 So, I would like to start as usual by thanking the organizers for inviting me here. It's a great place to be again, not great weather, unfortunately, this week, but hopefully we'll improve later. I will try to give a brief, the main part of this talk is about a recent experiment we did with the Google USB group improving MBL physics in superconducting qubits. And the idea is to use interacting photons as the main player here. So I will give a brief review of what interacting photons for quantum simulations are, and then I will get into the main part. This is the team, some of them are here, this is Jiravat, was mainly involved in this project on Jamfeng, they are sitting somewhere there, two of my students, and Victor, which is not here, he was also involved. And I would also like to thank the Google guys for the experimental part. Okay, so experimental quantum simulations, I mean state of the art and problems. We had this morning talks about ultra-cold atoms, it's one of the most popular platforms. Beautiful physics have been checked there in the sense of quantum phase transitions, both hybrid models, semi-hybrid models. Then there is the linear optics setups where people have been looking at topological physics and also application both assembling and chemistry. Of course, ion traps for ideal for digital quantum simulation and also quantum magnetism. And superconducting qubits is another platform which I would mainly focus on, like the theory things I'll be talking about. You have a kind of a picture in your head would be in these kind of setups. And ideally, if you look at controllability and scalability, to do not trivial physics you should like to be somewhere here for a fully fledged quantum simulator. Most of the experiments unfortunately go down here, but we have the issues of decoherence and interaction with the environment, but this is something that is especially the last few years has been shown that this might be possible to really do interesting physics somewhere here. So interacting photons. Interacting photons in this sense is maybe a hybrid between normal photons as we know them in linear optics and kind of the microwave version in superconducting qubits where you can get them really to strongly interact in a cavity QED or CQQD sense, as I will explain a little bit later. And if you want to look at it in some sort of a photon number versus interaction stress per photon diagram, this is where linear optics is and this is what we learn at school. Classical linear optics, again it's a big field for many years with many applications in technology as well. Quantum information and quantum science motivated the field of photon-photon quantum nonlinear optics. And the stuff I'll be talking about lying in this area, which is many body quantum nonlinear optics where we have high interaction strengths and large photon numbers as well. So an outlook of this talk, I will talk briefly how to engineer many body photonic states in quantum nonlinear setups for quantum simulation of in and out of equilibrium phases. I will briefly mention the tool we're using in this field, in this trade, it's open quantum systems approach, master equations, many body master equations, lots of numerics, especially in 1D, special network numerics. Then I will briefly mention the James Cummings Hubbard model as is known now, or the James Cummings Lattice model, and which is kind of generated out of this cavity QED arrays. I will mention what fractional quantum whole states of photons and I will mostly talk about this recent NBL stuff, many body localization. I won't have time to talk about slow light setups and quantum nonlinear optics setup based on Liedberg atoms, where you can do beautiful things like liquid photons, steering models, Wigner crystals, people in the audience that have been experts in this field, and I have no time either to talk about integrated optical systems for emulations, I wouldn't call it a quantum simulation of such, but they're interesting works. Okay, so strongly interacting photons, many body quantum nonlinear optics and quantum simulation. So as I mentioned, photons naturally do not interact, so it's kind of counterintuitive how one would think of them as the main player to simulate nature, condensed matter or even high energy physics, because they have no mass, they don't carry charge, and if you try to get two photons to talk to each other in low energies, it's almost impossible basically. So if you use a nonlinear material, then we have what's called the semi-classical regimes, you have some nonlinear effects, but again for that, for this optical care effect that is known in the field to be appreciable, you need to have billions, trillions of photons to work. If you want to get photons to interact in the quantum regime, you need to use a mediator, I don't know what's the mediator, it's an atom, but you need the photon as well to trap it and keep it for a long time in a, you know, localized in space. One way to do that in cavity QED is to use two very good, high reflective mirrors. You stick your photon in here, it bounces back and forth many times, millions of times, every time it goes through the atom, it interacts a little bit. So if the coherent interaction between the atom and the photon is larger than the loss, the times that the photon gets lost from the cavity or gets escaped, you just put the emission outside, then you are in what we call the strong coupling regime. Strong coupling regime is a very fundamental regime to do quantum technologies with light matter systems. It has been achieved in different platforms, the ones I've been talking about here is mainly for conducting qubits. So let's do a little bit more on the maths of this model. So you have the bosonic field of the photon operator in this language, which just describes one photon trapped into this cavity of the two mirrors. It has some frequency, the atom is in here, and you have the atom photon interaction, which is called, this is the James Cummings model. This is a very fundamental model in cavity QED and quantum optics. If you diagonalize this, you see you have the eigen energies dependent on the photon and this plus minus the coupling and the photon number. So as you go up, this is the ground state. This is the uncoupled states which correspond to E0 and G1, empty cavity, an atom excited or one photon at the ground state, and so on as you go up. Because of the interaction, these two states hybridize and you get these two eigen states. These are the real eigen states of the system now. I have ignored losses in this diagram. What losses do they broaden these levels? So if you are not in a strong coupling regime, you don't have this nice splitting. If you try to stick a photon in and kind of excite the system, you see that you have to be resonant with this transition from the ground state to what's called the one minus state. Now the second photon as it goes in, it will find a mismatch to the next available energy. That creates an effective non-linearity, a blockade effect. Like electron blockade, but now this is for photons. That's because of strong coupling. And this is the key behind this interacting photons for many-body simulations and many-body effects. So if you scale this up experimentally, how the systems look like in the microwave regime. This is a 15-site chip, the more recent one by Google. Most of my talk will be on the nine chip. You have 15 sites here, if you can count on each of them, you can model as a kind of a non-linear or cavity change-comics system. Now you can also do the same things in the optical regime, interfacing called atom of nanophotonics, and also in the ribbed, using ribbed atoms. Now, so assuming this is possible, let's try to build a model for this and see what kind of many-body physics we can do. You have some sort of lattice. Each of these lattice, these are the corresponding mirrors. As I said, they trap the light here, the photon, and you put your atomic interactions. Atoms can be superconducting artificial atoms or quantum dots in semiconductor. It could be different implementation. This is general at the moment. Then exploiting the photon blockade, maybe you can build some sort of lattice model or a James Cummings lattice model that has some sort of interesting ground states or many-body correlations. So the simplest Hamiltonian one can write, and this is what we did almost like exactly 10 years ago, was to take the James Cummings-Cabbage-GED type of Hamiltonian and just add a hopping term where photons can hope from one cabbage to the next. Basically, just from here to there and backwards. And you can write it in 2D or one-dimensional as well. Of course, one has to be careful here, because these are lossy systems. So what I say at the first part of the talk, I will ignore the losses. Then I will say, how do you can do driven dissipative simulations here? So by assuming a single two-level system, you write this Hamiltonian in its cavity. And then let's say we look at the ground state of this for a specific feeling fraction. So we put a few photons. Let's say we have 10 sites in one dimension. We put 10 photons. And then we diagonalize and we look at the ground state. And let's say we can change the hopping in this implementation actually. Usually in optical lattice, you change the hopping and the interaction. In our case, it's the detuning between the atom and the cavity. If you can detune, then the photon blockade becomes weaker and you have photons do not obey the photon blockade, where in the other regime, there is photon blockade in each of them. So at the photon blockade regime, if you look at the ground state, you have some sort of more like state, as you would expect, because photons are repelling each other. So if the repulsion is larger than the corresponding hopping, and you look at the fluctuations of what's this is the total number of excitations, atoms, and photons per side as a function of the detuning, you go via what's called something that looks like a phase transition and like a quantum phase transition. So that was kind of a preliminary observation back at 10 years ago, which kind of indicated that maybe we can do many metaphixes with interacting photons. If you go to two dimensions and you do the corresponding mean field theory, then you have the similar structure as in the Bose-Habbath model, where you have more lobes for specific field infractions with the extra ability that you can have the detuning as an extra dimension here. The detuning between the atom and the light, because the excitations here is what's what we're calling in cavity QD, dress states or atom photon excitations. So you can play with the nature of the excitations, be more atomic or more photonic, and have richer structures and interface diagrams out of equilibrium. Now, of course, if you can do that, then you can do spin models. You can simulate interesting generalized spin models, even in 2D in theory. All this is theory at the moment. So all you can do two-dimensional gauge fields by having two-dimensional cavity arrays somehow constructed where you can, as the photons hop around, they can pick up a non-trivial phase and implement a gauge field or a quantum whole state of light. I don't have time to go through the details of this. Perhaps we'll talk about it later. Experimentally, where this field is, so if you take recently, this is two years ago, this is just a unit cell, free of superconducting sites in this case, then by modulating the hopping elements between the photons between these three sites in time, this slightly different than the previous implementation here, which we modulate the laser phases, then you can create chiral states of photons that circulate around basically in this fundamental unit. In principle, it could be scaled up and do more larger site kind of quantum whole physics. And I will end with this first introductory part and say that although there are complementary advantages of using photons to do many body physics at the ground state and closed systems, ideally, this is an open system. By default, it's an open system. So ideally, we would like to look at driven dissipative and out-of-equilibrium phenomena. This is where it's real set would be. And this is where the field is really the last few years where you have drive, you have losses, and you look at out-of-equilibrium states, steady states. And it has been shown that you can get exotic states of exotic phases like photon crystallization or feminization beyond the expected by stability in this kind of systems. Experimentally, the most recent one is this 72 James Cummings Hubbard or cavity arrays model when you have, basically this is one dimension and each of these nodes correspond to a resonator with an artificial atom and transverse in this case. What these guys in Princeton did is they pumped from one side and they observed the dissipative phase transition in this 72 cavities lattice. There are more works in this direction, but I want to focus, as I said, on the NB also. Let me wrap up this part by saying that the richness of the system in out-of-equilibrium simulations is, except that it's naturally open, that you can play with the drives. You can have coherent drive in each of the lattice sites. You can have parametric drive where you pump in between and you have some sort of nonlinear processes that generate pairs of photons. Or you can have a weak, almost quantum or higher semi-classical driving situations. And then eventually what you need to do, you need to solve this massive mini-body master equation, which you have your Bose Hubbard physics here or the James Cummings Hubbard plus the dissipation from the atoms and the dissipation from the cavity. And look at either transit or steady-state physics. As I mentioned earlier, this is not the only way to go. You can have quantum nonlinear mini-body physics for photons in slow light setups and read back atoms. There have been experiments by various groups and also theory by people in the audience in doing interesting mini-body states here. I don't really have time on this, but it's also a nice system to do interesting sort and go-related states of light. So in summary, we have larger left scales of a law for efficient preparation, detection, and single-site addressing. This is something I should highlight compared to optical largescence that the cavities can be further apart. There is no optical wavelength limitation here. The state correlations are measured by using standard quantum optical techniques and driven dissipative nature is inherent here, so out of equilibrium should be doable. So to get into the second part, can we use the systems to do something beyond the ground state, like beyond what? Transitions or the steady-state physics. Can we do, like, the full spectrum? Can we look at mini-body localization, for example? So a brief introduction to MVL. So there is a fundamental assumption in statistical mechanics, quantum statistical mechanics, that all microstates associated with a given macro state have equal probability. And then this big ergodic hypothesis that says that the system, if it's ergodic, it explores all accessible microstates over time and I have a small, like, toy picture here, so you can see that in this case, you have breakdown of ergodicity in this kind of classical trajectory picture. The question is what happens when you really have quantum systems, interacting quantum systems? So the question was, if I have a closed mini-body quantum system, can it thermalize by its own without interacting with the outside? Or it doesn't. And when does it do it? Because quantum mechanics is unitary. If I start from a quantum state and I have unitary evolution, there is no reservoir, there's nothing outside, I will never thermalize. But if I look at the subsystem of the, of the, if I split my system into a smaller system and a bigger part, and I look at the local observables here, the local, the estimatrix, and I trace out, I use this reservoir, then I can, can I describe this as a thermal state? And when this hypothesis breaks down, which is also known as the Eigenstate Thermalization Hypothesis. This is a big question in quantum physics. We are not really quantum physics people in our group, but we wanted to check this from the interacting photos and quantum simulation perspective. And the story is the following. So, yes. So, I'm sorry, say again. How large is the system vision? Well, it can even be one particle actually, one site. So, now, pictorially how this looked like, the linear version of this, the easy, I mean not that easy. It took Anderson to solve it in 96-1, is that if you have non-interacting particles in a disorder lattice, then you can have what's called for large enough disorder and depends the dimension as well. In 3D, you have localized states, all like the states are localized. Now, it seems that this is true also for weak interactions, but where is really the limit when you put interaction? When do you go from a localization to the extended thermalized phase? So, in this kind of picture. Here's the localized phase and this is kind of the thermalized phase. And what are the implications in quantum information because we are talking about quantum many-body system, they're a massive system. So, if the system retains some memory, it doesn't really thermalize and then maybe we have a perfect quantum memory for example. So, but for that, you need to really look at the whole spectrum. So, it's a hard problem, specific for many-body physics. The recent experimental works in direction involved cold atoms, where they prepare some state and they look at the population in ballast. You put the atoms in half of the largest and you see how they spread. There's a line of works in theory and in using numerics trying to probe the problem, but remember this is exact diagonalization. We have to look at all the eigenstates, approximate methods do not work, so it's a hard problem. If you go beyond a few sites, it's a very difficult problem. So, what we tried here with the Google guys is we took the kind of older chip, now there's a 15 chip, this is the state of the art of, like at least publicly, nine sites. Now, instead of doing quantum computing, I mean, these guys look at this as qubits. We say, why don't you look at them as nonlinear harmonic oscillators? Because effectively the qubits are in the hardcore regime of a nonlinear harmonic oscillator. They really have microwave photons here in citations. So, you can write down a nice kind of boss-habo type of Hamiltonian with the frequency, the nonlinear part and the hopping. So, if you modulate the potential in a quasi-periodic sense, then you can actually, so that you can probe NBL physics here. And that's the last part. Before I get into that, I will explain how you do spectroscopy here. How do you really, because ideally we need to look at all the eigenstates, all the eigenspectrum. How do we extract the eigenspectrum here? The way to do it in this closed, very coherent system, we develop a kind of a novel method, kind of novel because it relates to early things in quantum mechanics, but nobody has really done it in this sense before. So, you have a many-body system. You poke it, you see how it vibrates basically. That's how we extract the eigenfrequency, the eigenenergizes. But we need to do this quantum mechanically. So, you start with some initial state, side zero, you let the system evolve and we would like to get these energies out. And also corresponding approaches, how to do that. So, let's look at some examples. So, you have the state it evolves. You want to calculate some sort of observables and follow the time evolution. If you can choose the right initial states, if you don't have for a general initial state, there will be energy differences here. So, if I have this experimentally somehow and I do it for you to transform, I get energy differences. How can I fix this kind of reference and get absolute energies and do my spectroscopy? The key for that is to choose your initial states cleverly. And as an example, this is a toy example, but just to illustrate the method, just two side-tied binding models you have hopping. And these are the operators that you measured. You could measure in an experiment. This is the initial states. If you put this in a cell state, you see the expectation value oscillates with a frequency difference. That's not good. This one is zero. It's not good. If you take this state with the superposition of the first tube bit or a first oscillator, then you see that for this one, again, you get energy difference, but for this one, you get absolute energies. So, if I have this expectation value somehow in my experiment, then I have the spectrum. This is kind of a toy example, but it was the same in the main body case. So, the protocol is the following. You have your lattice. This is the main body spectroscopy kind of part. You start, you put, let's say, one photon in the first side. You let it evolve. You do all your measurements. In this case, the operators look like a dagger plus a, this kind of quadrature operators. Then you put it in another side. You do the same. And you calculate as a function of time, you calculate the Fourier spectrum as in here, and you extract the eigeneresis. So, if you do this for this nine-side chip, what you get experimentally, you get this kind of picture of this. The first qubit, this is the ninth qubit, and this is basically where the corresponding resonances are, and this is where, how big the corresponding amplitude is. So, by superposing all this, we get one, two, three, four, five, six, nine, nine eigenenergizes. This is one photon, one particle in nine sides. So, we should get nine energies. That's exactly what we get in the experiment. Now, so it works, basically, and so on. If you put the interactions, again, it's a similar process. Now, to benchmark this, and see how good this spectroscopy method is, we decided in the linear physics, this is before we put the interaction, to try something kind of exciting and maybe non-trivial, which is the Harper model, or the butterfly Hamiltonian. In two-dimensional quantum-hole physics, we know that the corresponding spectrum looks like this famous butterfly as a function of the magnetic field. You can do similar physics by projecting this 2D into one-dimension, it's called the Harper model, where the magnetic field appears in the modulation of the lattice. So, I'll skip the mapping, but it's also the spectrum. So, this is the energy, this is the magnetic field. If you run it for 100 sides, you get the famous butterfly. Now, if you, we have nine sides. So, it's not really, it's a baby butterfly, in a sense, but still it looks like a butterfly. You have the head, this is the theoretical prediction, the tail and the wings. Doing the photon protocol, putting the photons then evolving and doing the spectroscopy, we get this. So, you get the Fourier amplitudes here. So, you get actually pretty close to a butterfly. So, the method works, the spectroscopy method works. If you look at the arrows, this is a mega-heads. I mean, this is slightly higher because of some systematic error in the experiment, but in general, it's pretty close to the theory. So, that's linear physics. To do interactions, really doing the main thing, you take the same Hamiltonian now, this one-dimensional Hamiltonian, we put the photon in the side, we let it evolve, et cetera, et cetera. But now, to do interactions, you need to put two photons or three photons to really access the interacting angle states. And the Hamiltonian we chose, is this obriod remodel with interactions. Why we chose the obriod remodel? Because we know, this is known for one dimension, where the ergodic phase and the localized phase are. So, if the disorder is smaller than 2j, then you have the ergodic, in the other case, you have the mnbl. So, we put now interactions and we checked how this breaks down and where, basically. And how do we do that? Because we have all the eigenstates. In this case, they're around, it's less than 100, so we can resolve them for the nine sides. And then we use, we do this participation ratio and we want to distinguish between, what's called the Galussian orthogonal ensemble, where you have the ergodic phase or the Poisson distribution, which is the localized phase. And this is the experimental result. Again, this is the Hamiltonian. You fix u over j at three and a half, which is reasonably large value. As you go up in the disorder, so this is zero disorder, as you go up, in the zero disorder, you have the embryo under the model and you have this, and you see how the GOE looks like. And as you increase, you go to Toro's Poisson distribution, which is shown here. Now, if you make cuts here, just to make this, remember for the linear case, delta by j2 is was there, the transition, but this is the interacting case. So, if you make a cut, then you get that these are the corresponding, the dotted lines are how they come from the Poisson and the GOE, and this is the experimental values on top. So, as we change between delta less than one to five over j, you go from a GOE to a Poisson distribution. And I just have like two minutes. That's not enough to check, I mean, the signature is there for the MBL, but we should actually look at the extent of the states, as well, because MBL says that the states, the eigen-energy in space, for example, if I take one and just pick any of the eigen-energy of the spectrum and I look how spread they are in the lattice site, I can calculate what's called the participation ratio for this, so PR space. The same thing I'm gonna do for the energy, which tells me, sorry, how many energy states are in each lattice site, and this is the reverse. One eigen-energy, how many lattice sites it goes. So, extracting again the eigen-energies from the spectroscopy, we did this and we saw the following. That, on this kind of the main result, that as you increase the disorder, you go from, this is the Obreon de Valle, and similar here, now here, we have the PR energy and this is the PR space. You have a kind of a two-photomobility edge as you increase the delta, first the edges go into the next phase, where the middle follows later on. And, okay, and the last slide is how efficient this method is in terms of really doing many body physics. As you increase the lattice sites, if you really want to extract the spectrum, at some point, the eigen-stands will get too close compared to your resolution, basically. And what's the resolution here? It's the coherence of the system. So, if you really have the thermodynamic limit, of course, everything fails, but the experiments are not really done in the thermodynamic limit. So, what we've done here, I mean, for 18 sites, which is twice as big, and keeping the same coherence time that the guys have in the Google chip, we've plotted the percentage of missing levels as a function of U and J, and we see that, so you get the maximum like 30% around here, but this is not too bad if you see how far the distribution that you get compared to the ideal distribution is. So, even at this kind of quite larger sites, which you have actually many, many levels that you miss, it will still work because at the end of statistics, what you're looking for. And I'm finished here. So, I hope I kind of gave you a taste of what one can do with interacting photons. You can look at the ground state physics. You can look at the full spectrum of many body models. You can do quantum physics. I didn't talk about this much. And I would like in the future to look more in the interface between topology interaction and disorder. Maybe look at, we're already looking at this, at flocade driving and independent stuff as we had in the morning as well by Dieter, and also hopefully try to get to test these proposals we have on the chips. And also we continue in this integrated chips and slow life stuff. So, I would like to thank the group again. And then we have openings as well. If anybody is interested, please write to me. Thank you.