 Okay, good afternoon, let's start. So, today we have a pleasure to have Professor Robin Santra from the University of Hamburg to give us a colloquium with the title Atoms in Intense Light Fields. Professor Santra has been a professor in Hamburg for a while, but before he has been in several institutions, born in the US, raised in Germany, started in Heidelberg. They had positions in the United States, in Argonne, in Heidelberg also, and then back to Germany where he's a professor. So, he's also here as part of this activity that we have together with IAEA, which is the School and Worship on Fundamental Methods for Atomic Molecular and Material Properties in Plasma Environments, and so we have here the pleasure to have the participants of that conference, including the directors, that we thank them to participate and organize it. Professor Santra has a very well-recognized figure in the field, he has many accomplishments, which let me read some of them to include the prediction of electronic whole dynamics and alignment in optical, strong field ionizations, and predictions that motivated and supported the first experiments at the X-ray free electron lasers, LCLS. He has been awarded the IUBAP Young Scientist Prize in Atomic Molecular and Optical Physics in 2007 for his outstanding contribution to the areas of physics, and has received the US Presidential Early Career Award, PECASE, and he's a fellow of the American Physical Society. So please join me to welcome Professor Santra. And introduction, thanks for having me. The great honor. So I'll talk about atoms in intense light fields, and intense in the context of this talk means that the electric field component of the optical, of the light field must be comparable in strength to the electric field that binds a typical valence electron in an atom. So you have such a strong electric field that typically you should doubt whether applying perturbation theory to describe the interaction of the light field with the atom is an adequate approach. And as I'll show you, it typically is not. Even though I'll show you also counter examples at quite high intensities as well. So we'll basically cover a wide range of photon energies starting at photon energies of about one electron volt. So basically in this region between the near infrared and the visible regime. And then we'll work our way up through the extreme ultraviolet all the way to hard X-rays at close to 10 keV. And we'll be investigating high intensity radiation atom interaction phenomena at these various colors of light. But before I go into details, let me briefly thank some of the people who have contributed to this work from my group here, members of my group at the Center for Free Electron Laser Science. I would like to highlight in particular Yijian Chen for the work in the first part of my presentation and Sankil Sun and Kudai Toyota for the second part of my presentation. We've been collaborating for a couple of years with Ference Krauss's group at the Max Planck Institute of Quantum Optics and Steve Leone at the University of California in Berkeley. Ziheng Lo at Nanyang Technological University in Singapore. Thomas Pfeiffer at the Max Planck Institute for Nuclear Physics. Michele Maier at the European Ex-FEL. And last but not least, Artem Rodenko and Daniel Rollis at Kansas State University. So we start with small photon energies, wavelengths of about 800 nanometers. As I said, this is basically this transition regime between near infrared and visible. And as we'll see, this is going to be a regime where the intensity that we'll consider is of the order of 10 to the 14, 10 to the 15 watts per square centimeters. If the electric field of the light field is indeed the same as the electric field that binds an electron in the hydrogen ground state, then your intensity is about 10 to the 16 watts per square centimeters. So I will not be talking exactly at an intensity where you exactly add one atomic unit, but a little bit below. And if you're in this regime a little bit below where the electric field is of the order of about the 10th of that interatomic electric field, where the light field is about the 10th of the interatomic electric field, we have a phenomenon that's typically called tunnelingization. And optical tunnelingization is something that actually depends on the way you describe the light-atom interaction. So typically, if you come from high energy physics, the natural way you would think of QED is that you have some kind of minimal coupling description, something like a del mu A mu type interaction between the electromagnetic field and the Dirac field, say. If you're doing non-relativistic theory, then you would also often start with minimal coupling but adopt the Coulomb gauge. If you use either of these descriptions, you wouldn't directly get to this picture of the tunnelingization. To get to this picture, you take yet another step. The alternative to what is known as the minimal coupling Coulomb gauge Hamiltonian is the so-called multipolar Hamiltonian. And in the dipole approximation, and here we're talking about long wavelengths, you know, 800 nanometers are 1,000 times bigger than the atom, so you can make the dipole approximation. And in the dipole approximation, the leading order interaction between the electrons and the electromagnetic field is just the coupling dipole operator dot laser electric field, yeah? Position scalar product with electric field. It looks very intuitive, but it requires some machinery to actually get to that point. Once you have it, then here you see the consequence. Let's assume the electron can respond very quickly to the presence of that electric field. So to put it differently, the response of the electron is in a sense fast on the time scale on which the electric field oscillates. At 800 nanometers, the period of one electric field cycle is about 2.7 femtoseconds. We're now considering situations where the electronic response is fast in comparison to that. Once we're in that regime, very rapid electronic response on the time scale of the optical cycle of the electromagnetic field, then this adiabatic picture underlying the tonal ionization model applies. So what is this? So the dashed line indicates, so to speak, the atomic potential that traps the electron, if it's bound, and let's say, bound in this quantum state at binding energy WB, and now you turn on your electric field and from the perspective of the electron, it's assumed to be quasi-static, right? You have adiabatic response. So at a given electric field, due to this position or dipole operator times electric field, your overall potential that the electron sees is shown by the solid line. So in one direction, the potential actually keeps going up, but in the other direction, this atomic potential basically has been pulled down, right? So the effective potential now is such that an electron bound at this level can tunnel through this barrier and in this way become ionized. That's the tonal ionization picture. It's, you can understand it strictly speaking only within this multipolar Hamiltonian framework. You don't have this picture of tonal ionization in the minimal coupling framework. Good, so tonal ionization. Let me tell you a bit about some things that have been discovered in the last eight years or so in connection with strong optical fields. So we're now considering a specific experiment which was carried out at the MPQ in Garching, the Max Planck Institute of Quantum Optics, where they exposed krypton atoms to a laser pulse, a near infrared laser pulse. At an intensity close to 10 to the 15 watts per square centimeters, such that the electric field of the light field is about the 10th of the intraatomic electric field. And the pulse, the infrared pulse that was used was quite remarkable. Not many groups can produce this. It was a pulse that was overall just four femtoseconds long. I told you, the period of the light field is 2.7 femtoseconds. The whole pulse was just four femtoseconds long, right? You have basically something like one or two cycles in this pulse and that's about it. Under certain conditions, they can even control how within the envelope of the pulse, the time evolution of the electric field takes place. So you can often control it, but in this specific experiment, the pulse duration was controlled, but the phase evolution of the electric field within the envelope was not. This will matter a bit for what I'll talk about in the following. So you have this very short pulse, which is only four femtoseconds long. Now let's look at this data. This scale is the delay between this optical pulse, which causes tonal ionization, which rips out an electron from the valence shell of krypton, and then you interrogate the whole formed in the valence shell through resonant excitation with an add a second pulse. So the add a second pulse is delayed relative to the infrared pulse, which does the tonal ionization. So here at a delay of zero, that's when, so to speak, the ionization takes place within that four femtosecond pulse. And out here, the add a second pulse looks at the ions that have been produced. And it looks at the electronic structure of the ions that have been produced. How? So you can see here the projection of these mountain ridges in the energy axis. So this axis here is energy, transition energy. Your add a second pulse is centered at an energy around ATEV, but because it's an add a second pulse, it has a spectrum that's about 20 EV wide. And you're basically checking which colors within this 20 EV bandwidth get absorbed by the ions produced through the tonal ionization process. Yeah, and so before, that means at negative time delays, there are no absorption lines because you haven't created a hole yet in the valence shell. And as soon as you've ripped out an electron from the valence shell, then these absorption lines appear. This line here on the left corresponds to a transition where you take an inner shell 3D electron with J equals five halves and you plug the hole if the hole happens to be in the 4P three halves level. So you must understand the valence shell of a krypton atom is a 4P shell, but if you realize that there's spin orbit coupling, the highest atomic orbit has J equals three halves and the next deeper one has J equals one half. There's spin orbit splitting of the 4P level. And so this corresponds to exciting into the hole if the hole happens to be in the topmost atomic orbital. The second line corresponds to exciting from 3D three halves into 4P one halves. The fact that you see the line tells you your ions have not been produced exclusively in the ground state. You have not removed the electron only from the topmost orbital, but in the tonal ionization process, you've also removed the electron with a certain probability from the next deeper one. That's why you see this absorption line. And then finally, this here corresponds to 3D three halves to 4P one half of 4P three halves. And what you observe very directly is that this evolves in time. What that tells you is again the ionizing pulse is basically over within four femtoseconds. This is this area. All what you see out here is the free evolution of the electron cloud of the atoms formed. If you've started with spherical ground state krypton atoms, zero angular momentum, you've ripped out an electron, you've done tonal ionization, and now you realize that this electron cloud, it breathes, it evolves, right? It does. Due to spin orbit coupling in the ion, you actually have dynamics in this electron cloud with a period of six femtoseconds. And because you probe this with about 100 second pulse, you can directly see the time evolution of the electron cloud in the valence shell. This is valence electron dynamics triggered by tonal ionization. Now, when we did the calculation at the time, we assumed that the pulse that is used, this is shown here as a dashed red curve. This is the time evolution of the electric field that we assumed. And we calculated basically the properties of the ions produced in this tonal ionization process. How do we do this? What we essentially do is we solve for the n electron atom, the time dependent trading equation, within an approximation. The approximation is we say we can write the n-body wave packet as a coherent superposition of the Hartree-Fock ground state of the atom, plus all so-called one particle, one whole configurations you can form when you do particle whole excitations relative to the Hartree-Fock ground state. So for the quantum chemistry in the audience, this is a time dependent configuration interaction singles framework that we've used. And basically you plot this answer for the wave packet into the time dependent trading equation. And this way you get equations of motion for the expansion coefficients and that's what you integrate numerically. Now, the experiment is an experiment that actually doesn't look at the electron that got ripped out by tonal ionization. It looks at the ion, it looks at the transition from inner shell into the hole. So what we really care about is not at the end of the day the n-body wave packet, but what we care about is the reduced density matrix associated with the hole. So we take this n-body wave packet, kept psi, which depends on time, form kept psi, bra psi to get the pure spend density matrix associated with the state. And then we trace this over the states of the photoelectron to get the reduced density matrix of the hole, of the ion that remains behind. And as you can see up here, let's look in the region after the ionizing pulses over. You see that most of the population, about 70% or so of the whole population is in the state where you removed an electron from the highest atomic orbital. The four p three halves, but specifically only with m plus and minus one half. Why do I emphasize m? Because if you have created a hole with a j of three halves, you have four m's available, right? You have m plus or minus three halves and m plus or minus one half. They have of course the same binding energy, but you do not populate them with the same probability. You've populated almost exclusively, as I indicated, the one with m plus or minus one half. And this one here at the bottom, which has just a couple of percent in population, that's the m plus or minus three halves, even though it has the same binding energy. But that comes from the selectivity that tunneling actually achieves, right? So to speak, if you have to go through a tunnel barrier, it's easier, it's faster for you to tunnel if your orbital is already aligned with the thinnest region of the barrier, right? And the m plus or minus three halves electrons effectively have to go through a wider tunneling barrier and their tunneling therefore is reduced, even though the binding energy is the same. And then you see here for the spin orbit excited state, this is this one here with a j of one half, that's again a sphere, that has a population of about a third. So this just tells you basically about the diagonal elements of the ion density matrix, the populations. But we get of course the coherences as well, the off diagonal element. And there's only one non-trivial off diagonal element that's basically correspond to the coherence between this state and this state. And this is the coherent superposition that gives rise to this electron cloud oscillation which we saw in the spectroscopy on the previous slide. When we calculate the degree of coherence between these two electronic states from the off diagonal element of the ion density matrix, then we find about a 60% degree of coherence if one uses the four femtosecond pulse that they adopted in the experiment. And the 60% degree of coherence is exactly what is consistent with the experimental data. If one had done this with a more conventional eight femtosecond pulse, the degree of coherence is less than 10%. And that's why only if you have already optical technology that allows you to create these few cycle laser pulses, do you even notice the dynamics going on in the electron cloud produced through tonal ionization? Now, one thing I'd like to point out if we go back to this upper panel, if you see how the whole population's built up. So the whole population builds up, basically whenever you go through an anti-node of the electric field, then you have a high tonal ionization probability. If whenever you're going through a node of the electric field, well then you're basically not tilting the potential and you're not tonal ionizing. So you should expect that in the tonal ionization model that you actually go up in steps. Whenever there is a node, there should be a step. You should no longer be ionizing. When you look carefully, there's always this overshooting. It goes up and then it goes down again. Now there's a step because we have an anti-node of the field, but then we overshoot and go down again. And at the time we saw this and thought, oh, that's cute to see, but because they had actually no control over the phase evolution of the electric field, we could see that in theory, but it wasn't seen in the experiment. It was seen, however, relatively recently in the experiment by a Steve Leonis group at Berkeley. They managed to get nicely phase-stabilized electric field, so that really reproducibly from shot to shot, the electric field evolves in a very controlled way relative to the envelope of the poles. And now you can really see this effect. You can see here in the experiment theory that this type of going up, but always overshooting and going down again is precisely what you see in experiment and theory. This is now not for Krypton anymore, but for Xenon, but the idea is the same. What's the meaning of this? What's going on? Why do we see this kind of funny overshooting? The analogy that I like to use is the following. Let's assume you have a spring, and if you're in the limit of hook flaw and you pull the spring and you let it go, then you go back to your equilibrium position, right? It's a nicely reversible process. Now take the spring, you pull it really hard and it breaks apart and then you have two pieces. This quantum system does both things always simultaneously. You always have a component which elastically flows back. So you're polarizing the electron cloud, you're pulling out the electron and a fraction of the electron cloud flows back into the hole. That's why you get overshooting. You overshoot because you pulled a bit of the electron cloud away, but once you go to the node of the electric field, that part of the electron cloud flows back. At the same time, you have an irreversible process which is the ripping apart of the spring, right? That's the tunnel ionization step. So you always get both together. One would think you get only one if you're in the really weak field limit and you get only the pure tunneling when you're in the really strong field limit. In fact, you always get both together. And at the end of the day, of course, there is... If all you're looking at is ions or holes produced, well, during the pulse, you're transiently forming a hole, right? You're pulling an electron out into a particle state, you've formed a transient hole so you can plug it transiently that gives rise to this maxima when the electron flows back and fills the hole, then basically this peak disappears. So the tunneling model would basically give something like this. You would have a step here, then you would have a step here, then you're done. And the polarization effect I just talked about is what leads to this overshooting. Okay, this type of time-dependent configuration interaction singles approach I've indicated in words, expanding the wave pack in terms of particle-hole configurations, with use for a number of optical strong field problems, but also for XUV-related high-intensity problems. Before I turn to that, I'd like to motivate this a little bit by discussing briefly this figure which was taken from this paper here. Add a second pulses. I said one can see the electron cloud move by using an add a second pulse to probe it. But how do you get add a second pulses in the first place? The way nowadays add a second pulses are generated is the process of high harmonic generation. And the process of high harmonic generation is actually intimately linked to the process of optical tunnelingization. What is happening is, you have your atoms, say, Krypton or Xenon or so, you apply your optical field, your tunnelingize, that is you rip out an electron, but it's an oscillatory field, right? It does this thing here. That means when the electron goes away, a certain fraction of the electron once the potential changes its sign gets pulled back and it gets accelerated in the electric field of the laser. That's why the optical field doesn't only ionize, it accelerates the electron. And when the electron then hits the ion, you can turn all the kinetic energy that the electron has acquired in the optical field into light. So high harmonic generation is just that. You use an optical field to tunnel ionize, you use the field to accelerate the electron, you use the field to bring the electron, at least with some probability back to the ion, and all the kinetic energy that the electron has acquired can be turned into the emission of a photon. So when you see high harmonic generation, you see how an atom gobbles up, say, 100 photons, fuses them together, and spits out one photon. So this is a spectrum calculated for high harmonic generation from atomic xenon. And in the experiment that we wanted to understand, they used, in this case, not these 800 nanometer standard laser pulses, but they used laser pulses of photon engines of about 0.75 EV. So when you detect a photon coming out at 75 EV, then you've done exactly what I just said. You've combined 100 photons and converted them into one photon at 75 EV. In this experiment, however, you see even photons out to 150 EV. When you look at this, just look at the red curve for the time being, because this looks like the standard high harmonic generation spectrum that people love and have been using for many years. This red curve, this shows what is known as the plateau. And if you try to understand harmonic generation perturbatively, you would think that different orders would somehow scale with a number of the order. That is particularly if you need 100 photons to create a 75 EV outgoing photon, or you need 200 incoming photons to create 150 EV outgoing photon, then you would think, ah, that must then be order 150, and this must be something like order 100. They're equally strong. That's why I said, this is highly non-perturbative. You can actually not describe this in any meaningful way using a perturbation expansion in terms of the light intensity, which is exactly the reason we simply solved the tiny penetrating equation numerically in order to accommodate such effects. But it's numerically expensive because why did these people use 0.75 EV photons? Somewhat counter-intuitive effect. So if you want to have higher harmonics coming out, so higher ex-EV photons coming out in this process, you actually need to use smaller infrared photons. Sounds odd, but the reason is maybe understandable if you think in terms of this oscillatory potential. If the potential oscillates nicely slowly, then the electron can roll down the hill for a longer time, and it can correspondingly roll in the opposite direction for a longer time. That means it can get accelerated more. You can acquire more kinetic energy if the field oscillates slowly. If it oscillates very rapidly, the electron gains hardly any kinetic energy. That means since the oscillation period is connected to the photon energy of the infrared photon, you want to use a lower infrared photon energy, not a higher one. That's exactly what they did. In the old experiments, they didn't even see any of these structures. Particularly what we wanted to understand and reproduce is this bump on top of the plateau. And that bump that had not been seen in the old HHG experiments, simply because their cutoff was about here. And in this experiment, they were able to push the cutoff out to 150 EV by using a lower infrared photon energy. And in this way, see in the experiment, this bump. And now I can tell you where's the bump from. So these three lines, they correspond to different approximations in the CI singles, in this configuration interaction singles. Essentially what we do, we set certain CI matrix elements, certain Hamiltonian matrix elements to 0. Most that are set to 0, so the simplest model is the red one, which looks like the standard models that people normally use. It excludes the ability of the particle that you generated through tunneling and the hole to exchange energy through Coulomb interactions. So we call that the intra-channel model. We do not allow, so to speak, the particle in the hole to form a nicely entangled pair. Because the Coulomb interactions that would allow the particle to change the state of the hole and vice versa have been turned off, then you get the simple plateau. But if you allow the particle in the hole to talk to one another in the continuum and to exchange energy and change the state of the hole through energy change of the particle, then suddenly this bump appears. So you have an enhancement in higher harmonic generation at an outgoing energy of about 100 eb. Where is that from? That brings us now to extreme ultraviolet physics. Imagine playing the higher harmonic generation process in reverse. So instead of having an electron coming in, accelerated by an optical field, and then emitting an XCV photon at 100 eV, take the xenon atom and come in with the 100 eV photon, and then the electron goes out. So you should see in the XCV absorption spectrum a big peak at 100 eV. In the photo absorption or photo ionization spectrum of atomic xenon. And this is known in the literature as the atomic giant diporessonance of xenon, the xenon 4D giant diporessonance. And now I'll talk more about XCV, no longer about optical. So Dave Eder and others had, but more than 50 years ago, actually discovered the giant diporessonance. This shows the xenon absorption cross-section as a function of the XCV photon energy. And you see here, this is a very broad resonance. It has a width of about 20 eV or so, again, centered at about 100 eV. And it's a resonance in the continuum. Because the ionization threshold, the energy it takes to remove an electron from the 4D shell is at about 70 eV. So here you see the two thresholds for the Jacobs 5 halves and the Jacobs 3 halves channels. The big bump is, however, higher in energy. So we're not talking about a Rittberg resonance. Most of you think of atoms, and you do say sodium through S to 3P or something like that, bound to bound transition. This is not what the giant diporessonance is. A giant diporessonance is a resonance in the continuum. 4D into the continuum, you're resonantly photoionizing, but with an enormous cross-section. And in the same issue of PRL in 1964, Cooper already came up with an explanation, sort of. This is shown here. So he took a simple approximation for the atomic potential. This was a Hartree-Slater type potential, VHSFR. So that's the atomic potential in which the electron moves. You start in the 4D level, and because of dipole selection rules, in this situation, you go predominantly into an f-wave final state. f-wave is L equals 3. So now we want to know what's the effective potential that such an electron sees, such an f-wave final state photo electron produced through XCV absorption. Well, what do you do? You take the atomic potential, and you add the centrifugal potential at L equals 3 to it. This is what is shown here. So this radius is the distance of the electron from the origin, which is where the nucleus sits. And you see that the effective potential has actually here a barrier as a function of electron distance, which comes precisely from this centrifugal potential. The dashed line indicates where your ionization threshold is. That is, if you excited the atom at an energy below 70EV, you would go into some bound Rittberg state and L equals 3 and f state that's bound in this potential. But if you're up here, then you are in the continuum, but you get what is known as a shape resonance, right? A resonance due to the presence of a potential barrier to which you basically have to tunnel out. The electron that you generate through the excitation, so this 4D into resonance state excitation, causes the electron to be trapped at the distance of about half a ball radius from the origin, which is also where the 4D electron is. So this is not like a Rittberg state. In a typical Rittberg state, the hole and the electron, they're kind of nicely separated. Here, this is not the case at all. They sit right on top of each other, which therefore is also the reason you should not be surprised that this model actually fails quite miserably. This is what you see here. So the theory that Cooper wrote down, based on this effective potential, sure enough, it gives you a resonance in the continuum above the ionization threshold, but you see it's peaked at 80EV and it's unbelievably narrow, and then you see the experimental resonance. So the simple mean field or one particle model doesn't work all that well. So you understand, therefore, that there are substantial interactions between particles and whole, and within CI singles one can at least recover some of that. Not fully, but you get then a resonance at 100EV with the right width. Only the cross-section is not at 30 megabarons in CI singles, but it's about 20 megabarons. But all the essentials are there in the model I'll describe to you. But what's the new thing that was discovered at high intensity? So we've moved away now from optical. We're now talking about shorter wavelengths, extreme ultraviolet, 100EV. We're now talking about a two-photon experiment at 100EV, which was only made possible by the development of free electron lasers operating in that regime. So we had enough intensity to explore the giant diaper resonance through a two-photon excitation or two-photon ionization scheme. Specifically, what we wanted to understand is whether this resonance, which is this big blob, whether in the language of resonance theory is actually one resonance state, or is there substructure maybe? And what I'll show you is with a nonlinear experiment enabled by high intensity XIV, we do find substructure. I won't go too much into the details. This is just to show you how basically this time-dependent configuration interaction singles compares with experiment for this specific situation, xenon giant diaper resonance. What's the experiment? Well, what you do is you have in the experiment two photon energies, 105EV and 140EV. That means you park yourself basically in the region here of the giant diaper resonance close to the peak and a little bit farther out here in the right-hand side, in the right wing of the giant diaper resonance. These two photon energies. And you're basically looking at the production of a photoelectron at an energy that's given by 1 times omega. That is 105EV, say, minus the binding energy. And you're also looking at photoelectrons that appear at 2 times omega minus the binding energy. So even though the first photon already ionizes the electron, there is a probability that that ionized electron absorbs the second photon. We call that process above threshold ionization. It's a two-photon process, but already the first photon ionized, but you can't absorb a second photon. So you see here this two-photon above threshold ionization signal within the solid line is the TDSS and the red dots, that's experimental data. And it's overall quite good. Now, based on that kind of, I would say, quite good agreement, we could calculate the full two-photon absorption cross-section. This is shown here in red. So we're tuning the photon energy across the giant diaper resonance. We're kind of with the first photon exciting this electron into this trapped state, this transiently trapped state, with the strong interaction between particle and hole. And then within 10 out of seconds or so, the electron flies away. But occasionally, it manages to absorb a second photon before it flies away. And that gives rise to this two-photon cross-section. If you look at this two-photon cross-section, which is also shown here in red in this inset, you see this doesn't actually look like a single Lorentzianism. I mean, you can already see that there is some kind of peak here near 80 EV. And there's also some kind of shoulder here near 110 EV. This is significant because in this naive picture that the giant diaper resonance corresponds to a single resonance pole, a single, if you want, Breitwigner type resonance, this two-photon cross-section should look totally different. We actually can calculate the two-photon cross-section very easily because we're resonant in this situation, which shows our photon energy to sit on the giant diaper resonance. And in that case, all you would do to calculate the two-photon cross-section naively is you would take the product of the one-photon cross section. This is the black giant diaper resonance cross-section that we get out of CI singles. And then you multiply that with a photo-absorption cross-section of an F wave quasi-bound state going into the continuum. I did just form the product of basically two one-photon cross-sections because your intermediate state is resonant. You probably realize that if your intermediate states are non-resonance, you have to perform some sum over intermediate states. But here, you've picked your intermediate state through your resonance condition. And what you would obtain is this blue curve with a single peak. What you have, however, is this double-peak structure. And to affirm that there are really two resonance states underlying the giant diaper resonance, we used the technique that Nimrod Moiseev who sits here in the audience has pioneered in quantum chemistry, which is a technique called complex scaling. Essentially what we're now doing, we're not doing, there's no light field anymore at this point. We're just diagonalizing the atomic Hamiltonian, no external field, but Coulomb interactions in place. We're just looking at the spectrum. But we're looking at the spectrum in the complex energy plane, right? Real part of the eigenenergy, imaginary part of the eigenenergy. And what happens effectively in complex scaling is if your real continuum states above the ionization threshold, they start here at the 4D ionization threshold. They would normally, in Hermitian quantum mechanics, form an energy continuum along the real energy axis. Complex scaling takes that spectrum and rotates it into the lower half of the complex energy plane, which is what happened here. And once the rotation angle is large enough, then you expose these individual resonance states. That's the power of complex scaling. You directly see the resonance states. There too, this R11 and R12, this, so to speak, gives rise to the bump on the left. And the two photon spectrum I showed to you in the previous slide. And this one gives rise to the shoulder on the right. These two, I say that two because only these two are dipole allowed from the ground state. In the process of this study, we also found two additional resonances, one at the total angle momentum of 3 and 5. And they are actually longer lift. They're closer to the real axis. I'm not aware that anyone has seen them experimentally, but they come out naturally of this type of study. Good. Let's increase the photon energy and let's increase the intensity. So I indicated that in order to be able to do this experiment at 100 EV, this two photon above threshold ionization experiment, one needed a free electron laser. Let's look a little bit at this brief history of X-ray intensity. This shows the peak brilliance. Let's call it peak intensity for our purposes. On the logarithmic scale, this is important to realize. And everything is normalized relative to Röntgen's tube from 1895. So this is our unit. The modern X-ray tubes, they increase the peak intensity relative to Röntgen's tube by about a factor of 10 to the 5. That sounds decent, but if you go to a third generation synchrotron radiation source, you're already at close to 10 to the 17 into the 18. That sounds large. Now you go to an X-ray free electron laser and you increase the peak intensity by another factor of a billion. It's very substantial, right? So we're now at 10 to the 27 relative to the Röntgen's tube in terms of peak intensity. So this increase in intensity relative to third generation synchrotron radiation sources comes mostly from two effects. One of them is that typically the X-ray pulses have about a million or so more photons in one pulse relative to a third generation source. And the pulses are typically a factor of 1,000 or so shorter. So you've squeezed the photons into a shorter bunch, which increases your intensity. So there's a very substantial intensities. And let me indicate briefly the type of applications that people use these sources for. I would say there are three important applications. One of them a true killer application, but three that really have kind of defined this field. One of them is what people call making a molecular movie. The idea here is the following. If you have a hard X-ray source, the natural thing to do with a hard X-ray source is to do some kind of diffraction experiment. You want to see atoms, right? Hard X-rays means 12 kV. You have a wavelength of one angstrom. A one angstrom just matches an interatomic distance in a material, in a molecule, whatever. And so you can do diffraction and see the atoms. But you have also very short pulses, I said. They're much shorter than what you get at a synchrotron. And in fact, at the free electron lasers today, you can routinely produce pulses that are only about 10 femtoseconds long. And most motions in molecules tend to be slower. So you can imagine you can take snapshots of motions in molecules with such sources. In these experiments, you don't necessarily want to use the high intensity. You want to use the fact that you have a short hard X-ray pulse. A very interesting application of X-ray free electron lasers totally needs the high intensity. And this is this field of single shot structure determination of biomolecules. You probably realize that the primary business of hard X-ray sources these days is protein crystallography. You want to understand the three-dimensional structure of proteins, because once you understand the three-dimensional structure, there's a chance that, for example, you can develop a drug that can act in the corresponding sites that you identify through your structural measurement. You basically want to have a strategy to fight certain illnesses at the end of the day. And the typical problem in crystallography is you need a crystal. And think of sodium chloride, piece of cake. You have a unit cell with just two atoms or so. But now imagine your individual molecules that you're supposed to repeat periodically in space have 100,000 atoms or maybe a million atoms. It's going to be relatively difficult to force these complex objects to grow a nice crystal. For smaller proteins, this is easier. But the bigger the proteins get, the less well-defined the structures become, the harder it becomes to actually grow crystals. So this field has basically, I think, become a really novel way how to look at protein structures by using the high intensity of the X-ray pulses to work with smaller crystals, say with only maybe 1,000 molecules altogether in the crystal. So that's what people often call a nanocrystal. It's less than a micron big. Or even pushing things towards single proteins. This little cartoon already shows, however, the issue. The X-ray pulses are short enough to prevent any atomic motion during the pulse. But because of the high intensity, you're not only forming a scattering pattern from that single exposure, but you're also changing the electronic structure. You're severely ionizing the system during the pulse. And later, the whole thing just makes puff and it breaks into pieces. And then in the next shot, you will bring in the next molecule. This is a destructive approach to determining structure. But protein crystallography is too. You're also frying your protein crystals at the storage room. They turn black and they form bubbles. You kill them. So it's always destructive. But here, you destroy the molecules basically in 10 femtoseconds and not over seconds. In any case, you get a pattern. And what is clear is, in order to understand the pattern, you have to have a clear understanding of the radiation damage. And I'll talk a little bit about this. The other area, the third area, which I think is an exciting area of application for X-ray free electron lasers, is this area of generating and probing extreme states of matter, which is the topic very much at the heart of this plasma workshop going on this week. And how do you actually know what's the state of a plasma? Often, you would use optical techniques. You would try to look into the plasma. But as you might realize, once the plasma is dense enough, the plasma frequency exceeds optical frequencies. In other words, an optical light pulse will not even penetrate. In other words, it's like a mirror. The plasma has become a mirror for your optical propulse so you can't look inside. So the way to look inside is by using propulsors that have higher frequencies. So this is a great opportunity for X-rays to peek inside very dense plasmas that are at basically solid density and where optical pulses would not be able to penetrate. And with X-ray pulses, you can look inside. The other thing is you can also use the X-ray pulses to generate such a plasma. And you can achieve relatively uniform heating, again, because the X-ray pulses manage to penetrate into the material. They're not reflected at the surface. And so temperatures that you can reach are in the region of maybe 100 EV or exceeding 100 EV, which corresponds to a million or so Kelvin. This was just to indicate why people think this whole development of X-ray-free electron lasers is actually quite cool. In the fall of 2009, when the LCLS, the first X-ray-free electron laser in the world at Slack National Accelerator Laboratory, when that was turned on, of course, the first thing to do was to try to see whether we actually understand what's happening. Are we able to model the radiation damage? We started with light atoms, put neon atoms into the beam. And then we saw, for example, neon 10 plus. So we stripped away all the electrons. Neon has 10 electrons and lost them all, at least in the focus. We saw things such as what we like to call beating the OJDK. So after you've made a 1S hole, then the hole can be filled within about two femtoseconds by OJDK. But because the intensity is so high, you can sometimes kick out the second 1S electron before the OJDK takes place. So you have a photo-absorption rate of about 1 per femtosecond, which is very substantial for X-rays, right? What we now want to focus on for the rest of the talk is heavy elements. Why heavy elements? A, they're a lot more complicated. And B, they do matter, in fact, for some of these bi-imaging applications. I'll show you results for Xenon, and I must admit it's not the Xenon that matters for bi-applications. But the computational challenge is the same for Xenon as, if you use, say, iodine or so, atoms that actually are used in bi-imaging applications for solving what is known as the phase problem. The phase problem refers to the fact that when someone measures a diffraction pattern, you cannot directly invert it. You can either try to find a model and fit the data, or they're actually systematic ways based on dispersion effects based on diffraction from heavy elements that one can use to directly invert. So solve the phase problem not through a fit, but through a true inversion. But again, these heavy elements are the elements that absorb the most. They have the highest X-ray absorption cross-section. So we want to know what these elements do. And I want to show to you that we're able to describe that, I would say, to some degree. So we developed a code called X-atom, and we've been extending that through the years, which basically is based on the notion that you're ripping out electrons one after the other. So you start with zero holes, then you form a set of one-hole configurations, then two-holes, three-holes, until maybe all the electrons are gone. And depending on the atom, you can end up with a pretty large number of such n-hole configurations. What we do is for every electronic state that we encountered during the polls, we basically solve an SCF problem at the Hartree-Slater level. This is not high-level electronic structure, but I would say it's a compromise to get something done for a very large number of states. So we recalculate orbitals all the time whenever we encounter a new electronic state of the atom. And for these electronic states, then we would calculate photonization cross-sections. So we can construct real continuum wave functions, in fact, on a numerical grid. We calculate OJDK rates, X-ray fluorescence rates, Compton cross-sections, and everything that you may want to know when you're dealing with hard X-rays. And then we're plugging all these pieces of information into a set of couple differential equations, so-called rate equations, which tell you the time evolution of the populations of the individual configurations. So at the beginning of the polls, you would assume that there are no holes. So all your population isn't that zero-hole configuration. And then as time progresses, you're generating more and more holes. And the probabilities tell you that time evolution. To give you a feeling for how big these rate equation systems are, so the number of active configurations that you need to take into consideration gives you the number of coupled rate equations. In carbon, if all you're doing is ripping away electrons, that's 27 coupled rate equations. In neon, it's 63. And in xenon, if we exclude the K and L shells, it's a million. It's going to get worse, but a million is good, as you'll see. That's the easy case. So xenon at a photon energy of 2KV. This is from LCLS. This shows you the experimental ion yields. Xenon 5 plus, 10 plus, 15 plus. Down here is 28 plus. Maybe I didn't mention yet, but you really ionize the systems, right? You ionize them with a very high degree of efficiency in a single ray, in a single pulse of light. The fact that you get a distribution, and not only one charge state, is simply because your X-ray beam has a spatial profile. In the center, you get the xenon 28 plus. In the wings, you get the xenon 5 plus. You basically, in your experiment, you're just collecting ions, producing different points in space. And in fact, we have to include all this stuff in the calculations as well. But you see, this works quite well. And please do notice that the ion yields and the intensity are both shown on a logarithmic scale. So when you go to low intensities, you see that all these curves, these ion yield curves, are linear. A linear curve in a doubly logarithmic plot means a power law. That means your ion yields scales with the intensity to some power. And as you go to the higher ion charge state, you see how the slope keeps increasing. So you have some law, intensity to the power of n, where n is connected to this increasing slope. And it basically indicates the number of photons you have to absorb in order to get to these charge states. So people look at these things and say, oh, here I had to absorb one photon, two photons, three photons. And so you can see it directly from these slope figures. Now let's look at a slightly more challenging situation. This is relatively recent. When the first LCLS experiments were done, they could focus the extra beam down to about one micron diameter. And they reached extra intensities of about 10 to the 18 watts per square centimeter. So this is probably part of the reason why you ionize so much. In the more recent experiments, now that the so-called coherent extra imaging beam line is available, you can focus the beam down to about 150 nanometers. And you get to peak intensities exceeding 10 to the 19 watts per square centimeters. And this is what is shown here. And at 5.5 kV, you see here that charge state distribution, the cutoff is at about 42 plus. So single shot, you're ionizing quite substantially. The black curve, that's the experimental ion yield. And the interesting structure that I'd like to point your attention to are these peaks here. Here's a peak. Here's a peak. Here's a peak. This type of triple peak structure, it's absent if we run our code the way I had described it to you. The code, the way I had described it, is A, non-relativistic, and B, it never includes resonances. And I mean bound to bound resonances. And so that's the green curve. It peaks about here, and then there's a cutoff at about 30. If we include relativity, but still no resonances, well, the curve even shifts downward. And it's still pretty structuralist. So all these structures probably have something to do with resonances. And indeed, when we do non-relativistic theory, but we include bound to bound excitations, then we get this magenta curve. And you see that already has some multiple peak structure, but it shifted relative to experiment. Once you include relativity and the resonances, then you have the peaks precisely where they are in the experiment. So what's going on? What's happening is the following. Your X-ray pulse, it strips electrons away. And your N equals 4 shell, of course, at the beginning is not available, because N equals 4 is occupied. But once you've stripped away enough electrons, new resonances appear. And you can resonantly excite in the first peak corresponds to resonantly exciting the 2S1 half into the N equals 4 shell. And in this way, excite auto-ionizing resonances. That's why you get an enhancement in the ionization. At a slightly higher charge state, it's the 2P1 half that suddenly can be resonantly excited into the N equals 4 shell. And now you get the next peak. And finally, you get the highest peak from resonantly exciting 2P3 halves. So you're kind of seeing spectroscopy here almost in the ion yields, because the system itself, it tunes itself to these various resonances as the ionization progresses. At the beginning, in the ground state, you don't have any of these resonances. They're formed, in a sense, due to the exposure to the poles. Let me show to you something about computational expense. This type of calculation, because we're now able to talk to the L shell, involves something like 24 million coupled rate equations. If we include relativity but still no resonances, it's about 5 billion. And to converge the relativistic calculations, including resonances, we have to go up to N-quam numbers of 30 and L-quam numbers of 7. And we estimate that this is a configuration space or set of rate equations of about 10 to the 68. You probably can suspect that we're not actually building up 10 to the 68 couple differential equations and integrate them directly. And if you're curious about it, please ask me later. Yes. Let me skip this and let me conclude. So I talked a little bit about whole dynamics that have been observed with adasecond technology, whole dynamics induced by optical tunnel ionization. And I've shown to you that now that we can really see what's, so to speak, happening during the ionization process, during the optical field, we can see this competition of reversible and irreversible processes in this regime. I showed to you that with nonlinear XCV spectroscopy, this two-photon above threshold ionization, we've discovered substructure in the xenon giant diporescence, which this giant diporescence has been investigated quite extensively now for 50 years. But only now that we can do nonlinear spectroscopy, we discovered the substructure. I talked a little bit, and at the end, about radiation damage at X-ray-free electron lasers. And understanding this is very important for applications of X-ray-free electron lasers, being able to quantitatively simulate what's actually happening. And I showed to you that because you basically absorb multiple photons, and whenever you create an initial hole, then the highly excited atom evaporates additional electrons through OJ cascades. This is why you form these high charge states. You're not forming xenon 40 plus by absorbing 40 photons. You're absorbing maybe 10 photons, and the rest is done by electron evaporation through OJ cascades. As I showed to you, in the regime we've considered, rate equations to successfully describe these processes. And in the latest data, we very clearly see the impact of both relativity and resonant effects. Thank you very much. Thank you very much, Professor Senter. It's very exciting presentation, and I'm sure there will be some questions. Very nice and very exciting to see how well Siri is doing in explaining the experiments. Now, I was trying to understand why, how does it happen that at such high energies, when you go to this really high quantum number and take a lot of states, how come that without higher excitations, time-dependent series works so well? Because you would think that doubly and triply and multiply-excited configurations would be, they are accessible at this energy range. But somehow, they are not important in reproducing the physics that you are studying. Do you have an explanation for that? Yeah, so I used time-pen CI singles only in the first part of the talk. Although I talked about when you ionize 40 electrons, it's not CI singles. We solve rate equations with n-hole configurations where n may be up to 40. And basically, we have transitions from a xenon 40 plus to 41 plus to 42 plus to 43 plus and so forth. So in that sense, the set of rate equations is either defined by the set, the number of holes you can form. And just from a combinatorial point of view, if you have a heavy element and you can form holes in all sorts of shells, you get relatively large spaces. If you track, in addition to the holes, also the states of the particles, when we photo ionize, we basically just let the particles go to infinity and we no longer follow them. That's why the configuration space is somewhat manageable. But if you also at least include the Rydberg state, suddenly your configuration space grows quite substantially. So there, we do have, in a sense, multiple excitations. But we don't treat them the way how you would mix in a CI sense. Configurations defined relative to a reference state. But here, all our configurations are, so to speak, meant to be approximations to a set of states, which we individually re-optimize. So we do as many re-optimizations of orbitals as we encounter states when solving our equations. So the CI singles, we use only for situations of single ionization and for nothing else. And for nothing else, would it actually work? There's another question over there. Yeah, thank you for the presentation. I went on a little train of thought in the beginning of your presentation and tried to follow me here, please. When you were talking about this overshoot and explain it with the springs there in a quantum mechanical way, I began to associate it with plasmons. That was the thing that came to mind. And when I think about the plasmon, I'm thinking about nanoparticles, where you could actually translate this whole story that you told, at least in a theoretical way, in some way. Do you know whether there has been work on this? And what is your thoughts about this new dimension, let's say? So I showed atomic problems mostly in order to create a certain degree of coherence and also because that's all I really understand. But in experimental science and experimental at a second science, people are already looking not necessarily at clusters, but at solids and are looking, for example, how, if you have such a few psychopaths, you have transient polarization in, say, an insulator and you're kind of transiently turning the insulator into something else. It's not exactly causing breakdown, at least some of the experimentalists would argue they don't have breakdown, irreversible damage. They think they're more in some kind of polarization regime in the solid. It's not the same as looking at plasmons because if you're starting with an insulator to begin with, you wouldn't have that. To what degree you have collective effects in these transient states, I don't know. The one connection I didn't make explicitly, which connects a little bit to plasmons, is in fact a giant dipper resonance. A giant dipper resonance has for many years been argued to be some kind of atomic manifestation of a collective excitation. This is mostly because if you think of a typical plasmon description, you really have an expansion in terms of entangled particle whole states, right? You cannot break it into, you cannot factorize it in the usual way if you have a plasmon. And that type of description is exactly what also fails. You cannot say you have a non-entangled particle whole pair, which works fine for Rydberg states, but doesn't work for the giant dipper resonance. So there are plasmonic aspects, let me put it like this. But the giant dipper resonance was never field driven. That was, so who knows? I think there are certainly opportunities to connect more to questions that salt state physicists have been asking. But it's also clear that a lot of the logic how people think in salt state physics comes from things like linear response theory. And that means perturbation theory. And a lot of that breaks down in the situations that are of interest in connection with strong fields. So if you're taking a solid and you're exposing it to field intensities of around 10 to the 14 watts per square centimeters, believe me, perturbation theory will not work. So it requires a bit different thinking, but it doesn't mean that all the concepts have to be abandoned, yes. Thank you. You're welcome. There's another question here. No, no, no, no, no. Here. No, he's bringing you the... Related to what Anna asked you about this single excitations determinants. So maybe a simple explanation is that electronic correlation is between introduced coupling within different hard to fork solutions, different determinants. Now the electronic correlation in the absence of the laser field is two electronic excitations. But in the case of the laser, the strong coupling is between single electronic excitations. And if the laser is so strong, these couplings actually by far are larger and stronger than the standard electronic correlation couplings. Yes. So in a sense, so what you're saying is if you're looking at the ground state and you want to describe the ground state better, the dominant effect comes from double excitations. But because we're looking at relatively large effects here, I mean, I guess for a chemist that's a big effect if you shift the ground state a little bit, but since here we're talking about relatively large field driven effects. In fact, it's true. The particle interactions are what dominate everything in the physics that I was talking about. Does it mean it can capture everything? Of course not. The situations where this fails and all the situations that you've looked at, for example, healing double excitations, you will never describe it within CI singles. I think that's clear. And actually what can happen is that if you are in a situation that when these double excitations, these authorization resonances are almost degenerated with the one electro excitations, then you can have a new type of physical phenomena that you cannot observe them in this way. Yeah, sure. I realize that. Okay, another question? No, he's bringing you the, because remember we're being recorded, so it's good. So I'm just confused about the statement that you said about protein crystallography by imposing the very strong pulsator. So every time you put a very intense pulse, so you excite the system and of course the configuration corresponding to that. And then the system don't allow to get relaxed and you again push it up. So it is out of equilibrium every time. So what kind of the configurations that we are talking about? So it's always the excitates. Active states, geometries. So you must understand that I'm not talking about the situation where the system gets hit over and over by a pulse. It gets hit by a pulse and then it's gone. And it's completely destroyed. The next time a pulse comes, a new sample gets hit. So of course, initially it's hopefully reason to be cold and then hopefully the right type of geometry that you actually care about. Yeah. But when the pulse hits and the pulse is 10 femtoseconds long, all hell breaks loose. But mostly electronically, what is diffraction for? Is it about wanting to know the electronic states or is diffraction your desire to, you wanna know where the atoms sit, right? So if the atoms don't move, then at least there's hope that this is a meaningful approach. And during the pulse, they largely don't move. The atoms that might move are the hydrogens but you don't see them anyways in X-ray scattering. So that's not such a big deal. And the main question is, are you interpreting your diffraction pattern correctly because you've perturbed the electronic structure? Your interest is in the atoms, as I said, atomic positions. Those you don't really perturb on the short time scales and later you don't look anymore. Even, you know, you just let the system explode and you no longer care. But because you changed electronic structure during the pulse and X-ray scattering is not scattering from nuclei. It's not neutron scattering, right? It's scattering from electrons. So then the perturbation of the electron cloud, so to speak, matters. And yeah, it's totally non-equilibrium. You're severely perturbing it. And that was a bit the purpose of why I wanted to demonstrate that to some degree I think we can compute some of what's going on. But I showed you atoms. I didn't show you proteins, right, so. Thank you. Yes. More questions? No, it's going to be here. Yes. I have a question about the xenon-distributed charge-state dissolution of the X-ray field experiment. Can you go back to the slide, please? You mean this one? No, with like xenon, yeah, that one, yeah. So if you have much higher peak fluence than the lower charge-state like xenon-5 plus is a little more anion-standing theory. So I was wondering what the ion density is in this experiment. The ion density is completely determined by the gas density. Okay. I mean, you just have gas with a certain number density, which is typically about 10 to the 14 or so per cubic centimeter. And this type of behavior, this type of saturation behavior, all that means is every atom is at least 5 plus. All the atoms become ionized, but your ion density is always limited by the neutral number density. Yeah, actually I was thinking of the gas density that is, yeah. That's what determines it. But again, all this comes from the wings. Right. In the beam, whereas all this comes from the center of the beam. So you have a very spatially inhomogeneous ion distribution. You don't have a mix spatially of 5 pluses and 10 pluses and so forth. Initially, they're all spatially separated. And then over time, because the system is of course not just ions, the electrons at some point actually will be trapped by the ion cloud, right? And then you start having collisions. That's right. So I was wondering if that could, I mean, there has something to do with collision alignsation or. The density, so from the estimates that we've made, the densities are too low for collision ionization to matter. Okay. I mean, this is not solid density. These are rather dilute gases. Okay. So. They're photon driven and as I like to say, electron evaporation driven. You also have multi-photonization in your model for this? Okay. So. Multi-photonization, if you understand multi-photonization by being able to see nonlinear effects, brings me back to what I tried to emphasize. You see nonlinear slopes. But the physics of this is a sequence of individual photo absorption events. So what people would call non-sequential multi-photon absorption is largely irrelevant in the extra regime. And that's because the photon energies are so high that you're always in resonance with a continuum state. So why optical laser people are so obsessed about non-sequential photo absorption is because that's all they can do. I'm serious because imagine you take a noble gas atom with an ionization potential of 15 ev and your photon energy is 1.5 ev. You're not be going through any real states for quite a while, right? The first excite state might be a 10 ev. So you will be going, in a sense, through a whole bunch of virtual states, which means these photons have to be absorbed simultaneously. It cannot be done sequentially. But here we are always, in a sense, resonant with something. So it doesn't matter, but it's still non-linear. It's still multi-photon physics. Sequential. It's sequential multi-photon physics. It's hard to see non-sequential multi-photon physics, even though some has been seen. But it's always weak. Question, or then? A very nice talk, Robin. I think I missed a key element in the very beginning. You talked about how you're going beyond perturbation theory. And for a lot of the experiments, a lot of the probe experiments, the pump was very non-perturbative. And I understood the model, that you had to model the dynamics of the electrons as the pump goes through. But let's just talk about the probe for a second. So this is the part I missed. Is the measurement in the end of a matter degree of freedom, or is it actually a measurement of the light that you're talking about here? So we're going to see. I mean, at the beginning, this trans-absorption experiment, this one here. Is this what you're asking about? Yeah, yeah, yeah. Yeah, this is measuring light. So how are you connecting the state of the electron now to the actual light field that's coming out after the probe? So, yeah, the pump was done non-perturbatively, but the theory of the absorption was done in first order perturbation theory for the XIV. And then you get directly an expression for the cross-section in terms of the density matrix elements that we compute, which are shown here. So from just this information, we can construct a cross-section. Okay, so the first bit is the non-perturbative. So there's never any, so there's never any, so as far as we're concerned, the classical description of the light is good enough here, effectively always. Yes, okay. Absolutely. Yeah, I didn't talk about anything that would require a non-classical description. I have a couple of naive questions from an outsider myself. What is wrong with the minimal coupling? Oh, there's absolutely nothing wrong with minimal coupling, but some people are emotionally upset when they have to worry about things such as a vector potential. And they really think one has to describe things in terms of gauge invariant quantities. They like electric fields and magnetic fields. And so if you write down the interaction Hamiltonian and minimal coupling, well, you have the vector potential and the scalar potential. And then I'm perfectly happy with that. And for extra purposes, actually that's the only thing that makes sense because multiple expansions are stupid in the extra regime. But people that come more from the optical regime, they like to argue that multiple Hamiltonian is just such a wonderful choice. But you know, it also has this advantage that sometimes for building physical intuition, it's sometimes more appropriate, but that doesn't make the other one wrong. And these are all equivalent, right? We're talking about basically gauge transformations you can perform on a Lagrangian and still the same physics. And you had this plot about how the intensity has been increasing over the years. Yes. 27 orders of magnitude or so. Yes. Is there any extrapolation? How far we can go in the future? So I think this is not the end, but I've heard is that there are ideas to boost this by a few more orders of magnitude with partly with things that can be done in the near future at X-ray FEL facilities. It has a lot to do with how these pulses are actually generated and there are options for some opportunities for improvement. And then there may be a next generation of FELs, but we're not talking about another factor for billion anymore. It doesn't look like that's possible. And I must also say, I think we're very close to a sweet spot for bi-imaging of single molecules because once your intensity is way too high, you're really blowing away all the electrons before you're forming any meaningful image. And then what's the point? I mean, it's cool to blow stuff up, I guess, just because it's an interesting physics, but in terms of the output that really people want and why, I guess, at the end of the day, taxpayers pay for it, it makes no sense. So I don't think one should be pushing this much further if bi-imaging is still considered a key application. I should also say one reason why people like this idea so much of using an FEL, probably many of you heard about this Nobel Prize in cryo-electron microscopy recently. And of course, that allows you to get very high resolution, kind of atomic level structures for proteins, but it's cryo, right? You freeze them. Here, there's an opportunity to see things in motion. So to speak, do the single pulse imaging thing combined with the molecular movie-making. So watching how processes unfold, you're not going to be doing any of that with cryo-EM. So I think that opportunities are great, but we're not actually there yet to do any of that. So there's no fundamental limit that you can set beyond this point, we cannot go... Well, let's put it like this. The limits are partly related to how many electrons you can squeeze into an electron bunch, and then there is a process, there's a plasma instability that's called micro-bunching due to the interaction of the light field, the synchrotron light that the electrons themselves generate, and undulate a field that they see as they move at relativistic speeds through this periodic array of dipomagnets. There's some sub-bunching, some pancake structures that appear, and they are then spaced all at about half a wavelength of the light that they're emitting. How many electrons you can squeeze into this is limited, and then also there are limits to how long you want to make the undulators. I think at the end of the day, that's why I said, I think we're not terribly far away from what is, with this type of technology, feasible. If people get away from standard accelerated technology, there are developments in plasma acceleration, laser-based plasma acceleration techniques. If you might be able to squeeze the acceleration step, if one can squeeze also the undulator step. People are talking about techniques that might bring the undulator sizes down to centimeters and not hundreds of meter undulator lengths, I mean the whole stretch of undulators, then suddenly you might be able to push this into a new regime, maybe. But I think we're close to the maximum. And the last question, how do you handle the 10 to the 68 configurations? Oh, thank you. Well, what we're effectively doing is we're using a Monte Carlo scheme, essentially. Let's imagine you have a tree of individual electronic configurations and at the top is our neutral ground state. That's the first one and then we calculate the various rates to go to a one plus here and say maybe with the one S-hole and the one plus with the two S-hole and the one plus with the two P-hole. And we calculate all the rates and then based on probabilities, we decide which path we take. Then from this state, we check the probabilities to go to the next states and we run through this many times and then by basically looking at many such individual realizations, if you want stochastic histories that the atoms run through in configuration space, then we build up the probabilities. So this approach will not work if one specific state in the space of 10 to the 68 configurations is what you're interested in and it has the probability of, say, 10 to the minus 15 or so, we'll never see it. We don't sample enough to ever see that. We're just sample, the reason this works is because all we're sampling is charged the distributions at the end of the day. So we're throwing a lot of this information together which the configurational space would contain. That simplifies things quite a bit. Impressive, very impressive. Okay, so before we finish, is the standard, the tradition here is that there are some refreshments outside so everybody can go outside to have some refreshments except for the diploma students who are here that I will ask them to come and talk to Professor Sandra to ask more questions without being intimidated by all the senior people like us. So let's thank Professor Sandra again for a great time. Thank you.