 Okay, so much. Should we start or are we ready? You are the, okay, much of you are the, you are the moderator, no? Yeah, I will take care of moderating the questions and everything. Yes. Okay, so should we start? Should we start? Okay, good evening everybody. It's my pleasure to welcome all of you to the ICTP colloquium by Professor Matsye Levenstein with the title, intriguing title, to thermalize or not to thermalize. That's the question. Professor Levenstein, he's a research professor at the Katalan Institution for Research and Advanced Studies, ICREA in Castel de Fels near Barcelona in Spain and at the Institute of Photonic Sciences. And his research interests are very wide, including physics of ultra cold gas, quantum information, statistical physics, mathematical physics, quantum optics, laser matter interactions. He's author of close to 500 articles and has received many international and national prizes, including the Humboldt Research Award, the Wilts Lambe Award, the Spanish Royal Physics Society Medal and is a fellow of American Physical Society, a member of academia, Europa, Europia, and so on. So it's a long list. I'm also told that apart from quantum physics, his other passion is music, in particular jazz music. And he's a acclaimed jazz writer and critic and author of a book, Polish Jazz Recordings and Beyond. And he has a huge collection of compact discs and denied records. And he has a close ICTP connection. He has visited us many times in the past. In 2012, he was one of the organizers of the workshop on quantum simulations with ultra-cold atoms. In 2013 workshop on ultra-cold atoms and gist theories, we can see very diverse topics connecting to ultra-cold atoms. So with this, I will leave the floor to Marcello to give the more scientific introduction to his work. Thank you, Ashish. Welcome everybody. I will not add much. I mean, there are just two communications which are very important. If you want to ask questions, please ask them in the Q&A, and then we will eventually stop Marcello if the question is urgent or refer to towards the end of the talk. So the topic that Marcello will present to us today, it's something which is as deep theoretical roots, but also nowadays experimental consequences in the field of quantum simulators and quantum computing. As you have seen from the abstract and from the talk, it actually, in those, and I think it represents really the breadth of Marcello's approach to theoretical physics, as you see words like gist theories, strongly correlated systems and quantum information all emerged within the same topic. But I will not, I don't want to spoil anything. And without further ado, I will let Marcello go ahead. Okay. Thank you very much, both Ashish and Marcello. And thank you very much the organizers of the colloquium for giving me opportunity to speak here. This is a great honor for me, of course. And I'm very happy to be here in this. I'm very sorry that I cannot be there in presence. I hope that soon this situation will change. So I will share the screen now and I will start my talk. Check that this sound sharing is okay, but it's probably okay. Okay. So can you see it? Perfect. Okay. So the title is to thermalize or not to thermalize paraphrasing the Hamlet, obviously. That is the question. I always start my seminars with all the slogans of organizations that are paying for my research in the quantum optics group at IKFO Institute of Photonic Sciences in Castle the Fells near Barcelona. And this organization played quite a lot, as you can see, because the group is huge. At this moment, if you include visiting people, which are more or less visiting very frequently and master students, it's 39 people. Some of them are experimentalists and they are really formally in the group and they work in other groups. I don't really run experiments. There is 10 females. It's very international, 14 counties. And we collaborate with many, many people that many of you know all over the world. This is a photo of the retreat of the group in Delta of Ebro in October this year. 25 or 7 people were there. Not everybody, but any. Okay. So the other thing on the dog is I will start to, of course, to tell you something about thermalization in the classical world and the relation to ergodicity. And I will talk a little about the subject which, so to say, calls for being mentioned in this context, which is related to the last Nobel Prize of George Parisi. I will talk a little about anomalous diffusion also. And then I will go to main subject, which is quantum thermalization. How does it happen? Why does it happen? And then I will talk about situations in which you do not have thermalization. You prevent thermalization. And there are few instances of that. Many body localization, absence of thermalization due to local symmetries like in lattice gauge theories. And recently discovered quantum many but discuss if I have time, I will also mention something which is the local specialty of Marchello in particular, dynamical purification in open systems. So thermalization in classical world. And in classical world, all of you know there are statistical ensembles which describe statistical mechanics. There is micro canonical ensemble which corresponds to strictly fixed number of particles in strictly fixed energy isolated system. There is canonical ensemble where the number of particles is strictly fixed but energy fluctuates and it's only fixed on average because of interaction with environment. And there is canonical ensemble where the number of particles and the energy can fluctuate due to exchange with environment. So they are fixed only in average. And this all goes back to Ludwig Boltzmann and Josiah Willard Gibbs, the fathers of the statistical mechanics. I mean, are there more ensembles? When I was taught the statistical mechanic by Professor Łukasz Tulski at Warsaw University in the 70s, of course, he already taught us about generalized Gibbs ensembles. So the ensembles in which not only number of particle and energy can fluctuate due to interaction with the environment, but there can be more conserved charges. Conserves in the sense that system plus environment conserves this thing, like I don't know a good momentum or electrical charge, charge for instance, but in the ensemble they can fluctuate because of exchange with the environment. An example of something which is I will, so this kind of ensembles were discovered recently in the context of quantum thermalization, but of course they are known since many years. Another example which you might not know what which is interesting is a so-called Maxwell-Diemann ensemble. This is an ensemble which is somehow a counterpart of canonical ensemble, but this time the energy is strictly fixed. You cannot exchange energy, energy is fixed and costs a lot to change the energy, but the number of particles fluctuates. It's cheap to add and subtract particles and this is an ensemble which the name comes from Maxwell, but it was introduced about 20 few years ago by Patrick Navez and my Warsaw colleagues and they called it fourth statistical ensemble. This is particularly good to describe fluctuations of particles in the Bose-Einstein condensate because the chemical potential in the Bose-Einstein condensate is zero, so it costs nothing to exchange particles, whereas it costs a lot to change the energy. Thermal systems call for ergodicity, so if you think about thermodynamic equilibrium it's of course only true in the hierarchical sense that pantera, everything is moving, but it's moving in such a way that the temporal averages are equal to ensemble averages over the thermodynamic ensembles. So thermodynamic system is ergodic when given equilibrium instance of the system, so one of the states in which the system is, it eventually visits or other possible equilibrium realization microstates of the same energy or of the same free energy or of the same thermodynamic potential. Typically approach to this kind of equilibrium, so the way the system fluctuates and so that the averages of temporal fluctuations approach average over the ensemble distribution, this particularly this approach is exponential in time when we observe the system. So now you can say okay let's take ferromagnetic visit non-ergodic in a sense, yes, because you have configuration of spins up let's say for easing spins and spins down and of course when you end up frozen in configuration with spin up you never visit the other configuration or hardly visit the other configuration. So you may say this is non-ergodic, there are two thermodynamic phases in the ferromagnetic in this case, but of course we talk about this phenomenon more as physicists ask about spontaneous symmetry break and not non-ergodic. Critical slowing down at the second order phase transition is also this kind of phenomenon that you could call in principle weak ergodicity breaking, why? Because the dynamics is very much slowed down here, you have a kind of memory, every single person with algebraic tails, with power low decades very very very slowly, there's no exponential approach to equilibrium. So you may say all right this is a situation indeed in which I cannot talk, I don't really visit all configuration accessible for a given energy or free energy because maybe I do it but I require so long time that it's completely impractical. But again about this kind of situations we physicists tend to talk in terms of critical slowing down and not in terms of ergodicity break. And then there are these other systems and here the situation is really different. I mean here the dynamics really may consist of entering in so many different thermodynamic in quotation mark phases that the dynamic stops to be ergodic and you really observe none ergodicity. And so let's talk about the example which are spin glasses. Spin glasses are known in condensed matter in solid state for many years but the first model which was very influential which has been in a sense introduced in the 75 is the model of Edwards and Anderson in which you have easing spins which interact on short range nearest neighbors with random couplings which can be ferromagnetic or anti-ferromagnetic or even sometimes zero. This is a caricature of real situation in solid state in which you have magnetic atoms that are interacting by some long range interaction that oscillate like cosine fermi vector divided by distance to the power three kind of dipole effect. But the atoms are in the random positions so we say efficiently this interaction between them are practically random. So Anderson and Edwards and geniuses of condensed matter physics said okay forget about it do it on the standard regular lattice but with random interactions for nearest neighbors. Unfortunately both of gentlemen are already have left us very recently some Edwards in 2015 and Philip Anderson in 2020. Another model which is very popular in the physics of spin glasses is the one introduced by David Sherrington and Scott Kirkpatrick in the same 75 a little later and this is David Sherrington at Oxford University and this is Scott Kirkpatrick who is actually more computer scientist working in IBM in New York. These people introduced the model which is very similar to Edward Anderson but which has easing spins that interact via infinite range interaction every spin interacts with every spin and therefore they could have solved this model using a kind of mean shield theory in more or less exact way not exactly but I will explain. The main thing that they use in order to find the solution of this model is a so-called replicatory which is simply a way of representing a logarithm of a given number x as a x to the power n minus one divided by n in the limit n equals zero it's called the replicatory because what they will do is they will take n copies of the system calculate the partition function foreign copies subtract one divide by n and analytically prolong the result from natural n to n equals zero and in this case the free energy per spin in this model becomes to be something like that so look the partition function is something that takes care about the thermal fluctuations but here it is averaged over the quench disorder describing the random couplings between the spin so this is somehow you free in this formula from the concrete realizations of this order you get the global information about the disorder system and this was published in this famous fissure of letter in 75 so the Hamiltonians as I'm saying easing spins with random couplings the probability of this random couplings was gaussian each coupling was statistically independent and it had the mean value j zero and the width j j however these numbers were scaled appropriately in order to have a proper thermodynamical limit the j zero was of the order of one over n and is a number of spins and j was equal j till that's over square root of n number of spins and in this model if you do this replicatory and you look for the so-called replica symmetric solution then you'll find out that there are three phases there is for large j zero so you are really biased toward the ferromagnetic couplings the system is ferromagnetic unless you go to very high temperature where in become paramagnetic ferromagnetic melts but if you go to low j zero and however low temperatures in comparison to the width of this distribution of random couplings then you enter a phase which is called spin glass phase in which the spins flows of freeze in a kind of random configuration determined by the disorder unfortunately it was already known in this in this paper that this replica symmetric theory cannot be right because it leads at low temperatures to negative entropy which is antisocial and it didn't also agree with the Monte Carlo simulation so there was a problem and then came the genius who was of Joe Parisi and he said okay you have to break the replica symmetry and you have to look for the solutions which are not symmetric with respect to replica in the replica theory important role is played by this matrix q alpha beta which is the average overlap which between configuration alpha for the steam and configuration beta for the spins which are the two possible frozen configuration in the glass state if they all if this all all the labs are equal this is a symmetric replica symmetric solution if they are not this is the breaking of replica symmetry and Parisi knew that it has to be like that but with the absolute genius he has guessed how one should do this replica symmetry break okay so this is a paper from which I learned about it this is a JT is a paper an order parameter for spin glasses turned out then to be a function on the interval zero x so if you the breaking of the replica symmetry if you take this matrix q alpha beta you have to do it into kind of hierarchical way so this is for eight by eight matrix you first divide it into blocks four by four you put some value on the off diagonal then you just divide the small blocks again in the house you put another value in the off diagonal and so on and so on you can do it up to infinity and then the results will be that the free energy for this system will be a functional of a function q of x divided on the interval zero one and this function is monotonous so you can invert it and the inverse function will be x of q q of x x of q you can call it probability of q because it is a function which has integral equal to one and this is really interpreted as a probability of overlap between the frozen spin configurations in the spin glass and this is what Paris really discovered and this is the essence of non-neurotic behavior of spin glasses you may think intuitively that the system cannot escape from deep minima in the hierarchical it is ordered energy landscape so in the states that the spins can freeze the distances between minima are given by ultra metric ultra metric distance whatever it means so there are a bigger and bigger by barriers between the further and further minima and the participation radio counts the number of states that are accessible from a given instance and of course the the deeper the energy you are the less configuration space is accessible to you so there is a direct non-neurotic aspect in spin glass and for this non-neurotic aspect this was instrumental for awarding the half of the Nobel prize in last year to the Paris not only that Paris he has done much more things in quantum field theory and statistical physics but this was one of the important parts this is georgio i would like to mention two of his collaborators who were essential exactly for the understanding of non-neurotic behavior of spin glasses of this fact that spin glasses can freeze in different configuration and then they essentially do not go to to visit other configurations in the accessible energetic alli at least face space mark me czar from a cold normal superior in Paris was essential for this and of course i have to mention here unfortunately the late michael angel virazoro michael angel was argentinian theoretician one of the most prominent theoreticians of of the 20s and 21st century had a lot of contribution to string theory and not only string theory but in particular also to statistical mechanics his friend he was very politically engaged he worked in argentina but we had to leave argentina for political reasons many times he worked in us he worked in france but very importantly he worked a lot of time in italy and in particular for seven years i think he was the director of the uh international center for theoretical physics interest dynamics of spin glasses must be of course very complex and very non-negotiate so if you think about it then you have to think that if panta ray for spin glass means really algebraic decays power laws and things like that i learned about dynamical spin glass theory many years ago from this paper of haim sompoliski and an etsy pelleus where they really show this very non-trivial scaling laws for decays in dynamics of spin glasses the fate of paris's solution is amazing actually because paris's solution has proven it was really guessed by the intuition of a genius but it was proven to be rigorous the first proof was in the perturbative limit so serano the dominicis which i had pleasure to know unfortunately the light serano the dominicis and imre kondor have proven that paris's solution is really a local minimum of the free energy in the replica trick theory of course what is minimum and maximum is a little tricky but anyway think physically it's a minimum free energy effectively okay so they proven that was local minimum and several years later a french mathematician michael talagrand proved that the parisifon is rigorous mathematically rigorous this was published in normals of mathematics in 2006 what about the fey uh the anderson model well in 2d 3d uh sorry in 3d and mar it is believed numeric says that there is a spin glass phase transition at finite at non-zero temperature in 2d people believe there is no uh spin glass transition at non-zero dot um uh temperature maybe there is the one at zero temperature uh there is nothing exotic in the sense that the one ground state exists up to a total spin flip like in ferromagnet that you flip all spins up down maybe there are domin worlds are non-trivial there is a droplet model of uh then uh fissure and dated hues uh in which indeed the domin walls in 3d are supposed to be even maybe fractal and very complicated but uh not much is known actually and not much rigorous is known uh there is a beautiful book that summarizing the problems of spin glasses and their complexity by dan steen and uh chak newman uh this book sorry was published two years ago it's kind of set of essays about complexity of spin glasses but i very strongly recommend it to those who want to get uh into this problem uh i was in uh after my phd my mentor unfortunately again the late fritz hacker uh told me that quantum optics is not a separate uh uh separate discipline of theoretical physics it's a part of statistical mechanics so i have i was supposed to start working on statistical mechanics and this is my second actually future of letter in my life where we were studying uh edward handerson model of spin glasses what we did here we we considered two replicas so replicas of spin sigma and sigma prime and averaged it on the with respect to random quench disorder and then uh calculated in this way effective probability distribution for the overlap variable which is sigma sigma prime so this overlap variable when it freezes thermo thermo magnetically it indicates the spin glass over there when it doesn't there is no spin glass uh funny well it's not funny enough there is a special issue of of the journal physics i devoted to the memory of fritz in which we are contributing with the hacker levenstein wilkins approach to spin glasses revisited where we kind of approach this method of averaging two replicas with the new techniques mathematically some kind of subtle point techniques and so on the results is not very satisfactory we see clearly phase transition in 3d and 4d but also in 2d which is not what one expect but still we also devoted this paper to the memory of another mentor of mine in disorder system marik ceplak who actually died in the last day of 2021 marik was working in particular on this sensitivity of boundary to boundary condition in these other systems and this is one of the techniques that we use in this new paper that is not yet in archives but it should be next week okay let me go now to another subject which is uh non-ergodic behavior in diffusion so if we talk about the diffusion we start with bronyan motion i mean this you've seen here bronyan particles making some nice trajectories which i can even plot to them this was obsessed first in the microscope by robert brown looking on the pollen of the plant clarkia pulchella in water and this is the famous clarkia pulchella and this is robert brown now what is bronyan motion again my mentor in diffusion and in bronyan motion was fritz so i mentioned him again who especially was a pioneer of systematic adiabatic elimination methods for stochastic processes including processes with non-additive stochastic noise and if you think about the diffusion of course you think about the fusion equation you have a probability distribution which evolves in time proportional to the second derivative multiplied by diffusion constant and the solution is a Gaussian which spreads in time linearly where this spreading or mean square displacement of this particle which probability distribution we are looking at goes linearly in time with the pre-factor which is proportional to diffusion coefficient d actually 2d times d okay so these are the properties so how do you measure these things you can measure it again either by looking on time average of the mean square displacement or making average of our ensembles of different bronyan particles both things agree because the this is an ergodic system it doesn't have stationary state and until i put it in some kind of loose potentials of the the bronyan particle doesn't go away somewhere but it has in a sense it goes to a stationary state or stationary equilibrium state in which the diffusion occurs like i say so the normal diffusion means means square displacement grows like time probability distribution is gaussian and everything is ergodic the average with respect to trajectories is equal to the average with respect to time uh so what happens if it doesn't happen this is a so-called normal diffusion in which the mean square displacement for instance goes like some power of time this can be super diffusive or sub diffusive depending whether this power is bigger or smaller than one and this is a three most important things of non-habilian sorry of anomalous diffusion and non-ergodicity so time averages are not equal to ensemble averages non-gaussianity the probability distribution doesn't have a gaussian shape and the correlations are non-zero between increments of the in the bronyan motion bronyan motion each step in principle is given by the independent random variable and that is aging and that is memory no longer system evolves for earlier time than the later times and where is anomalous diffusion is everywhere in particular very often in biology but also in soft matter in chemistry wherever you want in human behavior so we were exposed to this by uh uh our friends who were visiting who were in NIC for Giovanni Volpe the phd in NIC for and Janek there was frequent visiting professor at NIC for and these people started to work on classical diffusion in dynamical systems with multiplicative noise so the systems in which damping and diffusion constant depends on the position of the particle for instance and this leads to very very frequently to the anomalous type of diffusion and another person from xo and the group leader edict from maria merci apajaro who introduced us to this problem uh is these are specialists on single particle tracking so they look at diffusion and bronyan motion or anomalous diffusion but on the level of single particles they can see individual trajectories of single particles for long times but with good time resolution and very good special resolution so we are talking about the milliseconds and the dense of nanometers or something like that with lasers and things so both these people so they do really this kind of photos of the single trajectories and they can really analyze the properties of the single projectors again uh our first paper to this problem was the very simple model which still was kind of found its place in fissure letters we have proposed a model of random diffusivity so it's a motion of a particle in the random random media with quench disorder which is essentially installed in the diffusion constant so the particle goes from the regions where diffusion constant is big and then the motion is fast to the regions in which the diffusion constant is very small and this regions of small diffusion happens quite frequently with certain long tail in the probability distribution of the disorder and therefore you have a very long slowing down of the motion and this is what leads to subdivision in biological system it's clear i mean if you have a particles which have no biological role they do a normal bronion motion and don't care if you have particles that are enzymes or something like that that has to do something in biology then they very often stop and do something so the diffusion very frequently for biologically functional particles is subdivision and indeed in the second paper with experimental group of Maria Garcia Paraco we have studied these things compared our theoretical model of random diffusivity to experimental results these are experimental data and they see here that you have a non-ergodicity that the time average of trajectories is not equal to the average of an ensemble of trajectories you can measure the distribution of the fusion constant actually and you see that indeed it's peaked at some diffusion constant but it has a very long tail at very low diffusions and this is this tail which is responsible for sub-diffusing behavior and means square displacement which grows like e to the power beta with beta being 0.84 or something like that okay one of the things that we specialize in the recent years in the context of this is single trajectory characterization something dictated by this experiment they look at the single particle they measure the single trajectory for quite long time but not infinite time obviously and our my former student Gorka Munoz Gil was developed machine learning and artificial intelligent method to characterize diffusion on the basis of this single trajectory so in the real world we have models we have different features of trajectories we have microscopic models if you wish and so on we have trajectories that we get from data the problem is that the trajectories are not infinitely long they are very often noisy it's difficulty of sampling and it's very often non-ergodic and so on so on so the idea was to apply a single trajectory analyze and using artificial artificial intelligence something which goes beyond pure statistical approaches or by biasing inference fitting algorithms and things like that and this is what they did they did they really developed a method in the first paper random forest method and but then different ways of machine learning methods two characters in the diffusion anomalous diffusion actually in single trajectory this has led and so you can do different things in this approach you can ask yourself is this trajectory coming from a model of brownian motion or some other models like continuous time random walk or fractional brownian model motion or whatever so you can have an insight whether your data come from a given theoretical model you have an insight into what is the value of the anomalous exponent because you from single trajectory you get you can impose information about it and so on and so and okay i'm not going to go to the details of this machine learning approach but roughly speaking you train the neural network or artificial intelligence model on certain set of trajectories with a given property so it's a supervised learning and then you put a new trajectory and the machine tells you what it is what is the anomalous diffusion consonant this works so well that the boys have organized this anomalous diffusion challenge in the internet and there were like 14 groups from all over the world that were trying to compare their own best method of characterizing single trajectory trajectories in anomalous diffusion problems mostly with the machine learning method and this has led to this beautiful nature communication paper about objective comparison of methods through the code I have to go to quantum because I already speak too long quantum thermalization we go back what's happening in quantum situation very things are very similar but we will particularly look here at the close quantum system and here the question is how does the thermalization and if thermalization occur in the close thermal system this is a paper that was written by marcos regal and maxi mulshani and published in nature and this was a pioneering paper so they look on the close system and they said the following thing I mean if you take any observable aid in the states that evolves according to the hamiltonian close system then this evolution of the matrix elements is such of this average sorry such that you have here differences of the energy between the different energy eigenstates of the system the only thing that it can go to for long times is the time average so it goes to a so-called diagonal ensemble and this ensemble therefore if the system really thermalizes this diagonal ensemble has to remind something about micro-canonical ensemble this goes back to the so-called thermalization and against the thermalization hypothesis which rafi speaking is really the same thing that this is a hypothesis that tells you why an isolated quantum system can be described by the equilibrium statistical mechanics and this says rafi speaking something like that that this after this averaging this diagonal ensemble here gives me the averages that really for observables and low moments of these observables which remind the averages from the micro-canonical ensemble or from other statistical ensemble depending on what I have and as I'm saying this goes back to Mark Shrednitsky and just dodge Shrednitsky discovered it a little later but his paper was mostly re-influential and this is what they were doing with the bozehabad model I mean Rigault, Marcos Rigault and Maximal Shiny they were really looking on the quench system so some perturbation and they were looking how the average of the corresponding observables relaxed to the thermal micro-canonical value for the which is different here because it's a finite system from canonical but still they observed this thermalization okay they also discovered in this paper that if this happens nicely the system is non-integrable but if it's integrable then there is no convergence to micro-canonical ensemble and of course we know why because if you have many constants of motion then the system naturally should converge not to the micro-canonical or canonical ensemble but to generalize Gibbs ensemble and this is what they studied in the later papers and indeed this is the case if you have many constants of motion in quantum closed system then you will you will thermalize typically but you will thermalize to generalize Gibbs ensemble. Good. What about the equilibration scales in those close manipulations? This is still an open problem we have written of course these equilibration scales as you know sorry are related to the phasing so because any the only thing that you average over in the long time are these oscillating phases that come from Hamiltonian evolution so the e to the power different of energy level t times i this thing is the phasing nothing else so the question is how fast the phasing occurs many people were proposing different theories in this paper that we published few years ago with Arnau Riera and collaborators from Brazil and my student Christos and we somehow relate this whole theory to something that is now from quantum optics as collapse and the revivals of wave function which is really the phasing related to the fact that your initial state has energy levels which are in a certain band and the phasing of this within this band is what defines the time in which the averages approach the thermal level. Let me go to one phenomenon which is against thermalization and which was discovered a little later and this is many body localization. In order to talk of many body localization which occurs in disordered systems not only but mostly I will start reminding you what is Anderson localization. Anderson localization is a phenomenon proposed for electron by Philip W. Anderson and it is really a phenomenon that if the electron moves in a disordered medium in the medium consisting of scatterers that are for instance literally distributed randomly then the electronic wave function undergoes many scatterings gets many random phases in a sense and gets self-destructive interference so that the electron finally doesn't want to propagate in the medium but rather becomes localized okay so contrary to the periodic potential in which electron conducts and propagates very nicely in block wave function in disordered potential in 1d electron with arbitrary small disorder is localized exponentially on top of wave function the case exponentially in space in 2d it's also believed not rigorously proven but believed that it is localized perfectly although algebraic with algebraic tails in for low disorder and exponentially for high disorder and in 3d you need a certain amount of disorder to localize the particle fully so small disorder doesn't localize big disorder does and this has been observed in condensed matter but in a kind of dirty versions and very clean experiment had been recently done by people from ultra cold atoms in particular along spare and here Massimo Inguscio in Italy and that was experiments in lands in Florence uh you have this is experiment of of alias spare so what they what they create was they create uh both ancient condensate in the random potential the potential which was uh which was generated by the speckles of diffusing light essentially so the the intensity of the lights fluctuates randomly on a very small scale of micronscale or smaller in the space and the intensity causes a shift of atomic atoms that are uh energy shift of atoms that are uh and lighted with this light and this shift means that they did this this is like potential for atoms so atoms move in the random potential and indeed they get exponentially localized in this experiment the second experiment which was published in the same nature was by Inguscio here the idea was a little different they have a periodic lattice with periodic potential and they added to it in secondary small but in commensurate potential so the whole sum of these potentials was again the potentials were induced by this light shift effect but the sum of these potentials was quasi periodic and therefore quasi random and in this quasi random potential there is also localization which is very much analog to Anderson it's not exactly the same but it's very much analog so you can see that if there is no additional commensurate potential the atoms live in the extended block states if you wish so if you look at the momentum distribution it has peaks corresponding to block quasi momenta and it's uh the fraction whereas if the there is a this pseudo disorder the momentum distribution of atoms becomes very broadened and this is what reflects the localization what is many body localization many body localization by name is something similar but occurs in many body quantum system and typically in this order strongly disordered ones okay so they fail to reach thermal equilibrium and retain memory of its initial condition and local observables for practically infinite times whether it's really infinite times it's still an open question we don't know if this is really many body uh some kind of dynamical many body phase transition or no it can also occur in certain non-disorder system but mostly is related to dissolves I mean has been proposed in a kind of different manner uh by looking at metal insulator transition in weakly interacting many electron systems by um uh this gentle man body is actually and uh Igor Aleiner and also Giora Shlepnikov and others contribute of this particular description but it is maybe recently more known in the description of and the mentioned baby to Hughes uh who talks about many body localization transition there is a nice series of lectures in youtube by David on many body localization and thermalization I recommend it I learned a lot from that uh I'm sorry and what is important about the approach of Hughes in all the things is that Hughes uh in a sense considers in particular the case of strong disorder and he says if you think about spins in strong random magnetic field when the magnetic field takes a very strong value then the spin obviously uh aligns along this magnetic field this is the natural thing and this is like a local uh conserved quantity which happens in the system so he says many body localization is the kind of system in strongly disorder systems is a situation in which the system develops many local conserved quantities like constant of the motions which however are weakly related one with another which has an effect that the entanglement in such a system for instance if you take a block of the system and forget about the rest the entanglement of this block grows in time but grows very slowly logarithmically with the size of the block if system would have to thermalize the entanglement in the block should grow linearly in time and should lead the limit in which uh the entanglement the entropy of the block is uh extensive quantities proportional to the volume of the block good we have been working on this problem with the kubo zakczeski from krakow and piot sharan who is now a postdoc in ikvo and our main specialty is to really improve the exact realization method which is the essence to understand many body localization there are no at least for the disorder system there are no other techniques than exact organization to analyze it so what we did in this history of letter last year with kubo and with piot is really studies one of the models of and that's on a card saying which is j one there are two models of the spin second neighbor spin spin second neighbor interactions in random magnetic field and we have been able to improve existing numerical data and in this case we plot here a so-called tallest time whatever it is as a function of disorder all these lines which are below the orange line here means there is no dynamical many body localization transition and this peak that we found in the solutions for 22-24 spins this suggests that maybe there is so that's why it's very important to improve localization we also now apply all the techniques of machine learning for the uh uh many body localization particularly detect ergodic bubbles people believe that there's many body localization occurs like that the dead are regions in the space where thermalization occurs the so-called ergodic bubbles which are separated by the regions in which there is no ergodicity and there is really localization and this is what you can study with machine learning techniques which we did with another student of kuba atomek shaldra and kovina and kotman who is a student of me and tony asin at the equivalent who are experts on machine learning and essentially what they can do in these papers they can estimate histogram of those ergodic bubble sizes in the dynamics of the system for the systems with strong disorder or with quasi periodic disorder and then they see amazing the difference between true disorder and quasi periodic disorder and think like that another thing i want to mention is absence of thermalization and due to local symmetries which are in particular occurs in lattice gauge theory so this is another example people are working in quantum simulators of the lattice gauge theory is very intensive in our days this is one of the main subjects in quantum simulation this is another paper with kuba zacheski from kraka utitas chanda uh and luca talia kotso which is about confinement and lack of thermalization in the quench bosonic shfinger model why do we do the shfinger model is nothing more but quantum electrodynamics in one space and one time dimension so it's a u1 gauge theory which we propose to do with bosons on the one-dimensional lattice why with bosons because experimentalists laugh bosons it's much easier to work with bosons and this is why we propose it like that and of course titas is now here at this paper is still with krakow address but the next one i will mention will be already with uh with the 3 s address this was done with luca talia kotso who is in barcelona at this moment in uh university of barcelona so the model that he studied is that he introduces two bosonic fields on the lattice which corresponds to particle anti-particle and then interacting with u1 field which has its electric field operators and things like that it's u1 so it's abelian gauge theory which goes originally to this Lagrangian which is here up but i don't want to show it and then then you what you do is you quench you create a pair of particle anti-particle so this is like a meson and you let it go inside and then it will spread a little like in the leap linear bounce that it spreads with the finite velocity but that because of the confinement it turns back and informs a kind of region in which there is now a goddess city there is a lot of created electric flux tubes which should join these particles there is some lighter mesons going out to the outer space and there you can say something thermalizes but essentially it doesn't turn off this is a contaminated continuation of this paper in physical regulator and this one it's already well i guess with the abu salam international center address but also with the with the uh uh krakow address for titas titas was never in the info but he i consider him to be a member of our team because in a sense we shared the project of quanta uh quant the project in which also marchero was active uh between krakow and uh castle efforts and this is a similar model but now sorry um but now with the hicks interaction so now these bosons can interact by a contact interaction and then you can study in this model in this proposed quantum simulator uh transition from confined space to a hicks space and you can study the hicks mechanics so all the basics of the standard model there is a third paper which also comes soon with uh uh with the uh uh address of titas in trieste which is about the devil's darkest in uh in the model that is uh which is this is already in print in shippos maybe printed so it's a bosonic model in a dynamical lattice in which the spins meaning like z2 lattice gauge theory and this leads to very complicated but also with the interactions between the spins that form the z2 fields interaction is like in the spin spin and xxz model and leads to complicated phase diagram with phases that are incommensurate and are a sequence of super solid phases and sequence on crystalline multi-insulator phases which correspond to traves density waves and things like that i want to mention that i don't have time too much it's already one hour or more or less uh i have uh one two subject maybe to mention one is scars this was introduced in the context of quantum of quantum chaos so systems that are classically chaotic but we studied them in the quantum mechanical limit eric heller was talking about quantum scars so here you have an example from a review written by i think by uh my colleagues of quantum scars so if you have a standard chaotic trajectory in the classical bilia like this one then the wave function that corresponds to this state will look like that will also feel the whole bilia okay with the kind of random way of oscillations in the density but if you have a periodic trajectory like this one which is just uh a particle goes on the red line here and back now this will also spread a little as you see because of the presence of the chaos classically but will not feel the whole bilia and the same thing happens in the quantum mechanical dynamics the trajectories are somehow localized here in the middle they don't go and this is what eric called quantum uh scars and the people have then uh discovered this in the context of many body systems and the first experiment that showed it was experiment with rittberg atoms in optic traps where the hamiltonian is something like that and is the number of atoms or whether you have an atom in the rittberg state or not they interact by the long range dipole dipole interactions and they can have onset the chemical potential of energy if you wish and they can also have uh transverse sigma x interactions of transforming from one internal state to another and this was done for 51 quantum 51 cubic quantum simulator atom quantum simulator by misha looking marcus griner uh vladan vuletich and uh manuel andres collaboration between harvard and mit and this already in this model they realized that there are these crystalline state states corresponding to every second rittberg state every third every fourth and so on and then these configurations may have very long lasting oscillatory evolution and they indeed started to look at it with more care uh recently and publish this reporting in nature i guess where they study time dependent version of this hamiltonia and this has been uh and they see long lasting oscillations in the population of the rittberg states in different states in this system and this is interpreted as quantum uh many body scars which these oscillations of coquir and survive to very very long time this has been explained essentially by these people by zlatko papich uh dima abanin and maxim serbin uh in this paper which they call weak i've got this is breaking from quantum many body scars and i mean they explain it in a very similar way that eric has introduced it for the hamiltonian which is really a local uh z2 latis gauge theory hamiltonian more or less which is called this pxp whatever uh hamiltonian and this leads indeed to the scar states which are similar uh to the states that corresponds to no rittberg atom one rittberg atom no density charge density waste states and in this review they go indeed through the theory of the sinks these are the guys uh and one of the things that you see so the oscillations in time long quite a long time and you see that the point is that the scar states which you plot the spectrum and then you see that there is a set of states which have particularly big overlap with this very simple state zero one zero one zero one and these are the scar states you can also look at the and this is what we have been also looking in the recent paper we were looking also on this uh latis gauge theories but we realized with kuba zakczewski luca barbiero utzo and adi so it's collaboration with krakenzhe that you can have the scars and oscillations not only in the confined space where you maybe should expect them but also in the unconfined space and in the last paper we also uh use machine learning to discover scar states so we look at this different spectra and then as i told you we use certain cost function to recognize that the red states correspond to scars families which have big overlap in the certain symmetric initial state and that's why they lead to this oscillation in there and in the dynamics this is called weaker godicity breaking because this scar state i mean you have exponential number of states in the spectrum but only finite number extensive number of states you can classify as a scar states so they are in the measured theoretic sense rare but still they play important dynamics because they are similar to the states that you can prepare in an easy way in the system okay and this is my last slide and i have to do it because this is the guy who is behind everything and this is martello uh this is a beautiful paper about symmetries of dynamical purification in synthetic quantum matter so here they consider really open system with damping and decoherence and things like that and they point out that in a given quantum number sectors entropy can decrease as a function of time at least for certain time uh intervals signaling dynamical purification so it's a very rare phenomenon but again results with respect to symmetries you can see that the entropy in the system may may uh decrease or if you wish purification of the state which is this p here increasing time and then decrease but in the system that they consider which is again very similar to the spin chains with x y interaction this may happen and this is quite a general mechanism and i think i have conclusion to thermalize or not to thermalize so my father's advice was always do whatever you want you will regret anyway i have another advice do whatever you want you will enjoy anyway because both are interesting thermalizing and non-termalizing our way of enjoying is to go beyond physics and indeed some of you know that we have also program of trying to sonify quantum mechanics by putting quantum random processes into uh into sounds and there is a composer a postdoc in my group who does it and we even have a concert at the sonar festival one of the prominent electronic music festivals in Barcelona last fall when we have been presenting one hour music of this sort it was really translating with the help of quantum random generators from our collaborators in q-side company in xo collaboration of free improvising musicians with the composers like Andres Levin Richter one of the fathers of spanish electroacoustic music and you hear it of course it's a crazy music it's avant-garde music and it's based on randomness but if you like it you should you can enjoy it really there is the one hour youtube movie if you want from this concept which is called interpreting interpreting quantum randomness and with this i really think it's thank you very much for your attention sorry for being long thanks a lot magic for an amazing talk and for this last part which was really i mean a combination of polish humor and spectacular music thanks a lot so the field is open for questions and you can post them on q and a if you want or just write me directly otherwise i will actually start so so much i actually have a question on the first part of the seminar okay so out of this experiment and theoretical understanding that you have about these classical trajectories and kind of lead by flights i mean is there something which we can translate into the study of quantum trajectories that is also useful that's a very beautiful question the simple answer is i don't know yet but we are working on it so as you know there are several ways of incorporating trajectories or particle trajectories in quantum mechanics one is the thing which is which are quantum jumps essentially okay so in quantum optics you have this something which was developed by your friend solar and walls and monica martyre peter martyre i don't remember but mostly by dalibar and melmer and castin it's a so-called Monte Carlo wave function technique in which you evolve the wave function with the non-hermitian hamiltonian by monitoring probability of being ready to make a jump then you make a jump and then again non-hermitian hamiltonian and so on so in this case you get the description for instance of the resonance fluorescent process like in this paper actually this is what we do here exactly we take the so it's like a trajectory which comes from quantum process but it corresponds to a Poisson process of jumps there is another description of let's say resonance fluorescence for two level atoms which is based on vener quantum process which is also observable in quotation mark when you do the homodyne detection so there are several ways that both of these descriptions have the property that you average over trajectories you get exactly the solutions of the quantum mechanical density master equation for density matter so in sense they are equivalent but of course how what can you observe what can i don't know definitely you can simulate these trajectories and you can ask machine to this to characterize them for instance guess from single trajectory what are the what are the have you called them parameters of the system and things like this so this is what we start to work on another field is of course bomb trajectories okay i'm not great fan of bomb theory and i'm not crazy interpreting quantum mechanics because i'm too stupid for that but you might know that bomb trajectories can be also used as a computational tool and there are papers even in cheese of letter the guy whose name was Xavier Uriel sizing from Barcelona actually he proposed how to use the bomb trajectories for doing averaging of trajectories using for for doing the calculations in quantum mechanics so this is another thing that maybe you can use as again you generate in computer but you use machine learning to distinguish them and finally and finally i don't know but evidently we are interested in looking into it it's a very interesting direction so let me read off the first question so in the parisie solution asked by apollnero tan in the parisie solution you say that it was initially perturbative then eventually proven to be rigorous what okay no i said that i said that amazingly the parisie solution was proven to be rigorous so when you have a solid solution of a subtle point method you don't only find the solution of subtle point equation you have to find the gaussian fluctuations around it and prove that it is really a minimum or in this case marginal minimum and in the original pay until the paper of the dominicis and condor it was not not all the let's say hessian eigenvalues were considered or were investigated so i i was always thought that the second order so to say stability of the parisie solution lockout was proven by condor and the dominicis although some eigenvalues are already analyzed in original parisie pay okay so this is what i only said but then the proof of michel tarara is rigorous there is nothing perturbative there is nothing about local minimum it is a stable solution of the problem so it's an absolute minimum of the subtle of the subtle point equations so thanks i think please have my answer if you have more questions please post otherwise actually i also have another one so you mentioned when you study this quantum scars in in the gauge theories that you were not surprised to find them in the confining phase but you were surprised to find them in the deconfine phase and my question is do you think they were actually more i mean that then there was a question of stability so that they are more more stable in the confine phase and not in the deconfine but they can be there in both okay i shouldn't have said that because surprise is a relative subjective feeling and i can tell you that luca barbiero was more surprised finding that in the confine than in confine and this why he was more surprised i don't know because i have no possibility of knowing he's psycho so well but i mean i and it's hard to say but you at least okay the example that i showed before about this bosonic finger model shows that indeed there is something that leads not to thermalization in the confine phase because of confine and that i think it is clear because there are these local dimensions you keep it together and that it doesn't allow to go away so actualization i think it's clear so this is the only thing but frankly speaking i don't okay i i i don't have intuition there is another question now by tongli and the question is do we have to find all symmetries and then work with the ensemble with all the constants of motion and can we proceed our calculation with an ensemble with a lower number of constants of motion so this is more like about this sorry about this generalized gibts ensemble's question but not only okay i mean i think that the answer to this question depends a little on the system i would say but uh if you think about the system of coupled harmonic oscillators which is integrable obviously and if it happens so that the frequencies of these harmonic oscillators if you diagonalize the the Hamiltonian are incommensurate then the solutions of the thing will leave a multi-dimensional torus and we will of course cover this multi-dimensional torus with because the frequencies are incommensurate okay so in a sense you will thermalize on this torus even though the system obviously is integrable and obviously that you cannot forget any of these constants of motion problem but in practice you always have a situation that you know let's say three or four constants of the motion and uh and it's an ask to describe the statistical properties based on them i think so it's very much i think system dependent okay thanks so so i don't i cannot answer really this i would say that uh yeah i don't know i know there are works in which they try to exactly measure how many constants of the motion i have to include in general keeps ensemble in order to have good agreement and it depends on the system i would say there's no general i don't think there's general as you mentioned i mean also actually determine whether there are symmetries is principle don't agree but of course and the other thing is that yeah whether there are some exactly so then there is i think the last question then then we probably should take a short break and by shalazi is that the question is do classical and quantum spin glasses break the third law or of thermodynamics uh okay i didn't talk about quantum spin glasses because it's even much more complicated problem than classical uh now evidently spin glasses do have a property that the entropy they i mean if you understand by third law of thermodynamics the nurse nurse law that the entropy at zero temperature should be zero then it's true they break because there are many ground states and in quantum case probably also okay okay so there are many equivalent energetically states which correspond it's like the spin glass is a million of phases so obviously the entropy is um logarithm of million or something like that because there are so many ground states thanks so yeah in this sense it breaks okay but in a sense it's also trivial it's just that fact that you have a huge number of quasi-degenerated completely different and separated in the configuration space uh states which energetically are essentially the same but they look completely differently and they live in different parts of the configurations but okay thanks much i mean it's nothing trivial i agree it's not trigger but but you know what i mean that we can understand this thanks again machek i think now we should probably uh close out the the the question session and before our meeting with the students that will happen at 530 i will leave the word to atish for final no thank you very much for a very nice talk machek and as you know icdp has this tradition of interaction with students with the colloquium speakers which usually takes place in person and the idea is that they get the opportunity to talk to leading researchers you know without anybody else in the room yeah but now we will do this uh on zoom online so thank you very much for for your nice talk thank you thank you very much again thank you all machek we see each other in 10 minutes in the other the other zone okay thanks a lot again thank you goodbye