 uh, uh, driven-dissipative, uh, quantum systems. Okay. So why are we interested in, uh, driven-dissipative quantum dynamics? This, this is, should be an obvious, uh, have an obvious answer, right? So there have been many protocols in, uh, the quantum information literature that call for, uh, delicate engineering of the interaction between a system and its environment. And in particular, uh, the notion that, you know, an environment can have, you know, a stabilizing effect on, uh, certain, you know, quantum, certain quantum manifolds of states, for example, cat states. Um, and what I want to argue in this talk is that, uh, a lot of the inspiration for many of these autonomous quantum air correction protocols have often come from, uh, uh, work in the AMO literature a long time ago, uh, and, and so exactly solvable models have been used as, uh, basically inspiration for a lot of these, this, uh, uh, progress, right? So again, what, what am I referring to? The, there's the obvious, uh, example of, uh, driven-dissipative care resonator. Um, and here it's just, um, uh, a nonlinear, uh, electromagnetic mode. It has a, uh, a Hubbard interaction or a Cortic, uh, self-nonlinearity, uh, two photon driving and two photon loss, right? And as we all know, uh, this system is very favorable for, uh, stabilizing, uh, you know, non-trivial superpositions of, of, uh, distinct classical states, right? The anti-poetal coherent states in this case. And it's a nice fact about the master equation that I showed you in the previous slide that, uh, any, uh, superposition of such coherent states is a steady state of the master equation. So we would say that this is a decoherence-free, uh, subspace. And this gives this, uh, quantum memory its stability, uh, property. Right. So I want to take a step back here and I want to talk about, uh, enormous theoretical activity that's happened, uh, uh, studying these non-trivial, uh, driven distributive systems. So again, there's this, uh, two photon drive and two photon loss regime of a non-linear mode. This regime was first, uh, investigated by Carmichael Walls in 1988. Um, there's also the original version of this problem, which was solved in 1980, uh, by Drummond and Walls in, uh, Journal of Physics, um, at, uh, A. Um, and recently, uh, there's been a lot of very good work, uh, trying to combine these two regimes and basically understand the full, uh, effects of, uh, competing one and two photon drive and one and two photon dissipation. And I'll refer you to these, uh, PRAs in the, in, you know, by, you know, for example, a Chutes group. Um, and the reason why this, uh, these questions are very important, not just from a theoretical perspective is, you know, maybe we're missing some non-trivial, uh, steady state physics that was not uncovered, uh, in the previous works that I showed. Um, and so one thing I want to do is I want to, uh, look at this, take an even further step back, make this even more abstract, uh, and to, to, to see what, what do these works look like? So all these, this work that I've mentioned, uh, basically you can fit it into this paradigm where you, uh, have a quantum master equation and you want to find its steady state, its quantum steady state. And what these works do is they invent some sort of, uh, mapping to a classical random process or a classical, uh, probability distribution and, and basically look for some, uh, uh, symmetry in that problem and then solve that and then map it back to the quantum problem. Uh, so the, the, the most obvious example of what I'm talking about is say you have a quantum Van der Poel oscillator, but it's in this very simple regime where you just have incoherent, uh, gain and incoherent loss, uh, you know, you would spend a few seconds looking at this master equation and you could say, oh, well, I can solve for the steady state by looking at the diagonal part of the density matrix, right? And once you look at the diagonal part of the density matrix, you have a classical system, you have a bunch of kinetic, you have a kinetic equation with rates. And in fact, this kinetic equation satisfies detailed ballots. And so, uh, you can solve for, for the steady state that way and then, you know, you could solve the problem. Um, so yes, in general, this is, uh, a general framework that has been used a lot and very successful in the, uh, ammo literature, uh, over the past few decades. Um, so the reason why we're focusing on this is because this is exactly how, uh, typically these driven nonlinear cavity problems are solved. So if you look at the work of, uh, drum and walls in 1980, uh, what they did is they essentially invented a, uh, very complicated, uh, phase space distribution, the positive P, uh, phase space distribution. They mapped this quantum dynamics onto an effective, uh, diffusion process, classical diffusion process in four dimensions. Okay. And why did they do that? Well, because if you do this, there's this kind of mathematical accident that happens with this kind of driven care problem is that this, uh, diffusion process is completely reversible at the level of trajectory. So if you looked at, uh, the moments of, uh, the random process in the space space, and then you looked at the moments of the reverse process, they, those moments are exactly the same, uh, in the limit of, of, of a large number of trajectories. So, so this is a symmetry. And again, this leads to detailed balance. So you can solve this, uh, classical diffusion process on the right. So now this kind of leads to a lot of the work that has inspired a lot of the questions in my PhD. And also there's, you know, foundational work by Shtanagal, Ravel and Zola that I'm also going to talk about. It's basically what we're doing now, uh, like 30 years later, 40 years later is we're trying to find, um, uh, basically a, a, a quantum manifestation of the symmetry. So we don't want to look at these mathematical flukes. We want to look at the original Limblad master equation. Is there some quantum symmetry that, that basically this, this classical symmetry is a symptom of, or it's, it's indicating something, right? And it turns out this, the question of finding the corresponding quantum symmetry is actually non-trivial. Right. Because you can prove some, some sort of no-go theorem, which is basically for these driven distributive cavity problems, uh, any microscopic realization of this, this bit of system. So basically you're adding the environment back in, uh, must break, uh, time reversal symmetry, no matter how you construct one. So you can prove that the steady states of these systems are sufficiently complex, that there's no conventional notion of quantum time reversal symmetry in this problem. So what did people do? So, uh, there's this very, uh, good work by Sonogal, Ravel, and Zoller in new journal, physics 10 years ago, which basically, uh, finds a quantum symmetry that, uh, that, uh, is, is, is making this problem soluble. And what they do, so, so you can't look for conventional time reversal symmetry. So what they do is they look at a doubled version of the systems. They look at this very funny looking, uh, purification of the steady state density matrix, which we call a thermofield double state. And this purification depends on an anti-unitary operator T, right? And basically what Sonogal noticed, uh, 10 years ago is that if you measure, if you, if you consider a thought experiment where you measure cross-site correlation functions in this doubled, in this purified steady state, then those correlation functions are time reversal symmetric. So there is a symmetry once you look at, like, uh, basically a doubling of the system. And, well, this is exactly hidden time reversal symmetry. So what we've done is we've said, okay, this is a quantum, uh, symmetry. It makes the steady state problem solvable. And we call this a hidden time reversal symmetry because it does not correspond to any conventional microscopic, uh, TRS. Okay. Um, so yeah, so a lot of work has followed since that. So we, we published some, some work where we've actually used this quantum symmetry to actually extend the solvable regime of this, uh, uh, space, space of nonlinear cavity models. Uh, but we've also, uh, basically just proved, uh, the missing pieces in this whole picture. So we're putting it all together. So one, one thing that would be nice of this hidden time reversal symmetry as well, if you have a conventional time reversal symmetry, then it, then you better have a hidden one. Right. So one fact is that, well, hidden time reversal symmetry, uh, it's, yeah, it's, it's more general. So it's a general, it's a more abstract notion of time reversal symmetry that holds if you have conventional microscopic reversibility, but if you don't, it can still exist. For example, this class of problems. Um, another thing is, yes, this, the, the, the, the kind of the magic in, in the positive peat work from the 1980s is actually a symptom of hidden time reversal symmetry. So you can also prove another theorem that says, okay, if I have a restricted, um, class of, uh, you know, multi-mode, not, not even single-mode problems, you can have a lattice model, a Bosonic lattice model. And as long as those Bosonic modes are subject to single, uh, photon loss, then you can prove that any hidden time reversal symmetry of that problem, uh, will, will correspond to a reversibility of, of the corresponding classical diffusion process after you map to any, any kind of generalized P representation, you know, complex P, positive P, uh, and if you, if you have a well-defined Glauber P distribution, then that, that, that Facher-Poch equation will also be reversible. So this is, uh, basically guaranteed. You don't even have to check it. You don't have to do the mapping, right? And in fact, we actually just give a, uh, an explicit construction of the corresponding classical, uh, time reversal operation, uh, in the appendix of this, uh, BRX quantum work. Um, so yeah. Um, so that, that's already, uh, published. So now what I want to do is I want to look at, uh, going beyond this, uh, quantum optics work and solving, uh, many body systems, uh, lattice, lattice models, uh, using hidden time reversal symmetry. So to, to, to, to inspire this model, uh, so what I'm going to do, I'm going to introduce, uh, an exactly solvable many body system. And what we're going to do is we're going to dial down the complexity of the single site problem that we're looking at, right? So we're going to look at a two-photon driven, uh, care resonator with single-photon loss. And this is called a KPL, right? Care parametric oscillator, uh, also, also, you know, some people are calling it, you know, a parametron because it's, you know, it has, uh, classically bi-stable, uh, states in, in the large driving limit, uh, you know, you can go below and above threshold, you know, the standard theory, right? This has a Z2 symmetry. And so what we're going to introduce now is what we call a many body parametric oscillator or a large OPO. And in a large OPO, basically, it's the same as a single site OPO, except the discrete Z2 symmetry of the, the oscillator, basically the reflection in, in, in, in face space is generalized to a continuous orthogonal group. So this, this model has a continuous, uh, family of symmetries. And this is the model that, that we can solve. Uh, so basically, uh, there, there are two things to notice here. First of all, um, we have a global Hubbard, both Hubbard interaction. We don't have a local one. So, so basically the, the energy of a, of an eigenstate of the system, it just depends on, uh, just the total number of photons. Uh, we don't have hopping. We have kind of artificial hopping, which is, uh, induced by a pair driving. And, uh, we have this, uh, more physically realistic loss, which is where we have, uh, you know, individual Markovian loss on each mode. And interestingly enough, if the, if the loss rates are different, then you actually can't solve the system. So this is actually something that, that is needed. Um, yes. And so this has a continuous symmetry corresponding to the Z2, uh, symmetry in the, in the single-site problem. Um, so, so let's look at the simplest non-trivial case. So, so where there's no, uh, lattice structure in this problem, you just have two photon driving, single photon loss, and you have a global, um, Hubbard interaction. Okay. Uh, the reason why we'll, we'll focus on this is because this is actually quite reasonable to realize in a superconducting circuit. So the number of nonlinear elements that you would need to realize this, uh, uh, Lindblad master equation is, is three. It's constant. It doesn't scale with the number of, uh, with system size. Okay. So in particular, you can place, uh, a chain of linear, uh, resonators, or this could be a multi-mode resonator with, in first foster form. And, uh, to, to get a global Hubbard interaction, uh, you, you just put a, a transmon, or sorry, a, a Josephson junction in parallel with this. And then, uh, to get the, the two photon driving, you just, uh, put a flux tunable transmon in parallel. So you just have three junctions, and to, to increase the size of the system, we've, that just corresponds to increasing the number of, uh, linear modes in the chain. Um, so you might think, okay, this, this sounds very trivial. You know, there's no, this just has a global Hubbard interaction, coupling a bunch of, uh, lossy, uh, OPO's. However, the phase diagram of this system, so if you look at this system as a toy model of, of, you know, driven dissipative phase transitions, actually the most interesting physics occurs when, when you're not in like, you know, finite dimension, you know, one dimension, two dimension, three dimension. Um, and in particular, one of the really interesting things about this system is, uh, well, there's a non-trivial first-order phase transition. And actually, because we can interpolate between the few body limit and the many body limit in this model, it's exactly solvable, we can see where that phase transition is coming from. So this phase transition, here we're plotting the, uh, the average number of photons per mode. Uh, so that's this n bar variable, it's the, it's the mean density in thermodynamics, like language, right? We can see that that the space transition is coming from discreet, a multi-photon resonance is basically squashing together as n goes to infinity. And, and just giving you a very sharp, uh, just continuous, uh, density slip, jump in density. Um, another interesting thing is, well, you can see that, okay, if you do Gertz-Wieler mean field there, so if you do a self-consistent, uh, decoupling of the, the global Hubbard interaction, uh, you're gonna get, uh, basically, uh, possibly try, try stable solutions, but the exact solution always selects one such solution. And it's actually interesting to see that, as you can see, at moderate, uh, moderate system sizes, you have, like a kind of meta-stability of this high-density solution, right? So as you increase system size, that, that, that, the stability of the upper branch of mean field there basically becomes unstable, and you just drop, down, which is kind of interesting. And we predict, as n goes to infinity, you're the bi- as you go to the bifurcation, uh, you're gonna have the location of the, the phase transition. Um, right. So this, this, you know, resonance induced phase transition, well, the argument that I've shown previously on the previous slide, that's, that's the physics of very low dissipation, right? Low dissipation, you have these very sharply, defined, resonances, and so when you squash them together, sure, you might get a first-order phase transition. However, is this really an artifact of the zero, uh, dissipation limit, right? So one question you can ask is how robust is this phenomenon to dissipation strength? Because here, uh, dissipation is actually one one hundred of the, the Hubbard-Donnelly area. So this patient is extremely weak. Uh, and the answer is, well, it'll depend on the dimension, right? So in, in zero D, you have a very robust, uh, phase transition that extends to a finite kappa. In one D, you have, yes, it seems like this, this phase transition is an asymptotic property of a kappa going to zero. And 2D, you know, you can, you can conclude something different. Okay, so, the point is you, you can look at the dependence of the phase diagram on dimension, right? And to, to show that this is robust, for example, you can look at, the density susceptibility, right? So we can look at this region where this phase transition seems to be smoothing out, and we can plot the maximum, basically the derivative of, of the density as a function of detuning. So here, detuning could play the role of like a, a, a chemical potential, right? Uh, and, and you can see, as you approach some finite, uh, kappa, critical kappa, which is greater than zero, you get a nice, you know, one over, one over x, uh, divergence of, of, of the susceptibility. So it's like, you can get, start to look at critical exponents, right? So, basically, yes, this is a critical point, right? This is a quantum critical point in an exactly solvable, system, and you can see it by, you know, solving systems of up to, you know, 16,000, modes, right? Um, so, yeah, so it's so, and there's, there's another question you can ask. Again, this is all about this, the very simple, um, version of this bottle, where there's, there's a nice josicin circuit that realizes it. Um, so it seems that there are two phases, right? If you have a first order phase transition, there's a high density phase, a low density phase, then, then you have probably have two phases, right? The issue with this is that, if you look at this, phase diagram, you know, there's, there's low density here, and then there's also low density here. And so you can always continuously go from high to low density, right? In this phase diagram, so you might think that, okay, uh, you know, there's, there's one phase and it's all connected. So in order to resolve this question, you, you need to go to non mean field information. So you have to look at basically a correlation function. And one of the correlation functions that we can compute is the density density correlation function. So if you look at the density density correlation function, there is no ambiguity here. So the first order phase transition actually corresponds to, density fluctuation is becoming correlated or and going to being anti correlated. Right? And that notions is something that's well defined and serves to distinguish the phases even at high dissipation where everything is smoothed out. Right? Even in the crossover regime. So, so that's something that you can look at in this file. You can, you can compute any observable actually. That this is something that you kind of need to understand the harmonic function theory for. So there's this beautiful theory of harmonic analysis and and dimensions and and you can compute anything you want. In this case. So, so yeah, so I obviously there's a lot to talk about. You know, you can look at you can see low like features and you know, say a toroidal lattice around 200 sites. You can look at well first of all the single site version of the model. So the single site thing that inspires this model an OP a KPO has cat states. So this thing has an even better, I don't know if it's even better cat state it's a non locally distributed cat states a multi mode cat state a pair coherence state. Yeah, there's there's like mode selection and competition in this model. So you could hear here we're looking at like a 1D chain where we're increasing the the drive strength. So the ratio of drive to you is like almost a thousand or starts around 300 here and you can see that there are certain modes in momentum space that are being preferentially selected at high in the semi classical regime. So that's that's interesting. You can also look at density correlations there. There's a kind of super fluidity that emerges if you do a linear stability analysis it in the classical regime. So there's a nice exotic form of super fluidity here. And yet, so this is a toy model. So it's something that solve. So it's something it might be a nice playground to test ideas and learn something about non equilibrium quantum dynamics. Right. Well, not the dynamics the steady state everything is a steady state. Right. And we can take a step back and you know we can also solve another class of systems. I've censored this because we also have to publish this this year but I'm will be very excited to announce it maybe in a couple months. But yeah, they're they're multiple. This this is not a one off thing you can solve multiple different kinds of many body systems. So just to conclude, so what's what's the main takeaway from this talk? Well, what we want to say is, okay, there's an old there's an old fashioned way of doing things. Right. That's been very successful. Right. A lot of ammo work, you know, solving, for example, these driven descriptive care problems. They they they basically map Lindblad master equation to a classical master equation. Classical master equation has some form of reversibility. It's solvable. And this this is used to to understand our gain insights into the quantum master equation. And basically, all the models that I've talked about in this work have been solved or all the results I've shown are using a kind of inherently quantum symmetry that's hidden in like in the quantum master equation. Right. So this is independent of what we would say classical representations is more direct. Right. And and all these models have this very peculiar cross site correlation function symmetry. Yeah. So so just to conclude. Yeah. So we're using hidden time reversal symmetry to basically enhance our understanding of existing models and quantum optics. There's we've shown that there are some deep connections to like fundamental problems in ammo physics, for example, making a quantum notion of what it means for a master equation to be reversible or to have detailed balance. And yeah. So we're beginning to use these ideas to solve or uncover new many body systems that can be solved. Okay. Thank you, David. So we'll take some questions now. Thanks a lot, David. Very nice talk. I was wondering if you can explain again how this hidden time symmetry works and how the doubling works. Maybe I missed it, but yeah. That might be something we could quickly understand. Yeah. Yeah. It's a I kind of skipped over it. Yeah. Trying to get past the models here. Okay. You know, use my keyboard. Right. Sorry. This is very slow. That's okay. Okay. There we go. Yeah. So basically, yeah. No, it's it it just kept on going. I put too many anyways. Yeah. So so hidden time reversal symmetry is God damn it. Yeah. Yeah. So yeah. So so basically the way it works is instead of so first of all, it's a way to solve for the steady state of your original quantum master equation analytically. And basically the way this method works is you assume that this funny cross-site correlation function symmetry exists. And and and with respect to some time reversal operation T. So we saw that you can define a funny purification of the steady state density matrix which is T dependent. Right. You see just postulate that some T exists. If it exists, you get some basically some some very highly constrained system of equations that basically state that this correlation function have symmetry is enforced. Right. So so you basically what you do is you solve those equations and you get this thermo field double state. That that's how the method works. And then then you just have to trace out the the encella. Right. Does that make sense? It does make sense. I just have to look it up. I think but yeah, thanks a lot. Okay. Cool. Thanks. All right. We have a question. Okay. Let me do. Hey David, thanks a lot for the nice talk. Nicola Rock speaking. I'm an experimentalist. So I'm sorry if my question is completely dumb. It's regarding the your model, what you call the zero D model. So you have the if I understood, you have only a handful of of oscillators. Right. I think you said five or something like that. And I don't know what I remembered from column phase transition is that you need to be in the thermodynamic limit. I have a lot of degrees of freedom. And here you see that with only just, you know, few oscillators, you could see such effects. So just yeah, can you could you comment on that? Right. Yeah, no problem. Yeah. Where's the mini body basically? Yeah. Yeah. So so actually, this is actually a change. So there's actually an ellipsis here. So maybe I should emphasize that this is where the many body is many body phenomena is so they're actually n. If you have n modes in this array, then you have n linear resonators in this chain. So the number of nonlinear elements is staying constant in the thermodynamic limit. So it's easier to fabricate. Right. But you do it is a fundamentally multi mode problem. And all the data we're showing here for like, you know, systems with, you know, 150 modes up to, you know, 16,000 modes. Um, theoretically, but at the experimental level, yeah, it's because there's an ellipsis here. So so this is a chain of of linear modes. Is that is that clarify things? Yes. Thank you. Okay. Hello. Could you go to the bifurcation diagram thing? That's a few slides for. Yeah. So when it goes, you see the wiggles where it's going from the top one to the bottom one. Do you have any idea where those come from? Uh, no, we're just plotting some hyper geometric function. And we just, we were also plotting the quantum mean field solutions. And we just observed that the quantum exact solution sticks to one of them. Right. So, yeah. Yeah. Yeah. I don't have any deeper insight than that. Maybe it's it's hidden in the spec special function that we're plotting, I guess. Okay. Do we have any other questions? Okay. Then let's thank the speaker again.