 Okay, good afternoon everybody and I would like first to thank organizers for inviting me to such interesting conference and I will discuss a new type of lasing which could appear in case of when we want to achieve lasing with strongly interacting bosons. This would be a system which is not out of equilibrium when we have pump and dissipation of bosons, driven dissipative system and I will discuss the situation when we are very close to the condensation or lasing threshold, when the pump and dissipation are very close to each other. In fact, we are going to from below to the average condensation threshold and the concrete system I will be interested in would be exciton polaritons and semiconductor micro cavities and that system was already introduced at several times during this conference in particular by Alexei Kavokin and by Natasha Berlov but I still will say some words and I will try to avoid this system in general is complex because we have a system out of equilibrium, we have excitons, we have relaxation of excitons, we have formation of polaritons. So description of this system for example in means of generalized Gross-Petyevsky equation and Boltzmann equation for the excitons involves a lot of numerical conculations and we do need this numerical conculation if you want to describe real system. However, I will try to make a point, I will try to describe specific qualitative effects so I will consider very model system when this effect appears and this effect is called weak lasing and it is characterized by formation of states which are substantially many particle states they are different from single particle states and then I will discuss the recent observation of this effect in case of polarization degree of freedom of trapped condensate and in case of polariton superlattices. So exciton polaritons are combined quasi particles which are mixed excitons and photonic mode. Semiconductor micro cavity consists of two distributed break mirrors so I have two different semiconductors this gray and black which are different semiconductors say gallium arsenate, arsenide, gallium, minimum gallium arsenide and the thickness of this layer is say lambda over 2 so those are in fact mirrors, distributed break mirrors and then there is a defect which is for example several wavelengths and it is possible to put quantum wealth inside and they put quantum wealth exactly in anti-nodes of there is some cavity mode for photonic cavity which oscillates here, there is some oscillation here and then it is decaying opposite and you put they put quantum wealth exactly in anti-nodes and maximum of electric field of this cavity mode to achieve the strong coupling between this mode and excitons. So if I forgot about coupling I just put exciton level that would be in this scale just straight line exciton does have effective mass but this mass is very large so this is parabola of something like that and micro cavity mode is also parabola, parabola with respect to wave vectors and planes in horizontal direction. It's also parabola but this parabola is characterized by very small effective mass which is like 10 to the minus 4, 10 to the minus 4, 5 or 3 electron mass. Now if I turn the interaction between excitons and micro cavity modes on then I would have ribose splitting as I would have the formation of two new quasi particles which is shown by blue here and called exciton polaritons and this splitting would be a rabbi frequency usually this rabbi frequency is achieved about tens of milli eV like 10, 20 milli eV and this exciton shines light so in fact this photon and this micro cavity mode could escape and this is visible light so there is no spontaneous reaction of excitons to make the system useful we have to pump it and when you pump some excitons and then you have formation of polaritons. These polaritons are mixed quasi particles due to contribution of light they have very small effective mass and this effective mass is good because if you consider this system as equilibrium one you could suggest that for example if you get condensation of these particles corresponding costal established temperature would be would be high indeed you could reach in fact condensation at like in room temperature however this system is out of equilibrium and these particles this polaritons could relax because due to due to contribution of excitons they interacts with each other interacts with phonons still still there is some limitation and this relaxation in particular there is there is bottleneck region when velocity of polariton exceeds the sound velocity the sound velocity in the crystal they no longer can emit acoustical phonons there is some accumulation of particles here and then due to interaction there is still some flux at the bottom of this dispersion and one can speculate about condensation now the word exciton polariton condensation and exciton polariton lasing are used interchangeably and it depends mostly on how people want to sell their work I will consider this system as lasing so it is important that this effect was was experimentally achieved and one of the way to overcome this limitation of which imposed by energy momentum conservation is consider a dirty system consider the system with some disorders substantial disorder in particular the first observation of of this exciton polariton condensation was was observed in cadmium telur based at samples when when disorders strain disorder is strong also you you can also put some some superlatives on top of the structure and consider patterned structures so it is natural here to consider the system with with disorder and let's consider the case when you have the several condensation state localized for example and and what what is important for me will be that actually you have already interesting physical if you just consider two states and now it is also natural to assume that if you have different states in the system in disordered systems and different states would have would possess different lifetimes lifetimes with respect to escape from this micro cavity so so what happens in this case if you have strongly interacting bosons in this system when you have different states with different lifetimes what what what happens in this case and I will start with very very simple equations so I consider two two states and that would be the the amplitude of of condensate in each state it's complex number and it is complex number of subject of this two nonlinear equations so first I have this governing parameter g which is the difference of decay rate escape rate which is inverse lifetime of particles say average lifetime of particle in state one which is supposed to be the same in one and two and this w would be the pump rate and and then there is interaction of particles inside of each of each center which alpha is the interaction constant and then there is would be coupling between one and two and this coupling there are two couplings one is usual Josephson coherent coupling and then I would assume that there is also some dissipative coupling and this dissipative coupling is very important for the effects I am going to discuss now the same system you you you would discuss in terms of symmetric and anti so it should be a here as a subindex so and so in terms of symmetric and anti-symmetric states and in this basis you you immediately obtain that you have decoupled symmetric and anti-symmetric state and linear approximation and then you have coupling between these states and this coupling coupling appears due to interaction between particles you you you can see here that the energies of symmetric and anti-symmetric states are different symmetric states get smaller energies minus j over 2 the energy of anti-symmetric states is higher is plus j over 2 and they also have different lifetimes the lifetime of symmetric states is short so dissipation rate is strong as big gamma plus small gamma on the other hand the the lifetime of anti-symmetric states is is long and or dissipation rate is is weak so in fact we have some distribution of lifetimes so in place of having one single threshold we have some sort of threshold region of the width to gamma small okay so so this is what what are here in these equations what are not here what is not here is usual dissipative non-linearity you usually need for lasers for example forget about psi 2 consider just psi 1 then this equation would be bad for example if gamma is positive when you just below the threshold below the threshold then your psi would be zero you you do the system cannot sustain particles the pump is very small and the escape of particles very strong however if you cross the threshold and go to the region of negative gamma then the system would just explore the psi would go to infinity and usually when you make a laser theory you add some some dissipative non-linearity here so in place I don't want to consider for the moment this this dissipative non-linearity I would assume that it dissipative non-linearity smaller than non-linearity coherent due to interaction of particles it turns out that interaction of particle alone can stabilize the system so so so there are different reasons one we we want one why why we have can have different states with different lifetime one reason is dissipative coupling already discussed by Natasha Berlov consider for example now two condensation centers and now I want to to discuss the the emission of light from these centers now if I consider symmetric states then it would mean that deep phase difference between the between the centers is zero it means that there is a constructive interference of waves emitted by by different centers and that that that this constructive interference would increase the escape of particles on the other hand if I assume that the phase differences pi which corresponds to anti-symmetric states in these states these waves emitted by different centers would would cancel each other there would be destructive interference and that that is reflected by by the fact that in fact we have anti-symmetric states would have a longer lifetime than symmetric than polariton in anti-symmetric state would live longer in the cavity than the polariton put in in the symmetric state the the very fact that we have this sort of dissipative coupling between between two polariton condensate is not not specifically quantum or polariton effect we do know this from classical physics and well-known examples here two hugans clock which are two two oscillators which are which are could communicate to each other because they are connected to the same beam for example when we have oscillation of one it could emit sound waves and the other could emit sound waves now now if if they if they oscillate out of phase with pi phase difference the waves emitted by the waves emitted by by different oscillators destructively interfere and cancel each other so the dissipation of the system is reduced and that's why the the known effect exists that two two two two oscillator are synchronized out of phase and of course we can also put some string here and have normal uh jaws of some coupling between them now what happens in this system if i just go above the first threshold and if my my pump this g defined by g e i would still consider g positive parameter so i would be below the the average threshold in the system so gamma dissipation would be bigger than w however i would assume that this this gamma is smaller than gamma small in this case i am above so if i just consider anti symmetric states for this particular states i will be above the threshold and it would mean that for example if i put system in this state the system would explode the number of particle would grow exponentially with time and the system would go to infinity indeed such solution exists however this is the only trajectory that goes to infinity if i for example violate the this phase difference move it away from pi or i for example put different number of particles different number of particles in different centers the trajectory would would be close so i would have some dynamics of of dynamics with some finite occupation of of particles and in fact in this system i i never you should understand that there is some noise always some noise term so so existence of this divergent trajectory doesn't matter if you include noise you you immediately would go away from this trajectory where you go it turns out then in place of the of find finding this anti symmetric states you would have you would obtain two different states which we call big lasing state so so in fact it is good to consider the system as just non-linear dynamical system and what we are interested in in non-linear dynamical system we would study it phase portrait we we we need to look for attractors and first thing to do is just try to find the the fixed point and fixed point we just consider the solutions when this left hand side is zero and it turns out that there are two two solutions which we called big lasing solution they corresponds to some non-trivial occupations of of both centers and it also corresponds that one center is have smaller occupations than the other and this number of occupation is given by these expressions and this difference between between two centers is also non-trivial it is not exactly pi it is different from pi phase differences to two phi and it is somewhere between pi and pi over two so so here we have spontaneous symmetry breaking each each attractor provides spontaneous symmetry breaking so this system of equation is symmetric with respect to interchange of indices one and two however the solutions are not so let me illustrate what happens in this system if you just study to increase pumping below the so this parameter g would corresponds to the pumping increase of pumping would corresponds when I go down down in this figure so so first in fact what we what we find that this far this solutions I I just showed in this previous transparency they are unstable in fact well what the system forms that they form limit circle and I will tell something about about it but then this limit circle transforms to this symmetry breaking solution so characteristic of symmetry breaking solution is is we have breaking of disparity symmetry one center obtains highly highly populated the population in the other is small the phase difference is non-trivial which means that there is some flux of particles from one to two in this case and we have those several symmetry is broken and we closing state in particular there is parity symmetry broken and there is time reversal symmetry broken so for example if I consider the spectrum of light emitted from the system it it would that doesn't have the symmetry k to minus k so so this is what happens if I just go from from up to down so first I have this plenty of more or less chaotic emission then emission which consists a lot of peaks and distance between peaks is to our our period because my system here corresponds to to fix its attractor corresponds to limit circle the system rotates with some period which depends on the on the parameters of the system then period and period decreases the distance between peaks increases and finally we reach the situation when there is once one peak which corresponds to weak laser in the region now in fact there is some interesting physics happening happening here so in fact there is period doubling before cations so so if I change the parameter there is a period doubling before cations the system goes to limit circle and then then it doesn't close and the period doubles and then then in fact when when I go to to this region there is some chaotic behavior this period doubling doubling before cation are illustrated here in the in this figure for example this is emission of this so so this this statement system exhibits this this limit circle behavior corresponds to emission of so-called frequency comb this frequency comb is very very very useful technically because in in that atomic system are used for for atomic clocks and here for example we can achieve the regime when when distance between between peaks corresponds to terahertz region but but but what is also interesting is how the system evolves for example if I decrease the pumping there is this period doubling before cation and finally there is figure bound root to chaos also even in this in this regime when when there is a limit circle solution we can appreciate a symmetry because here on on on these figures we have emission spectrum from center one and here from center two we can appreciate that they are different qualitatively different so which corresponds to this broken symmetry between spontaneously broken symmetry between one and two so so this is the effect in a real system of course it is necessary to take into account what I mentioned is lasing non nonlinearity dissipating nonlinearity which is could be done by changing this parameter g which is difference between dissipation and pump we need to add nonlinearity which is proportional to the occupation of the system and this parameter eta should be compared to the interaction of particles parameter alpha in it is usually smaller than alpha and in this case it changes slightly the the the the dynamics of the system so if we increase pumping the system first is the condensate the first is formed in anti symmetric state or pi state and that that would be just simple condensation in this anti symmetric states and find at some at some pumping the states loses stability and there is formation of one of two weak lasing state and this verification reminds second order first transition when if you consider the order parameter is occupation difference between well this corresponding phase some complex number say difference of wave functions psi one and minus psi two okay so so now the same effect you can observe if you in place of considering two different centers you just consider one single condensate by concede but but but consider polarization of polarity so those polarity could be characterized by polarization right circular or left circular and we can write the same equation for example if you consider one is the right circular polarization and two is left circular polarization we would have slightly different equation because interactions that the terms would be slightly different because there is usually stronger portion of particles with the same polarization so for this parameter alpha one would be some positive parameter and there is some small attraction of of polaritons with opposite circle polarization so this parameter alpha two is about 10 percent by the magnitude of alpha one but it is negative so in this case if you consider polarization case the meaning of parameters gamma and and j it's some something different for example j in fact would would define the splitting between x and y linearly linearly polarized condensates and this is you this is known effect first predicted and discussed by Igor Lainer and Evgeny Ivchenko in 92 it appears because the symmetry in quantum well is usually low is usually c2v so so we don't have a rotational 90 degree and that's why x and y should be different also in real systems there are always some strains strain fields and they also lower the symmetry so it is it is not unusual to have splitting between x and y polarized light in micro cavity this dichroism and also if you have splitting between between between x and y we also in general should assume that they possess different lifetimes we should remember that this photon states reside inside the stopband of distributed Bragg mirrors so if I have two states in stopband inside the stopband the state which is close to the H would would have shorter lifetimes so so in general here we have also should have different lifetime of x and y polarized state also I will need this this picture it is known when in place of using two complex numbers we just make the matrix we make the matrix element of components of Pauli matrices and we obtained three real numbers and there is also total phase so so this is usually called as stock soap or anchor sphere and formation of for example pi condensate would corresponds to negative value of s6 so this is where I am pointing now this that would be formation of condensate in anti-symmetric states formation of weak lasing state would corresponds to the to the to the to the effect when occupation of one for example is bigger than the occupation of two or vice versa and it would corresponds to formation of condensate at some points in the north hemisphere or south hemisphere now this is the equation in terms the same equation but now written in terms of this pseudo spin vector as x and sy as z this is the solutions there are two weak lasing solutions one corresponds to positive value of of as that as that now is the difference between occupation of left circular right right circular polarized and left circular polarized and there there are some specific features of these states for example if I consider this components of this spin could be measured experimentally in fact what experiment is do they they measure difference between horizontal and vertical so they observe life horizontally polarized and vertically polarized and difference between this intensity would would relative difference between these two intensities would give me a sex if I want to calculate as why I just measure diagonal and anti-diagonal and finally as that would corresponds to to measuring of intensity of right circularly and left circularly polarized light now we can see here that vertically vertical component is always bigger than horizontal so as sex is always negative however for example if I have a weak lasing state if I have formation in north hemisphere I would have anti-diagonal strong anti-diagonal component of formation and and sorry this should be different indices formation of condensate in the south hemisphere would corresponds to formation of anti of the strong diagonal components now experiment of this type was done recently in the Jeremy Baumber group and in Cambridge so to top to observe this effect it is important it turns out to be important to to consider so-called strapped condensate condensate which are separated from reservoirs this interaction because interaction within within the condensate reservoirs contains this in this case what they do they create four four reservoirs these four points and then this excitons could transform into polarity in in form the condensate this here in the center so this contents condensate doesn't communicate with reservoirs and this is important for the observation of the effect also what they do this is in fact micro cavity membrane which is about 300 micrometers in diameter and in fact we can not only non-resonant non-resonantly excite the condensate we can also talk to it and excite by some pulses and indeed what they observe the absurd formation randomly of two v-clays in state with either left left on right circular component the formation of this state is random so if we turn off the laser and then we turn it on again the pump then we randomly generate condensate as it is seen here there is some trains of pulse excitation pulses and we have random either left or light formation of either left or light or right condensates also it is possible to reproduce limit circle solution and and everything how much time do i five thank you very much so this is fourth experiment which demonstrated the formation of v-clays in state and related spontaneously parity broken symmetry the other experiment corresponds to to periodic systems so if in place of let's forget now about polarization and consider different centers then we can consider for example chain of centers and we can reproduce the v-clays in solution when we start from ideal chain and but due to this uh before cation to v-clays in state we have a b a b with different occupation of a and b and it is possible to to study the stability of this state and this is some generalization for two-dimensional superlattices again we have we have here square square lattice with different occupation of nearest neighbors and this is how how it should emit light in the reciprocal space that would be a mission if we have just condensation in the age of brilliant zone with pi difference so if you just have pi difference between centers equal occupation you would have something like that v-clays in corresponds to appearance of apart from these peaks and ages of the old brilliant zone we have peak and k equal to zero also it is possible to consider honeycomb and again here we can appreciate that the symmetry k to minus k is broken and this formation of this v-clays in state was recently observed in one dimensional superlattices not exactly in semiconductor micro cavity by in zinc or micro road when which is uh put on top of of silicon uh substrate with some periodic uh grading and uh in this in this state indeed it was observed the formation of of light emitted with the same frequency but composed with peaks at the edges of brilliant zone and central k equal to zero peak functionally the the experiment is slightly slightly distorted by disorder so probably i will skip all this and uh i will conclude so v-clays in uh is uh effect which corresponds to formation of spontaneous symmetry breaking the formation of uh states uh in near the stress near the threshold which corresponds to uh to to to to spontaneous symmetry breaking in the system and and there are essentially many particle states and recently this effect of v-clays in was observed both in polarization state of single condensate and in superlattice of polarity on superlattice and uh i should acknowledge the collaboration of with eagerly laden and Boris Holt Schuller we did the the first work about theory of this v-clays in effect then the i acknowledge Sergei Flach and Cristian Rayanov we collaborated on the study of this frequency combination and um formation of limit circles there is ongoing collaboration with Alexey Kavokin, Sasha Podumny, Southampton, St. Petersburg, Timlew from Singapore and experimental group Jeremy Baumburg from Cambridge and Shanghai Chen from from China and that's that's about what i wanted to say about condensation of exciton polaritons and now i would like to say a few words about condensation of people so to have condensation of people it is necessary to have some uh outstanding human beings some outstanding man and uh i first uh met Boris in here in this very center it was in 93 it was and i had a privilege to and i was i was listening to his wonderful lectures on conductance fluctuations and it was it was easy to to see the condensation the people around Boris because you you could identify Boris very easily either here or in corridors on on the seaside there was also a cloud of people around him trying to convince him on something or and or arguing something and when you create uh i now i understand it is great responsibility when you create such a such a condensate of people and Boris told me that his first name comes from uh old synagogue in Prague and there is a legend uh folkloric uh related to this synagogue it's about golem so when when when uh ruby created a clay person and then put a life in it and this golem was actually a good guy it was a soldier it protected community however there are some various uh different variations of this legend for example there is clay boy which was a bad guy which was just gross and it was catastrophic consequences i particularly like the variation when ruby created a clay cow and then by sorcery put a life in it and all the community fit on this cow so i believe the condensate created by Boris is wonderful because we all fit on on Boris on his ideas on his knowledge on his uh on his wisdom as this conference demonstrates so Boris many happy renters and increase and proliferate your condensate of people thank you