Yeah, so there are people are rendering now. Yeah. Oh, wow. I see. You guys had a waiting. You are online. Okay. So we just wait. Okay. So it's nine. Good. Good morning. Good afternoon. Good night. Everybody and I just give the floor to Anna, which is quite late. So it's a pleasure to have this morning. This new session in this time. Crystal. So we're first speaking this morning. It's Norman now from the University of Berkeley, which is going to talk about periodically driven classical time. Crystal. Please go ahead and thank you. Wonderful. Yeah, I'd like to start by echoing the other speakers and thinking the organizers for putting together this super fun and an interesting program. So one thing you might have noticed is that my, my talk title has changed a little bit since, or quite a bit since, since it was posted online. And the reason for that is at the moment it's midnight in Berkeley or just past midnight in California. But I've spent quite a bit of today catching up on talks. I haven't finished watching all of them. I have to admit, but I've seen quite a few talks. And I have a sneaky suspicion that this particular topic. Of classical time crystals, maybe we'll generate more discussions. So I decided to pivot towards it. There are two parts in the talk broadly speaking. The first part is based on some work that's already published with Chaitanya. Leon Ballant. And the second part is some ongoing. It's been ongoing for a long time. So we hope to finish it soon. Work together with my student, Francisco Machado, my postdoc, Trin Taozhong, and Mike Zalotel. So I think we've been hearing lots of talks where the settings been sort of made clear, but just to make sure that everyone's on the same page. I wanted to sort of go through a little bit of my perspective and just give, you know, kind of specificity on what the question I'm going to ask is. Mainly the talk will be about classical systems, but I wanted to start with the quantum setting just to make sure we're all in the same page. And the setting specifically is periodically driven systems. Where we ask the question whether or not there can exist. A sub harmonic response. So here, this is just some operating response. So this is just some operating response. A sub harmonic response. So here this is just some operator, some position, some spin, for example, which is we're looking at it stroboscopically as a function of the driving period. We're specifically assuming that it's period doubled. So it'd be a new equals one half types of harmonic response. And the question always to keep in mind for this talk is basically what is tau. So in the sense of long range order, if we say that there's kind of temporal long range order and that the sub harmonic response has this analogy to what we'd like to think about with respect to kind of symmetry breaking and long range order, we're asking whether or not the auto correlation time for this particular type of sub harmonic response can be infinite or not. And the challenge in either the classical or quantum situation for a many body strongly interacting periodically driven system that kind of fights against this infinite auto correlation time, of course, is that one expects at late times that the system kind of slowly will absorb energy via interactions with the drive. And this goes typically these days under the moniker flow K heating. There are a number of strategies. I'm not going to be exhaustive. There's been lots of talks. So I've heard I've learned new stuff as well. But one strategy in the quantum setting goes under the name of many body localization in MVL systems. The idea is super simple that when uses disorder to induce localization would prevent the system from heating up. And it's sort of believed that in this setting of a time crystal so many body localized time crystal that indeed the auto correlation time goes to infinity. Of course, there are other restrictions. This ties the stability of the time crystal to the existence of many body localization, which at the moment, we think is true in 1D. It's not super clear whether it's true in dimensions greater than one. There's another strategy which goes into the name of pre thermalization. And here the basic idea is also very simple where one uses control of the driving frequency to make the absorption this flow K heating rate extremely slowly. So in this case, one doesn't have kind of a true maybe quotation marks time crystal in the sense that the auto correlation time is exponential in the drive frequency. So it's finite in the thermodynamic limit. But I would say kind of for all intensive purposes from an experimental perspective, it's impossible to tell the difference between a pre thermal and many body localized time crystal just by looking at this tau. There are also restrictions for pre thermal time crystals. One of the most well known ones of course is that it requires one to have finite temperature symmetry breaking and at least for short range interacting systems or local interactions. This means that the physics of this pre thermal time crystal is restricted to dimensions greater than one actually. So this was going to be the majority of my talk that I had planned in the abstract. And these is the theory in the experimental paper I was planning on talking about before deciding to pivot earlier this evening. Of course, the last and maybe most obvious strategy at some level is to use dissipation in the context specifically of quantum systems. This means a Hamiltonian plus some sort of Limbladian dynamics to allow for explicit cooling. But when it really thinks about this as coupling the quantum system to a path, then there's always the question of what happens, you know, how what's the interplay between the noise and the sub harmonic order that one expects in the time crystal. So the talk today is going to focus on it's all it's going to do is we're going to change the word quantum to classical. And these three particular strategies of these three particular strategies, we're going to ask a question about one of them, but we can emphasize that at least as far as we know, many body localization kind of doesn't have a strict classical analogy. Pre thermalization actually does have a kind of very nice and sort of precise strict classical analogy. And there's some really, really pretty papers from Mori looking at classical pre thermalization. But the specific context that I'm going to focus on today is again, dissipative classical systems. And I'm pretty specific by that. I mean that really we're going to think about a Hamiltonian system plus long given dynamics where in this case, you're really thinking about it again as finite temperature. So you do have dissipation in the long given dynamics, but that via fluctuation dissipation comes with noise. And the question we'd like to ask is, can one get a time crystal with an infinite auto correlation time in the setting of a classical system governed by Hamiltonian plus long given dynamics. Let me again emphasize, you know, these three strategies exist. They're not the only strategies. I mean, Angelo's talk was super interesting on how one could think about in classical systems as well. Good. So just as a little roadmap to be super specific, there's two parts of the talk. The first part is just going to be kind of trying the dumbest thing you could imagine. So let's take a look at a generic nonlinear classical one D system undergoing Hamiltonian and long given dynamics and see what one finds. And part two is we're going to be a little bit more clever about the Hamiltonian. So what's the setting for part one? It is, I think in some sense, the most natural and simple system one can consider. So we consider a one D array of parametrically periodically driven coupled nonlinear oscillators where there's nearest neighbor coupling between the oscillators, I and J. There's a drive at frequency omega D with amplitude delta. And the nonlinearity of the oscillator comes from the fact that it's a cost QI. Not just a Q squared. That's the Hamiltonian. But our goal is to understand if period doubly exists, the stability of period doubling relative to this periodic drive under long given dynamics. And in long given dynamics, I'm sure everyone's super familiar, but let me just be specific again. There are two terms in the long given dynamics in the sort of long given force. One of them is given by, you know, one of them is given by, excuse me, one of them is given by dissipation coming from a friction like term. So that strength is given by Ada. And of course it couples to Q dot the position, the derivative of position. And the second is kind of a stochastic noise. C of T, which you can really think about as being related to dissipation via fluctuation dissipation with a coefficient that's precisely the temperature of the system. So before even getting into the one D system, let me just remind us what the lore is. There is a long, long history on this problem. So if one imagines, for example, a single parametrically driven nonlinear oscillator, a single parametrically driven pendulum, this is the kind of zero dimensional starting point at T equals zero, there exists very well known sub harmonic responses. So these are Arnold tongues, not exactly the same thing, but a nice picture here. And the statement is that when one has a force pendulum at T equals zero, one can get oscillations at a fraction of the driving frequency. Here are some super simple numerics where the drive is in green and the pendulum response, for example, is in blue and it sort of doubles the period of the drive. If you just look at the position of the pendulum, then it goes from up to down, up to down, up to down as a function of time. And in this stroboscopic view that kind of looks like this, this strike pattern, but I'd like us to always, at least for the remainder of this talk, take sort of the stroboscopic even view where one only looks for the response at every other period. So when one sees sort of solid splotches of blue or red, that's kind of consistent with period doubling. And when one sees, for example, something that looks more white, that actually is going to be the absence of period doubling. Just to be clear how we're getting to this shade over here, if you start in the period-doubled phase for kind of a single, the parametrically driven pendulum for a single pendulum, and at t equals zero, there exist very, very sharp boundaries between regimes, for example, of period doubling and regimes where there's no subharmonic response. In our simulations, the main tuning parameter for us is super simple. It's the ratio of the damping, which was eta, to the driving amplitude of the periodic drive, which was delta. And we take that ratio to be lambda, and if one increases lambda from the previous plot where we once saw period doubling, then one naturally can see that at large lambda, the period is no longer locked to any period of the drive, and we can see, for example, that from this troposcopic viewpoint that there is no longer kind of this nice oscillation for the pendulum. So you've sort of melted away any of the harmonic responses from this. Again, all super well-known stories at t equals zero, the transition as a function of lambda at t equals zero between period doubling and this non- period-double behavior is sharp. But, as you might expect, at any finite temperature, these sharp boundaries between, for example, period-doubled and non- period-double behavior have to blur into a crossover. This sort of thermal noise leads to phase boundaries and this agrees with kind of very, very standard stat mech lore that in a zero-d system, one can't get a sharp phase transition at finite temperature in a zero-d system. Just to be super clear, the diagnostic of period-doubled that we're going to use is something that's used in the literature very often called the velocity parameter. The velocity parameter is just the fact that the derivative of the position knows about the resonant response of the system and if that resonant response of the system is at omega-d over 2, so when omega-q equals omega-d over 2, v is equal to zero and that means we're in the period-doubled phase and if v is finite that means we're no longer in the period-doubled phase. And doing numerics on a single pendulum using a simple Langevin stepper and varying lambda, you basically see that indeed, as you change lambda, v goes from kind of double to non-period-doubled behavior but it's clearly happening via crossover and this is consistent with the fact that there again is no sharp phase transition allowed for a single pendulum at finite temperature. So we can ask the question, what happens if we now go to the system of interest where we actually now think about a 1-d system of coupled pendulum? Again, still very classical, very simple physics. At t equals zero, it turns out the answer is still known and one has kind of similar behavior to that of the single pendulum case. I feel like in the literature it's more often called sub-harmonic entrainment in this sort of one-dimensional coupled case. But again, at zero temperature one can get, for example, a sub-harmonic period-doubled response or period-tripled response and there is a sharp phase transition between the kind of period-doubled phases and the non-period-doubled phases. At finite temperature it looks like if you do naive simulations that the behavior is kind of similar to the single pendulum case. So here now we're looking at, I don't know, a couple thousand pendula where the x-axis is pendulum sort of one-two indices of the pendulum index and the y-axis is time moving downwards and again at low values of lambda are tuning parameter at low values of the friction divided by the drive amplitude when sees period-doubled behavior at high values of lambda when sees not period-doubled behavior the melting of this period-doubled behavior and it looks like in intermediate values you get domain walls but maybe the crossover is kind of smooth. So the physics question that one can immediately ask is is there anything sharp for a 1D system of coupled non-linear pendula at finite temperature now like the 0D case is it just a crossover? Again in equilibrium a sharp phase transition is strictly forbidden in one dimension but we're not in equilibrium so maybe the situation is a little bit more rich and that's what I think I'll hope to convince you of. So it turns out one can again do the exact sort of same numerics. At high temperatures we're looking at the velocity order parameter average over the whole many body system as a function of this tuning parameter lambda. Again, at small lambda you see something that's consistent with period-doubling and at high lambda it looks like a pretty smooth crossover to the melting of the non-pure-doubled phase but something kind of interesting happens at low temperature in particular when one goes to low temperature one finds that this velocity order parameter when one changes lambda it looks like it sort of jumps on the way up and as a hysteresis loop on the way down and as you slow down the rate at which you change lambda as you increase the number of steps with which you sort of basically the number of steps of the launch bin stepper you find that this hysteresis loop gets smaller and smaller and smaller much like you'd kind of expect for a first order phase transition and as sufficiently sort of slow variation of lambda you basically find what looks at like a pretty sharp phase boundary emerging at low temperature so at any finite temperature in 0d it's always a crossover in 1d you might expect from equilibrium stat mech intuition that it's also always a crossover but it seems like we see a distinction between the physics at high temperature and low temperature so the question we can ask very simply is is there indeed kind of a sharp phase boundary emerging at low temperatures if we now if we now look at a different diagnostic the sort of nicest diagnostic I would say is to set up a very very simple domain wall and watch the subsequent dynamics so what we do here is we literally set up initial condition which is kind of trivial no period doubling on one side and period doubling on the other side and then we set the parameter lambda to be somewhere in the phase diagram and we ask what the subsequent dynamics look like and you can see when you're in the non-period doubled phase that eventually the sort of you know trivial phase sort of eats the time crystal and you of course sort of you know ultimately get to a situation where the full system is it lacks period doubling but if one now literally zooms in to a very very small splotch land is varying from 0.195 to about 0.215 so very very close to the phase transition you see that if lambda is just a little bit smaller than what we say is the critical point that when you start with these two regimes in fact the time crystal wins when you start just a little bit above then you see that the trivial phase wins and if you're right at the phase transition for many many millions of cycles of the launch of an evolution you basically see competition between a period double between the sort of domain wall for the period doubled and the non-period double phase so if you zoom in just a little bit further as a function of time it really does look like there is a sharp phase boundary emerging as a function of lambda and you can identify sort of this phase transition between period double to non-period double physics at some critical point in lambda so this kind of ends sort of you know what I think is this sort of a very sort of nice simple foray into the story of couple nonlinear oscillators where we think that the phase diagram computed numerically for this type of a system is that at high temperatures there is it's kind of like water in the PT phase diagram of water in some sense at high temperatures there is no phase transition you cannot walk as a function of lambda across a phase transition between the time crystal and the non-time crystal phase or the period doubled and the non-period double phase but at sufficiently low temperature and as you increase lambda you can in fact walk through a first order phase transition between the period double phase and the non-period double phase we believe that there is a line of first order phase transitions that basically ends at a critical point so you'll see actually the title of this slide at the moment still says time crystal ish and I should really emphasize that this ish refers to an answer to the question that I posed in the very first slide which is is the autocorrelation function of the time crystal infinite or not or is the time crystal only stable for maybe a long but sort of finite time so there's a phase transition here not because the lifetime of the time crystal goes from infinity to some finite number but because it goes from something that has exponential dependence to something again that's order one so it's this analytic dependence of tau on temperature that kind of is undergoing this first order phase transition across the transition and again let me emphasize what this means is that we don't really have an infinite time an infinite autocorrelation time time crystal in the sense that there's no kind of true long range order this tau this autocorrelation time is indeed activated so it is exponentially large in temperature with some delta you can think about it as some barrier that kind of depends on this lambda but again it doesn't have this infinite autocorrelation time so I would say that for part one the take-home message is pretty simple although the model we've been looking at has I think a very very interesting phase diagram and an interesting interaction driven sort of intrinsically non-equilibrium phase transition remember in equilibrium you're not allowed to have such a phase transition you don't have an infinite autocorrelation time so you don't really get a time crystal but in some sense we started out with a Hamiltonian that was the simplest possible Hamiltonian you could imagine coupled non-linear oscillators so we can ask a question if we're allowed to vary the Hamiltonian from part one can we be clever and actually realize a time crystal in this classical Hamiltonian plus Langevin situation that has an infinite autocorrelation time so the setting is still Hamiltonian plus Langevin but now we're going to get a little bit more clever with how we design the Hamiltonian and in particular that cleverness is going to borrow some inspiration from the land of cellular automata so I think everyone's familiar with pictures of cellular automata but I just want to give a very very intuitive connection between why it's kind of natural to think about the connection to cellular automata and the connection super simple we generally think of Langevin dynamics or kind of the Fokker Planck equation before that as some example of a continuous time Markov process where you have some probability distribution for Q and P position of momenta and the time dynamics of that are given by the Langevin dynamics the analogous construct for a cellular automata is kind of the discrete time version of the same of a very similar Markov process where again you have some probability distribution of sigmas and they undergo some transition matrix M to a new probability distribution at some time t plus one in a discrete fashion of course there's a small difference something I've been emphasizing throughout the talk but one important thing to consider from Langevin dynamics is that you naturally have errors that you should always think about there naturally being errors from the fluctuations intrinsic to finite temperature intrinsic to finite temperature noise so really the slightly more proper analogy is really to think about the connection between Langevin dynamics and cellular automata not being just a cellular automata but really a PCA or the literature calls a PCA which is a probabilistic cellular automata so the probabilistic cellular automata is the same thing as what we previously had except now the transition rule the transition matrix here is your old cellular automata transition matrix plus for example some random errors and this I think of as kind of the same similar as analogy to the random errors that one naturally gets from Langevin from finite temperature noise in the context of Langevin evolution so the question at this stage before you know using this intuition from a PCA to construct a Hamiltonian I want to first ask a question which is whether or not a PCA itself can support a time crystal in terms of there's also a really nice related work from a Pizzi-Nunen camp in the context of long-range PCAs here we're specifically thinking about locally interacting so nearest neighbor interacting probabilistic cellular automata but the key point is that any time crystal requires ergoticity breaking you have to break ergoticity because you have to remember the sub-harmonic orbit that you stay in in the NBL case you break it because NBL breaks it in the pre-thermal case you break it because you're pre-thermal and you know about symmetry breaking in one of the sectors the remarkable thing is that PCAs so probabilistic cellular automatas are super well known to support ergoticity without fine tuning the classical example of that is something called the two model or the northeast corner rule it says that this blue cellular automata cell it's a 2D cellular automata it says this blue cellular automata cell you know majority votes based on its north and east neighbors to figure out what its next transition rule is and the thing about the cellular automata is that there exists two stable distributions without the need for symmetry so you might say 2D two distributions this seems like a ferromagnet like if I think about like classical stat neck you know I can have a ferromagnet or easing symmetry breaking at finite temperatures in 2D but here it's much much stronger for the PCA construction because you don't need the symmetry you can have biased noise for example you don't need the symmetry but there are still two stable distributions in some sense the all up and the all down state but you don't need to have that easing symmetry to have this type of symmetry to have these two stable distributions just a couple more slides I'm almost done so just to give a sense you know just a super quick video of what happens here it's pretty boring it's not that interesting video you know it's a 2D grid of cellular automata and the two possible states of cellular automata and what I mean by stable distributions is just if I make a little chunk of purple and let the cellular automata evolve then it naturally sort of you know goes to the stable distribution and eats that purple away sort of error corrects in the same way that if you had all down as the ferromagnetic state a finite temperature fluctuation would never take you to all up so in the next setting in this particular setting it means that we can immediately basically a PCA that's also time crystal by simply taking the kind of flip version of the of the two model where you apply the same rule the same PCA rule but followed by a global spin flip so in this situation the dynamics look like a global spin flip so there's still a region it still gets eaten away as a function of time but now in principle just you know kind of trivially from proofs of the properties of this northeast corner rule when automatically that's a PCA time crystal with an infinite autocorrelation time which is robust to noise and local perturbations so the noisy version of the simulation also eats away at this and you have in principle an infinite autocorrelation time in this probabilistic cellular automata model in 2D it turns out I think that's already pretty cool but it turns out there's a kind of very very natural sorry you have two minutes last slide so it turns out that there's a supernatural way to translate PCA's to Hamiltonian simulations and in particular one can translate the PCA just kind of a Hamiltonian plus Langevin simulation there's a sketch of the simulation over here but what I'm showing over here is really numerics that look at a cellular automata with some error which means a probabilistic cellular automata and you can see that as a function of some pinning potential which is a property of the Hamiltonian and the Langevin numerics you indeed see the pinning of the magnetization which is consistent with the time crystal so this yeah I think is sort of a nice numerical demonstration that indeed we can simulate more or less an arbitrary probabilistic cellular automata with the Hamiltonian plus Langevin formulation let me just end here I won't go into further slides but let me just say that a very interesting question is whether or not the same thing holds in 1D can you have a Hamiltonian plus Langevin situation in 1D or a PCA in 1D that also is a time crystal in the same way we believe the answer is yes it's related to some beautiful work from Peter Gotch but yeah maybe I won't go into too much detail but we believe that the answer for 1D the specific the same question for 1D can also be answered in the affirmative with that let me just stop here and thank you for your attention okay thank you thank you very much there is a couple of questions sorry because we are a bit hard on time so Jamiya do you want to start with your question yes thank you so much now for the talk hi so for the first part do you have any type of Landau-Gainsborough picture normally this first order phase transition of course because you jump from a local minimum to a global minimum and one way in which you could get sensitivity is a final size scaling analysis because typically the time that you need to tunnel and to get the jump will start to increase exponentially with the height of the barrier that is the system size I'm saying that while you think about it because there was this nice work that was done with a motor model with noise and damping they have long range interactions so it's nothing to do with what you are talking about but they developed that at the end of their paper it's a phase where I don't know if you are aware of it yeah I'm actually not when I wasn't aware of that specific of specific that specific paper but the strategy that you said is super logical and actually we sort of it turns out you can kind of map the problem that we were looking at in the first part of the DC-driven version of a friction question of the Frankel-Kontorova model so I tried to do this kind of a land out Ginsburg analysis in this well a related analysis in the context of this DC-driven Frankel-Kontorova model but we weren't really able to make too much progress because it's kind of an intrinsically non-equilibrium phase transition so it sort of you know at least for us it sort of a little bit more that we kind of needed to use sort of non-equilibrium tools to even see the transition in a 1D system I think in the all-to-all coupled system you have this mean field limit so it's much more well controlled but in 1D because the transition is intrinsically non-equilibrium we couldn't quite use the same techniques as far as I understood thank you hi there another short yeah okay sorry because there is a lot of questions but we will not have time for all of them go ahead please okay in the formation of quantum time crystals we know that the central phenomenon is the spontaneous breaking of the time translation or symmetry of course we know that in the classical mechanics consequences of translation or any symmetry is very different because for example there is not a problem to have a single particle which reveals period doubling so do you agree that this is of course this many body synchronization in the classical system is very interesting out of question but do you agree that this is hard to say that in classical system we have a spontaneous breaking of the symmetry I literally can't agree more I mean I think that's exactly right so maybe let me just sharpen that a tiny bit let me give my own perspective on that in the quantum case we are used to really having a periodic system and thinking about this time translation discrete if the flow case system has a period that kind of varies you no longer have the physics the physics isn't stable to that but for example for this PCA or for the model we are starting out there is no notion if I had a little bit of a variation in the period as a function of time it's really that's right in the classical setting I think it's not really the kind of breaking of time translation symmetry but it's really related I think more globally to this question of whether or not you have ergodicity breaking or not but I totally agree okay let's take another couple of questions it's very interesting so is there please can you go ahead yes my question is about the symmetry there is no symmetry so hamiltonian is basically dependent on symmetry how could you say that it doesn't depend on symmetry I have to admit I didn't hear it super well but I can try to answer what I think I heard which was that the question was kind of if there's no symmetries in the hamiltonian how do you think about time crystals if you don't have a symmetry I think the point is just in some sense there's no sense in which the time evolution has to be truly periodic or like imagine that you have a period and you're more or less almost always the period is time t but sometimes the period of the evolution is t plus delta t sometimes t minus delta t so strictly there's no time translation symmetry and you can't break that symmetry but I think in the classical setting at least for either the PCA scenario or the long given dynamics of the nonlinear oscillators all you need is basically ergodicity so I think that's sort of what I mean okay, can I ask a quick question a simple question yeah so I have a question related to your first part I'm thinking that have you kind of simplified the model to explain your results for example maybe each nonlinear oscillator has by stability has some kind of by stable states then you can represent each nonlinear oscillator spin up and spin down then the time crystal states may be correspond to some kind of very magnetic or anti-ferromagnetic kind of state do you have that kind of simplified model? that's exactly right I think you have this by stability so I think maybe it's like figure two in our paper the fact that you have this by stability is the origin of this behavior but I think the question isn't that so certainly this by stability for a single parametrically driven oscillator underlies the time crystal but kind of what we were more interested in or at least what was a little bit more exciting that definitely is the underlying physics that by stability but the question is whether or not and you go to a finite temperature can you do a phase transition between this sort of by stable kind of time crystal and the non-time crystal phase because in equilibrium you would expect the answers no and I think in this case you basically see that the answer is yes and it's because you have a non-equilibrium system so you're not really governed by the same stat neck loss okay thank you thank you very much sorry because there are other questions but we don't have time so please I ask you to write it into the chat to Norman and we are going to enter Bedika Kemani please for this second talk so thank you Norman and it's my pleasure to introduce now Bedika Kemani who is going to talk if she has not changed the title which maybe she has many body is existing the NISC Arian all times okay can you can you see my slide yes you'll have to make it presentation oh I see you'll see part of it just one second yeah just one sec it's the time to study it's double you know yeah okay okay okay how about now is that good okay great yeah I didn't fully change my title I only dropped a part of it to make it less of a mouthful but the second clause is the same yeah thanks very much for inviting me today I'll be telling you about a recent proposal that we put out to realize the time crystal on Google's digital quantum simulator this is their so-called Sycamore chip that many of you may have heard about last year in the context of quantum computational supremacy or advantage but before diving in you know I really want to thank the organizers for putting together this really nice workshop while this beautiful physics of macroscopic oscillations and sub-hormonic responses has a long and storied history it's been very interesting to me to see the rich variety of new and different physical settings in which this phenomenon can show up and you know much of our modern world would be nowhere without the physics of macroscopic clocks but and I've really enjoyed hearing about how many interesting questions still remain to be explored in understanding these really fundamental phenomena so you know now my background is in condensed matter theory so when I say time crystals you know I specifically mean that I'm referring to many body time crystals right and in this workshop this context this question of you know how strict we should be in defining time crystals and you know which phenomena would correspondingly be included has has come up a few times during various talks so what I want to do is I want to start by clearly defining what a time crystal means to a many body physicist right and then I'll give you a brief summary of the experimental situation on time crystals and and motivate why we even need another time crystal experiment right because you might think that okay we already have all of these different experiments you know we had the different experiments from now three years ago so why is this proposal on on Sycamore's chip you know timely or relevant right and then I'll finally get into the details of the proposal itself and describe that okay and I want to just advertise that you know late in 2019 we put together a very detailed review where we discuss a lot of these questions especially you know how we should think about many body time crystals what are various like adjacent phenomena like that are similar in spirit to time crystals the experimental situation so so I won't have time to go into this in much detail in the talk but I but I urge you to look at this review if you're interested okay so let me let me start with this question of what what many body quantum time crystals are right and the very first thing as the name might suggest is that you know when I'm talking about a many body time crystal I really have in mind a system with many many couple degrees of freedom so this is not this is not a problem and that you can describe in terms of one mode or two coupled modes not just one or two differential equations but really coupled degrees of freedom all of which are strongly interacting with each other and all of which constitute the system right so in fact you know when I first arrived at Harvard in 2016 as a postdoc Bert Halperin asked me a question which was that you know we have macroscopic oscillations all the time indeed imagine that you had a macroscopic pendulum you know this is an actual pendulum clock or pendulum bob this is this has 10 to the 23 atoms you would have in your body for all practical purposes and now imagine sticking this pendulum in your favorite vacuum Belgium and making it frictionless and all of that stuff and then you would say that this pendulum actually should oscillate forever right so is that should we call that a many body time crystal and this was of course you know a key question and it deserves a careful answer and the point here is really should we think of this macroscopic pendulum from real material should we think of it as a single mode right the center of mass mode should we describe it as one mode in which case if we stick it in this frictionless bell jar indeed it will oscillate forever or should we think of this pendulum as actually having all of its numerous internal modes which will be present in any in any real material right and indeed if you describe the pendulum in terms of this multi mode coupled system of which the mode then by the second law of thermodynamics if we wait long enough then the energy will distribute itself from the center of mass mode onto all the other numerous internal modes and the pendulum will eventually come to rest right so now depending on the specific material of this pendulum and the stiffness of all the other modes this transfer of energy could take a very very long time this is the business of pre-thermal time crystals you know does this last forever or does this just last a long time but but as a matter of principle if you view this as a genuinely many body coupled object then this this will eventually come to rest okay so that's how the difference from a single mode describing the center of mass and a genuinely many body coupled problem appears in this in this simple example okay the second condition that I'll that I'll impose for the purpose of this talk is that my systems are non-dissipative okay so indeed as we saw in the last talk as we saw in Jameer's talk there have been many many beautiful examples of driven dissipative systems or just dissipative systems where you can get very interesting phenomena I just want to make a box about around you know what is what is strictly quantum what's strictly conservative and I want to make a clear separation between the types of quantum time crystals that we'll talk about and kind of say battery operated clocks right which are of course ubiquitous right so just to make this very clean separation our systems will be non-dissipative and specifically for a for a driven system like a periodically driven system like the ones we speak about this means that this driven system should not actually keep absorbing energy from the drive right there should be some mechanism to prevent this system from absorbing energy okay and then of course the defining property of time crystals is that this is a many body phase of matter which has long-range spatial temporal order right so I want to define this as a crisp asymptotic phase of matter I want to look in the limit of infinite sizes and infinite times and in these limits what spatial temporal order refers to is that you know if you measure a spatial correlation function arbitrarily far away the system shows long-range order and if you measure a temporal correlation function arbitrarily late in time this autocorrelator will show you periodic dependence okay so if you have a discrete time crystal then it shows this kind of period doubling period tripling whatever you want but in general it would have to show you some kind of periodic dependence at arbitrarily late times okay so really we want a many body system which is non-dissipative and has long-range spatial temporal order this is what I'm going to mean by a many body quantum time crystal and as we know you know using these definitions it was shown by various people and perhaps most crisply by Ushikawa and Watanabe that within the context of thermal equilibrium or ground states of many body systems you can't have these types of many body quantum time crystals and indeed this is what led to much of the work and much of the excitement on flow K or discrete time crystals which strictly lie outside the purview of these which are many body systems which lie which are strictly outside the purview of the systems that Ushikawa and Watanabe and Bruno, Nozie and others were thinking about right so for a flow K or discrete time crystal again I have in mind a many mode many body system here this Hamiltonian has a periodic dependence in time the observables in my system are going to show you some kind of you know they're going to show a sub harmonic response so the system has a period T observable to respond at multiples of that period the most common example is period doubling but you know in general you can have any multiples of the driving period or you can have time quasi crystals where you can get kind of non commensurate responses you can have time glasses there's multiple ways to break time translation symmetry right but we'll stick to time crystals okay to define a phase of matter we need the response to be robust and indeed this is this is where many body localization comes in because if you have a many body periodically driven system again just by the laws of thermodynamics you expect the system to actually heat up to infinite temperature in which case the system is non conservative and all correlation functions just look trivial okay so this so many body localization is is necessary to to prevent this heating to infinite temperature and then we again look for this the spatial temporal order okay so the the most familiar model where where this this shows up is you know the model we wrote down several years ago by now which is this kind of kicked icing chain right so we have some icing type interaction and then we have a rotation about the x-axis by tuning these parameters you know if you have only nearest neighbor icing interactions and these this rotation you can actually work out the phase diagram completely you get a number of different phases off which you know the time crystal is is one phase it's easiest to understand it in this in this limit of this perfect pie rotation so indeed if we fix this this rotation parameter to exactly enact a pie over two rotation then we have some diagonal z icing interactions and a perfect perfect flipping of the spins so then you know in this this trivially gives you period doubling right because if you start with some pattern of sigma z spins then you know one period later that pattern flips and then it flips back and so on okay so along this particular line which is the which is the pie over two the perfect pie rotation you know you trivially get period doubling and you also actually get glassy long range spatial order okay so if you if you go in and you look at the way to very crisply define this is to look at the eigen states of the floquet unitary operator and to measure long range correlation functions in all the different many body eigen states of this operator and then you'll find that arbitrarily long range correlation functions will will give you a non zero answer but that this correlation will be glossy in general it will have random sign okay so so fine you know this is this is well and good but of course the the interesting thing about this phase is that it's not just one fine-tuned point where I'm literally going and flipping all my spins but rather this this forms this robust extended phase of matter right so so I can I can perturb away from that point in various ways I can enact an imperfect rotation so I can I can rotate not by perfect pie but by pie minus some angle that's that's shown by this axis here and the the business of having a phase of matter is that even with those imperfections in the angle my response should be locked at period doubling and more generally you know I don't have to have only type interactions I can add in all sorts of longitudinal fields x y interactions all of these different things and and this phase persists as a robust phase of matter okay I should say that the really special thing about this formalism is that this time crystal phase is a paradigmatic example of a new type of eigenstate order based non-equilibrium phase of matter okay so this is a new entry in in quantum statistical mechanics and how we think about phases and phase structure there was once a poll that that that one of my collaborators giving this talk did you know he asked like an audience of experts in time crystals you know how many of them had heard the word eigenstate order a very very vanish very small subset and and this is really the key idea that allows us to crisply define this as a many body phase of matter you know one of the contributions of Oshikawa and Watanabe work was that they were able to very crisply give a definition of time crystals and then show that that does not exist in equilibrium so this eigenstate order where you go and look at individual eigenstates of this many body unitary and define long-range function functions in those gives you a similarly crisp way to define phases out of equilibrium and has crisp predictions for what the dynamical consequences of those of that eigenstate order will be so so good so then based on this kind of proposal to have the this this drive with dominantly ising interactions and imperfect rotations you know this is something that an experiment can easily do you can prepare some initial states see how it evolves under this type of drive and indeed there are kind of three three experiments that there've been many experiments where I picked the three that are kind of closest that you closest to this kind of proposal right and and you know these the two experiments that are most familiar are the ones in disorder nitrogen vacancy centers in diamond and and trapped ions in one dimension these are the 2017 papers but actually there was a one there's a different example just later on a on a spatially ordered no disorder no many body localization NMR crystal in in three dimensions and all three experiments followed a similar protocol despite having very different setups and all observe virtually indistinguishable signals okay so okay so if you had a super visual view of what was going on you would have said that you know actually every you know this this problem was this the winner of the problem was solved in 2017 you know we we had a fancy nature cover everything you know there's nothing left to do here right but actually what we've done in the in the years from 2017 till till now is to really go back and and try to understand these experiments in detail and to see you know what is it that they actually saw right because they clearly couldn't have been seeing how you localized time crystal because well one of the experiments on this on this NMR crystal it was it was a spatially ordered perfectly no disorder no nothing order crystal in three dimensions okay that's not a system that that has many body localization so you know here is the the current status of the of the experiment based on what we what we've understood since the original publications you know the diamond experiment already in the title of our paper that that's an experiment I was involved in in misha lukin's lab we already knew that that was not going to be an asymptotic time crystal the the interactions in the problem were too long-range for it to be many body localized so it was what's known as a critical time crystal which just means that the signal kind of decays power loss slowly in time okay so this was part of the title of the original paper it was never intended to be an asymptotic time crystal next experiment on the trap times in 1d that kind of was as close to the theory as possible because it was it was in in one dimension it had disorder but actually turned out that that that model also is an example of a pre-thermal time crystal this is a pre-thermal time crystal of the variant that that else bower and IOCRO done in 2017 and the basic idea of pre-thermalization here is that the system for a very long time looks as though it's governed by some kind of effective Hamiltonian and if that effective Hamiltonian has some kind of symmetry breaking or or approximate symmetry breaking then if you prepare very low temperature states for that effective for that effective Hamiltonian these can show order for a long period of time but if you prepare high temperature states of that effective Hamiltonian then those decay very quickly okay so this is an exact simulation of the exact experimental couplings in the experiment and what we found while writing this review is that you know some initial states these are the sort of these polarized initial states or near polarized initial states that were in the experiment can show you this very very long-lived signal but actually to prepare other initial states those that would be at high temperature then those would actually decay very quickly okay so this experiment having this disorder actually also was a type of pre-thermal time crystal and explain why in in just a moment okay and finally there's this the most puzzling of the experiments was this was this NMR spatially ordered solid and what we realized there is that the reason they were seeing the the signal that they were seeing is because this was a type of pre-thermal U1 time crystal is what we called it and what that means is that the system looked as if it had a quasi-conservation of magnetization so that's the U1 symmetry for a very very long period of time so if you measured a global observable like the magnetization because that's approximately conserved that looks like you know it has this long-lived signal but actually if you measured local spatial temporarily resolved auto-correlators which is what you know the MBL time crystal wants you to measure then you would see your signal decay very very quickly okay so this is all to say that you know if you just do experiments from a few you know from one or two initial states probing only one or two coarse-grained observables you can actually get very different very very similar looking signals even if the underlying physics is very different okay and one of the key questions that we have to sort out is the difference between a pre-thermal and asymptotic time crystal has to do with what happens at infinite temperature right but an actual experiment is never going to have a coherence time which is infinite an actual experiment is always going to be a finite time experiment so you have to wonder like if you only have access to finite times in experiments you know that's what this line is trying to show and the difference between pre-thermal and non-pre-thermal is what happens at very very late times and how do you actually tell those apart in a realistic experiment right and what this figure is showing is that by drilling into the physics of what causes that pre-thermal signal you actually can distinguish them even within the constraints of finite experimental lifetimes so how does this work for an MBL time crystal we know that you know any initial state you start with all the eigenstates of this flow-care unitary are localized all initial states are localized so any initial state that you start with will show you this kind of period for arbitrary late times okay whereas in this pre-thermal time crystal where you're using some type of effective Hamiltonian and low temperature with respect to that effective Hamiltonian some initial states you know which look like low temperature almost polarized initial states or near polarized initial states which is what these experiments consider those can show you a long-lived signal but actually if you started with just some random other initial state you would see even within your experimental lifetime that your signal decays very quickly okay so just by changing the initial states that you prepare and run in your experiment you can you can diagnose an MBL time crystal from a pre-thermal one even within the constraints of a finite experimental lifetime and likewise if you consider this pre-thermal U1 time crystal which is this time crystal that was there in the NMR experiment the reason that they saw that is because they were measuring a global Z auto-correlated they were measuring the polarization for the entire sample because that's what you can do in NMR it wasn't easy for them to have spatiotemporally resolved correlators on the other hand and that was showing this because that global polarization was coupling to this global conservation law but if they had measured site resolved spatiotemporal correlators they would have actually also seen a rapidly decaying signal okay so what these experiments did you know they're fascinating experiments and they're engineering feats and what they've done in us trying to really understand the physics of each of these experiments is they've helped us sharpen exactly what experimental capabilities we need to realize a bonafide asymptotic many body MBL time crystal okay so it's not just like we want a system that can realize it but we want it to also have capabilities that we can say that we want to realize it right so for instance we need the system to be large enough and generic enough to actually program any body physics we need the system to be to have a long enough coherence time so that we actually look at late time observables it can't be infinite but you know we needed to be at least long enough right and then you know from what I was saying right now to distinguish this time crystal from the pre thermal variants we need to have site resolved observable rather than just global ones we need to have the ability to tune and prepare different initial states and then finally for stabilizing MBL we need our interactions to be short-ranged enough you know these these diamond experiments and the NMR experiments had dipolar interactions in 3D which are just too long range to get to get MBL of any kind and finally this was the main ingredient that was absent in the trapdion setup is that we need disorder for this for this kicked icing type so we need disorder to be present in the icing even couplings if we only have disordered fields then what happens is that over 2 periods this disorder in the field can get echoed out so the system effectively looks very very weakly disordered and then that's not many body localized so this was just you know these are the different ingredients and then we can go through and look at the 3 different experiments and we find that actually like you know with present capabilities none of these experiments have kind of all the different features that you would like okay I should say that with the trapdion setup you know trapdions are a leading platform for building these digital universal simulators so you can try to get all of these different features but this requirement of icing even disorder it actually requires if you have these different couplings to get disorders in the couplings you need lasers and it from an engineering point of view it's certainly doable and there's an active effort to do it but it's challenging okay it's not easy to do so the main you know so this was the motivation for us to look at this when we read about Google's quantum supremacy experiment you know and we had sharpened this list of experimental desired experimental capabilities what we actually found is that Google's sycamore chip you know it's a super conducting based simulator it's a 2D array okay but you can actually create an effectively 1D system out of it this system you know you have all these different qubits and you can act with different gates to couple neighboring qubits you have a gate set that they're using and from the existing near-term capabilities of the sycamore chip what we found is that it actually checks all the boxes and meets all the requirements for you to realize this phase so I should say that this sycamore chip really represents a culmination of decades of or at least a decade or two of effort in pushing the boundary in the development of the designer many body quantum systems and it's not easy to do it but you know we're pushing the number of qubits we're pushing the error rates down and you know while the quest to build a universal quantum computer is still far in the future what this time personal proposal on Google's device shows is that the possibility of actually just doing treating these systems as new experimental platforms for many body physics and using their modern day capabilities not their capabilities of you know this will eventually be a universal quantum computer but the existing gate sets the existing noise levels can already allow us to probe interesting physics which no other platform can allow us to do specifically on Google's sycamore device here is their chip they have a natural gate set these are the so-called transmon gates they have tunable zz couplings tunable xx and yy couplings we can make these zz couplings dominant we can make disorder in the interactions as we wanted then we can have all sorts of on-site rotations so essentially this idea of having dominantly ising interactions followed by imperfect rotations exists in their current capabilities so that's what we did the details are not so important those are there in our paper but the point is that we can now look at a phase diagram based on their the couplings that are present in their experiment and we get a robust time crystal phase and this is a genuine robust time crystal phase which will show you all the different boxes unlike the previous experiments and then this can transition into different regimes and you can use the simulator to study the properties of these phases the properties of the transitions all of this stuff so coming back to our table even with the existing noise rates in these platforms you can look at signals for hundreds of periods we characterize this carefully this is 50 plus qubits so it's not the millions of qubits that present in the NV experiment but it's more than the 10 or so qubits in the trap-tion experiment the experiment has worked very hard to eliminate this kind of long-range crosstalk it naturally looks like nearest neighbor gates so it's short-range interacting you can get the sizing-even disorder you can make site-resolved measurements and prepare all sorts of initial states so these are kind of snapshots of what you might see on this experimental platform you know depending on which initial state you feed in you could actually just look at the state in time and see your time crystal show you this glossy spatial order and oscillations in time you would see your thermal phase quickly thermalize and you would see your MBL paramagnet just preserve local memory without oscillating so I'll conclude there but as a broader outlook I want to say that there's also lots of other questions on what we can do in terms of many-body physics in this era what types of new phenomena we should be exploring one example but actually a lot of different a lot of work in quantum dynamics now considers quantum circuits you know these to explore questions on chaos, on thermalization, on hydrodynamics entanglement dynamics and we're really getting to the point where you know we can go from thinking about condensed matter experiments in near equilibrium settings of time-independent Hamiltonians to these much broader classes of quantum experiments which are natural on these digital simulators so with that let me thank my collaborators and so yeah Thank you very much so time yeah let's have time for a couple of questions so Christoph you go ahead first okay thank you I would like to ask a question concerning many-body and few-body or many-body many degrees of freedom of few-body degrees of freedom in any time crystals what we need we need a kind of integrability either it is local integral of motion in the many-body localization case or any other let's say type of integrability and that also means that effectively the dynamics is always restricted to a few degrees of freedom as compared to the of course to a higher Hilbert space otherwise of course we would have some kind of a godly city and heating and so on do you agree? No I don't actually because if you take this effective L-bit Hamiltonian for your for many-body localized systems you know it still has yet there are these local integrals of motion but it actually still has interactions between all those local integrals of motion so if you start with if you look at entanglement dynamics in the problem starting from some product state what you'll find is that the entanglement grows very slowly so it will grow logarithmically in time but it actually eventually saturates to a volume loss state so yes there is a city breaking in terms of like local memory of the initial condition is preserved but over time the system still explores an extensively large Hilbert space volume law even if it grows very very slowly so having a volume law entanglement entropy which is exploring extensive Hilbert space is compatible with having local memory of initial conditions so these many-body localized systems are actually many-body and this is the main difference between Anderson localization and MBL because if you look at entanglement dynamics for Anderson localized systems the late time states remain an area law because that's the non-interacting limit but once you add in the many-body interactions it goes to a volume law so I would say that they are still actually extensively many-body systems okay thank you for your opinion yeah I have a short question I wanted to ask you also I know that there is now a lot of facts about using NISC because obviously it has a lot of advantages but nevertheless NISC is raising that you do want to have errors because there is no way to have errors you have to correct it okay and then you have your result so for some things I think this is fantastic but for instance in the platform that you were using why is it different from using I would say ultra-cold atoms in which you don't have to correct errors you have a many-body system and it's highly so it's of course it's not it's not universal but nobody cares at this moment if it's universal or not I would like to understand what is the point on you the point is just that to see this phase we need these different conditions and the only so the main thing that presents an obstacle with your usual we thought a lot we actually worked very hard in discussing with Emmanuel Bloch on his systems of both Hubbard systems and Fermi Hubbard systems the type of model that you need so of course lots of interesting simulations are done on cold atom systems right specifically for the time crystal we need this kind of dominantly and ising even disorder and this is hard to do on any of these cold atom platforms except for Rydberg atoms so Rydberg atoms actually meet all of these requirements because the Rydberg interactions are short-range it's 1 over R to the 6 they look like ising type interactions you can do these rotations so they meet all of these requirements but currently they're only limited by coherence time so we actually spent nearly a year trying to get this experiment to work in Emmanuel's lab but it's only 10 periods or so so indeed so if in the next 5 years their coherence time gets to hundreds of periods then we're fine and the Google experiment we're not actually doing active error correction yet so the effect of noise that we have will show up in terms of a decaying signal but we can characterize this decay very well because the noise is being calibrated very well and then we can use that to understand what kind of lifetimes we would get it's still a few hundred periods but in any case you have to calibrate the noise every time and so on so thank you very much Vika that was very nice talk let's move to the last talk because we are a little bit late so I have horrible chat person as you can see already late so if there are other questions which I know please use chat to ask Vika and thank you again to be here at this hour of the day for you which is quite late I appreciate it thanks so let's go to our last speaker of this first session which is Mali Efsegev he's going to talk about podonic temperature results correct? yes absolutely good morning I know that I appear on your screen like a Victoria but this is because I lost the zoom link but this is life and we can have to manage with it just a moment let me adjust everything and also to see that I can see the pointer ok so I'm going to talk to you about something completely different which is photonic tank crystals it's nothing to do with many body quantum it is completely classical completely Maxwell with the taste of quantum under spontaneous emission and stimulated emission and stuff which in my opinion if you see it right here it is the area that is about to explode in photonics trust me two years from now there will be hundreds of papers on this hopefully we'll have the first ones the first experiment so the idea is really to have some dielectric material that you change epsilon a lot by a large amplitude very fast in time sorry Marty sorry can you put the slide form because it appears small I think you should put a display I think on display settings I need to change something is this better well yes better better ok so thank you Anna so let's move a little bit so first of all the Technion team are my students Alex so the idea is this following when you think about the crystal any crystal what you really think of is some repeating structure some repetitive structure in space or in time it has a bend structure with forbidden bend gaps and so forth and when the crystal forms spontaneously like many crystals in nature they also form spontaneously then you have spontaneous symmetry breaking but sometimes you can have it artificially as we know very well photonic crystals even though they exist on some butterflies we actually fabricate them in the lab ordinary photonic crystals so I'm going to talk about the photonic crystal of different kinds photonic crystal in time ok so for example photonic crystals in space are structures 1d2d or 3d where the scale of the photonic crystals the periodicity is on the order on the order of half a wavelength or larger and what happens is that to get a full bend gap in a photonic crystal in space what you need is a contrast of the refractive index on the order of 1 usually about 2 or 3 this area exploded in 1987 with a seminal paper by Ilya Blonovich who predicted that if you have a photonic crystal in all 3d for both polarizations then you'll have inhibition of spontaneous emission that paper is considered a field opener and has 18,000 citations in google ok so I'm going to talk about now is about photonic time crystals which means that the epsilon is changing in time in a periodic fashion breaks time the bend gaps are in the momentum rather than the energy the system conserves momentum does not conserve energy and really what matters is that the time scale and the magnitude of the modulation and I'm going to try to cover some of these topics we are active in all of them topology of photonic time crystals bends and you will see the topological all of them spontaneous and stimulated emission in other words we took the question that Ilya Blonovich asked for spatial time spatial photonic crystals into time crystals and you will see that the answer is fascinating then we look at the Pursale radiation or better called Cherenkov radiation or radiation interaction with free electrons in this kind of a photonic time crystal and experimental observations we are well on our way probably I will not have much time to talk about systems photonic time crystals containing disorder and localization but we have a paper coming up in PRA exactly on this, on the theory of this so let's go first to the very basic time reflection and refraction so let's look first on the spatial interface spatial interface means that here you have some material with epsilon 1 and then you have something with epsilon 2 so comes a wave what happens to this wave the wave now is reflected some of it is refracted according to Snell's law and some of it is reflected ok and we all know for another reflections and everything now let's assume that instead what we have now is a temporal interface which means that the material is completely homogeneous in space translational symmetry in space imposes the conservation of momentum and then at some point in time you change epsilon abruptly like a wall ok like an interface so again you have time reflection and time refraction but since unfortunately we cannot come back in time at least I have not yet met anyone that came back in time the time reflection is in space it is caused by a temporal wall but causality imposes that you cannot come back in time but rather it is reflected in space ok so the implication is exactly as I've drawn so in other words you can you back reflect in space I think one of them is refract which I will show you in a few minutes that you will have a different the refracted wave will have a different frequency and also reflect back reflecting time which will change the frequency and also reflect back in space because it is caused by a temporal wall temporal change in the refractive index or in epsilon but we cannot reflect back in time because of causality so that's the idea I will tell you that what was done in experiment so far is time refraction has been shown it is really not extremely hard to show it but was shown only lately by several groups the first one I think was Bob Boyd from Ottawa but time reflection was shown only with water waves and in transmission lines so what you see is a very interesting experiment that my friend Matthias Fink did it where he dropped it he created a wave by putting inserting some little battery of the Eiffel Tower and then dropping the water tank and what you saw is exactly this that the waves went out and back by the time reflection but the rules of the game are completely different fundamentally between water waves and electromagnetic waves if nothing else because we are talking about tens of waves polarization and so on they make a big difference anyway but in electromagnetism a time reflection reflection has yet to be demonstrated it has never been observed so in order now in order to create a crystal what we need is to take these walls or these interfaces and repeat them periodically okay so in a spatial lattice like this let's say make a multi-layer system or one dimensional photonic crystal comes away part of that is reflected part of that is reflected and we have the resonance that we all understand so forth which really also talks about bend gaps and so on in a time lattice now we are talking about an incident wave and now you change epsilon in a periodic fashion you can do it in a cosine way or you can do it in a stepwise function either of them will do the same thing and what you will see in a moment is that what is conserved now because the material is completely homogeneous is k the wave the momentum which means the wave number and as we'll see in a moment there are some fundamental aspects having to do with the dispersion curve so what does it mean if you look at the modes then what you find the modes here is that the modes here we have two kinds of modes one of them diverges to this direction the other one diverges in this direction but the diverging tails have infinite energy therefore what we get is something that is localized in the bend gap but here it is really not the issue you always have something that blows up exponentially and also decays exponentially in the bend gap what really determines what you will see is causality so we can talk today about a space lattice spatial lattice a one-dimensional photonic crystal or a time lattice or something combined okay and there will have something that is completely different I'll come back to the bend gaps but if you want to look at what happens in a one-dimensional photonic crystal in space what you have is bend gaps in energy in other words are frequencies of the photons for which you cannot have nothing is transmitted everything is reflected okay in a temporal lattice in a time one-dimensional time crystal okay photonic time crystal what you have is bend gaps in momentum which means that waves that are coming here are exponentially exponentially amplified and also is exponentially suppressed those that are suppressed are there but they are not very interesting it's like any bend gap but those that are exponentially amplified are extremely interesting especially when you talk about vacuum fluctuations, spontaneous emission and so on and so forth and you can have something mixed which is also interesting in itself okay now let's see what happens now when you take a pulse and launch it into a photonic time crystal I remind you the medium is completely homogeneous so momentum is conserved because of transnational symmetry and let's start by looking at the central k momentum of the pulse in the bend in other words it's somewhere here let's see what happens so as you launch the pulse here now it enters the material it splits in two because it has experiences time refraction and time reflection okay splits in two it goes in opposite direction and the time it experiences the other interface it splits in two again so what you get is the first two split and then the other two again split and there's interesting physics in there too but let's see what happens now in the gap when you launch something that goes in the gap is exponentially amplified now if I make this time crystal very very long what you will see that light stops completely the pulse stops on its spot and the amplitude goes blows up exponentially okay now let us see now again go back to dispersion curve and let's see what happens so we calculated the exact phase for the bends now why is that this system really the way you have some pulses and you change epsilon in a b-model form in other words you change it from 1, 3, 1, 3 let's say as a square wave the shape doesn't matter much actually then this resembles a diatomic lattice a diatomic lattice everybody that is doing topological insulators or topological physics in large know that this is the simplest topological system which means that this really reminds us of the SSH model SSH, Schiffer's Eager model and it is it has topological aesthetics there's no topological transport in this because it's just 1D but it's still nonetheless interesting so that motivated us, motivated my students to go and look and calculate the one the analog of the very phase which is called the exact phase in this one dimensional topological system and look at the topology of the bends and the numbers of the exact phase are written here now what does it mean that these exact phases it means that if you look let's say this bend this part of this is winding which means that it has a non-zero topological geometric phase again you can see why you can see that if you go around the Brinois zone you undergo a twist here and a twist here so it's winding on the other end this one doesn't now this has interesting implications when you go and launch a pulse into it for example one of the implications now is what happens now here and the splitting between the light and the pulse that is refracted and reflected is determined by this topological phase exact phase in other words repeat the same experiment as before and just launch the pulse either in this bend or in this in this gap or in this gap or in the bends what will happen is that the relative phase between them between the refraction and reflection is determined by the topological geometric phase exact phase here is what happens on the what happens at an interface so we made an interface between two photonic time crystals the first one is associated with the with a bend that has a phase shift exact phase of pi and then after that a trivial bend let's say this is the first and this is the third bend in the dispersion curve that I showed you before so now we launch a pulse and we ask what happens so what happens now you've created a topological edge state an interface state very very similar to the edge state of an SSH model that we know from one-dimensional topological systems and here it looks here is how it looks it stays it is localized it gets stuck on the temporal interface and then here the amplitude goes exponentially so if I make this crystal longer and this crystal longer the amplitude here will grow with the size ok so the next question that we asked there are many questions some were classical some were quantum but here I want to focus on light-metre interaction so the next question is really let's take an atom and put it in a photonic time crystal what will happen to it in the bend and in the gap and the basic question is would spontaneous emission be suppressed like Yablonovich found in a time in a spatial photonic crystal or amplified because here the modes are diverging and then you can ask also what is the lifetime of an atom in the excited state will it be prolonged or not and the next one is really interesting in itself because here as I will show in a moment the spontaneous emission here will be amplified or the vacuum fluctuations also both of them will be amplified exponentially not drawing power from or energy from the atoms but drawing energy from the modulation so maybe you can actually design a laser where the laser perhaps starts by vacuum fluctuation or by some seed from an atom or dipolded that radiates but later on the energy everything is amplified exponentially and the energy actually is drawn from the modulation which does not have to be electromagnetic it can be anything it needs to be fast there are some conditions but you don't need light for it at all extremely interesting why non-resonant light matter interactions and you can make a laser that is tunable over many weapons irrespective of the frequency so I want to start again with Yablonovic so this is a seminar paper in 1987 when it both is said practically there in the ambition of spontaneous emission and then there was a race for people tried to do a threshold as laser and they tried really to make just the laser is sitting on a different mode and because all the spontaneous emission goes to that one mode that exists in the system a different mode in the system then the laser will be threshold less and this was the first 2D one was done by Amnon Yarib and his student Oscar Painter today a full professor at Caltech also and the 3D one to my knowledge the first one was done by Noda in 2000 and what we are asking is what will happen to spontaneous emission in photonic time crystals so actually this is a quite a question to solve so I want to draw again to the analogies in a special crystal we have now symmetry with a unit cell in space so the wave equation that describes it is written here I wrote it specifically in this form so we can compare the magnetic field in both of them which is simply easier to solve the modes here are block modes as we know very well the dispersion curve are what I've shown you earlier now when you look at the time crystal the equation looks similar but it is not the same notice the derivative zero in time derivative here of epsilon times the first derivative the eigen modes are now fluke modes and the dispersion curve which means that there are periodic functions with the periodicity of the modulation the dispersion curve is rotated by 90 degrees but there are additional spikes to it because in space you can deflect from x to minus x but in time it would be nice to deflect from t to minus t but unfortunately it doesn't happen ok so what's the implications for radiation sources so let's start classical to get intuition so we put the wave equation now and let's put the wave equation now with a source that is a plane wave with a particular k as a function of time and to solve this what you need is to look at the green function so we want first of all to find what happens to this plane wave when the illumination or the periodicity here is like a flash once boom and you solve it and I can tell you that the eigen modes in the bend here will look periodic and in the gap we have two of them both of them are physical exponentially amplifying and exponentially suppressed essentially became ok the important one is the one that is amplified now let's take this radiation source ok and put it and let's try several cases so the first one look at point dipole point dipole means that you really look at the superposition of all the plane waves that constitute a point source but let's say the fracture limited so plane waves so it decides will be half a wavelength ok then what will happen is let's put it and let's put it in several cases let's put it on the bend let's put it off the bend and then put it in the gap ok we do the same thing then with an extended dipole ok the results are essentially the same ok so it doesn't matter which one so the results are the same point A here on the dispersion curve you see this is the bend gap and this is the bend ok point A here is on the bend point B is off the bend so there will not be phase matching and point C and D are in the gap point C is on the dispersion curve in the gap and point D is off the dispersion curve in the gap so what we see is like this what we see is that if we look at points A and B on A is on the dispersion curve the energy grows nicely goes ok B B doesn't grow because it's not phase matched which means that what you are looking now is that the radiation source is not phase matched with the modulation so therefore the exchange energy and the amplitude remains always small but the more important one is really what happens in the gap because in the gap point C and D one of them on the dispersion curve both of them the energy of the electromagnetic field that is generated here we are talking about radiation field classical both of them are exponentially growing and what you see is the log of this was a linear scale here is the log so you see they are exponentially amplified yes you can see some modulation the modulation is not important the exponent of the growth is irrespective of the frequency it doesn't matter where you are on the spectrum or not so the next thing after that is to say ok we did it classical let's see what happens quantum because the mission is really a quantum phenomenon so the way to do it is actually quite tricky to say quite a few months to find the correct emiltonian you need to start with the eigen equation find the Lagrangian find the emiltonian then do second quantization and then after that so after that at the end you get the emiltonian for this what is actually not this this is the usual part of the emiltonian due to second quantization only thing that is interesting here is this but it's just a coefficient it doesn't matter so much the important one is this the important one is that you have generation of two photons simultaneously one of them with K the other one with minus K ok and now let us put inside it in atom or in atom in the excited set and ask what happens I will not show you all the calculations you have to believe me that they are correct but I will tell you that there are three basic cases and there is correspondence nice correspondence between classical and quantum ok so there are three three ones now oscillating linear growth and exponentially growing the oscillating one when you are off the dispersion curve there's no face matching and as we saw in the classical case it's just the the source of radiation the atom will exchange photons with the with the field practically with the population of the field then if you are on the bend the number of photons grows linearly it's a very interesting effect because what you have is really some chain when you exchange photons some of them are reabsorbed and re-emitted there's a whole scheme here and in the gap they are exponentially growing exponentially growing means that the number of photons in a mold is growing is amplified all the time by the modulation ok the difference between so the fundamental difference between spontaneous emission or emission in general in a photonic crystal in space and photonic time crystal is really energy conservation based on causality in photonic time crystals energy is not conserved in other words the frequency changes because of the modulation so the way to think about this is to think about the capacitor where you change epsilon so what happens when you do that in a capacitor you change the stored electromagnetic energy same happens here classically same happens here also quantum mechanically so you have quantum fluctuations that start from the modulation so you can trigger you can trigger the population in the molds either by quantum fluctuations or you can trigger them by the dipole emission always exponentially explodes in the in the gap regardless of frequency the spontaneous emission in in photonic time crystals which is also an interesting fascinating effect now if you sit on the dispersion curve I'll show you maybe in the previous slide ok ok let's look at the quantum picture of what happens when you are sitting right here and you are getting from point A close close close to the bandage ok and you do you solve the problem with the photonic now ok so what happens here is actually as you get closer and closer and closer to the bandage then the lifetime of the atom becomes extended and the radiation is suppressed but then as you cross a little bit into the band gap everything explodes you get exponential amplification of the number of photons in every mold and as a result of this the radiation field explodes the bandage which is most probably an exceptional point because it converts from something that slows down by virtue of the dispersion relation here and actually the density of states of momentum states into something that explodes exponentially and because of that we think about constructing a proper resonator and selecting the potential mold that will lead to photonic time crystal lasers now and now we also looked at what happens to free electrons which is Cherenkov for cell effect and what we found with the same Hamiltonian only now we have also this we have this term which is now for free electrons and we ask ourselves what will happen so we found three times three types of photons that are emitted regular Cherenkov photons momentum get photons and then also the photons that are emitted naturally something similar to the for cell effect ok by the periodicity in time and they actually interfere you want me to finish in a minute I will ok so they interfere and we found two regimes one of them is the free electrons emit radiation free electrons emit radiation in two regimes one of them is above the Cherenkov threshold and also below simultaneously and they are the radiation is emitted forward and backwards and if you are looking at the band gap the momentum look at what happens to the photons that are in the momentum gap the amplitude is increased exponentially in other words the number of photons exponentially in the quantum picture and the quantum picture gives not only enhancement but also until ever crossing I want to say maybe the two slides on experiments ok now to do everything that I told you it is not magic but it has two stringent requirements one of them is that the modulation is comparable at the rate of the electromagnetic waves that propagates in the medium ok so if you want to look at waves let's say in the near infrared at around one micron you will need to be able to change epsilon on the order of a single femtosecond not only that you need to change epsilon on the order of one so until recently it was impossible but now it is possible the idea is to use materials that have epsilon in zero and what you can see in a moment never mind how we do it what you see in a moment is something that we observed in experiments and we are looking now at something where a four beam is 1,300 nanometers near infrared and we use a pump beam that what it does it illuminates the material homogeneously uniformly everywhere and it has a very short pulse for on the order of four femtosecond that's a five femtosecond and we see rise time that is shorter than the time of the pulse at trigger extremely interesting it is smaller than four femtosecond which is our measurement device ok so in other words and the change in epsilon is on the order of one and here I will this is my last slide and I will say that in my opinion even though they are not spontaneously forming at least in the linear case in the nonlinear case they will also spontaneously form like solitons as we know very well from nonlinear optics but in the linear regime they are not spontaneously forming still there is a lot of new physics on light matter interaction exponentially amplifies spontaneous emission in the momentum gap we have many new features on Cherenkov for cell radiation topological aspects and interaction with disorder which is something that we have appeared coming out that within a year we will be able to demonstrate the first photonic tank thank you and I will be happy to answer questions thank you very much Moti there is a couple of questions we have time if we go a little bit fast so please is her thank you a little bit fast please why the special crystal and temporal crystal is so hard is it because of that there are of course special and temporal coherence in it just can you repeat it because I was not able to understand the speech something is wrong with your zoom I think maybe you can repeat it slowly why the special crystal and temporal crystal is so hard is it because of the special coherence or temporal coherence secondly my question is about the Hamiltonian which type of symmetry you have used to find the symmetry the Hamiltonian okay so the first answer if I understood correctly the question if you are asking me about coherence now keep in mind that the photonic crystals if I understood correctly Anna did I understand correctly the question it was also difficult for me to understand I have to say so if I understood if I understand correctly the question was is it is there any requirement of coherence so the answer to that is that what you need is coherence of the modulation the modulation needs to be periodic in time okay but it does not have to be another it will have a full spectrum so if you make the full spectrum not really stepwise but something that changes at the abatic let's say a bit inclined all the physics remains the same the key issue is that it cannot be slow it cannot be slow if the modulation instead of having like a wall in time if it is slow and it includes let's say 10 periods 10 periods of the electromagnetic waves that goes in the system then the time the band gap will be diminishingly small and you will not see anything this is so the really what's important is how abrupt is the modulation it does not have to be a stem function but it needs to be sharp okay that's the first one the second one regarding the Hamiltonian what we did we really follow the steps that we teach in quantum mechanics too and the symmetry that we used here the eigen modes are momentum modes this is really the important part and from there we found the fluke eigen modes because what you have is periodic symmetry in time the medium is completely homogeneous in space and the trick was really to understand first of all from intuition that because you must conserve momentum you have no choice but to emit to generate two photons at every instance so you emit pairs of photons k and minus k because the system must conserve momentum that's the trick and we are writing now the first paper on this business and there we will see all the derivation it's tedious but the intuition is actually simple just what I told you now okay thank you Moti thank you very much I'm afraid we have to stop here there is a break now also maybe you can ask Moti personally now because it's already 12 minutes after the break and so I'm afraid that we are we'll run a little bit of time so we resume in 20 minutes 20 minutes we start with Andreas Nulekamp see you thank you Anna best regards to Machik thank you Moti thanks can you hear me yeah should I try sharing my screen yes please thank you okay can you see okay I can see it slideshow mode slideshow mode I shouldn't share a different screen one second second you already started the sharing yeah I started but then I had to un-start in order to show that I can share my screen here can you see my laser pointer yes perfectly do you have audio or videos presentation no I only have an animated this should work this work in the past thank you very much thank you Andreas I think you can start sharing the screen yeah hello Andreas hello Anna so that works perfectly so the idea is I'm sorry because I'm sometimes I don't like to interrupt so it's better if I don't say anything but the idea is this 25 minutes talk that's five minutes question so this morning it didn't work exactly like that but let's try in the last two sessions yes yes yes so there's still people entering so let's start so good morning everybody again it's my pleasure to share this last session in the morning which starting with Andreas, Nun and Cam which is going to talk time crystals beyond the MLB so please Andreas can you share your one look and start all right thank you Anna good morning everybody so let me first start by thanking the organizers for putting it together the conference really reflecting nicely the breadth of the field so it's very nice to bring people together on this topic coming from very different directions and then let me first introduce my collaborators this is of course first and foremost Andrea Pizzi my student at Cambridge who did all this work and my collaborator Johannes Knoller who former colleague of mine now at TU Munich all right so let's look at the menu today what I want to talk about and I want to start with something which is possibly familiar to most of you but I will call the MBL many body localization paradigm so to period doubling and disorder at spin chains we've heard a little bit about this morning from Vidika about this but then I will really focus with the bulk of the time on this paper here of ours so higher order and fractional discrete time crystals in clean long range interacting systems and if time permits I will briefly say something about how we work on time crystals and quasi-time crystals in optical bosons but without further ado I will just start setting the stage and like with the other speakers it's nice to just flash out and say what we mean here I will be focusing on discrete time crystals so these are non-equilibrium phases of matter that break the discrete time translation symmetry of a periodic so aka low-key driving and more technically this was nicely flashed out for instance in the paper of Saros here there's a local observable with a quantum expectation value f of t so that in the thermodynamic limit and for a set of initial conditions we have the following properties discrete time symmetry breaking so although the Hamiltonian repeats itself after a time big t, the period to big t we don't come back with our expectation value of the local observable at this period and this response the sub-harmonic response is really not only fine-tuned in the parameters but it's very rigid so these oscillations do occur even if I change slightly the parameters in the Hamiltonian and perturb the Hamiltonian and third there should be persistence and this kind of sub-harmonic response should extend to infinite times so as you know as kind of out already in several other talks the devils of course in the details and we will revisit some of this now as we go forward one thing which is pretty clear with talking about a many-body system with interactions and this in the setting where we have a flow-k driving so periodic driving so the Hamiltonian is time-periodic as written very generally we have to face the scenario of heating to an infinite temperature state which does happen very generically and so how to avoid this which is sometimes called heat death and this Norman has also flashed out several categories how to avoid this here's my version of saying this you can throw in disorder into your problem then disorder and interactions together may lead to a phenomenon called many-body localization and that might be a source of non-agricidity on which you then can build a crystal a discrete time crystal and many papers do that a second option is to use dissipation here's an example from colleagues from Nottingham looking at this also very natural scenario so where you're trying to remove the energy which is and entropy to come back to as a non-infinite temperature state and stabilize an interesting non-trivial response that way I will focus on a scenario which is based on long range interactions and I thought Norman would have talked this morning about it but he changed his mind at least last minute so a long range interactions can lead to pre-thermal phases and that's the setting I will be talking about most but before doing that I will just start us off with a very familiar one namely going back to the seminal papers talking about an MBL based 2 DTC N equals 2 DTC also known as Pi spin glass so the setting is interacting a spin chain, spin wall half chain in the presence of a transverse field in one of the version of this there's binary driving so you divide the the period of your driving to two there's a first half way do a spin flip on all the spins and the second part where you combine interactions and disorder to stabilize that response and very schematically what happens is the following so you can start out and then stand this from an integrable or trivial point you switch off the interactions and go to a perfect spin flip then in pictures this looks like this so you start with you have initial condition all up then the spin flip drives all down after one iteration it goes all up all down so the magnetization shows these oscillations between plus one and minus one it's a period of two relative to the driving period of one so that is the the starting point for your DTC response the frequencies at one half of the driving frequency but of course that's trivial in a certain respect just the starting point you see that as you start to make the spin flip imperfect so you perturb the system you go away from this from this point and you add a bit of epsilon into the driving field then you start to see that again pictorially here you start to start all with all up but then you start to rotate around the spin in all kinds of directions and this particular response is lost now the non trivial bit is that this integral point can be made more rigid by turning on the interactions or here interactions and disorder and you end up with all up and then you start to make these errors these mistakes but the interactions do then counteract and stabilize a response at half the driving frequency this kind of sub harmonic response we're looking for which is robust to perturbations and then later on as a second criteria and persists for infinite times which you can't see from these pictures here even if they're just pictorial one nice way to look at it is you can calculate the Fourier transform of this magnetization and plot it as a function of frequency and of this mistake of this driving field driving strength so this is where your interactions off you just have one point where there's this response at half the frequency and if you turn interactions and disorder on then there's a whole plateau and this plateau means that the response is robust to these mistakes again this is here just pictorial so that just sets the stage of what I want to say now namely I want to focus now on a setting which is in many ways different but also shares a lot of this kind of physics so let me introduce our model we've been thinking about it's again it's a spin chain spin one half it's in a magnetic field the crucial difference to before is that here we're dealing with long range interactions and that's what's going to drive the whole difference in fact I will start to actually from a simple limit namely from the limit of all to all interactions so as indicated here we have a spin one half this is the sigma ij and it's coupled all to all the spin flip is indicated here by the second term and you see in this paper we're mostly focusing on continuous periodic driving in contrast to many of the other schemes which are thinking about binary driving and we think this is experimentally friendly we have looked at the case where there's binary driving and it works just very similarly this interaction term of all to all or a collective spin is very important in the following so this is an LMG model mesh of a glick model and it allows us crucially to have a semi-classical limit to the system and I will comment on going beyond this limit of collective spins a little bit later the key message I want to say out first is that this clean so no disorder system spin one half system sustains a whole zoo of different higher order and DTCs so with integer and surprisingly even with fractional ends and n being larger than the local Hilbert space dimension which is 2 if I then move on to beyond the LMG limit these will be pre thermo states so that's my main message let me know you walk you through what we actually did how do we do how do we study the system as I just said this in the simplest case is an all to all a collective spin system so we can make use of the Shringer mapping of a collective spin of angular momentum to how about dimer so to two bosonic modes which I call here spin spin up spin down with these are bosonic creation and elation operators with up and down so our Hamiltonian maps on to this kind of problem where we have not n spins but we have n bosons in a double well which are moving feel a onsite interaction with which is the first term here the direction term turns into an onsite interaction between the bosons and the second term the spin flip term turns into a tunneling between it for the bosons between the two wells as you may know looking at n bosons in a double well potential can be done the exact quantum dynamics can be done very efficiently and even for hundreds and thousands of bosons so that gives us one handle to do exact because we're going to find this symmetric part of the many body space for the spins allows us here to do efficient quantum dynamics and furthermore it gives naturally rise to a mean field limit so if you increase the number of bosons you come to a regime where the Gospelski equations become exact so these are these two these are coupled nonlinear equations which you can easily integrate for long times so that's the background what we use most of our methods let me now walk you through what we see so let's start from the simple state with all spins up and look at the we're going to look at more general states a bit later and study the dynamics of magnetization which is the collective spin model is the line to the z and in the bosonic language is the number difference between the two wells so here is the if you don't have any interactions then you expect this dynamics at stroboscopic times at least to be of a cosine and its frequency is linear in age in this driving field so if you plot the frequency as a function of spin flip you would expect straight lines as here now if the interactions turned on this becomes very rich so you end up with a very rich response this is the magnetization the Fourier transform of the magnetization of this system and you can actually nicely identify all kinds of dynamical phases here there's a microscopic self trapping for the bosons there's the 2 DTC 4 5 6 DTC you can zoom in to these regions here and find higher orders even more or here you find various fractional orders and my message is here really that these plateaus here that means that the NDC response is really robust to finite detunings finite changes in the spin flip strength and you see that many of these responses are some of which are higher order meaning that N is larger than 2 and some are even fractional as like they are not an integer number this kind of semi-classical mapping I talked about the magnetization before there's a second variable which is important which is a phase so if you think about the collective spin it has a Z projection as well as an angle on the Bloch sphere these two characterize the state of the dynamical system this kind of looking at it semi-classical dynamics is very useful and can characterize the system very well so here is the I start out here is the initial state of the system and now as I go through the various stroboscopic times I see the various red dots up here and I keep remain close to the transition this is how 2 DTC looks like I start again up here where the green star is and now I start to jump every time I every period I jump between two islands which I call here 0 and 1 up here down here up here down there up here down there and so on so these kind of semi-classical prokaryomaps which we have seen provide a very intuitive picture of this type of dynamics now let's move on to something a bit more exotic so the 4 DTC what does that look like well you might imagine so as a function of time the magnetization the stroboscopic magnetization goes 1, 0, minus 1, 0, 1 0, minus 1 etc you have a response at the quarter of the frequency of the driving frequency and again if you look into the prokaryomap it looks quite nice that you have these islands which are labeled here 0, 1, 2 and 3 and the system dynamics moves us every time period between these islands returning to the original one after 4 steps you also see nicely that the original Hamiltonian didn't have N symmetry so it cannot piggyback on any of that but it's really generally breaking this so you can go on you can look at N equal 8 and I want to just show the difference between what is 8 DTC and 8 3 DTC if you're 8 you leave me now or you can imagine that this is between 8 islands 0, 1, 2, 3 4, 5, 6, 7 you go around the clock as you move along and 8, 3 DTC which we call fractional here does it by having 8 islands but moving through them always in the 3 steps so it starts to move always 3 steps forward so after 8 steps after 8 driving periods it has completed 3 revolutions so that's what we call 8, 3, 8 over 3 DTC in this problem now of course there's many things to this was just basically the the main message but of course you have to check many things so there's stability to perturbed parameters so I showed you that it was stable under changes in the driving strength but you would like to see more I'm sure so here is the picture where we see how the state evolves in the whole plane of spin flip strength and interactions so the story is somewhat similar to what I introduced in the very beginning so here the point 0, point 25 which you could call an integral point the other response is trivial if you like this 4 DTC response and you see nicely that the interactions stabilize now a full phase a full connected part of the parameter regimes which show this type of response and you can also distinguish I point out two different phases here you can contrast it with namely a trivial phase and a thermal phase and in these gross BDFs equation you can also nice distinguish those two by looking at something called a decorrelator namely the sensitivity to change the initial state so that way you can even resolve differences in trivial and thermal states so in that sense there's a whole range of it is not a fine-tuned point but a robust phase where there's a finite region of interaction and spin flip strength where this response is stable now I hear you saying what about all the scaling with system size we look at that as well going back from the gross BDFs to exact numerics of the double well problem we look at what the exact quantum dynamics like in the symmetric sector of course as we increase the number of bosons or as we increase the number of spins they're interacting the response we are talking about here which is in the GPE equations becomes more and more stable as you can see here for the different examples of 3, 4, 5, 6, 7 and 8 so we do think that this response is into infinite times in the LMG of collective spin limit of course you can also see nicely how the different DDCs are converging at a different rate so if you're looking at the 2 DDC it converges pretty quickly its response converges pretty quickly for a very small number small system sizes so at the odd of 10 if you're looking at this 4 DDC at this higher orders then it takes much longer to converge and this may be the reason why this is much harder to observe numerically by exact diagonalization in more complicated settings. It's now really time to move away from this collective spin and go beyond the LMG limit so we are replacing now if we want to ask another question what happens if we replace the all to all by something more generic which is going to power law introduced here and is characterized by a component alpha or competing with a short range interaction of strength lambda so this breaks the full connectivity of the model although the LMG can be recovered if we put alpha and lambda to 0 the way we think about this is as we start from the collective limit and we are treating this as a system which has a finite range of functions and we are limiting ourselves to a spin wave approximation where it's just quadratic in the spin waves so focusing on the 4 DDC the higher order one we looked at before what we find is that if we do calculate the spin wave density as a function of time in this approximation which has been looked at in this paper down here then we do find as we vary the exponent alpha we find two different cases if it's small enough so close enough to 0 then we find that the spin wave density is nonzero but stays small on the length of time we can look at here numerically or if it's bigger then it starts to grow fast as I've shown up here so there seems to be a relatively sharp transition as we for fixed lambda move with increasing alpha this is shown here in this graph in the plane of alpha and alpha and of lambda this is the point the LMG point down here there we think the response is to infinite times and it starts to be pre-thermal arguably as we move away from that point and there's a transition then to as we move further out and indeed if you look at the response at the quarter of the drive frequency we find a very similar picture so we believe then that these DTCs I talked about before are survived the presence of power line even competing short range interactions to a certain degree and they survive as pre-thermal phases one thing I have not touched upon so far is perturbing the initial state I only talked about the fully polarized thermodynamic state if you want in the LMG limit I implicitly answered it maybe so namely these islands which I talked about before in the LMG limit these are stability islands as well or you can think about stability islands so if as long as you start inside them then you're going to see collect the kind of DTC response you've been seeing the problem is more complicated in the case of beyond LMG more parameters to think about one thing we did check there is whether this response is stable under injection of spin waves so we take a system where we now solve these spin wave equations for finite initial noise and we do find that this is under a certain range stable so numerically we can confirm that the response remains very similar qualitatively the same for finite initial spin wave density so this as much as I wanted to say about the spit I think I'm almost up in time just to flash out one or two minutes on the second project about time crystals and bosons here we're thinking about n-ultricold bosons in a ring lattice on L-sites and we're thinking about having three stages in the Hamiltonian hopping between here the red sites after one drive period from here to here to here to here or for the if you had the initial state on the blue side it would move from here to here to here to here so you have two entwined clocks which are moving counter counter rotating and the question here is whether this type of trivial response can be stabilized into a DTC so asking the same questions as before if you like what about if you start to move away from the trivial points here can interactions stabilize such a response and you'll likely say this answer will be affirmative especially if I throw in the information that this is a bosonic system of n particles on L-sites there is a highly complicated and weird mapping to a complicated all to all clock model similar in reverse to what I've talked before in this kind of Schringer mapping saves the day that we don't have heating and we have a response of DTC type just saying we similar steps as before we could do exact but we also do Gospodyevsky type we're looking at imbalances and deco relate us and find DTC phases and it again is a full phase starting from a trivial point full DTC phase is stabilized in the system so these kind of rotating clocks can be made into a stable L-half DTC and what's more if you throw in a potential term as well you not only have points where you have rigid sub harmonic response but you can have rigid sub harmonic response together with an additional incremented frequency which is moving about which we then call a quasi crystal incommensurate with the other frequency and again there's a trivial point which opens up into a many body phase as you'd introduce in this case contact interactions so with that I like to come to a close and just say I talked about these two projects here they have in common that there are clean systems no disorder needed there's no MBL to prevent the heat death instead there's the crucial long range interactions here and contact interactions between the bosons which map on to all to all for a clock model on this side it's very rich phenomenology with including exotic phases like NDTC, higher order DTCs and discrete time crystals in contrast to the MBL DTCs here we have a semi-classical limit which allows for efficient dynamics controlled approximations and contrasting different insights and nonetheless these are robust pre-thermal many body states which are whose robustness comes from the interactions between the inter-particle so just to close I said before this works closely with Andrea and Johannes we've worked on several other projects that we don't have time to talk about and I look very much forward to questions and comments thank you very much Andreas thank you very much there is a question from sorry Andreas something I did not get if I look at the finite state scaling of say the decay of the oscillation in the fractional and the integer steps is it the same or is it something between the fractional and the integers yeah actually I don't have a slide on that and I don't know if Andrea has actually how much she checked both so that's the one we have always looked at do you suspect there would be no no I have no idea I was just curious if there was something that was different or not that was my question I can't remember there was I don't have a slide or I don't have the data exactly here so can answer that exactly so Andreas also I think here and Johannes also in the chat so we can look at it again thank you thank you okay so I just wanted to make a short question so you have all these results that you kind of match with the gross beta ESC equation okay which is single particle the question so what is this interplay then between the long branch and the short branch because for me if you can reproduce with gross beta ESC which is a single particle thing the role of interactions I don't understand it completely okay so they are hidden in some way no in there well so do you do you refer to way thing so that sorry do you refer to the fact that so you're asking about the many body character or yes that's of course you know I guess with the video you saw this morning as well right so that this of course many discussions whether about what is a many body phase and where are many body effects crucial here here it's the fact that so if if you are thinking about long range what I said then it does map onto collective spin and you are in a regime where gross beta ESC becomes exact the so that's true as shown here the the the interactions of course still play a crucial role obviously right as you know they are come into play the non-linearity of the of course that is effective single particle level as we move beyond then this so that's true in this LMG limit as we move beyond the LMG limit in this part here this of course a completed general many body system and there you do not in this part here you extend explicitly into the form many body realm if you want but you're starting particularly of course in this quite nicely characterized limit where you have collective physics where you have a non-linear equation classical non-linear equation describing it we see this as similar not unlike actually know that in equilibrium the Ising transition right so if you're thinking about there them the phase transition of paramagnetic paramagnetic then you know we understand the physics of that transition usually by thinking about the mean field limit and studying you know a non-linear equation between the magnetization that's not only what we do right and then that does not mean that the transition itself or the states which are beyond are not interesting or not many body and there's many other things you can learn about going beyond mean field but it just gives you a good point a good starting point to understand the physics in this in this collective limit and do a lot of you know just gain a lot of understanding first before doing you know heavy numerics and really finding out where transitions are so we see it here a little bit similar as in that respect as well thank you very much so thanks Andreas for this very nice talk and let's go to the last talk coming by other which is going to talk about I think classical many body localization now not time crystals so maybe it will be a little bit can you hear me can you hear me yes okay great so let me just activate my timer that you don't have to reprimand me and okay so thank you everyone so I'm actually going to talk about many body time crystals and I'm going to actually adopt some sort of you know higher altitude viewpoint on the problem and try to make a little bit of order of how we interpret or understand the different statements that we also heard during the past two and a half days in some sense if you wish I'm trying to to connect to the statement about that you cannot see the forest of the trees so and it's probably is related to the discussion that I'm just hands with Andreas and before I start I would like to first thank Sao and Christoph for inviting me to give the talk here today I'm actually a condensed matter many body theorist we got to look into out of equilibrium responses of many body there is like a whole comprehensive list of publications that they bring as context to how we got to look at time translation symmetry breaking and I would like to before I go into the details for think a year long years long theory and experiment collaboration with our cheetah and our then joint students and Alexander Eichler that performed all of the experiments that will appear along the way okay so we're talking about a system we would like to actually view the point from a microscopic point of view and generally when we solve a many body system so from from a theoreticians viewpoint we can exactly diagonalize the system and then we will have some effective collective mode or an effective many body spectrum with associated Eigen states generally this many body system could be embedded in a larger ensemble of Eigen states that will then appear at very far detuned Eigen energies and will maybe then can be traced out effectively as some sort of dissipation and into this midst we want to now add time-dependent drive that you can also additionally think about some very massive additional degree of freedom that just moves with its Eigen frequency and shakes the system in time so as to what developed our higher altitude viewpoint was that we actually wanted to look at this generic light matter model the Dickey model so in 2014 we are as many body physicists we said hey what happens when we start to take many spins and couple them to an optical cavity this model was already proposed by Dickey in the 50s and was then solved by Hepp to show that there is actually a critical symmetry Z2 symmetry break Z2 Z2 second order phase transition sorry got stuck there and and technically this model got a new bliss because it can be realized in out of equilibrium system so in the rotating picture of light impinging on cold atoms trapped in an optical cavity so with this I refer also to the talks by and Angela so we looked into it and effectively what you can do and this model also then relates very strongly to the spin models that we saw during this conference you can actually make some generalized mean field description of the problem and you can then solve the problem and get this critical phase transition but on top of it you can also decide to study the problem as to adding a time-dependent interaction between the light and the spins and that tells you that you're actually now shaking effectively the spins collectively and this can then be realized as we also proposed again in this work by a time-dependent shaking of the intensity of the light as was then later realized the Humboldt experiments so effectively we were at that point in time technologically making our first steps into such an environment so we did add a little bit of dissipation but most of the statements I'm going to say are actually not arising due to dissipation this I will lucidate in a couple of minutes so let's start we first solve our mean field equation so I didn't write them here explicitly and we can write ourselves relative to positioning ourselves into the super radiant phase of the system and then applying such time-dependent drive and then you see that technically you will obtain something that looks like super radiance I will show to you in a second what that now means and on top of it you can also go to other regimes where the whole light field is oscillating widely but what is important for the conference topic is that when you Fourier transform this time oscillations you also then start to see response of the system in subarmonic resonances with additional sub peaks that are then incommensurate with any of the drives and are generated due to the interplay between the time-dependent drive and the nonlinearity imposed by the spin conservation of the problem looking at such time traces and taking their time average you can then draw a mean field phase diagram where what you can see over here is the ratio between the frequencies of this parametric drive and the amplitude of the parametric drive and what you can then obtain that there is actually a frequency that here when the amplitude of the parametric die of zero you can actually see that here we see the standard transition the super radiant phase transition but as we increase the drive the super radiance continues but starts to develop all sorts of funny looking of mean field observables when you look at the spin generalized spin projection along the z direction so we looked into it further and then also by the way here on the second panels you can actually see how many then subarmonic and additional harmonics you start to generate as you drive there so here is account sorry here is the amplitude and you can also count how many of them are being generated looking more closely and seeing what the spin actually does of the function of time you can actually see that the due to the coupling between light and matter the collective spin starts to tilt away from its quantization axis over here and then to the time the band drive it starts to process and it will perform limit cycles with many different frequencies on the block sphere generally when you're analyzing such models you can also then see the stability of your mean field assets by performing a fluctuation analysis of such a model and at that time we just looked at the stability analysis around the normal phase namely when interaction is negligible so we wrote down the Holstein-Primakov transformation of this creature generally looks like this when you expand around the normal phase the empty cavity and all the spins pointing down you start to see that you have two effective fluctuation modes that have a time-dependent coupling between them and with this by the way I would like to also refer to the talk by Jami where he actually did a similar analysis but also around the limit cycles of procession but this was was a more sophisticated analysis that was performed later on so generally when you analyze the stability then of these two coupled time-dependently coupled harmonic oscillators that arise from the fluctuations here of the light and matter you can actually form two normal modes of these creatures and what we find is that you have actually two parametric resonator normal modes that each of them can become stable or unstable in different situations and that actually tells you that our whole treatment of a mean field plus small fluctuation is only relevant where we have the overlap of the dark blue regions of both fluctuation oscillations and this is then these zebra lines highlighted over here and corresponds very well with the mean field treatment as you see over there so by combining these two we actually then predicted the emergence of a dynamical medibody normal phase which would have then stroboscopic light bursts as the matter starts to process in the time-dependent drive, the system so having done this analysis