 Can you see the screen? Yes. OK. So then it looks like it's time to start. Good morning. My name is Andreas Hemmerich from the Institute of Laser Physics at Hamburg University. I thank the organizers for having me here and for putting together such an interesting event. My talk will be on breaking of time translation symmetry and robust period doubling dynamics in a driven open atom cavity system. And I will ask myself whether this should be called a discrete dissipative time crystal. I want to start with introducing to you the team whose work I'm reporting here about. There's an experimental team in Hamburg with senior postdoc Hans Kessler. And two graduate students, Gustav Geoges and Fatemann Kronkamput. And we have theory support very nice here in the Institute by the group led by Ludwig Matai with two graduate students, Jim Koskulte and Gido Homan. And the theory team is enhanced by a former postdoc of Ludwig Jason Kosmer, who is, since a couple of months, a professor at the University of the Philippines. Well, if you Google the term time crystal, then basically you can get around this nature paper 2017, the quest to crystallize time, bizarre forms of matter called time crystals were supposed to be physically impossible now they are not. And this comes together with a pretty bizarre picture that also Frank Wojcik was showing in his colloquium yesterday. And even if you compare that to things like the little green thing in the lower right corner, which is a time crystal from the TV series Star Trek, it's still pretty bizarre, this picture. And the nature article comes with a quote that I have written up here saying, this is an intriguing development, but to some extent it's an abuse of the term. And that kind of signals to me that there is an open debate on what a time crystal really has to be until today. And it may not be wise to diversify this term in that situation, but this is exactly what I'm trying to do here talking about discrete dissipative time crystals. So I want to spend two transparencies to explain what I mean by that. And I'm starting with the original proposal by Frank Wojcik, which is simply a closed system in its ground state or in thermal equilibrium that provides them an observable that oscillates. And this is indeed an intriguing scenario. However, unfortunately, theory has told us immediately that it is impossible. And in an action to rescue the idea of time crystals, people have added AC drives to their systems, such that after an initial switch on phase, there's zero energy transfer, if average, over a driving period. And if then the system offers an observable that oscillates at a frequency different from integer multiples of the driving frequency, you have a discrete breaking of time translation symmetry. And it has been called a discrete time crystal. And one of the difficulties, particularly in experiments, is to prevent undesired dissipation that could destabilize this oscillation. There have been first experimental realizations. Sorry, this was a bit too fast. There is one by the Maryland group and one by the Harvard group in 2017. And some others have come later. So what I want to do is I want to add a bath to the system, such that now there is net energy transfer from the drive to the system to the bath. However, in a fashion that no entropy is generated. And again, there's an observable of the system that oscillates at a frequency different from the integer multiple of the driving frequency. And by engineering dissipation, one can stabilize this oscillation. And that's what I want to call a discrete dissipative time crystal. And you can take this a little further or if you want a little closer to the original proposal by Frank Wilczak, by replacing the AC drive by a DC drive. Still, you have net energy transfer from the driver to the bath, no entropy generated. And the system then oscillates with a system inherent frequency. And again, you have to engineer your dissipation to stabilize this oscillation. And then one could call that a continuous dissipative time crystal. There are deliverables that people ask for if you talk about time crystals. And I will try to show for most of them that they are in place for our system. This is robustness against noise of drive, thermal and quantum noise of the system. And typically, one expects that time crystal dynamics occurs by interaction and do spontaneous symmetry breaking. And there is an ongoing discussion on how many body systems should be to call it a time crystal, how many modes that should be allowed for describing the system and also how quantum is the system that's also a question I sometimes ask. Our system is composed of atoms and photons and an optical cavity where the atoms act to shape the photon state and the photons act back to shape the atomic state. And if you want the role of the cavity in a few words can be specified by saying that the photon storage increases the atom light coupling. There's multiple scattering of photons mediating interaction between the atoms. And in our specific system, the cavity allows for very long times between subsequent scattering processes because the lifetime of the photons bouncing back and forward between the mirrors is extremely large in our system. So in the many body viewpoint, in our systems, the interaction mediated by the photons is not only infinite range, but it is retarded infinite range due to the long lifetime of the photons in the cavity. And here I want to show you, oh, this is not working well. Here I want to show you our cavity at Hamburg in a more experimental physics type aspect. I basically want to focus on these two numbers. The bandwidth is 4.5 kilohertz, which is pretty much the same as the recoil frequency. And this shows you that there are similarly long timescales for the temporal evolution of the light field, the photon field on one side, and the matter field, the atoms on the other side. And that once more points at the circumstance that I mentioned before that the photons mediated a retarded infinite range interaction between the atoms. Now the atoms have to be fed with photons to get photons into the cavity. And this is done by coupling the atoms to a standing wave that's oriented perpendicular to the cavity axis, as I'm showing here in this little sketch. And for this system, already in an early paper, very insightful, Peter Domikos and Hamoud Rich have predicted that this should lead to a self-organization phase transition. And only a year later, this self-organization phase transition has been observed by the group of Blana Bulletich in 2003 with thermal atoms. And then quite a few years later, there has been an important paper by Tillman Esslinger's group, where they demonstrated this self-organization phase transition for the first time with both condensed atoms. And they were arguing that in this case, the phase transition is equivalent to the Hebleeb phase transition in the open DECA model. So what are the signatures of this DECA phase transition as I call it now? Let's start on the left panel here. This is a situation now that I address as the normal phase. When the atoms are coupled to a standing wave, the pump wave, but scattering of photons into the resonator, it's not possible. And this is understandable if you, well, realize that for every atom that, in principle, could scatter a photon onto the cavity axis, there is another atom, half an optical wavelength away along the z direction that would also scatter, however, with exactly opposite phase. And because of that, destructive interference prevents effective scattering. This comes up with the characteristic momentum spectrum that one can look at in experiments. This is these two black, these three black dots here. In the middle there, you have the k equals 0 component. This is the BEC component. And because of the coupling to the pump standing wave, you have two high order Bragg peaks at plus minus 2h per k. And now if you crank up the power above critical value, then the atoms self-organize into a Bragg grating that very efficiently allows scattering. Because now the Bragg condition is fulfilled, no destructive interference anymore. And if that happens, you get an intense light field in the cavity, which you can look at experimentally by just looking at the photons leaking out of the cavity up here. And there is a characteristic momentum spectrum associated with that situation where you have extra Bragg peaks here and here, that signal that you have a standing wave established in the cavity along the cavity axis. And this is what I denote as the density wave phase, where you have a matter grating in your cavity, stabilized by an intercavity light field. Now as I said before, the many-body perspective, photon mediator retodding long-range interaction between the atoms is in place here. And this is what leads to this phase transition. And one can see that there's, in fact, a retardation if you cross the phase boundary. And this is shown in this experimental plot where the blue curve sort of starts in the normal phase. There's no light in the cavity here. I'm plotting the intensity, intercavity intensity on the y-axis. And you see as you cross the equilibrium phase boundary, this gray bar here, nothing happens. The system has not yet realized that it should produce density wave state now. But if you further go in, at some point, the system does realize that and you are establishing a strong intercavity light field together with the matter-wave grating. And on the way back, this is the red curve tuning out of the DW phase again at some point you're crossing the phase transition boundary and end up back in the normal phase. And you can look at the different characteristic break spectra, momentum spectra with the different kinds of break peaks that I was discussing before. And for instance, as you go in here at this point, you still have no intercavity field. So you get this kind of momentum spectrum. Whereas once you're establishing an intercavity field, you see these additional black peaks. And after you go all the way around and you're back in the normal phase, you're back to both condensate with a few boggling of excitation on top. Okay, so that is a way to see that the response is retarded here. And now there's also spontaneous breaking of the two symmetry associated with this phase transition. And this is due to the fact that there are two gradings possible that both make the black condition work for efficient scattering. And these two cases differ by the fact that these lattices, these intercavity metagradings are shifted with respect to each other by half an optical wavelength. And this can be seen by looking at the light leaking out of the cavity here because there's a different phase between the outcoming photons here and the pump wave for the two cases. And it's easy to understand that the difference in the phase is simply pi. And that can be looked at by a heterodyne detector. And this is what I'm showing you here. So here at the beginning, we are in the density wave phase. You see the red curve here. The red data shows the intercavity light. So there is a light field in the cavity and you see that this is a heterodyne phase detection signal. You see the phase of that light that leaks out of the cavity is zero. And then we are going out of the density wave phase back to the normal phase by ramping down the pump strengths and return back going into the density wave phase again. And again, we get light in the cavity. And again, you see the phase is simply zero. But in other occasions, something else happens. You start again with zero phase and you end up, if you go in the second time with a situation where the phase is pi. So you see that there are two kinds of cases and you can make lots of measurements and finally put together histogram for the two cases. And this is what you see then in the sketch D here where you have a nearly 50% probability to be in one or the other case. So now how to prepare a discrete time crystal, a discrete dissipative time crystal actually. Well, we apply amplitude modulation of the pump in the density wave phase. So the drive is modulated in this fashion. There's two parameters, the modulation index at naught and the frequency omega D, which is in the range of a couple of kilohertz. So then this is, sorry, this is the center plot that I can show you. What I'm showing you here is we are ramping up the strengths of the pump wave in order to get into the density wave phase. This is happening here. And you see once you cross the dashed line you get power in the cavity. There's light leaking out of the cavity that you can detect. And the phase detector, the heterodyne detector shows that this is the case when the phase is zero. And then at this point, the modulation is turned on. And what you see is that there is an oscillation of the cavity intensity that goes to some maximal values and then all the way to zero. And again, spiking as you see here. And this is in phase with the modulation that you apply after the pump. However, as you can see the blue curve, there is the phase signal and that produces a period doubling as you can see. So as you see here, for different maxima of the intercavity intensity, you see different phases. The system oscillates between phase pi and phase zero a couple of times here. And one can also look at the momentum spectra as this happens at this point here, you are still without an intercavity field and not yet in the density wave phase. I saw the characteristic spectrum of the two break peaks up on the pump wave. Once you're in here, you are in the stationary density wave phase and you see these break peaks. And now as the system oscillates, you see that again in these pictures that there's a period doubling. You see that the system oscillates between the different broken symmetry states, if you want, back and forward. And you can calculate the single particle atomic density, according to these break spectra. And you see that for different maxima, for this maximum and this maximum, you see different pictures. So system oscillates between the two possible metagradings back and forward. Now also here, there is obviously spontaneous breaking out of the two symmetry in place. This can be seen by repeating this protocol. We are ramping up our pump strength and a certain point starting the modulation and we see our period doubling signal in the phase detector. And then we are going out of the density wave again. You see then the phase is completely undetermined and this is going up and down by two pi. And then we are going back in again and you see then this happens here for this trace, for this specific trace with the same phase. However, for other implementations, you see this happens with phase shifted by pi. And again, you could put together a histogram and see that this is nearly a 50-50 situation. Now, what are the dynamical regimes? Basically this, what I'm showing here on the right side is a kind of phase diagram with respect to two parameters, the modulation frequency, omega D and the modulation strength that we apply. And you see that there is an area where you can see color and the color is simply what you can call the relative crystalline fraction, which is nothing else, but the amplitude of the sub-harmonic peak in the normalized Fourier spectrum of the phase that we record with our Heterodyne detector. And this is a calculation, a mean field calculation actually by Jason Kosma. And you see that pretty much the place where we see the colors here, time crystal phase is predicted by mean field. Okay, so now I want to look a little bit closer to this phase diagram and position myself at this position here with the red crosses. And you see you're not yet into the, okay, you're not yet into the regime where something happens. Not actually, you simply see a constant phase here and no oscillation at half the frequency as you can see here from the spectrum. Now, if we move into the region where we see color here, things are different. And now we are in this situation, you see that the stem oscillates, shows a period doubling. And this is as shown in the spectrum where you see pronounced peak here at 0.5. This is half the frequency that we are modulating with. And you can also go on to the other side. And then the other side, you see that now the system oscillates with a higher frequency. You see basically there's some oscillation at omega D. The period doubling contribution is completely done. The robustness against driving noise is shown on this slide here. And simply what we're doing is adding noise to the modulation here. And well, this is getting stronger and stronger from A to B to C to D. And if you look at D, this is basically the trace of the modulated time of the modulation applied. And you can practically cannot recognize anymore that this is a regular modulation now. But still you see the period doubling response here. So very impressing. This is to us how stable this dynamics is against adding noise to the drive. And here we are sort of plotting the noise strength on the x-axis, which is simply a measurement of how much noise we apply. And we look at the crystalline fraction, how much of the period doubling signal do we still see? And you see that we can apply a lot of noise before this is going down. Now finally, I want to show you a calculation that has been done by Jason Cosmer, who analyzes the oscillation dynamics for different amounts of collisional action and if atom loss is introduced. So the collisional energy here, you see is zero for the blue and has some value of 0.08. Recall energy is for the green and a little larger 0.3 for the red. And in the black trace, there's in addition a loss rate introduced basically by hands to take care of the fact that in the experiment we are losing atoms as time evolves. And what you see is that, well, system can survive quite a bit of contact, interaction before it starts to suffer, before the oscillation starts to deco here. And this is seen in the red curve with 0.3 recall to body contact interaction. The system degrades here and you can also see that even better and if you want in the two point correlation function where you see that after 10 milliseconds for the red curve the correlation completely deco here. And also, if you add the timing for the loss of atoms, pretty much you see the curves that we see in the experiment. So that's it, I'm at my summary and what I've tried to show you is a discrete dissipative time crystal in an atom cavity system. I've shown you that we have spontaneous time translation symmetry breaking, there's period doubling dynamics and the oscillation you can see is robust against system and driving loss. Thank you for your attention. Thank you, Andreas. The talk is open for questions. Ordered. Thank you, Adress, and good to see you again. Thank you very much for the nice talk. I have a question. So in our original proposal of this time-driven Diki model, we saw three different phases that pop up. There was the standard normal phase, which you also see. There was the super radiant phase that you could also then time modulate and see then starting of small limit cycles that would show time period time doubling phase transition. And we dubbed another phase where you would also expect to see instabilities but then on average, the light is zero there but you would see that effectively the system processes and shows beating with generation of additional sub-requencies. Would you say that the third phase that you showed would actually correspond to that dynamical normal phase that we predicted? Well, that could be but I cannot make a clean statement because this, well, this is a situation that's close to a chaotic response in a way and it's hard to pin down now what it really is. It's obviously none of the other phases but this is something that we can, we have to spend more time, take in more data to really classify what kind of dynamics is happening there. Okay, so effectively it was corresponding to limit cycles on the effective block sphere. That was actually, it's generating higher harmonics through the non-linearity of the spin-danks conservation and the time-dependent driver which is expected at high driving aptitudes. But how to pin down, I mean, what's the signature that really would pin down that this is the phase that you're talking about? Ah, well, you would effectively see in your heterodyne you would see something very similar. You would actually see that your light pulses on and off. Which we are seeing. Only that it's basically rather at around omega D than omega D over two. And it is not exactly regular. There's... Yeah, you should have light polishing with many subharmonic generations over there because it effectively makes limit cycles on the block sphere. So it couples and decouples from the transverse cavity. And therefore you should see bursts of lights with some characteristic frequency appearing over there. I don't be happy to tell you more about it. Yeah, okay, so I mean... Okay, I think I postponed the other questions. We have a long break and maybe we can just... Those which are interested can follow in discussion and I hand over to Rosario. Okay, so just to say that we will start again at 11 o'clock. So because unfortunately the next speaker who was supposed to give a talk is not feeling well. So he had to cancel the talk. And so we resume at 11 with Anna Sampera. All the speakers of the next session that would like to... They did not do it, but would like to test the presentation. We can meet say, quarter of an hour before the beginning and we can make these checks. Okay, so rest, enjoy the rest and coffee. See you.