 or in a long time ago, and in 2008 she changed to what she is now, which is called Aalto University right now. She's also had a guest professorship at the T.H. Zurich, and as I said, she is a well-known theorist in the field of cold atoms. I actually worked with her very early on because she was the one that originated the idea of aris spectroscopy, probing superfluid pairing in atomic gases with aris spectroscopy. She had a pioneering paper on that with the beta solar, and then maybe I had a follow-up paper together, but she's doing lots of things. So today she will talk about quantum geometry. She's also doing experiments, so she managed to do both theory and experiments. So she's doing experiments on nanophotonics and mathematics, which is a different story. So it's quite impressive. She had a great breath. And I think I read that you are also... What was it I read that you are... You are here, when you read for the Medellin, grateful prize or something, or what was it? Head of the Millenium Prize Committee, yes. Yeah, which is probably made up of money, right? You should try to ask me questions. So with that, we go ahead and talk about the new stuff. Okay, thank you very much, Gerd, for this kind of invitation and the introduction. And as you said, I'm old, a collaborator of yours, and you are also old, but we are still doing it. So you will see, we are still doing it. Right, so let me try to share my screen. I'll open the slides. Can you see my slides? Yeah, nice. And also the laser point. Good. Okay. So I will indeed, as Gerd mentioned, talk about two quite distinct topics first. One is theory only, or I mean, we are doing only theory there, and it's very much related to quantum geometry and super fluidity of all kinds. And there we'll talk about these things and also the latest news, which is always very impressive. So let's go to our experiments where we have seen both inside condensation in an ultrafast room temperature system. It really is pushing the limits of both inside condensation and they are also the latest news you will see. Good. So, let's start from superconductivity. How useful it is, it's used for big magnets and quantum computers and so on. And wouldn't it be wonderful to have this in room temperature? Because for all these applications, the cooling takes actually a lot of energy. So there is a huge potential if we could bring superconductivity to room temperature. But we are not there after decades of research and it's very annoying because it's essentially just a factor of two, from 150 to 300, why we cannot do it. Usually factor of two is peanuts for a physical basis. So let's try to understand why is it so difficult. And you also know that superconductivity is a process I think on the same Cooper pairs. And Cooper pairs typically are born from some big interaction between the electrons. And those weak interactions are competing with kinetic energy, which is very big to the Fermi energy is very big in metals and so on. And the BCS theory then tells that from this competition, it's actually the TC critical temperature becomes exponentially suppressed. So here is interaction. So this thing is very small when interactions small. This is the density of states. So, maybe we could get rid of that kinetic energy part and let the interactions know it. And people have actually taught about it. That's so called flatband superconductivity. So now here, here's some basics from textbooks. If you have a periodic system, the solutions are block functions. Actually now this periodic part of the block function will be quite important in this talk. And then you get the band structure with some bandwidth. And if interactions are small compared to the bandwidth you are in the dispersive band limit. Then if they are much bigger than the bandwidth or maybe in the extreme case of having no bandwidth at all, you can call it a flatband. There, of course, group velocity is the one that this is zero. So flatbands are actually insulators and anything. What comes to the Cooper pairing temperature? Interestingly, in a flatband, it's not exponentially suppressed, like in a useful case. It becomes linearly proportional to the interaction and the volume of the flatband. So this has been predictive, but this is really just the pairing temperature. There is still something much more to understand. Before going to that, let's see what kind of flatbands we are actually interested in. Because you could also get a flatband by just pinning your particles so that they cannot move at all. But that's uninteresting. These interesting flatbands come from interference of wave functions. So for instance, if you have a lattice geometry, which is like this leaf lattice. And you can imagine that you have states where you have amplitude here with minus sign and here with plus sign. And now if the particle tries to tunnel to this side, this destructively interfere and there is no tunneling. So this means that the particles get kind of localized to these sides via interference. Another way of kind of producing flatbands is if you have a periodic system and then you make a super lattice. Then you have a smaller brilliance zone and the bands get folded and they repel each other and you may flatten the bands. These two are actually the same mechanism, but just seem more in the real space and in the reciprocal space. And then again, a landau levels is a well known example of flatband. So this kind of thing flatbands could be interesting for high temperature superconductivity. But I said there is a problem now in order to have a superconductivity, you have to have super current Meissner effects and so on. And super current is defined as the Cooper momentum times the super fluid weight. So this quantity has to be no zero. In conventional BCS theory, it's essentially related to the particle density divided by effective mass. And you see here what is effective mass is the band dispersion. So this would be zero at the flatband. So here we apparently have a big problem that they are pairs, but they don't move. Now, in my group, we got interested in this problem already some time ago and there is a series of work that I now summarize here. I would like to mention Sebastiano Beotan Longlian, who were very important in these early papers. And also Alexi Yulku, who is actually now a postdoc or who so you probably all know him. And especially lately we have done very nice work. Alexi has calculated these things for BC. And also other collaborators in particular the group of Sebastian Hooper from ETH. So what we did is we considered the Hobart model with hopings of the particles particles you can think about electrons or in ultra gold gases they would be some fermions in the lattice, and they interact at one side and the new, but the essential thing here is that I have this indices alpha and beta. This means that I have many orbitals in the unit cell. So this is really a multi band system. Many orbitals, so sites in the unit cell means many bands. And this is the index for that. And then we do BC as mean field theory for this model. And to see when the super current can go through we introduced a Cooper momentum which transforms the order parameter bearing gap like this. And then one can calculate the super fluid weight with various approaches here just two of them mentioned that we have used both of them. The essential result is this. So we saw that in the super fluid weight there is what we call the conventional part which is simply this a derivative of this person this would be zero in a flat band as I showed in this slide where I introduced the But then we notice that there is another contribution which had been not been noticed before and we call it geometric you will soon see why. And this one can be non zero also in a flat band and it's present only in the multi back case. And quite interestingly at certain limits, it becomes directly proportional to the interaction, just like the temperature and a quantity called quantum metric. So, what is this quantum metric well it is what it sounds. So, if you want to define an infinitesimal distance between two quantum states like this. You can express it as this kind of index or some. And here you have a quantity, but this one is what gates invariant and these people introduced a gates invariant version. And then they could write that the infinite decimal distance is this quantum metric times the change in the parameter. And this is very generic this parameter cake would be anything that parameterizes your quantum state, but if you want to think about lattice systems you can think that this is the block function and this is the lattice moment. But this quantum metric is actually part of a bigger object called quantum geometric tensor which looks like this. And it is its real part, while the imaginary part is the very good so well known famous very good. So this means that the quantum metric is actually connected to topology, why are being, you know, both of them are part of this object. And this quantum metric goes under some other names. Okay, so with this nice connection between the quantum metric and the very curvature. We can actually derive this kind of fundamental found lower bound for superfluid weight that there is always superfluid weight that means super current. If the band has a non zero chair number. And by the way this turn number here is a spare chair number for one spin, because this has been derived for time will also symmetric system but it's like you can think about having different sign magnetic fields for the two things. And then you have a chair. So this guarantees super fluidity. Okay, after that various groups have, including ours have proven this relation also via build mean field methods. So why, why is there this kind of bound that is related to topology and why there is at all some kind of transport. Cooper bears in these systems where actually single particles behave like an instrument. Well, you can look at this pictures. So here, I would have a system that I mentioned in the beginning where I localize particles very tightly on the lattice side so that they cannot move. So, then I have exponentially localized one year functions, and you can have a flat band here. And then you can have this other type of flat band, where the flatness come from interference, but the one year functions still overlap with each other, even quite a lot. And then if you introduce interactions between particles in this localized system, I mean one year function localized system at the particles would just move on site pairs which don't move at all. But then in the system where we have one year function overlap with interactions these interferences get kind of distorted and then the particles can actually move. So this is the thing behind and why this is related to topology is that it's known that topology is connected to how localized your one year functions can be. For instance, it has been shown that if you have a finite set number, then your one year functions must be non localized you cannot exponentially localized. So this makes more understandable this connection to topology. Okay, so this was very neat theory theory 2015. But of course there were not many experiments. I think actually none on platform superconductivity, but then came twisted pilot crafting and this of course changed the story very much. So, so here you see Alexi, whom you all know, I hope by now, and other co workers long from my group and demo and demo from a uvascular. So we started to look that, well, does this flat this quantum geometric superconductivity have any relevance to this twisted pilot graph in superconductivity, which was the big thing of 2018. And MIT, they, they saw that if you take two crafting flakes and twist them by a tiny angle, you get a system with this kind of large, you need some more pattern, and then certain conditions you can see superconductivity. And they also know that at the point where they see superconductivity here at the Fermi level they are almost flat pants. So we looked at this and evaluated the superfluid weight and calculated separately the conventional geometric parts, and also to be getting temperature which you can get from superfluid rate, and we saw that most probably this geometric term matters. So here is the interaction, and this is superfluid weight, and for two different type of bearings, we see that this pink area is the conventional part, what would come only from the conventional part because the bands are not completely flat. So there's a little bit of dispersion so there would be this conventional super fluidity as well. And then this blue one is the geometric part so depending on where you are in the interactions and we think it's probably somewhere here. It can be quite significant. So this conclusion was at the same time. And we made by another group. And then the results are highlighted in this news article, you can go over for it if you want to get some kind of easy to read the text about this. By now, we have already written a review article about these things so this is nice if you want to understand the quantum geometric superconductivity first the basics, and then how it can be relevant for the piston mobility systems. So this is not only considered a graphene systems but also explain how people are now creating these moir systems with ultra gold atom so this is a picture from from beautiful experiment with ultra gold gases and those of you who are from other gas is filled you can immediately interpret this and you see that for two different atomic states and you have lattices that are slightly tilted from each other and in this way you can do this more systems. Okay. Then I go to slightly different topic, this is still very similar in the sense that it's about this quantum mid geometry, but now we go from fermionic to bosonic systems. So, let's think about Bosch Einstein comments, do we see some same kind of effects there. This is a reminder of the basics of a usual Bosch Einstein condensate so if you have some kind of dispersion here for instance a quadratic dispersion that you would have for continuum system or close to some bandage. Of course the Bosch condensate would form in the in the lowest energy state. But in a weekly interacting system, it's very, very well known that you will have a so called quantum depletion or a finite amount of excitations because of interactions. And the excitation fraction is known within Bokeh-Leopold theory it looks like this. So there is dispersion interaction and the density. Fine. So then, now if you take a flat band, meaning that the effective mass goes to infinity, then this formula immediately gives you that the excitation fraction goes to infinity. So the poor bosons don't know where they would condense because every energy has to, I mean every momentum has the same energy. So in some sense these excitations immediately spread everywhere. There is no condensate. So it's true that in a single flat band, Bosch Einstein condensate is not possible. However, this is the kind of trivial flat band that I mentioned. But there are also these flat bands that come from interference and in those kind of systems it's known that there can be Bosch Einstein condensates. But it was not known like under what conditions they occur and what determines their stability and so on. And this is something that we have solved together with Alexi and Georg. So the theory is general there but I explained why the Kagome lattice. So Kagome is a nice lattice geometry because there is a flat band like this and then two dispersive bands and by chasing some parameters you can put the flat band either to the bottom or the top. So here for instance we put a flat band to the top and then of course Bosch a condensate forms here. So it's in a dispersive band. We can study the dispersive band case. And then we change so that the flat band is here at the bottom and then the condensate could form also, for instance in the K point of the brilliance zone. So here and now we are interested, okay, we have this condensate but how is the quantum depletion? How are these excitations around? How do they behave now in a flat band? And is the BC actually, when is it possible? Do we have finite sign velocity and so on. And interestingly, the result is that now these important quantities of BC depend on the quantum geometry. For instance speed of sound. In a dispersive system it's a square root of interaction times density. And now in a flat band it becomes linearly proportional to interaction. So even the interaction dependence is different. And then there is this quantum metric with the square root. And this we have tested with the metric tells whether you have a speed of sound. When it's finite you have a bit same similar fashion like finite quantum metric guarantees at the super current. So here it's the speed of sound but there is a square root dependence here. Then we calculated this excitation fraction. And that has a really remarkable behavior that for vanishingly small interactions it actually goes towards a constant that is determined just by quantum distances between the states in the flat band. So this is also kind of intuitive that like if the states would not be orthogonal at all. So this would be one, then you don't have any distance and this goes to zero. Sorry. Yeah this goes to one and then this goes to zero. Yeah, so basically the better way to explain it is that in the single band system we thought that okay the excitations go to other case states immediately because the energy is the same. But it's also important whether there are overlaps between the case states. If they are somehow not orthogonal at all then it's easy for the condensate to spread. But if there are some orthogonalities this kind of curtains the spreading of the condensate to the other case states. So that's why there is a dependence on the quantum distance. And you see it doesn't depend on interaction. So when we decrease interaction the flat band B C excitation fraction it goes towards a constant. While you know this person but it just goes to zero. So, so this means that if you now would you're somewhere here and if you would decrease the total number of the particles. Your excitation fraction would remain constant meaning that excitation fraction becomes bigger and bigger compared to the total number. So, indeed, in the flat band, one can reach the situations where quantum fluctuations and interaction effects are really prominent. Because with BC's it's often that the BC itself is a big mean field thing. And the quantum effects I mean they have been of course observed. Also in or push and so on but it's usually very tricky to see. But in flat bands, the quantum effects will be very prominent. Okay, we have also looked at quantum geometry and light matter interactions in the solid state light matter interactions context if somebody is interested. Right, so so then I will go to the news, saying that part of these things that I told are actually incomplete or even wrong. And this is always very interesting. So yeah, I would really like to highlight my student cook. And we had a pleasure to collaborate with the crew of Andre Bernwick, who could show these things with the complimentary metal. So this is a very fresh article in the archive. And now the key thing there is that there has been actually a problem in this connection between super fluidity and quantum geometry and it's a very subtle one. People have not noticed it because it's basically it's just one line in the supplementary of some some papers of our paper so so usually the devil is in the supplementary. So, there has been this prediction done by us that super fluid weight is proportional is given by the second derivative of the ground potential and produces this result where you have interaction and this quantum geometric term, which is an integral over the quantum metric. So this is the quantum metric. The problem here is that let's see if you have, for instance, this type of sleep lattice geometry that I saw. And here, let's assume that we keep always the tunneling between sites constant, but we move the lattice sites in space. But this would be something trivial it's like changing your furry conventions for a transfer conventions. So the super fluid weight is completely independent of that kind of moving of the orbitals. However, the quantum metric depends on this has been known. So now we are relating an orbital independent quantity to orbital dependent quantity so obviously there is a problem. And what is the reason for the problem. Well, there are assumptions but time reversal symmetry assumption is fine. There was in a previous literature in our papers assumption that one can make the order parameters always to be real. And actually, if they are always real, this is valid, but they are not always real, they can actually. So, the argument. I mean, this this shows what is the problem. So actually, the definition of super fluid weight is that you have to take total derivatives of the current contest, which is the partial very big minus some kind of term that contains further derivatives of the order parameters of the imaginary part of the order parameters. So there has been this argument that okay let's make all the order parameters real. So then this would anyway be zero and this term is not there. However, if you do such a transformation that makes the order parameters real, then you also change this one. So one should calculate this for the transport system this has not been done. So, we have now shown that one must use these equations and we also present some very nice methods of using them in convenient way. And using this equation, we can show that this actually the minimal quantum metric that determines super fluid, not just any quantum metric, because that it depends on the orbital position, but the one that has the smallest possible trace. Once that you can still connect super fluidity and quantum geometry. So this is really important. And now with this correct formulas we can we have, for instance, look that if you have the lead lattice where the band structure looks like this, and you open that you can open a gap there by tuning a certain parameter. So here is the critical temperature of superconductivity for different interactions for different gap openings. So, so when you have a gap, you have an isolated flat band and it's the slowest one. And now that it's actually beneficial to have a band touching. So this dirac come in here. And this is good news because in real systems you usually have band touchings you don't have isolated that fast. And then the general message, which already have we have made many times is this that in the small interaction regime, like here, you see how dramatically better this flat band results are compared to this gray line, which is a usual square lattice. So this would be a usual dispersion. It really goes orders and orders magnitude lower in this low interaction regime, while flat band superconductivity still gives a reasonably high critical temperature. Okay, so if you want to know about this quantum Jerry, Metri and superconductivity things you can read all our previous papers also the review paper they are fine for the qualitative use but then if you really want to calculate now you have to look at this latest paper it is completely correct for us. Okay, so that was all about the first at the theory part and now I go to our experiments. So, we have been interested in a plasmonic lattice systems, because there you can have light that is kind of bound into the nano scale and light matter interactions are very strong. So, if you have, for instance, these things here are nanoparticles that are about 100 nanometer size and these structures are made by even lithography. In this kind of structures you can have plasmonic excitations in the particle so this kind of surface plasmons that are partly light and partly electron motion. They create very intense fields here and they are in the nano scale, because they are partly electron motion, so you don't need to worry about diffraction anymore. And then if you put them periodically arranged of course you get band structures like here. So this is the in plane momentum and energy and you'll get clear band structures. Then the basic modes in these systems are so called surface lattice resonances and you can understand this plasmonic resonances at individual particles which I described they are like they act like little dipoles so there is oscillation and radiation. And when you arrange them by lambda from each other, of course you get interferences. So, the diffraction order of this periodic structure that is in plane is now important. It turns out that this diffraction order hybridizes with the single nanoparticle resonance. So the single nanoparticle resonance is very frog in spectrum. And then the diffraction order would be at this wavelength and there you get this kind of phano shape of a hybrid mode called SLR. And it is dispersive so you can give in plane momentum to it that corresponds to an angle, excitational emission and then it has higher energy. And importantly, there are these band structures with clear band edges. So we wanted to condense this SLR of excitations into the edge of this band. And where does our condensate sort of sit conceptually because there are many types of condensate that have light as part of them. There are photon condensates and then polarity condensates where you combine excitons and light into polarity. And the thermalization mechanism is quite different in each. It can be coulomb related relaxation or it may use vibrational states of the molecules. Like in the photon BC you emit and absorb photons and in between these processes you dampen energy to the vibrational decrease of freedom and in this way you kind of cool the photons to the lower states. So our BC, Plasmonic BC at the weak coupling regime is very similar to the photon BC but ours happens like three orders of magnitude faster. So the dynamics is really, really fast that's the difficulty actually there. And then we went to strong coupling regime and our BC there is somewhere between the polarity and the photon BC BC. So we have a few papers on this already. And first about the first one where Tomi Hakala and Antimojlan were really very important and also Alexi is on this paper. So we have this nanoparticle array which hosts this SLR modes. And then we combine it with a solution that contains dye molecules. So these dye molecules act like a bath that helps us to thermalize the photonic or Plasmonic excitations. And I mentioned this ultrafast time scales. We were thinking for a long time that how can we ever show that there is some kind of BC dynamics going on because everything happens in picosecond or faster. And then this was the smart solution that we detected the light that escapes from the system, especially resolved. And we pumped the system with the femtosecond laser only in one end of the sample. And it now makes these molecules excited. And then the molecules emit light to this SLR modes and add an energy where you have momentum so that the excitations start to propagate. And during the propagation this thermalization happens. So we can actually monitor what happens in time by imaging space. The important energies that you have to keep in mind to understand all this is this band-edge energy which we can easily tune by changing the period of the lattice. And then the molecule energies. So it has a big stock shift from between absorption and emission. And there is some energy where you basically don't have any absorption left but you have some emissions still. And here is a cartoon how this thermalization happens. So we have excited the molecules and now they are starting to emit. But we first tuned the band-edge to be sufficiently low. And now light is emitted to the SLR modes. Then it's reabsorbed. And before it's emitted again there is vibrational relaxation. So you go to low energy. And then you go to the ground state and you have this bosonic stimulation of the population there. And we see this spatially now. I talked about this spatial imaging. This spectrum resolved in space. And here really you start by excitation at the high energy and then the light kind of goes to what's low energies. And when it meets the band it condenses here. So that's a really neat way of seeing what happens. And a standard question now is that how is this different from lasing because what we get out is a coherent prime beam. And in this system it's very easy to show the difference. So now we put the band-edge even lower so that there is absolutely no emission. Sorry absorption left but there is some emission. So this point has a high gain. And then you see lasing immediately there. So you don't see this thermalization process. So in this way we can tune between easy and lazy. Okay so that was in the weak coupling. But we wanted to go to strong coupling because we thought that the phenomena will be even richer there. And there again Aro and Antti have been really important. And now we didn't do this propagation experiment. We just pumped all over and observed the light that comes. And we are now in the strong coupling regime. The molecules are no longer just a path. They actually hybridize with the SLR modes and make polarity. You can see it by the bending of this person here. So here is the exit on absorption and this person has been bent. And we saw actually quite different phenomena. We still see a busy but actually much nicer one. So quite funnily we have two thresholds. So this is the pump fluid. How much we pump the system. This is the output for the luminescence of the BC. So we see a first threshold where we get kind of something that looks like user lasing. And this we call polarity lasing. And then a second threshold. And in this threshold, we really see a very beautiful Bose-Einstein distribution of this thermal tail, these slopes, mattress room temperature and then the condensate at the lowest energy of the band. And in this regime, the spatial picture of the condensate looks like this. So it actually has some high intensity area in the middle and lower intensities at the edges. I will come back to this. But yeah, the point is that in this strong coupling regime, we actually see much more beautiful nice condensates. And that's what we have been working with ever since. So one obvious thing if you have a condensate is to look at the temporal and temporal coherence. And this is work by Antti Mojlan and others. And Antti is now a postdoc at ETH Surich. So he just left my group. So maybe many of you know that the story of long range order in two-diamond services is very complicated. In three-d, there is true long range order and it has been shown and seen, for instance, in atomic pieces. But in two-d, there was first this theorem by Mermin and Wagner that there should not be any order, thermal fluctuations prevented. However, Berezinski and Kostelis and Taules have shown that yes, there is quasi-long range order which decays only un-get-rightly. And this is related to vortex and anti-vortex pair. But these are equilibrium results. So for instance, our condensate, if it dies in one picosecond, well, it's certainly something between equilibrium and non-equilibrium. And many other condensates, these photonic and polaritonic ones are like that. So what's the story for non-equilibrium condensate? Well, there are at least two possibilities. People have shown that there could be still a non-equilibrium BKT transition with algebraic decay. So the exponents are a bit different in equilibrium. And then there is a completely different regime of parameters where you see a completely different phenomena, namely the KB set, a type of dynamical phase ordering where the decay of coherence would be stretched exponentially, again with some universal exponent. Maybe there would be even a crossover between these. So this is all very, very open still. So we wanted to see what happens in our system. And first of all, a clear message that we saw is that something very different happens for the polariton lasing the first threshold and then above the second threshold where we have BC. So for instance, piezo-coherence, we get a clear Gaussian decay for the polariton lasing and above the BC threshold. It's either power law or stretch exponential, but these are very hard to tell from each other. So we wouldn't make any claim except that it's not a Gaussian. So whether it's a power law or stretch exponential, I wouldn't play, but clearly non-Gaussian. And the same story in temporal coherence, very clear difference below the BC threshold. It's an exponential decay, which would be the usual laser theory. And then above, again, either power law or stretch exponential. So a clear transition in terms of the type of coherence happens, but this exponents that we got did not match with the known BKT or KBZ physics. So there might be that there is some new physics going on, or that we just didn't have a big enough system, either way. But it looks interesting this BC we see that you get some nice power law or stretch exponential behavior there. Yes. So then the last work about this BC is related to this, what I mentioned that there was this high-intensity area in the middle of our samples. Well, we never thought about it deeply before this study, where we wanted to go to study basically vector fields. So far, we had always pumped the system with one polarization, and that somehow triggers also the polarization of the condensate to be in a certain direction. So if you pump like this, we will see this. That's it. But of course, there is rich physics when you have really a vector field and we have polarization that offers us the kind of the shadow spin for the system. So we started to then pump with circular and polarized beam that we could see all kind of polarizations. And the polarization result detection and the result was completely surprising. So these ones, we understand this look like what we have seen before. Depending on the polarization, we see this high-intensity area in the middle. But then look at this filtering with left and right circular polarization. In one case, it's the middle of the sample that is bright and in the other, it's there. So something must be going on there. And from that, you can already think that, hmm, look, the difference between left and right circular polarization, it's just the pace here. So it seems as if going from the center of the sample to the edge, the phase is changing. Some phase is changing because that allows you to go from RCB to LCB. And indeed, that was the case. So in this work, we determined for the first time a BEC phase using a phase retrieval algorithm. And this now shows the phase. And indeed, there is a phase difference between the center part and the edges. And this explains these patterns. Why there is a phase? We don't exactly know. It could be some soliton or it's an overall function of the whole thing or something, but there is a phase. And it leads to these really nice patterns. So if this is now in the language of Stokes vector, if you have a constant phase, Stokes vector would look like this, but experiment gives us this. It's not the case. And then when we make this pi zero phase shift between the center and the edges, we see what the experiment is showing. And there are some domain walls here, but the structure as overall is not topological. But one could probably create also topological defects with more elaborate sounds. Okay. And then a couple of minutes do the last work. So this is not in the BEC regime. This is just lasing. But I wanted to show this because it's really promising considering the future. Because here we see clear effects of time reversal, symmetry breaking. We now did this kind of a nanoparticle arrays with magnetic materials. And there we collaborated with the group of Sebastian van Dijken. Their group is expert in nanomagnetism. So now the nanoparticles are made of cobalt and platinum. And we have a structure where we also have a cold layer underneath. This we didn't have before. And then we can magnetize these nanoparticles with external fields. And what we see is that when we get this system lasing, of course we have to again put dye molecules with the RCP pump. What we see is that depending on magnetic field, we see the lasing at slightly different energy. And we can also switch the lasing off and on by switching the magnetization. Very efficient. And the switching happens because the lasing threshold is slightly different or actually quite much different when we have up or down magnetization. So this leads to switching. And now what is the reason? Well, there is magnetic circular diaprosim in this system, but that's a tiny effect. It's below 1%. And here we see several percent shifts, for instance, in the threshold. So this required quite a deep theory to understand. So we calculated with a finite element method, the modes that this system has. So it has the nanoparticles and the gold layer down here. So the lasing mode looks like this. And then we notice that there are some kind of what we call hybrid modes where the nanoparticles and the surface plasmons of this gold layer kind of hybridize. They look like that. And essentially the lasing modes are very narrow and these hybrid modes are broad. And they are chiral. So this is the thing that the magnetic fields lifts the degeneracy of the mode. So why there is degeneracy to begin with is because it's XY symmetric system. But now you break time reversal symmetry and the digital mode split. So lasing mode split and the hybrid modes also split. And now our explanation is that these hybrid modes, they eat the gain available differently depending on whether it's a sigma plus or sigma minus or different magnetization. Because they overlap with the gain differently. And this explains the threshold difference. And indeed we see these two lasing modes that are split. So this is nice for switching of lasing, but what is important for the future is that exactly this kind of splitting of degenerate modes by magnetic field is needed if you want to make topological systems. So in these systems one could really do topological lasing with magnetic material. And not with artificial magnetic fields that have been mostly used in topological lasing. So I'm in the end now. So I told you that flatband superconductivity is possible because of quantum geometry and the same story for BC. It can be stable due to quantum theory and we have the new complete formula that you should use it to calculate these things. And then I talked about the BC in the plasmonic lattices and magnetic effects in the same systems. And obviously yes with flatbands the goal is to search whether this can really bring us the room temperature superconductivity. And in the plasmonic systems we very much want to study quantum geometry and topology in the future. Thank you very much.