 OK. Shall we start again? Hello? Is there any chairman clapping hands around? OK. Maybe several people have came to me a little bit. So without, let's say, a question mark in their heads of what do I mean by non-Marcovian, right? Because also, one has to say that the literature is not super, super clear about that. And let me clarify this a little bit. So there are, in my opinion, three main concepts of non-Marcovianity. One is, I would say, the traditional concept of when I was doing my PhD. My PhD was already on non-Marcovian open quantum systems. And by the way, everyone was asking me, why do you need that? So it was kind of, well, those were the times where people were thinking about open quantum systems in quantum optics. And in quantum optics, things are very, very Markovian. And at that time, by non-Marcovian, we were referring to those systems or to those dynamics where you cannot actually properly use the Limlan equation, right? So in that sense, this concept is very, very close to a paper by Gens Ayser, Dignasio, Sirac, where they actually propose a measure of non-Marcovianity that is related to whether your evolution can be properly described in terms of this dynamical semi-group or Limlan equation. So this is the first definition of non-Marcovianity. Let me not call it definition, but the first thing that people were thinking about when classifying the systems between non-Marcovian and Markovian. And then there was this, let's say, well, this emergence of non-Marcovianity measures. And I would say they are mostly divided into two main groups. One is that in which people consider non-Marcovian, those dynamics that cannot be described with a divisible map. And this was first proposed by Rivas, Huelga, and Plenio. And this is, by the way, what I was explaining to you earlier, the fact that, well, I think it's pretty useful. Non-divisible, I remind you, is whether when a map is divisible, you can always decompose it into little pieces, into little pieces, and more and more little pieces if you want. And each of these pieces is a dynamical map. And why is this important is because if you have a quantum information protocol and you have this property, you can actually use these little pieces in between operations over on your qubit. And it is important, again, that these are well-defined dynamical maps, because otherwise these guys would be crazy. They would do crazy things to your reduced density matrix. They would transform your reduced density matrix into something that is not a well-defined reduced density matrix. This is why it is so important to have pieces that are well-defined dynamical maps that preserve the nice properties of the reduced density matrix when applied to it. And then there is this other method, which is based on the backflow of information. And this was proposed by Breuer, Leine, and Philo from Freiburg and Finland. And it's based mostly on whether when your open quantum system is coupled to the environment, whether there is not only the information is flying or flowing from the system to the environment as it happens in many different cases, but also there is some backflow of the very same information back to the system during the evolution. Therefore, you see, you have the same names for three different things. And people then try to see what is the relationship between these concepts and so on. But I think the best thing to keep in mind is that they are somehow linked by the fact that, well, in general, non-Marcovian environments have a certain memory. And because they have a certain memory, there can be some backflow of information because the environment actually remembers. And in connection to this, the fact that you can divide into pieces, the map, it means that somehow the environment is recovering pretty well from the interaction with the system. And it is a good approximation to consider that from one piece of time to the next, the environment has recoiled back to its equilibrium state, roughly speaking. Please allow me the inconsistencies or the inaccuracies in this explanation. But this is just to, more or less, give you some physical ideas. So any questions so far or has this been clarifying or hopefully not un-clarifying at least? Okay, so let me go now to the next part we have, I think, 25 minutes until the coffee break, which is very encouraging. And, well, we have seen that we have different types of environments, but many, many of them can be described as a set of independent harmonic oscillators that are kind of connected to the system in this kind of a star configuration. Well, questions so far here, I added some slides to, well, trying to ask you questions. So what is the only thing we need to know so that we can properly describe our dynamics with a universal dynamical map? Well, it's actually that the initial condition is the correlated, although there are examples in which you can also derive a short of a map or a combination of maps for more complex, initially correlated states. But also you need to think that the universe is quantum. This is the only assumption that we have so far to describe everything in terms of a dynamical map, the fact that it's written in terms of cross operators and remember that these cross operators are related to the unitary evolution operator for both the system and the environment. So we need the environment to be quantum also. But, okay, we all agree that quantum theory so far has worked pretty well. Ah, why the harmonic environment model is so successful? Any ideas about that? Why Gaussian statistics are so nice? Well, this is related to the central limit theorem. So in a sense, when you have a complex system with many, many degrees of freedom and you go to the thermodynamic limit where you have many entities in your system, in this case in your environment, then the statistics tend to be Gaussian. And this is what lies below this harmonic, the success of the harmonic assumption. What, I mean, you know, when you think about a quantum information protocol and people are applying maps and maps and maps in between unitaries on the qubit, the assumption that is below that is that somehow we can describe this environment in terms of a dynamical semi-group. This is something that we just saw before. Well, just to catch up with you, well, when we have a closed system, this can be written in terms of a pure state, but here we need to deal with the reduced density matrix. This was meant to be a catch-up sliding between two blocks of the talk. By the way, just to mention here, before I was saying, look, if you have an environment and this is the total Hamiltonian, for some reason the laser is, I think it's, well, it's disappearing. Well, you can solve the full dynamics of the system and the environment with the total Schrodinger equation, but just to let you know that it could also be in the case that your environment and your system are described or have to be described not with a pure state psi of t, but with a reduced, not a reduced, but with a density matrix because it could be a mixed state. Well, why? Because you could initially start from your universe being composed of, still the open system and the environment, but the initial state that we would be considering, this is the total density matrix, being a mixed state that is still a correlated state between the system and the environment. And this is, by the way, the situation we were considering before that the environment is in a thermal state. It's actually exponential of minus beta, h, did I call this e? Sometimes I say e or b divided by the partition function, okay? Yes, yes, so far, yes. Indeed, there are many works in which people have considered more generally initial conditions. This only makes things more complicated. I'm gonna approach because some, yes. If you consider an initial entangled state between the environment and the system, things, I mean, you would have more complicated equations. So, well, you can still extract the reduced dynamics of the open system, but you would not, for instance, be able to describe everything in terms of a single dynamical map, for instance. Yes. Okay, but so far the situation I've been considering is kind of the quantum thermodynamical situation where you have an open system that is coupled to a reservoir, and this reservoir is in an equilibrium state, a thermal equilibrium state, for instance. And the only thing I'm saying here in this slide is, look, when you think about the total evolution, and you start from an initial state which is not a pure state, but a mixed state, you have to consider this von Neumann equation with the total Hamiltonian here. This is just what I was saying. Okay, and we have seen that we have this general form of the Hamiltonian in which the environment is Gaussian, has Gaussian statistics, and the point is, again, that we want to be able to solve the dynamics of the open system in its reduced Hilbert space, and we want to simply use as the statistical properties of the environment these correlation functions, in particular this second-order moment for the Gaussian environments. And now, okay, what do we do? How can we have access to the dynamical equations of the open system? Remember at the beginning I was telling you, well, in most of the cases, what you need is to make some sort of assumption that actually has to do with considering this coupling strength between the system and the environment to be weak, and what does it mean weak is what I will explain now in a minute. So remember, we have two different time scales that are relevant in our problem. So we have an environment relaxation time, which is the decay time of the correlation function, which is roughly speaking, the time that the environment takes to bounce back to its equilibrium state, to its give state. And then, as you will see later on, now it's hard to justify this without actually having seen the dynamical equations of the open system, but you have to believe me that the evolution or relaxation time of your open system is related to the inverse of the integral of the correlation function, which is kind of funny because this means that somehow the correlation function is going to be key to determine the two main time scales of the problem, namely the relaxation time scale of the environment and the relaxation time scale of the system. And by the way, I'm going to define a weak coupling parameter, which is G. I'm sorry that the laser is not really helpful, that is related to the relaxation time of the system. This is what we call the weak coupling parameter in which we will make perturbative expansions of our equations and therefore we will be able to have well-defined master equations for the reduced density operator of the system. And as you may imagine, there are three different, well, three different regimes. We are going to be focusing on the weak coupling and Markov limit. These two regimes correspond to cases in which there is a large separation of evolution time scales, in which we can basically assume that the environment is recovering very, very fast in the weak coupling case or almost instantaneously in the Markov limit from the interaction with the system. And later on, you will see how awful things become when you, well, at least you will get some ideas of how things become when you are in the strong coupling regime, yeah? Well, that's also a very good question, tricky one. I would say in the Markov limit, well, let me start with the weak coupling, okay? What do you assume is that, roughly speaking, you assume the born approximation. This born approximation is, as you will see later in a couple of slides, have to do with neglecting correlations between the system and the environment beyond second order in G, in this coupling parameter that we have there. So, this born approximation is allowing you to consider the correlations between the system and the environment that occur at very initial time, so to say, until the environment relaxes back to its equilibrium state. And actually, the Markov approximation is assuming even more, it's assuming that during the evolution, the total density matrix can more or less be approximated. This is also something that is assumed in quantum thermodynamics in the Markovian interpretation of quantum thermodynamics, can more or less be written as a product between the system and the environment state. And the environment is usually considered to be remaining in equilibrium. This is more or less the assumption that is behind quantum thermodynamics in the Markov limit, which is where the different laws of thermodynamics are justified. It's much stronger, of course, than the born approximation, but we will see this later on in more detail. Okay, so let me go a little bit to explain this. This is a little bit of a stupid slide. So, you visualize the open system is at the turtle and the environment is a bunny. And well, the nice thing about these approximations is that the dynamical map is reachable. You can really reach the master equation and the dynamical map. And actually in the Markov limit, in the limit where you have a Markov approximation, and actually another one that is called the secular approximation that we will discuss, we will not discuss in detail, but it's related to total energy conservation of the full system and the environment. The map will become a dynamical semi-group. And then now we connect a little bit with this concept of noise, right? Because well, so far we have been talking about master equations, the evolution of the reduced density matrix of the system. But just to let you know, we will also check a little bit this other way of expressing the reduced density matrix. Actually, this is called unraveling. So, this was a terminology that was coined by Carmichael in the 90s. You can also consider the reduced density matrix to be written or unraveled in terms of projections over stochastic wave vectors. So, the J stands for each of the stochastic trajectories in which you may unravel your reduced density matrix, and which in the Markov limit can be made to correspond to the trajectories that you obtain in each experimental run, only in the Markov limit. We will discuss a little bit also today. So, you see there are two different type of, or two different ways to access the open system dynamics, in particular the reduced density matrix. One is through computing the reduced density matrix, which by the way, just for those of you that are not so familiar, this allows you to calculate the quantum in value of any operator of the system A, just because this is written as a total trace over the environment and the system of the total density matrix times A, but this is equal to a trace over the system of a trace over the environment, because A is a system operator of Rho T, okay, and this simply is Rho S of T. So, you see that the reduced density matrix is actually allowing you to recover any quantum in value of a system operator A. And so you can compute this Rho S of T with master equation and with stochastic Rheninger equations that are driven by a noise. So, this is the connection with noise, okay? So, how can we access master equation just very briefly because, well, it's not a question today to come into details, but the idea is to remember this for Neumann equation for the total density matrix of the system and the environment. What you do is you expand it up to second order in the coupling parameter. So, every time there is an interaction Hamiltonian, you have to think that this is proportional in magnitude to G. So, this means that this term is of second order in G, at least, because you have twice the interaction Hamiltonian. I have changed a little bit the notation just to, I realize it's too confusing. This, well, the evolution with respect to the free Hamiltonian, I'm sorry about the laser, is written as Bt over A acting over the next operator, V tau Bt. So, this means that this equation, I have to ask for another laser, but, well, this equation is in principle dependent on some operators. So, the interaction Hamiltonian evolved in time with respect to the free part of the Hamiltonian, the Hs plus He, but here you have the total density operator, right? And by the way, what we are going to consider is the case where you have an interaction Hamiltonian which describes the different particles J coupled to the field. Like, for instance, when you have a photosynthetic complex of antenna molecules, J stands for each antenna molecule. And what you have is that, well, for instance, this is for the Fena-Matthew Orson complex of antenna molecules, found in part of bacteria. These guys are actually receiving light from the sun and they are transporting this light into a reaction center which should be somewhere here and within the complex. And this transport process is occurring in the presence of a phononic environment. So, this is the interaction Hamiltonian between the antenna molecules and the phononic environment where the phononic coupling operator is B, right? Environment coupling operator. And this is where the Born approximation is coming into play as we were mentioning before. So, as you can see, it's coming into play by replacing this total density matrix that corresponds to both the system and the environment by the correlated version of it. So, it's a soft version of this assumption where you assume that every time you are having such a decomposition and it's softer version because somehow you are making such a replacement in a term that is already of second order within an evolution equation. So, in a sense, this means that you are considering correlations up to second order if you like. You are neglecting whatever correlations because once they are plugged here, they would be a fourth order term, okay? Good, so, and then what you get is what is called a weak coupling master equation. It's kind of interesting because, well, here I have already replaced the form of the interaction Hamiltonian here and here, and this is how you have in this equation only the system operators, the system coupling operators, SJ and SL, and the environment coupling operators have gone through the trace to this correlation function which I should have written, but it's really looking like this. So, this CLJ of t is a trace over the environment of the initial state of the environment times BLTBJ0, where this guy is the exponential of minus IHETBL, I think this should be plus, right? So, this is a correlation function that, well, it's a second order moment if you like or a fluctuation of the environment coupling operators corresponding to the particle L and the particle J. So, it's a kind of, well, it's a kind of, well, it's a kind of, well, the particle L and the particle J. So, it's a kind of very, very nice object because it connects different particles. It connects different antenna molecules through the field. So, even if they were not connected through this coupling, through the Hopin process, they would be connected by the phononic field through the fact that you have these correlation functions. Well, that depends on different, that connects different particles. And as you can see, thanks to having considered the Born approximation, which is a kind of equivalent to a second order perturbative expansion, we have obtained a master equation in the reduced Hilbert space of the system. So, this equation is only living in the Hilbert space of the open quantum system because it only depends on reduced density matrix and operators of the open quantum system. And all the information of the environment, as I was announcing earlier, is encoded in this correlation function, which is the second order moment or a fluctuation of the environment coupling operators, connecting between different particles. And it's probably, I think, a good moment to stop for the coffee break. I'm correct. At 11? Yes? Okay, so, well, is there any question? We have three minutes for a question. It's probably more important to get the coffee, no? Okay, so we will resume in half an hour, I think. Thank you.