these to one of the most recent developments in the young bo cold so, what do you saw up to now is let's say the historical part of the cold, or how the cold was started and the tool to compute the quasi particles properties within GW or autom approximations and to describe exit inside the beta sala beta. So these was the first part which was developed in the cold tako čeenges, ki je počku, in nekaj bil jem so jelčne, pri çevesi happenje, nekaj imam vse šeždje. Eko se pošlobe. Espečno poradil bi si tudi. Priločil je andre marini, Claudio Dakkalite in Mirtagruhnik. To da si pošlobe kršt. Ne vse je podržilo lažen geničen bojte in počke obozopradi. Tkročen je soradnout. Implečaj počke obozopradi zač Washington Stryb. in človek mentioned for the Block-Way functions. So these two strategies can be used for different things among others to do what I call a real-time spectroscopy. So I will briefly discuss what is the concept of someday spectroscopy compared to let's say frequency space spectroscopy. So I start really from the beginning The spectroscopy generally think that we apply a perturbation in electric field to a material and the material reactor creating an internal polarization. and then the sum of the electric field and polarization is what is called the electric displacement. which relates to the change in the polarization Sveti dve obsezv, vzetnev, vsak so nekaj. Tko čekla tudi omožite otev, tebe vi nekih nepl. In pčelti, kjer ste prijeznavenost odstih tebe. En submerged zrednje, kar našli, ne bila, več da inleti prijel, sva je zrednje doker, in ne ne in vseh n gros je izgleda delerske reponse. Kratilj je to dve reponse. ki je tudi vse posledaj od razblizna, do vsega vsega vsega domena. Na konvoljucionu je to počke. Tudi obječno, ki je vsega polaizabilitva deločnega systema, ka je vsega in deločnega elektricija, ki je vsega, kaj smo način smo načine vsega vsega vsega. Počak je, da izbolo in posleda in plak je vsega, kaj je polaizacija, elektricije, zjavljantne, electric field, a zelo res replice elektricj skateli as electric function times the field. And there is a relation between epsilon omega, electric function and alpha omega, which is written here. So this is what one computes, what one uses to compute electric function in frequency space. Thisal focus, alpha omega can be expressed via skupaj s vzivom spremljama vzivom. Naredaj nekaj zračov, da je nekočo. Svečno je tako, da je dobrojo, a če je dobroj, če je površel dobran, če je bila okazena.  Zato ideje je, da imamo prejnega perturbačnja, v električnih plih, kaj imamo prejnega materijala. Vse objevamo nekaj prejnega, ki je prejnega. Vse objevamo prejnega, nekaj prejnega, nekaj prejnega perturbačnja, nekaj prejnega, nekaj prejnega, nekaj prejnega, nekaj prejnega. Vse povrčenim drugeh zabravno v pravdu včasneje verno, nekaj prejnega in nikaj prejnega privarčnje. Nekaj zelo, da jih imamo prejnega materijala in nekaj prejnega turist, kaj objevamo povračnja. I tem do državčenih jaj povralo pravdu in povralo pravdu. I se lahko bomo predmačiti implemented of this strategy? First of all, you choose the perturbation which is the probe of your material which can be delta like perturbation or perturbation which has a sim shape and you want to apply it to the material. The first thing that you realize is that if you apply the perturbation you break some of the symmetry of the system. Prepočaj, z simulacijenih, vse je zelo, da vse je vse je vse vse, je vse vse ZIPP, RT, vse ml, na krati, sa opotakiti vsofjambu. Vse je prišljno vse, da so nekaj ne potrebneri, na pravom zboju. I ga je vse zelo, da je tvoj ZIPP, R, T, ZIPP, N, L. Zato, kar je vse vse vse vse, zelo je veliko nismo, recommendations for the ones you used up to now before you had just YPP. And it means that the code will be compiled with some extra capabilities, which are related to the RT, real time project, or ML, non linear optics project.  So there are two different executables because there are two different strategies. And now in this lecture I will mostly focus on this strategy, the propagation of the density matrix, while after me, Claudio will describe mostly this strategy, so the propagation of the wave function. And basically after you have the time-dependent density matrix, so you know how it evolves in time under the action of the pulse, you construct the polarization, and then you process the polarization. So the third step is again to take the post-processing tool, Fourier transform the polarization and get the absorption spectra of the system. So as you can see it's a way to get the absorption, so in a sense it contains the same physics of the linear response, or even beyond, and it contains the same physics of the beta-salpeta, if you have the proper approximations. So this is the general idea, but Yambo is a mini-body perturbation theory code, or better an ab initio mini-body perturbation theory code. So like in the other tutorials one has to start from a density functional theory calculation and then use the wave functions and the energies to build up all the mini-body perturbation theory quantities. This holds for all the tutorials we did up to now. The main difference for real time is that now we also have to deal with an independent laser pulse. So as far as mini-body perturbation theory is concerned, the main difference is that you can still write a Dyson equation, for example, for the green function. It's exactly the same Dyson equation, which you saw before. But this time this green function is defined not on the standard real time axis, but on this finite contour in time. So which are the consequences of that? Well, the main consequence is that then you take this green function from the contour and you want to bring it back to the real axis because you want to deal with something which you can manage. And when you do that you end up with different components of the green function. So there is, you will see that here a lesser, greater and retarded in an advanced green function. So the label just depends on how the operators which defines the green function are ordered on the contour and then on how you bring them back to the real axis. So I don't want to enter in the details. It's just, I mean, a quick overview to understand what happens to mini-body perturbation theory. And in particular what you will have is that this Dyson equation will become basically a differential equation for the time propagation of these two quantities, the lesser and the greater green function. So this is the face of this differential equation. So there is the time evolution of this lesser and greater green function. There is the first piece which here is called the artrefochka-meltonian and the second piece is called the collision integral. And basically any approximation of the self-energy which is static can enter in this part of the equation and any approximation which is dynamical will enter in this part. So you see this thing is the complex object of the theory. It's called the collision integral and it contains an integral over time of the self-energy times again the green functions. And here if you take a static self-energy this integral will go to zero. So it's a very complex object, it's a two-time function but indeed we do not need often to know the two-time quantity. It's enough to know the time diagonal of this quantity which is indeed the density matrix of the system. So one can just take the time diagonal and write the equation of motion for the time diagonal. This equation of motion is not yet closed because here you have this collision integral which depends on the two-time green functions. But if one takes a static self-energy then the collision integral goes to zero. So neglecting the collision integral you have an equation of motion for the density matrix. It's a very simple equation of motion could have been derived directly starting from the density matrix and the R34c Hamiltonian for example. So in this way you saw even an exact equation which brings to that with some controlled approximation and you also have some way if you want to extend the equation of motion and to deal even with effects which are beyond the static approximation to the self-energy. So this is in detail the R34c Hamiltonian. There is the Bayer Hamiltonian which is the Bayer-Etonica-Miltonian and then the R34c self-energy which is a functional of the density matrix. And of course since we are in real time there is the external potential. So I used R34c up to now because it's how you find the description of the Caden of Bayer equations and the equation of motion for the density matrix in textbooks. But indeed if you want to run simulations in real system it's much better to use this HSEX approximation which is basically the same as R34c the only difference is that the fork term, the exchanger is statically squared. Ok, this is the equation of motion which I write here in the Abinish form. So it's exactly the same the time propagation of the density matrix and this time I write explicitly how the density matrix is propagated in the Jambo code. So the density matrix as you know is a function that is variable R and R prime in just one time. What we do, we project on the Konešam wave functions and here is where the DFT starting point enters and thus the density matrix becomes a matrix in the Konešam indexes or in the band indexes. So for each k-point there is n by m matrix where n and m are the number of bands you consider. So it runs both on the occupied states. Of course you need an initial condition so you assume that in the ground state you have just occupied you use the equilibrium occupation. So if you have a semiconductor you just have the valence band so when n is in valence this is 1, when n is in conduction this is 0 and the of diagonal terms are 0. And then you propagate. So here you have again this the equilibrium Hamiltonian plus the change in the R-th and exchange correlation part just the static part of exchange correlation and the pulse. So with respect to before I have just written here the variations of the exchange correlation function and R-th because the equilibrium part is conveniently put inside this equilibrium term. So in practice this equilibrium term is nothing but the band structure. So as it contains the gain values you can start from DFT so this is the mass structure of silicon for example you can start from the LDA band structure but it's a better starting point if you start from the quasi particle energies so in this case the blue band structure. As you see in silicon as in most semiconductors the main effect are the bands which I'm not sure you can see here. Then the variation of the many body effects which are contained here there is the classical part and the exchange correlation part they are both static and one can start to compare to what is done instead in time dependent density functional theory which is let's say more known so in density functional theory here instead of the self-energy you have the exchange correlation potential of TDDFT which in practice you also approximate with the alibatic exchange correlation potential and the main difference is that while the exchange correlation potential of TDDFT is local in space the self-energy can be non-local in space. Indeed the fork term of the artifox self-energy is strongly non-local and of course if you want to go beyond to put dynamical effects you can do that in principle so this term will not be zero but you have an expression to implicitly include them while for example in TDDFT there is not a systematic way to go beyond the adiabatic approximation. So again few more comments on the equation the first thing we have is an external field which in Yambo we express in this form so is what is called the length gauge or the p.e expression so this is basically the polarization of the system or this term times the electric field and as I said since we choose a perturbation we have an electric field which is polarized we break some of the symmetry of the system and this is represented here so this is the Brillouin zone of bulk silicon and in particular there are highlighted here these L points which are some of the special points of the Brillouin zone they are 18 total and they are all related by symmetries this means that when you do a DFT calculation quantum espresso will just compute the wave functions in one of these L points and to reconstruct the density it will use the symmetries that will get the wave functions at all the other L points so in the save folder that you usually have for your Yambo simulation you will have the wave function just at this point for example now you want to perform a simulation where there is an external field which is represented here and since there is an external field the effect of the field on this point or on this point is different in general so this will function here and apply a rotation to also know the wave function here this is the preliminary step of any real time simulation so after you do that you have your density matrix at all the needed k points and you can start the propagation as I said the equilibrium Hamiltonian contains the quasi particle energies so all the correlations from the ground state are here and this in the art in exchange correlation potential gives the dynamical correlations so these two terms they evolve in time because they are functionals of the density the art return gives the classical part of the of the electron interaction and it is somehow related or it contains the same physics of what we call the local fields in linear response and instead the self-energy which at the TDDFT level would have been simply the adiabatic correlation potential here we will approximate this variation of the sex potential it somehow contains the same physics of the electron interaction in the beta-salt-peta equation and then I will show in an example how this works in practice and then we can also add an extra term so here I squeezed the art and the static part of the self-energy in a single term to make space to this extra term which is a constant defacing of this density matrix so you can add either to it's a phenomenological term let's say to have a defacing term which is related to the smir in parameter you have used in linear response it can be just a constant term or you can even try to build it from first principle and it's related to the quasi-particle lifetimes in this case more comments on the classical field so the artery field can be in general divided in two terms the macroscopic one and the microscopic one the macroscopic one is just the average of your simulation box of the change in the artery potential and since it's the average over the supercell you immediately see that it is zero for any system with dimensionality lower than three and the reason of that so take this material for example boronitrite in two dimension so we do a simulation in the supercell so indeed we have replicas as we have seen in the other lectures ideally the exact simulation would be with the distance in between the two layers but if this distance goes to infinity you have an integral an average over this infinite supercell with the layer just in a specific point and this average gives zero so there is no contribution to the macroscopic field in 2D and the same arguments applies in 1D or in 0D and instead the microscopic part it contains the local fields and it's always there for the functionality or for 3D systems so what changes when you go to 3D so this is the bulk version of boronitrite in this case this macroscopic field is there and the problem is that it is non analytical in the density matrix or in the density in general after is indeed a functional of the density and it is problematic to be described it has something to do with the physics of longitudinal transverse splitting I'm not going into the details but I just want to mention that in practice this term is not directly included in the after term since it's not analytical so what one do is to take this away from the after term and put it back inside the external potential and then the effect is that instead of having the external potential one has here the total potential of the external world plus the induced polarization so let's try to see what's the meaning of this density matrix to understand the physics which describes one thing we can do we can take the equation of motion and expand in first order so this is the reason of the one year with respect to the applied field so if we apply to first order this term just has the density matrix here I have the equilibrium in Tornian and again I put the first order density matrix here I have the change in the self energy so if I put the equilibrium density matrix the change in the self energy it's zero if I evaluate it so I have to evaluate the self energy at the first order density matrix and then in the commutator I have to put the equilibrium density matrix here otherwise I have a second order term the field is by definition linear in itself and so it commutes again with the equilibrium density matrix so now that I did that I can also expand linearly this variation so I take just the functional derivative with respect to the density matrix times the change in the density matrix so this expression is exact for functions linear in rho and for example the R3 plus 6 in rho so it's an exact expression and I can rewrite this one you recognize is by definition the kernel of the better better equation so I take this one I plug it here I get this equation so the h equilibrium it just contains the quasi particle energy and since I have a commutator it becomes the difference and then this row equilibrium it just contains the occupations since I have a commutator I got the delta, the change in occupations and this is nothing but the better better equation so it's a derivation from the density matrix to the better better equation which is exactly the same of what Fulvio was showing yesterday I underline again that this is not the full response function L but is this L bar because we have the response to the total potential and not to the external one now what is one reason why this is instructive because now instead of staying in this space I can apply the rotation to the excitonic space to the row equation so the density matrix becomes a function of the exciton index and instead all this term is the better better which is diagonal in the excitonic space and it just contains the excitonic energy so it is replaced just by this term so and here you easily see that if you integrate this equation it just has a density matrix which oscillates back and forth in time with a characteristic frequency which is the excitonic frequency so to first order the effect of the a field applied to the system is just to create oscillations which are proportional to the excitonic energy or the excitonic energies if I have the proper self energy here or in any case the energies which are captured by this Hamiltonian ok, so if I have something which oscillates back and forth in time I can construct the polarization which will oscillate back and forth in time at this excitonic energy or say at the energies described by the by this Hamiltonian I can construct my absorption spectrum so the first thing I can try so this is what was shown in this paper nine years ago if I take the zero for the RT plus exchange correlation of self energy then the equation is super simple this density matrix which just oscillates at the quasi particle energies which are here so the polarization and if I take the Fourier transform which are the it is the black line it will be exactly as the independent particle or RPA response function so you see that there is an exact correspondence between the Fourier transform approach the black line and the frequency space approach which are the red circles and if I instead put this H6 Hamiltonian I have a polarization which is oscillating at the excitonic energy and the Fourier transform directly gives the excitonic spectrum so this is again exagonal boron nitrite which has a sharp exciton the experimental data in gray and you see that the independent particle one was missing completely this exciton when instead the time dependence simulation is capturing this effect so it's really an equation of motion which gives real time approach to the beta salpeta and a few more comments on this equation I mean it's the only equation we have to deal with and I'm always representing it in the slides and it is indeed the equation which is coded in YAMBO RT so for the external field I have shown you before this expression which is the length gauge so it's the external field e times the polarization in principle one could even use the a dot j so the so-called velocity gauge and I would say that both approaches has the advantages and the disadvantages in YAMBO we mostly deal with the length gauge although even the velocity gauge is implemented at least for some cases so one of the main disadvantage of the length gauge is that it needs the polarization and the expression which is shown here in polarization is valid only to first order so it's valid only to first order in the density matrix so I mean that if you apply the external field that you compute the first order density matrix in the external field then you get the first order polarization and you can construct the linear response of the system however if you want to go beyond you need something different which is the theory of the very phase that we will describe later on the other hand if one uses the velocity gauge this expression is in principle exact to all orders the problem is that when you introduce a vector potential in this equation then even the self energy and the density matrix but in particular the self energy needs to be gauged and then another issue the vector potential is numerically much less stable because there are some rules which numerically are almost never fulfilled and needs to be applied so this is a reason why to use the velocity gauges in practice is very difficult and you will see that to go beyond the linear regime the very phase polarization has been chosen in the jambo code so another comment is that the theoretical equation is this one but in practice we are dealing with a code so we have to deal with finite differences and the code you cannot directly code the time derivative so what you have you have a time propagation so a time step propagation so in practice the equation of motion is this one so if you want to know the density matrix at time t plus delta t you take the density matrix t plus you add some term which depends on this Hamiltonian so I stress this because there are I mean the real time propagation is a technical point in the simulation which is quite important and there are different schemes which one can approach so here I list three which are coded in jambo the whole level scheme which basically the simplest way to propagate the density matrix you just take this commutator and you use to time step the density matrix the exponential scheme which is a bit more accurate but even more demanding where instead of directly applying the commutator you use an exponential representation of this Hamiltonian operator and then what we call the inversion scheme which has this expression this expression comes from the fact that this is what I call an implicit scheme I'm not giving the details but just to know and it is even more complex that the exponential but it is more accurate I think that for linear response properties in practice we rely on this time stepping scheme let's say why for non-linear optics this time step scheme so this is how you propagate from step t to t plus dt with a single step let's say single step approach you can do even better if you go to multi step approaches which are represented here so the idea is that I want to go from t 2 times t plus dt but instead of using just the information as time t I would like to use the information at a time which is a bit later so this is a two step scheme because first then I have to propagate the density matrix with any of these schemes 2 times t plus dt and alpha then construct is functional and then propagate again so you will see in the input file of the simulation that you will have a choice for the kind of of a let's say single step scheme and a choice of the kind of multi step approach and regardless of which will be the scheme you choose the time step dt needs to carefully be chosen otherwise the simulation will diverge ok so with this I finish the first part on this idea of real time spectroscopy and I would like to stop a moment if you have questions on anything I've shown and then in case I will have more time about how to go beyond the static approximation to the self energy but first if you have any question sorry I said so it's not that it doesn't work in general let's say let me go back to this slide ok so here for example this is the case of silicon and the silicon is symmetric with respect to I think 90 degree rotations if I'm not wrong all these 8 L points are equivalent now if you apply an electric field along the specific direction say 1111 then some of the rotations for example the rotations in plane will leave the electric field unchanged and these are consistent with this equation you apply the symmetry and even this term it remains the same under the application of the symmetry and this symmetry is there instead the rotation which send let's say the field is in this direction in this direction the electric field is not invariant anymore against this rotation so in fact there are still some symmetries but less than the ones of the ground state in the case of silicon in particular there are in total 8 L points and if you choose this field along the 111 direction with 2 groups of L points the 2 one which are in black are one group and the other which are in blue are the other group so you have some symmetry but less than before I don't know if there is any I don't know if we compare with the equilibrium case is there any difference here I mean there is not really any difference in the in the in the ground state of the system if you want let me try suppose that you do a simulation with quantum express on black silicon and you choose a grid say 4 by 4 by 4 so in total there are 4 to the third k points in the Brillouin zone and then thanks to symmetry I think the k points are reduced let's say to 10 so you have 10 points in the IBZ 10 k points for the wave functions now you apply the electric field and you remove some of the system the symmetries and you end up with maybe 20 k points so you don't have all the k points it's not 4 to the third but you have a bit more than what you have in the beginning and this is something you always have to remember before starting a simulation so remove the symmetries and expand the wave functions it is done by this YPP tool so you don't really need to care in the result state another question there thanks, so my question is also about the external field so can you play with the maybe the strength of the field for example the energy of the excitation and then see the population change as a function of time so when I say here that you choose the perturbation so here I've designed a couple of perturbations but in practice in the input we have a list of parameters which define the shape so you could say delta sine field then the intensity which controls how strong is this field then the polarization which defines the direction of the field and then for example some of the fields not these two but so one of the most used indeed is a a field which is the convolution of the sine with a Gaussian it's a pulse and it has a duration then you can even define the duration of the pulse it depends a lot on what you want to do if you want to describe an experiment where they have a precise field or if you want to do real-time spectroscopy so for example for real-time spectroscopy this field is meant to be the probe so if you want a probe you want something which is broadband in frequency and if you want something which is broadband it is useful to have a delta function on the other hand when you will see the lecture on nonlinear optics if you want to see nonlinear optics so you want to know the response at a specific frequency so omega but then you want to know the response at 2 omega which is the second order term or 3 omega then it is useful to have a perturbation at a defined frequency so that you can distinguish I had a question with the last section where you had 3 cases with the commutator and the exponential and the inverse Ok, the time-stepping Yes, so is there any specific calculation in which we are supposed to apply these commutator for this kind of calculation expansion? Let's say that when you take the input file there will be an input variable where you have to define eulero, exp, inverse so you choose the kind of time propagation scheme and for the beginning you can leave it to the default the default usually for Yambartite uses the eulero time step combined with rk2 Runjekuta second order and it will work fine then in real life when you start to do simulations you can try to play and you will see that once the exponential or the inversion for example it is more stable so it allows you to have a bigger time step but it is also much more demanding so it depends a bit on I mean the system and you try and you see which is the best thing for you Ok, and I had another question with the pump probe so you explained about the probe part but what about the pump part of the lecture it was just about real time spectroscopy it's just the probe then I have a few slides about the pump probe just to give you a feeling but it is not something you will see in the tutorials so for now we just start to play with real time we put just one field and it's just the probe Thank you One question Ah, ok Do you want to take additional care with aliasing? Is this an issue here or does this not Yeah, since we are going from real time to omega, right and there might be some issues with aliasing of the Fourier transformation What do you mean by aliasing? High frequency terms, low frequency terms something missing, something cutting off Ok, so So I would say in general no and the reason is that first of all in this equation of motion there is a natural cutoff which is basically what you put here so here you select the number of states you include let's say I take three conduction band and three valence band and then the maximum energy difference which is included is the energy difference between the lowest band and the highest band possibly renormalized by this term and this is indeed one of the biggest advantages I would say of doing the propagation in this space compared to real space so if you are propagating real space then in principle you have any frequency and then you need a smaller time step and you have to care of this kind of things and then the other reason is that in any case it is much more dedicated to propagate the equation so you need a small time step for the propagation then if the time step is enough for a stable propagation is more than enough for the Fourier transform Indeed what we do in practice we propagate with a small time step and sometimes we store even the polarization with a bigger time step because we don't even need such an accurate time stepping You said that the time step has to be stable time step so how do you determine that whether the time step is stable or not stable? If you propagate and the time step is too big after a few time iteration we see not a number in the simulation and everything starts to diverge Ok Thank you very much It's really straightforward Ok, maybe I can go on with the second part I mean the second part is about ultrafast physics and it is just to give you a feeling of how all this could be principle extended to the description of pump and probe experiments You will not see that in the tutorial but it is something we are working on It's not in the GPL Because I mean we are developing still it we are there are many issues in this part it's much more involved complex the real time analysis but it's something that maybe in the future will be released in the GPL and if you wait and it's something which is anyway in the jumbo code So when I speak about pump and probe experiment I always start with this pictorial view of the human eye and it shows you that the human eye is the instrument we mostly use to explore the space around us and it is a very powerful instrument but of course it has limitation what on the special resolution and time resolution and so the idea is that we can use exploit technology to overcome these limitations So on the space side the first to do that was Galileo which is the creation of the telescope which allowed him in 1609 and 1610 to show to see for the first time the moons of Jupiter and since then the technology has grown a lot and then from the telescope we have been able to create the microscope and then all the instrument more sophisticated and now we are even able to see a picture of the atoms so this is a picture which is now even 9 years old a transmission electron microscopy image of graphene you really see the atoms in graphene On the time side the evolution took place much later so I start with this portrait of a running horse which was done in 1791 and here you see that there is a horse running on the floor with the legs stretched and indeed this is not the way the horse runs so this is not the way the legs moves when the horse is running and the reason why it is wrong in this portrait is because the humanite is not able to resolve the legs of a running horse at full speed so the first time we were able to see the legs of a horse it was in 1878 almost 100 years later and this is what is considered the first movie of the history so it's a series of 16 pictures of a running horse and you see that the situation which is the closest to the picture is maybe this one to the portrait but not all the legs of the horse are floating so you see there is a a difference in between the portrait and this gives you an idea of how taking pictures and using let's say trains of light or laser passes can be used to overcome the limitations of the humanite so this was the first result achieved more than 100 years ago and even on the time side the technology went on a lot and there have been a couple of Nobel prize in this history which was given to these three guys in 1949 for their ability to study extremely fast chemical reaction extremely fast here means 10 to the minus 6 seconds or 1 microsecond and then some years later 1999 another Nobel prize was awarded again in chemistry for the ability of studying chemical reaction in this time with femtosecond time resolution so from the microsecond up to the femtosecond 10 to the minus 15 indeed so it was a few hundred of femtosecond in this case and then you already see that if you take just the simple equation of motion for a classical atom so you plug here the mass of an atom and here the typical strain on interaction between atoms the time scale on which atoms move is around 100 femtosecond so with this technology we can see the motion of the atoms and then more recently so this was now 20 years ago people have been able to reach even the femtosecond time scale and even below and so we are really able to see the motion of the electrons so this representation of the kind of physics you can describe with full force short laser passes so arriving at this ultra short time scale so the first process is of course the absorption of the photons by the system and then the coherent dynamics so what we have seen before the fact that you send the pulse and the density matrix oscillates back in 14 time is related to the absorption of photons and the coherent dynamics and then there is a bunch of what I call dissipative process the electron-electron scattering electron photon scattering and other processes which are called dissipative process and they have something to do with the fact that the electrons in the material are interacting among themselves and with the phonos and so which requires dynamical self energies so the term which we have neglected before and then of course there are longer time scale processes which are also non-local in space which cannot be described instead in an abinish approach so what I want just to quickly show you now is that we have up to now described this part of the equation which describes the coherent evolution but in Yambo and again here is not yet available in the GPL distribution we are also working on the description of this term which takes into account all these processes so a bit of theory about that as I told you there are these two coupled equations for the green function with the collision integral which depends explicitly on the self energy and on the time green function and this set of equations are called Kadanov-Baim equations are exact if one could put the exact expression but in practice one cannot use the exact equations because they are too much demanding but anyway one can do controlled approximations to the Kadanov-Baim equations and the other remark is that they are computationally very demanding and one of the reason is that this is a function of two times so you just don't have to propagate on the time diagonal as we did up to now but you have to propagate both on t and t prime axes so the time propagation grows a lot I mean the cost grows a lot because the more you go up in time the more points you have to consider and moreover this collision integral depends on the history of the quantity you propagate so there is memory so it is a scheme with scales like t cube so before we saw that neglecting this term we can write an equation for the density matrix but if you include this term such equation is not closed anymore because it has this time integral and what one can do to close this equation is to introduce this gkba generalize Kadanov-Baim answers ok, I don't want to go in the details just to give you a feeling is that you represent this two time function via an expression where you put just the density matrix the time diagonal so you close the equation if you know the expression of this propagator which you can express in some way so in this case you obtain an equation which is closed for the density matrix and it's just an equation for rho and compared to the full Kadanov-Baim equation this equation still scales with t2 because you still have the memory in the collision integral and if you want an equation which scales linearly in time you have to neglect the memory and this is what we do in the jambo code so we use this approximation gkba, we neglect memory we use a specific expression which is the abinish expression for the propagator which enters here in the collision integral and then this equation let's now forget for a moment about the static correlation we've seen so far now they have this collision integral as a function of the time diagonal only and this is an equation for the occupations of the electrons on the material and in particular this term contains all the scattering processes so for example if you take the electron phonon self energy it will describe the evolution of electrons due to electron phonon and the equation as before can be expanded in the applied field and it becomes an equation which to first order describes the polarization what we've seen before and to second order really describes the evolution of the occupations so this is the physics factor so you apply the field you induce these oscillations then the field enters again it couples with the oscillations so you see this term this time and it creates a really electrons and then the collision integrals which takes this form it has a form of a semi-classical Boltzmann equation describes the time evolution ok, it is just to give an overview of the things which are going on in the code and a simple example of an application a pump and probe experiment where we have this time two laser pulses the first is the pump which is a resonance to the optical garp of silicon and excites electrons from the valence to the conduction band the second is a probe which photoemites from the conduction band and this is the intensity of the photoemission signal so it goes up in time with the pump pulse and then it goes down because electrons so the photoemission indeed is monitored from this valley and it goes down in time because the electrons from here they scatter down here and again there is the electric field so this is the picture you have seen so you break some of the symmetries of the material and here you see more in detail the effect so the electrons are injected in this L point so this one, but not in the L prime point so you see that there are electrons injected here but much less here and then they evolve due to electron phonos scattering so they scatter from here to here and then to the minimum of the conduction band ok, so this is just to give you a feeling of how the approach could be in principle extended to capture the physics of pump and bob experiments and I think I will stop here so I don't need you have