 Yes I'm really delighted to be here and I'm really thankful to organize this for setting such an interesting meeting up and for inviting me here. I also would like to join other speakers by saying happy birthday to Boris. We don't go with Boris as far as 26 years but a few years ago I came to ICTP for a conference but I haven't met Boris at the conference or at ICTP or even in Trieste. I met him what seemed to me as a middle of nowhere in the mountains very close to Slovenian border by accident and actually I thought that Boris is just a random Russian tourist so we got involved in some kind of meaning meaningless conversation. He was from New York, I was in Los Angeles so you know the small talk but when I left I was accompanied by a member of ICTP staff and he told me look the guy that we just met he's very famous. I'm telling you he is drowned I always see him with other very important people at ICTP you know being with them shaking hands etc etc and I even remember his name. His name is Arshavin. Obviously the guy was very heavy on football and my knowledge of football is rather rudimentary but the name sounds familiar so I said oh yeah even I know that Arshavin. Oh yes I think I've heard of that and it took me another 10 minutes like Arshavin. Wait a second. So I think Leonid Lidit suggested that we make a gallery of our associations of the associates that the speakers have with Boris you know wise owl, Shakespearean man, a Hollywood actor so this is my contribution to the to the gallery. Okay I'm going to talk about polariton condensates and actually modeling for polariton condensates and you have already heard some of the talks giving the basis of the physics of the polariton so I will only mention shortly what I really need to have to in order to model. I will discuss shortly various modeling approach mostly mean field. I will discuss the proposal that we put forward of how the vortices can be observed and various stages of the transition from integer vortices to fractional vortices can be observed by producing polariton lattices and then I'll show how these these ideas were then implemented in experiment and what we learn and then I'll go back and discuss the vortices lattices again. So in particular I want to comment and discuss how the sports the polariton sports actually couple and what the phase results from this coupling. I will revise therefore with these new ideas the previous experiments and if the time allows I will also talk about recent experiment on spin reversal in a trapped condensate and the condensate when the when the condensate is created away from the pumping pumping geometry. So the acknowledgments the theory on spin reversal was done with my master's student at Skaltech Kirill Kalinin with the Francetti my PhD student and postdoc from Cambridge who's now moved to Southampton. Madness born with a friend Francetti worked on some theory for the to explain the experiment. We did modeling with my colleague Jonathan Keeling from St Andrews University and there is a very close collaboration with two experimental groups that use the samples produced in University of Crete by Pavlos Savidis group and so this is not a photonics group in Cambridge led by Jeremy Baumburg. The key people here were postdocs and student Gabrielle Christman, Guelma Torsi, Peter Christofalini and the experimental group of hybrid photonics in Southampton led by Pavlos Lagudakis and two people here Hamid Ohadi and Alexis Asketopoulos. Hamid actually started as a postdoc with Pavlos then he was a postdoc with me funded by Skaltech and now he's a postdoc with Jeremy so as you can see there is a lot of a lot of flux fluxes of ideas between these groups. Okay the idea behind polaritone condensate is that the absorption of photon by semiconductor creates an exciton. Exiton emits photon but if the Bragg mirror's reflector is a place in resonance with the exciton then the photon get get reflected, reabsorbed, re-emitted and that creates quantum mechanically the superposition of states called polaritone. Polaritone is half light, half matter so it has the mass the effective mass is four or five orders of magnitude lower than the mass of the electron and since the BC temperature for the transition to BC is inversely proportional to the mass we expect that the temperature will be quite high in comparison with experiments on other BCs like in ultra-cold atoms. The schematics of the experiment is a quantum well surrounded by Bragg reflectors or another famous diagram is now compresses these layers huge layers of into these two plates just to show that the exciton and therefore polaritone is a two-dimensional object. In experiments that I'm going there are different experiments of how the pumping is being done how the particles are created. In the experiments I'm going to talk about there is an incoherent source of particles so the laser it is at a rather high energy creating the source of free carriers. These free carriers relax emit phonon scatter before they condense at the bottom of so-called lower polaritone bridge. The Bragg reflectors are not perfect so after bouncing back and forth the photon still escape but this is the curse and the blessing because it is the it has the continuation of the wave function of the condensate inside the cavity and so by imaging these emitted photons we can have all the information about the structure about the density energy of the of the condensate. So what are the essential properties? This is an interacting system in terms of the strength of the interaction these systems lie between liquid helium and ultra-cold BCs. The short lifetime as I mentioned but now it's even can be rather higher than 10 picoseconds in experiments by David Snook in Pittsburgh by simply growing the number of Bragg reflectors the lifetime has been increased to 100 picoseconds. That means that we dealing with essential non-equilibrium condensate that also has from the light it inherits the polarization state so we will talk about left and right circular polarized states but the coupling between mechanical strain the interfaces of the crystal actually leads to you in usual experimental Bragg symmetry and still favors the particular linear polarization so two components for the left and right circular states actually coupled together but there are some but the magnetic field for instance can change that. So our first attempt at modeling the system simply came from starting with equilibrium condensate the usual Gross-Petitayevsky equation with two rather phenomenological terms of three this is just the constant that represent the pumping so this is the usual Laplacian if we just take the low moment into consideration the repulsion due to the self-interaction some external potential that could include include also the repulsion with the non-condensed particles so there is the pumping all the detail of the pump is included in this in this general term there is the lifetime in the cavity of the Pelerotons and in some cases we would also use this inelastic scattering that leads to cubic non-miniarity cubic decay and then the details of the different models really just very of how much information we write into the pumping for the very simple model we can say that the pumping is just the constant and then we of course need this cubic non-miniarity to have the gain separation to gain saturation we can add the term which is like a Landau-Kalatnikov term for the for the superfluids that represent the energy redistribution between Pelerotons particles and non-condensed particles or it could be made more detailed by writing the rate equation for the reservoir of non-condensed particles it has the its own decay rate it has scattering to and from the reservoir into the condensate particles there is a pump and they there is even diffusion but it's been estimated that the diffusion of the reservoir cloud is rather small so this term typically is neglected so in this has come from the model first introduced by Walters and Carousel based on generic laser laser model so now we can have several regimes really if a mission follows the bare photon dispersion we refer to it as a regular lasing if a mission follows the lower Peleroton branch this is Peleroton condensation as I mentioned by changing the number of pregraph lectures the system can be made more at the equilibrium or less at the equilibrium so actually we should have some unified approach to discuss to describe the transition from normal lasers very non-equilibrium system all the way to equilibrium was Einstein condensate and we actually have you know had some ideas of how to do that so if one starts with a Maxwell and did it if one starts with Maxwell block equation for laser which can be reduced to complex with Hohenberg equation from this we actually can recover the complex Ginsburg-Landau it's rather rather standard standard scheme for deriving the complex Ginsburg-Landau in this context so it could be adapted to Peleroton systems so there is it's not just fully a phenomenological model it has some justification actually it's been a surprisingly accurate in modeling these condensates even more surprising because it's still rather phenomenological model we I will talk about hydrodynamics so I will just write this while I'll drop all the all the dimensions and use the simplest form where there is a linear pump and non- linear decay to illustrate the formation to illustrate the existence of vortices here so the usual model and transformation for the amplitude and the velocity reduces to the usual mass continuity and the integrated form of the Bernoulli equation which is almost classic except for this so-called quantum pressure term since the system is described by the by the wave function the quantized vortices naturally exist because as you go around in a closed contour the phase can only change by multiple of 2 pi and therefore they have to be a point inside the contour where the phase is 0 whereas or a way amplitude is 0 to compensate the phase singularity and then from these equations we again recover the usual equations the oil equation for the inviscid inviscid flow from this simple model we can actually get I will talk about the pump geometry so I will need to have some idea about the chemical potential density distribution and velocity and I'm not going to go into the detail in each time I mentioned some solutions but the idea is quite simple that if we look for the steady state and what I will mostly look for we drop the time dependent terms instead introducing therefore the chemical potential and from if from the symmetry of the system we can find the velocity at a particular point then from the first equation we know what the density is going to be from the second equation we know what we know what the chemical potential therefore for entire system is going to be and therefore we have then two equations to solve for density and velocity for a given chemical potential and this is usually done by some perturbation expansion with some asymptotic matching between different regimes of the solution okay so the vortices were observed in particular fractional vortices have been observed in the group of Constantinus Lagudakis and Grenoble where different materials so the experiments that I'm going to talk about use Gallium arsenide it's rather regular material you know there's imperfection are quite small Paola Savidis in Crete does really very excellent job in producing these accurate samples but the first experiments were done in cadmium telleride that has very disordered potential but what was noticed is that then the vortices during the condensate formation become pinned to the minimum of the density and that to the minimum of the of the density of this disordered potential and they can be visualized and for instance this is exactly the same position on the sample but now this is for left and right circularly polarized state separately and there is a vortex in one component but not in another and actually all different configuration of integer vortices and fractional vortices have been have been found in this simple sample of cadmium telleride so it's been established that the vortices such vortices exist and so then we looked at the modeling for the spin degree of freedom of these condensates again starting with the usual Hamiltonian for two component both guess from experiments it known that there is a tendency to buy exciton formation that gives the weakly attractive interaction between two different components about 10% of the repulsive interaction of the self self interaction within each component I will also to make to distinguish between or to to distinguish between the two components we can apply the magnetic field that this is just the Zeeman splitting that favors one polarization over another but in the usual crystal the symmetry the circular symmetry is broken so there are two equivalent axis and that brings about this term into the equation but on the interfaces between these crystals or in the presence of the mechanical stress actually this term will dominate that brings about the coupling between left and right so that now two components actually see the phases of each other through this kind of Josephson Josephson coupling so this is the model again dropping the simplest model dropping all the all the dimensions and now we have the interplay therefore between the between the magnetic field the effect of the magnetic field and the Josephson coupling between the components so we can understand how the system behaves by moving to two mode systems so neglecting all the spatial variations and writing the system in terms of the average density and the half of the difference between two densities and that reduces a global phase doesn't show up in the equation so we have the system of three ordinary differential equations to solve and for the Josephson regime when J is efficiently smaller than this coefficient which is about one of the 1.1 and the half of the density then we can actually reduce drop these terms solve the last equation for the average density and reduce therefore the system to the equation for driven damp pendulum once we have driven damp pendulum we know how the system behaves first of all there is a region of where we have fixed point for the relative phase there is a region where there is a limit cycle and then for some values for some relationship between these parameters we can have also the region of bistability Arnold's talk we don't have to just deal with this equation this is just the idea but if we solve the full full system of three equations we recover the full bifurcation diagram which is drawn again for the effect of the Josephson coupling magnetic field and if for instance I now now J is set by the structure itself so it's a constant so as I start increasing the magnetic field it's fixed point fixed point fixed point and then somewhere on this branch either through this part of this saddle no bifurcation or whole bifurcation we go into the limit cycle if the magnetic field is decreased then it's limit cycle limit cycle limit cycle and then through the hummer cling bifurcation we get to the to the fixed point so this all gave us idea how to actually observe and modify the vortices in this in the system so our proposal was to start pumping condensate polarity on condensate in the corners of equilateral triangle so if we create polaritons in these three corners each is created incoherently so they don't share the global phase but then we thought as soon as the polariton condensate is created the rough exchange of fluxes so each condensate now acts resonantly on another condensate so these all will be locked in phase I will show you then why we were actually overlooked one particular ingredient of the system so but that was the idea so they all look in the phase and then we have a simple superposition of wave functions even the linear sum of the wave function will produce the the lattice of vortices so this is alternating current plus minus plus plus minus plus 1 minus 1 plus 1 etc so this is how the vortex lattice is created there is no you know nothing interesting about that but then as the magnetic field is applied initially we have the lattice of integer vortices but as the magnetic field is applied we still have the the there is a decrease of the density of one component but still the integer lattice survives until there is this transition to the limit cycle to the desynchronized regime at which point only the lattice of the majority component survives and the second component has no vortices anymore it's just some oscillating state that averages to a non-vortex solution so that was our proposal actually it's never been published it's in the archive because it came back from PRL referees and all referees says you know we don't believe that this can be done experimentally so your proposal is flawed you know your theorists you know we don't don't think it can actually be done but I took this proposal to Jeremy and actually within the days you know they produced the vortex lattice the only correction that our proposal was to pump like 10 microns and Jeremy loved it but one micron radius of the pump was sufficient and indeed this this lattice has been generated again showing this plus minus plus minus alternating vortices but before so and then the idea came okay let's take a look at one sport let's take a look at two sports and see what what is there so this is the experiment on just one sport actually showing that there is some blue shift so the condensate is being formed on this hill as the as the ring actually condensate so this hill that you can see in black is actually the reservoir so this is the reservoir of non-condensed polaritons and then this the condensate is is the ring around the top and on this wave vector plot is it's also clear that actually there is expansion of this condensate that away from the pumping the sport has the constant velocity constant out propagating velocity and we had some so actually this problem can be solved exactly as I said with the asymptotic expansion and we actually did it to recover the full the full structure of the density and velocity depending on the system parameters but what I need to have for what is to come is just a simple estimate of the chemical potential so therefore I'll just take these two equations again I wrote it in continuity and Bernoulli and then if I have large pumping sport then at the center of this pumping sport the velocity is almost zero and the gradient of the velocity is almost zero so this expression and parentheses should become zero and therefore at the center of the sport we know approximately what the density is therefore we know what is the chemical potential is and then we can see go back to the equation and find to this equation and find therefore the velocity away from the pump should be given by square root of mu times the mass of the polariton and that's that's therefore gives us this expression for this wave vector in terms of the system system parameters okay what about two sports let me move to the two sports because that cause some that pose quite a few actually questions quite surprisingly so this is experiment again from from the Cambridge experiment when again there is a beam splitter that creates two condensates distance 20 microns apart and the pump for each the width of the pump is about one micron and what has been so then the there are two reservoirs these two heels created at the place where the pump is low two pumps are located and as you can see in this direction these two heels create what looks almost like the harmonic potential and indeed when energy is resolved along these distance the state of what looks like the quantum quantum oscillator quantum pendulum were obtained so we have this ground state the first excited state the next excited state and all these levels are equidistant there is an equal energy spacing between them and this is just the resolution of this of this line but now in two dimensions showing again the ground state the first excited state etc. another interesting thing was actually in this inset because it which shows that as the distance between the pumping sport increases the distance between energy level decreases and as we increase twice the distance between the sports the distance between the energy level decreases by a factor of two so how to explain that we explained that by again using the model and observing what we call the oscillation of the dark soliton between the peaks so I will explain what is shown so now this is theory this is numerical simulation again I take the line between two pumping spots and then resolve energy versus the distance again obtaining something that looks very much like experimental data but if the inexperiment all the states are average here I can resolve in time and then see what happens so again against this line but now I put it with 90 degrees and now you can see the black soliton just jumping between two maxima providing providing this pattern as we increase the energy of the pump we could also observe the turbulent excitations of this dark soliton of course when I say dark soliton it's not the dark it's not dark and it's not soliton because if we are not dealing with one-dimensional system it's two-dimensional system the governing equation is a complex Ginsberg-Landau so what is here is really more likely to be called traveling whole solution as the solution of the Ginsberg-Landau so it's a localized disturbance that appears and reappears but its action is to actually move the system through these through these energy levels to understand analytically what is going on I will look at slightly simplified problem in this problem it's one-dimensional problem there is a constant pump but to represent the reserve why it's a double Gaussian why double Gaussian so there is a minimum and there is a maximum because then everything can be sold analytically in terms of the error function and we can actually observe the formation of the traveling whole so this gives the velocity profile for various height of the Gaussian and then there is a criticality at which so now I show the density the travel whole is created you see the sudden drop in the density so this is now only up to this criticality we have an analytical solution and then there is traveling whole develops at the particular maximum of the velocity and this one dissimulation in time just showing the peak how the peak is created it decays and then the new one reappears it drifts away disappears etc so this is a simple simple model that actually captures this essential phenomena that as we increase the pumping intensity the velocity reaches the criticality and it creates the traveling whole and in a similar configuration it's also possible to create to create vortex anti vortex pair and then the but the the minimum part of the double Gaussian can be served as to actually trap this this vortex pair as a stationary solution but it's it's slightly different story okay so then the Southampton group they have done more also and another set of experiments but this time they just slightly about the threshold and then they observe that as they move if they change the distance between two pumping spots then they can have either destructive interference in between or constructive or destructive so they actually observe the single level instead of this graduation of levels there is just a single energy state it's a stationary state unlike the Cambridge experiment and then in Cambridge we did the same thing just just be about the threshold and indeed confirmed so that we have a single single energy level and that depending this time on the pumping intensity again we have the even or odd number of these blobs in this energy profile as and as the intensity of the pump is growing then we recover this harmonic oscillator and the highest intensity the more levels we're going to have so what is going on so why the first question was posed in this in this in this paper was so why do we have the stationary states because the destructive interference between two spots implies that these two condensates are not locked in phase they look with a pie different phase so how to explain to explain that and now so this was done again in this manuscript by Southampton and I think Yuri Rubo was responsible for very nice idea that what system is trying to do is trying to maximize the number of particles so if you just about the threshold it means that the first you're going to lays into the state that has the maximum number of particles total particles in the system and so and then a simple mass so if we take the sum of the two order parameters with a relative phase different theta and so now I use my analytical estimate for psi and evaluate this integral then we recover the function which is positive for density depends on the relative position of the pumping spots and their densities and then there is this Bessel function that depends on the distance times this parameter that depends on the chemical potential multiplied by the cosine of the difference of the phases therefore if the distance is such that for a given chemical potential the Bessel function is positive then two components lock in the phase so we face different zero there is in constructive interference in between but if this Bessel function is negative then they lock in with the pie phase difference and this is explained now the energy spacing in Cambridge experiments and it explains why we see these levels because for the high although we are quite above the threshold but again this system goes into the state depending on the chemical potential that correspond to the state so if we call the difference between the energy levels delta mu then we use this criterion to find out what how mu and a link together expand and tailor for the small energy distance and again and we see that indeed the energy spacing multiplied by the distance should be constant so if distant goes by a factor of 2 the energy spacing decreases by a factor of 2 so now I think we have the complete picture of what's going on and just to complete of what else have been done on two spots so we looked at the some discrepancy between spots and it turns out that the system can be phase locked but only if this difference in pumping is sufficiently small otherwise we simply have two different different chemical potentials for two different condensates and therefore during the average in experiment everything averages out to zero we don't see any so each condensate just acts according to its chemical potential and there are also these recent pulse excitation when the time evolution of the cloud has been time resolved and you can see either the propagation of the dark soliton or bright soliton depending on the intensity of the pumping spot again it makes it we understand why it happens because again depending on intensity the Bessel function for the given distance between pulsing between polaritons is either positive or negative so either we have constructive into a destructive interference in the middle of constructive and then as the time goes on that's indeed what what evolution of what we observe okay so it's all comes together so let me go back to this three pumping pump the geometry with the three pumps in Jeremy in Cambridge experiment we didn't see any other configuration just all condensate would always be in phase but with four components sorry but with four components there were two possibilities either all four components are in phase or there is a pie difference between the nearest neighbors and again if they all in phase then there is a constructive interference in the middle if they are not in phase then it's destructive the pumping spot and actually indeed observe these two configurations and now again once we know the criteria of how the condensates lock in the phase and indeed this corresponds to the situation when the pump the relationship between pumping and the distance between spot gives the negative Bessel function and therefore they lock with pi difference to maximize the number of particles in the system in Southampton group they redid this experiment for three pumping spots and observed that again there it's possible to have in this case two pi over three difference so that there is a winding as one goes around these three spots again this correspond to the case when this criterion gives the negative Bessel function so the number of particles is maximized when there is winding around the around these lattice latest points and numerical numerical simulation confirmed that okay there was some other work done on even more than two three or four condensates so this is for instance six condensates etc so it's an internet game but what is interesting here is that as one so previously I talked about the situation when you condense where you pump but it turns out that as the distance between the pump get close so if the distance becomes small then instead of condensing where the pump is the the polarity condenses in the middle and it's especially clearly seen on this on this perhaps on this picture when there are six pumping spots but the condensate is in the middle or here where the pump is in the ring but the condensate is in the middle okay so unlike so these were referred to as the lattice condensates and these were trapped condensates so there is a very sharp transition and we had some theoretical also explanation why it happens but what I would like to mention so that my last last topic to go back to the spinner to the spinner degree of freedom and there are these two recent experiments again in Southampton and Cambridge in Southampton where the condensate is created on the ring the ring it has small radius so actually the trapped state is in the middle and in the Cambridge in Cambridge the geometry was in four corners of the square again with the condensate in the middle in both cases Hamid Ohadi was doing experiment so that's why geometry perhaps is different but the idea is quite similar and what was observed and now there is this discrepancy between these two experiments that they resolved in both cases the degree of circular polarization and in Southampton when they recorded according to so there is this the increase in the pumping and this is the discrepancy between polarization the degree of circular polarization so at the threshold although they pump with almost the same intensity almost linearly there is about 60 80% of the condensate in one component in another and then as the power increases the pumping power increases it levers off until it becomes linearly but then it drops so there is a spin reversal it means that we pump with say predominantly left component but in this regime we have predominantly the right component in Cambridge experiment they've seen the same rise to 60 80% and then the leveling off but they've never observed this spin spin flip so we did theoretical investigation and we decided that it's really due to this discrepancy in the initial pumping pumping profile so we pump with slightly more one than another and this due to a very narrow focusing of the pumping being that actually makes this quite quite dramatically different and that difference create a kind of is human effect that now it prefers one polarization over another and the two polarizations are coupled with this Josephson Josephson coupling and again will write explicitly the rate equation so when we go back to two model model then for the relatively only right the equation for the relative phase and you can see that if the Z here left on the previous slide that is the half difference between the densities of two components so if we have an about linear polarization that is quite small then both of these terms and I'm pumping predominantly plus in comparison with the minus then both of these terms would be negative unless that changes its sign that changes its sign it means that at this point I would have to have spin flip and as in the case with with with Arnold's tongue that I described before we have the region where there is a fixed point when there is desynchronization so this is just the average over the oscillations and then fixed point again and so in this is the comparison of the experimental data and what our model gives us for moderate discrepancy between the pumping into the right and left so about 10 percent more is created in the right circular polarized light then in a left and as you can see qualitatively we're about the same so there is a rise to 80 percent and then there is a graduation and then at about twice 2.2 of the threshold we have the spin flip and then it goes back to linear polarization and now you can ask but why didn't we see why didn't they see it in Cambridge experiment and I would say that as we are very the discrepancy this point the point of transition moves further so which suggests that simply they pump and the discrepancy of their lenses are in a regime that the discrepancy that with which they feed the their condensate is higher and therefore this transition would be observed at much higher pumping powers that they didn't didn't really reach okay so in the conclusion what are the interesting things that can come out of this so I talked to you about modeling of Pleroton condensates I think I gave you some idea of how phenomenological complex Gainsbourg-Landau equation works actually it works as I said surprisingly well we need more experimental data and that's what we're working on now just again going back even to a single pump and actually tune the parameters and really understand you know which regime and change the regime to see how robust the model really is we also need to be more careful in modeling interaction and the non-condense cloud itself and also understand how spin dynamics work we also continue working on the pump condensate lattices and especially we interested in frustrated systems there are the questions concerning vortices in turbulent flow we also predicted and analyzed a little bit how the turbulent flow appears but more work needed on this front there is a lot of discussion on what is really non-linear linear effects in these systems so there you know that's another topic of discussion and finally we think of we put some proposals on applications for sensors for quantum circuits and that's so I thank you I think I should stop here the chairman is already standing so thank you