microscopic calculations of its current fluctuations. The very important thing is that you see now that the onset of power emission is slightly different than the one we had for DC transport. Before it was 400 microelectron volts. Now it's slightly off. And if we look closely, this is exactly the quantum of excitation of the photon into the detection line. And now if we take the microscopic calculations of the current fluctuations which are non-symmetrized, the emission noise, we indeed see that it feels very neatly. And the microscopic picture is that is that in order to be able to emit a photon into the environment, the quasi-particles need to have enough energy in order to do so. So of course, you will get the same rate of emission as you had as the rate of the current going through the device, but with an offset of exactly one quantum of excitation. So from this measure when we can check indeed from that the power emitted in vacuum is proportional to the emission noise of the current fluctuations of the conductor. Now we will use the NIN junction in order to tune the population of the electromagnetic environment. For that we just mean this is just power emitted by the show noise of the junction. When this is the differential power that we detect again at room temperature, it's the difference between the power emitted by the amplifier and the power emitted by the junction. So it fits, well, this is the curve we have. This is the expected show noise emitted by a normal state tunnel junction from which we can get, well, an interesting property, the electric temperature of the electrons, which is roughly 30 millikay. And the idea now is to look into how the SAS junction is changing energy with the environment when the environment has been tuned to these different emission population points. So when we do that, these are the traces we obtain. So this is the power when both sources are active and when we subtract the noise power of the chain. So we recover the curve we saw before. When we do nothing, it's just the emission noise. And as we increase the power emitted by the normal junction in this, there is a background power which is detected. And when we get close to the superconducting gap when we play the voltage in the SAS junction, we see that we are taking some power out of it. And the amount of power we take, it's larger and larger. It seems proportional to the available power to take from. So in order to make this more quantitatively, what we do is to subtract the power which is just present in the resonator. So this is the trace which describes the power exchanges between the SAS junction and its electromagnetic environment at finite population. So we add zero bias, this is just the emission race curve. And now that we see that for all the curves, the onset of absorption from the SAS junction starts at twice the gap minus the excitation quantum. All these negative power contribution is just a dissipation from the junction in the junction taking power from the environment. And this part is what globally, there is a net emission into the environment. So if we take all these curves, we can scale them into a single curve by using the formula I was presenting you before. So we should be able to describe these power exchanges with just these three quantities. So in order to extract the absorption noise of the quantum conductor, what we do is to take the data measured of the exchange power that finite population. We need to subtract the emission noise. This is just the exchange power measured at zero population. We need to weight it by this factor. This is just one plus the population that we can calibrate independently and finally divide by N. If we do that, all the traces fall into the same curve. Of course, when there is less signal, there is more noise, but they are sitting all in the same shape. And I have to say that in order to get these beautiful agreement between what we expect from microscopic calculations and our calibrations of the occupation number, we need to correct it by one dB, which is roughly 20%. So this is the trace we expect for the absorption noise and it's the same microscopic picture as we had for the emission is that indeed, when the bias is such that we are lying just beneath the transport gap by the amount of one photon, we can absorb this energy and dissipate it into the SIS junction. So I hope that I convinced you at this point that from just power exchanges between the quantum conductor and its electromagnetic environment, the fact that the electromagnetic environment is sitting so close to vacuum, it has enough asymmetry in its own voltage fluctuations to be able to extract something which is indeed asymmetric as well for the quantum conductor. But the point of having such a circuit is that it can be multiple purpose and we can do some other measurements that just in coherent power exchanges. So in between the double circulator and the quantum conductor, we have a 20 dB coupler, which is used to send a coherent tone which is provided by a network analyzer. This is a monochromatic excitation with a well-defined phase and then we can analyze from the reflection measurement what is the linear response of the SIS junction. This is the standard measurement of linear response. So of course you need to tune the excitation amplitude to a small enough value to have indeed a linear response, but with that we can measure independently the admittance of the conductor by a coherent linear response measurement. When we do that, we obtain this red line which looks like the DC conductance of the quantum conduct of the SIS junction, but with rather than have a sharp coherent peak, there is a sort of plateau. And if we understand the real part of the admittance as a function of the Cuba formula indeed, we expect to have this shape because we need to make the difference between these two curves. This difference becomes essentially constant at large enough voltage. This is just the normal state admittance and close to the current peak, it will be larger. So this is what we directly measure from the VNA and the black curve is the one we construct using a Cuba relation from the emission and the absorption noise we have measured before. So these are two completely different measurements. One is in coherent power exchanges between the conductor and its environment and the red curve is a linear response coherent measurement. And we see that at the end, the agreement is 10% roughly 0.5 dB. So it's even of the order of the calibration accuracy that we have in the line. Okay, so at this point, I would like to come back to the generalities of this Cuba relation. I mean, since it's so much related to the dissipation, it must be better than just a lowest order perturbation. If we take the simple system of the quantum conductor, which is a minimally coupled to an electromagnetic conductor via a radiative coupling I phi, well, you can extract these formulas from the lowest order perturbation theory. This is what essentially is done in the no memory calculation. It justifies that you can take these lowest order perturbation theory. Also the impedance mismatch is justifying that you can use it. But you see that the current fluctuations that you get describing these energy exchanges, other ones you have in the absence of the coupling. So it's not at all general. Because we know that we're missing lots of physics. When this coupling is big enough, either because current fluctuations or because of the flux fluctuations are big, we will have lots of things. When the flux fluctuations are big, we will have detection back action. When the current fluctuations are big enough, we will have memory effects building in the system. We can have, because of this frequency curvature between the frequencies that we are sending and the ones we are probing. So this is never captured by such a lowest order perturbation. But the thing is that the electromagnetic environment is a linear system. That means that the flux operator describing the dynamics of the electromagnetic operator can be split into essentially its continuum and we can specialize a detection frequency band. And of course, its own Hamiltonian can also be split as a sum over the continuum and a very narrow band detection circuit. So if we restrain the power exchanges to a very narrow band of observation of these power exchanges, we will end up with a vanishing measure flux fluctuations arising from the environment at which we are looking at the energy changes. And still the rest of the system gets decoupled and this is essentially the dress conductor by all the dynamics which are present in the system besides this. So unless we have a very, I would say singular electromagnetic environment, of course, if you have a singular electromagnetic environment, you will now look at the system as an open dissipative system. It will be a closed system and then you will do other sorts of, you will describe the physics differently. But if it's an open quantum system with no singularity response in the environment, this is listed. And then you can still use the same approach, but with just the coupling to this narrow band part. So the message here is that not all the power exchanges are given by this effect, but if you look at the power spectral density in a narrow band enough system, you can still define the energy exchanges in this narrow band as the weighted sum of the emission noise and absorption noise of the dressed system weighted by the absorption noise and emission noise of the voltage fluctuations of the electromagnetic environment. So we still have this result that the cubo formula is indeed describing the dissipation even in this strongly dressed conductor by the dynamics of its coupling to the environment. Okay, so I'm finished. I would like to sum up. So first thing I showed is that the non-symmetrized emission and absorption noise of a quantum conductor can be extracted simply from the power exchanges with a linear electromagnetic environment whose population can be calibrated. Then we could compare the cubo formula. We could test the cubo formula by comparing the asymmetric part of these current fluctuations to the linear response which was measured independently by a current tone. And finally, I stress that these power exchanges are linked to linear response as soon as the detection bandwidth is sufficiently small and there is a strong impedance mismatch between the quantum conductor and its electromagnetic environment which physically is just the conductor imposing its current to the detection and the detection exposing its voltage to the conductor. I would like to stress that this will stop working beyond this current source limit. As soon as the detection impedance is of the same order as the impedance of the quantum conductor, one cannot make this easy separation between current fluctuations injected and voltage fluctuations imposed. At this point, you need a self-consistency because essentially this is telling you that you cannot consider voltage fluctuations as arising from the electromagnetic environment alone. And then of course, if both are larger than the quantum of resistance, you will have heavy non-gaussian effects if the system is driven out of equilibrium. That's complicated. Also, I would like to stress that this simple formula, as I said, is only narrow band. But if you think of frequency conversion in the system, for example, if you have some sort of inelastic scattering at a given number of modes only. If you look locally at what happens to your conductor, if you're sitting into the middle band, some power will be transferred to the other modes. But if you integrate the power which is being carried by all these modes, you expect to have all of it originally. So I would expect to have some sum rules that hold for the non-linear conversion that arises from the higher order perturbation. I think I've never seen anything stated about if there are some possible sum rules. I think this is interesting in the, I would say, super non-linear dissipation you have in close to quantum phase transitions where you have huge cascades of energy transfer, yet the Josephson junction cannot dissipate anything. It's just shared amongst all other photons. So I guess that if you look at a wide band energy transfer, you should just recover all the power you were sending. That means that you will have some sum rules describing the non-linear conversion. So, yeah, I think that's everything I wanted to say. Of course, I would like to thank all our collaborators and I would like to thank Alot Fabien, who passed away two years, but from whom I learned everything I know what to do. So thank you very much. And if you have any questions, do not hesitate. Thank you very much, Carlos, for this beautiful experiment. Please, questions. Very interesting results, thanks for the talk. I have a few questions. One is about the Cooper pairs, like tunneling through enastically. Do you see a sign of that? Like, I think if we do not frustrate the squid, there are lots of them and they are very strong and it's a mess. You see all the imperfections of the sample. We did not design a sample in order to look at that. Essentially, you see all the ripples of the line. It's a mess. It's not made for that. If you want to do that properly, you engineer your electromagnetic environment on chip. This is what, it's not what we did here. You see the cavity filter is outside. There is some cable length between the cavity filter and the conductor. You will have lots of standing modes there, roughly at, I think it's 20 centimeters cable. So it will be in the 100 megahertz regime. They will be thermal populated. It's lots of structure, yes. Right, okay. And in your slide 19, like at low, at low voltage, at like, do you? 19. In principle, you should see the cooper pairs putting photons to that, your own chip cavity filter. Here you see, we did not exactly manage to frustrate it completely. You see a tiny peak. These are the cooper pairs. Right. Yes, that's what I was thinking. But what are you, so sorry, I didn't know. What do you kind of mean by the frustration? The squid, if you apply a magnetic field onto the squid, you will frustrate the Josephson current going through that. Right, okay. And of course, this is what we saw yesterday. The power you meet has the, the fluctuation strength of the source and the fluctuation strength of the source is parametrically changed by the frustration. Okay, great, got it, got it. Then I have one question about the, like you said that in the NIN case, you can actually map out the temperature of the electrons, but then in your SIS case, can you map out the quasi-particle temperature or something similar? The gap is so big that there is no thermal effect. So you don't see the zero temperature. I mean, of course, here there is some broadening, but again, this comes from the magnetic field which is killing the sharpness of the coherent peaks. So that's not a temperature effect. Yeah, I'm kind of just wondering what is the effective temperature of your- Quasi-particle bath? Of, let's say that, no, I mean, you have, like now you have an electromagnetic environment that is the photon-acid and tunneling. So what is the effective temperature of your, of that electromagnetic environment? That's my question. How close to zero is that? Okay, I mean, temperature, I mean, these are just electromagnetic power exchanges. So if you take the engineering conversion, whatever signal has a noise temperature, but this is not a physical temperature, this is just an easy way of describing power. But having a finite temperature means not only the power, its fluctuations, all of this. So we do not have a thermal source, we have a short noise source. This short noise source, in this limit, it's low impedance, it's very, very close to a chaotic source. So these are the calculations from a remember first author. But I mean, this, this is the short noise emitted at high frequency by the normal tunnel junction. And if the impedance is small, they will be essentially very, very close to a Gaussian. It's chaotic, the power gives you some number. From this number, you can express its own fluctuations. It looks like a black body source. But there's no reason, I mean, if you change the source, you would still be able to describe a power with temperature, but this is just playing with numbers. It wouldn't be able to predict what's going on at other correlation functions of the power. But here we just coupled to the lowest order. This is the point of having this impedance mismatch. So we do not even care about the actual state of the environment as soon as it is not singular. Thanks, very nice talk. I would have a question about the perspective. Yes. Could you think about, I guess, it's obvious, but how easy maybe to replace the SI's junction by a more exotic conductor? And is there any, because you have somehow a narrow band, a detection scheme, as you said, what can you learn? It's narrow band with respect to the non-linearity present into the system. If you have something as non-linear as the one you play with, 600 megahertz I think would be too wide. Yes. So my question is, how easy would it be to swap your SI's junction by a load? This is easy, but then you can learn something interesting. No, the problem. This is my question. Yes, I mean, I've always, I mean, there are these predictions from our Ash-Claire group where, I mean, that would be the sort of dynamical coulomb blockade when you change Gaussian fluctuations by non-Gaussian fluctuations, okay? That's the idea I guess you have in mind. Yes, that's interesting. The thing is that you are doomed to do that on chip if you wanna be, if you do not, I mean, you have too much losses to be doing that. So you will lose part of these non-linear fluctuations and they will be wasted in the lines here. I mean, in the calibration, at the end you always have insertion losses plus some return losses. You have all these ripples. It's difficult to be very quantitative and more importantly, you will be losing the non-Gaussian oscillator strength. I would say the interesting experiment is doing it on chip. I mean, I think Perti did some experiments I mean, long time ago doing this idea. I think that with what I understand now, if I did them, I would go with a various symmetric coupling scheme. I mean, here, the thing is that you want to impose the non-Gaussian current fluctuations into the detector. But you do not want the detector to have this effect. But then it's easy. You just use an impedance transformer. The impedance transformer will give rise to a big voltage here with a tiny current there and it will be the opposite. So for the equal amount of power exchange, this is reciprocal in power. It is not reciprocal in the two quadratures. So this is the way to go I think if you wanna do that. Thank you. Welcome. All right, thanks a lot. I think there is, maybe, quick. In the last slide you were saying that at the end, everything is going to be the same, but the modes are going to be dressed because of the coupling. And now you have to fix only in a narrow band. But I'm asking. If the band you fix into the detection is narrow band enough, you will have no perturbation from it into the dynamics. Okay, but let me ask. What do you mean by this dressing? Because I'm thinking in the AS4 diamagnetic term when you couple a qubit with a electromagnetic field, then what you have, it is not just dressing. It is something that can change substantially the nature of your modes. So you may not have even. Of course. So this is. The flux, the voltage fluctuations arising from that can have lot of current content from the source, of course. So it changes deeply the nature of it. It will become non-motion already. Of course. Yeah, I mean, this is the cause five you have in the spin boson or in the version in security. It's strongly non-motion. All right, let's thank Carlos again.