 talk is given by Schachlar. I'm inviting me to speak here. And I feel maybe I should also apologize because this might be the first talk this morning without DC-biased Josephson Junction. And it's the first time. I mean, we always work with DC-biased Josephson Junction. So it's also maybe doubly apologized because now this is the first experiment that we've done where I'll present some kind of unbiased Josephson Junction. So the topic is topological circuits. So it's more on topology. And so this work was done by Leo Peruchin, who's a student in my group. And it's kind of ideas that arose with my first student, Joel. And then we developed with postdoc Jean Damien. And then so it's a work that this is gonna be experimental work with some lots theory for us to understand what topology means and how to realize it in Josephson Junctions. And so I'd like to thank the whole group and especially Jean-Luc Smyr and Ramiro who helped a lot with experiments. And then we had a lot of help to understand the theory with Raphael León, Valar Fatemi, Landry Breto and Anton Akmerov. So this is kind of for the last, I'd say maybe 15, 20 years. Every time we have a talk about topology except maybe Wolfgang's, we didn't see some donuts. So the topology is kind of, yeah, we have a continuous deformation of some object and yet we have an invariant, something discrete which doesn't change. Okay, so here it's the number of holes. You can transform your donut and as long as you don't scrunch it up completely, you still have a hole. So that's your invariant. And so I would always see presentations about these donuts and then suddenly we go into some complicated theory of the invariance, some geometric theories so quite elegant but kind of much more mathematically involved in the donut. So I think there are examples of very physical donuts where you do have some topology and this is just the flux quantization in a superconducting ring. So you can see this as a real donut where if you field cool it, you can trap quantized flux. And if you change the, if you deform this donut, you keep the same number of flux quanta, it's just that the current in the loop will change. And if you imagine scrunching this donut completely, well, you exceed the critical current and then you have no more flux trapped in your loop. Okay, so this is kind of the physical consequence of equivalent to basically removing the hole from your edible donut. And I think Shapiro steps and what we've seen is also a case of topology where we can lock on to different steps, different values of an integer and these things, these steps are insensitive to the, in some large range to EJ to thermal noise and so on. So I think there are examples like this of topology which are maybe much older than what we're used to in recent times, but which are quite robust. These are very robust phenomena. And what's very interesting with topology is that since we have something that's discreet and that doesn't depend on physical parameters, in some sense it can only depend on fundamental constants. If you have a electrical system, for example, then you're limited to, here's the fundamental, you have the electric charge and you have H bar and you can make these quantities which have a physical interpretation and so this metrology triangle kind of gives you what transport quantities you can expect to quantize. Okay, so more recently, I mean this is the kind of topology that we've been excited about recently and it's quite new and so, and as we've seen it arises not from the physical samples such as the superconducting ring but here it's kind of the geometry of the manifold of the Hilbert space and the energy structure so here we have kind of the band structure for a material with some spin orbit coupling and it turns out depending on what kind of material if we have strong spin orbit coupling we can get what are called the topological insulators that we've seen about already and we can also get these whale semi-metals or Dirac semi-metals and so you can see that they have a quite kind of very distinct spectral signatures and these signatures have been measured with photo emission. Okay, so this is kind of I would say spectral proof of non-trivial topology in real physical condensed matter systems. Okay, and so this topology is supposed to be protected, we're supposed to see some very robust, for example, transport phenomena and the quantum Hall effect is a very good example where we can have extremely robust quantization but then with some of these other newer materials, I mean it seems that it's less precise, we don't have such good quantization and then the signatures, the transport signatures can even be ambiguous and this is because we have many different phenomena that can also occur. And so the question is, yeah, so I think, I don't know, there's some at least frustration from the experimentalist part where we were led to believe some of these states should be quite robust but in practice they're not. So the question is, something happened to my slide, but the question is what makes this topological state robust or not and I think it has to do with the role of the thermal, the thermal excitations and then I think the most important is this H junk that I got from Frank Wilhelm's presentation yesterday. You can always have terms in your Hamiltonian that you don't know about and these can open kind of gaps in your spectrum and ruin your topological protection. Okay, so for solids, for real solids, material fabrication is difficult, it's hard to synthesize these materials, they're sensitive to disorder, the topological phase might not be robust, there are many other terms in your Hamiltonian, H junk that you forget about and so the idea is can we do it with superconducting systems so this is something that people have looked at more recently and so they're well understood, they're coherent, we can make precise measurements in superconducting circuits and so this I think was the inspiration for all of our work was work realizing that you could map kind of these whale semi-metals to multi-terminal Josephson junction systems where you can get a very similar spectrum and then Wolfgang talked about that previously and so the problem with these systems is they're also a bit difficult to implement when we're using non-tunnel weak links. We have to make the dimensions quite small, the junctions have to be very close together, we have to have a few channels, these channels have to have high transmission so this is also quite difficult to realize experimentally and so the question was to go from multi-terminal SNS systems with Android bound states, small normal weak links to tunnel junctions and see how can we get some, can we obtain kind of a similar spectra, topological spectra in just tunnel junction based systems, okay, so that's the first question will be, can we make like a topologically non-trivial Josephson tunnel junction circuit and then are there signs that are not ambiguous? Can we obtain a really clear sign that it is different topologically and then is this, are these topological states robust? So that's kind of the questions that I'd like to kind of try to answer. So we've seen the Josephson junction and all the talks this morning, we have the charging energy and if we consider just the charging energy, we get these, this type of spectrum in the charge basis where we have these displaced parabolas and NG is the gate charge and as soon as we have some non-zero coupling between the charge states with the Josephson element, we open up a gap in the spectrum, okay? So this is kind of a trivial gap, okay? So if we now, let's try to find a topological Josephson junction circuit in the simplest way, so we just start adding now a junction, we now have a squid, we went from one junction to two, we have a squid and here, now we have some additional parameter, so this is kind of the first parameter we get that we can tune experimentally in addition to the gate charge, which is the flux in the loop, phi X. And so we can rewrite the Hamiltonian and we have this, we can rewrite it in this way where we sum, we do a sum in the complex plane of EjL and EjR and we have this phase factor and so if we actually make the squid totally symmetric, as you all know, we can actually tune out this Josephson element and we can obtain kind of degeneracies at phi X equals pi, okay? And so this we can interpret, so and if we make the junction, if we make the squid asymmetric, so here we've added 20% Josephson energy to the left junction, to the right junction, we see that we then open up a gap again and since trivially, since when we fabricate a squid, we will always have some disorder in the fabrication, we'll always open up a gap, okay? So this factor here, we can factor out this term here and then what we have is just this geometric kind of representation where we add a vector to a unit vector along the real axis and the thing is if we can close this kind of, we can close this structure on itself, this vector we can get back to zero, then we have a degeneracy, okay? So it has a very simple geometric interpretation, but the problem is that K is never exactly one. So to be able to construct this vector sum which gets back to zero, we'd have to have K equal to one, this is always kind of listed. So what we do, what we did is just to add a single junction, so this is what we call the bisquid, so you can see it as a squid, two squids or a tunable squid if you wish, but now we have two parameters, well, three in total, so the gate charge and then the two fluxes phi L and phi R which control now kind of add an extra dimension towards this sum that we can make, okay? There, so now we have the KLKR which will be fixed when we fabricate the sample and then we have the two phases, okay? So as you can imagine, so this gives you now the possibility to create triangles, so here in the case of a symmetric bisquid where all junctions are equal, we can construct this triangle in this fashion and if you calculate the energy dispersion, so this is that we fix the NG to 0.5 and look at the energy diagram as a function of the two fluxes, you actually, you get what we saw before for the whale semi-metal. And if you, now if you make a cut along NG at these degeneracy points, you actually see that you have, you've effectively eliminated the Josephson part from the whole circuit and it looks just like a pure capacitor, okay? And why are there two degeneracies? That's also quite clear geometrically. It's because you can close, if you fix the lengths of these two, the three sides of the triangle, you can actually close it in two ways. You can go around like this or around like this and that will give you the two degeneracies in the phase space, okay? So again, if you now vary the parameters, say you fabricate your sample, of course it's not symmetric, you see that you can still close the triangle because as long as KL plus KR, the magnitudes are larger than one, then you can close this triangle, okay? And so as you change the parameters for the Josephson junctions, the Josephson energies, you end up kind of moving around the positions of these degeneracies, but they're not lifted until you approach this critical point where you end up getting kind of a quadratic crossing. Yeah, we'll get there. So yeah, so this is the case where we're on the topologically trivial region where we have no gap in the spectrum. Okay, so you end up getting this phase diagram, which is quite neat where we have this topological region, a large region, and then if we make the sample too asymmetric, then we come out of this region, okay? So topological region can be seen as where we can close this triangle, and then the critical regime is when we have kind of a flat line, and then here we're in the gap phase. Okay, so I will skip this. Okay, so people have looked a lot at the topological properties of, or exploited the topological properties of similar circuits with Joseph's injunctions for a long time. One of the possible phenomena is charge pumping, and so I think we've worked on charge pumping for quite a while now, but it's still not at the point where the quantization is high enough or good enough for applications. And I think we've proposed ourselves a different circuit in which one could actually make an equivalent of a transconductance quantization. So it's a more complicated circuit where you could imagine closing this metrological triangle just with Joseph's injunctions. But again, this would be a difficult transport experiment. And so we felt that it would be better just to try to measure these degeneracies with spectroscopy. And again, this degeneracy, this is an intrinsic topological property of the system. It's not one that's induced by, say, some drive, okay? So the idea is to do microwave spectroscopy and try to detect these, to get some evidence of band crossings. And so what we did was a conventional microwave spectroscopy measurement where we couple the bisquid to an oscillator and then look at the dispersive shift, the coupling to the bisquid gives you some shift in the resonant frequency, which will also depend on the occupation of the level occupation of the state, of the bisquid. Okay, and so, and then what we can do is we can try to excite the transition. So this is a cut along the diagonal direction in the energy diagram where we have the two degeneracies. So these are the two degeneracies. And then the idea will be to try to excite this transition and then see the shift so we can probe that bisquid transition. Okay, so the model we use, the model we use I think will be discussed later, so Park et al in the session, in the, I think tomorrow. But the idea is to expand because we're weekly coupled, we can expand in the chart, in the voltage fluctuations in the resonator and then we can actually trace back the shift in the resonator frequency to the energy levels of the bisquid. All right, so at first we just measured a squid just to make sure everything was working fine. So if you have a squid, you can try to make it as symmetric as possible. So here I think this will be 5% asymmetry and so you expect kind of a gap spectrum, but which comes pretty close to a degeneracy. And if you consider kind of the two, the resonator level and then the transition, we expect it to come very close to zero in energy. And so if you look at the resonator shift, then you'll see that you get this kind of picture here. So depending on whether you're in the zero or one state. So the main limitation now to all of these techniques, I mean to spectroscopy in this way is that the two tone frequency shift actually vanishes at the degeneracy, okay? So from the start, I mean spectroscopy is quite powerful, but it turns out in these systems you can't excite the transition and that's kind of obvious because you're killing the Josephson part which allows you to actually couple the charge states. So you're killing that coupling so you can't actually drive the transition. So we'll be able to measure near the degeneracy but we won't be able to get there. Okay, so here's what the samples look like. So this is the resonator, co-planar waveguide resonator. We use the lambda mode and we will put the, so basically we'll drive at the resonator frequency and measure the shift in the resonator frequency and from this shift we can extract kind of the energy levels of the bisquid and these are the typical parameters we have in our experiments. Okay, so where do we put the bisquid or the squid? So this is first I'll show what happens with the squid. So we put the squid at the center of the line. So here you can see this is the squid right here. It's coupled near the, this is the center of the co-planar waveguide and then we have a flux line nearby. So wait, let me focus first of all. Yeah, here's the squid with the two junctions here and then this is where we apply our flux and it's a small junction. And of course we have an additional line here so we can both excite the transition and also apply the phase phi x, okay? So we inject the DC and the AC excitation through this line. Okay, so if we do now, in this sample we do a one tone, we sweep the gate voltage and what we would expect to see is kind of when the transition for the squid is above the resonator, we get kind of a negative shift here in the phase and then we get a positive shift when we're on the other side, okay? And that we can fit very well with our model and however when we do this, of course we see that it's not the charge, we have some charge noise and then we can calibrate that out. We can correct for this, there's slow drift and there's some jumps and we can discard measurements when there are jumps and correct the voltage so that we can correct for drift. And so once we've done this automatic calibration then kind of we can get a nice data to correct for the gate voltage fluctuations and so here we now do the two tone excitation and you see kind of this decrease, we approach a small transition when we get to NG equals 0.5, okay? So here's some data with the background removed and you can see kind of we can fit these lines over here which correspond to the squid transitions and I mean the interesting part is down here and you can see that we lose our signal as we approach this small gap and if we just, we can see that here, there's a faint line that's going down here and again the problem is that near, as we get to degeneracy, this matrix element connecting the two states vanishes. Okay, so if we just go down here and integrate for a very long time then we can kind of obtain, we can obtain the position for this transition as a function of the current and here we see that within our model, within what we can fit, here is the fit if we assumed it was perfectly symmetric and within our model we can say that it's basically 5% asymmetry between the two junctions. Okay, so I'll get now very quickly, so the bisquid, this is the interesting part, so here's what the bisquid looks like, so we now have three junctions, okay? The three junctions are here, two loops and then we have two lines now to be able to independently flex bias each of the loops and so it's a slightly more complicated setup and here's, so what we expect is that we should be able to have these two kinds of distinct excitation spectra, okay? So this is the transition energy in the two, in the gap and the topological regime and so you can see, I mean, very qualitatively, here we have kind of this, the curvature is different near pi and then we have the degeneracies for the topological regime, okay? So I will skip through some of the details but we have quasi-particle poisoning, so when we measure we see two quasi-particle states, we manage to get rid of this, so okay, so here's like a trace where we see the quasi-particle poisoning, we manage to get rid of this for some of our samples with gap engineering and unfortunately we haven't yet obtained a bisquit device where we've been able to get rid of quasi-particles, so what we have to do is we have, we can see both the quasi-particle states and so we have to, this is what you would get if you just did a raw trace and we have to actually select for the quasi-particle states so we can clean up this data and get some meaningful spectra, okay? So here's a device which is in the gaps regime, first of all, and so in the gaps regime what we expect is basically we have one minima here in the excitation energy, so this is, you can see like the minimum excitation energy doesn't go to zero and when we measure the two-tone spectra, for this sample you clearly see that we have, this is the region where this is analogous to the squid where we have a gap spectrum, a pi. Now if we produce now, we fabricate another device, the junctions are more symmetric and now we expect to be in the topological regime and what we see is now we have, we should have two points here, two points in the phase space where the transition energy is zero. And so this is the data for this sample and we clearly see kind of this, near pi we see the inverted, the negative curvature for the transition and then we have kind of the rest of this whale cone that we can see as we move away from pi for phi L equals phi R. Okay, we can create, we can fabricate another junction again and we see this one is less symmetric so you can see that the degeneracies start to approach each other and then, but we still have this region of negative curvature. Okay, so we have this regime where we have positive curvature and this corresponds to a gap by squid and then we have this regime with negative curvature so even though we, because of experimental difficulties we couldn't approach the degeneracy point as close as we would like but you can clearly see this qualitative feature which is this negative curvature that you can actually resolve quite clearly which is indicative of this topological regime. Okay, so this is a distinct signature of this regime and you could even, if you gave me a sample like this or gave my student and asked him, okay, I don't know what the symmetry of the junctions are, measure and tell me if it's in topological regime or not, you can actually do this easily with a single tone measurement so this is kind of the resonator shift without a second tone and so there are, okay. So these are two different samples and all you need to look for is if in one plaquette of the two currents, okay, these are the two currents we can apply to modulate these phases, do you have one blob or two blobs? So when you have two blobs that corresponds to the topological regime and these correspond really to the two minima, kind of the two minima you have in the Briand zone. Okay, so is this the time is to principle up? It's up? Yeah. Okay, so let me just finish then. So yes, so I think that it's not as robust because you have some H junk and this H junk might be inductances, it could be coupling to quasi-particles and I think this is something that we wanna look at but there's lots of stuff to do in these systems, lots of more complicated circuits to look at first we need to kind of improve the experiment, remove the quasi-particles and I think that yeah, there are many other things to do and eventually I think, I hope that eventually get the transport but again, I think spectroscopy is the first step before we try to do transport on some of these systems. So thank you. Yeah, thank you very much. Okay, so there's a very urgent question, Fido. Shall I correct me if I'm wrong? But I think the original proposal from Julie in a sort of was that you can only get topological if you have three independent phases unless you're dealing with topological superconductors. I mean, because when you have only two, I mean, these are the total charge number is zero. Yeah, so we have three because we have the gate charge. So we have the two fluxes and the gate. So in the multi-terminal circuits, you don't have gate, you just have phases. So you need three phases if you don't have this extra degree of freedom, which is the gate. Total charge number zero, no? Of the two, you can annihilate these two. Oh yes, exactly. So that was kind of when, yes. That's right, you can annihilate. I mean, in the picture I showed you of the spectra, basically in the different limits here, sorry. There we go, yeah. So this, you know, when we go from this to this with these points merge and then you start opening up a gap. So it's exactly, so this, if you had one more junction, what we call a tri-squid then you could actually merge these two points and then open up the gap continuously. Yeah, thank you very nice talk. I have a question about the transport. You mentioned that you were doing transport experiments because, I mean, spectroscopy is in principle not enough to be sure that you have the topological phase. Yes, so I think that spectroscopy is a first step, especially in these systems where, you know, we think we know everything in the Hamiltonian. I mean, with these superconducting circuits, that's the nice part. We can usually do spectroscopy very finely and really have nice fits to the model. So here the idea is, okay, let's try to do that in this circuit, which should be topological. You can fit the spectra. You have a model, you can fit to that. And then you can say it's consistent with a topological model. But then, of course, you have to ask yourself, did I miss, what did I miss in the Hamiltonian? And the thing is even if you do transport, you won't have, you know, unless all these energy scales that could open up a gap are very small compared to, you know, your quantization, the degree of quantization will also depend on whether it's topological or how topological it is. So I think that's the thing is that it's not like transport will also give you, there's sources of error also in transport. It's just that I think with the quantum hall effect, we've had decades and decades and it's quite good. I think that in decades, we will also have very good quantum spin hall insulators and all of these other systems which are not so robust maybe right now. This was the first, hey. So to overcome this problem that you have with exciting at very low frequencies, I'm wondering, can you go in the vicinity of your readout resonator and do a two photon transition, you know, like a side band? Yeah, so. This usually works in fluxonium qubits. You can see the really low frequencies if you do that. Okay, yeah. I mean, these are, this is really kind of quite recent and we really need to develop the data more, I mean, to the measurements more, but I think there's lots of things we wanna try to be able to probe there. And at the same time, I don't know, I mean, you have this fundamental limitation with this matrix element, which vanishes. Another question, Takis. Thank you for your talk. In your effective system, you don't really have H-physics if I'm right. So how can you compare with transport experiments where essentially everything is dominated by H-physics? Well, you expect that everything will be dominated by H-physics and maybe it's some day non-abelian excitations. Here you don't have that. Yeah, so I think that's open. You don't have this surface edge correspondence because you don't actually put your topologically non-trivial object in contact with something that is trivial which gives you these interesting H-states. But if you, like a lot of these phenomena, it's kind of you can still see transconductance quantization. So in this example that I showed you. So the question is, I don't think you need an edge, but there's probably some interpretation of this edge, bulk edge correspondence that you could somehow figure out. But yes, we don't have an edge that you can consider maybe putting, but we can tune in this tri-squid. We could tune between the two regimes. And then the question is in the critical regime, is there something, I don't know. Okay, let me know that in the quantum hall effect. Of course also you have topological arguments which does not, for the quantization, which does not rely on edge states, right? It's just, and this is similar here. And I think it's an open question. We keep discussing this, what the bulk edge correspondence here in this kind of systems is kind of open. Okay, yeah, Miku. Yeah, I have a couple of questions, but maybe I'll start with the one that's most interesting. So did I understand correctly that the bisquit is just like a stepping stone and then the sort of other device for the charge quantization is then the one that you're heading for, which is not exactly the bisquit or tri-squid, but it's more complicated. Or are you planning to use the bisquit somehow in another way, imploring a topology to get a quantum charge pumping? So frankly, I think it's important to understand why these things are not, why aren't they robust? So I don't think we've understand, we don't understand that exactly. I mean, I think, for example, if you, let's say, I mean, we can go one step further, add some junk terms and then try to say, we have a small gap. What is the, how well will we be quantized? Okay, so, I mean, these systems we know very well, we know the Hamiltonians very well, we can add all the parameters and then we can say like, okay, if I have this, I know how my spectrum will change and I can also calculate how my, what I would expect for, say, transconductance quantization or charge pumping. And so then just to see some idea, to get some idea of the spectroscopy, what could be the possible size of the gap and then say what could be the size of, you know, the, how well it's quantized. What about the, it seems that you are in charging regime, at least when you go to low EJ, you know, you will be in the charging regime. Is there like, is there a way, is there a way to use it in the charts, sort of quantization experiment device where you would be not in the charging regime, but somehow all the time dominated by EJ but still have like, you know, going around this topological points, for example, but not going into the points themselves. Yeah, so there's some, you know, this is my, my student came up with this make charge pumping great again. But so if you have, the bicycle is dual to the Cooper pair pump, if you, it's a dual graph, the circuit. So you can do charge pumping with the bicycle and there are some advantages, okay. And indeed these, these cars, you know, you, you, you can put yourself, I mean, you don't need to, here we're looking at spectroscopy, so we're always around NG equals 0.5, but you know, for some of these transport experiments, I mean, it's a, it's a different story where you want to kind of sweep some plane in the phase space that will go through, you know, NG equals five or through pi in the phases. I don't know if that- Yeah, you really quite answered the question. The question was that, can you in, for example, in this device or another device have this quantized charge pumping or quantized charge current in a regime where EC is, let's say always negligible compared to the EJ, instantaneous EJ of the system so that you would not be so, you know, spectroscopy will charge this person. Yeah, I mean, yeah, I mean, it's, it's, we have some margin on ECEJ, but you can't go like a factor 10. I mean, yes, I mean, it's, you won't be, it'll be quite difficult because then your gaps get small. I mean, you can, you have problems with adiabaticity in the limit, like as you would with other systems. So I don't know, yeah, I can't say what would be the best circuit with the best EJEC ratio to do charge pumping. All right, so maybe to continue the break. Thank you. So I think, yeah, another urgent question. If not, I think there's an announcement by one of the organizers, but let's first thank Shadlap for the talk. Yeah, there is two practical announcements.