 session. So my name is Christian Ast, and I'm from the Max Planck Institute for Solid State Research in Germany, and I will be your session chair this morning. The first speaker is Eugue Portier from SACLE. Please go ahead. Thank you, and thank you very much for waking up this morning. I'm going to present you experiments that we did in SACLE, and in fact this is a joint work with the next group of the next speaker Alfredo Levi Gallati. So in fact we will have a joint presentations. I will essentially present the experiment that we did in the SACLE, and this is essentially the thesis work of Cyril Metzger, who did his PhD, defended his PhD two months ago. Together with my dear colleagues Marcelo Goffman and Christian Abina, and with the technical help of Pascal Senard. The group of Alfredo Levi Gallati, we collaborated with two students, Sungun Park and Francisco Matute, and Leandro Tosi, who is now back in Bariloche, was postdoc in our group, and we worked with Andres Rednosso, and we all those experiments were made possible because of those fantastic nano wires that were grown in Copenhagen in the groups of Peter Krocstrup and Jesper Nugart. So we've heard a few times about Andre F. States during this conference. So in fact each time you put a coherent conductor between two superconductors, and you phase bias this structure, you have bound states for electrons within the coherent conductor, because electrons at energy below the superconducting gap cannot enter the superconducting electrodes. And those bound states are an energy which depends on the phase difference between the superconductors. And there are two simple limits that have been investigated in the past. The simplest one is the case of atomic contacts on which we did many experiments in the quantumics group by making break junctions in a superconducting film in the form of a suspended bridge that you can bend till you get one atom. And in this situation it's very simple. There is just one pair of Andre F. States in the gap, and the energy of those Andre F. States has a simple analytical dependence on the phase difference delta across the weak link and on the transmission of the channel that goes across the junction. Then there is an opposite limit which is a limit of diffusive SNS junctions. And in that case you can compute that there are many, many Andre F levels, such that in fact you have in practice a continuum of density of states at energies above a mini gap which depends on the phase difference. And this has been shown yesterday in the talk of Francesco Giazzotto. It's an experiment in the quantumics group where with an STM we measured the density of state in an aluminum wire which is connected to a loop of aluminum. And then in that case you see that the density of state which was measured by tunneling spectroscopy, so adding electrons at different energies. You see that the density of state is continuous and the mini gap here depends on the full shape, depends on the phase across the junction. What we did with atomic contact was a much more precise spectroscopy which is microwave spectroscopy. Typically the energy resolution of this technique is at least two orders of magnitude better than by tunneling spectroscopy. And we see here the spectrum near a phase difference of pi of an atomic contact junction in which you see the transition energy between the ground state and the excited state, so it's twice the Andre F energy. And it follows precisely this dependence here. And the line width is such that you can define the transmission with five digits. So in between those two limits one can wonder what happens if it's possible to understand intermediate situation. And this is why we started to measure weak links which are made with those very nice nano wires grown in a Copenhagen in which you have a very good and very perfect interface between a semiconductor, indium arsenide, and the thin layer of aluminum which in our case is all around the semiconductor. And the structure that we make and the one I'm going to show results on is precisely this sample here in which this nano wire has been, the aluminum of this nano wire has been etched away in a section which is about 500 nanometers long. And then this is placed between two superconducting contacts in fact which belong to a resonator out of niobium titanium nitride. And if you look at this structure from the side about that scale it looks like that. We have placed below a gate, a gold gate here. And then after etching the niobium titanium nitride and a little bit over etching the substrate you get that the gate here is just below the weak link at a distance of about 100 nanometers and they are separated by vacuum. And it turned out that this way of making the gate was allowing for a very good stability of the charges in the device because in fact the metallic gate below the weak link is screening the charge that could be at the surface of the substrate. So it's a situation which is very clean. The system has a finite length in contrast with atomic contact. It's tunable because of the gate and we can have just a few channels so it's much more simple than SNS junctions. So one could think that everything is simple in this situation. So in order to probe such a circuit we need to be able to phase bias the junction. And this is obtained by having the weak link here which is exactly in this position at the end of a resonator so that this is the central conductor of the Coplanda waveguide resonator that I will show just after. And this is a ground plane so we are making a loop here around okay and by applying a flux across this loop it's possible to phase bias the weak link. Okay so this is the end of a resonator so this picture is just here and it's a lambda over 4 resonator with a resonant frequency of 6.6 gigahertz and the coupling quality factor of 17,000. And this is probed by coupling this resonator to a bus and sending a microwave in and looking at what comes out. Okay so importantly we are able to change the chemical potential in the nano wire with this gate and it's also on this gate that we will apply microwave to excite the system. And this way instead of other systems in which the frequency is applied through the resonator here it's a very wide band we can as you will see excite transitions from typically 0 to 35 gigahertz. The coupling scheme is schematized here so in fact the resonator here at its end is shunted by the weak link so you have the inductance of this ending part of the central conductor here which is in parallel with the nano wire it looks like that and then when Andrea states here are occupied depending on their occupancy and their energy dependence it gives a shift to this equivalent LC resonator and this is what we measure. Okay so we sent a tone at a given frequency very close to 6.6 gigahertz then we amplify with the two part that we got from a Wheeler liver in the Lincoln lab and we measure the two quadratured i and q of the transmitted signal. So since the resonator frequency depends on the occupied Andrea's bound states when we induce transition we see a change in the frequency so we see a change in the transmission s to 1 so a change in i and q and this is what we monitor in the experiment. So I will show you directly some results one of the spectra that we have measured for a given gate voltage and I show you here the two quadratures that are measured so at each in each pixel in each pixel of this image we have measured during 150 millisecond and we have just made a difference between i and q when the drive is on and i the drive is off in fact we do a lock-in measurement. So the x-axis is a phase difference across a weak link tuned with a flux and here this is a frequency of the drive excitation applied on the gate. So what you see that it looks quite messy there are many many lines and I am going to show you that essentially we can understand what those lines are. So let me highlight some of the structures that we see we have typically red lines which look like what we obtained in atomic contact it's about sinusoidal line very regular then we have these bundles of four lines which cross at phase difference 0 or pi and have this funny funny shape here and then there are other lines which look like a bit the red one but sometimes have funny shapes okay and so the other lines that I have not highlighted are of one of those three kinds. This is other example taken at other gate voltages okay so once again these bundles of four lines that are highlighted in green regular line here and other which in that case also look quite regular and this is a third example here here you see this type of moustache here here also and then more regular line with a minimum at phase pi and the maximum at 0. Okay so what can we understand from the spectrum so let me give you in detail how we can give a metroscopic description of a weak link so we consider a normal region of a given length which is placed between two superconductor with two different phases so the intermediate region here is described by dispersion relation for electrons the superconductors have a gap okay and what happens if one electron comes from the middle part here and arrives at the superconductor at an energy below the gap it cannot enter as an electron but it can take a second electron and make a cooper pair which is described as having a hole reflected in the that travels in the opposite direction in this Andrea reflection process there is a phase difference between the electron and the hole which is given by the phase of the superconductor and the terms which depends on the energy of the electron now the same thing can happen on the other side and in which case the whole is reflected as an electron so you see that in the one turn here a cooper pair it's transferred from the left to the right so this is a state that carries a super current there is also in another system with a finite length a phase associated to the propagation of the electron and of the hole which have not exactly opposite k vectors so that if you sum the two phases you get a phase which can be recast in this simple form it's two times an effective length L divided divided by h bar vf or delta which looks like a coherence length but it's not the coherence length of the superconductor because here the Fermi velocity is that of the electrons which have travel in the middle part and this is very important okay so it's a high-bride quantity in the sense that it mixes the Fermi velocity of the middle part and the gap of the electrons okay and epsilon is the energy in units of delta so if you want to build constructive interference which is a condition for bound states you get a simple equation here that can be solved with a computer and this is a description for right moving electron here but I could have taken a left moving electron in which case I would have gotten the same expression but with the opposite sign in front of the face okay so if I solve this equation for weak links of different lengths this is what I get for the case of zero lengths essentially this is what we have for atomic contact and here I just show what happens that positive energy because the system is a symmetric in energy okay so you have you have a solution for right moving electrons and left moving whole and the opposite here and the two meet at the phase difference pi now if you increase the length interestingly you find that for some phases you have several solutions for this equation and the number of solutions increases with lambda now if you assume that there is some scattering and the simple way to describe that is by having a single scatterer at a given position in the weak link then the right moving electron and left moving electron are coupled to that at the point in the spectrum where the two lines two types of line were crossing you get a lifting of degeneracy and the bending like like here okay so this way you get the solution they have given before for zero length weak links but you can compute also for finite lengths solving this this transcendental equation shown here okay so what about the spin in this system in fact so long spin does not play any role because whether it's an election of spin up which is reflected in a hole or an election of spin down the equations are exactly the same which means that those two states here are completely spin degenerate so how can this spin degeneracy be lifted well it's like in atoms if you include a spin orbit interaction you get the lifted a lifting of the transition lines so in this manner why are you expect to have rush buspin orbit which can be recast in the Hamiltonian as a term which couples the k vector in the x direction and the spin in the transverse direction so if you assume just a 1d wire all the terms which depend on k in the y direction vanish and this is just splitting the two bands horizontally along the kx vector and it turns out that the propagating term here which depends on the sum of the k vectors here and here for this and ref reflection or here are exactly the same and there is no effect of spin orbit despite the fact that if you look at the dispersion relations the two spins are very well distinguished so in order to have an effect of a spin orbit you need to have at least two sub bands in which case you can consider the transverse kinetic energy and the confinement potential giving rise to a quantized k vector in the transverse direction and then this term here in the rash power Hamiltonian couple the spin and the k vector so that the two bands here that are crossing at points we are with different k vector and different spin couple and you get a bound structure in which the Fermi velocity at this energy here is different for different type of elections so here it's different which are essentially at this k vector with pin up and here if you follow it's been down and here they are spin down and here they become spin up so in fact in those bands you have a spin texture but not a given spin okay so if you play the same game as before given that the Fermi velocity are different for the different electron bands here you get that the effective legs lambda is different for electrons of spin up reflected in whole spin down or for electrons in spin essentially down reflected in holes here so that you get all together in the same weak link two diagrams which correspond to do different lengths is so without scattering you get this then if you have some scattering in the nowhere you leave the degeneracy and you get this structure okay and once again this can be obtained by solving this transcendental equation okay so this is the spectrum of the andrew state but what we are measuring in micro spectroscopy are transitions between the states okay so in this excitation representation where zero is the energy of the ground state so it's a condensate of Cooper pair you can make transitions which we call pair transition in which you break a Cooper pair and then you create two quasi particle that can be put either in the same manifold here for example the lowest one in the top manifold or they can be mixed per transition where you put one quasi particle in one level of the lowest manifold and the second one in the upper manifold and if you compute what this gives for the transition energies this is for this for this case here for this example this is the lowest pair transition here the upper pair transition and those mixed pair transition here make such bundles of lines now it turns out that it's important to consider situations in which the system is not in its ground state but it starts with some quasi particle occupying Andrea F levels so this is what we call quasi particle poisoning and in that case if you assume that there is one quasi particle in one level you can with microwave promote it to another level and you get here for example for two manifolds you get four possible transitions which are shown here and finally since here you have two levels which are close together you can make a flow energy transitions within a manifold okay intra manifold transition that's very low energy okay so now we can compare with with what we get in the spectra okay so once again this is here the spectrum of Andrea F level and here the corresponding transitions that you expect well you recognize you can recognize here at low energy this type of transitions here which are what we call single particle transition atomic like so promoting one electron from one Andrea F level to another then there is a very regular here per transition and the fact that we see both of them both type of transition mean that during the 150 millisecond of the measurement at each point sometimes the system is in its ground state sometimes there is poisoning and poisoning can occupy different Andrea F levels this is why we see all those transitions and if you look very closely at what happens here around this region so yeah well I was doing just one quadrature but in fact there is information on both and in fact you see better the lines in the Q quadrature here if you analyze precisely so the outer line correspond to transition between say the lowest level in the lowest manifold and the upper level in the opposite upper manifold and then the lowest line the lowest line correspond to transition which are from the upper here to the lower here okay and the middle line are just intermediate transition so from them you can compute the energy that you expect for transitions which are within the manifold and you get the dotted line here and in fact if you look closely there is some sign of these transitions at some in some regions here it's faint but it's visible okay so what I've shown you is that in this spectrum we understand these bundles of four line we understand the interim manifold transition we understand the per transition but we would expect to have above this lowest per transition mixed per transition okay in which you put one quasi particle in one manifold and the second one in the other and we don't and instead of that we see these lines here we don't which don't show any degeneracy at phase zero and pi and those are other examples that I have shown you okay so now you you get familiar with a single particle transition here we see two exemplaries of them and here one of them very well developed and each time we have a simple per transition and then those blue lines here okay so to have more insight of what those lines are let me show you the evolution of the spectrum of one of the spectra with gate voltage at a given phase which is given by this orange line here so this is the spectrum here correspond to this gate voltage and this is a large scan of gate voltage so you think that you see that things are moving around in a somewhat erratic way but what you can see that essentially the single particle transition which are in green and the lowest per transition which are which is in red they move opposite so when the per transition is going up the single particle transition is going down and if you understand that single particle transition are between the lowest and the upper manifold if the upper manifold and the lower manifold come together because the lowest comes up then the per the single particle transition goes to low energy so this is what you expect and then those lines that are in blue essentially they move a more or less parallel to the single to the per transition okay not everywhere but most of the places they move together which mean it has something to do with the per transition this is a spectrum at another position here and here spectra which are taken also in a in a narrow range of gate voltage where you see the same effect okay so here you have a single particle transition they go up with gate voltage while the per transition and the blue lines here come down so the interpretation we have of this is that those blue lines here are indeed mixed per transitions but we have to add one ingredient in the description of the system which is coulomb interaction because when we are making a transition in which two quasi particle are created since there is a finite length there is a finite dwell time and the two quasi particle interact and then depending on whether the two quasi particles that you create it would be in a system with simple spins are put in a singlet or a triplet state this coulomb interaction will have different intensities so in a system without spin orbit but with true spins you would expect that the transition in which you create the election in a triplet state will would have one energy and in the singlet another energy okay and in fact if you if you look at most of the spectra you see that there are three lines and one line above okay and the in the in the dependence here most of the time you have three lines which are close together and the first line which is further away so there is something left of this triplet singlet and this will be explained in detail in the talk of Alfredo just after okay so I will I will stop here so I've shown you that in finite length sweet links the spectra are much more rich that in in zero length sweet links there are many possible transitions but we can understand most of what is happening and so essentially spin orbit and coulomb interaction have to be taken into account and these results have been published in a series of papers here and you can also find them in the PhD thesis of a Ciel Metzger that you will find on our website and I must mention that there is some related work done in the group of Michel Devoray in Yale where they have essentially focused in manipulation of different transitions that you see in such spectra and this is something we have also done but I have not mentioned it here okay thank you for your attention okay thank you other questions thank you impressive talk how many modes do you have in your wire this one question and the other is have you consider a much more involved model from scattering as you have I mean it's been urban interactions so scattering could depend I mean in a way on only on the spin so how many channels we have so yeah you can try to guess it from the transition that you see and typically we think we have two three four sometimes for the type of spectra that I have shown here okay of course you can move with the gate and then get situation where you have much more okay then it becomes messy and also the phase polarization in that case is not perfect as it is here okay and as for the modelization the most complex that we have done was a type ending model so this is what Alfredo is going to present so I don't want to spoil this presentation okay more questions this is a bit of an invited question but can you tell us more about the the coherence or the time resolve or do you observe some coherence in the system okay so these are data which in that case I were taking in Yale so where they measure so rabbi oscillation for a pair transition here and they get the T1 of about 10 microseconds and T2 echo of 400 but Ramsey time was too small to be measured and I think here okay here I was moving in the wrong direction okay so those are rabbi oscillations also measured in in our experiments and this is manipulation of spin transition here using coherent Raman processes that were obtained in Yale and here is a sum up of their result with echo time which are between 10 nanoseconds and 500 nanoseconds and here an example of rabbi oscillation that we have measured at a given power for different single particle transition here and which show that so this is a chevron pattern across across this line and in fact if you move you see one of the arms of this transition which is a spin-conserving transition the other spin-conserving transition that the other end has almost the same rabbi frequency and the two intermediate one evolve more slowly showing that the matrix element is smaller but typically what you get for for this hundred qubits are relaxation time which are about 10 microseconds and echo time of hundred few hundred nanoseconds so it's very bad qubits sorry do you understand the limitation because we could imagine that spin transition should be long leave long coherence whatever but yes but as usual we spin you can have a nuclear spin that play a role and then we are foistening also that plays a role although this is not what limits here and the system is very rich there are many possible transitions so you can go if you are if you're in one state you can go to many other states yeah thank you have you tried I'm have you tried to incorporate a third subband because in these nanowires it seems from the chance for data that that actually there are more subbands than what you would expect so I have to spoil so this is a tie binding calculation and you see that when you move the gate voltage you get so this is the Andrews spectrum and these are the transition and when you increase the chemical potential you get more and more okay possible transitions and spectra which in fact resemble what what we get okay and you could perhaps I mean also for I mean because it seems that like contrary to what was stated originally in these nanowires it where they are actually many subbands playing a role and this is it depends on the gate voltage yeah you can go to a limit when you are very few already meet when you have a lot I don't show spectra in which you have lines all around that we have maybe if I can make a short very short and regarding the time domain following up the question of Nico do you see when you have sweet spots in the gate let's say you might have sweet spots when you change the gate an effect on the on the quickest time which might point to charge noise yes there is an effect okay it's better at the sweet spot okay okay it's not extraordinary okay thank you thank you okay I think we have to move on to the next speaker before we do that we thank you again