 Thank you. Thank you. Thank you, the organizers. So as you have seen, Luke already spoiled my talk. So thank you very much. I'm very happy to be here in presence again. Okay. So what I'm going to show is very much connected to what Luke has shown as complementary. It's a long-standing collaboration that we have with the Quantronic Group for more than 20 years. And this is part of European project in which we are working, which is called NQC, which is devoted to the ideas to use these 100 states for physical realization of a qubit. So very briefly, a motivation doesn't work. Okay. So let me give a brief motivation, already seen from the talk by Luke, that we are interested in this type of nanowires in which, well, they can be done with a high degree of perfection, with a very hard gap, proximity gap. You have seen already this in the talk by Luke. On the other hand, what is creating a revolution in the field of mesoscopic superconductivity is a possibility to explore these devices using secret techniques or microwaves. And these two things connect to a very old topic in mesoscopic superconductivity, which is the interplay between a superconductivity of pairing effects and interactions, coulomb interactions. And so this is having already introduced by Luke, but the old line of my talk is going to be, well, I'm going to have two parts, one in which I discussed the effects of interactions in a multi-channel nanowires junction. And the second one, if time allows, I give you some idea of the theory of superquity detection of these states, or not only the line shapes, but also the intensities, how we can understand them from the point of view of theory. So about the interplay between superconductivity and coulomb interactions. So this is all story, mesoscopic physics, I mean, and everything we know has been mainly focused on one model, which is the Anderson model. Anderson model with superconducting leads. And they are the main topic in the Anderson model is the interplay between the condor effect, so the fact that if you have an electron isolated in the dot, the spin is going to be screened by the spins, the anti-ferromagnetic coupling with the spins in the leads. And this competes with the paving correlations, which tends to destroy the condor correlations. So this interplay between condor and paving produces a transition, quantum phase transition, between a singlet phase and a doublet phase. And the Anderson model is what I understood have been calculated with many different techniques, including numerically exact techniques like energy. And with energy, well, you get the phase diagram for this parity phase diagram in the plane here, where they have the level position here and the coupling to the leads here, where you see, well, for phase difference equal to zero, you have this typical dome shape separating the doublet and the singlet, and when you have the phase difference equal to pi, you have a chimney-like shape separated the singlet and the doublet. And this has been, in fact, confirmed rather recently using this, well, secretive techniques in experiments from Delft, where we've been able, using these techniques and a transmon, like geometry, where the couple of quantum dots define on an indymars and a nanowire. They've been able to trace experimentally this phase diagram, no? It's a function of the dot level and function of the barriers, which are controlled by gates. Okay, but what is the general situation, in the case, like you have explained with the nanowire, is that you have a situation in which you have many channels, not like in the Anderson model, which you have one level only, so you have many channels. There is a finite length. They are a high transparency. It's not a dot, let's say. And Oakhouse already discussed, well, what is the effect of the length on the spectrum of these junctions? What is the effect of spin orbit? And now, well, we would like to ask the question, what is the effect of interactions? Even if it's not a dot, Coulomb interactions can have a role and this is what we want to address. I mean, these two effects have been already covered and discussed by Oak, so I'm going to focus mainly on the effect of interactions and, well, what I would like to convince you that there are noble effects beyond what is known from the Anderson model, which I call some for the friends. So already, Oak has explained what are the type of transitions that one can have within a non-interacting picture of the nanowire. So you have finite lengths, you have spin orbit, which splits the under-money force and so depending if you are initially in ground state or you are in a poison state, you can have these spare transitions or single particle transitions. Here, the transitions are the red ones, greens are the single particle transitions and we have these additional elusive mixed part transitions which are predicted by this non-interactive model where you excite a cooper pair and you place one class of particle in lower under-state and one class of particle in the upper under-state. So what has been possible with the recent experiment from Saclet is to map these transitions with a high stability as a function of the gate and be able to see the evolution more clearly. This is something that was not possible in the previous setup from Saclet. But here, this I just reminding you of the results that have been shown by Oak before. The blue lines here apparently evolve in a parallel way to the per transitions but do not behave as predicted by the non-interacting theory where you respect the generacy at phase equal to zero and phase equal to pi. Well, so something to do with interactions has to be playing here. Now, interactions in these devices we can expect to be very much screened because you have a back gate, some metallic gate. You have the leads which screen the interactions on the wire, the regions of the wire which are below the leads are highly screened. So we can think that the interactions are essentially present only in the normal region here but highly screened and we can model them by a counter light potential. Let's say only at the point with some intensity U zero. If we estimate this parameter U zero from a very simple approximation, let's say Thomas Fermi approximation, we need to know things like what's the Fermi wavelength here in the nanowire, what's the effective mass, what's the dielectric constant of the indium arsenide. If we take a reasonable parameters for our geometry or the geometry experiments, we find that the effective charge in energy here in this normal region is going to be rather small compared to the superconducting gap, like 10 times smaller than superconducting gaps of the order of 30 micro EDs. So this is the degree of interactions which is typically weak to produce zero pi transition like in the Anderson model. In the Anderson model, what you need interactions stronger than the gap to kill the condor effect and go to the odd state which where do we have the bias phase? Okay, so hand waving an argument that has already been put forward by UG is that if we have an excitation, here I showed the diagram for under states without spin orbit for a junction with the length of the order of coherent length, so we have 200 manifolds, but without splitting of spin orbit, I have, if you place two electrons there, two excitations there in the two under manifolds, well, you have to have in mind that these are states which are confined to the normal region. These states, the under states, decrease exponentially or inside the superconductors. So they are states which are confined. So if we estimate an exchange interaction direct for a magnetic interaction between the two arising from the Coulomb interaction, we would estimate that this exchange would be of the order of this parameter U0 volume and this gives something of the order of five giga. And this five giga with an isotropic exchange would split the lines into singlet and triplet with, well, this singlet is known to generate the triplet is for the generate. This is what you get from a hand waving argument, but it's not what is observed as we have shown what you're serving experiments that all lines are split. So if you want to go to be more ambitious and good to try to be more microscopic, which is not always a good thing to do, you have to go into a very complicated model, this tie binding model is, well, say the most accurate or microscopic model we could try to do in this system. And you have to have in mind that we need more than one channel because we need multi-channel effect is important. For instance, here we couple two, three chains like that and we have to add a final length because it's a final length, it's important for this. And then you write down Hamiltonian, the most general Hamiltonian you wish with potential electrostatic potential variation hoping in the X in the Y direction, spin orbit splitting, spin orbit tunneling. And you estimate all these parameters, you can estimate all these parameters from the discretization of the continuous model where you have effective mass estimation of a spin orbit in the Indian Mars and I which is known for this is of the order of 15 to 30 milli electron volt per nanometer, the size of the superconducting gap is only present there. And you have to add interactions in the normal region here, which you can use the unusual have a like interaction. Okay, but well this is plot which Luke have already shown but this isn't the non-interacting case. I mean the non-interacting case as who have already shown with this system where you can have up to three channels if you increase the chemical potential. So you start from a very pure doping and you start doping the system and you start populating the second sub-band. You will see that the spectrum under spectrum, the non-interactive under spectrum evolve like this. Evolve from a situation with the lower manifold and the upper, the second manifold have opposite phase dependence, so opposite curvature to a situation in which the lower and the second under manifold have the same curvature. This is because you start having a contribution from the second sub-band, essentially. These two are coming from separate channels. That's why they have the same phase modulation. And with this, even if it's a non-interacting, you can start having features which resemble what is in experiment because you see this is the situation where you have only one channel and this is the situation where you have two channels. You see these per transitions and these blue mixed transitions come closer and this single particle transition go up in energy. Okay, so, but in order to incorporate interactions, well, we can do something approximate first which is to take this infinite gap limit. So if you take this infinite gap limit, essentially, you project everything into the normal region. So you just take the four sides where we take four sides in the normal region. Why four? We take four because we need two channels and we need the final length. So this is like the minimal model in which we have these two effects, the final lengths and two channels. And then we project the paving interaction from the leads into the central region and we get effectively one paving term which is local, which is singlet paving. This comes from the singlet paving, the leads you project into the normal regions. It's a singlet paving. And then we also get what is a triplet paving, a triplet paving because we combine local paving with spin orbit. So this is like with the Majorana business that you induce combining spin orbit and paving. You induce triplet paving, but this is a time reversal symmetric paving. It's tripled but it's time reversal symmetric because spin orbit is time reversal symmetric. So this is still a time reversal symmetric. So such a model for parameters which resemble what is in the experiments gives this spectrum for transitions. The green lines are the single particle, the red ones are the triplet transitions and the blue ones are the mix. This is without interactions. When we include interactions, what we see is on the one side that the single particle transition go a bit up in energy and the per transition go a bit down in energy. Remember that this system with interaction wants to go into the odd state. So we favor the odd with respect to the even. That's why we have this energy, this lower of the per transition here. And if we compare the single particle transitions, we see that there is a generacy at phase zero and phase five. It's still there, it's protected by Kramer's theorem. They still have the time reversal symmetric, interactions are time reversal symmetric. Whereas in these mixed transitions, these are not protected, and this split, but do not split as what we expected with this hand wave argument with exchange, isotropic exchange in a way the combination of interactions and spin orbit makes that this effective change is not isotropic. And then you see that, okay, there is one line which goes up and that there are not three, the other three do not go down together. So we break for the general, for the general. If you want the phase diagram for this, the parity phase diagram for this four-side model looks like this, where you have, you see the phases which are called the zero phase, where we have a, the ground state is an even state and the pi phase where the ground state is an odd state. The position of the situation where you have the best agreement with experimental data is indicated here. So you see, we do not have any transition at this point. We have an even ground state. This is a big complicated evolution how the lines evolve with different interactions in the model, how they evolve with increasing the coulomb interaction, how they evolve with spin orbit. I think I'm going to keep the detailed discussion just so that without spin orbit and just with coulomb interactions, you see the splitting into singlet and triplet, as one would expect for the isotropic exchange, whereas when we switch on both interactions and coulomb interaction and spin orbit, you break the generality of the triplet lines and this is where we get good agreement with the experimental situation. If we want to get the better agreement with experimental, not using this minimal model, but realistic parameters, we go back to this divining model and in this divining model, we do an approximation. We cannot solve interactions exactly, but we can do an approximation for weak interactions, which consist in taking the non-interacting solutions of the Bogoliubov operators, of Bogoliubov quasi-particles as a basis and what we do is to write the fermionic operators in terms of the Bogoliubov operators and then what we do is to project the interaction into these basis of the Bogoliubov excitations, but it's on a truncated basis. We take a truncated basis in which we keep the ground state, we keep states with one excitation and we keep states with two excitations. We truncate this to some number of levels, which we call n-projection, which we keep small, and then we get this comparison with the experiment in which you can see how the lines in this multi-channel divining model evolve with increasing charging energy. So you can see that essentially the effect of interactions is that you increase the energy of these single particle transitions. The per transition goes down in energy a bit and the main effect is produced on the mixed per transitions. Mixed per transitions, which they generate here at phase zero and phase pi with interactions, they start to split. First, you clearly see that this one line which separates from the other three, this is singlet-triple-like, we see this camel-back shape, which is also present in interactions. So I think, well, the agreement is qualitatively okay for where we are happy with this level of agreement. Perfect agreement has not been obtained yet, but we are quite happy. We think we understand physically what is the meaning of all these lines, and well, we see that while single particle transition ships outward and keep the generacy, there's full breaking of the generacy of the mixed transitions. Well, this is something technical which is related to how the approximation converges with the number of states in the truncation of the effect in Hamiltonian. But how much time I have? Six minutes, perfect. So in these six minutes, I try to give an idea of the theory of detection. I mean, not only we are interested in understanding the line shapes, but we are also interested in understanding the intensities of the lines. So we did this work in 2010-20 with collaboration with the Sacle people where we analyzed the coupling of just a nanowire in a loop coupled to a microwave resonator. So the simplest theory would be I take the Hamiltonian of the nanowire with a phase bias and I expand to lowest order in the phase fluctuations. I get a coupling between the resonator and the loop like this. This is clearly not enough to describe what's the experimental results. We need to add or to go to second order in this coupling Hamiltonian. You have to take the second derivative of the Hamiltonian respect to the phase. Look here, this H prime is the current operator and this H double prime is like an inverse inductive operator which couples to the square of the fluctuations. And then we perform just a perturbation theory to get the coupling of how the levels of the under state levels and the resonators are modified. From the first order perturbation theory we get a contribution out of this second order term and with second order perturbation theory we get a contribution out of this first order term. And if we combine the two, we get what is measured in the experiment at the end which is the shift in the resonator frequency due to the population of a given under state. And this contains two terms. One term which is essentially the second derivative so the curvature of the level respect to the phase. And then another term which contains these energy denominators which are typically in dispersive chamber coming like measurements which contains these energy denominators and the matrix elements of the current operator. But we can have two regimes. Basically we have two regimes. One regime in which the resonator is largely detuned with respect to the under transition and the shift in the resonator is basically determined by the curvature of the level. And there is another what we call closer to resonance which is the dispersive regime where the shift in the resonator goes with these energy denominators in the James Cummings model. So we call this from adiabatic to dispersive readout. This is not only valid for the under state system but for any quantum circuit, let's say. There is another aspect of the theory which is how we drive the system. And there you can think of driving or sending the microwaves through the gate which is what is done in experiments from SACLEA or you can just modulate the flux in this loop. And by doing that, you induce different transitions. While driving with the gate, you typically break in the inversion symmetry and as you break the inversion symmetry, you allow for these pseudo-spin flips. Remember because of spin orbit here, spin is not conserved. It's pseudo-spin which is conserved if the system is in a symmetric respect to the inversion. But with breaking the inversion, you break the symmetry and you are allowed to do these spin-flint transitions. On the other hand, with the ideal flux modulation, you wouldn't conserve a spin. But nobody does this really. Because always in experiments, and this is a picture that took from this paper we have with the Yale people, there is always some elements which break the inversion symmetry like having leads, there are the gates, or even on the nanowire, we have the deposited superconducting layers which can break this inversion symmetry. So in general, the inversion symmetry is broken naturally here. So that's why, well, at the end, with this theory, we can fit not only the line shapes, but also the line intensities. This is an example that we take from this paper that already you have shown in which you can see that the lines goes from blue to red. So it means that the shift in the resonator goes to negative to positive and this is clearly going together with the curvature of state. So this is what we call the adiabatic regime because in these experiments, the resonator frequency was like at 3 GHz. So very much the tune with respect to these transitions which are at 50. So this is essentially dominated by the curvature. And, well, there are no selection rules because all transitions are possible because we are breaking this inversion symmetry in this. Just one thing, Norm, is saying that we have extended very recently the theory of detection, to the case of an interacting quantum dot. This is modeled by the Anderson model. This is a work we did in collaboration with James Pascal in Copenhagen. Essentially, we need to calculate what is the admittance of the dot coupled to the micro resonator. Essentially, it's the same as calculating the current correlation, the current susceptibility. And from this susceptibility, we can get both the shift in the resonator and the damping rate from the real and the imaginary part of this susceptibility. Well, this can be done analytically in this infinitive limit. I'm not going into the details. I just showed that well, these are the type of results we get for the shift in the resonator frequency when the system now is in an even state and the resonator crosses one of the 100 transitions. And these are the matrix elements that determine the susceptibility in a plane where you have a phase and you have the position of the level. You see here a region where there is no response because the system is in the odd state, basically. And just to flash this, because I want to tease a hook that we should do something with this data, I want to show that exactly people have already observed the transition, the zero pi transition in experiments. They have this singletone spectroscopy when you sweep the gate and you see a sudden jump in the response which is signaling this zero pi transition. And well, there's a similar result from the people in Gale which are going to be published. And I want to say that we can understand these results very nicely with a very simple model which is an Anderson model and have to include an ancillary level there for understanding also the two-tone spectroscopy but I don't have time here to discuss this in detail. So let me go to the conclusions and say, well, a combination of Ivory nanowires with thick equity are allowing us to understand the physics of the understates in this system with an unprecedented degree of perfection of accuracy which goes much beyond what you can do with transport. Well, we have, as you have shown, this experiment has revealed for the first time the fine structure like in atomic physics and the structure now for the under level coming from a splitting due to spin orbit. There are novel signatures due to interactions from the mixed transitions which go beyond the single-level answer model. And well, about theory of thick equity detection, we have developed the theory of adiabatic to dispersive readout and well, it started to work on this inductive response on interacting quantum dot. And well, there are of course many open issues understanding the origin of dynamics and dynamics of excess quasi-particles, extend the solution of the interacting model beyond the limit of weak interactions. Going to the topological regime is another dream in this field and now, acknowledgments were already shown by Uck. Let me mention Francisco was a leading force in this theory of interaction work. Sungun Park is a post-doc, a long-time post-doc in Madrid. Javier Sarejo was in Madrid for a short time and now he's in Cartagena and the Contronic people, you know, Cyril Metker, Leandro, Marcelo, Uck, Christian. Leandro is now in Bariloche. In Copenhagen, we had these people doing the nanowires and the Gens Pascal group. And I would like also to thank people in Yale, Max Hagen, Bara Fatemi, and Michelle Devoret with whom we also collaborated in this topic. Thank you for your attention. Okay, thank you, Alfredo. Are there questions? Yes. Thank you, Alfredo. I have a question regarding the coupling you worked out with the electromagnetic field. In principle, if you start from the minimal gauge-in-variant, let's say, light-matter interaction, especially in your case, since you have a gate, you should get both magnetic coupling and electrical coupling. I was wondering whether you had considered that and whether this would change, I mean, the predictions concerning the spectra. You mean for driving or for detection? No, we haven't considered, you mean capacity, including capacity coupling. We have not done it. I mean, in the case of the Anderson model, it's quite straightforward. I think there's not going to be much difference, but for the case of the nanowires, we haven't, it's much more involved. I mean, let's say, because you need to understand what's the charge correlated or slared, which are very much model-dependent, so I don't know. But it's an interesting issue, and people ask what could be the advantage of capacity with respect to inductive, and I don't have an answer. Maybe you have something. You have an idea. I know that essentially the inductive response is... More questions? Yeah, so a complementary question. What about the effect of the environment in the wire? You consider only the effect of the coulomb interaction, but maybe there is also some effect of the environment in the microscopic model of your wire. You mean the effect of the... Okay, so you mean the environment beyond the resonator, something like... Yeah, of course, we can incorporate in the theory some other spectrum, the spectral densities for modes that you can couple to, and this is going to affect the broadening. So, in principle, what we do when we try to fit that spectrum, we put some phenomenological broadening. I mean, the broadening of the lines they are in the spectrum is phenomenological. It's not from a microscopic model. We don't have yet full understanding of what are all the mechanisms that are limiting the... or what are determining the width of the lines. So we take this in a phenomenological way, but, of course, it could be nice... Yeah. Yeah, yeah, yeah. This would be a nice thing to do, but we haven't tried it yet. Hello. Just in the last slide, you mentioned that you're also interested in going into the topological regime, and I'm just wondering if you replace this nanowall with some like a nanowall with topological properties and your entrance level has some spin polarization. How does that affect the transition that you're allowed to have in this circuit Q and D setup? Yes, this is something that many people are trying, and the point is that the Majorana, if you say you want Majorana signature. Well, in these revisions, what you can expect is a signature of topologies that you induce topological superconductivity in the leads, and then you have Majoranas, two Majoranas. For instance, you have a junction, you have two Majoranas, but this is going to be typically very, very low frequency. The problem there is that you would only be able to detect them through the transitions and the transitions to higher energy levels. I think it can be done, but I don't know if it's going to be a complete demonstration of topology, because, well, for instance, we look at the spectrum we had, well, that Shala showed yesterday, that you have these two-tone spectroscopy which goes to zero at some point, apparently go to zero. You may have many, many different mechanisms which give you this signal, and it would not be a complete demonstration of topology. As you usually have with the Majorana business, there are many, many different possible scenarios in which you can have. Maybe you have some localized level there which is coupled and gives you very low zero energy level which is not the Majorana. So I think, well, people are trying to work on this, but I'm not sure that it's going to be definitely proof of anything. Fortunately. No braiding, yes. Okay, I think we have to close the discussion and move on to the coffee break. Okay, we, yes, we thank you again.