 record QD implementation of the boundaries and garden model please now okay better, okay Yes, thanks. Yeah, whereas we got so much used to the masks They are almost fixed on the place So thank you very much the organizers first of all for the Organization and invitation to this very nice meeting in the place that I like very much It's always a pleasure to come back Then of course I have to thank my collaborators. So it's a joint theory experiment work. So the experimental part Okay Yeah, okay, so the the main Experimental work was done by Sebastian Lerge who is who was a PhD student at the Neal Institute and so Two of his supervisors are here in the audience The theory part was done by Tosypoulk who was also a PhD student at Neal Institute with the search Florence and There was also Contribution from Isaac Simon and myself So Isaac is in Johannesburg and I'm in neighboring laboratory to the Neal Institute so I Was looking for a nice picture in Grenoble But I did not so I had little time and I just took the one that I had from my office window a few months ago So that's kind of Typical we are randomly chosen if you want not not specially tuned and not specially selected Okay, so the outline Well, I will give first some motivation which may be a little bit personal and Kind of subjective then I will describe the experiment system the experimental part Hopefully I will be able to do it correctly in if not there is Nikola and Olivier and Then I will discuss two theory aspects And there is of course a lot of overlap with the previous talk. So I will try to make connections Okay So motivation You have already seen boundary sign Gordon model. So we have this is the Lagrangian. This is basically an elastic string semi-infinite in this case and on the boundary we have this cosine Cosine term so this is called boundary sign Gordon if you put this cosine Inside the integral then this is called bulk sign Gordon. I will be speaking mostly about boundary So this model is well known. It's integrable and it has been solved exactly So the question is why should we bother studying it if everything is known about it? Well to my taste This model so this come this Lagrangian if you take it literally It's just a meaningless combination of letters because this Symbol J You don't know what it is unless the model is supplied by a short distance cutoff and Then many things depend actually In on the details of how how this short distance cutoff is implemented So it's better to say that this is not a model But it's rather a class of equivalence of models. So you have so you need always to start from Some latest model which is well defined at short distances And then when you go to low energies and large distances then you have Universal physics that comes out and this universal physics is somehow can be somehow extracted from this Boundary sign Gordon Lagrangian But the problem is that You so first of all there are phenomena that are not that Depend essentially on on this cutoff on this short distance physics and even if you study phenomena, which are supposed to become universal at this low energy scales How to control the so you have kind of asymptotic so this exact solution or this Integrability property that we heard about they predict The asymptotic infrared behavior of the model But how to control how well you are in this infrared limit and this you cannot Really do from the model from the model itself to control the theory you always have to come back to the latest model and So to give some examples So one example that we Recently did so we were studying the phase diagram of Offered Josephson junction chain Which so this is just the very standard Let me show you the so the very standard model for the Josephson junction chain without so all junctions identical Josephson energy Capacitance matrix so with the islands are coupled by the capacitance matrix you have the two capacitances in the problem so for this model you can want to study phase diagram and From if you think so this is supposed to be in the class of equivalents of bulk sign Gordon and So you have you know that there are two phases Superconducting phase insulating phase you know that the transition is Costal it's belongs to costal established University class you have scaling that is known since many years but if you try to establish to guess the phase diagram looking at this universal scaling trying to see which Which of the parameters of the latest model are renormalized in which way Then you would not guess the correct the correct form. So what we found is that this boundary is determined by some much more Stupid and dumb short-range effects then the which which dominate over the costal established scaling Even more Spectacular example is this recent story about Schmidt transition So what you guess from this universal scaling laws is this kind of phase diagram? so you have dimensionless resistance and the ratio e j to e c so from the scaling you would guess that your Boundary between the two phases would be vertical And so the experiment was done Which was supposed to be in according to this phase diagram in the insulating state They did not find the insulating state So they started to suspect that there is no phase transition at all but in fact if you Take this model and you simulate Taking into account all short distance effects then oops, sorry The phase diagram actually looks quite different. So this is a game Beyond the this usual sympathetic scaling that you would get from From the effective field theory and you see that the insulator phase is actually much more much more narrow than What you would think. Okay, so this is my motivation so the when you want to study something that some latest model that Boils down to a quantum field theory in the infrared limit then To control this infrared limit is not so easy. You cannot really control it based only on the quantum field theory Okay So now I go to the main topic So this is the experimental system and it's very very similar to What we saw from the previous talk. So there is a Josephson junction chain and In this part. So there is like 400 junctions which are large enough. So the cosine The Josephson cosine you can expand into Up to quadratic order and then you get just a harmonic chain Which you can think of transmission line which hosts linear modes So here there is our boundary impurity which is a squid So that you can control the Josephson so this remains a cosine and the coefficient in front of this cosine you can control by External magnetic field Then you do on the left side you have The external circuit where you send microwaves and you can measure their transmission So this the Hamiltonian of this is so if we just think about the chain and the boundary Just just this part. It's the harmonic part for the chain So there this small junction also has some capacitance and this capacitance goes into the harmonic part of the Hamiltonian And we have the the boundary the the cosine at the boundary. Yeah, sorry for the pointer I'm still high. I have to adjust to the speed of it. Maybe I don't know if I stand here. Maybe it's better Yeah, slightly better, okay. Yeah So the typical result from a measurement is this so you measure so you send photons into your transmission line so you send Incoming photon here you collect them here. So the transmission coefficient typically as a fun as a functional frequency has a form with some sharp dips and each deep is corresponds to standing wave in the chain And when you tune your flux Through the squid through the small squid you change the positions of these dips and also the widths and You can think of this as Modified so flux modifies this Josephson energy Which is standing here and this modifies the the modes of the chain and this is how you measure this So first of all, how are these modes made so if we take an infinite chain this harmonic Hamiltonian that I wrote on the previous slide and Just look for the linear modes Which are plain waves then we have a certain dispersion Which is linear only here at low frequencies and Then it deviates and saturates. So there is a natural cutoff In this latest model, which is called plasma frequency, which is 18 gigahertz here so this and When you take the experimental This experimental dips and you put them on the On the you just Put them on your plot as a function of the wave number you have this This dots and you can fit the dispersion relation and you get all the parameters of your chain So if you want to see how the they are spaced How is the quantization? Condition for the standing wave then you have to look at the boundary conditions of the at the ends of your chain So on the left we have this transmission line a 50 ohm and this is almost grounded on the right we have this element small junction and In the very first approximation you can think okay. Let us take it also quadratic Expand this cosine then we have something some effective boundary condition, which is sensitive to the flux and this boundary condition introduces a phase shift scattering phase shift and then this determines the positions of the of the modes and By measuring how the positions depend on the flux you can extract this phase shift and See how it behaves on the frequency. So this is the Plot of the scattering phase shift as a functional frequency for different fluxes So what does it tell us? the most important qualitative feature of this Curse is that they are very smooth and this means one thing here at the end you can think about a Of this small junction So we already heard in in the previous talks yesterday that you can see it can be a qubit or you can think of it as a Harmonic oscillator or a weakly a harmonic oscillator transmon Something that has its own energy levels and then if you if it has well defined energy levels then the It would produce Resonances for the modes of the chain if you have well defined resonances then your scattering phase shift Various very sharply when your frequency goes through the resonance. So the smooth nature of this Curve Tells us that there is basically no resonance that the It's completely over damped by the strong coupling to the modes of the chain and this is Where it is where this system is different from the works that The previous speaker Discussed so in particular the experiment in Manuchiran's group group so and then this means that we this is this system is Indeed more similar to the boundary sign Gordon where you have just the cosine you don't have any Internal dynamics associated to your boundary all the dynamics is in the chain and you just have this non-linear cosine at the boundary Okay, so scattering shifts that we can so now I will move more to theoretical part. So the first thing to Understand and to model is this scattering phase shift Which depends on the frequency and depends on the flux. So if we just take As I said Replace and replace this non-linear element by something effectively Linear then of course the phase shifts you can calculate exactly, but this is not this will not Correspond to the experimental result the effect of the non-linearity in the very first approximation you can think of Effective renormalization of your cosine by vacuum fluctuations so basically this Tells you that your Ej becomes renormalized it's that we call Ej star Renormalized by the vacuum fluctuations, which is well this kind of factor You see it like Dubai Waller factor. This is all comes from the same Physics renormalization by your harmonic environment So you can Put this this procedure into a self-consistent Into a self-consistent scheme, then it's called self-consistent harmonic approximation So basically this Delta Phi square you evaluate this by taking your by putting your linear element as As As some some effective inductance that is not exactly the bear one this is so this is known since long ago this procedure and you can view this as self-consistent Dressing of your self-energy or you can view it as the best Variational wave function from the Gaussian family for which you can From which you minimize the energy of the system, so The result is Not bad, so you see that the effect is strong enough we have a factor of So this the dashed line is the bear The bear ej that you would just guess from the squid From the Aron of bomb pre-factor So well, this is the actual ej star that you extract from the Positions of the pigs transfer translated into phase shifts translated into ej star So this is the the dots are the experimental curves, so we have This Magenta line is the self-consistent harmonic approximation Which breaks down? at some point when ej becomes too small The Dotted line is the universal scaling that you get from the boundary sine Gordon So our alpha here it's 0.3. So we are on the superconducting side of oops, sorry Superconducting side of the Schmidt transition. So here you see the behavior of the experiment of the experimental points it goes quite well with the With this universal boundary sine Gordon scaling breaks down at some Low frequencies low energies because of the finite temperature and the scaling is of course zero temperature result and at high energies it deviates because we have cut off So the theory you you really start feeling latest effects and this self-consistent Harmonica approximation of course takes care of all the latest effects correctly more Interesting is the decay. So this is precisely the The topic of the previous talk So you see we have as a function of flux and frequency the quality factor of the of the modes changes And this is the photon Down down conversion, which okay, so I don't have to explain what it is because the previous speaker did a very good job at this So how to model this? So here We want to take into account these latest effects and to implement the ultraviolet cutoff as it is So what do we do? So first of all So here we Introduce the greens function Of the face or face face correlator the transmission coefficient Into this in this transmission line you can relate it exactly to the greens function so the greens function Satisfies the Dyson equation in this Dyson equation here everything is Harmonic so C and L minus one are the capacitance and inverse inductance matrices of this whole whole thing So This is the damping introduced by the transmission line on this side and all the effect of the non-linearity is Sitting here in this self-energy Which is located only on this last side So this if we have the self-energy then we have this equation for the greens function. So the whole Interesting physics is in the self-energy. So what do we do with the self-energy? So we do We do perturbative calculation in the Josephson We this is because our the most Interesting that the strongest effect you see is near half flux Near half flux quantum where the Josephson is suppressed five minutes. Yeah, thank you So So we do Perturbative we have a perturbative expression for the self-energy up to the Up to second order So this Takes into account you see it still Takes into account in elastic processes to so Multi-photon production not just one to three but also one to five one to seven but to Perturbatively in Josephson This is controlled at high frequencies if the frequency is higher than ej star. This is what the previous speaker also Mentioned so we are on this high frequency side It's it can be easier easily generalized to finite temperatures and this is what we actually do. So here I wrote The zero temperature version, but you can put Keldish and put in put the finite temperature So we do self consistently again. It's a self consistent born approximation Because this greens function is the one that should be determined from this equation This defines the self consistency loop So it captures a lot of Useful effects. So it When you do it on the self consistent level You take into account the fact that the photons into which your decay. They are not the Bare photons, but they themselves are dressed and they themselves decay So you have this hierarchy of multi photon cascades Level this is important because we have a finite chain and the level discreteness matters More the more discreteness matters in this chain. So With this you can also describe many body localization or delocalization. So in our case, it's delocalization Because the peaks that we observe are smooth Lorentz and peaks in the in the transmission What this approximation misses it misses the quantum phase slips because it's perturbative in Ej So you do not go to the next minimum of the cosine so So this is the result for the flux near flux quantum, which is so well in the So this is the the parameter which controls So when we are within this Applicability regime, it's fine our theory Feeds the experiment reasonably well You see that this decay the frequency dependence is Rather exponential because we have linear scale here and logarithmic here and it's not at all the power law that you would expect from the From the boundary sine Gordon infrared asymptotics But so this is a success if you try to go to fluxes, which are smaller than Significantly smaller than half flux quantum Then you are out of the regime of applicability and then this self-consistent born second order it miserably fails You see there is so Search and to they even tried to push it to the third order But once your second order breaks once the leading order breaks you usually need all orders So the third order pulls you in the good direction, but does not cure the situation And in this regime, I think we don't have any Reliable theory available which would account both for the latest effects and for the all non perturbative stuff and This is all I wanted to tell you. Thank you very much. Thank you Okay The talk is open to discussion Thank you very much. So just to make sure and this then so you're claiming that what leads to the different Dependence of the decay rate makes it exponential not a power law is the dressing of the photons or what is the reason that you you get this deviation? You mean this exponential instead of power law. Yeah, this is Bad separation of scales e. J. Star and ultraviolet cutoff. We are not still not in the scaling regime Infraredo synthetics and see so what you need to get in this regime is I guess longer chains or Low longer chains and lower frequencies and I see okay. I don't know which one is more constrained Constrained for experimentalists I Wouldn't I would propose we thank the speaker. Thank you very much