 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 we actually then realize that we feel very uncomfortable with invoking all of this parametric oscillator know-how because we didn't feel like we understand sufficiently the individual resonator level okay so we got all the way back to the beginning and we looked at a simple linear parametric driven oscillator and that was already elucidated in this conference depending on the time-drive of this parametric time-drive you could have instabilities regions where the oscillator completely decouples and does not respond to the drive and regions where the drive enacts a resonance but this is not a standard resonant response to an external drive it's a resonance that is going on in the name of parametric resonance where actually you could have different conditions in the ratio between the bare frequency and the parametric drive and it will occur for even infinitesimal driving of this type we should all feel very comfortable with such an effect this is exactly what happens when we stand on a swing and we shake it so we actually at twice per circle per round trip of the swing we kick it twice with our body motion it's important to notice that this is a zero dissipation so dissipation does not play a role here in these instabilities and it's also then important to notice that if we want to go and understand what will happen over here the linear oscillator is then not enough to understand what's going on because for the linear oscillator all motions should go to infinity and we know that in a real system this cannot occur you could analyze it in the classical domain or in the quantum domain so here just for completeness I showed to you that if we take a quantization of such an equation and we also add say cubic nonlinearities to the problem you can quantize the fields and it would look effectively like a optical cavity or a quantum resonator that is driven by a two-photon drive and here specifically I write the Hamiltonian already in the rotating picture with respect to half the frequency of the drive so here you can already see that by moving to the rotating wave approximation you see the time translation symmetry breaking occur as a standard Z2 symmetry of the model so we are still here at the zero dissipation and we can really even draw the Ginsberg potential landscapes for such a nonlinear oscillator and you can really see the Z2 symmetry breaking and as we are in a rotating wave approximation this will occur due to the interplay between the two-photon drive and the nonlinearity so it's not because we have a dissipative system this physics this period doubling by four occasions they occur at the interplay between these drives and nonlinearity what does dissipation do so I mean there was this distinction of bin field versus dissipation versus many body so effectively until now from what we have seen we take a many body system it forms a collective mode an effective collective mode in our Dickey model it was actually two coupled effective modes that were coupled to one another and the period doubling in stabilities they seem to occur also for a single mode so effectively we have to kind of divide and conquer does the period doubling by four occasion occur because it's many body and what is kind of the game that we're trying to play here with so what I called the title of my talk classical many body time crystals but in fact you see that the period doubling by four occasion here is really not due to dissipation it's due to nonlinearities adding dissipation what does dissipation actually do in this context you can think that the resonator effectively weekly couples to a bath that has some noise statistics associated to the frequency distribution of all of the other modes that this mode has coupled to and if you then analyze the stability of this creature it will actually push the phase diagrams boundaries away from infinite decimal time-dependent drive but it will not stabilize it will not stabilize because the drive is to a two-photon drive and it kicks an infinite an exponential amount of energy into the system that cannot be stabilized by standard dissipation okay this is a ubiquitous effect in fact many many many applications occur here in this stabilized region where the parametric drive acts as effective nonlinear damping and then the motion of the oscillator subject to either thermal or quantum noise will be then squeezed and you can then use it for amplification applications moving beyond the threshold the system will undergo this time-translation symmetry breaking now in the presence of dissipation but we are aware that it's no longer because the dissipation is stabilizing it and the system effectively will now look if we look at it so here is the phase space distribution of an experiment measuring how the system then likes to appear at this type of histograms that we saw on the first day of the conference but of course if you don't measure it with a lock in amplifier you will see that the system undergoes this time-translation symmetry and the excitation will get back to itself after two cycles of the system I referred everyone to these two works over here and this would be probably a nice context for people studying these effects in different resonators so now the top view approach we set out in the beginning actually we were intrigued by these effects and we were intrigued by nonlinear physics involved in this and then we set out to look at actually the interplay between two different drive, parametric drive and the external drive but first I would like to first show to you what happens when you just isolate the single collective modes of an Avogadro number of atoms forming half a meter long guitar string and you can actually then drive the string externally or you can drive it parametrically like we saw in the equation of motion before and this is done by pulling and un-pulling over time over the effective tension that this material problem entails the effective equation of motion describing the system looks like this where the stretching and unstretching enters as this Matthew time-dependent drive over here and we have effective nonlinearities occurring due to the bending higher order bending moments of this beam this can be done on also essentially also on a carbon on a tube but this effective equation will be the same there as well we scan now the drive, the parametric drive over this highest instability mode and then indeed what you can see is this characteristic hysteresis cycles where here effectively the system undergoes a Z2 symmetry breaking to the time transition symmetry broken state and then obtains one high amplitude that would be that the system sits in one of these phase states and of course if you scan down you will see here a jump up and this allows you to then trace out the boundaries of this phase diagram you can also solve the steady state analysis of this creature and you can really obtain all of these stationary attractors in the rotating picture and mark all of the places where bifurcations occur technically we are now interested in this and studying what will happen or how long lived with this type of oscillation persist so effectively before the realization was in the collective mode over half a meter long guitar string you can also take an RLC electrical circuit that obeys exactly the same equation of motion as I showed here and then you can drive it also with noise so if you don't drive it with noise the system is effectively coupled to a zero temperature bath you would see that the system undergoes its time-translation symmetry breaking and effectively locks on to the parametric drive and stays there forever if you increase the noise what you see is that you start to get this telegraph noise the system starts to explore weak noise around its attractor and then gets activated to move to the other time the characteristic activation rate over here is what is going to actually kill your time-crystalline phase as it starts to then by being subject to coupling to the other modes in its environment and that then activates it to get a phase slip if you take such time traces and put them in in phase space the probability distribution function domain you really see that in the end the system regains our godicity and explores both time translation symmetry breaking attractors and technically if you want you can go more microscopic into it and then you can see that when you focus at weak noise activation you can transform such motion you can compare it to the spectral function response that you expect for the system and by this you can also then extract from the frequency at which the system responds to the noise and the width and then you can extract all of the the apunov exponents or even eigenvalues that are associated to fluctuations around this time-crystalline states so with this we saw actually in the conference that people kind of try to you know get at this problem and review it there were several references the history of first experimental realizations versus not I would argue that technically if we go all the way back to Faraday when he took all of his particles in his water glass they formed together a collective mode that then after the shaking of his hand like this formed again those modes that responded most strongly to the parametric drive and showed parametric resonance in translation symmetry breaking Riley already wrote an equation of motion similar to what I have described for this effective mode that appears and this leads to then Matthew type solutions to the problem this field has seen many many many realizations that actually every person that more or less looked at this problem wrote a patent and there are various applications for parametric resonances either for sensing or by actually encoding information in this phase information of the system it actually has even applications for waves of water driving oscillation motion of a rigid body and it has huge applications that even an infinitesimal drive can make these two boats or if it couples to another mode of the system it can do that to it so all in all we even teach a course on it this parametric resonance manifests in more or less all fields of physics it's ubiquitous and if I kind of want to go back to look at the forests I would say that what we're trying to do in this conference and now in this community is to try to engineer through the many body the different collective modes and then by seeing the many body physics see which collective modes respond to which drives under what's effective nonlinear coupling to other modes and what other effective dissipation channels that exist of course depending on the many body microscopic system we would see that maybe lint blood is no longer good enough description of the problem but all in all I would I wanted to use this talk in order to kind of make a divide and conquer that I would say the time translation symmetry breaking is of the parametric resonance type in optics people call it second harmonic down conversion so it appears in a variety of fields and as a society here in the time crystalline domain we need to stop and say okay what are we trying to then obtain with our many body realization and for this here I wanted to also highlight this experiment that Monk Dickman and collaborators from Korea showed where they took many atoms and shook them in time and then they also underwent this time translation symmetry breaking but then I ask what is different than this type of experiment in the guitar half a meter long guitar string and this is in the one minute that I have so this is kind of to show to you that even Bell Labs in the 50s and 60s people already found applications for such parametric instabilities and the same way that this was a pioneering commercial from Bell Labs I think that we are at the era that we need to go beyond this single collective mode and see where we can go so where can we go I use another two minutes Anna sorry we can see whether our equation is really susceptible to binary drives or cosine drives and this is not the case actually this is going under the name of Meissner driving in the older literature and the only instabilities and where they occur appear and manifest only at the commensuration of your floquet driving cell and the eigenfrequency of your collective modes so this is how what happens to these stability are not done as you anticipation you could add more drives and actually you can break the time transition symmetry breaking as a function of the tuning and this can give you additional funny steratic controls on how you can switch between which phase of your time crystal you fell into and this we could also try apply when this half a meter long realization and where actually it matches very well with what happens when the child wants to swing its swing and the parent tries to move the fear in the middle so hence the picture over here you can try to go quantum and you realize that all of this macroscopic it's called gas to liquid phase transition it's a first order type here from a low amplitude in your resonator to high amplitude in your resonator if you add quantum fluctuations it will make all of these first order transitions into crossovers including the one that we then showed with the phase symmetry breaking over here and as well as as the Z2 transitions all of this if you go deep non-linear is closely related or identical in fact to the recent realizations in devos group in Yale of cat qubits in driven RLC resonators so if you go deep quantum on this mode you will get an effective U photon qubit with a parity even odd realization of this system that seems to be very robust to noise because on its block sphere there are two coherent states embedding in the encoding if you go into using these phase states for logic actually the first Japanese computers were using C circuits and encoding bits in these two phase states we then went and looked at how we can activate and manipulate this type of states as as activation and fast activation for actual application of using this time transition broken states for encoding logic and then technically if you go beyond and you would couple many of these guys you will now be able to generate annealers and simulators that can be either classical or quantum in the disk notion or thereof so I'm using Star Trek's discovery again to say that if we want to go many body we need to have many of these guys and specifically we need to actually have that the system has many collective modes that each collective modes mode undergoes this time transition symmetry breaking and interacts with one another and effectively you can really think of it that if we have this Arnold now if we have many modes they will hybridize together they will form new normal most each will have its own instability low associated to it and we probably need to go to this region over here I finished sorry we actually went to explore it truly you could have strongly connected modes and then they would form new normal modes that would act in their own as a single collective mode even though they are now more of them together but if you have weak coupling and this instability lobes overlap this is where more stuff occurs you have 25 attractors to the problem and many many things can manifest here we went to explore the many body physics also with noise and here you also see bifurcations that lead to metastable states we call them ghosts because they actually participate in the noise activation in the system and this is actually something very interesting for the manifestation of all of this in as icing simulators so with this I want to summarize that I claim that the microscopic leads to effective collective modes that have nonlinear coupling to their environment the microscopic is seemingly interesting in order to then find out whether we have isolated modes or not nonlinear resonators show period doubling bifurcations and then we can ask ourselves do we have a many body period doubling where many modes undergo this translation and then interface with one another or do we have a single one that does that the role of dissipation is marginal and only pushes the threshold away and technically if you do couple more of these new things occur so with that I would like to thank you for your attention thank you for that your watch is not so precise sorry sorry so I have a question which has very little to do with your talk may I okay so I was wondering when I was listening between the difference between this classical so you argue that classical oscillators if you want driven oscillators okay display all these physics which I completely agree all these discrete time crystals if you want and I was wondering something holding this conference right and you are a major body physicist so maybe you would know I was thinking that the big difference between classical and quantum in case that they exist is that one should be able to see a kind of violation of very inequalities on times whether it's called legged inequalities okay I know it's not very related to your talk but this is what it came to my mind so I wanted to ask you this and exactly another thing which is also related so you were saying that this parametric driven systems have been using for metrology okay and then I was wondering which sounds sounds pretty reasonable so should that be the same if you are quantum and instead of having that measure and then feedback and so on just having a time crystal which can be used for quantum metrology okay so you asked two questions that I will try to answer as two different ones technically if you start to look at this out of equilibrium systems or you try to even derive in bloody and description of what's going on then we get very fast to also what Weda told us yesterday that we have actually something that will be jump kicks from the fluctuations that hit our system as well as something that gives it a lifetime okay the classical systems that I showed to you so for example half a meter long guitar string if you drive it in the way we drive it it's never going to decay it doesn't talk to anything else in the universe more or less okay so effectively it talks to a zero thermal bath right if you make this smaller and smaller your quantum fluctuations will activate transitions between the two all photo number minima of your system okay and that would probably be what happens there right I mean this would lead to these crossovers that I showed over here I'm going to try to scroll in parallel right so if you look over here so actually everything becomes a crossover at a certain point because inherently you have quantum jumps that then make it that it's hard for you to distinguish at which bifurcated minima of your Hamiltonian you're sitting at because the non-linearity is so strong the number of photons is so small relative to the symmetry breaking and then the quantum fluctuations will make you hop extremely extremely much so this would be what I would say is a direct manifestation of what quantum will do and therefore you will have very short lifetime for your phase states to to exist now relative to entanglement entanglement and belly inequality entanglement is always a question of relative to what right so I mean even when you drive the system and it's squeezed and you would look at the photons so vacuum photons in this squeezed parametric single oscillator then you squeeze quantum fluctuations under the parametric drive and then you will have actually a higher coincidence you see quantum already over there and now if you want to look into your many-body system then it depends it could easily be that you squeeze the collective mode in your many-body system and now you want to in space see how two spins are connected then you will have entanglement I cannot see it on time but anyway not on space but anyway so Taro wanted to ask something just a quick question in the telegraph noise that you showed the rate the transition rates from one side to the other is just thermal or there is something more because of the so this is a very good question we in the experiment we're deep classical so to say so I mean I think on the first day of the conference we have 10 to the 8 quanta if you wish and and therefore we have to induce noise externally so and then this is thermal noise and we use white noise for activation barrier questions colored noise is then becoming very interesting the rate of activation is intimately close to what people call Cramer's turnover but actually here it will not be Cramer's turnover because dissipation has a different role than in standard potentials of the system because in the rotating frame so this is something actually interesting to look into there is in these parametric resonators there's a work by Mark Dickman where he is showing that these systems have effectively to their activation rates and additional contribution that he denotes as quantum creep so there will be an additional quasi quantum effect appearing here as well so yes it's very interesting so if there is no more questions do you want to close the session? I would like to thank everybody and thank you Anna so thank you all the speakers I think we can get a break and resume at 2pm Central European time ok see you later bye Professor Liu can you hear me? are you connected? Professor Liu Professor Cosier can you hear me? ok thank you Professor Cosier would you like to check your presentation to share your screen for the lecture that you will have today? yes I've seen you are muted but you can ok see your slide in presentation mode it's ok thank you Professor Liu can you hear me? you are muted? no I can not it's unmuted, yeah great would you like to check your presentation? yes maybe share the screen yes thank you let's see thanks so much can you see that? yes presentation mode is perfect great thanks for everybody greetings to Sasha and Professor Fazil Sasha how do I pronounce your first name? Chris Dove it's the best way to call your name correct way thanks so much see you guys soon see so all the three speakers tested the connection I don't know if I should test the connections but I don't know I just joined he is ok I am muted ok so did you hi did you check your did you test the presentation already? or what's that? did you test the oh I'm here to test ok so I think we can try now just a moment sorry do I need to be oh I see share screen yeah I think it works ok sorry because they always ask this do you have videos or in your presentation or just slides? no videos ok so that's perfect ok and if I show it this way are you seeing the full powerpoint page with the is it in presentation mode I can't really tell now it's ok ok good and let me just also check can you see my mouse here if I do this yes you can you can see my mouse ok ok and I think we're set ok thanks a lot ok so I think we need minutes we started on time so far at the end it works very well also with timing and things I was a bit worried this morning but what happened was that multi seger was registered under a different name because I think he connected from the phone or something else so that's why we were not finding him in the list of participants because I think that he because he said that he lost his email and connected via Victoria and let's say account ok that was very ok so you know more than I do no I was just I got it but finally it was ok good so I will just disconnect for few minutes and then connect it to ok alright so maybe I will start share sure thanks much see so it's warmer I mean Pewdsworth I guess it's pretty warm in Italy right well actually not really in Trieste it's cold not yet I think at this moment I mean we are around in the morning 2 degrees above zero oh that's not that's ok a little bit cold that's like we are here so so I still had a fond memory of visiting Trieste hopefully we can have that opportunity in the future soon I hope so so all the currently you guys are still running program that's wonderful so are all the programs online or you still are? yeah so I think as far as I can say until August for sure August September and then we are evaluating if situation changes but it's complicated because all the people that come from outside have to go on quarantine so I mean having conferences is very difficult for many many different reasons so I think we can start so welcome everybody the last session of the conference and the first speaker is Vincent Liu from University of Peacebook so Vincent the screen is yours thanks so much I want to start by thanking the organizers especially Professor Sasha and Professor for inviting me to this very interesting conference on a special topic which is very timely also I had this opportunity to know many new friends on the screen to learn new physics and also meet new I mean old friends like L. Shapiro who's coming up next who have you know I've not seen for quite a while due to pandemic anyway today I'll talk about I will say I'll report two results one is actually on the Floqury Time Crystal without disorder I want to spend a few minutes on that and the second one is to propose a new idea on imaginary time crystal and this work is I mean this talk is based on work in collaboration with following people especially my post on collaborating in Shanghai I think it's in the audience and so I decided not to give any introduction I think many speakers in this conference ahead of me already did a great job also Frank Cloak and I understand from the test slides shown by L. Shapiro he will actually give more you know background regarding the origin I mean the origin of time crystal so I'm going to just go straight to the results we worked in the last couple of years so first I will talk about the clean time crystal then the second part I will talk about the imaginary time crystal and the macroscope level a temperature crystal so part one we come into this question by asking the following question is the concept of disorder or many body localization really necessary for the time crystal let me try to minimize the side view let's see somehow yes I can do that I think it is fine it's fine okay that's good alright so let me give my laser point back alright so then we seem to have found a model that indicates that you don't really need this so let me get to that point you know I just want to make a quick introduction to pave the way for me to introduce the model we are working on so think about it today like in 2018 or earlier at that time people already think about how to get around with local theory many great ideas by introducing the district slash flow query time crystal that means go beyond the local theory or by go beyond the assumption of local theory all those models as far as I know those pioneer work by those people seem to indicate the many body localization is a key element to stabilize the so called prismal time crystal phase then shortly after that the experiments in Maryland however confer on those predictions very remarkably I just want to point out that in the early experiment by Maryland they found indeed you saw this is essential however the experiment as everybody knew so far they have performed both cases with or without this order very recently actually this year the Maryland group actually together with Norma Yau and other collaborating in theory also did the case without this order I would not talk about that update so what is the particular model I want to actually introduce so the idea is actually based on the periodic driving so we have a given period like a T unlike the other models before us we do not have a disorder that means we can only have two time intervals in one period so in the first period T1 is a kind of like a Rabi oscillator so here we have a model based on the atoms the previous models are based on the localized spins so we try to do hard core bosons or fermions with the idea of code atoms later on I will talk about the realization how to use code atoms to realize this model so this is like hard core bosons there is a hidden term associated with the hard core condition which is not shown you can also think about it as code fermions so the first term is actually just switch between the leg A and leg B think about it you have two is a ladder with two lags you populate two lags however this one actually would change this population imbalance I would say between A and B so for that we define something like a pseudo spin polarization in the sense is the difference between the particle number between A chain leg A and leg B so this is the model in the time interval 2 this is actually a very typical along the chain tunneling hopping then we also have this nearest labor interaction on top of that there is a constant delta here this is not disordered so it's just like an energy imbalance which will be present in the model it's not so important in our model by the way and let me just get the results so we have done especially my collaborator I posted actually I've done this exact diagonalization in and also by the DMRG and both actually indicate you could have two phases this is the case without disorder by the way we never had any disorder in this model that's our goal to test with this possible to a time crystal phase as you can see that when the interaction is strong enough this is strong interaction case this is the weak interaction and indeed the period of the time crystal phase is locked to 2 pi in the friction space the spectral weight functions peaks at the pi in this axis of omega that actually indicates the time crystal satisfy all the criteria and people introduce like a rigidity distance all those conditions satisfied without disorder when the interaction is weak we did find that the perturbation away from the parameter in Hamiltonian you're actually going to see the period is going to go away from 2 pi that's not a time crystal phase there are some other results given the time because I want to focus more in the second part in the later I think in the 2018 it was paper by Yale group they also started a system with spatial order the crystal they also find the time crystal in our system there is no disorder so that result also seems to be consistent with the statement that the many body localization may not be so necessary for the time crystal phase then in the last few slides for this first part I'm going to ask question how do we realize in cold items other than the localize the speed models like trap irons systems and wait centers so here let me just propose one model and in the paper we actually talk about other ideas so let me just talk one of them just be focused not as actually we're going to it's possible to use the dipolar gases to test this idea so let me just tell you a little bit so this is the toy model and this side is more like a step two you know from time model to the realistic model and step three will be someone go ahead do the experiments that's a typical roadmap from idea to the observation and talk about the realistic model I think it's possible to use code I mean use optical lattices to form a ladder and the thing about the y direction is double wheel along the axis like one dimensional lattice this is rather versatile because you can change the lattice constants in space you know why much greater than x so that means kind of like it is not much interaction along the between the chain you have interaction along the chain that's exactly we need right here and then you wonder why you want to have dipole dipole gas especially you think about the polar molecules for those artists I guess look at the participants many of you may not so familiar with code add-in someone that are very familiar so let me just make a few remark that is the for the polar molecules without the presence of electric field the dipole moments are zero so in the presence of e-field you can induce the dipole moments in other sense you can induce this labor interaction so in this model we need this labor interaction but the dipole interaction with a little longer range will do that so by turning on and off the electric field you can turn it on off the Hamiltonian basically the interaction term that it means you can go into T2 or you can stay in T1 so that's actually the way to to drive the Hamiltonian steps and that actually there is the phase diagram associated with that so anyway so the take-home message for this part is you don't need it seems to me there's a particular model that shows you don't need many body localization ideas to stabilize the time crystal and also the long range interaction is not really a key in this problem this is just the nearest labor interaction here and then the possible ideas actually can be tested for example in the dipole gases all right so let me move on to the second part in the next 15 minutes also so I think it's 25 minutes right in total so the second part is actually there's a little bit more on the model level this is actually in close cooperation with the tie so we are asking the following question what is the thermo-dynamic analog of the time crystal in the original paper by Frank one of his paper I think he actually found out the possible idea of measuring time crystal so here we even try to generalize to the thermo-dynamic open-quantices that is actually an idea so let me first give you a few slides to heuristically introduce the idea to argue why a you know imaginary time crystal is inevitable by the physics we learned in the statistical field theory so let me start by asking the following question typically when you think about the temperature dependence of observables you might get into one of the top three situations for example you could have a observable such as resistivity in the metal that should grow as monotonically as functional temperature or you could have something go down monotonic in temperature like 1 over T for example like spin sensibility according to query Weiss law or you could have a situation that should peak such as heat capacity in the super free phase transition so that indicates something like phase transition yes then you wonder this actually never happened in the history in the past but you wonder whether it's possible to have a observable it's not monotonic but oscillating with some kind of period in temperature this is a temperature not time so that was actually kind of like the idea and that actually also relates to Frank's imaginary time quiz proposal if you comment in his paper at the end of this conclusion part I think in one of the papers that the reason it comes to the following slide that is actually we understood pretty well for example in the zero temperature a d-dimensional quantum matter it's actually mapped to a D plus one dimensional classical matter the famous example is the you know like a 2D ison model classical ison model is mapped to a 1D quantum transverse field ison model and those results it's well known add a finite temperature for a d-dimensional quantum thermal matter like a quantum sample the partition function can be written in two improvement forms as we know from this statistical mechanics and the statistical field theory it could be described by Hermit Tony in the deep dimension or by using the pass integral you can describe that by D plus one dimension Lagrangian you just go into the Euclidean space and by introducing the so-called imaginary time axis so that is actually the idea and the time limit is actually the inverse temperature beta right so that was actually well known result so having reminded all of us that you know that the dual theory between D plus one and D then you come back to here suppose I mean the D plus one dimension say XYZ you already discovered some many body states that breaks you know translational symmetry in the spatial axis Z suppose for example you can have a charged tensed wave we have many examples in solid state materials like that and they were also in other soft conditional systems as well then you think okay here by doing the mapping from the knowledge we learned in the statistical field theory you say okay ha ha we should have something along the two axis which is imaginary we should have something like a spontaneous symmetry breaking as well it's just like a Lagrangian you know to this Euclidean Lagrangian right so it's kind of like very equivalent mathematically so that was the idea so if you buy the idea you could have a spatial crystal structure that breaks the translational symmetry in Z then you should have something that breaks the translational symmetry in two so that was the idea so I think it's pretty convincing so the next is try to figure out a particular model to prove that it's not like a heuristic argument that will be in the next a few minutes I'm going to convince everybody with a particular model so that actually will be my remaining job so so far it's the very generic idea I think it's pretty robust or not regard so what is the model here we are trying to think about that there's a no go theorem for the time Chris I know that pretty well by now then that means the problem has to be time dependent should be beyond like Hamiltonian problem right so if you have Hamiltonian you think about it there will be a turn that time time dependent so here's our idea we're going to concede a time dependent the interaction so the time dependence enters the problem through the interaction not through the potential or through periodic driving not through the single particle turn that actually the main point and such interaction kind of like in Lamar Corvian for those people actually talking about it in the last few days inverse systems you could think about this interaction could it be you know kind of like the effective interaction could it be generated by carbon in the system to the environment you could have a atomic system covered to a photons and you could also encode that as you turn over the both on firm mixture etc and so this kind of interaction turn could happen and that we don't have the particular model yet that would be a more realistic down the road so far as like a time model so what is the main feature we would require from the study we made so number one so this potential this is the interaction potential it's very free tool by the way I I soon the V is local is like delta function in the position space so I suppress the spatial indices for the moment just focus on the time indices so it's we actually require from the model we studied this potential should have a minima at a certain time scale and that time scale actually will set a kind of like a lattice constant in time axis in the future and this oscillation is not very necessary from the model we studied we studied class models but otherwise to be more concrete we take this particular form which has some kind of oscillation and also some long distance decay second point I want to mention is that we have time dependent problem but we do not violated the translation symmetry because the time dependence is through the relative time so that's actually the key then this is the particular model it's a model in one plus one dimension is a hard to call both on system is a lattice in the X in the spatial dimension otherwise it's a continuum in the time space in the imaginary time space so it's one plus one dimension problem and for this particular problem we actually use this interaction I'm sorry time dependent interaction term described by this company strengths alpha so alpha indicates how strong this new term this new time dependent term is and in this one plus one dimension problem especially when we deal with particle bosons my collaborator was very able to solve that by quantum Monte Carlo simulation let me just highlight a few key results given the time we actually calculated for example the density density correlation function as defined here as function of time and space and here this one shows for the moment we're going to focus on the case time dependence the imaginary time dependence that means on the same site same lattice site but different time for the moment for this correlation then you did find that there are two situations two cases I would say two phases and like the black lines red lines there are weak interaction alpha is low and then for the blue and the purple as you see alpha is greater than like 0.2 and the higher then you see there is a kind of two different phases for the strong interaction you see a periodic oscillation this is a time axis imaginary time and you have some kind of like an emerging time scale come out of that so it seems to indicate a evidence to indicate that the spontaneous transnational symmetry breaking and indicates a true long range order by this case this is actually somehow seems to be a true long range order associated with this symmetry breaking and I'll come back to that moment it's actually it's continuing in the time but it's actually district in the axis so it's a joint continuous symmetry it's a continuous symmetry plus district symmetry so we'll talk about if you're interested in talking about whether it's possible you have a break in the continuous symmetry so what do we break I think is really still a partially discrete then you can actually extract this something like an order parameter by projecting this correlation function in time space on a particular Fourier component and you realize again that will confirm two phases and by the way this is the similar dynamic limit as you can see the horizontal axis is the inverse of the lens of the system the lattice size in the x dimension so the zero means you are in this one over L L goes to infinity as you can see when the interaction is strong enough that's like these two cases like blue case and the green and those the order parameter those numbers are finite and for other cases like weak interaction you see the order parameter for time crystal is zero and also in the spatial part like the structure factor in the spatial sense like S of R it also has a finite MS and this is the in tool space so M sub tool and MS is also finite so it's actually it's a space time locked crystal phase and then you may wonder the origin actually I already mentioned a little bit in the spatial crystal you require a minima in the interaction potential here we kind of like require a minima for the time independent potential that's actually the origin then you can also well because you know talking about the hard of the bosons you also think about in the 1D you have this quarter in long range order that may be super fluid right so we calculate the Grin's function as defined here it actually does show that actually has algebra decay a strong indication you have this one-dimensional quarter one-dimensional super fluid order exists very surprising however when we calculate the compressibility it's actually shows finite this is the case above a critical value this is above critical value in the sense you are in the time crystal phase the curve is finite in the sense you have energy gap so it's like a super free versus the gap so it's kind of like very you know intriguing not very conventional so this is the phase diagram you have time crystal phase when the interaction strength alpha is greater than alpha c and however it's actually a large iniquity which is very normal when the time-dependent interactions weak so we recover the well known result adding no interaction strength and that's actually a large iniquity with one gapless mode that's the typical associated with the super free case for the hard of the bosons and if you are in a weak interaction and what is the picture associated with the time crystal phase it's very remarkable especially at the half feeling and what do we find at any given time along the spatial axis you kind of like talking about the half feeling right you have like a particle number is like 0 1 0 1 0 1 pattern however we have this time crystal so it's like you have time lattice spacing you have time lattice constant at the second half of the time period you realize 0 1 become 1 0 so it's very interesting at one time it's like 0 1 0 1 0 1 along the x and then in the next time it's like 1 0 1 0 1 0 so it's kind of like you shift it and it's you have this kind of symmetry breaking along one direction like that way and you still have a remaining continuous translation invariance along this 45 degree direction and I think it's open the window which means I only have five minutes yes okay I'm going to try to finish in two minutes then and it seems to me this is a new phase it's not it's multi-insulator not a super free because multi-insulator full gap but you do not have an algebraic decay in this Green's function if it's full super free they should not have a gap so it's kind of like a very unusual I would say unconventional we still yet to discover more seems to be a third phase in the well-known both hybrid model so what about the temperature crystal idea so if you plot ms which is the charge density we'll order in a time I'm sorry in a spatial sense as a function of temperature then you realize in particular plot is one is function of the universe temperature you see a periodic oscillation with the period set by beta 0 and the beta 0 is roughly set by this minimum right here by this potential minimum in time space so you kind of have like a time lattice constant which is given beta 0 which is associated with the minimum of this potential right here and it's periodic and so I of course cheat a little bit it's not really temperature crystal is actually should be universe temperature crystal in a sense your action is a periodic in the universe temperature but otherwise we understand what that means so that's actually what it may mean and the imaginary time crystal is the microscopic mechanism you have this knowledge order seem to break in in the thermal dynamic level it manifests as a temperature crystal so that is actually an idea all right so I think with that I just want to summarize I talk about two ideas one you can have a clean quantum model to show the flow quay slash district time crystal without disorder many body localization that's actually the idea as the parting model shows second one I propose a model and for the kind of like a temperature crystal and with that I want to thank you for your attention thank you very much you have time for questions I think that links and Google was the first so okay thank you for very nice talk I just have a comment I think this non-microbial interaction you are looking for maybe already exist maybe you can look for some papers about giant atom or non-microbial giant atom there they use the superconducting artificial atom to capital surface and because the sound speeds are very slow and there will be some kind of a loop time delay loop then the atom state is kind of determined by the past the effect although the original model is time the original Hamiltonian is time independent but the effect when maybe it's the non-microbial interaction you are looking for wonderful I definitely should learn more from you that sounds like very exciting we should talk more so that's really a physical system I should definitely learn more it seems to me very reasonable and very exciting like you said the effective induced one is time dependent non-microbial do you have a minimum I'm not sure let's maybe talk more if an atomic decay doesn't work you need a minimum that's actually the key going down and up again so we will talk more, thank you let's keep in touch with email thank you Rosario yes I have a question about the imaginary time crystal if you do an LED continuation of the response function shouldn't you observe also a decad finite frequency in real time yes they this oscillation in the imaginary will give you like a decay if I understand if I do analytic continuation and I go from imaginary Matsubara to real time-greens function I think the same structure should give a structure at omega-zero some kind I don't know so if you have a decay in the imaginary axis and you do the weak rotation you would get the oscillation in real time so I think the I think the question is very profound so the question is whether you can get something in the real dynamics yes so I think it would be very interesting to see how this idea would be generalized while you're still not getting the local theory and other things and so I think it would be I think the answer should be yes if you think about the mathematic level right so that's actually my answer yeah thank you very much Vincent I think we are going to the next speaker thanks so much my pleasure I would like to introduce to you Alshapere who is going to talk about mechanical time crystal so Al, the screen is yours I think everybody hear me good okay okay well thank you Chishtof and all the other organizers it's very exciting to be here at this conference and I only wish I could have attended more of the talks because of the time difference I'm afraid I only been to the afternoon talks but I'm hoping to catch up on the morning talks the recorded versions of those so anyway but it's great to be a part of this conference I want to start off since it's kind of amazing how far we've come with time crystals we're approaching the 10th anniversary of the first two papers on the subject and I think it's appropriate to take a quick look back at the origins of time crystals so as far as I know the first mention of time crystals is due to Dr. Who of course and Dr. Who may already been mentioned in some of the talks but the time crystal plays a key role there in providing the power source for the TARDIS which is able to travel in time of course that's Dr. Who's time machine the term was used about the same time back in the 70s by biologist Arthur Winfrey to describe self-organizing oscillations and rhythms in biological systems so for example circadian rhythms or cardiac arrhythmias and these are all non-equilibrium driven systems so one could say we've in some sense returned to that original paradigm with all of the talk about flow-k systems and so forth but time crystals as we know them were born in the summer of 2010 and I must say it's one of the more exciting moments in my physics life I was visiting my former advisor and friend, good friend Frank Wilczek and Frank had been thinking about there we go about spontaneous synchronization of oscillators which is actually something that Arthur Winfrey also had thought about and I came to visit Frank in Cambridge and we took a long walk around fresh pond trying to sharpen questions and come up with physical examples and we focused in on the question which is a familiar question to us all now which is at the time maybe it was not so obvious to even pose this question which was can time translation symmetry be broken spontaneously in other words can the ground state of a system either classical or a system be time dependent well after a little bit of thought we realized that it was not going to be an easy symmetry to break that time translation symmetry is really a privileged type of symmetry that has that basically is built into Hamiltonian mechanics and a lot of the people who are familiar with so in fact you can easily show that a system described by a reasonable Hamiltonian cannot have a time dependent ground state and Frank mentioned this in his colloquium but it's such a simple argument it'll just take me 30 seconds to recap it so what we want suppose we're given a Hamiltonian and we want to look at the ground state of that Hamiltonian that means we want to minimize that we minimize the gradient of the Hamiltonian and we combine that with Hamilton's equations you can see instantly that this says that p dot and q dot must be zero whenever we're in a minimum energy energy solution so that gives us a classical no go theorem for time crystals because for a system governed by a Hamiltonian that for which we can apply Hamilton's equations one always finds that the minimum energy solution is time independent so that didn't stop us we said what about Lagrangians and it turns out you can actually get somewhere with Lagrangians so we we have a simple Lagrangian with it's key feature is it has this wrong sign kinetic term here and or unconventional sign let's call it and kappa here is some coefficient which I'm going to dial later on but we'll take kappa to be positive if that happens we have something like a double well kinetic term with a negative mass and it's analogous to the double well potential that one might find in more conventional systems except that it's a double well in the kinetic energy and or double well rather in the velocity so classically just as this would be true if this was a regular potential energy there are two minimum energy solutions and they spontaneously each of them spontaneously breaks parity symmetry of course quantum mechanically it's true that the true minimum is a super position of these two broken symmetry states and so at least in this simple system quantum mechanics undoes the symmetry breaking alright so to proceed we construct the energy out of this Lagrangian and the energy indeed has a double well form it looks almost the same as the Lagrangian except for this three here and it's minimized in the velocity by a non-zero two possible non-zero values of velocity one right moving and one left moving and so we have a whole continuum of solutions parameterized by the initial value x naught so this is what we mean by a time crystal and if you want your crystals to be periodic then you can just take x to be an angle variable okay so what about our theorem that we can't do this well here's how we get around it we'll be talking about loopholes later in the talk as well so in this case what's different from our assumptions in the theorem is that first of all momentum and velocity are not linearly related they're non-linearly related and that means for each value of the momentum there are up to three possible values of the velocity so if you convert our energy function on the previous page here into a Hamiltonian and write it as a function of p it's going to be a multi-valued function of p and you see there are three possible values of the momentum and the energy sorry of the velocity and consequently the energy and so we can also instantly see what the loophole in the theorem is which is this Hamiltonian is not differentiable at the minima and that was our key assumption that we made in order to show that the ground state was time independent so this is the type of Hamiltonian you want to be looking for if you want at least classically a ground state which is time dependent and it turns out to be it seems very strange from what we're used to from our paradigms but it's quite robust in fact generally for Hamiltonians with higher order terms in the velocities or rather for Lagrangians with higher order terms in the velocities one obtains Hamiltonians with these features that one has cusps at the minima so in fact any non-convex kinetic energy is going to lead to this type of behavior we call such systems mechanical time crystals and they evade the no-go theorems because the Hamiltonian is pathological just to recap it's singular with those cusps it's also non polynomial in the momentum and it's multivalued so it shouldn't be surprising that these Hamiltonians have some unexpected properties on the other hand the Lagrangian is well behaved and equations of motion in fact are solvable and even with the potential we can solve them but we will find that the solutions have some peculiar properties let me just make a brief aside I don't have much time to say this but just regarding what happens when you have terms which Hamiltonians or Lagrangians rather which are higher order polynomials in the velocity it turns out there's a very nice generalization and can be framed in terms of catastrophe theory so for the Hamiltonian we've been talking about at a particular value of kappa if I plot the energy surfaces here the Hamiltonian as a function of momentum and also as a function of that parameter kappa I sweep out exactly Rene Tom's swallowtail catastrophe surface and it's a beautiful observation that if one takes higher order polynomials in the velocity you in fact create the entire series principle series of catastrophes so this gives a very nice classification of mechanical time crystals alright so let's make our first generalization which is to add a potential energy term alright so we're going to add some v of x and we're going to probe this potential near to the minimum so we'll assume it's a quadratic potential and if you do that then here's our potential right here then and you write the energy as we did before then the variable X is going to try to minimize the kinetic and potential energy simultaneously but the problem is that these two conditions are incompatible because the kinetic energy tells us that the particle wants to be moving while the potential energy tells us the particle wants to be sitting still at a minimum so how do we resolve these this tension here even for non-minimal energy solutions like this one here there's a problem or at least an unusual feature which is that the equation of motion here which looks like it has a kind of a effective mass term depending on the velocity this equation of motion becomes problematic exactly at our critical value of the velocity because when you're at that velocity this coefficient this effective mass vanishes and any force at all that should be a v prime by the way any force at all is going to produce an infinite acceleration so let's explore these solutions a little more closely suppose we're near the bottom of the potential well and right at the bottom as we've said the velocity is going to want to be close to this critical value and in fact you can easily show at the turning points of this motion as it oscillates back and forth in the potential the velocity will be exactly equal to the critical value so you can our first hypothesis was that what happens at the turning points is that the particle attains that critical velocity and as we saw before that allows for an infinite acceleration so that the particle suddenly reverses course it doesn't slow down and stop it just keeps moving at that velocity and suddenly turns around like it hit a brick wall and it moves the other way so here's a plot versus time of this type of motion here it's basically a triangle wave and this is how we reconcile these two apparently contradictory conditions here that we at least accept at the turning points we are able to satisfy that we're at the minimum or very close to the minimum of the potential and we are moving at that critical velocity if we're very close to the bottom here then it turns out that the frequency of oscillation it grows with energy as sorry grows as your gap above the minimum energy decreases so as you approach that minimum energy point you approach an oscillator and oscillating with infinite frequency and this is quite a singular situation so I think to explore it further and to really understand what's happening at these turning points a regularization is required we'll find one by constructing a physical realization of this system the realization is going to be something like a hall conductor setup here sorry a two-dimensional conductor in a perpendicular magnetic field with tangential electric field and we can describe it by Lagrangian like this and let's see yes so f is effectively the derivative of this function f is the magnetic field and g is giving us our electrostatic potential okay and we're going to allow these fields to both be non-uniform so one finds these equations of motion and in the mu goes to zero limit this first equation reduces to y dot as g prime over f prime so the trick is we're going to choose f and g in a particular way a very particular way when we make the substitution y dot rather when we eliminate x out of this equation we'll get a Lagrangian for y that looks just like the time crystal Lagrangian that we've been talking about so we're going to make this very specific choice we'll take f to be this polynomial and g to be that polynomial again these are non-uniform potentials for electric and magnetic fields then the ratio of the derivatives gives us simply y dot equals x we needed this extreme fine tuning in order for this to work and of course you'll notice that this is well defined except at the point where f prime is zero that's the point of vanishing magnetic field but elsewhere in this limit y dot equals x that's the limit of mu goes to zero so if I plug this into our Lagrangian and set mu to zero and eliminate x using this equation here I get back the time crystal simple mechanical time crystal okay and when I turn mu back on it acts as a regulator and that's going to what it's going to do is it's going to keep us from having large velocities it's going to disfavor large velocities and it's going to to make the behavior turning points better behavior at least it's going to tell us what's happening allows to blow up what's happening near the turning points so with this regulator it turns out the velocity does not simply reverse at the turning points and follow this gold curve down here okay the blue one is our x variable and we're not going to worry about that anymore but this is what the solutions look like they climb for a while they reach some point along the potential they climb to the right here and then they suddenly spring back snap back to where they started okay it's like Sisyphus pushing his rock up the mountain and at the top it suddenly rolls back down to the bottom so we call this Sisyphus dynamics the motion appears quite jagged but at finite values of mu it's not everything smooths out here is I think the value of mu here is maybe one thousandth and here it's one tenth and you can see everything looks nice and smooth interesting point is that successive waveforms here it's not simply periodic like a triangle wave it's successive waveforms are not the same they oscillate actually between two types of forms and you can see it better if you enlarge the amplitude okay and what's actually happening here in this solution is we're exploring these two branches of the Hamiltonian of the as we move I think the steeper curve is moving along the the upper branch here and the flatter curve the less steep curve is the lower branch here so we're oscillating between branches you can find interesting dynamical phenomena as you vary parameters for example if you change the initial conditions you can see phenomena like period doubling starting to happen so we here we had an initial condition I don't think I wrote it down but the initial velocity here was one so oh no I guess this is this is the same picture here okay initial velocity is one and initial position is an arbitrary value of 0.7 and these are the conditions in X not in Y but it doesn't matter I'm just treating these as external parameters here so as or controllable parameters as you vary these initial conditions as you decrease this parameter 0.7 you can see that the two parts of the period here become more and more distinct and this wider part widens out even more and it flattens out close to the X axis okay so here's 0.651 and now we go to 0.650 and you can see that this this part of the trajectory here and goes back to the left side of the potential and then it comes back up makes two trips back up to the right and it goes back down on the right side before repeating so between here and there the period doubles if you continue not even by very much decreasing this parameter you can see we retain this kind of four phase period here alright and just a word about why we get period doubling turns out that this point here the transition between these two behaviors happens when the energy of the system passes through 0 and that's the point where we go from having two ranges of motion one on the right and one on the left in a single range where you go back and forth all the way from right to left and that's the reason for the period doubling alright so that is one very useful regularization of time crystals and I want to emphasize that this is not the only way to regularize I want to present another, show you another regularization it'll turn out that this leads to very different macroscopic behavior that we're just changing by a very small amount we're going to add a small term proportional to the acceleration squared to the Lagrangian here okay and Epsilon we're going to take to be quite small of order 10 to the minus 5 or 6 in the pictures I'm going to show you Kappa is again our other parameter so we're going to look at dependence on these parameters here and we'll find a very different behavior and I should give you a word of caution this is a singular perturbation we're adding by adding terms involving higher derivatives than first order derivatives we're introducing effectively new degrees of freedom and enlarging our phase space increasing the dimensionality of the phase space okay and this also as often happens with singular perturbations we'll introduce some instabilities so but let's proceed though let's take Epsilon to be one one thousandth and solve the equation of motion here which looks like what we had before except for the Epsilon term and you get exactly what I told you we were originally expecting for time crystals which is this triangle wave sort of pattern here all right with sudden reversals at the turning points well it looks nice and neat but if you look very closely here you'll see what looks like maybe some numerical errors near the top here where the curve wiggles a little but if you change the initial conditions you'll see that those wiggles blow up and we get some rather what we've done is uncovered some instabilities and these instabilities are persistent and generic and even chaotic so this was just changing the initial velocity from two down to one that we obtained this kind of picture here and if I plot the velocity on top of here this is the velocity as well as the position you can see the velocity is going absolutely haywire and if I continue varying that initial condition I get even more complicated and I can see that these two regularizations lead to very different outcomes very different macroscopic behavior both of these regularizations involve singular perturbations and add new degrees of freedom in the Sisyphus case it was this X degree of freedom that we added in the Sisyphus case the motion we found is discontinuous in the limit that the regulator goes away, but stable and periodic. In the higher derivative case, we found discontinuous velocities at the turning points and severe instabilities and chaotic behavior. Which is the right regularization scheme will depend on the microscopic properties of the system. You might be wary of introducing the higher derivative regulator, but as I will explain, this is not an unreasonable thing to do. I see that my speaking time is up. I will just flash through a couple more slides here. This is a spatial version of the mechanical time crystals here. These are materials that have ridges in them. They form spontaneously and let's see. Quantization, I will not talk about. I'll just say that it is possible, quite straightforward in fact, to quantize the Sisyphus regulated time crystal and you can calculate eigenfunctions and eigenvalues. And the interesting thing is the ground state wave function shows no sign of spontaneous symmetry breaking. In fact, it's just a symmetric superposition of left and right moving waves corresponding the two degenerate classical solutions. And we should not have expected in this case to find spontaneous symmetry breaking. Spontaneous symmetry breaking is a quantum field theoretic, not a quantum mechanical problem. And our field does not have enough degrees of freedom in order to see it. You need at least a one or two-dimensional field. I will mention that we've looked for possible quantum systems where you could find this type of energy curve or dispersion relation. In fact, such systems are known. One-dimensional cold atoms are the simplest of those in model by Gross-Peteevsky with a repulsive delta function potential, periodic delta function potential, has swallow tail shaped bands. If you make the potential attractive, they point downwards just like ours, although the ground state is not. So you could study this metastable band here and look for time crystalline behavior. And indeed you find a phenomena that might be interpretable in that way. These are soliton-trained states found in this type of band here, like band three experimentally. And I'll leave it as a question for the cold atom people, whether these really deserve to be called time crystals. Last thing I'll say is that in cosmology, one encounters Lagrangians, which also, which have very similar Lagrangians to what I've described. And in fact, they also include terms like acceleration squared terms. And so they may well be time crystals with a higher derivative regulator. And this in fact is the precise scenario for Starobinsky's original model of inflation. And so it's quite possible that the universe itself is a time crystal. All right. Last thing I'll say is, where do we expect time crystal Lagrangians to arise? Where do we expect higher derivative Lagrangians like this to arise is what the question really is because as we've seen this kind of behavior is generic when you have higher derivatives. Well, in effective field theories, we consider theories involving low dimension operators with coefficients that are fixed by experiment. And those in general will have higher derivative terms. And for large ranges of coefficients measured experimentally, we'll expect to get non-convex kinetic terms. What about the famous no-go theorems? Well, all of these will violate some of the hypotheses. For example, we'll have multi-valued Hamiltonians, which were not considered in the no-go papers. These Hamiltonians describe dynamics at least classically, which are kind of singular and the scaling properties are kind of unconventional as well. And also if you look at metastable ground states, like I mentioned in the case of cold atoms or non-equilibrium situations, you also have a way around. So these are just effective theories. They're not complete fundamental theories, but they're an enormous class. And they're quite, as I've said, generic when you have higher order kinetic terms. So we should be inspired, I hope, to look for more systems that break continuous time translation symmetry in this way. And I encourage everybody out there to go out and find some more. Thanks. Sorry that I went over. Thank you very much. You have time for one or maybe two short questions. Yes, Rosario. Yes, you said that in the quantum version, it does not show any trace of time crystal. My question is about the semi-classical regime. So do you observe any trace of this behavior? I don't know perhaps for a limited time. Yeah, that's a great question. Well, in the semi-classical regime does allow tunneling and tunneling is what causes these two broken symmetry states to superpose with each other. And that's what means that we lose the time crystal and behavior in the quantum case. But that's primarily, that's a quantum mechanical phenomenon in quantum field theories. It can be much harder to tunnel between vacua, as we know from standard examples of spontaneous symmetry breaking. And so that's the, in semi-classical quantum field theory, I think you can, that's where you may expect to find this time-dependent ground state phenomenon. Okay, the last question, Thomas. You seem to have these two time scales in the Sisyphus dynamic. So it's two different time scales, something I should be looking for in the experiment to find a system like that. So you mean the two time scales of the- Quick jumping and the slow return. Yes, so the quick jumping is your regulator time scale. That's introduced by your epsilon or your mu, the small parameter you introduced to smooth things out at the turning points. And so that is going to be governed in general, that's sort of your cutoff scale for the extra degrees of freedom that when they come into play, okay? But if I was an experimentalist, what I would look for is not just this funny jittering behavior near the ground state, but even at excited states, say involving solitons, semi-classical objects like solitons where you'd be able to describe it effectively by a classical theory. I would look for this sudden change of velocity either or sudden jumping. And I don't know if people have observed that in these soliton trains, my understanding is when they come to the end of the trap, they tend to break up. So it's hard to make measurements there, but that would- Thank you. Okay, thank you very much again, Al. So we have to move on. And the last speaker of the conference is Arkadius Kosscher from Max Planck Institute for Complex Systems. I recognize the screen is yours. Okay, hello and welcome. So let me show my screen now. Okay. Minimalize this one. Okay, so hello and welcome again. My name is Arkadius Kosscher and I'm currently affiliated at the Max Planck Institute in Dresden, Germany. So today, I would like to talk about condensimeter physics in temporal lattices. And in particular, I will show you how to realize a specific quantum simulator of flat-bent many-bent physics in separable temporal potensions with the MOPIS-3 topology. But first, let me thank my collaborators for this project, without whom this work would never come into being. And the most part of this talk, the main part of this talk is based on the following paper which we submitted to Arkadius just a few days ago. So comments are very welcome. So I will start with the structure and motivation behind this talk. First of all, throughout this paper, I will consider so-called gravitational bouncer model, which was introduced in 2015 by Dr. Stavsachan as the first manifestation of a discrete-time crystal. So let me mention here that this is also disorder-free model, like the one that was presented to us by Vincent Liu during the first part of his talk. And one of the greatest advantages of this model is that it allows for a drastic symmetry breaking, where symmetry-broken states evolve with a periodicity up to a hundred times longer than the periodicity of a drive. And as you remember, the big-time crystals were the topic of Peter Hannah Ford's talk on Monday. In this introductory part, I will also show you how the discrete-time translation symmetry can be broken in this model. Another important manifestation of this drastic symmetry breaking is that we can actually realize big crystalline structures in time. And as a specific example, I will show how to realize a system of properties associated to the dimension of time crystal, which lattice geometry can be designed with a great flexibility. And also I will show that we can have exotic tunable-long-range interactions with the system. These two features together make a unique platform for quantum simulations in time crystals of condensed metaphysics, both the known effects in condensed metaphysics, but the temporal analogs, and also the effects that go beyond the standard condensed metaphysics, which are also inactive. It's accessible by the standard quantum simulator platforms. And as a specific example, I will show how to realize a simulator of a flatbed-monotube body physics in the temporal lattices. At this point, let me also acknowledge that condensed metaphysics can also be studied in phase-space crystals, which was a topic of Lindsay and Gouw talk on Monday. If you have not seen it, I refer to his excellent review paper on this topic. So let me begin with one dimensional gravitational bouncer model, where ultra-cold atoms are bouncing on a mirror in the presence of a gravitational field. Let me first assume a classical description and assume that the system is non-interactive. For now, it turns out that in a classical case, without a drive, so while the mirror is static, all trajectories of classical particles are periodic in time. However, with a drive, when we turn on the mirror, let's say that the mirror oscillates at some frequency omega, only resonance trajectories are periodic. And this corresponds to a situation where a particle hit the mirror, which is temporarily at rest at its maximal or minimal position, so that there is no energy transfer. And in a classical case, we can have multiple resonances. A simple one-to-one resonance corresponds to a situation where we have one mirror oscillation per particle hit. This corresponds to a period T solution, but also we can have higher resonances, like the simplest one-to-one resonance, where there are two mirror oscillations per particle hit. And this, of course, corresponds to a period T, two-T solution. However, in a quantum case, all flock states must obey the symmetries of the Hamiltonian. So all flock states might be T-periodic as the Hamiltonian function. However, it turns out that flock states correspond to classical two-to-one resonance are shifted superpositions of two non-dispersive wave packets, which individually evolve with a period to T, but this superposition is too periodic, is T-periodic, because it should be so that it respects the symmetries of the Hamiltonian. So after the one period of evolution, the wave packet exchanged the positions so that the flock state has the periodicity of the Hamiltonian. Also, this is true for the interacting problem. However, for strongly attractive interactions, the particles prefer to group together in either one of the wave packets. So for large number of particles in the thermodynamic limit, the flock states is a superposition of two macroscopically different states, all of the flock states. So these are the Schroediger kets, which are, of course, very prone to perturbations. I'm sorry. And even an infinitesimal perturbation breaks this Schroediger ket, which collapses to one of the wave packets. And therefore, the symmetry, the time-translation symmetry is spontaneously broken. And this is in a full analogy with the formation of space crystals. So let me also say that this mechanism for spontaneously symmetry breaking in this model is robust for imperfections of the drive so that this time crystal is a phase, not just a fine-tuned point. For a stability diagram, I refer to the Veronica-Gullitz poster. And if you have not seen it, you can check in the air publication in the Journal of Physics. So as I have already mentioned, the high classical resonances allow for drastic time-translation symmetry breaking. So we could have big time crystals when the symmetry is broken from 20 to 100 times. And actually, it turns out that big time crystals are more feasible experimentally in this model than a pure double-time crystals. And this drastic symmetry breaking also allows us to create big crystalline structures in time so that we could study temporal analogs of condensed metaphysics. So in order to show how this crystalline structure emerges in time, I will quickly sketch the derivation of the effective Hamiltonian in the moving frame. Since a similar derivation was already presented to us yesterday by Julius Slavis, so therefore I will be brief so that we'll have more time for questions. And so it turns out that the gravitational boundary, the theoretical description of this problem is easier when we switch from the laboratory frame to a framework mirror-static but the gravitational field oscillates in time. This problem can be solved in a full quantum-mechanical way. However, for simplicity and the illustrative purposes, I will stick to the classical description. And turns out that the unperturbed problem is fully integrable and it can be solved using the action-angle variables i and theta. The action i is a concept of motion so that the unperturbed Hamiltonian can be expressed as a function of i only. And the angle theta is the periodic function which is defined between zero and two pi. Now, the full time-dependent Hamiltonian can be expressed in the same variables and to do so we need to decompose the Cartesian coordinate C as a Fourier series in the angles theta. And suppose now that we are at the general end-to-end resonance which means that the frequency of the classical particle motion is a multiplicative of the frequency of the drive. And in the moving frame which we go to by the renal transformation of the angle all the resonant projects are stationary. And therefore if we are interested in the new resonance motion that the effective Hamiltonian in the moving frame should be time-independent. And to show this effective Hamiltonian let's do the two steps. The first step we can neglect the highly oscillating phases in the time perturbation which is basically the rodative wave approximation and in the end we have a simple single cosine which is time-independent in the moving frame. Also we can expand the unperturbed part of the Hamiltonian up to the second order and in the result we obtain the effective time-independent Hamiltonian in the moving frame which has a form of a simple single part Hamiltonian in the presence of a the presence of a lattice. And indeed this crystal structure can be also seen in the exact first phase portrait of the problem with the full time-independent Hamiltonian. This first portrait shows the lattice structure of resonant islands and if those resonant islands are large enough then we can build on them in quantum mechanical vanu-states. These vanu-states in the lab frame corresponds to the wave packets evolving along resonant orbits which break the time periodicity of the drive. However for the particles the particles will tunnel between different neighboring vanu-states so that what we obtain is effectively the solid state behavior in the time domain. And now let me take a step further and consider a similar setup of two orthogonal oscillating mirrors. Since the mirrors are orthogonal the problem is is separable and we can repeat the previous derivation almost exactly. However we have now two independent degrees of freedom so that the effective Hamiltonian is two-dimensional. So that in the end we can obtain a very simple system which has the properties of a two-dimensional time crystal. The two angles theta x and theta y as previously are periodic functions that define between 0 to pi which of course defines the torus topology of a time lattice. And so let me represent the torus by a single square where the opposite size are associated with each other. In black I've plotted a sample trajectory of a classical particle so during the course of the time evolution it means the boundary and then respect the periodic boundary conditions and continues its motion. So in a result the particle is wrapped around the torus. And now let's consider a very similar system but now the two mirrors form a wedge of 45 degrees. Turns out that the system is also integrable and this is because when a particle hits the mirror it gets literally mirror reflected. It's trajectory is mirror reflected so that we can do the local mapping to the previous problem. And when the particle hits the mirror also the components of the Cartesian moment are exchanged. However, now the system is spatially bounded and it's twice smaller as in the previous case and it's constrained in the Cartesian space can be translated to the theta x theta y space and it induces this following condition. This condition actually cuts out the parts of the space and introduces two new boundaries. One of them is a hard boundary which corresponds to a pathological classical trajectory along the vertical mirror and the average boundary corresponds to the collision with the mirror and the topology of this space is probably better it's better to see the topology if we shift one of the angles by pie. So when a part classical path evolves it eventually hits the mirror and then due to the exchange of the Cartesian momenta it gets it gets to the other side and then the same in this direction. Here the angles represent the twisted boundary conditions and so we actually have a system with the mobius strip topology. You can see the details you could have seen the details on the archaic cross poster and if not you can see our recent preprint. But now let's consider two oscillating mirrors and the two non orthogonal mirrors oscillate then we have really an inseparable problem. As I mentioned we have a great flexibility in designing lattice geometries in them in the time dimension and actually this flexibility is probably greater than the engineering of optical lattices because we can have as many harmonic frequency as want by choosing different shaking protocols of the mirrors and one of the simplest non trivial examples are the homicamp lattice like on the left one left panel and the lip lattice so the lip lattice is a braver lattice with a free point basis and because of this free point basis the energy spectrum forms free energy bands one of them which one of them is flat so the particle in the lip lattice can tunnel between the sides in a one unit cell with amplitude G1 and between the sides belonging to the neighboring unit cells of the amplitude G2 and if the amplitudes G1 and G2 are different then we have the energy gap separation between the bands and this band gap will be important to us in a moment but first why are we interested in the flat band so first of all flat band has an enormous degeneracy and we can choose the eigen basis of this band in a form of localized eigen states which we can call the vanier states of the flat band and this vanier states are localized on a few lattice sides which makes actually this problem feasible experimentally so that we can load in the experiment atoms into the eigen states of the flat band and we can restrict our cells to this band only however if the system is non-interacting then there is completely no dynamics in the flat band because the group velocity vanishes identically however we can induce the dynamics back through the interactions and as I will show you in a moment these interactions could be quite unusual so that we and being tunable so that we obtain a quantum simulator of exotic many body phases and and as long as the interactions are smaller than the energy gaps then we don't have any band mixes so we can with a good approximation stay in the flat band so let's now consider the many body where the bosons are interacting through the zero range potential now if you expand the field operator in the eigen basis of the flat band then the most general form of the is the following where we have all possible two body processes with the amplitude UIGKL which is the integral of four vanier states in a normal case in a space crystals this integral U the case exponents with the distance between the vanier states but here we have a different situation on the temporal lattice in the lab frame is this vanier states of the flat band actually evolve periodically in the physical space and in terms of that even distance wave packets can physically cross at some moment of time which actually induces long crunch interactions in the model and in addition this long crunch interactions are tunable for the phase resonance namely let's consider the situation that we applied the magnetic field periodically at times when some specific wave packets overlap in the physical space then we can modulate the effective amplitude UIGK and therefore we can engineer some exotic microscopic processes in the temporal lattice however not every possible process can be realized because actually even if the wave packets cross in the physical space the coefficients U can be identically zero and this is due to the moment in conservation of the wave packets before and after the collision and therefore we cannot have density induced tunneling in our model on the flat band which processes are allowed so quite a few actually first of all if the scattering takes place between the same vanier states then we have a familiar on-site interaction which is trivial but also we can have long-range density-density interactions with different amplitudes depending on the lattice sites indexed and I think that most interestingly we can have long-distance simultaneous player tunnelings as uploaded in this figure which which conserves the center of mass of the tunneling pair I think that's all right Yes, yes, that's the last slide and thank you and just let me know that I don't plot all of the possible processes but only just a few for the administrative purposes and let me stress again that all of these processes all of these amplitudes are tunable so that we end up with a unique platform for quantum simulations in time crystals which can realize exotic many body phases the possible future directions that could be that could be that you want to analyze in the future include the constraint many binodynamics or exotic superfluids super solids but the least could go on and so in the conclusion I've showed that the gravitational bouncer model allows for the realization of big time crystals and for the study of condensed metaphysics in the time domain additionally we have a great flexibility in designing temporal lattice geometries we have a tunable long stretch interactions and exotic long distance hoplix the solvus features make a versatile platform for quantum simulations of many body physics which can go beyond the flat bed as we can engineer some more complicated lattices in time with this I would like to thank you for your attention thank you Eric so time for questions Thomas it's upload it's not a question ok thank you Rosario yeah I just wanted to ask if it's possible to have complex complex value for the hoppings well yes yes in principle it is also possible to have complex complex values of hoppings you can have some frustration induced in this lattice yes yes yes we have very good flexibility in designing the lattice geometries as I showed this in the examples but actually like this lattice geometry came from the shaking protocol of the mirror and as I said we can have almost as many harmonics harmonic frequencies as we want so we can engineer quite complicated potentials but also the frustration can be studied in the honeycomb lattice no ok do we have questions ok if not Eric thank you very much so this is that was the last talk of the conference so Rosario this is so yeah I just want to spend one minute just to thank many people so first of all I would like to thank all the people from the ICTS the technical assistant the section that helped us a lot especially Walter I mean I think we got excellent assistant and they were really devoted to solve all possible problems thanks a lot also to thank all the secretarial support Victoria Monique Adriana and especially Victoria for helping us and all the email exchange and fixing all details of the of the conference last but not least I would like to thank all the speakers and all participants for just giving us the opportunity to enjoy these three days I personally learned a lot of things I hope you also enjoyed typically you say that you know at some point travel safe home but we are all at home and so my wish is that we will be able to try to travel safely in the future to Trieste or somewhere else where you want to go and so this is what I wanted to say Christo I would like to thank first of all for the great idea of the conference this is perfect I hope that not only me but most of the participants also enjoy the conference and I would like to also thank all secretaries and all technical staff it was a great pleasure to collaborate with them many things thank you so all the best I hope to see you soon bye so by Christo at some point we have a chat in the next days so just to see great, see you bye