 Okay, shall we start again? Okay, now we have time until half past 12, perfect. Okay, so we were deriving a master equation that is living in the Hilbert space of the system. It depends on these correlation functions that I wrote here in the blackboard that connects different particles. I have to stress here that because this equation is approximated, it no longer in principle fulfills the nice properties that it should. In the sense that if we evolve the reduced density matrix through this equation, we may in some cases, in some instances, find that the reduced density matrix is no longer positive as it should be. And this is simply because we have obtained such time local master equation, which has the very same form as the first one I was showing early this morning, in an approximated way. Of course, when we numerically see that the reduced density matrix becomes negative, and this can be observed by the fact that for instance, some populations become negative, the diagonal elements of the density matrix become negative, and this is not physically a disaster because the diagonal elements of the reduced density matrix represents the occupation probabilities of the energy levels of the system if the density matrix is written in the eigenbasis of the system. So if we find that the reduced density matrix when running this equation becomes negative, this is a symptom of the fact that we are using it beyond its realm of the weak coupling approximation. It means that we are using this equation in a strong coupling situation, right? So this is kind of a symptom of using improperly this equation. Okay, but let me now derive the Markov limit. And before doing that, it is convenient, of course, to consider the particular form of the system Hamiltonian. And in particular, the system Hamiltonian for these particular complexes, photosynthetic complexes, contains some local terms that correspond to on-site interactions. So these would be terms that are proportional to some in J through all the molecules of some on-site energy sigma J dagger sigma J. That would be the first term in this Hamiltonian. But you may also have some Hopin term that I was also writing before that connects the site J with the site L through some coupling, well, through some Hopin coefficient. And this means that you have a Hamiltonian that is actually connecting different sites in the complex. And it turns out that it's convenient in order to reach the Markov limit properly to first diagonalize this Hamiltonian. And as a result, what we find is that the diagonal form of this Hamiltonian is written in terms of some eigen energies, epsilon A, and some eigen states A, which are called in the jargon of photosynthetic complexes and in quantum biology, they are called excitons. And they correspond to a superposition of local states. So these are non-local states that correspond to a combination of local states for each particle. So as you can see, these are extended states. So in this picture, you can see, well, this is a representation where you can see the location of the first exciton. And then you have a third and a fourth exciton that are extending actually over several antenna molecules. And now the interesting thing is that thanks to having this diagonal basis, we can actually rewrite the time evolution of the coupling operator that appears here in the master equation in terms of these bases. So now because you are considering an eigen basis, you can write the time evolution with respect to the system in terms of a phase with respect to the eigen energies, epsilon A and B. And now this operator is going to come in terms of a sum over eigen states A and eigen states B of this Limbland operator, I'm going to call it, that actually connects these different non-local eigen states. So it's kind of funny because we have written a coupling operator which is local, which is reflecting the coupling to the molecule J in terms of a combination of operators that reflect transitions between the system energy eigen states. And then in order to reach the Limbland form, or the Markov limit, what we need is to consider that actually this correlation function is very, very fast decaying. Remember that the Markov limit was related to considering that the evolution time of the environment, or the relaxation time of the environment, which is given by the decay time of this function, is much, much faster than the evolution time of the system, which is actually the evolution time, or the time that this, well, that this master equation takes to drive the open system towards an equilibrium state, or a steady state. And thanks to this assumption, we can actually extend the integration limit to infinity because we can actually neglect every effect that the initial structure of this function have in the integral. And this is precisely the Markov approximation that I was talking to you before. So in our previous form of the equation, we had that the integration limit was up to time t, but now we are really assuming that the correlation function is decaying very, very fast so that the initial structure of this coefficient is not important. We can really take its value in the infinite times. And then we plug also this time evolution form for the coupling operators of the system. And we consider this energy conservation approximation that I don't have time to talk to you in detail today, but as I said is related to imposing that the total energy of the system and the environment is conserved. And it's called the secular approximation. And once you do that, you obtain this Limbland equation. And this is what in the literature is called global Limbland equation. I had the question in the coffee break when I was asked what is the difference between a global Limbland equation and a local Limbland equation? Well, this is a global Limbland equation and it's called global because if you can observe these Limbland operators are really describing transitions between energy eigenstates, which are non-local states, are excitons for the case of, for instance, biological complexes. So it's kind of producing a dissipation that is referring or it's describing the transitions between the different energy levels of the system, of the open quantum system produced by the environment. And when you think about local equations, these are not so, well, these are equations in which this term is coming in the form, again, of a Limbland form like this one, but these Limbland operators are describing transitions between local eigenstates. For instance, for the molecules, this would be a sigma, so that would correspond to a case where you have, again, the same photosynthetic complex, but your Limbland operator describes a sum over particles j and l of some gamma jl of sigma j, I always forget, rho s of t, sigma l dagger plus the other term. Right? So this would be the, the Zipariff term corresponding to the local version of this Limbland equation. And now you tell me, but which one is the proper one? Well, the short answer to this question is that this equation, this global Limbland equation, is an equation that we have obtained step by step from first kind of very justified in each step. And mostly, it is thermodynamically consistent in the sense that one, when the environment is in a thermal state, and this is reflected in the form of these, of these correlation functions, it turns out that the steady state of this equation or one steady state of this equation is the thermal state of the system. So it properly describes equilibration, if you like. This is not so when you deal with this, this non-global or local Limbland equation. So the short answer to the question which one is better would be, okay, if you want to describe thermodynamics in terms of the dynamical process that carries your system to the thermal state when coupled to a thermal reservoir, then I would say this is the equation to use. And another justification for this equation is to say, okay, step by step, we have been able to justify how to derive such an equation. And as you can see, this equation has the very same Limbland form as we were seeing at the beginning of the talk, but it's actually, properly speaking, a Limbland equation because now the rates and the Limbland operators are actually time-independent. And this equation, by the way, just to connect with the jargon, it describes an evolution or it corresponds to a dynamical evolution described with, well, describes a dynamical semi-group and actually the corresponding map, the corresponding map is, well, have obeys the dynamical semi-group condition that we were writing before. And it's also quite interesting to see that the rates here in the dissipative term are actually including all the collective effects that come into play when you have a many-body open system. Namely, it really describes the connection between different particles mediated by the field through the fact that the correlation function is actually connecting different particles, L and J. And it really describes the fact that the dynamics, as I said before, is a dynamic that has to do with transitions occurring in the energy eigenbasis of the system. That is not necessarily the local basis. And the second thing to realize in this equation is that here, in the, let's say, in the unitary part of the evolution, this is the dissipative part and this is the part that is, in principle, reversible. So this is what produces the irreversibility. So this non-dissipative part is corrected with a term that has to do or is related to the environment. And in quantum optics, this term is actually called lam-shift. The only thing is that here, it will not necessarily correspond to a shift, or to, let's say, a renormalization of the energies because this term is more complicated than that. But it, for sure, represents a correction of the energy levels of the open system produced by the environment. Okay. And as I said before, this guy, this evolution equation corresponds to a universal dynamical map that is a dynamical semi-group and which has the property of being contracted, contractive, sorry. So what does this physically mean is that imagine that you have a qubit and a qubit, as you know, can live in a block sphere. So the states of a qubit can be represented as points in a sphere that is called the block sphere. And this represents the volume of available initial states for our qubit. And what this dynamical semi-group or Limbland equation will do will be to contract monotonically, namely without going, increasing at a certain point, but really, really shrinking, shrinking, shrinking in time, the volume of available states for our open quantum system. So this is, again, something that reminds you what does it mean to be irreversible, because it means that along the evolution you will have less and less and less possible states to access. Until in the end you reach a final volume. And some of you may ask me a volume, but don't you reach a point, a final state, steady state? Well, the general answer is not always. There might be cases in which you have some symmetries in your system. You have some conserved quantities in your system and your environment, which actually prevents you from really evolving into a fixed steady state independent from the initial state. So for these particular cases what we will find is that depending on the initial state, you will reach a steady state within this final volume. And this steady state will be dependent on the initial condition that you were considering. It's kind of funny because it's a, you could, I mean roughly speaking say it looks like a non-Markovian effect in a Markovian evolution, but it's not properly speaking a Markovian effect, it's simply that the symmetries in your system tell you, look you can go to this steady state if you start from this initial condition or to this steady state if you start from this other initial condition. But in that sense you say, okay you have a certain memory of your initial state, but the proper thing, way to say this is to say you have a dependency of your initial state. And well if the bath is in a thermal state, one of the states in this volume will for sure be the thermal state for the system. And this is because the thermal state of the system is a fixed point of the Limblan equation, the global Limblan equation I was showing you before. What does it mean is that if I plug this state on the right hand side of the master equation of the Limblan equation, it will be zero, right? So this means that this is a fixed point of the equation. Okay, but how can we know what I was saying before? How can we know whether we have a single steady state or several steady states which depend on the initial state that you were considering? How can we know if we have a final point or a final volume of steady states? The way to properly answer this question is by rewriting the master equation in terms of a vector form. Here I have vectorized the reduced density matrix. This means simply that I have rewritten, so I have been scattering in the whole blackboard. So in the matrix form the reduced density matrix of a qubit has the form rho 00, rho 01, rho 10, rho 11. And now in the vectorized form I simply write it like this, right? And then I rewrite my master equation in terms of this object that is called Limbladian. Sometimes it's called Liubilian because it's an object that was first described in the context of classical open systems and it was called back then Liubilian. But let me call it Limbladian because it really stands for a quantum situation. And you can rewrite it in terms of this big matrix that will be huge because, for instance, for a qubit it will be a four by four matrix. And now what you do is you diagonalize. This is a linear equation like a Heisenberg, sorry, like a Schrodinger equation. But the bad thing about this equation with respect to the Schrodinger equation is that this Limbladian is not a Hermitian matrix like a Hamiltonian. Instead it is a matrix that is symmetric but is non-Hermitian. This means that when we diagonalize it, we will find a set of right and left eigenstates. These are the right eigenstates v. And we will be able to write the solution of this system, linear system of equations, namely the reduced density matrix, as a combination of the right eigenstates times the exponential over the eigenvalues lambda j, which are, unfortunately, or fortunately I would say in this case, are complex quantities. So this is precisely because your Limbladian is not a Hermitian. And this is also precisely because these solutions are describing a contractive map, are describing a situation where this linear combination of solutions will decay in time and it will decay because the real part of the lambdas will be negative always, yes. Unitary. No, I don't think it's unitary. But it's not unitary. Right. It's really describing dissipation. Why? Because look, these lambdas here have complex quantities. They have a real part and they have an imaginary part. And the real part is always negative. This means that these combinations will decay in time. But there will be at least one of these j's of these eigenstates and eigenvalues that will not decay in time. And this corresponds to the steady state. Okay. So if you look at a nice picture to visualize this, let's see if this works. Yes. So this is the imaginary part of lambda and this is the real part. And there will be always at least one solution in which both the real and the imaginary part of lambda are zero. And this corresponds actually to the thermal state. And it might be the case that actually we say that this, let's say this steady state is degenerated. This corresponds to the case where more than one eigenvalue will have a zero real part. So this corresponds to the case where you are not evolving your initial volume into a point but into a finite volume. So I repeat. There are cases where you only have this steady state. You only have one eigenvalue with zero real part. And this is your thermal state. But there are other cases where you also have other eigenvalues with zero real part. And then your steady state will pretty much depend on your initial state. This is the case where you have symmetries and you have conservation of quantities and therefore you are not evolving your volume into a fixed point but into another little volume. And, of course, these other solutions will also appear. They have a finite value for the real part. So they will correspond to decaying terms in this decomposition. And, well, there is another interesting quantity here to extract from this spectral analysis, sorry, which is this gap here. So this gap here is telling you the distance between the, let's say, the zero real part eigenvalues and the eigenvalue with the smaller absolute value of real part. Because I repeat, the real part is always negative. And this guy is going to be very important, this particular value delta, because it will give you the slowest decaying time scale of your open system. So no matter what happens, the open system will not thermalize until the slowest term in this decomposition has decayed or will not have reached steady state. So this is a kind of a mathematical way to describe the, in particular, the steady state structure and the time evolution scales of your open system. Okay. And as I said before, if we have a single eigenstate with zero eigenvalue, then we know for sure that the system will thermalize always for whatever initial condition we may consider. Okay. So, very nice. We have a Limbland equation. It's positive, completely positive by construction here. I'd like to say something else, which is that interestingly, remember, we had a weak coupling master equation. And I told you, look, this master equation, it's kind of a bit tricky because since we have assumed the weak coupling approximation, it does not preserve complete positivity. So it really does not preserve, does not map the reduced density matrix into a reduced density matrix that is well defined mathematically. And now we have a Limbland equation which has been derived by using approximations that are even more strong than the weak coupling one because it really depends on the Markov approximation and the secular approximation. And this equation is behaving very nicely, much nicely than the weak coupling master equation. That's interesting because we have added more constraints and nonetheless, we have obtained a Limbland equation which is very well defined mathematically in the sense that it really evolves reduced density matrices into reduced density matrices. And this is a mathematical property. There is nothing else to extract from this. Again, we have as many decay channels here as system energy differences and the rates are time independent and they correspond to transitions between energy eigenstates. And just to give you an idea of how does it look like for a general open quantum system, well, if you have a two-level system, a single two-level system, then this means that, well, this should be the LAB corresponds to a transition between a spin up and spin down. In particular, if the Hamiltonian of the system is simply proportional to sigma plus sigma minus. But if we have a many body open system, this means that we have plenty of possible energy transitions, of course. And this means that this will scale, this equation will scale pretty awfully when considering the dynamics of a many body open quantum system. This is something that we were doing in a recent paper with Lincoln Carr and Daniel Jax. We were considering an ICIM model with only, I think, of the order of 1211 sites. And, well, that was more or less the limit that we could consider to be able to describe the system with such a master equation. And this is simply because you have to consider lots of energy transitions in your open system in this sum. You have to consider many different decay channels. It's called like this also. But I was telling you earlier that there is also the possibility of considering these stochastic equations where you reconstruct the dynamics of the reduced density matrix by summing up over different stochastic trajectories. And as it turns out, there are at least two different types of stochastic trajectories or two different families, so to say. One is that in which these stochastic trajectories are evolving in time and being continuously affected by a noise that I have called here set of T. But I'm not going to tell you any more about this today, but just to let you know that this quantum state diffusion corresponds to stochastic Schrodinger equations in which you have the evolution of each wave vector of the system being affected by continuously by a noise, which by the way is a Gaussian noise because we are talking about a Gaussian environment and it's a Gaussian noise because it has a Gaussian statistics and the autocorrelation function of this noise is the correlation function we were talking about. So once again, we see that no matter how we want to describe our open system, the only thing that we need is this correlation function. So this is another way of seeing again the same idea. And then you have these other guys here that I would like to tell you a bit about because this touches a bit the concept of how do we measure open systems and it also touches some ideas that are related to the foundations of quantum mechanics and which I always like to discuss. And this is a more phenomenological approach to describe that is, let's say in the Markov case was created or was envisioned to describe what happens when you are actually considering an experiment in which you have an atom coupled to the electromagnetic field and you are photon counting the photons that are being emitted by the atom and then people were saying, oh, look, I mean, then if the atom has emitted a photon and I detect this photon, this means that the atom has gone to the ground state, right? It has disexcited. And the collection of all these experimental runs in which people were photodetecting the atoms are the collection of quantum jump trajectories that when sum up allowed to recover they reduce the CT matrix. But let me explain a little bit more detail this idea of quantum jumps that I like to discuss with this because this goes back to the golden times of quantum mechanics and this guy so happy is Schrodinger and he was hating the idea of quantum jumps. He was hating the idea that nature would really produce such random events in which the atom would emit and decay and then you would detect and that was something he didn't like at all and he had a discussion with Born that was very, very like a gentleman discussion in which they were saying, yeah, they don't exist, they exist, blah, blah. And then it was so strong that Schrodinger was even saying that he didn't like at all this idea and if I have to put up with these dumb quantum jumps, I'm sorry that I ever had anything to do with quantum theory. So you can imagine that was very strong. And then this discussion was taking place in different meetings and there were different papers that were published also in the 50s several years afterwards. Schrodinger was publishing a couple of articles, other quantum jumps and it was now Max Born that was saying, that was discussing back and saying, pointing out that other guys, Born, Heisenberg, Poli, share my opinion. The quantum jumps exist. So there were basically two teams, quantum jumps don't exist. It was only this guy and then there was this other bunch of guys including Heisenberg, Einstein, Born, Poli saying yes, they exist. And then people in the 80s were very excited. There was even this New York Times, well, New, that was saying, oh, quantum jumps have been observed. This means they exist. And that was the result of very, very brilliant works in the group of Wine Land, Rainer Blood, and they were really observing quantum jumps in the lab. So they were really observing these sudden decays in the fluorescent signal of an atom. And this was an experiment that was published later on in Nature in 2007, in which I think in the group of Harosch where they were actually observing the state of atoms, of an atom, actually, because each of these atoms that are passing by a cavity, flying atoms, are actually like one experimental run, so to say. And what they were measuring is they were doing photo detection of the cavity field. And here they were measuring in one experimental run the probability that the flying atom is in the excited state, simply because in the moment they were clicking the detector of the cavity mode, they were actually detecting a photon in the cavity mode that was initially in the vacuum, they knew that the state of the atom had collapsed to the ground state. And when they were measuring or summing up many, many of these trajectories, this is I don't know how many trajectories, but more and more, they would obtain the curve corresponding to the quantum in value of the upper state of the atom, as reproduced with the reduced density matrix, which is evolved according to a Limbland equation. Nice. So with different experimental runs, they were summing them up, as you saw in this unraveling of the density matrix. And when summing up the results of all these experiments, they were obtaining the very same result as what you obtain with the reduced density matrix. And the theory behind that was derived by Dalibert and coworkers. They were actually saying, look, this is the Limbland equation, this is just another way of rewriting the Limbland equation. But now look at what they did. They kind of gathered some terms of the Limbladian of the dissipative parts in terms of an effective Hamiltonian that is now non-ermission, because it includes this term that is non-ermission. And they were writing the rest of the dissipative terms in this form. So they were saying, look, in an experiment, when you are not actually measuring any clique, what happens in the atom is that it's undergoing a deterministic and irreversible evolution according to this effective Hamiltonian. And when there is a clique, it means that this term is coming into play and there is a quantum jump that through this jump operator is projecting the state, for instance, to the ground state, in the case of a two-level atom. And as a result of, so this is an example of a quantum trajectory according to this scheme, this is a kind of a phenomenological scheme. This is an scheme in which they were saying, okay, each experimental run must look like this quantum jump trajectory, because if I sum up over all these trajectories, I obtain the result that I would obtain with the Limbladian equation, which I know that I experimentally obtained. This is the result that I obtained after summing up over all these stochastic trajectories. That is, sorry, again, the, well, the occupation probability of being in the upper state for the atom. Okay, so what is the answer to the question now? We seem to have seen that quantum jumps exist, but are they existing really? And this is why this is touching a bit with the foundations or these type of questions in quantum mechanics that not so many people ask anymore nowadays, because nowadays people are more interested about calculating things, but they still, I think, it's interesting to keep them in mind. The thing is that the fact that we in the lab measure quantum jumps with our photodetector does not necessarily mean that such quantum jumps have what people call an objective reality. This does not mean that the atom is really behaving like this. And the reason is the following. The reason is that depending on how I measure the atom, whether it's with a photon counter or with homodyne detection of the quadrators of the field, I would obtain a different type of trajectory. I would obtain a different type of information or unraveling of the atom. So the different informations or the different structure of the trajectories that I measure in my experiment, whether they are quantum jumps or quantum state diffusion, would correspond or would depend on the type of measurement that I'm doing. In the case of quantum jump, they correspond to photodetection. In the case of quantum state diffusion, namely trajectories that are driven continuously by a noise, they will correspond to what I obtain when I homodyne or heterodyne detect the quadrators of the field. So the answer of this polemic that was taken place more than a few years ago is we still don't know. We still don't know. And probably we will never be able to know because the information that we extract from a quantum mechanical system crucially depends on what measurement apparatus we are using. Exactly. But moreover, everything that I have explained to you so far, it's appropriate when the environment is Markovian. When actually the act of measuring or photodetecting a click does not have any consequence on the average dynamics of your open system. If you have a non-Markovian environment, I here put a liquid in between the detector and the atom, but it could also be the electromagnetic field within a photonic crystal. So this could be a photonic crystal in between. When I'm measuring the field, the electromagnetic field, I'm actually affecting on average or if you want the reduced density matrix of your open system. So whether I'm measuring or not will affect the reduced dynamics of the open system. And this is because there will be back action from the environment to the system. Remember, non-Markovian means somehow that the environment will not instantaneously, let's say, relax back to equilibrium, but there will be some information backflow from the environment into the system. And some of this information would precisely be, A, I've been measured. And then this means that now the trajectory of your open system will be conditioned on your measurement apparatus or your measurement process. So things get more complicated. So this means that in summary, first of all, a quantum system, it's like one of these statues that have different faces. That was an analogy that Howard Carmichael was telling me during lunch in a conference. And I like it very much because he was telling, look, depending on how you look at a quantum open system, it will have a different face. So depending on the angle in which you look photodetection, ombodine detection, it will have a different face. And on top of that, because usually you are measuring a quantum open system through its environment, for instance, through measuring the electromagnetic field, how this, let's say, how this medium in which your measure is in terms of being Markovian or non-Markovian will be important. Because if the environment is Markovian, there will be no back action of the measurement. But as I said before, if it's Markovian, non-Markovian, then actually the system will be altered by the measurement, even on average. Even, let's say, the evolution equation will no longer be, well, it will no longer be the evolution equation of the system coupled to the environment only. You have to also consider the measurement apparatus to obtain the reduced density matrix. Okay, so I see that I only have 10 minutes, so I was very, very optimistic with my program today. This means I only have time, I think, to touch the point number six, which is to tell you a little bit what happens when you go beyond the Markov limit or the Markov approximation. And maybe I will have time to discuss a little bit about the next point, but well, I'm happy to have questions later on. Okay, so let me remind you about this general structure of time scales and approximations. And remember that we have been so far talking only about Markov limit and weak coupling limit. But what happens here when you really cannot make a separation between time scales? Then things get more complicated. And, well, this is just to remind you that this is one of the cases where you have non-Markovian evolution. Again, this is when we have the light within a photonic crystal. And you have a very, very slowly decaying correlation function. But I also wanted to tell you that when you talk about, for instance, complex environments, like the ones that you find in biology, like in these photosynthetic complexes, and you look at the spectral density of these photosynthetic complexes, this spectral density is going to be very, very picked. And the correlation function is kind of a half or Fourier transform of this spectral density. And, well, as you know from Fourier transform, whenever you have a function here, which in frequencies is non-smooth, this means that in time, this guy will decay very, very slowly. So this will give rise to this highly non-flagged spectral densities will give rise to strong non-Markovian dynamics, which means that the relaxation time of the phonons will be very similar to the transport time between the photosynthetic antenna molecules. So this is another example of very, very strong non-Markovian effects. And then we say, okay, we cannot use perturbative expansion with respect to the weak coupling parameter. Can we still solve the dynamics of the open system in its or obtain the dynamical equations of the open system in its reduced Hilbert space without making any approximation or to put it in a different way? Can I still obtain the coefficients of this beautiful time local master equation that I was writing at the beginning that I know it has to look like this, but can I access these coefficients and limblant operators? Well, the point is that this can only be made in certain specific situations, in particular when you are talking about pure defacing, what is called pure defacing, which is a case in which the system and the interaction Hamiltonian are commuting with each other. This is, by the way, the case that most of the people consider when they want to describe the action of the environment in a qubit when they are talking about quantum information protocols. In this particular case, you can access the map exactly without any approximation in general. Then when you have a small number of excitations, when you have a Hamiltonian that conserves the number of particles of excitations, and you can actually consider yourself to be living in a small corner of the Hilbert space which is that in which there is a single particle coming into play. For instance, when you talk about a two level system that is connected to the vacuum field and they are interchanging a single quant of excitation, then you see that this is the master equation, and we know specifically how these decay rates look like in time. And of course, the third case is that in which we are talking about fully harmonic problems. Of course, when you have a quadratic Hamiltonian, so your system is also a harmonic oscillator with a quadratic Hamiltonian, and it's coupled to a quadratic environment through a linear type of interaction, then it's known since a few years ago in this original work from Juan Pablo Paz and Juan Zan, that you can really capture the master equation for this situation. You can really have access to the specific form of these rates for the master equations. It's a time-local master equation. You really can know these coefficients. And by the way, this is a case of a quantum Brownian motion, but not in general, not in general, because in general what will happen is that, well, it becomes very expensive to be stubborn enough to want to deal with reduced dynamical equations in the Hilbert space of the system. I don't have time to discuss here in detail, but what happens is that hierarchical structures appear both in the Heisenberg equations or master equations, and when you try to solve things with stochastic methods, including these non-Marcovian effects, like the stochastic Liouville for Neumann equation, then they have a bad convergence, particularly at long times. This is a slide to, well, where I have gathered non-comprehensively some of the reviews that have been created in the last few years to discuss the many, many, many, many different methods that people have thought about in different contexts and fields, including quantum chemistry, quantum optics, etc., to deal with this case of having non-Marcovian dynamics. And I think in the last four minutes, I'm going to skip these questions, and I'm going to go to the last, so I'm going to talk to you about a little bit about this point, this point seven to solve the full system and environment dynamics, because, okay, I remind you that our original idea was to say I'm going to, I'm going to really evolve or derive reduced equations in the Hilbert space of the system, but this brings a lot of problems, in particular to discuss what happens at long times and when the environment correlation time is very, very long. And, well, there are some people, including ourselves in my group, that have tried to tackle the full problem, including also the environment degrees of freedom. The point is that, well, in order to do that, you first have to, well, you first have to consider that your environment is usually composed of a continuum set of degrees of freedom, so instead of sums, you really have integrals over, for instance, momentum in the case of the electromagnetic field and in the case of the electromagnetic field or in the case of electrons, and you need to discretize this environment to pick from the environment the most relevant frequencies that are going to affect your open system dynamics, so you need to have a good method for discretizing your environment and only having NB environment oscillators, that can be, for instance, 100, 200, and with this, I have to be able to describe properly the full problem in the sense that I have to be able to describe the open system dynamics as though it was coupled to an infinite reservoir with infinite number of oscillators. I have to choose the most representative ones. And, well, but the problem is that even if I'm able to do that, and this is a problem that is not solved, there are many people that have tried to really select the best, the most important oscillators, but this is a problem that is not solved yet. Even if I do that, still, I have a Hilbert space that grows exponentially with the number of degrees of freedom. And that was, well, something that Dirac was pointing out, let's say, okay, we have beautiful Schrodinger equations to solve everything, we know how to do it, but the difficulty lies in that the equations are too complex to be solved. Actually, it means that the Hamiltonian is a matrix that is huge, and the wave function is expanded in a basis that is growing, that is living in a Hilbert space that is growing exponentially. People were talking about the catastrophe of dimension, and there was even people that were saying the wave function is not a legitimate scientific concept, because actually it's kind of a useless device to deal with systems that include a large number of particles. Well, just to give you an idea, and I think I only have time for two or three slides more, just to give you an idea when you have a classical state when there is no entanglement, the number of possible, well, way to write the total wave function is just a product of the wave functions of each of the open system and each of the oscillators. And this means that the number of such states scales linearly with the dimension of the open system and the dimension of the number of oscillators and the number of levels M within each oscillator, because an oscillator has an infinite number of levels, but I'm going to truncate this to just the M first ones, assuming that they are very close to the ground state. But if you have to deal with a quantum mechanical state, then you have to consider combinations of these basis states, wherein this coefficient here can no longer be decomposed in products of coefficients for each of these independent states. So this is a coefficient that is no longer, you can no longer write as Cj times Cn1 times Cn, et cetera, et cetera. This is what it means to have entanglement in your quantum mechanical case. And this means that the number of basis states in which you expand this state of your system is going to grow with your dimensionality exponentially. And just to give you an idea, this means that if you have, for instance, a two-level system coupled to one single oscillator with three internal levels, you will have six states in your Hilbert space and six classical basis states. But if you increase the number of oscillators, the Hilbert space dimension will increase much faster than the dimensionality of your classical representation, until when you have 50 oscillators, you will have to have over 10, over 24 basis states to be able to represent your quantum system. And the nice thing is that, well, we need to compress this information. The nice thing is that actually nature does not explore all possibilities. This is a silly cartoon. This tells you that basically nature likes to be not very entangled. In particular, ground states and states that are only very slightly excited with respect to the ground states are actually occupying a very small corner of the Hilbert space. They actually encode quite low entanglement. In general, there are exceptions to this. And this matrix product states representation that I'm going to talk very briefly in the last few minutes is actually taking into account this fact. So what they are doing is they are actually representing this coefficient in which you are expanding your state of the total system and environment. And they are representing them in terms of a product of matrices which have a particular dimension, which is called the bond dimension, which I like to call entanglement index because when the dimension of these matrices is equal to 1, this corresponds to a classical state because this means that these coefficients can be split into products of C numbers corresponding to the degrees of freedom of the atom, the degrees of freedom of the oscillator 1, oscillator 2, etc. But in the moment you start to want to encode more and more entanglement, these coefficients will become more complex, will no longer be describable in terms of products of the coefficients of each particle, and they will therefore be expressed as these products of matrices. But the trick here is that there is a condition for entanglement to remain pretty small and therefore to have a relatively small bond dimension for the matrices in this expansion, in this matrix product state, which is that they correspond to Hamiltonians that are local. So you need to have a system which has or is expressed as a local Hamiltonian. And these are my last two slides. Just to mention you that the problem of an open quantum system that is coupled to a reservoir of harmonic oscillators, it looks like a star and it looks therefore like a Hamiltonian that in a sense is no local because your system is coupled to each of the harmonic oscillators. But because you are having a quadratic reservoir, you can always make a unitary transformation of such a reservoir into a chain. And this chain now is going to look like a local Hamiltonian. It's going to be described like a local Hamiltonian, which makes the description in terms of matrix product state of the full problem pretty convenient in the sense that the entanglement will be under control. And together with Marie Carmen Van Nules, we were extending this idea to actually thermal states of the environment. I'm very sorry for having still stole from you six minutes from lunch. I guess we can use the last two minutes for questions. Otherwise, I'm happy to answer you one by one if you come to talk to me now. Thank you very much for your time.