 I couldn't let me just, you know, I'm doing something tricky because there was again another update in keynote or zoom and things stopped working. So I'm trying to get this window correct. Yeah, I believe you should be able to see my entire present at the moment. There's nothing hidden. Everything is in the slide here. So, yeah, good afternoon, everyone. First off, I want to thank the organizers for putting together what I expect to be a very interesting workshop covering a really bright broad range of topics. It's pity that I'm unable to be there amongst you. I suspect though I'm not sure that a number of old friends and colleagues are in the audience. So I greet them from here. I didn't see them in a long time. So, today, in this talk, I'm going to share our recent work studying the crossover from a perturbative to a non-perturbative regime of QED quantum electrodynamics. I'll be discussing that in the, in the context of an artificial Josephson junction based Adam coupled to a high impedance resonator. And the focus will be on the behavior of the spontaneous spontaneous emission dynamics of the Adam. So this is really in line and a continuation of our early work on quantum electrodynamics of superconducting materials. Some of the relevant published work is listed here. The work I'm going to talk about today should soon be online. It's not published yet. And that has been done with my postdoc kind of Sinha and my students say it can. Right. So here's an overview. The talk is roughly divided into two parts. In part one, I will revisit the problem of spontaneous emission in a resonator. I will discuss issues in modeling and computation of spontaneous emission dynamics in engineered media and point out some challenges in modeling spontaneous emission or the per cell rate in complex engineered media. So in part two, we'll be discussing the specific question of how spontaneous emission looks like in a non-perturbative regime of light matter coupling. And I will present a proposal for studying the crossover from the non-perturbative to a not a perturbative regime of light matter coupling using a high impedance cavity QED system. We will be using a theoretical framework based on a singular function expansion to calculate quantum dynamics and such resonators. In particular, we will see that this approach provides an exact set of hybridized modes for open cavity QED systems, providing some intuitive explanation of what's going on in dynamics. And this I will discuss an extension of this technique that currently we have for a one dimensional transmission line based systems to 3 plus 1D modeling of superconducting materials based on this great differential forms. Yeah. So, in free space, any atom in an excited state the case to its ground state, the probability of finding the atom in the excited state the case exponentially with a rate that depends on the third power of the frequency of transition. This is the archetypical problem of quantum electrodynamics, one that tells us about vacuum fluctuations of the electromagnetic field sense by the atom where it's sitting. So, we are going to talk be talking about the modification of this behavior, then we enclose this two level system in into a resonator, an effect that was first quantitatively studied by Ed Purcell in 1940s. And what is expected what we know is that the decay is still exponential in time. It's still in the right regime in the right regime, but the rate strongly depends on the frequency of excitation as shown here. And it's relative detuning from the cavity resonance frequencies later will discuss situations where this dynamics can substantially deviate from exponential decay, in particular in this. This is according to as a number of it, perturbative regime of Adam field coupling. So, thinking about this problem, especially in a complex cavity. I think that we should not be needing to worry about the Hilbert space of the electromagnetic degrees of freedom outside a given volume that contains all the material degrees of freedom. Yeah. Because one, the an excitation has left the atom and the cavity. There's no other material object nearby to scatter the photon back. And that will be also very convenient because it would save us a lot of Hilbert space resources about the degrees of freedom lying outside this volume. So, but how does one formulate a quantum field theory in a finite volume with trans transparent boundary conditions. Yeah, so I want, once light comes in from inside I wanted to leave as if, you know, this is just an imaginary surface. And if something comes from outside I wanted to enter this region. So, how, how do so in particular to, there are two questions, you know, if you can do it. How to do this in a gauge invariant way, because it turns out when you have, you know, you deal with finite quantum electrodynamic systems, imposing gauge invariance is tricky. And second question is what modes should be used to quantize such a theory. The finite computational domain for the computation of spontaneous emission dynamics or any other quantum dynamics of materials lying inside is important for certain situations and I want to give a couple, you know, very sort of specific cases. There is a specific instance where there is no clear boundary of the resonator or the middle is out when the emitter is outside the resonator structure or there's no clear boundary of the resonator. Yeah. And another way to think about such situations, as I'm showing here on the left. The more left corner is such situations involve overlapping resonances in the spectral domain. So you, you may have a sharp few sharp resonances as shown here. On top of a broader background, right. And what is such as here. And what is the spontaneous emission rate or dynamics, when I tuned the atomic transition frequency omega j between two high Q resonances, such as here, dominated by a broad non trivial background. How do we compute that right such situation may well prevail for instance in a dispersive readouts situation scenario. So things can be rather complex. There's a different perspective on this whole thing on this calculation in the spatial domain. If you tune the emitter in this in this, that is positioned here between two dielectric cylinders, if you wish or spheres. And then between two high Q resonances of this photonic molecule structure. The region is that spectral region is dominated by very lossy modes. A few of them near the frequency of the emitter are shown here. One would conclude the emission will be quasi isotropic. What happens though if two cavities have the right sizes and the gap between them is right. The mission turns out to be extremely narrow. Yeah, so I've specifically we have cooked up this situation to be in that prevail. So this is, you know, this, this can be understood as a collective interference effect of many broad resonances, each of which by themselves admit quasi isotropic. There's no single mode in the spectral region of the emitter that looks like this. Yeah. So, this, in a way, this is a multi mode interference effect and you'd like to capture this in the quantum dynamics of the emitter. Another, you know, relevant situation arises in in actual modeling of quantum computing chips. You know if you peek in inside of fridge like that you will see that your chip containing in superconducting cubits. It's basically embedded in a box in a 3D box and that box has resonances itself and there are chip modes and so on and you use end up with situations where those chip modes and box modes can interfere with what's going on on the chip. Something that you can take as a essentially crosstalk between between cubits on the chip. If you don't design these structures right. So it is important that you, you, you'd be able to model these structures like that so you need then all possible conserved all possible computational space and draw a boundary around this box to do some computation for quantum dynamics of what's going on inside the chip. Besides issues of computational efficiency the calculation of the poor cell decay rate in perturbation theory in coupling to all cavity modes is found to lead to divergences as first pointed out by these two experimental papers here. And this problem is reminiscent of divergent series in the Lamb shift that you know that is well known, the radiative correction due to vacuum fluctuations to the atomic transition frequency, except that these radiative corrections now have to be found not for fundamental particles, but for an oscillating condensate that sloshes back and forth across a thin insulating barrier in a piece of metal. Interestingly, you know that's so that's, that's, that's a complex problem. And interestingly, a semi classical formula for antenna loss gives pretty accurate results for spontaneous emission, finite and accurate, in contrast to a non-perturbative quantum theory of of the cavity and the qubit. Yeah. So we understand the resolution of this issue now through a manifestly gauge invariant calculation. Non-perturbative and light matter coupling for for those interested in the in the audience you can find some perspective on this problem in these two papers. Now, to be clear, this resolution is reached only within an internally self consistent effective quantum field theory that we know how to carry out consistently only in one plus one dimensional systems like transmission line systems. Couple two atoms. Yeah. So quantum editor dynamics of superconducting materials really still harbors a number of very interesting computational as well as fundamental problems. So I'm moving now to part two here. I just sort of summarized the state of thinking in QED of superconducting materials. I'm going to focus on a very specific problem. So consider now a very long cavity but with good enough mirrors that the modes are well resolved. So it is a hyphen as situation and we implant an atom at the position xj. Let's say on one end of the cavity. That is a transition frequency omega j and imagine we can increase the overall coupling of the atom to the modes and this G. Of course, the coupling of the atom to a cavity mode and generally depends on the atomic dipole dipole moment P12 and the spatial structure of the individual modes fine at the emitter position xj. But one one can globally increase the coupling to almost simply by increasing the size of the transition dipole moment P12 one one way of doing that. So this is a situation we have studied in a context of an optical cavity in in a 2014 paper shown here. So what we find there is that the coupling goes through. Those three couple important scales. There's a crossover between different behaviors of spontaneous emission dynamics of this two level system. So I'm plotting here, the occupation probability of the excited state of the atom. And as time progresses when you started in the excited state coupled to such a cavity with many modes and along cavity, a situation like here, spectrally the admins here, and I'm increasing increasing G. And we will start with, you know, this is sort of you get exponential decay. You get a small, much less than kappa, the nominal decay rate of the cavity. And we have an exponential decay of the atomic station with a well defined rate. And this rate is different from the free space spontaneous emission rate, the per cell decay rate. This is called the per cell decay rate. When G reaches a scale kappa and goes beyond it, one crosses over to an under damp decay so I'm assuming that the cavity that the atom can only decay through the cavity to continue here. So here's an under damp decay. This is a well known well studied strong coupling limit and the oscillation. The frequency scale here is approximately back in Robbie frequency G. Now, finally, when we G reaches a scale of the order of the mean free spacing Delta C, and beyond the dynamics crosses over to one of pulse emission. Here the physics is that of its one tip of spontaneous emission in the form of pulses that are emitted. The pulse travels over to the other mirror reflects gets reflected some of the case and comes back to the Adam and Robbie flops it again and so in as a result you get these revivals, if you wish to call it that way of the atomic dynamics. Sort of the characteristic timescale between the revivals is of the order of cavity round truth round trip time, and that's simply the inverse mean level spacing. So, now you start resolving this. So the this this the interesting this pulse emission regime or revival regime has been observed in the paint in Oscar painters group just last year. So, that's, that's really, you know, we, I, you can increase G further and you can hit G can hit the frequency of transition itself. And then omega j, and then we have the ultra strong coupling regime and I'm not going to be discussing that here. Yeah. But there's another interesting perspective on the same regime. It was also pointed out that in the, what, what, what I call the multi modes strong coupling rich team what PM maestro and and miser here denote as a super strong coupling regime. The cavity modes can be significantly modified in this region. So this paper studied atoms trapped in an optical lattice in a resonator, you know, very different system where G can be enhanced by the square root and N of the number of collocated atoms at a given space time point space point. So this year is somewhat different. You know, we have mobile atoms and so on and so forth, but the suggestion that the modes of an electromagnetic system that is not fixed is made here and that's very interesting. So, I'll come back to that in this talk. So, today I want to provide a unified perspective on this physics of crossovers. From a more from a perturbative to a non perturbative regime of coupling and then appropriate formulation of the problem of spontaneous emission dynamics that is intuitively clear. And addresses some of the challenges I discussed in quantum electrodynamics of superconducting materials I discussed before. And we want to do this within a finite volume with transparent boundary conditions. We can do this in the particular context of high impedance superconducting QED resonators such non perturbative regimes just discussed can be reached when the atom, a Josephson junction, Adam is coupled to an array of Josephson. Adam here is coupled to an array of Josephson junctions, the each which make a high impedance resonator. These junctions are larger in size than this one. So, we have an Adam couple to a resonator situation. The issue is a bit complex here we do not have a simple enough way to tune the dipole moment of the artificial Adam independent of the coupling to its environment, while only one atomic transition still stays the relevant one. So that's difficult but friends and colleagues, probably one who is sitting in the audience on Nicola Roche and others figured out starting around 2017 2018 how to get down to that regime. And I will state just that this can be done experimentally. And, and this helps this, you know, making this resonator high impedance line just slows down the light here. And that, you know, makes the cavity in a way appear longer, let's say. Yeah, so it brings that it that makes it possible to see this kind of non perturbative regime. Right. So here's the model. This is a very similar structure to the one studied in several experiments in recent years. The one main difference is the. So you have an Adam couple to a very long high impedance line made of Josephson junction and Josephson junction array, and that is coupled to a wave guide in this model here and infinite wave guide. And the one main difference here is the existence of a coupler. Yeah, which has a inductance and capacitance that is different than the inductance and capacitances of individual units making up the, the high impedance line. Adam plus coupler plus Josephson junction plus wave guide. Yeah. The transparent boundary condition here will be implemented after the last node right here of the of the high impedance resonator of after five and the node and yeah. And so it's the wave guide field. So the sort of the discussion starts with breaking down the wave guide field field into a right propagating and the left propagating modes. It is clear already here that the right propagating component carries signals from the cavity and left propagating component carries noise from the rest of the wave guide into the resonator. So then one can simply write the Heisenberg equations of motion for the reduced system of the Adam and the resonator only. And this includes that also one note of the wave guide. It turns out that I will call that the boundary or surface node. And this makes up this vector of quantum fields, Phi and to reduced. Yeah. So this only doesn't really, you can write these equations down. And this is the, this ability to break the wave guide modes into chiral components, provides you two components here that is only active in the last site and plus one which is corresponds with zero sites here. One provides dissipation turns out, and the other provides the corresponding noise coming to the wave guide. Yeah. So, then I'm skipping a lot of technical details here but it's rather straightforward the same set of equations you normally write down but you write this in this funny way. And simply integrate. So here, we will first neglect the nonlinearity of the emitter, which we are going to. We can take into account by multiple scale perturbation theory as was discussed in an earlier work from from my group. So, because it's a linear system you can integrate this out and simply find the solution by Laplace transform. And you see, the solution for the flux field on all the nodes is denoted by this sort of this bold notation here, it can be written as an integral over a piece that is a propagator. This is a standard propagator of the Josephson junction line coupled to a linearized Adam, but this piece now that is only active in this last site right here is includes the dissipation. And this propagator is now essentially the kernel that has a functional dependence on the Laplace coordinate s or the frequency, I capital Omega. So this multiplies this big term here, and you see that's simply the source term that you have seen in the previous equation here, the source term here. Plus the source term plus initial conditions of the reduced part of the system. Yeah. So, this this basic the noise turn is essentially there to thermalize the field inside the resonator to the right value, given the statistical properties of radiation and the wave type of. I want to also introduce this dimensionless parameter chi given in this way that's the coupler sort of how this coupler is different from the rest of the array. Yeah. And this guy. I think of this turns out as a, as a perturbation as a parameter of strength of coupling. And so, chi small is weak coupling or perturbative limit chi large is non perturbative limit. And chi was one, you know it's interesting, because then this become this unit becomes the same as the other in the array, and this is essentially the galvanic coupling limit that has been explored in experiments. So, in this form, the result is not very useful. The interpretable form can be obtained by using a singular function expansion of the kernel. This is a known not so very well known technique in classical electromagnetism and basically extended its application application to here to quantum fields. A. So you get the, these modes, the singular functions can can service modes turns out, and you can expand the fields at a given frequency s or I omega. Yeah. So let me discuss the meaning of these modes because it is rather strange. What we are talking to about here is the modes that are suitable to expand any mode that is excited by a source at frequency capital omega, you know, place that X, X prime here of by a source. They, those kinds of field in the far field outside in the wave guide, for instance, here's a 2D situation would obey asymptotic boundary conditions in this simple way, but they would be parametrically dependent on this capital frequency, frequency capital, omega. And so this parametric boundary conditions then give you a set of non Hermitian modes. And you can build the greens function to propagate any field from a source. Yeah, so this is what the idea behind this singular function expansion, we will use these parametric modes to expand this radiation. So here, a basic how these modes look for a basic situation of a transmission line cavity standard one with a qubit attached to it. These are exponentially these modes are trigonometric functions, but they exponentially increased towards the loss boundary and our constant in exponential, they're absolute values. Yeah, the important another important point is that these modes turn out to be modified by the placement of source, and you see this, this zero coupling cake was zero and non zero coupling, how much it can modify in the specially quantified situation. And this modification of modes is essentially what gives you finite by finite weights for spontaneous emission if you want to do a perturbative calculation with train normalized modes. We have some numerical results for this long cavity where we can get a glimpse into the physics as we tune coupling from perturbative to the non perturbative regime. Spectrally, we have n plus one modes for each frequent here, you know on the horizontal axis you see the atomic frequency omega a and the tune it. So these modes are modified they are pretty straight. And this is atomic mode that is goes through the resonator modes. We have these m plus one modes they form a band, and there's a bandage at the plasma frequency. Now the width of these, this is calculated the width of these resonances are also available but we are not including that here, because it's cluttering the behavior I want to discuss. So we see that the atom undergoes several onto crossings as it goes gets through the resonator modes they are very small so you don't see any modification in resonator modes. And the reason. So, on the lower plot, I'm plotting only the atomic mode. Yeah, and, and here is the position along the cavity along the resonator, and you see it's mostly localized at side zero where the atom is very flat here. But as I tuned the atomic frequency this modes, you know loses localization and in delocalizes or the system. You don't quite clearly see it here. You can see that in in in this inverse participation ratio, which is a measure of localization used by people studying disorder, but you can sort of use it to see very clearly the behavior. In this integration ratio is one, if the mode is fully localized so the atomic mode is localized but in small regions of atomic frequency, it gets delocalized is what you see here. Yeah, as you increase now the but I call the coupling to the full galvanic limit. What you see is now the atomic mode cannot be really distinguished anymore. All the modes of the resonator undergo some shift, tiny shifts, many more of the modes as the frequencies tuning across this diagonal here. The red line is just the closest mode to the atomic mode, right, and they are essentially pinned in regions of atomic frequency to these modes. And essentially the mode here atomic mode is doesn't really access it is melted into this background of band modes of the resonator, or in effect the band mode contains one more mode. So we just distributed and as a spatial domain, you see a complete delocalization of the atomic mode. It, of course, displays some interesting nodal structure, but outside. It's localized again, right, you see IPI equals one here right at the bandage, and the frequency also she sees a very strong glam shift as in addition. Yeah. So finally, I'm coming to the end I I'm out of time I see the, I want to show what I sort of the the basic non perturbative to perturbative. Sort of crossover in a very clean way. I'm plotting here the lamp the frequency of the atomic mode as I tuned the coupling Chi. Yeah. And this is coupling Chi goes one which is the cal one a coupling limit. What you see here is that the frequency of the atom computed by the singular function expansion in red, and the lamp shift and also calculate and perturbation by a where you calculate the lamp shift additional correction. And there are two ways of doing this but each of them convert sort of diverges strongly as you come to this number two bit of regime so I call this really, I mean, I refer to this as and the non perturbative regime of coupling. Similarly for per cell spontaneous emission rate of the atom. I'm comparing here. Again, as as I increase the coupling to standard formulas that are obtained through forming golden rule where the effective impedance of the array exactly calculated is used. And that is, I believe that my effective is in this dash line here in. And then the other limit is the one where we assume that the high impedance line is infinite length. Yeah, so there's no way guide that also misses the exact calculation quite a bit. And so you can say the number of it of regime really starts here. This is the exact calculation. Yeah. So they can also capture the dynamics. I'm out of time. So just, I guess, the last three slides. I can skip them safely because we have we have just, we have been, this is just new work, which which is not complete yet, but we are carrying this through three to three superconducting materials where we solve the exact order parameter equations and the light part and the max of the equations essentially with fields that are gauged invariant given in this way, and they are exactly hybridized. And you can use a formulation in terms of discrete differential forms to express these entire equations in two equations, which you can solve numerically and it's essentially in when you come down to transmission line limit of a three dimension structure of these equations fall back exactly onto a transmission line equation. So, so here's just, you know, how you can simulate a Josephson junction Adam inside a superconducting two dimensional cavity. You see them fields penetrating into the superconductor and so on. So this works, you can also capture flux quantization. So, so this brings me to the end of my talk. I'm three minutes over. I want to thank everyone for listening and also point out the people involved in this. This work kind of seen how we're starting as an assistant professor in the inverse of Arizona and say it can. And yeah, this concludes my talk. Thanks for this very impressive talk. It's a pity that we can't have you here, but we can have a bit of a discussion now. You couldn't hear. Oh, no, it's a pity that you're not here. Oh, yes, yes. Yeah, that for me. I tried without the mask. All right, I was really worried you guys couldn't hear anything. Sorry about that. We enjoyed it to be here. Thank you very much, Hakan for the nice presentation. You are him here. I'm wondering. Yeah, good to see you. I'm wondering about fluctuations say of the of the electromagnetic field. For example, have you have you explored this as well? How it changes when you go say in the in the strongly coupled regime when you have this hybridization basically how the the fluctuation properties change. So can you so I have to be very specific because we essentially calculated so one quantity we have calculated is the expectation. Let me one second. Do you see this slide? Yes. I skip the slide, but I guess this could be related to what you're asking me. I'm looking at the expectation value of the field, the occupation in a way, or the energy at the local atomic site. So expectation value of NA and we are able to compute this and you can see that, you know, I'm plotting here the corresponding modes and how they hybridize with the rest. You know, here it's only living in mode one in inside one but here's it's a short cavity here just for illustration. And and then So this is the spectrum of these transient oscillations. I'm not showing here the long time limit of state of state. I have some plot here that we have calculated in very extreme coupling limits. Let me show you that. So this is the long time limit of what I've shown you and the interesting thing that we observe here and these calculations are not complete. This is why I didn't show you still being. We are still discussing this. I'm showing here the exit the the energy in in on site atomic site as a function of time when it started initially with some energy there. And you see that in the long time limit, depending on the coupling, the status data is different. The status state when when you go to couple of small coupling kypo 001 is one example of it. It saturates and that value is the value of the thermal value of the wave thermal sort of the temperature of the wave guide here. We have chosen the temperature of the wave guide to be 50 milli Kelvin. But as you increase the coupling, the status state occupation, there is changes. Yeah, so clearly it looks like an effective thermalization to a status state value. But what is really happening is that thermalization does happen just happens to another mode, which is the localized and then you of course just look at the mode that is just a local site occupation, you see this thermalization that is not equal to the temperature outside, and you can plot, you cannot actually compute an effective temperature as a function of ky as temperature as the coupling goes down. It saturates with the wave guide as expected. So this is we are still wetting this calculation because we have don't have much exact to compare to another computational scheme. So this is why I didn't want to discuss this here, but we can I mean it's within the reach. Thank you very much. May I ask another question? This is a more speculative question. I mean, as you know, vacuum QED has the problem of renormalization. So I mean this is what Feynman has worked about. And my question, I mean this, then they somehow found and developed methods to deal with this renormalization problem. And my question is, I mean, is it conceivable to think again about this problem in this regime which is not accessible in vacuum QED. So do we, I mean, as I said, it's very speculative. I don't have any idea, but is this, I mean, at least conceivable to use the techniques that you developed in this strong ultra strong, super strong, extremely strong coupling regime to learn something about this field theory that we cannot say that we do not find in the weak coupling regime in terms of renormalization. Right. So the, you know, I will tell you an equally speculative story here about this. It's been in our minds since we started working on this problem in 2017. And I think that a number of other groups were thinking about the same thing. So yes, I believe one, you know, now we are looking at this problem of QED renormalization, not with individual electrons point like electrons, but with matter with solid state materials. And it is not clear whether how a good theory for that situation looks like. One day, you know, when we studied transmission line cavities just because there was a problem that, you know, computing per cell decay rate in that setting or lamb shifts. The general feeling was that this is effective field theory, you know, it's sub gap. It's only valid sub gap. Still, you can say, well, you know, imagine a superconductor whose effect, whose critical temperatures infinity, there's no problem imagining that. And there you can really see that these quantities still diverge. And if you take and take on this gauge and variant description gives you a convergent result, but it gives us nothing. It's not a fundamental theory, but it tells us why how we can get finite results. And the big issue was how to do this in 3D, because that's where we live. And this is, you know, is this attempt that we just started in here. This is where we want to go. And there, if we find the same kind of situation in 3D, then I would say it's going to be very interesting to study, go back to QED, fundamental issues QED, and study them in engineer setting. And theoretically as well, there's still a lot of unanswered questions here. It's very interesting. Thank you for that question. Just maybe a legitimate analogy question. So in ultra-cold atoms in two-body collisions, there was some really nice physics done with Feshbach resonances going from scattering to trapped states really close to the boundary, which seems, it just visually struck me as reminiscent of your highly hybridized states as the coupling is increased to the sort of asymptotically trapped long spatial, when you go, when you release the atom from the cavity from the quantized states in terms of frequency. So I was wondering if that analogy is valid or useful in any way. The analogy to collisional states of atoms where you can change the coupling there and go from trapped to scattering states. And there was some unitary physics that was done there. I see what you're saying. The analogy is, you know, you have a competition between localized physics and a continuum there also in collisions with atoms, right. And then to that extent, the analogy is similar, though the field there is an atomic field, a matter field, right, with that caveat. And but the physics that I'm discussing here is very analogous to spin boson physics, which is a very general model that has been studied in many, many contexts. In spin boson model, when the coupling of the impurity to the atom, oh, sorry, to the continuum increases, then, and you see, you know, strong coupling effects, let's say. The perspective that has not been studied a lot in that context is how, you know, that there are ways of thinking about the physics of the spin boson model with modes, effective modes that are created between the impurity and the bosonic bath. And, and probably, I'm sure someone has studied this coming up with effective modes of that structure of that spin boson model in various, you know, regimes of density of states. And you would see a very similar behavior there, probably as the coupling increases that an effective mode decays into the sort of spatially delocalizes into the continuum. And yeah, I think in the collisional, you know, for an electronic atomic field, you have probably similar effects and there's probably an analogy from the point of view of the equation describing the matter field in that context with the, sort of the hybridized field here. And maybe there's a value. Well, I guess all that I'm saying is there's a value to think in terms of hybridized modes. That's that's the one way of doing non-protobative calculations. I'm sorry. This answer was probably too long. Okay. I think that is enough. I think you again and all the other speakers of today's session.