 I'll just pose a Jen Nicovar from Royal Holloway University, who will be talking about thermalization of cold bosons by dynamic heat bath generation. Thank you. So today, indeed, I will talk about the non-strongly correlated system, but this is a virginal problem, as you will see, which, in principle, even independent of statistics. I will talk about thermalization of closed quantum system by mechanism-termed dynamical heat bath generation, DBG. And this work is more or less summarized in these two recent publications. It is actually the strong correlation group of Bonn, namely with Hans Crocher and Mauricio Trujillo Martinez. So it's an honor and pleasure to be here, to be invited here to celebrate his birthday. And indeed, in the condensed matter community, Trieste is indeed associated in many ways with peers who was driving force behind numerous workshops and schools, as we learned today from 1992 onwards. And he is basically endless and boundless energy and drive, I'm sure, inspired zillions of scientists from all over the world. And as you can see, it also inspired his colleagues not only in physics, but also in other creative ways. For example, these drawings from Alexei, kind of typical representations of peers, all the time, especially this one. This is really quintessential peers. Beware, I'm listening to your talk. One word wrong, and there is a number. OK, but peers, I did not put any year, because peers is time-independent. However, he is interested in many problems in physics, which are time-dependent, including non-equilibrium. So non-equilibrium is, Iman, his very broad range of interests. And that's actually how we work together on some non-equilibrium problems in strongly correlated system, in particular in quantum dots. And then I drifted away to... So I got interested in transport and just wanted to apply some of these ideas to actually non-correlated systems, for example, to bosons. And so, yeah, started collaborating with this bond group. And after some time, we actually arrived to a very general question, which can be asked, which is, can a closed quantum system thermalize? At first glance, this is a very strange question, because, as we know, a closed quantum system is described by pure state. And it evolves according to unitary time evolution, which prohibits maximization of entropy and thermalization is impossible. Yeah, so when system is not coupled to any thermal bus, it cannot thermalize, as we all know. And as an example here, I took a pool of cold atoms and this is because these systems are realized experimentally and they're supposed to be extremely well-isolated from any environment. However, experiments, these are our first experiments and some numerical simulations show that actually thermalization is possible in such systems. And then there is a question, why is it possible? Even independent of our statistics. But these experiments are in the cold bosons. So there are several ideas on the market. Why is it possible? And one of them is referred to as ETH. She has nothing to do with Zurich, but means Eigenstate, just abbreviation for Eigenstate Thermalization Hypothesis. It's a very popular conjecture nowadays, but still it remains conjecture. And I just wanted to briefly kind of bring you through these ideas just to show the limitations of this conjecture. Actually, the idea is very simple. We all know about ergodic classical ergodic problem from our first lecture in any statistical physics course. So these people, so this back to 90s, to Deutsche and Sridnitsky. So these people wanted basically to formulate a quantum analog of basically classical ergodic problem. So as we know, a classical ergodic problem is impossible to solve. And I mean to prove, right? In general case, only a couple of particular cases. And so they had this idea that maybe for quantum system, so for quantum case, it's actually easier. And indeed, so their conjecture goes along these following lines. So when we have system, yeah? So some closed system in some pure state, initial pure state. So let's, we can always expand it over some basis. And there's a basis we choose, for example, energy eigenstates. This psi n's, these are energy eigenstates. C n's are the coefficients, as usual normalized. So all is fine. And then, so when we want to follow the time evolution of the state, we just need to add an exponent here, yeah? According to the Schrodinger equation. So now what we need to do, so what we want to prove kind of is a long time average of some physical observable. For example, it's called A is equivalent to a micro canonical average. So in quantum mechanics, we know what time, so the time dependence of physical observable is. This is just that, right? Where A and M, the symmetric elements with this energy eigenstates. And if you look closely at the sum, so time dependence is only here, right? It can be split into a diagonal sum and of diagonal plus of diagonal terms. But then when we calculate the long time average, we need to divide by T, remember, and send the T to infinity. And then the off diagonal terms will give this oscillating terms, which are vanishing in the long time limit. And that's why so the long time average of a physical observable will have just this simple expression. And now there is however a small problem with this expression because this expression depends on CNs. And CNs are this expansion coefficients of their initial state, right? So it means that the long time average depends on initial state. It should not be the case, right? When the system is thermalized. And that's why, so there is this discussion so that this kind of, this is referred to as a so-called diagonal ensemble. So the discussion follows along the following way. So that in order to fulfill the condition that this doesn't depend on this coefficients, basically in very stupid way to see it, we need to take this out. And if you want to take this out, we need to assume that this distribution of this diagonal terms is very narrow. And this would follow from actually a very narrow distribution of the eigen values of the energy, E. So if E ends, E ends have a narrow distribution around some average state, E, yeah? So then this A and N will be also narrow distributed and so on. And then this long time average does not depend on the initial conditions anymore. And then actually it is trivially shown that it is equivalent to a micro canonical average. So here Teta is the number again, number of eigen states in a very narrow area in energy. Okay, and then they say that's fine. Then it means that thermalization occurs at the level of individual eigen state of a given Hamiltonian. However, so there are several assumptions made during this proof. And first of all, the non-degeneracy of majority of many body eigen states, because otherwise all this of diagonal terms will not vanish. Then the narrow distribution of eigen energies. This is another assumption crucial for their derivation. And then another assumption that within this narrow energy range, yeah? So all this kind of diagonal elements of physical observables should be kind of also very close to each other. So this ETH hypothesis lies on the very narrow kind of slice of the hyper surface. So there are many problems. And because of these limitations, it is very difficult to formulate some general kind of a concept of which system it is applied and how each system will thermalize. So that's why it's only possible to check each time numerically what happens. And moreover, we also notice even some contradictions here because once there is a narrow distribution of, for example, of this eigen energies, right? So then you cannot actually put this diagonal of diagonal elements to zero in the long time limit. Okay, so we wanted to consider a more kind of general case. First of all, what if ETH, this ETH conditions are not fulfilled and then can a system thermalize? And if yes, then how? Because ETH doesn't tell you how the system thermalize. So basically what happens dynamically to the system? What are the leading time scales when the closed system thermalizes? It just tells you if it thermalizes, then why, right? So we want to ask a much more broad question. And for, just as an example, we chose the simplest non-equilibrium system, which is both Josephson junction. First of all, it is isolated. It is out of equilibrium. And this is a weakly interacting bosons, which are in principle not so difficult to treat. So basically the only condition we have is that our system is macroscopic. The Hilbert space dimension is very large, which is true already for number of particles, even 25, it's already very large. And then initially it doesn't need to be in this very narrow distribution of eigenstates, but on the contrary, very broad distribution initially usually follows for this kind of systems. So okay, so what we have is called condensate in a double well potential. This is how both of Josephson junction is realized in experiments. And what we do initially at time zero is just quench the Josephson junction. And the coupling basically between the wells. So and then we see if the system thermalize or not and how. And we showed that actually the system does thermalize under certain conditions and it proceeds during the stimulation or weight through the thermal state. It proceeds through several stages, which I will describe a little bit in more detail later, but basically what happens is that the system generates its own bus dynamically as a result of a parametric resonance. So let me tell you more details of how we do the calculations. Okay, first of all we, okay, so we need to choose a basis and the basis for the problem is their exact as single particle eigenstates of the double well. So basically one takes a double well potential source as a single particle Schwerdinger equation and here is your basis. Yeah, but again, what is very important about the system, again, that it is not very large so that the levels are discrete. So discreteness is very crucial point for our description and for our mechanism of thermalization. Okay, so this will be the basis and then we just expand the bosonic field operator over this basis in principle. So when number of levels is infinite, this is exact, but of course we have to do some approximations namely we truncate the basis basically at seven levels. So two lowest levels we assume are filled with condensate and the higher levels in this double well potential are described quantum mechanically by these B operators. So these are basically quasi particles, excitations out of condensate. And then we just pluck this expansion in the general Hamiltonian with the contact interaction described by the constant G and then derive all kinds of terms. Sorry, let me. So there will be of course terms associated with the condensate and interactions and this quasi particles and interactions between them, all kinds of. Then interactions between quasi particles and condensed particles and all kinds of mixed terms. Then in addition to that, there will be Josephson typical Josephson term plus Josephson renormalized term basically Josephson assisted quasi particle assisted tunneling in and there will be also in elastic collisions which will be responsible for eventual thermalization if at all. And then what we do, we derive so in time. Yeah, so we keep her the two times and the derive equation of motions for example, condensate amplitudes. I just show you just one example for condensate amplitude and you will see it will be renormalized by all this normal and anomalous Green's functions of the quasi particles. And then we solve them self consistently. And let's look at the results. So what is plotted here is their so-called population imbalance. So it's a difference, population difference between the two wells. So and one is population of one well and two is the population of the second well normalized by the total particle number which stays constant. So, and what this, you can see first of all that this is indeed the system thermalizes nicely. Yeah, because actually we will see why because this quasi particle excited because of the quasi particle collisions. And we also did the logarithm plot of this population imbalance and you see it's perfectly fitted by a linear. So which means there is exponential approach of the thermal state. So basically now if we return to this picture what is plotted here is from this time. So there is several times scales. So initially no quasi particles are excited and there are just semi-classical Josephson oscillations between the wells. So here what is plotted here is from this from this time scale when the quasi particles are excited and then contribute to eventual thermalization of the system. And there is this vertical line which I will explain a bit later. So in order to understand better what is going on we also plotting the occupation numbers of the levels in the double well. Okay, so what we see again, so if we in this regime which is kind of blue, yeah, regime we see that these levels are not occupied. So here there are no quasi particles excited. Then there is regime two between tau C and tau F which is characterized by an abrupt excitations of those quasi particles. Yeah, you see that it's very strong. So one, two, three, four, five. One, two, three, four, five. And the brown curve is the sum of all five quasi particles levels. And then you see there is a third regime which shows actually thermalization and the slow relaxation to a stationary state. So in order to better understand the physics of the system one actually needs to do spectral analysis. Namely we need to do a Fourier transform of the two-time Green's function. So this is the Fourier transform of the spectral function with respect. So we introduce Wigner coordinates as usual. So this is Fourier transform with respect to the difference of times, right? And this is for large times in a second Wigner coordinate. So basically this gives us the spectrum, yeah, for large times. Spectrum of this quasi particle subsystem. And remember we had five levels. So as expected we see actually five peaks and this are renormalized the Rabi frequencies. What is extremely surprising is when you take actually condensate in this phase where it's not, in the stage where it's not normalized yet and just do a Fourier transform, this is a red line. It actually knows pretty well about this kind of peaks. So you can almost overlap them. And this means that the quasi particle subsystem and this condensed subsystem, they're extremely strongly coupled in this regime, in this regime here. However, when you Fourier transform the condensate part, yeah, it later times, it completely, you see it is completely decoupled from quasi particles. It means so that we got from, from effectively extremely strongly coupled regime for condensate and the quasi particle subsystems where you cannot separate them and you cannot tell the two subsystems from each other to a weakly coupled regime when they effectively decoupled. Yeah, and then once they become effectively decoupled, they begin to serve a thermal bus for each other. Okay, so each, so for example quasi particle subsystem acts as a thermal bus for condensate and chronicononical ensemble and vice versa. Okay, so and then we check that the system thermalizes by checking how it approaches the BOSA distribution with time. Okay, so this was a bit speculative, but however, so there is this inflation model for the universe and basically the idea is that the inflation happened because of this highly excited in-front or in-flaton field. And this is kind of similar to us how is a preparation of system in non-equilibrium state. And in-flaton field is actually very analogous to the BOSA-Einstein condensate. And all this density, oscillations and the elementary particle energies, they actually resulted as a, they came as a result of parametric resonance. And we have very similar effect. We have parametric resonance of BOSA, so Einstein-Judgeson oscillations and the quasi-particle ruby excitation energies. And then there is decoupling of in-flaton by expansion and we also have freeze out or quasi separation of those quasi-particles from condensate due to energy conservation. Okay, so this was just for fun. And to conclude, so we show basically the way how the closed quantum system can thermalize and how it goes through several stages during this thermalization and that the bus is not to be there, even it does not need to be there even at initially, how the system kind of generates its own bus. And so there are at least three regimes associated with this behavior. And so as an outlook, we can expand this model to lattices and to generic many particle systems and apply to realistic trap potentials. Okay, thank you for your attention.