 So we are now at the second talk of this last morning session and the speaker is Amir Burshtain and you will tell us about inelastic decay from integrability. I would say just please go ahead. Okay, so you can hear me, right? Okay, so hi. My name is Amir. I'm from the group of Moshe Goldstein. So first of all, again, let me thank the organizers for this wonderful opportunity to tell you about the recent work. So this talk is going to have two parts. In the first part, I'll tell you about your motivation and about some experiments from the Manchin Group that we've been involved with. And then I'll tell you how we can calculate some quantities that relate to these experiments. So let's begin with our motivation. This is something that we've heard about several times in this week, but I'd like to rephrase it in our terms, which relates to our work. So the question we started from is whether we can observe a significant inelastic decay in a cavity QD system. So for example, if you think about an atom in a cavity and you have an electron at the ground state, so an incoming photon will excite the electron and when the electron decays back to the ground state, it usually emits a single photon. This corresponds to elastic scattering, and the question is, can you have inelastic scattering? For example, can you have three outgoing photons whose frequencies sum up to the incoming frequency? And the answer is that this is possible in principle, but rarely observed because what parameterizes the light metal coupling is the fine structure of constant alpha, which is a small number, so the probability to have such a process is very small. But this would not be the case if you have an effective system which you have an effective alpha on the order of one. And as we've heard, we can do this by replacing the cavity whose impedance is the vacuum impedance z naught. We can replace this cavity by some medium or different impedance z, and then we would have an effective alpha which is proportional to this impedance. And if this impedance gets close to the resistive quantum or even exceeds it, you could expect to see inelastic scattering. So we collaborated with the modern sharing group to do just that. So here you can see a photo and a sketch of the device where we will place the cavity with a transmission line and also replace the atom with an artificial atom, which is a squid. So you can see your antenna for spectroscopy here. It is capacitively coupled to the transmission line, and here you have your squid. Now, in order to get the high impedance, again, as was discussed, we use not a regular transmission line, but rather a Josephson array, so you can see the individual Josephson junctions here in this photo. And the high kinetic inductance of the junction is what leads to a high impedance, which would induce a high effective alpha. So we can expect to see significant inelastic scattering in the system, and in fact, this is what we saw. So in our previous work, we considered a squid in the transmission regime. And what we showed in terms of theory, we showed that we can indeed get significant decay rates for microwave photons in this transmission line due to their interactions with facelifts that occur in this transplant. So we developed a theoretical framework to calculate this cross-section, and actually I have a poster outside about it. So if you're interested about it, I'll be very happy to tell you more about this. It might be useful in other contexts. And this theory agrees very nicely with the experiment, so this is measurements from the Manachaean group, which shows very large decay probabilities for a single photon in a single round trip in the array. So the important thing to take from this figure is that you have very large decay probabilities, which is, again, something you usually do not see in cavity QD-like systems, and this is a rare demonstration of that. But it's not just an exotic effect. This is actually a tool that we can use to probe many body physics. And I'd like to tell you a little bit about how it relates to the dissipative quantum phase transition, which deals with a system very much like the one we discussed before, where we have some joules in junction that is coupled to a dissipative environment. So what Schmidelblutd I predicted almost four years ago is that you would have a quantum phase transition. Can you please move the... can I move it? Oh, I can move it. Yeah, okay. Yeah, I think it's fine, right? Okay, thank you. Okay, so the prediction was that... So the prediction was that you would have a quantum phase transition depending on the value of the normalized impedance of this dissipative environment. So if your normalized impedance is below one, you are in the superconducting state, where the cosine operator of the junction at the end is a relevant operator, and you have a relevant argiscale in the system. In that case, the phase is localized around one of the minimum of the potential, and you can say that in that case, your boundary in the system will look like an inductor. On the other hand, when the normalized impedance is above one, you are in the insulating state, where your cosine is an irrelevant operator, you do not have an argiscale in that case. Your phase can diffuse all of the potential due to phase slips, and the boundary looks like a capacitor. So that was a prediction from almost four years ago, and there have been several walks that tried to observe this, some of which by Hakone, which is here in the audience. But overall, this is hard to observe this transition, and this is mainly due to the fact that you have to apply DC measurements to see this, and these DC measurements might interfere with the state of the system. And in fact, this has sparked a recent debate in the last two years with, again, papers by some of the people here in the audience. Now our proposition to attack this problem is to use the fact that the system is sensitive to single photons in order to learn about its state while interfering with it in the most minimal way possible. So the idea is to basically measure the properties of the photons, namely the decay rates and mode shifts of the single photons in order to learn about the state of the system and check whether the power laws and scaling laws predicted by the theory agree with the measurements. So here we choose to work in the Cooper-Pedro's regime where E c is much better than E j. So we can look at, indeed, at these two quantities. So the first thing we can look at is the decay rates, which is supposed to go as a power law that depends on the impedance, but this parallel diverges for impedances, normalized impedances, which are below one, which indicates that indeed you have a irrelevant RG scale in the system. On the other hand, this parallel is not a verge for a normalized impedance above one, which indicates that the RG scale is irrelevant in that case. So unfortunately, the Menendrara group cannot measure decay rates all the way down to zero frequency, but they can measure intermediate frequencies, and you can use that to see the different trends. So here you can see the decay rates as a function of frequency for several values of the impedance, and you can see a different trend between impedances which are above one, and this is the, excuse me, below one, which are the blue and yellow here, which trend towards the divergence as opposed to the purple one, for example, that has a different trend, and this is an impedance above one. What they can look at at zero frequency, or low frequency, not zero, but low frequency, is mode shifts, and what the mode shifts will tell us, they will tell us about the type of boundary we have in the system. So we have a capacitive boundary on one end, so if we are in the superconducting state and the other boundary looks like an inductor, then we would expect a mode shift of a phase shift of pi half, whereas if we are in the incidentally state, we're expecting to see a mode shift, a phase shift of zero. So by looking at these quantities, we can learn about the transition. So this is just a tease. This is the work that we're wrapping up right now, so stay tuned for the paper. Now, what I'd like to tell you about is how we can do some theory in order to understand the system in the full frequency regime. So what do you mean by that? So we have this result from perturbation theory, and perturbation theory is good above the artist scale, so it is supposed to be good for all frequencies when z is above one, but when z is below one, perturbation theory breaks down below an artist scale, but we want a solution for this regime. So I'd like to tell you how we can achieve that, obtain the solution using the fact that our system is integrable. So this takes me to the second part of my talk, where I'll tell you how we can calculate these decay rates and mode shifts for the entire frequency range. So let's look at our system. So our system is, again, we have some Josephson junction that is coupled to a transmission line for our purposes. You can see the Lagrangian of the system here, and this system is called the Bondi-Sygon model, which we'll also hear about in the next talk. Now, in terms of what interests us, so let's first of all consider this model classically. So we can write down the differential equation of motion, which describes this system, and we have some nonlinear differential equation, and the solutions of this differential equation are actually well known, and they are called solitons, which are windings of the phase going from 0 to plus or minus 2 pi, as you can see in red and blue here, and there is another type of solution called breathers, which appears only for specific values and impedance, and this is, you can see it here in the green. So this is individual solutions of this equation of motion, so we can ask, can we get solutions which are comprised of multiple solitons or multiple solitons of breathers? So, naively, it seems quite complex because this is a nonlinear differential equation, so we do not expect to have a superposition or generally expect to have a complicated type of scattering, but it turns out that actually scattering in the system is quite simple because this system is integrable, and this means that it has an extensive number of conservation laws that highly restrict the type of interactions that we can see in the system. So for example, if you consider the scattering of two breathers or another that propagate along the array and meet up at some point, then when they come out of the scattering, they will maintain their individual shapes, but can only gather some phase shift. Now, more generally, the scattering is purely elastic in the sense that if we consider the scattering of any two solutions with one another, then we have conservation of the set of momenta, so it means that we do not have just conservation of P1 plus P2, but actually we have conservation of P1 and P2, so this would hold for conservation for scattering of two solutions and also for scattering of n solutions with one another. So this would be the case for scattering in the bulk and also reflections of the boundary that we have in the system. So that is the classical theory, and when we go to the quantized theory, we have to take these solutions and propose them to be field excitations and the same picture holds. So we have three types of excitations corresponding to the classical solutions in the system. Here theta is some parameter and technically it's called the rapidity. And again, the same picture holds. So if you consider the scattering of n excitations with one another, then the scattering is purely elastic, so the individual momenta or individual rapidities are conserved, and the only thing that can happen is some momentum-dependent phase shift. So this is the case for scattering in the bulk and also for reflections of the boundary. And this model has been studied extensively and actually one can obtain exact expressions for the scattering reflection matrices only from the integrability of the model. So there have been several walks that worked out these matrices. Okay, so now we can go back to our original questions. So remember that we're interested in obtaining expressions for the mode shifts and decay rates of the photons, but this seems quite contradictory to what I was just telling you about because I was telling you about the system in which scattering are purely elastic. So the question is how is it possible or is it possible to have inelastic scattering in such a system? And the answer is that this is possible because the scatterings are purely elastic in the basis of the sultans and breathers, but they could be inelastic in the basis of the photons because these photons depend non-linearly on the sultans and breathers. And this is underlined by the special z equals half case at which we can actually refirminize the field. So essentially what we can do, we can introduce a fermionic field which is proportional to the exponent of the bosonic field and this fermionic field exactly creates a sultan in the system. So the interaction can be inelastic in the basis of the bosons of the photons here, but even though it is inelastic in the basis of the fermions or the sultans. So this is the case for z equals half where we actually have an explicit relation between the bosons and the sultans and breathers. We don't have such a nice relation for other bosons of z, but the same principle holds. Now it's not just that inelastic decay is possible on the basis of the photons, but we can actually use the integrability of the model and the strong analytical tools that it provides to calculate the decay rate of the photons. So generally to calculate decay rates we have to calculate some sort of correlation functions. For example, we can calculate the photonic propagator which defines a reflection coefficient from which we can read off the mode shifts and the decay rates. So to calculate correlation functions we can use the fact that the sultans and breathers define a complete set of states which we can insert between two operators in some correlation function and then the only thing we need to know are these F functions called form factors which are matrix elements of the operators between the vacuum state and the state of N excitations of N sultans and breathers and these form factors same as for the scattering of the reflection matrices, these form factors are well known for this boundary signal model and can be derived exactly only from the integrability of the model. So now we have everything we need to calculate the decay rates, so we can now go to our results. So starting from the total decay rate, so as I said this can be obtained by calculating a photonic propagator and there is some heavy lifting to do here in order to calculate these integrals so I'm not going into the details but this can be done and we did this. So what we get is this result so this shows you the decay rate as a function of frequency in low-glog scale for several values of the impedance and you can see that the exact solution which is here in solid lines agrees very nicely with perturbation theory for frequencies above the artist scale. We get the same parallel that we had from the perturbation theory and perturbation theory is no longer valid below the artist scale so we need the exact solution to complete the picture and the impedance independent parallel that shows the decay rate goes to zero at low frequencies. Now we can actually go one step further than that and calculate not only the total decay rate which tells us the rate at which a frequency omega k splits to any combination of frequencies we can actually also calculate the decay spectrum which tells the rate at which a frequency omega k splits to a specific omega k prime plus some other frequencies to conserve energy and the decay spectrum obeys a sum rule which relates it to the total decay rate it is given by a three-point correlation function so that is slightly more involved work which is still, we're still in progress this is something we're wrapping up for now we can show the result for z equals half where we can actually obtain a closed exact expression for this decay spectrum and the other values are now in progress. Okay so allow me to briefly conclude so I told you how single photon splitting is possible in a strictly QD environment and how it can be used to observe many body physics and specifically the Schmidt-Buglead transition by measuring the decay rates and mold shifts of the photons and I told you how even though our model is integrable and therefore the scatterings are purely elastic it is still possible to get elastic scattering in the basis of the photons and we can use the integrability to obtain an exact expression for these decay rates so these are two works that we're wrapping up right now so stay tuned for the papers and thank you for listening Thank you Amir Okay, questions On slide six you had this division between insulating and superconductor so in my view the definition in insulating state that it's like a capacity boundary condition it's not really accurate because you can have very infrequent facelifts so basically it's still inductor and only infrequently facelifts so I think for this you need like a running phase completely but it's not needed for the insulating state I'm not sure I understood I mean okay, when So the question is whether I think the capacitance is the right model for the insulating state is that there? Yeah, okay Yeah, I mean the capacity does not take into account the facelifts that's correct but I mean so the fact that you have facelift means you have a well-defined charge, right? So if you have a well-defined charge so we can... If I understand the series correctly then if you have one facelift in one day then it's an insulator basically Well I mean, okay so if you think about it as a capacitive element so this means you have localized charge right, so you have localized cooper pairs so this would correspond to a phase which is not well-defined, right? So I think it does take into account I think this might be a bit more technical discussion I think it all goes back to time scales and the question is what you call an insulator Yes, thanks for the nice talk I would have a more experiment-oriented question In your second part in this whenever you use integrability how much of the... What is the assumption of the spectral function of the bath? Can you really take into account the plasma frequency as we call it or are you assuming a simple omic bath or can you comment on that? Thank you for the question so first of all we're working in the cooper pairs regime and this solution is exact in the limit where E c is infinity so we completely eliminate the now you still have some final bandwidth you still have some bandwidth to your system and we can extract this bandwidth the only energy scale we have in this solution from the integrability is the RG scale and the RG scale relates to the bandwidth so your Ej star is Ej over the bandwidth so you can extract it so basically you need the bandwidth in order to calculate this RG scale to put it into the calculation so you can extract it from perturbation theory and in fact you need to extract it so what you can do you can compare your exact solution with your perturbative solution and then extract the find what is this so here you have some coefficient which tells you exactly what you need to put into to account for this final bandwidth and you can find this coefficient from this comparison from comparing the exact solution to the perturbative solution so you get exactly the parallel from the exact solution but you don't know where to put it it can be shifted up or down and what tells you where to put it is exactly this bandwidth now one thing that is not taken into account here is temperature, this solution is for zero temperature that's also something we should note we're looking into some ways to extend this to to find the temperature you can do this using both answers but yeah, that's pretty much it okay other questions I think if not then we want to thank the speaker once more