 Vseh je to, da je, da jazem skupaj, zelo skupaj, in zelo skupaj v težkjih vseh, zelo skupaj, in vseh skupaj, zelo skupaj, in vseh skupaj, in naredaj vseh, da nam je polarizacija. To je vseh, ki je materija, na zelo skupaj. Tko je polarizacija, izgleda, da je nasličen, Here, for the moment, sorry for the notation that is different from Davide. It's called alpha in the Davide I will call chi, because the nonlinear community is chi. It's chi in all the books and all the textbooks. Then you read paper that the notation is different, so for me it's chi. The polar adjacent can be expanded in term of the total field E. And everything that is linear optics is here, that you see in the first days. You can have a material within in principalization, but it's not the case now in this lecture. And everything that is nonlinear is this term that are beyond the first one. Now, in practice, what it means, nonlinear. In a day life, I would like to introduce you what are nonlinear phenomena, because normally in day life you don't see any nonlinear phenomena in optics. And the reason is simple, because to get a nonlinear response for a material, you need very strong laser source. And for this reason, for this reason, the first experiment in nonlinear optics, date 1961, it was performed by Franken just one year after the discovery of laser. These are the three guys that developed the field, Blomberg, Schaulow and SIGBA. And also you can find a very interesting lecture of Blomberg on nonlinear spectroscopy on YouTube. The simple example to visualize a nonlinear response on material is to change the color of light. Now imagine you have a laser of a given color, red, yellow, what do you want? And you send this laser field, like this one that I have, on a material. The light that is reflected, transmitted from the material, it will be of the same color of the incoming light. So if I send green, you will see green, less intense maybe, but you will see green light. There is no way to change color in linear optics. The first phenomena that you can see in nonlinear is exactly in state the phenomena where you change the color of light. So you send on a material light that is red, you absorb two photons, and then you emit one photon that is a different color, that is green. This is a typical nonlinear phenomena. It's one of the phenomena, there are many others, but this is the very simplest one to visualize. The first was measured in the experiment. So where you can find the nonlinear spectroscopy, the nonlinear phenomena in experiment in real life. But the first example is very simple, is exactly this one, the laser pointer. So if you go to the supermarket, you buy your laser, you can buy red, green, then you go home, imagine you want to open them, and you open the green one. And you discover that the green laser pointer has a laser that is this one, that has a frequency that is red. This is the wavelength of the red light. But then there is a crystal here that changes the color of the light from red to green, and that's the reason you see green light from the green laser pointer. I have to say that this light is a bit outdated, because recently Nikij Osbram developed a green laser, and that is also the price of green laser pointer drop down at the same price of the red. But at the beginning they were very expensive because they used this technique here. So the first laser pointer built like this, the new ones are really a green laser directive. But there are also many applications only in spectroscopy, in physics, in research. I will want to show you some example before discussing how we want to calculate this. This is an example from a resident experiment in 2014 on mollibrandizol fight and 2D material. So you take this 2D material and you take a picture with your camera and you see that it's light blue. And you say, oh, it's nice. It's light blue, this mollibrandizol fight. But then you take, imagine you have a camera, they can take a picture with the nonlinear response. It means that you send the light with a given frequency and you measure the response at two times the frequency. So you are really probing the nonlinear response of this material. And what you will see is something like this, that in reality this was not a single flake, but it was composed of many flakes and if you color the flakes according to the intensity of the light, you can also see the orientation. And this is because nonlinear response is sensitive to the angle between light and the crystal orientation, while linear is not. So this is the typical application of characterization of 2D materials. Another application that was, I want to show you that it's nice, it's on excitons. So you saw in the nanotubes. So in the talk of Ludger, for example, the ethics on behavior like a nitrogen atom, so you have 1S level, 2P and so on. This is in nanotubes. But for this level there are some selection rules. So for example with light you can excite this first level, but you cannot excite the 2P because it's forbidden. But if you go beyond the linear response, you can absorb 2 photons, like in this experiment, you can excite also this level here and then the system decay and the mid-light from here. And this is exactly a spectral 2 photon absorption of a nanotube that show for the first time where is the 2P level of the exciton. And then also they can show the 1S from the luminescence. So with this experiment they want to show that the excitons are present and are here and instead if there was no exciton in the conduction band you would not see this beautiful spot. The experiment was very nice, but also, to be honest, they were also very lucky because the selection rules they applied to explain this experiment, they were wrong and after some years people noticed this, it was a dress allows and dress allows. But anyway they were lucky because this level was invisible in 2 photon absorption by coincidence, so not by selection rules. And there are other applications that are this here. Again, when you have the response to the second harmonic generation, so this factor you send laser with a frequency and you generate photon with 2 time the frequency, this phenomena is different from zero only if you don't have inversion symmetry. If you have inversion symmetry in your material, the response of second harmonic generation will be zero. And this was used, for example, to measure the number of layers. Now, imagine you have deposited some layers of material, if you have only one layer, is the inversion symmetry is broken, so you have a response. If you have two layers, there is inversion symmetry, response is zero. Three layers you have a response, four layers you don't have a response and so on. So it's a way to visualize number of layers or you can visualize the stacking of material and so on. But you can also probe symmetry not only of the lattice, but also of electrons. Imagine you have a material that, in principle, was inversion symmetry, but in certain moment electron decided to break this inversion symmetry. For example, for magnetization, when you create a ferromagnet, all the spin go in one direction, so inversion symmetry is not valid anymore. And in this case, really you can probe magnetic transition with second harmonic generation. This is a typical probe of magnetic transition. So it's zero when you have paramagnetism and it's different from zero when you broke the inversion symmetry. And then there are some more fancy application. You can use nonlinear optics in the inverse way. So you send the laser at die frequency and you generate a lot of photons at lower frequency. Like in this case, this one laser that is green and then create many photons that are red. And this is used to create entagled photons for quantum communication, quantum information, different experiments that use nonlinear crystal in this way. Or there are some application in biology that are nice. You can take a nano crystal that has a strong nonlinear response. You attach the nano crystal to a protein you put in a living system and then you can follow the protein in a simple way. You send the red laser on your living system and you see where the system responds in green. And because biological material doesn't have a nonlinear response you see only the response from these crystals. And so you can see the protein and move and so on. The nice point of this technique is that different from green foreign protein there is no quenching so it's always stable the nonlinear response. And then there are also the sometimes there are the dark side of nonlinear response. There are some applications where you can transmit nonlinear response. And one example is optical fiber. So you want to transmit your signal from here to another place and you want to transmit as far as possible. So you have your optical fiber you want to send a pulse inside the optical fiber as strong as possible in such a way that it arrives far, far away. But there is a phenomenon called sun focusing. So it means the beam will start to focus in the center of the optical fiber and at the end it will burn it. And this is due to the nonlinear response of the optical fiber and unfortunately at the present nonmetal is known for increasing the self-focus limit of the optical fiber. This is from Photonic Encyclopedia. So it is an intrinsic limit of materials the number of light that it can transport due to the nonlinear response. This was an introduction to show you some phenomenon of nonlinear response. And now how we calculate nonlinear response in Jambu. In parts of the edition by Davide, but the idea is this one. So in this case we choose an external perturbation instead of propagating density matrix now I will comment on this point we propagate the Schrodinger equation clearly not the full Schrodinger equation it's an effective Schrodinger equation propagating time, we calculate the polarization and from the Fourier transform of the polarization we can get linear response, but also nonlinear old and nonlinear response that you want. Now everything seems easy but there is some technicality and some physics that you need to know in order to implement this equation that I want to show you. And the first point is that how you calculate polarization. Because the polarization is an object that is not easy to define for molecules of finite system but for periodic system the definition was unknown for many years even textbooks was not a definition of the polarization and let's see why. So the polarization for finite system is something easy, you integrate the dipole the average of the dipole and divided by the volume of the cell so you get the polarization. But how you define a polarization in a periodic system in a bulk system. Now in the literature people propose different solutions for example you can say OK I can average the dipole of the sample divided by the volume of the sample I can use the dipole of the cell divided by the volume of the cell or I can use the dipole matrix element that you can calculate with yaml. Not good. If you average on the sample the effect of the dipole cancel inside the sample you just get a surface contribution of the polarization. So this definition is in principle good but you need to simulate an infinite piece of material and just to have the contribution of the surface why you want the contribution from the bulk. So no way to this. The second idea is to do the average of the dipole and divide it by the volume of the unit cell is completely arbitrary because the unit cell is arbitrary. So imagine a material like this you have a positive charge here and a negative charge here you can change this as a unit cell and then repeat periodically and the dipole is zero but you can also change d1 is a valid unit cell and the dipole is defined in this way. So the dipole in the unit cell can not be used. And if you are thinking about periodic system where you can use the relation that is for localized charges you have to make that this doesn't hold for solids because solids are we function of the localized. And finally the third way that is the use of dipole matrix element is very bad because intra band sample in the final they diverge at the band crossing so one band touch and so on also only they say that they generate. For sure you can use them for linear optics for the linear response because you just need balance conduction but beyond linear response they cannot use it anymore. So how we calculate polarization? To calculate polarization we use an approach that is based on Berry phase now I don't know many of you are familiar with Berry phase so I will make a short introduction and to see where it comes from. Now the idea of Berry phase this is the picture of Berry that also got the HG novel is not so complicated so imagine a valamiltonia that depend from a parameter see a parameter in the Hamiltonian so you can for each value of the parameter you can diagonalize the Hamiltonian and get wave function and again values and also the again values. OK, that's fine. Then you change the value of your parameter for example from xi1 to xi2 and you diagonalize again your Hamiltonia you get new wave function new again values and so on and now you can ask the question how much the phase of the wave function change when I change this parameter from xi1 to xi2 so you can define the change of the phase of the wave function that is the subject here that is similar in geometry when you have two vectors and you want to define the angle between the vectors is the scalar product divided by the modulus and you can ask how much the phase change in reality this question is not the correct question because as you know from quantum mechanics the phase is arbitrary so you can have an arbitrary phase here when you diagonalize Hamiltonia so the phase change is arbitrary too to make no sense but what to discover Barry was that if you consider the phase change in a closed path like this one so the phase change when you do a path in a parameter space this object is gaug invariant it means that if you attach a random phase to this wave function here since it appears as brine cat the random phase cancel and you got a new object that is gaug invariant and so it means that it is a possible observable because in physics when you have an object that is gaug invariant usually it is an observable but it is a bit exotic because it is an observable that it is not expressed in term in a median operator so it is a bit why and why it exists because the fact that there is a parameter and you can also do the limiting continuous difference it means that your Hamiltonian is not isolated so this object is like the copy with the paper of Barry with the rest of the universe if you have a really isolated Hamiltonian and you take an account of all the degree of freedom you will not have the very phase but if your Hamiltonian is coupled with something you can have observable that are not ok there are some example in physics like the random bomb effect the molecular random bomb effect or the effect in transport that you can remap in this phenomenon of an Hamiltonian coupled with an external parameter but now how this is connect with polarization to see the connection with polarization we come back to the electronic periodic system when you have an electronic periodic system you have the wave function to obey to the Bourbon-Karman boundary condition so you have a periodic part and a phase that depends on k and you can then you solve your one-party or Schrodinger equation with these orbitals and use the block theorem you can show that this corresponds to the solution of this equation for the u where k act as a parameter for the periodic function k enter if you put in this wave function here enter as a parameter on the Hamiltonian so you can ask the question ok, if k is like a parameter if I do a loop in the k-space which is the observable that I will get and seems incredible but ok what you will get if you do a loop in the k-space the observable that you will get and this was demonstrated in 1993 but King's-Mittenbander demonstration is also quite easy they passed through the varnier function to demonstrate that this is exactly the fish of polarization and the oh sorry, I put two times this line the video of this object is that is a bulk quantity so it doesn't depend from the surface the time delivery is give the current and reproduce the polarizability to all the order so it's exactly the way how you should calculate the polarization when you know the wave function then they also propose a discretized version of this object and extend it to the three dimensional material so you have different line and you have a region of different line now we have a formula for the polarization how you go and how you are going to use it the idea is that once you have the polarization you can also try to find the question of motion because you define a Lagrangian from the Lagrangian they define the Hamiltonian and from the Hamilton, and from I will skip all this step about this reference at the end you find the Schoeninger question for solids where the coupling with the external field is defined by the derivative in K at the end what we what you can show that the equivalent, the dipole in a solid is nothing at this derivative ok, the polarization is expressed with the formula that I showed before these are all the ingredients that you need you need the coupling with the external field and the way to calculate the polarization and now then you can do all the response function you want I will show you a typical example also taken from some other code now this is an example of how people do in open system but in periodic system more or less we do the same now so imagine you have a molecule and you excite with a field that is like this or monochromatic how we do you get the polarization oscillate on the frequency of the laser but then there are other smaller oscillations for a few time the frequency and this is the nonlinear response of the molecule this is a typical example from TDFT or octopus or fiesta so this is the way we simulate the system and it's nonlinear response but now ok, that way as I told you how I want to simulate nonlinear response I also have to speak about which are the advantage why we want to use this approach so the first of all as I told you because the coupling is correct, I mean we have the coupling with all the orders of the external field so you can simulate all the response funds and every time you have a phenomenon that is coerent with the external field beyond the linear response you need an approach like this so maybe we can find another solution but for the moment you need an approach like this with the very face and moreover there are the advantages to do this simulation of nonlinear response in real time the advantage is that when you have the polarization ok, the linear response all this nonlinear coefficient they represent a very complicated phenomena for example the second harmonic generation means that yet you can have a laser two times two fields with the same frequency or you have two fields with different frequencies so the possible interference phenomena between the external fields are many are very different and in the experiment they do trade with some frequency generation different frequency generation so they put a laser with a frequency a laser with another frequency and then they measure the response in the outcome frequency and if you have the third order the experiments start to be even more complicated you can have some or two laser difference with another, two laser have the same frequency so the number of possibilities is very, very large here is a laser where you absorb three-fourth and so on and there are experiments even more complicated there is this four-way mixing where they put three lasers and then they measure the response in the fourth or the two-fourth on spectroscope so the combination that in the experiment are very large and the advantage of a real-time approach is that you get directed polarization and so you can try to extract all the different coefficients for the different possibilities also you can simulate the laser in the experiment another advantage that I like is that you have only one question so the question is always the same and all the correlation effect are in the Hamiltonia so if you choose an Hamiltonia you put some correlation effect if you choose another Hamiltonia you have different correlation effect Hamiltonia determines all effect that you want to put from the correlation point of view we are not playing go beyond this and the other advantage of to have a simulation in real-time is that you can go in nonperturbative phenomena like for example, I-armonie generation where you put the laser at a given frequency and you are able to generate a different many-many frequency like 14 times, 20 times the frequency that you put at the entrance in your solids in other code in the atomic motion like in octobuses and now I want to show you an example of real-time simulation so I will that I took from YouTube to give you an idea so this is the absorption they put the laser with the frequency that is exactly resonant to this absorption and so you can see as density oscillate when it is excited by the laser field and you can do the same in solids but I don't have a video yet clearly in this case but also in the case I will discuss after the density continue to oscillate because we have an amyltonia disarmidian so there is not damping, not the phasing so at present this is a TDFT calculation we can clamp the ions yes it's nice let's discuss a bit of the amyltonia as I told you you want to do time-dependent shot in great question so all the correlation effect are in the amyltonia so let's see which amyltonian we can use in part it was shown by David also from full so the first amyltonian you can use step 0 start from the coneshama amyltonia that is this one here and fix the density ground state density and in this case you are just propagating an independent particle amyltonia and if you do your propagation time and you analyze the result you can see what you get is independent particle linear and non-linear optics but then you can go beyond for example including GW correction what are GW corrections in real time GW corrections are nothing else that a shift of the eigenvalues so it's like if you add the disoperator to the previous amyltonian and you change the eigenvalues again you can propagate in time and you can get linear and non-linear response I had to comment on this that GW correction for linear response is a shift but when you go beyond the linear response it's not a rigid shift anymore so this is an example of this secular monogeneration in aluminium and cyanide and this one you include the GW correction but in the other case it changes even more the shape of the spectre another effect that you can include is independent R3 in this case you say ok I start from my conmichana amyltonia but I let the density of the RT term evolve in time so you evolve the wave function the wave function generate a new density the density generate a new R3 that move by 5 wave function in non-linear response this is a second monogeneration cadium telenite without and with local field effect and this is the experiment so it works you can go beyond this approach and say no ok let's try to evolve also the exchange correlation potential and this is the creativity of the TDFT so you can you evolve the wave function you update both for the potential and so and this is what you get in the DFT it works very well for at least standard DFT very well for molecules but it's not so effective for solids but there are different reasons but I like the fact that the Runge-Gross theorem doesn't guarantee that this theorem is correct for solids so in principle you should go beyond the simple density in the code we implement also other terms beyond the TDFT for example which you include another term that depends on the polarization so you can have an Hamiltonian that depends on the density and from the polarization and this is if you want the minimal ingredient to get excitons you're going to use another way but this is a very atrofined and elegant way without including divergences in the theory and so on with this approach you can also include longer range part that was accent in artery exiton in the real-time propagation and finally you can also derive an Hamiltonian from many body as was shown by Davide so you can say that your Hamiltonian is the Hamiltonian GW that I showed you before plus the ex self-energy that is written in this way where this is the variation of the density matrix and the real-time propagation of this Hamiltonian is exactly equivalent to the leader where I took the picture from Andrea and but when you propagate in real-time coupled with a very phase polarization and so on you can also get the effect of exiton in non-linear response I show you an example boronitride because we show only example on this material in this school no story so this is exagonal boronitride and this is the linear response the electric constant at independent particle this is when you include local field effect that doesn't do so much this is the second harmonic generation that resonate at the linear frequency and the alpha of the frequency also here local field effect doesn't do so much and this is the Hamiltonian Hamiltonian in connecham plus the term of artery then you turn on GW GW shift rigidly the electric constant but change the shape of the second harmonic generation and then you get here and then you can turn on other term in the Hamiltonian like the screen that change and you get exiton so you get your beautiful exiton in linear response you get UC action in non-linear response that are replicated because in non-linear response there is a resonance at omega and alpha at 2 omega finally I want to spend a couple of slides on the phasing as I told you so if you excite the system this will oscillate forever so you have to introduce the phasing term that as the one was introduced by Davide this phasing term is nothing else the term that in the wave function formula is that bring back the wave function to the ground state so after a while you excite that there is a term and say ok let's go back to the ground state in other code this term is done in post processing even if this is much more difficult for non-linear response at least for my point of view and the effect is this one you introduce the phasing term that was a very long time so your system oscillate a lot it means that you get very sharp peak when you do the Fourier transport you introduce the phasing term with a short time short and short it means that you Fourier transform your peak begin broader and broader so this is really the equivalent between the broadening and the defacing time that is the same in density matrix and you show the parameter yeah, it is a parameter but just to show that it is a parameter that count because for the spectra you will get at the end no, it is like a neurotic you put a parameter for sure you can derive from self energy and so on but otherwise you put like a neurotic and finally a comment on how we extract on linear response as Davide told we excite with the sinus electric field so then we suppose that the polarization oscillate at the frequency of the exciting sinus plus all the possible multiples so we excite omega, 2 omega, 3 omega and so on we have the defacing term so that they kill the oscillation of the auto oscillation of the system after some while we analyze the spectrum to extract all the coefficient of the polarization and this give us the response 2 times the frequency and so on this is for the second half second or third harmonic generation so we use these answers but for other response function we have to find new strategy I will show you an example and to finish I will show you a couple of applications this is an example of non-linear response in molybeno diesel fight the red line is the independent particle and the green is when you turn off the exciton and the pointer is the experiment so the agreement is not so bad the interesting point is that the exciton double the intensity of the non-linear response on a material like this so it was very interesting let's say that then for the experimental point of view to get the absolute intensity is very difficult the first experiment got order of magnitude different one from the other another example that I want to show you is how to extract for example a coefficient like this one imagine that you want to extract the third order response at the same frequency of the exciting elasens that is the chifoton assort so how you do a simple strategy that is a research on extrapolation you do different simulation with a field E with a field E divided by 2 and then combining the result of these three you can cancel this term and get exactly the coefficient you want this is the strategy to extract coefficient at higher order so perform different simulation of different intensity then recombine them with the correct coefficient in such a way to cancel the term that you are not interested in and the result is again sorry it is a bulk the bulk are two levels two that are dark and two that are bright and the bright can be excited with the linear optics but the dark are visible only if you absorb two photons and this is the spectra of linear assortion and this is the spectra of two photon assortion and this one is the experiment performed in Montpellier in 2016 I think that's all I want to acknowledge some people that collaborate in this works with the Francois Achim Myrta Silvan and also I would like to leave you on some references that you can find interesting is the reference for non-linear optics this is very nice for response functions also there is the demonstration how the beta salpita reduced to a nitrogen like equation when you have parabolic bands the one at Ludger show this is for the very face in the electronic structure but it is also free lecture by Resta and this is the general electrodynamics of solids