 What is yours? OK, thank you. So good afternoon, everyone. I'm really glad to be able to present to you today the work we have been doing on trying to understand the different non-marcopian memory effects we can have in open quantum systems by using tensor networks methods. But before to go into that, I want to give a bit of motivation for the result in the work I'm going to present to you today. And the motivation comes, obviously, from biology, where in the last decades, there have been experiments showing that you can have in photosynthetic system current transport of excitons even at final temperature, which is something that is extremely interesting. It's still debated whether or not this quantum currents is responsible or is useful or is really important for the function of photosynthesis, but the fact is that it's there. And so it's interesting to try to understand why there. And so in this figure, what you can see on the left part of it is basically a reaction center of a photosynthetic system where you have different pigment protein that are embedded into a scaffolding of a mass, basically of protein that are interacting with them until they're basically bridging vibrations over different pigments. And the right part of this figure is basically a zoom in on those pigments. And what happens here in the reaction center, for example, that you have your atonic excitation that can hop between the different sites of the different pigments to the different sites that you can have here. And this process of multiple looping is done by having different feedback. And this is a mechanism with this scaffolding of proteins. And so it's a very interesting mechanism that basically means that those feedbacks basically means that we are what we call a non-Marcovian environment that has vibrational modes that are interacting with the pigment there coming from all of these environments of protein, non-Marcovian. And so I'm going to give you a quick explanation of what that means. So basically when we speak about environment, there are two kinds. So the Marcovian one, which is memoryless, which means that the characteristic timescale of evolution of your environment is extremely short compared to the timescale of your system, which means that if through the interaction between your system and your environment some excitation is created in the environment, it will basically propagate away and dissipate in the environment without ever interacting with it. And in the non-Marcovian case, which can happen when this timescale of the dynamic of your environment is of the same order of magnitude than the timescale of your system, then you can have basically interaction created by the system into the environment, basically coming back a bit later on a different part of your system influencing it. This can happen. Basically, those two limits can correspond for the Marcovian one when you have a very weak coupling between your stamina environment. The non-Marcovian one can happen when you have a strong coupling. And when you try to describe the reduced density matrix of your system with master equation, the Marcovian case is what we call time-local master equation, meaning that there is only the instantaneous state of the environment that matters. Whereas the non-Marcovian one is a non-local master equation where you integrate over the past time to take into account this memory and this past interaction with the system. OK. And so what we are going to do today and what I'm going to present to you is the way we have done a model of a non-Marcovian interaction between the stamina environment. And so we start with something simple, where we have different sites that are arranged in space with distance r. These sites correspond, for example, with the pigments that I've talked about earlier. So each of them have a specific energy. You can add a coherent coupling between the different sites, which means that if you put an excitation in one of them, then it will pop on the different sites. And you can do that. And then all of that into a common environment, a Bosnian environment that represents the different vibrational modes that you can have interacting with those sites. So basically, it's a formal environment. And then we can couple those sites with these environments. All the sites couple with the same strength, this GK there. But there is a difference for each site, which is that you can re-add a phase that depends on the position of those different sites in space. So basically, we encode the spatial structure of our system in the interaction with the environment. OK. And so what we are going to do is to try to write a wave function for this system and the environment together. Do the time evolution and look at the dynamic. And to do that and to make things simple, in the Hamiltonian, I just showed you the environment is made of a continuum of modes of independent modes. So we can't basically, merrily, use to do a description of a continuous number of modes if we want to use wave function. So we could do one thing. We could sample those modes and say, OK, we are only going to keep the mode of a specific wave number. But then there is some kind of arbitrary choice to make there. And so we use a different technique, which is that we do a unidirectional transformation based on orthogonal polynomial that will transform these continuous environments into a chain, basically. So it's a chain mapping there, where now we have a new set of modes that are interacting together and still interacting with the environment. And so here what we do is that we separate, for example, the left moving modes and the right moving modes of the original environment. We do the chain mapping, we obtain two different chains that are coupled, where the different modes of those chains are coupled with the environment. So we have a new coupling now. And so if we look at the Hamiltonian of that, it's basically that we have this new set of modes labeled with this n label. We can create excitations in this chain. And the excitation basically can hold on the different sides of the chains. And we have still our system that couples through the different modes of this chain, but with a new coupling constant, which is this gamma n of r. OK, and the other thing we do with that is the wave function of the system and the environment that we are going to write. To make the numerical time evolution tractable, we need to do something. We need to use an N that which is going to be a tensor network. Because if we add a couple of system sites, for example, but a hundred of environment modes, then the dimension of the dual respect is going to explode because everything goes exponentially. And so we need to restrict the possibility of states that we are going to have. And by using a tensor network representation, we can basically go from something that grows exponentially to something that grows polynomially, and that is not working. And so here I give you basically a very quick introduction to the diagrammatic notation of tensor networks, where basically to make things more visual and more quickly understandable instead of only writing equations, we use this notation where when you have an object that has different dimensions, different index, for example, of vector, so those different dimensions are represented by a leg onto a given geometric object. And so a matrix there will have two indices, so it would have two free legs. And when you want to basically sum of an index, for example, when you do a dot product there, the representation of that would just be to connect through the leg that you have to two different objects. And so an object with no leg within the scalar, and so here you see that if you take two vectors, basically sum over the index, so do here for vector a vector be the sum of a i of a i b i within the scalar, and the presentation is of this form. And the same way if you want to do a matrix, and if you want to apply a matrix to a vector, you would just take a matrix that has two legs, connect a leg of a vector to the leg of a matrix, and that's what's done here. And at the end you obtain an object that has only one open leg left, which is a vector, et cetera, et cetera. You can generalize that. And so here is the tensor that we are using, which is something called a matrix product state where we replace the general form of the wave function by a product of matrices of small tensors. And so this gives us local representation of our system. And so there we have basically, for example, the tensors that correspond to the system where we'd have a leg that has the dimension of the local in the space of our system, and connects to all the tensors that represent the environment, because I had basically two chain environments by my chain mapping. I have two sets of tensors there where, for example, this leg would represent the local in the space dimension of my environment. And then I need to print my Hamiltonian into a local form to be able to apply that to this wave function and to perform them the time evolution. And so we are able to do this in that case, and so we are able to represent this Hamiltonian to what is called the matrix product operator. All right, so that was for the basic method that we are using. And now the results. And so the first result that when we look at those new coupling constants that we have after doing the chain mapping, and we are looking at the zero temperature, they have a very nice structure. So if you have two sides only, and you put your first site at a position x equals zero and your second site at the position r. And you want to look, for example, at the coupling of the site at position r with the different modes of your chain environment. If, for example, r is equal to 20, you can see that you have a maximum of the coupling that is around there, which is between it mean that your site at position 20 will couple mostly to the region of the chain around the mode n equals 10. And basically this diagonal line there of maximum coupling mean that general site, system site that is at position x in space will couple mostly to the part of the chain that is that L over two, x over two, which creates basically kind of correlation between the spatial structure of your system and what is happening in the environment because then a site at r equals 20 will, for example, interact mostly with 10, the modes on 10 in the chain and then create the excitation over there. And so you have this correlation with reality between what is happening to the system and what is happening to the environment that you're going to directly leave there. Okay, so if we look at the dynamics now of what will happen in the system, if we take only two sides and we could initially an excitation there that is in the upper and the jagged state of your system. So delocalize over the two sides. The solid blue line there shows a limit that is called the spin boson model, which is well known and where basically we will know that if you put something into the highest in the jagged state it will just dissipate to the lowest in the jagged state due to the environment and then it will just decay there and decay from high energy to low energy. And so that's what we see with the blue solid line. And with our model there, where, for example, the sites are separated by a distance r equals 40, we see that initially does exactly the same thing. It decays to the lowest in the jagged state. But at a time that this commensurate with the separation of the two sides, we have a revival of the population of our higher state there. And so basically the system goes back to its earlier states through this revival. So we can understand where this revival is coming from because we have access to what's happening in the environment. So here in this figure on the right part is basically the orange dashed line of before that just rotated everything from 19 degrees to the time axis aligned there with this time axis of the left part of the figure which represents the occupation of the different modes in my chain map environment. And so what you see here basically is for the different modes of the chain on the X axis, what are the occupation number now and so you see that initially the environment is empty. You start to populate the origin of the chain which is mostly coupled to the first side of your system and then there around n equals 20, you start to populate the part of the chain that is coupled to the second side. And over time, you have this excitation that start to travel along the chain until it reached the origin of the chain that is coupled to the other side and that corresponds to the revival. So what's happening there basically that you are putting, you have transferred information about the previous state of the system in the bath, in the form of those populations that are traveling and basically transferring information into the system at a later time. And we can play a little bit with that. So this is exactly the same kind of system but with a distance that is smaller. So instead of having r equals 40 here, we have r equal 10. And so we see that you start to create population here around n equals five, which makes sense then and this population of modes in your chain, in your environment start to propagate there exactly as it did before and create a revival. But then when it's reflected at the origin of the chain, it's going to propagate and then cross the part of the chain that is connected mostly to the other side of the system, part of excitation is going to be transmitting, another part of the excitation going to be reflected back to the origin of the chain. And so you have those multiple reflection, internal reflection basically inside of the chain that are going to be revised to basically periodic revival of the population which is highly harmonic here and with decreasing amplitude because you lose part of the information is transmitting back. So we have been able to extend that to finite temperature where basically we still have those revivals but as we increase the temperature because of the thermal excitation because of the thermal population in the environment at some point everything gets into the noise of the thermal population and you almost don't see the revival anywhere but it's still survived at finite temperature and quite high temperature. And so if you look at what happened in the environment there you see basically the same behavior as before where you have something that propagates towards the origin gets reflected and it arrives to this revival but everything is way more populated because of thermal excitation. Okay, and so to sum up all of that basically here we have a spatially extended system where the different sides of the system are all in a common environment and we are able to do a matrix product state representation of this system and environment via chain mapping and when we do that we see that we start to have spatially correlated environment where the different parts of the environment interact mostly to different parts of the chain and by doing that we can see that we have non-Markovian memory effect there that we can trace back to this transfer of population both at zero and finite temperature and what we are working on at the moment is to have a look at what it does on multi-side dynamics and with different topologies and try to see if this could be applied to all the biological system where there's been hypothesized that quantum effect could play a role and I can be out of state-of-the-art proteins. So thank you very much for your attention and I'm looking forward to your questions.