 You don't need to change it anymore. You see Priyanne, so Panduari. Sorry for that. Yeah. OK, so I think we start while people still come in, because it's already a bit too late. And so the next talk is given by Denis Villon. And you have 30 minutes, and I try to give you some signal. Yes, yes. Five minutes earlier. So thank you. Good morning, everyone. As for many of you, it's my first post-COVID conference. I'm super happy to be here. And I warmly thank the organizers for this kind invitation. The story I will tell you today is about a very simple circuit that emits quantum microwaves. More precisely, it's a Josephson junction. Simply a voltage bias in series with an LC resonator. And for a proper choice of the voltage here, you get some emission of bunches of an exact number of photons in a continuous way. OK, the reason we tell you this story is because we think it is interesting in itself because it enlarges the QED subject to out of equilibrium situation. But of course, it also helps building the quantum computer, solving the humidity and the global warming problem. But this is not the only reason I'm telling you all this. OK, so the PI of this work is not me. It was our friend Fabien Portier, who passed away one year and a half ago. The work was mainly done by his last PhD students and post-doc, Ambrace Peugeot and Gerbo Ménard. And it was done in strong collaboration with the groups of Joachim Zoncorot in Ulm. And many of the simulation I will show have been done by Bjorn and Cyprien here. Here I have only put the members of the CEA members that have participated to this precise work. But I know that there were many more collaborators of Fabien, even in this room. And I say to them, hello for him. OK, that said, no, someone has moved the mouse away from me. OK, yes. So the context of the study is the quantum transport and the production of quantum microwaves. So the problem is that you push on cooper pairs with a voltage source across quantum-coerent conductor. And you will transform these microscopic excitation QV here into collective excitation that we call photons in the environmental impedance seen by the quantum conductor. And the relevant questions are how many photons, at which speed you produce them, what is the photon statistics, are the photons entangled, and so on. Good. The quantum-coerent conductor we consider is a simple Johnson-Johnson, SIS-Johnson. And I recall here the ID characteristic. Basically, you have a supercurrent branch at zero voltage simply because cooper pairs can tunnel from the left to the right at the Fermi level without any dissipation. Now, the situation is even more simpler when you put a DC bias because nothing happened. You have no current at all. Why? Because if there were some current, the energy delivered by the voltage to EV would have to be dissipated somewhere. And there is a gap of excitation. You cannot dissipate, so no current. So to have an interesting situation, you have to add something like this single LC mode in the circuit here. And in that case, when the voltage is equal, it's such that the energy to EV 1 delivered by the battery is actually the energy of a photon. You produce one photon in the resonator like this, and you get some current. And you can do the same reasoning when the voltage V reached the value for two photons, three photons in the resonator. And for any integral number of photons, you will have a resonance with a multiplet of photons being created per cooper pair passing across the junction. In order to understand the ingredients that govern this physics, I use this analogy of an excited atom in a box with a box mode resonant with an electronic transition and an impedance of the order of the vacuum impedance z0. And for this relaxation to occur and a photon being produced in the box, you need two things. You need, first, a matrix element that depends on the atom itself. You need overlap between the electronic wave functions in the atom. This is the first ingredient. And the second ingredient is that your box has to be able to accept this energy in form of a photon. And this is given by the ratio of the impedance of the mode divided by the quantum of resistance h over e square. And these forms, well known fine structure constants of 1 over 137 for this matter-like coupling. In our case of the circuit, the role of the atom is played by the Josephson junction. It's not pre-energized because the energy is hidden in the battery here. The box is a box, an LC mode. And once again, you have the same two ingredients. You have the Josephson energy that tells you how Cooper pairs can be transformed from one electron to the other. And on the box side, you have the impedance of the box, square root of L overseas that tells you how much the photon will be created easily or not. So this is, again, the ratio of the mode impedance to, this time, the superconducting quantum of resistance involving the charge of a Cooper pair. And the big difference between the two cases is that here alpha is nature-given, whereas here it's rather easy to play with alpha and have a fine structure constant of 1 or even larger if you want. OK, so Fabien has covered many chapters over the last 10 years with many collaborators. And the first chapter was to prove that, indeed, there is the same number of photons produced as the number of Cooper pairs going through the junction. And this was done by Max Ofain when he was postdoc with Fabien. And Max was measuring the real impedance in situ, as seen from the junction. And then he could measure both the current and the microwave emission produced and check that the rate of Cooper pairs was matching the rate of photons both experimentally and with the theory. So then several chapters were covered that I don't want to detail today. I come with the last chapter that we will cover, which is this emission of multiplets of photons when the fine structure constant is 1. So I will tell you about the model, how we have implemented the circuit, the amplitude of the emission of these photons. And I will give you a proof of the k granularity, at least when the power emitted is low. So what's the Hamiltonian of the system? Up to the vacuum energy, it's h bar omega times the number of photons in the resonator, if you believe in photons. Plus the Josephson energy involving the cosine of the superconducting phase across the Josephson junction. Now, we can use the fact that the sum of all the voltages of all the generalized phase across the loop sum up to 0 and replace the phase of the junction by the phase imposed by the bias, which is linearly time dependent, you see here. And the phase across the resonator, which is nothing but the position operator A plus A dagger, times this matter like coupling coefficient that I was mentioning entering inside the cosine with the square root. So it's a very highly nonlinear Hamiltonian. But if I ask you what is the dynamics from this Hamiltonian, you have to be a very good terrorist to understand from this form what's going on. So we prefer to make a rotating wave approximation around the different VK voltage at which you have this production of k-photon multiplets. And in that case, following a recipe that was given in a paper from Ashish Clair's group, you obtain a very compact and readable form of this Hamiltonian. So first thing you see here that k-photon multiplets will be created to have this A dagger to the power of k. And with the Hamiltonian conjugate, it's just parametric drive at order k. So first, second, you see that the amplitude of production will be proportional to the Josephson energy here, OK? Slightly renormalized by the matter like coupling constant, and also proportional to the k over 2 power of alpha. So better to have a high alpha if you want to favor the production of these multiplets. Now, I have forgotten an important term here, the last operator, BK, which complicates the situation. Actually, BK is the identity matrix up to a large number of photons if alpha is small. But here, we want alpha big. So it's a big correction. And actually, its analytic form is known. It's simply a diagonal matrix in the Fox state basis involving some generalized Laguerre polynomials, OK? And what it does, this BK operator, is canceling the climbing of the harmonic oscillator ladder k-steps by k-step at some n. On the higher the alpha, the lower the n at which this cancellation will occur. You see that we have two very strong non-linearities in this system. OK, that said, you can also write the Hamiltonian for a voltage slightly different from the resonant voltage, and you have a slow time-dependent term here in the evolution. Moreover, we won't study a closed system. We will make a hole in the resonator so that the energy can leak out, OK? And this leak will be characterized by the energy-decorate kappa that defines the quality factor of the resonator. OK, and then I stress that this Hamiltonian is almost exact, apart from the rotating wave approximation. So it's at all orders in alpha. And it will so include a lot of feedback from the field already present in the resonator on the probability to have Cooper-Perce tunneling and more photons emitted. So this is the thing. And what we can do with these two objects is some Linn-Blatt master equation simulation, including the quantum regression theorem to have the quantum statistics of the photons going out here. OK, for those interested in fabrication, we want high Z, high L, low C. So we first fabricate on a low epsilon substrate. Second, because we want low C, we don't make any C. And we just do our resonator for a spiral coil of 60 nanohundry. And the capacitance is simply the stray capacitance to ground of the spiral coil. Like this, we reach a frequency of 4.4 gigahertz and an impedance of 2 kilohertz, which gives an alpha of almost 1, OK, here. The quality factor of this resonator is not at all governed on chip. It depends on the microwave elements that we put in series with the chip. And in all what I will say, it's between 36 and 72. So it's slow in order to favor the output of the photons of their exit of the resonator, OK? Instead of making a single Josephson junction, we make a squid in order to tune the Josephson energy, which is the parameter that governs the amplitude with which you produce these photons. OK, a bit more detail. You see that we have to connect the inside of the spiral coil to the outside. And for this, we deposit a brick of plastic on top of which we make an aluminum bridge. Here you see the detail of the squid. We here one of the two junctions, smaller than 100 by 100 nanometer and with a capacitance of one femtophora. Good, then we place this sample at the bottom plate of a dilution refrigerator. We thread a flux through the loop by a current biased small coil in the fridge. Then we connect to a DC line, which is simply a voltage divider, heavily filtered, and connected to the resonator through a bias T so that the DC current can flow through the Josephson junction. And all the microwave is directed to a measuring line through these right capacitance. To this measuring line is at the geometry of a hamburger brand and twist setup. It's a trick known from the 90s that allows to measure a properly very, very weak signal and to get rid of the added noise by the amplifiers because you duplicate your signals on two lines and then you amplify it twice on the two lines. And of course, the noise of your amplifiers is uncorrelated and you can calculate autocorrelation by measuring intercorrelations on two lines and you get rid of lots of the noise. And then what you measure, you have two possible setup. Either you use power detector to know the power arriving on the two lines and you digitize in time at one gigasample per second this thing, these signals, or you use a homodyne detection with two simple mixers and digitize at the same speed. Good. So I want to go into the details which are a bit tricky on how you get all the correlation functions of a quantum field coming out of a resonator using this setup. But believe me, there is a recipe. You have to alternate on and off period of time where you establish VK or not or set the voltage to zero to make the difference between certain correlated noise to subtract certain correlated noise on the two lines. And there is a recipe published by Andreas Varas Groups in 12 years ago by Dasilba and you can follow it and perform the right calculation to get these correlators. For instance, G1, which is simply the Fourier transform of the power spectral density emitted or the G2 correlators that will give you the statistics of the field. So I skip all the details. We operate by Fourier transform or multiplication inverse Fourier transform in parallel in C using multiprocessing. And we have a duty cycle of about one half. Half of the time we take data, half of the time we compute things. Okay, and this is the first result we have observed. So four years ago in the first run, we were measuring the light emitted in the full bandwidth of the resonator and simply scanning the voltage. And you see that where we expect at the voltage V1, V2 up to V5, we observe the peaks of emission corresponding to one, two, three, four, five photons. And be careful, the sixth peak is not at all K equals six. It's not at the right place and it is due to a spurious mode of the system that we could get rid of in the second run. Okay, good, we see the phenomenon. The small parent is this here. When you scan, so here it's just for V1. We are on the first peak, one photon per couple per pair. And you see that we scan the bias voltage across the mode that is also measured in situ here. We see a monochromatic line following the V equal h nu over two line. And this line is not absolutely monochromatic, of course. You see it's width here. It's about three nanovolts, which corresponds to five megahertz when translated with this formula. And this is simply the residual voltage noise of the source between 20 and 30 milli-K. Okay, so the first thing we try to see is whether we understand or not the power spectral density which is emitted by the system at the different voltages for K equal one to six. So this is a second run at a rather large EJ value so that there are a lot of photon emitted. And here the dots are the measured spectral densities in photon per second and per hertz of bandwidth. And so these are the dots actually. And they are compared in black with what we expect without any fitting parameters. And you see that it corresponds more or less except for K equal one. For the very same reason I've just told you, there is voltage noise. And here the simulation assumes no voltage noise at all. Okay, but so often the peaks are, the experimental peaks are a bit smaller than the, or significantly smaller than the simulation. But when they are smaller, they are wider. But all what counts is not the shape of the peak, but the amount of photons to the integral that is shown here in Sian below each peaks. Note also that we have to subtract the spurious contribution of another mode in the environment for K equal six. But when we count this number of photons per second emitted, this is what we get. So here you have a comparison between the data and the simulations without any noise by a Lindblatt-Master equation. So you see that we vary the Josephson energy by almost one order of magnitude, okay? And over more than three orders of magnitude, we have a reasonable agreement for this photon rate on the six peaks of emission. So here you see the heat in absolute value. So we emit between one million and one billion of electrons per second. And in terms of occupation in the resonator, it's super small. You see that the number of photons inside the resonator, in average, is between 10 minus three and at most two, three photons. Even when you emit one billion of photons, because the Q is low. So the energy goes out very quickly. Good. Now, could we reproduce these results by simple formulas rather than using the heavy simulation and so on? No. Actually, if you take the dynamical Coulomb blockade theory of the Purcell formula we were told about on Monday, it doesn't work. Of course, it's very simple. You can plot it. These are the dashed lines here that you see. And at the beginning, of course, it works very well for a very weak emission, but very rapidly, it doesn't work anymore. And for instance, this simple formula doesn't predict at all that there will be crossing and that it's possible to emit more power at V3 than at V1. This is never predicted by this simple theory. So you have to, because alpha is big, because the number of photons in the resonator is not zero, you have to rely on full quantum simulation. What about the granularity? Okay, because I told you that we have to, for the moment, I'm telling you there is one or two or three photons per group per pair, but there is no proof up to now. So can we get a proof? So for this, we need to do the statistics. Okay, so here I start with the statistics of cooper pairs. So the best way to characterize it is to compute the final factors, which are simply the variance of the number of cooper pairs having crossed the junction in time t, divide it by the average of this number. Okay, and this is equal to a certain number. And now if you look at the photons, if you believe me that there are, for instance, three photons per or K photons per cooper pair, now the final factor for the photons now will be the same variance over average, but for a new variable Kn, and it will be K times the final factor for cooper pairs. So if we can measure a final factor of K, we have a proof of the granularity. Of course, it assumes that the tunneling events will be Poissonian, okay, that the final factor for cooper pairs will be one, so that you get K for the photons. So let's count the photons, but unfortunately no white band microwave parametric amplifiers, no white band photon counter exists for microwave photons. So the two doesn't exist, and we will have to calculate the final factor from the correlators that we can measure on the light going out. Fortunately, the translation between the g2 function and the final factor is known. You just have to integrate the g2 function above one, multiply by the rate of photons, and add one to get the final factor of the signal. The g2 being the same as I have told you before. So here are the results. So here are the g2 function as a function of time for K equal one, two, three, four, okay, at a rather large ej value, okay, and you see already that, immediately that for K equal one, we have anti-buttons and for K equal two, three, four, the photons are bunched. From this, we calculate the integral from zero to pi as indicated by the final factor formula, and we have to plot the result. But we did this in a random number three, where there was a voltage noise, much more voltage noise than in the previous case. Instead of three nanovolts, we had 87 nanovolts, and this has a big impact on both the number of photons in the resonator and the final factor, and it has to be included in the simulation, and for that, you'd better know Cyprian and Gorn, otherwise your data never fit with the theory. So they could simulate the number of photons in the resonator, which is the quantity that we measure, okay, as a function of ej. So thanks to their calibration curve, we put our, we place our data at ej values corresponding to the number of photons that we measure in the resonator, and like this, we could obtain this fair agreement between the measure that the final factors in dot with the simulated ones in presence of the voltage noise. So you see that the prediction tells you that, indeed, the final factor will reach one, two, three, four, only at very low emission power, and that then, very rapidly, it will depart from these asymptotic values, so we were able to measure only down to these amount, it's more than one day of averaging per point, and you see that we approach the value of one, two, three, and four from the top by reducing ej, even if for k equal one, we have a final factor of 0.7, you see, but okay, we, at least we do understand, okay, so we have a kind of signature of this k-photon production of multiplets, so it's done, so I skip the discussion of why, why it's going up and down, in all directions like this, so if you're interested, ask me, but we also do understand why, and it's written in the Hamiltonian, when it goes up and down like this, like crazy, okay, so this I skip, and I just come to my take home message, we have used a very simple circuit, a battery-biased Josephson junction with a simply high impedance resonator, we could observe the multiplet photon emission of this circuit in the strong coupling regime, we have a quantitative understanding of the photon rates emitted, and a signature of the granularity of the photon emission. If you're interested in this subject, there are other related work in optics and in microwave photonics in the group of Chris Wilson, where they produce triplets by a more conventional way consisting in applying microwave pump, and there is also an additional work not published yet which shows tens of these peaks at high values, but in a way which is not fully understood. And then, on a broader view, I want to say that here what is interesting is not this particular subject, but we have ways to extend circuit QED in a steady auto-equilibrium situation with super-strong non-linearities and strong coupling between the electrons or the charge carriers and the light, and it's a test bench, please play with it. Thank you for your attention. Yeah, thank you very much. So we have time for some questions. I guess I potentially have several, but I'll start with one. Have you thought about splitting the photons with multiple resonators? Yes, this was one of the chapter where we have used two resonators instead of one, and we could demonstrate two continuous beams of entangled photons at two different colors on two different lines. Okay, with a decent degree of entanglement of at the db level in terms of the two most squeezing, so not super high. Other question, yes. So this is an invited question. Why does it go up and down? Ah. Okay, so we were not understanding at all on the experimental side why it was going up and down and the people from Harim's group told us just to look at the Hamiltonian. It's written in the Hamiltonian. So what happened is that we can have two different situations, one where the field already present in the resonator stimulates Cooper paternally, and another situation where this field blocks the thing. And why? Because as written in the Hamiltonian, you have this term here which is, which contains a stimulated emission, a dagger to the power of K. So the more photons you have in the resonator, the easier it is to produce more photons. So it produces over, it produces bunching of the bunches here. So a super procedure, you see. And then the second non-linear term here blocks the production at a certain level. And you have this competition with the two levels which is different for the different case. And this is why you have this variation, this trans variation that we observe. The representation of this operator in terms of Bessel functions, K-th order, Bessel functions which are normal ordered and the non-linearity of the Bessel functions basically what is expressed here. And that gives this up and down and these feedback. Thank you. Okay, so time for another short question. If somebody, yeah, I knew that. Have you looked at also higher order statistics of the noise? So like in our photon triplet paper, we see lots of patterns in the skew and interesting things. We thought that G2, it's already one day of averaging to get a G2 when at low emission. Okay, theoretically it's done. Okay, then yeah, let's thank the me again for the talk. Thank you very much. Thank you. Thank you.