 So thank you very much. It's a tremendous pleasure to be here, so I'm delighted to come to yet another party, lots of birthdays this year it seems. So I wanted to follow on actually from a really nice talk that you just heard with a kind of a different view of sort of a broader class of Pilaritan systems and just to sort of lay out the talk. I thought what I should do is to start by saying a little bit about a microscopic model for Pilaritan conversations in quasi-equilibrium theory. Just to give you a bit of background, because what I really want to talk about in this talk is systems which actually have other interactions in them beyond the interactions that we classically think about with Pilaritons. Nicely described. And so in particular there are many systems that one can make condensate itself potentially in effect, some which have been achieved, which involve quite strong coupling to the electron-phonon interaction. And there's a question about what that does. And in particular can you condense into systems which actually have a phone on side dance, so if you have a strong coupling there what would that mean? And then I'd like to spend quite a bit of time talking about cavity-coupled Rydberg atoms which have the interesting effect that in the excited states the excitons which you've created or the Rydberg states of the atom have strong and long-range interactions. And because of long-range interactions there's going to be competition between the kind of superfluid or Pilaritan state that was just talked about in the last talk and actually some kind of quantum crystal and what might that look like. So quick reminder, without any worries about Copenhagen and other statistical interpretations, the kind of systems that we can eventually talk about are typically 2D. You have excitons which are typically confined formally to a quantum well or some array of quantum wells. They're pretty much dispersionless on the scale on which the light modes in the cavity move. You have a confined 2-dimensional like photon mode and by tuning the mirrors correctly you can bring this photon mode more or less into residence with those two. And that of course means that as quantum mechanical objects these two things, if there's a dipole moment to connect these to mix these then there's some splitting here which is the ruby splitting which can be quite substantial in some systems. And as you've just heard, at least formally ignoring inelastic processes, decays, whatever. And I will say assuming the rotating wave approximation which is actually critical because it means that there's actually a conservation law built into the Hamiltonian. So assuming that and that's generally a good approximation but not exact, then you have this kind of picture. The polariton wave function is a mixture of exciton and photon with that mixture depending on the detuning of the level so it varies enormously across here. Furthermore the nice thing about the system is you get to look at the occupation of these states because the mirrors are not perfect. Eventually the component will tunnel out and you'll be able to observe an emitted photon and if the mirrors are flat the transverse momentum is conserved and therefore the angle of emission can be converted into a momentum in here. And the reason that, no this goes back a long way, but the reason that this is interesting from the point of view of condensates is that down here you have a branch with a very light effective mass, typically around 10 to the minus 4 of the electron mass, you also have roughly the characteristic coherent scale, scales inversely with that. So relatively dilute gases down here can in principle have quite strong on sets of coherence. So what's been seen run through some experiments, indeed this is momentum distribution that you get to see by looking at the light so this is a paper going back in 2006. If you look for the, if you look now for what happens to the dispersion of that lower polariton branch when you go into a putative condensed phase you begin to see something which looks quasi linear as if a buggy-lubar spectrum. You can look at real space images and demonstrate coherence seeing fringes and you can also see vortices, in fact you can see half vortices because there's a polarization degree of freedom here as well and that's also locked by the transition. Because it's 2D you might expect to see power law correlations and you can find those, you can find ways of setting up condensates that move and here's a condensate moving and there's some kind of object in the way here and you can see going from super flow around the object to ejecting vortices and and there's solitons and then you can do fancier experiments by pumping pairs of condensates and having them coupled and looking at the dynamics of how that happens. So there's been quite a lot of work much more than this actually which has happened really over the last decade. So for what's useful I want to move a little bit on to theory space about how I like to think about these and I will say right at the start that I don't like to think about polaritons as particles and I'd like to explain why. And not for the same reasons you heard in the last talk although I agree with that also. It's actually the fact that there are very strong renormalizations of the exciton and photon component in this and I'd rather like to think about models which actually have both degrees of freedom explicitly. So let's imagine just as a theorist that you have a bunch of dots coupled in a cavity and each dot can have an exciton or not. And if you're in a regime where a double occupancy doesn't happen the natural model to describe this is actually a spin model where an empty state is spin down and a singly occupied state is spin up. And of course you flip the spins by absorption emission of a photon. So this is the simplest Hamiltonian you can write down for this. This has a single cavity mode at a frequency omega and principle there's a dispersion as a set of modes. There should be a K index on here but it doesn't matter. This is the energies associated with these spins. Typically there's very little dispersion in here. There may be quite a bit of disorder so it's often useful to think about this in a kind of a site based index. Here is the Rabi coupling between this and this is the interconversion term as I say written in the rotating wave approximation. And the consequence of writing it in the rotating wave approximation is that the total number of excitations is conserved. If you go outside the rotating wave approximation and that actually begins to be important in very strong coupling then you need to be a little bit more careful with that. There's a parameter in here which I call n but it really is simply associated with the fact that the wavelength of light is very big and the size of an exciton is of all to the bar radius or whatever the trap it's in and that's very small. And so typically there are a large number of exciton states which are available within a wavelength of light so n is a very large number. So as a theorist when you see this Hamiltonian when n is large you know what the ground state is. It's a coherent state. It's just a large n limit of this and so here is a generic coherent state, a coherent state of photons. This is a coherent state of spins. You know if you turn spins into fermion pairs you get something which looks exactly like BCS and you can also go through the standard BCS style argument with all of this and you get a transition temperature which depends on the coupling constant and generically on the density of states. So this is what the mean field theory looks like. You can follow that mean field theory, do that mean field theory to find out doping and so this is concentration. This is the total number of excitations. We don't know whether the excitations are photon like or exciton like. You start off with a lower enough polariton branch as you add particles to the system. We're here to find out temperature. Here's the chemical potential. Chemical potential rises. At this point it hits the bottom of the band and at that point the spectrum renormalizes. The upper polariton starts to push up. The chemical potential clamps. You get coherent light emission associated with this. This is a conversate appearing here and then there's of course a ghost branch that comes down here. Built into the model by the way of course is a parameter which we tend to call detuning which is the difference between the photon frequency and the exciton frequency. Here I've had them tuned together. It's worth discussing what happens when you tune the photon mode way above the exciton. Then intuitively what happens when you do that is that you're going to have excitons coupled by virtual transitions through the photons. As that happens you actually get a model which is kind of more familiar from Bose Hubbard. You get the appearance of mottlums. Let me just go through this rather busy fugue graph about what happens as you detune. Here I'm just plotting the chemical potential which sits in the polariton band again as a function of pumping. If you look down at the bottom I've broken the wave function up and because this is by the way a coherent state. This is entirely classical. I'm in the coherent state limit here at least in the theory. Here what happens is that the exciton computation is here and the photon curve is up here. What you see is that they run in parallel but after a while of course you can't put too many excitons into the system. So it saturates but the photon number is allowed to grow without bounds. As you start detuning what happens of course is that the ground state becomes more exciton like and less photon like. But interestingly if the detuning exceeds a critical number which is actually 2 something odd happens. The exciton number grows and grows and grows because of course the photon is far above it and you're filling mostly exciton states. So it's really an exciton cognizant there's very little photon occupation. But you notice interestingly that the exciton cognizant grows all the way it gets to a half in these units so this is fully occupied. So this is just a field band and at that point of course you can't have a cognizant because there are no excitations. And the photon occupation goes to zero and then you begin to get a photon cognizant growing on the other side. So this is just the appearance of a modeler because you've got you've you've spin half is like a Bose Hubbard model with infinite u. And so in terms of a detuning parameter there's actually a mod insulating phase which is actually in the model at least it's hard to see experimentally. And then there's a super radiant phase which is sort of outside. A bit more background that was mean field what happens if you go beyond mean field going beyond mean field. Of course what does it mean you actually have to include the finite k modes the boggle above modes and to incorporate all of those. So if you look at what happens this is the dispersion in the normal state in the you know at this leading order one loop level. This is what the dispersion curves look like now so these are the boggle above mode this is the lower and the upper mode is still here. And as you go through this line this the spectrum changes you can actually do this with disorder it doesn't matter I'm worried so much about that. But of course those fluctuations then are normally populated thermally and that would be in the low density region the dominant method of producing decoherence when you go to a finite temperature. Because when I do mean field the transition temperature just depends on the rubby coupling it doesn't depend on the effective mass at all. And I told you that the reason for believing that we should be using polaritons to make condensates was because their effective mass is very light so where did that go. And it's quite clear that the let me just run through all of this. So the mean field transition is a function of density scaled by something like you know the characteristic size of an exit on the ball radius is going to be some curve that comes up like this. And if you went and did however your standard theory for the cost of a stylist like theory for a transition here of course you get a BC transition which would be roughly linear as a function temperature as a function of density. So down here in the dilute limit of course you're dominated by phase decoherence associated with the acoustic modes but once that transition temperature exceeds the coupling constant you're dominated by decoherence associated with thermally populating the upper polariton line. So where's that crossover? Interestingly in these experiments well this line is very steep because the effective mass is very small. So the scale is roughly determined by taking the binding energy of the exit on the ratio of that to the ridboat coupling so this is a number which is of order one. Sometimes it's bigger than that sometimes it's of order five but then there's an effective mass ratio so this number typically turns out to be about ten to the minus four. So only very dilute systems are in this region and it's actually very hard to do experiments in this region. Most experiments are in that range. So that's an argument by the way for a theory which is of course based on long range interactions which is then very stable essentially close to a coherent state. And taking that one step further that provides a justification for what everybody does in practice which is to use Gross-Piedevsky in these systems. And you can add to this of course decay processes which allow things to decay into the bath and things to come back in. And what that does in the end is it adds lifetimes to the quasi-particles and gives you a damp driven Gross-Piedevsky equation to describe the dynamics of all of these systems. And that is often the most effective way to describe real experiments. So how am I doing? I started a little late so I'm going to keep going. I'll keep this piece in. So that was really just intended to be a bit of background. And now I want to talk about some recent stuff we've done on two separate problems. One some work actually with Jonathan Keeling's group on what happens if you've got systems with strong electron phonon coupling. So why might that be interesting? Well let me just describe the model I want to talk about. I'll call it Dickey Holstein. So it's the same one we saw before. There's a cavity, there's an exciton, there's a ruby coupling. But in particular I want to think about systems which may be made with Frenkel excitons on organic molecules. And there are many systems which are like that which can also be put into cavity, couples. Organic molecules typically can have very large ruby couplings. But they also typically have very strong local phonon modes. And so I've added in here just a local Holstein coupling on the mode. And the parameters that describe that are of course the frequency of the phonon and some dimensionless coupling strength. And as you know, if you have an exciton coupled to signalling the photon, the effect of this is that the exciton will develop side bands associated with adding and subtracting phonons. So rather than have a single set of excitonic modes, I've got excitons with side bands. And I've got all of those side bands and the question is can you have conversation into a phonon replica? So the way we've treated this is to do essentially a mean field treatment for the photons arguing that that's a reasonable thing to do because of the very long wavelength and the long range coupling. But we do numerical diagonalizations on the phonon. So you can break this up and you can just diagonalize the phonon states and work through that. So what happens? So the first thing that's interesting if you look at this. So this is now a phase transit boundary with temperature and an excitation number. I've gone of course to very high excitations. It's not particularly physical to go up here. Just to notice that one of the effects of doing this is because you can actually get re-entered mottlopes. So the blue curve was the one I showed you before. If you add electron phonon coupling in here, actually quite strong electron phonon coupling, I would say S equals 2 is pretty big for a Hoang Ries parameter. Then what happens is that you shrink this regime in response to the stabilizer if you like the mod-insulating states. But if you look carefully, and in particular if you look down at the bottom of this band, so this is raising the chemical potential just beginning to fill it, you get some re-entrance associated with this. So in particular if you sit here and raise the temperature, you go from an empty state to something which actually wants to go into a condensed state, and what's happening here. And that's really related to the problem I told you one should worry about, is I've got a whole bunch of phonon replicas. And of course at a finite temperature I have phonon replicas which exist below the exciton line, because I can steal a phonon, a thermally occupied phonon from the system and use it to dress the exciton. So I can go on the anti-stokes side at finite temperature, and that's actually what's happening here. So another way to look at that curve is, you know, plot the density along this axis, so coming from zero now, not from a half, but so there's no excitations along here. This is the chemical potential rising at some finite temperature. And now we've plotted all of the phonon lines together with the replicas. And so this is zero phonon, one phonon, two phonon, three phonon, four phonon lines. And, you know, you notice that something odd happens here, because of course there's in principle an infinite number of replicas going all the way down here at any finite temperature. You know, they're bosonic in some sense. So the chemical potential seems to however to go through them. So what happens, and actually the interesting thing is that there's kind of a conspiracy, is for most of these level crossings, what happens as the chemical potential goes through, it turns out that the photon component of the wave function vanishes at this crossing point. So that means that the excitons, of course, can't be multiply-occupied, they really spins. The photons could be multiply-occupied, but if there's no phonon weight when the level crossing happens, there's no transition. And so on the right-hand side, what you're really looking at is what happens to the photon number as you raise the temperature. Not surprisingly, when the temperature is small, you can only add photons. You have to go on the stokes side. But as the temperature grows, then what happens is that there are photons around in the system that can actually be absorbed. So the line effectively shifts down. You have a thermally populated state, and that thermally populated state, at least at the mean field level, can condense. So that's a proposal. It's not being seen, but one can look for it. I think there's actually some interesting data from Stefan Keener-Cohen on this, but I would show something similar, but my current interpretation of that is that that's probably dynamics, and I think you would agree with that. So now I want to finish by moving to a different problem where you don't have local interactions with phonons, but you actually have interactions between excitons. Of course, the basic model I talk about has on-site interactions with excitons. You don't put two of them in the same place. But you can have systems where the excitons do actually have long-range interactions, and one way of thinking about that is with Rydberg atoms. So when you make an excitation of an atom to a very large n-value, the atom itself gets very big, and it interacts with other excited interactions on a long-range effect. So that produces what people like to think of as a blockade effect, and that's certainly been seen in Rydberg systems, and there's work under foot to build, to put Rydberg atoms and to put them actually into microcavities. So you can also make Rydberg atoms in solids. This is some really lovely work on the classic textbook exciton material, cuprous oxide, which has the famous yellow and green exciton series, and here you can see it, you go from n equals two, and then you look up at the end here, and you can see the next piece of this. This goes all the way around to about n equals 25 or so, where you can see these in this really lovely material. And here's the blockade effect that as you increase the laser intensity, if you look at these large n-levels, what you're discovering is that the absorption cross-section goes down, because the presence of states which are already occupied moves them off resonance. So this is not work done in a cavity, but there's I think work under foot going to try and put cavities around this, so it looks plausible to be able to do this. So what about a model? So the model I'd like to propose again is to take the previous one. So this is same Hamiltonian I've been showing all the way through. The thing that I've done in here is just to write this, I'm sorry, not as a spin model, but to take the spins and represent them as two fermions with a constraint. This is mathematically equivalent. And so if you like A and B are the lower and upper levels, but you can only have one site occupied by as a result of this constraint. So this is SZ, this is S plus, this is S minus. And then what we're going to do is that we're going to add an interaction in the upper state. So the upper A state, if it's occupied, then there's interaction. There's no interaction if the system is not occupied. And because I'm not smart enough to solve the general problem, I'm going to mostly simplify by talking about a lattice model with nearest neighbor interactions and then generalize it to long range. But the kind of things that you would think about here if you just look at this is now it's quite clear that there's a possibility of an instability of the superfluid state because the superfluid state, super radiant state assumes that the excitations are spread out across all of the system. And there's an alternative state which would take the same number of excitations and put them one at a time per blockade radius and produce some kind of crystal. So there has to be some kind of transition about this to try and understand this. A few technical points. If you actually take a long range repulsive interaction one over R to the sixth, that has some unpleasant behavior because you can't get to the final answer from a weak coupling instability. As you know from any kind of theory of fermionic gases, if you have a in Fourier space, if you is always positive, then there is no weak coupling instability because the compressibility is positive at any queue. Same problem you have if you just want to get to a vignette crystal of the electron gas starting from weak coupling, you actually have to start by having a magnetic instability creating spins and then going through that. There's another one which is kind of interesting here and you will see is that of course if you look at the super radiant state of these two things, it has of course amplitude and phase modes. The phase modes are neutral and therefore they don't construct with density. The amplitude mode is also neutral and it doesn't mix with density and all of the action only happens at finite queue. So what do we do? Well, here's the machinery. You construct an action, it has in it excitons and photons. What we would like to do is we would like to integrate out the excitons, leaving the photons there but also introduce collective variables which you can really think of as density weights, I'll call them phi, psi for photons, phi for the density weights. At the mean field level you find the stationarity of the action, then you look at fluctuations about that and we'll see what we get. So let me remind you again of mean field. You've seen this already, no interactions, what happens well depending on the detuning. There's a phase diagram, his temperature between condensed and normal state or if you look at it in a way that you may be more familiar, chemical potential rising up here, then the condensed state is up here, this is increasing coupling constant, this is the mott-lobe with nothing in it, this is the mott-lobe with one particle per site in it and here's the condensed region which sort of enters down in this space. What happens if I add just a uniform repulsive potential at the mean field level, then what actually happens is that you split the mott-lobes. So this is really just a trivial effect because what happens is you have the possibility of using a blue shift produced by interactions to shift the excitons closer into resonance with photons so then you get a regime out here where even for weak coupling you can go down to that. Now let's look at the fluctuation spectrum, let's look at the excitation spectrum. Here's in the normal state again, the upper and lower polariton branches. These come as we know from the interaction between dispersionless excitons and the photon modes. In the coherent state depending on the detuning you get these two branches and they don't actually cross here, there's just a very small gap at this point. So this again is no interactions, what happens with interactions? Again at the mean field level when it turns out in the normal state the excitation spectrum is completely unmodified and the reason for that is that you can't couple q equals 0 superfluid fluctuations to q equals 0 density fluctuations. However if you go into the condensed state and now I'll just, this is actually just in the nearest neighbor model on lattice then what you find is that in the condensed state then because you can actually have the mixing of the amplitude, then you get a phase and amplitude modes and out here at some large q which really corresponds to q equals pi in units of the lattice you can have an instability. So here are the two Mott lobes now and now what you find is looking in this regime here there's a condensed phase out here but there's an instability line so this is associated with a soft mode and inside this instability line it looks like you're developing density fluctuations at finite q. So this is just saying that the condensate is unstable to density fluctuations. As you can diagnose that instability just by changing the bandwidth this is probably not too relevant. Let's now go to the next stage of doing a calculation where I look for a ground state which can have this kind of staggered mean field. So in this one so then I look for the, so now I've again done mean field but I've got a mean field which is staggered so as well as having a Mott lobe at 0 and a Mott lobe at 1 this is a Mott lobe at a half, so this is a half field lattice. And at that level if you look at that what you discover is that this state its energy cross is relative to the condensed photon like state along this green line but this is a first order transition this is level crossing. But of course the next thing you have to do is you have to go and look at the normal state the so-called normal state here this is a Mott state and look at the phonon dispersion curves of that and figure out whether there are any instabilities there. And then you, so then the dispersion requires an extra branch and that extra branch of course in the Mott state there is a gap but then there's a new mode which appears in here which is actually associated with the, eventually with the super fluid order and that softens along this green line. So what have we got? So at this level we have a mean field analysis for the normal state, a mean field analysis for the condensed state and they seem to be stable in here. Along the green line you have a point where the Mott state develops an instability to a photon super fluid order and on this purple line coming in you have a point where the photon coherent state develops an instability towards charge order. Good, thanks, I'm close. So at this level of approximation one has in between this some kind of super solid. Just to check that there are no particular artifacts about looking for the order you can do a generalized variational Monte Carlo to allow the order to develop on different points and that more or less confirms this. The next thing one might do is to start adding long range interactions and as you add longer range interactions you expect to get on a lattice well, you expect to get some kind of devil's staircase and indeed you begin to reproduce that. So you get a self similar branch appearing in here. This is yet another piece of a Mott lobe which is appearing down here but this is one half, this is two thirds and in principle there's a whole bunch of things which are sort of down there. So it seems that at least on a lattice one has a possibility of a phase with both super fluid and charge order. There are now three acoustic modes in it. There's two sound modes and one bogey-luboff. It has two amplitude modes. It has an upper polariton and a charge density wave amplitude mode. The amplitude modes mix and the sound modes don't but it's not a gauge theory. There's no Higgs-ish stuff here and maybe this is a Cold-Atom version of Nibium-Dicelenide which is one of these things which has a mix of this but I don't know whether this phase continues to be present without the lattice and I think there's a sort of big open question about whether that really makes sense. This is very much built in to the lattice model. So the last slide but two because I have some small things to say. Just to thank, over the years all of the people who worked on this and many colleagues that we've collaborated with, particularly the work that I just showed you was done by, unless you have lost them, Jonathan Keeling and Justina Quick who I seem to have lost off this list. That's embarrassing. And Sahin Orreja on the organic materials and most recently on Rydberg Atoms with a graduate student in Chicago, Alex Edelman. So I'm done with this piece of the talk but I have a remark or two because this is a birthday. So, Ariel, I'm afraid you've seen this. Ariel knows the punchline because I used it on him last year. So there's a very famous speech or many famous speeches in Shakespeare but one is about the Seven Ages of Man and I think there are really seven ages and there's the infant show and tell as an undergraduate, the school boy he was sort of dragging to school carrying his satchel and that's really, of course, the graduate school. Then there's the lover, presumably a sister professor at that stage. Can imagine somebody as a lover at this rate. Then, of course, there's the soldier, the full professor charging forward, bombastic all of those. And, of course, what happens after that? What happens after that, of course, is the distinguished professor. So as you said, the distinguished professor is the fifth age and actually what did Shakespeare have to say about this age? The justice in fair round belly with good cape online to somebody who likes their food happens to all of us. Eyes severe, well it is in this one, not so much in that one but certainly a beard of formal cut and full of wise sores and modern instances. So I wish you a very extended fifth age because sixth and seventh aren't really so great. But there's nothing about how long this should last and so indeed I wish you another 60 years of great contributions to science. So thank you very much. Thank you.