 Has the conference started yet? This morning. Oh, yeah, I didn't consider getting up much earlier Yeah, it's already kind of early you are online Can can the audience hear us? Yes. Yes now we are That's why we are maintaining dignified silence Maybe you can mention we're just letting people in Okay, yes Okay, welcome everybody. I think We'll start in a couple of minutes. We are just waiting for all the participants to join us. So just Bear with us for a couple of minutes. Yeah, I think maybe maybe we can start now So we are very pleased to welcome professor Frank. We'll check here today This colloquium is on time crystals past present and future and it's related to the To the conference on time crystals, which is taking place online at ICTP this week Co-organized by my colleague here, Professor Rosario Fazio and Professor Sasha Kristoff from Jagiellonian University in Krakow And it's co-sponsored by the Trieste Institute for the theory of quantum technologies So I think I'm sure all of you here are familiar with Frank Wilczek he is a Co-recipient of the Nobel Prize in physics in 2004 along with David Gross and David Pulitzer for the discovery of asymptotic freedom in the theory of strong reactions It's one of the great landmarks in modern theoretical physics And then of course Frank is very good at naming things He has many interesting names to his credit including axioms and anions and now time crystals but of course in all these behind these names he actually pioneered completely new fields for example axioms is now very you know very important potential candidate for dark matter in cosmology Aeneons is a big field in condensed matter physics and now today we are going to hear about time crystals. So he has this impressive breadth of contributions to physics. I can also say that he is an inspiring teacher When I was a student at Princeton, he was a professor at the Institute for Advanced Study I didn't have the opportunity to take any courses from him or work with him. I was working on string theory but I recall that Frank used to have this very nice post-dinner evening meetings at his home Just to discuss about whatever was occupying his mind at the time. I remember at that time it was about various phase or something like that and it was it was held at this famous address in Princeton is 112 Mercer Street which is the old house of Einstein's It was quite an experience as a graduate student to go and you know listen to the known interesting just to bounce off ideas So we are very happy to have Frank with us today And I will leave the introduction about his colloquium to Professor Thank you, Adish Just very good remarks. I mean, of course we are all thrilled to hear Professor Wilczek This colloquium is about time crystals As Adish said is connected to a also connected to a conference is Running in these days from today till wednesday and organized by this of sucking myself here at ICTP on a topic which was initiated by Professor Wilczek almost 10 years ago a couple of papers that really Put the seed on this and Let people think about this new state of matter, but But during this Times a lot of things came out, especially very interesting Experiments and now it's not only just a theoretical question, but It's confirmed state of matter and I will just pass The floor to Christoph soccer before Stuffing The colloquium Christoph Thank you. I would like to say that I learned about the works on time crystals by Frank Wilczek and our chapter in 2012 shortly before they were published because my colleague was invited to write a viewpoint on time crystals To American physical society and it was a unique experience when reading articles you realize that uh That your eyes are opening to a whole new direction of research And of course, we know that in order to conduct scientific research, we need seeds And We know that Frank Wilczek is a person who's constantly providing the seeds. So that's all thank you Well, okay, I'm on I guess So thank you very much. Thank you for these very kind introductions and my topic today And it's nice to be joining you even if virtually I'd love to be in sunny Italy. I'm here and uh, well sunny but but cold Concord, Massachusetts, and it's Not not the same Today, I'm going to be talking about time crystals as advertised and this is a subject that Got it A distinct identity I would say About 10 years ago as has been mentioned but has roots That go back much further and I'll talk a bit about the past I'll talk about what's Just a couple of highlights of what's happening Now you can hear much more about that in the conference And I'll I'll present some ideas about where I think The subject might develop in the future So, uh, the Tao of Tao Time translation symmetry needs a name that's shorter than time translation symmetry So I call it Tao And that enables this little joke that We can talk about the general state the general position of time translation symmetry in physics And symmetry is a big theme in modern physics, of course And time translation symmetry Is a profound widely occurring symmetry If you think about it, it's the symmetry that says that there are Permanent physical laws Uh, it's also by Nurtis theorem connected to the conservation of energy And uh, so it's it's a very very Profound feature of the natural world that Another Theme of modern physics that's been extremely fertile is spontaneous symmetry breaking though, uh We know many systems where the equations have more symmetry than the stable solutions and Under A wide variety of circumstances that mismatch between the symmetry of the equations and the symmetry of a given solution Which might be a material Has a whole series of consequences that are very rich having to do with the existence of phase transitions the existence of soft modes And others which we'll come to discuss so, uh We have these two big themes What could be more natural than to join them? however Explicit focus on the issue of spontaneous tau breaking Or what come to be known as time crystals is a recent development And the fruitfulness of that union is only becoming widely appreciated Recently And why did that happen? Why did it take so long? Well, uh spontaneous tau breaking brings in subtleties it's not just another symmetry that it's not a routine generalization of Spontaneous symmetry breaking to take it into to apply it to tau The analytic mechanics of time crystals looks problematic So if you try to write a Hamiltonian in particular For a time crystal And look at the Hamiltonian equations. I'll actually write this down below If you look at the Hamiltonian equations They are equations that are what's called gradient Descent gradient equations that is the time derivatives of p and q Are related to the gradient of the Hamiltonian in phase space and uh At the minimum you would expect the gradient to zero to be zero and so nothing happens To get around that you need to have something strange And uh At least spiritually and vaguely related to that energetic criteria for many body spontaneous symmetry breaking Uh are not straightforward to apply to tau breaking because tau breaking through its Connection to the conservation of energy means that energy is not conserved And so saying you're trying to minimize the energy is uh is a little questionable So those are formal problems. I think much more alarming when you start to think about Spontaneous tau breaking is that it seems to be in great tension with Uh conventional wisdoms that are very Seem very uh Wise very well established Uh one is that in a quantum mechanical ground state nothing happens The time evolution of a quantum mechanical Ground state is or any eigen state is just that it it gets multiplied by a time dependent phase And of course if the wave function is multiplied by a phase that really has no physical Significance Uh also, uh There's a strong intuition that nothing happens in equilibrium states even if we're moving we want to move outside the framework of Zero temperature to non-zero temperature There's the strong intuition that nothing can happen in thermal equilibrium states that's that's the statement that there's no perpetual motion Nevertheless nevertheless There's been a hunt for time crystals Uh over the last few years that's actually been very exciting and very fruitful Uh, this this was uh a so-called advent Sir television series in uh in sweden called the hunt for time crystals had nothing really very Tenuous connection to the time crystals. We know and love in physics But was very popular and a lot of fun and you can see Uh that it's the name is very evocative for many people Okay, so so now let me uh launch into the body of the colloquium talking about uh time time crystals past and I'm going to focus on Uh Josephson effects because well, there's among the most beautiful effects in physics The josephson effect itself goes back to the early sixties but Generalizations are very widespread and and and large classes of modern time crystals At least there is there a spirit a an family resemblance to josephson effects And and as you'll see it's very instructive to work out the formal description of the josephson effects in a way that addresses Uh the fundamental objections if you like to to time crystals The counterintuitive aspects we just mentioned So, uh, let me remind you what the josephson effect is uh In cartoon form. We have the junction between two superconductors with what's called a weak link So that whereas in a bulk superconductor, there's a well-defined phase When you have a weak link the phase can be different on the superconducting Phase can be different on the two sides And what's found experimentally is that when you apply a constant voltage across such a junction You get not a constant current but An oscillating current an ac current So this is a search. This is a situation in which you have a time independent Set of equations a time independent physical setup that gives a time dependent response Uh We can describe this very compactly and very elegantly Uh with a Lagrangian that captures the relevant physics so, uh, there's a phase number Conjugacy between the number uh the at n actually is the difference in the number of cooper pairs on the two sides and The difference in phases delta is conjugate to that There's a coupling term which says that there's an energetic cost To having but a finite energetic cost to having a difference in phase across across the junction between the two superconductors and then finally there's a Uh an energetic cost a simple voltage if you if uh because there's a voltage if you move A pair from one side to the other and that's it Okay, so those are the relevant effects And then if those are the relevant, uh physical Circumstances that are sort of necessary and sufficient to understand the josephson effect If we vary this Lagrangian with respect to delta We learn that we have a current which is j zero sine delta And if we vary with respect to n We learn that the time derivative of delta is two e v over h bar And so putting those together we see that we have a time dependent current in response to a Time independent voltage so in this Lagrangian nothing depends on time, but all the solutions do So tau you can see that time translation symmetry is broken here because if you Change t by a constant the argument of the sign changes And we also have a soft mode a kind of Nambu goldstone mode Namely this phase delta zero Is something that's kind of the residual of the time translation the original time translation symmetry and as Uh, if you made a time translation, you would in effect change delta zero and that so Uh, that's why there are solutions for every value of delta zero Delta zero they're related to each other by the broken symmetry of time translation Instead so when the symmetry is broken you get a whole family of solutions by starting with any one of them And there's a soft mode that interpolates in between the solutions Uh, it's instructive because of the difficulties that with canonical form with Hamiltonians that I mentioned to see how that's avoided in this circumstance And it's instructive also because if we want to quantize the josephson junction if we're dealing with Cases in which the the phase Is not a completely well defined if we're dealing with small superconductors where the phase Significantly fluctuates we should quantize the whole thing Uh, and then we need a Hamiltonian So to get a Hamiltonian. Well, it's straightforward given the Lagrangian And we identify the canonical momentum by varying with respect to delta dot using delta as the dynamical variable And then the Hamiltonian is this extremely simple Hamiltonian. That's linear in the canonical momentum and Sure enough all solutions are oscillatory It's the same equations basically and you get the same solutions Uh If you write down the schrodinger equation for this, uh, you get a first order partial differential equation because the the pi sub delta turns into a derivative with respect to delta A partial derivative with respect to delta and first order differential equations can be solved completely There's no no necessity for Fancy analysis going into energy eigenvalues. In fact, that's a very awkward way to analyze the system You can just write down the solution for any given set of initial values and see see how it evolves and Use the method of characteristics to see that the classical solutions underpin the quantum mechanical behavior This in fact beautifully embodies a framework that's much more general that I'll discuss at the end Because I think this is a case where the past Coins to the future when you think about it Correctly or think about it and From the right from a fruitful direction Just a quick question. So it's the Hamiltonian is not positive definitive. No. Well, yes, we're coming to that So so what happened to that big problem that I mentioned that if you take this gradient flow and minimize It seems to imply that q dot and p dot are zero Because when you go to the minimum the gradient will be zero And the answer is there's no minimum This this Hamiltonian is Linear in the momentum so the momentum can can fall as low as you want And there's no minimum to the energy and so what so what As long as it's a closed system, we don't have to worry about it's catastrophically If it doesn't matter that it doesn't have a ground state Whatever whatever state it's in it evolves in a perfectly nonsingular way so Okay, so let me move on from there to discuss a little bit about time crystals present and again you can Learn much more about what's going on in this vibrant subject at at the conference And it looks like a very exciting lineup and certainly a lot is going on But I'll just mention a couple of highlights So some of the most interesting time crystals involved driven systems with What's called sub harmonic response? so the drive Hamiltonian Does not have complete time translation symmetry, but has time translation symmetry under a discreet translation Discrete version of time translation symmetry So if you like the drive itself is a kind of crystal but But the the system that we'd like to call the time crystal in this context Has less time translation symmetry than the equations that govern it Now sub harmonic response of driven systems is a very well studied phenomena with enormous literature Going back at least to faraday Faraday did experiments in which he looked at the surface of A bowl of mercury that he shook up and down in the presence of a magnetic field and found a sub harmonic response of of its hydrodynamic modes And there are many many other circumstances in fluid mechanics and geophysics and so forth where You find sub harmonic response but What's new and different Is that we can only really claim to have a distinct state of matter here When there is a clean separation between the system and the drive You have to have part of it that's you want to call the time crystal that's broken the symmetry and another part That's the drive But if you can't make a clean separation between them, it's just a system that has less less Time translation symmetry than then then you thought and then it's not it's not Isn't there's nothing spontaneous about it. It's just a system that that is oscillating at a certain rate So in order to identify the system as separate from the drive We and and to get the consequences that we normally associate with spontaneous symmetry breaking We need to demand that our system ceases to exchange energy and entropy with the drive Then we can say that there's a system here and a drive there and they're different things And it turns out that that that separation is enough to In the presence of spontaneous symmetry breaking fully realize the combination of rigidity and softness that characterizes spontaneous symmetry breaking and makes it a powerful tool in Different parts of modern physics By rigidity, I mean things like that we can have sharp phase transitions into and out of The symmetry broken states as we vary parameters. So we get sudden changes in in the symmetry Uh And also that the faith the the states that we get are robust under local perturbations. They have the kind of long range order here in time that Is maintained even if we allow small regular irregularities in um In the exact nature of the drive or Or other yeah, and in or uh in in the external world On the other hand, we have softness Uh because there if you make Make transformations that uh restore the symmetry on a large scale gradually Uh So ideally non-locally you could just change the whole material into it's as we saw in the josephson effect change Change the the overall phase all at once then that costs no energy It gives you takes you from a solution to another solution And it's always perfectly fine if you change it gradually that will cost little energy And the system will be sensitive to perturbations of that kind So this is the philosophy that gives named google stone modes In particle physics and condensed matter and and when you have Gauge interactions long range interactions on top of this it also gives you the higgs phenomenon and so forth so we have a combination of robustness under a wide range of perturbations and sharp phase transitions, but on the other hand sensitivity to certain kinds of Long-range global perturbations now Again all all this these wonders occur only if you have a clean separation between system and drive And it was not entirely clear prior to 2017 that this kind of system drive separation could happen it's Not true of the systems that faraday studied and is not true of many of the other systems that have been studied In the intervening years. It's not even necessarily true clear of the josephson junctions Without taking special precautions If you have a voltage that means you have a battery somewhere and the battery can run down so You need you need to You need to make sure that it's really a time independent system and whatever whatever is causing the Voltage to stay constant is really not affecting the dynamics of the junction Okay, in any case It was not entirely clear prior to 2017 that this kind of system drive separation could happen And then came a dramatic issue of nature magazine Which made me really happy Which you see here where time crystals are on the cover uh, I should Tell you that this is not a picture of a time crystal at all not even remotely Uh, it's an it's an artist's inspiration somehow of a mystical thing which is some kind of combination of time and crystals and That has very little to do with the actual experiments, but inside the inside this issue. There were two wonderful experiments that showed that you could have these kinds of so-called driven systems so-called floquet time crystals In which there was a clean separation between the drive and the system of interest and uh, and you got some of the Characteristic signatures of spontaneous symmetry breaking. So there were two discovery papers There was a so-called news and views which was a scientific Appreciation by an expert. There was there were also and then there was also a very large More popular description of this so it was an issue that was just chock full of time crystallology and The the basic recipe is deceptively simple For making these things one uses a two-step stroboscopic drive In one step we apply what's called the many-body localization Hamiltonian that Uh Whose eigenstates are locally ordered with respect to the z direction of spin. So Roughly speaking it's kind of uh, A disordered ferromagnet where you have Ferromagnetism of an unusual nature where There are conserved pieces In different places. It's kind of ferromagnet that does not have normal thermodynamic properties It's very hard to transfer heat from one place to another And in the second step one flips all those spins So the special feature of many-body localization, which is very which is important to getting this separation Is that it can reach a State of equilibrium where it does not keep heating up as you apply the drive Uh, it it settles into a sort of non-ergodic state where That it no longer can absorb heat it reaches a kind of equilibrium. That's non-trivial Where it no longer can absorb heat from from the drive And in the second step, so it's ordered and as ordering with respect to Preferred direction the z direction and in the second step one uses nuclear magnetic resonance to flip all the spins so There's a two step drive and one step we establish the state and the other one in the other step we flip all the spins and Because the spins are flipped The result of this cycle is not the same as as uh, as what you started with you have A flip and you change the periodicity of the the periodicity of the response is twice as large as the periodicity of the The period is twice as large. I guess the periodicity is quicker less than uh, if uh, then the Then the drive so, uh for physically realizable states Then the ones that have these ordered arrangements of spins Time translation under one cycle period is broken on the other hand time translation symmetry through two cycle periods remains valid We have what's called period doubling Uh Choi at all even in this pioneering experiments reported also that we're using a different protocol They'd have period tripling in the intervening years many many more elaborate patterns of Time translation symmetry breaking have been observed and So here is the ax is a numerical simulation and then The actual data looks something like that and so you have the A non-zero expectation value of the spin in the z direction which flips Half as fast as the drive Is is as the drives period And this behavior Can be is robust under changes in the Flipping protocol so that it's not precisely flipping by uh by pie Nevertheless So it doesn't directly flip up to down. It's a little bit off. Nevertheless, the time crystal is robust against that that Uh for a range and also the amount of coupling Can be varied and you see phase transitions between the time crystal with this Doubling of the behavior of the order parameter And more conventional thermal states or trivial response where it follows the drive Directly and there are sharp phase transitions that Distinguish these as a function of the parameters Yeah, I should uh Yeah, I should have mentioned let me go back to it for a moment A little bit more about the two experiments Uh one of them Involved uh a handful. I think I was 11 or 12 Different atoms that whose interactions you could control very precisely using Lasers And so you had a Hamiltonian which was suggested by theorists and was the basis of those numerical simulations and and basically the experiment was just a simulator which Realized those Hamiltonians by clever manipulations of the atoms using Lasers to put them into levels where they could exchange energy or not and so forth Uh, the other was a more naturalistic experiment in what in using what are called nv centers Uh nitrogen vacancy centers in diamond and in this case they were it's a A more natural system where you doped things in uh to a diamond dope nitrogen in and uh that presumed that Those vacancies have effectively dangling spins which interact with each other And it turns out that those For reasons that are not completely understood because this was not Simulating a known Hamiltonian, but a natural system that was built for other reasons That that this behaved as a time crystal too So as I was mentioning since then many other time crystals have been explored Uh, it's important to note that different investigators have interpreted the concept more or less strictly So When we say that it's that we that we want to insulate from the external world We don't want to we have to when we have a state of matter. This is a very general Consideration when we have a state of matter We uh that exists under certain circumstances for a long period of time and how you strictly define the circumstances And how long the period of time you wait is is very much Uh Up to you and and and people can define things in different ways Uh, otherwise the only stable state of matter would be a black hole Because eventually things will the protons will decay and your your system will decay will Spontaneously collapse into a black hole. You have to uh, you have to use things that are relative to human human times or laboratory timescales and And a certain range of perturbations that you allow Or or shield again or protect against Okay So so people have used different definitions of time crystals if you use a very strict definition Of what spontaneous symmetry breaking means then you get more consequences But there are fewer examples if you use a looser definition Then you have more examples, but the the uh These the strong consequences of spontaneous symmetry breaking get diluted Okay, so uh here i'll just so again if you want to learn more about this thriving subject There's a conference going on and it's all around you There's tremendous vibrancy on the internet You can you can tap into Here i'll just share one of the latest which is very exciting and Uh Brings in spatial dynamics This is uh just appeared from Groups in and located in germany and poland Uh where they they have a system uh with a Magnons that condense into a kind of spin wave That evolves with time That uh can be considered as a Uh A spacetime crystal so it evolves not only in space but also in time and I should be able to show the movie Do this right? Yeah, there we go So they actually have A picture where you can see And this is very exciting as we'll see because when you have Uh time crystals that are that also have spatial structure a whole new set of questions arise And opportunities Okay, let me see Good, so let me then say a few words about the the future Uh, and this is going to be a very biased View of the future. It's basically things that I've been interested in recently and I'm developing but they're There are actually no proper papers on it yet and uh, so Uh caviar at empty or bio beware But I think these are very promising ideas and directions One of them. I've already alluded to the idea of sensitive modes. So what are time crystals good for? Their rigidity Their insensitivity to perturbations Uh could be used I think plausibly in the future to make better clocks better atomic clocks better along along If the improvements could be made along several axes one is absolute accuracy one is making things that are smaller and tougher uh As you know atomic clocks are a big field that uh power the gps system among others And making ones that are smaller and portable would be a very important thing As well as making things that are more accurate So, uh, it's very intuitively plausible that by Hooking together many atoms You can make things that are more robust than and Still have the advantages of single atoms But another direction that that would uh exploit the rigidity having to do with Spontaneous symmetry rigging, but there's also the question of uh, it's rigid rigid against some kinds of perturbations But unusually soft against other perturbations the ones that tend the more global trans transformations that tend to restore the symmetry so, uh We know how powerful that can be in the context of super fluidity and superconductivity Where you can derive all the main phenomena Starting with the notion of spontaneous symmetry rigging the main macroscopic phenomenon The essence of the matter is exploration of the vacuum manifold through a long wavelength almost non-local perturbations Now, uh To make Though uh, conventional namboo goldstone modes, uh, you need To be able to make perturbations that are very gradual and it would seem that if you have, uh Discrete symmetries being broken as in the, uh Floquetime crystals, uh, that This this this couldn't be applied. However, there's another way to get soft modes Which let me exemplify in the icing ferromagnet uh And basically the ideas instead of moving individual spins gradually you move the boundary between spin Spin up and spin down Gradually by small perturbations So, uh, here's a concrete example So we take an icing ferromagnet and then So it's symmetric between spin up and spin down. It chooses one of those directions Impose a global field a global b field to make the chosen state Energetically unfavorable so your system chooses a direction And then you apply apply a little field to say that you chose wrong That's the wrong direction and then it's then it's metastable It would like to point in the other direction But it has a hard time doing that because, uh Because you have to flip all the spins and the interactions are local and that's Not easy to for them to all cooperate and flip at the same time There's a big barrier to it. In fact an infinite barrier in the infinite volume limit Uh, but what you can do is give it some help by, uh Flipping some finite region of spins And then the question is does this region grow or not? And the energetic differences between growing and not are very small and that's connected to the fact that you can Make continuous changes in the shape Or more or less continuous changes in the area And see that there is a very small energetic cost to, uh, to those kinds of small changes so, uh Well, I won't I won't attempt to to Because of the time limitations. I won't go into great detail great detail on this but, uh You just take that kind of issue in space And turn it into time where we have a kind of icing behavior More like an icing ferro magnet in time where the flips the things were flipping up and down as you saw and, uh what you can do is make A magnetic field which favors the other directions of the spins an oscillating magnet and, uh, and then a finite region where The system does what it what it really wants to do energetically and then see if it grows and so you can reproduce all that uh, it's kind of, um hyper cooling, uh super cooling and, uh Nucleation in the context of time crystals, I believe and because the energetics near this, uh Transition are controlled by the fact that you're trying to restore a symmetry We can make universal statements about The relevant modes Another possibility That arises when you have spontaneous imagery breaking is the possibility of topology When you have defects and if you have Time crystals now that are also have spatial structure Uh You or even if they don't didn't have spatial structure, but but it's easier to visualize if you have spatial structure You can imagine introducing defects Uh, you can imagine for instance. Okay, so our our pump is going through a certain number of cycles In general, but you might in a small region of space have the pump doing one extra cycle Or one less cycle that introduces a defect Into the into the time Uh crystal how does it respond does it respond in a way that is stable as it goes forwards in time? What are these kinds of topological quasi particles that Settle out or are they reproducible? Very little is known about this, but it seems like a very interesting kind of question How can we make it if they are reproducible? Can we make a gas of them? What are their properties? Okay, so that's that's one idea for the future another one that I I'd like to share is also Quite nice. I think so this was a beautiful picture that caught my eye. This is supposed to be a picture of a proton Uh But wait a minute. Okay, so proton is supposed to be made of three quarks. We understand that but but then What what is this beautiful picture mean where you see things happening inside the proton? what And we have some intuition that there are quantum fluctuations going on and and things are moving around but On the other hand if we go back to our textbook of quantum mechanics or just look at a proton nothing is happening right, so So, uh What are these pictures mean? We think they mean something and more more more over in chemistry chemists are very fond of drawing pictures and molecules with balls and sticks and things moving around and resonant resonant resonances Uh resonating valence bonds that percolate around what what does that mean in the context of ground states or Things that have had in some sense aren't changing in time Okay, so the intuition I think behind this movie and many related pictures Is that the quantum state can be described and it's advantageous to describe it as a superposition of quasi-orthogonal states that in some sense evolve classically So we take not the usual analysis of a wave function in terms of energy eigen states where nothing happens But take the wave function as decomposed into a state into a frame that itself evolves in time so states that are that have a kind of uh coherence to them a classical behavior that are quasi-orthogonal that we can understand And when we're making these pictures we're making pictures of a typical such state that's That's the now can this intuition be made precise and may be useful And I of course, I wouldn't be talking about this if I didn't think the answer was yes Uh, first, let's think about it in a slightly more formal way Uh, what we'd like to say is that we describe the path integral. Uh, that's uh applying to our quantum system, uh, and We want to we want to decompose it into a collection of dominant paths That are not necessarily time independent And but when but one changes into the other as a function in other words, we have kind of time crystals that That are the a better way to analyze it and this is very much the case in the josephson effect and the josephson effect That the uh as if you quantize that the energy eigenstates are of very little interest what that what and the the the thing or or I shouldn't say that they're of interest, but they're not the natural way to analyze it The natural way to analyze it is in terms of the time dependent solutions that we know as the josephson effect that people see in laboratories And and and and perturb around those To build in the quantum fluctuations so Now there's a Beautiful, but I think underappreciated way uh due to kutman and von neumann of recasting classical mechanics into the same Framework, I'm sorry. I duplicated the work The the the word form and framework this should be you should choose either form or framework But in any case Framework where as we have come to know and love in quantum theory with wave functions Probabilities strobing their equations and so forth. Let me show you how that works. In fact, I'll show you a generalization of the kutman von neumann trick that Takes even dissipate of quantum mechanics into the framework of a dissipative classical mechanics into The framework of quantum mechanics so Just consider a dynamical system a classical system that that's you Have in this form. So hamiltonian form is fine, but it's not necessary hamiltonians are Hamiltonian formulas and gives you this but you could you have other things that uh, for instance, I studied in chaos theory or Vander paul oscillator and other things that that Are analyzed this or fall into this category, but aren't hamiltonian and what you do is Make these quantum mechanical equations of motion of a system Uh, so you elevate the y's to operators in hilbert space that commute with each other You can get however this equation of motion as a quantum mechanical equations of motion By introducing conjugate variables to the y's so effectively momenta and then uh using this hamiltonian Or actually you can also add to this hamiltonian any function of y because That will commute with y's the y's and give you the same equations of motion But this one's good enough to get those equations of motion And so we've embedded our classical system into a Full quantum mechanical system as wave functions shorting your equation, uh, and so forth So at the price of expanding the formalism and using a rather unusual hamiltonian that's linear in the momenta When even these particular momenta Exactly what we found in the in their joseph's in effect We've embedded our classical dynamical system into a conventional quantum system the special feature of wave functions in y space is that they don't spread in fact they Evolve according to the classical evolution. That's how we arranged it. So they can be The things that That beautiful picture was a picture of it's a picture of a classical approximation Or a classical model of a quantum mechanical system Promoted in but we can now promote it into a full quantum model and embed it as the start of a well-defined approximation scheme so in this way time crystals become The foundation of everything as Of describing an alternative way of describing Quantum mechanical systems that can can be advantageous. I think in a wide variety of circumstances okay, so With that I will just Conclude I've given you a very personal sampling of a few of the highlights of time crystallology I hope I've at least conveyed the sense that It's an exciting subject and the hunt continues in laboratories computers and minds. Thank you So thank you very much. Thank you for a beautiful talk. So I think I don't know probably it's easier to start from the panel if you have question. Otherwise, there are several questions that are Going up. Should I should I should I I can Yeah, no, no, I can I can read the The question, okay, I'll leave up. I'll leave this slide up anyone because I think it's a very charming What? Okay So there is the first question by James singled by so He would like to Some clarification would like to understand better why Justice and juncture is not the perfect example of a time please. So let me see understood this way I think it is a very good example Actually, okay, but some people some people claim that it's not truly an equilibrium system because you have a voltage Uh And to be sure if you plug into a conventional circuit with a battery the battery runs down and there's dissipation and there's resistance somewhere Uh, but if you take it on its own as an ideal Lagrangian, there's no question that it It you you get time dependent solutions to time independent equations and not only for the ground state but for all solutions and the And and that's a system that can be very very well approximated in a laboratory So yes, it might eventually dissipate But it takes a very long time if you take appropriate precautions And In other systems Then sort of electrical is I mean the josephson phenomena Uh is much more So similar equations govern other phenomena and magnetic systems in Not only in superconductors, but in superfluids and and many other circumstances And in those circumstances The question of dissipation may be even less pertinent. So I think it's a I think I think it's a perfectly fine example And uh, it's only disadvantages that it's old and familiar and that's uh Uh, but but also at some point maybe maybe it's maybe it's you know, maybe it's uh You know the kind of perturbations it's robust against In the case of josephson junction Maybe is not as large as you would like because after all it is electrons that are electrically coupled and and so so But but but it's a pretty good example Yeah, so in this respect may you comment also because lasers have equations that are close to The time dependent josephson effect, right? Yeah, I think there's a similar that's right. There's a similar discussion for lasers although I think they are uh It's harder to make them clean cleanly non dissipative. I would say But if you could you uh, you might expect to get sharp phase transitions and the combination of rigidity and softness that's characteristic of spontaneous symmetry breaking There's but but I But it takes special precautions. I mean so so I mean there's a large literature on of course on laser phenomena of all kinds and there's there are strong analogies to phase transitions but uh well, I Think they could be sharpened. Yes. I do. I do think they could be sharpened and That that would be a very interesting thing to do, but I should say, you know, my my command of that literature is is not Professional so there may there may be a lot of things that people have done that I'm not fully aware of So there is another question by uh, jaffari can a timescale for a time crystal be as small as a plank Uh, well I don't know That uh, that brings in a whole raft of other questions. Uh, maybe maybe The appropriate response I can give is that Of course, you know at the very very broadest level we can say that uh, the big bang is a kind of Spontaneous breaking of time translation symmetry because we have the the equations the underlying equations as In most formulations that we know of not only within the standard model, but also in string theory Basically have time translation symmetry And yet the solution we live in does not an asymmetry and uh And actually there are some identifiable soft modes steve Weinberg has emphasized this and his approach to cosmology So, uh, so there's a little bit of that if you talk about cyclic universes. It's also possible but another possibility That's that that's more novel but much more speculative is that There may have been in the early universe all kinds of things might have happened in the early universe We just don't in the very early universe. We just don't know there might have been a period where you had not smooth evolution But a very rapidly oscillating behavior taking over for a while So a time crystal in the in the in the more usual sense where you have many oscillations But uh It's it's also interesting to speculate that under extreme conditions Of curvature So, you know at the center of a black hole for instance as well as in the early universe You might have new phases of spacetime that are crystalline in some sense and if Because you're talking about spacetime. It would be very natural very unnatural to just have crystals that are not also time crystals So so yeah, so I I think there's there's uh There is a small literature, but but it's an interesting possibility to consider time crystallization of spacetime itself And possible application areas would be black holes in the early universe Yeah, so, okay, so probably I take the chance to ask myself a question Is it possible? I mean, what would think probably an application of time pressure would be in synchronization, just, you know, you know what? Synchronization or just trying to synchronize different. Yeah, well Self-correcting ways to synchronize things Yeah, well in the in the context of of clocks for instance if you have something that's uh Robust to perturbations you can use it to feed back into the things that Drive atomic clocks and make corrections if they if they if they wander a little bit So that that that's that that's the atomic clocks. I mean, uh Several different applications have been speculated about in the literature I'd be very interested to hear at this conference what some of the more some more details But uh, one one possibility is that eventually when we have quantum computers, they're going to need to I mean all all computers need clocks that you know that do Operations where where they have to be synchronized across different modules And if you want to have clocks that are gentle and don't Uh interfere with the quantum mechanical behavior, you probably want to use some kind of time crystal clock if and uh, these Magnon clocks might have technological potential Certainly the authors of This paper and also there was another paper Recently from Helsinki Their low temperature group Mentioned that they have applications in mind, but but the papers Well, I didn't read the papers closely enough to see exactly what they had in mind If I'm not sure you can find out from those papers exactly what they have in mind, but you should be asking them not me I don't know they uh But so it's early days. I don't think we know What if any technological applications they will be but I think because of This combination of robustness to most perturbations But sensitivity to Selected kinds of perturbations Uh, those those are very potentially useful properties sensitivity for sensors obviously and robustness for keeping stable time And uh being able to synchronize reliably Adish Yeah, just to follow up on this Josephson junction question. So So it looks like the trend crystals have nothing to do with many body physics or even quantum You can have a completely classical system with just a single degree of freedom Well, uh, well, I mean the jokes. Well the joke As I said, uh, the people you can Different investigators have used different definitions or different different, uh, criteria for, uh Uh, what they mean by spontaneous symmetry breaking and Depending on how strict your definition is you'll have more or less Of the desirable features of or the interesting universal features of robustness and softness Now, of course the josephson effect is a many body is definitely is a many body effect It's just that we're focusing on collective, uh degrees of freedom that, uh Uh Can be treated To a good approximation, uh On their own, uh So and that's what makes it robust if it really is, you know, the delta is the phase of an enormous thing Enormous number of electron physical So, uh, so it's still it is a many it still is a quantum and is very much quantum mechanical There were h bars all over the place but, uh Yeah, but I I do think your remark is well taken that, uh, it certainly is possible and fruitful to, uh Think about, uh, classical dynamical systems that have this property Of, uh, solutions with less symmetry than the equations. Uh, of course, you know That's a huge subject in itself and a huge classical subject with the whole business that goes into chaos and, uh Nonlinear circuits and all kind of yeah and limit cycles. It's you know, it's a tremendously rich subject, uh, but most of that has to do with, uh dissipative systems and The kind of robustness and soft mode structure they have is rather different However, uh, you know, it's it's not An entirely different universe and I think one can one can cross fertilize between Those different ideas So this one this one more question Sorry, I just read it. I I'm not sure I really understand very well But say can a time crystal have a space crystal equivalent in a sense that you can do rotations between and just Transform one in the other. I think it relates to well, yes Often there's locking between space and time symmetry. So if you think about it in a traveling wave The traveling wave at any point changes in time at any Time changes as a function of space, but if you move space and time together in the right way It's symmetric. So you can have these locked symmetries and that's uh, that's definitely possible and there's There's a kind of you can imagine a four-dimensional classification of possible crystalline patterns And things like that. So in fact, that's where I started as a kind of formal investigation in group theory as to possible patterns, but the dynamics I think is has been has been especially fruitful And and and for the patterns by the way, there's a very highly developed literature Uh Of of of classifying that and so called bifurcation theory. Well, you see what how one pattern can change into another So there's a there's a very well developed rich mathematical literature Just ready for use. I think in the time crystal Take me In the time crystal context Yeah, so I should apologize Now because then the the q of the unanswered question is growing much faster than You know the speed at which I'm able to read. So I'm not sure we are going to Okay, so uh, tomas fischer Ask about If you say something more about, you know, you mentioned about the A clean distinction between the system and the drive. So Yeah, you just say so why is the drive? What is the system? It's just a purely theoretical point of view that can be checked or not No, no, no, it's it's not it's much more than theoretical It's I apologize with tomas. I'm condensing the because it was a long I actually can I can see the question here. Uh, the But it's it's not It's it's not it's not conventional at all. It's uh The thing is, okay, you put you have a drive and you have a system and they start by exchanging energy and the drive In general because of friction, uh, if you don't pump energy out of the system You'll just have the the the system heating up If you do take things out of the system, it's not closed and and you know, it's it's uh You shouldn't be treating it as a closed system. Uh, so the criterion is that you have you the the system settles down Into a state where it's no longer exchanging energy and entropy with the drive so Mathematically, it's as if you have an external a time dependent Hamiltonian of of a closed system It's implemented through a drive, but the drive is not exchanging entropy and energy with the system And so it settles down into the system settles down into a kind of autonomous behavior as if it just has a time dependent Hamiltonian and uh, the objective Correlate of that is That the system doesn't keep heating up that it that it forms that it falls into a reproducible equilibrium And it's it's properties uh Don't change anymore And uh, so so it it has settled down into a well-defined state And as you saw that can be phased transitions and So, let's see. Let's also ask Oh, sorry, sorry Go ahead. No, go ahead. Okay, uh, let's see. Let's ask, uh, is block oscillation in crystals can be considered Another example. Oh somebody, you know, somebody once told me what block oscillations are, but I forgot So I can't answer the question Yeah Okay, uh, I don't know but the general answer is to all these questions though is uh that If you use a very strict definition of time crystals, then you'll have fewer examples, but more consequences If you use a very loose definition You'll have hordes of examples, but very they'll have very little in common And they to me the dividing line is you want you really want to have this kind of robustness against a large class of perturbations and softness against some other chance Class of transformations that are connected to the symmetry breaking. So another At least you you tell me when we should stop. I don't know. I mean that we can go like this like this So another question is that in your talk, you put stress on lack of dissipation by our Chuck There are a lot of recent works on disability time crystal. Can you please comment on them? Well, as I said, if you if you, uh If you lower the criteria Then there are more examples, but fewer universal properties I would have to look at them sort of on a case-by-case basis to see where they fall Yeah, that's that's what I'll say, you know that If you use a looser enough definition, then Then you you you get back to Faraday and And that that's fine, but you should you should realize that You're not necessarily getting the full Uh Weight or implications of spontaneous symmetry breaking or because Really your system is not The closed system includes both the drive and what you call the system. So it just has less symmetry It's not a matter of breaking spontaneous. It just has less symmetry so So that that that's the issue whether whether you can identify a separate system that has less symmetry But is separated from from the drive and Maybe you can loosen that criterion to a Quite a bit and still preserve the good properties, but That would have to be examined case by case Okay probably I should apologize for all the people that could not Then I could not read the questions, but I think there are Constraints time constraints I got already emails from Icts system that We cannot go on so I really would like to thank for the To thank you a lot for the very nice and inspiring A lot. Yeah, thanks. Thanks for your interest and it was a real pleasure Thanks, and thank you very much. Yeah, and it's been uh, it's you know, it really woke me up this morning in a nice way well, okay Hope to see you in ictp in person. Maybe sometime in the future and Okay, yeah, and I guess now we're going to adjourn to another another room and then yes Yeah, so if you like maybe a few minutes like what do you think like 15 30 in about Yeah, a 30 minute break 30 minute. No. No. No. I was saying 10 minute break for eight minutes 10 minute break A 10 minute break is less wonderful, but I'll also do that. Sure Okay, I think the idea is that you yeah, just to interact with our students and then I will leave you All right. Very good. Thank you. All right. Bye for now. Okay. Thank you Bye. Bye